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Physical Review Letters

Time-Dependent Optimized Effective Potential

Carsten A. Ullrich, U. J. Gossmann, Eberhard K. U. Gross

Published Feb 6, 1995• 234 citations
Publisher

Abstract

Given an expression for the quantum mechanical action $A[{\ensuremath{\varphi}}_{j\ensuremath{\sigma}}]$ of an $N$-electron system as a functional of $N$ time-dependent spin orbitals, we present a method of constructing the variationally best local time-dependent single-particle potentials ${v}_{\ensuremath{\sigma}}(\mathrm{r}t)$ which, when inserted in time-dependent single-particle Schr\"odinger equations for the spin-up and spin-down electrons yield orbitals ${{\ensuremath{\varphi}}_{j\ensuremath{\sigma}}(\mathrm{r}t)}$ that make $A[{\ensuremath{\varphi}}_{j\ensuremath{\sigma}}]$ stationary. We also propose a simplification of this scheme leading to a time-dependent generalization of the static optimized effective potentials recently introduced by Krieger, Li, and Iafrate [Phys. Lett. A 146, 256 (1990)]

DOI
10.1103/physrevlett.74.872

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Given an expression for the quantum mechanical action A[\ensuremathφj\ensuremathσ]A[{\ensuremath{\varphi}}_{j\ensuremath{\sigma}}]A[\ensuremathφj\ensuremathσ​] of an NNN-electron system as a functional of NNN time-dependent spin orbitals, we present a method of constructing the variationally best local time-dependent single-particle potentials v\ensuremathσ(rt){v}_{\ensuremath{\sigma}}(\mathrm{r}t)v\ensuremathσ​(rt) which, when inserted in time-dependent single-particle Schr"odinger equations for the spin-up and spin-down electrons yield orbitals \ensuremathφj\ensuremathσ(rt){{\ensuremath{\varphi}}_{j\ensuremath{\sigma}}(\mathrm{r}t)}\ensuremathφj\ensuremathσ​(rt) that make A[\ensuremathφj\ensuremathσ]A[{\ensuremath{\varphi}}_{j\ensuremath{\sigma}}]A[\ensuremathφj\ensuremathσ​] stationary. We also propose a simplification of this scheme leading to a time-dependent generalization of the static optimized effective potentials recently introduced by Krieger, Li, and Iafrate [Phys. Lett. A 146, 256 (1990)]
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.
We derive a variation equation for a time-dependent effective potential that is local both in time and in space and show that this potential generates and propagates spin orbitals with which a many-particle, time-dependent, multiconfigurational wave function $\mathrm{\ensuremath{\Psi}}(t)$ is constructed. Within this approach, the wave function and the effective potential are determined simultaneously in a self-consistent manner both in time-independent and in time-dependent cases. We also derive an equation that determines a real-valued effective potential and show that the equation establishes a relation among the first-order and second-order reduced density matrices and the electron repulsion integrals in the spin-orbital representation. By introducing the first-order density equation, we show that the variation equation for the effective potential can be simplified for an exact wave function. We also show that we can derive an effective potential with which a ground-state wave function that fulfills the Brillouin-Brueckner condition is constructed and that we can derive the effective potential proposed by Slater, with which a ground-state wave function represented by the multiconfiguration expansion is calculated, when an additional constraint is imposed on the varitation equation.
A density-functional formalism comparable to the Hohenberg-Kohn-Sham theory of the ground state is developed for arbitrary time-dependent systems. It is proven that the single-particle potential $v(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}t)$ leading to a given vvv-representable density $n(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}t)$ is uniquely determined so that the corresponding map $v\ensuremath{\rightarrow}n$ is invertible. On the basis of this theorem, three schemes are derived to calculate the density: a set of hydrodynamical equations, a stationary action principle, and an effective single-particle Schr\"odinger equation.
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.
By minimizing the difference between the left- and the right-hand sides of the many-body time-dependent Schr\"odinger equation with the Slater-determinant wave function, we derive a nonadiabatic and self-interaction-free time-dependent single-particle effective potential, which is the generalization to the time-dependent case of the so-called localized Hartree-Fock potential. The new potential can be efficiently used within the framework of the time-dependent density-functional theory as we demonstrate by the evaluation of the wave vector and frequency-dependent exchange kernel ${f}_{x}^{h}(q,\ensuremath{\omega})$ of the homogeneous electron gas. This is found to be nonsingular and causal, and it satisfies the positiveness of the dissipation, in contrast to the earlier known kernel from the first-order perturbation theory, which makes our fxh{f}_{x}^{h}fxh​ promising for applications.
The previous time-dependent variation-perturbation theory, which is derived from the conventional time-dependent perturbation equations, contains two inconvenient features. When the perturbation becomes static, the wave function so obtained does not satisfy the time-dependent Schr\"odinger equation, and the perturbation equations cannot be reduced to the ordinary time-independent perturbation equations. To remove these difficulties, we introduce a perturbed time-dependent phase factor in deriving a new set of perturbation equations. We then construct the variation-perturbation functionals as in the previous theory; the perturbed wave functions are solved by minimizing these functionals. The same method is also applied to many-electron systems, using the coupled and uncoupled Hartree-Fock scheme.