Performance of Delta-Coupled-Cluster Methods for Calculations of

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Quantum Electronic Structure

On the Performance of Delta-Coupled-Cluster Methods for Calculations of Core-Ionization Energies of First-Row Elements Xuechen Zheng, and Lan Cheng J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.9b00568 • Publication Date (Web): 31 Jul 2019 Downloaded from pubs.acs.org on August 3, 2019

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Journal of Chemical Theory and Computation

On the Performance of Delta-Coupled-Cluster Methods for Calculations of Core-Ionization Energies of First-Row Elements Xuechen Zheng and Lan Cheng∗ Department of Chemistry, The Johns Hopkins University, Baltimore, Maryland 21218, USA E-mail: [email protected] ‘ Abstract A thorough study of the performance of delta-coupled-cluster (∆CC) methods for calculations of core-ionization energies for elements of the first long row of the periodic table is reported. Inspired by the core-valence separation (CVS) scheme in response theories, a simple CVS scheme of excluding the vacant core orbital from the CC treatment has been adopted to solve the convergence problem of the CC equations for core-ionized states. Dynamic correlation effects have been shown to make important contributions to the computed core-ionization energies, especially to chemical shifts of these quantities. Maximum absolute error (MaxAE) and standard deviation (SD) of delta-Hartree-Fock results for chemical shifts of core-ionization energies with respect to the corresponding experimental values amount to more than 1.7 eV and 0.6 eV, respectively. In contrast, the inclusion of electron correlation in ∆CC singles and doubles augmented with a non-iterative triples correction [∆CCSD(T)] method significantly reduces the corresponding deviations to around 0.3 eV and 0.1 eV. With the consideration

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of basis-set effects and the corrections to the CVS approximation, ∆CCSD(T) has been shown to provide highly accurate results for absolute values of core-ionization energies, with MaxAE of 0.22 eV and SD of 0.13 eV. To further demonstrate the usefulness of ∆CCSD(T), calculations of carbon K-edge ionization energies of ethyl trifluoroacetate, a molecule of significant interest to the study of X-ray spectroscopy and dynamics, are reported.

1

Introduction

Recent advances in synchrotrons, 1–4 free-electron lasers 5–12 as well as light sources based on high-harmonic generation 13–17 have significantly enhanced time and energy resolution of X-ray beamlines. The availability of these short-pulse X-ray sources have enabled studies of ultrafast nuclear and electronic dynamics. 18–20 Computational predictions of energies and spectra for core-ionized or core-excited states play an important role in studies of X-ray spectroscopy and dynamics, in particular when experimental characterization of these transient species with core holes is challenging. To be useful in facilitating or even guiding experiments, it is desirable to obtain computational accuracy comparable to or even higher than the resolution of experimental spectra, i.e., with errors below 0.2 eV for computed coreionization or excitation energies of first-row elements, or at least for chemical shifts of these quantities. Development of quantum-chemical methods for calculations of core-ionized and excited states that can provide such an accuracy constitutes a major challenge and is an active area of research. 21,22 Many computational methods widely used in calculations of core-level spectroscopy are based on response theories. These include configuration interaction methods, 23–26 timedependent density-functional theory (TDDFT), 21,27–30 coupled-cluster linear response theory [and the closely related equation-of-motion coupled-cluster (EOM-CC) methods], 31–36 symmetry-adapted-cluster configuration interaction (SAC-CI), 37,38 a multireference generalization of EOM-CC (MR-EOM-CC), 39 Green’s function approaches 40–42 including the alge2


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braic diagrammatic construction (ADC) methods. 43–45 These methods first treat the ground state of a molecule and then compute the difference between ground state and targeted coreionized or core-excited state. Response theories benefit from the cancellation of electroncorrelation effects that are transferable from ground state to excited state. Computation of transition probabilities in response theories is also convenient. However, it is in general a challenging task for response theories to obtain highly accurate core-ionization or coreexcitation energies. Since creation of a core hole induces strong relaxation of wavefunction, the difference between ground state and core-ionized or excited state, which is directly targeted in response theories, is large in terms of absolute magnitude. Hence it is difficult to obtain high accuracy. Most response theories apply a linear wave operator to transform ground-state wavefunction into the wavefunction of core-ionized or excited state, which exhibits rather slow convergence in the treatment of wavefunction relaxation. For example, the errors of EOM-CC singles and doubles (EOM-CCSD) results for K-edge ionization energies of first-row elements have been shown to be 1-3 eV. Triples contributions and in some cases even quadruples contributions are required to obtain quantitative results. 35 We mention that MRCC methods aiming to treat both wavefunction relaxation and electron correlation via exponential parametrizations of wave operator 46–50 have emerged as promising methods and have been under developments. The alternative to response theories is to explicitly optimize the wavefunction of coreionized or excite states. Methods in this category include delta-self-consistent-field (∆SCF) methods, 51–56 the orthogonality-constrained SCF methods, 57–59 the delta-complete-activespace SCF (∆CASSCF) methods, 53,60 the static exchange method, 61 and the nonorthogonal configuration interaction method. 62,63 These methods perform separate wavefunction optimizations for ground and core-ionized or core-excited states. The relaxation of wavefunction thus is explicitly taken into account. The delta-Hartree-Fock (∆HF) method can often provide absolute values of core-ionization energies closer to experimental values than those from EOM-CCSD calculations. However, it can be seen from available results 55 and will also be

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shown in detail in the present study that ∆HF results exhibit significant errors for chemical shifts of core-ionization energies. In other words, the well-established importance of electron correlation in calculations of relative energies, e.g., bond energies and heats of formation, is also present in calculations of relative shifts of core-ionization or core-excitation energies. Therefore, it is essential to take electron correlation into account to obtain accurate results. However, although Delta-Møller-Plesset second-order perturbation theory (∆MP2) 64–71 calculations of core-ionization energies have been reported quite extensively, the study of delta-coupled-cluster (∆CC) methods aiming at accurate calculations of core-ionized or excitation energies is still scarce. 50 Nevertheless, core-ionized states obtained from ∆CCSD calculations have been used as reference states in calculations of core-excited states, 31 Xray emission spectra, 72 as well as core-ionized shake-up states. 73 The limited usage of ∆CC methods may be attributed to the convergence difficulty of CC equations for core-ionized or core-excited state due to the presence of a core orbital in the virtual space. 31 A double excitation consisting of an excitation from an occupied orbital to the unoccupied core orbital and an excitation from an occupied orbital to a high-lying virtual orbital may have a very small orbital energy difference, which leads to numerical instability in solution of CC amplitude equations. Conceptually, this problem arises from couplings of targeted core-ionized or core-excited state with high-lying valence continuum states that are nearly degenerate. We mention that effective-core-potential (ECP) based ∆CC calculations of core-ionization energies circumvent this problem by not working with explicit core orbitals. 74,75 Consequently, ECP-based CC calculations cannot provide accurate results for absolute values of core-ionization or core-excitation energies, although the computed chemical shifts of these quantities appear to be quite accurate. It should be noted that a similar problem exists in response theories. The convergence of excited-state eigenvalue equations can be plagued by (near-) degeneracy of targeted core-ionized or excited state with high-lying valence excited states. The core-valence separation (CVS) scheme has emerged as a promising approach to solve this problem within the framework of response theories. 76 In the CVS scheme, the pure

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valence excitations are excluded from excited-state eigenvalue equations by construction, and the resulting excited-state equations only involve excitations containing the core orbital(s) of interest. Note that treatment of core-ionized or excited states, which belong to high energy resonance states, in principle requires the use of non-Hermitian quantum mechanics. 77 (See Ref. 78 for a recent review of non-Hermitian quantum-mechanical methods for treating resonance states.) However, thanks to the localized nature of core hole(s), it is sufficient to explicitly exclude the continuum part by construction and to focus on the localized part of the wavefunctions of core-ionized or excited states, in order to obtain accurate energies and spectra for these states. The present work aims at a thorough study of the performance of ∆CC methods in calculations of core-ionization energies. Inspired by the CVS scheme in response theories, a simple approach that keeps the unoccupied core orbital frozen in CC treatment has been adopted to deal with the convergence difficulty of CC equations for core-ionized state. Extensive benchmark calculations of K-edge ionization energies for elements in the first long row of the periodic table are reported to demonstrate the accuracy of ∆CCSD and ∆CCSD with a non-iterative triples correction [CCSD(T)] methods. ∆CCSD(T) results for carbon K-edge ionization energies in ethyl trifluoroacetate are also reported to further demonstrate the applicability of ∆CCSD(T). Theory and computational details are presented in Section 2, while computational results are reported and discussed in Section 3. Finally, a summary and an outlook are given in Section 4.

2

Theory and computational details

In delta-coupled-cluster (∆CC) calculations of core-ionization energies, the CC wavefunction ˆ

of a core-ionized state |Ψcore,CC i is obtained by applying an exponential wave operator eT to the Hartree-Fock (HF) wavefunction of the core-ionized state |Ψcore,HF i, i.e, ˆ

|Ψcore,CC i = eT |Ψcore,HF i. 5


(1)


The HF wavefunction for the core-ionized state |Ψcore,HF i is obtained from a direct optimization with the constraint that the target core orbital is kept unoccupied during the HF iterations. The present implementation first carries out HF calculations of spherical atoms to obtain atomic orbitals, among which the target core orbital is identified. During the molecular HF calculation, the molecular orbital that has the maximum overlap with the target core orbital is kept unoccupied. Thanks to the localized nature of core holes, HF calculations of core-ionized states reported here have been converged using standard SCF techniques. In calculation of a core ionization from symmetrically equivalent atoms (F2 , N2 , CO2 , C2 H2 and C2 H4 ), the superposition of atomic HF density matrices with an atomic core orbital excluded in one of the equivalent atoms is used as the initial guess for the HF calculation. This leads to a symmetry-breaking HF solution with the core hole localized in that atom. It has been found necessary to work with a localized core hole, in order for the HF calculation of the ionized state to capture most of the orbital relaxation induced by the core hole. In contrast, if one of the delocalized core molecular orbitals is kept vacant in these molecule, each atom only sees half of the core hole and the corresponding HF calculation does not fully relax the orbitals. Consequently, the ∆HF calculation with a delocalized core hole tends to grossly overestimate the core ionization energy (by more than 10 eV). The excitations into the vacant core orbital make large contributions in the CC treatment. The calculations using localized core holes can alleviate this problem by allowing more thorough treatment of orbital relaxation. The necessity of using localzed core holes can also be inferred from results in the literature. 55 We should mention that in principle core-ionized states with delocalized core holes can be obtained as linear combinations of the ones with localized core holes. Since the difference between energies of these states is insignificant, we shall use energies of the states with localized core holes in our discussion. As in standard CC methods, the cluster operator Tˆ is a linear combination of elementary excitation operators weighted by cluster amplitudes. In CC singles and doubles (CCSD)

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method, the cluster operator comprises single and double excitations

Tˆ =

X

tai a†a ai +

ia

1 X ab † † t a a aj ai . 4 ijab ij a b

(2)

Here we follow the standard convention, in which {i,j,...} and {a,b,...} denote occupied and virtual orbitals, respectively. In the present context, the virtual space comprises the ordinary unoccupied orbitals av ’s as well as a vacant core orbital ac , i.e., {a, b, ...} = {ac } ∪ {av , bv , ...}. The convergence difficulty that plagues the solution of CCSD amplitude equations for a coreionized state arises from double excitations of the type a†ac a†bv aj ai involving the unoccupied core orbital ac . Orbital energy differences |εj + εi − εac − εbv | may be very small, which leads to unphysically large cluster amplitudes and hence divergence of CCSD amplitude equations. Inspired by the success of the core-valence separation (CVS) scheme in response theories, we have adopted a simple scheme of freezing the unoccupied core orbital in CC treatment to solve this convergence problem. Conceptually, similar to the CVS scheme in response theories, this approach decouples target core-ionized state from valence continuum states by construction. ∆CC methods with this approximation will be referred to as CVS-∆CC methods. A CVS-∆CCSD calculation of a core-ionized state converges as rapidly as a CCSD calculation for ground state. The computational cost of a CVS-∆CCSD calculation thus is similar to a spin-unrestricted HF based CCSD calculation (3-4 times higher than a spinrestricted HF based CCSD calculation).

We mention that the present CVS scheme is in

the same spirit as the scheme of Ref. 31 , which excludes excitations involving the unoccupied core orbital except those of single excitations that are important due to the use of the orbitals of the neutral molecule, and excitations that have large contributions due to the use of delocalized symmetric core holes. Since our calculations use HF orbitals of core-ionized state as well as localized core holes, these excitations are not as important in the present calculations. To assess the accuracy of this CVS approximation, ∆CC calculations including excitations involving ac that have orbital energy differences |εi + εj − εac − εb | greater than

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a chosen threshold δ have been performed and compared with CVS-∆CC calculations. The δ values of 5.0, 3.0, and 2.0 Hartree have been used to study the sensitivity of errors of the CVS approximation with respect to the choice of the threshold. All CCSD 79 and CCSD(T) 80 calculations presented here have been performed using CFOUR 81–83 program package. Scalar-relativistic effects have been treated using the spinfree exact two-component theory in its one-electron variant (SFX2C-1e) 84,85 with correlationconsistent basis sets 86,87 recontracted for the SFX2C-1e scheme (available on www.cfour.de). We have adopted the same benchmark set as in Ref. 35 comprising twenty-one C, N, O, F 1s ionization energies in fourteen molecules with well-established experimental values in the literature. 88 We refer the readers to the supplementary material of Ref. 35 for the geometries of these molecules used in the calculations (experimental equilibrium geometries for diatomic molecules and SFX2C-1e-CCSD(T)/cc-pCVQZ geometries for polyatomic molecules). CVS∆CCSD and CCSD(T) calculations have been performed using cc-pVXZ (X=T, Q, 5) and cc-pCVXZ (X=T, Q, 5) basis sets. Estimates of basis-set limit have been obtained by using standard extrapolation schemes for HF 89 and electron-correlation 90 energies. Corrections to the CVS approximation have been obtained as the differences between CVS-∆CCSD(T) results and ∆CCSD(T) results with δ threshold values of 5.0, 3.0, and 2.0 Hartree using cc-pVTZ and cc-pCVTZ basis sets. High-level correlation contributions have been studied by means of CC singles doubles and triples (CCSDT) 91,92 and CCSDT augmented with a non-iterative quadruples correction [CCSDT(Q)] 93 calculations with cc-pVTZ basis sets for seven small molecules in the benchmark set using the MRCC 94–96 program package. Carbon and nitrogen K-edge ionization energies of CH3 CN and CH3 NC have been studied with the emphasis on relative shifts of carbon 1s ionization energies, for which available experimental results are inconsistent among themselves 97,98 and are also inconsistent with previous calculations. 35 Finally, to further demonstrate the usefulness of CVS-∆CCSD(T), we report CVS-∆CCSD(T)/cc-pVTZ calculations for carbon 1s ionization energies of ethyl trifluoroacetate (CF3 COOCH2 CH3 ), the so-called "ESCA" molecule that is of significant interest to

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the study of electronic spectroscopy, 99,100 and compare the results with corresponding experimental values. 101 The calculations have been carried out for both the Cs and C1 conformers of ethyl trifluoroacetate. The geometries of these two conformers have been documented in the supplementary material.

3 3.1

Results and discussions A benchmark delta-coupled-cluster study of core-ionization energies of first-row elements

The benchmark study aims to assess the performance of ∆CC methods in calculations of K-edge ionization energies of first-row elements. Electron-correlation contributions, basisset effects, as well as corrections to the core-valence separation (CVS) approximation in the calculations of both absolute values and chemical shifts of these ionization energies have been carefully studied. In the following we first analyze the importance of these contributions and then compare the computational results with the corresponding experimental values. Emphasis has been placed on the significance of electron-correlation contributions to chemical shifts, the relevance of corrections to the CVS approximation, and the overall performance of ∆CCSD(T).

3.1.1

Significance of electron-correlation contributions

Electron-correlation contributions to absolute values of core-ionization energies can be seen in Table 1 as the difference between ∆HF and CVS-∆CCSD(T) values. The magnitude of electron-correlation contributions range from 0.02 eV (in case of N2 ) to 1.93 eV (in case of CO2 ) with an averaged value of 0.53 eV. Given that absolute values of K-edge ionization energies of first-row elements studied here range from 290 to 700 eV, electron-correlation contributions seem to account for only a small fraction of total value. However, the significance of electron correlation lies in chemical shifts of these core-ionization energies. As 9



shown in Table 2, electron-correlation contributions to chemical shifts typically amount to several tenths of eV with the largest value being 1.49 eV for the middle nitrogen in N2 O. As these chemical shifts range from a few tenths to several eV, the percentage of electroncorrelation contributions amount to 20% in average. It is absolutely necessary to take into account electron correlation to obtain a favorable comparison between computed chemical shifts of core-ionization energies and the corresponding experimental results. As shown in Table 2, mean absolute error (MAE) of ∆HF results for chemical shifts amounts to more than 0.5 eV and 20% of the total value. Maximum absolute error (MaxAE) of ∆HF results occurs for the middle nitrogen of N2 O, for which the error of 1.61 eV of the ∆HF method is more than 60% of the total shift (2.6 eV). The inclusion of electron-correlation contributions significantly improves the agreement between computed and experimental values. MAE of ∆CCSD(T) results is only 0.09 eV with MaxAE being 0.15 eV. The importance of electroncorrelation contributions in calculations of chemical shifts for core-ionization energies is thus clearly established. Triples corrections are often important when aiming at high accuracy. For example, the triples correction to 1s ionization energy of the middle nitrogen in N2 O amounts to 0.35 eV. On the other hand, contributions from high-level correlation effects [those beyond CCSD(T)] are insignificant. As shown in Table 3, the largest full triples correction [the difference between CCSDT and CCSD(T)] and the largest quadruples correction [the difference between CCSDT(Q) and CCSDT] amount to only 0.05 eV and 0.04 eV, respectively. ∆CCSD(T) thus is a promising candidate for practical calculations of core-ionization energies.

3.1.2

Corrections to the CVS approximation

In the core-valence separation (CVS) scheme proposed here for ∆CC methods, excitations involving the unoccupied core orbital are excluded in CC treatment. This incomplete treatment of electron correlation for core-ionized states results in an overestimation of energies of core-ionized states and hence a systematic overestimation of core-ionization energies. The

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magnitude of errors introduced by the CVS approximation have been estimated as the difference between ∆CCSD(T) and CVS-∆CCSD(T) results using cc-pVTZ and cc-pCVTZ basis sets. The convergence difficulty of CC calculations of core-ionized states is alleviated by excluding excitations with orbital energy differences below a threshold. The sensitivity of the results with respect to the value of the threshold has been studied using threshold values of 5.0, 3.0, and 2.0 Hartree. As shown in Table 4, the differences between the results obtained with threshold values of 3.0 and 2.0 Hartree are less than 0.02 eV and thus insignificant for the present purpose. The rapid convergence with respect to threshold values may be attributed to that these excitations are associated with rather small integrals involving both core and valence orbitals. This also means that the contributions to the bound part of wavefunction of core-ionized state are converged with threshold value of 2.0 Hartree in these calculations. The use of smaller threshold values lying out of this convergence region has been seen to cause convergence difficulty for some of test molecules and also change corresponding results significantly. This can be attributed to spurious couplings with valence continuum states. Thus it is not recommended to use smaller threshold values in these calculations. In the following discussions, we will use the results obtained using the threshold of 2.0 Hartree. As shown in Table 4, corrections to the CVS approximation obtained using cc-pCVTZ basis are around 0.3 eV, 0.4 eV, 0.4 eV, and 0.5 eV for 1s ionization energies of C, N, O, F, respectively. The inclusion of the corrections to the CVS approximation thus is important when targeting accurate absolute values of core-ionization energies. It should be noted that these corrections are of atomic nature; the corresponding contributions to relative shifts are less than 0.05 eV. We mention that contributions of core-correlating functions to these corrections (the difference between cc-pCVTZ and cc-pVTZ results) amount to around 0.1 eV.

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3.1.3

Basis-set effects

As shown in Table 5, CVS-∆CCSD(T) core-ionization energies tend to decrease with the enlargement of basis sets. All results converge smoothly with the systematic increase of cardinal number of basis sets. Within the series of cc-pVXZ sets, the differences between cc-pV5Z and cc-pVTZ results are less than 0.31 eV. The convergence of the cc-pCVXZ series seems faster, with differences between cc-pCV5Z and cc-pCVTZ results less than 0.17 eV. It should be noted that contributions from core correlating functions (the difference between cc-pCVXZ and cc-pVXZ results) amount to around 0.4 eV and are often more pronounced than the variation of the results with the increase of cardinal number of basis sets.

3.1.4

Comparison with experiment

Comparison to experimental values shows that CVS-∆CCSD(T)/cc-pCV∞Z results in Table 5 consistently overestimate core-ionization energies by several tenths of eV. This is anticipated due to the neglect of electron-correlation contributions from the unoccupied core orbital in calculations of core-ionized states. The inclusion of corrections to the CVS approximation (taken from the last column of Table 4) significantly reduces the errors. Our best results summarized in Table 5 as "CVS-∆CCSD(T)/cc-pCV∞Z+∆CVS" agree very well with the experimental values, with maximum absolute error (MaxAE), mean absolute error (MAE), and standard deviation (SD) being 0.22 eV, 0.11 eV, and 0.13 eV, respectively. All CVS-∆CCSD(T)/cc-pCV∞Z+∆CVS values are consistently higher than the experimental values. This can readily be attributed to the incomplete treatment of corrections to the CVS approximation, in which only triple-zeta basis sets have been used and excitations with small denominators have to be excluded. The accuracy of ∆CCSD(T) is similar to that of equation-of-motion ionization potential CC singles doubles triples and quadruples (EOMIP-CCSDTQ), 35 which is a vastly more expensive method. We emphasize that ∆CC methods benefit from full consideration of wavefunction relaxation through separation optimization of ground state and core-ionized states. In contrast, EOMIP-CC methods 12


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account for relaxation of the wavefunction through configuration interaction using on the similarity-transformed Hamiltonian, which exhibits relatively slow convergence. The middle nitrogen in N2 O is an exceedingly difficult case in EOMIP-CC calculations with the triples and quadruples corrections being as large as -1.83 and -0.31 eV, respectively. In contrast, it is straightforward to obtain accurate results using ∆CC methods. This verifies that the difficulty of EOMIP-CC in accurate calculations of this specific core-ionization energy can be attributed to strong relaxation of wavefunction. Interestingly, CVS-∆CCSD(T)/cc-pVTZ calculations provide very good results for coreionization energies presented here. The omission of core-correlating functions overestimates ground-state energies more than energies of core-ionized states and thus tends to underestimate core-ionization energies. At the same time, the CVS approximation tends to overestimate core-ionization energies. Therefore, the good performance of CVS-∆CCSD(T)/ccpVTZ can be attributed to a systematic cancellation between contributions from corecorrelating functions and corrections to the CVS approximation. Further, as shown in Table 6, neither the effects of core-correlating functions nor the CVS corrections make significant contributions to chemical shifts. Therefore, CVS-∆CCSD(T)/cc-pVTZ may be recommended as a practical method of choice for accurate calculations of core-ionization energies of first-row elements.

3.2

Carbon and nitrogen 1s ionization energies in CH3 CN and CH3 NC

C and N 1s ionization energies of CH3 CN and CH3 NC computed at the CVS-∆CCSD(T) level are summarized in Table 7. Of particular interest here are relative shifts between carbon 1s ionization energies of the two carbon atoms in CH3 CN and those in CH3 NC. As shown in Table 7, two available experiments 97,98 are not consistent with each other, while neither of them completely agree with previous EOMIP-CCSDT calculations. 35 More specifically, EOMIP-CCSDT results for CH3 CN are consistent with the experimental results in Ref., 97 but not with Ref. 98 It is the other way around for CH3 NC. The present ∆CCSD(T) results 13



are consistent with EOMIP-CCSDT results. The ionization energy for the cynide carbon in CH3 CN is slightly higher than that for the methyl carbon (consistent with the experiment in Ref. 98 ). The ionization energy of the methyl carbon in CH3 NC is about 1 eV higher than that in the NC group (consistent with the experiment in Ref. 97 ). Due to the efficiency of CCSD(T), we are able to thoroughly consider basis-set effects in the present calculations. It is safe to conclude that computational results from ∆CCSD(T) calculations are consistent with those from EOMIP-CCSDT calculations, and the remaining error in computations would be insignificant. An experimental re-investigation is thus recommended to resolve these inconsistencies among experiments and also between experiment and computation.

3.3

Carbon 1s ionization energies of ethyl trifluoroacetate

Ethyl trifluoroacetate (CF3 COOCH2 CH3 ) has been designed to have four carbon atoms with distinct chemical environments, and has been used extensively in the study of spectroscopic techniques. 99,100 Ethyl trifluoroacetate exhibits extraordinarily large relative shifts for carbon 1s ionization energies, e.g., the relative shift between the CF3 carbon and the methyl carbon is as large as around 8 eV. 101 This molecule has recently been used in a study of core-hole induced dynamics, 102 and is also a good candidate molecule for studying two-site double-core-hole states.

∆HF, CVS-∆CCSD, and CVS-∆CCSD(T) results using cc-pVTZ basis for carbon K-edge ionization energies of ethyl trifluoroacetate in both the C1 and Cs conformers are summarized in Table 8. The C1 and Cs conformers have very similar carbon 1s ionization energies with difference below 0.05 eV for all carbon edges. As expected, the ∆HF results agree reasonably well with the experimental values in terms of absolute values, but exhibit significant errors in terms of relative shifts. For example, the electron-correlation contribution to the relative shift between the carbonyl carbon and the methyl carbon amounts to more than 1 eV, which accounts for more than 25% of the total value. CVS-∆CCSD(T) values agree very 14


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well with the corresponding experimental results. 101 Maximum absolute deviations between computation an experiment are 0.39 eV for absolute core-ionization energies and 0.12 eV for relative shifts. CVS-∆CCSD(T)/cc-pVTZ seems to be a very promising candidate for accurate prediction of core-ionization energies for molecules of moderate size.

4

Conclusion

In this paper we report benchmark delta-coupled-cluster (∆CC) calculations of K-edge ionization energies for first-row elements C, N, O, and F with a careful analysis of electroncorrelation and basis-set effects as well as corrections to the core-valence separation approximation. The ∆CCSD(T) method, with the explicit consideration of wavefunction relaxation, has been shown to provide core-ionization energies as accurate as those from equation-ofmotion CC calculations with the inclusion of quadruple excitations. Maximum absolute error and standard deviation of ∆CCSD(T) results for absolute values of core-ionization energies with respect to the corresponding experimental values only amount to 0.22 and 0.13 eV. The electron-correlation contributions have been demonstrated to play a vital role in accurate calculations of chemical shifts of core-ionization energies. Maximum absolute error (and standard deviation) of computed chemical shifts is reduced from the ∆HF value of 1.7 eV ( and 0.6 eV) to the ∆CCSD(T) value of 0.2 eV (and 0.1 eV). Based on the excellent performance of ∆CC methods in treatment of wavefunction relaxation of single core-ionized states, a future direction is to apply ∆CC methods to calculations of double-core-ionization energies, for which wavefunction relaxation due to the presence of two core holes poses a formidable challenge for response theories.

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Table 1: K-edge ionization energies (in eV) of first-row elements in bold letter marked with a star. The errors of ionization energies with respect to experimental values are enclosed in parentheses. MaxAE, MAE, and SD refer to maximum absolute error, mean absolute error, and standard deviation, respectively. Scalar-relativistic effects have been taken into account using the spin-free exact two-component theory in its one-electron variant. The cc-pVTZ basis sets have been used in all calculations presented here. ∆HF

∆CCSD

∆CCSD(T)

Experiment 88

C∗ O

296.83 (0.62)

296.24 (0.03)

296.18 (-0.03)

296.21

CO∗

542.14 (-0.41)

542.74 (0.19)

542.86 (0.31)

542.55

F∗2

695.73 (0.96)

696.99 (0.30)

697.09 (0.40)

696.69

HF∗

694.03 (-0.20)

694.63 (0.40)

694.69 (0.46)

694.23

N∗2

410.00 (0.02)

410.01 (0.03)

410.02 (0.04)

409.98

C∗ H4

290.90 (-0.01)

290.73 (-0.18)

290.73 (-0.18)

290.91

H2 O ∗

539.73 (-0.17)

540.07 (0.17)

540.12 (0.22)

539.90

C∗ H2 O

294.20 (-0.27)

294.37 (-0.10)

294.36 (-0.11)

294.47

CH2 O∗

538.79 (-0.69)

539.58 (0.10)

539.71 (0.23)

539.48

C∗2 H2

291.07 (-0.07)

291.21 (0.07)

291.24 (0.10)

291.14

C∗2 H4

290.61 (-0.21)

290.70 (-0.12)

290.73 (-0.09)

290.82

C∗ O2

299.46 (1.77)

297.92 (0.23)

297.53 (-0.16)

297.69

CO∗2

541.29 (-0.05)

541.47 (0.13)

541.53 (0.19)

541.34

NNO∗

541.10 (-0.32)

541.59 (0.17)

541.77 (0.35)

541.42

NN∗ O

414.23 (1.64)

413.11 (0.52)

412.76 (0.17)

412.59

N∗ NO

409.10 (0.39)

408.94 (0.23)

408.87 (0.16)

408.71

N∗ H3

405.61 (0.05)

405.63 (0.07)

405.66 (0.10)

405.56

HC∗ N

293.26 (-0.14)

293.35 (-0.05)

293.35 (-0.05)

293.40

HCN∗

406.61 (-0.17)

406.85 (0.07)

406.91 (0.13)

406.78

C∗ H3 OH

292.55 (0.12)

292.32 (-0.11)

292.30 (-0.13)

292.43

CH3 O∗ H

538.79 (-0.32)

539.27 (0.16)

539.35 (0.24)

539.11

MaxAE

1.77

0.52

0.46

n/a

MAE

0.41

0.17

0.18

n/a

SD

0.65

0.21

0.22

n/a

16


Page 17 of 38 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 2: Chemical shifts of 1s ionization energies (in eV) for the atoms in bold letter marked with a star. The 1s ionization energies of carbon in CO, oxygen in CO, nitrogen in N2 , and fluorine in F2 are taken as the reference values in the calculations of relative shifts. The deviations of computed chemical shifts with respect to experimental values are enclosed in parentheses. MaxAE, MAE, and SD refer to maximum absolute error, mean absolute error, and standard deviation, respectively. Scalar-relativistic effects have been taken into account using the spin-free exact two-component theory in its one-electron variant. The cc-pVTZ basis sets have been used in all calculations presented here. ∆HF

∆CCSD

∆CCSD(T)

Experiment 88

HF∗

-1.70 (0.76)

-2.36 (0.10)

-2.40 (0.06)

-2.46

C∗ H4

-5.92 (-0.62)

-5.52 (-0.22)

-5.45 (-0.15)

-5.30

H2 O ∗

-2.40 (0.25)

-2.68 (-0.03)

-2.74 (-0.09)

-2.65

C∗ H2 O

-2.62 (-0.88)

-1.87 (-0.13)

-1.82 (-0.08)

-1.74

CH2 O∗

-3.35 (-0.28)

-3.16 (-0.09)

-3.15 (-0.08)

-3.07

C∗2 H2

-5.75 (-0.68)

-5.03 (0.04)

-4.94 (0.13)

-5.07

C∗2 H4

-6.22 (-0.83)

-5.54 (-0.15)

-5.45 (-0.06)

-5.39

C∗ O2

2.64 (1.16)

1.68 (0.20)

1.36 (-0.12)

1.48

CO∗2

-0.85 (0.36)

-1.27 (-0.06)

-1.33 (-0.12)

-1.21

NNO∗

-1.04 (0.09)

-1.15 (-0.02)

-1.10 (0.03)

-1.13

NN∗ O

4.22 (1.61)

3.10 (0.49)

2.74 (0.13)

2.61

N∗ NO

-0.90 (0.37)

-1.07 (0.20)

-1.15 (0.12)

-1.27

N∗ H3

-4.40 (0.02)

-4.38 (0.04)

-4.36 (0.06)

-4.42

HC∗ N

-3.57 (-0.76)

-2.90 (-0.09)

-2.83 (-0.02)

-2.81

HCN∗

-3.40 (-0.20)

-3.16 (0.04)

-3.11 (0.09)

-3.20

C∗ H3 OH

-4.27 (-0.49)

-3.93 (-0.15)

-3.88 (-0.10)

-3.78

CH3 O∗ H

-3.35 (0.09)

-3.48 (-0.04)

-3.51 (-0.07)

-3.44

MaxAE

1.61

0.49

0.15

n/a

MAE

0.56

0.12

0.09

n/a

SD

0.71

0.17

0.10

n/a

17



Page 18 of 38

Table 3: 1s ionization energies (in eV) for the atoms in bold letter marked with a star. δT refers to energy differences between ∆CCSDT and ∆CCSD(T). δ(Q) refers to energy differences between ∆CCSDT(Q) and ∆CCSDT. Scalar-relativistic effects have been taken into account using the spin-free exact two-component theory in its oneelectron variant. The cc-pVTZ basis sets have been used in all calculations presented here. ∆CCSD(T)

δT

δ(Q)

C∗ O

296.18

0.05

-0.04

CO∗

542.86

-0.01

0.01

F∗2

697.09

-0.03

0.02

HF∗

694.69

0.00

0.01

N∗2

410.02

-0.05

0.00

C∗ H4

290.73

0.00

0.00

H2 O∗

540.12

0.00

0.00

N ∗ H3

405.66

0.00

0.00

18


Page 19 of 38 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 4: Corrections to the core-valence separation (CVS) approximation (in eV) obtained as the differences between ∆CCSD(T) and CVS-∆CCSD(T) results. Excitations with orbital energy differences smaller than the cutoff threshold have been excluded in CC calculations of core-ionized states. cc-pVTZ

cc-pCVTZ

cutoff threshold (in Hartree)

cutoff threshold (in Hartree)

5.0

3.0

2.0

5.0

3.0

2.0

C∗ O

-0.13

-0.23

-0.23

-0.21

-0.30

-0.30

CO∗

-0.33

-0.32

-0.32

-0.43

-0.43

-0.43

F∗2

-0.34

-0.34

-0.34

-0.50

-0.50

-0.50

HF∗

-0.32

-0.32

-0.32

-0.47

-0.47

-0.47

N∗2

-0.32

-0.33

-0.33

-0.40

-0.40

-0.40

C∗ H4

-0.16

-0.27

-0.26

-0.20

-0.30

-0.30

H2 O ∗

-0.32

-0.30

-0.30

-0.42

-0.42

-0.42

C∗ H2 O

-0.17

-0.28

-0.29

-0.21

-0.31

-0.30

CH2 O∗

-0.35

-0.35

-0.36

-0.43

-0.43

-0.43

C∗2 H2

-0.17

-0.29

-0.28

-0.21

-0.33

-0.32

C∗2 H4

-0.17

-0.28

-0.28

-0.21

-0.31

-0.31

C∗ O2

-0.15

-0.25

-0.25

-0.22

-0.29

-0.29

CO∗2

-0.32

-0.32

-0.30

-0.42

-0.42

-0.42

NNO∗

-0.31

-0.30

-0.29

-0.41

-0.41

-0.41

NN∗ O

-0.31

-0.33

-0.33

-0.37

-0.38

-0.39

N∗ NO

-0.30

-0.31

-0.31

-0.38

-0.38

-0.38

N ∗ H3

-0.30

-0.30

-0.30

-0.36

-0.36

-0.36

HC∗ N

-0.17

-0.28

-0.28

-0.21

-0.32

-0.32

HCN∗

-0.33

-0.33

-0.35

-0.40

-0.40

-0.40

C∗ H3 OH

-0.16

-0.27

-0.27

-0.20

-0.30

-0.30

CH3 O∗ H

-0.32

-0.31

-0.31

-0.42

-0.42

-0.42

19



Page 20 of 38

Table 5: ∆CCSD(T) and experimental values for K-edge ionization energies (in eV) of the atoms in bold letter marked with a star. The cc-pCV∞Z results are the estimate of basis-set limit obtained by extrapolation of the cc-pCVXZ (X=T, Q, 5) results. ∆CVS refers to corrections to the core-valence separation approximation taken from the last column of Table 4. MaxAE, MAE, and SD refer to maximum absolute error, mean absolute error, and standard deviation with respect to experimental values, respectively. Scalar-relativistic effects have been taken into account using the spin-free exact twocomponent theory in its one-electron variant. cc-pVXZ T

Q

cc-pCVXZ 5

T

Q

5

cc-pCV∞Z ∞

+∆CVS

Exp. 88

C∗ O

296.18 296.17 296.17

296.69 296.64 296.64 296.65

296.35

296.21

∗

542.86 542.67 542.61

543.24 543.12 543.09 543.07

542.64

542.55

F∗2

697.09 696.87 696.78

697.38 697.29 697.28 697.30

696.80

696.69

HF∗

694.69 694.50 694.43

695.00 694.92 694.92 694.92

694.45

694.23

N∗2

410.02 409.96 409.94

410.47 410.41 410.40 410.41

410.01

409.98

C H4

290.73 290.67 290.68

291.25 291.13 291.10 291.08

290.79

290.91

∗

H2 O

540.12 539.99 539.96

540.49 540.43 540.42 540.42

540.00

539.90

C ∗ H2 O

294.36 294.37 294.39

294.89 294.84 294.84 294.85

294.54

294.47

CH2 O∗

539.71 539.54 539.48

540.09 539.98 539.95 539.94

539.51

539.48

C∗2 H2

291.24 291.18 291.19

291.76 291.66 291.64 291.64

291.31

291.14

C∗2 H4 C∗ O2 CO∗2 ∗

290.73 290.68 290.69

291.25 291.14 291.13 291.12

290.81

290.82

297.53 297.57 297.60

298.06 298.04 298.06 298.08

297.79

297.69

541.53 541.39 541.35

541.91 541.84 541.82 541.82

541.40

541.34

541.77 541.62 541.58

542.14 542.07 542.05 542.05

541.63

541.42

CO

∗

NNO ∗

NN O

412.76 412.72 412.73

413.21 413.18 413.18 413.20

412.81

412.59

∗

N NO

408.87 408.82 408.81

409.31 409.27 409.27 409.29

408.91

408.71

N ∗ H3

405.66 405.59 405.60

406.11 406.04 406.03 406.03

405.67

405.56

HC∗ N

293.35 293.32 293.34

293.88 293.80 293.79 293.80

293.48

293.40

HCN∗

406.91 406.83 406.80

407.37 407.28 407.26 407.25

406.86

406.78

C H3 OH 292.30 292.30 292.33

292.83 292.76 292.76 292.76

292.46

292.43

CH3 O∗ H 539.35 539.19 539.15

539.71 539.63 539.61 539.60

539.18

539.11

∗

MaxAE

0.46

0.27

0.23

0.77

0.69

0.69

0.69

0.22

n/a

MAE

0.18

0.11

0.08

0.55

0.48

0.47

0.47

0.11

n/a

SD

0.22

0.13

0.10

0.58

0.50

0.49

0.50

0.13

n/a

20


Page 21 of 38 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 6: ∆CCSD(T) chemical shifts of K-edge ionization energies of the atoms in bold letter marked with a star. MaxAE, MAE, and SD refer to maximum absolute error, mean absolute error, and standard deviation with respect to experimental values, respectively. ∆CVS refers to corrections to the core-valence separation approximation, which are taken from the last column of Table 4. Scalar-relativistic effects have been taken into account using the spin-free exact two-component theory in its one-electron variant. cc-pVXZ T

Q

cc-pCVXZ 5

T

Q

5

cc-pCV∞Z ∞

+∆CVS

Exp. 88

HF∗

-2.40 -2.37 -2.35

-2.38 -2.36 -2.36 -2.37

-2.34

-2.46

C ∗ H4

-5.45 -5.51 -5.49

-5.44 -5.51 -5.54 -5.57

-5.56

-5.30

H 2 O∗

-2.74 -2.69 -2.65

-2.75 -2.69 -2.66 -2.65

-2.63

-2.65

C∗ H2 O

-1.82 -1.80 -1.78

-1.80 -1.80 -1.80 -1.81

-1.80

-1.74

CH2 O∗

-3.15 -3.14 -3.13

-3.15 -3.14 -3.13 -3.13

-3.13

-3.07

C∗2 H2

-4.94 -4.99 -4.98

-4.93 -4.98 -5.00 -5.02

-5.04

-5.07

C∗2 H4

-5.45 -5.49 -5.49

-5.44 -5.49 -5.51 -5.54

-5.54

-5.39

C∗ O2

1.36

1.37

1.43

1.44

1.48

CO∗2

-1.33 -1.28 -1.26

-1.32 -1.28 -1.26 -1.25

-1.24

-1.21

NNO∗

-1.10 -1.06 -1.04

-1.09 -1.05 -1.04 -1.02

-1.00

-1.13

NN∗ O

2.74

2.75

2.79

2.80

2.61

N∗ NO

-1.15 -1.14 -1.13

-1.15 -1.14 -1.13 -1.12

-1.10

-1.27

N∗ H3

-4.36 -4.37 -4.34

-4.35 -4.37 -4.37 -4.37

-4.34

-4.42

HC∗ N

-2.83 -2.85 -2.83

-2.81 -2.84 -2.85 -2.86

-2.87

-2.81

HCN∗

-3.11 -3.13 -3.14

-3.10 -3.13 -3.14 -3.15

-3.15

-3.20

C∗ H3 OH -3.88 -3.88 -3.84

-3.86 -3.88 -3.88 -3.89

-3.89

-3.78

CH3 O∗ H -3.51 -3.49 -3.47

-3.53 -3.49 -3.48 -3.47

-3.46

-3.44

MaxAE

0.15

0.21

0.19

0.14

0.21

0.24 0.27

0.26

n/a

MAE

0.09

0.09

0.08

0.09

0.09

0.09 0.09

0.09

n/a

SD

0.10

0.10

0.10

0.10

0.10

0.11 0.11

0.12

n/a

1.39

2.77

1.43

2.79

1.40

2.77

21

1.42

2.78



Page 22 of 38

Table 7: ∆CCSD(T), CVS-EOMIP-CCSDT, and experimental values of C and N 1s ionization energies of CH3 CN and CH3 NC (in eV). The cc-pCV∞Z results are the estimate of basis-set limit obtained by extrapolation of the cc-pCVXZ (X=T, Q, 5) results. ∆CVS refers to corrections to the core-valence separation approximation, which are taken from the last column of Table 4. Scalar-relativistic effects have been taken into account using the spin-free exact two-component theory in its one-electron variant.

Exp. 97

Exp. 98

CVS-∆CCSD(T)/cc-pCV∞Z

CVS-EOMIP-

+∆CVS (cc-pCVTZ, δ = 2.0)

CCSDT/cc-pCVTZ 35

C∗ H3 CN

293.1

292.98

292.73

292.81

CH3 C∗ N

293.2

292.45

292.82

292.90

CH3 CN∗

405.9

405.64

405.66

405.71

CH3 NC∗

293.8

292.37

292.45

292.35

C∗ H3 NC

293.1

293.35

293.38

293.41

CH3 N∗ C

407.1

406.67

406.75

407.02

22


Page 23 of 38 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 8: CVS-∆CCSD(T)/cc-pVTZ results for C 1s ionization energies (in eV) of ethyl trifluoroacetate (CF3 COOCH2 CH3 ). Relative shifts with respect to the methyl carbon are enclosed in the parentheses. Scalar-relativistic effects have been taken into account using the spin-free exact two-component theory in its one-electron variant. C in CH3

C in OCH2

C in COO

C in CF3

Cs geometry ∆HF

291.37 (0.0)

293.20 (1.82)

296.78 (5.41)

299.64 (8.27)

∆CCSD

291.20 (0.0)

292.89 (1.69)

295.71 (4.51)

298.74 (7.54)

∆CCSD(T)

291.24 (0.0)

292.88 (1.65)

295.56 (4.33)

298.58 (7.34)

C1 geometry ∆HF

291.29 (0.0)

293.19 (1.90)

296.80 (5.52)

299.64 (8.35)

∆CCSD

291.12 (0.0)

292.89 (1.77)

295.73 (4.61)

298.73 (7.61)

∆CCSD(T)

291.12 (0.0)

292.85 (1.73)

295.48 (4.26)

298.54 (7.42)

Experiment 101

291.47 (0.0)

293.19 (1.72)

295.80 (4.33)

298.93 (7.46)

23



Acknowledgement L. C. is grateful to Johns Hopkins University for the start-up fund. All calculations presented here have been carried out using computational facilities of Maryland Advanced Research Computing Center (MARCC). The authors are indebted to Jaime Combariza at MARCC for assistance in performing the computations.

Supporting Information Available The geometrical parameters for the C1 and Cs conformers of ethyl trifluoroacetate used in the present study have been compiled and given in the supplementary material. This material is available free of charge via the Internet at http://pubs.acs.org/.

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Performance of Delta-Coupled-Cluster Methods for Calculations of

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