Atom-Based Strong Correlation Method: An Orbital Selection Algorithm

Article Cite This: J. Phys. Chem. A XXXX, XXX, XXX-XXX

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Atom-Based Strong Correlation Method: An Orbital Selection Algorithm Aaron C. West* Department of Chemistry, Iowa State University, Ames, Iowa 50011-3111, United States ABSTRACT: The present study proposes a methodology that advances the selection of initial orbitals for subsequent use in correlation calculations. The initial orbital sets used herein are split-localized orbitals that span a full-valence orbital space and were developed in a previous study (J. Chem. Phys. 2013, 139, 234107) in order to reveal the bonding patterns of molecules in a specific, quantitative manner. On the basis of the quantitative chemical features of these localized orbitals, this new method systematically extracts orbital sets and assigns excitation levels that systematically recover strong correlation with smaller numbers of configurations than can be achieved with traditional as well as nontraditional correlation methods. Moreover, this method not only provides organized initial orbitals for correlation calculations but also results in compact configuration interaction expansions via the use of the split-localized orbitals.

1. INTRODUCTION In quantum chemistry to date, affordable and accurate electronic structure methods often require initial guess molecular orbitals. Over the years, theory and atomic computations led to the concept of and quantification of the minimal basis set (MBS) for atoms. In fact, the MBS concept extends to molecules. For low energy electronic states, the electron distribution primarily resides in the full-valence orbitals, which by definition have a dimension no larger than the total number of MBS orbitals on all atoms in a given molecule. Furthermore, when obtained from an energy optimization, the full-valence, MBS orbitals can be exactly expressed1−4 in terms of deformed atomic-like orbitals, which are coined the quasi-atomic orbitals. While weakly correlating orbitals are clearly important for recovering dynamic correlation, quantitatively accurate MBS orbitals retain the most importance (see Introduction and section IIA of ref 5). Given such a validation, many analyses attempt to construct and then resolve the molecular orbitals into internal and external orbital sets where the internal orbital set is defined as the molecular orbitals whose energy contributions dominate the bonding energy. The internal orbital set often but not always has the MBS dimension. Of course, such a resolution is approximate when an approximate wave function generates the given orbital set for a particular molecular problem. As a result, quantum chemists devote much time to creating, examining, and sorting qualitative MBS orbitals for subsequent use and very often further refinement in more accurate wave functions, such as configuration interaction (CI) calculations or multiconfigurational selfconsistent field (MCSCF)6 wave functions, which include the full optimized reaction space (FORS),1−3,7 complete active space self-consistent field (CASSCF),8−10 and occupation restricted multiple active space (ORMAS)12,11 methods. When followed by standard localization procedures (e.g., Boys13 or Edmiston−Ruedenberg14 localizations), the Hartree−Fock (HF) energy optimization sometimes provides guess occupied orbitals for subsequent use in correlated methods. However, the HF bonding and lone pair occupied orbitals © XXXX American Chemical Society

frequently prove unsuitable for a number of reasons. For example, a subsequent MCSCF energy optimization might lead to occupied active orbitals that include correlating orbitals outside the MBS in an improper manner.15−17 As another example, when a particular minimum energy pathway (MEP) is studied, wellchosen MCSCF active orbitals might prove to be insufficiently flexible6,18 to produce the correct qualitative features across the entire MEP. Furthermore, it is well-known that the HF energy optimization cannot provide suitable unoccupied orbitals of the MBS-type. Over the years, researchers have tried to recover suitable unoccupied orbitals in an affordable manner.19−29 One such approach affordably recovers suitable unoccupied orbitals through the use of orbital overlaps. Using a saturated set of optimized uncontracted one-center Gaussian primitives, extremely accurate free-atom self-consistent field (SCF) calculations yield the accurate atomic minimal basis set (AAMBS)23−25,28,29 orbitals, which form quantitative MBS orbitals on each atom. Orbital overlaps are formed between the interatomically orthogonalized AAMBS orbitals and the virtual orbitals from the HF energy optimization. By application of the singular value decomposition (SVD) to these orbital overlaps and subsequent transformation of the virtuals, the valence virtual orbitals (VVOs)23−25,28,29 are formed. The HF occupied orbitals and VVOs approximate the AAMBS orbitals and are expressed in the current working atomic orbital basis.30 Together, the HF occupied orbitals and the VVOs closely approximate the full-valence orbital space from a FORS calculation (by “closely approximate”, see section V.A.2 of ref 5 for a variety of properties). For example, when used in a full-valence CI calculation, these MBS orbitals recover ∼80% of the energy from the HF to the FORS wave function.28 While the VVOs span a MBS orbital set, quasi-atomic orbitals (QUAOs)5,31−33 must be determined in terms of the Received: August 24, 2017 Revised: October 20, 2017

A

DOI: 10.1021/acs.jpca.7b08482 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry A

is divided up into system and environment blocks. Computationally manageable numbers of electronic basis “block” states are formed through state decimation while iteratively growing and shrinking the orbital blocks. However, for large complex systems, smaller block sizes can yield less accurate energies, and for a given molecular orbital basis, smaller block sizes have larger amounts of interblock coupling, or quantum mechanical “entanglement”. The DMRG method can actually be expressed in terms of the matrix product states (MPS) method, which in turn falls under the more general tensor network states (TNS) approach.53 As a result of these advances, many research groups have started to investigate how to automatically select active orbitals (for example, see reference54) using TNS-type methods. However, it is well-known that the block size, orbital order, and individual orbital character all determine the convergence, accuracy, and computational expense of TNStype methods, such as DMRG (for example, see Section 4.4.6 of ref 53 and references therein). In regards to these matters, the present study provides several improvements: (1) how to better select orbitals to reduce block entanglement, (2) computationally cheaper diagonalizations in the initialization steps, and (3) the reduction of the number of block states while maintaining an accurate energy. However, while all these CI-based methods certainly reduce the number of configurations, a better approach requires the coupling of a localized orbital basis to provide local correlation methods, which were actually introduced in the 1960s.55−57 The OQUAO and SLMO localized orbital sets provide an opportunity to design a specific CI method that systematically reduces the full-valence configuration expansion size through quantitatively constructed and quantitatively chosen SLMOs. As will be shown in the Methods, the present study combines the information in the OQUAOs and SLMOs in order to directly remove ineffective configurations by identifying ineffective orbital pair excitations. In future studies, this localized orbital approach will be applied in CI calculations to reduce the CI expansion size in a systematic manner. This approach makes some advancement toward solving the strong correlation problem. To validate the present study does not require exact minimum geometries. In addition, this study does not seek to resolve property discontinuities when the active space qualitatively changes over a range of geometries.18,58 These problems are independent of the present analysis, which allows for a generalized approach at any geometry where restricted Hartree−Fock (RHF) or restricted open-shell Hartree−Fock (ROHF) is applicable.

working orbital basis. The HF occupied orbitals and VVOs frequently form the internal orbital set. For all the internal orbitals, overlaps are formed to the AAMBS orbitals for a single atom A. This rectangular overlap matrix is diagonalized, and the resulting transform yields the QUAOs for atom A. Repeating this procedure for each atom followed by orthogonalization gives orthogonal QUAOs between all atoms. However, the formation of the QUAOs leaves arbitrary intra-atomic orbital mixings. In previous work,34,35 a process termed orientation defines specific intra-atomic orbital mixings that result in quantitatively hybridized QUAOs, which then actually appear as hybridized orbitals (i.e., sp, sp2, sp3, etc.) that point in the directions of the atoms of the given molecule. By design, the OQUAO bond orders (BOs)36,37 quantitatively indicate population shifts between atoms, and these population shifts represent how electron charge accumulates in bonds between atoms. In terms of interference energy contributions that result in bonding,38 the OQUAO BOs pAa,Bb contribute as follows: ⎤ ⎡ ∇2 pAa,Bb ⎢⟨Aa| − |Bb⟩ + ⟨Aa|V |Bb⟩⎥ 2 ⎦ ⎣

(0)

where V represents various one- and two-electron potential operators and where |Aa⟩ denotes OQUAO a on atom A. The OQUAO BOs alone obviously lack some bonding information since the energy contributions consist of BOs and occupations that are scaled by integrals, which depend on interatomic distance between the OQUAOs. Because the kinetic energy contribution typically dominates the interference energy contribution, the OQUAO kinetic energy integrals ∇2 |Bb⟩ and 2 ∇2 pAa,Bb ⟨Aa| − 2 |Bb⟩

(KEIs) ⟨Aa| −

OQUAO kinetic bond orders

(KBOs) provide a computationally inexpensive energy-based measurement of bonding that also takes into account interatomic distance dependence. The OQUAOs provide important bonding information. However, slightly less localized orbital sets have additional advantages, such as the effective generation of nondynamic correlation. Separate localizations in the occupied orbital and VVO spaces provide a means to quantatively approximate lone pair, bonding, and antibonding orbitals. Bytautas et al.39 first suggested the name “split-localized” orbitals and demonstrated that these types of orbitals can yield a more compact CI expansion than natural orbitals (NOs) and thus a more rapid CI convergence. In previous studies, a specific fast algorithm was developed to create split-localized molecular orbitals (SLMOs)5,31−33 from OQUAOs. The previously developed fast algorithm that creates SLMOs should prove useful if these localized orbitals can be coupled to a variety of different CI methods. Many CI methods exist that, in some manner, reduce the number of configurations relative to a complete active space. “Traditional” methods include but are not limited to the following: restricted active space,40 restricted configuration interaction,41,42 quasi-complete active space,43 general active space,44 macroconfigurational,45 ORMAS,12,11 and correlating participating orbitals46−48 methods. More recent approaches are largely based on the renormalization group approach. White49,50 proposed the density matrix renormalization group (DMRG) method, which was subsequently introduced51,52 into quantum chemistry. The DMRG method approximates the full configuration Hilbert space of a given orbital space. In the DMRG method, a given orbital space

2. METHODS 2.1. Localization Procedure. The present work utilizes several kinds of localized orbitals. Several previous papers5,23,28,31,34 describe the procedures that form these various localized orbitals. The present study uses orbitals from HF wave functions, so the HF localized orbital procedure is now summarized. Extremely accurate free-atom SCF calculations result in the AAMBS orbitals,23−25,28 which are stored in Fortran code in arrays for all atoms up through Xenon. The AAMBS orbitals have quantitative s-, p-, and d-character. The energy optimized HF occupied orbitals form part of the MBS orbital set. To extract the remaining MBS orbitals, overlaps are formed between the interatomically orthogonalized AAMBS orbitals and the virtual orbitals. These overlaps are diagonalized, and the resulting orbital transform produces the VVOs.23−25,28 Together, the HF occupied orbitals and the VVOs form an internal orbital set. Between the AAMBS orbitals on each atom B


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Table 1. Automatic Extraction of Each SLMOb Group, Definition of Each SLMO Group in Terms of OQUAOsc, and the Reference Electron Countd of each SLMO Group for Carbonate Anion Minimum Geometrya

and all the internal orbitals, orbital overlaps are formed and separately diagonalized for each atom. The internal orbitals are transformed to approximate the AAMBS orbitals on each atom while still spanning the molecular orbital space. Then, these quasi-atomic internal orbitals are orthogonalized to preserve the maximal AAMBS character (see appendix of ref 31). This process results in the canonical QUAOs.5,31 As previously described in the Introduction and several references,5,34 further intra-atomic orbital mixings quantitatively hybridize the canonical QUAOs into OQUAOs.34,35 The orientation procedure compacts the first-order density bonding interactions. In other words, through covalent binding between atoms in the first-order density, the OQUAOs have as quantitatively few interatomic bond orders (BOs)36,37 as possible based on a fourth power maximization algorithm. From the OQAUOs, the SLMOs5,31 are formed. The HF occupied orbitals and HF VVOs serve as starting orbitals. Each starting orbital is projected onto each OQUAO. As described in refs 5 and 31, the fourth power sum of all the aforementioned projections is maximized separately for the HF occupied orbitals (one orbital space) and for the HF VVOs (another orbital space). Thus, for each separate orbital space, the resulting SLMOs span as few OQAUOs as possible under the constraints of orbital orthogonality and of the orbital division from the given energy optimization itself. 2.2. Orbital Selection Algorithm. Let ψj denote the SLMOs. Let φi denote the OQUAOs. The orthonormal transformation Tij relates the two orbital sets by ψj =

∑ φiTij

KCij =

(2)

n

(3)

4 In this context, kurtosis is defined as n n ⎛ |T | − M ⎞ 4 1 ij ⎟ ∑∑⎜ σ N j=1 i=1 ⎝ ⎠

(5)

n

1 ∑ ∑ (KCij 2)(KCij 2 − 1) N j=1 i=1

(6)

At this stage, kurtosis OQUAOs explicitly define each SLMO. Next, each preliminary orbital group is formed from SLMOs whereby each SLMO is composed of the same type of and same number of kurtosis OQUAOs. Now, in order to connect very important orbital groups that might be separated in the preliminary group assignment (e.g., see later the π system of carbonate anion), two alternatives exist. For the first alternative, preliminary orbital groups are collected into orbital groups based on common kurtosis OQUAOs. In other words, the same kurtosis OQUAO cannot occur in more than one orbital group. So, different orbital groups never hold the same kurtosis OQUAO. For the second alternative, each SLMO is placed into its own one orbital group, and later on the method of assigning excitations between groups then connects the important groups. The present study uses a combination of both alternatives to form the overall scheme. The first alternative serves two purposes: (A) to separate out lone pair, radical orbitals, and collect SLMOs with common OQUAOs into contiguous order

3 Calculate the standard deviation σ: σ=

σ

n

1+

where n stands for the total number of valence MBS molecular orbitals and N = n × n. n

|Tij| − M

6 Store the smallest cutoff value T cut = |Tij| that corresponds to the smallest KCij value >1. For each SLMO ψj, perform the following: 1 Store the kurtosis OQUAO index i where |Tij| ≥ Tcut. Kurtosis measurements yield information about both the peak and the tail of a distribution. Within the particular usage of kurtosis in the present study, the kurtosis OQUAOs individually increase the total kurtosis while the nonkurtosis OQUAOs individually decrease the total kurtosis about the value +1.59 This idea is borne out in eq 6.

n

1 ∑ ∑ (|Tij| − M)2 N j=1 i=1

2 2 2 2 2 2 2 2 2 0 2 0 2 0 2 0

The three oxygen atoms are distinguished by indices of 2, 3, and 4. SLMO refers to split-localized molecular orbital. cOQUAOs refers to oriented quasi-atomic orbitals. dReference electron count gives the integer occupation for the orbital group.

1 Sort Tij by descending magnitude. 2 Calculate the mean M: n

reference electron count

O3sS O2sS O4sS O3pS O2pS O4pS O2c1π, C1o2o3o4π O4c1π, C1o2o3o4π O3c1π, C1o2o3o4π C1o2o3o4π, O2c1π, O3c1π, O4c1π O2c1σ, C1o2σ C1o2σ, O2c1σ O3c1σ, C1o3σ C1o3σ, O3c1σ O4c1σ, C1o4σ C1o4σ, O4c1σ

b

Since the SLMOs are determined by a fourth power maximization algorithm that disproportionates the transformation Tij elements for each SLMO ψj, a few distinct OQUAOs can be assigned to the SLMOs by calculating the fourth power (i.e., kurtosis) contribution. Later, these distinct OQUAOs are referred to as the kurtosis OQUAOs. These distinct OQUAOs are the statistical outliers of all OQUAOs that make up each given SLMO. For all SLMOs ψj, perform the following:

1 M = ∑ ∑ |Tij| N j=1 i=1

OQUAOs

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 a

(1)

i

group

(4)

5 For each OQUAO i, calculate the kurtosis contribution KCij: C


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Figure 1. Split-localized molecular orbitals (SLMOs) and oriented quasi-atomic orbitals (OQUAOs) of carbonate anion minimum geometry. Only the symmetry-unique orbitals are shown. The first row contains the lone pair OQUAOs. (The lone pair SLMOs are similar and are not shown.) The second row contains the OQUAOs. The third row contains the SLMOs.

Figure 2. Number of split-localized molecular orbital (SLMO) group pairs versus the kinetic energy integral magnitude (|KEI|) in hartree for carbonate anion minimum geometry: (a) group pairs with intra-atomic excitations, (b) group pairs of interatomic doubly occupied orbitals, and (c) group pairs of intra-atomic doubly occupied orbitals.

to larger numbers of configurations. In contrast, for the preliminary orbital grouping, each orbital group consists of N SLMOs where each SLMO consists of M kurtosis OQUAOs. Here, N and M are integers that may or may not be equal. For example, bond-antibond pairs such as the σ SLMO and

and (B) to extract any clear-cut pairs of bond-antibond SLMOs. With the common OQUAO collection, the first alternative leads to larger orbital groups, which is beneficial for collecting all relevant SLMOs into a single orbital group (e.g., full π system of carbonate anion). However, that approach leads D


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a Place doubly occupied SLMOs with a single kurtosis OQUAO first in the list. b Sort the SLMOs from the previous step by descending occupation of the corresponding OQUAOs. c Sort the remaining singly occupied SLMOs with a single kurtosis OQUAO in a similar manner. 6 Apply the second alternative orbital orbital grouping (i.e., use all one orbital groups). 7 For each orbital group, calculate the integer occupation. These integer occupations represent reference electron counts when no intergroup excitations are allowed. For larger molecules, the above OQUAO occupation sort is useful in order to remove configurations by positioning orbitals with the most lone pair character first in the list. If desired, the first orbitals in the list may be assigned as doubly occupied inactive orbitals in the CI procedure. At this stage, a list of unique orbital group pairs is constructed that indicates allowed excitations between two groups. First, “simple” bond-antibond SLMO pairs have the same two kurtosis OQUAOs; so, these SLMO pairs are added to this list with double excitations. Second, since SLMOs on the same atom are extremely important in correlation,4 intra-atomic group pairs are added to the unique orbital group pair list where the two intra-atomic SLMOs have at least one kurtosis OQUAO on the same atom. For any two orbital groups on the same atom, up to double excitations are allowed. Third, as explained in the Introduction, the interatomic KEIs convey a measure of the distance among the interatomic orbital groups, and kurtosis may be applied to the interatomic KEIs in order to give the most important interatomic group pairs the largest computationally affordable excitations in the correlation calculation. Here, two interatomic SLMOs have no kurtosis OQUAOs on the same atom. The present study uses the largest KEI magnitude found between all the kurtosis OQUAOs for a given pair of SLMOs. Add the most important interatomic group pairs to the unique orbital group pair list as

Figure 3. Definitions for all atom indices for the hexane minimum geometry.

the σ* SLMO each have the same two kurtosis OQUAOs. On the other hand, for a more complicated example like the π system of carbonate anion, the preliminary group assignment has four kurtosis OQUAOs that contribute to one delocalized SLMO and three pairs of those same four kurtosis OQUAOs that contribute to three other SLMOs. After the formation of preliminary orbital groups, the application of the first alternative with common OQUAO collection, and the removal of the groupings due to the common OQUAOs, the second alternative of orbital group formation serves two purposes: (A) to recover excitations for more complicated and delocalized SLMOs through KEIs and (B) to further reduce the number of configurations relative to the use of common OQUAOs to form groups. At this point, perform the following: 1 Form preliminary orbital groups. 2 Apply the first alternative orbital grouping with common OQUAO collection. 3 Remove the groupings that result from the common OQUAO collection but keep the orbital reordering. 4 Sort the current orbital groups by increasing number of SLMOs in each group. 5 Sort the one orbital groups (if any) as follows:

Figure 4. Split-localized molecular orbitals (SLMOs) of hexane minimum geometry. Only the bin-unique SLMOs are shown (see Methods section). The first nonsubscript number gives the bin-unique number shown in Table 2. The atom subscript labels are given in Figure 3. E


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Table 2. Automatic Extraction of each SLMOb Group for Hexane Minimum Geometrya

follows: 1 Set EXCUR = EXINTER+1 where EXINTER = 2. 2 Set EXCUR = EXCUR−1. 3 Sort the current interatomic |KEI| collection by descending magnitude. 4 Apply kurtosis to the current interatomic |KEI| collection. 5 For any kurtosis contributions >1, add those interatomic group pairs to the unique orbital group pair list with the current excitation level EXCUR. 6 For any interatomic group pairs added to the list in the previous step, remove the corresponding |KEI| values from the current interatomic |KEI| collection. Cycle back to step 2 until EXCUR = 0. At this stage, taking into account all the various reordering processes, reorder the SLMOs into the new contiguous orbital group order. For the given program, create lists for the following items: 1 Starting orbital numbers for the orbital groups. 2 The allowed minimum and maximum electron counts for each orbital group. 3 The reference electron counts for each orbital group. 4 The unique orbital group pair list that indicates which groups are allowed excitations. 5 The excitation levels for the group pairs in the unique orbital group pair list.

3. RESULTS AND DISCUSSION All computations were done with General Atomic and Molecular Electronic Structure System (GAMESS) software.60,61 Unless otherwise explicitly noted in the Results and Discussion section, for each molecule, minimum HF geometries were located and verified by nuclear Hessian analysis, and the cc-pVTZ basis62 was used. As described in the Methods section, the localization procedure and orbital selection algorithm were then performed. The orbital contour surfaces displayed in the subsequent figures correspond to absolute orbital values of 0.1 (e/bohr3)1/2.

group

bin-unique number

OQUAOs

reference electron count

35 36 29 30 37 38 23 24 25 26 9 10 11 12 13 14 33 34 17 18 19 20 5 6 7 8 15 16 31 32 1 2 3 4 21 22 27 28

1 1 2 2 2 2 3 3 3 3 4 4 4 4 4 4 4 4 5 5 5 5 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7

C8c11σ, C11c8σ C11c8σ, C8c11σ C11c14σ, C14c11σ C14c11σ, C11c14σ C8c5σ, C5c8σ C5c8σ, C8c5σ C17c14σ, C14c17σ C14c17σ, C17c14σ C1c5σ, C5c1σ C5c1σ, C1c5σ C11h12σ, H12c11σ H12c11σ, C11h12σ C8h10σ, H10c8σ H10c8σ, C8h10σ C11h13σ, H13c11σ H13c11σ, C11h13σ C8h9σ, H9c8σ H9c8σ, C8h9σ C17h18σ, H18c17σ H18c17σ, C17h18σ C1h2σ, H2c1σ H2c1σ, C1h2σ C5h6σ, H6c5σ H6c5σ, C5h6σ C14h16σ, H16c14σ H16c14σ, C14h16σ C5h7σ, H7c5σ H7c5σ, C5h7σ C14h15σ, H15c14σ H15c14σ, C14h15σ C17h19σ, H19c17σ H19c17σ, C17h19σ C1h3σ, H3c1σ H3c1σ, C1h3σ C17h20σ, H20c17σ H20c17σ, C17h20σ C1h4σ, H4c1σ H4c1σ, C1h4σ

2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0

a The second column shows the bin-uniquec SLMOs involved in bondantibond groups shown in Figure 4 and shown in Figure 5 by decreasing kinetic energy integral magnitude (|KEI|). The definition of each SLMO group is given in terms of OQUAOsd. The reference electron counte for each SLMO group is given. bSLMO refers to split-localized molecular orbital. cThe term bin-unique refers to unique orbital group pairs for degenerate |KEI| values (see in the beginning of Section 3). d OQUAOs refers to oriented quasi-atomic orbitals. eReference electron count gives the integer occupation for the orbital group.

In order to present the orbital results in an orderly fashion, labels are created for both the OQUAOs and SLMOs. These labels have no impact on how the kurtosis contribution identif ies and relates the SLMOs and OQUAOs. Furthermore, these labels do not effect how kurtosis assigns excitations between SLMO groups. The OQUAO labels are defined as follows. The first atomic symbol is given in upper case and gives the central, or home, atom of the given OQUAO. If the OQUAO has a substantial (i.e., 0.4) bond order to and is thus directed toward another OQUAO on a different atom, then a second atomic symbol is given in lower case for that different atom, which is called a

Figure 5. Bond−antibond excitation category for hexane minimum geometry. The number of split-localized molecular orbital (SLMO) group pairs versus the kinetic energy integral magnitude (|KEI|) in hartree is shown. The percent of bond-antibond group pairs is shown out of the total number of SLMO group pairs that can be assigned excitations. F


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Figure 6. Number of split-localized molecular orbital (SLMO) group pairs versus the kinetic energy integral magnitude (|KEI|) in hartree for hexane minimum geometry: (a) intra-atomic excitation category, (b) interatomic excitation category, and (c) no excitation category. The percent of group pairs is shown out of the total number of SLMO group pairs that can be assigned excitations.

can be defined in terms of intra-atomic and interatomic because the SLMOs are defined in terms of the kurtosis OQUAOs and because each OQUAO is unambiguously assigned5 a single atom through the singular value decomposition that initially forms the QUAOs. The interatomic excitations and no excitation categories result from applying kurtosis to the |KEI|. For any histograms given, the number of SLMO group pairs versus a given |KEI| are formed. For each particular histogram, 1000 histogram bins are taken for the given |KEI| domain. The histogram bin size distinguishes the number of SLMO groups for each of the degenerate |KEI| values. If the bin size is not small enough, small discrepancies could slightly change the overall appearance of counts in the histogram, but the bin size choice has no affect the assigned excitation level. In order to check that the bin size was sufficiently small, the uniqueness of SLMO group pairs was explicitly analyzed and determined. Truly unique SLMO group pairs are denoted as “bin-unique” SLMO group pairs. In order to convey an idea of the percent of excitations allowed out of the total possible excitations, the group pairs are added up from the following categories: bond−antibond excitation, intra-atomic excitation, interatomic excitation, and no excitation. A percent is given for each molecule for each of these four categories relative to this total group pair count. 3.1. Carbonate Anion CO32−. Table 1 gives the SLMOs in terms of the corresponding kurtosis OQUAOs. Through kurtosis, the following assignments occur. Each SLMO lone pair has a single corresponding kurtosis OQUAO lone pair. For each orbital pair that has a substantial bond order, each SLMO OACσ and each SLMO COAσ* consist of the two

partner atom. If the OQUAO has multiple partner atoms, then multiple lower case atomic symbols follow the first upper case atomic symbol. The subscripts A, B, C, ... or 1, 2, 3, ... distinguish different atoms with the same element symbol (if any). For OQUAOs with substantial bond orders, the characterizing symbols σ or π appear after the atomic symbols. For OQUAOs without substantial bond orders, the characterizing symbols S for lone pair, rd for radical, etc., appear after the home atom symbol. For lone pairs the dominant hybridization fraction (e.g., s, p, etc.) with respect to the AAMBS orbitals precedes the S symbol. For a given SLMO, combining all the upper case atom indices f rom the kurtosis OQUAO labels in the order of the kurtosis contributions defines the corresponding SLMO. In cases where the SLMO is composed of OQUAOs whereby the labels have only two atom indices, exchange of the atom labels in the SLMO label does not matter in terms of identif ying the given SLMO. In order to gather information about allowed or prohibited excitations, histograms are formed using |KEI| values. All |KEI| are in units of hartree. Up to double excitations are allowed only between compatible SLMOs. So, two doubly occupied SLMOs have no allowed excitations. Likewise, two unoccupied SLMOs have no allowed excitations. Histograms are possible for the following categories: bond−antibond excitation, intra-atomic excitation, interatomic excitation, no excitation, intra-atomic doubly occupieds, intraatomic unoccupieds, interatomic doubly occupieds, and interatomic unoccupieds. As explained in the Methods section, the bond−antibond excitation category results from locating two SLMOs with the same two kurtosis OQUAOs. Categories G


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Figure 7. Number of split-localized molecular orbital (SLMO) group pairs versus the kinetic energy integral magnitude (|KEI|) in hartree for hexane minimum geometry: (a) intra-atomic doubly occupieds category, (b) intra-atomic unoccupieds category, (c) interatomic doubly occupieds category, and (d) interatomic unoccupieds category.

Figure 8. Definitions for all atom indices for the porphin minimum geometry at the MP2-RHF/cc-pVTZ level of theory.

Figure 9. Occupied π split-localized molecular orbitals (SLMOs) of porphin minimum geometry at the MP2-RHF/cc-pVTZ level of theory. The atom subscript labels are given in Figure 8.

kurtosis OQUAOs CoAσ and OAcσ. Each OACπ consists of two kurtosis OQUAOs OAcπ and CoAoBoCπ. Figure 1 shows the symmetry-unique OQUAOs and SLMOs, which include the delocalized SLMO COAOBOCπ. As shown in Table 1, the tenth SLMO COAOBOCπ consists of four kurtosis OQUAOs: CoAoBoCπ, OAcπ, OBcπ, and OCcπ. In particular, the carbonate

anion example explicitly illustrates how to identify delocalized SLMOs in a systematic manner. In this case, the first alternative orbital grouping with common OQUAO collection recovers the π system in a single orbital group. On the other hand, if H


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Figure 10. The 11 unoccupied π split-localized molecular orbitals (SLMOs) of porphin minimum geometry at the MP2-RHF/cc-pVTZ level of theory. The atom subscript labels are given in Figure 8.

Figure 11. Occupied π 2nd split-localized molecular orbitals (SLMOs) from the 2nd split-localization of porphin minimum geometry at the MP2-RHF/cc-pVTZ level of theory. The atom subscript labels are given in Figure 8. Figure 13. The 2nd bond bond−antibond excitation category from the 2nd split-localization for porphin minimum geometry at the MP2RHF/cc-pVTZ level of theory. The number of 2nd split-localized molecular orbital (SLMO) group pairs versus the kinetic energy integral magnitude (|KEI|) in hartree is shown. The percent of bondantibond group pairs is shown out of the total number of SLMO group pairs that can be assigned excitations.

the groupings due to common OQUAOs are then removed, that action breaks up the π system. If this orbital grouping is followed by the second alternative orbital grouping along with the various excitation assignments via kurtosis, then that sequence provides a means to recover excitations between the different π SLMOs with less expense. However, since carbonate is such a small molecule, all the π orbitals receive intra-atomic excitations. At this point, the carbonate anion serves as an example to explicitly check and show how the algorithm works to identify the various allowed or prohibited excitation categories. For the carbonate anion, the following excitations are assigned. For the OACσ and COAσ* SLMOs, the algorithm yields three bond-antibond SLMO group pairs with |KEI| = 1.13297.

Figure 12. Unoccupied π 2nd split-localized molecular orbitals (SLMOs) from the 2nd split-localization of porphin minimum geometry at the MP2-RHF/cc-pVTZ level of theory. The atom subscript labels are given in Figure 8. I


Article

The Journal of Physical Chemistry A Table 3. Automatic Extraction of Each 2nd SLMOb Group for Porphin Minimum Geometry at the MP2-RHF/cc-pVTZ Level of Theorya group

bin-unique number

OQUAOs

1 2 3 24 25 26 4

A A B B C C D

5

D

8

D

11

D

6

E

7

E

9

F

10

F

12

G

14

G

15

G

16

G

20

H

21

H

22

H

23

H

18

I

19

I

13

J

17

J

31 32 33 34 71 72 103 104 27 28 29 30 63 64 65 66 37 38 43

1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3

N4pS N3pS N2c7c8π, C8c23c16n2π, C7c24c15n2π N1c5c6π, C5c22c13n1π, C6c21c14n1π N3c9c11π, C9c22n3π, C11c24n3π N4c10c12π, C10c21n4π, C12c23n4π C24c7c11π, C7c24c15n2π, C11c24n3π, C9c22n3π, C8c23c16n2π C22c5c9π, C5c22c13n1π, C9c22n3π, C11c24n3π, C6c21c14n1π C23c8c12π, C8c23c16n2π, C12c23n4π, C10c21n4π C7c24c15n2π C21c6c10π, C6c21c14n1π, C10c21n4π, C12c23n4π, C5c22c13n1π C15c16c7π, C16c15c8π, C7c24c15n2π, C8c23c16n2π C14c13c6π, C13c14c5π, C5c22c13n1π, C6c21c14n1π C17c19π, C19c17π, C9c22n3π, C11c24n3π C18c20π, C20c18π, C10c21n4π, C12c23n4π C8c23c16n2π, C23c8c12π, N2c7c8π, C16c15c8π, C24c7c11π, C15c16c7π C6c21c14n1π, C21c6c10π, N1c5c6π, C14c13c6π, C22c5c9π, C13c14c5π C5c22c13n1π, C22c5c9π, N1c5c6π, C13c14c5π, C21c6c10π, C14c13c6π C7c24c15n2π, C24c7c11π, N2c7c8π, C15c16c7π, C23c8c12π, C16c15c8π C12c23n4π, C23c8c12π, N4c10c12π, C20c18π, C21c6c10π C9c22n3π, C22c5c9π, N3c9c11π, C17c19π, C24c7c11π C11c24n3π, C24c7c11π, N3c9c11π, C19c17π, C22c5c9π C10c21n4π, C21c6c10π, N4c10c12π, C18c20π, C23c8c12π C14c13c6π, C13c14c5π, C21c6c10π, C22c5c9π C15c16c7π, C16c15c8π, C24c7c11π, C23c8c12π C19c17π, C17c19π, C24c7c11π, C22c5c9π C20c18π, C18c20π, C23c8c12π, C21c6c10π N3c11σ, C11n3σ C11n3σ, N3c11σ N4c12σ, C12n4σ C12n4σ, N4c12σ N3c9σ, C9n3σ C9n3σ, N3c9σ N4c10σ, C10n4σ C10n4σ, N4c10σ N2c7σ, C7n2σ C7n2σ, N2c7σ N1c5σ, C5n1σ C5n1σ, N1c5σ N2c8σ, C8n2σ C8n2σ, N2c8σ N1c6σ, C6n1σ C6n1σ, N1c6σ C5c22σ, C22c5σ C22c5σ, C5c22σ C8c23σ, C23c8σ

Table 3. continued

reference electron count 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 J

group

bin-unique number

44 49 50 77 78 35 36 51 52 79 80 101 102 45 46 47 48 39 40 41 42 91 92 93 94 95 96 97 98 55 56 73 74 99 100 109 110 59 60 61 62 67 68 69 70 75 76 107 108 53 54 57 58 81 82 83 84 85 86 87 88 89

3 3 3 3 3 4 4 4 4 4 4 4 4 5 5 5 5 6 6 6 6 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 9 9 9 9 10 10 10 10 10 10 10 10 11 11 11 11 11 11 11 11 12 12 12 12 12

OQUAOs C23c8σ, C8c23σ C6c21σ, C21c6σ C21c6σ, C6c21σ C7c24σ, C24c7σ C24c7σ, C7c24σ C24c11σ, C11c24σ C11c24σ, C24c11σ C23c12σ, C12c23σ C12c23σ, C23c12σ C22c9σ, C9c22σ C9c22σ, C22c9σ C21c10σ, C10c21σ C10c21σ, C21c10σ C19c17σ, C17c19σ C17c19σ, C19c17σ C20c18σ, C18c20σ C18c20σ, C20c18σ C15c16σ, C16c15σ C16c15σ, C15c16σ C14c13σ, C13c14σ C14c13σ, C13c14σ C6c14σ, C14c6σ C14c6σ, C6c14σ C7c15σ, C15c7σ C15c7σ, C7c15σ C8c16σ, C16c8σ C16c8σ, C8c16σ C5c13σ, C13c5σ C13c5σ, C5c13σ C17c9σ, C9c17σ C9c17σ, C17c9σ C18c10σ, C10c18σ C10c18σ, C18c10σ C19c11σ, C11c19σ C11c19σ, C19c11σ C20c12σ, C12c20σ C12c20σ, C20c12σ N2h37σ, H37n2σ H37n2σ, N2h37σ N1h38σ, H38n1σ H38n1σ, N1h38σ C17h29σ, H29c17σ H29c17σ, C17h29σ C20h32σ, H32c20σ H32c20σ, C20h32σ C18h30σ, H30c18σ H30c18σ, C18h30σ C19h31σ, H31c19σ H31c19σ, C19h31σ C13h25σ, H25c13σ H25c13σ, C13h25σ C16h28σ, H28c16σ H28c16σ, C16h28σ C14h26σ, H26c14σ H26c14σ, C14h26σ C15h27σ, H27c15σ H27c15σ, C15h27σ C21h34σ, H34c21σ H34c21σ, C21h34σ C23h36σ, H36c23σ H36c23σ, C23h36σ C22h33σ, H33c22σ

reference electron count 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2


Article

The Journal of Physical Chemistry A Table 3. continued group

bin-unique number

90 105 106

12 12 12

OQUAOs H33c22σ, C22h33σ C24h35σ, H35c24σ H35c24σ, C24h35σ

three (OAsS ,OBsS ), six (OApS ,OBCπ), and six (OAsS ,OBCπ). Because of the small size and closeness of the |KEI| for the last three SLMO group pairs, Figure 2b) shows these groups pairs together in a total count of 15. Through the definitions of interatomic and intra-atomic in the present work, the carbonate anion has zero group pairs with all interatomic unoccupied orbitals. In Figure 2c), the intra-atomic doubly occupied group pairs are shown from largest to smallest |KEI| as three (OAsS ,OACσ), three (OACπ,OBCπ), three (OACσ,OBCσ), three (OACσ,OACπ), six (OACπ,OBCσ), three (OApS ,OACσ), three (OAsS ,OACπ), three (OAsS ,OApS ), and three (OApS ,OACπ). Once again, because of the small size and closeness of the |KEI|, some group pairs are shown together in the peaks in Figure 2c). The intra-atomic unoccupied SLMO group pairs are three (COAσ*, COBσ*) and three (COAσ*, COAOBOCπ). The carbonate anion has the following excitation percents: 6.25% for bond-antibond, 68.75% for intra-atomic, 12.5% for interatomic, and 12.5% for no excitation. In other words, the |KEI| criterion excludes excitations between 12.5% group pairs. 3.2. Hexane C6H14. Figure 3 gives the atom indices that are used in the subsequent tables and figures for hexane minimum geometry. Figure 4 gives the bin-unique SLMOs for hexane, which has no lone pairs and consists of CCσ and CHσ bonds with 19 substantial bond orders that are close to 1 in the OQUAO basis. Figure 5 and Table 2 complement each other. Figure 5 shows the results from the bond-antibond excitation category, and Table 2 gives the SLMOs in terms of the corresponding kurtosis OQUAOs. As described above, Table 2 additionally identifies the bin-unique SLMOs involved in SLMO

reference electron count 0 2 0

a

The second column shows the bin-uniquec 2nd SLMOs involved in bond-antibond groups by decreasing kinetic energy integral magnitude (|KEI|). The unique lone pair and π orbitals are kept separate and labeled by letters. The definition of each 2nd SLMO group is given in terms of OQUAOsd. The reference electron counte for each 2nd SLMO group is given. bSLMO refers to 2nd split-localized molecular orbital. cThe term bin-unique refers to unique orbital group pairs for degenerate |KEI| values (see in the beginning of Section 3). d OQUAOs refers to oriented quasi-atomic orbitals. eReference electron count gives the integer occupation for the orbital group.

Figure 2a) holds the number of SLMO group pairs with intraatomic excitations. From the largest to smallest |KEI|, the SLMO group pairs are identified as three (OAsS ,COAσ*), three (OACπ,COAOBOCπ), six (OACσ,COBσ*), and 21 other pairs. Interatomic excitations are assigned to the six SLMO group pairs of (OApS ,COBσ*) with |KEI| = 0.403008. The six SLMO group pairs of (OAsS ,COBσ*) are not assigned any excitations with |KEI| = 0.0649048. The following excitations are not allowed in any form. Figure 2b) exhibits the number of SLMO group pairs of interatomic doubly occupied orbitals. From largest to smallest |KEI|, these SLMO group pairs are listed as six (OApS ,OBCσ), six (O A sS ,O B Cσ), six (O A sS ,O B pS ), three (O A pS ,O B pS ),

Figure 14. Number of 2nd split-localized molecular orbital (SLMO) group pairs from the 2nd split-localization versus the kinetic energy integral magnitude (|KEI|) in hartree for porphin minimum geometry at the MP2-RHF/cc-pVTZ level of theory: (a) intra-atomic excitation category, (b) interatomic excitation category, and (c) no excitation category. The percent of group pairs is shown out of the total number of SLMO group pairs that can be assigned excitations. K


Article


Figure 15. Number of 2nd split-localized molecular orbital (SLMO) group pairs from the 2nd split-localization versus the kinetic energy integral magnitude (|KEI|) in hartree for porphin minimum geometry at the MP2-RHF/cc-pVTZ level of theory: (a) intra-atomic doubly occupieds category, (b) intra-atomic unoccupieds category, (c) interatomic doubly occupieds category, and (d) interatomic unoccupieds category.

bond-antibond group pairs for degenerate |KEI| values, which are shown in Figure 5. The middle CCσ bond pair has the largest |KEI| ≈ 0.75. For the other CCσ bond pairs, the uniquebin numbers 2 and 3 (see Table 2 and Figure 4) closely follow with |KEI| ≈ 0.74 in Figure 5. The CHσ bond pairs all have |KEI| ≈ 0.59. In Figure 5 for |KEI| < 0.6, four bin-unique peaks exist with group pair counts of 4,2,4, and 4 from largest to smallest |KEI|. Again, Table 2 identifies the bin-unique SLMOs involved here. Figure 6 shows the results for the intra-atomic, interatomic, and no excitation categories in panels a, b, and c, respectively. Each category in Figure 6 does not overlap in |KEI| values. The intra-atomic excitation category has the largest |KEI|. The interatomic excitation category has the second largest |KEI|. The no excitation category has the smallest |KEI|. As shown in Figure 6, applying kurtosis to the |KEI| results in the division of about 1/2 an order of magnitude in |KEI| between the interatomic and no excitation categories. Figure 7 shows the results for the rest of the prohibited excitation categories. Since hexane contains no lone pair orbitals or singly occupied orbitals, some categories are identical by definition. The intra-atomic doubly occupieds category and intra-atomic unoccupieds category are identical. Likewise, the interatomic doubly occupieds category and interatomic unoccupieds category are identical. As anticipated from physical distance, the intraatomic doubly occupieds have large |KEI| relative to the |KEI| of the interatomic doubly occupieds. The hexane molecule has the following excitation percents: 5.26% for bond-antibond, 19.94% for intra-atomic, 27.15% for interatomic, and 47.65% for no excitation. In other words, the

|KEI| criterion excludes excitations between 47.65% group pairs. These excitation percents are also displayed in Figures 5 and 6. 3.3. Porphin. In the case of porphin, a RHF geometry minimization fails to give a qualitatively appropriate geometry.63 The minimum geometry of porphin was located and verified by nuclear Hessian with closed-shell second-order Møller−Plesset perturbation theory.64−66 Figure 8 gives the atom indices that are used in the subsequent tables and figures for porphin. In this molecule, the HF split-localization procedure actually results in 13 occupied and 11 unoccupied π SLMOs. Since the split-localization process does not mix occupied and unoccupied SLMOs, the 11 unoccupied π-symmetry SLMOs are all unique. Figure 9 shows the 6 unique out of 13 occupied π SLMOs, and Figure 10 shows the 11 unique unoccupied π SLMOs. The last row of Figure 9 displays the very delocalized π-symmetry SLMO C5 C 6C 7C 8C 9C 10C 11 C12C 13C 14 C15C 16 C 17 C18C 19 C 20 . The split-localization process leaves this π-symmetry SLMO more diffuse for the expense of compacting the rest of the π SLMOs. However, this delocalization can be easily improved by allowing this single diffuse π SLMO to mix with the unoccupied SLMOs (i.e., by performing a second split-localization where the allowed orbital rotations are redefined with this modification). Figure 11 shows the 5 unique out of 12 occupied π SLMOs from the second split-localization, and Figure 12 shows the 4 unique out of 12 π SLMOs of mixed occupation from the second splitlocalization. The diffuse SLMO is no longer present. Figure 13 and Table 3 complement each other. Figure 13 shows the results from the bond-antibond excitation category, and Table 3 gives the SLMOs in terms of the corresponding kurtosis OQUAOs and also defines the bin-unique SLMOs. L


The Journal of Physical Chemistry A In Table 3 under the bin-unique column, the unique SLMOs are indicated by letters for the lone pair and π-symmetry SLMOs and by numbers for the σ SLMOs. Figure 14 shows the results for the intra-atomic, interatomic, and no excitation categories in panels a, b, and c, respectively, and Figure 15 shows the results for the rest of the prohibited excitation categories. Throughout Figures 14 and 15, a large number of group pair counts occur at 0 |KEI|. This result happens because of different orbital symmetry. These exact counts can be verified by hand if desired. Porphin has the following excitation percents: 1.39% for bond-antibond, 14.90% for intra-atomic, 17.91% for interatomic, and 65.81% for no excitation. In other words, the |KEI| criterion excludes excitations between 65.81% group pairs. These excitation percents are also displayed in Figures 13 and 14.

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ACKNOWLEDGMENTS

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REFERENCES

A.C.W. gives a very special thanks to Theresa Windus for a myriad of discussions on active spaces over the last 11 years that forms the basis of the present research. Likewise, discussions over the last 11 years with George S. Schoendorff made this work possible. A.C.W. very much thanks Klaus Ruedenberg and Mike Schmidt for their patience and for instruction on localized orbitals. The present work was supported by the National Science Foundation under the Grants CHE-1147446 and CHE-1565888 to Iowa State University.

(1) Ruedenberg, K.; Schmidt, M. W.; Gilbert, M. M.; Elbert, S. T. Are atoms intrinsic to molecular electronic wavefunctions? I. The full optimized reaction space (FORS) model. Chem. Phys. 1982, 71, 41− 49. (2) Ruedenberg, K.; Schmidt, M. W.; Gilbert, M. M. Are atoms intrinsic to molecular electronic wavefunctions? II. Analysis for FORS orbitals. Chem. Phys. 1982, 71, 51−64. (3) Ruedenberg, K.; Schmidt, M. W.; Gilbert, M. M.; Elbert, S. T. Are atoms intrinsic to molecular electronic wavefunctions? III. Analysis of FORS configurations. Chem. Phys. 1982, 71, 65−78. (4) Schmidt, M. W.; Lam, M. T. B.; Elbert, S. T.; Ruedenberg, K. Chemical binding and electron correlation in diatomic molecules as described by the FORS model and the FORS-IACC model. Theor. Chim. Acta 1985, 68, 69−86. (5) West, A. C.; Schmidt, M. W.; Gordon, M. S.; Ruedenberg, K. A comprehensive analysis of molecule-intrinsic quasi-atomic, bonding, and correlating orbitals. I. Hartree-Fock wave functions. J. Chem. Phys. 2013, 139, 234107. (6) Shepard, R. The multiconfiguration self-consistent field method. Adv. Chem. Phys. 1987, 69, 63−200. (7) Ruedenberg, K.; Sundberg, K. R. MCSCF studies of chemical reactions: natural reaction orbitals and localized reaction orbitals. Quantum Sci. 1976, 505−515. (8) Siegbahn, P.; Heiberg, A.; Roos, B.; Levy, B. A comparison of the super-CI and the Newton-Raphson scheme in the complete active space SCF method. Phys. Scr. 1980, 21, 323−327. (9) Roos, B. The complete active space self-consistent field method and its applications in electronic structure calculations. Adv. Chem. Phys. 1987, 69, 399−445. (10) Roos, B. O. The multiconfigurational (MC) self-consistent field (SCF) theory. Lect. Notes Chem. 1992, 58, 177−251. (11) Ivanic, J. Direct configuration interaction and multiconfigurational self-consistent-field method for multiple active spaces with variable occupations. I. Method. J. Chem. Phys. 2003, 119, 9364−9376. (12) Ivanic, J. Direct configuration interaction and multiconfigurational self-consistent-field method for multiple active spaces with variable occupations. II. Application to oxoMn(salen) and N2O4. J. Chem. Phys. 2003, 119, 9377−9385. (13) Boys, S. F. Localized orbitals and localized adjustment functions. Quantum Theory At., Mol., Solid State. 1966, 253−262. (14) Edmiston, C.; Ruedenberg, K. Localized atomic and molecular orbitals. Rev. Mod. Phys. 1963, 35, 457−465. (15) West, A. C.; Kretchmer, J. S.; Sellner, B.; Park, K.; Hase, W. L.; Lischka, H.; Windus, T. L. O(3P) + C2H4 Potential Energy Surface: Study at the Multireference Level. J. Phys. Chem. A 2009, 113, 12663− 12674. (16) West, A. C.; Lynch, J. D.; Sellner, B.; Lischka, H.; Hase, W. L.; Windus, T. L. O + C2H4 potential energy surface: lowest-lying singlet at the multireference level. Theor. Chem. Acc. 2012, 131, 1−14. (17) West, A. C.; Lynch, J. D.; Sellner, B.; Lischka, H.; Hase, W. L.; Windus, T. L. O + C2H4 potential energy surface: excited states and biradicals at the multireference level. Theor. Chem. Acc. 2012, 131, 1− 15.

4. CONCLUSIONS As in previous studies,1−3,5,29,31−35 the molecular wave function is expressed in terms of deformed and hybridized quasi-atomic orbitals as well as split-localized orbitals. The present study successfully develops a scheme to recover strong correlation more cheaply from an internal orbital space than other methods by unbiasedly selecting excitations that lead to the most important configurations. In the present scheme, the application of kurtosis to an appropriate orbital transformation uniquely defines the split-localized orbitals in terms of a small number of deformed quasi-atomic orbitals, which are termed kurtosis orbitals. In turn, the kurtosis orbitals provide a simple and effective means to organize the split-localized orbitals. Furthermore, the application of kurtosis to a distance dependent criterion (e.g., kinetic energy integral in the present implementation) subsequently provides a means to systematically select the most important excitations among the split-localized orbitals for use in any occupation restricted configuration interaction method. Applications to carbonate anion, hexane, and porphin explicitly show how this scheme categorizes the orbitals and selects only the most important excitations in a full valence orbital space. In terms of future work, the criterion for excitation selection can be not only improved via fourth power adaptation but also easily extended to open-shell systems. Each open-shell might lead to more than one dominant configuration. Because the distance dependence of such open-shell orbital pairs is clearly different than excitations between occupied and unoccupied orbital pairs, kurtosis should be separately applied among all singly occupied orbitals in order to differentiate the most important groups of singly occupied orbitals out of all the singly occupied orbitals. In addition, the original ORMAS implementation (a specialized MCSCF approach) enumerates through electron distributions based on the allowed minimum and maximum electron counts for each group. On this basis, the present excitation scheme can be readily extended into the ORMAS framework in future work.

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AUTHOR INFORMATION

Corresponding Author

*(A.C.W.) E-mail: [email protected]. ORCID

Aaron C. West: 0000-0001-5372-8458 Notes

The author declares no competing financial interest. M


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