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Oxford University Press eBooks

Properties of the time-dependent xc potential

Carsten A. Ullrich

Published Dec 22, 2011• 0 citations
Publisher

Abstract

This chapter reviews the properties of the time-dependent exchange-correlation (xc) potential. It starts with a discussion of the meaning of the universal xc functional, and then covers a number of exact conditions: the adiabatic limit, the zero-force theorem, freedom of self-interaction, sum rules, and various scaling properties. Next, Galiean invariance of the xc potential and the harmonic potential theorem are discussed. The xc potential has the intriguing property that it has a memory of the previous history of the system. The ramifications of this memory dependence are discussed, as well as causality properties. Another property is the dependence on the initial state of the system. Then, a time-dependent variational principle is presented which yields the xc potential, provided that one pays attention to certain boundary conditions. Finally, it is shown that the time-dependent xc potential has a discontinuity upon change of particle number.

DOI
10.1093/acprof:oso/9780199563029.003.0006

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This chapter reviews the properties of the time-dependent exchange-correlation (xc) potential. It starts with a discussion of the meaning of the universal xc functional, and then covers a number of exact conditions: the adiabatic limit, the zero-force theorem, freedom of self-interaction, sum rules, and various scaling properties. Next, Galiean invariance of the xc potential and the harmonic potential theorem are discussed. The xc potential has the intriguing property that it has a memory of the previous history of the system. The ramifications of this memory dependence are discussed, as well as causality properties. Another property is the dependence on the initial state of the system. Then, a time-dependent variational principle is presented which yields the xc potential, provided that one pays attention to certain boundary conditions. Finally, it is shown that the time-dependent xc potential has a discontinuity upon change of particle number.
The key quantity in the formulation of linear-response theory within time-dependent density-functional theory is the so-called exchange-correlation (xc) kernel. This chapter discusses the properties of the xc kernel. In general, it is a complex and frequency-dependent object, and a functional of the ground-state density. Several exact properties such as the zero-force theorem, static and high-frequency limit, and the long-range behaviour are discussed. The xc kernel can be related to a time-dependent variational principle provided one pays attention to certain time-dependent boundary conditions to ensure causality. The adiabatic approximation of the xc kernel is discussed. Finally, the third section of the chapter reviews the xc kernel of the homogeneous electron liquid. Many of its exact properties are covered, such as the high- and low-frequency limit and the long-wavevector limit. Three widely used parametrizations are presented and the analytical behaviour in the complex frequency plane is investigated.
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.
The optimized effective potential (OEP) method is a formally exact approach to treat orbital-dependent exchange-correlation functionals. The first part of this chapter reviews the static OEP method. Orbital-dependent exchange-correlation functionals are implicit density functionals. The associated exchange-correlation potential is constructed via an integral equation. In particular, the exact exchange potential is defined in this way, starting from the Fock exchange energy functional. The resulting exchange potential has the correct asymptotic behaviour, and has a discontinuity as a function of particle number. Several properties and applications of the static OEP are reviewed. The second part of the chapter discusses the time-dependent OEP method. Again, the time-dependent exchange-correlation potential is constructed with an integral equation, which is derived from a variational principle. Two case studies are discussed, which illustrate the memory dependence and the discontinuity of the exchange correlation potential. The linear-response limit of the time-dependent OEP is introduced.
This chapter presents various techniques from many-body theory to construct time-dependent exchange-correlation (xc) functionals. The first section introduces a perturbation theory along the so-called adiabatic connection. This leads to a perturbative expansion of the xc potential, with exact exchange representing the lowest order. The following section introduces nonequilibrium Green's functions along the Keldysh contour. A Keldysh action principle is formulated which yields a formally exact representation of the time-dependent xc potential. As an alternative, the xc potential is expressed through the so-called Sham-Schlüter equation. The third section establishes a connection between the xc kernels of time-dependent density-functional theory in linear response and many-body perturbation theory. A diagrammatic expansion of the xc kernel is given. Finally, an expression for the xc kernel is derived starting from the Bethe-Salpeter equation. This xc kernel gives excellent agreement with experimental optical absorption spectra of insulators, including excitonic features.
In this work, nonadiabatic exchange-correlation (XC) potentials for time-dependent density-functional theory (TDDFT) for strongly correlated materials are derived in the limits of strong and weak correlations. After summarizing some essentials of the available dynamical mean-field theory (DMFT) XC potentials valid for these systems, we present details of the Sham–Schluter equation approach that we use to obtain, in principle, an exact XC potential from a many-body theory solution for the nonequilibrium electron self-energy. We derive the XC potentials for the one-band Hubbard model in the limits of weak and strong on-site Coulomb repulsion. To test the accuracy of the obtained potentials, we compare the TDDFT results obtained with these potentials with the corresponding nonequilibrium DMFT solution for the one-band Hubbard model and find that the agreement between the solutions is rather good. We also discuss possible directions to obtain a universal XC potential that would be appropriate for the case of intermediate interaction strengths, i.e., a nonadiabatic potential that can be used to perform TDDFT analysis of nonequilibrium phenomena, such as transport and other ultrafast properties of materials with any strength of electron correlation at any value in the applied perturbing field.
The exact exchange-correlation (xc) potential of time-dependent density functional theory has been shown to have striking features. For example, step and peak features are generically found when the system is far from its ground-state, and these depend nonlocally on the density in space and time. We analyze the xc potential by decomposing it into kinetic and interaction components and comparing each with their exact-adiabatic counterparts, for a range of dynamical situations in model one-dimensional two-electron systems. We find that often, but not always, the kinetic contribution is largely responsible for these features that are missed by the adiabatic approximation. The adiabatic approximation often makes a smaller error for the interaction component, which we write in two parts, one being the Coulomb potential due to the time-dependent xc hole. Non-adiabatic features of the kinetic component were also larger than those of the interaction component in cases that we studied when there is negligible step structure. In ground-state situations, step and peak structures arise in cases of static correlation, when more than one determinant is essential to describe the interacting state. We investigate the time-dependent natural orbital occupation numbers and find the corresponding relation between these and the dynamical step is more complex than for the ground-state case.