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2015 Valence Virtual Orbitals: An unambiguous ab initio quantification of the LUMO concept Michael Schmidt Iowa State University, [email protected]

Emily A. Hull Iowa State University, [email protected]

Theresa Lynn Windus Iowa State University, [email protected]

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This Article is brought to you for free and open access by the Chemistry at Iowa State University Digital Repository. It has been accepted for inclusion in Chemistry Publications by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. Valence Virtual Orbitals: An unambiguous ab initio quantification of the LUMO concept

Abstract Many chemical concepts hinge on the notion of an orbital called the lowest unoccupied molecular orbital, or LUMO. This hypothetical orbital and the much more concrete highest occupied molecular orbital (HOMO) constitute the two “frontier orbitals”, which rationalize a great deal of chemistry. A viable LUMO candidate should have a sensible energy value, a realistic shape with amplitude on those atoms where electron attachment or reduction or excitation processes occur, and often an antibonding correspondence to one of the highest occupied MOs. Unfortunately, today’s quantum chemistry calculations do not yield useful empty orbitals. Instead, the empty canonical orbitals form a large sea of orbitals, where the interesting valence antibonds are scrambled with the basis set’s polarization and diffuse augmentations. The UMOL is thus lost within a continuum associated with a detached electron, as well as many Rydberg excited states. A suitable alternative to the canonical orbitals is proposed, namely, the valence virtual orbitals. VVOs are found by a simple algorithm based on singular value decomposition, which allows for the extraction of all valence-like orbitals from the large empty canonical orbital space. VVOs are found to be nearly independent of the working basis set. The utility of VVOs is demonstrated for construction of qualitative MO diagrams, for prediction of valence excited states, and as starting orbitals for more sophisticated calculations. This suggests that VVOs are a suitable realization of the LUMO, LUMO + 1, ... concept. VVO generation requires no expert knowledge, as the number of VVOs sought is found by counting s-block atoms as having only a valence s orbital, transition metals as having valence s and d, and main group atoms as being valence s and p elements. Closed shell, open shell, or multireference wave functions and elements up to xenon may be used in the present program.

Disciplines Materials Chemistry | Other Chemistry | Physical Chemistry

Comments Reprinted (adapted) with permission from J. Phys. Chem. A, 2015, 119 (41), pp 10408–10427. Copyright 2015 American Chemical Society.

This article is available at Iowa State University Digital Repository: http://lib.dr.iastate.edu/chem_pubs/930 Article

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Valence Virtual Orbitals: An Unambiguous ab Initio Quantification of the LUMO Concept Michael W. Schmidt,* Emily A. Hull, and Theresa L. Windus

Department of Chemistry, Iowa State University, Ames, Iowa 50011, United States

ABSTRACT: Many chemical concepts hinge on the notion of an orbital called the lowest unoccupied molecular orbital, or LUMO. This hypothetical orbital and the much more concrete highest occupied molecular orbital (HOMO) constitute the two “frontier orbitals”, which rationalize a great deal of chemistry. A viable LUMO candidate should have a sensible energy value, a realistic shape with amplitude on those atoms where electron attachment or reduction or excitation processes occur, and often an antibonding correspondence to one of the highest occupied MOs. Unfortunately, today’s quantum chemistry calculations do not yield useful empty orbitals. Instead, the empty canonical orbitals form a large sea of orbitals, where the interesting valence antibonds are scrambled with the basis set’s polarization and diffuse augmentations. The LUMO is thus lost within a continuum associated with a detached electron, as well as many Rydberg excited states. A suitable alternative to the canonical orbitals is proposed, namely, the valence virtual orbitals. VVOs are found by a simple algorithm based on singular value decomposition, which allows for the extraction of all valence-like orbitals from the large empty canonical orbital space. VVOs are found to be nearly independent of the working basis set. The utility of VVOs is demonstrated for construction of qualitative MO diagrams, for prediction of valence excited states, and as starting orbitals for more sophisticated calculations. This suggests that VVOs are a suitable realization of the LUMO, LUMO + 1, ... concept. VVO generation requires no expert knowledge, as the number of VVOs sought is found by counting s-block atoms as having only a valence s orbital, transition metals as having valence s and d, and main group atoms as being valence s and p elements. Closed shell, open shell, or multireference wave functions and elements up to xenon may be used in the present program.

I. INTRODUCTION lowest empty canonical orbitals often have little valence I.A. Motivation and Outline. All chemists know, deep in antibonding character, as will be shown below. This is well their souls, that molecules and extended systems are composed recognized among theoretical chemists, as may be seen from of atoms. Furthermore, computational methods for molecules the number of papers seeking alternative empty orbitals usually involve building up molecular wave functions using (reviewed just below). The poor connection between the atom-centered basis functions (plane-wave calculations fa- LUMO concept and the lowest empty canonical orbital seems mously have difficulty building up the density peaks at atomic to be less well understood in the wider chemical community. nuclei). The purpose of this paper is to demonstrate that useful, Consequently, many concepts in chemistry are based upon chemically relevant, and interpretable unoccupied orbitals can be obtained from MO calculations in a simple, robust, and single-particle molecular orbital (MO) theory, using linear 3 combinations of atomic orbitals (LCAO) to expand the MOs. automated way, from any type of SCF or MCSCF calculation. These alternative empty MOs are called the valence virtual The valence electrons of stable molecules usually occupy 4 bonding linear combinations of atomic orbitals (AO) and orbitals (VVOs). nonbonded lone pairs, often delocalized across broad regions of Accordingly, the main body of this paper deliberately avoids use of the familiar acronym LUMO. Instead, for precision, the the molecule, while mainly avoiding the use of antibonding AO fi combinations. The exact electron count determines the rst few empty canonical MOs are designated LCMO, LCMO + 1, LCMO + 2, ... to avoid confusion with members of the set occupancy. For example, O2 occupies all available bonding orbitals and then half-fills a π* antibond. Electron poor systems of all valence virtual orbitals termed LVVO, LVVO + 1, etc. might not utilize all available bonding levels. Of particular These two sets are compared and evaluated for their suitability conceptual importance are the two frontier orbitals,1,2 known as as the conceptual LUMO, LUMO + 1, ... orbitals. the highest occupied and lowest unoccupied MOs (HOMO The outline of the paper is as follows. The second half of the and LUMO), and their energy difference, known as the “band introduction is a review of various procedures proposed by gap”. The frontier orbitals are well-known to play a key role in theoretical chemists to improve upon the empty CMOs. Section II presents the equations for VVO generation in a explaining chemical reactivity patterns. 4 Quantum chemistry calculations rather naturally produce the simpler form than a decade ago, since the present paper is not HOMO along with all lower energy occupied orbitals (HOMO − 1, ...), as the occupied canonical orbitals. The term Received: July 16, 2015 “canonical” means those orbitals that diagonalize the Fock Revised: September 21, 2015 operator of self-consistent field (SCF) theory. However, the Published: October 2, 2015

© 2015 American Chemical Society 10408 DOI: 10.1021/acs.jpca.5b06893 J. Phys. Chem. A 2015, 119, 10408−10427 The Journal of Physical Chemistry A Article concerned with the closely related atom localized orbitals appropriate for spectroscopy or for electron correlation known as QUAMBOs4 and QUAOs.5,6 Section III describes recovery.16 Writing the usual closed shell Fock operator with the characteristics and applications of VVOs. The VVOs are six user selectable coefficients in front of its usual kinetic shown to be nearly independent of the quantitative basis set in energy, nuclear−electron attraction, and Coulomb and which a calculation is performed, in their shape and in their exchange operators, energy. The VVO energies allow the construction of a full MO CORE CORE diagram including all antibonding MOs. Such MO diagrams are FaTbVcJ=+NE + + dK found to be qualitatively similar for Hartree−Fock and DFT VALENCE VALENCE ++eJ fK calculations but not quantitatively identical in energy. The LVVO is found to predict the correct shape for an excited facilitates discussion of some additional energy operator fi electron in the rst excited states of molecules with low-lying schemes. Of course, the canonical virtual orbitals are obtained states (for example, dyes). Frequent comparisons of VVOs to from the usual Fock operator (namely, a = b =1,c = e =2,d = f the usual empty CMOs are made along the way. VVOs are also = −1 for closed shell systems). Rather than remove electron shown to be excellent starting orbitals for more sophisticated density when creating Coulomb and exchange operators, as in calculations that begin to occupy antibonding orbitals, such as the various aforementioned “hole operator” schemes, Cooper multiconfigurational SCF (MCSCF). The paper ends with a and Pounder17 suggested increasing the strength of the nuclear short summary in section IV. attraction, namely, to increase the coefficient b. A number of − The valence virtual orbitals address only those aspects of workers18 20 have used the exchange operator because the φ 2 → φ 2 chemistry that involve the valence orbitals of the atoms Hamiltonian matrix element for the double excitation i a comprising the molecule. Quite often molecular excited states is given by the exchange integral Kia (i is an occupied MO, a is have Rydberg character, involving spatially large, nonvalence an empty MO). Maximization of exchange interactions should 7 shapes. On rare occasions, such as dipole bound anions or improve singles and doubles configuration interaction (CI-SD) 8 solvated electrons, even the ground state may be nonvalence in energies and is accomplished by diagonalization of the valence character: all such situations are beyond the scope of this paper. exchange operator (a = b = c = d = e = 0 with f = −1). It is most I.B. Review of Schemes To Improve upon the CMOs. practical to use the total valence exchange operator instead of There is a long history of seeking virtual orbitals different from working with one occupied orbital at a time. Finally, the K- the canonical molecular orbitals that are the usual direct output orbitals proposed by Feller and Davidson21 use the closely of molecular orbital calculations. These are grouped into four related operator 0.06F − KVALENCE, namely, a = b = 0.06, c = e = main classes below: diagonalization of an energy operator, 0.12, d = −0.06, and f = −1.06, and are one of the best virtual diagonalization of a molecular density matrix, direct imposition orbital choices for producing compact CI expansions. Anyone of atomic valence character (which includes the present work), wishing to experiment with energy operators constructed with and atomic sub-block density diagonalization. Often the goal the six coefficients above can do so with the GAMESS program. has been to obtain better convergence of configuration The computational cost of all these procedures is very low, 9 interaction (CI) energies (e.g., Boys’ 1960 “oscillator orbitals”, typically just a single SCF iteration’s time to build the desired 10 based on dipole moments ). So the first two categories do not energy operator, plus one diagonalization in the virtual space. necessarily separate the virtual space into two separate Only one of several studies of the rate of CI convergence using subspaces, namely, valence plus nonvalence. They are none- different virtual orbitals is mentioned here,22 to illustrate that all theless historically important and are briefly reviewed, before schemes in this “energy operator” category are expected to be the two groups of methods intended to cleanly separate valence more effective in truncated CI calculations than use of the virtuals from all other virtuals. The nonvalence empty orbitals canonical virtual orbitals. are called the external orbitals in this paper. A second major category of virtual orbital adjustment Diagonalization of Fock-like or exchange-like operators involves diagonalization of some molecular density matrix. within the canonical virtual orbital space of a converged SCF Requiring the density diagonalization to occur only within the calculation leads to altered virtuals, without any change to the SCF calculation’s virtual block leaves the occupied SCF orbitals molecule’s occupied orbitals.11,12 The improved virtual orbitals unchanged. The resulting weakly occupied natural orbitals with (IVOs) are obtained by diagonalizing a Fock operator the larger occupation numbers will contain the important corresponding to the removal of one electron from the highest antibonding valence orbitals but will also include in−out or occupied orbital.13 Modified virtual orbitals (MVOs) are angular correlating shapes as well. Since the idea is to have a obtained from a Fock matrix after removal of all valence reasonable but economical procedure, the density matrix is electrons, to further increase the “Coulombic attraction” of the usually obtained at a doubles level of theory. To name just a virtual space into the “valence region” of the molecule.14 few, the density might be obtained using second order − Averaged virtual orbitals (AVOs) remove one electron, but perturbation theory (MP2)23,24 or perhaps CI-SD.25 27 It is spread this hole in the electron density out evenly over all natural to consider more rapidly obtainable density matrices as occupied valence orbitals.15 Two schemes suggested by these well. In the limit of a small molecule, with at least as many procedures are available in the GAMESS program, namely, valence electrons as there are valence orbitals, one might simply removal of the user’s choice of electrons from several of the do a high-spin-multiplicity open shell SCF calculation, highest occupied orbitals (often 6−10, so more than 1 as for obtaining all valence orbitals, since all are occupied (e.g., σ* IVOs but usually not all valence electrons as for MVOs), or else H2O in a quintet state, to occupy both OH orbitals). In to remove half of the valence electrons (one electron from larger molecules, one may average the density matrices every valence orbital, not just one electron as for AVOs). It obtained from several SCF calculations (ground state and a should also be noted that alternative Fock operators few excited states) as a means to occupy more of the valence corresponding to Hamiltonians with an unchanged number of orbitals.28 A systematic procedure for obtaining those few electrons can be formulated to give virtual orbitals more empty valence orbitals involved in “strong correlation” is to

10409 DOI: 10.1021/acs.jpca.5b06893 J. Phys. Chem. A 2015, 119, 10408−10427 The Journal of Physical Chemistry A Article extract natural orbitals from the total density matrix of spin- identifying (somewhat distorted) atom centered entities within unrestricted open shell wave functions29,30 (known as the the molecular wave function, the AIM theory is not orbital UNO-CAS method). based but rather partitions the molecule’s electron density into The two diagonalization schemes (energy-type or density- atomic regions. type) just discussed often produce within their “lower energy” In the present work, the external virtual orbital space is of or “greater occupancy” orbitals some additional orbitals that do little interest, after it has been successfully separated from the not strictly speaking have valence character. Many of the most valence virtual space. However, selection of atomically or at important empty orbitals will in fact be valence orbitals, such as least regionally localized orbitals within the external space σ* or π*, but other orbitals with just one new node (such as facilitates the truncation of effort needed for recovery of in−out correlating orbitals for lone pairs) will also occur. While dynamical electron correlation energy. Consequently the important for improved CI convergence, perhaps, such orbitals literature on this topic is now quite large, from which a very with radial or angular correlating nodes are not necessarily very few references are given here for the interested reader. A interesting for valence chemistries. Therefore, it is desirable to particularly relevant paper is that of Subotnik et al., whose seek a procedure that (i) cleanly separates the canonical virtual procedure40 first projects out the valence virtual space by a space into valence and external subspaces, (ii) finds all valence process that falls in the third category mentioned here and then virtual orbitals, and (iii) does so without requiring complicated localizes the remaining “hard virtuals”. Very often, orthogon- algorithms or long computer times. ality is sacrificed in order to obtain a much higher degree of The third group of methods targets the identification of only localization, as pioneered in Pulay’s projected atomic orbitals.52 valence virtual orbitals. It is apparent that the objective can be Two very recent attempts to localize the external space with achieved by projecting some prototype collection of the atomic retention of orbital orthogonality use an SVD procedure53 or a valence orbitals onto the molecular virtual space. Projections fourth power variant54,55 of the Pipek−Mezey localization. were first used quite some time ago,31,32 albeit only after In closing, note that several of the procedures just reviewed, fi including ref 3, have been adapted to the study of periodic explorations of the rst two diagonalization schemes were tried. − Justification for the idea that valence atomic orbitals are the systems,56 58 even for plane wave calculations. predominant contributors to valence molecular orbitals comes from projecting free atom SCF orbitals onto full valence type II. GENERATION OF VALENCE VIRTUAL ORBITALS MCSCF wave functions: overlaps between the free atom AOs Two types of orbitals3,5 with predominantly atomic character and their projections onto the molecular orbitals remain at or can be extracted from the occupied and virtual orbital spaces of 33 above 0.99. The present scheme for the projection of valence molecular orbital calculations. One set is very atomic in nature: orbitals out of the virtual space of closed shell functions to yield these are termed the quasi-atomic minimal basis orbitals 3 atom localized orbitals was presented in 2004. This algorithm (QUAMBOs or QUAOs), which are localized onto individual is recast in a simpler form in the present paper, as an instance of atoms and are as close as possible to the s, p, d orbitals of the the singular value decomposition (SVD). The SVD procedure free atoms. The other set is molecular in nature: the occupied (also known in quantum chemistry as the method of canonical orbitals of the Hartree−Fock wave function corresponding orbitals) consists of basis rotations in two supplemented by valence virtual orbitals (VVOs), which are different orbital spaces to bring the bases into maximum unoccupied and typically antibonding valence molecular coincidence with each other; namely, it is a mutual projection − orbitals. This second set (occupied plus VVOs) should, to of each space onto the other.34 38 Many others have used the maximum extent possible, be formed from the core and similar projections of AOs onto molecular orbitals to produce valence orbitals of the atoms comprising the molecule. The the extracted polarized atomic orbitals,39,40 enveloping localized number of orbitals in both sets is equal to the sum of core plus orbitals,41 intrinsic minimal atomic basis,42 molecule-adapted valence orbitals for all atoms. Both sets (QUAMBOs or atomic orbitals,43 or intrinsic atomic orbitals.44 A great many occupied plus VVOs) exactly span the converged occupied interesting chemical applications of the intrinsic atomic orbitals orbitals of the molecular SCF and the same valence subspace of were presented in the latter paper.44 The equivalence of the the molecule’s virtual orbitals. orbitals in ref 44 to those in refs 3 and 5 has been shown.45 II.A. Separation of the VVO Space. The present paper Recent work at this university has generalized the present involves only applications for the VVOs, which are hoped to work’s methodology3 to all types of SCF wave functions, provide a quantification of the frontier orbital concept. including multiconfigurational MCSCF,5,6 and demonstrated Applications for the atom-localized QUAMBO orbitals are useful applications6 to chemical bonding and charge population given elsewhere.5,6 In case only VVOs are needed, the analyses. QUAMBO/VVO algorithm of Lu et al.3 can be greatly Perhaps the most commonly applied procedure for simplified, as shown below. The algorithm is also now generating chemically meaningful virtual orbitals is the natural recognized as an instance of the singular value decomposition − − bond order analysis.46 48 Its key step is diagonalization of the (SVD), whose mathematics is discussed elsewhere.34 38 density matrix in atomic sub-blocks, producing atom-localized The algorithm’s first input is a set of ordinary canonical hybrid valence orbitals. Antibonding combinations of the molecular orbitals (CMOs) from some kind of self-consistent atomic hybrids may then be generated to form a valence field (SCF) calculation, such as closed shell or high- or low-spin virtual orbital space similar to the results obtained in the coupled open shell SCF or even multiconfigurational SCF. The present paper. Other methods in this fourth and final virtual subspace of the CMOs is given as a linear combination fi ff ϕ ∑ χ classi cation of valence orbital extraction are the e ective of atomic orbitals (LCAO) expansion, v = μ μCμν. The user 49 atomic orbital method and the adaptive natural density may freely choose their favorite quantitative basis set χμ for the partitioning (AdNDP) method.50 molecular calculation. The goal is to divide this molecular The famous “atoms in molecules” theory of Bader51 is not virtual space into an internal (valence-like) part and its external categorized here. While certainly well within the spirit of orthogonal counterpart.

10410 DOI: 10.1021/acs.jpca.5b06893 J. Phys. Chem. A 2015, 119, 10408−10427 The Journal of Physical Chemistry A Article

The second input is a set of accurate atomic minimal basis set application of the rotation T to the original CMO virtual space orbitals (AAMBS) for every atom in the molecule, denoted separates it into the desired internal and external orbital spaces: collectively as Ai,orasAAa when referring to a particular atom A with its various occupied orbitals a (index a includes both ψw = ∑ ϕvTvw core and valence, but not empty atomic orbitals): v

A a If the diagonalization (or SVD) routine orders λ into AAa = ∑ χν Aν fi ψ descending order, the rst w are the desired internal orbital ν space while the remainder are the external orbital space. The The exact nature of the AAa orbitals is given in the following number of internal orbitals created is the sum of all AAMBS section II.B. For the moment, it is only important to know that orbitals minus the number of orbitals occupied in the molecular A ff each of these is expanded in an auxiliary basis set χν, centered wave function. A correct program will exhibit a clear drop o in a λ − on just one atom A. The atomic orbital expansions, Aν, are from near unity to around 0.1 0.2 at the boundary of these pregenerated and internally stored in the program. two virtual subspaces.5 3 The original paper did not explicitly identify the algorithm As an example, the closed shell wave function for H2SO4 has as an SVD and therefore did not stress that the AAMBS side of 25 occupied molecular orbitals, and 2 × 1+9+4× 5=31 the SVD must be preorthogonalized, perhaps by a symmetric AAMBS orbitals. When these 31 (preorthogonalized) atomic orthogonalization, orbitals are used in the SVD process, 31 − 25 = 6 values of λ are found near unity, and their 6 columns of T generate the * −1/2 Aj ==⟨|∑ ASiij() S ij AA ij⟩ desired antibonding valence virtual space. The number of i external orbitals depends on the user’s choice of the molecular basis set: 25 of the remaining columns of T have small but where {A } is the union of all A . Note that each diagonal i Aa nonzero singular values, while the remainder vanish since the block of S is a unit matrix by virtue of the orthogonality of any number of nonzero SVD diagonal elements of a rectangular atom’s AAMBS orbitals, but S contains nonzero interatomic matrix is equal to its smaller dimension. overlaps. The ∗ emphasizes the requirement of orthogonality The final step in preparation of the valence virtual orbitals for the A* orbitals (i.e., ∗ is not complex conjugation). Since (VVOs) is a “pseudocanonicalization” to produce uniquely the SVD step below deals with the AAMBS side and CMO side defined orbitals with an energy estimate. The pseudocanonical- as two entire function spaces, the preorthogonalization ization is accomplished by diagonalization of the relevant Fock procedure chosen does not matter unless one is also interested 4,5,59 operator, within its internal virtual and its external virtual in forming the atomically localized QUAMBOs. Note that the blocks, separately. The pseudocanonical internal orbitals are the CMO side of the SVD below is already orthonormal. desired VVOs, which Steps i and ii of the algorithm3 represent a singular value ’ decomposition of the overlap between the molecule’s virtual (a) are as valence-like as possible, due to the SVD s maximal orbitals and the atoms’ preorthogonalized AAMBS space, alignment of the CMO virtual orbital space to the AAMBS, * * † A * “ ” svj =⟨ϕχχv |ACAj ⟩=∑∑ νμ ⟨μν | ⟩ νj (b) have a pseudoeigenvalue that represents an energy μν estimate for each VVO, namely, their Fock operator expectation values, Note that it must be possible to compute the rectangular (c) possess the full symmetry of the molecule (assignable to overlap matrix between two different sets of Gaussian type some irreducible representation of the point group), and orbitals (GTOs), shown in the center of this formula. The SVD (d) are typically delocalized across the entire molecule, just is this generalized eigenvalue problem, as occupied canonical orbitals are. T†∗sU= λ The program creates a complete set of MOs by joining together three subspaces: the converged occupied SCF orbitals, The orthogonal transformations T and U represent rotations then the pseudocanonical VVOs, and finally the pseudocanon- within the virtual orbital space and the AAMBS space, ical external molecular orbitals. Because the occupied orbitals respectively, to create new functions within each space that are obtained by closed or open shell SCF, the only nonzero are parallel to each other, with overlaps λ. The number of Fock elements are those connecting the VVO and external nonzero λ is at most (and usually equal to) the smaller orbital spaces (full diagonalization of the entire virtual block of dimension of the molecular virtual space and the AAMBS space the Fock operator just regenerates the virtual CMOs of the (the latter should be smaller). Clearly, the SVD between the SCF calculation). core and valence atomic orbitals A* and the CMO space must Implementation of VVO generation is straightforward. The pick out the most atomic-like orbitals lying in the latter space. only unusual requirements are storage of the AAMBS orbitals Alternatively, the SVD equation may be squared, to create an for all atoms and the ability to compute overlap integrals ordinary eigenvalue problem, between their GTO expansions and the molecular basis set. (TsU†∗)( TsU †∗ ) †== TBT †λ2 , where B = s ∗ ( s ∗† ) The computational requirements are just a few matrix multiplications and diagonalizations, so VVOs may be extracted which is step ii in the original algorithm.3 Diagonalization of the after a molecular SCF calculation converges, in far less time symmetric matrix B means that an SVD subroutine need not be than a single SCF cycle. available. II.B. Specification of the AAMBS Atomic Orbitals. The In the present work, which does not involve the use of atom- original algorithm3 suggested using atomic orbitals expanded in localized QUAMBO orbitals, steps iii−vi in the original the same basis set that is chosen for the molecular calculation. algorithm which generate QUAMBOs may be skipped. Instead, This is impractical, since there are a very large number of

10411 DOI: 10.1021/acs.jpca.5b06893 J. Phys. Chem. A 2015, 119, 10408−10427 The Journal of Physical Chemistry A Article

Table 1. Summary of Configurations and Terms Used in the State-Averaged MCSCF Calculations Used To Prepare the a Transition Metal Atomic Orbitals

atom term term ΔE atom term term ΔE Sc d1s22D◁ d2s14F33Yc d1s22D◁ d2s14F31 Ti d2s23F◁ d3s15F19Zrb d2s23F◁ d3s15F14 Vd3s24F◁ d4s16D 6 Nb d3s24Fd4s16D◁ 3 Crb d4s25Dd5s17S◁ 22 Mob d4s25Dd5s17S◁ 31 Mn d5s26S◁ d6s16D49Tcd5s26S◁ d6s16D7 Fe d6s25D◁ d7s15F20Rub d6s25Dd7s15F◁ 21 Co d7s24F◁ d8s14F10Rhd8s14F◁ d9s02D10 Ni d8s23F◁ d9s13D 1 Pd d9s13Dd10s01S◁ 19 Cu d9s22Dd10s12S◁ 32 Agc d9s22Dd10s12S◁ 87 Zn d10s21S◁ Cd d10s21S◁ aAn arrow indicates the lower energy L−S term, by experiment.61 The experimental energy separation ΔE between the lowest J levels of these L−S terms is given in kcal/mol. bExperimentally, there is a second L−S term in the lower energy configuration which lies between the two states that are averaged: Cr and Mo’sd5s15S, Zr’sd2s23P, and Ru’sd6s23F. These terms are ignored since the intent is to average configurations. cExperimentally, Y’ss2p12P and Ag’sd10p12P states lie between the two L−S states averaged but are ignored since the intent is to obtain s and d orbitals. possible basis sets, although one can imagine generating each configurations and terms chosen for the averaging, guided by atom’s orbitals by “on the fly” atomic SCF calculations. A more experimental data for neutral atoms.61 The two chosen L−S practical solution is to choose some particular basis set to terms were averaged with equal weights, and 5-fold radial expand the atoms, once and for all, storing the resulting core degeneracy was imposed on the d orbitals. and valence atomic orbitals. This choice leads to the The metal calculations are technically complex and are aforementioned requirement to compute overlaps between described briefly. Since the two terms to be averaged sometimes the AAMBS’ Gaussians and the molecule’s Gaussians. have different spin multiplicities and usually possess compli- The accurate atomic minimal basis set (AAMBS) orbitals are cated angular momentum couplings, a full CI determinant62 chosen to be nonrelativistic SCF orbitals or state-averaged based state-averaged MCSCF program was used to perform the multiconfigurational SCF (SA-MCSCF) orbitals, for neutral atomic calculations. State-specific true single configuration atoms. At the present time, all atoms are stored down to Xe (Z calculations are easily done with the occupation restricted = 54). Beyond Xe, atomic orbital size changes due to scalar multiple active space (ORMAS) determinant program,63 which relativity would have to be considered. The number of AAMBS can choose a specific occupancy such as s1dm+1. Analogous state orbitals must be defined: Alkali and alkali earths are considered averaged SCF-level calculations are not always possible. Even if to be s-block, with valence ns orbitals. Main group elements are ORMAS is used to rigorously select only two of the three considered p-block, with valence ns and np orbitals. Transition possible electron configurations, a given L−S term still might metals are considered d-block, with valence (n − 1)d and ns. All not be single configurational. Consider the yttrium atom, using filled (core) orbitals below these valence AOs are also part of ORMAS to select only s2d1 and s1d2 determinants, thus each atom’s AAMBS. Of course, the working basis set for ignoring s0d3 determinants. The s2d1 configuration contains accurate molecular calculations should contain additional only a 2D term, which is Y’s ground state. The higher s1d2 functions for certain higher energy orbitals that mix to some configuration contains sufficient determinants to form 4F (the extent into the molecule’s occupied orbitals, for example, the np configuration’s lowest L−S term), along with 2F, 4P, 2P, 2G, 2D, orbitals of alkali or transition metals or the nd functions that and 2S terms. Since 2D appears again, state averaging yields a may be important in hypervalent main group compounds. At true single configuration 4F term but a two configuration 2D present, such functions are not part of the AAMBS. term. Since even using ORMAS cannot preclude some The s-block and p-block AAMBS orbitals are obtained by multiconfigurational character, the metals were computed closed- or open-shell SCF calculations on their free atom using all determinants arising from all valence electrons in a ground states, imposing radial degeneracy on the p orbitals. It is (s, d) active space, thus allowing up to three configurations to well-known that the radial size of p orbitals is insensitive to the mix. So the actual yttrium AAMBS orbitals also involved L−S coupling for open shell pm configurations, so the ground determinants from s0d3 which contains (among other L−S state term is used. terms) an additional 2D and 4F term, so that the Y atom’s SA- Transition metals, however, require more consideration. It is MCSCF contains mixing of three 2D terms and two 4F terms. well-known that molecular chemistry may involve several low- However, the orbital averaging is over the lowest 2D and 4F lying configurations: (n − 1)dm, ns2;(n − 1)dm+1, ns1;or(n − states, which are largely but not exactly single configurational in 1)dm+2, ns0. These atomic configurations have markedly character. different radial expectation values (sizes) for the d shells, It is hoped that the valence d and s orbital sizes that result since the extent of screening by the smaller (n − 1)d AO from these configurationally averaged metal atom calculations increases with its increasing occupancy.60 Accordingly, the are similar to the orbitals belonging to a metal found in a AAMBS chosen for the metals is an average of the two lowest molecule. The ground state of metal atoms in the gas phase is atomic configurations: usually but not always the gaseous state often (n − 1)dm, ns2. The decrease in relevance of this metal’s two lowest energy configurations are (n − 1)dm, ns2 and configuration after incorporation into molecules and the (n − 1)dm+1, ns1. The SA-MCSCF calculations averaged two difficulty of dividing the valence orbitals from other orbitals Russell−Saunders terms, using the lowest term from the two for atoms found deep in the periodic table have been discussed − lowest energy configurations. Table 1 gives details about the by Schwarz.64 66

10412 DOI: 10.1021/acs.jpca.5b06893 J. Phys. Chem. A 2015, 119, 10408−10427 The Journal of Physical Chemistry A Article

It remains to specify the GTO expansions used for each Molecular geometries are mostly chosen to be tightly atom. It is desirable to use a large number of Gaussians, to optimized RHF/ACCT structures. In a few instances, effectively be at the atomic basis set limit, so that the structures found by RHF/ACCT are not as close to experiment calculations are essentially free of yet another basis set as would be desired. Thus, DFT with the TPSS functional was −1 truncation choice. Accordingly very large expansions in terms used for MnO4 as RHF has too short MnO bonds, full of GTO primitives are used. The s- and p-block elements up to valence space multiconfigurational self-consistent field Ne (Z = 18) use even-tempered (ET) Gaussians,67 after which (MCSCF)93 was used for dioxirane as RHF has a too short well-tempered basis set (WTBS) Gaussians are used, up to Xe OO distance, an 8 e- in 8 orbital MCSCF was used for o- − (Z = 54).68 71 The s-block and p-block AAMBS orbitals are the benzyne as RHF has a too short CC triple bond, second order following: H−He, 8s; Li−Be, 14s; B−Ne, 14s,7p; Na−Mg, perturbation theory (MP2)94 was used for 2-norbornyl cation 18s,9p; and Al−Ar, 18s,12p ET primitives; and K−Ca, 26s,16p; as RHF has too long nonclassical CC bonds, MP2 was used for − − − fi Ga Kr, 26s,20p,14d; Rb Sr, 28s24p22d; and In Xe, P4 as RHF has too short PP distances, and nally DFT with the 28s,23p,17d WTBS primitives. The metals use Sc−Ni, B3LYP functional was used for ferrocene as RHF has too long 26s,17p,13d; Cu−Zn, 26s,17p,14d; Y−Tc, 27s,20p,17d; and metal/ring distances. In the structures chosen, bond distances Ru−Cd, 28s,20p,17d WTBS primitives.71 All of these primitive are typically within about 0.02 or 0.03 Å of available sets approach the ground state SCF energies to within 1 experimental data. mhartree or better. A general contraction of these SCF or SA- Even when a higher level geometry is used, orbital results are MCSCF orbitals to the minimal basis set sizes defined above mostly from closed shell RHF wave functions, although some produced the final AAMBS orbitals. closed shell density functional theory (DFT) results are Future work could extend the VVO methodology to heavy presented, using the functionals PZ81,95 PBE,96 PBE0,97 atom all electron calculations which require scalar relativity B3LYP,98,99 wB97,100 or TPSS.101,102 The collective term (beyond Xe). For instance, the molecule might be treated by DFA (density functional approximation) is used for calculations the infinite order two-component72 or a finite order Douglas− − with these approximate functionals. The collective term KS Kroll−Hess approximation.73 75 The inclusion of scalar (Kohn−Sham) is used to compare DFA eigenvalue data to relativity is well-known to result in the contraction of s orbitals, those obtained using the exact (largely unknown) functional or a smaller contraction of p orbitals, and expansion of d or f some almost exact functional.103,104 For purposes of evaluating orbitals.76,77 It seems likely these shape changes will require the utility of VVOs in predicting the shape of valence excited storing an alternative set of AAMBS orbitals, prepared by state orbitals, the natural orbitals of the relaxed density matrix including IOTC-level relativity in the same type of atomic specific to each excited state are obtained by time-dependent − calculation just described. A future extension to relativistic DFT105 107 (TD-DFT) using the B3LYP functional, and the 78,79 model core potential calculations should then be feasible: Tamm−Dancoff approximation.105 The choice of closed shell the semicore and valence orbitals retained in any MCP examples in the rest of this paper is just for simplicity: VVOs of calculation should have the correct radial shape so that one similar characteristics may be readily obtained for both low- and would simply omit any core orbitals dropped by the MCP from high-spin coupled open shell SCF wave functions or after the AAMBS side of the SVD. Extension to relativistic or MCSCF. nonrelativistic effective core potentials80 is problematic, since III.B. Basis Set Dependence of the CMO and VVO ECP orbitals are typically nodeless and therefore do not closely Energies. Koopmans’ theorem108 asserts that the occupied resemble the true radial shapes of valence atomic orbitals. orbitals of a molecule correspond roughly to ionization II.C. Implementation. The generation of VVOs (and also potentials from each occupied orbital, affording a qualitative QUAMBOs) for any molecule containing the elements H−Xe explanation of photoelectron spectra. That is, the energies of is implemented in the GAMESS quantum chemistry pack- the canonical HOMO, HOMO − 1, ... of a neutral molecule M 81−83 age as a user selectable option. The generation of VVOs is provide information about the energies of the cations M+1. This switched on by a single keyword. The process is entirely is well accepted by the community and is borne out by data automatic, since no user assistance is needed to count the shown below: the HOMO, HOMO − 1, ... concept is number of core and valence orbitals (as in the H2SO4 example concretely realized by the occupied canonical SCF (or DFA) given above). orbitals. VVOs may be obtained for closed shell RHF, high- and low- Textbooks discussing the Hartree−Fock equations also spin open shell ROHF, and MCSCF wave functions. The contain a Koopmans’ theorem interpretation of the unoccupied number of VVOs to be found decreases as more orbitals are canonical orbitals as being related to ionic states M−1. However, occupied (the sum of occupied and virtual valence molecular again correctly, this is nearly universally accompanied by the orbitals always equals the total number of occupied orbitals statement that the eigenvalues of the lowest canonical from every atom). Localization of the VVOs is also possible molecular orbitals LCMO, LCMO + 1, ... are not even (see section III.C below). qualitatively useful for estimating electron affinities. Nonethe- less, it is still common to encounter the terminology HOMO− III. APPLICATIONS INVOLVING VALENCE VIRTUAL LUMO gap, although only the occupied eigenvalues are ORBITALS qualitatively useful. III.A. Methods. Orbital results given in this paper are from It is easy to give a physical reason for the poor valence closed shell self-consistent field calculations (RHF) with the properties of many unoccupied CMOs. As already mentioned, a − aug-cc-pVTZ basis set84 87 (ACCT), unless otherwise conventional interpretation for the empty orbitals is as a place indicated. Since no ACCT all-electron basis set exists for to attach an additional electron, meaning they can represent (N iodine and ruthenium, a similar TZ-quality Sapporo basis + 1) electron states. The anions of closed shell molecules 88−90 set was used for all atoms in IF3 and Ru(bpy)3. Orbitals typically have only loosely bound electrons, at best. are drawn with the MacMolPlt visualization program.91,92 Alternatively, virtual orbitals provide approximate excited states

10413 DOI: 10.1021/acs.jpca.5b06893 J. Phys. Chem. A 2015, 119, 10408−10427 The Journal of Physical Chemistry A Article

Figure 1. Energies of RHF canonical MOs (in color) and valence virtual orbitals (in black) for five different basis sets. Eight of the panels used the same basis sets, as shown in the upper left panel. The energy scale increases going to the right, in hartree units. See text for discussion of each molecule. For ethane, only canonical MO positions are shown, and its RHF results are compared to DFT. The DFT LCMO of ethane (TPSS functional) is depicted at a reduced contour value of 0.025 bohr−3/2 appropriate for a low amplitude, spatially large MO.

M*, for example, by singly exciting electrons into them (N canonical virtuals can only be valence antibonding orbitals and electron states). Some of the M−1 or M* states correspond to may be used as such. The quantitative comparison of empty CMOs to VVOs in Rydberg states, or even to a completely detached electron plus − MorM+1, and the kinetic energy of any such free electron is this section uses four Pople-style basis sets110 112 and one not quantized. Whether viewed as representing M−1 or M*, essentially saturated basis set. The first basis is the very small 6- such orbitals have physical meaning as scattering resonances 31G, containing only s,p functions. This is improved by adding and have been described as “being infinitely extended, with only polarization to all atoms, namely, 6-31G(d,p). The next two a few orthogonality wiggles in the molecular neighborhood”.103 improvements consist of either adding more polarization or else ff In other words, the canonical Fock spectrum of a neutral di use s,p functions: 6-31G(3df,2p) or 6-31++G(d,p). Generally speaking the polarization improvement provides a molecule, like the textbook H atom case, should contain a ff ’ continuum of unbound electron states, which are asymptoti- larger energy lowering, but the di use basis lower CMO eigenvalues group closer to those of the final basis set, which is cally plane waves, starting at energy zero. Note that this is true − aug-cc-pVQZ84 86 (called ACCQ herein). This final basis is for all neutral molecules so that the LCMO spectrum in large very large, containing both diffuse augmentation and numerous basis sets starts from zero, as will be demonstrated below. It polarization functions up to the level of g AOs. These five basis should be remarked that the Gaussian orbitals used in everyday sets are used in all panels of Figure 1, in the same order and as quantum chemistry applications are not particularly good plane −1 labeled in the upper left panel. MnO4 is an exception, as waves! described below. This section focuses on the VVO energies, Of course, other orbitals in the virtual space do have valence while their shapes are considered in the following section III.C. character, and their extraction from this fairly uninteresting The first two panels in Figure 1 show only the Fock and continuum, and utility in chemistry, is the point of this paper. TPSS canonical orbital spectra of ethane. Ethane has rather As we will see, in some cases like C60 or indigo, there may be uninteresting valence σ* antibonding orbitals, whose VVO one (or more) empty Hartree−Fock CMOs below the energies are not included in Figure 1, to concentrate on the continuum starting from zero, and these few discrete virtuals characteristics typical of the canonical orbital eigenvalue have valence character. Note that smaller basis sets may not spectrum of most molecules. Note that the occupied orbitals contain any diffuse GTOs so that their virtual orbitals may readily converge with respect to basis set: two of the five lines 109 exhibit reasonable valence character. In the limit of a shown for C2H6 below zero are doubly degenerate, accounting minimal basis set (such as is used in extended Huckel theory), for all seven valence pairs. On the other hand, it is clear that the

10414 DOI: 10.1021/acs.jpca.5b06893 J. Phys. Chem. A 2015, 119, 10408−10427 The Journal of Physical Chemistry A Article

−1 unoccupied CMOs change greatly with the basis set and do in The small anion MnO4 has occupied orbitals at higher fact start to resemble a continuum for ACCQ. By happen- energy than typical neutrals, just as cations have deeper than fi − ’ stance, the LCMO in all ve basis sets has symmetry type a1g, usual HOMO, HOMO 1, etc. The transition metal s 6-31G but for the two final bases with diffuse functions, this orbital contains s, p, and d functions. Polarization is shown in its panel becomes spatially very large. This is because the plane waves by specifying these for its two atoms (Mn, O) in that order. with kinetic energy just above zero are being mimicked by the The diffuse augmentation of the Pople-style sets is created by large (diffuse) GTOs added in the final two basis sets. Note even-tempered ratios.114 Although there are drop-offs in the that use of TPSS density functional theory also reaches a position of the anion’s LCMO eigenvalue as the basis grows, continuum virtual space limit for large bases, but the energy even the very large ACCQ basis set’s first canonical eigenvalue is well above zero, at +0.114 hartree. This is due to the negative values do shift: the occupied orbitals are all raised substantially, − and the LCMO for the final two bases with diffuse functions charge, as a true continuum for M/e type virtuals can only be drops slightly below zero. The occurrence of some empty obtained by using GTOs which are far more diffuse than canonical orbitals below zero is not uncommon in DFA normal. The smallest exponent found in the ACCQ basis for calculations. This paper does not explore in depth what DFA Mn is an s function with exponent 0.0128. Adding an extra p functional might be quantitatively the most accurate: the GTO shell to the central metal with the small exponent 0.001 remaining panels in Figure 1 show eigenvalues and inserts a new triply degenerate t2 LCMO at +0.034 hartree. If pseudoeigenvalues from the Hartree−Fock operator. the extra function is instead an s GTO with the even smaller Diborane’s CMO spectrum in the bottom panel of the first exponent 0.0001, the new LCMO is a singly degenerate a1 column of Figure 1 is similar to ethane, except the symmetry orbital at +0.016 hartree. As in this case, the lowest virtual CMOs of other anions seldom possess useful valence type of the LCMO now changes (as indicated) with the basis −1 set, particularly when diffuse functions are added. Of the eight antibonding character. In contrast, the six VVOs of MnO4 + have similar pseudoeigenvalues and shapes for all bases. The e, molecules shown in Figure 1, only ethane and H3O have the t2, and a1 symmetry VVOs represent antibonding combinations same symmetry for the LCMO for all bases. As noted above, fi this is just accidental for ethane, as its LCMO undergoes a big of metal d orbitals with the oxygens: the rst degenerate pair spatial expansion. The pseudoeigenvalue for each VVO is involves antibonding interactions with O lone pairs, while the top four are MnO σ* in nature. placed on top of the CMO spectrum using thick black lines. fi Notice that the VVO positions converge quickly with respect to The nal column of Figure 1 returns to neutral molecules, the basis set improvements! Their shapes also remain nearly which were selected as likely to contain a low-lying unoccupied orbital. In the singlet carbene HCCl, there should be a p orbital identical and, if visualized, are seen to be of antibonding valence at carbon, perpendicular to the molecule. The first four basis character. Removal of the seven VVOs from the virtual space sets do show such an a″ symmetry shape, of valence 2p size. means that the remaining external orbitals of diborane shift However, the largest basis once again has its LCMO being very around somewhat. This is not illustrated, and for the largest diffuse and even the incorrect a′ symmetry. In contrast, the atomic bases, the external pseudoeigenvalues remain a near- three VVOs are very sensible. All basis sets show the same continuum, just missing seven values. σ* fi + orbitals, in ascending order: the C 2p, CCl , and nally the The middle column of Figure 1 contains ions. For H3O , the CH σ*. small size means that the LCMO is a symmetric combination 115 σ* Dioxirane (H2CO2) is important in ozonolysis and has a (a1 symmetry) of all three OH antibonds. Even for the low-lying OO σ* orbital due to ring strain, which is occupied by largest basis set, the LCMO looks similar to the LVVO, almost 0.1 e− in a full valence MCSCF wave function. Thus, the although the latter clearly has a more systematically converging LCMO of the first three basis sets are this b symmetry OO σ*, energy. Thus, the Koopmans’ theorem argument that the 2 + but as usual for the two largest basis sets, the LCMO loses its virtual orbitals of M can be interpreted as electron attachment valence character. In energy order, the five VVOs are the OO sites holds up, at least for the few CMO levels appearing below σ* σ* fi , an antisymmetric combination of the two CO , zero. The higher CMO levels begin to ll in a continuum above symmetric and antisymmetric CH σ*, and finally the symmetric zero, as it remains true that many of the attached electron states CO σ* combination. correspond to an unbound additional electron scattering from Pyrazine116 has low-lying π* orbitals, so this moiety is often + H3O . The two VVOs are the a1 and degenerate e symmetry incorporated into donor/acceptor charge transfer compounds, σ* combinations of the OH antibonds and, as for neutrals, have although curiously, the isolated molecule has only a very small similar energy and shapes no matter what basis is used. The electron affinity. Nonetheless, small basis sets have two unique tendency for small cations to exhibit valence character in their low-lying CMOs, which are its π* accepting orbitals, with a 13 low-lying CMOs provided the motivation for IVO and noticeable gap before reaching the remaining CMOs. These 14 MVO types of alternate virtual orbitals. two orbitals are swamped out and lost in the final two basis sets The much larger 2-norbornyl cation (in its nonclassical which add diffuse orbitals in the same energy range. The first 113 geometry ) has its charge spread over three carbon atoms, so three VVOs are π* orbitals (two are much lower than the now the LCMO in the two largest basis sets is a spatially diffuse third), beneath the ten σ* antibonds, for all basis sets. orbital (and has different symmetry than the three smaller sets). For cations or for molecules with a low-lying unoccupied As is typical of cations, the unoccupied CMO spectrum begins valence orbital and particularly when using modest basis sets, below zero. The HOMO and the first two VVOs are chemically the LCMO may have a shape corresponding to the expected interesting and will be discussed and illustrated below in section low-lying valence orbital. The utility of the next higher CMO, III.C. This molecule also contains numerous CC and CH σ* namely, the LCMO + 1, is minimal (although pyrazine has two antibonds, forming a dense cluster of black lines, with the useful empty CMOs in small bases). Thus, several decades ago highest VVO being the σ* of its shortest CC bond (at the base when use of small basis sets was the norm, the LCMO could of the three-membered ring). often be taken for the empty frontier orbital. Modern quantum

10415 DOI: 10.1021/acs.jpca.5b06893 J. Phys. Chem. A 2015, 119, 10408−10427 The Journal of Physical Chemistry A Article chemistry applications routinely use large basis sets and clearly cannot be expected to yield meaningful CMOs, as spatially large empty CMOs commonly occur. An illustration of what is “ ” meant by spatially large CMOs is given for C2H6 in Figure 1, and additional examples will be presented below. In contrast, all VVOs are found to occur at nearly the same pseudoeigenvalue, no matter what basis set is used and no matter what the molecular charge is. The shapes of VVOs, too, are also nearly independent of the basis set. The set of VVOs represents all missing valence antibonding orbitals, not just the lowest one, as will be convincingly demonstrated in the next section. The basis set dependence of the lowest unoccupied canonical orbital’s eigenvalue is not well-known within the chemical community perhaps because workers may use only a single basis set for any given theoretical study. However, the results in Figure 1 are unquestionable. As the basis set grows, the LCMO of neutral and anion systems is expected to drop to zero. This has been noted before, in atoms and small hydrides,117 and clearly is a very general result, which casts profound doubt on any physical significance for the LCMO. III.C. VVO Shapes. A few illustrations are given in this section to prove the assertion that VVOs possess antibonding valence character. Additional examples are given in subsequent sections which focus on possible applications for VVOs. VVO shapes are found to be almost entirely independent of the basis set, so only ACCT orbitals are shown in the remainder of this paper. Because of the “pseudocanonicalization”, which assigns a Figure 2. This figure and all others (unless otherwise indicated) uses a diagonal Fock (or DFA) operator energy to each VVO, the contour increment appropriate to valence-sized orbitals, namely, 0.05 VVOs are usually delocalized across the entire molecule. bohr−3/2, which is the only contour visible in the three-dimensional Since the number of VVOs is similar to the number of plots and is the increment in these two-dimensional slices. Electron occupied valence orbitals, there is no trouble using ordinary occupancies (2 or 0) and some symmetry labels are shown. See text for localization algorithms to localize the VVOs. This is in contrast discussion of the three molecules. to the great difficulty of localizing within the external virtual − space, which contains many orbitals from every atom.40,53 55 As implemented in GAMESS, any localization takes care to nonclassical CCC triangle involving three p orbitals, one from leave the total wave function invariant: the filled, partially filled each carbon. The LVVO and LVVO + 1 are antibonding in the in the case of open shell wave functions, and virtual valence same region. Symmetry constrained localization, if it were done, orbitals are localized separately. Preserving the occupied orbital would remove most of the amplitudes on more remote atoms, space(s) means the localized orbitals represent the same total such as can be seen on the top hydrogen atoms in the LVVO wave function while often providing greater interpretation. If (middle image). the molecule of interest does not possess any inherent The hypervalent IF3 compound is shown in the center of delocalization (aromatic rings, three-center bonds, ...), the Figure 2. Counting the 3 lone pairs at each F atom, there are a result of localizing the occupied orbitals will often be two- total of 16 valence orbitals in IF3, only two of which are center bonds and one-center lone pairs. In this case, the unoccupied VVOs. Three of the canonical occupied (labeled localization of the VVO space results in a set of two-center with occupation 2) and both VVOs (occupation 0) are shown antibonds, with one antibond for every bond. in the top row for IF3. The canonical occupied orbitals mix up For certain applications, such as using VVOs as starting the orbitals corresponding to the two center IF bond and the orbitals for MCSCF (see below), it is convenient to localize as three center FIF bonds. Symmetry constrained localization, much as possible but without destroying orbital symmetry. This illustrated in the next row, clarifies the bonding. The two VVOs “symmetry constrained” localization is available as a user option are of different symmetry, so symmetry constrained localization in GAMESS. The localization of VVOs, possibly with such an in the virtual space does not change them. However, the imposed symmetry constraint, is available for all popular occupied orbitals separate into a two-center IF bond, which is localization procedures: Edmiston/Ruedenberg self-en- paired with the a1 symmetry VVO, both shown at the right. The ergy,118,119 “Boys” dipole,120,121 and Pipek/Mezey popula- three orbitals on the left correspond to a typical three-center, tion122 criterions. The Edmiston/Ruedenberg procedure is four-electron bond:123 an axial 5p of iodine bonds to a fluorine used for all localized orbital results presented here. hybrid with a phase change at I caused by its 5p (symmetry b2), For cases where the unoccupied frontier orbital(s) are followed by a nonbonding orbital with small amplitude on expected to be clear-cut, the pseudocanonical VVOs themselves iodine which is a symmetric combination of the ligand hybrids; may already be fairly well localized. This is the case for the 2- both are occupied. The final three-center orbital is the norbornyl cation, as can be seen in the top row of Figure 2. antibonding counterpart of the first, and this VVO is not After filling all ordinary two-center bonds in the rest of the occupied. Two more occupied localized orbitals at iodine are of molecule, there are two electrons left over. This pair occupies interest but are not shown: there is a 5p perpendicular to the the HOMO of the cation, which is a bonding orbital in the plane and a largely 5s lone pair in-plane. The latter orbital, in

10416 DOI: 10.1021/acs.jpca.5b06893 J. Phys. Chem. A 2015, 119, 10408−10427 The Journal of Physical Chemistry A Article accordance with the VSEPR model,124 is responsible for the slight bending of the IFI axis toward the equatorial F. fi 125−127 The nal example in Figure 2 is tetrahedral P4, seen in the bottom row of Figure 2. The P4 is oriented with a triangular face at the bottom, and one atom at the top, so the bottom right corner is the projection of atoms above and below the plane drawn. VVOs possess the full symmetry of the molecule so that the six PP antibonds transform as two triply degenerate levels: t1 at +0.138 and t2 at +0.213 hartree for RHF/ACCT. Thus, the VVOs are completely delocalized. This is a case where full localization (without symmetry constraint) produces the most chemically relevant picture: the occupied orbitals give four equivalent lone pairs and six identical curved bond pairs. The localization of the VVOs gives six equivalent antibonds, which are also curved outward due to ring strain. Note that under unconstrained localization in singly bonded molecules, VVOs localize to one two-center antibond for every localized two-center bond, whereas lone pairs are found only in the occupied space. Because P4 possesses only lone pairs and two center bonds, its orbitals are not as interesting as the 2- Figure 3. MO diagram of D5d ferrocene, with orbital energies from +1 norbornyl ion or IF3, apart from localization clearly displaying B3LYP, given in hartree units. Red levels indicate the most relevant its ring strain in curved bonds and antibonds. orbitals composed of metal s and d orbitals and ring π and π* orbitals. The examples in this section show that even for a large basis Species names (in blue) separate occupied orbitals from the VVOs. The inset illustrates the highest red VVO: Fe 4s antibonding to ring π. set like ACCT, VVOs are found to be the same size as occupied ’ − valence orbitals, to be antibonding combinations of various Ferrocene s relevant orbitals are its occupied CMOs a1g ( 0.408), a2u (−0.359), e (−0.267), e (−0.248), a (−0.227), e (−0.198) and valence atomic orbitals, and to provide all of the antibonds that 1g 1u 1g 2g its empty VVOs e1g (+0.005), e2u (+0.094), e2g (+0.107), a1g (+0.199). should exist. Doubly degenerate orbitals are shown as pairs, but the 5-fold III.D. MO Diagrams. Since the two previous sections degeneracy of the metal 3d is not indicated. indicate both VVO shapes and VVO energy values are chemically sensible, as well as basis set independent, it is The orbitals shown in red in Figure 3 form the ferrocene reasonable to combine the ordinary occupied CMOs (HOMO, molecule: π levels from Cp· and the valence s/d from Fe. The HOMO − 1, ...) and the unoccupied VVOs to construct MO high symmetry of ferrocene leads to a relatively small number diagrams of bonding and antibonding levels. Four examples are of metal/ring interactions, shown as the red diagonal lines. considered in this section. These interactions are in perfect accord with the first 128 The famous molecule ferrocene contains sufficiently explanation of the occupied orbitals by MO theory.135 That interesting bonding interactions to serve as a first example. report was entirely based on group theory rather than any 129 The staggered D5d conformation found in the solid state is actual calculation in 1953! Many of the twinned π levels have ff used to enable comparison to recent photoelectron spectra of di erent symmetry types than the metal levels in D5d and thus the solid.130,131 Figure 3 shows a ferrocene MO diagram, do not interact with the metal. Of particular interest are the constructed with B3LYP occupied orbitals as well as the VVOs symmetric combination of the lowest ring π and the metal’s4s that pseudocanonicalize the same DFA operator. An SCF-level and 3d0, which all have a1g symmetry in D5d: the 4s makes RHF Fock operator produces qualitatively identical MO shapes bonding and antibonding combination with the ring, with the but orders the occupied orbitals differently, whereas B3LYP latter being the highest interesting VVO (shown in Figure 3’s gives good agreement with experimentally known occupied inset). Interestingly, the metal 3d0 scarcely interacts, in spite of orbital orders130,131 (and to similar DFA calculations130,131 that having the correct symmetry, and so lies at a nearly unchanged were used to interpret the experimental results). RHF also gives energy in the molecule. The formally symmetric combination of three nearly isoenergetic doubly degenerate levels as the first the middle π level of Cp· has the correct orbital symmetry to 132 ’ three unoccupied VVOs, while B3LYP separates out the e1g interact with Fe sdxz and dyz pair (e1g) in a bonding and as the ferrocene LVVO. antibonding fashion: the latter is the LVVO of ferrocene. The Many diagrams for ferrocene construct this molecule from formally antisymmetric combination of the middle two π levels − Fe2+ and two cyclopentadienyl anions (Cp ), but in accordance has no orbital on Fe to interact with and so just creates an with section III.B above, the Fe2+ cation orbitals are very low occupied degenerate level at slightly lower energy in ferrocene. − and the Cp anion orbitals are too high. Figure 3 suggests that The highest π* of Cp· just twins in ferrocene. The rest of the using open shell B3LYP energies for neutral Fe and Cp· is a VVO spectrum is evidently sensible: all antibonding metal/π more reasonable way to construct the MO diagram for orbitals lie below the many twins of the Cp· σ* VVOs. ferrocene.133 The Cp· moieties are far enough apart that A very recent XANES experiment131 interprets the pre-edge formal bonding and antibonding combinations (with respect to feature in the photoelectron spectrum of ferrocene as due to an the inversion center) of its orbitals have almost the same e1g LUMO. The B3LYP/ACCT e1g symmetry LVVO reported energy: these are slightly split at the very bottom, but their here, arising from the antibonding interaction between the fi · twinning often results in only thicker lines, for example, in the metal dxz/dyz and the partly lled HOMO of Cp , agrees with numerous orbitals at the top of the ferrocene diagram. Note this experimental inference.131 fi that a successful energy analysis of ferrocene has been made C60 is a case where the nature of the rst two unoccupied using either neutral Fe/2Cp· or ionic Fe2+/2Cp−1 fragments.134 orbitals is well established. Alkali doping results136 showed

10417 DOI: 10.1021/acs.jpca.5b06893 J. Phys. Chem. A 2015, 119, 10408−10427 The Journal of Physical Chemistry A Article fi 97 − K3C60 is conducting while K6C60 is insulating, so clearly the rst The hybrid GGA functional PBE0 (25% Hartree Fock unoccupied level must be triply degenerate. Elementary group exchange) just interpolates between RHF and PBE. The range- theory137 using the eleven L = 5 spherical harmonics (h-type separated GGA named ωB97100 has an orbital spectrum much − AOs) correctly predicts a HOMO of symmetry hu (in the point closer to Hartree Fock than the other DFA calculations group Ih), a lowest empty orbital of t1u, but the remaining t2u shown. Clearly the gap between HOMO and LVVO (lowest level found in the ungerade h spherical harmonics is not the VVO) varies greatly in Figure 4. However, the order of both π π* second lowest empty level of C60. Huckel theory and also band occupied and unoccupied is unchanged except for a minor structure calculations136 established the two lowest empty levels reordering of two very close lying π and two very close π*. The fi fi as rst t1u and then t1g. This is con rmed by recent high level quantitative values of the occupied eigenvalues and unoccupied calculations138 of the electronic excitation energies, where the VVO pseudoeigenvalues thus depend on the nature of the DFA 1 lowest Gg state arises from HOMO to t1u excitations, with operator used, but their energy order and shapes do not. In somewhat higher states arising from excitations involving the particular, the LVVO and LVVO + 1 are consistently and t1g. correctly predicted to be of t1u and t1g symmetry no matter The full set of VVOs below 0.4 hartree for C60 is shown in what functional is used. Figure 4. As expected, the LVVO and LVVO + 1 levels are the The valence-like t1u LCMO of C60 is also shown in Figure 4, as a blue line. Clearly, compared to HF, practical DFA calculations show a sharp drop in the LCMO energy, in contradistinction to a rise in the HOMO energy. For RHF there is no other empty CMO below zero, and there is only one more for the range separated ωB97, while the other DFA calculations have multiple empty CMO levels below zero. These shifts are discussed at the end of this section, after the remaining chemical examples. For the moment, this is expected behavior: most DFA calculations show a “DFA upshift” in their occupied orbitals, including the highest ones; this means DFA estimates of the ionization potential are usually too small. Empty DFA orbitals incur roughly the same amount of “DFA upshift” as for the occupied orbitals so that the band gap may be reasonably well predicted. Hartree−Fock levels for empty orbitals tend to be much too high, so in spite of also suffering a “DFA upshift”, the lowest empty DFA orbitals appear below those of HF. −2 Another icosahedral symmetry molecule, namely, B12H12 , has a 4-fold degenerate HOMO as well as a 4-fold degenerate LVVO! Conceivably, this is unique, as degeneracies of 4 or 5 can only occur in the rare icosahedral point groups. The high π π* degeneracy of both frontier orbitals is a result anticipated in a Figure 4. MO diagram of Ih symmetry C60. The 30 and 30 are − group theoretical/simple MO calculation on B12 icosahedra shown in red. Closed shell Hartree Fock and several DFT functionals 139 fi are used to assign the occupied orbital energies, and the valence virtual performed in 1955. To be speci c, for RHF/ACCT, the − pseudoeigenvalues, in hartree units, from ACCT. The orbital ordering HOMO is at 0.065 (gg symmetry), while the LVVO is at a is essentially unchanged from the labels at left except as indicated. substantially higher value of +0.553 hartree (gu). Both HOMO Variation of the HOMO and LVVO energy is discussed in the text. and LVVO orbitals have amplitude only within the B12 The blue line indicates the position of the LCMO orbital, whose shape polyhedron itself but not on the axial BH bonds (not is very similar to the LVVO in C60. In the Ih group, a, t, g, h symmetry illustrated). The LVVO shows no trace of diffuse spatial orbitals are 1-, 3-, 4-, 5-fold degenerate, respectively. character. Note that only very large basis sets yield a HOMO energy below zero in this doubly charged anion. π* fi t1u and t1g, followed by all remaining orbitals: all thirty of The nal example in this subsection is an orbital symmetry these appear before any σ* VVO occurs. Most of the highest diagram, showing the correlation of orbital energies along a ’ π occupied SCF orbitals are C60 s thirty orbitals, but the lowest reaction path. Figure 5 is a quantitative version of Figure 40 σ ’ ff ’ of these intermingle with some of the highest orbitals. C60 s found in Woodward and Ho mann s famous book on Hartree−Fock LCMO is shown with a blue line in Figure 4. conservation of orbital symmetry.2 The dihydrogen exchange ε − 2 The LCMO occurs at = 0.0232 hartree, has the correct t1u reaction between ethane and ethylene is symmetry allowed but symmetry, and is only slightly larger spatially than the LVVO was later shown to have a substantial energy barrier.140 The which has a nearly identical pseudoeigenvalue. However, the original WH diagram was drawn qualitatively, using labels S and ’ rest of C60 s unoccupied CMO level diagram consists of A for symmetry or antisymmetry with respect to a mirror plane numerous diffuse orbitals and resembles a continuum starting perpendicular to both CC bonds. Lacking any computation, the at zero energy (see ref 138 for a typical C60 CMO spectrum). original WH diagram simply showed three horizontal lines In addition to ordinary closed shell Hartree−Fock, Figure 4 (constant orbital energy) in the occupied space and three more shows the VVO positions (and the t1u LCMO level) for various in the virtual space. Figure 5 shows that two of the three types of density functional theory. Two pure DFA functionals occupied orbitals involved in the hydrogen exchange actually have nearly the same orbital energies: the GGA PBE96 and the rise in energy substantially near the transition state (see the metaGGA TPSS.101 Both have substantially raised occupied solid lines). This is easy to understand from the elongation of levels (compared to RHF) and substantially lowered VVOs. the reacting CH bonds at the saddle point. Of course, the

10418 DOI: 10.1021/acs.jpca.5b06893 J. Phys. Chem. A 2015, 119, 10408−10427 The Journal of Physical Chemistry A Article

Hartree−Fock due to the N − 1 nature of the KS potential compared to the N-electron mean field potential of Hartree− Fock. Their conclusion is the KS HOMO should be a good match to the experimental IP. Additionally, they show that molecules with low-lying states should (and do) possess one or more KS orbitals below zero energy. The lowest orbital occurs at a position above the HOMO that accurately matches the optical gap, which is probably a more precise term than the equivalent term band gap. In regards the topic of the next section, they also demonstrate that TD-DFT calculations within KS theory are expected to show excited states as almost clean occupied to empty individual single excitations. A number of these worker’s expected results are met by the SAOP functional,141 which gives near KS quality results. In practice, most DFA calculations differ from true KS calculations because the DFA exchange-correlation potential is shifted upward by a few eV.103,104 This shift in potential leads to what is called throughout this paper the “DFA upshift” in the eigenvalues, seen for C60 in Figure 4. Because the empty RHF eigenvalue (LVVO or LCMO) is much too high, even with some “DFA upshift”, the empty DFA orbitals occur lower than for RHF. However, the empty DFA orbitals are reasonably Figure 5. Orbital correlation diagram for the dihydrogen exchange “ ” reaction between ethane and ethylene. RHF/ACCT has a barrier of 82 positioned relative to the occupied levels, as both upshifts · −1/2 should be similar. To be somewhat more quantitative, the kcal/mol. The saddle point, at path distance 0.0 bohr amu , has D2h − symmetry (imaginary frequency 2334 cm 1 is shown in the inset) . following experimental facts for C60 may be considered (see The orbital energies (given in hartree) are colored according to their Figure 4, whose scale is in hartree units). The C60 ionization 142 symmetry type in the C2v symmetry of the reaction path: a1 = blue, a2 potential is 7.58 eV, or 0.28 hartree: clearly the RHF = red, b1 = purple, b2 = green. Solid lines represent the six reacting HOMO eigenvalue lies reasonably close to the negative of this orbitals and correspond to the original Woodward−Hoffmann IP, while the pure DFA eigenvalues (PBE and TPSS) are ’ diagram. WH s labels of S/A for symmetric/antisymmetric correspond “upshifted” considerably. Note that DFA calculations with to a1/b2 symmetry (blue/green). All other valence orbitals are shown inclusion of “exact exchange” through either range separation as dashed lines. (ωB97) or hybridization (PBE0) interpolate between RHF and ffi pure DFA. The electron a nity of C60 is measured as 2.689 143 antibonding orbitals become somewhat less antibonding by the eV, or 0.10 hartree. This value has little to do with any − same cause. Neither variance in energy during the reaction is orbital position of neutral C60, since the EA of a neutral sufficient to cause an orbital crossing, so the reaction remains corresponds most closely to the IP of its anion,103 but no data −1 “allowed”. Note that all spectator orbitals (namely, the CC σ in Figure 4 are for C60 . The band gap of C60 measured in and σ* and non-reacting CH σ and σ*) are also shown in films is 1.7 eV 144 (0.062 hartree) and in liquids 1.82 eV 138 Figure 5, as dotted lines. (0.067 hartree), values that agree fairly well with the separation ’ The examples in this section were chosen to be cases where between the HOMO and the LVVO (or C60 s very similar the nature of the lowest unoccupied orbital was already well- valence LCMO) for both pure DFAs, an indication that both established. VVOs matched those expectations and also the occupied and empty orbital have the same “DFA upshift”. provided sufficiently accurate predictions for all remaining Clearly, research into more meaningful eigenvalues from valence antibonding levels to enable the construction of DFA calculations is important but is not the central theme of “semiquantitative” full MO diagrams. this paper, which is concerned with the separation of the empty Before leaving the topic of MO diagrams, a few remarks orbital space into valence and nonvalence orbitals. Other about the accuracy of both HOMO and LVVO eigenvalues are workers often attempt to correct what this paper terms very in order. Of course, the quality of the VVO pseudoeigenvalues nonspecifically as a “DFA upshift” by addressing the self- cannot be expected to be any better than the usual Koopmans’ interaction error. This can be reduced by hybrids or range- theorem level of accuracy for occupied orbitals. For example, separation, but there are efforts to develop new functionals with − the HOMO eigenvalue only approximates an experimental IP self-interaction corrections.117,145 147 Other recent papers on 148 value. In addition, the C60 example in this section shows both DFA eigenvalues include a survey of DFA IPs and band gaps occupied and VVO energy values depend strongly on the DFA suggesting TD-DFT corrections may improve accuracy; the functional. But one can certainly hope that choosing any utility149 of KS HOMO positions for Koopmans’-type IPs; particular functional would predict trends in HOMO/LVVO discussion150 of the LCMO’s relationship to EAs; and an positions within a series of related molecules. evaluation151 of range-separation on HOMO and LCMO Readers interested in the prospects for more quantitative positions. VVO pseudoeigenvalues may consult some of the recent III.E. Valence Excited States. Frontier orbitals may also be literature. Of particular note is recent work by Baerends and co- expected to give insight into low-lying valence states of workers103,104 on the physical significance of unoccupied true molecules. For example, the ground state’s unoccupied orbitals Kohn−Sham orbitals (KS, as opposed to conventional DFA) in might be some indication as to the nature of the lowest valence relation to electron affinities and the optical gap. They expect state, upon excitation of an electron out of the occupied space the true behavior of KS orbitals to differ fundamentally from into the empty space. This can be explored by comparing the

10419 DOI: 10.1021/acs.jpca.5b06893 J. Phys. Chem. A 2015, 119, 10408−10427 The Journal of Physical Chemistry A Article

Table 2. Excitation Energies to the Lowest Singlet and Triplet States (in eV) and Oscillator Strengths f, Computed at the TD- a DFT/B3LYP/ACCT Level at RHF/ACCT Geometries Except As Indicated

no. of empty valence CMOs no. of VVOS with ε < 0 TD-DFT/B3LYP states

molecule charge RHF B3LYP RHF B3LYP E(S1) f(S1) E(T1) b Ru(bpy)3 +2 8 15 27 57 2.58 0.000

S7 2.99 0.125 blueOLEDc 0 0 8 0 15 2.67 0.000

S9 4.10 0.046 p-litmusd 0 0 3 0 8 4.19 0.006 3.91

S3 4.73 0.039 indigoe 0 1 3 0 6 2.62 0.395 1.44 luciferinf 0 0 3 0 7 4.25 0.366 3.18 arsenicing 0 0 4 0 6 3.83 0.033 3.40 3TCh 0 0 1 0 4 4.72 0.005 3.73

S2 4.92 0.133 thymine 0 0 1 0 3 4.93 0.000

S2 5.33 0.149 q-litmusd −1 0 1 0 0 2.37 0.007 2.02

S8 3.63 0.210 b-litmusd −1 0 1 0 0 1.51 0.000 1.49 orangei −1 0 1 0 0 2.38 0.001 2.05

S4 3.31 0.996 cAMPj −1 0 2 0 0 3.69 0.000 pentobarbk −1 0 0 0 0 3.75 0.001 benzylic 8l −1 0 2 0 0 1.26 0.002 0.84 a Where S1 has no appreciable intensity, the lowest state with some intensity at this level of theory is also listed. The number of unoccupied ground state canonical molecular orbitals with apparent valence character is given, as well as the number of ground state valence virtual orbitals with pseudoeigenvalues less than zero, at either RHF or B3LYP levels. bGeometry optimization by MP2/SPK-DZP. Results in table increase the basis to ff c SPK-TZP (no di use). The blue OLED O2S(C6H4-DMAC-DPS)2 of Zhang, Q.; Li, B.; Huang, S.; Nomura, H.; Tanaka, H.; Adachi, C. Nat. Photonics 2014, 8, 326−332. Geometry optimization was by B3LYP/ACCD, and because of the large size of this system, the excited state calculations also use ACCD rather than ACCT. dProtonated, quinoidal, and benzoquinodal forms of phenolphthalein. See Kunimoto, K.-K.; Sugiura, H.; Kato, T.; Senda, H.; Kuwae, A.; Hanai, K. Spectrochim. Acta 2001, A57, 265−271 for practical reasons the ACCD basis was used for RHF geometry optimization. eIndigo dye, see refs 153−155. fFirefly luciferin, from White, E. H.; Capra, F.; McElroy, W. D. J. Am. Chem. Soc. 1961, 83, 2402−2403. gStereoisomer (S)-arsenicin A, see refs 157−159. hLamivudine, aka 3TC, CAS no. 134678-17-4. iMethyl orange dye, CAS no. 547-58-0. jCyclic adenosine monophosphate, CAS no. 60-92-4. kPentobarbital (barbituate), CAS no. 76-74-4. lAnion number 8 of Perrotta, R. R.; Winter, A. H.; Coldren, W. H.; Falvey, D. E. J. Am. Chem. Soc. 2011, 133, 15553−15558). RHF/ACCD geometry was used. natural orbitals (both hole and particle) found for each excited the neutral and anionic molecules of Table 2, B3LYP yields one state to the occupied CMOs and to either unoccupied CMOs or more empty CMOs with apparent valence character. To or VVOs. understand the commonplace B3LYP prediction of valence Because the chemical community has adopted time-depend- LCMOs in Table 2, recall the discussion at the end of the ent density functional theory with the B3LYP functional as the previous section, regarding the expected behavior of KS most common treatment of excited states, this section presents calculations and thus many DFA calculations too. mainly TD-DFT results (within the Tamm/Dancoff approx- Figure 6 illustrates these remarks for the neutral thymine imation105) using the B3LYP functional and the ACCT basis molecule.152 RHF and the 100% Hartree−Fock exchange set. One set of results is presented for the equivalent wave PZ8195 functional clearly have diffuse LCMO orbitals, which function theory level: TD-HF (which is also known as singly bear no resemblance whatsoever to the TD-DFT natural orbital fi excited con guration interaction, or CIS). The molecules are which accepts the excited electron in the S1 state (shown at chosen to have low-lying valence-type excited states, mainly of lower right). As one might expect, the ground state’s lowest π* character, and are fairly large in size. Some were designed to valence virtual orbital (LVVO) does predict nicely the shape of have low-lying valence triplet states. One cation was considered, the excited state’s orbital (the LVVO for RHF, PZ81, and along with several neutrals and anions. Table 2 summarizes the B3LYP all look the same, so only the last is illustrated, bottom molecules considered: only pentobarbital anion lacks a S1 state row center). The B3LYP LCMO takes on valence character, so with valence character. like the LVVO, the B3LYP LCMO also correctly predicts the S1 In keeping with results presented above, one expects the state’s orbital shape. The apparent correctness of the thymine lowest empty Hartree−Fock canonical MOs in a basis set B3LYP LCMO (lower left) is expected behavior for KS containing diffuse functions to be spatially large except possibly calculations in molecules with a low-lying excited state103,104 for cations or if there is an exceptionally low-lying valence state and also occurs in most DFA applications. This result for (see C60). The second column of Table 2 shows that to be the thymine is quite typical for most other molecules in Table 2. case for all but one of its neutral and anionic systems: RHF Note that valence character for empty DFA canonical orbitals does indeed predict a spatially large nonvalence lowest does not occur in all molecules: see for example the spatially canonical molecular orbital (LCMO), assessed qualitatively by large TPSS-level LCMO for C2H6 in Figure 1. Of course, visualizing them,91 except for indigo. However, in all but one of ethane lacks low-lying valence excited states.

10420 DOI: 10.1021/acs.jpca.5b06893 J. Phys. Chem. A 2015, 119, 10408−10427 The Journal of Physical Chemistry A Article

Figure 6. Thymine orbitals. The LCMOs for RHF or for the pure correlation DFT functional PZ81 shown at the top are clearly similar. The B3LYP calculation’s LCMO + 1 at upper right resembles these Figure 7. Ground state and first excited state orbitals of indigo, treated two, but the B3LYP LCMO shown at lower left has valence character. fi This valence LCMO closely resembles the LVVO at bottom center, by RHF/ACCT and CIS/ACCT, respectively. The rst empty CMO closely resembles the first VVO. Higher canonical orbitals are spatially and both empty B3LYP orbitals are good predictors of the shape of the fi π* natural orbital to which the electron is excited in the first singlet state, large, as shown for LCMO + 1, while the rst six VVOs are various shown at lower right. The three valence orbitals at the bottom are shapes like the lowest two shown here. The natural orbitals of the − drawn at 0.05 bohr 3/2, while the three diffuse orbitals at the top are lowest singlet state of indigo show S1 is well described as a HOMO to shown at the reduced amplitude value of 0.025 bohr−3/2. LVVO (or LCMO) transition. Orbital energy values are in hartree, and occupation numbers are in electron units. The amplitude shown for the spatially large LCMO + 1 is reduced to 0.01 bohr−3/2, while all valence orbitals are drawn at 0.05 bohr−3/2. The only exception in Table 2 to finding a valence LCMO in B3LYP ground states is the relatively small pentobarbital anion, π which has only three nonconjugated CO occupied orbitals, in A more interesting example is arsenicin, As4O3(CH2)3, the addition to being negatively charged. The spatially large LCMO first molecule containing more than one As to be isolated from − for B3LYP-level pentobarbital does successfully predict the a living organism (a sponge found in New Caledonia).157 159 shape of the natural orbital of its S1 electronic state, which is The molecule can be considered to be derived from found to possess Rydberg character. adamantane, by substituting its CH groups by As and half its Two of the remaining examples of valence states from Table CH2 groups by O. It is also related to the laboratory compound 2 will now be discussed in more detail. Note that natural As4O6. The experimental spectrum shows four peaks in the orbitals of each specific excited state are generated to convert visible region, albeit “in the absence of an obvious the possibly numerous excitation amplitudes for computations chromophore”.158 However, the observed transitions were with CMOs into a single orbital from which the electron is explained by TD-DFT calculations, as due to excitation into σ* removed and another orbital which accepts it. orbitals. The B3LYP functional (and others) were found to give As the next example, consider the blue dye molecule a good match to experimental observations. Results in the indigo,153 which has very low-lying excited states.154,155 In fact, present work are obtained by TD-DFT using the B3LYP this means that even for RHF the LCMO has valence character, functional and the larger ACCT basis set. The LVVO and as shown in Figure 7. By TD-HF (aka CIS) with the ACCT LVVO + 1 of arsenicin are illustrated in Figure 8, along with the − basis set, the S1 state is found to lie at 3.77 eV, not surprisingly HOMO and HOMO 1. The four peaks in the experimental less accurate than B3LYP’s 2.62 eV compared to experiment’s spectrum (according to computed intensities) are identified as 156 2.28 eV. Note that the LVVO is almost identical to the S1 (3.82 eV), S4 (4.66 eV), S10 (5.21 eV), and S13 (5.56 eV). S1 LCMO. The Hartree−Fock LCMO and LVVO are both is a clear-cut example of a HOMO to LVVO excitation, while excellent predictors of the shape of the upper orbital in the S1 the next peak in the spectrum involves four orbitals but with state, which differs only slightly from them, at the atom the primary character being HOMO − 1 to LVVO + 1. This indicated by the red arrow in Figure 7. Not surprisingly, the state also slightly depopulates the HOMO by placing about 0.2 HOMO of the ground state is also very similar to the orbital e− in an orbital resembling the LVVO (these two are not shown fi from which the electron is excited in S1. Note that the LCMO + to keep the gure simple). Like indigo, this is a molecule where 1 lies at an almost identical energy to the LCMO and is now a canonical virtuals also do well: the B3LYP LCMO and LCMO spatially large orbital. No illustrations are provided in Figure 7 + 1 (not illustrated) both closely resemble the LVVO and for states above S1, but a short description follows. There are LVVO + 1. many densely spaced and spatially very large orbitals above the In general, the remaining molecules in Table 2 have both the LCMO + 1, whereas the first six VVOs are various kinds of π* LCMO and LVVO from B3LYP calculations appearing to be fi σ* orbitals, followed by the rst valence . The S2 excited state nearly identical, as was explicitly shown for thymine (in Figure (4.51 eV by TD-HF/ACCT) involves four natural orbitals, with 6). Also, as for all three examples shown, the two fractionally − electron occupancies 1.788, 1.265 (this is the HOMO 1), occupied natural orbitals of the S1 states are quite well 0.735 (this resembles the LVVO), and 0.212 (this resembles predicted by the shapes of the HOMO and LCMO/LVVO. ’ the LVVO + 1). The S3 state (4.63 eV by TD-HF/ACCT) is n Typically, the S2 state s natural orbitals correspond to the → π* type, with occupations 1.814, 1.197, 0.802, and 0.183 excitation from HOMO − 1 to the LCMO/LVVO. Higher where once again the more important accepting orbital excitation energies (S3 on up) do utilize the LVVO + 1 as an resembles the Hartree−Fock LCMO/LVVO. upper orbital but frequently involve sufficient energy to

10421 DOI: 10.1021/acs.jpca.5b06893 J. Phys. Chem. A 2015, 119, 10408−10427 The Journal of Physical Chemistry A Article

the VVO orbital shapes: if one freezes the occupied orbitals at their SCF shapes but optimizes the three weakly occupied orbitals for HNO, the CAS-SCF energy is −129.9753 hartree. Thus, the missing 10.5% of the MCSCF energy is partitioned as 8.0% due to relaxation (optimization) of the three VVOs, with 2.5% due to relaxation of the six occupied SCF orbitals of HNO. This is reasonable, as the occupied orbitals of the SCF may variationally optimize their usage of the diffuse or polarization parts of the working basis set during the SCF step. In contrast, the singular value decomposition selects VVOs based on their closeness to occupied atomic orbitals, and thus the VVOs have little contribution from the diffuse/ polarization basis set extensions. Two additional examples, similar to those given elsewhere,5 are diborane and dioxirane. Full valence CI within a valence orbital space containing all occupied MOs and all VVOs recovers 86.3% and 87.3% of the optimized full valence CAS- SCF energies for diborane (14 e− in 12ϕ) or dioxirane (18 e− in 14ϕ), respectively. It is reasonable to consider the union of occupied SCF orbitals with VVOs, followed by a full CI in those orbitals, as “a poor man’s MCSCF”. In cases where the molecule is too large to permit full valence MCSCF calculations, VVOs can still guide the calculation in 160,161 ’ two ways. For o-benzyne a reasonable active space is Figure 8. Comparison of the ground state s B3LYP/ACCT occupied obvious on chemical grounds: the three π and three π* orbitals CMOs and empty VVOs in As4O3(CH2)3 to the natural orbitals of the 1 1 perpendicular to the ring plus the in-plane π and π*. Here, the TDDFT BS1 and AS4 valence excited states. Orbital energies are in hartree for the former, while the latter are labeled by their electron symmetry constrained localization, applied separately in the occupied and VVO spaces, generates excellent starting orbitals. occupation numbers. The view is along the C2 axis of the molecule. The initial MCSCF iteration using these localized starting orbitals already recovers 83% of the correlation energy scramble its shape with the LVVO and to have mixed contained in this eight electron, eight orbital active space. Of occupancies. Typically, the higher states have four natural course there are also cases where the active space is much more orbitals with mixed occupancies and resemble combinations of difficult to anticipate from chemical intuition. One can simply the ground state’s HOMO − 1, HOMO, LVVO, and LVVO + generate and visualize VVOs to consider the low energy VVOs 1. Arsenicin is a notable counterexample, as its S4 state can be as active orbital candidates. If the active space is still not qualitatively considered as a simple HOMO − 1 to LVVO + 1 obvious, one can perform a limited CI calculation, such as single excitation. singles and doubles only, from all occupied orbitals into all Clearly the chemical community’s oft-made choice of B3LYP VVOs, which is feasible for rather large molecules. The resulting TD-DFT calculations for large molecule excited state natural orbitals of this CI-SD will reveal those orbitals whose predictions is related to the circumstance that the ground occupation deviates significantly from 2.0 or 0.0: these ’ state s LCMO will often have valence character, when the S1 fractionally occupied orbitals constitute an automatically state is a valence state, whereas RHF often does not. The generated set of “active orbitals” for the system’s multi- LVVO from either Hartree−Fock or DFA calculations is configurational wave function. One can repeat this process at capable of predicting the lowest valence state, even in cases like several geometries typical of the PES being examined to ensure fi pentobarbital anion where S1 is not a valence state, but higher an entire set of important active orbitals is identi ed. states are. Of course, full valence MCSCF is not the only kind of wave III.F. VVOs as MCSCF Starting Orbitals. In addition to function that involves the occupation of antibonding levels. The the conceptual value of VVOs, discussed above, the VVOs are GVB perfect pairing (PP) wave function is another example. also found to have practical use in designing and carrying out This wave function contains pairs of orbitals (geminals) in multireference calculations. which a bonding orbital is doubly excited into its antibonding As shown elsewhere,5 the occupied SCF orbitals plus the counterpart. This is precisely the type of orbital pairs found VVOs form a very good approximation to the full valence space when localizing the occupied and the VVO spaces separately. MCSCF wave function. Performing a full CI calculation within Consequently, the glycine example proposed as a GVB-PP(10) the valence orbital space recovers around 80−90% of the convergence test162 is readily convergent from VVOs, after correlation energy found after CAS-SCF orbital optimization. preparing ten perfectly paired starting geminals by matching This is impressive, considering the O(N3)effort in preparing each of the ten localized closed shell bonding orbitals with its VVOs, compared to the O(N5) integral transformation work in corresponding ten antibonding orbitals, obtained by localizing running a MCSCF program. As an example, the SCF energy of the VVOs. The RHF energy of glycine is −282.8373 hartree, HNO is −129.8516 hartree, the full valence CI using its closed the initial iteration with the ten starting geminals is −282.9475 shell VVOs is −129.9652 hartree, and a fully relaxed CASSCF hartree, and convergence to the optimal GVB-PP(10) energy of is −129.9784 hartree (energies are from the aug-cc-pVQZ basis −283.0194 hartree is smooth. The initial geminals generated set). The VVOs thus provide 89.5% of the near-degeneracy from VVOs thus provided some 60% of the GVB-PP(10) correlation recovered by the CAS-SCF. Most of the defect is in correlation energy.

10422 DOI: 10.1021/acs.jpca.5b06893 J. Phys. Chem. A 2015, 119, 10408−10427 The Journal of Physical Chemistry A Article IV. SUMMARY ■ REFERENCES This paper has illustrated the previously known result (at least (1) Fukui, K.; Yonezawa, T.; Shingu, H. A Molecular Orbital Theory among theoretical chemists) that for most neutral molecule of Reactivity in Aromatic Hydrocarbons. J. Chem. Phys. 1952, 20, 722− SCF calculations, the lowest canonical MO changes shape with 725. (2) Woodward, R. B.; Hoffmann, R. The Conservation of Orbital an increasing basis set, becoming spatially large and moving to Symmetry; Verlag Chemie GmbH: Weinheim, Germany, 1971. an energy near zero. The LCMO therefore does not (3) Lu, W. C.; Wang, C. Z.; Schmidt, M. W.; Bytautas, L.; Ho, K. M.; correspond well to the LUMO concept in chemistry. The Ruedenberg, K. Molecule Intrinsic Minimal Basis Sets. I. 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10426 DOI: 10.1021/acs.jpca.5b06893 J. Phys. Chem. A 2015, 119, 10408−10427 The Journal of Physical Chemistry A Article ical Possessing Strong Anti-acute Promelocytic Leukemia Cell Line Activity. Organometallics 2012, 31, 1808−1816. (160) Wentrup, C. The Benzyne Story. Aust. J. Chem. 2010, 63, 979− 986. (161) Groner, P.; Kukolich, S. G. Equilibrium Structure of Gas Phase o-Benzyne. J. Mol. Struct. 2006, 780−781, 178−181. (162) Muller, R. P.; Langlois, J.-M.; Ringnalda, M. N.; Friesner, R. A.; Goddard, W. A. A Generalized Direct Inversion in the Iterative Subspace Approach for Generalized Valence Bond Wave Functions. J. Chem. Phys. 1994, 100, 1226−1235.

10427 DOI: 10.1021/acs.jpca.5b06893 J. Phys. Chem. A 2015, 119, 10408−10427