AB INITIO METHODS IN COMPUTATIONAL QUANTUM CHEMISTRY

Transcription

1 AB INITIO METHODS IN COMPUTATIONAL QUANTUM CHEMISTRY Aneesh. M.H A theoretical study on the regioselectivity of electrophilic reactions of heterosubstituted allyl systems Thesis. Department of Chemistry, University of Calicut, 2012

2 CHAPTER I AB INITIO METHODS IN COMPUTATIONAL QUANTUM CHEMISTRY Contents 1.1 Introduction to Computational Chemistry 1.2 Molecular Mechanics/Molecular Dynamics Methods 1.3 Electronic Structure Methods 1.4 An Overview of ab initio Methods Hartree-Fock (HF) method The HF limit and the concept of electron correlation Post-HF methods Configuration interaction methods Perturbative theories Coupled cluster methods Multi-configuration (MC) methods Composite methods Methods for excited states Multirefernece methods r 12 methods 1.5 Quantum Mechanics Molecular Mechanics (QM-MM) Methods 1.6 Semi Empirical Methods 1.7 Density Functional Theory (DFT) 1.8 Basis Sets Exponential type orbitals Gaussian type orbitals (GTO) Even - tempered basis sets Well - tempered basis sets Universal basis sets Geometrical basis sets Contracted basis sets Segmented contractions Polarization functions Diffuse functions STO-nG basis sets Pople style basis sets General contractions Basis set super position error (BSSE) Two electron basis functions 1.9 Model Chemistry Appendix 1: Atomic Units REFERENCES 0

3 CHAPTER I AB INITIO METHODS IN COMPUTATIONAL QUANTUM CHEMISTRY 1.1 Introduction to Computational Chemistry Although much of its discovery process is descriptive and qualitative, chemistry is fundamentally a quantitative science. It serves a wide range of human needs, activities and concerns. Chemistry attained the status of a quantitative science through the development of one major discipline in science, namely, quantum mechanics. The basic principles of quantum mechanics were formulated in the 1920s through the works of Heisenberg, Born, Schrödinger, Dirac and many others. Though, in principle, the equations of quantum mechanics could be applied to any system to produce exact quantitative results, in practice, it was not so. Exact solutions of the equations were possible only for certain one electron systems. Even with the advent of the so called approximation methods the situation didn t improve dramatically due to the formidable mathematics involved in solving the equations. This is quite evident from the famous quote made by Dirac in The fundamental laws necessary for the mathematical treatment of large parts of Physics and the whole of Chemistry are thus fully known, and the difficulty lies only in the fact that application of these laws leads to equations that are too complex to be solved. A dramatic leap from this almost stagnant situation was made possible through the advent of digital computers in the 1950s. Efficient algorithms were written which helped to solve the formidable mathematical equations with relative ease. The benefits were evident not only in solving equations of quantum mechanics but also in solving the equations of classical mechanics. 1

4 This fusion between radically different disciplines paved the way for a new branch in Chemistry, Computational Chemistry. Since then computational science in general has made and continues to make its rapid expansion, both in terms of hardware and software technologies. Every step forward in computational science has its beneficial impact on Chemistry. Thus, computational chemistry can be broadly defined as the branch of chemistry that uses computers to generate information that is complementary to experimental data on the structures, properties and reactions of substances. Computational chemistry has also sought to devise and to implement quantitative algorithms for organizing massive amounts of data from the laboratory and for predicting the course and extent of chemical phenomena in situations that are difficult or even impossible to observe directly. Over the last 50 years, methods have evolved from those that were used to study 1-and 2-atom systems in 1928, through those that were used to study 2- to 5-atom systems in 1970, to the present programs that produce useful quantitative results for molecules with up to 10 to 20 atoms. With cruder models of the atom (for instance, simpler approximations as in molecular mechanics) it is now possible to model biological molecules with thousands of atoms. As a result of the revolutionary expansion in the breadth and capability of theoretical and computational chemistry in recent decades, these fields have acquired the power to resolve pressing problems both of a fundamental scientific character and of clearly practical interest. 1 Computational chemistry also underpins rational drug design, contributes to the selection and synthesis of new compounds and guides the design of catalysts. Computational Chemistry has two broad areas: Molecular mechanics /molecular dynamics methods which use the principles of classical mechanics and electronic structure methods which use the principles of quantum mechanics. 2

5 1.2 Molecular Mechanics/ Molecular Dynamics Methods Molecular mechanics and molecular dynamics (MM/MD) refer to methods for computing certain molecular properties, particularly molecular structure and relative energy. They both typically use fairly simple potential energy functions that are derived from classical mechanics (e.g., a parabolic function to calculate the energy required to stretch or to compress a chemical bond). In addition, they both rely on parameters that are derived either from experiment (e.g., infrared spectroscopy and X-ray crystallography) or from quantum mechanics-based calculations (e.g., high-level ab initio molecular orbital calculations). A collection of potential energy functions and the associated parameters that are employed for molecular mechanics/ molecular dynamics calculations is frequently referred to as a "force field"; thus, calculations that utilize the MM/MD approach are often referred to as empirical force field calculations. The molecular mechanics method is generally employed to compute the relative energies of different geometries (conformations) of the same molecule that arise from rotations about chemical bonds as well as relative energies of intermolecular complexes. Often, energy minima are sought; thus, the molecular mechanics method is frequently coupled with optimization procedures. On the other hand, in the molecular dynamics method, Newton's equations of motion are solved by using the gradient of the above-mentioned potential energy function (force field) to compute the dynamic trajectory of a molecule or of an ensemble of molecules. Both the MM and MD methods have found widespread use in the modeling of bio-molecules, for which quantum mechanical calculations are simply not practical due to the overwhelming number of particles involved. 3

6 1.3 Electronic Structure Methods These methods which use the principles of quantum mechanics can be classified into two categories: ab initio methods and semi empirical (SE) methods. 1.4 An Overview of ab initio Methods The term ab initio comes from the Latin words for from the beginning. Ab initio methods involve quantum mechanical calculations which are derived directly from theoretical principles (only mathematical approximations are involved), with no inclusion of experimental data. In this section, we attempt to highlight the basic principles involved in the different ab initio methods qualitatively without discussing much of the mathematical details. Nearly all ab initio quantum chemical methods attempt to solve the eigen value problem defined by the time independent Schrödinger equation, ĤΨ = EΨ (1.1) where, Ψ is the total molecular wavefunction (a function of electronic and nuclear coordinates) and Ĥ is the molecular Hamiltonian. For an n electron molecule having M nuclei, Ĥ in SI units is given by equation 1.2a Ĥ= ħ 2M + ħ !" 4! + # 4 $ # #" (%.&') Ĥ in atomic units is given in equation 1.2b (see appendix 1 for details). In the remainder of the thesis, atomic units (a.u.) are used. Ĥ= M 2!" 1! * + ( ) )>( $ () (%.&,) 4

7 The molecular Hamiltonian is formidable enough to strike terror in the heart of any quantum chemist. Fortunately, there exists a highly accurate simplifying approximation of neglecting the kinetic Hamiltonian of nuclei (the first sum of terms in equation 1.2) in comparison with the kinetic Hamiltonian of the electrons - the Born-Oppenheimer approximation. 2 After applying the above approximation equation 1.1 can be written as (-. / )Ψ / =3Ψ / (1.3) where, -. / and 0 11 are given by equations 1.4 and / = 2 + +!" 1! (%.5) 0 11 = # $ # #" (%.6) E gives the total energy including nuclear repulsion under Born-Oppenheimer approximation. The variables in equation 1.3 are the electronic coordinates. The inter nuclear distances, $ #, in equation 1.3 are not treated as variables but are kept fixed at some constant values (this is referred to as fixed nuclear framework), while the electronic motions are calculated using equation 1.3. Of course, there is infinite number of such fixed nuclear configurations each with a constant 0 11 and for each of them we may solve the electronic Schrödinger equation 1.3 to get a set of electronic wavefunctions and corresponding electronic energies. Thus, the electronic wavefunctions and energies depend parametrically on the nuclear configuration defined by If 0 11 is omitted from 1.3, there is not going to be any change in the value of Ψ / (It can be proved that the omission of a constant term, C, from the Hamiltonian does not affect the wavefunctions and simply decreases each energy eigen value by C). Thus if 0 11 is omitted, 1.3 becomes, -. / Ψ / =3 / Ψ / (1.6) 5

8 such that 3=3 / ( 1.7 ) The Ψ /, thus obtained by solving equation 1.6 is taken to be the exact wavefunction for practical purposes even though the equation involves the Born-Oppenheimer approximation (keeping nuclei fixed in space with respect to the electrons) and neglects all relativistic effects. The inter-electronic repulsion terms, r ij s, in the Hamiltonian (see equation 1.4) make exact solution of the electronic Schrödinger equation impossible for many electron systems Hartree - Fock (HF) method The HF method occupies a central position as far as all ab initio techniques are considered. All the other electronic structure methods (including the SE methods and Density Functional Theory (DFT) based methods) stem from the HF method for one or other reasons. In the HF approximation, the electronic Hamiltonian (1.4) is replaced with an approximate Hamiltonian identical in the first two terms, but with an effective potential V HF instead of the exact r ij terms. 3,4 This means that the Coulombic electron electron repulsion is not specifically taken into account. However its net effect is included in the calculation in an average manner through the central field approximation. The method was made vastly more applicable with the introduction of the Roothan formalism. 5 In 1951, Roothan showed that there exists a procedure by which a matrix representation of the HF problem may be solved iteratively by approximating orbitals as linear combinations of a finite set of basis functions (known functions used to represent an unknown function). In the same year, Ruedenberg devised mathematical techniques for evaluating integrals over a basis of atomic orbitals. 6 The concept of basis sets introduces a degree of chemical intuition into quantum chemistry by way of almost universally known orbital concept. A summary of Roothan-Hall-HF formalism is highlighted below as a separate segment. 6

9 Summary of the derivation of the Roothan-Hall equations 1. The total wavefunction Ψ of a closed shell molecule having n electrons is expressed as a Slater determinant of spin MOs (9 i s) and is designated here as Ψ :; 9 (1) 9AAAA(1) 9 9 (2) 9AAAA(2) 9 (2) Ψ :; (1,2,,n)= 9 (3) 9AAAA(3) 9 (3)! : : : : : 9 (D) 9AAAA(D) 9 (D) AAAAAA(1) B AAAAAA(2) 9 AAAAAA(3) B : : : : : 9 AAAAAA(D) B where, 9 stands for i th MO with F spin and 9H G for i th MO with I spin. ( 1.8 ) Since each orbital can accommodate two electrons D B 2 spatial orbitals only are required. 1, 2, 3, n in parentheses stand for the coordinates of electrons 1, 2, 3,, n respectively. where, 2. The electronic energy is given by the average value postulate (assuming Ψ :; to be normalized). <3>=KΨ :; L-. / LΨ :; M ( 1.9 ) 3. Substituting the Slater determinant Ψ :; (1.8) and the explicit form of the Hamiltonian -. / {equation (1.4)}, the energy can be shown to be (after much algebraic manipulation) Y B & Y B & Y B & Q RS =&T UU +(&V UW X UW ) U% - =[9 (1)\ ] ^_ U% W% (%.%Z) \9 (1)`+[9 (1)\ ]a b c^b \9 (1)` (%.%%) d! =[9 (1)9 (1)\ \9 c! (2)9! (2)` (1.12) e_ f! =[9 (1)9! (2)g 1 g9 (2)9! (1)` (%.%h) We can assign the following physical meanings to each of the above terms (1.11 to 1.13). - corresponds to the electronic energy of a single electron moving simply under the attraction of a nuclear core with all the other electrons stripped away. d! which is called the coulomb integral represents the electrostatic repulsion between an electron in 9 and another in 9!. f!, which is called the exchange integral can be regarded as a kind of correction to 7

10 d! reducing the effect of d!. We could consider the summed (2d! f! ) terms to be the true coulombic repulsion corrected to electron spin. Substituting equations 1.11 to 1.13 in 1.10, 3 / can be conveniently written as a sum of three terms 3 / =i (%.%5) where, i ; kinetic energy due to all the n electrons.(resulting from the first term in 1.11) 0 1 ; potential energy due to nuclear electronic coulombic attraction (resulting from the second term in 1.11) and 0 ; potential energy due to inter electronic repulsion (due to the combined effect of coulomb and exchange integrals. 4. The HF method looks for those MOs (9 s) that minimize the variational integral (equation 1.9). Minimizing Q RS in equation 1.10 with respect to 9 s [to find the best 9 s ] gives a set n HF one electron equations. jk(1)9 (1)= 9 (1) (1.15) where, jk is the Fock operator which contains additional terms due to electron spin compared to the Hamiltonian operator and is a one electron energy. jk(m)= - + { 2do! (m) ḟ! (m)} (%.%q)! do! is the coulomb operator and ḟ! is the exchange operator. 5. Unlike the Schrödinger equation, the HF equations are not quite eigenvalue equations. This is because of the fact that the Fock operator jk(m) itself depends on 9 : in a true eigenvalue equation the operator can be written without reference to the function on which it acts. As such the HF equations were not very useful for molecular calculations for mainly two reasons: they do not prescribe mathematically viable procedure for getting the initial guesses for the MO functions 9 and the wavefunction may be so complicated that they contribute nothing to a qualitative understanding of the electron distribution. A key development that helped to make feasible the calculation of accurate SCF wavefunctions was Roothan s proposal to 8

11 expand the spatial orbitals 9 s as linear combinations of a set of one electron basis functions. Substituting into the HF equations the Roothan-Hall linear combination of m basis functions χ s 9 =t s s χ s (%.%u) Substituting for the MO s (in 1.15) the linear combinations of basis functions (1.17) gives the Roothan Hall equations (after some mathematical manipulations) which can be written in the matrix form as, j j.. j w t t.. t w j v j.. j w t : : : : xy t.. t w : : : : z j w j w.. j ww t w t w.. t ww or F C = S C ε { {.. { w t t.. t w { =v {.. { w t : : : : xy t.. t w 0 : : : : zv.. 0 x (%.%}) : : : 0 { w { w.. { ww t w t w.. t ww ww Where, F is the Fock matrix consisting of the one electron Fock operators, C matrix consisting of the coefficients in the linear combinations of basis functions in the different MOs and S is the overlap matrix and ε is the diagonal matrix The procedure for diagonalizing the fock matrix F and extracting the MO coefficients and eigen values 1. The overlap matrix S is calculated and used to calculate an orthogonalising matrix ~ ]½ 2. ~ ]½ is used to convert F to F The transformed Fock matrix F satisfies F = C ε (C ) -1 The overlap matrix S is readily calculated, so if F can be calculated, it can be transformed to F, which can be diagonalised to give C and ε [ε yields the MO energy levels ε i s] 3. Transformations of C to C gives the coefficients c si in the expansion of the MOs 9 s in terms of the basis functions χ s s. 9

12 From a practical standpoint, the central field approximation and the basis set approximation allow an initial guess at the electron configuration (usually a Slater determinant) of the chemical system to produce a guess at the effective potential, which can then be used as an improvement to the guess of the orbitals and so on. The iterative calculation attains a self consistency, once the orbitals and the potential no longer change. The built-in advantage of the HF method is that since it involves a variational calculation, there is a guarantee that the energy computed is never going to be less than the actual ground state energy of the system. At first, HF-SCF calculations restricted the energies of α and β spin orbitals to be identical as implied by most simple models of chemical systems. This type of calculation is called restricted HF calculation (RHF). Although RHF wavefunction is generally used for closed-shell states, two different approaches are widely used for open-shell states. In the Restricted open-shell Hartree Fock (ROHF) method, electrons that are paired with each other are given the same spatial function. For example, the ROHF wavefunction for the ground state of Li atom is given by the Slater determinant 1 1AAA 2. The interaction of the 2sα and 1sα electrons differs from that between the 2sα and 1sβ electrons and it seems to be reasonable to give the two 1s electrons slightly different spatial orbitals (1s and 1s ). This is the methodology in selecting an unrestricted HF (UHF) wavefunction. Thus in the example of the Li atom the UHF wavefunction is 1 1 AAAA 2, where 1s 1s. 7 The UHF wavefunction gives a slightly lower energy than the ROHF wavefunction and is much more useful in predicting esr spectra. The main problem with the UHF wavefunction is that it is not an eigen function of the square of the spin operator,, whereas, the true wavefunction and the ROHF wavefunction are eigen functions of. 10

13 1.4.2 The HF limit and the concept of electron correlation The conceptual underpinnings of HF-SCF are both its strength and its downfall. Because of the central field imposed by the HF potential, each inter electronic interaction is not explicitly taken into account, instead it is treated only in an average manner. Therefore, there is a finite probability that two electrons will occupy the same point in space, which is unrealistic. For high accuracy, it is necessary to describe properly how the motion of each electron is affected by that of the other electrons. This is the electron correlation problem 8, which is often the single largest source of error in quantum chemical calculations. There are two types of correlation: Fermi correlation arising from symmetry and Coulomb correlation arising from electron electron repulsion. Higher levels of theory attempts to resolve this issue by recovering the correlation energy, which is defined as E correlation = E actual E HF-limit ( ) where, E HF-limit is the HF energy of the system obtained with an infinite basis set. E correlation is always negative because the HF energy is an upper bound to the exact energy (as guaranteed by the variation theorem). Most of the standard high-level techniques in quantum chemistry utilize the HF approximation as a starting point and then attempt to correlate electrons by more rigorous computational methods Post - HF methods Generally, there are two classes of methods employed to recover the majority of correlation energy - variational and perturbational. Variational methods treat the exact solution as a linear combination of discrete solution sets, while perturbative methods separate the problem into an exactly solvable part and a difficult part with no general analytic solution. The most popular variational post-hf method is called the Configuration interaction (CI) and the two important perturbative post-hf methods are the Many Body Perturbation Theory (MBPT) and the Coupled Cluster (CC) method. 11

14 Configuration interaction methods Configuration interaction (CI) methods begin by noting that exact wavefunction Ψ cannot be expressed as a single determinant as HF theory assumes. 9,10 The standard CI method relies on partitioning the exact wavefunction Ψ into a selected collection of Ψ s and their corresponding coefficients c i s with a view to including the interaction energy due to excited states. Ψ= t 0 Ψ 0 + m t m Ψ m (%.&Z) where, Ψ 0 is the HF determinant (equation 1.8) and each Ψ i is a Slater determinantal antisymmetric function corresponding to an excited state configuration obtained by replacing one or more occupied orbitals 9 s within the HF determinant with a virtual orbital. If all the possible excited configurations are considered, the calculation is referred to as a full CI (FCI). Such a calculation is variational and size consistent,* but extremely expensive and therefore impractical for all but very smallest systems. Practical CI methods augment the HF by adding only a limited set of substitutions, truncating the CI expansion at some level of substitution. For example, the CI-singles (CIS) method 11 adds single excitations (a virtual orbital, say, 9, replaces an occupied orbital 9 within the HF determinant) to the HF determinant, CI-doubles (CID) adds double excitations (two occupied orbitals are replaced by virtual orbitals), CI-singles doubles (CISD) adds singles and doubles excitations, CI-singles doubles triples (CISDT) adds singles, doubles and triples excitations and so on. 12,13,14 The Davidson correction can be used to estimate a correction to the CISD energy to account for higher excitations. 15 A disadvantage of all these limited CI variants is that they are not size consistent. CISD, which is typically augmented with triple and/or quadruple excitations in some fashion is known as the Quadratic Configuration Interaction (QCISD) 16 and was developed to correct this 12

15 deficiency. QCISD (T) adds triple substitutions to QCISD, providing an even greater accuracy. 16 Similarly, QCISD (TQ) adds both triples and quadruples from the full expansion to QCISD Perturbative theories Many body perturbation theory (MBPT) was the first of the perturbative theories to evolve. MBPT is a way to account for electron correlation by treating it as a perturbation to the HF wave function. It is a rather straightforward application of simple perturbation theory. Using MBPT, we wish to solve the eigen value problem Ĥ Ψ = Ĥ 0 L kL D = E Ψ (1.21) where, we know the eigen functions and eigen values of Ĥ 0 Ĥ = E ( ) and will systematically improve the eigen functions and eigen values of Ĥ 0 such that they approach those of Ĥ. This can be accomplished by producing a set of separable equations that represent various orders of a Maclaurin series: by solving each equation, the exact energy can be determined by summing the values E = E 0 + E (1) + E (2) + + E (n) ( ) In the Møller Plesset perturbation theory 17 (generally referred to by the acronym MPn, where, n is the order at which the perturbation theory is truncated), Ĥ 0 is defined as the sum of the one electron Fock operators. Ĥ 0 = Σ F i ( ) where, F i is the Fock operator acting on the i th electron. MP0 would use the electronic energy obtained by simply summing the HF one electron energies (first sum in equation 1.10). This ignores interelectronic repulsion except for refusing to allow more than two electrons in the same spatial MO. MP1 corresponds to MP0 corrected with the Coulomb and exchange integrals J and K, i.e., MP1 is just the HF energy. 13

16 The variant of MP theory which is truncated at the second order is known as MP2. 18,19,20,21,22 MP2 remains a very useful tool in a computational chemist s tool box, for it can successfully model a wide variety of systems and optimize geometries quite accurately. Higher level MP orders (MP3 23,24, MP4 25, and MP5 26 ) are available for cases where the second order solution of MP2 is inadequate. In practice, however, only MP4 sees wide use. MP3 is usually not sufficient to handle cases where MP2 does poorly, and it seldom offers improvements over MP2 which are commensurate with the additional computational cost. Since MPn methodology is not variational, it is possible that such methods (especially MP2) overestimate the correlation energy. However, this rarely happens in practice because of the basis set limitations. At the MP4 level, integrals involving triply and quadruply excited determinants appear. The evaluation of the terms involving triples is the most costly (triples doubles the time) and scales as N 7. If we ignore the triples, the method scales more favourably and this choice is typically abbreviated MP4(SDQ). 27, Coupled cluster methods One of the mathematically elegant techniques for estimating the electron correlation energy is the coupled cluster theory. 29,30,31 Since its introduction in the late 1960s, it has emerged as perhaps the most reliable method for the prediction of molecular properties. The theory begins by suggesting that the exact wave function may be described as Ψ = e T Ψ 0 ( ) in which, Ψ 0, is a suitable reference function, usually the HF function. T is an excitation operator called the cluster operator. The effect of e T operator is to express Ψ as a linear combination of Slater determinants that include Ψ 0 and all possible excitations of electrons from occupied to virtual spin orbitals. T = T 1 + T T n (1.26) 14

17 The exponential form of the operator makes coupled cluster (CC) theory unique among all techniques in quantum chemistry. Operating on the HF wave function with T is equivalent to doing a full CI. But CC method is superior to CI when we try to truncate T at some point. For example, suppose that we consider only the double excitation operator, i.e., T =T 2, then the method is called coupled cluster doubles (CCD) or coupled-pair manyelectron theory (CPMET). 32 = (1+ i + Š! + _ Ψ CCD = _ Ψ HF + ) Ψ HF (1.27)! The first two terms, (1 + T 2 ), define the CID method and the remaining terms involve products of excitation operators. Each application of T 2 generates double excitations, so the square of T 2, generates quadruple excitations and the cube of T 2 generates hextuple excitations and so on, thus making it size consistent. In fact, the failure to include these excitations makes CID non size consistent. Studies show that CCD theory gives results close to those of a Møller-Plesset perturbation treatment to fourth order in the space of double and quadruple substitutions MP4(DQ). Methods for analytic evaluation of the CCD gradient have been developed by Bartlett et.al. 33 Approximations to the CCD method include, the linearized CCD (LCCD), which uses the approximation _ T 1 + T 2 ), the coupled electron-pair approximation (CEPA) and independent electron-pair approximation (IEPA). The next step in improving the CCD method is to include the operator T1 and take T = T 1 + T 2 in e T : this gives the CC singles and doubles (CCSD) method. 34,35,36 With T = T 1 + T 2 + T 3, one obtains the CC singles, doubles, triples (CCSDT) method. 37 CCSDT calculations give very accurate results for correlation energies but are very demanding computationally and are only feasible for small molecules with small basis sets. Instead of the very demanding CCSDT, one often performs the CC perturbative triples 15

18 {CCSD(T)} which uses the perturbation theory to compute the connected triples. 38 It is one of the most popular ab initio methods. One goal of ab initio quantum chemistry is to be predictive. Predictive quantum chemistry requires a very accurate inclusion of the essential effects of electron correlation. CC theory offers a novel, elegant approach for correlation that has had a dramatic impact on the field in the past two decades and is destined to have important impact in the future. 39 The application of orbital based CC theory to the calculation of molecular energies and properties is recently reviewed by Helgaker et.al. 40 Hybrid methods incorporating the spirits of CI and perturbative methods are also reported. The CIS(D) which uses configuration interaction singles with perturbative doubles is an example. 41,42 A modification of the CIS(D) abbreviated as PR-CIS(Ds) (partially renormalized variant of configuration interaction singles with perturbative doubles and extra singles) has been suggested recently. 43 Two factors that should be mentioned with post-hf calculations are of whether a method is size-consistent and/or whether it is variational. A size consistent quantum mechanical method is one for which the energy and hence the energy error in the calculation increase in proportion to the size of the molecule. It is important whenever calculations on molecules of substantially different sizes are to be compared. A special case of size consistency is that of infinitely separated systems. When dealing with such a system, a method is said to be size-consistent if it gives the energy of a collection of n widely separated atoms or molecules as being n times the energy of one of them. HF& FCI - size-consistent and variational. MPn - size-consistent but not variational. CID - not size-consistent but variational. CCD - size-consistent but not variational. 16

19 Multi-configuration (MC) methods In configuration interaction (CI), the trial wavefunction is written as a linear combination of determinants with the expansion coefficients determined by requiring that the energy should be a minimum. The MO s used for building the excited state Slater determinants are taken from a HF calculation and held fixed. The MCSCF 44,45,46,47,48 on the other hand, can be considered as a CI, where not only the coefficients in the linear combination of determinants are optimized by the variational principle, but also the MO s used for constructing the determinants are made optimum. The MCSCF optimization is iterative just like the SCF procedure (if the multiconfiguration is only one, it is simply HF). Since the number of MCSCF iterations required for achieving convergence tends to increase with the number of configurations included, the size of MCSCF wavefunction that can be treated is somewhat smaller than that for CI methods. 49 MCSCF methods are rarely used for calculating large fractions of the correlation energy. The orbital relaxation usually does not recover much electron correlation, it is more efficient to include additional determinants and keep the MOs fixed (CI) if the interest is just in obtaining a large fraction of the correlation energy. An example where MCSCF method can be used to good effect is in dealing with the structure of ozone. O O O O O O Figure 1.1: Resonating structures of ozone Ozone has two non-equivalent resonating structures. A RHF single determinant wavefunction is inadequate in this case. The simplest MCSCF for ozone will contain two configurational state functions (CSFs) with the optimum MOs and configurational weights determined by the variational principle. The CSFs entering an MCSCF expansion are pure spin states, and 17

20 MCSCF wavefunctions therefore do not suffer from the problem of spin contamination. The major problem with MCSCF methods is selecting the necessary configurations to include the property of interest. One of the popular approaches is the complete active space SCF (CASSCF) method 50 {also called full optimized reaction space (FORS)}. Here the selection of configurations is done by partitioning the MOs into active and inactive spaces. The active MOs will typically be of some of the highest occupied and some of the lowest unoccupied MOs from a RHF calculation. The active MOs either have 2 or 0 electrons, i.e. always doubly occupied or empty. Within the active MOs a full CI is performed and all the proper symmetry adapted configurations are included in the MCSCF optimization. Which MOs to include in the active space must be decided manually, by considering the problem at hand and the computational expense. A common notation is [n,m] CASSCF, indicating that n electrons are distributed in all possible ways in m orbitals. As for any full CI expansion, the CASSCF becomes unmanageably large even for quite small active spaces. A variation of CASSCF is the restricted active space SCF (RASSCF) method. 51. Here the active MOs are further divided into three sections, RAS1 (MOs which are doubly occupied in the HF reference determinant) RAS2 (has both occupied and unoccupied orbitals) and RAS3 (MOs which are empty in the HF determinant). RAS3 CAS All excitations RAS2 RAS1 Figure 1.2: Partitioning of MOs in CASSCF and RASSCF methods 18

21 An FCI in RAS2, a maximum of two electrons excited from RAS1 and a maximum of two electrons excited to the RAS3 space. In essence, a typical RASSCF procedure thus generates configurations of a full CI in a small number of MOs (RAS2) and a CISD in a somewhat larger MO space (RAS1 and RAS2). The number of singlet CSFs generated for an [n,n] CASSCF wavefunction shows a factorial increase. Table 1.1: No. of CSFs as a function of no. of electrons in a CASSCF method N No. of CSFs Composite methods The highest level ab initio techniques are not applicable (in a practical point of view) to even medium sized molecules. Two different classes of approximations (compromise) to overcome this difficulty are basis set extrapolation techniques and the design of composite methods (which include some kind of empiricism). The former, which is more accurate among the two classes, is based on CCSD(T) calculations using very large correlation consistent basis sets extrapolated to the complete basis set limit with addition of corrections for some smaller effects not included in the calculations such as core-valence effects, relativistic effects and atomic spin-orbit effects. This type of approach is limited to smaller molecules because of the use of very large basis sets. The later methods are widely used for the calculation of thermo-chemical data. They combine methods with a high level of theory and 19

22 a small basis set with methods that employ lower levels of theory with larger basis sets. They are commonly used to calculate thermodynamic quantities such as enthalpies of formation, atomization energies, ionization energies and electron affinities. They aim for chemical accuracy which is usually defined as within 1 kcal/mol of the experimental value. The most popular among them are the Gaussian-n (Gn) theories, which employ a set of calculations with different levels of accuracy and basis sets with a goal of approaching the exact energy. In the Gn approach, a high level correlation calculation {e.g. QCISD(T) and CCSD(T)} with a moderate sized basis set is combined with energies from lower level calculations (e.g. MP4 and MP2) with larger basis sets to approximate the energies of more expensive calculations. In addition, several molecule-independent empirical parameters {higher level correction terms (HLC terms)} are included to estimate remaining deficiencies, assuming that they are systematic. The different steps involved in the G1 52, G2 53, G3 54 and G4 55 theories are listed below. An intermediate approach, referred to as correlation consistent composite approach (ccca) with no parametrization has recently been introduced. 56 Other composite techniques related to Gn methods include, the complete basis set (CBS) methods 57 and the multi coefficient methods. 58,59 Steps involved in G2 theory 1. E + = E[MP4/6-311+G(d,p)] E[MP4/6-311G(d,p)] 2. E 2df = E[MP4/6-311G(2df,p)] E[MP4/6-311G(d,p)] 3. E QCI = E[QCISD(T)/ 6-311G(d,p)] E[MP4/6-311G(d,p)] 4. E +3df,2p = E[MP2/6-311+G(3df,2p)] E[MP2/6-311G(2df,p)] E[MP2/6-311+G(d,p)] + E[MP2/6-311G(d,p) 5. 3 HLC = N β 0.19 N α 6. 3 G2 = E[MP4/6-311G(d,p)] + E + + E 2df + E QCI + E +3df,2p + 3 HLC 20

23 Steps involved in G3 theory 1. HF/6-31G(d) geometry optimization 2. Zero point vibrational energy (ZPVE) from HF/6-31G(d) frequencies 3. MP2(Full)/ 6-31G(d) geometry optimization (all subsequent calculations use this geometry) 4. E[MP4/ 6-31+G(d)] - E[MP4/ 6-31G(d)] 5. E[MP4/ 6-31G(2df,p)] - E[MP4/ 6-31G(d)] 6. E[QCISD(T)/ 6-31G(d)] - E[MP4/ 6-31G(d)] 7. E[MP2(Full)/G3 large] - E[MP2/ 6-31G(2df,p)] - E[MP2/ 6-31+G(d)] - E[MP2/ 6-31G(d)] 8. E HLC = (number of valence electron pairs) (number of unpaired valence electrons). 9. E G3 = (2) + E[MP4/ 6-31G(d)] + (4) + (5) + (6) + (7) + (8) Steps involved in G4 theory 1. B3LYP/6-31G(2df,p) geometry optimization 2. ZPVE from B3LYP/6-31G(2df,p) frequencies scaled by a factor E HF/aug-cc-pVnZ = E HF/limit + B ]α where, α is an adjustable parameter and n the number of contractions in the valence shell of the basis set. 4. E + = E[MP4/ 6-31+G(d)] - E[MP4/ 6-31G(d)] 5. E (2df,p) = E[MP4/ 6-31G(2df,p)] - E[MP4/ 6-31G(d)] 6. E CC = E[CCSD(T)/ 6-31G(d)] - E[MP4/ 6-31G(d)] 7. E (G3 large XP) = E[MP2(Full)/G3 large XP)] - E[MP2/ 6-31G(2df,p)] - E[MP2/ 6-31+G(d)] + E[MP2/ 6-31G(d)] 8. E (combined) = E[MP4/ 6-31G(d)] + E + + E (2df,p) + E CC + E (G3 large XP) + {E HF/limit E[HF/G3 large XP]}+ E (SO), where, E (SO) is the spin-orbit correction. 9. E e (G4) = E (combined) - E HLC, where, E HLC is the same as that of G3 10. E 0 (G4) = E e (G4) + E(ZPE). 21

24 The rationale behind these model chemistries is determined by a fitting to a library of compounds called the test set. A test set is a compilation of accurate experimental data. The first in this series was the G2 test set of 125 energies. 53 This was followed by the G2/97 (301 energies) 60, G3/99 (376 energies) 61 and G3/05 (454 energies) 62 test sets. Each succeeding test set includes energies from the preceding test sets and additional species of larger sizes and of different types. The test sets contain thermochemical data such as enthalpies of formation, ionization potentials, electron affinities and proton affinities chosen based on listed accuracy of + 1 kcal/mol or better. The latest test set G3/05 contains 270 enthalpies of formation, 105 ionisation energies, 63 electron affinities, 10 proton affinities and 6 hydrogen bonded complexes. When the G3 theory was originally published, it was assessed on the G2/97 test set and was found to have an average absolute deviation of 1.02 kcal/mol from experiment. The two succeeding test sets, G3/99 and G3/05 gave average absolute deviations of 1.07 and 1.13 kcal/mol respectively for the G3 theory. G4 theory provides significant improvement and this is evident from the average absolute deviation of 0.83 kcal/mol from experiment Methods for excited states Ground state wave functions can very often be expressed in terms of a single Slater determinant formed from variationally optimized MO s, with possible accounting for electron correlation taken thereafter. However, the problem of variational collapse typically prevents an equivalent SCF description for excited states. That is, any attempt to optimize the occupied MO s with respect to the energy will necessarily return the wave function to that of the ground state. 63 This limitation of traditional quantum chemistry led researchers to develop new techniques for electronically excited states. This is absolutely essential since many reactions are photochemically activated (leading to electronically excited states) rather than thermally activated. ab initio methods used to simulate electronically excited states include CIS, 22

25 time dependent HF and Green function methods. Even though a large variety of efficient quantum chemical methods is available for electronic ground-state calculations, the situation is much more difficult when excited-state energy surfaces and surface crossings should be treated. Analytic energy derivatives and nonadiabatic coupling vectors are crucial tools for the determination of energy minima, saddle points and conical intersections Multireference methods Chemically speaking, a multireference system is a molecule or reaction in which the one electron approximation is not only quantitatively but also qualitatively wrong. Practically, all reactions that involve bond breaking are of this type, as are some low lying excited states of many molecules. In such situations, the typical HF wavefunction is not a good description of the system, and something more complex is necessary to reproduce the chemistry. The CI methods consider only CSFs generated by exciting electrons from a single determinant. This corresponds to having a HF type wavefunction as the reference. However, a MCSCF wavefunction may also be chosen as the reference. In that case a CISD involves excitations of one or two electrons out of all the determinants which enter the MCSCF. This method is referred to as a multi-reference configuration interaction [MRCI] method. 64 Compared to the single reference CISD, the number of configurations is increased by a factor roughly equal to the number of configurations included in the MCSCF. Large scale MRCI wavefunctions [many configurations in the MCSCF] can generate very accurate wavefunctions, but are also computationally very intensive. Variations of MRCI include the MRCISD where single and double excitations are also considered. Since MRCI methods truncate the CI expansion, they are not size-consistent. State specific MRCC (SS-MRCC) methods applying the principles of coupled cluster theory on the multireference determinants has also been reported. 65 They are shown to be 23

26 size consistent. The size consistency of various multireference methods is analysed by Ruttink et.al r 12 methods The Hartree-Fock (HF) approximation uses an effective potential (a smooth potential as a consequence of the central field approximation) to replace the inter-electronic terms (r ij s) in the Hamiltonian. It has been long acknowledged that the true Hamiltonian form displays singularities in the coulomb potential at the points where electrons would exist at the same position. These are called coalescence points, and they imply that there will be irregularities in the first and higher derivatives of the wavefunction with respect to the interelectronic distance r 12 as shown by Kato in A smooth potential (like that used in HF) will not intrinsically model this cusp and therefore will show slow convergence behavior. All electron correlation methods based on expanding the N-electron wavefunction in terms of Slater determinants built from the HF determinants suffer from this problem of agonizingly slow convergence. Literally millions or billions of determinants are required for obtaining results which in an absolute sense are close to the exact results. It would therefore seem natural that the interelectronic distance would be a necessary variable for describing electron correlation. Methods based on this idea has been known since the early days of quantum mechanics. Hylleraas used a trial wavefunction consisting of an orbital product times an expansion in electron coordinates, such as Ž(, ) = ] ec e ] _ c _ t /w ( + ) ( ) / w /w (1.28) for attacking the two electron He atom problem, variationally optimized Ž(, ) by hand and yielded an energy within 0.6 kcal/mol of the exact non-relativistic value (it would require 30 terms in a CI calculation and a computer to match that feat). Calculations incorporating more complicated expansions have achieved results approaching the exact non-relativistic energy of He atom to within kcal/mol. 68 Hylleraas type 24

27 wavefunctions used for H 2 molecule produces an energy within 10-9 a.u., which is more accurate than that can be determined experimentally. 69 Unfortunately, Hylleraas type wavefunctions became impractical for systems containing more than 3-4 electrons. For such systems, ways to explicitly include terms in the wavefunctions which are linear in the interelectronic distances have been developed by Kutzelnigg and coworkers. 70 The first order correlation to the HF wavefunction involves only doubly excited determinants (as in MP2 theory). In r 12 methods additional terms are included which essentially are the HF determinants multiplied with r ij factors. For example, Ž ce_ = +!!! +!!! (1.29) Such r 12 wavefunctions may then be used in connection with the CI, MBPT or CC methods. These methods show significant promise and are in frequent use for achieving greater accuracy. The recent versions of explicitly correlated coupled-cluster (CC-R12 or F12) methods include CCSD-R12 (up to double excitations) 71, CCSDT-R12 (up to triple excitations) and CCSDTQ-R12 (up to quadruple excitations) 72. Using the Slater-type function exp( r 12 ) as a correlation function, a CC-R12 method can provide the aug-cc-pv5z-quality results of the conventional CC method of the same excitation rank using only the aug-cc-pvtz basiss set. 1.5 Quantum Mechanics- Molecular Mechanics (QM-MM) Methods Force field methods are inherently unable to describe the details of bond breaking/ forming reactions, since there is an extensive rearrangement of the electrons, which is neglected in the classical model. If the system of interest is too large to treat entirely by electronic structure methods, there are two possible approximate methods that can be used. In some cases the system can be pruned to a size that can be treated, by replacing unimportant parts of the molecule by smaller model groups, e.g. substitution of a hydrogen or methyl group for a phenyl ring. For studying enzymes, however, it is usually 25

28 assumed that the whole system is important for holding the active size in the proper arrangement and the backbone conformation may change during the reaction. Hybrid methods have been designed for modeling such cases, where the active size is calculated by electronic structure methods (usually semiempirical, low level ab initio or DFT methods), while the backbone is calculated by a force field method. Such methods are often denoted QM-MM methods. 73 The new ONIOM (our own n-layered integrated molecular orbital and molecular mechanics) approach has been proposed and shown to be successful in reproducing benchmark calculations and experimental results. 74 The main problem with QM-MM schemes is deciding how the two parts should be connected. Partial charges on the MM atoms can be incorporated into the electronic HF equations, analogously to nuclear charges (i.e. adding V ne -like terms to the one electron matrix elements, and the QM atoms thus feel the electric potential due to all the MM atoms. 1.6 Semi Empirical Methods Semi Empirical (SE) methods were developed to treat large molecules because of the difficulties met with in applying the ab initio methods to such molecules. SE methods also attempt a quantum mechanical description of electrons based on similar principles as ab initio, but with many (more) approximations built into the equations to make calculations go faster. Some of the commonly used approximations include the zero differential overlap approximation (i.e. overlap matrix is reduced to unit matrix), setting oneelectron integrals involving three centres to zero and neglecting many three and four centre 2-electron integrals. These approximations considerably decrease the complexities in the computation process. However, to compensate for these approximations, parameterization (design of computational equations or input parameters) based on experimental (empirical) data is often done. The advantages include; calculations are faster than ab initio, larger systems can be handled and the results often agree well 26

29 with experimental values because of parameterization. But the major disadvantage of such methods is that they are only reliable for systems for which a method is parameterized. Some of the widely used SE methods include HUCKEL method, Austin Model 1 (AM1), Parametric Method 3 (PM3) and methods based on neglect of differential overlap (MNDO, CNDO, MINDO, CINDO etc.). 1.7 Density Functional Theory (DFT) One of the most important and difficult problems in chemistry is the accurate calculation of molecular properties from first principles. Chemists aim to calculate the energy of molecules to within chemical accuracy (<1 kcal/mol), which is required to predict reaction rates. Exact first-principles calculations of molecular properties using the wavefunction theory (solving the Schrödinger equation using the FCI method and computing the 3N dimensional wavefunction) are currently intractable because their computational cost grows exponentially with both the number of atoms and basis set size. The difficulty of solving this problem exactly has led to the development of several approximate methods for first-principles quantum chemistry. The Density Functional Theory (DFT) is one among them. The foundation of DFT lies in the fact that it is not necessary to solve the Schrödinger equation and determine the 3N dimensional wave function in order to compute the ground state energy of a system. Instead the energy and other properties can be expressed as functionals of electron density; a three dimensional entity. Since the computational method used in the present work is DFT based, a detailed account of DFT methods and concepts is included in the second chapter. 1.8 Basis Sets Another approximation involved in all ab initio methods is the introduction of a basis set (see Roothan-Hall-HF formalism). A basis set is defined as a set of known functions used to represent an unknown function such as a molecular 27

30 orbital. The quantum chemistry literature has a plethora of basis sets and a comprehensive review is beyond the scope of the present writing. In this section, a simplified introduction to basis sets is attempted with a view to highlighting a historical sketch of the development of basis sets and the merits and demerits of different types of basis sets in use. Special care has been given to include the original reference as far as possible. The first step in any ab initio method of atoms or molecules is the selection of a trial atomic or molecular wavefunction, which is usually built from a set of one-electron one-center functions (usually called atomic orbitals) and then making improvements (depending on the method in use) to produce the best possible final wavefunction. Since molecular properties are computed from this function it is always very crucial to select the initial guess in a judicious manner. There is controversy that whether the individual functions in a basis set can be called atomic orbitals (AOs) or not. Strictly speaking, AOs are solutions of the HF equations for the atom, i.e. wave functions for a single electron in the atom. Anything else is not really an atomic orbital. Later on, the term AO was replaced by "basis function". Hence, a basis set may be defined as a collection of basis functions (not really AOs) used to construct an AO or a MO of a multi electron system. Thus, it can be generally represented as: here 9 =t s s χ s (%.& ) varies from 1 to m. Here, 9 is the i th MO, t s are the coefficients of linear combination, χ s is the s th basis function and m is the number of basis functions. The choice of a basis set must be guided by the consideration of two mutually opposing factors - desired accuracy in the results and computational costs. The basis set should be flexible enough to produce good results over a wide range of molecular geometries and sufficiently small so that the problem is 28

31 computationally easy and economically reasonable. The enormous computational cost required when a large basis set is used forces us to truncate the basis set at some convenient point on practical grounds. The truncation of the basis set is an important source of error in molecular electronic structure calculations. Most recent work has recognized the need for a more systematic approach to the problem of constructing basis sets with the aim of reducing the basis set truncation error in molecular electronic structure calculations. In practice, most popular computer programs for ab initio calculations contain internally defined basis sets from which the user must select an appropriate one, always creating a chance of a subjective error. It may be possible in the future to have programs make an informed decision about the choice of basis set based on the results of thousands of previous calculations which are accessible in a data base, but that time has not yet come. The responsibility still rests with the program user. The available basis functions can be grouped into two major classes - Exponential Type Orbitals (ETOs) and Gaussian Type Orbitals (GTOs) Exponential type orbitals The only exactly known wavefunctions are the hydrogenic AOs which are one electron functions and can be written in the form, Ψ,/,w (,θ,φ)=$,/ () /,w (θ,φ) (1.30) where, the $,/ () is the radial part which consists of the Lauguerre polynomials and the /,w (θ,φ) is the angular part which is a spherical harmonics function. Naturally, one will be tempted to go for these functions (with appropriate modifications) when looking for a set of functions to describe the wavefunction of a multi-electron atom or a molecule. The multicentre integral evaluations using these kinds of functions (as required by HF- Roothan-Hall formalism) are mathematically formidable. In 1930, Slater 29

32 proposed modifications of the hydrogen wavefunctions by replacing the radial part by a much simpler function which has the functional form, $(;D,ζ)=2ζ ( ½) n(2d )!p ]½ ] ]ζœ (1.31) where, n*(the effective quantum number) and ζ = a] (where, Z being the atomic number and S being the screening constant) are parameters determined by Slater s rules. 75 He formulated the rules by fitting of mathematical expressions to the numerical data for the radial distributions for the electrons in many electron atoms. 76,77 During the same period Zener showed that the wave functions for the atoms Be-Ne, can be written as simple analytic expressions with several parameters whose best values being determined by the variation method. 78 Afterwards, Roothaan and Bagus wrote an SCF code for atoms under the LCAO approximation. They introduced the functions χ(r,θ,φ;ζ,n,l,m)=n ] /,w (θ,φ) ]ζœ (1.32) where, n, l and m being the principal, azimuthal and magnetic quantum numbers, respectively and ζ is called the orbital exponent. The values are determined by performing variational HF calculations for atoms using the exponents as variational parameters. The exponent values which give the lowest energy are the best at least for the atom. In general, these functions are named Slater type orbitals (STOs). They satisfy the nuclear cusp condition because of their exponential relationship with the nucleus-electron distance. This fact allows the STO basis functions to reproduce correctly the behaviour of electrons in the regions near the nucleus. They also decay properly in the regions far away from the nucleus, in a similar way to that of the hydrogen atomic orbitals. Thus STOs show excellent behavior in the near and far regions of the atomic nucleus. A basis set constructed using STO basis functions is known as an STO basis set. For the atoms from He to Kr, basis sets of STO functions were optimized with respect to the atomic energy by 30

33 Clementi. 79 This work was improved and extended until the Xe atom by Clementi and Roetti 80 and until the Rn atom by McLean and McLean. 81 They also pointed out that if a single STO (per sub shell) fails to reproduce the exact chemistry, the quality of the results can be improved by representing each sub shell by proper linear combinations of more than one STO s. For example a 2s sub shell in an atom can be represented using two STO s as, STO basis sets can be classified as minimal or single zeta (SZ), double zeta (DZ), Triple zeta (TZ) and so on depending on the number of STOs used to represent each sub shell in an atom or a molecule. Minimal basis set: a set containing only enough functions to contain all the electrons of the neutral atom(s). DZ basis set: there are two STOs for each sub shell, such that they have identical n and l but different ζ values. The basis function with larger ζ value is a tighter function and the one with smaller exponent is more diffuse. A DZ basis set offers more flexibility to the wavefunction and can be used to represent different types of bonding better than a SZ basis set (say σ and π bonds formed by the same atom in a molecule). TZ basis set: Three times as many functions as in the minimal basis set. Quadruple zeta (QZ) and Quintuple zeta (5Z, not QZ) are also created on similar arguments. Split-valence basis set: Often it takes too much effort to calculate a doublezeta for every orbital. Since only valence electrons are involved in bonding, doubling can be restricted to only the valence orbitals. This is the principle behind a split-valence basis set. Split valence basis sets can further be categorized as valence double zeta (VDZ), valence triple zeta (VTZ) etc. based on number of STO s used to represent the valence orbitals. For 31

34 example, the VDZ basis set uses a SZ for the core electrons and a DZ for valence electrons. Unfortunately, the initial and successful work developed in the atomic calculations with STO functions was almost stopped for a long period. The exponential functions on r 2 (named Gaussian type orbitals or GTO) started to become the standard basis sets employed in the calculation of the molecular electronic structure. This regression in the use of the STOs was due to the fact that such orbital products on different atoms were difficult to manipulate for the evaluation of two-centre integrals (due to cumbersome orbital translations involving slowly convergent infinite sums). Later, efforts were undertaken to solve the problems associated with integral evaluation and the interest in STOs was renewed (of course due to their superiority compared with GTO s). Earlier reported basis sets 6 were improved by re-optimizing the exponents taking advantage of the present computational resources and new optimization algorithms. 82 Optimization criteria different from the energy minimization have also been proposed and used for obtaining basis sets. Such methods include the maximum overlap between the functions of a large reference basis set and the functions of the optimized basis set 83, the minimal geometrical distance between the sub space associated to the reference basis set and the optimized one 84, the minimization of the mono and bi electronic contributions to the total atomic energy 85 and the Generator Coordinate HF method. 86 These types of basis sets have been applied to the calculations of molecular systems. The basis set optimization demands a high computational cost which increases with the size of the considered basis set. To overcome this difficulty alternative ways for generating basis sets with a limited lower number of variational parameters were proposed. Slater functions with a common exponent for different principal quantum numbers was one of them. That is 32

35 1s, 2s and 2p sub shells were represented as linear combinations of STOs with the same exponent. The use of only one exponent reduces the number of integrals to be evaluated in the electronic structure calculations. Such STO basis sets are known as Single Exponent Slater Functions (SESF). 87 Another attempt in this direction was the introduction of STO basis sets with noninteger quantum numbers. Roothaan-Hartree-Fock calculations were carried out for the ground states of the atoms from helium to xenon, 88 for the singly charged cations Li + to Cs + and anions H - to I -89 and for the atoms Cs (Z = 55) through Lr (Z = 103) 90 using a minimal basis set of Slater-type functions whose principal quantum numbers were allowed to take variationally optimal non-integer values. The resulting energies are substantially superior to those obtained previously under the usual restriction that principal quantum numbers be positive integers. Considering the improvement in the total energy produced by the non-integer quantum numbers and the decreasing of variational parameters as in SESF basis sets, a mixed strategy has been proposed. For the atoms He to Ar, a new type of double-zeta basis set that combines the use of non-integer principal quantum numbers and the use of common exponents in Slater-type functions has been suggested. 91 The new double-zeta non-integer SESFs (NSESF) basis sets result in an improvement of the energy and a reduction of the computational time. An alternate method which uses exponential type orbitals (ETO) other than STOs has been proposed and found to produce significant improvement. Modified Bessel functions (BTO), has been used for this purpose by Filter and Steinborn. 92 The main properties of these BTO functions are related with the computation of the integrals. A two center distribution can be written as a multiple one-center distribution, whereas a four-center integral can be considered the sum of two-center Coulomb integrals. These two features together with the simple form of their Fourier transform allow to evaluate all the required molecular integrals. At DZ level, BTOs yield almost the same 33

36 accuracy as STOs, while at SZ level, STO basis sets are significantly better than BTOs. Recently, a method called Coulomb operator resolutions was suggested which enables the exponential type orbital translations to be completely avoided in ab initio molecular electronic structure calculations. 93 Coulomb operator resolutions provide an excellent approximation that reduces the two electronmulti center integrals to a sum of one electron overlap-like integral products, which involve orbitals on at most two centers. The convergence using this procedure has been shown to rapid in all cases. 94 Four center STO molecular integrals have been solved using this method for the H 2 molecule dimer (van der Waals complex) in a very recent report. 95 Numerical values for the four center terms in the H 2 molecule dimer agree well with complete ab initio results obtained using very large Gaussian basis sets. These recent works offer promising breakthroughs in basis set theory and it is hoped that more powerful basis sets based on STOs will be developed in future which will be useful to reproduce the chemistry more accurately at a lower cost Gaussian type orbitals (GTO) The use of STOs is still limited to atomic and diatomic systems where high accuracy is required and to certain semi empirical methods where all the three and four center integrals are neglected. Boys and McWeeny first advocated the use of Gaussian-type basis functions on the practical ground that all of the integrals required for a molecular calculation could be easily and efficiently evaluated. 96,97 The functional form of a Gaussian Type Orbital (GTO) in cartesian coordinates is χ(x,y,z;ζ,i,j,k)=n ª «! ]ζœ_ (1.33) where, N is a normalization constant, ζ is called " the exponent". The i, j, and k are not quantum numbers but non-negative integers (indices) that dictate the nature of the orbital in a cartesian sense and =ª +«+. The sum of the 34

37 indices, L=i+j+k, is used analogously to the angular momentum quantum number for atoms, to mark the functions as s-type (L=0), p-type (L=1), d-type (L=2) f-type (L=3) etc. When all three of these indices are zero, the GTO has spherical symmetry, and is called an s-type GTO. When exactly one of the indices is one, the function has axial symmetry about a single cartesian axis and is called a p-type GTO. There are three possible choices named as p x, p y and p z GTO s. When the sum of the indices is two, the orbital is called a d- type GTO. But here there are six possible combinations of index values (i,j,k) that can sum to two. This leads to six cartesian pre-factors x 2, y 2, z 2, xy, xz and yz. These six functions are called the cartesian d functions. In the solution of the Schrödinger equation for the H atom, only five functions of the d-type are required to span all possible values of the z-component of orbital angular momentum for l=2. These functions are usually xy, xz, yz, (x 2 - y 2 ) and (2z 2 - x 2 - y 2 ). The first three of these canonical d functions are common with the cartesian d functions, while the later two can be derived as linear combinations of the cartesian d functions. A remaining combination that can be formed from the cartesian d functions is ª +«+, which is actually an s-type GTO (since it has spherical symmetry). Different Gaussian basis sets adopt different conventions with respect to their d functions; some use all six Cartesian d functions, others prefer to reduce the total basis set size and use the five linear combinations. The extra function is used to represent an s orbital of higher angular momentum than that of the d functions (since this combination has the same exponent as the other d-type functions and a d orbital is more diffuse than an s orbital). Examination of f-type functions shows that there are 10 possible cartesian Gaussians, which introduce 4p x, 4p y and 4p z type contamination. However, there are only 7 linearly independent f- type functions. This is a major headache since some programs remove these spurious functions and some do not. Of course, the results obtained with all 35

38 possible cartesian Gaussians will be different from those obtained with a reduced set. 98 Cartesian GTOs are used in molecular calculations because the multicenter integrals are easily evaluated due to the Gaussian theorem that allows to express the product of two Gaussian functions centered in two different points of space as another Gaussian function centered in a third point located on the line that joins the two initial points. The main difference of a GTO from an STO is that ], the pre-exponential factor, is dropped, the r in the exponential function is squared, and angular momentum part is a simple function of cartesian coordinates. Calling these functions GTOs is probably a misnomer, since they are not really orbitals. They are simpler functions. In recent literature, they are frequently called Gaussian primitives. The absence of ] factor restricts single Gaussian primitive to approximating only 1s, 2p, 3d, 4f... orbitals. It was done for practical reasons, namely, for fast integral calculations. However, combinations of Gaussians are able to approximate correct nodal properties of atomic orbitals by taking them with different signs. Because a Gaussian has the wrong behavior both near the nucleus and far from the nucleus, it was clear that many more Gaussians would be required to describe an atomic orbital than if Slater- type orbital (STO) basis functions were used. To ease this problem, several Gaussian primitives are often grouped together to form what are known as contracted GTOs (CGTOs). The term contraction means "a linear combination of Gaussian primitives {or more often called primitive GTO (PGTO)s} to be used as basis function.". A CGTO can be represented as χ(cgto)= χ (±²i³) (%.h5) 36

39 Such a basis function will have its coefficients and exponents fixed { and ζ i in χ (±²i³)}. Early work on contracted functions dealt with Gaussian lobe functions in which, for example, a p orbital was approximated by differences of s orbitals slightly displaced from each other. 99 For higher angular momentum, the number of terms required, and the loss of accuracy in computing the integrals, made the method unmanageable. Clementi and Davis extended the use of contracted functions to include Cartesian Gaussians. 100 Although individual integrals with Cartesian Gaussians are somewhat more complicated than with Gaussian lobes, the ease in extending the basis set to higher angular momentum has made this procedure the method of choice. Fortunately, the ratio of the number of Gaussians to the number of STO s required to obtain comparable accuracy is not as large as originally believed. Although 4 Gaussians are needed to get within 1 mh (1 mh = 6.27 kcal/mol) of the exact energy for the hydrogen atom, for atoms further down the periodic table, such as argon, the ratio is about CGTOs have other advantages as well and on practical grounds they have been used extensively. So a detailed discussion of CGTOs is presented later in this section. An important task in the use of GTOs (whether they are used as such or as CGTOs) is to derive a good set of PGTOs. Gaussian primitives are usually obtained from quantum calculations on atoms (i.e. Hartree-Fock or Hartree- Fock plus some correlated calculations, e.g. CI). Typically, the exponents (ζ) are varied until the lowest total energy of the atom is achieved (as in the optimization of STOs). The first optimized Gaussian set for atomic SCF energies was published by Huzinaga. 101 Later, van Duijneveldt extended Huzinaga's work increasing the sizes of the basis set analyzed up to (14s 9p). 102 Ever since, over the years several GTO sets have been optimized. The related papers are consolidated in a table elsewhere. 103 The methodology used in these works was based on the variational determination of the exponents 37

40 with respect to the HF atomic energies. The (10s 6p) and (14s 9p) basis imply a space of 16 and 23 dimensions, respectively, in which a local energy minimum is looked for. The difficulties found in the optimization of individual exponents of large basis sets are associated to the smoothness or non existing minima of the energy surface. Therefore, no individually optimized exponent sets larger than van Duijneveldt s have been produced Even-tempered basis sets When the basis set becomes large, the optimization problem is no longer easy. The basis functions start to become linearly dependent and the energy becomes a very flat function of the exponents. Furthermore, the multiple local minima problem is encountered. An analysis of the basis sets which have been optimized by variational methods reveals that the ratio between two successive exponents is nearly a constant. Taking this ratio to be constant reduces the optimization problem to only two parameters for each type of basis function independently of the size of the set. Such basis sets were labeled even-tempered basis sets with the i th exponent given as ζ =FI where F D I are optimized for a given type of function (s, p, d etc.) and nuclear charge (or atom). 104 The advantage of such basis sets is that it is easy to generate a sequence of basis sets which are guaranteed to converge towards a complete basis. This is useful if the attempt is to extrapolate a given property to the basis set limit. It was later discovered that the optimum F D I costants to a good approximation can be written as functions of the size of the basis set, N (the number of primitives) allowing extrapolation to very large basis sets without the need for re-optimization Well-tempered basis sets Even tempered basis sets have the same ratio between exponents over the whole range. From chemical considerations it is usually preferable to cover the valence region better than the core region. This may be achieved by well- 38

41 tempered basis sets. 106 The idea is similar to the even-tempered basis sets, the exponents are generated by a suitable formula containing only a few parameters to be optimized. The exponents in a well-tempered basis of size N (number of PGTOs) are generated as ζ =FI ] µ1+ ¹, where, i = 1, 2,,N (1.35) 1 δ The four parameters are F,I, D δ are optimized for each atom. The exponents are the same for all types of angular momentum functions. Thus s, p, d and higher angular momentum functions have the same radial part Universal basis sets Conventionally the basis sets employed in electronic structure calculations have a small number of functions in order to restrict the computational requirements. However, for achieving high accuracy the basis set size should be large to provide the much needed flexibility. Silver and Nieuwpoort proved that optimization becomes a less important step in deriving larger basis sets. Thus they introduced the concept of universal basis set as the basis set that might satisfactorily describe several atoms. 107 Following this idea Wilson and coworkers suggested the use of even-tempered STO and GTO basis sets large enough to reproduce the atomic energies of the elements belonging to the first row. 108 For example, in their initial work they generated a universal STO basis set, nine s and six p functions, for the B to Ne atoms. The atomic energies obtained thus have an accuracy of mhartrees. The great advantage of these universal basis sets is the transferability of the integrals. As a consequence the use of the same basis set for each atom in the molecular system and the computation and storage of integrals is simplified and reduced. Such universal Gaussian basis sets may substantially reduce the computational work required for the calculation of molecular integrals in ab initio MO calculations. 39

42 Geometrical basis sets Clementi and Corongiuz combined the ideas of even-tempered exponents and universal basis sets along with six constraints to produce a new type of basis set, which they called geometrical, for use in large molecule calculations. 109 All atoms from H to Sr are represented by the same set of exponents (although differing numbers of s, p, and d functions appear). In a generator coordinate version of the HF equations, the obtained equations are numerically integrated. From the discretization points used to perform this integration Trisic et al. defined the exponents of the Gaussian basis sets. Since there is no need for a variational optimization of the basis functions, the same discretization points are valid for different atoms, and from them universal Gaussian basis sets are defined. The accuracy obtained for the atomic energies of Li to Ne atoms varies between 24 and 100 mhartrees, with respect to corresponding STO basis sets Contracted basis sets One disadvantage of all energy optimized basis sets is the fact that they primarily depend on the wavefunction in the region of the inner shell electrons. The 1s electrons account for a large part of the total energy and minimizing the energy will tend to make the basis set optimum for the core electrons, and less than optimum for the valence electrons. However, chemistry is mainly dependent on the valence electrons. Furthermore, many properties (for example, polarisability depend on the wavefunction tail (far from the nucleus), which energetically, is unimportant. This fact that many basis functions go into describing the energetically important, but chemically unimportant, core electrons is the foundation for contracted basis sets. Contraction is especially useful for orbitals describing the inner (core) electrons, since they require a relatively large number of functions for representing the wavefunction cusp near the nucleus, and further more are 40

43 largely independent of the environment. Contracting a basis set will always increase the energy, since it is a restriction of the number of variational parameters and makes the basis set less flexible, but will reduce the computational cost significantly. A CGTO is defined as a linear combination of PGTOs (see equation.1.34). The molecular orbitals (the initial guess at the molecular wavefunction as required by the Roothan- Hall procedure) will be chosen as linear combinations of the CGTOs whose coefficients will be obtained in the electronic molecular calculation. χ(mo)= χ (º²i³) (%.hq) However, neither the exponents of the PGTOs nor the coefficients involved in the CGTO definition are modified during the calculation. The cost of a computational calculation depends on the number of PGTOs and CGTOs in the following way. For gradient searches for optimum structures (geometry optimization) the cost can be dominated by the time to do the basic integrals. The computational effort (i.e. "CPU time") for calculating integrals in the Hartree-Fock procedure depends upon the 4 th power in the number of PGTOs. This advocates smaller primitive sets for geometry optimization. Calculations of the spectrum and other properties by perturbation theory or configuration interaction depend on the number of CGTOs, but are relatively independent of the number of PGTOs. Hence, in this step there is a strong motivation for using larger numbers of primitives to produce better contracted basis functions. Also, the storage required for integrals (when direct SCF is not used) is proportional to the number of basis functions i.e. CGTOs (not primitives). Frequently the disk storage and not the CPU time is a limiting factor. So it is always advisable to do the geometry search with a smaller basis set and then calculate the energy, at the selected geometries, with a more elaborate basis. 41

44 Two main schemes of contraction have been proposed: the segmented contraction 110 and the general contraction. 111 In segmented contraction, every PGTO only contributes to one CGTO, which implies that each PGTO has only a significant weight in the description of only one atomic orbital. In the general contraction scheme, however, all the given PGTOs in the basis set of a given l symmetry contribute to all the CGTOs of this symmetry. The segmented contractions are far more popular than general contractions. The reason for their popularity is not that they are better, but simply, that the most popular ab initio packages do not implement efficient integral calculations with general contractions. The computer code to perform integral calculations with general contractions is much more complex than that for the segmented case Segmented contractions The segmented basis sets are usually structured in such a way that the most diffuse primitives (primitives with the smallest exponent) are left uncontracted (i.e. one primitive per basis function). More compact primitives (i.e. those with larger exponents) are taken with their coefficients from atomic Hartree-Fock calculations and one or more contractions are formed. Then the contractions are renormalized. The primitives are chosen from different shells of functions to which the Cartesian Gaussians are grouped. The s-shell is a collection of s type Gaussians; p-shell is a collection of p-type Gaussians; d- shell is a collection of d-type Gaussians; and so on. Of course, combining primitives belonging to different shells within the same contraction does not make sense because primitives from different shells are orthogonal. But many basis sets use the same exponents for functions corresponding to the same principal quantum number, i.e., electronic shell. For the group of basis functions in which s and p type functions share the same exponents, the term sp shell is used. 42

45 The early Gaussian contractions were obtained by a least square fit to STOs. The number of CGTOs (not primitives) used for representing a single STO (i.e. zeta) was a measure of the goodness of the set. Based on the number of CGTOs used for representing a single STO, the basis sets can be classified as minimal or single zeta (SZ), double zeta (DZ), triple zeta (TZ), quadruple zeta (QZ), etc. (similar to the classification of STO basis sets). In the minimal basis set (i.e. SZ) only one basis function (CGTO) per STO is used. DZ sets have two CGTOs per STO, TZ sets have three CGTOs per STO and so on. Since valence orbitals of atoms are more affected by forming a bond than the inner (core) orbitals, more CGTOs are assigned frequently to describe valence orbitals. This prompted development of split-valence (SV) basis sets, i.e., basis sets in which more CGTOs are used to describe valence orbitals than core orbitals. That more basis functions are assigned to valence orbitals does not mean the valence orbitals incorporate more primitives (PGTOs). Frequently, the core orbitals are long contractions consisting of many primitive Gaussians to represent well the "cusp" of s type function at the position of the nucleus. Split valence basis sets can further be categorized as valence double zeta (VDZ), valence triple zeta (VTZ) etc. based on number of CGTO s used to represent the valence orbitals. For example, the VDZ basis set uses one CGTO per core electron and two CGTOs per valence electron Polarization functions Most Gaussian primitive sets are constructed by optimization of the Hartree- Fock energy of the atom. As already remarked, this choice will place heavy emphasis on representing the core orbitals, as these orbitals contribute most of the total energy of an atom. The most obvious defect is that atomic calculations only define functions with the same L as the occupied orbitals, e.g., s and p for carbon. But during molecule formation these atomic orbitals 43

46 undergo distortion. As atoms are brought close together, their charge distribution causes a polarization effect (the positive charge is drawn to one side while the negative charge is drawn to the other) which distorts the shape of the atomic orbitals. In this case, 's' orbitals begin to have a little of the 'p' flavor and 'p' orbitals begin to have a little of the 'd' flavor. This is not satisfactorily represented using atomic Gaussian sets. For use in a molecular calculation, these sets may be supplemented with additional functions of higher angular momentum. Such functions are called polarization functions. (a) (b) Figure 1. 3: (a) mixing of s and p (b) mixing of p and d The exponents for polarization functions cannot be derived from HF calculations for the atom, since they are not populated. However, they can be estimated from variational calculations on atoms with correlated wavefunctions or from variational HF calculations on molecular systems where the HF energy depends on polarization functions. In practice, however, these exponents are also estimated "using well established rules of thumb or by explicit optimization". Augmentation with polarization functions is denoted by the letter P in the "zeta" terminology. Thus, DZP means double-zeta plus polarization, TZP stands for triple-zeta plus polarization, etc. Sometimes the number of 44

47 polarization functions is given, e.g. TZDP, TZ2P, TZ+2P stands for triplezeta plus double polarization Diffuse functions In chemistry, we are mainly concerned with the valence electrons which interact with other molecules. However, many of the basis sets we have talked about previously concentrate on the main energy located in the inner shell electrons. This is the main area under the wave function curve. In the graphic below, this area is that to the left of the dotted line. Normally the tail (the area to the right of the dotted line), is not really a factor in calculations. Figure 1.4: General nature of a plot of wavefunction vs distance from the nucleus However, when an atom is in an anion or in an excited state, the loosely bond electrons, which are responsible for the energy in the tail of the wave function, become much more important. To compensate for this area, we can use diffuse functions. These basis sets utilize very small exponents to clarify the properties of the tail. 45