Time-dependent density-functional theory for strongly interacting electrons

Abstract

We consider an analytically solvable model of two interacting electrons that allows for the calculation of the exact exchange-correlation kernel of time-dependent density functional theory. This kernel, as well as the corresponding density response function, is studied in the limit of large repulsive interactions between the electrons and we give analytical results for these quantities as an asymptotic expansion in powers of the square root of the interaction strength. We find that in the strong interaction limit the three leading terms in the expansion of the kernel act instantaneously while memory terms only appear in the next orders. We further derive an alternative expansion for the kernel in the strong interaction limit on the basis of the theory developed previously [Phys. Chem. Chem. Phys. 18, 21092 (2016)] using the formalism of strictly correlated electrons in the adiabatic approximation. We find that the first two leading terms in this series, corresponding to the strictly correlated limit and its zero-point vibration correction, coincide with the two leading terms of the exact expansion. We finally analyze the spatial nonlocality of these terms and show when the adiabatic approximation breaks down. The ability to reproduce the exact kernel in the strong interaction limit indicates that the adiabatic strictly correlated electron formalism is useful for studying the density response and excitation properties of other systems with strong electronic interactions.

Strictly-correlated-electron approach to excitation energies of dissociating molecules

“…The ASCE kernel was shown to reproduce the so-called zeropoint oscillation part of the spectrum. The attainment of this exact result shows that the ASCE kernel becomes exact in the this regime as we also concluded from earlier work [7]. However, most current interest in molecular dissociation in chemistry is devoted to the complementary regime of large bond distance at intermediate interaction strength.…”

“…To correct these problems within the present formalism a natural way to proceed would be include ZPE and higher order kernels in the expansion of the Hxc kernel as was done in Ref. [7]. However, that work showed that the extra terms lead to worse approximation than just the ASCE approximation for low interaction strengths, as is typical for an asymptotic expansion.…”

“…The main question is, however, how reliable the asymptotic expansions in Eqs. (7) and (8) are for values close to the physically relevant interaction strenght λ = 1. Since the expansion is asymptotic, retaining higher order terms typically worsen the approximation unless we increase the value of λ.…”

“…We will not investigate this limit in detail; a homogenous QR at various interaction strength has been studied in detail in Ref. [7]. In Fig.2 we plot the ground state and the first few excited state KS orbitals.…”

“…The difference a − l+1 (q) − a + l (q) is exponentially small in the large q limit [13] a − l+1 (q) − a + l (q) = 2 4l+5 l! We note that in our previous work [7] we denoted a − l+1 by a − l in the asymptotic formula Eq. (A1) which amounts to a different labeling convention for the characteristic values.…”