Kohn-Sham Density Functional Theory: Application to Metal Clusters and Nanoparticles

Transcription

1 Kohn-Sham Density Functional Theory: Application to Metal Clusters and Nanoparticles A Dissertation SUBMITTED TO THE FACULTY OF UNIVERSITY OF MINNESOTA by Kaining Duanmu IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY Advisor: Regents Professor Donald G. Truhlar September 2016

2 Kaining Duanmu 2016

3 Acknowledgments I want to thank my parents, who have supported me all the time and will be supporting me in the future. I want to particularly thank Professor Donald Truhlar, who has been helping me in both research and life with his knowledge, wisdom, and kindness. This dissertation would not be possible without his supervision. I am grateful to my coauthors on published work, as listed in the preface. I also want to thank the professors who taught me academic courses and faculty members serving on my preliminary and final oral exams Professor Laura Gagliardi, Professor Christopher Cramer, Professor Ilja Siepmann, Professor Sanford Lipsky, Professor Aaron Massari, and Professor Chris Hogan. I want to thank all collaborators and friends who supported me throughout the PhD program, especially those in my research group Run Li, Xuefei Xu, Bo Wang, Haoyu Yu, Pragya Verma, Rubén Meana Pañeda, Yuliya Paukku and Junwei Bao. I am grateful for their advice, criticism and encouragement. i

4 Dedication This dissertation is dedicated to my parents, who have always loved me and supported me. ii

5 Abstract Metal clusters and nanoparticles are important in chemistry and materials science because of their unique properties that are different from those of bulk materials. The most widely used theoretical method for studying metal clusters and nanoparticles is Kohn Sham density functional theory. This method would be exact if the exact exchange-correlation (xc) functional were known and were used; however, the exact xc functional is unknown and probably unknowable, therefore one must use approximate xc functionals. The continued development of xc functionals enables us to apply the theory to the more and more chemistry problems, and it also leads to questions: Which of the xc functionals are reliable for metal clusters? Which ingredient in the xc functionals is necessary for accurately predicting the properties of metals clusters? This thesis provides high-level benchmarking calculations, as well as validation of various xc functionals for silver and magnesium clusters and nanoparticles. This thesis also shows the new charge model for silver clusters and its application in CO adsorption. iii

6 Table of Contents List of Tables List of Figures vi viii Chapter 1. Introduction Density Functional Theory Exchange-Correlation Functionals Organization of The Dissertation 3 Chapter 2. Partial Ionic Character beyond the Pauling Paradigm: Metal Nanoparticles Introduction Theoretical and Computational Methods Results and Discussions Conclusions Acknowledgments 17 Chapter 3. Validation of Methods for Computational Catalyst Design: Geometries, Structures, and Energies of Neutral and Charged Silver Clusters Introduction Computational Details Test of method for generating reference data Reference Data Density Functional Calculations Structure Searching 27 iv

7 3.3. Results and Discussion Conclusions Acknowledgments 35 Chapter 4. Geometries, Binding Energies, Ionization Potentials, and Electron Affinities of Metal Clusters: Mg n 0,±1, n = Introduction Computational Methods Results and Discussion Testing CC Approaches and Generating Reference Data Kohn-Sham Density Functional Calculations Conclusions Acknowledgments 64 Chapter 5. Thermodynamics of Metal Nanoparticles: Energies and Enthalpies of Formation of Magnesium Clusters and Nanoparticles as Large as 1.3 nm Introduction Computational Methods Results and Discussion Geometries of Magnesium Clusters Cohesive Energy Benchmarking Benchmark Cohesive Energies for n = Density Functional Calculations Mg Conclusions Acknowledgments 84 Bibliography 92 v

8 List of Tables Table 1.1. xc Functionals Appearing in the Thesis and Their References. 5 Table 2.1. Mean Unsigned Errors (MUE) in Magnitude of Dipole Moments (Debye). 18 Table 2.2. Binding Energies of CO on Silver Monomer and Dimer with Different Charge States. 19 Table 2.3. CM5M Charges and Corresponding Binding Energies for Silver Clusters with Two or Three Different Binding Sites. 20 Table 3.1. CCSD(T) Calculated Internuclear Distances (Å) of Ag 2 and Ag Table 3.2. Percentage of Hartree-Fock Exchange and Mean Unsigned Error (MUE) of 14 Bond Lengths for Each xc Functional. 37 Table 3.3. The Number of Wrong Predictions (out of 15) of the Lowest-Energy Isomer and the Dependence on Kinetic Energy Density for Each xc Functional. 38 Table 3.4. Energies (kcal/mol) for Neutral and Charged Silver Clusters. 39 Table 3.5. Adiabatic IPs and DEs and Vertical IPs and DEs (kcal/mol) of Silver Clusters Calculated with M Table 3.6. Mean Unsigned Error for Each Energetic Quantity (kcal/mol). 42 Table 3.7. Total Mean Unsigned Error of Energy (kcal/mol) for Each xc Functional. 43 Table 4.1. Bond Length (Å) of Mg2 Optimized with Coupled-Cluster Methods. 65 Table 4.2. Bond Lengths (Å) of Small Magnesium Clusters Optimized with Coupled-Cluster Methods. 66 Table 4.3. Cohesive Energy per Atom (ev). 67 Table 4.4. Cohesive Energy per Atom (ev). 68 Table 4.5. NBO Bond Order of the Shortest Bond. 69 Table 4.6. Adiabatic Ionization Potentials (ev). 70 Table 4.7. Adiabatic Electron Affinities (ev). 71 vi

9 Table 4.8. Percentage of Hartree-Fock Exchange and MUE for Bond Lengths (Å). 72 Table 4.9. MUEs for Energetic Quantities (ev). 73 Table 5.1. Cohesive Energy per Atom (kcal/mol). 85 Table 5.2. Cohesive Energy per Atom (kcal/mol) Calculated with the CC + ΔMP2 Method. 86 Table 5.3. Enthalpy of Atomization and Heat of Formation (kcal/mol) at 0 K and 1 bar. 87 Table 5.4. MUE and MSE (kcal/mol) for Cohesive Energies per Atom of Mg vii

10 List of Figures Figure 2.1. Comparison of dipole moments. 21 Figure 2.2. CM5M partial atomic charges and binding energies for C2 structure of Ag17 (left) and Cs structure of Ag18 (right). 22 Figure 2.3. CM5M partial atomic charges and binding energies for four large size silver clusters 23 Figure 3.1. Neutral and charged silver clusters used in geometry studies. 44 Figure 3.2. Lowest-energy isomers for neutral and charged Ag n (n = 3 7). 45 Figure D and 2D silver clusters. The structure on the left is the lowestenergy 3D structure, and the one on the right is a competing low-energy 2D structure. 46 Figure 4.1. Structures of the neutral, cationic, and anionic magnesium clusters. 74 Figure 5.1. Geometries of Mgn, n = 2 19 and Figure 5.2. Diameters of Mgn, n = 2 19 and Figure 5.3. Cohesive energy per atom of Mgn. 91 viii

11 Preface The following are citations for previously published works reproduced in this thesis: Chapter 2. Adapted from Duanmu, K; Truhlar D. G. Partial Ionic Character Beyond the Pauling Paradigm: Metal Nanoparticles. J. Phys. Chem. C 2014, 18, American Chemical Society. Chapter 3. Adapted from Duanmu, K; Truhlar D. G. Validation of Methods for Computational Catalyst Design: Geometries, Structures, and Energies of Neutral and Charged Silver Clusters,. J. Phys. Chem. C 2015, 119, American Chemical Society. Chapter 4. Adapted from Duanmu, K.; Roberto-Neto, O.; Machado, F. B. C.; Hansen, J. A.; Shen, J.; Piecuch, P.; Truhlar D. G. Geometries, Binding Energies, Ionization Potentials, and Electron Affinities of Metal Clusters: Mg 0,±1 n, n = 1 7. J. Phys. Chem. C 2016, 120, American Chemical Society. Chapter 5. Adapted from Duanmu, K.; Friedrich, J.; Truhlar D. G. Thermodynamics of Metal Nanoparticles: Energies and Enthalpies of Formation of Magnesium Clusters and Nanoparticles as Large as 1.3 nm. Submitted to J. Phys. Chem. C 2016 American Chemical Society. ix

12 Chapter 1 Introduction 1.1 Kohn-Sham Density Functional Theory Kohn-Sham density functional theory 1 (KS-DFT) is based on the Hohenberg-Kohn theorem 2 that the energy of a system E[ρ] can be determined by its electron density ρ (r), where r denotes the spatial coordinates. KS-DFT partitions the electronic energy as follows: E[ρ] = ENe [ρ] + T[ρ] + J[ρ] + Exc[ρ] (1.1) where ENe, T, and J are the nuclei-electron attraction, electronic kinetic energy, and classical electron-electron repulsion respectively, and Exc is called exchange-correlation energy. The Kohn-Sham equation can be derived by using the variational method: N ( Z A A r ia 1 2 i 2 + ρ(r 2) dr r 2 + V xc (r 1 )) χ i = ε i χ i 12 (1.2) The four terms in the parenthesis are the energy operators for nuclei-electron energy, kinetic energy, electron-electron repulsive energy and exchange-correlation energy. The exchange-correlation potential is defined as: V xc = δe xc δρ (1.3) The Kohn-Sham equations can be solved by using the self-consistent field (SCF) method. 1

13 1.2 Exchange-Correlation Functionals In KS-DFT, all the terms except the exchange-correlation energy term can directly derived, and the theory would be exact if the exact exchange-correlation (xc) functional were known and were used; however, the exact xc functional is unknown and probably unknowable, therefore one must use approximate xc functionals. The quality of the xc functional determines the accuracy of the KS-DFT method. In general, xc functionals can be categorized by their ingredients. Local xc functionals: LSDA: A functional that only depends on spin-up and spin-down electron densities (ρ σ, σ = α, β) is called a local-spin-density approximation (LSDA). GGA: A functional that depends on the ρ σ and the magnitudes of their gradients ρ σ is called a gradient approximation (GA). A traditional GA, called a generalized GA (GGA), approximates exchange and dynamical correlation separately and has a separable form for exchange. Exchange is density-based exchange (which includes static correlation). NGA: A GA that uses a nonseparable functional form that approximates exchange and correlation together without separating them is called nonseparable gradient approximation (NGA). meta-gga and meta NGA: If the xc functional is also dependent on kinetic energy densities (τ σ ), it is called a meta-gga or meta-nga. 2

14 RS functional: A functional that employs a different treatment of the short-range and long-range regions of interelectronic separation is called range-separated (RS) functional. Hybrid exchange-correlation functionals: Hartree-Fock (HF) exchange is nonlocal, so a functional with HF exchange is a nonlocal functional. Global-hybrid functional: A functional that includes a certain percentage of Hartree- Fock exchange is called global-hybrid functional. RS hybrid functional: The short-range and long-range regions of interelectronic separation are treated by mixing different percentages of Hartree-Fock exchange with local exchange. Such a functional is called an RS hybrid functional. DFT-D: A functional that corrects the density functional energy with a post-scf molecular-mechanics damped-dispersion energy is called DFT-D. All the xc functionals appearing in this thesis and their references are gathered in Table Organization of the Dissertation Chapters 2 and 3 present DFT studies of silver clusters. Chapter 2 proposes a new charge model for silver clusters, which is validated against Kohn-Sham calculations of dipole moments. This model is found to be helpful for understanding the site dependence of CO molecular adsorption. Chapter 3 tests the accuracy of xc functionals for Ag n, Ag + n, and Ag n, n = 1 7, in three respects: geometries (quantitative prediction of 3

15 internuclear distances), structures (the nature of the lowest-energy structure, for example, whether it is planar or nonplanar), and energies. Chapters 4 and 5 present DFT studies of magnesium clusters. Chapter 4 shows the computations of the equilibrium geometries, binding energies, adiabatic ionization potentials, and adiabatic electron affinities for neutral and singly charged magnesium clusters, 0,±1 Mg n, n = 1 7. The xc functionals are validated against high-level computations by several coupled-cluster methods with single, double, and triple excitations. Chapter 5 extends the benchmark database for Mgn with n up to 19, by using a CCSD(T) scheme with an MP2/CBS correction. We validate exchange correlation functionals into the nanoparticle regime, and use the two best-validated functionals to calculate the enthalpy of formation of Mg28, which has a diameter of 1.30 nm. We also calculate the enthalpy of formation of all Mg clusters and nanoparticles from Mg2 to Mg19. 4

16 Table 1.1 xc Functionals Appearig in the Thesis and Their References. type functional references type functional references LSDA GVWN3 1,3,4 global hybrid GGA B1LYP 5 GGA BLYP 6,7 B3LYP 8 BP86 9,10 B MOHLYP 20 B mpwlyp 7,13 B3PW91 14 OreLYP 15,16 BHandH 17 mpwpw91 13,18 BHandHLYP 17 PBE 19 mpwlyp1m 20 PBEsol 21 mpw3lyp 22 PW91 18 O3LYP 23 RPBE 24 PBE0 25 SOGGA 26 SOGGA11-X 27 NGA GAM 28 global hybrid meta-gga BMK 29 N12 30 M05 31 meta-gga M06-L 32 M05-2X 33,34 TPSS 35 M06 36 revtpss 37 M06-2X 36 τ-hcth 38 M06-HF 39 meta-nga MN12-L 40 M08-HX 41 RS meta-gga M11-L 42 M08-SO 41 PW6B95 43 PWB6K 43 TPSSh 44 TPSS1KCIS 37,44,45 τ-hcthhyb 38 RS hybrid GGA ωb97x 46 CAM-B3LYP 47 HSE06 48,49 RS hybrid meta-gga M11 50 RS hybrid NGA N12-SX 51 RS hybrid meta-nga MN12-SX 51 RS hybrid GGA-D ωb97x-d 52 5

17 Chapter 2. Partial Ionic Character beyond the Pauling Paradigm: Metal Nanoparticles 2.1 Introduction The partial ionic nature of chemical bonds has long been understood in terms proposed by Pauling. 53 He supposed that a bond between two identical atoms could be considered as a normal covalent bond; whereas if one atom is more electronegative than another, there will be some ionic character within the bond. This paradigm has become a dogmatic way of thinking of chemical bonds, so it might be surprising to see our calculations (details below) showing that the equilibrium C2v structure of Ag 4 has a dipole moment of 3.4 D, larger than those 54 of some polar organic molecules, such as the similarly shaped formaldehyde with a dipole moment of 2.3 D or thioformaldehyde (1.7 D). This unusual large dipole moment indicates polar charge distributions in homonuclear metal clusters that take us beyond the Pauling paradigm. In order to find an accurate description of the charge distribution on metal clusters, we developed a simple charge model for silver clusters (as a prototype) in which charge distributions are predicted simply from geometry, and we note that an accurate charge model is very useful for understanding CO binding energy. There is a large experimental and theoretical literature on gold (especially) and silver nanoparticles being used as catalysts in the carbon monoxide oxidation reaction. The first step of the reaction is the adsorption of CO on the catalyst, and an important question about this process is which adsorption site on the catalyst is most preferred and 6

18 why. Most theoretical studies of nanoparticle metal catalysts have focused on important factors such as the support 66,67 and the size, 66,68 although there has been some discussion of the charge effect as well. For example, Solov yov et al. 69 and Donoso et al. 70 studied the charge distributions in cationic and neutral alkali metal clusters. Zhou et al. 71 calculated the binding energies of CO with neutral and charged silver clusters and found that positively charged clusters tend to have higher binding energies. Neumaier et al. 72 studied the binding energies of CO on gold and gold-silver cations and showed that adsorption sites with lower electron density tend to have higher binding energies. The Zhou et al. study shows that the overall charge state of a silver cluster significantly affects the CO binding energy, and the Neumaier et al. study shows that the local electron density, which is conceptually very close to partial atomic charge, largely affects the CO binding energy as a function of adsorption site. Our focus here is on partial atomic charges even in neutral metal clusters, and by applying our new charge model, we further show that partial atomic charge is one of the major determinants of preferential binding sites of CO on the metal particles. Charge models may be sorted into four classes. 73 Class I charges are derived without reference to quantum mechanics, e.g., by assigning charges based on geometries and experimental multipole moments. Class II charges are based on atomic population analysis of wave functions or reference wave functions obtained from quantum mechanical calculations, yielding procedures such as Hirshfeld population analysis, 74 Mulliken population analysis (MPA), 75 and natural bond orbital population analysis (NPA). 76,77 The Hirshfeld method, based on a so called shareholder analogy, has the advantage of being formally independent of the locations of the basis functions, 7

19 provided that the charge density is well converged, but it systematically underestimates charges. 78 Class III charges, such as CHELPG 79,80 and Merz-Kollman-Singh (MKS) 81,82 charges, are those fitted to reproduce a calculated physical observable, usually the electrostatic potential (ESP), from an approximate electronic structure calculation. Class IV charges are parametrized mappings of class II charges to new charges that reproduce experimental or high-level quantum mechanical observables, such as dipole moments. The most broadly applicable class IV charges are CM5 charges. 83 The CM5 model makes up on average for the systematic underestimate of the polarity of heteronuclear bonds by the Hirshfeld method, but for homonuclear clusters, CM5 charges are the same as Hirshfeld charges. For silver clusters, there is not enough experimental data to assign class I charges, and one is limited to the quantum mechanical approaches of classes II, III, and IV, but all previous models of these types have deficiencies for silver clusters. For example, traditional class II charges, such as MPA charges, are unstable for extended basis sets. Hirshfeld charges, although reasonably independent of basis set, tend to underestimate multipole moments, and class III charges suffer from ill conditioning for buried atoms in molecules 80,84 or extended structures, such as systems with periodic boundary conditions or large clusters. For silver clusters, we found that although CHELPG and MKS charges reproduce the dipole moments well, they often give very different partial atomic charges even though the only difference between the two methods is that they use different points in space for ESP fitting; furthermore, we found that the buried atoms in large clusters are usually assigned unphysically large charges by these two methods. 8

20 Therefore, we need a new accurate charge model to describe the uneven charge distribution on metal clusters. In the present article dipole moments are calculated by Kohn-Sham density functional theory. Kohn-Sham density functional theory has been well validated against experimental results for dipole moments and is known to give reasonably accurate dipole moments for molecules both without and with metal atoms We further validated Kohn-Sham density functional theory for small metal clusters by carrying out high-level wave function calculations, in particular CCSD(T)/aug-cc-pVQZ-PP 91 calculations. The resulting dipole moments of the C2v structures of Ag3 and Ag4 are 0.86 D and 4.20 D, while the Kohn-Sham density functional dipole moments of these two clusters, averaged over four exchange-correlation functionals, are 0.83 D (3% lower) and 3.44 D (18% lower) respectively. With the unknown exact exchange-correlation functional, Kohn-Sham density functional would give the exact charge distribution and dipole moment since they are one-electron properties, and the evidence is that the results are reasonably accurate with available approximate exchange-correlation functionals. 2.2 Theoretical and Computational Methods The CCSD(T) method with aug-cc-pvqz-pp basis set is used for validating dipole moments of C2v structure of Ag3 and C2v structure of Ag4. The optimizations of silver clusters for parametrization, and all binding energy calculations were performed using Kohn-Sham density functional theory with the N12 exchange-correlation functional, which is a nonseparable gradient approximation. The basis set is G(2df,2p) 92 for carbon and oxygen atoms, jun-cc-pvtz-pp 93,94 for silver in molecules containing up to 9

21 seven silver atoms, and cc-pvdz-pp 91 for silver in molecules containing more than seven silver atoms. All Kohn-Sham calculations are carried out with a locally modified version (MNGFM v6.4) of Gaussian ,96 The CCSD(T) calculations were carried out with MOLPRO software package. 97, Results and Discussions Charge Model 5 (CM5) maps Class II charges from Hirshfeld population analysis onto class IV charges using equations (2.1)-(2.3): q k CM5 = q k HPA + T kk B kk k k (2.1) B kk = exp[ α(r kk R Zk R Zk )] (2.2) T kk = D Zk Z k (2.3) where q k CM5, q k HPA, Z k, and R Zk are respectively the CM5 and Hirshfeld charge, the atomic number, and the covalent radius 99 of atom k, r kk is the distance between atoms k and k, and α, T kk, and D Zk Z are the model s parameters. Note that B k kk can be interpreted as the Pauling bond order of the k, k atom pair. 100,101 These equations show that CM5 attempts to correct systematic errors in Hirshfeld population analysis by transferring charges between atoms with finite bond orders and different atomic numbers, but this leads to no correction for homonuclear clusters. For Figure 2.1, we performed Kohn-Sham density functional calculations for an equilibrium C5v structure of Ag6 and a non-equilibrium structure of Ag8 created by adding one atom to an equilibrium D5h structure of Ag7. The Ag6 structure has one fivecoordinated atom and five 10

22 three-coordinated atoms; because of the symmetry there is a unique partial atomic charge assignment that reproduces the dipole moment. The charge on the central atom must be , whereas the Hirshfeld charge is 0.055, which yields the wrong direction of the dipole. The Ag7 structure has zero dipole moment, but when an extra atom is added, the Kohn-Sham dipole moment increases to 1.15 D, whereas the Hirshfeld dipole moment is 1.72 D, with nearly a 50% error. The negative charge is on the side of the extra atom, and the positive charge is on the Ag7 side. These examples show that a negative correction on multi-coordinated atoms may give more accurate dipole moments. Here we propose a way to correct this systematic error by making a pairwise correction dependent on the coordination number (N); adding such a pairwise correction to CM5 yields a model we call CM5M (CM5 for metals). The required new equations are T kk = { D Z k Z k (Z k Z k ) λ Zk [tanh (ν Zk N Zk ) tanh(ν Zk N Zk )] (Z k = Z k ) (2.4) N Zk = exp [ α (r kk R Zk R Zk )] (Z k = Z k ) k k (2.5) Where the quantities λ Zk and ν Zk are new parameters to be fit. Notice that T kk is nearly proportional to the difference of coordination numbers for a small difference of coordination numbers, but then it saturates and approaches a limit; this allows us to treat large coordination numbers (some of the clusters in this study have N as large as 12). We take α = Å -1 as in CM5. We optimized the two parameters against the dipole moments of 39 silver clusters. Seventeen of these are the structures in references 102 and 103 that have nonzero dipole 11

23 moments; we optimized the geometries of these structures with Kohn-Sham density functional calculations. The other 22 structures are nonequilibrium structures created by adding a chain of one or two silver atoms to the optimized ones because the added atoms have smaller coordination numbers than the atoms in the big clusters, and this increases the diversity of our training set. The largest optimized structures are Ag14-18 clusters, which have a width range of nm; the largest non-equilibrium structures are Ag14-18 with a chain of two atoms sticking to them, which have a width range of nm, where the width is defined as the largest Ag-to-Ag distance plus twice the van der Waals radius (with the latter from Ref. 99). The parameters λ and ν were optimized for these structures (labeled J = 1 to 39) to minimize the error function: 4 39 χ = (μ CM5M IJ μ IJ I=1 J=1 ref ) 2 (2.6) where μ ref is the dipole moment calculated from the Kohn-Sham density, and μ CM5M is the dipole moment calculated from the CM5M partial atomic charges. The first sum in equation (2.6) indicates that we averaged the results calculated by Kohn-Sham density functional calculations with four exchange-correlation functionals, namely, N12, M06- L, M06, and TPSS. The final parameters are λ Ag = 0.080, and ν Ag = In Table 2.1 we can see that minimum-basis-set Mulliken population analysis (MBS) 104,105 and natural bond orbital population analysis (NPA), which are both class II charge models, do not reproduce the accurate dipole moments. Although the class III models (CHELPG and MKS) have small errors in dipole moments, they are still not generally applicable models because of the deficiencies we discussed above (problems that become more serious as the size of the particle increases). The class IV model that 12

24 was our starting point, namely CM5, has errors intermediate to the other two types of models. The new model CM5M has physical charges, the quality of dipole moments is as good as class III models, and the method is straightforwardly applicable even to buried atoms, extended systems, and calculations with periodic boundary conditions. Next we consider the interaction of CO with silver clusters. The binding energy ( H 0 ) of CO on a silver cluster Ag n is: H 0 = E(Ag n )+ E(CO) E(Ag n CO) (2.7) where the energies (E) include electronic energy, nuclear repulsion, and zero point vibrational energy. The partial atomic charges and binding energies of all structures are tabulated in supporting information. We first studied neutral, cationic, and anionic silver monomers and dimers. Since bond lengths of silver dimers with different charge states are very close, the charge state can be considered as essentially the only factor determining different CO binding energies. Each of these clusters has one unique binding site, so the atomic charges can be determined by symmetry from the total charge, e.g. the atomic charge on each atom + of Ag 2 is 0.5. Table 2.2 shows that the positively charged silver monomer and dimer have the highest binding energies, while the negatively charged silver monomer and dimer have the lowest binding energies. The results in Table 2.2 suggest that positively charged sites tend to have higher binding energies. Next we studied nine neutral Agn clusters with n = 4 8 and 10, where each of the clusters has two or three different binding sites. The cluster size and symmetry, the CM5M partial atomic charge on the binding site, and the binding energy are listed in Table 2.3. We postulate that the more positively charged atoms tend to have higher 13

25 binding energies, and we find that six out of nine cases strictly follow the postulate. While the C2v structure of Ag5 and the C3v structure of Ag7 do not strictly follow the postulate, each of the two clusters has three binding sites, and the most negative atoms do have the smallest binding energies, although the other two sites have the reversed order. Only the C5v structure of Ag6 has the opposite trend to our postulate for the relationship between charge and binding energy. Based on the results in Table 2.3, we can see that there is a trend, although not a strict rule, that in a silver cluster, atoms with more positive partial atomic charges are more likely to be sites of higher binding energy. Finally, we studied large neutral silver clusters: Ag9, Ag11, Ag17, and Ag18, in which each cluster has more than three different binding sites. Figure 2.2 shows the atomic charges and binding energies on different binding sites of Ag17 and Ag18. For each cluster, the number of unique binding sites is less than the number of atoms because of symmetry. Structures of these clusters include the icosahedral Ag13 structure as a substructure, which is slightly distorted by the additional atoms. The Ag13 structure has one center atom and twelve atoms surrounding the center one. In Ag17, atoms 2, 3, 4, and 17 can be viewed as extra atoms added to Ag13; in Ag18, atoms 2, 3, 4, 17, and 18 can be viewed as extra atoms added to Ag13. Topographically, if the surface atoms of Ag13 are apex atoms, these extra atoms flatten the nearby surfaces, thus making nearby atoms less sharp and more like edge atoms or face atoms. We noticed that positive charges are more likely to be on apex atoms, while negative charges are more likely to be on edge and face atoms. For example, in Ag17, the most two positive charges are on atom 9 and 13, which are apex atoms; the two most negative charges are on atom 6 and 7, which are on much flatter surfaces. For Ag18, the most two positive charges are on 14

26 atom 9 and 14; the most two negative charges are on atom 1 and 12, which still follow the trend. Our finding parallels Neumaier s et al. 72 calculations of electron densities and CO binding energies on Au In their studies, the structure of Au 21 is a tetrahedron of 20 atoms with one extra atom binding to it, so there are 5 apex atoms (four vertexes of the tetrahedron and the extra atom), 12 edge atoms and 4 face atoms. They found the 5 apex atoms have the lowest electron densities, while the 4 face atoms have the highest electron densities. Coordination number (N) is also related to the topography. Edge and face atoms are in general bonded to more atoms than apex atoms, and thus they have larger coordination numbers. For example, in Ag17, atom 6 has N = 6.0, whereas atom 9 has N = 4.8. In Ag18, atom 1 has N = 6.1, while atom 9 has N = 4.9 (N is calculated using eq 5). This correlation is consistent with our new charge model, which has a negative correction on multi-coordinated atoms. We diagrammed all binding energies and partial atomic charges on corresponding atoms in Figure 2.3, which shows a nearly linear dependence of binding energies on charges, and the trends we found in the smaller clusters are also followed in large silver clusters. Zhou et al. 71 performed DFT studies of CO binding energies of neutral and charged Ag n with n = 1 7. They found that clusters with positive charge states have the highest binding energies, while clusters with negative charge states have the lowest binding energies. Their work show that the overall charge state of a silver cluster significantly affects the CO binding energies, and our work further shows that the partial atomic charge is a major determining factor for binding site preference. We also studied the charges on the adsorbed CO molecules and found that for small silver clusters, there is a general trend that the more charge transferred from CO to silver 15

27 clusters, the higher the binding energy is. However for large silver clusters with more than three binding sites (Ag9, Ag11, Ag17, and Ag18), there is no linear correlation between charges on adsorbed CO and binding energies. Figure 3.3 shows that a change in partial atomic charge equal to a small fraction of the unit charge (i.e., a small fraction of the charge on a proton) yields a significant change in binding energy. For example, in Ag17, the charges of the most positive atom and the most negative atom differ by only 0.079, while the two corresponding binding energies differ by 12.4 kcal/mol; in Ag18, a difference of in charge causes a difference of 13.4 kcal/mol in binding energy. Therefore it is necessary that a charge model, to be maximally useful, should be accurate to within a few hundredths of a charge unit. In addition to providing an understanding of preferred binding sites, the new charge model is expected to be useful for catalyst design; for example, one could use a support to make coinage metals more positive and increase their capabilities to adsorb CO. All other things being equal, coarse surfaces of particles not only increase the ratio of surface to volume a well-known effect, but also they create more active binding sites for CO because bumpy surfaces have more peaky atoms with more positive atomic charges which is a previously unappreciated aspect Conclusions We found that homonuclear silver clusters have very uneven charge distributions. We proposed a Class IV charge model, CM5M that yields accurate charges by considering both Hirshfeld population analysis and the coordination numbers of the 16

28 atoms in the cluster. CM5M is validated by showing that it yields accurate dipole moments for silver clusters. Applying CM5M to the study of interaction of CO and silver clusters, we find that silver atoms with more positive charges are more likely to have higher binding energies, and an accurate charge model is essential to correctly understand the connections between charges and CO binding energies. Charges are important for binding and hence for catalysis, and coordination number is one of the factors that helps determines charges. It would be interesting to in future work to study the effect on the charge distributions of placing the metal clusters on an oxide substrate as is employed in many catalytic setups. We believe that the role of the uneven charge distribution in homonuclear metal particles has not been widely enough appreciated, and we hope the present findings will change that situation. The results have widespread implications for understanding binding by subnano and nano metal particles, not just for silver but for all metal clusters Acknowledgments This research was sponsored in part by the Inorganometallic Catalyst Design Center and supported in part by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Chemical Sciences under Award DE-SC

29 Table 2.1. Mean Unsigned Errors (MUE) in Magnitude of Dipole Moments (Debye). charge model CM5M CM5 CHELPG MKS MBS NPA MUE 0.17 (13%) 0.35 (27%) 0.25 (20%) 0.12 (9%) 0.54 (42%) 1.30 (100%) *The Percentage Errors are also Calculated using the Averaged Dipole Moment Over all 39 Structures, μ=1.29 D. 18

30 Table 2.2. Binding Energies of CO on Silver Monomer and Dimer with Different Charge States. complex binding energy (kcal/mol) AgCO AgCO 6.8 AgCO 4.5 Ag 2 CO Ag 2 CO 12.6 Ag 2 CO

31 Table 2.3. CM5M Charges and Corresponding Binding Energies for Silver Clusters with Two or Three Different Binding Sites. cluster Ag4 Ag5 Ag6 Ag6 Ag6 Ag7 Ag7 Ag8 Ag10 symmetry D2h C2v C3v C5v C2v D5h C3v D2d D2d CM5M charge on the binding atom binding energy (kcal/mol)

32 Figure 2.1. Comparison of dipole moments. The dipole moment calculated from the electron density with the N12 exchange-correlation functional is μ ref, and the dipole moment calculated by Hirshfeld population analysis is μ HPA. The arrows show the direction of the accurate dipole moment; in the case on the left the central Ag atom is above the plane of the page, and the arrow (whose tipped side is the negative direction) points toward the viewer. μ ref = 0.47 D μ HPA = μ ref = 1.15 D μ HPA = 1.72 D 21

33 Figure 2.2. CM5M partial atomic charges and binding energies for C2 structure of Ag17 (left) and Cs structure of Ag18 (right). Atomic charges are in red color, binding energies (kcal/mol) are in blue color. 22

34 Δ H 0 (kcal/mol) R²= 0.79 R²= 0.84 R²= R²= 0.74 Ag9 Ag11 Ag17 Ag Charge Figure 2.3. CM5M partial atomic charges and binding energies for four large size silver clusters. R 2 denotes the square of the correlation coefficient. 23

35 Chapter 3 Validation of Methods for Computational Catalyst Design: Geometries, Structures, and Energies of Neutral and Charged Silver Clusters 3.1. Introduction Metal clusters and nanoparticles are important in chemistry and materials science because of their unique properties that are different from those of bulk materials Silver clusters and nanoparticles are of special interest for their roles in photography 109 and catalysis, 110,111 and a variety of experimental techniques and quantum mechanical computations 91,102,103, have been used to study the properties of silver clusters. Cluster properties vary significantly with size and structure, 106,108,110, ,120,122, which is one reason why they are so interesting. To understand the properties of silver clusters, we need to study both geometrical and electronic structures. One theoretical method widely used for studying silver clusters is Kohn Sham density functional theory, 1 although its accuracy for metal clusters is less well validated than its accuracy for many other properties. Kohn-Sham theory would be exact if the exact exchange-correlation (xc) functional were known and were used; however, the exact xc functional is unknown and probably unknowable. Therefore one must use approximate xc functionals, which leads to questions: Which, if any, of the xc functionals are reliable for metal clusters? Which of them give the most accurate results for metal clusters? The answers are important in the short term for guiding applications and in the long term for designing better density functionals. 24

36 In recent decades, a number of new xc functionals have been developed. Our group tested xc functionals against reliable experimental data for 3d and 4d transition metal atoms, ions, and compounds In the present work, we choose a few of the more successful and more popular xc functionals from the literature and assess their reliability for small silver clusters in three respects: (1) geometries, (2) structures, and (3) energies. (1) Geometries determined from experiments are limited to Ag 2 and Ag ,123 Based on available validation tests, 91,125,126 we chose coupled cluster theory with single and double excitations and quasiperturbative connected triple excitations (CCSD(T)) 141 to optimize the structures of isomers of Ag 2 4, Ag + 2 3, and Ag 2 3. We used data from experiments and CCSD(T) calculations as reference data to assess the xc functionals in terms of the mean unsigned error of bond lengths. (2) Structures of small silver clusters are well established. For example, the structure of silver trimer is a Jahn-Teller-distorted C 2v obtuse isosceles triangle, and the structure of silver tetramer is a D 2h rhombus. Joint theoretical and experimental investigations 112,113,120 identified the lowest-energy isomers for Ag n, Ag n +, and Ag n (3 n 7) clusters. We performed a systematic search for these cluster sizes to find out not only the lowest-energy isomers but all possible isomers, and then we tested which of the xc functionals are able to predict the lowest-energy isomers correctly. (3) Experimental values are available for dissociation energies of Ag n + and Ag n (n = 2-7) 116,117, vertical ionization potentials of Ag , and vertical detachment energies of Ag We calculated all these types of energies with each xc functional and then compared them to the reference data from the experiment and to our CCSD(T) calculations on small neutral clusters. 25

37 3.2. Computational Details In our calculations, all silver clusters are in the lowest possible spin states (singlets and doublets) except for the D 3h structure of Ag 3, which is in triplet spin state Test of method for generating reference data For reference data, we used CCSD(T), as explained in section 3.1. All of our CCSD(T) calculations are carried out with the Molpro software package. 97,98 Peterson et al. 91 developed pseudopotential correlation-consistent polarized valence basis sets (cc-pvnz-pp, n = D,T,Q,5), diffuse-function-augmented versions (aug-ccpvnz-pp, n = D, T,Q,5) of these basis sets, and polarized outer-core and valence versions (cc-pwcvnz-pp, n = D,T,Q,5). For silver these are based on small-core (28 electrons replaced by pseudopotential) relativistic pseudopotentials. 142 Peterson et al. and Huang et al. 126 did CCSD(T) calculations for Ag 2 and Ag 2 respectively with polarized valence basis sets and a correction (ΔCV) for core-valence correlation as the lowering in energy when upgrading the calculation to a polarized outer-core and valence basis set (i.e., in the notation below, from pv to pwcv). Table 3.1 shows the results (from the literature as indicated in the table, although we did repeat all the calcuations) of CCSD(T) geometry optimizations with various basis sets on Ag 2 and Ag 2, indicating that the aug-cc-pvqz-pp basis set gives converged geometries at the polarized valence level, and cc-pwcvdz-pp is already converged for the effect of ΔCV on geometry. Spin-orbit (SO) interaction was also calculated by Peterson et al. using internally contracted multireference configuration interaction (MR-CISD) calculations 143 with augcc-pvtz-pp basis set for Ag 2, and the results in Table 3.1 indicate that the effect of SO 26

38 coupling decreases the equilibrium interatomic distance (re) but only by only Å. 91 Therefore we didn t include SO interaction in the rest of our calculations Reference Data As a result of the tests in section 3.2.1, we used the aug-cc-pvqz-pp basis set in our polarized valence CCSD(T) calculations, and the cc-pwcvdz-pp basis set to calculate the core-valence correlation effect Density Functional Calculations In our Kohn-Sham calculations, we used a small-core relativistic pseudopotential 142 based basis set. We tested correlation-consistent basis sets ((aug)-cc-p(wc)vnz-pp)). as well as efficient diffuse-function-augmented basis sets, prefixed with maug, may, jun as explained previously. 93,94 We found that Kohn-Sham calculations are not very sensitive to core functions in basis sets, but are relatively more sensitive to diffuse functions. Considering both accuracy and cost, we finally chose the jun-cc-pvtz-pp basis set. We use pruned grids with 99 radical shells and 590 angular points per shell (keyword ultrafine in Gaussian 09). All Kohn-Sham calculations are carried out with a locally modified version, MNGFM-6.4, of Gaussian , Structure Searching In order to systematically study Ag 5 7 clusters, we use the coalescence kick algorithm 144 to systematically search for all possible isomers of each size of cluster. This method randomly generates a large number of structures by generating and pushing 27

39 atoms from random directions toward the center, and then these structures are optimized to nearest local minima or saddle point structures using the Kohn-Sham method. We first search for structures of neutral clusters, and then we use them as starting structures to further search for cationic and anionic structures. In our studies of neutral and charged silver clusters, we found that N12 is very good for geometries, with a mean unsigned error of bond length of only Å, so we used N12 in all the searches. For all clusters with one imaginary frequency (saddle point structures), we also perform minimum energy path calculations to find their connections to local minima Results and Discussion First we test the accuracy of xc functionals for geometries. Internuclear distances of Ag 2 and Ag 2 have been determined from spectroscopy experiments. 121,122 We use CCSD(T)/aug-cc-pVQZ-PP plus core-valence correlation functions (CCSD(T)/ccpwCVDZ-PP) method to optimize the lowest-energy isomers of Ag + 2, Ag 3, Ag + 3, Ag 3, Ag 4, as well as other isomers: a D h structure of Ag 3, a D h structure of Ag 3 +, a D 3h structure of Ag 3, and a C 2v structure of Ag 4. Figure 3.1 shows the structures of 11 neutral and charged silver clusters used as reference data for geometries. Since the D 2h structure of Ag 4 has two different bond lengths, the C 2v structure of Ag 4 has three different bond lengths, and each of other structures has one unique bond length, there are 14 bond lengths in total. We optimized these structures using the Kohn-Sham method with 42 xc functionals, compared the 14 bond lengths with reference data from experiments and CCSD(T) calculations. 28

40 One thing that needs to be mentioned is that the structure of Ag3 is not an equilateral triangle because of John-Teller effect. Previous work predicts that the structure has C2v symmetry with bond angle between 65 and 80 degrees, 133,134 and we found most of our DFT calculations agree with the prediction. The accuracy of xc functionals for geometry in terms of mean unsigned error of bond length is shown in Table 3.2 along with the percentage of Hartree-Fock exchange. We can see that although there are less local functionals than hybrid functionals in the test set (19 local functionals and 23 hybrid functionals), seven out of the ten best functionals are local functionals, and the best two functionals, TPSS and N12, are both local functionals. Only three of the ten best functionals are hybrid functionals, and two of these have relatively low percentages of Hartree-Fock exchange, in particular 10% and 15%. We conclude that including Hartree-Fock exchange in xc functionals is generally not good for geometry optimization of silver clusters, and this is consistent with our experience on other metals. Next we examine how well the xc functionals are able to predict the correct structures. There is more than one isomer for each silver cluster containing more than two atoms, so it is important to see if the xc functionals are able to correctly predict the lowest-energy isomer for each cluster. The lowest-energy isomers of neutral, cationic, and anionic silver clusters be ascertained from previous experiments and theoretical studies. For neutral clusters, optical absorption was measured experimentally and also calculated with time-dependent density functional theory; 112 the comparison of experiment to the calculations identified the lowest-energy isomers for n = 3 7. For cationic clusters, collision cross-sections were measured from ion mobility experiments and compared with results calculated by Kohn-Sham calculations 113 so that the lowest- 29

41 + energy isomers of Ag 3 7 clusters were identified. For anionic clusters, transition energies were measured from photoelectron spectra experiments 120 and were compared to theoretical calculations; 145 the main spectra peaks agree well for particular Ag 3 7 clusters, which were therefore judged to be the lowest-energy isomers for each size of anionic silver clusters. So the determination of all the clusters are based on the comparison of one feature measured from experiments and calculated from computations. One common feature of the findings is that for all neutral, cationic, and anionic silver clusters with n 7, one predominant isomer could be identified for each size of cluster; with n > 7, there might be more than one isomer coexisting so that the feature measured from experiments (such as optical absorption spectrum, collision cross section, or photoelectron spectrum) cannot be attributed to only one isomer; thus the lowest-energy isomers cannot be identified in this way for higher n. Lowest-energy isomers for neutral and charged Ag n (n = 3 7) are shown in Figure 3.2. As we mentioned before, there is more than one isomer for each size of cluster. For example, we find that there are at least 10 other local minimum structures for Ag 7. Therefore, we take some low-energy local minima structures optimized with the N12 functional, and we calculate the electronic energies with various xc functionals to see which of them are able to correctly predict the lowest-energy structures. There are 15 lowest-energy isomers for neutral and charged Ag n (n = 3 7) determined as described above, and we tested 42 exchange-correlation functionals, so we have a total of 630 theoretical predictions of the lowest-energy isomers, and we find that 121 of the 630 predictions are incorrect. First question will be which structures are most likely to be predicted incorrectly. We find that the wrong predictions mainly occur 30

42 for Ag + 6, Ag 6, Ag + 7, and Ag 7 ; there are 82 wrong predictions for these four clusters, which is 68% of the total number of wrong predictions. A key point here is that for all these four cases there is a close competition between 2D and 3D structures: the four lowest-energy isomers are three dimensional structures, but all the wrong predictions yield planar two dimensional structures as the lowest-energy isomers. Figure 3.3 shows the 3D lowest-energy isomers of Ag + 6, Ag + 6, Ag 7 and Ag 7, as well as the 2D competing low-energy isomers. The first 13 entries of Table 3.3 are the functionals that are able to predict all the lowest-energy isomers correctly. In the full set of xc functionals that we tested, 14 depend on kinetic energy density and 28 do not, while nine out of the 14 functionals that depend on kinetic energy density are exact for all structures, which suggests kinetic energy density makes the xc functionals more likely to be correct for predicting lowestenergy isomers. Similar finding were reported in previous work on gold clusters; in particular Mantina et al. 146, Johansson et al. 147 and Ferrighi et al. 148 studied the 2D-to- + 3D structural transitions (as functions of n) for singly charged gold clusters Au n and Ag n, and in this context Ferrighi et al. provided an analysis of the connection of the kinetic energy density dependence to predictions of the 3D-2D competition. The gold clusters tend to be planar structures at small size but to be compact 3D structures at large + size. Experiments indicate that n = 8 for Au n and n = 12 for Au n are the sizes where the 2D to 3D transitions take place. Computational studies revealed that xc functionals with kinetic density dependence are more successful in predicting the correct transitions because the kinetic energy density favors the compact 3D structures. In our studies for silver, which is in the same column of the periodic table as gold, n = 5 and n = 6 are the 31

43 sizes where the 2D-to-3D transitions occur for Ag n + and Ag n, respectively, and xc functionals dependent on kinetic energy density are more likely to correctly predict the lowest-energy isomers near the transitions. Finally, we studied the reliability of the density functionals for reaction energies. These reaction energies are categorized as reaction energies of the neutral clusters (Ag 1 4 ), dissociation energies of cationic and anionic silver clusters (Ag n + and Ag n, n = 2-7), vertical ionization potentials(ag 1 7 ), and vertical detachment energies (Ag 1 7 ). We use CCSD(T) calculation results as the reference data for the reaction energies of the neutral clusters. There was a multiple-collision induced dissociation (MCID) experiment of cationic silver clusters 116, further analysis of the experimental data generates the dissociation energies of singly charged cationic silver clusters Ag n +. The fragmentation channels of singly charged cationic silver clusters have also been studied 115,116. In the case of Ag n + (n = 2-7) clusters, the fragmentation channel for each cluster is almost exclusive: even-numbered clusters decay by emission of a silver monomer, while odd-numbered clusters decay by emission of a silver dimer. Therefore we can consider the dissociation energies are exclusively monomer-loss reaction energies for even-numbered clusters (Ag + n Ag + Ag + n 1, n = even number) and dimerloss reaction energies for odd-numbered clusters (Ag + n Ag 2 + Ag + n 2, n = odd number). Collision-induced dissociation (CID) experiment was performed on anionic silver clusters 117. What is similar to cationic silver clusters, the predominant fragmentation channel is atom-loss for even-numbered anionic clusters and dimer-loss for odd-numbered anionic clusters, and a fitting scheme single-channel threshold fits is used for calculating the energies. However, the atom-loss fragmentation channel is not 32

44 negligible for odd-numbered anionic clusters, so another scheme competitive CID threshold fits was used in the paper to calculate the dissociation energies of oddnumbered anionic clusters through two fragmentation channels. We take the dissociation energies of single-channel threshold fits for even-numbered clusters and competitive CID threshold fits for odd-numbered clusters which are more accurate as our reference data, and finally use experimentally measured monomer-loss dissociation energies for even-numbered silver anions (Ag n Ag + Ag n 1, n = even number) and both monomer-loss and dimer-loss dissociation energies for odd-numbered silver anions (Ag n Ag + Ag n 1, and Ag n Ag 2 + Ag n 2, n = odd number) as reference data. Dissociation energies from experiments (D 0 ) cannot exclude zero-point energies (ZPE). In order to compare our calculations with experiments, we calculate zero-point energies with N12 and subtract that from experimental data to get D e. Vertical ionization potentials (VIP) and vertical detachment energies (VDE) of silver clusters were measured from experiments 118,119,120. With the known lowest-energy isomers in Figure 3.1, we calculated VIP and VDE of Ag 1 7 clusters. There are 39 reaction energies of all types mentioned above. Here we show the experimental and the best theoretically calculated energies of neutral clusters, dissociation energies of charged clusters, VIPs and VDEs in Table 3.4 and Table 3.5, and we also show DFT calculated adiabatic ionization potentials (AIP) and adiabatic detachment energies (ADE) in Table 3.5 because those are more physically relevant to experiments. We summarize the mean unsigned error (MUE) for each energetic quantity in Table 3.6. As we can see, different functionals are good at differenct energy quantities: 33

45 BHandH, TPSS and PBE are the best for the energies of neutral clusters; M06, τ- HCTHhyb and BMK are the best for the dissociation energies of cationic and anionic silver clusters; M06, MN12-L and B3LYP are the best for VIP; SOGGA, SOGGA11-X and mpw3lyp are the best for VDE. We summarize the total MUE and the ingredients of functionals in Table 3.7. The best functionals in total are M06, PBE and BMK. Of the functionals the number of wrong structure predictions (from Table 3.3) is zero for M06, three for PBE, and one for BMK. As we can see, the accuracy of a functional for energies is not dependent on Hartree-Fock exchange or kinetic energy density. In the top ten functionals, half of them are hybrid functionals and half of them are non-hybrid functionals; half of them are meta-functionals and half of them have no kinetic energy density dependence Conclusions Neutral and singly charged small silver clusters (Ag n, Ag + n, and Ag n, n = 1-7) have been studied. We systematically searched for all possible isomers for clusters with n = 5-7. We assess the reliability of 42 xc functionals in terms of structures, geometries, and energies. We find non-hybrid functionals are generally better for geometries, seven out of 10 best functionals for geometries are non-hybrid functionals, and the best three are TPSS, N12 and BHandH. We also find that functionals with kinetic energy density dependence are more likely to be correct for structures. In our study, nine out of the 14 exchange-correlation functionals including kinetic energy density predict all 15 structures for n = 3-7 correctly, whereas only 4 of the 28 exchange-correlation functionals that do not include kinetic energy density predict them all correctly. Energies 34

46 are categorized as reaction energies of neutral Ag 1 4 clusters, dissociation energies of cationic and anionic silver clusters (Ag n + and Ag n, n = 2-7), vertical ionization potentials (Ag 1 7 ), and vertical detachment energies (Ag 1 7 ). The best functionals for the total energetic quantities are M06, PBE and BMK. The accuracy for energies is not much affected by the ingredients of functionals. These findings could be useful for computational catalyst design Acknowledgments I am grateful to Boris Averkiev for providing the coalescence kick program and to Run Li for early-stage exploratory calculations on Ag clusters. This work was supported in part by the Inorganometallic Catalyst Design Center, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences under Award DE-SC This research was performed in part using the computing resources of the MSCF in EMSL, a national scientific user facility sponsored by the U.S. DOE, Office of Basic Energy Research and located at PNNL and also using the computing resources of Minnesota Supercomputing Institute. 35

47 Table 3.1. CCSD(T) Calculated Internuclear Distances (Å) of Ag 2 and Ag 2. Basis set a Ag 2 b Ag 2 only valence polarization aug-cc-pvdz-pp aug-cc-pvtz-pp aug-cc-pvqz-pp aug-cc-pv5z-pp ΔCV cc-pwcvdz-pp cc-pwcvtz-pp cc-pwcvqz-pp cc-pwcv5z-pp ΔSO aug-cc-pvtz-pp Expt. c a Reference 91 Reference 126 c References 121,122. In reference 122, the internuclear distance of Ag 2 was determined to be r e Å, where r e is the internuclear distance of Ag 2. The reference paper used an uncertain assignment r e = 2.480Å. Here we substitute r e with the more accurate value Å and get the internuclear distance of Å for Ag 2. 36

48 Table 3.2. Percentage of Hartree-Fock Exchange and Mean Unsigned Error (MUE) of 14 Bond Lengths for Each xc Functional. a MUE of bond functional X b functional length (Å) X b MUE of bond length (Å) TPSS M06-L N HSE BHandH M11-L revtpss mpw3lyp TPSSh mpwlyp BP M PW B3LYP MN12-L MPWLYP1M PBE B τ-hcthhyb BLYP PBEsol RPBE MN12-SX GVWN SOGGA B1LYP τ-hcth SOGGA11-X M BMK mpwpw OreLYP N12-SX O3LYP CAM-B3LYP BHandHLYP B3PW B PBE M TPSS1KCIS MOHLYP a Listed in order of increasing MUE before rounding b Percentage of Hartree-Fock exchange 37

49 Table 3.3. The Number of Wrong Predictions (out of 15) of the Lowest-Energy Isomer and the Dependence on Kinetic Energy Density for Each xc Functional. functional dependence on kinetic energy density # of wrong predictions functional dependence on kinetic energy density # of wrong predictions BHandH No 0 mpwpw91 No 3 M05 Yes 0 M11 Yes 3 M06 Yes 0 PBE No 3 M06-L Yes 0 PW91 No 3 MN12-L Yes 0 CAM-B3LYP No 4 MN12-SX Yes 0 SOGGA11-X No 4 PBE0 No 0 τ-hcthhyb Yes 5 PBEsol No 0 τ-hcth Yes 5 revtpss Yes 0 mpw3lyp No 5 SOGGA No 0 RPBE No 5 TPSS Yes 0 B97-1 No 5 TPSS1KCIS Yes 0 BHandHLYP No 5 TPSSh Yes 0 mpwlyp No 6 BMK Yes 1 mpwlyp1m No 6 GVWN3 No 1 OreLYP No 6 HSE06 No 1 O3LYP No 6 M11-L Yes 1 B97-3 No 6 N12-SX No 1 BLYP No 7 N12 No 2 B3LYP No 7 BP86 No 3 B1LYP No 7 B3PW91 No 3 MOHLYP No 7 38

50 Table 3.4. Energies (kcal/mol) for Neutral and Charged Silver Clusters. M06 CCSD(T) Expt. (D 0 ) a Expt. (D e ) b Ag 2 2Ag C 2v -Ag 3 Ag 2 + Ag C 2v -Ag 3 3Ag C 2v -Ag 3 D h -Ag D 2h -Ag 4 C 2v -Ag 3 + Ag D 2h -Ag 4 2Ag D 2h -Ag 4 4Ag D 2h -Ag 4 C 2v -Ag Ag + 2 Ag + Ag Ag + 3 Ag 2 + Ag Ag Ag + Ag Ag Ag 2 + Ag Ag Ag + Ag Ag Ag 2 + Ag Ag 2 Ag + Ag Ag 3 Ag + Ag Ag 3 Ag 2 + Ag Ag 4 Ag + Ag Ag 4 Ag 2 + Ag Ag 5 Ag + Ag Ag 5 Ag 2 + Ag Ag 6 Ag + Ag Ag 6 Ag 2 + Ag Ag 7 Ag + Ag Ag 7 Ag 2 + Ag a The dissociation energies (D 0 ) are from reference 116,

51 b The dissociation energies of Ag2 and Ag3 are from reference 121 and 123 respectively. The dissociation energies (D e ) of cationic and anionic clusters are calculated by excluding the zero point energies from D 0. 40

52 Table 3.5. Adiabatic IPs and DEs and Vertical IPs and DEs (kcal/mol) of Silver Clusters Calculated with M06. AIP VIP Expt. VIP 118 ADE VDE Expt. VDE 120 Ag Ag Ag Ag Ag Ag Ag Ag Ag Ag Ag Ag Ag Ag

53 Table 3.6. Mean Unsigned Error for Each Energetic Quantity (kcal/mol). a functional neutral dissociation VIP VDE functional neutral dissociation VIP VDE M mpw3lyp PBE SOGGA BMK CAM- B3LYP τ-hcthhyb MN12-SX PW mpwlyp TPSS mpwlyp1m mpwpw RPBE TPSSh M N B3LYP HSE M11-L BP BLYP PBE MN12-L revtpss B N12-SX B1LYP SOGGA11-X B TPSS1KCIS OreLYP B3PW BHandHLYP M06-L M BHandH O3LYP PBEsol MOHLYP τ-hcth GVWN a Listed in order of increasing total MUE before rounding. 42

54 Table 3.7. Total Mean Unsigned Error of Energy (kcal/mol) for Each xc Functional. a dependence MUE of dependence MUE of functional X b functional X c on τ σ energy on τ σ energy M06 27 Yes 3.8 mpw3lyp 0 No 6.6 PBE 0 No 3.9 SOGGA 0 No 6.7 BMK 42 Yes 4.0 CAM- 19 B3LYP 65 No 6.7 τ-hcthhyb 15 Yes 4.2 MN12-SX 0-25 Yes 6.9 PW91 0 No 4.2 mpwlyp 0 No 6.9 TPSS 0 Yes 4.3 mpwlyp1m 5 No 6.9 mpwpw91 0 No 4.5 RPBE 0 No 7.1 TPSSh 10 Yes 4.7 M05 28 Yes 7.1 N12 0 No 4.7 B3LYP 20 No 7.3 HSE d No 4.9 M11-L 0 Yes 7.5 BP86 0 No 4.9 BLYP 0 No 8.2 PBE0 25 No 5.1 MN12-L 0 Yes 8.6 revtpss 0 Yes 5.3 B No 8.9 N12-SX 25 0 No 5.3 B1LYP 25 No 9.0 SOGGA11-X No 5.4 B No 9.5 TPSS1KCIS 13 Yes 5.6 OreLYP 0 No 11.5 B3PW91 20 No 5.7 BHandHLYP 50 No 11.7 M06-L 0 Yes 6.0 M Yes 12.3 BHandH 50 No 6.1 O3LYP No 12.5 PBEsol 0 No 6.4 MOHLYP 0 No 14.1 τ-hcth 0 Yes 6.4 GVWN3 0 No 18.0 a Listed in order of increasing MUE before rounding b Percentage of Hartree-Fock exchange c Kinetic energy density d When a range of x is given, the first value is for small interelectronic separation, and the second is for large interelectronic separation. 43

55 Figure 3.1. Neutral and charged silver clusters used in geometry studies. 44

56 Figure 3.2. Lowest-energy isomers for neutral and charged Ag n (n = 3 7). 45

57 Ag 6 + Ag 6 Ag 7 + Ag 7 Figure 3.3 3D and 2D silver clusters. The structure on the left is the lowest-energy 3D structure, and the one on the right is a competing low-energy 2D structure. 46

58 Chapter 4 Geometries, Binding Energies, Ionization Potentials, and Electron Affinities of Metal Clusters:, n = Introduction Understanding the dependence of structural, electronic, energetic, optical, and magnetic properties of atomic and molecular clusters on the system size is one of the most important aspects of cluster science This, in particular, applies to clusters formed by divalent metals, which have been of fundamental interest for quite some time In this category, clusters of magnesium atoms, which have an s 2 valence electron configuration, are especially interesting, since the smallest Mg cluster, namely Mg2, is a system characterized by the very weak binding of the van der Waals type governed by a delicate balance of dispersion and exchange repulsion forces, 180,181 which cannot be captured by the Hartree-Fock or other low-order theory treatments, whereas larger Mgn species are covalently bound and characterized by an increase in the degree of s-p hybridization with the number of atoms in the cluster, 176 while having significant contributions from pairwise non-additive many-body interactions Furthermore, one observes an unusually large and not always monotonic variation of bonding properties when going from the smaller magnesium clusters toward the larger ones and the bulk limit, and this non-monotonic trend constitutes a challenge for electronic structure methods. Another challenge is the rapidly varying cohesive energy per atom in the smaller Mgn species; it is about 4 and 10 times larger in the Mg3 and Mg4 clusters 47

59 than in Mg Another aspect that makes the magnesium and other Group II elements challenging for quantum chemistry is that their ground states display significant multiconfigurational character due to the quasi-degeneracy of the valence s and p subshells, which is further enhanced by cluster formation, as in the classic case of the Be atom, and the dimer 189 and trimer species of Be, which require a high-level treatment of electron correlation effects. Similarly strong multi-configurational character of the ground-state wave function was also found in our earlier studies of the neutral and charged Al2-9 clusters. 195 Several experimental and theoretical studies of magnesium clusters of varying size have already been reported. Experimental structures and electronic properties of smallto-large magnesium clusters have been obtained using gas-phase spectroscopy, 180,196 Raman matrix spectroscopy, 197 photoelectron and mass spectroscopy techniques, electron ionization in supersonic expansion chambers, 201 and high-resolution transmission electron microscopy. 202 These experiments have probed a number of neutral clusters containing up to over 100 atoms and anions of up to 35 atoms. However, the only accurate, isomer-specific information provided by the experiments is the bond length and binding energy of Mg2 and the photoelectron spectra of small magnesium anions; the existing spectra of larger clusters cannot be attributed to individual isomers. Theoretical investigations of magnesium clusters reported to date involve numerous Kohn-Sham density functional theory (DFT) 1 calculations exploiting various exchangecorrelation (xc) functionals 163,165,168, 197, and several ab initio wave function computations using Hartree Fock theory, 187 Møller-Plesset (MP) perturbation theory, 215 including the second-order MP and fourth-order MP4 treatments, 176,186,188 48

60 complete-active-space self-consistent-field (CASSCF) 216 approach, 182,217 the coupledcluster (CC) method 218,219 using a popular quasiperturbative CCSD(T) 141,220 approximation, 176,181,183,185,187, 217 and the multireference configuration interaction (MRCI) method. 182 A van der Waals potential for Mg2, in which damped dispersion contributions based on C6, C8, and C10 coefficients were added to a repulsive potential based on Hartree Fock calculations, has been reported as well. 221 A few of the past computational studies have attempted to examine structural and energetic properties of the neutral and charged magnesium clusters as functions of the number of Mg atoms in the cluster. For example, Ref. 187 reported calculations of cohesive energies of Mgn clusters with n = 2 22 using a combination of Hartree Fock theory and the method of increments based on CCSD(T) computations for small clusters, and Refs. 209 and 210 reported DFT studies of structures and cohesive energies of neutral and anionic Mgn clusters with n = Reference 214 reported analogous DFT calculations of neutral and cationic Mgn clusters with n up to 30. However, none of the previously published studies have considered high-level (CCSD(T) or better) wave function calculations for magnesium clusters with more than four atoms, and none of the previous DFT studies of magnesium clusters have attempted to systematically calibrate the DFT results against high-level CC calculations. In this article, we apply several CC methods with single, double, and triple excitations and larger correlation-consistent basis sets to compute accurate structures of neutral and singly charged Mg2-4 clusters and to determine energetic properties, including cohesive energies, adiabatic ionization potentials, and adiabatic electron affinities of neutral and singly charged clusters Mg 0,±1 n, n = 1 7. The size range of the 49

61 magnesium clusters encompasses the transition from the van der Waals, dispersion-type bonding in Mg2 to covalent bonding in Mg4 and larger clusters. It also encompasses the transition from 2D to 3D structures, which occurs around Mg4. The results of our highlevel CC calculations and the available experimental data provide us with the necessary reference information to assess the performance of 39 xc functionals, so that we can make useful recommendations about which functionals are best suited for calculating the structural and energetic properties of magnesium clusters Computational Methods Most of the wave function computations that serve in this article as reference data for benchmarking DFT are based on the widely used CCSD(T) method, in which a quasiperturbative noniterative correction due to connected triply excited clusters that represents an improvement over the earlier CCSD+T(CCSD) CCSD[T] approach 222 is added to the energy obtained in CCSD 223,224 calculations. This method sometimes called the gold standard of quantum chemistry offers reasonably high accuracy for the majority of closed-shell and high-spin, open-shell polyatomic systems near their equilibrium geometries. However, we have found that in the case of Mg2, the bond length calculated by CCSD(T) and, to a lesser degree, the CCSD(T) binding energy are not in good agreement with the available experimental values. To understand whether these discrepancies are due to the noniterative quasiperturbative treatment of the connected triple excitations by the CCSD(T) approach and to examine if the inadequacy of CCSD(T) in the case of the magnesium dimer extends to other neutral and charged Mgn species, we have performed a number of additional CC calculations with singly, 50

62 doubly, and triply excited clusters, ranging from the iterative treatments of singles, doubles, and triples offered by the full CCSDT 225,226 and active-space CCSDt 227 methods, to the robust non-iterative corrections due to all or some triples to the CCSD and CCSDt energies defining the CR-CC(2,3) and CC(t;3) approaches. The full CCSDT and active-space CCSDt calculations, and the CC(t;3) computations have been limited to the magnesium dimer. They have been used in this work to judge the performance of the more practical CR-CC(2,3) method, which can be applied to larger molecular clusters, and to provide insights into problems encountered in the CCSD(T) calculations for Mg2. The most accurate CR-CC(2,3) calculations, represented by variant D of CR-CC(2,3), which we elaborate on some more below and which are of the nearly full CCSDT or CC(t;3) quality in the all-electron calculations for Mg2, have been performed for the magnesium dimer and trimer, and their ions, allowing us to calibrate the CCSD(T) results, which we have subsequently obtained for all Mgn systems with n = 1-7 and their ions. The active-space CCSDt approach is based on selecting the dominant triply excited amplitudes within the full CCSDT scheme using active orbitals, following arguments originating from state-specific multi-reference CC considerations employing a singlereference formalism. 235,236 This enables one to replace the prohibitively expensive computational steps of full CCSDT that scale as no 3 nu 5, where no and nu are, respectively, the numbers of occupied and unoccupied orbitals used in the post-scf calculations, by more manageable NoNuno 2 nu 4 steps, where No (< no) and Nu (<< nu) designate the numbers of the active occupied and active unoccupied orbitals used to select the triples within CCSDT. The CCSDt calculations with No active occupied and Nu active 51

63 unoccupied orbitals are designated in this work as CCSDt{No,Nu}. We use the CCSDt approach in its own right and, also, to provide the high-quality cluster amplitudes for the subsequent CC(t;3) calculations. A different philosophy in approximating the triples is offered by the CR-CC(2,3) methodology, where, in analogy to the conventional CCSD(T) approach and its CCSD[T] predecessor, one adds a correction due to connected triply excited clusters to the CCSD energy. The difference between CCSD(T) or CCSD[T] and CR-CC(2,3) lies in the definition of the connected triples correction, which in the CR-CC(2,3) case uses the complete form of the triply excited moments of the CCSD equations (projections of the CCSD equations on triply excited determinants 237 ) rather than the lowest-order contributions to these moments that are used in CCSD(T) and CCSD[T]. Moreover, in the CR-CC(2,3) approach, one renormalizes the connected triples correction to CCSD through the use of the left eigenstate of the similarity-transformed Hamiltonian of the CC theory, which adds the necessary flexibility in handling systems with more multireference character, where CCSD(T) and CCSD[T] encounter difficulties (see Refs. 237 and 238 for other forms of renormalized triples corrections). We distinguish between a full variant of CR-CC(2,3), originally labeled CR-CC(2,3),D and abbreviated here as CR-CC(2,3) without any additional symbols, where the relevant perturbative denominator in the definition of the triples correction to CCSD is represented by the Epstein-Nesbet-like expression, and its so-called CR-CC(2,3),A approximation, abbreviated in this work as CCSD(2)T [since the CR-CC(2,3),A method is equivalent to the CCSD(2)T approach of Ref. 239 when the canonical Hartree-Fock orbitals are employed], where one uses the more traditional Møller-Plesset-type formula. The CR- 52

64 CC(2,3) and CCSD(2)T calculations have the same iterative no 2 nu 4 and noniterative no 3 nu 4 computational steps as CCSD(T), but the CPU time required by the CR-CC(2,3) and CCSD(2)T computations is typically twice as long as that of the corresponding CCSD(T) run due to additional left eigenstate CCSD iterations and a more complete treatment of the triples energy correction as compared to the (T) formula of CCSD(T). There exist other noniterative corrections due to triples using the left eigenstate of the similarity-transformed Hamiltonian of CCSD, such as those described in Refs. 240, 241, 242, and 243, but as shown, for example, in Refs. 243 and 244, none of them is as accurate as variant D of CR-CC(2,3), so they are not considered in the present work. The iterative perturbative approximations to CCSDT, such as CCSDT-1 (see Ref. 245), are not used in this study either, since they are not robust enough in multi-reference situations and, more importantly, they use iterative no 3 nu 4 steps that make the calculations much more expensive than in the case of the CCSD(T), CCSD(2)T, and CR- CC(2,3) approaches. The CR-CC(2,3) and CCSD(2)T approaches are more robust than the CCSD(T) method and its generally less accurate CCSD[T] predecessor, especially for multireference single-bond rearrangements, but there are cases where the coupling of singles and doubles with the triples in the CC equations is substantial enough to cause problems to noniterative corrections to CCSD energies, even the CR-CC(2,3) ones. 233,234 If we are to rely on the single-reference CC methodology and remain within the manifold of excitations up to triples, the most accurate description of the coupling of singles and doubles with the triples is offered by the full CCSDT approach. However, as pointed out above, the full CCSDT method is usually too expensive to be practical. The active-space 53

65 CCSDt approach describes the coupling of singles and doubles with the most important triples selected with the help of active orbitals, but it misses some dynamical correlation effects due to triples that are outside the subset of triples selected via active orbitals. The most promising recent solution to this problem is represented by the aforementioned CC(t;3) approach, which uses the CR-CC(2,3)-style expressions to correct the activespace CCSDt energies, which describe the dominant triple excitations and which include the most essential information about the coupling of singles and doubles with the triples, for the subset of triples missing in the CCSDt description. As shown in our initial studies (and as confirmed in the present work using magnesium dimer as an example), the CC(t;3) approach is as accurate as its full CCSDT parent, while being more practical, since it replaces the iterative no 3 nu 5 steps of CCSDT by the CCSD-type NoNuno 2 nu 4 steps of CCSDt and the CR-CC(2,3)-type noniterative no 3 nu 4 steps associated with the correction due to triples missing in CCSDt. As in the case of the underlying CCSDt calculations, the CC(t;3) computations with No active occupied and Nu active unoccupied orbitals are designated as CC(t;3){No,Nu}. The structures of the magnesium clusters considered in the present study are shown in Figure 4.1. As previously reported, 209,214 magnesium clusters with less than 18 atoms have low-spin ground states. Therefore, we used a spin multiplicity of 1 for neutral clusters and 2 for the charged ones. The higher symmetry structures of the magnesium clusters may break symmetry due to Jahn-Teller (JT) effects or limitations of the employed wave functions, 246 so no symmetry restrictions were imposed at first. However, most clusters were eventually optimized to high-symmetry structures as 54

66 shown in Figure 4.1. The initial cluster geometries were manually constructed based on structures found in the literature. 213,214 All of the CC calculations reported in this work employed restricted Hartree Fock reference functions for singlets and restricted open-shell Hartree Fock reference functions for doublets. We used aug-cc-pvnz and aug-cc-pcvnz (n = T, Q) correlationconsistent basis sets, 247 where the calculations with pvnz basis sets correlated only valence electrons and the calculations with the pcvnz basis sets, which include additional core-valence (CV) polarization functions correlated all electrons. Only the larger all-electron aug-cc-pcvtz basis set was used in the reported DFT calculations. In the case of the DFT computations, the geometries were optimized with all functionals studied for neutral and charged Mg2-4 clusters, but for clusters with 5-7 atoms, we used the geometries optimized with PWB6K; this functional was selected because the various DFT calculations for systems with 2 4 atoms show that it is the most accurate functional for the geometries of Mgn species. In the case of the CCSDt, CC(t;3), and CCSDT calculations for Mg2 and the CR-CC(2,3) calculations for Mg2, Mg3, and their ions, we optimized the geometries at the respective CC levels using numerical gradients. The CCSD(T) calculations for the larger Mgn clusters with n = 5-7 used the PWB6K-optimized geometries. All results in this paper are electronic equilibrium quantities, i.e., they do not include vibrational or rotational effects (neither zero-point nor thermal). All CCSD(T) and DFT calculations were carried out with a locally modified version, MN-GFM6.4, of Gaussian. 95,96 The CR-CC(2,3) calculations were performed with the computer codes described in Refs. 228, 229, 230, 231, and 248 that form part of the 55

67 GAMESS package. 249,250 The CCSDt and CC(t;3) calculations were performed with the standalone codes developed in Refs. 232 and 233, which are interfaced with the Hartree Fock and atomic orbital integral and integral transformation routines available in GAMESS. The full CCSDT calculations were performed with NWCHEM Results and Discussion Testing CC Approaches and Generating Reference Data The results in Table 4.1 show that there is a large effect on the Mg2 bond distance from adding core-valence polarization functions to the basis set and correlating all electrons in the calculations. For example, the distance calculated by CCSD(T) drops by Å when we compare the valence-electron aug-cc-pvtz result with its all-electron aug-cc-pcvtz counterpart. For this reason, we draw conclusions about the performance of CC methods for this molecule from the all-electron calculations employing the augcc-pcvtz basis set. The experimental equilibrium bond length in Mg2 is Å, but Table 4.1 shows that the conventional CCSD(T) method fails to reproduce this value, giving a rather large error of Å. The full CCSDT method performs much better than CCSD(T), giving a small error of Å, which indicates that the complete and iterative treatment of triple excitations considerably improves the accuracy of the CC results for Mg2, but CCSDT is not an affordable method for larger clusters, so we need to find less expensive methods, while preserving its accuracy. For this reason, we consider three other CC methods in Table 4.1, including the iterative active-space CCSDt approach and the 56

68 noniterative corrections to CCSD and CCSDt energies represented by the CCSD(2)T, CR-CC(2,3), and CC(t;3) approximations, which are all less expensive than CCSDT. When the aug-cc-pcvtz basis set is employed and when all electrons are correlated, CCSDt is less accurate than CCSD(T). CCSD(2)T is not good enough to be of further interest to us either, but CR-CC(2,3) is almost as accurate as CCSDT, considerably improving the CCSD(T) results and producing an error of only Å. The CC(t;3) method, which corrects the CCSDt results for the missing dynamical triples, is more accurate than the methods to the left of it in Table 4.1, producing a virtually perfect agreement with CCSDT, but it is also more expensive than other ways of approximating the triples considered in this work. Based on examining the accuracy vs computational cost and the ease of use, we choose CR-CC(2,3) and CCSD(T) as our main CC methods to optimize the structures and determine the energetics of other Mgn clusters with n = 2 4 and their ions. The very good agreement of CR-CC(2,3) results for Mg2 with our highest-level CC(t;3) and CCSDT calculations and experimental data for the same species makes us believe that we can use CR-CC(2,3) to judge the performance of CCSD(T) for other magnesium clusters and their ions. Table 4.2 shows bond lengths in Mg n 0,±1, n = 2 4. Because of symmetry, all the clusters have one unique bond except Mg 4 which has two unique bonds R1 and R2 as we have shown in Figure 4.1. We find that the core-valence correlation effects affect the calculations for other clusters as much as for Mg2, so using augmented core-valence basis sets and correlating all electrons is very important for the larger clusters too. On 57

69 the other hand, although the CCSD(T) bond length in Mg2 differs from the corresponding CR-CC(2,3) and CCSDT values by and Å, respectively, when the aug-cc-pcvtz basis set is employed and all electrons are correlated, the discrepancies between CCSD(T) and other higher-level CC methods considered in this work considerably drop down for the five other species with n = 2 and 3 considered in Table 4.2. Indeed, the largest deviation among the all-electron CCSD(T)/aug-cc-pCVTZ, CR-CC(2,3)/aug-cc-pCVTZ, and CCSDT/aug-cc-pCVTZ results for the bond lengths in the 0,±1 Mg n species with n = 2 and 3 other than Mg2 is only Å. For Mg + 2, the results obtained with CCSD(T), CR-CC(2,3), and CCSDT agree to within an impressive Å. Based on the above observations and considering the facts that the Mg2 dimer is characterized by a very weak non-covalent bond which originates from a delicate balance of high-order dispersion and exchange interactions and that other magnesium clusters, being more covalently bound, are less challenging in this regard, it is quite likely that the poorer performance of the CCSD(T) approach observed for Mg2 is limited only to Mg2. The excellent agreement between the CR-CC(2,3) and CCSD(T) data for the Mg3 clusters and their ions allows us to conclude that although CCSD(T) is inadequate to describe the magnesium dimer, it is sufficiently accurate to assess performance of various DFT functionals in clusters containing three or more magnesium atoms. We also believe that the results of our all-electron CCSD(T)/aug-cc-pCVTZ calculations are more accurate than those reported in the previous studies. For example, Ref. 181 reported all-electron CCSD(T)/aug-cc-pVQZ calculations for the neutral Mgn species with n = 2 4. As shown in Table 4.2, calculations of this kind tend to underestimate the Mg Mg distances and is not a good scheme. Ref. 185 used the non- 58

70 augmented cc-pvtz basis set, correlating valence electrons only. The results in Table 4.2 show that this is not enough to obtain an accurate description of the bond length in Mg4. Further examination of Table 4.2 shows that the effect of adding core polarization functions is larger than the effect of increasing the basis from triplet zeta to quadruple zeta. Consider, for example, the four CCSD(T) cases where we have aug-cc-pcvqz results. For these four cases, the median absolute change upon decreasing the basis set size to aug-cc-pvqz is Å, whereas the median absolute change upon decreasing the basis set size to aug-cc-pcvtz is only Å. For comparison, the median absolute error in decreasing the basis set size all the way to aug-cc-pvtz is Å. 0, 1 Table 4.3 shows the cohesive energies per atom for Mg, n = 2 4. As expected, the cohesive energy of the weakly bound magnesium dimer is extremely small, but even the smallest changes in the cluster size, from Mg2 to Mg3 and Mg4, and small changes in the charge state, from the neutral Mg2 to Mg2 + and Mg2 ions, result in massive changes in cohesive energies. For example, the cohesive energies per atom characterizing Mg + 2, Mg 2, and Mg 3 are approximately 24, 8, and 4 times larger than in the case of Mg2. Combining these results with the results in Table 4.2, which shows that the CCSD(T) optimizations for the 0,±1 Mg n species with n = 2 and 3 other than Mg2 (including Mg3, which represents a transition from the van der Waals to covalent bonding) agree much better with higher-level CC methods, such as CR-CC(2,3) and CCSDT, than in the case of the Mg2 dimer, with the results presented in Table 4.3 confirms our earlier observation that it is only the weakly bound Mg2 species that cannot be accurately described by CCSD(T). For all other species listed in Tables 4.2 and 4.3, CCSD(T) is in n 59

71 almost perfect agreement with the available CR-CC(2,3) and CCSDT data. The previous studies reported in Refs. 181 and 185 exploited CCSD(T) as well. As already alluded to above, Ref. 181 reported all-electron CCSD(T)/aug-cc-pVQZ calculations, but, as pointed out in the previous paragraph, such calculations tend to underestimate the Mg Mg bond distances, overestimating the cohesive energies as a consequence. Ref. 185 divided the contributions to the cohesive energy for Mg4 into three portions, namely ev extrapolated from a series of calculations using the cc-pvnz (n = T, Q, 5) basis sets augmented by additional diffuse functions, ev representing an estimate of the core-valence correlation effects, and ev representing an estimate of higherorder, post-ccsd(t), correlation effects. The sum of the three contributions gives ev per atom, with the uncertainty estimated at ±0.015 ev. We can see that our allelectron CCSD(T)/aug-cc-pCVTZ value of the cohesive energy per atom in Mg4 agrees with this result almost perfectly. The geometries of Mg5-7 clusters are optimized with PWB6K/aug-cc-pCVTZ, and the cohesive energies calculated with CCSD(T) are shown in Table 4.4. Since the augcc-pcvtz basis set is expensive for large clusters while the core-valence correlation is not negligible, we correct the CCSD(T)/aug-cc-pVTZ energy with core-valence (CV) correlation correction defined as CV = E[CCSD(T)/cc-pCVDZ] E[CCSD(T)/cc-pVDZ] (4.1) Table 4.4 shows that this approach reproduced the cohesive energies for Mg 5, Mg + 5, Mg 5 and Mg 6 very well, so it is safe to be used to obtain the benchmark method for the rest of the clusters. 60

72 We also calculated the NBO bond orders 76,252,253,254 of the shortest bonds in neutral Mg2-7 clusters corresponding to a PWB6K wave function (see Table 4.5). Mg2 and Mg3 have very small bond orders, so their bonds, especially in the case of Mg2, are close to van der Waals type, but the shortest bonds in Mg4-7 are more like regular covalent bonds, because their bond orders are close to 1. The transition from van der Waals to covalent bonding is consistent with our previous discussion. We have also calculated the adiabatic ionization potential (E(Mg + n ) E(Mg n )) and the adiabatic electron affinity (E(Mg n ) E(Mg n )) of the magnesium clusters. The adiabatic ionization potential (IP) measures the energy difference between the ground state of the neutral and the ionized cluster at their optimized geometries. The adiabatic electron affinity (EA) is calculated as the difference in the total energies between the ground state of the anion and the neutral cluster, again with each system at its equilibrium geometry and with no vibrational energy contributions. We find the EA for Mg to be a small negative value, which means Mg is an unstable species (resonance), so we do not use it as a benchmark. The results are shown in Tables 4.6 and 4.7. In conclusion, we tested various CC methods and correlation-consistent basis sets. We find that CCSD(T) is not accurate enough for optimizing the geometry of Mg2 and determining its reliable energetics. Because of very weak bonding in Mg2, higher-level CC methods are needed to improve the CCSD(T) results in this case. On the other hand, CCSD(T) seems reliable for larger clusters. All the CC calculations are sensitive to the quality of the basis set; in particular, the core-valence correlation effect is large and should be taken into account. 61

73 Next, we will use the best available data as the benchmarks for testing and validating xc functionals, which means we will use experimental data if it s available, and when there is no experimental data we will use data calculated with the highest-level CC method and the largest basis set. The benchmark data include the bond lengths of neutral and charged Mg2-4, cohesive energies of neutral and charged Mg2-7, and adiabatic ionization potentials of Mg1-7 and adiabatic electron affinities of Mg Kohn-Sham Density Functional Calculations First, we will examine the accuracy of xc functionals for geometries. Table 4.8 shows + the mean unsigned error (MUE) for bond lengths of neutral and ionic Mg2-4. Only Mg 4 has two unique bond lengths and each of the other eight clusters has only one unique bond length, so the MUE is averaged over 10 bond lengths. The table shows that hybrid functionals generally perform better than local functionals. There are 23 hybrid functionals and 16 local functionals in total, but the top 10 are all hybrid functionals. As examples of the improvement offered by Hartree-Fock exchange, the MUE of PBE0 is Å less than that of PBE, the MUE of B3LYP is Å less than that of BLYP, and the MUE of M06-2X is Å less than M06-L. Table 4.9 shows the MUEs of cohesive energies, adiabatic ionization potentials, and adiabatic electronic affinities and the total MUE averaged across all three categories of 31 energies. According to the total MUE, hybrid functionals are generally better again; in particular the table shows that seven of the ten top functionals are hybrid functionals. Furthermore those hybrid functionals that are good for energies are also usually good for 62

74 geometries. Four hybrid functionals, namely, PWB6K, PW6B95, M11, and SOGGA11- X, are among the top six in both Table 4.8 and Table Conclusions Several CC methods with polarized valence and polarized core valence correlationconsistent basis sets were employed to calculate geometries and energies of neutral and singly charged magnesium clusters, Mgn, n = 1 7. The methods included CCSD(T) (all species), CCSD(2)T and CR-CC(2,3) (n = 1 3 species), and CCSDt, CC{t;3}, and CCSDT (n = 1 and 2 species). We have found that CCSD(T) is not accurate enough for providing accurate geometry and binding energy of Mg2, which is a challenging weakly bound species that needs higher-level ab initio treatments, such as those represented by CR-CC(2,3), CC(t;3), and full CCSDT, but CCSD(T) is a reasonable method for obtaining accurate benchmark information for the geometries and energetics of larger magnesium clusters with three or more atoms that have an increasingly covalent character. We have also demonstrated that core polarization effects are important. For example, for the four cases where we calculated the valence CCSD(T)/aug-cc-pVQZ as well as all-electron CCSD(T)/aug-cc-pCVQZ geometries, the median difference in Mg- Mg distance was found to be 0.03 Å. Our calculations clearly indicate that hybrid exchange-correlation functionals perform better than local functionals for both geometries and energies. The overall best functionals of the 39 tested in this work are PW6B95, SOGGA11-X, M11 and PWB6K, which are among the top six functionals in accuracy for both geometries and energies. 63

75 4.5. Acknowledgments I am grateful to all of my collaborators in this work: Orlando Roberto-Neto, Francisco B. C. Machado, Jared A. Hansen, Jun Shen and Piotr Piecuch. This work is supported as part of the Inorganometallic Catalysis Design Center, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences under award DE-SC , and the Chemical Sciences, Geosciences and Biosciences Division, Office of Basic Energy Sciences, Office of Science, U.S. Department of Energy under award DE-FG02-01ER ORN and FBCM wish to thank the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) and Fundação de Amparo àpesquisa do Estado de São Paulo (FAPESP). Computational resources were provided by the University of Minnesota Supercomputing Insitute and by the Molecular Science Computing Facility in the William R. Wiley Environmental Molecular Sciences Laboratory of Pacific Northwest National Laboratory sponsored by the U. S. Department of Energy. 64

76 Table 4.1. Bond Length (Å) of Mg2 Optimized with Coupled-Cluster Methods. a basis set CCSD(T) CCSDt{2,2} CCSDt{2,6} CCSD(2)T aug-cc-pvtz aug-cc-pvqz aug-cc-pcvtz basis set CR-CC(2,3) CC(t;3){2,2} CC(t;3){2,6} CCSDT aug-cc-pvtz aug-cc-pvqz aug-cc-pcvtz a The experimental value is Å

77 Table 4.2. Bond Lengths (Å) of Small Magnesium Clusters Optimized with Coupled- Cluster Methods. method Mg 2 + Mg 2 Mg 2 Mg 3 + Mg 3 Mg 3 Mg 4 CCSD(T)/ aug-cc-pvtz aug-cc-pvqz aug-cc-pcvtz aug-cc-pcvqz aug-cc-pvqz cc-pvtz CR-CC(2,3)/ aug-cc-pcvtz CCSDT/ aug-cc-pcvtz expt method Mg + 4 _R1 Mg + 4 _R2 Mg 4 CCSD(T)/ aug-cc-pvtz aug-cc-pvqz aug-cc-pcvtz

78 Table 4.3. Cohesive Energy per Atom (ev). method Mg 2 + Mg 2 Mg 2 Mg 3 + Mg 3 Mg 3 Mg 4 CCSD(T)/ aug-cc-pvtz aug-cc-pvqz aug-cc-pcvtz aug-cc-pcvqz CR-CC(2,3)/ aug-cc-pcvtz CCSDT/ aug-cc-pcvtz CCSD(T) CCSD(T) ±0.015 expt method + Mg 4 Mg 4 CCSD(T)/ aug-cc-pvtz aug-cc-pvqz aug-cc-pcvtz

79 Table 4.4. Cohesive Energy per Atom (ev). CCSD(T)/ Mg 5 + Mg 5 Mg 5 Mg 6 + Mg 6 Mg 6 Mg 7 + Mg 7 Mg 7 aug-cc-pvtz aug-cc-pvqz aug-cc-pcvtz aug-cc-pvtz +ΔCV

80 Table 4.5. NBO Bond Order of the Shortest Bond. Mg2 Mg3 Mg4 Mg5 Mg6 Mg7 bond order

81 Table 4.6. Adiabatic Ionization Potentials (ev). Mg Mg2 Mg3 Mg4 Mg5 Mg6 Mg7 CCSD(T)/ aug-cc-pvtz aug-cc-pvqz aug-cc-pcvtz aug-cc-pvtz+δcv CR-CC(2,3)/ aug-cc-pcvtz expt

82 Table 4.7. Adiabatic Electron Affinities (ev). Mg2 Mg3 Mg4 Mg5 Mg6 Mg7 CCSD(T)/ aug-cc-pvtz aug-cc-pvqz aug-cc-pcvtz aug-cc-pvtz+δcv CR-CC(2,3)/ aug-cc-pcvtz expt

83 Table 4.8. Percentage of Hartree-Fock Exchange and MUE for Bond Lengths (Å). a functional X b MUE functional X MUE PWB6K τ-hcthhyb PW6B RPBE PBE mpw3lyp HSE PBEsol M B3LYP SOGGA11-X mpwlyp1m M M08-SO M06-2X SOGGA TPSSh mpwlyp B N12-SX BP M PW ωb97x mpwpw BLYP τ-hcth M05-2X PBE N M08-HX MOHLYP revtpss M06-L ωb97x-d GVWN TPSS M06-HF GAM a Listed in order of increasing MUE before rounding b Percentage of Hartree-Fock exchange 72

84 Table 4.9. MUEs for Energetic Quantities (ev). a functional cohesive cohesive IP EA total functional energy energy IP EA total PW6B M05-2X BP TPSSh SOGGA11-X ωb97x M revtpss M08-SO TPSS PWB6K τ-hcth τ-hcthhyb PBEsol mpwpw B3LYP RPBE mpw3lyp M08-HX N12-SX HSE SOGGA PBE GAM PBE mpwlyp PW mpwlyp1m M MOHLYP M06-2X BLYP ωb97x-d M06-HF B M06-L M GVWN N a Listed in order of increasing total MUE before rounding. 73

85 Figure 4.1. Structures of the neutral, cationic, and anionic magnesium clusters. 74

86 Chapter 5 Thermodynamics of Metal Nanoparticles: Energies and Enthalpies of Formation of Magnesium Clusters and Nanoparticles as Large as 1.3 nm 5.1 Introduction In the previous study 256 of magnesium clusters 0,±1 Mg n with n = 1 7 provided equilibrium geometries, binding energies, adiabatic ionization potentials, and adiabatic electron affinities for neutral and singly charged magnesium clusters by using coupledcluster (CC) methods 218 and Kohn-Sham density functional theory (KS-DFT). 1 The most popular CC method, CCSD(T), 141 which includes an iterative treatment of single and double excitations and a quasiperturbative approximation for the connected triple excitations, is widely used for benchmarking closed-shell and high-spin, open-shell polyatomic systems near their equilibrium geometries, and the previous study 256 showed that this method is sufficiently accurate for the geometries of magnesium clusters except the weakly bound Mg2, which requires a fuller treatment of connected triple excitations, such as the CR-CC(2,3), CC(t;3) or CCSDT 225 methods. The cost of CCSD(T) calculations scales steeply with system size, with an asymptotic scaling of N 7, where N is the number of atoms. Kohn-Sham density functional theory is more affordable, scaling asymptotically as N 3 or N 4, depending on the functional and algorithm, but the accuracy depends on the quality of the exchange-correlation (xc) functional. By comparing geometries and energies calculated by the Kohn-Sham density functional method with 39 exchange-correlation functionals to a reliable benchmark data set 75

87 consisting of experimental data and coupled-cluster calculations, it was concluded 256 that hybrid xc functionals generally have higher accuracy for Mg clusters than local ones, with the hybrid functionals PW6B95, SOGGA11-X, M11, and PWB6K being the most accurate on average for geometries and all categories of energies. The asymptotic cost scaling of calculations with hybrid xc functionals is only N 4, so they remain affordable up to very large systems. Nevertheless the calculations remain demanding, in part because the number of isomers of metal clusters increases dramatically as the size of the magnesium cluster grows and makes searches for the lowest-energy structure difficult. 257 The question remains though of whether these conclusions about xc functionals still hold for larger clusters, in which the metal atoms have higher average coordination numbers. To answer the question, we here extend the benchmark data to larger magnesium clusters. Since CCSD(T) has historically been too expensive to be applied to larger clusters, we need to use efficient strategies to extend its applicability. A number of different kinds of strategies have been exploited for this purpose; this includes fragmentation of molecular orbitals (FMO), 258,259 the divide-and-conquer scheme, the cluster-in-molecule method (CIM) with recently introduced variants DECIM and SECIM, 263,264,265 localized-molecular-orbital correlation methods, and the incremental scheme of Stoll, 272 which is based on the Bethe-Goldstone expansion, 273,274,275 and is conceptually similar to the FMO scheme. Here we use a CCSD(T) scheme with an MP2/CBS correction on clusters as large as Mg19, which has a diameter of 1.05 nm (the diameter is defined as the largest Mg-Mg distance in the cluster plus two times the van der Waals radius (1.73 Å) 276 of Mg) and hence is in the realm of 76

88 nanoparticles. The resulting CCSD(T) energetics are not only useful as benchmarks for testing xc functionals, but also useful in their own right since thermodynamic properties of the clusters examined here are not available experimentally, and in fact thermodynamic data on metal nanoparticles is scarce in general. Furthermore, we use the best-validated density functional from the studies of Mg2 Mg19 to produce reliable thermodynamic data for an even larger nanoparticle, namely Mg28. We calculated the singlet-to-triplet excitation energies for all clusters and found they are all positive, meaning singlet is the ground spin state for them, so all the clusters studied in the paper are closed-shell singlets Computational Methods The density functional calculations for n = 2 7 clusters are taken from a previous paper, 256 and the data for n 8 are new in this work. The previous work 256 showed that PWB6K is the best xc functional for optimizing the geometries of magnesium clusters, with a mean unsigned error for bond lengths of only Å with the aug-cc-pcvtz 277 basis set. We find we can obtain very close density functional bond lengths for neutral clusters with the smaller cc-pvdz basis set so we decide to optimize the geometries of Mg8-19 at PWB6K/cc-pVDZ level, then calculate the cohesive energies with various xc functionals and the cc-pvtz basis set. To provide thermodynamic reference data, we also calculate the enthalpies of formation. The electronic energies are taken from the best available benchmark data, and the zero-point energies are calculated using KS-DFT method with PWB6K functional at the pressure of 1 bar and scaled by using the scale factor obtained from the ZPVE15 77

89 database in ref So the enthalpy of atomization at 0 K can be calculated by adding the zero-point energies to the electronic energies, and the heat of formation can be calculated by subtracting the enthalpy of atomization from n times the enthalpy of formation of one Mg atom (34.87 kcal/mol per atom 279,280 ). All density functional calculations are carried out with a locally modified version, MN-GFM6.4, of Gaussian ,96 The coupled cluster and MP2 calculations were performed with TURBOMOLE Results and Discussions Geometries of Magnesium Clusters The geometries for Mgn with n = 2 7 are well determined from earlier experimental and theoretical studies and are taken from ref. 256; however they are not well established for n > 7. In the present work for n = 8 19, we searched for the lowestenergy isomers by optimizing clusters with various initial guesses for the geometries reported in the previous theoretical literature on magnesium clusters. 175,206,207,214 The resulting most stable geometries that we found are shown in Figure 5.1. One noteworthy issue is that the geometries of Mg16 and Mg19 in the literature have Cs and C2v symmetries, but we find there are imaginary frequencies in those high-symmetry geometries that lead to small distortions to lower-symmetry geometries. To give perspective about the sizes of the clusters, we show the diameters as a function of n in Figure 5.2. We can see that the diameter increases non-monotonically with size because some clusters with larger n have more closely packed geometries than some clusters with smaller n. The diameters of all n 11 clusters are more than 1 nm. 78

90 Cohesive Energy Benchmarking We found a straightforward coupled cluster treatment to be very difficult, due to the basis set requirements. Medium-sized basis sets suffered significantly from basis set effects and only very large core-valence basis sets of enormous size including the correlation of the 2s 2p orbitals in the correlation treatment could provide accurate results. After studying a variety of treatments, we decided to use a method we shall denote as CC+ΔMP2, where the energy E is given by the following formula E = E(CCSD(T)/cc-pVDZ) E(MP2/cc-pVDZ) + E(MP2-F12/cc-pCVQZ-F12) (5.1) where MP2-F involves the addition of explicitly correlated two-electron basis functions to a conventional one-electron basis in a Moller-Plesset 215 2nd order perturbation theory calculation, and cc-pvdz and cc-pcvqz-f12 denote the oneelectron basis sets. Note that basis sets with pv in the name are designed for treating polarization and correlation of only the valence electrons, whereas basis sets with pcv in the name are designed for treating the polarization and correlation of both core and valence electrons. Accordingly we only correlate a single orbital per Mg atom in the coupled cluster treatment; the core electrons are correlated at the MP2 level; therefore it is very efficient. With this approach it is possible to overcome the problem with the slow basis set convergence of coupled cluster methods. We tested CC+ΔMP2 against full CCSD(T)(F12*)/cc-pCVQZ-F12 calculations for Mg2-Mg7, where the CCSD(T)(F12*) method 283 involves the addition of explicitly correlated two-electron basis functions to a conventional one-electron basis in a CCSD(T) calculation. We found an unsigned deviation (between CC+ΔMP2 and CCSD(T)(F12*)/cc-pCVQZ-F12) in the cohesive energies per atom of only 0.05 ev. 79

91 Table 5.1 compares CCSD(T)/aug-cc-pCVTZ [where "basis sets with both aug and pcv in the name contain a large number of diffuse basis functions as well as core and valence basis functions], CCSD(T)(F12*)/cc-pCVQZ-F12, and CC+ΔMP2 to experimental data for Mg2 and to higher-level CR-CC(2,3)/aug-cc-pCVTZ data for Mg3. The CCSD(T)/aug-cc-pCVTZ and CCSD(T)(F12*)/cc-pCVQZ-F12, calculations were carried out with only the 1s orbitals frozen. We found empirically that CCSD(T)/aug-ccpCVTZ agrees best with the most accurate data. If we compare the most affordable method, CC+ΔMP2, to experimental data for Mg2, to CR-CC(2,3) data for Mg3, and to CCSD(T)/aug-cc-pCVTZ for Mg4 7, we find a mean unsigned deviation of only 0.41 kcal/mol. We expect most density functionals to have errors larger than this, so this method should be accurate enough to benchmark density functional calculations for larger clusters Benchmark Cohesive Energies for n = 8 19 Table 5.2 gives the cohesive energy per atom for Mg8-19 calculated by CC+ΔMP2 method (except Mg11 is omitted because of convergence difficulty in the wave function calculations). The data in Table 5.2 is to be used as benchmark data for validating the accuracy of xc functionals. Table 5.3 shows the results for enthalpies of atomization and heats of formation, which are from the CC+ΔMP2 electronic structure calculations (except Mg11, whose electronic energy is from PW6B95 calculation) with vibrational contributions obtained from scaled 278 density functional frequencies Density Functional Calculations 80

92 Details of the methods and exchange-correlation functionals used in the density functional calculations are given in the Methods section at the end of the paper. In the previous studies 256 0,±1 on Mg n, n = 1 7, we evaluated xc functionals on the basis of 31 0, 1 energetic data in three categories: cohesive energies per atom for Mg, n = 1 7, n adiabatic ionization potentials (IP) for Mg1-7 and adiabatic electron affinities (EA) for Mg2-7, and we also considered bond length prediction. We concluded that PW6B95, SOGGA11-X, M11, and PWB6K are among the top six xc functionals for both geometries and energies, and thus our expectation prior to the present work was that they are the best choices of functionals for Mg clusters. In the present article we focus on cohesive energies for larger neutral magnesium clusters and nanoparticles. For each of 39 xc functionals, we evaluated the mean signed error (MSE) and mean unsigned error (MUE) in cohesive energy per atom for all neutral clusters and nanoparticles with n = The 39 functionals are all defined with references in the Methods section; some of the functionals are local and in particular have no Hartree-Fock exchange; other functionals are nonlocal because they replace some of the local exchange with nonlocal Hartree-Fock exchange. The results of the testing and the percentage X of Hartree-Fock exchange for each functional are in Table 5.4. The functionals are listed in order of increasing MUE. From Table 5.4, we can see that hybrid functionals (i.e., those with nonzero X) are generally more accurate than local ones; 12 of the top 15 functionals are hybrid functionals. The hybrid functionals PW6B95, PWB6K, and SOGGA11-X, whose good performance on geometries and energies of both neutral and charged smaller clusters is mentioned above, are also distinguished here as being among the top 6 in Table 5.4. This 81

93 makes them the final overall best choices for calculations on Mg clusters and nanoparticles. The M11 functional is ranked 12th in Table 5.4, but, considering the performance for geometries in previous study, we still recommend it as a good functional for Mg clusters. If one is only interested in energetics on the larger clusters, we also note the good performance of RPBE, BP86, and mpwpw91. Note that for most functionals in Table 5.4, the MSE equals the MUE. This implies the same sign of the error for every size cluster. We note that SOGGA11-X is a very simple functional, with the same ingredients as the popular B3LYP functional; i.e., it is a hybrid GGA, which means it includes density, density gradient, and Hartree-Fock exchange. In contrast, PW6B95, PWB6K, and M11 are hybrid meta functionals, which also include kinetic energy density Mg28 Here we use the top two functionals of Table 5.4 to predict thermodynamic data for a large cluster, Mg28, with a C6v-symmetry geometry 175 and a diameter of 1.30 nm. The enthalpy of formation calculated at 0 K with RPBE and PW6B95 are kcal/mol and kcal/mol, and the heats of formation are kcal/mol and kcal/mol respectively. To tighten up the prediction, we consider Figure 5.3 and analyze the mean signed errors (MSEs) for cohesive energy per atom for these two functionals. The MSE for PW6B95 is 0.55 kcal/mol for n =2 19 clusters, exactly the same as the MUE, which means PW6B95 always overestimates the cohesive energies. The MUE and MSE for RPBE are 0.45 kcal/mol and 0.43 kcal/mol for n =2 9 clusters, however they are

94 kcal/mol and kcal/mol for n =10 19 clusters, which means RPBE almost always overestimates cohesive energies for small clusters but systematically underestimates cohesive energies for large clusters. For n = 17 19, we find that E = 0.6E(PW6B95) + 0.4E(RPBE) (5.2) has a mean signed error of only 0.2 kcal/mol. Using this semiempirical linear combination, we estimate the heat of formation of Mg28 to be 424 kcal/mol with an estimated uncertainty of 13 kcal/mol based on the variation of the optimum coefficients with n Conclusions. This study begins by validating the methodology. First, to determine whether the best xc functionals for small magnesium clusters are also accurate for large clusters, we extended the benchmark database by performing CCSD(T) calculations and an MP2/CBS correction. We show that the CCSD(T) scheme with an MP2/CBS correction can reproduce accurate cohesive energies for small magnesium clusters, and this scheme is also applicable to nanoscale particles. Second, by comparing Kohn Sham density functional calculations with the extended benchmark database, we find, similar to our previous finding on small clusters, that hybrid functionals are generally more accurate than local functionals for energies; in particular, 12 of the 15 top-performing functionals are hybrid functionals. The hybrid functionals PW6B95, PWB6K, and SOGGA11-X are the overall most highly recommended for geometries and energies of Mg clusters; M11 is less accurate for energies but still a recommended functional for Mg clusters. Then we apply the validated methods to determine heats of formation that are unavailable from experiment for large clusters and small nanoparticles of Mg, in particular for Mg3 83

95 Mg19 and Mg28. Reliable thermodynamic data on size-selected metal nanoparticles is hard to come by, either experimentally or theoretically, but it is badly needed to support applications in catalysis, electrochemistry, and other technologies. The work reported here obtains such data by using a combination of high-level wave function calculations and powerful modern density functionals; these methods open new capabilities for nanothermodynamics. We believe that this is the first time that such accurate thermodynamic data has ever been obtained for metal nanoparticles, either theoretically to experimentally, and it shows the ability of theory to contribute quantitatively to the emerging field of theoretical design of single-site catalysts of size-selected metal nanoparticles Acknowledgments I am grateful for my collaborator Joachim Friedrich. This work is supported as part of the Inorganometallic Catalysis Design Center, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences under Award DE-SC We also acknowledge the use of computer resources at Minnesota Supercomputing Institute and NERSC. 84

96 Table 5.1. Cohesive Energy per Atom (kcal/mol). a method Mg2 Mg3 Mg4 Mg5 Mg6 Mg7 CR-CC(2,3)/aug-cc-pCVTZ CCSD(T)/aug-cc-pCVTZ CCSD(T)(F12*)/cc-pCVQZ-F CC+ΔMP expt a All values in this table are based on equilibrium dissociation energies without vibrational contributions. 85

97 Table 5.2. Cohesive Energy per Atom (kcal/mol) Calculated a with the CC + ΔMP2 Method. cluster Mg8 Mg9 Mg10 Mg12 Mg13 Mg14 Mg15 Mg16 Mg17 Mg18 Mg19 cohesive energy a All values in this table are based on equilibrium dissociation energies without vibrational contributions. 86

98 Table 5.3. Enthalpy of Atomization and Heat of Formation (kcal/mol) at 0 K and 1 bar. cluster Mg2 Mg3 Mg4 Mg5 Mg6 Mg7 enthalpy of atomization heat of formation cluster Mg8 Mg9 Mg10 Mg11 Mg12 Mg13 enthalpy of atomization heat of formation cluster Mg14 Mg15 Mg16 Mg17 Mg18 Mg19 enthalpy of atomization heat of formation

99 a Table 5.4. MUE and MSE (kcal/mol) for Cohesive Energies per Atom of Mg2-19. functional X b MUE MSE functional X b MUE MSE PW6B B RPBE M BP TPSS PWB6K revtpss mpwpw M05-2X SOGGA11-X M06-HF HSE N PBE GAM M PBEsol M08-SO MOHLYP M08-HX M06-L M MPW3LYP τ-hcthhyb N12-SX ωb97x-d mpwlyp M06-2X B3LYP τ-hcth mpwlyp1m PW SOGGA PBE GVWN TPSSh BLYP ωb97x a Functionals are listed in order of increasing MUE before rounding. All values in this table are based on equilibrium dissociation energies without vibrational contributions. b Percentage of Hartree-Fock exchange; when a range is indicated the first value applies to small interelectronic distances and the second to large ones. 88

100 Figure 5.1. Geometries of Mgn, n = 2 19 and

101 Figure 5.2. Diameters of Mgn, n = 2 19 and

102 Figure 5.3. Cohesive energy per atom of Mgn. 91