Physical Properties, Exciton Analysis, and Visualization of Core-Excited States: An Intermediate State Representation Approach.

Article pubs.acs.org/JCTC

Physical Properties, Exciton Analysis, and Visualization of CoreExcited States: An Intermediate State Representation Approach Jan Wenzel* and Andreas Dreuw* Interdisciplinary Center for Scientific Computing, University of Heidelberg, Im Neuenheimer Feld 368, 69120 Heidelberg, Germany ABSTRACT: The theoretical simulation of X-ray absorption spectra is in general a challenging task. However, for small and medium-sized organic molecules, the algebraic diagrammatic construction scheme (ADC) for the polarization operator in combination with the core−valence separation approximation (CVS) has proven to yield core-excitation energies and transition moments with almost quantitative accuracy allowing for reliable construction of X-ray absorption spectra. Still, to understand core-excitation processes in detail, it is not sufficient to only compute energies, but also properties like static dipole moments and state densities are important as they provide deeper insight into the nature of core-excited states. Here, we present for the first time an implementation of the intermediate state representation (ISR) approach in combination with the CVS approximation (CVS-ISR), which gives, in combination with the CVS-ADC method, direct access to core-excited state properties. The performance of the CVS-ADC/CVSISR approach is demonstrated by means of small- and medium-sized organic molecules. Besides the calculation of core-excited state dipole moments, advanced analyses of core-excited state densities are performed using descriptors like exciton sizes and distances. Plotting electron and hole densities helps to determine the character of the state, and in particular, the investigation of detachment/attachment densities provides information about orbital relaxation effects that are crucial for understanding core excitations.

1. INTRODUCTION In recent years, X-ray absorption spectroscopic (XAS) techniques like near edge X-ray absorption fine structure (NEXAFS) spectroscopy are commonly used in surface science, medical biological research, or in the field of organic electronics.1−9 With the help of modern synchrotron soft beam sources, molecules absorb high-energy X-rays resulting in spatially localized core-excited states, in which electrons are promoted from the 1s orbitals to the virtual level (K-edge). These quasi-bound states are metastable and undergo fast decay processes like Auger or interatomic Coulombic decay (ICD).10−15 In practice, these techniques are used to investigate, for example, the electronic structure of molecules and the orientation between adsorbed substances on metal surfaces or to analyze the band levels of conducting materials.16−24 The theoretical description of core-excited states is crucial for a proper interpretation of X-ray absorption (XA) spectra and for the prediction of unknown molecular species.25 Since the excitation of core electrons requires a large amount of energy, the generated core−hole interaction induces a rearrangement of the valence electrons due to the reduced shielding of the nuclei.23,26−30 This effect can be understood as an orbital relaxation effect, which leads to a lowering of the excitation energy of the final state compared to cases where no rearrangement of the valence electrons occurs. Therefore, a © XXXX American Chemical Society

quantum chemical method for the calculation of core excitations requires a treatment of these relaxation effects. Nowadays, a plethora of quantum chemical approaches is available for the calculation of core-excited states, for example, the coupled cluster (CC)31−34 family, like equation-of-motion coupled cluster (EOM-CC),35−39 the symmetry-adapted cluster configuration interaction (SAC-CI),40−45 or the approximate coupled cluster scheme of second order (CC2).46 CC2 can be combined with a complex polarization propagator (CPP) approach.32,33,47 Other methods, like the GW approximation to the Bethe−Salpeter equation 48 and the static-exchange (STEX)49 approach also provide accurate results. Large systems are usually treated with time-dependent density functional theory (TD-DFT)8,50−54 in a reasonable computational time. Recently, we have presented an efficient implementation21−23 of the restricted and unrestricted algebraic-diagrammatic construction scheme (ADC)55−57 up to third order exploiting the core−valence separation (CVS)4,58−62 approximation. Generally, ADC is a quantum chemical method for the calculation of excitation energies and transition moments originating from many-body Green’s function theory using diagrammatic perturbation theory of the polarization propagator in combination with the typical Møller−Plesset63 partitioning of the Hamiltonian. At second and third order, Received: December 8, 2015

A

DOI: 10.1021/acs.jctc.5b01161 J. Chem. Theory Comput. XXXX, XXX, XXX−XXX

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Journal of Chemical Theory and Computation

the 1TDM is directly available using the ADC method, the 1DDM or rather excited-state density (DM) can only be obtained via the intermediate state representation (ISR) approach.56,75,76 Here, the ADC secular matrix is formulated with respect to an explicitly constructed basis set of intermediate states leading to a representation of the excited state wave function and straightforwardly to excited-state properties, if the corresponding one-particle operator is available. In this work, the CVS approximation is applied for the first time to the ISR formalism of a general one-particle operator, resulting in the CVS-ISR approach. This gives access to coreexcited state properties at the CVS-ADC level. For demonstrating the capability of the new CVS-ADC/CVS-ISR approach, calculations of static dipole moments and core-excited state densities have been performed for a set of representative molecules. The paper is structured as follows. In the following Section 2, the theoretical basis of the ADC method, CVS approximation, and ISR approach are shortly outlined, followed by implementation notes and computational details in Section 3. In Section 4, the results of calculations of physical properties and analyses of the exciton wave functions are presented. Finally, Section 5 provides concluding remarks.

small- and medium-sized molecules of up to 25 atoms can be treated. Since the method is size consistent and ab initio, it can be seen as an almost black-box method for the calculation of excited states.25,57,64,65 To solve the ADC eigenvalue problem, one has to use iterative diagonalization schemes, which usually provide the lowest excited states. To obtain core-excited states directly, which are located in the high-energy X-ray region of the spectrum, the combination of ADC with the CVS approximation (CVS-ADC) is necessary to avoid the calculation of all underlying valence-excited states. The CVS approximation is based on the effective decoupling of the valence-excited states from the core-excited ones. An important advantage of CVS-ADC beyond strict second order is the indirect inclusion of orbital relaxation effects via double amplitudes. Hence, a proper description of core-excited states is straightforwardly possible. It could be demonstrated already a few times that the extended variant CVS-ADC(2)-x provides excellent agreement with experimental data of excitation energies and oscillator strengths.21−23,60,61 Especially, in combination with the 6-311++G**66−68 basis set, an averaged error of the excitation energies of only about 0.1% is found.21−23 However, to understand core-excitation processes in detail, it is not sufficient to only analyze excitation energies and the transition moments of core-excited states. Excited-state attributes, like static dipole moments or excited-state densities, are also important physical properties that give deep insight into their nature. A proper description of static excited-state dipole moments, for example, is useful to analyze solvent or environmental effects that are mainly influenced by Coulomb interactions and charge distributions of the molecule. However, it is very challenging to determine dipole moments of coreexcited systems experimentally due to the short lifetime of a core excitation. Therefore, experimental data of static dipole moments of core-excited states are not available yet. For the proper characterization of an electronically excited state, it can be advantageous to look at state densities rather than considering molecular orbitals (MO) because excited-state vectors mostly consist of a mixture of different MO transitions.69 Especially for excited-state vectors, where no dominant amplitude can be identified, the determination of their character can be difficult in the MO picture. In such cases, the evaluation and visualization of electron detachment/ attachment (D/A) densities70 based on the one particle difference density matrix (1DDM) between ground and excited state or the respective hole/electron (h/e) densities,71 which are derived from the one-particle transition density matrix (1TDM),72 provide an unambiguous picture of the excitation process. Additionally, the (D/A) densities contain information about orbital relaxation effects, which can be visualized by plotting the (D/A) densities or can be quantified by calculating the electronic promotion numbers.72,73 This is an important interpretation tool for core-excited states. A deeper evaluation of the 1TDM and 1DDM also provides access to exciton sizes, like hole sizes or distances between hole and electron densities. Such descriptors can be calculated straightforwardly by applying statistical analysis in terms of multipole moments of the state densities.69,74 Generally, excited-state properties can be readily deduced from excited-state wave functions.56 In the case of ADC, the direct computation of excitation energies and transition moments is inherent, but it is not possible to determine excited-state wave functions and their properties directly. While

2. THEORY AND IMPLEMENTATION Detailed descriptions of the ADC method, ISR approach, and CVS approximation are given in the literature.21−23,55−60,75−77 Here, for clarity and completeness, the general derivations of these concepts are briefly summarized. 2.1. Algebraic Diagrammatic Construction Scheme of the Polarization Propagator. Within the ISR, the excited state wave function Ψn is represented in a complete basis of intermediate states (IS) |Ψ̃I⟩ according to Ψn =

∑ XnJ |ΨĨ ⟩ (1)

J

where XnJ denotes eigenvectors of the ADC matrix. The IS {Ψ̃I} values are constructed from a set of so-called correlated states {Ψ0J } that are generated by applying an excitation operator (Ĉ ) to the correlated ground state wave function Ψ0.

Ψ 0J = Ĉ Ψ0

(2)

This correlated state basis has to be orthonormalized via the Gram−Schmidt procedure, resulting in an intermediate state basis that is used to establish a matrix of the Hamiltonian shifted by the exact ground state energy E0: MIJ = ⟨ΨĨ |Ĥ − E0|Ψ̃J ⟩

(3)

Using this matrix, the corresponding Hermitian eigenvalue problem MX = XΩ,

X†X = 1

(4)

has to be solved, where the excitation energies Ωn = En − E0 are calculated as eigenvalues, and the eigenvectors are directly related to the transition amplitudes. Therefore, properties like transition dipole moments are calculated directly using

f n = ⟨Ψñ |μ|̂ Ψ0⟩

(5)

where μ̂ represents the dipole operator. The solution of the Hermitian eigenvalue problem is obtained via typical iterative B


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Using expression 7, the matrix element of the one-particle operator D̂ with respect to many-particle states can generally be written as

diagonalization schemes, for example, the Davidson algorithm, yielding the energetically lowest eigenvalues.78 Since the exact ground state wave function is generally unknown and the configuration space has to be restricted, the ADC matrix elements and transition moments are approximated via truncation of the configuration space and using perturbation theory, that is, Møller−Plesset63 partitioning of the Hamiltonian.

⟨Ψ|I D̂ |ΨJ ⟩ = ⟨Ψ|I ∑ drscr†cs|ΨJ ⟩ r ,s

γ II(r , r′) = n

∫ ΨI(r , r2 , ..., rn)ΨI(r′, r2 , ..., rn)dr2 ... drn

can be constructed, where ri corresponds to the spatial and spin coordinates of the ith electron. Then, the total density of a coreexcited state can be calculated as the diagonal part of the density matrix as ρ II (r ) = γ II(r , r )

(13)

The density matrix as well as the one-particle transition density matrix provide access to a magnitude of descriptors, for example, exciton particle or hole sizes. A detailed description of the density analysis tools, which are employed in this work, can be found in the literature.69,71,73,74 2.3. Core−Valence Separation Approximation. The calculation of core-excited states is generally tedious because they are located in the high X-ray region of the spectrum. Therefore, standard iterative diagonalization schemes that are applied to solve the ADC eigenvalue problem (eq 4) cannot be used because they are designed to calculate the lowest excited states. Hence, one has to compute all excited states energetically below the core excitations, which demands a drastic computational effort. The CVS approximation exploits the very weak coupling of core-excited states to valence-excited ones. The Coulomb couplings are very small and vanish practically. Hence, they can be neglected and set a priori to zero

(7)

†

(8)

where X⃗ n is the n eigenvector of the ADC matrix, and the matrix D̃ is the representation of D̂ in the ISR basis according to th

D̃IJ = ⟨ΨĨ |D̂ |Ψ̃J ⟩

(11)

(12)

where drs denotes the one-particle matrix elements associated with D̂ , and c†r cs describes the particle creation and annihilation operators, respectively. Similar to the derivation shown in Section 2.1, the excited state property in the ISR/ADC picture is given as

Dn = ⟨X⃗ n |D̃ |X⃗ n⟩

∑ drsρrs

If J = I, ρrs describes the one-particle density of a state, while the transition density between two states ρIJ is provided if I ≠ J. Thus, it is possible to extract equations for the one-particle (transition) density matrices from the above ISR expressions for the calculation of excited state densities. Generally, a oneparticle density matrix of the state I of the type

∑ drscr†cs r ,s

=

r ,s

(6)

The ADC(2) and ADC(3) matrices consist of four different blocks due to the configuration space with the typical structure of particles (p) and holes (h): p-h,p-h; p-h,2p-2h; 2p-2h,p-h; 2p-2h,2p-2h. For the ADC(2) method, two variants exist: the strict version ADC(2)-s and the extended variant ADC(2)-x. In ADC(2)-s, the matrix elements of the 2p-2h,2p-2h block are expanded only in zeroth order, while in ADC(2)-x, this block comprises also the first-order terms. The explicit equations of matrix elements of M up to ADC(3) are given in the literature.77 2.2. Intermediate State Representation of a General One-Particle Operator. Besides the derivation of the ADC matrix elements and transition moments, the ISR approach also provides an elegant way to obtain physical properties of electronically excited molecules as shown by Schirmer and Trofimov.56 Starting with a general one-particle operator D̂ , for example, dipole moment, the corresponding property Dn of the nth excited state can be evaluated using Dn = ⟨Ψn|D̂ |Ψn⟩ with D̂ =

∑ drs⟨Ψ|I cr†cs|ΨJ⟩ r ,s

M = M(0) + M(1) + M(2) + M(3) + ... f = f (0) + f (1) + f (2) + f (3) + ...

=

(9)

⟨Ip|qr ⟩ = ⟨pI |qr ⟩ = ⟨pq|Ir ⟩ = ⟨pq|rI ⟩ = 0

Analogous to eq 6, D̃ IJ is described via perturbation expansion. At the second-order level, the following contributions have to be considered:56

⟨IJ |pq⟩ = ⟨pq|IJ ⟩ = 0 ⟨IJ |Kp⟩ = ⟨IJ |pK ⟩ = ⟨Ip|JK ⟩ = ⟨pI |JK ⟩ = 0

Small letters p, q, and r refer to general valence orbitals, while capital letters correspond to core orbitals. Numerically, the occupied index in the p-h configurations and one index in the 2p-2h configurations is restricted to a core orbital. As a result, only matrix elements corresponding to MIa,Kc (p-h,p-h block), MIjab,Kc (2p-2h,p-h block), MIa,Klcd (p-h,2p-2h block), and MIjab,Klcd (2p-2h,2p-2h block) need to be considered, where I, J, and K represent core orbitals, while j and l describe occupied valence orbitals and a, b, c, and d virtual orbitals. Consequently, the core-excitation space can be treated independently, and the diagonalization of the CVS-ADC secular matrix is computationally much faster compared to the conventional ADC approach. The final CVS-ADC working equations up to third

̃ = D11 ̃ (0) + D11 ̃ (1) + D11 ̃ (2) D11 ̃ = D12 ̃ (0) + D12 ̃ (1) D12 (0) D̃22 = D̃22

(14)

(10)

At the first-order treatment of ADC, the ADC(1) eigenvectors ̃ (1) are combined with the D̃ (0) 11 and D11 property matrices. Note that all first-order contributions of the ISR property matrix vanish. At the third-order ADC level, no expressions of the ISR property matrices are available yet. To overcome this issue, the CVS-ADC(3) excitation vectors are contracted with the second-order expression (eq 10). C


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Journal of Chemical Theory and Computation order can be found in the literature.23 In this work, the CVS approximation is applied, for the first time, to the ISR of a general one-particle operator. In Appendix A, the final CVS-ISR working equations are given and discussed. 2.4. Role of Relaxation Effects. Orbital relaxation effects play important roles for the description of core-excited states. Electrons that occupy 1s orbitals are strongly bound, and hence, a large amount of energy is needed to detach an electron from a core level. As a consequence, the nucleus loses substantial effective shielding that induces a rearrangement of the valence electrons.26−30 This orbital relaxation, that is, a formal contraction of the wave function, leads to a significant lowering of the energy of the final state. Former work23 has been concerned with a detailed discussion of relaxation effects and the theoretical inclusion within the CVS-ADC method via couplings to doubly, or generally, higher excited amplitudes and provided an explanation of the accuracy of the CVS-ADC methods at different order. Since CVS-ADC(1), which for the energies is identical to configuration interaction singles with the CVS approximation (CVS-CIS), does not contain higher excited configurations than singles, and orbital relaxation or polarization effects are not included at this level of theory. As a consequence, excitation energies are strongly overestimated. CVS-ADC(2)-x provides the best agreement with experimental data because fortuitous error compensation is inherent at this level of theory, leading to an improved description of relaxation and other many-body effects. At the CVS-ADC(2)-s and CVSADC(3) levels, this error compensation is broken due to a decreased effective coupling between singly and doubly excited configurations. Hence, relaxation effects are less considered, leading in a general to a slight overestimation of core-excitation energies at these CVS-ADC levels. So far, the amount of doubly excited amplitudes (R2) has been used as a rough indirect indicator to measure the influence of orbital relaxation effects.21−23,69 However, using the R2 value is theoretically not fully justified because it simply counts the number of doubly excited amplitudes, while the relaxation effects are included in the couplings between singles and doubles. In addition, the quantitative interpretation of the R2 value is only valid for the CVS-ADC method itself; thus, there is no possibility for a comparison with other methods. In this work, alternative quantification tools are presented based on the core-excited state density, which gives also access to visualize these effects. These tools are (D/A) densities, corresponding promotion numbers p DA , and exciton sizes.69,71,73,74 Since the (D/A) densities are generally independent of the employed quantum chemical method, a direct comparison between different methods is quantitatively possible.

required to perform symmetry-aware calculations. Oscillator strengths are calculated via the transition density matrix form of the spectral amplitudes, which are derived from the ADC formalism after applying the CVS approximation. Note that at the third-order ADC level, no algebraic expressions for the spectral amplitudes are available yet. To overcome this issue, the CVS-ADC(3) excitation vectors are contracted with the second-order expression of the spectral amplitudes. The final working equations of the transition moments are thus identical to the ones of CVS-ADC(2) and can be found in the literature.60 The same applies for the ISR equations for a general one-particle operator as explained in Section 2.2. The accuracy of the transition moments of this approximation for valence- and core-excited states has been validated in previous work, and for consistency, the method for calculating oscillator strengths and excited state properties at the third-order ADC level is therefore denoted as CVS-ADC(3,2).23,65 Descriptors like exciton sizes and promotion numbers have been calculated using the libwfa69,71,73,74 package in the Q-Chem 4.3 program that has been combined with our implementation of the CVS-ADC/CVS-ISR methods. 3.2. Computational Details. The ground state structures of the small molecules NH3, H2O, CH2CHF, and the CH3 radical have been optimized at the CCSD31 level using the def2-QZVPPD82 basis set. Ground state geometry optimizations of acenaphthenequinone (ANQ), phenol, and 2,2′bithiophene (BT) used for the PES scans (Section 4.4) have been performed at the level of second-order Møller−Plesset perturbation theory (MP2)63 employing the 6-311++G**66−68 basis set. For comparison with former results,21−23 the structure of ANQ (used in Section 4.2) has been optimized at the MP2 level in combination with the resolution-of-the-identity (RI)83,84 approximation and the def2-TZVPP85 basis with the respective auxiliary TZVPP basis set.86 The ground state geometry of cytosine has also been optimized at the MP2 level of theory, but the SV(P)87 basis set has been employed to be in accordance with former work.69 To save computational time, point group symmetry has been exploited in all calculations of ANQ (C2v) and BT (C2). The CCSD calculations have been performed using the Q-Chem 4.3 program,80 while for the geometry optimizations at the RI-MP2 level the TURBOMOLE 6.3.7 program88 has been used. For the open-shell systems, the unrestricted variants of CCSD and MP2 have been used. Core excitations have been calculated with our implementation of CVS-ADC in the Q-Chem 4.3 program up to third order of perturbation theory. To obtain the core-excited state properties, our implementation of the CVS-ISR of a general one-particle operator has been used. The calculations of the methyl radical have been performed using the unrestricted variant of CVS-ADC (CVS-UADC).22 The core-excited states at the SAC-CI SD-R level have been computed using the implementation available in Gaussian 09.89,90 Henceforward, we neglect to mention the SD-R approximation explicitly in this article. If not otherwise stated, the calculations of core-excited states have been performed employing the Cartesian 6D/10F version of the 6-311++G** basis. Furthermore, note that relativistic effects are not included within the CVS-ADC approaches. However, relativistic effects have only a minor influence on XA spectra of light elements.60 All computed values shown in this work are absolute without any constant shift, as is typically done for core-excitation spectra.8

3. COMPUTATIONAL DETAILS AND IMPLEMENTATION 3.1. Implementation Notes. CVS-ADC up to third order and the CVS-ISR for a general one particle operator up to second order have been implemented in the adcman79 program available in the Q-Chem 4.380 program package. The CVS-ISR formalism has been implemented as the oneparticle density matrix of a core-excited state. All tensor operations that are required to solve the CVS-ADC/CVS-ISR eigenvalue problems are performed using the parallelized general purpose tensor library libtensor.81 This library fully supports point group, spin, and integral symmetry. Hence, no symmetry-adapted versions of the ADC equations are D


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4. CORE-EXCITED STATE PROPERTIES AT THE CVS-ADC/ISR LEVEL In this section, the application of the CVS-ISR approach is presented by means of a variety of example systems. The first part concentrates on core-excited state dipole moments at the CVS-ADC/ISR level with a comparison to SAC-CI results. Therefore, the small molecules ammonia (NH3), fluoroethene (CH2CHF), and water (H2O) have been chosen. The focus of the subsequent sections will be on the analysis of relaxation effects provided at different CVS-ADC orders followed by a demonstration of investigating core-excited state densities by using powerful descriptors.69,71,73,74 The DNA base cytosine, acenaphthenequinone (ANQ), CH2CHF, NH3, H2O, and CH3 have been selected for this analysis. These results are compared at different CVS-ADC levels by plotting (D/A) as well as (h/e) densities and are quantified by means of exciton sizes as well as promotion numbers. As the last step, trends of core-excited state properties along potential energy surfaces (PES) are analyzed. For this purpose, two representatives from the field of organic electronics have been chosen as well as one from the field of technical chemistry: ANQ as a model system for an electron acceptor, 2,2′-bithiophene (BT) for an electron donor, and phenol as a typical precursor in polymer synthesis. Regarding ANQ and phenol, the PES along the distances of the C−O bonds are evaluated, while the torsion around the central dihedral angle that connects the two thiophene rings is analyzed in the case of BT. The structures of all molecules investigated in this work are given in Figure 1.

to provide accurate ADC eigenvectors for core-excited states, in principle, one can expect the same accuracy for the CVS-ISR approach. For this investigation, a set of only small molecules has been chosen due to the computational limitations of the SAC-CI method. These are the carbon 1s excitations of fluoroethene, the nitrogen 1s excitations of ammonia, and the oxygen 1s excitations of water. Table 1 summarizes the results for the excited state dipole moments and excitation energies, μex and ωex, obtained at different CVS-ADC levels compared to results at the SAC-CI level of theory. The quality of the core-excitation energies obtained at different CVS-ADC levels have been discussed in detail in previous work.21−23 It could be shown that the most accurate results are calculated at the CVS-ADC(2)-x level using at least an augmented triple-ζ basis set due to fortuitous error compensation of basis set truncation, neglect of relativistic effects, electron correlation, and orbital relaxation. The other CVS-ADC methods overestimate the core-excitation energies compared to the experiment because this error compensation is broken.23 The present study confirms these investigations. Going to the C 1s excitations of fluoroethene, the CVSADC(2)-x results are in perfect agreement with the experiment, while CVS-ADC(2)-s and CVS-ADC(3) slightly overestimate the energies. CVS-ADC(1) (having only p-h contributions) provides a strong overestimation due to the lack of ground state correlation and of the description of orbital relaxation effects. The same applies for the core-excitation energies of the N 1s and O 1s excitations of ammonia and water, respectively. Note that the overestimation rises with the atomic number of the respective atom due to stronger influences of the relaxation effects. Similar to CVS-ADC(2)-x, the SAC-CI method provides accurate excitation energies compared to the experiment. While CVS-ADC(2)-x tends to slightly underestimate the core-excitation energies, SAC-CI overestimates them a little. In the case of the N 1s excitations of ammonia, for example, the core-excitation energies at the SAC-CI level are about 1 eV too high, while they are about 1 eV too low at the CVS-ADC(2)-x level. The numerical accuracy provided by both methods is also influenced by the core-types, respectively. The heavier the atom from which the 1s excitation originates, the larger the absolute error. Let us turn to the static dipole moments of the core-excited states. Starting with the C 1s excitation of fluoroethene, μex of the states S1 and S3 provided at the CVS-ADC(2)-x, CVSADC(3,2), and SAC-CI levels are in a good agreement with each other. The difference in the core-S1 μex value between CVS-ADC(2)-x and SAC-CI is 0.31 D, while CVS-ADC(3,2) and SAC-CI differ only by 0.18 D. CVS-ADC(1) and CVSADC(2)-s overestimate the dipole moments of the core S1 and S3 states by about 1.0 and 0.5 D, respectively. On the other hand, the μex value of the dark core-S2 state provides a different picture. While CVS-ADC(2)-x and CVS-ADC(3,2) are again at the same level, the SAC-CI value of μex is around 1.0 D above the CVS-ADC(2)-x value. The CVS-ADC(2)-s value is slightly above the one provided by CVS-ADC(2)-x, and the dipole moment obtained by CVS-ADC(1) is only 0.32 D above the SAC-CI result. In the next example, the N 1s excitations of ammonia, there is again a good agreement between SAC-CI and CVS-ADC(2)x/CVS-ADC(3,2) results with one exception (S3). The difference between SAC-CI and CVS-ADC(2)-x values are only 0.07, 0.10, 0.05, and 0.07 D for the S1, S2, S4 and S5 states,

Figure 1. Structures of ammonia (NH3), water (H2O), cytosine, fluoroethene (CH2CHF), acenaphthenequinone (ANQ), 2,2′-bithiophene (BT), phenol, and the methyl radical (CH3). Nuclear coordinates of phenol, BT, and ANQ that are investigated in Section 4.4 are marked in red.

4.1. Static Dipole Moments of Core-Excited States. Let us start with calculations of static dipole moments (μex) of coreexcited states. Due to the lack of experimental data, proper benchmark is challenging. Using the Full-CI (FCI) approach is computationally not possible; thus, the SAC-CI method has been chosen. This method is well known to provide accurate results for core-excited states as well as their excited state properties.89,91−94 However, SAC-CI cannot be used as real benchmark method for our CVS-ADC calculations because both methods are at similar level of theory with similar expected accuracy. The SAC-CI results, however, help to demonstrate the correct behavior of the CVS-ADC/ISR approach. The numerical accuracy of the standard ISR has already been verified a few times for valence-excited states.56,79,95 Since the CVS approximation has been proven E


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Table 1. Comparison of Excitation Energies (ωex) and Excited State Dipole Moments (μex) of the First Three Core-Excited Singlet States of CH2CHF (C 1s) and the First Five Core-Excited Singlet States of NH3 (N 1s) and H2O (O 1s), Calculated Using CVS-ADC at First (1), Strict Second (2s), Extended Second (2x), and Third (3) Order as well as SAC-CIa ωex [eV] state

SAC-CI

1

2s

μex [D]

2x

96,97

3

S1 S2 S3

286.04 288.52 288.18

295.39 300.15 296.72

288.73 290.21 291.01

285.05 287.09 287.33

S1 S2 S3 S4 S5

401.54 403.24 404.84 405.64 406.25

416.09 416.47 419.40 421.29 421.88

402.50 403.98 405.84 406.73 407.13

399.93 401.67 403.30 404.03 404.62

S1 S2 S3 S4 S5

534.95 536.74 539.48 539.65 539.79

551.19 551.83 556.58 556.21 558.01

534.83 536.42 539.25 539.76 539.62

532.90 534.76 537.42 537.70 537.71

experiment

CH2CHF C 1s 287.17 285.00 289.88  289.07 287.10 NH3 N 1s 404.10 400.66 405.71 402.33 407.27 402.86 408.30 403.50 408.96  H2O O 1s 538.44 534.00 540.05 535.90 543.06 543.01 537.00 543.69

SAC-CI

1

2s

2x

3

3.04 3.28 1.76

4.02 2.96 2.35

3.43 4.77 2.24

2.73 4.31 1.55

2.86 4.17 1.62

0.97 3.15 4.04 3.69 1.40

1.88 1.98 6.09 1.84 2.97

1.20 3.39 2.48 3.72 0.63

1.04 3.25 3.39 3.74 1.33

1.19 2.87 4.67 3.23 1.87

1.25 0.90 3.83 2.88 2.23

1.61 0.22 6.26 2.19 0.62

1.70 1.31 0.71 2.69 1.77

1.38 1.18 2.35 2.76 2.26

1.35 0.70 5.35 2.60 1.75

a

The 6-311++G** basis set has been employed for all calculations. Experimental data of the excitation energies are given for comparison.96,97 The degenerate double π-states of NH3 are given only once (State 2 and 4). Note that the energetic order of states sometimes differs between the methods. For a direct comparison, the order of states at the CVS-ADC(2)-x level is fixed, while the ones obtained with the other methods are sorted corresponding to the CVS-ADC(2)-x result.

Table 2. Comparison of Excitation Energies (ωex), Amount of Doubly Excited Amplitudes (R2), and Various Descriptorsa of Respective First Core-Excited Singlet State of CH2CHF (C 1s), NH3 (N 1s), H2O (O 1s), and the CH3 radical (C 1s) Calculated Using CVS-ADC at Different Orders with the 6-311++G** Basis Set method

ωex [eV]

R2 [%]

pDA

CVS-ADC(1) CVS-ADC(2)-s CVS-ADC(2)-x CVS-ADC(3,2)

295.39 288.73 285.05 287.17

 11.64 23.89 19.55

1.00 1.69 1.87 1.75


416.09 402.50 399.93 404.10

 18.47 22.19 17.15

1.00 1.84 1.90 1.79


551.19 534.83 532.90 538.44

 19.06 20.68 15.53

1.00 1.84 1.85 1.73

CVS-UADC(1) CVS-UADC(2)-s CVS-UADC(2)-x CVS-UADC(3,2)

288.47 284.00 281.44 282.65

 6.20 13.85 10.58

1.00 1.47 1.71 1.61

σD [Å] CH2CHF C 1s 0.17 0.97 1.00 0.92 NH3 N 1s 0.14 0.95 0.89 0.83 H2O O 1s 0.12 0.84 0.79 0.72 CH3 C 1s 0.17 0.78 0.89 0.82

σh [Å]

σA [Å]

σe [Å]

dD→A [Å]

dh→e [Å]

0.17 0.17 0.17 0.17

1.24 1.29 1.20 1.15

1.24 1.46 1.45 1.37

0.61 0.31 0.19 0.22

0.61 0.64 0.59 0.61

0.14 0.14 0.14 0.14

1.81 2.11 1.89 1.75

1.81 2.72 2.50 2.24

0.77 0.34 0.31 0.34

0.77 0.72 0.70 0.72

0.12 0.12 0.12 0.12

1.48 1.90 1.67 1.48

1.48 2.40 2.13 1.82

0.80 0.44 0.40 0.42

0.80 0.91 0.86 0.84

0.17 0.17 0.17 0.17

0.82 0.89 0.89 0.84

0.82 0.98 1.04 0.97

0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00

pDA is the promotion number, σ describes the exciton size, and d refers to the distance between the charges, where D and A correspond to the detachment/attachment densities and h and e to the hole/electron densities, respectively. Further explanations can be found in the literature.69,71,73,74

a

respectively, while the difference in dipole moments for the S3 state is 0.65 D. The CVS-ADC(2)-s and CVS-ADC(3,2) values are shifted without any identifiable regularity, sometimes underestimating and sometimes overestimating the CVSADC(2)-x results slightly or strongly. The same applies for the CVS-ADC(1) results, where the deviations to the other approaches is larger.

The last example is the O 1s core-excitation spectrum of water. Again, the results look very similar compared to the other examples. Here, the differences in the μex values between CVS-ADC(2)-x and SAC-CI are 0.13, 0.28, 1.48, 0.12, and 0.03 D for the first five core-excited singlet states. Hence, there is again one exception (S3), for which the deviations between this methods are significantly larger compared to the others. The F


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Figure 2. Comparison of hole(red)/electron(blue) densities based on the 1TDM with detachment(red)/attachment(blue) densities based on the 1DDM of the first C 1s core-excited singlet state of CH2CHF calculated at different CVS-ADC levels and employing the 6-311++G** basis set. The isosurfaces of the densities were rendered with the isovalues 0.0256 e (opaque), 0.0064 e (colored transparent), and 0.0016 e (transparent). The promotion numbers pDA and the respective exciton sizes σx (Å) are shown. The “∗” means that in the case of the electron densities only the CVSADC(1) isosurface and σe value are shown; the shapes of the electron isosurfaces at the other CVS-ADC levels are very similar.

value at the CVS-ADC(2)-x level is the highest. For C 1s excitations, CVS-ADC(3,2) provides a larger value than CVSADC(2)-s, but in the case of excitations from heavier atoms, the opposite is the case. Looking at the promotion numbers pDA based on the (D/A) densities, the same trend can be identified. The pDA value at the CVS-ADC(1) level is 1.0 in every case because no higher excited configurations are available, which would provide orbital relaxation and other many body effects. Hence, the 1DDM and the 1TDM are identical in the case of CVS-ADC(1), and thus, the respective descriptors and properties based on these densities are equal, too. CVS-ADC(2)-x results exhibit the highest promotion numbers. All pDA values are larger than 1.7 at the CVS-ADC(2)-x level indicating strong orbital relaxation effects. To put this in relation, typical pDA values for “normal” low-lying valence-excited states at the ADC(2) level are below 1.7.73 The trend of the promotion numbers going from the lighter carbon to the heavier oxygen atom does not behave as expected. Since relaxation effects are stronger for core excitations from heavier atoms, one would expect larger promotion numbers. This is not the case. From carbon to nitrogen (CH2CHF to NH3), the promotion number rises slightly from 1.87 to 1.9 at the CVS-ADC(2)-x level but decreases in the water example to 1.85. The R2 values fall straight from carbon to oxygen. Obviously, promotion numbers are not a reasonable tool to capture this effect because they just count the number of detached and attached electrons due to an excitation process based on an initial and final state picture. In other words, the promotion number describes the final results after the relaxation, which must neither necessarily depend on the excited core type nor on the dynamics of the process itself. This should remind us that orbital relaxation effects are only included indirectly within the CVS-ADC method via couplings to higher excited amplitudes. The stronger influence of heavier cores on the created holes can be observed by analyzing the σh values based on the 1TDM. All investigated core-excited states are characterized by an excitation from one contracted 1s orbital located on one specific atom. Hence, the hole sizes shrink from 0.17 to 0.12 Å going from carbon to oxygen due to stronger Coulomb attraction.

CVS-ADC(2)-s and CVS-ADC(3,2) results again overestimate or underestimate the dipole moments compared to the CVSADC(2)-x results, but for this system, the differences are only minor. Solely, the S3 state shows a completely different behavior provided by the investigated methods. Overall, this study demonstrates the accuracy of the CVSADC/CVS-ISR approaches in spite of the lack of experimental data. A qualitative and quantitative agreement of static coreexcited state dipole moments between the CVS-ADC(2)-x and SAC-CI levels of theory is given, while the CVS-ADC(3,2) and CVS-ADC(2)-s results often show deviations. CVS-ADC(1) values strongly differ compared to the others. The good agreement of CVS-ADC(2)-x with the SAC-CI method was to be expected, because at this CVS-ADC level, very accurate excitation energies and transition moments compared to experimental data are obtained. Hence, the CVS-ADC(2)-x eigenvectors are well suited to provide adequate excited state properties compared to the ones obtained at the other CVSADC levels. Larger discrepancies between SAC-CI and CVSADC(2)-x results, for example, the S3 states of the ammonia and water examples, cannot be explained clearly. This might be due to slight differences within the excited-state vectors provided at the various levels of theory. 4.2. Analysis of Core-Excited State Densities: Quantification and Visualization of Relaxation Effects. In the following section, visualization and quantification of relaxation effects in the set of the representative small molecules CH2CHF, H2O, NH3, and the CH3 radical are discussed. The first core-excited singlet state of these molecules have been calculated using CVS-ADC(1) to CVS-ADC(3,2) (in the case of CH3, the unrestricted variant has been used), and the results are summarized in Table 2. The excitation energies for all examples follow the same trend as explained in Section 2.4: CVS-ADC(1) strongly overestimates ωex, while CVS-ADC(2)-s and CVS-ADC(3,2) provide an only slight overestimation compared to the CVS-ADC(2)-x method that yields an almost perfect agreement with experimental values. Furthermore, the amount of doubly excited amplitudes (R2) follows qualitatively the trend of the ωex values. With less doubly excited amplitudes coupled to the core-excited state within the ADC schemes, the excitation energy is higher. In all investigated cases, the R2 G


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Table 3. Comparison of Excitation Energies (ωex), Oscillator Strengths (fosc), Amount of Doubly Excited Amplitudes (R2), Character (γ), and Various Descriptorsa of the First Five and Three Core-Excited Singlet States with B2 Symmetry of ANQ (C 1s and O 1s) and the First Five Core-Excited Singlet States of Cytosine (O 1s)b state

ωex [eV]

fosc

R2 [%]

pDA

σD

11B2 21B2 31B2 41B2 51B2

284.40 284.78 284.81 284.96 285.27

0.038 0.041 0.086 0.036 0.024

25.53 25.29 25.21 26.45 24.41

1.99 1.96 1.94 1.97 2.18

1.75 2.56 1.87 2.58 1.55

11B2 21B2 31B2

529.45 533.08 534.78

0.069 0.000 0.001

22.70 26.94 27.00

1.85 2.05 2.23

1.75 2.04 2.18

S1 S2 S3 S4 S5

531.04 532.31 532.63 533.59 533.64

0.027 0.001 0.000 0.000 0.000

22.16 25.83 23.70 24.10 23.86

1.95 2.21 2.18 2.22 2.23

1.15 1.52 1.51 1.62 1.65

σh

σA

ANQ C 1s 1.19 2.06 2.41 2.57 1.31 2.18 2.43 2.57 0.19 1.91 ANQ O 1s 1.48 1.96 1.48 2.60 1.48 2.79 Cytosine O 1s 0.12 1.64 0.12 2.11 0.12 2.57 0.12 3.32 0.12 3.43

σe

dD→A

dh→e

dex

Reh

γ

2.51 2.58 2.45 2.53 2.25

0.05 0.06 0.40 0.19 0.43

0.22 0.11 1.06 0.42 1.00

2.22 2.38 2.44 2.60 2.47

0.47 0.55 0.45 0.47 0.00

1s-π* 1s-π* 1s-π* 1s-π* 1s-π*

1.95 2.67 2.68

0.37 1.01 1.29

0.90 2.67 3.64

2.05 4.22 4.65

0.46 -0.16 0.13

1s-π* 1s-π* 1s-π*

1.74 1.99 3.10 4.27 4.17

0.58 1.09 0.98 1.20 1.51

1.36 3.07 2.46 3.32 3.98

2.22 3.66 3.96 5.41 5.76

0.00 0.00 0.00 0.00 0.00

1s-π* 1s-π* 1s-Dip. 1s- Ryd. 1s-Ryd.

pDA is the promotion number, σ describes the exciton size, and d refers to the distance between the charges, where D and A correspond to the detachment/attachment densities and h and e to the hole/electron densities, respectively. dex further denotes the dynamic charge separation, and Reh corresponds to the Pearson correlation coefficient. Further explanations can be found in the literature.69,71,73,74 bResults are computed using CVSADC(2)-x and the 6-311++G** basis set. Exciton sizes and distances are given in Å. a

because the exciton sizes only describe the extent of the exciton defined as the root-mean square deviation of the respective position operators.69 This also explains why the σA values are smaller than the σe values despite the contribution located at the C2H atom. Promotion numbers are thus better suited for a general quantification of relaxation effects. Comparing the center of charge exciton sizes with each other reveals that dD→A is significantly smaller than dh→e for all investigated systems using higher order CVS-ADC methods (Table 2). This can also be explained by looking at the (D/A) and (h/e) densities. As mentioned before, the considered states contain only excitations from one contracted 1s orbital; thus, the hole density is localized on one atom. The strong relaxation of the detachment density leads to a spatial extent of the density over the whole molecule; thus, the center of charge of the detachment density is shifted in the direction of the center of the molecule. Since the attachment density is only slightly influenced and still delocalized over the molecule, the distance of the center of charges of the (D/A) density is reduced. Before we go on with larger systems, note that R2 and pDA values of the organic radical CH3 are smaller compared to the closed-shell systems. A former investigation revealed that this is often the case for core-excited states with s,π* character of small open-shell systems.22 This can be explained with the local character of this excitation indicated by its exciton size d (center of charges) of almost 0.00, independent of the applied analysis. At the CVS-ADC(2)-x level, the dexc value, which includes the dynamic charge separation contribution, is 1.04 Å, significantly larger than the values based on the center of charge picture. 4.3. Analysis of Core-Excited State Densities: Characterization of Core-Excited States of Medium-Sized Molecules. In this section, the analysis of core-excited state densities is extended to medium-sized molecules, that is, ANQ and cytosine. The theoretical XA spectra of both systems have been investigated recently at the CVS-ADC(2)-x level, showing excellent agreement with experimental data.4,21,69 Here, the

However, promotion numbers are an appropriate tool for the description of the strength of the final orbital relaxation process. Figure 2 visualizes the differences obtained at the investigated CVS-ADC levels. The hole and electron densities based on the 1TDM look practically identical at all CVS-ADC levels; thus, only one example is shown. The first C 1s excited singlet state of CH2CHF is characterized by the promotion of one electron from the doubly hydrogen substituted carbon atom (C2H) to a delocalized π* level. The small hole size of 0.17 Å indicates the excitation from one contracted carbon 1s orbital. As mentioned before, CVS-ADC(1) provides no difference between (D/A) or (h/e) density properties. For the higher order CVS-ADC methods, the influence of relaxation effects is apparently visible. Especially, the detachment densities exhibit strong relaxation effects, while the attachment densities contain only slight effects compared to the (h/e) densities. Looking at the detachment densities, all higher-order CVS-ADC methods generally show the same effect: the density is expanded from the C2H atom to the neighboring atoms, leading to a strong diffuse delocalization over the whole molecule. The density is mostly relaxed to the other carbon atom, but the two connected hydrogen atoms also gain density. The attachment density is mainly located on the C2H atom. Visually, the delocalization of the (D/A) densities is largest at the CVS-ADC(2)-x level. This can also be quantified by using the exciton sizes. In the case of CVS-ADC(2)-x, σD is 1.00 Å, almost 6 times larger than σh. On the other side, σA is 1.20 Å, about 0.25 Å smaller than σe. Generally, σe is slightly larger than σA, while σD is much larger than σh for all investigated systems (Table 2). Hence, the relaxation effects of core excitations lead to a strong expansion of the detachment density and a slight contraction of the attachment density. These effects are largest using CVS-ADC(2)-x. Generally, one has to compare the exciton sizes based on the 1DDM with the ones based on the 1TDM to recognize the influence of relaxation effects to the (D/A) densities. σD at the CVSADC(2)-s level is larger than at the CVS-ADC(3,2) level, although the effects are visually larger at the third-order level H


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equivalent carbon atoms. This is close to the center of charge of the attachment density, which is delocalized over the molecule. Therefore, one has to be careful with the interpretation of exciton sizes in the case of molecules that contain equivalent atoms due to point group symmetry. However, the promotion numbers and a direct comparison between 1DDM and 1TDM properties are still valid tools for analyzing the influence of relaxation effects. Both O 1s and C 1s excitations of ANQ exhibit large pDA values around 2.0 and higher, indicating very strong relaxation effects. Correlation effects, calculated as the correlation coefficient Reh,69 are also influenced by symmetric equivalent atoms. The Reh values are ≠ 0 if the state is characterized by linear combinations of equivalent 1s orbitals. In cases where only the electron is promoted from a single 1s orbital, for example, the O 1s excitations of cytosine or state 51B2 of the C 1s excitation of ANQ, Reh is exactly zero. This is due to the fact that the excitation proceeds from a very compact localized orbital, while in cases of equivalent atoms, the excitation is delocalized. Before we go on to the next section, the former investigations69 of the O 1s core-excitations of cytosine are extended by the information provided from the 1DDM analysis. The pDA values are around 2.0 or even larger, revealing strong relaxation effects. The hole size is 0.12 Å, constant for all core-excited states, because there is only a single oxygen atom. Similar to the results shown in Section 4.2, the σD values are at least 10 times larger due to the strong expansion of the detachment density. In Figure 4, the (D/A) and (h/e) densities of the first, third, and fourth core-excited singlet states of cytosine are plotted. In particular, the detachment densities reveal a strong expansion of the hole due to relaxation effects. The dominant part of the density is relaxed to the neighboring carbon and nitrogen atoms, but there is also a noticeable contribution delocalized over the molecule. Furthermore, this illustration demonstrates the powerful advantage of plotting (h/e) and (D/A) densities to characterize the core-excited states over looking at MO amplitudes. The O 1s excited S1 state is obviously an excitation from the 1s orbital to a π* level. The amplitude with the highest contribution to this state (MO1 to MO48) has only a percentage of 21.3%. It is very hard to characterize states being represented by such strong mixtures of amplitudes. Looking at the (h/e) or (D/A) densities is thus strongly advantageous because all contributions are collected in one picture. From inspecting the densities (Figure 4), it seems that the O 1s core-S3 state is a dipole bound state because its electron density is located around the excited oxygen exhibiting diffuse character in space. An example for a Rydberg state is the coreS4 state with a diffuse electron density, which is delocalized over and beyond the whole molecule in space. Together with the visible interpretation of the density plots, the σe and σA values also give hints for the character of the state. The two s,π* states (S1 and S2) exhibit σe values smaller than 2.0 Å, while the dipole bound S3 state is 3.10 Å larger because it has a diffuse character. Straightforwardly, the Rydberg-type states (S4 and S5), which are even more diffuse than the dipole bound state, provide electron sizes larger than 4.0 Å. A similar trend can be identified for the dynamic exciton distance dex, where the two Rydbergtype states exhibit the largest values. 4.4. Trends of Core-Excited State Properties along Potential Energy Surfaces. In this section, properties of core-excited states are computed along relaxed scans of ground state potential energy surfaces (PES) of the representative

density analysis tools are used to provide deeper insight into the XA spectra. Table 3 contains the calculated data for the C 1s and O 1s excitations of ANQ and the O 1s excitations of cytosine. The largest difference between the C 1s excitation of ANQ and the core excitations discussed in Section 4.2 is the number of atoms of the same type from which core electrons are excited. ANQ contains 12 carbon atoms, and due to the C2v symmetry, some of the carbon atoms are equivalent.21 As a consequence, the 1s orbitals form linear combinations that describe the core-excited states. An example is shown in Figure 3, where the hole and detachment densities of the 21B2 C 1s core-excited state are illustrated.

Figure 3. Comparison of hole and detachment densities of the 21B2 C 1s core-excited state of ANQ computed at the CVS-ADC(2)-x/6-311+ +G** level. The isosurfaces are rendered with the isovalues 0.0128 e (opaque), 0.0064 e (colored transparent), and 0.0008 e (transparent).

This state is characterized by electron promotion from the linear combination of two symmetrical identical carbon atoms (C4) to a π* level. The hole density is strongly localized on these C4 atoms, while the detachment density shows the typical relaxation effects with the expansion of the density to the neighboring atoms. Due to the point group symmetry, the detachment density is also mostly located on both C4 atoms. Considering that real systems are not perfectly symmetric, electrons are promoted from one specific 1s orbital located on one atom in experiments due to the strong contracted and localized nature of the core orbitals. For the theoretical description of excitation energies and oscillator strengths, exploiting point group symmetry accompanied by linear combinations of 1s orbitals is not problematic, leading to perfect agreement with experimental spectra.21,22 These states can be interpreted as excitations located on all symmetrically identical atoms. However, exciton sizes might provide a wrong picture of the real situation, especially in the case of descriptors that contain information about the generated electron hole. The σh values of the first five 1B2 states of ANQ (C 1s) are different depending on the distance or spacing between the symmetric identical carbon atoms. The 21B2 state, for example, exhibits a large σh value of 2.41 Å. Since the exciton hole size is defined as the root-mean-square deviation of the position operator of the hole (center of charges),69 this value fits perfectly to the intramolecular distance of the two C4 atoms of 4.81 Å, which is almost twice the σh value. In the case of the 51B2 state, σh is 0.19 Å, indicating an excitation from a single 1s orbital located on one carbon atom, which is the one in the center of the molecule. The influence of the linear combinations of equivalent carbon atoms is also observable in the exciton distances dD→A, which are very small with values less than 0.5 Å. These values originate from the fact that the center of charge of the hole is located between the two I


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Figure 4. Comparison of electron/hole densities based on the 1TDM with detachment/attachment densities based on the 1DDM of the first, third, and fourth O 1s core-excited singlet states of cytosine computed at the CVS-ADC(2)-x/6-311++G** level. The isosurfaces are rendered with the isovalues 0.0064 e (opaque), 0.0016 e (colored transparent), and 0.0004 e (transparent).

medium-sized model systems phenol, ANQ, and BT using the CVS-ADC(2)-x/CVS-ISR approach. In these calculations, the chosen reaction coordinate was constrained, while all other coordinates were allowed to relax freely. While phenol is a common molecule in the chemical industry and is mostly used as a precursor for synthetic materials, ANQ and BT are model systems from the field of organic electronics. Derivatives of BT are well suited as electron donors in organic semiconducting materials, while ANQ is a simple model for electron accepting molecules like perylenetetracarboxylic dianhydride (PTCDA).21,98,99 Note that the PES have been optimized in the ground state at the MP2 level, and no core-excited state geometry optimizations have been performed. 4.4.1. Dependence of the O 1s Core-Excited State Dipole Moment on the C−O Bond Distance in Phenol. Let us start with the scan of the C−O bond length (rCO) of phenol that has been calculated in 0.05 Å steps from 1.10 to 1.65 Å. The potential in the S0 has the typical shape of a Morse potential, as expected for a diatomic internuclear separation (Figure 5). The relative core-excited state energy of the bright O 1s excited 11A′ state is 533.84 eV at the minimum S0 geometry. The shape of the corresponding potential is strongly dissociating, indicating an unstable situation. At small C−O distances, the relative energy of the O 1s core-11A′ state rises strongly; at larger distances from 1.37 to 1.55 Å, the energy decreases slightly. It

Figure 5. Relaxed PES scan along the stretching of the C−O bond rCO of phenol in the electronic ground state S0 computed at the MP2/6311++G** level of theory. The vertical O 1s excited 11A′ state is calculated using the respective ground state geometries and the CVSADC(2)-x method in combination with the 6-311++G** basis set. (a) Relative energies Erel; S0 minimum at 1.37 Å is set to zero, (b) static dipole moments μ of the S0 and O 1s excited 11A′ states, and (c) different exciton sizes of the 11A′ state based on the 1DDM, that is, σD, σA, and dD→A.

seems that the potential converges against a stationary point. Depending on the lifetime of the O 1s excited 11A′ state, one should expect an OH radical dissociation after core-excitation. Now, let us take a look at the core-excited state dipole moment μex along the distance between C and O (Figure 5b). J


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the bond starts to occur; thus, the density located at the disconnected carbon atom is enhanced. On the other side, less density is expanded to the hydrogen atom of the OH group. As a consequence, the excited electron is delocalized over the carbon atom and the OH group, which leads to a compensation of the partial charges. Hence, the core-excited state dipole moment decreases. Furthermore, Figure 5c shows the trends of exciton sizes related to the 1DDM analysis as a function of rCO. These trends underline the influence of the breaking of the C−O bond. From 1.10 to 1.50 Å, the exciton size rises, while at an internuclear distance of 1.55 Å, it starts to shrink. Before the bond breaking occurs, the extent of the detachment and attachment parts increases because the excitation is mostly localized on the OH group and only a minor part of the density is located at the aromatic ring. At larger distances, the density at the carbon atom expands; thus, the extent of the exciton decreases. 4.4.2. O 1s Core-Excited State Dipole Moment along the Symmetric CO Stretch Motion in ANQ. The dipole moments of the O 1s core-excited states of ANQ along the symmetric stretch motion of the two CO bond distances of ANQ (Figure 1) are investigated in a bond length range from 1.10 to 1.55 Å. This coordinate is of special interest because the O 1s XA spectrum is dominated by one broad peak that has been identified as the O 1s excited 11B2 state.21,101 In contrast to the phenol example, this system exhibits two CO double bonds, which are equivalent due to the C2v point group symmetry. Figure 7 summarizes the results. The ground state potential as well as the potential of the O 1s excited 11B2 state have a typical Morse-type shape, but the latter is less steep than the one of the S0. Hence, the coreexcitation energies decrease about 4.09 eV going from small to larger CO distances. The (D/A) densities show the same state character at all CO distances; thus, only the (D/A) densities at the S0 minimum at rCO = 1.21 Å are shown in Figure 6. In contrast to the phenol example, the static dipole moments in both S0 and O 1s excited 11B2 states follow the same trend. From small rCO values, the dipole moments increase to a maximum at 1.55 Å. A further difference in the phenol investigation is that the absolute μex values are in this case smaller than the ground state ones. For example, at the CO distance of 1.21 Å, the core-excited state dipole moment is only 2.79 D, while the S0 one is 6.09 D. These findings can again be explained by inspecting the (D/A) densities. Taking into account the symmetric treatment of the equivalent CO bonds, the core excitation is localized on the whole CO bonds and not at the oxygen atoms themselves due to the double bond character. This leads to a strong delocalization of the excited electron accompanied by relaxation effects. Hence, the polarity of the CO bonds decreases in the O 1s excited 11B2 state, leading to a decrease in μex compared to the ground state. 4.4.3. Core-Excited State Properties of Bithiophene along the Torsion around the Central Dihedral Angle. As the last example, core-excited state properties of 2,2′-bithiophene (BT) are studied along the torsion around the central dihedral angle β (S−C−C−S) (Figure 1) in a range from 0° to 180°. This dihedral angle plays an important role for the absorption and emission properties in the ultraviolet region of bithiophenes. Former work showed that the most important reaction coordinate, along which energy transfer and charge transfer processes in oligothiophene-based organic electronic devices

The dipole moment in the ground state at the relaxed MP2 level describes a polynomial behavior. The minimum is around the S0 geometry with a value of almost 1.40 D in agreement with the experimental value of 1.22 D.100 The trend of the coreexcited state dipole moment looks very different compared to the ground state one. At first, the absolute value of μex is 4.23 D larger at the S0 minimum than the ground state dipole moment. This indicates enhanced polarity due to the core hole. Second, from 1.10 to 1.45 Å, the dipole moment rises from 4.5 to 6.2 D, which agrees with the expectation if two partial charges are moved to larger distances. However, at 1.50 Å, the dipole moment breaks down steeply to almost zero at an internuclear distance of 1.65 Å. To understand this phenomena, it is helpful to look at the (D/A) densities (Figure 6). The shapes of the

Figure 6. Detachment(red)/attachment(blue) densities based on the 1DDM of the 11B2 O 1s core-excited singlet state of ANQ, the 21B S 1s core-excited singlet state of BT, and the 11A′ O 1s core-excited singlet state of phenol calculated at the CVS-ADC(2)-x/6-311++G** level. The isosurfaces are rendered with the isovalues 0.0128 e (opaque), 0.0032 e (colored transparent), and 0.0008 e (transparent) in the case of ANQ and BT, while the respective isovalues for phenol are 0.0256, 0.0064, and 0.0016 e. The structures of BT and ANQ are in their energetic S0 minimum, and for comparison, two structures of phenol are shown, exhibiting a C−O bond length of 1.37 Å (energetic S0 minimum) and 1.60 Å.

detachment density at rCO values of 1.37 and 1.65 Å, respectively, are almost identical. Due to relaxation effects, the density is strongly broadened around the oxygen atom by extending the hole to the neighboring atoms. The attachment densities, in contrast, differ from each other. While at smaller C−O distances the core excitation is mostly localized at the OH group, at larger C−O distances of 1.60 Å, the breaking of K


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Figure 7. Relaxed ground state PES scan along the parallel stretching of the two CO bonds rCO of ANQ computed at the MP2/6-311+ +G** level of theory. The vertical O 1s excited 11B2 states are calculated using the respective ground state geometries and the CVSADC(2)-x method in combination with the 6-311++G** basis set. (a) Relative energies Erel; energy of the S0 minimum at 1.21 Å is set to zero and (b) static dipole moments μ of the S0 and core-excited 11B2 states.

Figure 8. Relaxed PES scan along the dihedral angle β of BT in the electronic ground state S0 computed at the MP2/6-311++G** level of theory. The vertical S 1s core-excited 21B2 state is calculated using the respective ground state geometries and the CVS-ADC(2)-x method in combination with the 6-311++G** basis set. (a) Relative energies Erel; energy of the S0 minimum at 139.79° is set to zero, (b) static dipole moments μ of S0 and the S 1s core-excited 21B2 state, and (c) different exciton sizes of the core-21B2 state based on the 1DDM, that is, σD, σA, and dD→A.

occur, is dominated by this dihedral angle.99 While BT is nonplanar in the ground state, the first bright valence-excited state with a large oscillator strength has a planar structure because the lowest unoccupied MO (LUMO) has a binding character between the two thiophene rings. Here, the influence of the torsion on the bright S 1s core-excited 21B2 state is analyzed. The results along the scan are summarized in Figure 8. Note that C2 point group symmetry has been exploited in all calculations. The two sulfur atoms are thus equivalent, and the

S 1s excited states are given as linear combinations of the two 1s orbitals of the sulfur atoms. The PES of the ground state along the central torsion angle exhibits the typical shape for BT systems with three local maxima and two local minima at the MP2 level.102−104 The absolute minimum is located at 139.8°. Since the torsion of β is L


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valence space. After solving the CVS-ADC eigenvalue problem, the calculation of the state properties is straightforward using the CVS-ADC eigenvectors in their intermediate state basis to evaluate expectation values of one-particle operators. This implementation in the adcman program features the calculation of static dipole moments as well as total densities of core-excited states. The combination of the CVS-ISR formalism with density analysis tools, that is, the libwfa package in the Q-Chem program, gives access to advanced state properties like exciton sizes or promotion numbers that further help to interpret XA spectra and provide deeper insight into the nature of core-excited states. To demonstrate the performance of the CVS-ADC/CVS-ISR approach, a variety of model systems have been chosen. At first, the accuracy of the CVS-ISR approach has been analyzed by comparing core-excited state dipole moments at different CVSADC levels with SAC-CI results. As expected, the largest accuracy is given at the CVS-ADC(2)-x level, which is well known to provide highly accurate core-excitation energies and spectral features. While qualitative and quantitative agreement between CVS-ADC(2)-x and SAC-CI results are obtained, CVS-ADC(3,2) and CVS-ADC(2)-s often show slight deviations and at the CVS-ADC(1) level; the deviation is large compared to the other methods. This investigation revealed once more that the CVS-ADC(2)-x method is well balanced for the calculation of core-excited states and their properties. Furthermore, the influence of orbital relaxation effects, which play important roles for a proper description of core excitations, has been further investigated in this work. Theoretically, the contributions of these effects to the CVS-ADC formalism have been identified in previous publications, but a proper tool to quantify or visualize these influences has not been available so far. Since the CVS-ISR approach gives access to core-excited state densities, the subsequent decomposition of the difference density matrix into detachment/attachment (D/A) densities is a useful procedure that provides the possibility of visualizing orbital relaxation effects. By extracting the respective promotion numbers from the (D/A) densities, the relaxation effects can be quantified. The interpretation of these numbers is independent of the employed quantum chemical method; thus, they can be used to compare the treatment of relaxation effects at different levels of theory. Since a promotion number of 1.0 indicates no treatment of relaxation effects, which is typical for uncorrelated quantum chemical methods like CIS, values larger than 1.0 indicate these many-body effects. In the case of CVS-ADC(2)x, the promotion numbers are larger than 1.7 (on average around 2.0), which indicates a huge contribution for singly core-excited states. By comparing the trends of the promotion numbers at different CVS-ADC levels, a clear relation between the promotion numbers and the excitation energies could be revealed. A further tool for the analysis of core-excited states are properties based on the 1TDM. Since hole and electron (h/e) densities based on the 1TDM do not contain relaxation effects, they can be seen as a description of the vertical excitation process itself. A direct visual comparison between the (D/A) and (h/e) can thus be used to identify relaxation effects. Typically, the detachment density shows strong relaxation effects, where the core hole is expanded to neighboring atoms, while the attachment density is less influenced by relaxation effects. Besides the visualization of the respective densities, exciton sizes based on either 1DDM or 1TDM have been

around a single bond, the corresponding energy barrier is very low with a maximum value of 0.11 eV at 0°. Looking at the relative energy of the S 1s excited core-21B2 state, the energy barrier is 0.06 eV and thus lower compared to the ground state. There are also three local maxima and two local minima, whose corresponding β values are slightly shifted compared to S0, except for the maximum located at 90°. This indicates that the potential of the S 1s excited 21B2 state along the torsional mode is mostly dominated by the valence electrons, and thus, the influence of the torsion around β to the core-21B2 state and vice versa is negligible. Inspection of the (D/A) densities of this core-excited state (Figure 6) gives a hint of a reason because the attachment density is delocalized over the thiophene rings due to the C2 point group symmetry and exhibits no binding character along the central C−C bond. Turning to the static dipole moments along the torsional mode (Figure 8b), the one of the ground state differs strongly from the one of the S 1s excited 21B2 state. The absolute values of the dipole moments are small, ranging from 0.22 to 0.52 D in the ground state, while in the core-21B2 state, μex has values between 0.02 and 2.84 D. From 0° to 75°, the dipole moment in the S0 remains practically constant and drops between 75° and 120° to 0.22 D. Furthermore, it exhibits a small maximum around 160°. On the other side, μex increases almost constantly from 0° to 60° and then falls to almost 0 D at 180°. Note that the trend of the dipole moment in the S0 at the MP2/6-311+ +G** level differs from former calculations in the literature, where a smaller cc-pVDZ basis set has been employed.105 Since the distance between the two sulfur atoms (rSS) increases going from β values of 0° to 180°, the dipole moment in the S0 should constantly decrease toward zero at 180°, where the charges totally compensate each other. It seems that the diffuse and polarization functions included in the 6-311++G** basis set enhance the description of the dipole moment in the ground state. The observed trends in the core-excited state dipole moment can again be explained by means of the exciton sizes and the C2 point group symmetry of the molecule (Figure 8c). As mentioned before, the internuclear distance of the two sulfur atoms increases during the torsion around β, which can be seen by looking at the trends of σD and σA. Due to relaxation effects that expand the hole density to the neighboring atoms, σD only increases from 2.00 to 2.36 Å, while rSS ranges between 3.31 Å at 0° and 4.38 Å at 180°. The trend of the distance of the center of charges (dD→A) provides the same shape as μex. The maximum is located at β = 60°, and then dD→A decreases to almost zero at 180°, where the two sulfur atoms are opposite to each other. Thus, the partial charges compensate each other. Due to the delocalization of the (D/A) densities, μex and dD→A increase until β = 60°. Eventually, a strong influence of the torsion around β on the dipole moment and polarization of BT in the S 1s excited core-21B2 state is observed that differs strongly from the electronic ground state.

5. CONCLUSION In this work, the first implementation of the intermediate state representation approach for a general one-particle operator at the CVS-ADC level has been reported for the calculation of core-excited state properties. Therefore, the core−valence separation approximation has been applied to the ISR working equations, leading to the CVS-ADC/CVS-ISR approach. Due to this approximation, the size of the ISR matrix is reduced, and the core-excited states are rigorously decoupled from the M


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Journal of Chemical Theory and Computation employed to characterize core-excited states. By comparing hole and electron sizes based on both kinds of densities, further information about relaxation effects and the extent of the core excitation can be learned. In the case of the molecules investigated in this work, the hole density is strongly expanded in space by relaxation effects, while the electron (or rather attachment) density slightly contracts. In addition to the characterization of relaxation effects, analysis of state and transition densities help determining state characters, which is often challenging because an excited state vector often contains a magnitude of different amplitudes. Plotting the (h/e) or (D/A) densities provides an unique and clear picture of the core-excitation process that helps to distinguish between Rydberg, dipole-bound, or s,π*-states. Analyzing the electron sizes of the core excitations further helps in the characterization. Regarding the cytosine example, the electron sizes grow by about 2.0 Å going from s,π* states to dipole-bound states and Rydberg-type states. Note that in the case of equivalent core orbitals due to point group symmetry, exciton sizes based on the center of charge are strongly influenced by occurring linear combinations. Hole sizes, for example, depend on the distances between the equivalent atoms and their position in space, shifting the center of charges between the equivalent atoms. This results in large hole sizes. Dynamic charge separation and correlation effects of coreexcited states have been investigated as well. These effects are also influenced by point group symmetry. While core excitations from a single atom exhibit no correlation effects due to the localized contracted 1s orbital, excitations described via linear combinations of equivalent 1s orbitals show slight contributions. To demonstrate the range of applicability of the CVS-ISR approach, core-excited state properties along relevant nuclear coordinates have been investigated using model systems from the field of organic electronics and technical chemistry. Here, using exciton sizes and plotting (D/A) densities help to understand the behavior of core-excited state dipole moments along the coordinates of interest. Especially, the torsion around the central dihedral angle of bithiophene has a strong influence on the dipole moment of the bright S 1s excited state. Eventually, the CVS-ADC/CVS-ISR approach successfully extends the interpretation of core-excited states by giving access to a magnitude of core-excited state properties. Especially at the CVS-ADC(2)-x level in combination with a diffuse triple-ζ basis set, which is already known to provide very accurate coreexcitation energies and oscillator strengths, the approach can be seen as an almost black-box method that provides deep insight into the nature of core-excited states.

t pqrs =

⟨pq rs⟩ * = −trspq ϵr + ϵs − ϵp − ϵq

(15)

t *pqrs =

⟨rs pq⟩ ϵr + ϵs − ϵp − ϵq

(16)

are used, where ϵ denotes the HF orbital energies, and ⟨pq∥rs ⟩ = ⟨pq|rs ⟩ − ⟨pq|sr ⟩ describes the antisymmetrized Coulomb integrals. As explained in Section 2, the explicit ISR configuration space comprises the p-h and 2p-2h states at the second- and third-order level. Therefore, the matrix elements of the p-h,p-h and 2p-2h,2p-2h diagonal blocks as well as the respective coupling blocks have to be considered at zeroth and second order. Note that all first-order contributions are zero (1) (1) DIa(1), Jb = DIa(1), Klcd = DIjab , Kc = DIjab , Klcd = 0

(17)

A.1. Zeroth Order

DIa(0), Jb = δIJdab − δabdJI

(18)

The zeroth-order contributions of the coupling blocks are reduced from four terms to only two by applying the CVS approximation. The terms that vanish contain δco expressions, where c describes the core space and o the valence occupied space. These terms are always 0 because contents belonging to these different spaces cannot be equal: DIa(0), Klcd = δadδIK dlc − δacδIK dld

(19)

(0) (0) DIjab , Kc = DIa , Klcd (h.c .) = δbcδIK dja − δacδIK djb

(20)

The same applies for the zeroth-order contributions to the 2p-2h block. Here, eight terms are reduced to six by applying the CVS approximation: (0) DIjab , Klcd = δIK δlj(δbddac − δbcdad − δaddbc + δacdbd)

− δacδbd(δljdIK + δIK dlj)

(21)

A.2. Second Order 3

DIa(2), Jb =

∑ DIa(2,,Jbx)

(22)

x=1

Without the CVS approximation, there are seven second-order terms contributing to the p-h block. Since four of them contain Coulomb integrals of the types described in eq 14, they are reduced to three terms given as DIa(2,1) , Jb = − δIJ ∑ (ρkb dak + ρak dkb)

APPENDIX A. Second-Order Intermediate State Representation of a General One-Particle Operator in Combination with the Core−Valence Separation Approximation In the following section, the explicit matrix elements of a general one-particle operator with respect to the second-order ISR after applying the CVS approximation are given. The derivation and the original ISR expressions without the CVS approximation can be found in literature.56 I, J, K, ... label core orbitals and i, j, k, ... refer to valence occupied ones. Furthermore, virtual unoccupied orbitals are denoted as a, b, c, ..., while p, q, r, and s describe general orbitals. The oneparticle matrix elements are denoted as drs. For simplification, the short-hand notations

(23)

k

1 * * DIa(2,2) , Jb = − δIJ ∑ (tklcdtkladdcb + tklcdtklbddac) 4 cdkl ⎛1 ⎜ DIa(2,3) = − δ IJ ⎜ , Jb ⎝2

(24)

⎞

* dcd − ∑ tklbdtmlad * dmk ⎟⎟ ∑ tklbctklad cdkl

dklm

⎠

(25)

with ρpq describing the second-order corrections to the oneparticle ground state density matrix.56 The following secondorder contributions to the coupling blocks originate from eight terms that are reduced to three after applying the CVS approximation because they contain the types of Coulomb terms that are set to zero (eq 14) N


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DIa(2), Klcd = δacδIK ∑ tljdbdbj − δadδIK ∑ tljcbdbj bj

bj

− δIK ∑ t jlcddaj (26)

j (2) * * DIjab , Kc = δacδIK ∑ t jlbdddl − δbcδIK ∑ t jladddl dl

dl

* dcl − δIK ∑ tljab l

■

(27)

AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (J.W.). *E-mail: [email protected] (A.D.). Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS Deeply grateful, we thank our deceased friend and colleague Michael Wormit in memoriam. Without him, this project would not have been possible. Financial funding from the Deutsche Forschungsgemeinschaft Forschergruppe 1789 “Intermolecular and Interatomic Coulombic Decay” is gratefully acknowledged. J.W. acknowledges the Heidelberg Graduate School of Mathematical and Computational Methods for the Sciences for their support.

■

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