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arXiv:cond-mat/0503635 · Physical Review B

Generic Galilean-invariant exchange-correlation functionals with quantum memory

Yair Kurzweil, Roi Baer

Published Jul 6, 2005• 29 citations
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Abstract

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.

DOI
10.1103/physrevb.72.035106

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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.
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.
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).
Using Landau Fermi-liquid theory we derive a nonlinear nonadiabatic approximation for the exchange-correlation (xc) vector potential defined by the xc stress tensor. The stress tensor is a local nonlinear functional of two basic variables: the displacement vector and the second-rank tensor which describes the evolution of momentum in a local frame moving with Eulerian velocity. For irrotational motion and the equilibrium initial state the dependence on the tensor variable reduces to that on the metrics generated by dynamical deformation of the system.
We propose a memory form of exchange-correlation potential vXC(r,t){v}_{\mathrm{XC}}(\mathbf{r},t)vXC​(r,t) for time-dependent interacting many-particle systems. Unlike previous memory-XC potentials, our vXC{v}_{\mathrm{XC}}vXC​ is not limited to the linear response regime. The proposed form of vXC{v}_{\mathrm{XC}}vXC​ is a generalized local-density approximation chosen so as to satisfy the nonlinear harmonic potential theorem and Newton's third law. For the case of the inhomogeneous electron gas, we give an explicit prescription for vXC{v}_{\mathrm{XC}}vXC​ based solely on an existing parametrization of the linear XC response kernel ${f}_{\mathrm{XC}}^{\mathrm{hom}}(n,\ensuremath{\omega})$ of the uniform gas. Application to quantum wells seems promising.
Frequency dependent exchange-correlation kernels for time-dependent density functional theory can be used to construct approximate exchange-correlation potentials. The resulting potentials are usually not translationally covariant nor do they obey the so-called zero-force condition. These two basic symmetry requirements are essential for using the potentials in actual applications (even in the linear regime). We provide two pragmatic methods for fully imposing these conditions for both linear and nonlinear regimes. As an example, we take the Gross and Kohn frequency dependent XC functional [Phys. Rev. Lett. 55, 2850 (1985)], correct it, and numerically test it on a sodium metal cluster. Violation of the basic symmetries causes instabilities or spurious low frequency modes.