Relativistic ab Initio Accurate Atomic Minimal Basis Sets: Quantitative

Apr 13, 2017 - basis set that can be transformed into orthogonal oriented quasi- atomic orbitals .... was used for the non- relativistic AAMBS orbital...
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Relativistic ab Initio Accurate Atomic Minimal Basis Sets: Quantitative LUMOs and Oriented Quasi-Atomic Orbitals for the Elements Li−Xe George Schoendorff,*,† Aaron C. West,‡ Michael W. Schmidt,‡ Klaus Ruedenberg,‡ Angela K. Wilson,† and Mark S. Gordon‡ †

Department of Chemistry, Michigan State University, East Lansing, Michigan 48824-1322, United States Department of Chemistry, Iowa State University, Ames, Iowa 50011-3111, United States

‡

S Supporting Information *

ABSTRACT: Valence virtual orbitals (VVOs) are a quantitative and basis set independent method for extracting chemically meaningful lowest unoccupied molecular orbitals (LUMOs). The VVOs are formed based on a singular value decomposition (SVD) with respect to precomputed and internally stored ab initio accurate atomic minimal basis sets (AAMBS) for the atoms. The occupied molecular orbitals and VVOs together form a minimal basis set that can be transformed into orthogonal oriented quasiatomic orbitals (OQUAOs) that provide a quantitative description of the bonding in a molecular environment. In the present work, relativistic AAMBS are developed that span the full valence orbital space. The impact of using full valence AAMBS for the formation of the VVOs and OQUAOs and the resulting bonding analysis is demonstrated with applications to the cuprous chloride, scandium monofluoride, and nickel silicide diatomic molecules. virtual orbitals (IVOs),3 modified virtual orbitals (MVOs),4 and averaged virtual orbitals (AVOs)5 based on diagonalization of exchange-like or Fock-like operators in the canonical virtual orbital space.6,7 Alternatively, low occupancy orbitals resembling LUMOs may be obtained from diagonalization of the density matrix obtained from post-Hartree−Fock methods such as MP28,9 or CISD,10−12 albeit at an increased cost due to the inclusion of at least double excitations in the correlated method. An alternative and computationally economical method is to extract orbitals with atomic-like character from the occupied and virtual orbital spaces from a molecular calculation. The present work concerns two types of modified orbitals: valence virtual orbitals (VVOs) and orthogonal oriented quasi-atomic orbitals (OQUAOs). The VVOs are predominantly molecular in nature and supplement the wave function optimized occupied orbitals to form a complete set of minimal basis set (MBS) orbitals.13−15 In order to extract the remaining unoccupied MBS orbitals, orbital overlaps are formed between the orthogonalized ab initio accurate atomic minimal basis set (AAMBS) orbitals and the virtual orbitals of the energy optimized wave function. These overlaps between the virtual orbitals and the AAMBS are diagonalized via a singular value decomposition (SVD).16−19 The resulting transformation yields the remainder of the MBS orbitals known as the valence virtual orbitals (VVOs) as described in refs 13 and 15. The other

1. INTRODUCTION The concept of orbitals is fundamental to chemists’ understanding of the reactivity and general behavior of any chemical system. Frontier orbitals such as the highest occupied and lowest unoccupied molecular orbitals (HOMOs and LUMOs) are often used to rationalize chemical behavior.1,2 The conventional picture of atomic orbitals is based upon single particle solutions of the Schrödinger equation, and the resulting atomic orbitals are used to construct molecular orbitals through a linear combination of atomic orbitals (LCAO). Molecular quantum mechanical computational methods usually use atomcentered basis functions to construct the atomic orbitals and subsequent molecular orbitals. The core orbitals often are atomic-like due to their small radial extent relative to the valence atomic orbitals, while the valence orbitals tend to become delocalized across the molecule upon the formation of molecular orbitals and subsequent canonicalization. The canonical orbitals obtained from molecular quantum mechanical methods provide a well-defined HOMO, whereas the LUMOs are not well-defined. Rather, the LUMOs are basis set dependent since they are composed of the leftover basis functions and constrained to be orthogonal to the occupied canonical orbitals. As the size of the basis set increases, the canonical LUMOs become increasingly diffuse with an orbital energy that approaches zero. The canonical LUMOs are thus not defined to give an atomic-like appearance, nor do they have chemically meaningful orbital energies. A number of methods have been proposed to recover chemically meaningful LUMOs from the canonical virtual orbitals including the improved © XXXX American Chemical Society

Received: February 27, 2017 Revised: April 12, 2017 Published: April 13, 2017 A

DOI: 10.1021/acs.jpca.7b01916 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry A

each Russell−Saunders term was weighted equally and 5-fold radial degeneracy was imposed on the d orbitals. Often, the terms included in the state-averaged densities of the transition metals have differing multiplicities as well as differing orbital occupations. Thus, state-averaging over the Russell−Saunders terms was achieved using the full configuration interaction (CI), determinant-based30 MCSCF program in GAMESS.22−24 Orbital occupancy was enforced using the occupation restricted multiple active space (ORMAS) method.31,32 When computing the state-averaged density all possible determinants were allowed to mix, although only the selected L−S terms were included in the density. This inclusion of all determinants results in multireference character being incorporated in the state-averaged density. For example, only the d7s2 and d8s1 determinants of cobalt were computed while the d9 determinants were neglected. This results in 4F, 4P, 2G, 2 2 P, H, and 2D terms arising from the d7s2 configuration and 4F, 4 P, 2F, 2D, 2P, and 2G terms arising from the d8s1 configuration. While all possible determinants were computed for the selected L−S terms through the orbital occupation restrictions of the ORMAS method, only the 4F states arising from each configuration were included in the state-averaged density. Thus, both 4F states include multiconfigurational mixing of the d7s2 and d8s1 configurations. The ground and low-lying excited states used in the SAMCSCF calculations are shown in Table 1.33 When scalar relativistic effects are included in the calculations, the computed ground states always correspond to the experimentally determined ground states as shown in Table 1. With the exception of the group 11 and group 12 metals, at least one excited term was included in the state-averaged computations. However, the inclusion of scalar relativistic effects results in additional Russell−Saunders terms that are close in energy to the excited states or even lower in energy than the excited states used in the prior nonrelativistic implementation of the AAMBS.15 The energy separation ΔE of the lowest J value of the ground state and the lowest J value of the highest energy excited state used in the state averaging is also given in Table 1. The highest energy separations occur for manganese, nickel, and palladium. The 49 kcal mol−1 energy separation between the ground and excited state of manganese is due to the stability of the half-filled 3d shell of the 3d54s2 ground state configuration compared to the 3d64s1 configuration of the first excited state. The large energy separation for nickel and palladium is due to the inclusion of the 3P term corresponding to an excited (n − 1)d8s2 configuration in the state-averaged density. The unoccupied valence np orbitals were optimized in a second step after the energy optimization of the occupied AAMBS orbitals was completed. The optimized orbitals were used for a subsequent ORMAS SA-MCSCF calculation wherein an electron was promoted to the np orbitals. During the optimization of the np orbitals, the previously optimized occupied orbitals were frozen so that the optimization of the np orbitals was not at the expense of the quality of the occupied orbitals. For the given atoms, the lowest-lying virtual canonical orbitals of the neutral atoms coincidentally correspond to the np orbitals, providing a suitable choice of initial np orbitals for the optimization of the remaining valence orbitals. Details of the L−S terms and configurations used in the optimization of the valence np orbitals are shown in Table 2. In most cases, an electron was taken from the ns orbital of the ground state configuration and promoted to the np orbital

orbitals, OQUAOs, are minimal basis set orbitals that are constructed with the West, Schmidt, Gordon, and Ruedenberg (WSGR) procedure outlined in the Methods section.20,21 The resulting OQUAOs are atomic-like orbitals that result from minimizing the number of significant off-diagonal terms in the density matrix. The original implementation of the AAMBS orbitals used to construct the VVOs suggested that during a molecular computation additional “on-the-fly” atomic energy minimizations be performed for each atom in order to provide atomic orbitals in the given atomic orbital (AO) basis set.13,14 However, the AAMBS orbitals were subsequently developed using a large even-tempered Gaussian basis and internally stored in arrays in code format.15 Using this large Gaussian basis provides highly accurate atomic orbitals, which are mapped onto orbitals in the working basis of the given molecular computation through relatively inexpensive overlap integrals. The basis of the given computation is referred to as the working basis. The even-tempered Gaussian basis set was used for the nonrelativistic AAMBS orbitals for H−Ar, while a well-tempered basis set (WTBS) was used for the nonrelativistic AAMBS orbitals for K−Xe. These nonrelativistic AAMBS orbitals for H−Xe have been previously implemented for use in General Atomic and Molecular Electronic Structure System (GAMESS).15,22−25 The nonrelativistic AAMBS orbitals include the following valence orbital subspaces: ns for the sblock metals, ns and (n − 1)d for the transition metals, and ns and np for the p-block elements. In contrast, the AAMBS orbitals presented here are relativistic orbitals that span the full valence space.

2. METHODS 2.1. AAMBS Orbitals. A relativistic version of the AAMBS orbitals has been developed with a full valence space for all elements Li−Xe. Scalar relativistic effects are included via the infinite order two-component relativistic method.26−29 The pblock AAMBS orbitals were obtained with open- or closed-shell self-consistent-field (SCF) or generalized valence bond (GVB) calculations for the neutral atoms. Since there is little variation in the radial extent of the p orbitals due to L−S coupling for open shell np m configurations, only the ground state configurations were computed enforcing radial degeneracy for the p orbitals. Thus, the 2P, 3P, 4S, 3P, 2P, and 1S states were computed for the icosagens, crystallogens, pnictogens, chalcogens, halogens, and noble gases, respectively. Because of the unoccupied valence p orbitals of the neutral transition metal elements, the AAMBS orbitals for the transition metals were developed in a two-step process. The occupied valence orbitals, i.e., ns and (n − 1)d, as well as the core orbitals were optimized, followed by optimization of the unoccupied valence np orbitals. In contrast to the p orbitals, the radial extent of the d orbitals of the transition metals is more sensitive to the choice of electronic configuration. Transition metals also tend to have numerous low-lying excited states due to the partial occupation of the d-shell, and these low-lying excited states are known to be relevant for molecular chemistry. Thus, computation of a number of electronic configurations was necessary and spanned the (n − 1)dm, ns2; (n − 1)dm+1, ns1; and (n − 1)dm+2, ns0 configurations when computing the AAMBS core and occupied valence orbitals. To this end, stateaveraged multiconfiguational self-consistent field (SA-MCSCF) calculations were performed to obtain the AAMBS orbitals for the transition metals. State averaging was performed such that B

DOI: 10.1021/acs.jpca.7b01916 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry A Table 1. Summary of the Terms and Configurations Used in the State-Averaged Multiconfigurational Self-Consistent Field (SA-MCSCF) Calculations To Optimize the Occupied Atomic Orbitals of the Transition Metal Atomsa atom

ground state

Sc Ti V Cr

2

D d1s2 3 F d2s2 4 F d3s2 7 S d5s1

Mn Fe Co Ni

6

Cu Zn Y

2

Zr

3

F d2s2

Nb Mo Tc Ru

6

4 1

Rh Pd

4

Ag Cd

2

S d5s2 D d6s2 4 F d7s2 3 F d8s2 5

S d10s1 S d10s2 2 D d1s2

excited state(s) 4

F d2s1 5 F d3s1 6 D d4s1 5 S d5s1 5 D d4s2 6 D d6s1 5 F d7s1 4 F d8s1 3 D d9s1 3 P d8s2

Table 2. Summary of the Terms and Configurations Used To Optimize the Valence Orbitals Unoccupied in the Atomic Ground State, i.e., the np Orbitals, for the Transition Metalsa

ΔE

atom

33 19 6 22

Sc Ti V Cr Mn Fe Co Ni Cu Zn Yb Zr Nb Mo Tc Ru Rh Pdc Ag Cd

49 20 10 45

1

Dds 7 S d5s1 6 S d5s2 5 F d7s1

F d8s1 1 S d10s0

1

4

2 1

Fds 4 P d2s1 3 P d2s2 5 F d3s1 4 F d3s2 5 D d4s2 6 D d6s1 5 D d6s2 5 P d7s1 2 D d9s0 3 D d9s1 3 F d8s2 3 P d8s2

43 14 3 31 7 23

L−S terms 4

F, 4D, 4P 5 G, 5F, 5D, 6 G, 6D, 6F, 7 P 8 P 7 D, 7F, 7P 6 F, 6D, 6G, 5 D, 5G, 5F, 2 P 3 P 2 P 5 G, 5F, 5D, 6 F, 6P, 6D 7 P 8 P 5 D, 5F, 5G, 4 D, 4G, 4F, 3 P, 3F, 3D 2 P 3 P

5

S, 5D, 5P 6 D, 6P, 6S

6

P, 6D, 6S 5 P, 5D

5

S, 5D, 5P

5

S, 5D, 5P P, 4D, 4S

4

configuration

ΔEEX

ΔEGS

d1s1p1 d2s1p1 d3s1p1 d5s0p1 d5s1p1 d6s1p1 d7s1p1 d8s1p1 d10s0p1 d10s1p1 d0s2p1 d2s1p1 d4s0p1 d5s0p1 d5s1p1 d7s0p1 d8s0p1 d9s0p1 d10s0p1 d10s1p1

8 34 41

53 79 88 67 53 68 152 126 87 92 30 73 74 73 47 111 117 115 84 86

12 84 52

31 5

35 39 18

a In this table, all energy differences are given for the lowest J levels. ΔEEX is the experimental energy difference in kcal mol−1 between the lowest L−S term of the resulting excited states above and the highest energy excited state, and ΔEGS is the experimental energy difference in kcal mol−1 between the ground state and the highest energy state L−S term.33 bThe d0s2p1 configuration was used for Y rather than the d1s1p1 configuration for the optimization of the 5p orbitals since the d0s2p1 configuration is 12.6 kcal mol−1 lower in energy than d1s1p1 configuration. cThe d9s0p1 configuration was used since the ground state of Pd is d10 and thus has no s electron to excite into the p orbitals.

10 100

S d10s1 S d10s2

a

The computed ground state corresponds to the experimentally determined ground state when relativistic effects are included. In this table, all energy differences are given for the lowest J levels. ΔE is the experimental energy separation in kcal mol−1 between the ground state L−S term and the highest energy state L−S term used in the state averaging.33

electrons in the d shell, i.e., elements with 30 determinants from the (n − 1)dxnsy−1np1 configuration. While the energy difference with respect to the ground state is sizable, the energy range of the excited state terms is much smaller and similar to the energy difference between the terms used in the SAMCSCF calculations on the ns and (n − 1)d orbitals (Table 1). Thus, all terms arising from the (n − 1)dxnsy−1np1 configuration were included in the constrained SA-MCSCF optimization of the np orbitals. The AAMBS orbitals for s-block elements were developed in a manner analogous to the method employed for the transition metals. The optimization was a two-step process in which the occupied orbitals of the ground state were energy optimized. The lighter alkali elements then were supplemented by an np shell. For the elements Ca and higher, the ground state was augmented by np and (n − 1)d orbitals. The ground state 2S and 1S terms were computed at the restricted open-shell Hartree−Fock (ROHF) and restricted Hartree−Fock (RHF) levels, respectively. The orbitals optimized in this manner were then frozen in the subsequent SA-MCSCF calculations to produce the np and (n − 1)d orbitals. When computing the excited states, an electron was promoted from the ns orbital to the np and (n − 1)d orbitals. Details of the configurations computed for the optimization of the unoccupied valence orbitals are shown in Table 3 along with the experimentally determined energy differences33 between the ground state and

resulting in a (n − 1)dxnsy−1np1 configuration. Occupying the np orbitals in this manner generally results in the lowest energy terms involving np orbitals. Only two exceptions were made in the choice of configuration for the optimization of the np orbitals. The d0s2p1 configuration was used for yttrium rather than the d1s1p1 configuration since the lowest term resulting from the d0s2p1 configuration is 12.6 kcal mol−1 lower in energy than the lowest term arising from the d1s1p1 configuration. The other exception occurred with palladium wherein the d9s0p1 configuration was used. In this case, the ground state of palladium is d10 and has no electrons in the 5s orbital. The (n − 1)dxnsy−1np1 electronic configuration results in at most 30 determinants giving rise to the L−S terms listed in Table 2. The experimentally determined excited state energy differences33 ΔEEX between the lowest J values of the lowest and highest energy terms resulting from the (n − 1)dxnsy−1np1 configuration are shown in Table 2 as well as the energy difference ΔEGS between the lowest J values of the ground state and the highest energy term arising from the (n − 1)dxnsy−1np1 configuration. The difference in energy with respect to the ground state is usually in excess of 50 kcal mol−1 with the largest differences occurring for elements with 2, 3, 7, or 8 C

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the excited state energies. As the α parameter of the WTBS recursion formula (eqs 1 and 2) is the most diffuse exponent, the WTBS parameters were reoptimized, allowing for sufficiently diffuse exponents to allow convergence of the 2P and 3P states of Na and Mg, respectively.

Table 3. Summary of the Terms and Configurations Used To Optimize the Valence Orbitals Unoccupied in the Atomic Ground State, i.e., the np and (n − 1)d Orbitals, for the sBlocka atom Li Be Na Mg K Ca Rb Sr

L−S terms P s0p1 3 P s1p1 2 P s0p1 3 P s1p1 2 P s0p1 2 D d1s0p0 3 P s1p1 3 D d1s1p0 2 P s0p1 2 D d1s0p0 3 P s1p1 3 D d1s1p0

ΔEEX

ΔEGS

24

43 63 48 62 62

2

ζN = α

⎛ ⎛ k ⎞δ ⎞ ζN − k + 1 = ζN − k + 2β ⎜⎜1 + γ ⎜ ⎟ ⎟⎟ ⎝N⎠ ⎠ ⎝

(1)

k = 2, ..., N (2) 36

15

58

19

55

11

52

When reoptimizing the WTBS parameters, Molpro was used, employing the Broyden−Fletcher−Goldfarb−Shanno (BFGS)37−40 algorithm with an orbital gradient convergence threshold of 10−6 and SCF energy convergence threshold of 10−10 Eh; all atomic orbitals were symmetry equivalent, and spherical harmonic functions were employed. The final step in AAMBS generation is contraction of the WTBS Gaussian functions just described into minimal basis sets. Modifications to the nonrelativistic WTBS exponents are given in the Supporting Information. The AAMBS contractions are easily obtained from state-averaged calculations on the indicated atomic states, carried out with the IOTC relativistic method26−29 for atoms starting at lithium. 2.3. WSGR Localization Procedure. The occupied orbitals of the wave function together with the VVOs form a MBS orbital set, which often represents an internal orbital space.13−15 In order to form quasi-atomic orbitals (QUAOs), the orbital overlaps are formed between the internal orbitals and the intra-atomic orthogonal, interatomic nonorthogonal AAMBS orbitals for each atom. For each atom, these overlaps are diagonalized. Then, in the basis of the given computation, i.e., the working basis, the internal orbitals are transformed to closely approximate the AAMBS orbitals of atom A. The QUAOs for all atoms are then symmetrically orthogonalized (i.e., SO-QUAOs). In order to more closely resemble the AAMBS, the SO-QUAOs are further modified as described in the appendix of a previous paper.21 From the orthogonalized QUAOs, the orthogonal oriented quasi-atomic orbitals (OQUAOs) are formed. The OQUAOs embody the quantitative realization of the orbital hybridization concept. The orientation process uses the first-order density, which encompasses the majority of that quantitative hybridization. As described in refs 42 and 43, the orientation algorithm results in a few large bond order magnitudes that correspond to the dominant interactions between the OQUAOs,42,43 which then actually appear as the hybrid orbitals that a chemist readily interprets as sp, sp2, etc., orbitals. For the OQUAOs, a labeling scheme is employed and lists one to two atomic elements, an orbital symmetry, and an orbital type.44 The OQUAO is largely centered on the first atomic element symbol, which is given in uppercase. If the OQUAO is involved in bonding, then that OQUAO is directed toward and has a substantial bond order with another “complementary” OQUAO. In that case, the labeling scheme lists a second atomic element in lowercase. Lastly, the orbital symmetry and type are given. For example, a σ bonding OQUAO of NH3 centered on nitrogen and directed to hydrogen #1 (H1) would be denoted as Nh1σ, and the complementary OQUAO on H1 would be H1nσ. If the orbital occupation is near 2.0, then the orbital type is denoted as a lone pair S . Radicals are denoted with rd if the occupation of the OQUAO is near 1.0, whereas nonbonding orbitals with a low

a In this table, all energy differences are given for the lowest J levels. ΔEEX is the experimental energy difference in kcal mol−1 between the lowest L−S term of the resulting excited states above and the highest energy excited state, and ΔEGS is the experimental energy difference in kcal mol−1 between the ground state and the highest energy state L−S Term.33

the highest excited state (ΔEGS) and the energy range of the excited states (ΔEEX). The energy range of the excited state terms is less than 25 kcal mol−1, with the 2P and 3P states lower in energy than the 2D and 3D states. Likewise, the energy range of all states included in the development of the AAMBS orbitals for the s-block is 63 kcal mol−1 or less. When (n − 1)d orbitals were required to produce a full valence set of AAMBS orbitals, excited state terms arising from the occupation of the (n − 1)d and np orbitals were state-averaged with equal weights, accounting for the differing degeneracies of the P and D terms. 2.2. Well-Tempered Basis Sets. Details of the basis sets used in the construction of the AAMBS are available in Table S1 of the Supporting Information. The nonrelativistic welltempered basis sets of Huzinaga et al. were employed.25,34 Reoptimization of the WTBS parameters was not required to account for changes in the radial extent of the AAMBS functions due to relativistic effects as it has been shown that the nonrelativistic WTBS parameters are generally suitable for most elements up to Xe as long as enough functions are present to account for the relativistic contraction of the s orbitals.35 Reoptimization of the WTBS parameters was avoided by the extension of the WTBS exponent formula to tighter functions, resulting in saturated basis sets suitable for relativistic calculations. Rb−Xe in particular required additional tight functions that were added until the change in atomic energies of the neutral atoms was less than 1 μEh. As the original WTBS parameters were optimized for the ground states of the neutral atoms, augmentation of s- and d-block metals with diffuse d and p exponents was necessary to properly describe the valence orbitals that are unoccupied in the ground state of the neutral atoms. Atomic energies were computed for the lowest excited states of the atoms that result from promotion of an ns electron into the np or (n − 1)d orbitals of the s- and d-block metals. Diffuse p or d functions were added from those available in the WTBS formula, until the excited state atomic energies were converged to 1 μEh. In the cases of Na and Mg, addition of p primitives with even the most diffuse exponents generated by the original WTBS parameters did not result in convergence of D

DOI: 10.1021/acs.jpca.7b01916 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry A occupation number are denoted as nb. Thus, the lone pair on nitrogen in NH3 would be labeled as NpS while a radical sp hybrid orbital centered on the oxygen of the (2,2,6,6tetramethyl-piperidin-1-yl)oxyl (TEMPO) free radical would be Oprd. The hybridization of each OQUAO can be characterized based on the percent of s, p, and d character. For this purpose, the percent of each orbital type is determined relative to the AAMBS orbitals. A future publication will address specifics to assigning labels and calculating percentages of orbital types.

3. RESULTS AND DISCUSSION Three diatomic molecules are used to illustrate the effects of using AAMBS with a full valence space on the resulting bonding analysis, e.g., by inclusion of the np orbitals into the MBS of the bonding analysis for transition metal species. Cuprous chloride was chosen to illustrate the effects of the np orbitals on the bonding with late transition metals while scandium monofluoride shows the effects of the np orbitals on bonding by early transition metals. Finally, nickel silicide is used to demonstrate the changes in the electronic structure on either side of the shoulder present in the ground state potential energy curve.45 Orbital results presented in the following applications were obtained using the GAMESS22−24 quantum chemistry software package, and all orbitals were drawn with the MacMolPlt46,47 visualization program using a contour value of 0.025 bohr−3/2 for the CuCl orbitals and 0.050 bohr−3/2 for all other orbitals. Calculations on diatomic molecules were performed at the experimentally determined equilibrium bond lengths.48−50 The infinite order two-component (IOTC)26−29 relativistic method was employed to account for scalar relativistic effects. The ccpV(T+d)Z-DK correlation consistent basis sets were used for the 3p elements (Si and Cl)51 while the cc-pVTZ-DK basis sets were used for all other atoms.52−55 RHF calculations were performed for 1Σ+ ground states of CuCl and ScF to illustrate the effect of the full valence space in the AAMBS orbitals on both late and early transition metals. The full optimized reaction space multireference self-consistent field (FORSMCSCF)56−58 method was employed for calculations of the 1 + Σ ground state of NiSi as described previously45 to show the impact of using a full valence space on the interpretation of the bonding and characterization of the shoulder on the ground state potential energy curve. 3.1. 1Σ+ Ground State of Cuprous Chloride. The 1Σ+ ground state of the cuprous chloride diatomic molecule is dominated by an ionic configuration corresponding to Cu+(3dδ43dπ43dσ2)Cl−(3s23pσ23pπ4).59 However, rather than a complete electron transfer from copper to chlorine, CuCl can also be described as a singly bonded system with a σ-bond between a pz or sp hybrid orbital on chlorine and an s or sd hybrid orbital on copper. Both descriptions can be rationalized using a truncated valence space where the copper 4p orbitals play no role. Figure 1A shows a bonding analysis in terms of the OQUAOs in which the copper 4p orbitals are excluded from the active space and the subsequent localization and orientation. There are eight lone pairs corresponding to the full 3d shell on copper (CudS ) as well as the 3pπ orbitals on chlorine (ClpS ) and an sp hybrid orbital on the chlorine atom (ClpS ). A single bond is formed between the 4s orbital on copper and the other sp hybrid orbital on chlorine with a bond order of 0.69. The computed dipole moment of CuCl is 6.18 D, suggesting a largely ionic bond; the occupations of the bonding

Figure 1. Orthogonal oriented quasi-atomic orbitals (OQUAOs) of the 1Σ+ ground state of the cuprous chloride diatomic molecule without (A) and with (B) the inclusion of the 4p orbitals in the accurate atomic minimal basis set (AAMBS) and in the bonding analysis for copper. Occupations are shown beside each orbital, and bond orders are shown between pairs of bonding orbitals.

OQUAOs support a primarily ionic interpretation of the bonding with 1.73 electrons in the chlorine sp bonding orbital and only 0.27 electrons in the copper 4s orbital. The bonding picture changes dramatically upon inclusion of the copper 4p orbitals in the valence space (Figure 1B). When a full valence space is employed, only the copper 3d orbitals remain as lone pairs while all other orbitals in the valence space take part in bonding, resulting in a total of four bonding interactions. The largest bond order remains the σ-bond through sp hybridization on copper with an increased bond order of 0.76. In the absence of the 4p orbitals, the Cuclσ OQUAO is 98% s and 2% d, whereas inclusion of the 4p orbitals in the valence space results in a hybrid orbital that is 72% s, 26% p, and 2% d. An additional σ-bonding interaction also occurs between another pair of OQUAOs with a bond order of 0.31. This pair of OQUAOs largely remains the empty E

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The Journal of Physical Chemistry A CupS orbital with a hybridization of 29% s and 71% p and the lone pair ClpS orbital with a hybridization of 47% s and 53% p. The polarity of this σ-bond, as well as the remaining empty and lone pair character of the participating orbitals, is also reflected in the relative occupations of the OQUAOs with 0.05 electrons in the Cuclσ (Cupnb) orbital and 1.95 electrons in the Clcuσ (CupS ) orbital. The remaining bonding interactions result from degenerate π-bonds between the 4pπ and 3pπ orbitals on copper and chlorine, respectively. While the π electrons remain localized primarily on chlorine with 1.92 electrons in each of the 3pπ orbitals on chlorine (Clcuπ) and 0.08 electrons in each of the 4pπ orbitals on copper (Cuclπ), the diffuse nature of the 4p orbitals reveals additional bonding in the form of degenerate π-bonds with bond orders of 0.40 via back-donation from chlorine to copper. The differences between a truncated valence space and a full valence space are substantial. With a truncated valence space in which the 4p orbitals are omitted, CuCl is predicted to have a single σ-bond with a bond order of 0.69. Addition of the 4p orbitals to make a full valence space results in two σ-bonding interactions and two π-bonding interactions with a total bond order of 1.87. The σ-bonding interactions are enhanced by sp hybridization involving the 4s and the 4pσ orbitals on copper, and the π back-donation is possible only with the inclusion of the 4pπ orbitals on copper in the valence space. 3.2. 1Σ+ Ground State of Scandium Monofluoride. Scandium is an early transition metal, and thus conventional wisdom might suggest that unsaturated scandium complexes do not require the 4p orbitals to adequately describe the molecular ground electronic states. The 1Σ+ ground state of the scandium monofluoride diatomic molecule then can serve as a gauge of the importance of the np orbitals to the electronic structure of early transition metal complexes. However, the lack of saturation of the d-shell results in numerous low-lying excited states that are typical of unsaturated early transition metal complexes. In fact, the first triplet excited state of ScF, the 3Δ state, is so close to the ground state that simply changing the polarity of the bond changes the order of the states as can be seen by comparison with the isoelectronic species TiO.60−66 A bonding analysis based on the OQUAOs of ScF is shown in Figure 2, and an analogous bonding analysis for the 1Σ+ excited state of TiO is included in the Supporting Information. The OQUAOs obtained with a truncated valence space and with a full valence space are shown in Figures 2A and 2B, respectively. In contrast to CuCl the overall interpretation of the bonding in the 1Σ+ ground state of ScF does not change much when the 4p orbitals on scandium are included. In both cases there are four bonding interactions: two σ- and two πbonding interactions. The π-bonds are due to donation to the 3dπ orbitals on scandium (Scfπ) from the 2pπ orbitals on fluorine (Fscπ) with no hybridization of either pair of orbitals. There is a transfer of only 0.07 electrons to each of the 3dπ orbitals on scandium from the 2pπ lone pairs on fluorine, yet each of the resulting π-bonds has a bond order of 0.38. Both of the σ-bonds involve interactions with sp hybrid orbitals on fluorine that interact with sd hybrid orbitals on scandium in the truncated valence space. The sd hybrid orbitals on scandium (Scfσ) are composed of 25% s and 75% d character for the stronger σ-bond (bond order = 0.61) and 75% s and 25% d character for the weaker σ-bond (bond order = 0.11). Because of the large difference in electronegativity between scandium and fluorine, the occupations of the OQUAOs are small for the σ-bonding OQUAOs centered on scandium, e.g., only 0.21

Figure 2. Orthogonal oriented quasi-atomic orbitals (OQUAOs) of the 1Σ+ ground state of the scandium monofluoride diatomic molecule without (A) and with (B) the inclusion of the 4p orbitals in the accurate atomic minimal basis set (AAMBS) and in the bonding analysis for scandium. Occupations are shown beside each orbital, and bond orders are shown between pairs of bonding orbitals. OQUAOs with an occupation number less than 0.01 electrons are not shown.

electrons for the strongest σ-bond. The other Scfσ orbital has an occupation of only 0.01 electrons while its complementary OQUAO FScσ remains essentially a lone pair with an occupation of 1.99 electrons. Inclusion of the 4p orbitals into the AAMBS for scandium (Figure 2B) results in bonding that is qualitatively unchanged overall. However, all of the bonds have increased bond orders due to the hybridization with the 4p orbitals on scandium. For example, the bond order of the strongest σ-bond increases from 0.61 to 0.68. The OQUAO on scandium changes from a hybrid with 25% s and 75% d character to having 22% s, 22% p, and 56% d character. This change in hybridization results in an OQUAO that is significantly polarized toward fluorine. F

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The Journal of Physical Chemistry A Likewise, the bond order of the weaker σ-bond is more than doubled by the inclusion of 62% p character in the OQUAO centered on scandium that is possible with the full valence AAMBS. In fact, this OQUAO changes from an sd hybrid to a pd hybrid when a full valence space is used for the AAMBS, resulting in an OQUAO that is significantly more polarized. The bond orders of the π-bonds are also increased by hybridization of the 3dπ orbitals with the 4pπ orbitals. While the Fscπ OQUAOs on fluorine remain pure p orbitals, the π OQUAOs centered on scandium mix in 28% p character in order to polarize the dπ orbitals on scandium. This hybridization facilitates additional electron transfer from fluorine to scandium, thereby increasing the bond order of each π-bond from 0.38 to 0.45 despite the Scfπ and Fscπ orbitals largely retaining unoccupied and lone pair character, respectively. Thus, while the 4p orbitals on scandium do not directly take part in bonding in such an unsaturated complex, the hybridization that the inclusion of the 4p orbitals in the AAMBS allows results in OQUAOs that are polarized to maximize electron transfer from their complementary Fscσ and Fscπ OQUAOs resulting in increased bond orders. 3.3. Nickel Silicide. The ground and excited states of nickel silicide have been characterized by a number of computational methods,50,67−70 yet only recently has agreement with experiment been obtained.45 A prior bonding analysis of NiSi was performed using the intrinsic localized density analysis (ILDA)42,43 based on Edmiston−Ruedenberg localized orbitals71 at the computed equilibrium geometry of the 1Σ+ ground state.45 It was determined that NiSi had one σ-bond formed from an sd hybrid orbital on nickel and the 3pσ orbital on silicon. Two degenerate π-bonds were also formed between the 3dπ and 3pπ orbitals on nickel and silicon, respectively, resulting in a total bond order of 2.41. Additionally, a shoulder in the potential energy curve was observed and ascribed to a change in the character of the σ-bond wherein the s character of the sd hybrid orbital on nickel increased as the bond was stretched in order to maximize overlap with the 3pσ orbital on silicon. This analysis was performed based on a CASSCF wave function that omitted the 4p orbitals on nickel due to their small occupation numbers at all internuclear distances. However, the previous examples of CuCl and ScF show that non-negligible bond orders can be obtained despite small occupation numbers. Here, the nature of the bonding of the ground state of nickel silicide is re-examined with the inclusion of the 4p orbitals into the active space and also in the AAMBS used for nickel. The bonding analysis is performed at both the computed equilibrium bond length (2.086 Å) and for an internuclear distance of 2.5 Å so that the nature of the bonding on either side of the shoulder in the potential energy curve of the 1Σ+ ground state may be assessed. The OQUAOs for both the computed equilibrium bond length and the stretched bond length of 2.5 Å are shown in Figures 3A and 3B, respectively. At the equilibrium geometry nickel silicide has five nonbonding orbitals. Three of the nonbonding OQUAOs are NidS lone pair orbitals each with occupations in excess of 1.9 electrons. Both of the dδ OQUAOs have 100% d character with no hybridization, while the dσ OQUAO has 10% s character and 90% d character. The other two nonbonding OQUAOs correspond to the 4pπ orbitals on nickel (Nipnb) with each having 90% p character and occupations of 0.06 electrons. As with the prior bonding analysis, the σ-bond consists of an sd hybrid orbital on nickel with 86% s character, while the σ-bonding OQUAO on silicon

Figure 3. Orthogonal oriented quasi-atomic orbitals (OQUAOs) of nickel silicide on either side of the shoulder in the 1Σ+ ground state potential energy curve. The OQUAOs in (A) were computed at the CASSCF ground state equilibrium geometry,45 and the OQUAOs in (B) were computed at an internuclear distance of 2.5 Å. Occupations are shown beside each orbital, and bond orders are shown between pairs of bonding orbitals.

is an sp hybrid with 24% s and 74% p character resulting in a bond with a bond order of 0.95. There is an additional σ-bond formed upon the inclusion of the 4p orbitals in the nickel AAMBS that was not observed previously.45 The Nisiσ OQUAO on nickel is primarily a p orbital with 92% p character while its complementary OQUAO is an sp hybrid orbital with 67% s character on silicon. With the exclusion of the 4p orbitals in the nickel valence space, the Siniσ OQUAO would be a lone pair 3s orbital. However, the hybridization that results from the inclusion of the valence np orbitals in the AAMBS for nickel allows for the transfer of 0.23 electrons to nickel resulting in a second σ-bond with a bond order of 0.61. As with the prior bonding analysis, the π-bonds are formed between pπ orbitals on silicon and dπ orbitals on nickel. However, the Nisiπ dπ orbitals have 9% p character allowing for polarization due to hybridization resulting from the np orbitals G

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The Journal of Physical Chemistry A

(n − 1)d orbitals were included in the AAMBS for alkali and alkaline earth metals resulting in full valence AAMBS for all atoms up to xenon. The AAMBS provide a means for extracting basis set independent and chemically relevant LUMOs from the sea of leftover basis functions from quantum mechanical calculations13−15 and a means to produce orthogonal oriented quasi-atomic orbitals for use in bonding analyses.20,21 In the present work, the relativistic full valence AAMBS were used to characterize the bonding of the cuprous chloride, scandium monofluoride, and nickel silicide diatomic molecules, and a comparison was made with the bonding analysis that results from a truncated valence space when the np orbitals are omitted from the AAMBS. While the np orbitals of the transition metals have small occupations for the particular molecules in the present study, the presence of the np orbitals in the AAMBS for the transition metals allows for increased bond orders due to back-donation of electron density from the ligand to the empty p orbitals on late transition metals as exemplified by the degenerate π bonds in CuCl. Early transition metals may not require the np orbitals for back-donation, but the presence of the p orbitals in the AAMBS increases the effects of hybridization. The ScF molecule illustrates the effects of hybridization with the most pronounced effect being the increased bond orders primarily due to the pd hybridization of the π-bonding OQUAOs centered on scandium, and the bonding analysis of NiSi at the equilibrium and nonequilibrium distances shows that the pd hybridization facilitates the breaking of the π-bonds leading to the 3F ground state of neutral Ni upon dissociation.

in the AAMBS. Overall, the total bond order at the ground state equilibrium geometry of nickel silicide increases from 2.41 to 2.90 upon inclusion of the 4p orbitals in the AAMBS for nickel. This increase in bond order is primarily due to the second σbonding interaction that is made possible by the hybridization of the 4pσ with the 4s orbital on nickel. The OQUAOs at the stretched bond length of 2.5 Å are shown in Figure 3B. The dominant σ-bond maintains a high bond order of 0.83, but this is achieved via sp hybridization of the Nisiσ OQUAO as opposed to sd hybridization due to the larger radial extent of the 4p orbitals compared with the 3d orbitals. Likewise, the second σ-bonding interaction maintains nearly the same bond order as at the equilibrium geometry despite the increased internuclear distance. This result is due to increased back-donation from silicon such that the occupation of the second Nisiσ OQUAO centered on nickel nearly doubles from 0.23 to 0.45 electrons when compared to the occupation at the equilibrium geometry. There are still five nonbonding OQUAOs, but the hybridizations and occupations change significantly beyond the shoulder in the potential energy curve. While there are still the three NidS lone pair orbitals, the two nonbonding p orbitals take on 50% d character and the occupation increases from 0.06 to 0.55 electrons each. The increased occupation of the nonbonding orbitals is a result of electron transfer from the Nisiπ orbitals as the bond is stretched. The largest difference between the bonding at the equilibrium geometry and at the stretched bond length beyond the shoulder occurs with the π-bonds. Just as the nonbonding pπ orbitals on nickel hybridize with the dπ orbitals and increase their occupations, the dπ bonding Nisiπ OQUAOs also increase the degree of hybridization to have 49% p character and 51% d character. This hybridization facilitates interactions with the complementary Siniπ orbitals at long bond lengths due to the increased radial extent of the 4p orbitals relative to the 3d orbitals. The occupations of the π-bonding OQUAOs also change at the stretched bond length; most of the electron density goes into the nonbonding π OQUAOs on nickel rather than to the p orbitals on silicon, with the Ni π nonbonding orbitals having increasing occupation to become NidS lone pair orbitals as the bond is stretched to eventually produce neutral Ni in the 3F ground state with a 3d84s2 configuration. Stretching the Ni−Si bond from the computed equilibrium bond length to 2.5 Å results in a reduction of the bond order of the degenerate π-bonds from 0.67 to 0.38. As the bond is stretched, the nonbonding π orbitals on nickel transform from orbitals with mostly p character to d orbitals while the πbonding OQUAOs transform from having primarily d character to primarily p character as the occupation is reduced. Thus, the 3d84s2 and 3s23p2 ground state configurations of nickel and silicon, respectively, are recovered at the dissociation limit. As with the previous examples of CuCl and ScF, a complete description of molecular bonds in NiSi cannot be made with a truncated valence space as hybridization with the np orbitals is essential to describe the bonding at the equilibrium geometry and especially at nonequilibrium geometries.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpca.7b01916. Well-tempered basis set parameters; bonding analysis of the 1Σ+state of TiO (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (G.S.). ORCID

George Schoendorff: 0000-0001-8624-5217 Klaus Ruedenberg: 0000-0003-4834-0314 Mark S. Gordon: 0000-0001-6893-553X Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS G.S. thanks Tom Cundari for discussions related to the applications presented in this work. M.W.S. thanks Mariusz Klobukowski for providing details about the nonrelativistic WTBS basis sets that were extended to additional valence orbitals in this work. This project was funded by the National Science Foundation under Grants CHE-1362479, DMR1636557, CHE-1147446, and CHE-1565888. Computational resources were provided via the NSF Major Research Instrumentation program under Grant CHE-1531468.

4. CONCLUSIONS Relativistic ab initio AAMBS solutions have been obtained for the elements lithium through xenon. The AAMBS were developed to be relevant in a molecular environment by state averaging over the ground state and low-lying excited states. Additionally, the np orbitals were included in the development of the AAMBS for the transition metals while both the np and

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REFERENCES

(1) Fukui, K.; Yonezawa, T.; Shingu, H. A molecular orbital theory of reactivity in aromatic hydrocarbons. J. Chem. Phys. 1952, 20, 722−725.

H

DOI: 10.1021/acs.jpca.7b01916 J. Phys. Chem. A XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry A (2) Woodward, R. B.; Hoffman, R. The Conservation of Orbital Symmetry; Verlag Chemie GmbH: Weinheim, Germany, 1971. (3) Hunt, W. J.; Goddard, W. A. Excited states of H2O using improved virtual orbitals. Chem. Phys. Lett. 1969, 3, 414−418. (4) Bauschlicher, C. W. The construction of modified virtual orbitals (MVOs) which are suited for configuration interaction calculations. J. Chem. Phys. 1980, 72, 880−885. (5) Mogensen, B. J.; Rettrup, S. Average virtual orbitals in configuration interaction studies with application to the low-lying singlet states of the carbon monoxide and acetone molecules. Int. J. Quantum Chem. 1992, 44, 1045−1056. (6) Huzinaga, S.; Arnau, C. Virtual orbitals in Hartree-Fock theory. J. Chem. Phys. 1971, 54, 1948−1951. (7) Beebe, N. H. F. Modification of virtual orbitals. Int. J. Quantum Chem. 1979, 15, 589−600. (8) Jensen, H. J. A.; Jorgensen, P.; Agren, H.; Olsen, J. Second-order Møller-Plesset perturbation theory as a configuration and orbital generator in multiconfigurational self-consistent field calculation. J. Chem. Phys. 1988, 88, 3834−3839. (9) Caballol, R.; Malrieu, J.-P. Improved non-valence virtual orbitals for CI calculations. Chem. Phys. 1990, 140, 7−18. (10) Bender, C. F.; Davidson, E. R. A natural orbital based energy calculation for helium hydride and lithium hydride. J. Phys. Chem. 1966, 70, 2675−2685. (11) Jafri, J. A.; Whitten, J. L. Iterative natural orbitals for configuration interaction using perturbation theory. Theoret. Chim. Acta 1977, 44, 305−313. (12) Abrams, M. L.; Sherrill, C. D. Natural orbitals as substitutes for optimized orbitals in complete active space wavefunctions. Chem. Phys. Lett. 2004, 395, 227−232. (13) Lu, W. C.; Wang, C. Z.; Schmidt, M. W.; Bytautas, L.; Ho, K. M.; Ruedenberg, K. Molecule intrinsic minimal basis sets. I. Exact resolution of ab initio optimized molecular orbitals in terms of deformed atomic minimal-basis orbitals. J. Chem. Phys. 2004, 120, 2629−2637. (14) Lu, W. C.; Wang, C. Z.; Schmidt, M. W.; Bytautas, L.; Ho, K. M.; Ruedenberg, K. Molecule intrinsic minimal basis sets. II. Bonding analyses for Si4H6 and Si2 to Si10. J. Chem. Phys. 2004, 120, 2638− 2651. (15) Schmidt, M. W.; Hull, E. A.; Windus, T. L. Valence virtual orbitals: An unambiguous ab initio quantification of the LUMO concept. J. Phys. Chem. A 2015, 119, 10408−10427. (16) See appendix of: Amos, A. T.; Hall, G. G. Single determinant wave functions. Proc. R. Soc. London, Ser. A 1961, 263, 483−493. (17) King, H. F.; Stanton, R. E.; Kim, H.; Wyatt, R. E.; Parr, R. G. Corresponding orbitals and the nonorthogonality problem in molecular quantum mechanics. J. Chem. Phys. 1967, 47, 1936−1941. (18) Stewart, G. W. On the early history of the singular value decomposition. SIAM Rev. 1993, 35, 551−566. (19) See appendix A of: Sax, A. F. Localization of molecular orbitals on fragments. J. Comput. Chem. 2012, 33, 1495−1510. (20) West, A. C.; Schmidt, M. W.; Gordon, M. S.; Ruedenberg, K. A comprehensive analysis of molecule intrinsic quasi-atomic, bonding, and correlating orbitals. I. Hartree-Fock wave functions. J. Chem. Phys. 2013, 139, 234107. (21) West, A. C.; Schmidt, M. W.; Gordon, M. S.; Ruedenberg, K. A comprehensive analysis of molecule intrinsic quasi-atomic orbitals. II. Strongly correlated MCSCF wave functions. J. Phys. Chem. A 2015, 119, 10360−10367. (22) Schmidt, M. W.; Baldridge, K. K.; Boatz, J. A.; Elbert, S. T.; Gordon, M. S.; Jensen, J. H.; Koseki, S.; Matsunaga, N.; Nguyen, K. A.; Su, S.; et al. General atomic and molecular electronic structure system. J. Comput. Chem. 1993, 14, 1347−1363. (23) Gordon, M. S.; Schmidt, M. W. Advances in electronic structure theory: GAMESS a decade later. In Theory and Applications of Computational Chemistry: The First Forty Years; Dykstra, C. E., Frenking, G., Kim, K. S., Scuseria, G. E., Eds.; Elsevier: Amsterdam, 2005; pp 1167−1189.

(24) GAMESS: http://www.msg.chem.iastate.edu/GAMESS/ GAMESS.html. (25) Huzinaga, S.; Klobukowski, M. Well-tempered Gaussian basis sets for the calculation of matrix Hartree-Fock wavefunctions. Chem. Phys. Lett. 1993, 212, 260−264. (26) Barysz, M.; Sadlej, A. J. Infinite-order two-component theory for relativistic quantum chemistry. J. Chem. Phys. 2002, 116, 2696−2704. (27) Kedziera, D.; Barysz, M. Two-component relativistic methods for the heaviest elements. J. Chem. Phys. 2004, 121, 6719−6727. (28) Kedziera, D.; Barysz, M.; Sadlej, A. J. Expectation values in spinaveraged Douglas-Kroll and infinite-order relativistic methods. Struct. Chem. 2004, 15, 369−377. (29) Barysz, M.; Mentel, L.; Leszczynski, J. Recovering fourcomponent solutions by the inverse transformation of the infiniteorder two-component wave functions. J. Chem. Phys. 2009, 130, 164114. (30) Ivanic, J.; Ruedenberg, K. Identification of deadwood in configuration interaction spaces through general direct configuration interaction. Theor. Chem. Acc. 2001, 106, 339−351. (31) Ivanic, J. Configuration interaction and multiconfigurational selfconsistent field method for multiple active spaces with variable occupations. I. Method. J. Chem. Phys. 2003, 119, 9364−9376. (32) Ivanic, J. Configuration interaction and multiconfigurational selfconsistent field method for multiple active spaces with variable occupations. II. Application to oxoMn(salen) and N2O4. J. Chem. Phys. 2003, 119, 9377−9385. (33) Kramida, A.; Ralchenko, Y.; Reader, J.; NIST ASD Team (2016). NIST Atomic Spectra Database (ver. 5.3) [Online]. Available: http://physics.nist.gov/asd; National Institute of Standards and Technology: Gaithersburg, MD. (34) Huzinaga, S.; Miguel, B. A comparison of the geometrical sequence formula and the well-tempered formulas for generating GTO basis orbital exponents. Chem. Phys. Lett. 1990, 175, 289−291. (35) Matsuoka, O.; Huzinaga, S. Relativistic well-tempered Gaussian basis sets. Chem. Phys. Lett. 1987, 140, 567−571. (36) Werner, H.-J.; Knowles, P. J.; Knizia, G.; Manby, F. R.; Schütz, M. MOLPRO, version 2010.1, a package of ab initio programs. See http://www.molpro.net, 2010. (37) Broyden, C. G. The convergence of a class of double-rank minimization algorithms. II. The new algorithm. IMA J Appl Math 1970, 6, 222−231. (38) Fletcher, R. A new approach to variable metric methods. Comput. J. 1970, 13, 317−322. (39) Goldfarb, D. A family of variable-metric methods derived by variational means. Math Comp. 1970, 24, 23−26. (40) Shanno, D. F. Conditioning of quai-Newton methods for function minimization. Math Comp. 1970, 24, 647−656. (41) Lu, W. C.; Wang, C. Z.; Chan, T. L.; Ruedenberg, K.; Ho, K. M. Representation of electronic structures in crystals in terms of highly localized quasiatomic minimal basis orbitals. Phys. Rev. B: Condens. Matter Mater. Phys. 2004, 70, 041101. (42) Ivanic, J.; Atchity, G. J.; Ruedenberg, K. Intrinsic local constituents of molecular electronic wave functions. I. Exact representation of the density matrix in terms of chemically deformed and oriented atomic minimal basis set orbitals. Theor. Chem. Acc. 2008, 120, 281−294. (43) Ivanic, J.; Ruedenberg, K. Intrinsic local constituents of molecular electronic wave functiona. II. Electronic structure analyses in terms of intrinsic oriented quasi-atomic molecular orbitals for the molecules FOOH, H2BH2BH2, H2CO and the isomerization HNO → NOH. Theor. Chem. Acc. 2008, 120, 295−305. (44) West, A. C.; Schmidt, M. W.; Gordon, M. S.; Ruedenberg, K. A comprehensive analysis in terms of molecule-intrinsic, quasi-atomic orbitals. III. The covalent bonding structure of urea. J. Phys. Chem. A 2015, 119, 10368−10375. (45) Schoendorff, G.; Morris, A. R.; Hu, E. D.; Wilson, A. K. A computational study on the ground and excited states of nickel silicide. J. Phys. Chem. A 2015, 119, 9630−9635. I

DOI: 10.1021/acs.jpca.7b01916 J. Phys. Chem. A XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry A (46) Bode, B. M.; Gordon, M. S. MacMolPlt: A graphical user interface for GAMESS. J. Mol. Graphics Modell. 1998, 16, 133. (47) MacMolPlt: http://brettbode.github.io/wxmacmolplt/. (48) Manson, E. L.; De Lucia, F. C.; Gordy, W. Millimeter- and submillimeter-wave spectrum and molecular constants of cuprous chloride. J. Chem. Phys. 1975, 62, 1040−1043. (49) McLeod, D., Jr.; Weltner, W., Jr. Spectroscopy and FranckCondon factors in scandium fluoride in neon matrices at 4 K. J. Phys. Chem. 1966, 70, 3293−3300. (50) Lindholm, N. F.; Brugh, D. J.; Rothschopf, G. K.; Sickafoose, S. M.; Morse, M. D. Optical spectroscopy of jet-cooled NiSi. J. Chem. Phys. 2003, 118, 2190−2196. (51) Dunning, T. H., Jr.; Peterson, K. A.; Wilson, A. K. Gaussian basis sets for use in correlated molecular calculations. X. The atoms aluminum through argon revisited. J. Chem. Phys. 2001, 114, 9244− 9253. (52) Balabanov, N. B.; Peterson, K. A. Systematically convergent basis sets for transition metals. I. All-electron correlation consistent basis sets for the 3d elements Sc-Zn. J. Chem. Phys. 2005, 123, 064107. (53) Dunning, T. H., Jr. Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen. J. Chem. Phys. 1989, 90, 1007−1023. (54) Feller, D. The role of databases in support of computational chemistry calculations. J. Comput. Chem. 1996, 17, 1571−1586. (55) Schuchardt, K. L.; Didier, B. T.; Elsethagen, T.; Sun, L.; Gurumoorthi, V.; Chase, J.; Li, J.; Windus, T. L. Basis set exchange: A community database for computational sciences. J. Chem. Inf. Model. 2007, 47, 1045−1052. (56) Ruedenberg, K.; Sundberg, K. R. In Quantum Science; Calais, J.L., Goscinski, O., Lindenberg, J., Ö hrm, Y., Eds.; Plenum: New York, 1976; pp 505−515. (57) Siegbahn, P.; Heiberg, A.; Roos, B.; Levy, B. A comparison of the super-CI and the Newton-Raphson scheme in the complete active space SCF method. Phys. Scr. 1980, 21, 323−327. (58) Schmidt, M. W.; Gordon, M. S. The construction and interpretation of MCSCF wavefunctions. Annu. Rev. Phys. Chem. 1998, 49, 233−266. (59) Ramirez-Solis, A.; Schamps, J.; Delaval, J. M. MCSCF+MRCI calculation of diagonal and transition dipole moments in CuCl. Chem. Phys. Lett. 1992, 188, 599−603. (60) Carlson, K. D.; Moser, C. Electronic structure and ground state of scandium monofluoride. J. Chem. Phys. 1967, 46, 35−46. (61) Barrow, R. F.; Gissane, W. J. M. The ground states of scandium and yttrium monofluorides. Proc. Phys. Soc., London 1964, 84, 615− 616. (62) Barrow, R. F.; Bastin, M. W.; Moore, D. L. G.; Pott, C. J. Electronic states of gaseous fluorides of scandium, yttrium and lanthanum. Nature 1967, 215, 1072−1073. (63) Scott, P. R.; Richards, W. G. A reassignment of molecular orbital configurations of the electronic states of ScF. Chem. Phys. Lett. 1974, 28, 101−103. (64) Carlson, K. D.; Nesbet, R. K. Wavefunctions and binding energies of the titanium monoxide molecule. J. Chem. Phys. 1964, 41, 1051−1062. (65) Phillips, J. G. Satellite bands of the γ′-system of titanium dioxide. Astrophys. J. 1971, 169, 185−189. (66) Pathak, C. M.; Palmer, H. B. New electronic transitions in TiO. J. Mol. Spectrosc. 1970, 33, 137−146. (67) Andriotis, A. N.; Menon, M.; Froudakis, G. E.; Fthenakis, Z.; Lowther, J. E. A tight-binding molecular dynamics study of NimSin binary clusters. Chem. Phys. Lett. 1998, 292, 487−492. (68) Butler, K. T.; Harding, J. H. A computational investigation of nickel (silicides) as potential contact layers for silicon photovoltaic cells. J. Phys.: Condens. Matter 2013, 25, 395003. (69) Haberlandt, H. Pseudopotential MRD-CI calculations of nickelcontaining molecules. II. The electronic 1Σ+ ground state and 20 lowlying excited states of the NiSi molecule. Chem. Phys. 1989, 138, 315− 325.

(70) Shim, I.; Gingerich, K. A. Electronic states and nature of bonding of the molecule NiSi by all electron ab initio HF-CI and CASSCF calculations. Z. Phys. D: At., Mol. Clusters 1990, 16, 141−148. (71) Edmiston, C.; Ruedenberg, K. Localized atomic and molecular orbitals. Rev. Mod. Phys. 1963, 35, 457−464.

J

DOI: 10.1021/acs.jpca.7b01916 J. Phys. Chem. A XXXX, XXX, XXX−XXX