E. Engel

From Explicit to Implicit Density Functionals

The concept of orbital‐ and eigenvalue‐dependent exchange‐correlation (xc) energy functionals is reviewed. We show how such functionals can be derived in a systematic fashion via a perturbation expansion, utilizing the Kohn–Sham system as a noninteracting reference system. We demonstrate that the second‐order contribution to this expansion of the xc‐energy functional includes the leading term of the van der Waals interaction. The optimized‐potential method (OPM), which allows the calculation of the multiplicative xc‐potential corresponding to an orbital‐ and eigenvalue‐dependent xc‐energy functional via an integral equation, is discussed in detail. We examine an approximate analytical solution of the OPM integral equation, pointing out that, for eigenvalue‐dependent functionals, the three paths used in the literature for the derivation of this approximation yield different results. Finally, a number of illustrative results, both for the exchange‐only limit and for the combination of the exact exchange with various correlation functionals, are given. © 1999 John Wiley & Sons, Inc. J Comput Chem 20: 31–50, 1999

Four-component relativistic density functional calculations of heavy diatomic molecules

We perform accurate four-component calculations for heavy closed-shell diatomic molecules in the framework of relativistic density functional theory using local and gradient corrected density functional schemes. As examples we have chosen Cu2, Ag2, Au2, Tl2, Pb2, Bi2, and Pt2. The potential energy curves show the quality, and the discrepancies of the density functionals unscreened from any approximation of the relativistic effects.

Random-phase-approximation-based correlation energy functionals: Benchmark results for atoms

The random phase approximation for the correlation energy functional of the density functional theory has recently attracted renewed interest. Formulated in terms of the Kohn-Sham orbitals and eigenvalues, it promises to resolve some of the fundamental limitations of the local density and generalized gradient approximations, as, for instance, their inability to account for dispersion forces. First results for atoms, however, indicate that the random phase approximation overestimates correlation effects as much as the orbital-dependent functional obtained by a second order perturbation expansion on the basis of the Kohn-Sham Hamiltonian. In this contribution, three simple extensions of the random phase approximation are examined; (a) its augmentation by a local density approximation for short-range correlation, (b) its combination with the second order exchange term, and (c) its combination with a partial resummation of the perturbation series including the second order exchange. It is found that the ground state and correlation energies as well as the ionization potentials resulting from the extensions (a) and (c) for closed subshell atoms are clearly superior to those obtained with the unmodified random phase approximation. Quite some effort is made to ensure highly converged data, so that the results may serve as benchmark data. The numerical techniques developed in this context, in particular, for the inherent frequency integration, should also be useful for applications of random phase approximation-type functionals to more complex systems.

Asymptotic properties of the exchange energy density and the exchange potential of finite systems: relevance for generalized gradient approximations

It is shown that generalized gradient approximations (GGAs) for exchange only, due to their very limited form, quite generally can not simultaneously reproduce both the asymptotic forms of the exchange energy density and the exchange potential of finite systems. Furthermore, mechanisms making GGAs formally approach at least one of these asymptotic forms do not improve the corresponding quantity in the relevant part of the asymptotic regime of atoms. By constructing a GGA which leads to superior atomic exchange energies compared to all GGAs heretofore but does not reproduce the asymptotic form of the exact exchange energy density it is demonstrated that this property is not important for obtaining extremely accurate atomic exchange energies. We conclude that GGAs by their very concept are not suited to reproduce these asymptotic properties of finite systems. As a byproduct of our discussion we present a particularly simple and direct proof of the well known asymptotic structure of the exchange potential of finite spherical systems.

Orbital-Dependent Functionals for the Exchange-Correlation Energy: A Third Generation of Density Functionals

This chapter is devoted to orbital-dependent exchange-correlation (xc) functionals, a concept that has attracted more and more attention during the last ten years. After a few preliminary remarks, which clarify the scope of this review and introduce the basic notation, some motivation will be given why such implicit density functionals are of definite interest, in spite of the fact that one has to cope with additional complications (compared to the standard xc-functionals). The basic idea of orbital-dependent xc-functionals is then illustrated by the simplest and, at the same time, most important functional of this type, the exact exchange of density functional theory (DFT - for a review see e.g. [1], or the chapter by J. Perdew and S. Kurth in this volume).

Second-order Kohn-Sham perturbation theory: Correlation potential for atoms in a cavity

Second-order perturbation theory based on the Kohn-Sham Hamiltonian leads to an implicit density functional for the correlation energy E(c) (MP2), which is explicitly dependent on both occupied and unoccupied Kohn-Sham single-particle orbitals and energies. The corresponding correlation potential v(c) (MP2), which has to be evaluated by the optimized potential method, was found to be divergent in the asymptotic region of atoms, if positive-energy continuum states are included in the calculation [Facco Bonetti et al., Phys. Rev. Lett. 86, 2241 (2001)]. On the other hand, Niquet et al., [J. Chem. Phys. 118, 9504 (2003)] showed that v(c) (MP2) has the same asymptotic -alpha(2r(4)) behavior as the exact correlation potential, if the system under study has a discrete spectrum only. In this work we study v(c) (MP2) for atoms in a spherical cavity within a basis-set-free finite differences approach, ensuring a completely discrete spectrum by requiring hard-wall boundary conditions at the cavity radius. Choosing this radius sufficiently large, one can devise a numerical continuation procedure which allows to normalize v(c) (MP2) consistent with the standard choice v(c)(r-->infinity)=0 for free atoms, without modifying the potential in the chemically relevant region. An important prerequisite for the success of this scheme is the inclusion of very high-energy virtual states. Using this technique, we have calculated v(c) (MP2) for all closed-shell and spherical open-shell atoms up to argon. One finds that v(c) (MP2) reproduces the shell structure of the exact correlation potential very well but consistently overestimates the corresponding shell oscillations. In the case of spin-polarized atoms one observes a strong interrelation between the correlation potentials of the two spin channels, which is completely absent for standard density functionals. However, our results also demonstrate that E(c) (MP2) can only serve as a first step towards the construction of a suitable implicit correlation functional: The fundamental variational instability of this functional is recovered for beryllium, for which a breakdown of the self-consistent Kohn-Sham iteration is observed. Moreover, even for those atoms for which the self-consistent iteration is stable, the results indicate that the inclusion of v(c) (MP2) in the total Kohn-Sham potential does not lead to an improvement compared to the complete neglect of the correlation potential.

Density functional approach to quantumhadrodynamics: Theoretical foundations and construction of extended thomas-fermi models

Abstract We outline the density functional approach to the strong interaction model of quantumhadrodynamics. In particular the extension of the Hohenberg-Kohn theorem to this situation is demonstrated. On the practical level we derive the gradient expansion of the noninteracting kinetic energy to second order in h 2 , including the effects of full four-vector meson exchange as well as vacuum contributions, and discuss the variational equations of the corresponding extended Thomas-Fermi model.

Kohn-Sham perturbation theory: simple solution to variational instability of second order correlation energy functional.

The orbital-dependent correlation energy functional resulting from second order Kohn-Sham perturbation theory leads to atomic correlation potentials with correct shell structure and asymptotic behavior. The absolute magnitude of the exact correlation potential, however, is greatly overestimated. In addition, this functional is variationally instable, which shows up for systems with nearly degenerate highest occupied and lowest unoccupied levels like Be. In this contribution we examine the simplest resummation of the Kohn-Sham perturbation series which has the potential to resolve both the inaccuracy and the instability problem of the second order expression. This resummation includes only the hole-hole terms of the Epstein-Nesbet series of diagrams, which has the advantage that the resulting functional is computationally as efficient as the pure second order expression. The hole-hole Epstein-Nesbet functional is tested for a number of atoms and ions. It is found to reproduce correlation and ground state energies with an accuracy comparable to that of state-of-the-art generalized gradient approximations. The correlation potential, on the other hand, is dramatically improved compared to that obtained from generalized gradient approximations. The same applies to all quantities directly related to the potential, as, for instance, Kohn-Sham eigenvalues and excitation energies. Most importantly, however, the hole-hole Epstein-Nesbet functional turned out to be variationally stable for all neutral as well as all singly and doubly ionized atoms considered so far, including the case of Be.

Exchange effects in low energy electron impact ionization of the inner and outer shells of argon

First order distorted wave Born approximation (DWBA) triple differential cross sections are reported for low-energy electron-impact ionization of the inner 3s and outer 3p shells of argon. Previous DWBA works have demonstrated that experiment and theory are not in accord for low energy ionization of inert gases and here we investigate the importance of exchange scattering. Different approximations for treating exchange scattering are investigated. It is shown that exchange scattering is particularly important for 3s ionization. Even with a proper treatment of exchange, the first order calculations are still not in satisfactory agreement with experiment. Consequently higher order effects will have to be included to achieve a satisfactory description of the low-energy ionization process. We also investigated both the Hartree-Fock and optimized potential methods for calculating atomic wavefunctions and static potentials and found that both methods produced almost the same cross sections.

Time-Dependent Density Functional Calculation of e-H Scattering

Phase shifts for single-channel elastic electron-atom scattering are derived from time-dependent density functional theory. The H- ion is placed in a spherical box, its discrete spectrum found, and phase shifts deduced. Exact exchange yields an excellent approximation to the ground-state Kohn-Sham potential, while the adiabatic local density approximation yields good singlet and triplet phase shifts.