The calculation of adiabatic-connection curves from full configuration-interaction densities: Two-electron systems

Abstract

The Lieb formulation of density-functional theory is briefly reviewed and its straightforward generalization to arbitrary electron-electron interaction strengths discussed, leading to the introduction of density-fixed and potential-fixed adiabatic connections. An iterative scheme for the calculation of the Lieb functionals under the appropriate constraints is outlined following the direct optimization approach of Wu and Yang [J. Chem. Phys. 118, 2498 (2003)]. First- and second-order optimization schemes for the calculation of accurate adiabatic-connection integrands are investigated and compared; the latter is preferred both in terms of computational efficiency and accuracy. The scheme is applicable to systems of any number of electrons. However, to determine the accuracy that may be achieved, the present work focuses on two-electron systems for which a number of simplifications may be exploited. The procedure is applied to the helium isoelectronic series and the H(2) molecule. The resulting adiabatic-connection curves yield the full configuration-interaction exchange-correlation energies extrapolated to the basis-set limit. The relationship between the Kohn-Sham and natural orbitals as functions of the electron-electron interaction strength is explored in detail for H(2). The accuracy with which the exchange-correlation contributions to the modified local potential can be determined is discussed. The new accurate adiabatic-connection curves are then compared with some recently investigated approximate forms calculated using accurate full configuration-interaction input data. This study demonstrates that the adiabatic-connection integrand may be determined accurately and efficiently, providing important insights into the link between the Kohn-Sham and traditional quantum-chemical treatments of the exchange-correlation problem in electronic-structure theory.

Alternative Representations of the Correlation Energy in Density‐Functional Theory: A Kinetic‐Energy Based Adiabatic Connection

“…Nonetheless, it is clear that the PBE integrands and associated correlation energies tend towards too negative values with increasing Z, noting that the PBE correlation energy for Ne 8+ in this basis set is already 5 mE h below the estimated basis-set limit value of Ref. 14 For the beryllium isoelectronic series, a more pronounced failure is observed for the PBE approximation as Z increases-see, for example, Ne 6+ in the right-hand panel of Figure 1.…”

“…As well as providing a formal justification for these functionals, the AC can be used as a tool to study the behaviour of the exact functional-see, for example, Refs. [12][13][14] From this perspective, alternative models for the challenging exchange-correlation energy have been proposed [15][16][17][18] and tested against accurate ab initio models. The utility of the AC formalism in this context stems from the fact that it provides a direct and simple bridge between the Kohn-Sham model system of non-interacting particles (described by a single Slater determinant) and the complex physical interacting system (described by the full configuration-interaction wave function), at constant electronic density.…”

ACKS2: Atom-condensed Kohn-Sham DFT approximated to second order

“…This approach is analogous to Lieb's formulation of DFT. 97,98 The Legendre transform of the energy of a system of non-interacting fermions was already introduced in the adiabatic connection in DFT [128][129][130] and was recently also implemented in a numerical algorithm by Wu and Yang. 131 We are interested in the Kohn-Sham kinetic energy of the population-constrained density, ρ pop , which is always v-representable.…”

Excitation energies along a range-separated adiabatic connection

“…(10)] are then calculated using the FCI method. Several accurate ground-state calculations have been performed in the past along the standard adiabatic connection [62][63][64][65][66][67] and rangeseparated adiabatic connections [1,6,[67][68][69] for small atomic and molecular systems, but accurate calculations of excited-state energies along adiabatic connections are very scarce-see, however, Refs. 62, 70.…”

“…Eq. (6)] are determined at the full configuration-interaction (FCI) level using Lieb's Legendre-transform approach [61][62][63]. The excited-state energies of the long-range interacting Hamiltonian along the adiabatic connection [cf.…”