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Galilean-Invariant Exchange-Correlation Functionals with Quantum Memory

Roi Baer, Yair Kurzweil

Published Nov 1, 2003• 0 citations
OpenAlex

Abstract

A framework for variational time-dependent current-density-functional theory beyond linear response and adiabatic approximations is given. The method allows incorporation of quantum memory into exchange-correlation functionals which respect Vignale’s “Galilean invariance” (GI) principle (Phys. Rev. Lett. 74, 3233 (1995)). The free variables of the theory are the electron velocity-field and density. Gauge-invariant time-dependent (TD) Kohn-Sham equations are derived. The action functionals allow memory effects and are expressed in terms Lagrangian quantities, to ensure GI. General closed-form expressions for the corresponding scalar and vector potentials are derived and applied within a specific functional, based on the homogeneous electron gas (HEG). The scalar potential depends on the future velocity field while the vector potential depends on both future and past velocity fields. These future dependencies naturally arise when non-local temporal effects are allowed and do not violate physical causality.

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A framework for variational time-dependent current-density-functional theory beyond linear response and adiabatic approximations is given. The method allows incorporation of quantum memory into exchange-correlation functionals which respect Vignale’s “Galilean invariance” (GI) principle (Phys. Rev. Lett. 74, 3233 (1995)). The free variables of the theory are the electron velocity-field and density. Gauge-invariant time-dependent (TD) Kohn-Sham equations are derived. The action functionals allow memory effects and are expressed in terms Lagrangian quantities, to ensure GI. General closed-form expressions for the corresponding scalar and vector potentials are derived and applied within a specific functional, based on the homogeneous electron gas (HEG). The scalar potential depends on the future velocity field while the vector potential depends on both future and past velocity fields. These future dependencies naturally arise when non-local temporal effects are allowed and do not violate physical causality.
Most present applications of time-dependent density functional theory use adiabatic functionals, i.e., the effective potential at time t is determined solely by the density at the same time. This paper discusses a method that aims to go beyond this approximation, by incorporating "memory" effects: the derived exchange-correlation potential will depend not only on present densities but also on the past. In order to ensure the potentials are causal, we formulate the action on the Keldysh contour for electrons in electromagnetic fields, from which we derive suitable Kohn-Sham equations. The exchange-correlation action is now a functional of the electron density and velocity field. A specific action functional is constructed which is Galilean invariant and yields a causal exchange-correlation vector potential for the Kohn-Sham equations incorporating memory effects. We show explicitly that the net exchange-correlation Lorentz force is zero. The potential is consistent with known dynamical properties of the homogeneous electron gas (in the linear response limit).
Today, most application of time-dependent density functional theory (TDDFT) use adiabatic exchange-correlation (XC) potentials that do not take into account non-local temporal effects. Incorporating such "memory" terms into XC potentials is complicated by the constraint that the derived force and torque densities must integrate to zero at every instance. This requirement can be met by deriving the potentials from an XC action that is Galilean in-variant (GI). We develop a class of simple but flexible forms for an action that respect these constraints. The basic idea is to formulate the action in terms of the Eularian-Lagrangian transformation (ELT) metric tensor, which is itself GI. The general form of the XC potentials in this class is then derived and the linear response limit is derived as well.
Today, most applications of time-dependent density-functional theory use adiabatic exchange-correlation (XC) potentials that do not take into account nonlocal temporal effects. Incorporating such ``memory'' terms into XC potentials is complicated by the constraint that the derived force and torque densities must integrate to zero at every instance. This requirement can be met by deriving the potentials from an XC action that is both rotationally and translationally invariant, i.e., Galilean invariant (GI). We develop a class of simple but flexible forms for an action that respect these constraints. The basic idea is to formulate the action in terms of the Eularian-Lagrangian transformation metric tensor, which is itself GI. The general form of the XC potentials in this class is then derived, and the linear response limit is derived as well.
The Runge-Gross [E. Runge and E. K. U. Gross, Phys. Rev. Lett. 52, 997 (1984)] action functional of time-dependent density-functional theory leads to a well-known causality paradox; that is, a perturbation of the electronic density in the future affects the response of the system in the present. This paradox is known to be caused by an inconsistent application of the Dirac-Frenkel variational principle. In view of the recent solutions to this problem, the action functional employed by Runge and Gross in their formulation of time-dependent density-functional theory is analyzed in the context of the Keldysh contour technique. The time-dependent electronic density and the concept of causality are extended to the contour. We derive a variational equation that obeys causality and relates the exchange-correlation potential with its kernel and the functional derivative of the exchange-correlation action functional with respect to the density. It is shown that the adiabatic local-density approximation is a consistent solution of this equation and that the time-dependent optimized potential method can also be derived from it. The formalism presented here can be used to find new approximation methods for the exchange-correlation potential and to avoid the causality dilemma.
This chapter discusses time-dependent current-density-functional theory (TDCDFT). It first asks under what circumstances one should choose the current density as basic variable, instead of the particle density. There are several reasons: a current-based formulation is more suitable for constructing functionals beyond the adiabatic approximation. Furthermore, the current is the natural variable to describe coupling to electromagnetic fields and vector potentials. The basic existence theorems of TDCDFT are proved, and several exact properties and the linear-response regime are discussed. The second half of the chapter deals with the so-called Vignale-Kohn functional, which is a local and frequency-dependent approximation to the exchange-correlation vector potential. The Vignale-Kohn functional is expressed in terms of viscoelastic stresses in the electron liquid. Several applications in the linear-response regime and in real time are presented. The Vignale-Kohn functional works well for describing dissipative effects in collective, plasmon-like electron dynamics.
Time-dependent density functional theory, though exact in theory, is in practice applied in an adiabatic approximation using exchange-correlation functionals with only local temporal dependence. Simultaneously, the exact correlation potential formally depends on, among other quantities, the time-history of electron densities. Here we develop a framework to learn neural network models of correlation functionals that feature explicit memory-dependence. Our framework includes two different approaches, both linked through their use of adjoint-based optimization. One approach decouples the learning of the functional from inversion to find ground truth values of the correlation potential. The other approach learns the functional directly without requiring inversion. We apply these methods to modeling the electron dynamics of two-electron systems in two spatial dimensions. In both cases, our methods yield correlation functionals with low test set propagation error, outperforming standard local density and generalized gradient approximation functionals by one to two orders of magnitude. Overall, our framework points at one strategy to move beyond the adiabatic approximation and develop memory-dependent correlation functionals that yield accurate propagation for excited state and/or non-equilibrium dynamics.