Generalised adiabatic connection in ensemble density-functional theory for excited states: example of the H₂ molecule

Abstract

A generalized adiabatic connection for ensembles (GACE) is presented. In contrast to the traditional adiabatic connection formulation, both ensemble weights and interaction strength can vary along a GACE path while the ensemble density is held fixed. The theory is presented for non-degenerate two-state ensembles but it can in principle be extended to any ensemble of fractionally occupied excited states. Within such a formalism an exact expression for the ensemble exchange-correlation density-functional energy, in terms of the conventional ground-state exchange-correlation energy, is obtained by integration over the ensemble weight. Stringent constraints on the functional are thus obtained when expanding the ensemble exchange-correlation energy through second order in the ensemble weight. For illustration purposes, the analytical derivation of the GACE is presented for the H 2 model system in a minimal basis, leading thus to a simple density-functional approximation to the ensemble exchange-correlation energy. Encouraging results were obtained with this approximation for the description in a large basis of the first 1 Σ + g excitation in H 2 upon bond stretching. Finally, a range-dependent GACE has been derived, providing thus a pathway to the development of a rigorous state-average multi-determinant density-functional theory.

Excited states from range-separated density-functional perturbation theory

“…We now give the behaviors of the zeroth+first-order energies with respect µ near the KS system (µ = 0) and near the physical system (µ → ∞), which are useful to understand the numerical results in Section IV. The total energies up to the first order of the perturbation theory are given by the expectation value of the full Hamiltonian over the zeroth-order wave functions in Eq (11). Using the Taylor expansion of the wave function Ψ…”

Calculating excitation energies by extrapolation along adiabatic connections

“…Indeed, several time-independent DFT approaches for calculating excitation energies exist and are currently being developed. These include ensemble DFT [7][8][9][10][11][12], ∆SCF [13][14][15][16][17] and related methods [18][19][20][21], or perturbation theory [22][23][24][25] along the standard adiabatic connection using the noninteracting Kohn-Sham (KS) Hamiltonian as the zero-order Hamiltonian.…”

Communication: Kohn-Sham theory for excited states of Coulomb systems

“…3-5), although several time-independent approaches have also been given to treat excited states within DFT. [6][7][8][9][10][11][12][13][14][15][16][17][18] The subspace theory of Theophilou 6 and its generalization by Gross, Oliveira, and Kohn 7 are complicated by the requirement that a whole ensemble of states has to be considered. Individual excitedstates can be targeted using time-independent approaches based on the adiabatic connection 19 or the constrained search.…”

“…(Nevertheless, approximate functionals for this bifunctional approach and the ensemble theory have been developed for and applied to atomic and molecular systems. [9][10][11]17,18 ) The time-independent Kohn-Sham theory that we give in this communication has the advantage that it employs a unifunctional.…”

“…The asymptotic decay of the excited-state density is governed by the ionization energy. As the excited-state density is the same for any α, the ionization energy is independent of α and constraint (17) follows. To prove relations (18)- (20), the derivation presented in Ref.…”