SYSTEMATIC APPROACHES TO PREDICTIVE COMPUTATIONAL CHEMISTRY USING THE CORRELATION CONSISTENT BASIS SETS. Brian P. Prascher, B.S.

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1 SYSTEMATIC APPROACHES TO PREDICTIVE COMPUTATIONAL CHEMISTRY USING THE CORRELATION CONSISTENT BASIS SETS Brian P. Prascher, B.S. Dissertation Prepared for the Degree of DOCTOR OF PHILOSOPHY UNIVERSITY OF NORTH TEXAS May 2009 APPROVED: Angela K. Wilson, Major Professor Weston T. Borden, Committee Member Thomas R. Cundari, Committee Member Martin Schwartz, Committee Member Michael G. Richmond, Chair of the Chemistry Department Michael Monticino, Interim Dean of the Robert B. Toulouse School of Graduate Studies

2 Prascher, Brian P., Systematic Approaches to Predictive Computational Chemistry using the Correlation Consistent Basis Sets. Doctor of Philosophy (Physical Chemistry), May 2009, 275 pp., 49 tables, 28 illustrations, references, 307 titles. The development of the correlation consistent basis sets, cc-pvnz (where n = D, T, Q, etc.) have allowed for the systematic elucidation of the intrinsic accuracy of ab initio quantum chemical methods. In density functional theory (DFT), where the cc-pvnz basis sets are not necessarily optimal in their current form, the elucidation of the intrinsic accuracy of DFT methods cannot always be accomplished. This dissertation outlines investigations into the basis set requirements for DFT and how the intrinsic accuracy of DFT methods may be determined with a prescription involving recontraction of the cc-pvnz basis sets for specific density functionals. Next, the development and benchmarks of a set of cc-pvnz basis sets designed for the s-block atoms lithium, beryllium, sodium, and magnesium are presented. Computed atomic and molecular properties agree well with reliable experimental data, demonstrating the accuracy of these new s-block basis sets. In addition to the development of cc-pvnz basis sets, the development of a new, efficient formulism of the correlation consistent Composite Approach (ccca) using the resolution of the identity (RI) approximation is employed. The new formulism, denoted RI-ccCA, has marked efficiency in terms of computational time and storage, compared with the ccca formulism, without the introduction of significant error. Finally, this dissertation reports three separate investigations of the properties of FOOF-like, germanium arsenide, and silicon hydride/halide molecules using high accuracy ab initio methods and the cc-pvnz basis sets.

3 Copyright 2009 by Brian P. Prascher ii

4 ACKNOWLEDGEMENTS I would like to acknowledge Jesus Christ as my source of strength in all things. ( I have strength for all things in Christ who empowers me, Philippians 4:13) Further, the love, support, and encouragement of my wife and best friend, Laura; my beautiful daughters, Emma and Charlotte; my Mom and Dad; and Lew and Diane, has been paramount for me during my graduate career. I wish to thank my colleagues during my time in the Wilson Group (in order of appearance): Xuelin Wang, Scott Yockel, James Seals, Ray Bell, Ben Mintz, Pankaj Sinha, John Determan, Nathan DeYonker, Sammer Tekarli, Gavin Williams, Brent Wilson, Gbenga Oyedepo, Kameron Jorgensen, and Marie Majkut, for constantly challenging me with questions about computational chemistry and, in so doing, kept my knowledge of the subject up to par. In particular, I want to thank and acknowledge Brent Wilson and Gavin Williams for their contributions to projects discussed in this dissertation. Many thanks and acknowledgements also go to the numerous undergraduates that served in the Wilson Group, particularly Rebecca Lucente Schultz, Arjun Kavi, and Jeremy Lai for their contributions to work presented herein. I also want to thank the professors of the Chemistry Department, who challenged and molded me into the academic that I am today. Finally, I would like to give a heart felt thanks to Angela Wilson, my academic mother and mentor, whose open door, yet hands off approach gave me room to spread my wings. iii

5 TABLE OF CONTENTS Page ACKNOWLEDGEMENTS... iii LIST OF TABLES... vii LIST OF ILLUSTRATIONS... xi Chapter 1. INTRODUCTION COMPUTATIONAL QUANTUM CHEMISTRY The Schrödinger Equation Ab initio Methodology The Hartree Fock Approximation Configuration Interaction Coupled Cluster Theory Many Body Perturbation Theory Density Functional Theory The Hohenberg Kohn Theorems The Kohn Sham Method Basis Set Theory Model Chemistries CORRELATION CONSISTENT BASIS SETS Introduction Valence and Tight d Basis Sets Augmented Basis Sets Core Valence Basis Sets Scalar Relativistic Basis Sets SYSTEMATIC TRUNCATION OF THE CORRELATION CONSISTENT BASIS SETS IN DENSITY FUNCTIONAL THEORY CALCULATIONS Introduction Computational Methodology Results and Discussion Atoms Homonuclear Diatomics iv

6 4.3.3 CH 4, SiH 4, NH 3, PH 3, H 2 O, and H 2 S Computational Time Savings Conclusions SYSTEMATIC RECONTRACTION OF THE CORRELATION CONSISTENT BASIS SETS FOR DENSITY FUNCTIONAL THEORY CALCULATIONS Introduction Computational Methodology Results and Discussion Atoms Molecules Kohn Sham Limits Basis Set Superposition Error Diffuse Functions Conclusions THE DEVELOPMENT OF S BLOCK CORRELATION CONSISTENT BASIS SETS Introduction Computational Methodology Basis Set Construction Tight d Functions for Inner Valence Correlation Core Valence Functions for Sub Valence Correlation Recontracted Basis Sets for Scalar Relativistic Computations Benchmark Computations Ionization Potentials and Electron Affinities Optimized Geometries and Vibrational Frequencies Thermochemistry Conclusions THE RESOLUTION OF THE IDENTITY APPROXIMATION APPLIED TO THE CORRELATION CONSISTENT COMPOSITE APPROACH Introduction Computational Methodology Results and Discussion RI ccca Implementation Auxiliary Basis Sets for RI ccca Energetic Properties Computational Cost Conclusions A SYSTEMATIC INVESTIGATION OF DIHALOGEN μ DICHALCOGENIDES v

7 8.1 Introduction Computational Methodology Results and Discussion Stability of X 2 A 2 Compounds XAAX Structures Vibrational Frequency Analyses Anomeric Effects Conclusions A SYSTEMATIC INVESTIGATION OF GERMANIUM ARSENIDES Introduction Computational Methodology Results and Discussion Optimized Structures Classical Barriers to Isomerization Thermochemistry and Relative Stabilities Conclusions A SYSTEMATIC INVESTIGATION OF SILICON HYDRIDES AND HALIDES Introduction Computational Methodology Optimized Structures SiH, SiF, and SiCl SiH 2, SiF 2, SiCl 2, SiHF, and SiHCl SiH 3, SiF 3, SiCl 3, SiH 2 F, SiH 2 Cl, SiHF 2, and SiHCl SiH 4, SiF 4, SiCl 4, SiH 3 F, SiH 3 Cl, SiH 2 F 2, SiH 2 Cl 2, SiHF 3, and SiHCl Thermochemistry Atomization Energies and Enthalpies of Formation Dissociation Reaction Enthalpies Conclusions CONCLUDING REMARKS REFERENCES vi

8 LIST OF TABLES Table 3.1 Table 4.1 Table 4.2 Table 4.3 Table 4.4 Table 4.5 The composition of the correlation consistent basis sets for the first three rows of the main group Total energies (E h ) of the carbon and oxygen atoms computed with full and truncated basis sets; energy differences are listed in me h Ionization potentials (ev) of the carbon and oxygen atoms computed with full and truncated basis sets; relative differences are listed below the full basis set values Electron affinities (ev) of the carbon and oxygen atoms computed with full, truncated, and augmented truncated basis sets (aug); relative differences are listed below the full basis set values Total energies (E h ) of the C 2 and O 2 molecules computed with full and truncated basis sets; energy differences are listed in me h BLYP vertical ionization potentials (ev), electron affinities (ev), atomization energies (kcal/mol), and optimized bond lengths (Å) of the C 2 and O 2 molecules computed with full and truncated basis sets; electron affinities include diffuse functions and relative energy differences are listed below the full basis set values Table 4.6 B3LYP atomization energies (kcal/mol) and optimized bond lengths (Å) for CH 4 and SiH 4 (both 1 A 1 ); relative differences are listed below the full basis set values Table 4.7 Table 4.8 Table 4.9 Table 5.1 B3LYP optimized geometries (Å and degrees) for NH 3 and PH 3 (both 1 A 1 ); relative differences are listed below the full basis set values B3LYP optimized geometries (Å and degrees) for H 2 O and H 2 S (both 1 A 1 ); relative differences are listed below the full basis set values Percent CPU time and average time saved computing B3LYP single point energies with truncated basis sets, relative to the full basis sets A comparison of the BLYP and B3LYP total energies (E h ) of the first row atoms (including hydrogen) computed with the original and recontracted cc pvnz basis sets; relative differences to the original basis sets are shown in me h under [rc] vii

9 Table 5.2 Table 5.3 Table 5.4 Table 5.5 Table 5.6 Table 5.7 Table 5.8 A comparison of the BLYP and B3LYP ionization potentials and electron affinities (ev) of the first row atoms (including hydrogen) computed with the original and recontracted cc pvnz basis sets; relative differences to the original basis sets are shown under [rc] A comparison of the BLYP and B3LYP total energies (E h ), ionization potentials, and electron affinities (ev) of the first row dimers (including hydrogen) computed with the original and recontracted cc pvnz basis sets; relative differences to the original basis sets are shown in me h (total energies) and ev under [rc] The BLYP and B3LYP optimized geometries (Å and degrees) of several first row molecules computed with the original and recontracted basis sets; relative differences to the original basis set are shown under [rc] The BLYP and B3LYP atomization energies (kcal/mol) of several first row molecules computed with the original and recontracted basis sets; relative differences to the original basis set are shown under [rc] Two and three point extrapolated BLYP and B3LYP Kohn Sham limits of atomization energies (kcal/mol) computed with the original and recontracted basis sets. Non monotonic behavior is denoted by The BLYP and B3LYP atomization energies (kcal/mol) and their extrapolated Kohn Sham limits of four molecules corrected for basis set superposition error (BSSE) A comparison of BLYP and B3LYP atomization energies (kcal/mol) and their extrapolated Kohn Sham values using the aug cc pvnz, aug cc pvnz[rc], and augcc pvnz[rc] basis sets corrected for basis set superposition error (BSSE) Table 6.1 The composition of the correlation consistent basis sets for s block atoms Table 6.2 Table 6.3 Table 6.4 Table 6.5 CCSD(T) ionization potentials and electron affinities of Li, Be, Na, and Mg; the tight d correlation consistent basis sets for Na and Mg have been used CCSD(T) optimized geometries (R e, Å) of some s block molecules computed with various families of correlation consistent basis sets CCSD(T) harmonic vibrational frequencies (Å) of some s block molecules computed with various families of correlation consistent basis sets CCSD(T) standard state enthalpies of formation at K (kcal/mol) for some s block molecules computed with various families of correlation consistent basis sets; tight d functions have been used for the Na and Mg atoms viii

10 Table 7.1 Table 7.2 Table 7.3 Differences between ccca and RI ccca total energies (me h ), relative CPU times, and relative disk space using different combinations of RI SCF and RI MP2 auxiliary basis sets. a Total energies (E h ) computed with ccca, RI ccca, and RI ccca+l; differences (me h ) are relative to the ccca energies Enthalpies of formation (ΔH f ) at 298 K (kcal/mol) computed with ccca, RI ccca, and RI ccca+l Table 7.4 CPU times and disk space usage of RI ccca and RI ccca+l relative to ccca Table 8.1 Table 8.2 Table 8.3 Table 8.4 Table 8.5 Table 8.6 Table 9.1 Table 9.2 Table 9.3 Table 9.4 Stabilities (kcal/mol) of the different isomers of X 2 A 2 molecules relative to the gauche isomer, XAAX, computed with B3LYP and CCSD(T) B3LYP and CCSD(T) optimized geometries (Å and degrees) of the fluorine species using various correlation consistent basis sets B3LYP and CCSD(T) optimized geometries (Å and degrees) of the chlorine species using various correlation consistent basis sets B3LYP and CCSD(T) optimized geometries (Å and degrees) of the bromine species using various correlation consistent basis sets B3LYP and CCSD(T) harmonic vibrational frequencies (cm 1 ) computed with the aug cc pvtz basis set A comparison of CCSD(T) harmonic and anharmonic vibrational frequencies (cm 1 ) of FSSF computed with the cc pvdz basis set Non relativistic and spin orbit (SO) total energies (E h ) of some main group atoms computed at the MCSCF/cc pvtz level of theory; SO corrections are given in kcal/mol A comparison of the CCSD(T) optimized geometries (Å and degrees) for the L GeAs and GeAs L isomers; previously reported geometries are included for comparison A comparison of the B3LYP optimized geometries (Å and degrees) for the L GeAs and GeAs L isomers; previously reported geometries are included for comparison CCSD(T) and B3LYP classical reaction barriers (kcal/mol) for the forward (f) and reverse (r) reactions; previously reported reaction barriers are included for comparison ix

11 Table 9.5 Table 9.6 Table 9.7 Non relativistic (NR), Cowan Griffin (CG), and Douglas Kroll (DK) enthalpies of formation at 298 K (kcal/mol) of the L GeAs isomers computed with CCSD(T) and B3LYP; spin orbit effects have been included in the relativistic enthalpies Non relativistic (NR), Cowan Griffin (CG), and Douglas Kroll (DK) enthalpies of formation at 298 K (kcal/mol) of the GeAs L isomers computed with CCSD(T) and B3LYP; spin orbit effects have been included in the relativistic enthalpies CCSD(T) and B3LYP enthalpies of isomerization (kcal/mol) from L GeAs to GeAs L at 298 K. Various basis set levels and CBS limits are represented as well as nonrelativistic (NR) and relativistic energies using the Cowan Griffin (CG) and Douglas Kroll (DK) approach Table 10.1 CCSD(T) optimized geometries (Å) of SiH, SiF, and SiCl Table 10.2 Table 10.3 CCSD(T) optimized geometries (Å and degrees) of SiH 2, SiF 2, SiCl 2, SiHF, and SiHCl CCSD(T) optimized geometries (Å and degrees) of SiH 3, SiF 3, SiCl 3, SiH 2 F, SiH 2 Cl, SiHF 2, and SiHCl Table 10.4 CCSD(T) optimized geometries (Å and degrees) of SiH 4, SiF 4, SiCl 4, SiHF 3, SiH 3 F, SiHCl 3, SiH 3 Cl, SiH 2 F 2, and SiH 2 Cl Table 10.5 Table 10.6 Table 10.7 Table 10.8 Table 10.9 CCSD(T) and ccca atomization energies (kcal/mol) and extrapolated energies; atomic spin orbit corrections have been applied to the CCSD(T) results CCSD(T) and ccca enthalpies of formation at 298 K (kcal/mol) and extrapolated energies; atomic spin orbit corrections have been applied to the CCSD(T) results CCSD(T) and ccca enthalpies of dissociation at K (kcal/mol) of the silicon hydrides, computed with the aug cc pv(n+d)z basis sets; the mean absolute deviation (MAD) is relative to the experimental values CCSD(T) and ccca enthalpies of dissociation at K (kcal/mol) of the silicon fluorides, computed with the aug cc pv(n+d)z basis sets; the mean absolute deviation (MAD) is relative to the experimental values CCSD(T) and ccca enthalpies of dissociation at K (kcal/mol) of the silicon chlorides, computed with the aug cc pv(n+d)z basis sets; the mean absolute deviation (MAD) is relative to the experimental values x

12 LIST OF ILLUSTRATIONS Figure 2.1 Figure 2.2 Figure 2.3 Figure 2.4 Figure 2.5 Figure 4.1 Figure 4.2 Figure 4.3 Figure 5.1 Figure 5.2 Representations of restricted closed shell (I), restricted open shell (II), and unrestricted (III) wavefunctions A comparison of the dissociation of H 2 using a single determinant (SCF) and a two determinant (MCSCF) wavefunction Examples of singly, doubly, and triply excited configurations of the HF reference wavefunction Plots comparing the behavior of a single STO to a single GTO (left) and that of a single STO with a linear combination (LC) of three GTOs (right). The units are arbitrary A conceptual view of basis set size versus level of sophistication in correlated ab initio methods. The number of determinants with respect to excitation level (n), number of electrons (N), and basis set size (K) is given by the formula on the lower axis BLYP ionization potentials of the oxygen atom computed with truncated basis sets (hash marks) and plotted against the full basis set values. The inset shows more detail at the quadruple ζ level. Points denoted l > 3 are the truncated basis sets with only s, p, d, and f basis functions BLYP ionization potentials of the O 2 molecule ( 3 Σ g ) computed with truncated basis sets (hash marks) and plotted against the full basis set values. Points denoted l > 3 are the truncated basis sets with only s, p, d, and f basis functions Comparisons of CPU time savings (closed circles), ionization potentials (triangles), electron affinities (upside down triangles), atomization energies (open squares), and total energies (closed squares) between the full and truncated cc pv5z basis set using B3LYP. Truncation levels: 0 is the full basis, 1 is cc pv5z( 1h ; 1g), 2 is cc pv5z( 1h1g ; 1g1f), etc The BLYP potential energy curve of O 2 ( 3 Σ g ) computed with the original and recontracted correlation consistent basis sets A comparison of the BLYP atomization energies computed with the original and recontracted correlation consistent basis sets xi

13 Figure 5.3 Figure 5.4 Figure 6.1 Figure 6.2 Figure 6.3 Figure 6.4 Figure 6.5 Figure 6.6 Figure 7.1 Figure 7.2 Figure 7.3 Figure 8.1 Figure 8.2 A comparison of BLYP atomization energies computed with the original and recontracted correlation consistent basis sets, and the recontracted basis sets with basis set superposition error (BSSE) removed A comparison of BLYP atomization energies computed with the pc n, aug cc pvnz, and aug cc pvnz[rc] basis sets with basis set superposition error (BSSE) removed The incremental CISD correlation energy lowering of the Mg atom due to each polarization function Comparisons of the Na atom valence (cc pvnz) exponents with the core valence (cc pcvnz) and weighted core valence (cc pwcvnz) exponents; the exponents are grouped by angular momentum The spacing of the d exponents in the cc pvnz basis sets compared with those in the cc pv(n+d)z basis sets CCSD(T) valence correlation energies recovered by the cc pvnz (V/V), cc pcvnz (V/CV), and cc pwcvnz (V/wCV) basis sets CCSD(T) core valence plus valence correlation energies recovered by the cc pvnz (CV/V), cc pcvnz (CV/CV), and cc pwcvnz (CV/wCV); the 1s orbitals of Na and Mg were kept frozen in each calculation Comparisons of the scalar relativistic corrections to the atomic Hartree Fock energy using the original contractions (diamonds) and the Douglas Kroll contractions (squares) with increasing basis set size CPU times of RI ccca (top) and RI ccca+l (bottom) versus ccca; each data point represents a molecule in the test set Disk space usage of RI ccca (top) and RI ccca+l (bottom) versus ccca; each data point represents a molecule in the test set Correlation between molecular size and average CPU time and disk space saved using RI ccca and RI ccca+l, relative to ccca The orbital representation of anomeric delocalization in XAAX systems: the lone pair on a bridging A atom delocalizes into the adjacent σ*(ax) orbital The lowest unoccupied CISD natural orbitals of FSSF: both contour plots are F S σ* orbitals (14A, left; 13B, right). The perspective is down the z axis (iso contour value: 0.095) xii

14 Figure 8.3 Figure 9.1 Figure 9.2 Figure 10.1 Figure 10.2 The molecules FSSF (left), ClSSCl (center), and BrSSBr (right), and the lone pair orbitals responsible for anomeric delocalization. The prespective is down the z axis (iso contour values: 0.06, 0.03, and 0.03, respectively) A schematic of the forward (f) and reverse (r) reaction coordinates of L migration from the germanium atom to the arsenic atom A side by side comparison of non relativistic and Douglas Kroll (DK) relativistic CCSD(T) and B3LYP relative energies from mixed Gaussian/exponential extrapolations Plots comparing the CCSD(T) optimized geometries of SiH, SiF, and SiCl using four different families of correlation consistent basis sets Plots comparing the CCSD(T) enthalpies of formation at 298 K of the silicon fluorides using four families of correlation consistent basis sets xiii

15 CHAPTER 1 INTRODUCTION The practice of chemistry, in one form or another, has been around for millennia, but only in the past two centuries has the practice of chemistry evolved from archaic alchemy to a pure and applied science. Modern chemistry is able to probe the structure, microscopic properties, and reactions of atoms and molecules through systematic laboratory methods that include synthetic techniques, such as directly reacting a substance with another substance in a controlled environment; instrumental techniques, including the myriad of spectroscopic techniques; and, more recently, computational techniques. With the advent of quantum mechanics in the early 20 th century, the way in which chemical properties traditionally are elucidated changed dramatically. Schrödinger s wave mechanics, simultaneously introduced alongside Heisenberg s matrix mechanics, offered ways of calculating both static and dynamic properties of quantized systems. Following the construction of the first electronic computer, chemists and physicists began to submit atomic and molecular properties for calculation using Schrödinger s and Dirac s mathematics. The amount of information acquired about chemical properties during the early years of quantum mechanics was not substantial, limited primarily by the computational expense of both the underlying mathematics and the physical computational resources (i.e. storage, time, and software). The general organization of this dissertation is such that systematically approaching a problem with computational chemistry is covered in detail, from the selection of a 1

16 computational model (Chapter 2) and basis sets (Chapter 3); the development of systematic basis sets for density functional theory (Chapters 4 and 5) and ab initio methods (Chapter 6); to the application and benchmarking of these basis sets (Chapters 7 10). First, a basic working knowledge of the mathematics behind computational chemistry, specifically ab initio methods and density functional methods, is presented in some detail in Chapter 2. From there, Chapter 3 discusses a special class of basis sets, called the correlation consistent basis sets, which are the focal point of this dissertation as every subsequent chapter employs them. In Chapters 4 and 5, work is described that has been done to gain a greater understanding of the basis set requirements in density functional theory calculations. By examining the effects of systematically truncating and recontracting the existing correlation consistent basis sets, a set of compact, fine tuned basis sets is presented that provides accurate results relative to the original correlation consistent basis sets at a reduced computational cost. Further, the recontracted basis sets, augmented with diffuse functions and a correction for basis set superposition error, restore the smooth, monotonic behavior inherent to the correlation consistent basis sets in Kohn Sham calculations. Chapter 6 discusses the development and benchmarks of new correlation consistent basis sets for the s block atoms lithium, beryllium, sodium, and magnesium. As discussed in the chapter, these atoms present a unique challenge in the development of accurate basis sets due to their electronic structure. Chapter 7 describes the implementation of the resolution of the identity approximation into the correlation consistent composite approach. Specifically, this chapter presents the implementation and discusses the need for specialized correlation consistent basis sets 2

17 optimized for the resolution of the identity. The chapter is concluded by demonstrating the incredible efficiency of resolution of the identity methods, which are the key to accessing larger and larger electronic systems on current computational hardware. The last three chapters of this dissertation are benchmark/predictive studies employing the correlation consistent basis sets in FOOF like molecules, germanium arsenides, and silicon hydrides/halides. The calculated structures, spectroscopy, and thermochemistry of many molecules is presented and discussed along side experimental data, where available. In the work on FOOF like molecules, the electronic structure is specifically focused upon due to the unique geometry of these molecules stemming from their electronic structure. All of the chemical properties presented for the germanium arsenides are purely predictive since these molecules have never been reported in the literature. Finally, the silicon molecule benchmarks critically examine how the correlation consistent composite approach performs compared with coupled cluster theory and present high level ab initio predictions of geometries, spectroscopic properties, and thermochemistry of several transient silicon hydrides and halides. 3

18 CHAPTER 2 COMPUTATIONAL QUANTUM CHEMISTRY 2.1 The Schrödinger Equation The central goal of computational chemistry is to model chemical systems by computing their properties using the Schrödinger equation. 1 4 The Schrödinger equation is an eigenequation in which an energy operator, the Hamiltonian, operates on a wavefunction describing the electronic and nuclear structure of a given system to produce the system s total energy as an eigenvalue. There is no formal proof of the Schrödinger equation, but the fact that highly accurate solutions to this equation agree with experimental observations leads to its continued use. Since chemical properties are usually more dependent on the electronic, rather than the nuclear structure of a chemical system, the time independent form of the electronic Schrödinger equation within the Born Oppenheimer (BO) approximation 5 is often employed: Ψ Ψ (2.1) In the equation above, is the Hamiltonian, Ψ is the electronic wavefunction, and is the total electronic energy of the wavefunction. The Hamiltonian is, in atomic units: 6, (2.2) where is the number of electrons interacting with nuclei. The first term of the Hamiltonian is the kinetic energy of the electrons, where is the Laplacian; the second term is the nuclearelectron attraction operator, where is the charge on nucleus and is the distance 4

19 between electron and nucleus ; and the third term is the electron electron repulsion, where is the distance between two electrons. The Hamiltonian is more compactly written as, where is the kinetic term, is the nuclear electron term, and is the electron electron term. Once the nuclear electron potential (plus any external potential) has been specified, the Hamiltonian is uniquely defined for a given electronic system. The time independent wavefunction Ψ is a mathematical function of the electron positions within a field of fixed nuclei. All information regarding the static discrete energy states of the electrons is contained within this wavefunction and is extracted using the energy operators of the Hamiltonian. 6,7 In general, the wavefunction is complex valued and has no direct physical interpretation, but, the squared modulus of the wavefunction, Ψ, is the electron density (discussed later), which is observable by experiment using x ray photoelectron spectroscopy. Considering this, the wavefunction itself may be interpreted from a statistical point of view as a position function of the electrons. Despite the statistical interpretation, the wavefunction may not be used to find the exact position of the electrons a consequence of the Heisenberg uncertainty principle. 8 The wavefunction must have various mathematical properties so that it remains physically viable (i.e. spans a Hilbert space): 1) it must be single valued at all points in space, 2) it must be continuous at all points in space, 3) it must be zero valued at the boundaries (usually taken to be infinity), 4) its derivatives at the boundaries must also be zero valued, and 5) the wavefunction must be square integrable. 6 Further, the electronic wavefunction must be antisymmetric (change sign) with respect to the interchange of two electron coordinates. 6,7 That is to say, the wavefunction must obey the Pauli exclusion principle, 9,10 5

20 Ψ,,,,,, Ψ,,,,,, (2.3) One method of writing an N electron wavefunction with the necessary anti symmetry, Ψ, is the Slater determinant: 7,11 Ψ 1! 1!! 1 (2.4) where the columns represent electron orbitals and the rows represent electrons. The factor 1! is a normalization constant, 1 is the parity of the th term, and is the permutation operator, which interchanges the orbital indices. The term in brackets is a product wavefunction called the Hartree product From linear algebra, it is known that the properties of a determinant are such that the interchange of two rows (electrons in this case) leads to a change of sign in the value of the determinant. 15 Thus, the necessary anti symmetry is taken care of naturally using a determinant as the wavefunction. The form of the individual electron orbitals may be any mathematical function that obeys the five conditions previously discussed, and are usually taken from Gaussian type basis sets (discussed later). A subtle point worth making here is that the non relativistic Hamiltonian does not intrinsically account for the spin of the electrons. This phenomenon is accounted for a posteriori by using generalized electron coordinates of the form, where contains the spatial components and the spin component. The generalized coordinates used in (2.4) lead to spin orbitals, which may be separated into their spatial and spin components as: (2.5) The spin functions and are orthonormal,. When the Hamiltonian (more 6

21 specifically, the operator) is applied to the determinant wavefunction, the anti symmetry, along with the orthogonal nature of the spin orbitals produces an exchange interaction. This interaction (also called exchange correlation or same spin correlation) only occurs between electrons of the same spin, has no classical analog, and lowers the classical Coulomb repulsion energy a wavefunction without anti symmetry (i.e. a Hartree product). Spin anti symmetry is one of the underlying phenomenon responsible for Hund s rules. 16,17 The Hamiltonian operator contains two 2 particle interaction terms: the term and the term. The nuclear positions are fixed in the BO approximation, reducing to a 1 electron problem. However, remains a 2 electron term. It is well known that there is no closed form solution to the interacting 3 particle (or higher) problem, and thus no closed form solution to the electronic Schrödinger equation for more than one electron. This is the central problem of computational quantum chemistry. 6,7 To overcome the intractable problem of finding wavefunctions that satisfy the manyelectron Schrödinger equation, approximate wavefunctions may be constructed using variational calculus. Once constructed, the energy of an approximate wavefunction may be computed as the expectation value of the Hamiltonian, Ψ Ψ d d Ψ,, Ψ,, (2.6) where the wavefunction is assumed to be normalized, Ψ Ψ 1. To obtain the best approximation to the exact solution of the Schrödinger equation, an approximate wavefunction is constructed with variational parameters that may be adjusted until the expectation value of the energy is minimized. Once minimized, the wavefunction is the best solution (in a variational sense) to the Schrödinger equation. The question then arises as to how the variational 7

22 parameters are chosen and how the initial wavefunction is constructed. The answer to this question leads to the many approximate methods of constructing many electron wavefunctions and basis set theory. There are two categories of computational methods that will be focused on in this dissertation: ab initio methods and density functional theory. Basis set theory is discussed later in this chapter. 2.2 Ab initio Methodology Ab initio, or from the beginning, methods are so called because only fundamental physical laws and constants (e.g. Coulomb s law, the elementary charge, the masses of subatomic particles, Planck s constant, the speed of light, etc.) are employed in them, and thus, no bias from experimental data. 6,18,19 Ab initio methods first construct an approximate wavefunction as a reference, then use that reference in more rigorous methods to approximate the exact solution to the Schrödinger equation. The first ab initio approximation to the solution of the Schrödinger equation is the Hartree Fock approximation, 12 14,20,21 which recasts the many electron Hamiltonian as a sum of one electron operators and an average electron electron repulsion potential. This procedure typically accounts for greater than 99% of the exact energy obtained with the Schrödinger equation. However, as a result of this approximation, the Hartree Fock method recovers all but the opposite spin correlation energy (hereafter referred to as just correlation energy), which is physically interpreted as the instantaneous changes in the motions of electrons due to repulsions by other electrons. The correlation energy is defined as the difference between the exact ( ) and Hartree Fock ( HF ) energies: 7,22 8

23 C HF (2.7) Although a small fraction of the total energy, the correlation energy is responsible for the quantitative and, in many cases, qualitative description of chemical properties. Computing the correlation energy is computationally demanding, and in order to obtain all of the correlation energy for an electronic system, every electron must be correlated or allowed to interact with every other electron simultaneously in the wavefunction. In practice, this corresponds to a full configuration interaction (FCI) 7,23 25 treatment of the system, which is only possible to employ, using current computing technology, on systems of a few electrons (i.e. ten or less). The task of approximating a FCI treatment is the so called N electron problem, 18 and correlated ab initio methods are distinguished from one another in how they recover the correlation energy from a HF reference wavefunction. 7,19,22 Correlated ab initio methods become more computationally demanding as the number of electrons increases, and so, there are two approximations that are often employed to reduce the computational demand: 1) the frozen core approximation and 2) truncation of the correlation space. The frozen core approximation assumes that chemical properties are dominated by valence electron effects, and, thus, only valence electrons need to be correlated. The remaining core electrons are left frozen, that is, uncorrelated. Truncation of the correlation space involves only correlating a certain number of electrons simultaneously. For example, it is well known that the largest contributor to correlation energy is the electron pair energy, followed by the single electron correlation energy. Thus, some correlated ab initio methods are able to recover a significant amount of the total correlation energy by only correlating up to two electrons simultaneously. 19 9

24 2.2.1 The Hartree Fock Approximation Since the Schrödinger equation may not be solved exactly as an eigenvalue equation for a many electron system, the Hartree Fock (HF) method is invoked as a first approximation to the exact many electron wavefunction. Starting from a trial one electron orbitals, the HF method seeks to find an optimal set of one electron orbitals that minimize the energy expectation value of the determinant. 7 In molecular HF calculations, each one electron orbital is approximated as a linear combination of fixed one electron functions. This is the so called linear combination of atomic orbitals (LCAO) theory, 7,18 in which the molecular orbitals (MOs) forming are taken to be a weighted sum of fixed atomic orbitals (AOs): (2.8) The coefficients may be varied so that the energy expectation value reaches a minimum, leading to the variational nature of the HF method. In this way, the HF wavefunction is known to be the best possible wavefunction constructed from the set of AOs when the total energy has been minimized. The energy produced by the HF procedure will always be an upper bound to the exact ground state energy. The actual HF approximation replaces the exact electron electron repulsion operator with an average effective potential (called the HF potential) by fixing an electron s coordinates and computing its interaction with the other 1 electrons. To arrive at this potential, consider the electronic Hamiltonian recast in terms of a one electron core Hamiltonian operator and the term, taking : 10

25 1 2 (2.9) 1 2 According to Slater s rules, 7,26 the electron electron repulsion operator connects all pairs of electrons in a Slater determinant such that: 1 2 Ψ Ψ 1 2 (2.10) where the following shorthand has been introduced: 1 (2.11) The permutation operator interchanges the orbital indices of electrons and, a consequence of the anti symmetry of the Slater determinant. The first integral that results is the classical Coulomb repulsion integral, denoted, while the second integral is the aforementioned exchange interaction between two electrons, denoted, and only arises when electrons and have the same spin. Now, we define two operators that correspond to the values of and : d (2.12) d Note that these new operators are one electron operators, and each represents a piece of the average electron electron interaction experienced by electron in the field of electron. By summing over all the occupied MOs, we obtain the HF potential for a single electron: 11

26 HF (2.13) Next, a Fock operator for a single electron 21 is defined and a new one electron Hamiltonian is written in terms of these Fock operators: HF (2.14) The operator is often referred to as the HF, or zeroth order, 22 Hamiltonian. Thus, the HF approximation is to turn the complicated, many electron Schrödinger equation into a oneelectron problem. As a consequence of replacing the exact electron electron repulsion with the average HF potential, the correlation of the electrons is lost. The HF approximation leads to a new set of Schrödinger like equations called the HF equations. The Fock operator operates on a given orbital to produce an eigenvalue the orbital energy within the averaged out electron electron repulsion: (2.15) The HF orbitals are constrained to be orthogonal, and the orbital energy is (2.16) According to Koopmans theorem, 27 this orbital energy may be interpreted as the ionization potential of the corresponding orbital. However, Koopmans theorem neglects both the correlated motions of electrons and orbital relaxation upon ionization. 7 To solve the HF equations, (2.8) is inserted into (2.15): 7,22,28 12

27 (2.17) d d Note that the integral on the right hand side is not the Kronecker delta since the AOs are allowed to overlap. If we define the Fock integral on the left hand side as an element of a Fock matrix, ; the overlap integral on the right hand side as an element of an overlap matrix, ; and write the coefficients as to form the coefficient matrix, ; then (2.17) is written in matrix notation as: (2.18) Solving (2.18) is equivalent to solving each of the HF equations simultaneously. We are interested in obtaining a matrix such that (2.18) holds. This leads to an iterative procedure for solving (2.18) called the self consistent field (SCF). The column vectors of, called the canonical HF orbitals, are the MOs of the optimal wavefunction. These orbitals are not unique and may be mixed among themselves to form other solutions to the HF equations with the same total energy. The Slater determinant formed from the set of HF orbitals is the HF wavefunction, denoted Ψ. Applying the full Hamiltonian to the HF wavefunction gives the HF energy: HF 1 2 (2.19) There are three types of wavefunctions that the SCF procedure may produce, depending on the electronic structure of the system of interest: 1) a restricted closed shell, 2) a restricted open shell, or 3) an unrestricted wavefunction (cf. Figure 2.1). 7,18,19 A closed shell wavefunction 13

28 4 4 α 4 β 4 α 4 β α 3 β α 2 β α 1 β Ψ I Ψ II Ψ III Figure 2.1 Representations of restricted closed shell (I), restricted open shell (II), and unrestricted (III) wavefunctions. is one in which each electron is paired up with another electron of the opposite spin, whereas an open shell wavefunction contains one or more unpaired electrons. In the restricted wavefunction, each pair of electrons occupies a single spatial orbital, while in the unrestricted case, the α electrons occupy different spatial orbitals than the corresponding β electrons. The restricted open shell case uses restricted orbitals for the electron pairs and unrestricted orbitals for the unpaired electrons. In the unrestricted case, the β orbitals are higher in energy relative to the α orbitals due to spin polarization of the α electrons. The advantage of using an unrestricted wavefunction is a lower energy, and, hence, better wavefunction. However, the consequence of the electron spins not being perfectly paired leads to a phenomenon called spin contamination, such that the unrestricted wavefunction is not a pure spin state. 7,22 The proper selection of a restricted versus unrestricted HF wavefunction is crucial to exploiting the size consistency of the HF method. 7,22 A method is size consistent if it correctly describes the dissociation of a bond, e.g. 14

29 H HH 2 H (2.20) For example, if a restricted closed shell wavefunction is used to dissociate H 2, the dissociation energy will be too high (see Figure 2.2), but an unrestricted wavefunction will give the qualitatively correct dissociation behavior into two hydrogen atoms. The restricted wavefunction does not correctly dissociation the H 2 bond because the wavefunction always forces the two electrons in the system to occupy the same spatial orbital, despite the fact that, at long bond lengths, the occupied orbitals are localized on each hydrogen atom and are spatially very different. Further, there are ionic terms corresponding to H + H + in the restricted wavefunction that do not disappear as the bond elongates, which lead to the inflated dissociation energy of the SCF curve in Figure 2.2. The unrestricted wavefunction will dissociate the H 2 molecule correctly, but the wavefunction at the dissociation limit will be spin contaminated by the triplet configuration. The general consequence of size consistency at the HF level is that a restricted wavefunction should be used when dissociation results in fragments that can be properly described by restricted wavefunctions, while an unrestricted (or restricted open shell) wavefunction should be used when the dissociation fragments are open shells. A closely related concept that has been alluded to, but not explicitly discussed is the use of more than one determinant to describe the reference electronic state. The HF formulism introduced thus far has only used a single determinant in the total wavefunction. There are many systems in which the electronic state may not be qualitatively described by a single determinant (e.g. atoms and molecules with orbital degeneracies, open shell singlets, etc.) and a multi determinant method is needed. In these special cases, multi configuration selfconsistent field (MCSCF) is employed to obtain the HF wavefunction. The MCSCF wavefunction 15

30 0.100 E(H 2 ) 2E(H) / hartree SCF MCSCF R(HH) / angstrom Figure 2.2 A comparison of the dissociation of H 2 using a single determinant (SCF) and a two determinant (MCSCF) wavefunction. is constructed as a linear combination of HF determinants that each describe a different pure electronic state, Ψ Ψ (2.21) where both the coefficients and determinants Ψ are simultaneously optimized subject to the constraints Ψ Ψ and Ψ Ψ 1. The MCSCF energy expectation value is minimized using the HF operator. By introducing more determinants in the reference wavefunction, so called static correlation is introduced, leading to a lower energy solution that the single reference wavefunction. Static correlation is the interaction between two or more configurations that arises when the ground state wavefunction is written as in (2.21) Configuration Interaction As already discussed, the HF orbitals may be mixed among themselves without affecting 16

31 a change in the total energy. 7,22 Thus, it can be said that the occupied HF orbitals form an orbital basis for the single determinant ground state wavefunction. However, the entire collection of HF orbitals, occupied and virtual (unoccupied), form an orbital basis for the fullyinteracting ground state wavefunction. Another way to look at it is to consider the entire set of HF orbitals, arranged from lowest to highest orbital energy. By the aufbau principle, the HF ground state will have the electrons filling up the lowest energy orbitals. But, this is only one possible configuration of the electrons. In fact, there are possible configurations (where is the number of one electron basis functions and is the number of electrons). These other configurations (Slater determinants) form a basis for the exact N electron wavefunction. Determining how these configurations interact with each other leads to the method of configuration interaction (CI). 7,19,22 25 The fully interacting wavefunction may be written as a linear combination of excited Slater determinants formed from the HF orbitals: Φ Ψ Ψ 1 4 Ψ 1 36 Ψ (2.22) Here, Ψ represents the HF reference wavefunction, while Ψ, Ψ, and Ψ represent connected singly, doubly, and triply excited determinants formed from the HF reference, respectively (the term connected will be discussed in subsection 2.2.3). A few examples of these types of excited determinants are shown in Figure 2.3. The set is the occupied ( ) orbitals and are the virtual ( ) orbitals. The script notation Ψ denotes the th occupied orbital is excited to the th virtual orbital, etc., and the factors 1/4 and 1/36 ensure that no excited configuration enters the overall wavefunction more than once. Finally, the sets 17

32 7 Occupied Virtual Ψ 0 Ψ Ψ Ψ Figure 2.3 Examples of singly, doubly, and triply excited configurations of the HF reference wavefunction.,, and are called the singles, doubles, and triples amplitudes, respectively. The energy of the CI wavefunction is, recalling the definition in (2.7): Φ Φ HF C Φ (2.23) The correlation energy C is obtained by left projecting (2.23) onto the HF wavefunction: C Ψ Φ Ψ HF Φ (2.24) C Ψ Φ 1 HF Ψ Ψ 1 4 Ψ Ψ Assuming intermediate normalization, Ψ Φ 1, which implies that 1, the CI correlation energy C is written: 18

33 C Ψ Ψ 1 4 Ψ Ψ 1 36 Ψ Ψ (2.25) As a consequence of Brillouin s theorem, 7,22 which states that the HF wavefunction does not mix with its singly excited determinants, the first summation in (2.25) equals zero. Also, according to Slater s rules, 26 the Hamiltonian does not connect determinants that vary by more than two permutations of the electrons. Thus, the triply excited, and higher, determinants drop out of the expansion in (2.25) leaving: C 1 4 Ψ Ψ (2.26) Now, all that must be obtained to find the correlation energy of the fully interacting wavefunction is the set of doubles amplitudes. Equation (2.26) demonstrates why two electron correlation is the predominate contributor to the total correlation energy. In practice, obtaining the doubles amplitudes is non trivial since they are coupled to not only the other doubles amplitudes, but also to the reference, singles, triples, and quadruples amplitudes. This means that in order to compute the correlation energy, all of the singly, doubly, triply, and quadruply excited determinants must be formed from the HF reference. Even more complication arises since the reference, singles, triples, etc. amplitudes are also coupled to other excited determinants. As an example, if (2.24) is left projected by an arbitrary singly excited determinant Ψ, the result would be: C Ψ HF Ψ 1 4 Ψ Ψ 1 36 Ψ Ψ (2.27) Note that, even though (2.27) is an expression for the amplitude, the presence of the other 19

34 singles amplitudes in this equation precludes that a method of obtaining each of the sets of amplitudes simultaneously is needed. This leads to the construction of the FCI matrix, which, in general, is very large and requires computing expectation values over all possible combinations of configurations. 7,19,22 For this reason, the FCI method is computationally demanding and can only be performed on systems with a few electrons. Instead of constructing the entire FCI matrix, the frozen core and truncated wavefunction approximations discussed earlier may be employed. If the correlated wavefunction of (2.22) is truncated after, say, the doubly excited determinants, then the size of the CI matrix is reduced to a much more tractable form, called CI singles and doubles (CISD). There are two major problems in truncated CI theory. One of these problems is the lack of size consistency, despite the correct choice of a restricted or unrestricted HF reference. Truncated CI wavefunctions do not have multiplicative separability, and, thus, not all excitations necessary to properly describe bond dissociation are included when two fragments are pulled apart. The other problem with truncated CI theory is the lack of size extensivity. 29 A correlated method is size extensive if the correlation energy recovered scales properly with the number of electrons. Further, size extensivity ensures that the same amount of correlation energy is recovered everywhere on the potential energy surface of an electronic system. The lack of size extensivity in truncated CI methods stems from the fact that disconnected excitations (discussed in the following section) do not enter the correlated wavefunction Coupled Cluster Theory Instead of constructing all or some of the possible excited determinants from the HF 20

35 reference wavefunction, it is possible to recover correlation energy by writing the correlated wavefunction in terms of cluster functions. 30 A cluster function is a term added to the HF wavefunction that correlates electrons. For example, consider a reference wavefunction describing a system containing three electrons, Ψ 1 3! (2.28) A cluster function that correlates electrons in orbitals and may be constructed as follows:, 1 2 (2.29) Here, and have the same meaning as in subsection 2.2.2, and is a doubles cluster amplitude. If this cluster function is added to the orbital product, resembling a sort of CI expansion, and replaced in the reference wavefunction as follows, a correlated wavefunction results: Φ, (2.30) Φ Ψ 1 2 The correlated wavefunction in (2.30) includes only pair correlation stemming from electrons occupying the orbitals and. If cluster functions correlating all possible orbital combinations (not just pairs) were included in the reference wavefunction, then the FCI wavefunction would result. 30 This process is called the coupled cluster (CC) method It can be shown that the CC wavefunction may be written in an exponential ansatz: 21

36 Φ CC Ψ 1! Ψ (2.31) Note that the Taylor series expansion of has been included for later discussion. The cluster operator is composed of single ( ), double ( ), etc., up through N tuple ( ) cluster functions, each defined by how they operate on the reference wavefunction: Ψ Ψ (2.32) Ψ 1 4 Ψ Ψ 1! Ψ If the cluster operator were applied directly to the reference wavefunction, the FCI wavefunction (2.22) would result. However, the principle difference between CI and CC theories is the exponential ansatz of (2.31), which gives multiplicative separability to the CC wavefunction. 29,37 Truncated CC methods are defined by restricting the cluster operator. For example, the singles and doubles coupled cluster (CCSD) wavefunction is determined by setting and inserting into the exponential ansatz (2.31): Φ CCSD 1! Ψ (2.33) Φ CCSD Ψ. Consider the system composed of three electrons in (2.28); the highest possible excitation is a 22

37 triple excitation, thus, only cluster operators and products of cluster operators that correspond to, at most, a triple excitation will be included in the CCSD wavefunction. 30,38 In this case, (2.33) then reduces to Φ CCSD Ψ (2.34) Φ CCSD Φ CISD Ψ. It is clear from this equation that the CCSD wavefunction includes more configurations than the CISD wavefunction, and hence, includes more correlation than the CISD wavefunction. In fact, not only do single and double excitations enter the CCSD wavefunction, but triple excitations enter through the disconnected excitation terms and. Overall, the exponential ansatz allows for disconnected excitations to enter truncated CC wavefunctions, which are the reason that CC methods are both size consistent and size extensive. 30 Despite the recovery of more correlation in the wavefunction compared with truncated CI methods, the most popular implementations of CC are not variational. 30 Although, it is possible evaluate the truncated CC energy in a manner similar to that of truncated CI. Many conventional CC methods exploit simplifications that arise from using a similarity transformed Hamiltonian,. This transformed Hamiltonian comes from the bra representation of (2.31), whereby the inverse exponential arises as the adjoint of. Due to Slater s rules, 26 naturally truncates at quadruple excitations making it more computationally efficient than computing the full CC matrix (analogous to the FCI matrix) introduced in subsection As a consequence of employing the transformed Hamiltonian, both the variational and hermitian nature of the CC equations are lost, but the resultant energy eigenvalues of 23

38 are, typically, very close to those of, which is motivation for the continued use of this implementation of CC theory Many Body Perturbation Theory The correlated methods discussed up to this point have used different configurations formed from the HF orbitals as a basis for constructing correlated wavefunctions. However, another method of obtaining correlation energy is many body perturbation theory (MBPT). 7,19,22,34 Perturbation methods add corrections (perturbations) to the Hamiltonian and wavefunction, which, in turn, give rise to a perturbed total energy: (2.35) Φ Ψ Ψ Ψ Here, Ψ and are the th order wavefunction and energy, respectively, of the th eigenstate of the given system. Inserting these perturbed quantities into the Schrödinger equation and collecting in orders of yields the th order perturbed Schrödinger equation: Ψ Ψ Ψ (2.36) Next, MBPT, like the CI and CC methods, assumes that the eigenstates of the zeroth order wavefunction form a basis for the th order wavefunction such that Ψ Ψ (2.37) If the eigenstates of the zeroth order wavefunction are orthonormal, Ψ Ψ, then 24

39 to find the th order energy, left project (2.36) by the zeroth order eigenstate to obtain Ψ Ψ (2.38) Now, it appears that the th order energy may be calculated from the 1 order wavefunction, but Wigner has shown that the th order wavefunction may be used to obtain up to the energy up to order ,39 To obtain a given order wavefunction, we need to calculate the set of coefficients, which are obtained by inserting (2.37) into (2.36), leftprojecting by an arbitrary eigenstate of the zeroth order wavefunction, and using algebra to clean up the resulting equation. Because this dissertation will only discuss MBPT results up through second order, the derivation of the necessary quantities for obtaining the secondorder energy is now presented. According to Wigner s theorem, the first order wavefunction is all that is necessary for obtaining the second order energy. Thus, inserting (2.37) into (2.36), where 1, and leftprojecting by Ψ yields Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ (2.39) With some algebra and recalling the orthogonality of the eigenstates of the zeroth order wavefunction, the coefficient for the th component of the first order wavefunction is Ψ Ψ (2.40) Inserting (2.40) into (2.37), then into (2.38) gives the second order energy as Ψ Ψ Ψ Ψ (2.41) Now, consider the form of the perturbation operator. If the operator is the HF 25

40 operator (2.14) then the perturbation is defined using (2.9), (2.14), and (2.35): 1 2 (2.42) The zeroth order wavefunction is taken to be the HF wavefunction, and the different eigenstates are the excited configurations of the HF wavefunction. This form of MBPT is known as Møller Plesset perturbation theory (MPn, where n denotes the order of perturbation). 7,19,22,40 Using (2.36), the zeroth order ground state energy is just the sum of the HF orbital energies, Ψ Ψ (2.43) The first order correction to the energy is found using (2.10) and (2.38): Ψ Ψ 1 2 (2.44) Note that the total energy to first order is the HF energy (2.19), MP HF. This very important result demonstrates that the HF wavefunction is stable to first order, which is proof that mixing the HF orbitals among themselves will not lower the total energy. As a result, eigenstates outside the HF ground state must be considered to compute the second order energy. Further, note that the first order energy correction effectively removes the doublecounting of the electron electron interactions inherent to the zeroth order energy. In the energy expression of (2.41), the eigenstates that correspond to Ψ must be doubly excited configurations of the HF wavefunction since 1) the HF wavefunction is stable to first order and 2) the singles do not mix with the HF ground state by Brillouin s theorem. Thus, using script notation defined earlier, the second order energy is 26

41 1 4 Ψ Ψ (2.45) Here, and are the orbital energies of the and occupied orbitals connected by to the virtual orbitals and, whose energies are and defined by (2.15). Again it is shown that the predominate contributor to the correlation energy is the double excitations (specifically the pair correlation) since it is the leading correction to the HF energy. Perturbative methods, in general, are not variational, and, thus, the energies obtained are not upper bounds to the exact energy. 22 However, MP2 recovers a significant fraction of the total correlation energy at a lower computational cost than both CISD and CCSD. Further, MPn is both size consistent and size extensive, making it a more desirable method than CISD. The size consistency of MPn, like CI and CC methods, hinges on the proper selection of the underlying HF wavefunction. If the zeroth order wavefunction is sufficient for describing the electronic state, then the MPn energies will be size consistent. In homolytic bond cleavage or dissociation of a molecule into open shell fragments, MPn, as discussed so far, will lose size consistency since the underlying reference wavefunction is not size consistent. 2.3 Density Functional Theory Instead of deriving electronic properties from a wavefunction, it is possible to derive electronic properties from an electron density. The concept of using an electron density in place of the wavefunction to solve the Schrödinger equation was explored by Thomas, Fermi, and Dirac from using a simplified quantum model called the homogeneous electron gas It was not until 1964, however, that the formal proof of principle was introduced by 27

42 Hohenberg and Kohn, 44 the landmark theorems of whom have become the basis for a modern quantum model called density functional theory (DFT) There are several fundamental differences between ab initio methodology and DFT. For one, the wavefunction cannot simply be replaced by the electron density in the Schrödinger equation and be expected to produce meaningful information. This is because the Hamiltonian is specified using the positions of the electrons, the information of which is not explicitly available in the electron density. Thus, the whole of DFT is dependent upon designing density functionals, which are operators analogous to and that act on densities instead of wavefunctions to extract the energy of the electronic system. Before discussing density functionals in detail, it is pertinent to define the concept of an electron density and discuss its properties. In the sections that follow, the fundamental theorems comprising atomic and molecular DFT are discussed. As already discussed, the wavefunction is a function describing the spatial and spin coordinates of the electrons in an electronic system. The distribution function or density of electrons is generally defined by ; Ψ,, Ψ,, (2.46) where the primed and unprimed electron coordinates represent two independent sets. 45,49 If the primed and unprimed sets are equal, then is just the squared modulus of the wavefunction, and the integration of (2.46) over all coordinates is unity (assuming normalization). Equation (2.46) is defined as the reduced density function (RDF) of order N. The two independent sets are interpreted as coordinates for an element of the reduced density matrix (RDM) of order N. 50 Other important RDFs are the first and second order RDFs, defined respectively as 28

43 ; d d Ψ,, Ψ,, (2.47) ; 1 d 2 d Ψ,,, Ψ,,, When and, both RDFs of (2.47) represent diagonal elements of their respective RDMs (corresponding to physical observables) such that ; d d Ψ (2.48) ; 1 2 / d d Ψ,, where is a spin orbital, and, is a spin geminal, or two electron function. Both the set of spin orbitals and spin geminals form orthonormal sets. The product of ; and a coordinate element is statistically equivalent to the probability of finding any electron within the coordinate element. If all possible electron coordinates are integrated over, then the result for is the number of electrons, while the result for is the number of unique electron pairs: d ; d (2.49) / d d ; d d,, 1 2 Using algebra and the results of (2.49), a relationship between and may be established: ; 2 1 d ; (2.50) Equation (2.50) is an important property that stems from a more general observation that firstorder RDFs may be used to construct higher order RDMs. 49,50 29

44 The diagonal first and second order RDFs can be further simplified to their respective spatial densities, denoted, by integrating out the spin component: 45,48 d ; d d d Ψ (2.51), d d ; 1 d 2 d d d Ψ This assumes that the electron spin has been included a posteriori as in (2.5). A similar process may be used to obtain the spin densities and, by integrating out the spatial components. If the orbitals are defined by the LCAO theory (2.8), then may be written in terms of AOs as (2.52) where is an element of the charge density matrix. 7 The electron label is spurious since the anti symmetric nature of the wavefunction does not allow individual electrons to be distinguished from one another, and will be dispensed with in the following sections. Equation (2.52) is the fundamental quantity that will be focused on in the Hohenberg Kohn and Kohn Sham theorems below The Hohenberg Kohn Theorems Much of the following derivations follow directly from the original paper of Hohenberg and Kohn. 51 Recall that the Hamiltonian (2.2) for a given electronic system is uniquely defined for a given nuclear electron potential, and thus, so are the wavefunction Ψ and energy. The extension that uniquely defines the density and that, in turn, uniquely 30

45 defines is the first Hohenberg Kohn (HK) theorem. To prove this, first consider two systems with different potentials: and. Now, assume that the potentials and produce the same density. Since the Hamiltonians and are uniquely defined, it follows that Ψ Ψ. The respective energies are Ψ Ψ Ψ Ψ Ψ Ψ (2.53) Ψ Ψ Ψ Ψ Ψ Ψ The integrals on the right hand side of the above equations may be further simplified to: Ψ Ψ d (2.54) Ψ Ψ d where the spatial density is used instead of the total density since the potential only depends on the spatial coordinates. Inserting (2.54) into (2.53) and adding the resulting equations together gives a contradiction,, disproving that two different potentials can lead to the same density. It is said that is a unique functional of, where a functional is a mathematical function that maps one function to another, and is written as. The implication of the first HK theorem is that given any density, the exact nuclear electron potential can be deduced. The second HK theorem demonstrates a variational approach to minimizing the total energy with respect to the electron density. Written as a density functional, the energy is Ψ Ψ Ψ Ψ Ψ Ψ (2.55) d where Ψ Ψ is a universal energy functional (universal in the sense that it 31

46 may be applied to any number of electrons in any potential) describing the kinetic and Coulomb energies. The total density is used in the expression since the term contains the exchange interaction. If the number of particles in the wavefunction remains constant (i.e. equation (2.51) is enforced), then Ψ may be varied to minimize the energy. If a trial wavefunction Ψ is based on a different potential than the ground state wavefunction Ψ, then d d (2.56) in which is the density of Ψ and is the density of Ψ. This equation demonstrates that the energy of an electron density may be minimized by varying the density arbitrarily with respect to a given potential. For convenience, the classical Coulomb repulsion is separated from the universal functional since it may be directly evaluated with one electron densities: 1 2 d d (2.57) Here, is another universal functional of the density that contains information on the kinetic, exchange, and correlation energies. The universal energy formula of Hohenberg and Kohn is written as follows, without any recourse as to the form of. d 1 2 (2.58) Given a trial density and the exact form of, the HK theorems demonstrate that the exact, non relativistic energy of an electronic system may be obtained by variationally minimizing the energy with respect to the density. In practice, a density must have certain properties, namely it must be N representable and V representable. 45,46 For an electron density to be N representable, there must be some anti symmetric wavefunction that corresponds to 32

47 it; while to be V representable, the density must be N representable and correspond to a unique potential (via the first HK theorem). Parr and Yang give a proof that one electron densities are automatically N representable if they can be decomposed as in (2.48) with occupation numbers between 0 and 1 (that is, the density obeys the Pauli principle). 45 By extension, two electron densities are also N representable by (2.50) if the one electron density is N representable. The conditions for a density to be V representable are not known, and, as a result, the variational nature of the second HK theorem is not strictly enforceable The Kohn Sham Method Following the seminal paper of Hohenberg and Kohn, Kohn and Sham formulated the HK theorems into a self consistent procedure entirely analogous to the HF SCF procedure with respect to one electron densities. 52 The first step in the Kohn Sham (KS) procedure is to recast the universal functional in terms of directly calculable quantities; writing it as a sum of the remaining energy terms, KS XC (2.59) where KS is the KS kinetic energy functional and XC is an exchange correlation (XC) functional. The kinetic energy functional KS is chosen to be that of the non interacting electron model, similar to the HF approximation: KS 1 2 d 1 2 d (2.60) Because a non interacting model is assumed, the kinetic energy of KS is deficient such that, KS Δ (2.61) 33

48 in which is the (unknown) exact kinetic energy functional and the residual Δ 0. The use of KS removes sole dependency of the model on the density and introduces orbitals called the KS orbitals. It is worth noting that (2.60) is not the only form that the kinetic energy functional may take. As Kohn and Sham and others have discussed, the use of orbitals in this manner helps to properly describe the shell structures of atoms. Using equations (2.59), (2.60), and (2.58), a KS equation can be written using the density functionals of each energy contribution as d XC (2.62) where is the KS orbital energy and the fourth term in the parentheses is the exchangecorrelation potential for a single electron. Solving the KS equations becomes an iterative procedure since the KS energy is a functional of the density: KS KS 1 2 XC (2.63) For the KS energy functional to be exact, the form of the XC functional must include not only exchange and correlation effects, but also the kinetic energy residual: 45,46 XC X C Δ (2.64) The energy expression of (2.64) is the attention of most modern DFT research. The exchange term X has a form that is known exactly (from the HF approximation), however, the form of the correlation functional C is not known. In practice, the exact HF exchange is only used in a class of density functionals called hybrid functionals, and even then, generally no more than 50% of the exact exchange is implemented. Another class of density functionals called pure functionals do not use HF exchange at all. When approximate exchange functionals are used in 34

49 place of the exact HF exchange, self interaction occurs, and must be corrected. 45,46 Selfinteraction is the spurious interaction of an electron with itself that occurs when the Coulomb and exchange energies do not exactly cancel as they would, cf. (2.13), if the exact HF exchange were employed. In using any class of density functionals to evaluate the KS equations, the KS orbitals do not retain any physical meaning, only a mathematical convenience for expressing the electron density. 45,48 As a result, Koopmans theorem does not strictly apply. Further, the variational nature of the second HK theorem is lost since the exact energy functional is not known, nor are the conditions for V representability. Despite these facts, the KS procedure does provide a much faster method of including (approximate) correlation effects in atoms, molecules, and extended systems without the computational cost associated with correlated ab initio methods. 2.4 Basis Set Theory Broadly speaking, a basis set is a collection of fixed elementary functions, called basis functions, the linear combination of which will describe any function in a given function space. 15 A basis set is called complete if it spans all space for any function, and orthogonal if the basis functions are linearly independent. In computational quantum chemistry, a basis set is a set of fixed elementary functions that are used to construct the electronic wavefunction. 6,7,18,19,22 As already discussed, the basis set must have the five characteristics of a physically viable wavefunction. It is known that the exact radial solution to the Schrödinger equation for the hydrogen 35

50 atom takes the form of a combination of Laguerre and Legendre polynomials in spherical coordinates. 6 In quantum chemistry, these functions are called Slater type orbitals (STOs): 6 Ψ,,, (2.65) Here, is an integer called the principle quantum number and is the exponential parameter that may assume any positive value and determines the radial extent of the wavefunction. The angular dependence, and hence, the orbital and magnetic quantum numbers (the integers and, respectively) of the exact solution enters through the spherical harmonic. The values of determine the shape of the orbital ( 0 is an s type function, 1 is a p type function, 2 is a d type function, etc.), while the value of takes values as, and determines the spatial orientation of the orbital. The number of radial nodes in a wavefunction is 1 and the number of angular nodes is. Since the many electron Schrödinger equation has no closed form solution, it is assumed that a wavefunction describing two or more electrons may be written as a linear combination of these one electron, or hydrogen like STO solutions since the Laguerre and Legendre polynomials form a complete set. This assumption is entirely analogous to the expansion of an electric potential in terms of Legendre polynomials in classical electrodynamics. 53,54 A many electron wavefunction expanded in a basis set of STOs is quite accurate, having the correct short range and long range behavior about the nucleus. 22 Variational methods employing STO basis sets produce ground state energies that quickly converge as the number of STOs increases. However, the 2 electron integrals that arise due to the use of STOs in solving the many electron Schrödinger equation are non trivial in molecular computations and must be solved using numerical techniques. 55 While many efficient numerical techniques exist for computing integrals, the computational demand dramatically increases 36

51 with the number of electrons in the system. As a result, computations involving STOs are generally reserved for atoms and small molecules (i.e. diatomics). For convenience, a Gaussian type orbital (GTO) is typically used in place of an STO This idea was introduced by Boys, 56,58 who postulated that the radial part of an STO could be represented, to a tolerable degree of accuracy, as a linear combination of GTOs, (2.66) where is a contraction coefficient and is the GTO exponential parameter. In fact, only an infinite sum of GTOs will exactly describe an STO. The summation above is called a contracted basis function or simply contraction, while the individual exponential functions comprising the sum are called primitive basis functions or just primitives. A wavefunction expanded in a basis set of GTOs exhibits some fundamental problems when compared with a wavefunction expanded in a basis set of STOs. First, the short range behavior of a primitive GTO varies quite markedly from that of an STO and results in an incorrect description of the wavefunction at the nuclear cusp. 22 Second, the long range behavior of a primitive GTO is such that it decays more rapidly than the corresponding STO. Both of these behavioral differences, demonstrated in Figure 2.4, may be circumvented (within tolerable accuracy) by using a large number of GTOs in the basis set. For example, the cusp may better be described by using more high exponent or tight functions, while the long range behavior is better described by more low exponent or diffuse functions in (2.66). The tradeoff between using a larger basis set composed of GTOs, compared with a smaller STO basis set, is the ease of computing 2 electron integrals. Whether employed in atomic or molecular computations, a GTO basis set produces 2 electron integrals that may be 37

52 Value of Function STO GTO Value of Function STO LC GTO Distance from Nucleus Distance from Nucleus Figure 2.4 Plots comparing the behavior of a single STO to a single GTO (left) and that of a single STO with a linear combination (LC) of three GTOs (right). The units are arbitrary. reduced to error functions through either a Fourier or Laplace transform. 55 The bottleneck of either of these transformed integrals is less computationally demanding than the numerical techniques required to compute 2 electron integrals using STOs. There are other mathematical tricks, such as projective or decomposition techniques, that may be employed to lower the computational bottleneck of the 2 electron integrals. One type of projective technique called the resolution of the identity is discussed in Chapter 7. The basis sets discussed in this dissertation are atom centered, as opposed to a common origin or between nuclei (bond centered), and are GTO based. The atom centered GTOs are optimized to describe the AOs of a free atom. When employed in molecular calculations, the LCAO theory is invoked. The question arises as to how many basis functions, and what values, should be chosen to form the basis set. The answer is dependent upon the system to be described and the degree of accuracy sought for the many electron wavefunction. For example, a free carbon atom may be described by three basis functions: two s type functions ( 0) for the 1s and 2s orbitals and a single p type function ( 1) for the 2p 38

53 orbital. This is an example of a minimal basis set, where there are just enough functions to cover the occupied AOs of each atom. Each occupied orbital is described by a contracted function rather than a single GTO due to the deficiencies of GTOs described earlier. The problem with employing a minimal basis set is two fold: 1) the basis set is not flexible enough to describe the many possible chemical environments an atom might be in and 2) employing a minimal basis set in correlated ab initio methods does not recover much correlation energy. 7 The electronic structure of a free atom is much different than the same atom in a molecule, and the electronic structure from atom to atom varies markedly as well. Thus, it is imperative to have a basis set that will be flexible enough, that is, have basis functions describing the proper orbital space, from one chemical environment to another. Consider a π bond between two carbon atoms, formed by the overlap of their 2p orbitals. If d type functions are added to the carbon basis sets, then some of the d orbitals will mix with the p orbitals by symmetry and cause the overlap forming the π bond to increase; a sort of p d hybridization occurs, lowering the bond energy. 18 These additional d type basis functions are called polarization functions because they polarize (distort) the 2p orbitals. It is typical of basis sets to include at least some polarization functions since these functions aid in the qualitative and quantitative description of chemical bonds. Polarization functions also become important in systematically recovering correlation energy (discussed in Chapter 3). An investigation reported by Schwartz on two electron systems demonstrated that the MP2 correlation energy is proportional to the highest orbital quantum number in the basis set. 59 This observation lends credence to the use of polarization functions with values well beyond the minimal basis set. 39

54 2.5 Model Chemistries The discussion thus far has covered different types of computational methods that construct approximate many electron wavefunctions (or densities) and basis sets that can be used to expand such wavefunctions. A dichotomy arises as to which computational method should be selected for a particular chemical system and which basis set will provide the most accurate expansion of the wavefunction. Ideally, one would like to pair the FCI method with a basis set that spans all possible AO space. This would give the exact solution to the nonrelativistic Schrödinger equation within the BO approximation. However, a basis set that spans all possible AO space requires an infinite linear combination of GTOs, making the approach intractable. Instead, a compromise must be made as to how many and what type of basis functions need to be employed to properly describe the electronic structure. Further, a compromise as to which computational method is appropriate and applicable to the system of interest must be made. Hehre et al. coined the term model chemistry, or level of theory to describe the coupling of a computational method and a given basis set. 18 This concept is graphically represented in Figure 2.5. Since doubles correlation is the dominant fraction of the total correlation energy, compared with other excitation levels, ab initio methods such as CISD, CCSD, and MP2 are generally sufficient and tractable for systems up to 25 non hydrogen atoms and a modest sized basis set (the inclusion of singles in addition to doubles does not appreciably increase the computational cost). In some electronic systems, triple and quadruple excitations make appreciable contributions to the total correlation energy. To recover triples and some quadruples, one may use the CCSDT method (note that CISDT is not usually employed since it is 40

55 HF Limit Complete Basis Set Limit Exact Basis Set Size Applicable Model Chemistries Full CI No Correlation 0 Excited Determinants 2 Figure 2.5 A conceptual view of basis set size versus level of sophistication in correlated ab initio methods. The number of determinants with respect to excitation level (n), number of electrons (N), and basis set size (K) is given by the formula on the lower axis. not size consistent). However, the computational expense of such methods dramatically increases with the number of electrons in the system. To avoid using CCSDT, the perturbative correction to triple excitations, CCSD(T), is used, which recovers a significant portion of the triples correlation energy without the computational overhead of a full CCSDT calculation. As already discussed, a basis set beyond the minimal set of AOs is usually employed in ab initio methods. This is generally true of DFT methods, as discussed in Chapter 4. The composition of a basis set varies from system to system and chemical property to chemical property. For example, a polarized basis set optimized with HF will not have basis functions optimal for recovering correlation energy if employed in a correlated ab initio model chemistry. As a result, it is desirable to have a basis set with basis functions optimized for both HF and correlated ab initio methods. The correlation consistent basis sets are of this type and are discussed in detail in Chapter 3. Further, the computational methods discussed so far have 41

56 been non relativistic, and if relativistic effects are to be included, a basis set optimized with a non relativistic method will not recover all possible relativistic effects. It is pertinent that the basis set employed in relativistic computations have core AOs optimized for the relativistic Hamiltonian employed since the major fraction of relativistic effects come from these orbitals. Relativistic basis sets are discussed in detail in Chapters 3 and 6. Finally, there is a special class of model chemistries called composite methods in which individually energy components of the total energy of a system are computed by different levels of theory and specialized basis sets in an effort to achieve an approximation to a higher (or intractable) level of theory. 18,19 Often time, for example, only the valence correlation energy is computed, but the correlation between core and valence electrons (called core valence correlation) is important for accurately computing thermochemical properties. Further, most computational methods are non relativistic, but atoms from the second row of the main group, and heavier, have significant relativistic effects. To include these two energy components, a composite method may compute the core valence energy and relativistic corrections at a low level of theory like MP2 using specialized basis sets. These individual energy components are then added to a previously computed reference energy that was computed at a higher level of theory. There are several composite methods in popular use including the Gaussian n (Gn), complete basis set (CBS n), high accuracy extrapolated ab initio thermochemistry (HEAT), 70 Weizmann n (Wn), 71,72 and correlation consistent Composite Approach (ccca) methods. Chapter 7 discusses the ccca formulism in detail, including how the method recovers corevalence and relativistic effects. 42

57 CHAPTER 3 CORRELATION CONSISTENT BASIS SETS 3.1 Introduction Having established the basics of basis set theory in computational quantum chemistry in the previous chapter, a specific class of basis sets called the correlation consistent polarized valence basis sets (cc pvnz, where n = D, T, Q, etc.) are discussed at length in this chapter. The correlation consistent basis sets were introduced by Dunning as a means to systematically recover correlation energy in correlated ab initio computations. The novel approach of designing basis sets to help recover correlation energy was first studied by Ahlrichs et al and by Almlöf and Taylor. 113,114 Dunning s approach was similar to that of Almlöf and Taylor in which 1) high angular momentum functions (d, f, g, etc.) are employed beyond the minimal basis set and 2) basis functions are grouped according to their contribution to the total correlation energy. Further, the optimized atomic orbitals (AOs) in the basis set are taken to be the natural orbitals from correlated ab initio wavefunctions. The natural orbitals (NOs) are obtained by diagonalizing the correlated density matrix, 7,22,50 and have the intrinsic property of forming a basis in which the correlated wavefunction converges most rapidly (as opposed to, say, the molecular orbital basis). The primary difference between the basis sets developed by Almlöf and Taylor and those of Dunning is the smaller sets of functions used by Dunning. In fact, Dunning s basis sets recover more than 99% of the correlation energy obtained by the basis sets of Almlöf and Taylor, yet contain less basis functions. 43

58 The correlation consistent basis sets for the main group atoms were constructed using Gaussian type orbital (GTO) basis sets developed by van Duijneveldt. 115 The AOs represented in each basis set were taken from non relativistic multi configuration self consistent field (MCSCF) calculations, yielding a set of AOs that adequately describe the valence orbital degeneracy inherent to most atoms. Each AO is represented by a general contraction of GTOs, 114,116,117 in which all symmetry equivalent basis functions are grouped in the contraction. From this point on, this minimal set of AOs will be called the Hartree Fock (HF) basis set. Within each family of correlation consistent basis sets, there is a hierarchy called the ζ level, where the quality and size of the basis set increases with the ζ level. More than that, the ζ level tells how many valence basis functions are used per atom. For example, a double ζ basis set contains two functions per valence orbital; a triple ζ basis set contains three functions; etc. The underlying principle of the ζ level is that more functions in the valence space give more flexibility to the basis set. The core orbitals are always described by a single contracted GTO at each ζ level. Because only the valence orbitals are described by multiple functions, the correlation consistent basis sets are often referred to as split valence basis sets. Table 3.1 lists the composition of the correlation consistent basis sets for main group atoms (including H and He) according to each ζ level. Note that the primitive, or uncontracted, basis functions are denoted by parentheses, while the contracted functions are denoted by brackets. The notation 3s 2p 1d indicates that three s type functions, two p type functions, and one d type function comprise the basis set, and are not to be confused with the principle and orbital quantum numbers. The following sections describe the process by which the correlation consistent basis sets were developed by Dunning and coworkers, including basis sets for valence only 44

59 Table 3.1 The composition of the correlation consistent basis sets for the first three rows of the main group. Row / Atoms ζ level Primitives Polarization Contracted Basis Set H, He D ( 4s ) ( 1p ) [ 2s 1p ] T ( 5s ), ( 6s ) a ( 2p 1d ) [ 3s 2p 1d ] Q ( 6s ), ( 7s ) a ( 3p 2d 1f ) [ 4s 3p 2d 1f ] 5 ( 8s ) ( 4p 3d 2f 1g ) [ 5s 4p 3d 2f 1g ] 1 st / B Ne D ( 9s 4p ) ( 1d ) [ 3s 2p 1d ] T ( 10s 5p ) ( 2d 1f ) [ 4s 3p 2d 1f ] Q ( 12s 6p ) ( 3d 2f 1g ) [ 5s 4p 3d 2f 1g ] 5 ( 14s 8p ) ( 4d 3f 2g 1h ) [ 6s 5p 4d 3f 2g 1h ] 2 nd / Al Ar D ( 12s 8p ) ( 1d ) [ 4s 3p 1d ] T ( 15s 9p ) ( 2d 1f ) [ 5s 4p 2d 1f ] Q ( 16s 11p ) ( 3d 2f 1g ) [ 6s 5p 3d 2f 1g ] 5 ( 20s 12p ) ( 4d 3f 2g 1h ) [ 7s 6p 4d 3f 2g 1h ] 3 rd / Ga Kr D ( 14s 11p 5d ) ( 1d ) [ 5s 4p 2d ] T ( 20s 13p 8d ) ( 2d 1f ) [ 6s 5p 3d 1f ] Q ( 21s 16p 11d ) ( 3d 2f 1g ) [ 7s 6p 4d 2f 1g ] 5 ( 26s 17p 12d ) ( 4d 3f 2g 1h ) [ 8s 7p 5d 3f 2g 1h ] a. Indicates the primitive set for He. correlation, long range correlation, core valence correlation, and the recovery of scalar relativistic effects. 3.2 Valence and Tight d Basis Sets As already discussed, the HF basis set is insufficient for recovering correlation energy in correlated ab initio computations. Further, high angular momentum (polarization) functions are necessary for the qualitative and quantitative description of chemical bonds and necessary for expedient recovery of all possible correlation energy by a given ab initio method. Polarization 45

60 functions optimized using HF will aid in the non correlated description of chemical bonds, however, HF polarization functions are far from optimal as virtual orbitals in correlated methods. Therefore, the polarization functions of the correlation consistent basis sets were optimized as an even tempered expansion using singles and doubles configuration interaction (CISD). An even tempered expansion of basis functions is a method of determining the exponential parameters using a geometric series: 116,118, 1,2, (3.1) The parameters and are unique for each group of basis functions with the same value and are varied so as to minimize the correlation energy. The advantages of using (3.1) instead of explicit optimization of each polarization function are 1) there are only two parameters to optimize for the entire group of functions and 2) the polarization functions are evenly distributed in log space, providing even coverage of the orbital space. 22 The high angular functions were optimized using (3.1) in groups according to their values, and were added to the HF basis set of each atom, as suggested by Schwartz, 59 until the correlation energy did not change appreciably. By adding functions this way, the correlation space for each particular value is saturated. Dunning noted that certain functions within each group contributed similar amounts of correlation energy. Specifically, it was noted that the first d function contributed most to the correlation energy; while the second d and first f functions contributed similar amounts to the correlation energy; the third d, second f, and first g functions contributed similar amounts to the correlation energy; etc. From this, Dunning suggested that high angular functions should be added to the HF basis sets in a correlation consistent manner, that is, in shells as ( 1d ), ( 2d 1f ), ( 3d 2f 1g ), etc. In this way, the 46

61 correlation energy is systematically recovered with each additional shell of high angular functions. Further tests were performed by explicitly optimizing the high angular functions within each group, but no appreciable change was noted in the correlation energy, so the even tempered scheme was kept. Even tempered optimizations of s and p functions in addition to the HF basis sets were also performed with respect to the correlation energy. Dunning found that uncontracting the outermost s and p functions had the same effect of lowering the correlation energy as explicit optimization. The split valence nature of the correlation consistent basis sets stems from the number of s and p functions that are uncontracted: a single s and p function are uncontracted at the double ζ level, two s and p functions are uncontracted at the triple ζ level, etc. (cf. Table 3.1) After being introduced, studies by Bauschlicher and Partridge 119 and Martin et al employing the cc pvnz basis sets showed that including an additional tight d function significantly improved the computed atomization energies of molecules containing second row atoms. While the most dramatic improvements were shown for energetics of sulfur species, Bauschlicher and Partridge also showed that the atomization energies of SiH 3 and SiH 4 were affected by 1.1 and 1.6 kcal/mol, respectively, when a tight d function was employed in the triple ζ basis set. The role of an additional tight d function in second row atoms was commented on by Woon and Dunning in their original paper, however, the need for such a function did not become evident until the properties of a wider variety of molecular species than were in the original benchmarking papers were studied. 123 As a result, Dunning, Peterson, and Wilson modified the sets, developing the tight d correlation consistent basis sets, cc pv(n+d)z, 124 and in numerous studies, have demonstrated the importance of the augmented 47

62 tight d basis sets in coupled cluster and density functional theory studies. 101, The correlation consistent build up of high angular functions with increasing ζ level leads to smooth, monotonic behavior of total energies. As the basis set becomes saturated at high ζ levels, more and more of the AO space is covered by the basis functions, resulting in a nearly complete basis set. Once the basis set reaches completeness, any remaining error between a computed chemical property and a reliable experimental measurement is due solely to the computational method. As the ζ level increases, the total energy computed with the cc pvnz basis sets asymptotically approaches the complete basis set (CBS) limit. 93 The exact mathematical form of the monotonic behavior is not known, but resembles an exponential type decay. To this end, several empirical extrapolation techniques have been developed to approximate the CBS limit. 85,95, Two popular extrapolation schemes are below: CBS (3.2) CBS (3.3) Equation (3.2) is a standard exponential fit due to Feller, 132 while (3.3) is a mixed Gaussian/exponential fit due to Woon, Dunning, and Peterson. 85,95 In each equation, and are fitting parameters, is the chemical property (i.e. energy) computed at the th ζ level ( 2 for double ζ, 3 for triple ζ, etc.), and CBS is the estimated CBS limit. Both equations are analytic for three points, but are numerical for any more. Taking to be the lowest ζ level in a series of three points, the analytic forms of the parameters in (3.2) are 48

63 CBS 2 (3.4) 2 ln while those in (3.3) are CBS (3.5) Another extrapolation formula in common use is that of Halkier et al., 133 a two point scheme based on the convergence of correlation energies in atoms: CBS 1 1 (3.6) in which takes on the same meaning as in (3.2) and (3.3). The virtue of using (3.6) over (3.2) or (3.3) is that it only requires two points to estimate the CBS limit. However, it is not possible to estimate higher ζ levels with (3.6) as it is with (3.2) and (3.3). Further, using one of the exponential type extrapolation formulas will sometimes provide a better approximation to the true CBS limit since more points are used in the fit. 49

64 3.3 Augmented Basis Sets In order to quantitatively describe chemistry stemming from long range phenomena such as van der Waals interactions, Rydberg states, multipole moments, polarizabilities, hydrogen bonding, and the binding of excess electrons, diffuse functions must be employed in the basis set. Dunning and coworkers have developed augmented correlation consistent basis sets (aug cc pvnz) describing such chemical phenomena. 86,88,91,93,95,99,100 The diffuse functions added at each ζ level are comprised of a single function of each value present. For example, to the double ζ basis set is added a set of ( 1s 1p 1d ) functions, to the triple ζ basis set is added a set of ( 1s 1p 1d 1f ) functions, etc. The diffuse s and p functions were optimized on top of the HF basis set in MCSCF calculations of the atomic anions, while the high angular functions were optimized with CISD on top of the polarization functions already present. 3.4 Core Valence Basis Sets Valence only, or valence valence (VV) correlation is generally accepted to be adequate for accurately computing most chemical properties since the qualitative and quantative description of many chemical properties relies on changes in the valence electronic structure. As already discussed, the cc pvnz and aug cc pvnz basis sets recover this correlation systematically with increasing ζ level. Also well known is that in order to achieve ±1.0 kcal/mol accuracy, or better, in thermodynamic properties, core valence (CV) correlation must also be included. 93,134 The cc pvnz and aug cc pvnz basis sets are not optimal for the recovery of CV correlation and were reoptimized specifically for the recovery of CV correlation by Woon, 50

65 Peterson, and Dunning. 81,98 There are two types of CV correlation consistent basis sets, the original CV basis sets (cc pcvnz) 98 and the weighted core valence (wcv) correlation consistent basis sets (cc pwcvnz). 81 The principle difference between the CV and wcv basis sets is that the former are designed to recover more core core (CC) correlation energy, while the wcv basis sets are intentionally biased towards CV (almost no CC) correlation energy. The optimization of the core correlation (tight) functions proceeded by adding H/He like basis sets for correlating the 1s electrons to the first row, main group atoms. Woon and Dunning found that adding groups of tight functions as ( 1s 1p ) to the double ζ basis set, ( 2s 2p 1d ) to the triple ζ basis set, etc. resulted in smooth, monotonic behavior of the CC, CV, and VV energies, as opposed to the erratic behavior of these energies when the valence only cc pvnz basis sets were used. The following formula was used to optimize the core correlating functions: Δ CISD CC CV VV CISD CC CISD CV CISD VV (3.7) The weight,, in the above equation is taken to be 1.00 in the cc pcvnz basis sets, and to be 0.01 in the cc pwcvnz basis sets from an in depth study of CC and CV effects of first and second row molecules by Peterson and Dunning. 81 It has been noted by Peterson 81,93 that both the cc pcvnz and cc pwcvnz basis sets converge on similar CBS limits, although the cc pwcvnz basis sets tend to produce lower energies in valence only calculations because of the more diffuse nature of the weighted core correlation functions. 3.5 Scalar Relativistic Basis Sets As discussed in section 2.4, basis sets optimized with non relativistic methods are not 51

66 optimal for use in relativistic calculations. The correlation consistent basis sets discussed so far were all developed using non relativistic formulisms of HF and CISD. One of the most common relativistic models is the spin free Douglas Kroll (DK) Hamiltonian, which approximates the full, four component Dirac equation 138,139 by a transformation of the electron orbitals. The resulting orbitals are then used in a subsequent HF or correlated ab initio calculation to include scalar relativistic effects such as the mass velocity and Darwin corrections, the two largest contributors to the relativistic energy. 137,140,141 These corrections are termed scalar relativistic since they do not include inner products containing spin components (i.e. spin orbit coupling, spin spin coupling, etc.) The largest effect of scalar relativity occurs in the core orbitals, which contract inward toward the nucleus due to their increased kinetic energy. Because the core orbitals of the cc pvnz basis sets are represented by a single contracted GTO, there is no flexibility in the core to change under the DK transformation, resulting in a loss of accuracy within the relativistic model. Thus, the core contractions must be reoptimized. Instead of recontracting only the core orbitals, all of the contractions are typically reoptimized to account for variations in the valence orbitals due to fluctuations in the core. The first to report on correlation consistent basis sets optimized for scalar relativistic computations, denoted cc pvnz DK, were de Jong et al. 142 Their scheme for recontracting the cc pvnz basis sets is as follows: 1) remove the polarization functions (d, f, g, and h) from the cc pvnz basis sets, and uncontract the s and p basis set; 2) run a HF calculation using the uncontracted sp basis set and the DK Hamiltonian; and 3) the optimized s and p contraction coefficients are then determined by a population analysis. A similar scheme is used in Chapter 6, in which DK optimized basis sets for the s block atoms are reported. 52

67 CHAPTER 4 SYSTEMATIC TRUNCATION OF THE CORRELATION CONSISTENT BASIS SETS IN DENSITY FUNCTIONAL THEORY CALCULATIONS 4.1 Introduction The plethora of density functionals available 46 has brought about a tremendous challenge in applying density functional theory (DFT) 44,45,52 to the study of chemical systems, namely, the choice of a density functional that is suitable for the problem of interest. Unfortunately, no simple means currently exists to aid in density functional selection, barring whether or not a density functional was specifically developed for a particular problem, and reliance upon benchmark studies, when available, is essential. However, it is of great interest to have a simple, efficient means to gauge the intrinsic error of a density functional. To accomplish this, a first step is to develop a greater understanding of the behavior of density functionals with respect to the basis set. As shown in Chapter 3, the correlation consistent basis sets exhibit smooth, monotonic convergence in energetic properties with increasing basis set size, and that the complete basis set (CBS) limit may be numerically approximated through the use of various extrapolation schemes. Unfortunately, it is not clear whether or not such an approach is possible for DFT calculations since the correlation consistent basis sets were developed specifically for ab initio This entire chapter is adapted from B.P. Prascher, B.R. Wilson, and A.K. Wilson, Behavior of Density Functionals with Respect to Basis Set. VI. Truncation of the Correlation Consistent Basis Sets. J. Chem. Phys. 2007, 127, , with permission from the American Institute of Physics. 53

68 methods. Energetic and structural properties are generally known to converge more quickly with respect to increasing basis set size for DFT than for ab initio methods. 125,126, In fact, it is commonly assumed that small changes in energetic properties beyond the triple ζ basis set level typically occur, but recent work has demonstrated that significant changes in energetic properties sometime do occur beyond the triple ζ level. 148,149 Further, recent work has shown non monotonic behavior of energetic properties can occur with respect to increasing basis set size in DFT calculations. 125,126, This observation has been made using a number of molecules and density functionals, with no obvious trends appearing as to when the nonmonotonic behavior does or does not occur. When monotonic behavior does occur, then it has been shown that various extrapolation formulas can be used to estimate the CBS limit, or in the case of density functional theory, the Kohn Sham (KS) limit. 144,145, However, with nonmonotonic behavior, such an approach is not reliable. To date, no single solution to the nonmonotonic problem has evolved; rather, a number of possible solutions have emerged. Provided a reasonable grid size is utilized for the numerical calculation of the DFT energy, grid selection does not seem to impact the basis set convergence. 156 Other factors that influence convergence include basis set superposition error 152 and basis set contraction, 156 which, when combined, have been shown to lead to monotonic convergence (discussed in the next chapter). 148 It is important to note that systematic convergence to the KS limit should not necessarily be expected from the correlation consistent basis sets as they were explicitly developed for ab initio methods. However, Jensen has developed a series of polarization consistent basis sets (pc n, where n = 1 4) explicitly for DFT, and non monotonic behavior 54

69 using various density functionals is still observed for these basis sets. 155 Ab initio methods rely directly on the inclusion of many excited determinants, constructed from high angular momentum functions, while density functionals do not. High angular momentum functions in DFT computations have the same effect as in Hartree Fock (HF), that is, to polarize the electron density. As a result, DFT computations are less basis set dependent than ab initio computations, and DFT computations, with systematic increases in basis set size, tend to converge quickly. Rapid convergence towards the KS limit is desirable, but the high angular momentum functions included in many basis sets may not have a significant effect on a computed property. Thus, these functions create unnecessary overhead in terms of computational expense and, as shown in this chapter, may be removed without the introduction of significant error. Earlier work has utilized truncated or auxiliary basis sets as a means to reduce computational cost in DFT calculations. For example, the resolution of the identity Coulomb approximation (RI J) has been studied by Skylaris et al. 159 and by Head Gordon et al. 150 In the RI J approximation, an auxiliary basis set is used to compute the Coulomb integrals, which can reduce the computational cost without introducing errors of more than a few millihartrees in the total energy. 159 Head Gordon et al. have also investigated a dual basis approach, first proposed by Van Alsenoy and Vahtras et al. 160,161 in which a truncated form of a basis set, such as cc pvtz without f functions or the inner d functions, is used to construct an initial approximation to the full cc pvtz basis density. Subsequent parameterization is included to help account for the difference in densities between the full cc pvtz basis set and the truncated basis set. 151 Earlier 55

70 work by Jensen examined truncation of core basis functions from double and triple ζ polarized basis sets (DZP and TZP, respectively) as well as the pc n basis sets. 147 The conclusions from his investigation indicate that basis sets truncated of certain core functions lead to results similar to those obtained when the full basis sets are employed. However, truncation of core functions is not suitable for the accurate determination of KS limits due to the incomplete coverage of the exponent space. The present investigation examines the truncation of valence basis functions from the correlation consistent basis sets without parameterization, and does not suffer from reduced coverage of the necessary exponent space since neither the s or p functions are removed from the HF basis set. By analyzing the change in computed properties as basis functions are systematically truncated, the necessity of high angular momentum functions becomes clear, as was demonstrated in a similar study by Mintz et al. in which truncated correlation consistent basis sets were studied in ab initio methods. 162,163 Understanding how high angular momentum functions impact not only computed properties, but also basis set convergence, will aid in designing basis sets for DFT computations. 4.2 Computational Methodology Two popular density functionals have been included in this study, including pure and hybrid density functionals. The pure density functional is the gradient corrected Becke (B) 164 exchange functional, coupled with the Lee Yang Parr (LYP) 165 gradient corrected correlation functional, BLYP. For the hybrid DFT computations, the Becke three parameter scheme (B3) 166 is employed with the LYP correlation functional, utilizing the third parameterization of the 56

71 Vosko, Wilk, and Nusair local correlation functional (VWN 3). 167 The correlation consistent basis sets, cc pvnz, through quintuple ζ quality have been employed. Systematic truncation of these basis sets follows that of the scheme published by Mintz et al. for the hydrogen atom, 162,163 but is extended to include non hydrogen atoms. The truncation scheme involves removing the highest value basis function first, then the most diffuse function from the 1 shell, then the second most diffuse function from the 1 shell, etc. The truncation continues through the polarization functions, but does not include the contracted s and p functions. The notation used is as follows: the full triple ζ basis is denoted as usual: cc pvtz; the first truncation of this basis is denoted cc pvtz( 1f); the second truncation is denoted cc pvtz( 1f 1d); and the third truncation is denoted cc pvtz( 1f 2d) which reflects the removal of one f function and the two most diffuse d functions (for cc pvtz, this means removal of all of the d functions). It is implied that the first truncation of a specific shell is the most diffuse (i.e. lowest exponent). In atomic computations it is common practice to perform a symmetry averaging of degenerate electronic states since this leads to a more physically sound wavefunction. If the resulting wavefunction is spherically symmetric ( 0), then basis functions of angular momentum not present in the filled, state averaged orbitals can be separated from the total wavefunction without consequence to the total energy. This phenomenon has been investigated in great detail by Bauschlicher and Taylor. 168 This chapter strictly employs single reference wavefunctions in all computations. Atomic symmetry breaking of the type used here has been discussed in rigorous detail elsewhere, and is often necessary to correctly compute electronic properties. 46,169 As atom centered basis sets are typically used in molecular 57

72 computations, where atoms are not spherically symmetric, broken symmetry in atomic computations is an appropriate means to effectively gauge basis set truncation effects. Vertical ionization potentials and electron affinities have been determined using experimental geometries for the neutral molecules taken from the National Institute of Standards and Technology (NIST) Webbook The geometry was fixed so that the impact of removing high angular momentum functions could be gauged without a geometry optimization affecting the energy. Optimized geometries and their atomization energies are reported after gauging the impact of truncation. The use of diffuse functions in the basis set has been restricted to only the s and p functions here, which are taken from the augmented (aug cc pvnz) basis sets, and have been employed in computations of electron affinities. 173 Atoms of the first and second rows, their dimers, and the molecules CH 4, SiH 4, NH 3, PH 3, H 2 O, and H 2 S have been investigated. All DFT computations of this study have been performed using the Gaussian software suite of programs. 174 The default pruned grid (75,302) 175 has been used in all computations, and total energies in all computations are converged to 10 8 E h, or better. 4.3 Results and Discussion Atoms Total energies for the first row atoms carbon and oxygen are given in Table 4.1. Ideally, the energy difference between a truncated basis and the full basis set should not be more than 1.0 me h (approximately 1.0 kcal/mol) since this is the typical accuracy sought for computed energetics when compared with experiment. As shown in the table, truncation effects in BLYP 58

73 computations are similar to those in B3LYP computations. The largest difference between BLYP and B3LYP as a result of truncation occurs for carbon and is me h at the cc pvtz( 1f 2d) level. Comparing carbon to oxygen, the effect of truncation on the total energy of oxygen is almost twice that of carbon. For example, at the cc pv5z( 1h 2g 3f 4d) level for carbon, the change in the BLYP energy with respect to the full basis set is me h, while for the oxygen atom the change is me h. The greatest difference between the truncated and full basis sets occurs for the fully truncated quintuple ζ basis set, cc pv5z( 1h 2g 3f 4d), for all atoms investigated. Generally, the fully truncated cc pv5z value for the total energy is comparable to the full cc pvqz basis set value for atoms with non spherical wavefunctions. Computed ionization potentials and electron affinities of carbon and oxygen are shown in Table 4.2 and Table 4.3, respectively. It is immediately obvious that electron affinities are not as affected by truncation as ionization potentials (note that the electron affinities have been computed using the s and p diffuse functions from the aug cc pvnz basis sets). In the oxygen atom, significant changes (i.e. more than 0.04 ev, or 1.0 kcal/mol) in the computed ionization potential do not occur until f functions are completely removed. The carbon atom ionization potentials show little change relative to the full basis, even with complete truncation of all high angular momentum functions. Electron affinities, on the other hand, do not change until d functions are truncated, and even then, the differences between the full and truncated basis sets are less than 0.04 ev. There is little dependence on the density functional used since truncation effects appear at the same level for B3LYP and BLYP. For example, the ionization potential of oxygen begins to be affected by truncation at the cc pvtz( 1f), cc pvqz( 1g 1f), and cc pv5z( 1h 2g 1f) levels in both BLYP and B3LYP. A similar trend results for electron affinities. 59

74 Figure 4.1 is a plot of the full basis set versus truncated basis set ionization potential of oxygen determined with BLYP. Because basis functions of 3 do not significantly affect the computed ionization potentials of atoms, a plot of each basis set containing only s, p, d, and f functions is included for comparison in Figure 4.1. The 3 values coincide well with the full basis set values. The inset in the figure demonstrates the effect of different truncation levels at the cc pvqz level on computed ionization potentials. Based on these observations for first row atomic properties, a basis set for use in DFT computations need only contain up through f functions to achieve full cc pvnz basis set results for total energies, ionization potentials, and electron affinities within 1.0 kcal/mol for atoms Homonuclear Diatomics The BLYP and B3LYP total energies for the first row diatomics C 2 and O 2 are shown in Table 4.4. In general, truncation affects the diatomic total energies by almost an order of magnitude more than in atoms. This is expected, since the high angular momentum functions are necessary for a better description of bonding and anti bonding orbitals, specifically d and f functions which polarize p functions in σ and π type bonds. This is readily seen for each diatomic investigated, where truncation of f functions from the cc pv5z basis set typically causes the difference in total energy relative to the full basis to increase on the order of me h, and the truncation of d functions increases on the order of tens of me h. Table 4.5 lists computed ionization potentials, electron affinities, and optimized bond lengths for the first row dimers with BLYP. The ionization potentials remain unaffected by truncation until d functions have been truncated, but the electron affinities are affected with truncation of the f functions. 60

75 Figure 4.1 BLYP ionization potentials of the oxygen atom computed with truncated basis sets (hash marks) and plotted against the full basis set values. The inset shows more detail at the quadruple ζ level. Points denoted l > 3 are the truncated basis sets with only s, p, d, and f basis functions. The general trend in truncation of d functions affecting ionization potentials and f functions affecting electron affinities remains consistent. Note how truncation affects ionization potentials and electron affinities in diatomics as compared with atoms. In atoms, ionization potentials computed with a truncated basis are lower than the full basis set results, but are higher for diatomics (cf. Figure 4.1 and Figure 4.2). The electron affinities of atoms computed with a truncated basis are typically higher than the full basis result, and in diatomics they are lower than the full basis set result, with the exception being when d functions are truncated. Truncation leads to a systematic change in the computed property for atoms, but this trend is not observed for diatomics. Figure 4.2 demonstrates the impact of truncation upon the 61

76 Figure 4.2 BLYP ionization potentials of the O 2 molecule ( 3 Σ g ) computed with truncated basis sets (hash marks) and plotted against the full basis set values. Points denoted l > 3 are the truncated basis sets with only s, p, d, and f basis functions. computed ionization potential of the oxygen molecule. It is clear that at least some of the f functions must remain to maintain atomization energies that are within 1.0 kcal/mol of the full basis set values. An example of how f functions impact atomization energies is shown by the truncation at the double and triple ζ levels for the oxygen molecule in Table 4.5. The removal of the only f function from cc pvtz lowers the computed atomization energy by 1.4 kcal/mol and the removal of the most diffuse d function increases this deviation almost an order of magnitude. At the quadruple ζ level, removing one f function significantly increases the deviation from the full basis more so than the removal of the higher g function. Truncation through the most diffuse f function of the quintuple ζ basis 62

77 does not significantly alter the computed atomization energy, but removing the second f function does. It is clear that the f shell must be retained in the homonuclear diatomics to minimize the deviation of the truncated basis atomization energy from that of the full basis. Optimized bond lengths follow a trend similar to that of molecular electron affinities in that they depend heavily on the d functions in the basis set. Target accuracy for truncated basis set bond lengths is 0.01 Å relative to the bond length determined using the full basis set. The removal of the d function from the double ζ basis set in a BLYP computation of the oxygen molecule affect the optimized bond length by Å. At the triple ζ level, truncation of the most diffuse d function significantly affects the bond length and removal of the entire d shell causes the bond length to deviate by Å from the full basis set result. The most diffuse d function of the quadruple and quintuple ζ basis sets does not significantly affect the bond length upon truncation, but removal of any of the other d functions from these bases significantly impacts the computed value. These deviations in truncated basis set bond lengths for BLYP are also typical in B3LYP computations CH 4, SiH 4, NH 3, PH 3, H 2 O, and H 2 S Table 4.6, Table 4.7, and Table 4.8 provide B3LYP optimized geometries for CH 4, NH 3, H 2 O, SiH 4, PH 3, and H 2 S. The truncation notation used in these tables has two terms. The first represents the truncation of the non hydrogen basis set, and the second represents the truncation of the hydrogen basis set. For example, at the triple ζ level, 1f 2d ; 1d 2p denotes truncation of one f and two d functions from the non hydrogen basis and one d and two p functions from the hydrogen atom basis set. Both truncations are performed simultaneously. 63

78 Included in the tables for CH 4 and SiH 4 are atomization energies. Table 4.6 shows that the atomization energies in heteronuclear molecules are sensitive to the basis set truncation, varying more than 1.0 kcal/mol when truncation of d functions from the non hydrogen basis (including p functions from the hydrogen basis) occurs. Compared with the atomization energies for homonuclear diatomics, the atomization energies for larger heteronuclear molecules is less sensitive to basis set truncation overall. In fact, the atomization energy of the oxygen molecule suffers almost a 30.0 kcal/mol deviation from the full basis set result when truncation of all of the high angular momentum functions from each basis set is done. For methane, full truncation of the basis sets only results in a 10.0 kcal/mol deviation from the full basis set atomization energy. The atomization energy for silane is comparable to that of oxygen in that the largest angular momentum functions required are f functions in the non hydrogen basis set to maintain energies within 1.0 kcal/mol of those of the full basis set. Therefore, f functions remain necessary in the non hydrogen basis set for computed atomization energies. In molecules composed of first row atoms, the computed bond lengths do not vary by more than 0.01 Å when the d and p shells are truncated from the non hydrogen and hydrogen basis sets, respectively. In contrast, molecules which include a second row atom do depend on the d and p shells from the non hydrogen and hydrogen basis sets, respectively, to avoid deviating by more than 0.01 Å from the full basis set bond length. This is not surprising considering that the second row atoms are known to form bonds with more 3p character than 3s, and the 3p orbitals are polarized by d functions present in the basis set. Further, angles of second row systems are not as dependent on the d and p functions of the non hydrogen and hydrogen basis sets, respectively. This is the direct result of more p character in the bonding 64

79 molecular orbitals for these compounds (cf. Table 4.7 and where the H P H angle is 93 and the H N H angle is 106 ; the H S H angle is 92 and the H O H angle is 104 ). As a result of the different bonding characteristics of first and second row systems, truncation of the first row non hydrogen/hydrogen basis sets of their f/d functions the second row nonhydrogen/hydrogen basis sets of their d/p functions affects bond angles more than 0.1. Basis set truncation is therefore only effective in keeping the accuracy of the full basis set in bond lengths and angles (within 0.01 Å and 0.1 ) when the d and p shells are present in non hydrogen and hydrogen basis sets, respectively, for second row systems; and when f and d functions are present, respectively, in first row systems Computational Time Savings In atomic computations, it is observed that truncating most of the high angular momentum functions from the quintuple ζ basis could produce total energies that are comparable with the full quadruple ζ basis energies at a fraction of the computational cost. In heteronuclear molecules, if the basis set is no longer truncated when the computed angles begin to vary by more than 0.1 from the full basis set result, then the computational cost is significantly reduced without the introduction of significant error compared with the full basis. The benefits are two fold: 1) computational time can be reduced significantly and 2) both the bond length and the angle will be within 0.01 Å and 0.1, respectively, from the full basis set results. Bond lengths computed with truncated basis sets remain steadily within 0.01 Å of the full basis. Table 4.9 lists the percent central processing unit (CPU) time saved in single point computations for the heteronuclear molecules, and Figure 4.3 illustrates how much time is 65

80 saved with respect to energetic properties at various truncation levels of the cc pv5z basis set. The percent CPU time saved over the full basis computation escalates quickly as high angular momentum functions are removed, reaching more than 50% when the highest angular momentum functions are removed from the cc pvqz and cc pv5z basis sets. CPU time savings are greater than 80% at the cc pv5z level when the two highest angular momentum shells are removed from both the non hydrogen and the hydrogen basis. Since truncation impacts molecular ionization potentials, electron affinities, and atomization energies when truncating the f functions, by only removing the higher angular momentum functions, the percent time saved in computing these properties can be as large as 90% without deviating more than 1.0 kcal/mol (0.04 ev) from the full basis set results. Further, in bond lengths (without considering angles) where the d shell is important for the non hydrogen basis, truncating the higher angular momentum shells saves over 95% of the computational time without deviating from the full basis set bond length by more than 0.01 Å. 4.4 Conclusions Truncation of the correlation consistent basis sets in DFT is an excellent way to save computational time and resources when computing various energetic properties without the introduction of significant error. In the atoms and molecules studied, truncation of high angular functions such as g and h functions affects ionization potentials and election affinities by less than 0.01 ev, and affects atomization energies by less than 1.0 kcal/mol. Typical CPU time savings upon truncation of only g and h functions from non hydrogen and hydrogen basis sets are on the order of 60 70%. 66

81 Unlike ab initio studies of cc pvnz basis set truncation, 162,163 DFT allows for the truncation of not only the hydrogen basis but also the non hydrogen atom basis set. Simultaneous truncation of both the hydrogen f and g functions and non hydrogen atom g and h functions from the cc pv5z basis set impacts optimized geometries by less than 0.01 Å in bond lengths and 0.1 in angles as compared with the full basis geometry. In all molecules investigated here, the removal of f functions from the non hydrogen basis with simultaneous removal of d functions from the hydrogen basis does not affect bond lengths by more than 0.01 Å and affects angles by no more than The geometries of molecules composed of first row atoms are not affected as greatly as those composed of second row atoms. With truncation of all but the s, p, d, and f functions from the correlation consistent basis sets, full basis set values for total energies, atomization energies, ionization potentials, electron affinities (using s and p diffuse functions), and geometries are accessible (within 1.0 kcal/mol in energetics and 0.01 Å/0.1 in bond lengths/angles) with a large reduction in CPU computational overhead on the order of 70 90%. The trends of truncation versus property hold for both functionals investigated here. When the correlation consistent basis sets of doublethrough quintuple ζ quality are employed in DFT computations, much computational overhead can be alleviated by truncating the basis sets of all but the s, p, d, and f functions without the introduction of significant error in energetics or geometries compared with the full basis sets. 67

82 Figure 4.3 Comparisons of CPU time savings (closed circles), ionization potentials (triangles), electron affinities (upside down triangles), atomization energies (open squares), and total energies (closed squares) between the full and truncated cc pv5z basis set using B3LYP. Truncation levels: 0 is the full basis, 1 is cc pv5z( 1h ; 1g), 2 is cc pv5z( 1h1g ; 1g1f), etc. 68

83 Table 4.1 Total energies (E h ) of the carbon and oxygen atoms computed with full and truncated basis sets; energy differences are listed in me h. BLYP B3LYP Atom Basis Set Truncation Energy Δ Energy Δ C ( 3 P) cc pvdz none d cc pvtz none f f 1d f 2d cc pvqz none g g 1f g 2f g 2f 1d g 2f 2d g 2f 3d cc pv5z none h h 1g h 2g h 2g 1f h 2g 2f h 2g 3f h 2g 3f 1d h 2g 3f 2d h 2g 3f 3d h 2g 3f 4d O ( 3 P) cc pvdz none d cc pvtz none f f 1d f 2d cc pvqz none g (continued on next page) 69

84 Table 4.1 (continued) BLYP B3LYP Atom Basis Set Truncation Energy Δ Energy Δ O ( 3 P) cc pvqz 1g 1f g 2f g 2f 1d g 2f 2d g 2f 3d cc pv5z none h h 1g h 2g h 2g 1f h 2g 2f h 2g 3f h 2g 3f 1d h 2g 3f 2d h 2g 3f 3d h 2g 3f 4d

85 Table 4.2 Ionization potentials (ev) of the carbon and oxygen atoms computed with full and truncated basis sets; relative differences are listed below the full basis set values. BLYP B3LYP Basis Set Truncation C ( 3 P) O ( 3 P) C ( 3 P) O ( 3 P) cc pvdz none d cc pvtz none f f 1d f 2d cc pvqz none g 1g 1f g 2f g 2f 1d g 2f 2d g 2f 3d cc pv5z none h 1h 1g 1h 2g 1h 2g 1f h 2g 2f h 2g 3f h 2g 3f 1d h 2g 3f 2d h 2g 3f 3d h 2g 3f 4d

86 Table 4.3 Electron affinities (ev) of the carbon and oxygen atoms computed with full, truncated, and augmented truncated basis sets (aug); relative differences are listed below the full basis set values. BLYP B3LYP Atom Basis Set Truncation EA EA(aug) EA EA(aug) C ( 3 P) cc pvdz none d cc pvtz none f f 1d f 2d cc pvqz none g 1g 1f 1g 2f g 2f 1d g 2f 2d g 2f 3d cc pv5z none h 1h 1g 1h 2g 1h 2g 1f 1h 2g 2f 1h 2g 3f 1h 2g 3f 1d 1h 2g 3f 2d h 2g 3f 3d h 2g 3f 4d O ( 3 P) cc pvdz none d cc pvtz none f f 1d f 2d cc pvqz none (continued on next page) 72

87 Table 4.3 (continued) BLYP B3LYP Atom Basis Set Truncation EA EA(aug) EA EA(aug) O ( 3 P) cc pvqz 1g 1g 1f g 2f g 2f 1d g 2f 2d g 2f 3d cc pv5z none h 1h 1g 1h 2g 1h 2g 1f 1h 2g 2f 1h 2g 3f h 2g 3f 1d h 2g 3f 2d h 2g 3f 3d h 2g 3f 4d

88 Table 4.4 Total energies (E h ) of the C 2 and O 2 molecules computed with full and truncated basis sets; energy differences are listed in me h. BLYP B3LYP Atom Basis Set Truncation Energy Δ Energy Δ C 2 ( 1 Σ g ) cc pvdz none d cc pvtz none f f 1d f 2d cc pvqz none g g 1f g 2f g 2f 1d g 2f 2d g 2f 3d cc pv5z none h h 1g h 2g h 2g 1f h 2g 2f h 2g 3f h 2g 3f 1d h 2g 3f 2d h 2g 3f 3d h 2g 3f 4d O 2 ( 3 Σ g ) cc pvdz none d cc pvtz none f f 1d f 2d cc pvqz none g (continued on next page) 74

89 Table 4.4 (continued) BLYP B3LYP Atom Basis Set Truncation Energy Δ Energy Δ O 2 ( 3 Σ g ) cc pvqz 1g 1f g 2f g 2f 1d g 2f 2d g 2f 3d cc pv5z none h h 1g h 2g h 2g 1f h 2g 2f h 2g 3f h 2g 3f 1d h 2g 3f 2d h 2g 3f 3d h 2g 3f 4d

90 Table 4.5 BLYP vertical ionization potentials (ev), electron affinities (ev), atomization energies (kcal/mol), and optimized bond lengths (Å) of the C 2 and O 2 molecules computed with full and truncated basis sets; electron affinities include diffuse functions and relative energy differences are listed below the full basis set values. Molecule Basis Set Truncation IP EA(aug) D 0 R e C 2 ( 1 Σ g ) cc pvdz none d cc pvtz none f f 1d f 2d cc pvqz none g 0.1 1g 1f g 2f g 2f 1d g 2f 2d 8.0 1g 2f 3d cc pv5z none h 1h 1g 0.1 1h 2g 0.2 1h 2g 1f 0.2 1h 2g 2f h 2g 3f h 2g 3f 1d h 2g 3f 2d h 2g 3f 3d h 2g 3f 4d O 2 ( 3 Σ g ) cc pvdz none d cc pvtz none f f 1d f 2d cc pvqz none g 0.2 (continued on next page) 76

91 Table 4.5 (continued) Molecule Basis Set Truncation IP EA(aug) D 0 R e O 2 ( 3 Σ g ) cc pvqz 1g 1f 0.7 1g 2f g 2f 1d g 2f 2d g 2f 3d cc pv5z none h 1h 1g 0.1 1h 2g 0.2 1h 2g 1f 0.1 1h 2g 2f h 2g 3f h 2g 3f 1d h 2g 3f 2d h 2g 3f 3d h 2g 3f 4d

92 Table 4.6 B3LYP atomization energies (kcal/mol) and optimized bond lengths (Å) for CH 4 and SiH 4 (both 1 A 1 ); relative differences are listed below the full basis set values. CH 4 SiH 4 Basis Set Truncation ΣD 0 R e ΣD 0 R e cc pvdz none d ; 1p cc pvtz none f ; 1d f 1d ; 1d 1p f 2d ; 1d 2p cc pvqz none g ; 1f g 1f ; 1f 1d g 2f ; 1f 2d g 2f 1d ; 1f 2d 1p g 2f 2d ; 1f 2d 2p g 2f 3d ; 1f 2d 3p cc pv5z none h ; 1g 1h 1g ; 1g 1f 0.1 1h 2g ; 1g 2f h 2g 1f ; 1g 2f 1d h 2g 2f ; 1g 2f 2d h 2g 3f ; 1g 2f 3d h 2g 3f 1d ; 1g 2f 3d 1p h 2g 3f 2d ; 1g 2f 3d 2p h 2g 3f 3d ; 1g 2f 3d 3p h 2g 3f 4d ; 1g 2f 3d 4p

93 Table 4.7 B3LYP optimized geometries (Å and degrees) for NH 3 and PH 3 (both 1 A 1 ); relative differences are listed below the full basis set values. NH 3 PH 3 Basis Set Truncation R e A HNH R e A HPH cc pvdz none d ; 1p cc pvtz none f ; 1d f 1d ; 1d 1p f 2d ; 1d 2p cc pvqz none g ; 1f 1g 1f ; 1f 1d g 2f ; 1f 2d g 2f 1d ; 1f 2d 1p g 2f 2d ; 1f 2d 2p g 2f 3d ; 1f 2d 3p cc pv5z none h ; 1g 1h 1g ; 1g 1f 1h 2g ; 1g 2f 1h 2g 1f ; 1g 2f 1d h 2g 2f ; 1g 2f 2d h 2g 3f ; 1g 2f 3d h 2g 3f 1d ; 1g 2f 3d 1p h 2g 3f 2d ; 1g 2f 3d 2p h 2g 3f 3d ; 1g 2f 3d 3p h 2g 3f 4d ; 1g 2f 3d 4p

94 Table 4.8 B3LYP optimized geometries (Å and degrees) for H 2 O and H 2 S (both 1 A 1 ); relative differences are listed below the full basis set values. H 2 O H 2 S Basis Set Truncation R e A HOH R e A HSH cc pvdz none d ; 1p cc pvtz none f ; 1d f 1d ; 1d 1p f 2d ; 1d 2p cc pvqz none g ; 1f g 1f ; 1f 1d g 2f ; 1f 2d g 2f 1d ; 1f 2d 1p g 2f 2d ; 1f 2d 2p g 2f 3d ; 1f 2d 3p cc pv5z none h ; 1g 1h 1g ; 1g 1f 1h 2g ; 1g 2f h 2g 1f ; 1g 2f 1d h 2g 2f ; 1g 2f 2d h 2g 3f ; 1g 2f 3d h 2g 3f 1d ; 1g 2f 3d 1p h 2g 3f 2d ; 1g 2f 3d 2p h 2g 3f 3d ; 1g 2f 3d 3p h 2g 3f 4d ; 1g 2f 3d 4p

95 Table 4.9 Percent CPU time and average time saved computing B3LYP single point energies with truncated basis sets, relative to the full basis sets. Basis Set Truncation CH 4 NH 3 H 2 O SiH 4 PH 3 H 2 S Average cc pvdz none 1d ; 1p cc pvtz none 1f ; 1d f 1d ; 1d 1p f 2d ; 1d 2p cc pvqz none 1g ; 1f g 1f ; 1f 1d g 2f ; 1f 2d g 2f 1d ; 1f 2d 1p g 2f 2d ; 1f 2d 2p g 2f 3d ; 1f 2d 3p cc pv5z none 1h ; 1g h 1g ; 1g 1f h 2g ; 1g 2f h 2g 1f ; 1g 2f 1d h 2g 2f ; 1g 2f 2d h 2g 3f ; 1g 2f 3d h 2g 3f 1d ; 1g 2f 3d 1p h 2g 3f 2d ; 1g 2f 3d 2p h 2g 3f 3d ; 1g 2f 3d 3p h 2g 3f 4d ; 1g 2f 3d 4p

96 CHAPTER 5 SYSTEMATIC RECONTRACTION OF THE CORRELATION CONSISTENT BASIS SETS FOR DENSITY FUNCTIONAL THEORY CALCULATIONS 5.1 Introduction With correlated ab initio approaches, such as coupled cluster theory with single, double, and perturbative triple excitations, CCSD(T), 35,38 a high level of accuracy in properties such as energetics can be achieved (e.g., atomization energies within 1.0 kcal/mol from reliable experiment) when paired with a large basis set. However, CCSD(T) becomes so costly in terms of computational time, memory, and disk space that this method is generally restricted to molecules composed of no more than non hydrogen atoms with a modest sized basis set. A practical alternative is density functional theory (DFT), 44,45,52 which includes correlation energy without the additional computational overhead of correlated ab initio methods. One of the challenges in using DFT is determining the best and most appropriate functional for a problem of interest. There are hundreds of density functionals available, each designed for different tasks or with respect to a test set of atoms and molecules. Because of the increasing number of density functionals available and the popularity of DFT, a quick and efficient means to determine the intrinsic error of a density functional is needed. The use of systematically constructed basis sets has proven to be a valuable tool in This entire chapter is adapted from B.P Prascher and A.K. Wilson, The Behaviour of Density Functionals with Respect to Basis Set. V. Recontraction of Correlation Consistent Basis Sets. Mol. Phys. 2007, 105(19 22), 2899, with permission from Taylor & Francis. 82

97 determining the successes and failures of ab initio methods. 122,133,176 For example, the monotonic convergence towards the complete basis set (CBS) limit exhibited by a number of molecular properties computed with the correlation consistent basis sets in Hartree Fock (HF), DFT, and electron correlation methods, as the basis set size increases, allows for the systematic elucidation of the intrinsic error of ab initio methods. Unlike numerical methods for determining the HF CBS limit, which tend to be computationally expensive for systems larger than two atoms, accurate approximations of the CBS limit by extrapolating a series of properties computed with the correlation consistent basis sets can be done at the HF level for molecules of many atoms. Recently, a set of polarization consistent basis sets (pc n, where n = 1 4) specifically optimized for DFT were developed ,147 Wang and Wilson showed that using the pc n basis sets in DFT calculations do not necessarily result in an improvement in the accuracy of properties computed using the correlation consistent basis sets. 155,156 For example, Wang and Wilson showed that the performance of the correlation consistent basis sets is similar to that of the pc n basis sets within 0.10 kcal/mol in computed energetics for a test set of 17 first row, main group molecules. 155 Yet, the correlation consistent basis sets have fewer basis functions than the comparable polarization consistent basis set beyond the double ζ level. The impact of employing segmented contraction versus general contraction schemes in DFT has been investigated by Jensen. 147 A segmented contraction does not include all the basis functions of a given symmetry type in a basis set, whereas a general contraction does. 19 Jensen concluded that core functions are typically not important for either valence orbital descriptions or for energetic properties computed with DFT that are largely impacted by the valence space 83

98 such as ionization potentials and atomization energies. This means that if a segmented contraction scheme is used, high exponent basis functions need not be included in the contracted valence functions; or, anaglogously, if a general contraction scheme is used, the core functions do not need large contraction coefficients in the valence basis functions. In a subsequent paper, Jensen discussed how the contraction of a basis set (independent of the type of contraction scheme used) in DFT computations tends to impact computed properties more than the choice of primitive functions. 146 The overall conclusion of his papers was that socalled core pruned basis sets should be sufficient for valence energetic properties in DFT computations, and that optimization of the contraction coefficients rather than the primitive basis functions leads to a greater impact on the energetic properties. As a result of their construction, the contracted valence basis functions in the correlation consistent basis sets have small weights on the core functions, making them a type of core pruned basis sets. Another important effect to examine when studying atomization energies is basis set superposition error (BSSE). 152, In the linear combination of atomic orbitals (LCAO) approximation, the interaction between two subunits is described by the overlap of their respective basis sets. In the overlap region, there is a better description of the orbital space than within the individual subunits. This leads to interaction energies that are too high, compared with a more even description of the orbital space. To account for this effect, the ex post facto Boys Bernardi correction is typically employed: 178 AB AB A B (5.1) where AB represents the dissociation energy of the system composed of subunits A and B (or atomization energy of a molecule). The terms A and B are total energies of the 84

99 respective subunits computed in the presence of the basis sets of both subunits. Applying (5.1) is a way of accounting for the over described regions of space by the overlapping basis sets; in effect, smoothing out the basis set description so that the entire molecule is evenly described by the basis set. As the basis set increases in size (i.e. becomes more complete) the BSSE decreases. For example, the BSSE for the CO 2 atomization energy in BLYP computations was 7.11 kcal/mol at the cc pvdz level, 2.25 kcal/mol at the cc pvtz level, and 1.36 kcal/mol at the quadruple ζ level. 152 It has also been reported that correcting for BSSE in computations with the cc pvnz basis sets can lead to monotonic behavior in computed atomization energies. 156 Based on the observations of Jensen regarding basis set contractions in DFT, the success of recontracting the correlation consistent basis sets for the recovery of scalar relativistic effects (discussed in Chapter 3), and in continuing with the theme of Chapter 4 in determining the basis sets requirements for DFT, this chapter reports on reoptimization of the contractions of the correlation consistent basis sets specifically for DFT computations. The effect on the convergent behavior of various energetic properties including atomization energies, vertical ionization potentials, and vertical electron affinities is examined in addition to its impact on optimized geometries. Further, the effects of incorporating diffuse functions in the basis sets and the removal of BSSE are examined. 5.2 Computational Methodology As in Chapter 4, the Becke 1988 gradient corrected exchange functional (B) 164 has been coupled with the Lee Yang Parr correlation functional (LYP) 165 in pure DFT computations, and the Becke three parameter exchange scheme (B3) 166 has been coupled with the LYP functional 85

100 in hybrid DFT computations. In the latter case, the third parameterization of the Vosko Wilk Nusair local correlation functional (VWN 3) 167 is used. The correlation consistent basis sets, cc pvnz, were employed from double to quintuple ζ quality (n = D, T, Q, and 5). Recontraction of the s and p functions in these basis sets was performed in a similar manner to that described in Chapter 3: 1) all high angular momentum functions were removed from the basis set; 2) the s and p functions were uncontracted; 3) computations of atomic energies with BLYP and B3LYP were performed; and 4) the atomic orbital (AO) coefficients from Kohn Sham (KS) orbital analyses were then taken as the new contraction coefficients. After replacing the high angular momentum functions (d, f, etc.) from the original cc pvnz basis sets, the new basis sets are denoted cc pvnz[rc], where rc stands for recontracted. Diffuse functions have also been added to these recontracted basis sets to form augmented basis sets; the diffuse functions are unaltered from the original aug cc pvnz basis sets. Recontraction of the correlation consistent basis sets was performed with the Molpro software package 180 using a variable grid size to achieve better than me h convergence in the total atomic energy. The Gaussian 03 software suite 174 was then employed with the default grid (75,302) to perform the BLYP and B3LYP computations with the recontracted basis sets. All computations were performed using the spin unrestricted formalism, and the impact of BSSE has been examined. Two extrapolation schemes including the Halkier et al. two point (3.6) 133 and Feller three point exponential approach (3.2) 132 were employed in estimating KS limits. 86

101 5.3 Results and Discussion For oxygen and fluorine, there are two sets of contraction coefficients possible for the 2p manifold: the singly occupied p eigenstate or the doubly occupied p eigenstate. For example, the oxygen p manifold has the electronic configuration, which leads to two possibilities for contraction coefficients since and are degenerate. The properties computed in this investigation used contracted coefficients from the doubly occupied eigenstates since they produce lower atomic energies. Several energetic properties were computed and examined using the recontracted correlation consistent basis sets including vertical ionization potentials, electron affinities (vertical attachment energies). The effect of adding diffuse functions to the recontracted basis sets has been examined with respect to electron affinities and atomization energies. Recontracted basis sets that include the diffuse functions of the original aug cc pvnz basis sets 95,99,100 were used. Molecules from the test set of Wang et al. 152, have been included in this study, specifically H 2, N 2, F 2, HF, CO, CO 2, O 3, H 2 O, and CH 3 OH. In addition, several molecules have been added to augment the set, including O 2, Ne 2, CH, NH, OH, CH 2 (singlet and triplet states), NH 3, CH 4, (H 2 O) 2, and (HF) Atoms Table 5.1 displays the changes in total energies of hydrogen and the first row atoms B Ne using the recontracted basis sets compared with the original basis sets. The change in energy is greatest at the double and triple ζ levels and tends to increase as the atom size 87

102 increases. The energy changes are smaller for B3LYP than for BLYP, with the largest changes occurring at the double ζ level for both methods. For example, the total energy for the carbon atom is me h lower with the recontracted basis sets as compared with the total energy arising from original basis set using BLYP, but is only me h lower using B3LYP. The differences in energy grow to and me h for BLYP and B3LYP, respectively, for the neon atom. Table 5.1 also helps to gauge the importance of specific density functional optimizations of the contraction. The B3LYP recontraction of the basis sets, denoted cc pvnz[b3lyp], was utilized to compute BLYP energies, and the BLYP recontraction, denoted cc pvnz[blyp], was used to calculate B3LYP energies. Overall, when the same functional is used for both the recontraction and the calculation of the total energy, lower energies are obtained. The exceptions occur for the B3LYP/cc pvdz[blyp] calculations for oxygen and fluorine, where the total energies are lower than those obtained using the B3LYP recontraction by and me h, respectively. The largest differences in total energies for the B3LYP and BLYP recontraction schemes are observed for the BLYP calculations, with differences of and me h, for oxygen and fluorine, respectively. The ionization potentials and electron affinities of the first row atoms B Ne and H using the double and triple ζ basis sets are given in Table 5.2. Overall, comparing the cc pvnz basis sets to the recontracted sets shows very little difference in ionization potentials for each atom. The largest differences occur at the double ζ level. For oxygen and fluorine, the impact of recontraction increases the ionization potential by 0.06 and 0.07 ev, respectively, using BLYP. These increases are above the threshold of chemical accuracy (0.04 ev). Using B3LYP, the 88

103 oxygen and fluorine ionization potentials are increased by 0.04 and 0.05 ev, respectively, by recontraction. The magnitude of changes between the original basis sets and the recontracted sets is smaller using B3LYP than using BLYP. Electron affinities listed in Table 5.2 are computed using the diffuse functions from the respective aug cc pvnz basis sets. Electron affinities are typically not affected by recontraction as much as ionization potentials, and do not display differences as compared with the original basis sets beyond 0.02 ev (approximately 0.5 me h ). In general, atomic electron affinities are affected by recontraction only at the double ζ level. Note that Ne, which does not bind an electron, has been included for comparison with the other first row atoms. Both BLYP and B3LYP correctly predict a negative value for the Ne electron affinity Molecules Table 5.3 lists total energies, vertical ionization potentials, and electron affinities (vertical attachment energies) for H 2, B 2, C 2, N 2, O 2, F 2, and Ne 2 computed with fixed geometries from experiment: r HH = Å, r BB = Å, r CC = Å, r NN = Å, and r OO = Å, r FF = Å, and r NeNe = Å. 170 In general, the largest change between the original and recontracted basis sets occurs at the double ζ level. The changes in the BLYP total energies due to recontraction are, for example 3.704, 5.088, 7.719, , 6.412, 9.457, and me h, respectively, for H 2, B 2, C 2, N 2, O 2, F 2, and Ne 2. Compared with atomic computations, the ionization potentials for these molecules tend to be less sensitive to recontraction of the basis set, and vary at most between the original and recontracted basis sets in Table 5.3 by 0.15 ev with BLYP and 0.11 ev with B3LYP (each occurring in the Ne 2 89

104 complex). Electron affinities included in Table 5.3 are also affected the most at the double ζ level, with the exception of H 2 at the quintuple ζ level. The electron affinities of N 2 and Ne 2 are included for the sake of comparison despite the fact that these systems do not stably bind an electron. However, both BLYP and B3LYP are qualitatively correct in predicting a negative value for the electron affinity of these two systems. The double and triple ζ optimized geometries of the molecule test set used are shown in Table 5.4. The largest deviation in bond lengths and angles occurs at the double ζ level for each molecule of Table 5.4, but are typically less than 0.01 Å and 0.1, respectively. Any observed changes at the quadruple and quintuple ζ levels are not considered significant and are not included in Table 5.4. The bond lengths reported for (H 2 O) 2 and (HF) 2 are the intermolecular hydrogen bonds. The difference between the original and recontracted double ζ bond lengths are Å and Å for BLYP and B3LYP, respectively. The largest differences between the original and recontracted basis set bond lengths and angles in the molecules of Table 5.4 are consistently at the double ζ level for both BLYP and B3LYP. Bond lengths tend to vary less than 0.01 Å at the double ζ level, except in a few cases: Ne 2, (H 2 O) 2, and (HF) 2, which are all weakly bound systems. Angles tend to vary only on the order of a few tenths of a degree between the two families of basis sets. To further illustrate how geometries are affected little by recontraction, Figure 5.1 shows a comparison between the O 2 Σ potential energy curve computed with the original and recontracted cc pvdz basis set. Note that the minima reside in the same location (a consequence of the geometry not being significantly affected by recontraction, even at the double ζ level), but energies vary by almost 0.01 E h. The energy difference between the two 90

105 Figure 5.1 The BLYP potential energy curve of O 2 ( 3 Σ g ) computed with the original and recontracted correlation consistent basis sets. curves increases as the inter nuclear distance increases. Atomization energies (ΣD 0 ) are tabulated in Table 5.5 for BLYP and B3LYP. In general, atomization energies are affected most at the double ζ level. For example, the atomization energy of O 2 is impacted by 1.55 kcal/mol using B3LYP and 2.02 kcal/mol using BLYP at the double ζ level. For O 3, the impact of recontraction on the computed atomization energy is 2.81 kcal/mol using B3LYP and 3.59 kcal/mol using BLYP at the double ζ level. Other molecules like NH 3, CH 4, and CH 3 OH have atomization energies that are affected by more than 2.0 kcal/mol using both BLYP and B3LYP at the double ζ level, with CH 4 affected most at 3.91 kcal/mol using BLYP. As shown in Table 5.5 and Table 5.6, not all atomization energies using the cc pvnz basis sets converge monotonically as the basis set size is increased, a point which was investigated by Wang et al. 125,154,156 Employing the recontracted cc pvnz basis sets does not remedy non 91

106 Figure 5.2 A comparison of the BLYP atomization energies computed with the original and recontracted correlation consistent basis sets. monotonic behavior for all of the molecules. However, for some species, as shown in Figure 5.2 for CO 2 and O 3, the recontracted basis sets can change the behavior observed for the atomization energies as the basis set size is increased. For example, the atomization energies of CO 2 at the triple and quadruple ζ levels using the original cc pvnz basis sets change by almost 0.5 kcal/mol, whereas the energies resulting from use of the equivalent recontracted basis sets only change by almost 0.05 kcal/mol. For O 3 atomization energies computed with the cc pvnz basis sets, the behavior is oscillatory. Using the cc pvnz[rc] basis sets, the oscillatory behavior is not observed, and the behavior resembles that of CO 2. Often, as evidenced by Figure 5.2, a series of properties calculated at the double through quadruple ζ levels exhibits monotonic behavior respect to increasing basis set size Kohn Sham Limits Table 5.6 lists extrapolated atomization energies for each molecule in Table 5.5 using 92

107 the two and three point approaches described earlier. In the two point scheme, the first extrapolation is performed using double and triple ζ points, (DT), while the second extrapolation uses the triple and quadruple ζ points, (TQ). In the three point scheme, the first extrapolation uses double through quadruple ζ points, (DTQ), and the second uses triple through quintuple ζ points, (TQ5). In general, KS limits should not be calculated for each extrapolation approach for all of the molecules in Table 5.6 because of the non monotonic convergence of the atomization energies with respect to increasing basis set size. Out of the 20 molecules represented, the atomization energies of H 2, N 2, O 2, F 2, CO, CO 2, O 3, CH 4, and (H 2 O) 2 cannot be extrapolated using all four extrapolation techniques employed. Typically, if non monotonic behavior is observed in BLYP, then it is also observed in B3LYP. The obvious exceptions to this trend in Table 5.6 are N 2, O 2, F 2, and O 3. The N 2 molecule is unique in that its B3LYP atomization energies can be extrapolated with all four extrapolation approaches, but those of BLYP cannot be extrapolated with all four schemes. The F 2 molecule is even more peculiar since only the BLYP atomization energies employing the original cc pvnz basis sets can be extrapolated. Recontraction affects the computed F 2 atomization energy ies such that the extrapolations of the BLYP values are not possible beyond (DT). Further, all B3LYP computations on the F 2 atomization energy cannot be extrapolated due to non monotonic behavior beyond the (DT) scheme. The largest difference between the original and recontracted KS limit extrapolations occurs for the (DT) scheme for most molecules studied. For O 3, CH 4, and NH 3, the differences are 1.99, 1.57 and 1.26 kcal/mol for BLYP and 1.54, 1.16, and 1.02 kcal/mol for B3LYP, 93

108 respectively. This is expected since the largest impact on energetics due to recontraction occurs at the double ζ level. The largest molecule of this study, CH 3 OH, varies in (DT) extrapolations of its computed atomization energy by 0.91 kcal/mol and 0.71 kcal/mol at the BLYP and B3LYP levels, respectively. In BLYP computations on Ne 2, only the double ζ level atomization energy is affected by recontraction (cf. Table 5.5). Therefore, the Ne 2 molecule experiences the smallest difference between the cc pvnz and cc pvnz[rc] basis set extrapolated energies, with differences ranging from 0.01 to 0.33 kcal/mol in magnitude Basis Set Superposition Error As discussed, BSSE can be corrected using the Boys Bernardi counterpoise correction. Table 5.7 and Figure 5.3 demonstrate how counterpoise corrections can be important in restoring monotonic behavior expected with the correlation consistent basis sets using BLYP. Recall from Table 5.6 that BLYP atomization energies of N 2, F 2, and O 3 computed with cc pvnz or cc pvnz[rc] were not extrapolated due to non monotonic behavior except with the (DT) scheme. Correcting each ζ level for BSSE improves the behavior so that it is monotonic for N 2 with both the original and recontracted basis sets (cf. Table 5.7). For F 2, employing the counterpoise correction does not alleviate the non monotonic behavior in the recontracted basis set computations; it also does not affect any convergent behavior of the original basis set computations. The O 3 computed atomization energies, when corrected for BSSE, become monotonic only in the original basis set computations. Unfortunately, including a counterpoise correction in CO computations with BLYP does not result in monotonic behavior. Nonmonotonic behavior, even after a counterpoise correction has been applied, is not surprising in 94

109 light of the work of Wang et al., 152,156 who also found that some first row molecules did not display monotonic behavior after accounting for BSSE (e.g. CO). Accounting for BSSE using the counterpoise correction lowers the atomization energy at the double and triple ζ levels for each of the five systems in Figure 5.3. For F 2, the difference between the BLYP/cc pvdz[rc] energy and the cc pvdz[rc] energy including a counterpoise correction is almost 4.5 kcal/mol, while in CO and O 3 the difference is almost 2.5 kcal/mol. At the triple ζ level, F 2 has an energy difference of almost 2.0 kcal/mol when counterpoisecorrected; and at the quadruple ζ level, the difference is almost 1.0 kcal/mol. The N 2 molecule displays the smallest difference between the cc pvnz[rc] atomization energies and their respective counterpoise corrected values Diffuse Functions The impact of including diffuse functions in BLYP and B3LYP computations employing the cc pvnz basis sets has been investigated by Wang. 156 Eight molecules that do not display monotonic behavior with either the original or recontracted cc pvnz basis sets have been reexamined after the addition of diffuse functions to the recontracted basis sets. The BLYP and B3LYP computed atomization energies of H 2, N 2, O 2, F 2, CO, CO 2, O 3, and CH 4 using the aug cc pvnz[rc] basis sets with and without a counterpoise correction for BSSE are included in Table 5.8 and are compared with aug cc pvnz results. It is evident that the inclusion of diffuse functions improves the convergence of atomization energies as compared with the original cc pvnz basis sets (cf. Table 5.6), which is in accord with the conclusions of Wang. 156 However, including diffuse functions for each of the 95

110 eight molecules of Table 5.8 does not completely alleviate non monotonic behavior as the basis set size increases. Specifically, O 2, F 2, and O 3 with BLYP and B3LYP do not display monotonic behavior, nor does CH 4 with BLYP when the aug cc pvnz basis sets are utilized. For each of these systems with the exception of F 2, the convergence problem in both BLYP and B3LYP is eliminated by using the aug cc pvnz[rc] basis sets. Further employing a counterpoise correction to the F 2 molecule alleviates non monotonic behavior. It is now clear that simply employing diffuse functions does not correct non monotonic behavior for any of the first row systems studied. Rather, recontraction, in addition, to diffuse functions and a correction for BSSE is needed to achieve monotonic behavior with the correlation consistent basis sets. Figure 5.4 shows the atomization energies of H 2, N 2, F 2, CO, CO 2, and O 3 computed with the augmented and recontracted basis sets (a counterpoise correction is added to the latter) compared with the pc n basis sets. The pc n data is from Wang, 156 which does not include O 2 or CH 4. Examining the plots, it is evident that the three different basis set families converge on similar KS limits. The molecule F 2 stands out in that it is the system in which the pc n basis sets display erratic behavior, while the correlation consistent basis sets result in monotonic convergence with respect to increasing basis set size. 5.4 Conclusions Fine tuning the cc pvnz basis sets for DFT through optimization of the contraction coefficients for BLYP and B3LYP has lead to a revised family of cc pvnz basis sets, which are referred to as cc pvnz[rc] basis sets. For atomic energies, the most significant impact of recontraction occurs at the double ζ level and, overall, is on the order of several me h for each 96

111 first row atom and increases as the atom size increases. Atomic ionization potentials and electron affinities are also affected most at the double ζ level with recontraction of the basis sets. Several small first row molecules have been used in the benchmarking of these recontracted basis sets and show that the double and triple ζ level ionization potentials and atomization energies are most affected (with and without a correction for BSSE). Overall, optimized bond lengths are not affected by recontraction by more than 0.01 Å at any ζ level. When non monotonic behavior is observed in computed atomization energies, recontraction alone does not provide a means to restoring monotonic behavior when employing the cc pvnz basis sets. Previous studies have shown that both the contraction of the cc pvnz basis sets and BSSE effects can affect the convergence of computed properties with increasing basis set size. When a counterpoise correction is applied to the original cc pvnz basis sets, for example, it has been shown to give monotonic behavior in some molecules where computational atomization energies converge non monotonically with increasing basis set size. Applying the Boys Bernardi counterpoise correction to the cc pvnz[rc] basis sets can result in monotonic behavior for some molecules that do not converge monotonically when the standard correlation consistent basis sets are used. In comparing the counterpoise corrected aug cc pvnz[rc] basis sets to the pc n basis sets, the recontracted sets exhibit monotonic behavior for some molecular properties where the pc n sets do not (i.e. the atomization energy of F 2 ). In general, the recontracted basis sets perform as well as the pc n sets for the molecules investigated, despite the fact that the pc n primitive basis functions are explicitly optimized for DFT, while only the contraction coefficients of the cc pvnz[rc] set have been optimized for DFT. 97

112 Employing the Boys Bernardi counterpoise correction with these recontracted basis sets can correct non monotonic behavior in energetic properties at the double, triple, and quadruple ζ levels. When a counterpoise correction on the cc pvnz[rc] basis sets does not alleviate non monotonic behavior, the addition of diffuse functions acts to correct nonconvergent behavior in most systems of this study. However, it is necessary to also include the Boys Bernardi scheme with the aug cc pvnz[rc] basis sets in the case of F 2 to achieve monotonic behavior with BLYP and B3LYP atomization energies. Thus, all three corrections: recontracting the basis sets, adding diffuse functions, and including the counterpoise correction can alleviate non monotonic behavior in the molecules of this study. Further, it is recommended that the cc pvnz[rc] basis sets be used in BLYP and B3LYP computations due to their smaller size as compared with the pc n basis sets. 98

113 Figure 5.3 A comparison of BLYP atomization energies computed with the original and recontracted correlation consistent basis sets, and the recontracted basis sets with basis set superposition error (BSSE) removed. 99

114 Figure 5.4 A comparison of BLYP atomization energies computed with the pc n, aug cc pvnz, and aug cc pvnz[rc] basis sets with basis set superposition error (BSSE) removed. 100

115 Table 5.1 A comparison of the BLYP and B3LYP total energies (E h ) of the first row atoms (including hydrogen) computed with the original and recontracted cc pvnz basis sets; relative differences to the original basis sets are shown in me h under [rc]. BLYP B3LYP Atom ζ cc pvnz [rc] [B3LYP] b cc pvnz [rc] [BLYP] b H a D ( 1 S) T Q B D ( 2 P) T Q C a D ( 3 P) T Q N a D ( 4 S) T Q O a D ( 3 P) T Q F a D ( 2 P) T Q Ne D ( 1 S) T Q a. The original cc pvnz energies come from Wang, X., The Performance of Density Functionals with Respect to the Correlation Consistent Basis Sets. Ph.D. thesis, University of North Texas (2004). b. The recontracted set employed was optimized for the bracketed density functional, and the differences are relative to the [rc] values. 101

116 Table 5.2 A comparison of the BLYP and B3LYP ionization potentials and electron affinities (ev) of the first row atoms (including hydrogen) computed with the original and recontracted cc pvnz basis sets; relative differences to the original basis sets are shown under [rc]. BLYP B3LYP Atom ζ IP IP[rc] EA a EA[rc] a IP IP[rc] EA a EA[rc] a H D T B D T C D T N D T O D T F D T Ne D T a. Diffuse functions from the aug cc pvnz basis sets have been included in these calculations. 102

117 Table 5.3 A comparison of the BLYP and B3LYP total energies (E h ), ionization potentials, and electron affinities (ev) of the first row dimers (including hydrogen) computed with the original and recontracted cc pvnz basis sets; relative differences to the original basis sets are shown in me h (total energies) and ev under [rc]. Method Molecule ζ Energy [rc] IP IP[rc] EA a EA[rc] a BLYP H 2 D T Q B 2 D T Q C 2 D T Q N 2 D T Q O 2 D T Q F 2 D T Q Ne 2 D T Q B3LYP H 2 D T Q (continued on next page) 103

118 Table 5.3 (continued) Method Molecule ζ Energy [rc] IP IP[rc] EA a EA[rc] a B3LYP B 2 D T Q C 2 D T Q N 2 D T Q O 2 D T Q F 2 D T Q Ne 2 D T Q a. Diffuse functions from the aug cc pvnz basis sets have been included in these calculations. 104

119 Table 5.4 The BLYP and B3LYP optimized geometries (Å and degrees) of several first row molecules computed with the original and recontracted basis sets; relative differences to the original basis set are shown under [rc]. BLYP B3LYP Molecule ζ cc pvnz [rc] cc pvnz [rc] a H 2 R e D T a N 2 R e D T O 2 R e D T a F 2 R e D T Ne 2 R e D T CH R e D T NH R e D T OH R e D T HF a R e D T CH 2 ( 1 A 1 ) R e D T A D T CH 2 ( 3 B 1 ) R e D T A D T CO a R e D T a CO 2 R e D T (continued on next page) 105

120 Table 5.4 (continued) BLYP B3LYP Molecule ζ cc pvnz [rc] cc pvnz [rc] a O 3 R e D T A D T H 2 O a R e D T A D T NH 3 R e D T A HNH D T CH 4 R e D T (H 2 O) 2 R e (O H) D C i T (HF) 2 R e (F H) D C 2h T CH 3 OH a R e (OH) D C s T A(COH) D T a. The original cc pvnz energies come from Wang, X., The Performance of Density Functionals with Respect to the Correlation Consistent Basis Sets. Ph.D. thesis, University of North Texas (2004). 106

121 Table 5.5 The BLYP and B3LYP atomization energies (kcal/mol) of several first row molecules computed with the original and recontracted basis sets; relative differences to the original basis set are shown under [rc]. BLYP B3LYP Molecule ζ ΣD 0 [rc] ΣD 0 [rc] a H 2 D T Q a N 2 D T Q O 2 D T Q a F 2 D T Q Ne 2 D T Q CH D T Q NH D T Q OH D T Q (continued on next page) 107

122 Table 5.5 (continued) BLYP B3LYP Molecule Ζ ΣD 0 [rc] ΣD 0 [rc] OH HF a D T Q CH 2 ( 1 A 1 ) D T Q CH 2 ( 3 B 1 ) D T Q CO a D T Q a CO 2 D T Q a O 3 D T Q H 2 O a D T Q NH 3 D T Q (continued on next page) 108

123 Table 5.5 (continued) BLYP B3LYP Molecule Ζ ΣD 0 [rc] ΣD 0 [rc] CH 4 D T Q (H 2 O) 2 D T Q (HF) 2 D T Q CH 3 OH a D T Q a. The original cc pvnz energies come from Wang, X., The Performance of Density Functionals with Respect to the Correlation Consistent Basis Sets. Ph.D. thesis, University of North Texas (2004). 109

124 Table 5.6 Two and three point extrapolated BLYP and B3LYP Kohn Sham limits of atomization energies (kcal/mol) computed with the original and recontracted basis sets. Nonmonotonic behavior is denoted by. BLYP B3LYP Molecule Extrapolation ΣD 0 [rc] ΣD 0 [rc] H 2 P (DT) a P (TQ) a P (DTQ) P (TQ5) N 2 P (DT) a P (TQ) a P (DTQ) a P (TQ5) O 2 P (DT) P (TQ) P (DTQ) P (TQ5) F 2 P (DT) a P (TQ) P (DTQ) P (TQ5) Ne 2 P (DT) P (TQ) P (DTQ) P (TQ5) CH P (DT) P (TQ) P (DTQ) P (TQ5) NH P (DT) P (TQ) P (DTQ) P (TQ5) OH P (DT) P (TQ) P (DTQ) (continued on next page) 110

125 Table 5.6 (continued) BLYP B3LYP Molecule Extrapolation ΣD 0 [rc] ΣD 0 [rc] OH P (TQ5) HF P (DT) a P (TQ) a P (DTQ) a P (TQ5) CH 2 ( 1 A 1 ) P (DT) P (TQ) P (DTQ) P (TQ5) CH 2 ( 3 B 1 ) P (DT) P (TQ) P (DTQ) P (TQ5) CO P (DT) a P (TQ) a P (DTQ) P (TQ5) CO 2 P (DT) a P (TQ) a P (DTQ) P (TQ5) O 3 P (DT) a P (TQ) a P (DTQ) P (TQ5) H 2 O P (DT) a P (TQ) a P (DTQ) a P (TQ5) NH 3 P (DT) P (TQ) P (DTQ) P (TQ5) (continued on next page) 111

126 Table 5.6 (continued) BLYP B3LYP Molecule Extrapolation ΣD 0 [rc] ΣD 0 [rc] CH 4 P (DT) P (TQ) P (DTQ) P (TQ5) (H 2 O) 2 P (DT) P (TQ) P (DTQ) P (TQ5) (HF) 2 P (DT) P (TQ) P (DTQ) P (TQ5) CH 3 OH P (DT) a P (TQ) a P (DTQ) a P (TQ5) a. The original cc pvnz energies come from Wang, X., The Performance of Density Functionals with Respect to the Correlation Consistent Basis Sets. Ph.D. thesis, University of North Texas (2004). 112

127 Table 5.7 The BLYP and B3LYP atomization energies (kcal/mol) and their extrapolated Kohn Sham limits of four molecules corrected for basis set superposition error (BSSE). cc pvnz cc pvnz[rc] Molecule ζ / P ΣD 0 BSSE ΣD 0 BSSE a N 2 D T Q P (DT) P (TQ) P (DTQ) P (TQ5) F 2 D T Q P (DT) P (TQ) P (DTQ) P (TQ5) CO a D T Q P (DT) P (TQ) P (DTQ) P (TQ5) O 3 D T Q P (DT) P (TQ) P (DTQ) P (TQ5) a. The original cc pvnz energies come from Wang, X., The Performance of Density Functionals with Respect to the Correlation Consistent Basis Sets. Ph.D. thesis, University of North Texas (2004). 113

128 Table 5.8 A comparison of BLYP and B3LYP atomization energies (kcal/mol) and their extrapolated Kohn Sham values using the aug cc pvnz, aug cc pvnz[rc], and aug cc pvnz[rc] basis sets corrected for basis set superposition error (BSSE). BLYP B3LYP Molecule ζ / P ΣD 0 [rc] [rc] BSSE ΣD 0 [rc] [rc] BSSE a H 2 D T Q P (DT) P (TQ) P (DTQ) P (TQ5) a N 2 D T Q P (DT) P (TQ) P (DTQ) P (TQ5) O 2 D T Q P (DT) P (TQ) P (DTQ) P (TQ5) a F 2 D T Q P (DT) P (TQ) P (DTQ) (continued on next page) 114

129 Table 5.8 (continued) BLYP B3LYP Molecule ζ / P ΣD 0 [rc] [rc] BSSE ΣD 0 [rc] [rc] BSSE a F 2 P (TQ5) CO a D T Q P (DT) P (TQ) P (DTQ) P (TQ5) a CO 2 D T Q P (DT) P (TQ) P (DTQ) P (TQ5) a O 3 D T Q P (DT) P (TQ) P (DTQ) P (TQ5) CH 4 D T Q P (DT) P (TQ) P (DTQ) P (TQ5) a. The original cc pvnz energies come from Wang, X., The Performance of Density Functionals with Respect to the Correlation Consistent Basis Sets. Ph.D. thesis, University of North Texas (2004). 115

130 CHAPTER 6 THE DEVELOPMENT OF S BLOCK CORRELATION CONSISTENT BASIS SETS 6.1 Introduction In correlated ab initio methodology, achieving the exact solution to the non relativistic Schrödinger equation requires two criteria: 1) a complete N electron treatment and 2) a complete (infinite) basis set. 7,18 The former can be achieved through a full configuration interaction (FCI) treatment, which is rarely applicable to any but the smallest of electronic systems (i.e. less than ten electrons). Even in small systems where FCI can be applied, the use of modest sized basis sets is a limiting factor in achieving the exact solution of the Schrödinger equation since FCI scales factorially in basis set size. Further, the use of an infinite basis set is computationally infeasible. In a practical sense, the N electron treatment and basis set must be chosen considering both the desired level of accuracy and the computational feasibility. As already discussed in Chapter 2, the use of an infinite basis set is desirable but infeasible, and the best method of approximating the exact solution to the Schrödinger equation would be to select as large a basis set as possible. However, this quickly limits both the system size and the N electron treatment that can be employed. The correlation consistent, polarized valance basis sets (cc pvnz) were specifically designed so that ab initio methods recover correlation energy more quickly than other basis sets (e.g. basis sets developed with Hartree Fock). Further, the correlation consistent basis sets offer a way to circumvent the Work reported in this chapter was performed in collaboration with Kirk A. Peterson, David E. Woon, and Thom H. Dunning, Jr. 116

131 infeasibility of infinite basis sets by systematically approaching the complete basis set (CBS) limit as the ζ level, or cardinal number, increases. Correlation consistent basis sets have been developed for much of the main group, including all electron basis sets for the first three rows and pseudopotential basis sets for the fourth and fifth rows Further, correlation consistent basis sets have been developed for the 3d and 4d transition metals, while development of basis sets for the heavier transition metals continues. There are several families of correlation consistent basis sets beyond the standard valence sets, including the tight d basis sets for inner valence correlation in second row, main group atoms [cc pv(n+d)z]; 101 the augmented basis sets for long range interactions [aug cc pvnz]; 95,99,100 the core valence basis sets for sub valence correlation energy [cc pcvnz and cc pwcvnz]; 81,98,103 and the Douglas Kroll (DK) basis sets for accurate recovery of scalar relativistic energy [cc pvnz DK]. 142 Finally, there has been recent development of correlation consistent basis sets for resolution of the identity (RI) and explicitly correlated methods. 104,181 Correlation consistent basis sets for Li, Be, Na, and Mg 93 have been available for some time via the extensible Basis Set Exchange. 182,183 However, only unpublished basis sets through quadruple ζ quality have ever been available. Further, no tight d basis sets are available through the Basis Set Exchange for the Na and Mg atoms. The goal of this chapter is to present the correlation consistent basis sets for Li, Be, Na, and Mg, while at the same time revisiting the construction of their polarization functions (d, f, g, and h) using the robust Broyden Fletcher Goldfarb Shanno (BFGS) 184 numerical optimization technique. The preexisting Hartree Fock (HF) sp basis sets are left intact, and sp sets for the quintuple ζ level are presented. The new basis sets formed from the BFGS optimized polarization functions have already found a niche in 117

132 benchmarking the correlation consistent Composite Approach (ccca) and studying the ab initio potential energy surfaces of BeOH and MgOH. 185,186 The first half of this chapter discusses the details of building the different families of correlation consistent basis sets for s block atoms, while the second half focuses on various benchmark calculations of atomic and molecular properties utilizing these new basis sets. In particular, double through quintuple ζ basis sets are presented for standard valence, tight d valence (for Na and Mg only), diffuse, core valence, and DK scalar relativistic computations. 6.2 Computational Methodology The construction of correlation consistent basis sets follows that of previous work. First, a set of HF atomic orbitals are developed, then shells of correlation consistent polarization functions are developed. The development of the HF basis sets for Li, Be, Na, and Mg differs from that of the main group atoms in that there are no filled valence p orbitals. However, the inclusion of optimized p orbitals in the valence space is paramount to accurately describing the bonding environments of these atoms in molecules. The valence p functions were optimized using the excited states of the atoms ( for Li, Na and for Be, Mg) in multi configuration self consistent field (MCSCF) calculations. The HF basis set was then contracted using the general contraction scheme (see Chapter 3). The distinguishing characteristic of the Li, Be, Na, and Mg basis sets, compared with those of the main group atoms, is the contracted valence p function is unoccupied in the ground state. The number and type of primitive and contracted basis functions used for Li, Be, Na, and Mg are listed in Table

133 Table 6.1 The composition of the correlation consistent basis sets for s block atoms. Atom cc pvdz cc pvtz cc pvqz cc pv5z Li (9s 4p 1d) [3s 2p 1d] (11s 5p 2d 1f) [4s 3p 2d 1f] (12s 6p 3d 2f 1g) [5s 4p 3d 2f 1g] (14s 8p 4d 3f 2g 1h) [6s 5p 4d 3f 2g 1h] Be (9s 4p 1d) [3s 2p 1d] (11s 5p 2d 1f) [4s 3p 2d 1f] (12s 6p 3d 2f 1g) [5s 4p 3d 2f 1g] (14s 8p 4d 3f 2g 1h) [6s 5p 4d 3f 2g 1h] Na (12s 8p 1d) [4s 3p 1d] (16s 10p 2d 1f) [5s 4p 2d 1f] (19s 12p 3d 2f 1g) [6s 5p 3d 2f 1g] (20s 14p 4d 3f 2g 1h) [7s 6p 4d 3f 2g 1h] Mg (12s 8p 1d) [4s 3p 1d] (15s 10p 2d 1f) [5s 4p 2d 1f] (16s 12p 3d 2f 1g) [6s 5p 3d 2f 1g] (20s 14p 4d 3f 2g 1h) [7s 6p 4d 3f 2g 1h] The polarization functions (d, f, g, and h) are optimized for the atoms using singles and doubles configuration interaction (CISD). A challenge in developing polarization functions with CISD for Li and Na is the one electron valence space, since no correlation energy is recovered in valence CISD with only one electron. To circumvent the one electron valence, the Li and Na atom polarization functions were optimized in Li 2 and Na 2 CISD calculations, analogous to the way the hydrogen basis sets were originally developed. 94 The polarization functions of Be and Mg were optimized correlating only the two valence electrons. The polarization functions of each angular quantum number were optimized using the even tempered expansion scheme (3.1). The polarization functions are grouped together in shells, where each function contributes a similar amount of correlation energy (cf. Figure 6.1). Diffuse functions have been developed to augment the new correlation consistent basis sets for Li, Be, Na, and Mg. Following the prescription established in developing basis sets for main group atoms, a single diffuse function has been added for each angular momentum group in the valence basis set. For example, a set of ( 1s 1p 1d ) diffuse functions is added to the 119

134 double ζ basis, a set of ( 1s 1p 1d 1f ) functions is added to the triple ζ basis, and so on. The diffuse functions are optimized in atomic anion calculations. All calculations were performed using the Molpro software package. 180 Energy gradients were converged to 10 4 E h or better in all geometry optimizations, leading to a precision of at least 10 7 E h in total energies. The reported vibrational frequencies and zero point energies were computed using the harmonic oscillator approximation. Thermochemical values were computed using enthalpies of formation from the National Institute of Standards and Technology (NIST) database Basis Set Construction Tight d Functions for Inner Valence Correlation To construct the cc pv(n+d)z basis sets, a single tight (high exponent) d function was added to the double, triple, quadruple, and quintuple ζ cc pvnz basis sets. All of the d functions at each basis set level were then reoptimized in CISD calculations (using the Na 2 molecule, so as to avoid the one electron valence as previously discussed, and the Mg atom). The exponents of the cc pvnz and cc pv(n+d)z basis sets are plotted in Figure 6.3 for comparison. It can be seen from the figure that there is little change in the original valence functions upon optimization with a tight d function, and that the additional d function moves to describe more of the inner valence space with increasing basis set size. 120

135 ΔE corr (millihartree) d 4f 5g 6h 0.01 Mg Number of Functions Figure 6.1 The incremental CISD correlation energy lowering of the Mg atom due to each polarization function Core Valence Functions for Sub Valence Correlation The importance of valence correlation energy cannot be over emphasized in the accurate calculation of atomic and molecular properties. Moreover, the importance of corevalence (CV) correlation energy (the energy of interaction between valence and sub valence electrons) is also to be emphasized, especially when accuracy below 1.0 kcal/mol in atomic and molecular properties (i.e thermochemistry) is required. For that reason, CV basis sets for Li, Be, Na, and Mg that monotonically approach the CBS limit are introduced. The CV basis sets are optimized in an analogous manner as those for main group atoms using atomic CISD calculations using (3.7). For the Li and Be atoms, all of the electrons are correlated, but the 1s orbitals of Na and Mg are constrained to be doubly occupied (frozen) in each correlated calculation. Following the core valence prescription previously discussed in Chapter 3, shells of tight functions are added to the cc pvnz basis sets of Li and Be as follows: 121

136 h g Valence Core Valence h g Valence Weighted Core Valence f f d d p p s s Primitive Exponents Primitive Exponents Figure 6.2 Comparisons of the Na atom valence (cc pvnz) exponents with the core valence (cc pcvnz) and weighted core valence (cc pwcvnz) exponents; the exponents are grouped by angular momentum. ( 1s 1p 1d ) to cc pvdz; ( 2s 2p 2d 1f ) to cc pvtz; ( 3s 3p 3d 2f 1g ) to cc pvqz; and ( 4s 4p 4d 3f 2g 1h ) to cc pv5z. Similarly, shells of tight functions are added to the double through quadruple ζ basis sets of Na and Mg. To avoid near linear dependence in the s functions at the quintuple ζ level for Na and Mg, an additional four s type functions are uncontracted from the HF basis set and are taken to be the tight s functions for CV correlation. That is, the ( 20s 14p ) primitive set is recontracted to ( 10s 6p ). The usual ( 4p 4d 3f 2g 1h ) tight functions are then added. As a consequence, the cc pcv5z and cc pwcv5z basis sets do not contain a contracted s type function for the 3s orbital of Na and Mg in the HF basis set. Plots comparing the exponents of the cc pvnz and cc pcvnz basis sets are found in Figure 6.2, while plots comparing the amount of valence correlation energy recovered by the 122

137 5Z Basis Set QZ TZ DZ cc pvnz cc pv(n+d)z Primitive Exponent Figure 6.3 The spacing of the d exponents in the cc pvnz basis sets compared with those in the cc pv(n+d)z basis sets. cc pvnz, cc pcvnz, and cc pwcvnz basis sets are found in Figure 6.4. Figure 6.5 demonstrates the necessity of using basis sets designed for CV correlation. It is shown in atomic calculations that if the valence only basis sets are used, a loss of more than 20 me h (12.5 kcal/mol) in the CV energy of Li and Be results, while greater than 100 me h (62.5 kcal/mol) is lost in Na and greater than 150 me h (94.0 kcal/mol) is lost in Mg calculations Recontracted Basis Sets for Scalar Relativistic Computations The necessity of systematically altering a basis set to accommodate a relativistic Hamiltonian has been thoroughly discussed by de Jong et al. with regard to the DK Hamiltonian. 142 The problem in using basis sets contracted with non relativistic methods is the improper description of the radial contraction that occurs in core s and p type orbitals. 123

138 ΔE corr (millihartrees) Be V/V D T Q 5 V/CV V/wCV ΔE corr (millihartrees) Mg V/V V/CV V/wCV D T Q 5 Basis Set (cc pvnz) Basis Set (cc pvnz) Figure 6.4 CCSD(T) valence correlation energies recovered by the cc pvnz (V/V), cc pcvnz (V/CV), and cc pwcvnz (V/wCV) basis sets. Specifically, atomic contractions optimized with non relativistic HF produce core s and p orbitals that are too diffuse. The use of these diffuse core orbitals results in scalar relativistic energies that are too high. To that end, a family of DK contracted basis sets for Li, Be, Na, and Mg are introduced. To optimize the s and p type contractions for use with the DK Hamiltonian, the steps outlined in section 5 of Chapter 3 have been employed. While the differences between the amount of scalar relativistic energy recovered with non relativistic and scalar relativistic basis sets is small for Li and Be (less than 10 4 E h ; less than 0.1 kcal/mol), the difference compounded over several atoms in a molecule, say, will become significant (i.e. greater than 1.0 kcal/mol). Further, as Figure 6.6 shows, the scalar relativistic energy recovered by the original contractions of the correlation consistent basis sets actually decreases as the basis set increases in size for Li and Be. The DK contracted basis sets increase the amount of scalar relativistic energy recovered with increasing basis set size. In the Na and Mg atoms, the use of the DK contractions is necessary to achieve kcal/mol accuracy since the 124

139 difference between the original and recontracted scalar relativistic energies are on the order of 14 kcal/mol for Na and 22 kcal/mol for Mg. 6.4 Benchmark Computations Ionization Potentials and Electron Affinities To demonstrate the ability of the correlation consistent basis sets to correctly describe the frontier orbital properties of the alkali and alkaline earth metal atoms, ionization potentials and electron affinities have been computed and are listed in Table 6.2. The first and second ionization potentials and electron affinities are computed in the typical manner. Due to the one electron valence of Li and Na, the ionization potentials are the same as the HF computed ionization potentials, and it is seen that there is very little difference between successive ζ levels. Further, comparing the Li and Na aug cc pv5z ionization potentials with experiment shows errors of ev and ev, respectively. It is only when CV correlation is included do the ionization potentials of these two atoms compare well with experiment. At the cc pcvdz level, for example, the difference between including CV correlation and not is enough to bring the Li ionization potential within 0.04 ev of experiment. Using the cc pwcvdz basis set instead of the cc pcvdz basis set makes a larger impact on the computed ionization potentials of Li, Be, Na, and Mg. For example, employing the cc pcvdz basis set in computing the Li ionization potential shows an error relative to experiment of ev, but the cc pwcvdz basis set shows and error of ev. This observation is typical for the rest of the atomic ionization potentials of Table 6.2: the cc pwcvnz basis sets give results closer to the experimental value when 125

140 ΔE corr (millihartrees) Li CV/V CV/CV CV/wCV D T Q 5 ΔE corr (millihartrees) Be CV/V CV/CV CV/wCV D T Q 5 Basis Set (cc pvnz) Basis Set (cc pvnz) Na Mg ΔE corr (millihartrees) CV/V CV/CV CV/wCV D T Q 5 ΔE corr (millihartrees) CV/V CV/CV CV/wCV D T Q 5 Basis Set (cc pvnz) Basis Set (cc pvnz) Figure 6.5 CCSD(T) core valence plus valence correlation energies recovered by the cc pvnz (CV/V), cc pcvnz (CV/CV), and cc pwcvnz (CV/wCV); the 1s orbitals of Na and Mg were kept frozen in each calculation. compared with the cc pcvnz basis sets. Further, it has been remarked previously that both basis sets will converge on the same CBS limit for properties of interest, and it is shown to be true for the Li, Be, Na, and Mg atoms. The electron affinities of Be and Mg are not reported since it is generally accepted that a bound electron should not be stable on these atoms. The use of diffuse functions in the computation of anionic properties is vital to achieving results that compare with experiment. 126

141 ΔE rel (millihartrees) Li D T Q 5 ΔE rel (millihartrees) Be D T Q 5 Basis Set (cc pvnz) Basis Set (cc pvnz) ΔE rel (millihartrees) Na D T Q 5 ΔE rel (millihartrees) Mg D T Q 5 Basis Set (cc pvnz) Basis Set (cc pvnz) Figure 6.6 Comparisons of the scalar relativistic corrections to the atomic Hartree Fock energy using the original contractions (diamonds) and the Douglas Kroll contractions (squares) with increasing basis set size. This claim is supported by the fact that, at the cc pvdz level, the Li and Na electron affinities are in error by ev and ev, respectively. When the diffuse functions are included (aug cc pvdz), the errors in the Li and Na electron affinities drop to ev and ev, respectively. Further, the inclusion of core valence correlation does not have a significant impact on the electron affinities of Li or Na at the triple ζ level or higher. In fact, the electron affinities of the cc pv5z, cc pcv5z, and cc pwcv5z basis sets are quite similar. 127

142 6.4.2 Optimized Geometries and Vibrational Frequencies The optimized geometries of several di and triatomics and their corresponding harmonic vibrational frequencies computed with the cc pv(t+d)z basis set are listed in Table 6.3 and Table 6.4, respectively. The target accuracy of computed structures relative to experiment is 0.01 Å. The impact of core valence correlation is readily seen in the optimized geometries of the diatomics. For example, the bond length of Li 2 (the smallest system of Table 6.3) at the cc pv5z level deviates from experiment by Å, while the corresponding corevalence basis sets only deviate by Å. A similar trend is observed in the bond lengths of LiF, Na 2, NaF, and MgF. Interestingly, it is the standard valence basis sets that more accurately predict the bond length of MgO as opposed to the core valence basis sets. The difference between experiment and theory at the cc pv5z level is Å, but is Å at the cc pcv5z level (0.012 Å at the cc pwcv5z level). Examining the bond lengths of the cc pwcvnz basis sets more closely reveals that the bond length has a very low error at the double ζ level, but diverges from the experimental value as the basis set increases in size. Another basis set effect that is observed is the general improvement in the convergence of the bond lengths with the cc pv(n+d)z basis sets relative to the cc pvnz basis sets. The largest tight d effects observed are in the bond lengths of MgO and MgF. Using the cc pvdz basis set, the bond lengths of MgO and MgF are Å and Å, respectively, which decrease to Å and Å, respectively, when the cc pv(d+d)z basis set is used. At the higher ζ levels, the changes in the bond length due to the tight d function are less than 0.01 Å. A similar effect of the tight d function is seen in the aug cc pvdz/aug cc pv(d+d)z bond lengths of MgO and MgF, and in the cc pvdz/cc pv(d+d)z bond lengths of MgF

143 6.4.3 Thermochemistry The computed enthalpies of formation for a set of twelve s block molecules are found in Table 6.5. The desired accuracy of computed enthalpies of formation with respect to experiment is ±1.0 kcal/mol. It is readily seen that the Li 2 enthalpies of formation are within 1.0 kcal/mol of experiment at the cc pvtz and aug cc pvtz levels and higher, while the Na 2 enthalpies are within 1.0 kcal/mol of experiment at all ζ levels with each family of basis sets. Using the cc pvnz and aug cc pvnz basis sets, only the NaF molecule reaches 1.0 kcal/mol of experiment at the quintuple ζ level; the other molecules are still outside the desired accuracy range. The overall effect of adding the diffuse functions is to accelerate the convergence of the enthalpies of formation. For example, in LiF, NaF, MgF, BeF 2, and MgF 2, the cc pvdz and aug ccpvdz enthalpies of formation differ by more than 10 kcal/mol, with the diffuse functions aiding convergence towards the experimental value. Adding core valence correlation does have a noticeable impact on the enthalpies of formation in the larger basis sets (i.e. quadruple and quintuple ζ levels). For example, at the cc pcv5z and cc pwcv5z levels, the BeO enthalpy of formation differs from the cc pv5z enthalpy by 1.65 kcal/mol and 1.83 kcal/mol, respectively. Still other noticeable differences are between the BeF 2 and MgF 2 enthalpies of formation at the cc pcv5z and cc pwcv5z levels with respect to the cc pv5z level. The BeF 2 enthalpy of formation computed with the cc pcv5z and cc pwcv5z basis sets, for example, is different from the cc pv5z enthalpy by 1.92 kcal/mol and 2.12 kcal/mol, respectively. Using the largest basis sets to compute the enthalpies of formation of Li 2, LiF, BeO, and BeF, gives results that more closely agree with experiment than the other basis set families. 129

144 However, the enthalpies of formation of NaF and MgF 2 computed with the cc pcvnz and cc pwcvnz basis sets are slightly better (relative to experiment) than those computed with the aug cc pcvnz and aug cc pwcvnz basis sets. 6.5 Conclusions The unpublished basis sets for Li, Be, Na, and Mg have been reexamined using robust numerical optimization techniques, and new official correlation consistent basis sets for these four atoms are presented. Specifically, new cc pvnz and aug cc pvnz basis sets of double, triple, quadruple, and quintuple ζ quality have been developed and benchmarked. In addition, a set of tight d basis sets, cc pv(n+d)z, have been developed for Na and Mg in correlated valence calculations. For subvalence correlation, new core valence and weighted core valence (cc pcvnz and cc pwcvnz) basis sets have been developed. Benchmark calculations of atomic CV correlation energies demonstrate the necessity of including core valence functions when subvalence correlation is needed (i.e. for high accuracy thermochemistry). Benchmarks of molecular properties also demonstrate the necessity of CV correlation, especially in Li and Na, whose one electron valence is not a large enough valence space for correlated molecular calculations when 0.01 Å and 1.0 kcal/mol accuracy is desired. Finally, the HF sp basis sets have been recontracted for scalar relativistic DK calculations, resulting in cc pvnz DK basis sets that consistently recover more scalar relativistic energy than the original HF contractions at each basis set level. 130

145 Table 6.2 CCSD(T) ionization potentials and electron affinities of Li, Be, Na, and Mg; the tight d correlation consistent basis sets for Na and Mg have been used. Li Be Na Mg Basis IP EA IP (1) IP (2) IP EA IP (1) IP (2) cc pvdz cc pvtz cc pvqz cc pv5z aug cc pvdz aug cc pvtz aug cc pvqz aug cc pv5z cc pcvdz cc pcvtz cc pcvqz cc pcv5z cc pwcvdz cc pwcvtz cc pwcvqz cc pwcv5z aug cc pwcvdz aug cc pwcvtz aug cc pwcvqz aug cc pwcv5z Experiment a a. Martin, W. C.; Wiese, W. L. Atomic Spectroscopy. in Atomic, Molecular, & Optical Physics Handbook; Drake, G. W. F., Ed.; American Institute of Physics: Woodbury, NY, 1996; pp

146 Table 6.3 CCSD(T) optimized geometries (R e, Å) of some s block molecules computed with various families of correlation consistent basis sets. Molecule ζ VnZ V(n+d)Z CVnZ wcvnz avnz av(n+d)z acvnz awcvnz Expt. Li 2 D T Q a LiF D T Q a BeO D T Q a BeF D T Q a Na 2 D T Q a NaF D T Q a MgO D T Q a MgF D T Q a (continued on next page) 132

147 Table 6.3 (continued) Molecule ζ VnZ V(n+d)Z CVnZ wcvnz avnz av(n+d)z acvnz awcvnz Expt. BeH 2 D (D h ) T Q BeF 2 D (D h ) T Q MgH 2 D (D h ) T Q MgF 2 D (D h ) T Q a. Constants of Diatomic Molecules; Number 69 ed.; Huber, K. P.; Hertzberg, G., Eds.; National Institute of Standards and Technology: Gaithersburg, MD,

148 Table 6.4 CCSD(T) harmonic vibrational frequencies (Å) of some s block molecules computed with various families of correlation consistent basis sets. Molecule VnZ V(n+d)Z CVnZ wcvnz avnz av(n+d)z acvnz awcvnz Expt. Li a LiF a BeO a BeF a Na a NaF a MgO a MgF a BeH b (D h ) b BeF c (D h ) c c MgH d (D h ) d MgF e (D h ) e a. Constants of Diatomic Molecules; Number 69 ed.; Huber, K. P.; Hertzberg, G., Eds.; National Institute of Standards and Technology: Gaithersburg, MD, b. Wang, X.; Andrews, L. Inorg. Chem. 2005, 44, 610. c. Yu, S.; Shayesteh, A.; Bernath, P. F.; Koput, J. J. Chem. Phys. 2005, 123, d. Wang, X.; Andrews, L. J.Phys. Chem. A 2004, 108, e. Snelson, A. J.Phys. Chem. 1966, 70,

149 Table 6.5 CCSD(T) standard state enthalpies of formation at K (kcal/mol) for some s block molecules computed with various families of correlation consistent basis sets; tight d functions have been used for the Na and Mg atoms. Molecule ζ VnZ CVnZ wcvnz avnz acvnz awcvnz Expt. a Li 2 D T Q LiF D T Q BeO D T Q BeF D T Q Na 2 D T Q NaF D T Q MgO D T Q MgF D T Q (continued on next page) 135

150 Table 6.5 (continued) Molecule ζ VnZ CVnZ wcvnz avnz acvnz awcvnz Expt. a MgF BeH 2 D (D h ) T Q BeF 2 D (D h ) T Q MgH 2 D (D h ) T Q MgF 2 D (D h ) T Q a. Chase, M. W. J. Phys. Chem. Ref. Data 1998, 9,

151 CHAPTER 7 THE RESOLUTION OF THE IDENTITY APPROXIMATION APPLIED TO THE CORRELATION CONSISTENT COMPOSITE APPROACH 7.1 Introduction In correlated ab initio methods, the number of four index two electron Coulomb integrals (2.11) increases dramatically as the number of electrons increases. 7,19,22,55 The computation and storage of these integrals is the bottleneck in applying correlated methods, and so it is highly desirable to have efficient ways of computing and storing them. Typically, integral screening techniques 187 are employed to avoid computing very small (i.e E h ) integrals, which helps reduce the number of four index integrals stored. To deal efficiently with the other integrals, a novel approach is to employ a projective technique. The resolution of the identity (RI) is a well known projective technique from vector algebra. 188 Consider a ket vector expanded in a complete, orthonormal basis : (7.1) By projecting onto an arbitrary basis vector in, the component of along that vector is resolved: (7.2) Insertion of (7.2) into (7.1) yields the following result: Work reported in this chapter was performed in collaboration with Jeremy D. Lai, and has been submitted for publication in the Journal of Chemical Physics. 137

152 (7.3) where the summation in parentheses apparently takes the form of an algebraic 1 : (7.4) The object in (7.4) has several names in the literature including identity operator, completeness relation, and unit dyadic, the latter of which is used throughout this chapter. The summation terms are projection operators instead of scalar products, since the ket is written before the bra. When (7.4) operates on a ket vector, the result is the same ket represented in whatever basis set comprises the unit dyadic, provided that basis is complete. This is why this mathematical construct is called the resolution of the identity, because it resolves a vector in one basis to another. For example, a unit dyadic in the basis may be constructed as (7.5) then, since, the vector may be resolved in the new basis using (7.1): (7.6) The components of in are now defined in terms of its components in and the overlap between the two basis sets. More generally, if the domain of the bra vector in (7.4) is different from the domain in the ket, then the resulting operator is the Dirac delta function: 7 138

153 (7.7) Instead of resolving vectors from one basis to another, the Dirac delta function resolves a function in one domain into another as follows: d (7.8) As a sidebar, the definition of the Dirac delta function in (7.7) is entirely analogous to the definition of the first order reduced density function in (2.47). The Dirac delta function is related to the following Fourier integral, where is the momentum vector: d, (7.9) which is why RI methods are sometimes referred to as pseudospectral methods in the literature. 189 Finally, it is readily seen from (7.9) that the inverse Fourier transform of the Dirac delta function, and, by extension the unit dyadic, is unity. Boys and Shavitt first introduced the idea of reducing the computational cost of fourindex integrals using the RI technique, 190 while Löwdin discussed the use of RI in terms of describing quantum phenomena. 191 Since then, the RI approximation has been employed in density functional theory (DFT), 161,192 Hartree Fock self consistent field (HF SCF), 160 secondorder Møller Plesset perturbation theory (MP2), 160,193 and coupled cluster theory. 194 For a review of the RI approximation in computational chemistry, see Kendall and Früchtl. 190 Following the derivations of Vahtras et al. with regards to various RI approximations, 160 any four index integral can be approximated as a linear combination of three or two index integrals by insertion of the unit dyadic. Vahtras et al. showed that describing charge distributions with the RI approximation leads to three different formulas for approximating 139

154 four index Coulomb integrals: (7.10) (7.11) (7.12) in which the basis sets denoted by,,, are called auxiliary basis sets. The bracketed terms on the right hand side are three center overlap integrals, the terms are two center overlap integrals (in matrix form) between two auxiliary basis sets, and the terms are two center, two electron Coulomb integrals (also in matrix form). Equations (7.10) and (7.11) are called the S and SVS approximations, respectively, in which the individual charge distributions are projected onto an auxiliary basis set. Note that the auxiliary basis set for the bra vector is not necessarily the same used for the ket vector in the S and SVS approximations, which is why the additional overlap integrals enter the equations. Equation (7.12) is called the V approximation, in which the Coulomb operator is projected onto an auxiliary basis set. The V approximation more closely approximates the exact four index integral in practice, and, consequently, is the most common RI approximation in use for approximating the four index integrals in SCF, DFT, and MP2 calculations. In more recent implementations of RI methods, a single auxiliary basis is used instead of multiple auxiliary basis sets as indicated by the equations above. When a single auxiliary basis set is used in the V approximation, the number of four index integrals is reduced from to approximately, where is the number of orbital basis functions and is the number of auxiliary basis functions. So long as does not exceed, there will be an effective 140

155 reduction in the number, and hence, the computational scaling of four index integrals. This is the primary motivation behind employing the RI approximation in correlated ab initio methods, where the number of four index integrals over excited determinants becomes very large. In DFT methods, the RI approximation takes on a slightly different role, namely, that of projecting the exchange correlation functional, which may not have a trivial integral form, onto an auxiliary basis set in which integrals over the exchange correlation functional become more tractable. In practice, a different auxiliary basis set is used for each computational method, each having been optimized specifically for that method s ansatz. For example, a density fit or Coulomb fit auxiliary basis set might be used for all or some of the HF SCF integrals, while a different auxiliary basis set is used for correlated integrals that arise in an MP2 calculation employing those HF orbitals. An additional means to reduce the computational expense of ab initio methods is via composite methods (discussed in Chapter 2). These approaches approximate a high level correlated ab initio/large basis set calculation with a series of lower level/small basis set calculations. Examples of composite methods include the Gaussian n methods (Gn), Weizmann n methods (Wn), 71,72 the high accuracy extrapolated ab initio thermochemistry (HEAT) method, 70 complete basis set (CBS n) methods, and the recently developed correlation consistent Composite Approach (ccca) In general, each composite method listed varies in the target accuracy with respect to reliable experimental data and in formulation. The ccca method makes use of the well known monotonic behavior of the correlation consistent basis sets, cc pvnz, with respect to increasing basis set size for energetic properties (e.g., thermochemical properties, atomization energies), and considers 141

156 additional effects such as high order correlation effects, core valence effects, and scalar relativistic effects. Further, ccca has been applied to over 700 main group molecules, resulting in an overall mean absolute deviation (MAD) of approximately 1 kcal/mol from reliable experiment The use of composite methodology may be able to approximate high level computational methods with a lower computational expense, however, further modifications to the handling of four index integrals can be introduced to further reduce the computational scaling. In this Chapter, the RI approximation is applied to the ccca method and the resulting formulism (RI ccca) is discussed. The accuracy of properties computed with RI ccca is compared with those computed with the original formulation of ccca. Further, relative timings and storage requirements are presented for RI ccca and are compared with the original ccca formulation. 7.2 Computational Methodology The set of molecules examined is a subset of the G2/97 test set 61 containing 102 closed shell systems, including both first and second row, main group atoms. All calculations were performed with the Molpro software package 180 without employing symmetry and using the default semi direct algorithm. Energy gradients and hessians were converged to at least 0.1 me h in the geometry optimizations and frequency calculations. Following the formulation for the ccca method, the geometry optimization of each molecule is performed using Becke s 3 parameter exchange scheme coupled with the Lee Yang Parr correlation functional, B3LYP, 165,166 and the cc pvtz basis set. Harmonic frequencies are then computed at the same 142

157 level of theory and scaled by to account for anharmonicity. 73,74 In order to compare energetic properties without bias from a change in the geometry, we have used the same B3LYP/cc pvtz geometries and scaled harmonic frequencies in each of the different ccca formulations discussed in the next section. From the B3LYP/cc pvtz geometry, the ccca reference energy (E ref ), higher order correlation (E CC ), core valence correlation (E CV ), and scalar relativistic (E SR ) corrections are computed as follows: MP2/aug cc pv Z (7.13) CCSD T /cc pvtz MP2/cc pvtz MP2 full /aug cc pcvtz MP2 fc /aug cc pvtz DK MP2/cc pvtz DK MP2/cc pvtz The reference energy, MP2/aug cc pv Z (where aug cc pv Z denotes the CBS limit), is extrapolated using the mixed Gaussian/exponential formula (3.3) from double, triple, and quadruple ζ single point energies. In the core valence step, MP2(full) correlates the standard atomic valence plus the (n 1)s and (n 1)p orbitals for main group atoms, while the MP2(fc) denotes using a frozen core of (n 1)s and (n 1)p electrons. The scalar relativistic correction employs the second order, spin free Douglas Kroll (DK) Hamiltonian 135,136 and the DKrecontracted correlation consistent basis sets. 142 Spin orbit (SO) effects (not included in the DK Hamiltonian) have been included a posteriori for the atoms. 74 Where thermochemical properties have been computed, atomic enthalpies of formation and thermal corrections have been taken from CODATA, 195 except for the Si atom, 143

158 which has been taken from Karton and Martin Results and Discussion RI ccca Implementation An RI based ccca approach has been constructed by substituting the conventional SCF and MP2 steps in ccca with their RI analogs. The general formulation is listed below, and is hereafter denoted RI ccca: RI MP2/aug cc pv Z RI (7.14) Δ Δ Δ CCSD T /cc pvtz RI RI MP2/cc pvtz RI RI MP2 full /aug cc pcvtz RI RI MP2/aug cc pvtz RI DK RI MP2/cc pvtz DK RI RI MP2/cc pvtz RI Δ Δ Δ Each of the conventional MP2 steps of (7.13) have been replaced with RI MP2. Also, the underlying SCF reference calculations have been replaced with RI SCF. For the CCSD(T) step, two different routes have been examined: one in which conventional CCSD(T) is used and another in which the density fit (DF, which is entirely analgous to the RI approximation) local CCSD(T) method, denoted DF LCCSD(T), of Schütz and Werner is used. In the latter part of this section, the difference between employing the conventional CCSD(T) method in RI ccca versus the DF LCCSD(T) method in RI ccca (denoted RI ccca+l for clarity) is analyzed and discussed. 144

159 7.3.2 Auxiliary Basis Sets for RI ccca To determine the optimal auxiliary basis sets to be used in the RI SCF and RI MP2 calculations, a comparison has been made between RI ccca, which utilizes conventional CCSD(T), and ccca employing several different auxiliary basis sets. Utilimately, the auxiliary basis sets should be chosen so that the convergent nature of the correlation consistent basis sets remains intact, and so that the RI ccca energies are as close to the ccca energies as possible. In the RI SCF steps, the use of completely unconctracted cc pvnz basis sets one ζ level higher than the orbital basis set, denoted cc pv(n+1)z(unc), and the basis sets of Weigend, denoted JKFIT, 200 have been studied. (It is common practice when using RI methods to assume that an uncontracted basis set that is larger than the orbital basis set is an adequate auxiliary basis set; i.e. if a cc pvtz orbital basis set is used, then the uncontracted cc pvqz basis set is assumed to be adequate as an auxiliary basis set.) The JKFIT basis sets are studied here because they have been reported to reproduce conventional SCF atomic energies within 120 μe h at any ζ level through quadruple ζ. 200 Further, the use of the cc pv(n+1)z(unc) and the cc pvnz RI basis sets of Weigend et al. 181 have been studied as potential auxiliary basis sets in the RI MP2 steps of RI ccca. The latter auxiliary sets were optimized specifically for use with the correlation consistent basis sets, which is why they were selected, and have been reported to reproduce conventional MP2 correlation energies withing 101 μe h per atom in molecules containing first and second row atoms. 181 The results of the test calculations described above for H 2 P +, Cl, CH 3, NH 2, OH, SiH 3, PH 2, SH, CN, NH + 4, H 3 O +, C 2 H + 3, SiH + 5, PH + 4, H 3 S +, and H 2 Cl + are listed in Table 7.1 according to 145

160 the auxiliary basis sets used. The relative energy differences of RI ccca, as compared with ccca, are shown in the table, as well as the relative computational cost of RI ccca in terms of CPU time and disk space. Overall, a computational savings is observed in each RI implementation of Table 7.1, except for a few isolated cases (i.e. Cl, OH, and SH ). A mean absolute deviation (MAD) of 213 me h is observed in the test set when the cc pv(n+1)z(unc) auxiliary set is used. Employing the JKFIT auxiliary basis sets in the RI SCF reference calculation and using the cc pv(n+1)z(unc) auxiliary sets in the RI MP2 step lowers the MAD of the energies dramatically (compared with the previous approach) to me h. Thus, the JKFIT sets have demonstrated much better accuracy than the cc pv(n+1)z(unc) set as the auxiliary basis set in RI SCF calculations. Comparing the different auxiliary basis sets used in the RI MP2 step shows the ccpvnz RI basis sets of Weigend et al. to have the lower MAD in total energies (0.270 me h ), relative to ccca. In light of these test calculations, it is recommended that the RI ccca implementation use the JKFIT auxiliary basis sets in the RI SCF reference calculations and the cc pvnz RI basis sets in the RI MP2 and RI LCCSD(T) steps of (7.14). This recommended formulation is employed from here on in this investigation Energetic Properties Total energies computed with ccca, RI ccca, and RI ccca+l listed in Table 7.2. The largest difference in total energies between ccca and RI ccca is me h which occurs for SiCl 4, while the smallest difference between the two methods is me h, which occurs for H 2. The average difference in total energies between ccca and RI ccca for the entire test set is me h. The total energy differences between RI ccca+l and ccca are almost an order of 146

161 magnitude larger than those of RI ccca and ccca. The average total energy difference between RI ccca+l and ccca is me h, with the worst difference being me h for ClF 3. Examining each additive component (included in the supplemental material of the submitted manuscript) 201 of the RI ccca reveals that the largest differences relative to the ccca method occur in the core valence and scalar relativistic steps. There is no total energy difference larger than 1.0 kcal/mol between the original ccca coupled cluster step and the RI ccca coupled cluster step in any of the molecules studied. Enthalpies of formation computed at 298 K are listed in Table 7.3. Overall, good agreement is observed between ccca and RI ccca. The largest deviation of the RI ccca enthalpies of formation from those of ccca, not surprisingly, comes from SiCl 4 and is 1.39 kcal/mol. The MAD of RI ccca enthalpies of formation from ccca for the entire test set is 0.27 kcal/mol. The minimum and maximum deviations from experiment are similar between RI ccca and ccca, and the MAD is 1.00 and 1.03 kcal/mol, respectively, for each method. The average difference between ccca and RI ccca+l enthalpies of formation is an order of magnitude larger than those of RI ccca and ccca at 2.63 kcal/mol, with the largest deviation being 8.93 kcal/mol from ClF 3. Compared with experiment, RI ccca+l has a MAD of 2.30 kcal/mol, and deviates the worst by 9.20 kcal/mo in the enthalpy of formation of SiF Computational Cost The CPU time and disk space savings of RI ccca and RI ccca+l, relative to ccca, are provided in Table 7.4. The relative CPU times range from 10% more computational time (worst case: FH) to 94% less time (best case: C 2 H 6 ), with an average CPU time savings of 72% compared 147

162 1.0E E+04 RI ccca CPU Time (s) 1.0E E E E E E E E E E+05 ccca CPU Time (s) 1.0E+05 RI ccca+l CPU Time (s) 1.0E E E E E E E E E E E+05 ccca CPU Time (s) Figure 7.1 CPU times of RI ccca (top) and RI ccca+l (bottom) versus ccca; each data point represents a molecule in the test set. with ccca. Employing RI ccca+l affords a slightly better average CPU time savings of 76% and 97% savings in the best case, but increases the computational cost to 129% in the worst case (H 2 ). Comparing RI ccca to RI ccca+l, we observe an average CPU time savings of 28% using RI ccca+l, and in the best case, an 86% CPU time reduction is observed using RI ccca+l. Overall, for the test set we have employed, the RI ccca+l method affords lower CPU times. 148

163 Further, dramatic reductions in the disk space requirements are observed, ranging from 84% to 99% for RI ccca, compared with ccca, with an average disk space savings of 97%; and ranging from 77% to almost 100% for RI ccca+l, with an average savings of 96%. Comparing RI ccca to RI ccca+l, an average 17% less disk space is required in the latter. In the largest molecule examined, benzene, the disk space saved using RI ccca and RI ccca+l is 97% and 98%, respectively. Further, for the series CH 3 Cl, CH 3 CH 2 Cl, and CH 3 CH 2 CH 2 Cl, the CPU time saved employing RI ccca over ccca is 87%, 89%, and 93%, respectively, while using RI ccca on each molecule affords a 98% disk space savings over ccca. Using RI ccca+l on the same series produces CPU time savings of 84%, 90%, and 96%%, respectively, with disk space savings of 99% over ccca. The CPU time and disk space savings of RI ccca and RI ccca+l, relative to ccca, are reflected in Figure 7.1 and Figure 7.2, respectively. Figure 7.1 clearly demonstrates that the use of the RI approximation does not always result in a computational time savings, as is seen in the left most three points in the plots (cf. the Cl atom and OH and SH diatomics of Table 7.1). Employing the RI ccca+l method exacerbates this phenomenon, as is seen in the bottom plot of Figure 7.1 in which the left most three points are above the line indicating a CPU time increase relative to ccca. However, as Figure 7.1 shows, there is clearly more of a computational time savings using RI ccca+l over RI ccca since the calculations that require the most computational time are lower on the vertical axis of the bottom plot than in the top plot. As Figure 7.2 demonstrates, there is always a clear disk space savings when RI ccca or RIccCA+L is used, compared with ccca. 149

164 1.0E E+05 RI ccca Disk Usage (MB) 1.0E E E E E E E E E E E E+06 ccca Disk Usage (MB) 1.0E+06 RI ccca+l Disk Usage (MB) 1.0E E E E E E E E E E E E E+06 ccca Disk Usage (MB) Figure 7.2 Disk space usage of RI ccca (top) and RI ccca+l (bottom) versus ccca; each data point represents a molecule in the test set. In general, it is observed that there is a correlation between molecular size and amount of computational resources needed for RI ccca and RI ccca+l, compared with ccca, which is demonstrated in Figure 7.3. The values shown are averages over each group of molecules containing a certain number of atoms. There is a clear increase in both CPU and disk space savings as molecular size increases. The cross over point at which the RI ccca+l method 150

165 Savings Relative to ccca 100% 95% 90% 85% 80% 75% 70% 65% 60% 55% 50% CPU (RI ccca) CPU (RI ccca+l) Disk (RI ccca) Disk (RI ccca+l) Number of Atoms Figure 7.3 Correlation between molecular size and average CPU time and disk space saved using RI ccca and RI ccca+l, relative to ccca. becomes more efficient in CPU time is observed to be between three and four atoms in a molecule, while the point is between four and five atoms in terms of disk space. Practical experience has shown that the computational bottlenecks in the original ccca formulation are the MP2/aug cc pvqz step in the calculation of the reference energy and the MP2(full)/aug cc pcvtz core valence step. Although the CCSD(T) step formally scales as, versus the scaling of MP2 ( is the number of basis functions), the drastic difference in the size of the basis sets used between the CCSD(T)/cc pvtz and the MP2/aug cc pvqz steps makes the MP2 step more computationally demanding. In the core valence step, the correlation of the (n 1)s and (n 1)p orbitals precludes this step as more computationally demanding than the CCSD(T) step. An analysis of each step in RI ccca with the corresponding 151

166 step in the original ccca implementation shows that every molecule of this study has a significant CPU time reduction in the MP2/aug cc pvqz step the average CPU reduction being 91%. Further, 95 of the 102 molecules of this study have a significant CPU time reduction in the MP2(full)/aug cc pcvtz step as well, with an average CPU time reduction of 77%. Thus, both bottlenecks of the original ccca implementation are greatly reduced when the RI approximation is invoked. 7.4 Conclusions A new formulation of the ccca method employing the RI approximation in the MP2 steps has been developed and benchmarked for total energies, atomization energies, and enthalpies of formation for a test set containing 102 molecules containing first and second row, main group atoms. This new formulation of ccca, denoted RI ccca, reproduces total energies within me h, on average, compared to the original implementation of ccca. Further, the average error in enthalpies of formation introduced by the RI approximation in ccca is 0.27 kcal/mol. There are dramatic CPU time and disk space reductions afforded by using RI ccca over ccca, both of which correlate with molecular size. The average CPU time reduction by RI ccca over ccca for our test set is 72%, while the average disk space reduction is 97%. If the conventional CCSD(T) step of ccca is replaced with DF LCCSD(T), then the computational cost of RI ccca may be further reduced when larger molecules are to be investigated. It has been shown here that employing RI ccca+l in molecules composed of four, or more, atoms affords a greater computational savings in terms of CPU time and disk space than with RI ccca: the average CPU time reduction for RI ccca+l being 76% with a disk space reduction of 96%, 152

167 relative to ccca. However, employing RI ccca+l results in an average deviation from the ccca total energies of me h and 2.63 kcal/mol in enthalpies of formation. In summary, the RI ccca method is recommended as a means to significantly reducing the computational cost of ccca calculations, thus accessing much larger molecules in correlated ab initio computations. The original ccca method is limited severely by bottlenecks in the MP2/aug cc pvqz reference energy step and MP2(full)/aug cc pcvtz core valence energy step; RI ccca has demonstrated that these bottlenecks can successfully be overcome without the introduction of significant error. When even larger molecules (i.e. 14, or more, atoms) are to be studied, RI ccca+l affords greater computational cost reduction, but at a significant (i.e. >1.0 kcal/mol) cost in accuracy relative to ccca. 153

168 Table 7.1 Differences between ccca and RI ccca total energies (me h ), relative CPU times, and relative disk space using different combinations of RI SCF and RI MP2 auxiliary basis sets. a RI SCF: cc pv(n+1)z (unc) JKFIT JKFIT RI MP2: cc pv(n+1)z (unc) cc pv(n+1)z (unc) cc pvnz RI Molecule ΔE ΔCPU b ΔDisk ΔE ΔCPU ΔDisk ΔE ΔCPU ΔDisk H 2 P % 97% % 97% % 97% Cl % 66% % 67% % 67% CH % 98% % 98% % 98% H 2 N % 97% % 97% % 97% HO % 89% % 89% % 89% H 3 Si % 98% % 98% % 98% H 2 P % 97% % 97% % 97% HS % 89% % 89% % 89% CN % 93% % 93% % 93% H 4 N % 99% % 99% % 99% H 3 O % 98% % 98% % 98% + C 2 H % 98% % 98% % 98% H 5 Si % 98% % 98% % 98% H 4 P % 98% % 98% % 98% H 3 S % 98% % 98% % 98% + ClH % 97% % 97% % 97% MAD Average % 94% % 94% % 94% Worst % 66% % 67% % 67% Best % 99% % 99% % 99% a. The designation unc denotes an uncontracted basis set, while n+1 indicates the use of the next highest ζ level basis set as an auxiliary basis set; diffuse or core valence functions where employed as appropriate. b. A negative CPU time indicates a savings, while a positive CPU time indicates additional computational cost relative to the ccca formulation. 154

169 Table 7.2 Total energies (E h ) computed with ccca, RI ccca, and RI ccca+l; differences (me h ) are relative to the ccca energies. Molecule ccca RI ccca Δ RI ccca+l Δ H FH CO N ClH F CS SiO FCl P Cl CH 2 ( 1 A 1 ) H 2 O HCN SiH 2 ( 1 A 1 ) H 2 S CO N 2 O O HOCl F 2 O OCS ( 1 Σ + ) SO ClNO CS NH C 2 H CH 2 O PH H 2 O C 2 N 2 (cyanogen) CF 2 O BF (continued on next page) 155

170 Table 7.2 (continued) Molecule ccca RI ccca Δ RI ccca+l Δ NF AlF PF ClF BCl AlCl CH SiH H 2 CCO HCOOH CH 3 Cl CH 2 F CHF CH 2 Cl CF SiF CHCl CCl SiCl C 2 H CH 3 OH H 4 N CH 3 CN CH 2 CHF CH 3 SH C 2 H 2 O 2 (glyoxal) CH 2 CHCl CF 3 CN C 2 F C 2 Cl CH 3 NH C 3 H 4 (propyne) C 3 H 4 (allene) C 3 H 4 (cyclopropene) (continued on next page) 156

171 Table 7.2 (continued) Molecule ccca RI ccca Δ RI ccca+l Δ CH 3 CHO C 2 H 4 O (oxirane) CH 2 CHCN C 2 H 4 S (thiirane) CH 3 NO CH 3 ONO CH 3 COF CH 3 COCl C 2 H C 2 H 5 N (aziridine) CH 3 SiH CH 3 COOH Si 2 H CH 3 CH 2 Cl C 3 H 6 (propene) C 3 H 6 (cyclopropane) CH 3 CH 2 SH CH 3 SCH C 4 H 4 O (furan) C 4 H 4 S (thiophene) C 4 H 6 (trans butadiene) C 4 H 6 (methylene cyclopropane) C 4 H 6 (bicyclo[1.1.0]butane) C 4 H 6 (cyclobutene) CH 3 COCH C 4 H 5 N (pyrrole) CH 3 CH 2 CH CH 3 CH 2 CH 2 Cl C 4 H 8 (cyclobutane) C 4 H 8 (isobutene) CH 3 CHOHCH C 6 H 6 (benzene) N(CH 3 ) CH 3 OCH 2 CH (continued on next page) 157

172 Table 7.2 (continued) Molecule ccca RI ccca Δ RI ccca+l Δ C 5 H 8 (spiropentane) C 4 H 10 (trans butane) C 4 H 10 (isobutane) Average Best Worst

173 Table 7.3 Enthalpies of formation (ΔH f ) at 298 K (kcal/mol) computed with ccca, RI ccca, and RI ccca+l. Molecule Expt. ccca Δ expt RI ccca Δ ccca Δ expt RI ccca+l Δ ccca Δ expt H FH CO N ClH F CS SiO FCl P Cl CH 2 ( 1 A 1 ) H 2 O HCN SiH 2 ( 1 A 1 ) H 2 S CO N 2 O O HOCl F 2 O OCS ( 1 Σ + ) SO (continued on next page) 159

174 Table 7.3 (continued) Molecule Expt. ccca Δ expt RI ccca Δ ccca Δ expt RI ccca+l Δ ccca Δ expt ClNO CS NH C 2 H CH 2 O PH H 2 O C 2 N 2 (cyanogen) CF 2 O BF NF AlF PF ClF BCl AlCl CH SiH H 2 CCO HCOOH CH 3 Cl CH 2 F CHF (continued on next page) 160

175 Table 7.3 (continued) Molecule Expt. ccca Δ expt RI ccca Δ ccca Δ expt RI ccca+l Δ ccca Δ expt CH 2 Cl CF SiF CHCl CCl SiCl C 2 H CH 3 OH H 4 N CH 3 CN CH 2 CHF CH 3 SH C 2 H 2 O 2 (glyoxal) CH 2 CHCl CF 3 CN C 2 F C 2 Cl CH 3 NH C 3 H 4 (propyne) C 3 H 4 (allene) C 3 H 4 (cyclopropene) CH 3 CHO C 2 H 4 O (oxirane) (continued on next page) 161

176 Table 7.3 (continued) Molecule Expt. ccca Δ expt RI ccca Δ ccca Δ expt RI ccca+l Δ ccca Δ expt CH 2 CHCN C 2 H 4 S (thiirane) CH 3 NO CH 3 ONO CH 3 COF CH 3 COCl C 2 H C 2 H 5 N (aziridine) CH 3 SiH CH 3 COOH Si 2 H CH 3 CH 2 Cl C 3 H 6 (propene) C 3 H 6 (cyclopropane) CH 3 CH 2 SH CH 3 SCH C 4 H 4 O (furan) C 4 H 4 S (thiophene) C 4 H 6 (trans butadiene) C 4 H 6 (methylene cyclopropane) C 4 H 6 (bicyclo[1.1.0]butane) C 4 H 6 (cyclobutene) CH 3 COCH (continued on next page) 162

177 Table 7.3 (continued) Molecule Expt. ccca Δ expt RI ccca Δ ccca Δ expt RI ccca+l Δ ccca Δ expt C 4 H 5 N (pyrrole) CH 3 CH 2 CH CH 3 CH 2 CH 2 Cl C 4 H 8 (cyclobutane) C 4 H 8 (isobutene) CH 3 CHOHCH C 6 H 6 (benzene) N(CH 3 ) CH 3 OCH 2 CH C 5 H 8 (spiropentane) C 4 H 10 (trans butane) C 4 H 10 (isobutane) Average Minimum Maximum MAD

178 Table 7.4 CPU times and disk space usage of RI ccca and RI ccca+l relative to ccca. Molecule CPU Disk RI ccca RI ccca+l RI ccca RI ccca+l H 2 32% 129% 87% 91% FH 10% 58% 84% 84% CO 45% 24% 90% 83% N 2 42% 19% 90% 83% ClH 2% 36% 85% 79% F 2 40% 17% 89% 80% CS 46% 27% 91% 81% SiO 46% 25% 91% 78% FCl 47% 31% 89% 79% P 2 57% 42% 90% 79% Cl 2 56% 44% 88% 77% CH 2 ( 1 A 1 ) 66% 55% 96% 96% H 2 O 66% 55% 96% 96% HCN 57% 44% 94% 89% SiH 2 ( 1 A 1 ) 61% 50% 97% 95% H 2 S 66% 56% 96% 95% CO 2 70% 59% 93% 87% N 2 O 60% 65% 96% 93% O 3 62% 69% 97% 96% HOCl 58% 64% 97% 97% F 2 O 59% 76% 97% 97% OCS ( 1 Σ + ) 62% 68% 96% 93% SO 2 71% 77% 97% 96% ClNO 60% 71% 97% 96% CS 2 64% 70% 96% 92% NH 3 86% 83% 98% 99% C 2 H 2 75% 70% 96% 93% CH 2 O 80% 77% 97% 96% PH 3 83% 79% 98% 98% H 2 O 2 84% 83% 98% 98% C 2 N 2 (cyanogen) 66% 79% 97% 96% CF 2 O 65% 85% 97% 97% BF 3 64% 87% 97% 97% (continued on next page) 164

179 Table 7.4 (continued) Molecule CPU Disk RI ccca RI ccca+l RI ccca RI ccca+l NF 3 61% 87% 98% 98% AlF 3 64% 86% 97% 96% PF 3 66% 87% 97% 98% ClF 3 56% 87% 97% 97% BCl 3 68% 88% 96% 97% AlCl 3 66% 86% 96% 96% CH 4 88% 86% 99% 99% SiH 4 86% 83% 98% 98% H 2 CCO 74% 82% 98% 98% HCOOH 74% 83% 98% 98% CH 3 Cl 77% 84% 98% 99% CH 2 F 2 72% 85% 98% 99% CHF 3 67% 89% 98% 99% CH 2 Cl 2 72% 85% 98% 99% CF 4 63% 92% 98% 98% SiF 4 63% 91% 97% 98% CHCl 3 69% 91% 97% 98% CCl 4 66% 92% 97% 98% SiCl 4 69% 93% 96% 98% C 2 H 4 89% 89% 98% 98% CH 3 OH 91% 91% 98% 99% H 4 N 2 91% 92% 98% 99% CH 3 CN 79% 87% 98% 99% CH 2 CHF 74% 81% 98% 99% CH 3 SH 80% 87% 98% 99% C 2 H 2 O 2 (glyoxal) 72% 88% 98% 99% CH 2 CHCl 76% 83% 98% 99% CF 3 CN 66% 95% 98% 99% C 2 F 4 63% 95% 97% 98% C 2 Cl 4 68% 93% 96% 97% CH 3 NH 2 85% 89% 99% 99% C 3 H 4 (propyne) 83% 91% 99% 99% C 3 H 4 (allene) 84% 92% 98% 99% (continued on next page) 165

180 Table 7.4 (continued) Molecule CPU Disk RI ccca RI ccca+l RI ccca RI ccca+l C 3 H 4 (cyclopropene) 84% 91% 98% 99% CH 3 CHO 80% 89% 98% 99% C 2 H 4 O (oxirane) 82% 91% 98% 99% CH 2 CHCN 78% 91% 98% 99% C 2 H 4 S (thiirane) 80% 89% 98% 99% CH 3 NO 2 72% 90% 98% 99% CH 3 ONO 74% 91% 98% 99% CH 3 COF 73% 88% 98% 99% CH 3 COCl 75% 89% 98% 99% C 2 H 6 94% 96% 99% 99% C 2 H 5 N (aziridine) 83% 92% 99% 99% CH 3 SiH 3 84% 90% 99% 99% CH 3 COOH 75% 90% 98% 99% Si 2 H 6 87% 92% 98% 99% CH 3 CH 2 Cl 81% 90% 98% 99% C 3 H 6 (propene) 85% 94% 99% 100% C 3 H 6 (cyclopropane) 85% 94% 99% 100% CH 3 CH 2 SH 80% 90% 98% 100% CH 3 SCH 3 83% 92% 98% 100% C 4 H 4 O (furan) 73% 83% 98% 98% C 4 H 4 S (thiophene) 75% 86% 98% 99% C 4 H 6 (trans butadiene) 85% 92% 97% 99% C 4 H 6 (methylene cyclopropane) 88% 93% 98% 99% C 4 H 6 (bicyclo[1.1.0]butane) 87% 92% 98% 99% C 4 H 6 (cyclobutene) 86% 89% 98% 99% CH 3 COCH 3 79% 92% 99% 100% C 4 H 5 N (pyrrole) 75% 87% 98% 99% CH 3 CH 2 CH 3 89% 93% 98% 99% CH 3 CH 2 CH 2 Cl 91% 96% 98% 99% C 4 H 8 (cyclobutane) 88% 94% 98% 99% C 4 H 8 (isobutene) 90% 96% 98% 99% CH 3 CHOHCH 3 90% 96% 98% 99% C 6 H 6 (benzene) 88% 94% 97% 98% (continued on next page) 166

181 Table 7.4 (continued) CPU Disk Molecule RI ccca RI ccca+l RI ccca RI ccca+l C 5 H 8 (spiropentane) 88% 96% 98% 99% C 4 H 10 (trans butane) 89% 97% 98% 99% C 4 H 10 (isobutane) 90% 97% 98% 99% Average 72% 76% 97% 96% Best 94% 97% 99% 100% Worst 10% 129% 84% 77% 167

182 CHAPTER 8 A SYSTEMATIC INVESTIGATION OF DIHALOGEN μ DICHALCOGENIDES 8.1 Introduction Mixed halogen chalcogen species, in particular the fluorine and oxygen containing class of compounds, have drawn much attention in the literature over the past seven decades both for their applications and unique bonding. Molecules containing a fluorine oxygen bond, such as FOOF, have many applications, such as a near room temperature fluorinating agent for various actinides and xenon Sometimes referred to as oxyfluorides, these compounds have also been reported to be of use in various rocket fuels. 206 They are also speculated to participate in various reactions of atmospheric importance, such as the depletion of ozone by a photolysis mechanism that produces FO radicals. 207,208 The sulfur analog of FOOF, FSSF, has also been implicated in the depletion of ozone. 209 Other halogenated persulfides, ClSSCl, BrSSBr, ClSeSeCl, and BrSeSeBr, are used extensively as plasma etchants in manufacturing various microelectronic devices Disulfide and diselenide bonds are particularly important in biological systems, specifically cystine rich proteins in the case of disulfides. Molecules with disulfide bridges are studied to gain insight into the relationship between the strength of the S S bond with respect to varying the terminal substituents. An understanding of this relationship can provide insight into the structure activity relationship of proteins, especially in This entire chapter is adapted from B.P. Prascher and A.K. Wilson, A Computational Study of Dihalogen μdichalcogenides: XAAX (X = F, Cl, Br; A = S, Se), J. Mo. Struct. THEOCHEM, 2007, 814(1 3), 1, with permission from Elsevier. 168

183 proteins where a disulfide bond is involved in the chemistry of the active site or if the disulfide bond stabilizes the tertiary structure of the protein While the applications of FOOF and a few of its sulfur analogs are known in the literature, there has been disagreement between computational and experimental geometries. In the earliest microwave spectroscopy experiment reported by Jackson 219 on FOOF, a short O O bond length resembling that of O 2 and a long F O bond relative to FO were observed. Subsequent reports in the literature fail to reproduce Jackson s geometry, instead reporting a peroxide like (longer) O O bond and shorter F O bonds. A careful electron diffraction study undertaken by Hedberg and coworkers 220 verified the structure Jackson had observed, and noted that a better understanding of the structure of FOOF is crucial to elucidating the mechanism by which its fluorinating abilities arise. The structure of FOOF has F O bond lengths of Å, 219,220 longer than that of FO at Å. 221 FOOF also has an O O bond length of Å, 219,220 resembling the oxygen molecule, Å, 222,223 and not a typical peroxide ( Å). 224 From microwave spectroscopy, it is known that FSSF also has a short A A bond and long A X bonds Another structural anomaly of XSSX species is the increase in the S S bond length when the halogen size is increased, 229 a trend which is observed in XOOX species also. 222,230 The structural anomalies of FOOF have been addressed in much detail elsewhere 207 as the result of overlap between an oxygen lone pair orbital and the adjacent unoccupied F O σ* orbital a phenomenon called anomeric delocalization (see Figure 8.1). This phenomenon is not unique to the FOOF molecule, but also exists in molecules such as α fluoroethers 231 and other molecules where highly electronegative atoms are in close proximity to one another. 169

184 Figure 8.1 The orbital representation of anomeric delocalization in XAAX systems: the lone pair on a bridging A atom delocalizes into the adjacent σ*(ax) orbital. From an experimental standpoint, anomeric effects can account for the shortened O O bond and lengthened F O bond, while, computationally, anomeric effects can present significant challenges to predicting structures accurately, even for some of the most sophisticated theoretical approaches. 232 Computational methods such as multi reference configuration interaction (MRCI) and coupled cluster (CC) with basis sets of varying sizes have been employed to study the FOOF molecule. 232 Additionally, composite approaches such as Gaussian 1 and Gaussian 2 (G1 and G2, respectively) have been used to study FOOF. 233 Both density functional theory (DFT) and CC methods reproduce the experimental geometry with the least amount of error compared with the other methods listed above. 232 Absolute errors of Å in bond lengths and in angles occur with various pure and hybrid density functionals, while errors of Å in bond lengths and in angles occur with coupled cluster computations employing various basis sets. Small basis sets (i.e. the double ζ level), coupled with electron correlation methods, have been shown to fortuitously yield accurate structures of FOOF compared with experiment. 232,234,235 Structures with deviations from experiment less than 0.01 Å compared 170

185 have been computed in the past by adding non optimized polarization 235 and diffuse functions 236 to a double ζ basis set. An improvement in the accuracy of the O O bond length is typically observed with a loss of accuracy in the O F bond length using this approach. Computational studies with smaller, more rigid basis sets, such as that of Scuseria 237 where the polarization functions of the oxygen and fluorine basis sets were constrained to have equal exponents, tend to give results closer to experiment. This work, and others, 232,235 do not condone the use of such nonstandard approaches to predictive quantum chemistry since the method cannot be readily applied to study other molecular systems. Instead, one would like to apply a standard and systematic approach to studying similar systems. In this chapter, the heavier analogs of the FOOF molecule with the structure XAAX (X = F, Cl, Br and A = S, Se) are investigated. When this investigation was published, no systematic ab initio or DFT study on the structure, frequencies, and anomeric effects had been performed on the sulfur and selenium XAAX analogs of FOOF, nor had CC computations on the relative stabilities of the XAAX conformation compared with other A 2 X 2 isomers been reported. 8.2 Computational Methodology Both hybrid DFT and CC methods have been utilized for this work. DFT is particularly attractive because of its formal scaling (where is the number of basis functions). The DFT method employed is Becke s three parameter exchange scheme (B3) 166 coupled with the Lee Yang Parr (LYP) correlation functional. 165 The third parameterization of Vosko, Wilk, and Nusair correlation functional (VWN 3) has been used. The coupled cluster method including singles, doubles, and non iterative, perturbative triples is also employed, CCSD(T). 31,38 171

186 A series of natural orbital (NO) analyses using singles and doubles configuration interaction (CISD) has been performed. The CISD NOs provide insight into the amount of multireference character of the system and the most important orbitals involved in computing the dynamic correlation energy. The NOs reported here also demonstrate how well correlated methods, specifically truncated ab initio methods, describe XAAX systems with respect to anomeric delocalization. The correlation consistent, polarized valence basis sets, cc pvnz (where n = D, T, and Q) have been employed in this study. In computations involving second row atoms, the tight d correlation consistent basis sets, cc pv(n+d)z, 101 were employed. Additionally, the augmented correlation consistent basis sets, aug cc pvnz and aug cc pv(n+d)z, 95,99,100 were used for describing long range effects, specifically lone pair interactions. This investigation is concerned with the chemistry of disulfide and diselenide compounds, following the results of experimental work regarding the stability of the conformations of X 2 A 2 compounds. 215,216,225, Structures have been fully optimized at each level of theory and basis set, utilizing symmetry where present. The B3LYP calculations were performed using the Gaussian 03 software suite, 174 while the Molpro software package 180 was used for CCSD(T) and CISD computations. Harmonic vibrational frequencies were computed analytically with B3LYP and numerically with CCSD(T) at the aug cc pvtz level. Computed anharmonic frequencies are included for comparison

187 8.3 Results and Discussion Stability of X 2 A 2 Compounds Following the reported stabilities of several conformations of FOOF by Kraka et al., 232 computations have been performed at the aug cc pvtz level to compare the stabilities of the linear, cis, trans, and X 2 AA conformations of the heavier analogs with their gauche forms. The computed relative stabilities (in kcal/mol) are tabulated in Table 8.1 for both B3LYP and CCSD(T). It is found that all the conformations are significantly less stable than the gauche form. For F 2 S 2, the B3LYP value shows FSSF to be more stable than F 2 SS, but CCSD(T) does not. This apparent contradiction to what is known experimentally regarding FSSF being more stable than F 2 SS 215,216,225 has been investigated further through a series of computations using the augmented double through quadruple ζ basis sets with both methods. As the basis set size increases, B3LYP predicts that FSSF is the more stable compound. The CCSD(T) computations with increasing basis set size show F 2 SS as the more stable conformation. However, when a thermal correction (for 298 K) is applied to the CCSD(T) electronic energy, FSSF is shown to be more stable than the F 2 SS conformation by 0.04 kcal/mol. Other isomers of the XAAX type molecules have been investigated previously, specifically the XAXA and AXXA isomers of iodine oxides. 243,244 We have briefly examined these two isomeric forms here using second order Møller Plesset perturbation theory (MP2) and the aug cc pvtz basis set. MP2 has been used since CCSD(T) computations without symmetry can become time intensive, and MP2 will recover a significant percent of the CCSD(T) correlation energy. Further, test computations on the cis, trans, and linear forms have shown MP2 relative 173

188 stabilities to be comparable to CCSD(T) stability energies. For most XAAX compounds of this study, the corresponding XAXA compound is found to dissociate into two XA fragments as doublet electronic states. Only in the case of FSFS does the molecule dissociate into FSF ( ) + S ( ). Similarly, the AXXA compounds display no bound minima as tetraatomic molecules, but dissociate into two XA fragments, each with a doublet electron configuration XAAX Structures Optimized geometries are listed in Table 8.2, Table 8.3, and Table 8.4 for the fluorine, chlorine, and bromine species, respectively. For the fluorine compounds, relative to experiment both B3LYP and CCSD(T) result in the least error at the aug cc pvqz level as compared with the other basis sets the errors being Å and Å, respectively. The CCSD(T) S F bond length is in good agreement with experiment, with a difference of only 0.01 Å at the triple ζ level. CCSD(T) only slightly outperforms B3LYP for the bond lengths and angles. B3LYP dihedral angles tend to converge to the experimental value while CCSD(T) dihedrals converge away from experiment. In FSeSeF, the optimized Se Se bond length converges away from the experimental value for both methods and their smallest errors relative to experiment are Å for B3LYP and Å for CCSD(T) at the cc pvdz level. Just as in the case of FSSF, the Se F bonds with CCSD(T) are within chemical accuracy by the triple ζ basis set. The dihedral in FSeSeF is closest to the experimental value for both methods when the smallest basis set is employed. The differences between the computed angles and the experimental angle range from for B3LYP and from for CCSD(T), where an experimental value of 100 has been reported by Haas. 245 In either the case of B3LYP or CCSD(T), adding diffuse functions does not 174

189 significantly affect the bond lengths, but does affect the angles and dihedral angles. The inclusion of the tight d function in FSSF is important, especially at the double and triple ζ levels where the bond length can be affected by up to Å and Å, respectively. For the chlorine species, ClSSCl, the optimized S S and S Cl bond lengths in ClSSCl move towards experiment as the basis set size is increased for both CCSD(T) and B3LYP. The results are within 0.01 Å with CCSD(T) at the cc pv(q+d)z level. Angles are closer to experiment with CCSD(T) than with B3LYP at the largest basis set level. Dihedrals show a similar trend as that of the angles. The tight d basis set is again noted to improve upon the description of bond lengths and angles as compared with the standard correlation consistent basis sets for both B3LYP and CCSD(T), which suggests that core polarization can be important in DFT. 126,246 For example, with B3LYP, the error in the bond lengths is lowered by Å Å at the double ζ level when a tight d function is added to the valence and augmented basis sets. For ClSeSeCl, the B3LYP computations using the quadruple ζ basis sets (original and tight d) converge to similar values for the Se Cl bond length. It is observed that the addition of the tight d function will not significantly affect bond lengths in DFT computations since this method is typically not very basis set dependent. CCSD(T) is more basis set dependent, and we observe that the addition of a tight d function also has little effect on the computed bond lengths of ClSeSeCl. Optimized bond lengths for S S and S Br in BrSSBr are within 0.01 Å of experiment when computed with the CCSD(T) method, however, the basis set level at which the errors are lowest varies. Table 8.4 shows that for cc pvqz and aug cc pvqz, the CCSD(T) deviation from experiment in the S S bond is Å and Å, respectively. The valence and augmented tight d sets deviate by Å and Å, respectively, at the triple ζ level. Computed B3LYP 175

190 S S bond lengths and bond angles converge away from the experimental values when the basis set size is increased. Additionally, the tight d function helps to accelerate this convergence away from the experimental values. The opposite trend is observed in the S Br bond length with B3LYP: the larger basis sets yield the more accurate geometries. In BrSeSeBr, as seen in ClSeSeCl, there is no effect of adding diffuse functions since the valence and augmented sets converge on the same values at the quadruple ζ level. The B3LYP computations show a deviation of Å in the Se Se bond and Å in the Se Br bond relative to experiment at the quadruple ζ level of both families of basis sets, while CCSD(T) shows deviations of approximately Å and Å in the Se Se bond and Se Br bond, respectively. The addition of diffuse functions does not significantly affect these results, which is not surprising since the cc pvqz basis set already contains functions that are rather diffuse Vibrational Frequency Analyses Computed harmonic infrared (IR) vibrational frequencies at the aug cc pvtz level for the six molecules in this investigation are presented in Table 8.5. The assignments of computed normal modes concur with experimental assignments. There is an artifact in the S Br asymmetric stretch in which the computed frequencies deviate from the experimental frequencies of Frenzel et al. 241 by 39.6 cm 1 and 67.3 cm 1 for B3LYP and CCSD(T), respectively. The magnitude of the deviation between theory and experiment in this particular case not consistent with computed results of the other normal modes for this compound and the results of two other experimental determinations of the spectrum of BrSSBr have been included for comparison. An explanation of the discrepancy between later experiments and that of Frenzel 176

191 et al. is offered by Forneris et al. 239 as due to liquid Br 2 dissolved in the BrSSBr sample from which the spectrum was taken. The dissolved Br 2 impurity has the effect of dampening the observed S Br asymmetric stretch frequency considerably. To compare our computations with experimental IR spectra, computations of the anharmonic corrections to the normal modes of FSSF have been performed at the CCSD(T)/cc pvdz level 242 and documented in Table 8.6. The largest difference between the harmonic and anharmonic frequencies in FSSF occurs in the S S stretching mode and has a magnitude of 11.5 cm 1. The correction for anharmonicity to the zero point energy is computed to be kcal/mol. From these anharmonic computations and the low deviation of the computed harmonic frequencies from the experimental values, it is postulated that the anharmonic corrections for the other dihalo μ dichalcogenides investigated here will be negligible. As a result, the computed harmonic frequencies, as well as geometries, may be compared to the experimental values without the introduction of significant error Anomeric Effects The two lowest unoccupied NOs for FSSF are displayed in Figure 8.2. These orbitals represent F S σ* orbitals with equal occupation numbers of This indicates that the heaviest contributors to the dynamic correlation energy are excited configurations involving these σ* orbitals. This observation is in accord with the conclusion reported by Kraka et al. that the unique geometry of FOOF is a result of anomeric delocalization. 232 These F S σ* orbitals are formally unoccupied in the ground state, but symmetry allows for overlap between them and adjacent lone pairs. The result is a destabilization of the F S bond and a stabilization of the S S 177

192 Figure 8.2 The lowest unoccupied CISD natural orbitals of FSSF: both contour plots are F S σ* orbitals (14A, left; 13B, right). The perspective is down the z axis (iso contour value: 0.095). bond, which accounts for the experimentally observed S 2 like bond in FSSF and the elongated F S bonds. Further analysis of the truncated CI wavefunction indicates that the weighting coefficient of the HF reference state is 0.925, which indicates some multi reference character in the system. Previous analyses of the FOOF molecule by Kraka et al. 232 and Feller et al. 234 concluded that advanced ab initio methods that include excited configurations through quadruple excitations would be important in the quantitative description of the FOOF structure and the exact wavefunction. These highly excited configurations, they claimed, would be important because the anomeric delocalization in FOOF involves two lone pairs simultaneously delocalized. The wavefunction that would describe such delocalization is a quadruply excited configuration. Considering the results obtained in this investigation do not include quadruplyexcited configurations, there is evidence for anomeric effects in FSSF, but the impact on the overall structure is diminished compared with FOOF. This is evidenced by the computed geometries at the CCSD(T) level, where systematic improvements in the basis set lead to lower 178

193 Figure 8.3 The molecules FSSF (left), ClSSCl (center), and BrSSBr (right), and the lone pair orbitals responsible for anomeric delocalization. The prespective is down the z axis (iso contour values: 0.06, 0.03, and 0.03, respectively). deviations from experiment. Figure 8.3 shows plots of the lone pair orbitals involved in anomeric delocalization in FSSF, ClSSCl, and BrSSBr. Analysis of the overlap between the lone pair orbitals and the corresponding σ* orbitals demonstrates that as the halogen increases in size, the lone pair orbitals do not have the radial extent to significantly overlap with the σ* orbitals and anomeric delocalization is diminished. This is a plausible explanation for the change in the geometry of XSSX to an S2 type bond length and XS type halogen sulfur bond length when the halogen increases in size. Inspection of the computed and experimental geometries of FSeSeF indicates that anomeric effects are markedly decreased compared with those of FSSF. The Se Se bond measures 2.25 Å, compared with Å in the selenium dimer.170 Additionally, the Se F bond distance is 1.77 Å, while the SeF molecule has a bond length of Å.170 Since the chalcogen bond length is not shortened, nor is the chalcogen halogen bond lengthened as considerably as in FOOF or FSSF, it is clear that anomeric effects do not play a significant role in determining the 179

194 structure of the selenium compounds in this investigation. The unique structure of these compounds is due to the presence of anomeric delocalization, and this delocalization is directly related to the radial extent of the lone pair orbitals. The other factor in determining the impact of anomeric effects in these species is symmetry. In computations performed at the CCSD(T)/aug cc pv(t+d)z level on trans FSSF (C 2h ), the S S bond extends from Å to Å, and the S F bond recedes from Å to Å. The lone pairs cannot overlap with the F S σ* orbitals in this conformation, thus leaving the structure with a typical disulfide bond length and a S F bond length resembling that of diatomic SF. This is not new to the literature, as it has been reported by Samdal et al. 229 for a variety of dihalo μ dichalcogenides computed at the HF/6 31G* level. The present investigation reports for the first time correlated computations demonstrating that anomeric effects diminish as the dihedral angle changes. Kraka et al. 232 also performed similar computations with B3LYP and reported that trans FOOF has a peroxide like structure with O F bond lengths similar to that of molecular FO. 8.4 Conclusions For the sulfur systems in this investigation, geometries accurate to Å are possible with CCSD(T) and Å with B3LYP when the correlation consistent basis sets of quadruple ζ quality are employed. The largest deviations, relative to experimental values occur for the geometries of FSSF. The geometries of the selenium compounds optimized with CCSD(T) deviate by no more than Å in bond lengths, while B3LYP geometries deviate by as much as Å in bond lengths relative to experiment when computed with the cc pvqz basis set. 180

195 Bond angles deviate up to 5.07 in CCSD(T)/cc pvqz computations with respect to experiment, however, overall CCSD(T) angles result in less deviation from experimental values as compared with B3LYP. The use of the cc pv(n+d)z basis sets improves the computed geometries at the doubleand triple ζ basis set levels for the CCSD(T) geometries, and, as expected, only slightly improves B3LYP optimized geometries. The inclusion of diffuse functions in the basis has little impact on the geometry of the species investigated here. For the computed harmonic vibrational frequencies, the maximum deviation from experimental values is almost 67 cm 1, which is attributed to damping from Br 2 impurities in one of the experimental studies referenced on BrSSBr. Other compounds have computed frequencies that deviate no more than 25 cm 1. Computed anharmonic effects of the FSSF molecule revealed that their impact on bond lengths and harmonic frequencies is negligible. Anomeric effects have been investigated by others as the source of the unique geometry of the FOOF molecule. The current investigation shows that the impact of anomeric delocalization on FSSF is decreased as compared with the impact on FOOF, and that it has almost no impact on the structure of FSeSeF. Upon increasing the halogen size to chlorine and bromine, the radial extent of the lone pair orbitals is not sufficient for significant overlap with the appropriate σ* orbitals, and thus anomeric effects on the geometric structure are almost nonexistent. When FSSF is rotated into the trans conformation, anomeric delocalization cannot occur due to broken symmetry. The natural consequence of this is an increase in S S bond length and a decrease in the S F bond length. 181

196 Table 8.1 Stabilities (kcal/mol) of the different isomers of X 2 A 2 molecules relative to the gauche isomer, XAAX, computed with B3LYP and CCSD(T). System Method Linear cis trans X 2 AA F 2 S 2 B3LYP CCSD(T) a Cl 2 S 2 B3LYP CCSD(T) Br 2 S 2 B3LYP CCSD(T) b F 2 Se 2 B3LYP CCSD(T) Cl 2 Se 2 B3LYP CCSD(T) b Br 2 Se 2 B3LYP CCSD(T) b a. Including a thermal correction to 298 K in F 2 SS and FSSF shows FSSF to be the more stable isomer. b. Optimization leads to dissociation: X 2 A 2 2X + A

197 Table 8.2 B3LYP and CCSD(T) optimized geometries (Å and degrees) of the fluorine species using various correlation consistent basis sets. B3LYP CCSD(T) FSSF ζ VnZ V(n+d)Z avnz av(n+d)z VnZ V(n+d)Z avnz av(n+d)z Expt. R e (AA) D T Q a R e (AF) D T Q a α D T Q a δ D T Q a FSeSeF R e (AA) D T Q b R e (AF) D T Q b α D T Q b δ D T Q b a. Kuczkowski, R. L. J. Am. Chem. Soc. 1964, 86, b. Haas, A. J. Fluor. Chem. 1986, 32,

198 Table 8.3 B3LYP and CCSD(T) optimized geometries (Å and degrees) of the chlorine species using various correlation consistent basis sets. B3LYP CCSD(T) ClSSCl ζ VnZ V(n+d)Z avnz av(n+d)z VnZ V(n+d)Z avnz av(n+d)z Expt. R e (AA) D T Q a R e (ACl) D T Q a Α D T Q a Δ D T Q a ClSeSeCl R e (AA) D T Q b R e (ACl) D T Q b α D T Q b δ D T Q b a. Marsden, C. J.; Brown, R. D.; Godfrey, P. D. J. Chem. Soc. Chem. Comm. 1979, 399. b. Forneris, R.; Hennies, C. E. J. Mol. Struct. 1970, 5,

199 Table 8.4 B3LYP and CCSD(T) optimized geometries (Å and degrees) of the bromine species using various correlation consistent basis sets. B3LYP CCSD(T) BrSSBr ζ VnZ V(n+d)Z avnz av(n+d)z VnZ V(n+d)Z avnz av(n+d)z Expt. R e (AA) D T Q a R e (ABr) D T Q a α D T Q a δ D T Q a BrSeSeBr R e (AA) D T Q b R e (ABr) D T Q b α D T Q b δ D T Q b a. Hirota, E. Bull. Chem. Soc. Jpn. 1958, 31, 130. b. Forneris, R.; Hennies, C. E. J. Mol. Struct. 1970, 5,

200 Table 8.5 B3LYP and CCSD(T) harmonic vibrational frequencies (cm 1 ) computed with the aug cc pvtz basis set. Molecule Method Torsion A A X Bend A A X Bend A X Asymm. Stretch A X Symm. Stretch A A Stretch FSSF B3LYP CCSD(T) Expt. a ClSSCl B3LYP CCSD(T) Expt. b Expt. c Expt. d (430) BrSSBr B3LYP CCSD(T) Expt. b Expt. c Expt. d FSeSeF B3LYP CCSD(T) Expt. ClSeSeCl B3LYP CCSD(T) Expt. c Expt. d BrSeSeBr B3LYP CCSD(T) Expt. c e 118 e Expt. d a. Brown, R. D.; Pez, G. P. Spectroc. Acta 1970, 26A, b. Frenzel, C. A.; Blick, K. E. J. Chem. Phys. 1971, 55, c. Forneris, R.; Hennies, C. E. J. Mol. Struct. 1970, 5, 449. d. Frankiss, S. G. J. Mol. Struct. 1968, 2, 271. e. Hendra, P. J.; Park, P. J. D. J. Chem. Soc. A 1968,

201 Table 8.6 A comparison of CCSD(T) harmonic and anharmonic vibrational frequencies (cm 1 ) of FSSF computed with the cc pvdz basis set. Mode ω e ν 0 Difference Torsion S S F Bend S S F Bend S F Asymm. Stretch S F Symm. Stretch S S Stretch

202 CHAPTER 9 A SYSTEMATIC INVESTIGATION OF GERMANIUM ARSENIDES 9.1 Introduction Compounds containing heavy group 14 and 15 elements, especially compounds with multiple bonds, have received considerable attention over the past three decades by both experimentalists and theoreticians The surge in interest in these compounds came after the synthesis of several non transient multiply bonded phosphorus and silicon systems in 1981, and these species have been the subject of several reviews. 250,252, Before that time, it was believed that multiple bonds between heavy main group elements would likely not exist due to the so called double bond rule, which stated that elements such as silicon and phosphorus would exhibit weak π bonds formed by their p orbitals. 259,264 Subsequent attempts to synthesize these types of compounds failed since the molecules would oligomerize in solution. The oligomerization was found to be reduced by incorporating large, bulky substituents on the heavy main group atoms. 250,260 Since that time, several compounds that have multiply bonded group 14 and 15 elements with low coordination numbers (i.e. P 2, Si 2, SiC, PC, etc.) have been reported. However, compounds that are not found in earlier reviews are those with germanium multiply bonded to an arsenic atom so called arsagermenes and arsagermynes (germanium doubly and triply bonded to arsenic, respectively). While it is purported that these types of systems should exist, and could be synthesized via similar mechanisms as analogous silicon This entire chapter is adapted from B.P. Prascher, A.M. Kavi, B. Mintz, S.M. Yockel, and A.K. Wilson, Theoretical Investigation of the Germanium Arsenides, Chem. Phys., 2008, 353(1 3), 209, with permission from Elsevier. 188

203 phosphide systems, no such studies have been reported. Germanium arsenides are important in various research fields. Although gallium arsenides are routinely encountered in solid state research, germanium arsenide materials such as GeAsS glasses and MGeAs 2 (M = Zn, Cd) are of interest for their nonlinear optical properties and electrical properties. Germanium arsenide materials have been studied both by experimentalists and by theoreticians for their applications in solar cells and as ternary semiconductors. 247,248 Another example of an exotic germanium arsenide is the material (Ge 38 As 8 ) 8+ 8 I, which has been characterized by x ray and neutron diffraction, and is a known N type semiconductor. 253 The only previous computational investigation of the germanium arsenide molecules considered in the present study was performed by Lai et al. 258 They utilized both correlated ab initio methodology and density functional theory (DFT) with the G(d,p) and G(3df,3pd) basis sets (triple ζ quality basis sets with diffuse and polarization functions) to describe the structures, bonding characteristics, and relative stabilities of L GeAs and GeAs L molecules. They computed that the most stable L GeAs molecule of their test set should be H 2 N GeAs, while the most stable GeAs L molecule should be GeAs AlH 2. Further, Lai et al. predicted with DFT that, for L = F, OH, and CH 3, the GeAs L isomer should be more stable than L GeAs, while their ab initio predictions showed the opposite (i.e. the L GeAs isomer is more stable for these L groups). This apparent discrepancy between DFT and correlated ab initio methodology has prompted this systematic re investigation of Lai s molecules. One of the differences between the present investigation and that of Lai et al. 258 is the use of the correlation consistent basis sets, cc pvnz The systematic construction of the 189

204 correlation consistent basis sets leads to monotonic convergence towards the complete basis set (CBS) limit of many energetic properties as the basis set size increases 93 a property not inherent to the basis sets used by Lai et al. 265 The present investigation reexamines the GeAs L and L GeAs molecules studied by Lai et al. 258 (where L = H, Li, Na, BeH, MgH, BH 2, AlH 2, CH 3, SiH 3, NH 2, PH 2, OH, SH, F, and Cl; and has been extended to include Br) Reported are transition state energies, enthalpies of formation at 298 K, and enthalpies of isomerization at 298 K. Typically, the threshold for chemical accuracy of main group molecules is considered to be 1.0 kcal/mol. To achieve this threshold of chemical accuracy for third row, main group systems, it has been shown that scalar relativistic and spinorbit effects must be included. 62, Since this is a predictive investigation into the properties of third row, main group molecules yet to be characterized by experiment, relativistic effects cannot be ignored. 9.2 Computational Methodology Coupled cluster with single, double, and non iterative, perturbative triple excitations, CCSD(T), is used. 31,38 In addition, the hybrid three parameter exchange functional of Becke (B3) 166 coupled with the Lee Yang Parr (LYP) 165 correlation functional, B3LYP, has been employed. In all B3LYP computations, the third parameterization of the Vosko Wilk Nusair local correlation functional (VWN 3) is invoked. 167 The correlation consistent basis sets of double through quadruple ζ quality have been employed. For germanium arsenides that contain an alkali or alkaline earth metal, new cc pvnz basis sets discussed in Chapter 6 have been used; while for atoms in the second row of 190

205 the periodic table (Na, Mg, and Al Ar), the tight d correlation consistent basis sets, cc pv(n+d)z 101 have been employed. To compute CBS limits, the exponential (3.2) and mixed Gaussian/exponential (3.3) extrapolation schemes have been employed. 85,95,132 The geometries of the L GeAs and GeAs L systems were fully optimized using both B3LYP and CCSD(T) at each basis set level. Zero point energies were computed from cc pvtz harmonic vibrational frequencies using each method. Classical barriers were computed from transition state structures with exactly one imaginary frequency along the isomerization pathway. Enthalpies of formation and enthalpies of isomerization were determined using the equations of Ockterski 272 from atomic enthalpies of formation at 100 kpa and 298 K. 195,273 All B3LYP computations were performed using the Gaussian 03 software suite, 174 while CCSD(T) computations were performed using the Molpro software package. 180 Scalar relativistic corrections for both the CCSD(T) and B3YLP energetics were determined using both the Cowan Griffin (CG) 274 and the second order, spin free Douglas Kroll (DK) Hamiltonians The DK approach is variational, while the CG (perturbative) approach is not. 141,275 Relativistic corrections were computed using the B3LYP/cc pvtz geometry at each basis set level, since scalar relativistic effects are not very sensitive to changes in the geometry. 276 Spin orbit (SO) coupling of the L GeAs and GeAs L molecules is neglected because they are closed shell molecules. First order SO corrections (see Table 9.1) are included a posteriori in the atomic energies since the spin free implementations of the CG and DK Hamiltonians do not include it intrinsically. These SO corrections were computed using multiconfiguration self consistent field (MCSCF) wavefunctions and the Breit Pauli spin orbit Hamiltonian 180,277 implemented in the Molpro software package. 191

206 9.3 Results and Discussion Optimized Structures The bonding characteristics of L GeAs and GeAs L systems have been discussed in some detail by Lai et al. 258 The structures of L GeAs systems are linear, or very close to linear, and exhibit a triple bond between the germanium and arsenic atoms not unlike a cyanide group. However, the GeAs L systems exhibit a strongly bent structure as opposed to the linear structure of an isocyanide molecule. Instead, there is a double bond between the germanium and arsenic atoms, with a lone pair on the arsenic atom, resembling an extremely bent iminetype structure. The decrease in bond order in migrating the L group from the germanium to the arsenic atom is reinforced by the increase in the Ge As bond length. For example (cf. Table 9.2 and Table 9.3), in the optimized structures of the L GeAs molecules of this study, the Ge As bond length falls in the range of Å for CCSD(T) and Å for B3LYP. Further evidence to support a triple bond between germanium and arsenic in the L GeAs molecules is that the bond lengths resemble that of the arsenic dimer, which has a formal triple bond and bond length of Å. 170 The Ge As bond length in the optimized structures of the GeAs L molecules falls in the range of Å for CCSD(T) and Å for B3LYP, resembling that of the GeAs diatomic at Å. 270 The Ge As L angle remains rather small, even when L groups that are highly electropositive or highly electronegative are used, making the bonding characteristics of germanium arsenides unique among main group molecules. In Table 9.2 and Table 9.3 the CCSD(T) and B3LYP optimized geometries for the 192

207 different L groups of this study are listed. The results of Lai et al. 258, which are included for comparison in the tables, were computed using CCSD(T)/6 311G++(3df,3pd)// B3LYP/6 311G++(d,p). Overall, for the L GeAs molecules optimized with CCSD(T), the L Ge and Ge As bond lengths differ up to Å and Å, respectively, when compared with the results of Lai et al. The same bond lengths differ up to Å and Å, respectively, when optimized with B3LYP. The largest differences between the bond lengths of Lai et al. and this study are in the CCSD(T) geometries of Li GeAs, Na GeAs, HBe GeAs, and HMg GeAs. The small difference between the B3LYP results and those previously published reaffirms the fact that B3LYP bond lengths for these species are not very basis set dependent. The average difference between results reported here and those of Lai et al. are Å and Å for the L Ge bond length and Å and Å for the Ge As bond length, for CCSD(T) and B3LYP, respectively. In the GeAs L molecules, the variations between previously published bond lengths and those of this study are more dramatic. The CCSD(T) optimized As L and Ge As bond lengths differ the most from previous work 258 up to Å and Å, respectively. However, the B3LYP As L and Ge As bond lengths show less dramatic differences with respective variations up to Å and Å. The largest differences between both the CCSD(T) and B3LYP bond lengths and those of Lai et al. 258 arise in the GeAs AlH 2 molecule. Also, the increased variation in bond lengths between the B3LYP results and those previously published indicate that B3LYP is more sensitive to the choice of basis set for GeAs L systems, compared with the L GeAs systems. Bond angles are affected more than bond lengths in both the L GeAs and GeAs L systems, however, the qualitative structure remains the same as previously reported the 193

208 L GeAs systems are linear in X Ge As (within 5.0 ), where X is the non hydrogen atom of the L group, and GeAs L systems are severely bent in the Ge As X angle. In both the CCSD(T) and B3LYP optimized structures of the L GeAs molecules, the differences in the bond angles are in the range , compared with Lai et al., 258 but the differences in the bond angles of the GeAs L molecules are in the range of The largest differences between the computed bond angles and those of Lai et al. occur in the GeAs OH molecule with CCSD(T), and in the GeAs AlH 2 molecule with both CCSD(T) and B3LYP. Upon closer examination, the B3LYP/ G(d,p) structures of the HO GeAs and GeAs OH isomers that Lai et al. reported were reproduced, and it was found that explicit optimization with CCSD(T)/cc pvnz, as performed in this investigation, is responsible for the large variation in the bond angle. Comparing the CCSD(T) to the B3LYP optimized geometries, the largest differences between the two methods are: in the Ge F bond length (0.056 Å at the cc pvqz level); in the As OH bond length (0.186 Å at the cc pvtz level); and in the Ge As bond length of the L GeAs isomers, the largest differences between the two methods are Å (L GeAs isomers) in HBe GeAs at the cc pvtz and cc pvqz levels and Å (GeAs L isomers) in GeAs Br at the cc pvqz level. Further, the X Ge As and Ge As X bond angles vary between the two methods up to 3.7 and 38.1, respectively. In general, CCSD(T) and B3LYP agree on the linearity of the L GeAs molecules, however, the ranges of the bond lengths and angles of the GeAs L molecules are significantly different between the two methods. The average differences in the geometries of CCSD(T) compared with B3LYP are Å in the Ge L bond, Å in the Ge As bond of the L GeAs isomers, Å in the As L bond, Å in the Ge As bond of the GeAs L isomers, 0.28 in the L Ge As angle, and 5.23 in the Ge As L angle. 194

209 Figure 9.1 A schematic of the forward (f) and reverse (r) reaction coordinates of L migration from the germanium atom to the arsenic atom Classical Barriers to Isomerization The barriers to isomerization (migration) of the L group from the germanium terminus to the arsenic terminus have been computed and are listed in Table 9.4. Each of the energies, including the CBS limits, are computed at 0 K, with a zero point vibrational energy correction, but without tunneling effects, which are only expected to be important for hydrogen migration. Both the forward (f) and reverse (r) reactions are represented (cf. Figure 9.1). The minimum structures are connected by a single transition state (the highest energy structure along the lowest energy pathway connecting L GeAs and GeAs L), which is verified by exactly one imaginary (negative) vibrational frequency along the isomerization pathway. These reported double through quadruple ζ classical barriers do not show monotonic behavior with respect to increasing basis set size, yet a CBS value is still reported because the electronic energies (the energy from which the barriers are computed) do converge monotonically. There are some notable trends in the forward energy of activation (E act ) listed in Table 9.4. First, in the alkali metal and alkaline earth metal hydrides, which require the least energy to migrate from the germanium atom to the arsenic atom, Li GeAs only requires 1.02 kcal/mol and decreases to 0.43 kcal/mol for Na GeAs with the CCSD(T)/quadruple ζ level of theory, while E act 195