Core-valence Separation Research Articles

The quest for best suited references for configuration interaction singles calculations of core excited states.

Near edge X-ray absorption fine structure (NEXAFS) simulations based on the conventional configuration interaction singles (CIS) lead to excitation energies, which are systematically blue shifted. Using a (restricted) open shell core hole reference instead of the Hartree Fock (HF) ground state orbitals improves (Decleva et al., Chem. Phys., 1992, 168, 51) excitation energies and the shape of the spectra significantly. In this work, we systematically vary the underlying SCF approaches, that is, based on HF or density functional theory, to identify best suited reference orbitals using a series of small test molecules. We compare the energies of the K edges and NEXAFS spectra to experimental data. The main improvement compared to conventional CIS, that is, using HF ground state orbitals, is due to the electrostatic influence of the core hole. Different SCF approaches, density functionals, or the use of fractional occupations lead only to comparably small changes. Furthermore, to account for bigger systems, we adapt the core-valence separation for our approach. We demonstrate that the good quality of the spectrum is not influenced by this approximation when used together with the non-separated ground state wave function. Simultaneously, the computational demands are reduced remarkably. © 2016 Wiley Periodicals, Inc.

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.

Communication: X-ray absorption spectra and core-ionization potentials within a core-valence separated coupled cluster framework.

We present a simple scheme to compute X-ray absorption spectra (e.g., near-edge absorption fine structure) and core ionisation energies within coupled cluster linear response theory. The approach exploits the so-called core-valence separation to effectively reduce the excitation space to processes involving at least one core orbital, and it can be easily implemented within any pre-existing coupled cluster code for low energy states. We further develop a perturbation correction that incorporates the effect of the excluded part of the excitation space. The correction is shown to be highly accurate. Test results are presented for a set of molecular systems for which well converged results in full space could be generated at the coupled cluster singles and doubles level of theory only, but the scheme is straightforwardly generalizable to all members of the coupled cluster hierarchy of approximations, including CC3.

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.

Calculating X-ray Absorption Spectra of Open-Shell Molecules with the Unrestricted Algebraic-Diagrammatic Construction Scheme for the Polarization Propagator.

X-ray absorption spectroscopy (XAS) is a powerful tool that provides information about the electronic structure of molecules via excitation of electrons from the K-shell core region to the unoccupied molecular levels. These high-lying electronic core-excited states can be accurately calculated using the algebraic-diagrammatic construction scheme of second order ADC(2) by applying the core-valence separation (CVS) approximation to the ADC(2) working equations. For the first time, an efficient implementation of an unrestricted CVS-ADC(2) variant CVS-UADC(2) is presented for the calculation of open-shell molecules by treating α and β spins separately from each other. The potential of the CVS-UADC(2) method is demonstrated with a set of small organic radicals by comparison with standard TD-DFT/B3LYP values and experimental data. It turns out that the extended variant CVS-UADC(2)-x, in particular, provides the most accurate results with errors of only 0.1% compared to experimental values. This remarkable agreement justifies the prediction of yet nonrecorded experimental XAS spectra like the one of the anthracene cation. The cation exhibits additional peaks due to the half-filled single-occupied molecular orbital, which may help to distinguish cation from the neutral species.

Calculating core-level excitations and X-ray absorption spectra of medium-sized closed-shell molecules with the algebraic-diagrammatic construction scheme for the polarization propagator.

Core-level excitations are generated by absorption of high-energy radiation such as X-rays. To describe these energetically high-lying excited states theoretically, we have implemented a variant of the algebraic-diagrammatic construction scheme of second-order ADC(2) by applying the core-valence separation (CVS) approximation to the ADC(2) working equations. Besides excitation energies, the CVS-ADC(2) method also provides access to properties of core-excited states, thereby allowing for the calculation of X-ray absorption spectra. To demonstrate the potential of our implementation of CVS-ADC(2), we have chosen medium-sized molecules as examples that have either biological importance or find application in organic electronics. The calculated results of CVS-ADC(2) are compared with standard TD-DFT/B3LYP values and experimental data. In particular, the extended variant, CVS-ADC(2)-x, provides the most accurate results, and the agreement between the calculated values and experiment is remarkable.

Local Hartree-Fock orbitals using a three-level optimization strategy for the energy

Using the three-level energy optimization procedure combined with a refined version of the least-change strategy for the orbitals--where an explicit localization is performed at the valence basis level--it is shown how to more efficiently determine a set of local Hartree-Fock orbitals. Further, a core-valence separation of the least-change occupied orbital space is introduced. Numerical results comparing valence basis localized orbitals and canonical molecular orbitals as starting guesses for the full basis localization are presented. The results show that the localization of the occupied orbitals may be performed at a small computational cost if valence basis localized orbitals are used as a starting guess. For the unoccupied space, about half the number of iterations are required if valence localized orbitals are used as a starting guess compared to a canonical set of unoccupied Hartree-Fock orbitals. Different local minima may be obtained when different starting guesses are used. However, the different minima all correspond to orbitals with approximately the same locality.

Communication: A density functional with accurate fractional-charge and fractional-spin behaviour for s-electrons

We develop a new density-functional approach combining physical insight from chemical structure with treatment of multi-reference character by real-space modeling of the exchange-correlation hole. We are able to recover, for the first time, correct fractional-charge and fractional-spin behaviour for atoms of groups 1 and 2. Based on Becke's non-dynamical correlation functional [A. D. Becke, J. Chem. Phys. 119, 2972 (2003)] and explicitly accounting for core-valence separation and pairing effects, this method is able to accurately describe dissociation and strong correlation in s-shell many-electron systems.

Accurate Relativistic Small-Core Pseudopotentials for Actinides. Energy Adjustment for Uranium and First Applications to Uranium Hydride

The options to adjust accurate relativistic energy-consistent pseudopotentials for actinides are explored using uranium as an example. The choice of the reference data and the core-valence separation is discussed in view of a targeted accuracy of 0.04 eV or better in atomic energy differences such as excitation energies and ionization potentials. A new small-core pseudopotential attributing 60 electrons to the core has been generated by an energy adjustment to state-averaged multiconfiguration Dirac-Hartree-Fock/Dirac-Coulomb-Breit Fermi nucleus reference data of 100 nonrelativistic configurations of U to U(7+) corresponding to 30190 reference J levels. At the finite-difference multiconfiguration Hartree-Fock level the mean absolute errors are 0.002 and 0.024 eV for the configurations and J levels, respectively. A first molecular application to uranium monohydride UH yields very satisfactory agreement with results from all-electron calculations based on the Douglas-Kroll-Hess Hamiltonian.

Analysis of the segmented contraction of basis functions using density matrix theory

A particular formulation based on density matrix (DM) theory at the Hartree-Fock level of theory and the description of the atomic orbitals as integral transforms is introduced. This formulation leads to a continuous representation of the density matrices as functions of a generator coordinate and to the possibility of plotting either the continuous or discrete density matrices as functions of the exponents of primitive Gaussian basis functions. The analysis of these diagrams provides useful information allowing: (a) the determination of the most important primitives for a given orbital, (b) the core-valence separation, and (c) support for the development of contracted basis sets by the segmented method.

An intermediate state representation approach to K-shell ionization in molecules. II. Computational tests

We report on a new implementation and first numerical tests of the fourth-order algebraic–diagramatic construction [ADC(4)] propagator method for K-shell ionization in molecules. The theory, which has been presented in a preceding paper (paper I) [J. Chem. Phys. 115, 10621 (2001)], is based on an intermediate state representation (ISR) or non-Dyson reformulation of the general ADC approach and uses the core–valence separation approximation to specialize the method to the case of core-level ionization. The ISR form offers the possibility to go beyond the finite perturbation-theoretical expressions of the previous (strict) ADC(4) scheme, and several such modifications are considered. The general aim of the present development is to establish a practical “frozen” orbital method that can be applied equally well to systems with and without equivalent core levels. A set of small model calculations has been carried out on the 1s ionization in CO and N2 allowing us to compare the ADC results with those of large-scale configuration interaction treatments and analyze the performance of several ADC(4) modifications in describing the 1s ionization energies as a function of the bond lengths. These studies clearly demonstrate the need for improving the previous (strict) ADC(4) scheme. In the theoretical description of the C1s and N1s ionization, the proposed modifications lead to a substantial improvement, whereas in the case of the O1s ionization further developments appear to be necessary. In another set of calculations using reasonably large basis sets, the improved ADC(4) method is applied to the vibrational structure in the CO C1s and N2 N1s ionization spectra. The results are in very good agreement with the findings of recent high-resolution measurements. An interesting feature is the different behavior of the g and u components of the N1s ionization energy curves of N2 leading to different vibrational line intensity distributions.

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.

Does the topological approach characterize the hydrogen bond?

The topological analysis of the electron localization function has been applied to complexes representative of the weak, medium and strong hydrogen bond. For both the weak and the medium hydrogen bonds, the number of basins in the complexes is the sum of those of the moieties. In this case, the formation of a weak or a medium hydrogen-bonded complex does not involve a chemical reaction. In the weak hydrogen bond case, the reduction of the localization domain yields two domains in the first step, which can be partitioned afterwards into valence and core domains. In contrast, for medium complexes the core–valence separation is the first event which occurs during the reduction process and therefore the complex should be considered as a single molecular species. Moreover, the analysis of the basin population variance indicates in this case a noticeable delocalization between the V(A, H) and V(B) basins. Finally, the symmetrical strong hydrogen bond has a protonated basin V(H) at the bond midpoint. Such a topology corresponds to an incomplete proton transfer and to a rather covalent bond.

Core-level electronic spectra in ADC(2) approximation for polarization propagator: Carbon monoxide and nitrogen molecules

An effective ab initio approach to core-level electronic spectral studies is discussed. The approach uses polarization propagator theory in a second-order algebraic diagram construction ADC(2) approximation for calculating the characteristics of electron transitions; it also uses the linear vibronic model LVM for investigating the vibrational structure of transitions. The core excitation specialization of ADC(2) is achieved by introducing the core valence separation (CVS) approximation. K-excitation spectra of CO and N2 molecules are calculated to examine the potential of the approach. The calculated spectra and the available experimental data are analyzed to characterize the method. A number of additional facts of methodological and practical value are found, and new transitions are predicted. It is concluded that ADC(2)/CVS/LVM is a promising approach to problem solving in core level spectroscopy, which requires qualitatively reliable theoretical estimations.

Quantum Monte Carlo calculation of Compton profiles of solid lithium

Recent high resolution Compton scattering experiments in lithium have shown significant discrepancies with conventional band theoretical results. We present a pseudopotential quantum Monte Carlo study of electron-electron and electron-ion correlation effects on the momentum distribution of lithium. We compute the correlation correction to the valence Compton profiles obtained within Kohn-Sham density functional theory in the local density approximation and determine that electronic correlation does not account for the discrepancy with the experimental results. Our calculations lead do different conclusions than recent GW studies and indicate that other effects (thermal disorder, core-valence separation etc.) must be invoked to explain the discrepancy with experiments.

VB-DFT: a nonempirical hybrid method combining valence bond theory and density functional energies

A nonempirical procedure is presented for combining the valence bond (VB) methodology with density functional theory (DFT). The method, called VB-DFT, is based on partitioning the system into two coupled electronic subsets, one treated by DFT and the other by VB theory. The DFT energy component includes the exchange correlation potential while the coupling involves electron repulsion and exchange interactions. Tests and applications show that the VB-DFT hybrid method has the qualities of a VB method along with a quantitative advantage over VB-only methods with traditional core–valence separation.

Monte Carlo study of core–valence separation schemes

All-electron partitioning of wavefunctions into products ΨcoreΨval of core and valence parts in orbital space results in the loss of core-valence antisymmetry, uncorrelation of motion of core and valence electrons, and core–valence overlap. These effects are studied with the variational Monte Carlo method using appropriately designed wavefunctions for the first-row atoms and cations. It is shown that the loss of antisymmetry with respect to an interchange of core and valence electrons is a dominant effect which increases rapidly through the row, while the effect of decorrelating the core and valence electrons is significantly smaller. Orthogonality of the core and valence parts partially substitutes the exclusion principle and is absolutely necessary for meaningful calculations with partitioned wavefunctions. Core–valence overlap may lead to nonsensical values of the total energy. Even relatively crude core–valence partitioned wavefunctions generally can estimate ionization potentials and electron affinities with better accuracy than that of the traditional, non-partitioned ones, provided that they achieve maximum separation (independence) of core and valence shells accompanied by high internal flexibility of Ψcore and Ψval. Our best core–valence partitioned wavefunction of that kind estimates the ionization potentials with an accuracy comparable to the most accurate theoretical determinations in the literature.

Shell structure of atoms

The average local electrostatic potential function, V(r) ϱ(r) , is calculated for 87 atoms and the corresponding monopositive and mononegative ions in the ground state, using the nonrelativistic average-over-configuration numerical Hartree-Fock density. It is found in general that the shell boundaries are expressed as the successively increasing maxima in V(r) ϱ(r) and the outermost maximum presents good approximate estimates of the core-valence separation in atoms. The reason for such behaviour is explained analytically. The proposed function is identified as the inverse of charge capacity per unit volume which attains a maximum at the shell boundary.

1994 Polanyi Award Lecture Concept of active electrons in chemistry

The notion of division between active and dormant electrons has been well received and widely used in the chemists' way of thinking. The core–valence separation in atoms is the best-known example. This paper describes a theoretical and computational method called the model potential method, which deals only with active electrons in molecular and solid state calculations. The method is capable of reaching computational accuracy of testing the validity of the separation of active and dormant electrons in individual cases. Keywords: separability of electrons, model potential method, valence orbitals, relativistic effects.

Atomic shell structure in Hartree—Fock theory

The radial density distribution function computed from the spherically averaged numerical Hartree—Fock density is found to reveal distinct topological features characteristic of the shell structure of atoms beyond Ar for which hitherto reported studies based on the analytical Hartree—Fock density do not show the formal shell structure. Based on the numerical evidence, the location of the midpoint corresponding to the last two successive points of inflection in the distribution function is proposed to define the core—valence separation for heavier atoms ( Z⩾19).