Generator coordinate method in time-dependent density-functional theory: Memory made simple

Abstract

The generator coordinate (GC) method is a variational approach to the quantum many-body problem in which interacting many-body wave functions are constructed as superpositions of (generally nonorthogonal) eigenstates of auxiliary Hamiltonians containing a deformation parameter. This paper presents a time-dependent extension of the GC method as a new approach to improve existing approximations of the exchange-correlation (XC) potential in time-dependent density-functional theory (TDDFT). The time-dependent GC method is shown to be a conceptually and computationally simple tool to build memory effects into any existing adiabatic XC potential. As an illustration, the method is applied to driven parametric oscillations of two interacting electrons in a harmonic potential (Hooke's atom). It is demonstrated that a proper choice of time-dependent generator coordinates in conjunction with the adiabatic local-density approximation reproduces the exact linear and nonlinear two-electron dynamics quite accurately, including features associated with double excitations that cannot be captured by TDDFT in the adiabatic approximation.

Physical signatures of discontinuities of the time-dependent exchange–correlation potential

“…One way to build memory in the functional has been outlined in reference [21]. Here, however, we will focus on aspects where memory effects are not expected to be crucial.…”

Configuration mixing within the energy density functional formalism: Removing spurious contributions from nondiagonal energy kernels

“…Indeed, the nuclear MR-EDF method is outside the scope of existing implementations of the Hohenberg-Kohn theorem, even of those dealing specifically with symmetry breaking issues [17,18,19,20]. As far as motivating MR-EDF calculations based on the GCM, hybrid approaches used in electronic systems that combine a density functional for diagonal energy kernels and (sometimes scaled) matrix elements of the bare Coulomb force for off-diagonal ones [21] have to be ruled out because of the different nature of the nuclear force and the more complex correlations it induces in the nuclear medium. There is a consensus among practitioners that the SR-EDF should be recovered for any diagonal energy kernel appearing in the MR-EDF and that the non-diagonal kernels should be extrapolated from the SR-EDF on the basis of the Generalized Wick Theorem (GWT) [22] which provides an efficient tool to evaluate the matrix elements of any operator between two different product states.…”

Exchange and correlation in molecular wire conductance: Nonlocality is the key

“…The GCM can also be used to describe correlated dynamics. 93 Here one writes the time-dependent GCM wavefunction, Ψ(t), as (8) where the time evolution of the coefficients, c(t; η), is governed by the time-dependent WH (TD-WH) equation…”

“…90,91 More recently, the GCM has been used to make connections between DFT and wavefunction-based approaches to correlation. 92,93 For a time independent problem, the fundamental idea is to write the target wavefunction, Ψ, as a linear transformation of a continuous set of states:…”