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Oxford University Press eBooks

The time-dependent optimized effective potential

Carsten A. Ullrich

Published Dec 22, 2011• 8 citations
Publisher

Abstract

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.

DOI
10.1093/acprof:oso/9780199563029.003.0011

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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.
A diagrammatic formalism for taking functional derivatives from the ab initio optimized effective potential, many-body perturbation theory (OEP-MBPT), exchange-correlation potentials is developed. Using this technique the frequency-independent (adiabatic approximation) kernel, corresponding to the second-order approximation OEP-MBPT(2) exchange-correlation potential is derived. The structure of the kernel is discussed and some initial numerical tests presented. The complexity of the computational procedure raises the question of what is warranted for a rigorous, ab initio approach that retains the one-particle structure of density functional theory compared to wave function approaches for excited states.
The integral equation for the time-dependent optimized effective potential (TDOEP) in time-dependent density-functional theory is transformed into a set of partial-differential equations. These equations only involve occupied Kohn-Sham orbitals and orbital shifts resulting from the difference between the exchange-correlation potential and the orbital-dependent potential. Due to the success of an analog scheme in the static case, a scheme that propagates orbitals and orbital shifts in real time is a natural candidate for an exact solution of the TDOEP equation. We investigate the numerical stability of such a scheme. An approximation beyond the Krieger-Li-Iafrate approximation for the time-dependent exchange-correlation potential is analyzed.
The memory effect in time-dependent density functional theory (TDDFT) is important in simulating many time-dependent physical processes, and its implementation in real time has been a longstanding challenge, thus limiting most of TDDFT applications to either adiabatic or linear-response regime. In this paper, we conduct the non-adiabatic calculations for a one-dimensional two-electron Helium model in a triplet state using the recently formulated Sturm-Liouville-type time-local equation for the time-dependent optimized effective potential (TDOEP) with the exact exchange functional, and the results agree with the exact time-dependent Schrodinger equation solutions. It is also found that the time-dependent dipole moment and probability density calculated from the TDOEP approach are more accurate than those from the adiabatic time-dependent Krieger-Li-Iafrate (TDKLI) approximation and the adiabatic local spin density approximation. Specifically, the non-adiabatic and memory-dependent terms in the time-local TDOEP equation correctly describe the time-dependent structure of exchange-correlation potential and yield the probability density evolution. These findings should provide important insights toward future studies on memory effects in TDDFT.
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.
Within density-functional theory a time-dependent perturbation theory is derived that yields exact orbital-dependent expressions for the time-dependent exchange-correlation potential in the form of a perturbation series. This leads to an exact time-dependent Kohn-Sham scheme based on perturbation theory. In first order an ``exchange-only'' Kohn-Sham procedure is obtained, that is a generalization of a recently presented time-dependent optimized potential method. The exact, in higher order, but quite complicated expressions for the exchange-correlation potential may serve as a starting point for the construction of approximate exchange-correlation potentials. An adiabatic connection or coupling constant path between a time-dependent physical system and its corresponding Kohn-Sham system is introduced.
A long-standing challenge in the time-dependent density functional theory is to efficiently solve the exact time-dependent optimized effective potential (TDOEP) integral equation derived from orbital-dependent functionals, especially for the study of nonadiabatic dynamics in time-dependent external fields. In this Letter, we formulate a completely equivalent time-local TDOEP equation that admits a unique real-time solution in terms of time-dependent Kohn-Sham and effective memory orbitals. The time-local formulation is numerically implemented, with the incorporation of exponential memory loss to address the unaccounted for correlation component in the exact-exchange-only functional, to enable the study of the many-electron dynamics of a one-dimensional hydrogen chain. It is shown that the long time behavior of the electric dipole converges correctly and the zero-force theorem is fulfilled in the current implementation.