Localized Intrinsic Valence Virtual Orbitals as a Tool for the Automatic

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Localized Intrinsic Valence Virtual Orbitals as a Tool for the Automatic Classification of Core Excited States Wallace D Derricotte, and Francesco A. Evangelista J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.7b00493 • Publication Date (Web): 10 Nov 2017 Downloaded from http://pubs.acs.org on November 14, 2017

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Journal of Chemical Theory and Computation

Localized Intrinsic Valence Virtual Orbitals as a Tool for the Automatic Classification of Core Excited States Wallace D. Derricotte∗ and Francesco A. Evangelista∗ Department of Chemistry and Cherry L. Emerson Center for Scientific Computation, Emory University, Atlanta, GA, 30322 E-mail: [email protected]; [email protected]

Abstract Accurate assignments of the unoccupied molecular orbitals involved in electronic excitations are crucial to the interpretation of experimental spectra. Here we present an automated approach to 5

the orbital assignment of excited states by introducing a unique orbital basis known as localized intrinsic valence virtual orbitals (LIVVOs), which are a special case of the previously reported valence virtual orbitals. The LIVVOs are used to quantify the local contributions to particle orbitals from orthogonality-constrained density functional theory, providing an assignment with atomic-level/angular momentum shell specificity. This localized set also allows us to define the total valence character of

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an excited state. We highlight the utility of our approach by studying the local orbital changes in core-excited states at the sulfur K-edge of ethanethiol and benzenethiol as well as the oxygen K-edge spectrum of the water monomer and dimer.

1. INTRODUCTION Any interpretation of electronic spectroscopy depends upon assigning features of the experimental spectrum 15

to discrete electronic transitions calculated using excited state quantum chemistry methods. This requires characterization of the molecular orbitals involved in the discrete transition. Unfortunately, the canonical virtual molecular orbitals (MOs) produced in quantum chemistry calculations, either from Hartree–Fock or density functional theory (DFT), are a large and diverse set, where the orbitals that are useful for describing valence excitation processes include contributions from Rydberg and continuum states. 1 Transitions are

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usually assigned by visual inspection of the contributing virtual molecular orbitals, which requires that the orbitals have a sensible shape such that the orbital character is comprehensible to the user. Often, when the MOs within this large set do not have well defined orbital character, orbital labels are used instead 1 ACS Paragon Plus Environment


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(e.g. LUMO, LUMO + 1, etc.). These approaches are somewhat unsatisfactory. The assignment of orbital character based on MO plots is potentially ambiguous. Furthermore, characterizing a transition using orbital 25

labels is problematic because canonical virtual orbitals are strongly depend on the basis set and exchangecorrelation functional. A unique branch of electronic spectroscopy involving the excitation of core electrons using X-ray photons called near-edge X-ray absorption fine structure (NEXAFS) spectroscopy has garnered particular interest due to advancements in high-resolution tunable synchrotron light sources. NEXAFS spectroscopy has be-

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come an indispensable experimental tool for probing local electronic and geometrical information in a wide variety of molecular systems. 2–5 The significant features of the NEXAFS spectrum arise from core to unoccupied valence orbital transitions. 6 The atomic nature of the core orbitals involved in these transitions (1s, 2s, 2p, etc.) makes NEXAFS a spectroscopic technique for elucidating local electronic structure information. Quantum chemistry calculations serve as a vital tool for gleaning meaningful electronic structure information

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from NEXAFS spectra including the nature of unoccupied orbitals, orbital mixing effects, and covalent character. 3,6–9 A myriad of theoretical methods have been developed for the computation of core-excited states, ranging from rigorous methods focused on high accuracy 10–27 to computationally efficient methods focused on applications to larger systems. 28–40 The primary motivation underpinning these theoretical developments has been to obtain more accurate excitation energies and/or oscillator strengths. In contrast, the character

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of the wave function is analyzed in a more cursory fashion. A recent work by the group of Dreuw 41–43 has taken a significant step toward a more rigorous description of core-excited states. Using the one-particle transition density matrix and the one-particle difference density matrix, these authors were able to quantify the effects of relaxation, estimate multireference character, and provide information on the exciton size. These methods are a significant step toward a robust characterization

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of excited-states. Myhre, Coriani, and Koch 26 also recently used visualization of the one-particle difference density matrix in order to track the localization of core-excited states, relying on analysis of the excitation amplitudes of coupled cluster response calculations to assign the character of core-valence transitions. Although a fair amount of attention has been put forth to add a breadth of different descriptors for excited states, little attention has been put toward trying to improve upon the method of assigning orbital

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character. Valence orbitals that are chemically meaningful can assist in the analysis of excited states. One approach toward obtaining such a set is to project the virtual space onto a set of atomic valence orbitals. This solution hinges on the assumption that valence atomic orbitals are the dominant contributors to the valence molecular orbitals. This assumption was justified in the 1980s by Rudenberg and coworkers by obtaining overlaps close to unity between free-atom AOs and their projection onto the molecular orbitals of MCSCF

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wave functions. 44 Perhaps, the most popular method to analyze wave functions in terms of atomic valence contributions is Weinhold’s natural atomic/bond orbital (NAO/NBO) analysis. 45,46 This method partitions the full density matrix into atomic sub-blocks that are diagonalized in order to produce atom-localized hybrid valence orbitals. AO projection based methods have been presented throughout the literature including polarized atomic orbitals, 47,48 enveloping localized orbitals, 49 intrinsic minimal atomic basis orbitals, 50

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molecule-adapted atomic orbitals, 51 and intrinsic atomic orbitals. 52 The approach presented in this work is based on the valence virtual orbitals (VVOs) scheme. 53 VVOs are built by bringing virtual valence MOs into maximum coincidence with a minimal atomic basis set via a unitary transformation. The VVOs have been used with success to identify the valence LUMO in a variety of molecular systems, 1 to characterize bond breaking/formation along dissociation pathways, 54 and as a set of starting orbitals for multi-configurational

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SCF calculations. 55 VVOs can be easily localized and provide an accurate representation of the localized valence antibonding orbitals in a molecule. In this work we report a new method for automated classification of virtual orbitals in electronic excited states by quantifying the local contributions using a localized set of VVOs. To generate the VVOs, an accurate atomic minimal basis set (AAMBS) that spans the atomic valence space is required. We propose

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to represent the AAMBS using a minimal basis set of quasiatomic orbitals. 53 These AAMBS are generated using projection operators according to Knizia’s intrinsic atomic orbitals (IAOs) procedure. 52 The VVOs are then localized using the same approach used to build intrinsic bond orbitals (IBOs), 52 which maximizes a quartic function of the number of electrons located on the IAOs. The combination of VVOs projected onto a localized intrinsic orbital basis is abbreviated as LIVVOs. To compute excitation energies we employ our

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recently proposed orthogonality-constrained DFT (OCDFT) approach 56 generalized to target a manifold of states. 57 OCDFT is a variational time-independent DFT method built upon a generalized Kohn-Sham scheme. In OCDFT excited state energies are approximated by a Kohn–Sham procedure augmented with orthogonality constraints between ground and excited state wave functions. This new classification method is applied to the core-excited states involved in NEXAFS spectroscopy.

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We utilize LIVVOs to quantify local contributions to analyze substituent effects in thiols (ethanethiol, benzenethiol) and intermolecular interactions in the water dimer. The thiols were chosen as an example of molecules where the relevant excited states have significant contributions from more than one valence antibonding MO. Previous studies have not fully characterized the antibonding character of orbitals involved in core-valence excitations. 58–60 We show that we are able to provide insight into the nature of these excited

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states using LIVVOs. Hydrogen bonding interactions in water have received plenty of attention within the X-ray absorption community. 61–67 Studies have shown that NEXAFS spectral changes upon the formation



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as a linear combination of atom-centered basis functions {χµ }

|φa i =

N AO X

|χµ iCµa ,

a = 1, . . . , Nvir

(1)

µ=1

where Cµa is the MO coefficient matrix, while NAO and Nvir are the number of atomic and virtual orbitals, respectively. As shown in Figure 1A, the diffuse nature of the virtual CMOs complicates the interpretation of their valence character. To identify the valence character of the CMOs, we perform a VVO analysis 1,74,75 using the intrinsic 105

atomic orbitals of Knizia 52 as the underlying accurate atomic minimal basis set. IAOs are a set of orthonormal polarized atomic orbitals that can exactly express the occupied molecular orbitals of the ground state Kohn-Sham (KS) determinant |Φi. IAOs are expressed in terms of the atomic orbital basis as

|ψρIAO i =

N AO X

|χµ iC˜µρ ,

ρ = 1, . . . , NIAO

(2)

µ=1

where C˜µρ is the IAO coefficient matrix and NIAO is the number of IAOs. As shown in Figure 1B, in the case of water there are 7 IAOs that correspond to the oxygen atom 1s, 2s, and 2p shells and a single 1s 110

orbital for each hydrogen atom. Next, the overlap matrix (S) of the canonical virtual orbitals with the IAOs ψρIAO is evaluated: (S)aρ = hφa |ψρIAO i

(3)

and as suggested in Ref. 1, a singular value decomposition (SVD) is performed on S:

S = UσV†

(4)

to yield orthogonal transformation matrices U and V. These matrices are rotations of the virtual space and the IAO space, respectively, that bring the two sets of orbitals into maximum coincidence. Note that 115

our goal is to identify virtual orbitals that maximally overlap with atomic functions (1s for the first period, 1s 2s 2p for the second period, etc.), which here are represented with the IAOs. The IAOs, which number NIAO , by construction span the union of atomic functions of all atoms. Therefore, the number of valence virtual orbitals (NVVO ) is given by the number of IAOs minus the number of occupied molecular (Nocc ) orbitals: NVVO = NIAO − Nocc . Assuming that the singular value decomposition orders the singular values

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(σv ) in descending order, then the VVOs are obtained by transforming the canonical virtual orbitals with



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the first NVVO columns of the matrix U: |ψvVVO i

=

N vir X

|φa iUav ,

v = 1, . . . , NVVO

(5)

a=1

As pointed out to us by one of the reviewers, there is an alternative and equivalent definition of VVOs. The VVOs may also be expressed as the null space of the overlap of the IAOs with the canonical occupied orbitals. From this perspective, then the VVOs are the part of the IAOs that lie outside the space of occupied 125

orbitals, and it is possible to compute them even without knowledge of the virtual orbitals. The VVOs for water shown in Figure 1C, have valence character and correspond to the bonding and ∗ antibonding combination of localized σO–H orbitals. Following localization using the approach used by Knizia

to produce Intrinsic Bonding Orbitals (IBO), 52 we arrive at a set of localized VVOs that span the antibonding interactions of the molecular environment. We refer to this localized set as localized intrinsic valence virtual 130

∗ orbitals (LIVVOs). In the case of water, following localization of the VVOs we obtain the σO–H orbitals

drawn in Figure 1D. Note that other approaches may be used to define charges that enter in an orbital localization scheme. However, a recent study by Lehtola and Jonsson 76 has shown that although charges estimated with different methods may differ significantly, the corresponding localized orbitals are very similar. From our experience, 135

Pipek–Mezey localization 77 with Mulliken charges 78,79 produces a set of localized VVOs that is nearly identical to that produced using IBO-style localization. This observation is also confirmed by other recent works that compared Pipek–Mezey localization with different partial charges. 76,80 2.2. Determination of the character of the IAOs and LIVVOs. The goal of our analysis is to express excited states using the LIVVO basis. In addition, we are also interested in determining the dominant atomic

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character of each excited state. To this end, we have automated the analysis of the atomic character of the LIVVOs. Our approach starts with the assignment of the atomic character of each IAO via a Mulliken population analysis. 78,79 Despite the fact that Mulliken’s population analysis is strongly dependent on the basis set, the method of assignment proposed here appears to be insensitive to this choice. This is due to the fact that IAOs are mostly composed of basis functions centered on one atom. An example that shows

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the independence of the assignment of the IAOs on the choice of the basis set is reported in this section. In Sec. 4.3. we further assess the dependence of the LIVVO assignment of particle orbitals. For a given IAO, ψρIAO , the gross population on angular momentum shell l of atom A [GPAl ] is obtained as partial sums of


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population matrices (C˜µρ Sµν C˜νρ ):

GPAl (ρ) =

AO X X N

C˜µρ Sµν C˜νρ

(6)

µ∈Al ν=1

where the sum over µ is restricted to all functions centered on atom A with angular momentum quantum 150

number l. We define the character of ψρIAO as the pair atom/shell (A,l) for which the absolute value of GPAl (ρ) is maximum: character(ψρIAO ) = arg max |GPAl (ρ)| (A,l)

(7)

To determine the principal quantum number of IAOs centered on the same atom and with identical angular quantum number (e.g. O 1s and 2s), we use the diagonal Fock matrix elements in the IAO basis (ǫIAO ) ρ

ǫIAO = ψρIAO fˆ ψρIAO ρ

(8)

where fˆ is the ground state DFT Fock operator. The principal quantum number is determined by sorting 155

the IAO with identical character according the value of ǫIAO . ρ Values of GPAl and ǫIAO for the water example using a range of basis sets are reported in Table 1. ρ The assignment of IAOs using the gross population is straightforward since there is always a dominant contribution from a single atom/shell, independently of basis set. For example, in the case of ψ6IAO , the largest contribution to GPAl is from the H2 s orbitals, followed by smaller contributions from the O1 s and

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p orbitals, and H3 s orbitals. Table 1 also shows that the principal quantum number within the orbitals with O1s character can be clearly discerned using the quantity ǫIAO . For example, independently of the basis ρ set, the values of ǫIAO for IAOs ψ1IAO and ψ2IAO are approximately −20 Eh and −1 Eh (using the B3LYP ρ functional), which leads to the assignment of these two orbitals to the 1s and 2s shells, respectively. In the case of water, when eq (7) is applied to the IAOs we obtain the assignment reported in Figure 1B.

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Once the atomic character of each IAO is determined, we characterize the atomic contributions to each LIVVO. For each LIVVO, ψlLIVVO , we evaluate the overlap with all the IAOs (ψρIAO ): ′ Slρ = |hψlLIVVO |ψρIAO i|2

(9)

The initial step in classifying the LIVVOs is to determine their overall orbital character (σ,π,. . . ). A LIVVO ′ is classified as σ ∗ if the largest elements of Slρ include contributions from IAOs with s and p character. ′ Similarly, a LIVVO is assigned π ∗ character if the largest elements of Slρ arise exclusively from p-type IAOs. 170

′ The atoms involved in the LIVVO are identified by the character of the largest elements of Slρ . For this



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Table 1 Atom-centered gross orbital populations [GPAl ], and corresponding diagonal Fock matrix elements (ǫIAO , in ρ units of Eh ) for the seven IAOs of water computed using the B3LYP functional and various basis sets.

O1s O1p H2s H3s ǫIAO ρ O1s O1p H2s H3s ǫIAO ρ O1s O1p H2s H3s ǫIAO ρ O1s O1p H2s H3s ǫIAO ρ O1s O1p H2s H3s ǫIAO ρ O1s O1p H2s H3s ǫIAO ρ

ψ1IAO

ψ2IAO

ψ3IAO

ψ4IAO

ψ5IAO

ψ6IAO

ψ7IAO

1.00 0.00 0.00 0.00

1.15 0.00 −0.07 −0.07

cc-pVDZ 0.00 0.00 1.04 1.09 −0.02 −0.05 −0.02 −0.05

0.00 1.00 0.00 0.00

−0.08 −0.07 1.17 −0.03

−0.08 −0.07 −0.03 1.17

−20.02

−0.97

−0.54

−0.49

−0.51

−0.18

−0.18

1.00 0.00 0.00 0.00

1.16 0.00 −0.08 −0.08

cc-pVTZ 0.00 0.00 1.04 1.11 −0.02 −0.05 −0.02 −0.05

0.00 1.00 0.00 0.00

−0.08 −0.07 1.18 −0.03

−0.08 −0.07 −0.03 1.18

−19.87

−0.98

−0.55

−0.50

−0.53

−0.20

−0.20

1.00 0.00 0.00 0.00

1.15 0.00 −0.08 −0.08

cc-pVQZ 0.01 0.00 0.99 1.07 0.00 −0.04 0.00 −0.04

0.00 1.00 0.00 0.00

−0.05 −0.06 1.13 −0.04

−0.05 −0.06 −0.04 1.13

−19.85

−0.99

−0.56

−0.51

−0.54

−0.20

−0.20

1.00 0.00 0.00 0.00

1.15 0.00 −0.08 −0.08

aug-cc-pVDZ 0.01 0.00 1.02 1.08 −0.02 −0.05 −0.02 −0.05

0.00 1.00 0.00 0.00

−0.06 −0.06 1.15 −0.04

−0.06 −0.06 −0.04 1.15

−19.89

−1.01

−0.58

−0.54

−0.56

−0.22

−0.22

1.00 0.00 0.00 0.00

1.16 0.00 −0.08 −0.08

aug-cc-pVTZ 0.01 0.00 1.04 1.11 −0.02 −0.05 −0.02 −0.05

0.00 1.00 0.00 0.00

−0.08 −0.07 1.18 −0.03

−0.08 −0.07 −0.03 1.18

−19.89

−1.01

−0.58

−0.54

−0.56

−0.22

−0.22

1.00 0.00 0.00 0.00

1.23 0.01 −0.09 −0.09

aug-cc-pVQZ −0.02 0.00 1.06 1.22 −0.02 −0.10 −0.02 −0.10

0.00 1.01 0.00 0.00

−0.14 −0.14 1.37 −0.06

−0.14 −0.14 −0.06 1.37

−19.87

−1.00

−0.58

−0.56

−0.22

−0.22

−0.53

purpose, we only include those IAOs whose overlap with a given LIVVO is greater than a threshold, which in the following examples is set to 0.1. To illustrate this analysis we consider the case of acrolein (C3 H4 O). The following LIVVOs are charac-


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with each LIVVO (ψlLIVVO ): (n)

LIVVO 2 Ωpl = |hφ(n) i| . p |ψl

(11)

(n)

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The individual overlaps Ωpl are used to assign the character of the particle orbital to the l-th LIVVO. In val,(n)

addition, we also define the total valence character tp with all LIVVOs: tval,(n) = p

for any given particle orbital as the sum its overlap

NLIVVO X

(n)

Ωpl .

(12)

l=1

Since the LIVVOs give an accurate representation of valence orbitals in the molecule, it is possible to quantify the most important local contributions to the particle orbital. While this offers a very robust description for 200

localized valence particle orbitals, it is important to note that the particle orbital describing the excited state can also be fairly diffuse with respect to the molecular environment. Such an orbital will obviously have relatively low overlap with the LIVVOs described here. This can be a desirable feature of our method as it allows for immediate identification of spectral contributions that arise from more diffuse orbitals. However, any further classification of the diffuse nature or Rydberg character of virtual orbitals is outside the scope

205

of the current study. 3. COMPUTATIONAL DETAILS OCDFT is currently implemented as a plugin for the PSI4 81 ab initio quantum chemistry package. Core excitation energies and oscillator strengths for the thiols and water monomer and dimer are obtained using the B3LYP 82–86 functional and an augmented triple zeta correlation-consistent basis set (jun-cc-pVTZ).

210

This basis set is obtained by deleting diffuse functions 87 from the highest subshell of every atom in the aug-cc-pVTZ basis set 88–90 (e.g. diffuse f functions of C). It was chosen because it has the advantage of reducing the computational cost and improving the convergence of the SCF procedure with respect to the fully augmented basis. 87 The two-electron integrals are approximated using density-fitting 91–96 with the jun-cc-pVTZ-JKFIT auxiliary fitting basis. 97,98 Basis set dependence of the LIVVO analysis is investigated

215

using the cc-pVXZ and aug-cc-pVXZ (X=D,T,Q) family of basis sets. 88–90 Intrinsic atomic orbitals were obtained using Robert Parrish’s implementation in PSI4. 99 The IAO scheme utilizes the MINAO basis set as used in Ref. 52, which consists of the contracted functions of the cc-pVTZ basis. The geometries for ethanethiol, benzenethiol, water monomer, and water dimer were all optimized at the B3LYP/aug-cc-pVTZ level of theory.

220

For ethanethiol and benzenethiol we computed the first ten sulphur K-edge transitions,

while we computed six and twelve oxygen K-edge transitions for the water monomer and dimer, respectively. Note that the cost of computing n core-excited states using OCDFT is approximately n times the cost of a ground state DFT computation.


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Peak intensities for each transition from the ground state to excited state (n) are based on the oscillator strength (fosc ):

fosc =

225

2 |µn0 |2 ωn 3

(13)

where ωn is the excitation energy calculated in OCDFT and µn0 is the transition dipole moment vector. OCDFT does not provide a direct route toward computation of the transition dipole moments, however, these can be approximated using the noninteracting Kohn–Sham determinants: µn0 = hΦ(n) |r|Φ(0) i

(14)

where Φ(n) is the KS determinant for the n-th excited state and r is the position vector. Spectroscopic broadening effects are simulated by representing each transition using a Gaussian lineshape with a FWHM 230

of 0.4 eV. 4. RESULTS In the following two subsection we present applications of our OCDFT and LIVVO analysis to study substituent effect in the spectrum of thiols and the water monomer and dimer. These applications are followed by an analysis of the robustness of the LIVVO assignments with respect to the choice of basis and density

235

functionals used in the OCDFT computation. 4.1. Analysis of substituent effects in the spectra of thiols. Our first application of the LIVVO analysis for the classification of NEXAFS spectral features focuses on the effect of substituents on peak position and intensity. In this section, we compare the sulfur K-edge of ethanethiol (C2 H5 SH) and benzenethiol (C6 H5 SH). The gas-phase experimental NEXAFS spectra for both thiols (from the work of Behyan, Hu,

240

and Urquhart, Ref. 60) are compared to the OCDFT spectra in Figure 2. To show convergence of the near-edge spectrum with respect to the number of states, we report OCDFT spectra computed with 5 and 10 states. Figure 2 shows that for both compounds the first five states account for the leading contributions of the near-edge spectrum. States 6–10 lie more than 3 eV above the first K-edge state and give a smaller contribution to the absorption profile. For example, in the case of benzenethiol, states 6–10 contribute to

245

only 28 % of the absorption intensity.

This figure also reports the molecular structures and numbering

scheme adopted in this work. Table 2 reports a comparison of experimental and OCDFT excitation energies, OCDFT oscillator strengths, the LIVVO assignments, and total valence character for each state. The gas-phase experimental excitation energies shown in Table 2 for ethanethiol represents the three




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By analyzing the overlap of the particle orbitals with all orbitals in the LIVVO basis, it is possible to quantify the degree of localization along each bond and discern the origin of some features of the ethanethiol 275

and benzenethiol spectra. Our analysis focuses on the lowest five particle states since these are the ones with the largest contributions from valence orbitals. The lowest energy transition in the NEXAFS spectrum ∗ of ethanethiol and benzenethiol occurs at 2472.9 eV, and was attributed mostly to a S 1s → σS–H transition ∗ transition. 60 The LIVVO assignments, displayed in Figure 3, with a “weak” contribution from a S 1s → σS–C ∗ show that in both cases the first state (corresponding to the particle orbital φ1p ) can be assigned to σS–H

280

∗ (59%) and σS–C (22%) character. Going from ethanethiol to benzenethiol there is a slight increase in the ∗ degree of localization along the thiol bond. This is evidenced by the σS–H character of the first excited state

increasing from 59% in ethanethiol to 61% in benzenethiol. The second peak in the NEXAFS spectra of the thiols occurs at 2473.9 eV for ethanethiol and at 2474.2 eV for benzenethiol. Simple inspection of the particle orbitals (φ2p ) shows orbital character along both the 285

S–H and S–C bonds. The LIVVO assignments show that over 50% of the character of the particle orbital can ∗ ∗ be assigned to the σS−C orbital in each molecule, while less than 17% can be assigned to the σS−H orbital.

Comparing our results with the HF-STEX results of Behyan et al., 59 our assignments agree well for the case of ethanethiol. However, the OCDFT assignment is at odds with theirs for benzenethiol. Behyan et al. 59 attributed this peak feature to transitions involving π ∗ orbitals on the phenyl ring. The discrepancy in this 290

assignment likely stems from the differences in the energy ordering of the virtual orbitals between HF and DFT. The third peak in the spectrum for both thiols is a combination of three low-intensity transitions. Of these, the most intense transition is predicted by OCDFT to be at 2476.2 eV for ethanethiol and 2476.0 eV for benzenethiol and in both cases corresponds to the particle orbital φ4p . Considering the total valence

295

characters shown in Figure 3 and the oscillator strengths shown in Table 2, we see a clear trend relating these two quantities. In the case of ethanethiol, the low valence character (≤ 28%) of transitions 3–5 suggests that the smaller oscillator strength of the third peak is due to the diffuse mixed Rydberg character of particle orbitals φ3p , φ4p , and φ5p . In the case of benzenethiol, the valence character of states φ3p and φ4p is high (85% and 86%, respectively),

300

but these orbitals are mostly localized on the phenyl ring and give small contributions to the transition ∗ ∗ 4 dipole moment. State φ3p is a mixture of πC 2,3,4 and πC4,5,6 at 53% and 32% respectively, while state φp is a ∗ ∗ ∗ mixture of πC 1,2,6 , πC2,3,4 , and πC4,5,6 at 52%, 17%, and 17%, respectively. This analysis suggests that the

small overlap of these states with the sulfur atom is the cause of the small intensity of the corresponding transitions. State φ5p of benzenethiol has a much lower total valence character (16%), however, since it has


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∗ 7% overlap with the σS−H thiol bond orbital, the corresponding transition still has a modest intensity. We

should note that thanks to the LIVVO analysis we were able to quantify the weak local contributions to the states that contribute to the third peak. Previous studies 59 were unable to assess the orbital character of this set of transitions. 4.2. Signatures of hydrogen bonding in the NEXAFS spectrum of the water dimer. Due to the 310

critical role it plays in many processes in nature, hydrogen bonding is one of the most important noncovalent interactions in chemistry. 105–109 The water dimer (H2 O)2 is one of the simplest—yet most important— examples of hydrogen bonding in chemistry and has garnered plenty of attention from both theory and experiment. 110–121 This simple dimer is the basic building block of the structures of liquid water and ice, which have both been studied extensively with NEXAFS spectroscopy to uncover their underlying hydrogen

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bond coordination. 122–124 A host of computational methods have been applied to study the effects of hydrogen bonding on the NEXAFS spectrum of water, including transition potential DFT (TPDFT), 64–66,125–128 coupled-cluster singles and doubles (CCSD), 10,129,130 and complex polarization propagator DFT (CPPDFT). 131 Here, we utilize our LIVVO analysis in order to quantify these effects on the NEXAFS spectrum. Figure 4 shows the OCDFT NEXAFS spectrum for water monomer and the lowest-energy configuration

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of the water dimer. For the water monomer we also report the gas-phase experimental spectrum. 132 To facilitate our analysis, we separate the dimer spectrum into contributions from the hole localized on the oxygen accepting the hydrogen bond (O1 , denoted as dimer-A) and a spectrum for the hole localized on the oxygen donating the hydrogen bond (O4 , denoted as dimer-D). These partial contributions are shown in panels B and C of Figure 4 and are obtained by separating transitions that involve the oxygen donor or

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acceptor 1s core orbital. Table 3 shows all of the calculated excitation energies, oscillator strengths and final LIVVO assignments for each state. Water monomer. The OCDFT computed NEXAFS spectrum of water monomer at the O K-edge, shown in Figure 4, is defined by three characteristic peaks at 533.7 eV, 535.4 eV, and 537.3 eV respectively. OCDFT also predicts a fourth peak; however, the corresponding state has less than 5% total valence character and

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is found at higher energy (539.4 eV). The positions of the three lowest peaks are in excellent agreement with the gas-phase experimental peak features which occur at 534.0 eV, 535.9 eV, and 537.0 eV. 132 The first peak at 533.7 eV has a relative intensity of 0.44 and is characterized by the particle orbital φ1p shown in Figure 5A, while the second peak at 535.4 eV is the most intense of the spectrum and is characterized by the particle orbital φ2p . Two Rydberg transitions at 537.3 eV and 537.8 eV contribute to the third peak in the

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spectrum and are accompanied by a higher energy shoulder feature at 538.5 eV. High-level CCSD calculations performed by Fransson and co-workers 130 also provide a good basis for comparison. The CCSD spectrum




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Table 3. Analyzing the contributions of each LIVVO to the respective particle orbitals reveals that for this ∗ ∗ case, both the σO 1 −H2 and σO1 −H3 LIVVOs contribute equally to each excited state. This is a characteristic

feature of particle orbitals in the spectrum of the water monomer and will be useful in the comparison with 350

the water dimer spectrum. Water dimer, hydrogen bond acceptor. The spectra for localized excitations in the water dimer shown in panels B and C of Figure 4 allow us to discern features that arise from hydrogen bonding in the accepting and donating water molecules. Comparing panels A (monomer) and B (dimer-A) of Figure 4, the two spectra are very similar with three distinct peaks. The first two core excitations for the dimer-A spectrum occur at

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534.0 eV and 535.8 eV and exhibit a 0.3 and 0.4 eV blueshift, respectively, when compared to the monomer. The first two transitions of the dimer-A are also found to have very close oscillator strengths in the monomer. The similarities between these transitions becomes obvious when comparing the particle orbitals for the monomer and the dimer (shown in Figure 5). The particle orbitals for the first two transitions in the dimerA spectrum have no contribution from the donating water molecule and are localized exclusively on the

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accepting water molecule. In addition, the particle orbitals φ1p and φ2p for the dimer-A look nearly identical to those for the water monomer. These two particle orbitals are localized on the hydrogen atoms of the acceptor water and thus are weakly perturbed by the hydrogen bond. In contrast, the three higher energy transitions show some degree of delocalization onto the donating water molecule. The LIVVOs can be used in an effort to quantify the degree of localization/delocalization in the particle

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∗ orbitals involved. Water dimer contains four LIVVOs, two that are localized on the accepting water (σO 1 −H2 ∗ ∗ and σO 1 −H3 ), one that is localized on the free OH bond in the donor water (σO4 −H6 ), and one that represents ∗ the hydrogen bonding interaction between the two water molecules (σO While similar in orbital 4 −H5 ). ∗ character to the other LIVVOs, σO 4 −H5 has contributions from the accepting water molecule which gives

us a direct way to track the overlap of an excited state orbital with O-H antibonding interaction from the 370

hydrogen bond. The values reported in Figure 5B correspond to the percentage of overlap between each water dimer LIVVO and the particle orbitals contributing to the dimer-A spectrum. Considering that orbitals φ1p and φ2p are heavily localized on the accepting water, it is unsurprising that the first two states have large and ∗ ∗ nearly equal contributions from the LIVVOs localized on the accepting water (σO 1 −H2 and σO1 −H3 ). The

375

total valence character of these states (84% and 77%) is similar to the total valence character of the first two particle orbitals in the monomer spectrum (85% and 75%), implying that the presence of the donor water has a negligible effect on φ1p and φ2p . It is encouraging to see that our LIVVO analysis parallels the uniformity seen in these states with regard to their orbital character, peak position, and peak intensity.



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The three particle orbitals corresponding to higher energy transitions in the dimer-A spectrum (φ3p , φ4p , 380

∗ ∗ φ5p ) have significant overlap with σO 4 −H5 and σO4 −H6 . It is in this higher energy region of the spectrum

where we see the most significant distinctions between the dimer-A and monomer spectrum. In the monomer spectrum state 5 gives a shoulder feature at higher energy, while in the dimer-A this state is close in energy to state 6. Our analysis based on LIVVOs shows that states 3–6 experience a stark increase in total valence character from 0%, 0%, 12%, and 5%, respectively in the monomer to 22%, 8%, 29%, and 11%, respectively 385

in the dimer-A. This drastic increase in valence character is caused by interaction with the donor water, as demonstrated by the fact that the particle orbitals for states 3–6 contain important contributions mostly from LIVVOs localized on the donor water. Water dimer, hydrogen bond donor. In contrast to dimer-A, the spectrum for dimer-D bears little resemblance to the monomer spectrum. Although there are still three distinct peak features, the oscillator

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strength of all transitions drop significantly. We also observe a redshift of all three peaks. The first peak exhibits a mild shift of 0.2 eV while the higher energy peaks show more dramatic shifts of 0.7 eV and 1.0 eV respectively. The stark differences in the dimer-D spectrum, can be rationalized using our LIVVO analysis. The φ1p orbital has similar character in the monomer and dimer-D; however, in the latter case it is perturbed by the hydrogen bond and localizes along the free OH group. This phenomenon has been observed in previous

395

studies, 62,130 but the degree of localization has not been rigorously quantified. The LIVVO overlaps reported in Figure 5C for φ1p shows that 53% of the orbital character lies along the ∗ σO 4 −H6 orbital with only 15% localized along the hydrogen bond and 10% of the orbital character coming

from LIVVOs localized on the accepting water molecule. For φ2p , a similar localization is observed, however, a significantly higher percentage (36%) of orbital character is accounted for by LIVVOs on the accepting water. 400

This sharing of electron density with the accepting water molecule is indicative of the effect of hydrogen bonding on the dimer spectrum and has a direct correlation to the changes in intensity that we see in the simulated NEXAFS spectrum. Table 3 shows a drop in fosc when comparing the monomer to dimer-D, with a modest decrease for state 1 (0.00811 → 0.00605) and a far more drastic decrease for state 2 (0.01839 → 0.00878). This intensity drop can be attributed to delocalization of the particle orbital and correlates well

405

with the population of LIVVOs on the accepting water molecule (10% and 36%, respectively). In conclusion, LIVVOs help to address the origin behind the stark differences in the dimer-A and dimerD spectra. Perhaps the most revealing signature of hydrogen bonding in the water dimer is the decrease in intensity of the lowest two peaks in the dimer-D contribution. Previous computational studies of the water dimer have reasoned that the differences in these two spectra arise from the polarization of the orbitals in

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water. 62,130 Our LIVVO analysis shows that the interaction of the two monomers causes the particle orbitals


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to delocalize over both waters, and effect that is more pronounced for the water donating the hydrogen bond. As a consequence, the NEXAFS spectrum will show more marked changes for the water molecule donating a hydrogen bond than for the one accepting it. 4.3. Basis Set and Functional Dependence of LIVVO Analysis. We conclude this section by assessing 415

the dependence of the LIVVO analysis on the choice of basis set and density functional. To fully address this issue of basis set dependence, there are three aspects of the method that must be tested: 1) How consistent are the LIVVO assignments, i.e. is there significant fluctuation in the LIVVO/IAO overlaps (eq 9)? 2) Are the LIVVOs assigned to the excited state consistent? and 3) How much fluctuation is there in the particle orbital/LIVVO overlaps (eq 11)?

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To address the three points raised above, we consider the sulphur K-edge spectrum of ethanethiol. First, we take a look at the assignment of the atomic character of the first two LIVVOs (σS∗3 −H9 ,σS∗3 −C2 ). As shown in Table 4, the overlaps of each LIVVO with eight of the IAOs in ethanethiol are extremely consistent, regardless of the choice of basis set, only deviating by a maximum of 0.01. Next we evaluate if the LIVVO assignments of the particle orbital are consistent for different basis sets. (n)

425

(n)

In the last two columns of Table 4 we report the quantity Ωpl , that is, the overlap of a particle orbital [φp ] with each LIVVO (l ∈ {σS∗3 −H9 , σS∗3 −C2 }). For ethanethiol, φ1p was assigned primarily to the σS∗3 −H9 LIVVO with a weaker contribution from σS∗3 −C2 while φ2p received a complementary assignment. Regardless of the choice of basis set, the LIVVO classification of both excited states is consistent. In all cases, φ1p has greater than 57% overlap with the σS∗3 −H9 LIVVO and 25% or less overlap with the σS∗3 −C2 LIVVO. Similarly, φ2p has

430

57% or greater overlap with σS∗3 −C2 and 25% or less overlap with σS∗3 −H9 . The characterization of the orbital character of excited states is by far the most important feature of this technique, therefore, the consistency seen in the data across different basis sets is very encouraging. Lastly, we will address the consistency of the LIVVO/particle orbital overlap quantity presented in eq 11 with respect to choice of basis set. Table 4 shows that this value can potentially fluctuate ± 10% with

435

the choice of basis set. However, since we have shown that the formulation of the LIVVOs has a negligible dependence on the choice of basis set, we conclude that the basis set dependence seen in these values can solely be attributed to the basis set dependence of the particle orbitals. Although these values can change fairly significantly, it is encouraging that they do not affect the overall interpretation of the excited state. In addition to basis set dependence of the LIVVO analysis, it is important to note that the particle

440

orbitals and their assignment will also depend on the choice of the density functional. The issue of functional dependence of OCDFT excitation energies has been investigated both for valence 56 and core-excited states. 104 For example, a comparison of the C and O K-edge spectra of CO computed with OCDFT and a


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range of functionals showed that relative peak positions are largely insensitive to the amount of Hartree–Fock exchange included in the functional. This is contrary to what is observed for core-excited states computed 445

with time-dependent DFT. 133 Here we extend our previous analysis by considering both the sensitivity of the excitation energy and the LIVVO analysis. In Table 5 we report a comparison of the LIVVO analysis of the first five O K-edge states for each oxygen in the water dimer. We compare the B3LYP hybrid functional to the PBE pure functional, 134 the PBE0 hybrid functional, 135,136 and the long-range corrected hybrid functional ωB97X. 137 In general, we

450

find that the predicted excitation energies are consistent across functionals. When the excitation energies are shifted with respect to the lowest peak (state 1 of dimer-D), the discrepancy between the relative excitation energies as predicted by the B3LYP, PBE, and PBE0 functionals is smaller than 0.2 eV. Results for the ωB97X functional are also consistent with those from B3LYP, but the relative excitation energies for state 5 of dimer-A and states 2 and 5 of dimer-D show differences in the relative excitation energies up to 0.5 eV.

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This difference is likely due to the improved treatment of the long-range part of the exchange-correlation potential in ωB97X. The LIVVO assignment is also consistent across all functionals considered here. A ∗ noticeable exception is the percentage of σO 4 −H5 character assigned to the first particle state of the dimer-D

(52–55 % for B3LYP, PBE, and PBE0 vs. 62% for ωB97X). Other significant differences in the LIVVO assignments are also observed for states that have smaller valence character (3–5), where, for example, the 460

contributions of a given LIVVO to a particle orbital may vary by as much as 6%. Overall, the very good agreement of the OCDFT excitation energies and the LIVVO analysis corroborates our previous findings and attests to the robustness of this methodology. 5. CONCLUSIONS In this work, we present an automated method for the characterization of core-valence excited states. Our

465

approach utilizes intrinsic atomic orbitals (IAOs) to derive a set of localized intrinsic valence virtual orbitals (LIVVOs) and assign their orbital character. LIVVOs are in turn used to classify particle orbitals calculated using orthogonality constrained density functional theory (OCDFT). For molecular orbitals with dominant valence character, this analysis provides keen insight into the localized σ ∗ and π ∗ orbitals that contribute to it. For example, in the first two core excited states of

470

ethanethiol, an electron is promoted to virtual orbitals that span both the S–C and S–H bonds. These contributions are difficult to discern by visual inspections of the MOs. For the first state, our analysis reveals that this transition involves primarily the thiol bond. Instead, for the second state we find that the particle orbital is dominated by a contributions from a C-S σ ∗ orbital. We have also shown how LIVVOs may be useful to quantify differences in excited states due to changes in



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the molecular environment. For example, in the case of the water dimer we analyze the correlation between orbital localization due to hydrogen bonding and spectral features. The NEXAFS spectrum of water dimer at the oxygen K-edge is known to have distinct contributions from the oxygens at water donating (dimer-D) or accepting (dimer-A) the hydrogen bond. Analysis of the two lowest particle orbitals for excitations on dimer-A show equal localization along intramolecular OH bonds. However, the corresponding excitations

480

for dimer-D are significantly different. In the lowest energy state, 70% of the electron is localized on the dimer-D, and this percentage decreases to 34% in the second state. These changes in degree of localization explain changes in intensity in the oxygen K-edge spectrum predicted by theory. We have also considered the basis set dependence of the LIVVOs and the orbital analysis. Due to the robustness of the intrinsic atomic orbitals, the LIVVOs and our analysis of the orbital character are largely

485

insensitive to the basis set size. As for other ∆SCF methods, a major advantage of our OCDFT/LIVVO approach is its low computational cost. This is dominated by the OCDFT step, which has a cost equal to that of one ground state DFT computation times the number of excited states targeted. Hence, a wide range of future applications may be envisioned of the OCDFT/LIVVO method to systems significantly larger than those studied here.

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For example, we are currently investigating the use of OCDFT to compute the carbon K-edge spectrum of organic molecules chemisorbed on semiconductor nanoparticles. In this case, the LIVVOs will be helpful to quantify the participation of the the adsorbate and surface atoms in the features of NEXAFS spectrum. Also, while our LIVVO analysis is applied here within the context of orthogonality constrained density functional theory, the method is general and can be used to decompose any set of virtual orbitals. Overall, our

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LIVVO-based analysis of particle orbitals is a first step toward creating a general, robust, and automatic method to assign the character of excited states.


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ACKNOWLEDGMENTS This work was supported by start-up funds provided by Emory University. We would like to thank Dr. Robert M. Parrish for kindly providing us with the implementation of intrinsic atomic and bond orbitals in 500

Psi4. W.D.D. is supported by the National Science Foundation Graduate Research Fellowship under Grant No. 0000048655. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. ASSOCIATED CONTENT Supporting information. Cartesian coordinates of all molecules (text file) and plots of all the LIVVOs

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for ethanethiol and benzenethiol (pdf). This information is available free of charge via the Internet at http://pubs.acs.org

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Table 2 Transition energies (unshifted), oscillator strengths (fosc ), and LIVVO assignments for all states calculated in the NEXAFS spectrum of ethanethiol and benzenethiol. The numbering scheme used to classify the particle orbitals is reported in Figure 2. Experimental energies and assignments are taken from Ref. 60.

State

Energy (eV)

1

2472.9

OCDFT fosc LIVVO Assignment

0.00219

Ethanethiol 59% σS∗3 −H9

2

2473.9

0.00114

3

2475.6

0.00044

4

2476.2

0.00072

5

2476.1

0.00047

6

2476.2

0.00003

7

2477.0

0.00018

8

2477.3

0.00004

9

2477.3

0.00031

10

2477.4

0.00030

22% σS∗3 −C2 57% σS∗3 −C2 17% σS∗3 −H9 ∗ 9% σC 2 −H8 ∗ 7% σC 2 −H7 6% σS∗3 −H9 7% σS∗3 −C2 ∗ 5% σC 2 −H8 ∗ 6% σC1 −H6 ∗ 4% σC 2 −H7 ∗ 5% σC 1 −H5 ∗ 4% σC 2 −H7 ∗ 2% σC1 −C2 ∗ 1% σC 1 −H6 ∗ 1% σC 1 −H5 ∗ 5% σC 2 −H7 ∗ 3% σC 2 −S3

1

2472.9

0.00231

Benzenethiol 61% σS∗7 −H8

Experiment Energy (eV) Assignment

2472.2

∗ Weak σS−C

2473.1

∗ σS−C

2474.9

Poorly Defined

2472.4

∗ σS−H

21% σS∗7 −C3 2

2474.2

0.00105

3

2475.0

0.00005

4

2475.2

0.00009

5

2476.0

0.00056

6

2476.4

0.00003

7

2477.4

0.00074

8

2476.9

0.00043

9

2477.4

0.00023

10

2477.4

0.00018

54% σS∗7 −C3 16% σS∗7 −H8 ∗ 53% πC 2,3,4 ∗ 32% πC 4,5,6 ∗ 52% πC 1,2,6 ∗ 17% πC 2,3,4 ∗ 17% πC 4,5,6 7% σS∗7 −H8 ∗ 4% σC 4 −H13 ∗ 6% σC2 −H9 ∗ 3% σC 4 −H13 ∗ 4% πC 4,5,6 ∗ 2% πC2,3,4 ∗ 7% σC 3 −S7 ∗ 4% σC5 −H12 ∗ 4% σC 6 −H11 ∗ 3% σC 1 −H10 ∗ 3% σC5 −H12 ∗ 4% σC 6 −H11

∗ σS−H

∗ Weak σS−C

2473.5

∗ Weak πC=C

2475.3

∗ σS−C



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Table 3 Transition energies (unshifted), oscillator strengths (fosc ), and LIVVO assignments for all states calculated in the NEXAFS spectrum of the water monomer, dimer-A, and dimer-D. The numbering scheme used to classify the particle orbitals is reported in Figure 4. State

Energy (eV)

1

533.7

fosc Monomer 0.00811

LIVVO Assignment

∗ 42% σO 1 −H2 ∗ 42% σO 1 −H3

2

535.4

0.01839

∗ 38% σO 1 −H2

3 4 5

537.3 537.8 538.5

0.00785 0.00257 0.00167

Rydberg/Diffuse Rydberg/Diffuse ∗ 6% σO 1 −H2

∗ 38% σO 1 −H3

∗ 6% σO 1 −H3

6

539.4

0.00548

∗ 2% σO 1 −H2

1

534.0

Dimer-A 0.00737

2

535.8

0.01862

∗ 39% σO 1 −H2

3

537.7

0.00578

∗ 13% σO 4 −H5

∗ 2% σO 1 −H3 ∗ 41% σO 1 −H2 ∗ 41% σO 1 −H3 ∗ 39% σO 1 −H3 ∗ 9% σO 4 −H6

4

538.0

0.00510

∗ 5% σO 4 −H6 ∗ 3% σO 4 −H5

5

539.3

0.00277

∗ 12% σO 4 −H6 ∗ 8% σO 4 −H5 ∗ 4% σO 1 −H2 ∗ 4% σO 1 −H3

6

539.0

0.00146

1

533.5

Dimer-D 0.00605

2

534.9

0.00878

∗ 5% σO 1 −H2 ∗ 5% σO 1 −H3 ∗ 53% σO 4 −H6 ∗ 15% σO 4 −H5 ∗ 23% σO 4 −H6 ∗ 18% σO 1 −H2 ∗ 18% σO 1 −H3

3

536.1

0.00575

∗ 3% σO 1 −H2 ∗ 3% σO 1 −H3

4

536.6

0.00283

∗ 2% σO 4 −H5

5

537.3

0.00624

∗ 26% σO 4 −H5 ∗ 4% σO 1 −H2 ∗ 4% σO 1 −H3

6

537.8

0.00210

∗ 12% σO 1 −H2 ∗ 12% σO 1 −H3


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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(n)

Table 4 Basis set dependence of the IAO assignment of LIVVOs and overlap [Ωpl ] of the σS∗3 −H9 and σS∗3 −C2 LIVVOs with the first two particle orbitals of ethanethiol. LIVVO

ψ7IAO (C22s )

ψ9IAO (C22p )

IAO (C2 ) ψ10 2p

IAO (S3 ) ψ13 3s

σS∗3 −H9 σS∗3 −C2

0.00 0.09

0.00 0.33

0.00 0.09

0.08 0.07

σS∗3 −H9 σS∗3 −C2

0.00 0.09

0.00 0.33

0.00 0.08

σS∗3 −H9 σS∗3 −C2

0.00 0.09

0.00 0.33

0.00 0.08

σS∗3 −H9 σS∗3 −C2

0.00 0.09

0.00 0.33

0.00 0.08

σS∗3 −H9 σS∗3 −C2

0.00 0.09

0.00 0.33

0.00 0.08

σS∗3 −H9 σS∗3 −C2

0.00 0.08

0.00 0.33

0.00 0.09

IAO (S3 ) ψ17 2p

cc-pVDZ 0.23 0.00 cc-pVTZ 0.08 0.22 0.07 0.00 cc-pVQZ 0.08 0.22 0.07 0.00 aug-cc-pVDZ 0.08 0.22 0.07 0.00 aug-cc-pVTZ 0.08 0.22 0.07 0.00 aug-cc-pVQZ 0.07 0.23 0.07 0.00

(1)

(2)

IAO (S3 ) ψ18 3p

IAO (S3 ) ψ19 3p

IAO (H9 ) ψ25 1s

Ωpl

Ωpl

0.00 0.32

0.15 0.09

0.54 0.00

63.7 22.6

22.3 67.1

0.00 0.32

0.15 0.09

0.54 0.00

59.3 24.6

23.9 61.5

0.00 0.32

0.15 0.09

0.54 0.00

57.8 24.3

22.8 59.2

0.00 0.32

0.16 0.09

0.54 0.00

60.8 15.3

11.1 63.1

0.00 0.32

0.15 0.09

0.54 0.00

57.9 19.2

14.7 59.1

0.00 0.32

0.16 0.09

0.54 0.00

56.5 20.4

15.7 57.3

Table 5 Transition energies and LIVVO assignments for all states calculated in the NEXAFS spectrum of the water dimer-A and dimer-D using various density functionals and the jun-cc-pVTZ basis set.

State

B3LYP Energy (eV)

1

534.0

2

535.8

3

537.7

4

537.9

5

539.3

1

533.5

2

534.9

3

536.1

4

536.6

5

537.3

LIVVO

PBE Energy (eV)

41% 41% 39% 39% 13% 9% 5% 3% 12% 8%

∗ σO 1 −H2 ∗ σO 1 −H3 ∗ σO 1 −H2 ∗ σO 1 −H3 ∗ σO 4 −H5 ∗ σO 4 −H6 ∗ σO 4 −H6 ∗ σO4 −H5 ∗ σO 4 −H6 ∗ σO 4 −H5

533.1

53% 15% 23% 18% 3% 3% 2% 1% 26% 4%

∗ σO 4 −H6 ∗ σO 4 −H5 ∗ σO 4 −H6 ∗ σO 1 −H2 ∗ σO 1 −H2 ∗ σO 1 −H3 ∗ σO 4 −H5 ∗ σO1 −H3 ∗ σO 4 −H5 ∗ σO 4 −H6

532.5

534.8 536.7 537.1 538.3

533.9 535.2 535.6 536.3

41% 41% 38% 38% 8% 5% 9% 8% 14% 7% 52% 15% 26% 18% 3% 3% 3% 1% 29% 4%

LIVVO

PBE0 Energy (eV)

Dimer-A ∗ σO 1 −H2 ∗ σO 1 −H3 ∗ σO 1 −H2 ∗ σO 1 −H3 ∗ σO 4 −H5 ∗ σO 4 −H6 ∗ σO 4 −H5 ∗ σO4 −H6 ∗ σO 4 −H6 ∗ σO 4 −H5 Dimer-D ∗ σO 4 −H6 ∗ σO 4 −H5 ∗ σO 4 −H6 ∗ σO 1 −H2 ∗ σO 1 −H2 ∗ σO 1 −H3 ∗ σO 4 −H5 ∗ σO1 −H3 ∗ σO 4 −H5 ∗ σO 4 −H6

533.4 535.2 537.1 537.2 538.7

532.9 534.4 535.5 535.9 536.7


LIVVO

ωB97X Energy (eV)

41% 41% 38% 38% 13% 8% 3% 2% 13% 8%

∗ σO 1 −H2 ∗ σO 1 −H3 ∗ σO 1 −H2 ∗ σO 1 −H3 ∗ σO4 −H5 ∗ σO 4 −H6 ∗ σO 4 −H6 ∗ σO 4 −H5 ∗ σO4 −H6 ∗ σO 4 −H5

534.8

55% 15% 19% 18% 2% 2% 1% 1% 24% 4%

∗ σO 4 −H6 ∗ σO 4 −H5 ∗ σO 4 −H6 ∗ σO 1 −H2 ∗ σO 1 −H2 ∗ σO 1 −H3 ∗ σO4 −H5 ∗ σO 1 −H3 ∗ σO 4 −H5 ∗ σO 4 −H6

534.4

536.7 538.9 538.9 540.6

536.1 537.2 537.7 538.6

LIVVO

41% 41% 39% 39% 16% 9% > 1% > 1% 10% 8%

∗ σO 1 −H2 ∗ σO 1 −H3 ∗ σO 1 −H2 ∗ σO 1 −H3 ∗ σO4 −H5 ∗ σO 4 −H6 ∗ σO 4 −H6 ∗ σO 4 −H5 ∗ σO4 −H5 ∗ σO 4 −H6

61% 14% 18% 16% 2% 2% 1% 1% 25% 5%

∗ σO 4 −H6 ∗ σO 4 −H5 ∗ σO 1 −H2 ∗ σO 4 −H5 ∗ σO 1 −H2 ∗ σO 1 −H3 ∗ σO1 −H3 ∗ σO 1 −H2 ∗ σO 4 −H5 ∗ σO 4 −H6

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