Density‐matrix propagation driven by semiclassical correlation

Abstract

Methods based on propagation of the one-body reduced density-matrix hold much promise for the simulation of correlated many-electron dynamics far from equilibrium, but difficulties with finding good approximations for the interaction term in its equation of motion have so far impeded their application. These difficulties include the violation of fundamental physical principles such as energy conservation, positivity conditions on the density, or unchanging natural orbital occupation numbers. We review some of the recent efforts to confront these problems, and explore a semiclassical approximation for electron correlation coupled to time-dependent Hartree-Fock propagation. We find that this approach captures changing occupation numbers, and excitations to doubly-excited states, improving over TDHF and adiabatic approximations in density-matrix propagation. However, it does not guarantee N -representability of the density-matrix, consequently resulting sometimes in violation of positivity conditions, even though a purely semiclassical treatment preserves these conditions.

Density-Matrix Coupled Time-Dependent Exchange-Correlation Functional Approximations

“…Approximations in time-dependent RDMFT still struggle with much of the same problems that adiabatic approximations in TDDFT struggle with. 36 The point of this paper is instead to focus on TDDFT with KS propagation, whereρ 1 is used only to provide an approximation for v T C . Contrary to RDMFT,ρ 1 does not need to be N -representable as it is not intended to be a physical quantity, but instead its purpose is as a calculation tool for our KS system.…”

“…In this situation, the KS propagation is redundant, as all the information comes from the RDMFT. Approximations in time-dependent RDMFT still struggle with much of the same memory problems with which adiabatic approximations in TDDFT struggle . The point of this paper is to focus on TDDFT with KS propagation, where ρ̃ 1 is used only to provide an approximation for v C T .…”

The BBGKY hierarchy for ultracold bosonic systems

“…The Bogoliubov backaction method [59][60][61][62][63][64] as well as non-commuting cumulants [65,66] constitute alternative but conceptually similar approaches. Recently, novel approaches using semi-classical correlations [67] or solving a time-dependent variational optimization problem [68] have been pursued. At this point, it shall be noted that while there are numerous theoretical works on the BBGKY hierarchy and its truncation, the literature on the accuracy and numerical stability of this approach in dependence on the truncation order by explicit simulations is limited to the best of our knowledge [67][68][69][70][71][72][73].…”

“…Recently, novel approaches using semi-classical correlations [67] or solving a time-dependent variational optimization problem [68] have been pursued. At this point, it shall be noted that while there are numerous theoretical works on the BBGKY hierarchy and its truncation, the literature on the accuracy and numerical stability of this approach in dependence on the truncation order by explicit simulations is limited to the best of our knowledge [67][68][69][70][71][72][73]. The comprehensive study [69] unravels that instabilities as a consequence of the non-linear closure approximation can occur and lead to unphysical states, i.e.…”

The BBGKY hierarchy for ultracold bosonic systems: II. Applications

“…While there are numerous theoretical works on the BBGKY hierarchy and its truncation (see the references in [1] as well as e.g. [12] for an overview), the literature on the accuracy and stability of the truncated BBGKY equations of motion (EOM) in dependence on the truncation order by explicit numerical simulations is -to the best of our knowledge -limited [13][14][15][16][17][18][19]. Actually, most studies deal with fermions (for bosons, see [20][21][22] as well the BBGKY-related approaches [23][24][25][26][27][28][29][30]) and are based on the truncation of the BBGKY hierarchy after the second order.…”