Core-valence Separation Approximation Research Articles

Cluster Perturbation Theory for Core Excited States and Core Ionization Potentials Using Core-Valence Separation.

The development of accurate and fast computational procedures for the ab initio calculation of X-ray spectroscopies is paramount to facilitate theoretical analysis of modern X-ray experiments on molecules. Herein, we present the extension of Cluster Perturbation theory to comprehend the calculation of core excited states and core ionization potentials using the core-valence separation approximation, which has seen widespread success for various quantum chemistry methods. We derive the theoretical framework for introducing core-valence separation into Cluster Perturbation series for excitation energies and display the performance of the methodology in S(D) orbital excitation spaces. The obtained core excitation energies on a test set of medium sized organic molecules show that carbon, nitrogen, and oxygen K-edge excitation energies can be determined with errors below 2 eV relative to the CCSD reference results using the developed CPS(D) excitation energy models which can be used for systems way beyond the reach of conventional CCSD.

GUGA-based MRCI approach with core-valence separation approximation (CVS) for the calculation of the core-excited states of molecules.

We develop and demonstrate how to use the Graphical Unitary Group Approach (GUGA)-based MRCISD with Core-Valence Separation (CVS) approximation to compute the core-excited states. First, perform a normal Self-Consistent-Field (SCF) or valence MCSCF calculation to optimize the molecular orbitals. Second, rotate the optimized target core orbitals and append to the active space, form an extended CVS active space, and perform a CVS-MCSCF calculation for core-excited states. Finally, construct the CVS-MRCISD expansion space and perform a CVS-MRCISD calculation to optimize the CI coefficients based on the variational method. The CVS approximation with GUGA-based methods can be implemented by flexible truncation of the Distinct Row Table. Eliminating the valence-excited configurations from the CVS-MRCISD expansion space can prevent variational collapse in the Davidson iteration diagonalization. The accuracy of the CVS-MRCISD scheme was investigated for excitation energies and compared with that of the CVS-MCSCF and CVS-CASPT2 methods using the same active space. The results show that CVS-MRCISD is capable of reproducing well-matched vertical core excitation energies that are consistent with experiments by combining large basis sets and a rational reference space. The calculation results also highlight the fact that the dynamic correlation between electrons makes an undeniable contribution in core-excited states.

Electron-Affinity Time-Dependent Density Functional Theory: Formalism and Applications to Core-Excited States.

The lack of particle-hole attraction and orbital relaxation within time-dependent density functional theory (TDDFT) lead to extreme errors in the prediction of K-edge X-ray absorption spectra (XAS). We derive a linear-response formalism that uses optimized orbitals of the n - 1-electron system as the reference, building orbital relaxation and a proper hole into the initial density. Our approach is an exact generalization of the static-exchange approximation that ameliorates the particle-hole interaction error associated with the adiabatic approximation and reduces errors in TDDFT XAS by orders of magnitude. With a statistical performance of just 0.5 eV root-mean-square error and the same computational scaling as TDDFT under the core-valence separation approximation, we anticipate that this approach will be of great utility in XAS calculations of large systems.

Simulating X‐ray absorption spectra with complete active space self‐consistent field linear response methods

AbstractIn this work, two approaches for simulating X‐ray absorption (XA) spectra with the complete active space self‐consistent field (CASSCF) linear response (LR) method are introduced. The first approach employs the well‐known core‐valence separation (CVS) approximation, which is predominantly used by many other electronic structure methods for simulating X‐ray spectra. The second ansatz uses the harmonic Davidson algorithm for finding interior eigenvalues that lie close to a target excitation energy shift and virtually solves a shifted‐and‐inverted (S&I) generalized eigenvalue problem. LR‐CASSCF K‐edge transition energies are systematically blueshifted though have consistently smaller errors than those of the CAS or restricted active space (RAS) configuration interaction (CI) methods. For simple molecules at which the core hole can only be created at a single site, the state‐specific RASSCF or n‐electron valence second‐order perturbation theory/RASCI gave more accurate principal K‐edge excitation energies. If the core hole can be created at multiple sites, the LR‐CASSCF approaches perform much better than RASSCF. Moreover, we observed that the LR‐CASSCF variants were the only MR methods discussed here that predicted correctly the order of O K‐edge features in the ozone molecule and the permanganate ion. The peak separation of edge features in ozone was as accurate as with equation‐of‐motion coupled cluster singles and doubles. The error of the CVS approximation turned out to be very system dependent and in some cases amounted up to 1.0 eV for the K‐edge excitation energies. Those CVS errors are still acceptable if one considers the observed deviation from the experimental reference by 5–11 eV. The deviations made in the XAS intensities were even more pronounced. CVS and the full S&I oscillator strengths could differ even by a factor of 2.8. Since the S&I approach is at least as efficient as the LR‐CASSCF method for valence excitations, future endeavors to improve the accuracy by accounting for dynamic correlation could be built on top of the full S&I approach.

Quantifying the error of the core-valence separation approximation.

For the calculation of core-excited states probed through X-ray absorption spectroscopy, the core-valence separation (CVS) scheme has become a vital tool. This approach allows us to target such states with high specificity, albeit introducing an error. We report the implementation of a post-processing step for CVS excitations obtained within the algebraic-diagrammatic construction scheme for the polarization propagator, which removes this error. Based on this, we provide a detailed analysis of the CVS scheme, identifying its accuracy to be dominated by an error balance between two neglected couplings, one between core and valence single excitations and the other between single and double core excitations. The selection of the basis set is shown to be vital for a proper description of both couplings, with tight polarizing functions being necessary for a good balance of errors. The CVS error is confirmed to be stable across multiple systems, with an element-specific spread for K-edge spectrum calculations of only about ±0.02 eV. A systematic lowering of the CVS error by 0.02 eV-0.03 eV is noted when considering excitations to extremely diffuse states, emulating ionization.

Hetero-site Double Core Ionization Energies with Sub-electronvolt Accuracy from Delta-Coupled-Cluster Calculations.

Benchmark scalar-relativistic delta-coupled-cluster calculations of hetero-site double core ionization energies of small molecules containing second-row elements are reported. The present study has focused on the high-spin triplet components of two-site double core-ionized states, which are single reference in character and consistent with the use of standard coupled-cluster methods. Contributions to computed double core ionization energies from electron-correlation and basis-set effects as well as corrections to the core-valence separation approximation have been analyzed. On the basis of systematic convergence of computational results with respect to these effects, delta-coupled-cluster calculations have been shown to be capable of providing accurate double core ionization energies with remaining errors estimated to be below 0.3 eV, and thus they are recommended for use to facilitate experimental studies of two-site double core-ionized states that are involved in X-ray pump/X-ray probe studies of electronic and molecular dynamics following inner shell ionization or excitation.

Third-order algebraic diagrammatic construction theory for electron attachment and ionization energies: Conventional and Green's function implementation.

We present implementation of second- and third-order algebraic diagrammatic construction (ADC) theory for efficient and accurate computations of molecular electron affinities (EA), ionization potentials (IP), and densities of states [EA-/IP-ADC(n), n = 2, 3]. Our work utilizes the non-Dyson formulation of ADC for the single-particle propagator and reports working equations and benchmark results for the EA-ADC(2) and EA-ADC(3) approximations. We describe two algorithms for solving EA-/IP-ADC equations: (i) conventional algorithm that uses iterative diagonalization techniques to compute low-energy EA, IP, and density of states and (ii) Green's function algorithm (GF-ADC) that solves a system of linear equations to compute density of states directly for a specified spectral region. To assess the accuracy of EA-ADC(2) and EA-ADC(3), we benchmark their performance for a set of atoms, small molecules, and five DNA/RNA nucleobases. As our next step, we demonstrate the efficiency of our GF-ADC implementation by computing core-level K-, L-, and M-shell ionization energies of a zinc atom without introducing the core-valence separation approximation. Finally, we use EA- and IP-ADC methods to compute the bandgaps of equally spaced hydrogen chains Hn with n up to 150, providing their estimates near thermodynamic limit. Our results demonstrate that EA-/IP-ADC(n) (n = 2, 3) methods are efficient and accurate alternatives to widely used electronic structure methods for simulations of electron attachment and ionization properties.

X-ray and UV Spectra of Glycine within Coupled Cluster Linear Response Theory.

The coupled cluster models CCSD and CC3 are used to investigate the (core) excited states and ionization energies of glycine in the gas phase. Excited states and ionization energies in the UV spectral range are calculated using a standard coupled cluster linear response, while core-level excited states and ionization potentials are calculated using the core-valence separation approximation. The temperature dependence from different conformers is also assessed.

Performance of Delta-Coupled-Cluster Methods for Calculations of Core-Ionization Energies of First-Row Elements.

A thorough study of the performance of delta-coupled-cluster (ΔCC) 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. The 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 and 0.6 eV, respectively. In contrast, the inclusion of electron correlation in ΔCC singles and doubles augmented with a noniterative triples correction [ΔCCSD(T)] method significantly reduces the corresponding deviations to around 0.3 and 0.1 eV. With the consideration of basis set effects and the corrections to the CVS approximation, ΔCCSD(T) has been shown to provide highly accurate results for absolute values of core-ionization energies, with a MaxAE of 0.22 eV and SD of 0.13 eV. To further demonstrate the usefulness of ΔCCSD(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.

Simulating X-ray Absorption Spectra with Linear-Response Density Cumulant Theory.

We present a new approach for simulating X-ray absorption spectra based on linear-response density cumulant theory (LR-DCT) [ Copan , A. V. ; Sokolov , A. Yu. J. Chem. Theory Comput. 2018 , 14 , 4097 - 4108 ]. Our new method combines the LR-ODC-12 formulation of LR-DCT with core-valence separation approximation (CVS) that allows us to efficiently access high-energy core-excited states. We describe our computer implementation of the CVS-approximated LR-ODC-12 method (CVS-ODC-12) and benchmark its performance by comparing simulated X-ray absorption spectra to those obtained from experiment for several small molecules. Our results demonstrate that the CVS-ODC-12 method shows good agreement with experiment for relative spacings between transitions and their intensities, but the excitation energies are systematically overestimated. When compared to results from excited-state coupled cluster methods with single and double excitations, the CVS-ODC-12 method shows a similar performance for intensities and peak separations, while coupled cluster spectra are less shifted, relative to experiment. An important advantage of CVS-ODC-12 is that its excitation energies are computed by diagonalizing a Hermitian matrix, which enables efficient computation of transition intensities.

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

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/CVS-ISR 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.

Analysis and comparison of CVS-ADC approaches up to third order for the calculation of core-excited states.

The extended second order algebraic-diagrammatic construction (ADC(2)-x) scheme for the polarization operator in combination with core-valence separation (CVS) approximation is well known to be a powerful quantum chemical method for the calculation of core-excited states and the description of X-ray absorption spectra. For the first time, the implementation and results of the third order approach CVS-ADC(3) are reported. Therefore, the CVS approximation has been applied to the ADC(3) working equations and the resulting terms have been implemented efficiently in the adcman program. By treating the α and β spins separately from each other, the unrestricted variant CVS-UADC(3) for the treatment of open-shell systems has been implemented as well. The performance and accuracy of the CVS-ADC(3) method are demonstrated with respect to a set of small and middle-sized organic molecules. Therefore, the results obtained at the CVS-ADC(3) level are compared with CVS-ADC(2)-x values as well as experimental data by calculating complete basis set limits. The influence of basis sets is further investigated by employing a large set of different basis sets. Besides the accuracy of core-excitation energies and oscillator strengths, the importance of cartesian basis functions and the treatment of orbital relaxation effects are analyzed in this work as well as computational timings. It turns out that at the CVS-ADC(3) level, the results are not further improved compared to CVS-ADC(2)-x and experimental data, because the fortuitous error compensation inherent in the CVS-ADC(2)-x approach is broken. While CVS-ADC(3) overestimates the core excitation energies on average by 0.61% ± 0.31%, CVS-ADC(2)-x provides an averaged underestimation of -0.22% ± 0.12%. Eventually, the best agreement with experiments can be achieved using the CVS-ADC(2)-x method in combination with a diffuse cartesian basis set at least at the triple-ζ level.

An intermediate state representation approach to K-shell ionization in molecules. I. Theory

The general intermediate state representation (ISR) for single-electron ionization is adapted to the case of K-shell (or core-level) ionization in molecules. The development is based on the so-called core–valence separation (CVS) approximation leading to a considerable simplification of the ISR secular equations. Using the CVS approximation the core-level ISR can be formulated entirely in terms of the intermediate states of the valence electron excitation problem, which allows one to construct consistent nth-order approximation schemes for the (single-hole) ionization energies by a specific extension of the (n−2)-nd order ISR approximation for electronic excitation. In particular, the CVS-ISR concept is used to derive a consistent fourth-order approximation for core-level ionization based on the existing second-order algebraic-diagrammatic construction [ADC(2)] approximation to electron excitation. The computational scheme combines the diagonalization of a Hermitian secular matrix with finite perturbation expansions for the secular matrix elements. The explicit configuration space is spanned by one-hole (1h), two-hole-one-particle (2h-1p), and (3h-2p) ionic states with exactly one hole in the core-level shell of interest, while the configurations considered implicitly via perturbation theory extend to the class of 5h-4p states. A characteristic of the method is that the dominant valence electron relaxation effect is accounted for at the post-Hartree–Fock (HF) level. This calls for the relatively high order of perturbation-theoretical consistency, but avoids, on the other hand, the necessity of a localized (symmetry breaking) one-particle representation in the case of molecules with equivalent 1s orbitals. The method is size consistent and thus suitable for applications to large systems.