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arXiv:1301.4645 · Physical Review B

Time-dependent effective potential and exchange kernel of homogeneous electron gas

V. U. Nazarov

Published Apr 19, 2013• 9 citations
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Abstract

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 ${f}_{x}^{h}$ promising for applications.

DOI
10.1103/physrevb.87.165125

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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 fxh(q,\ensuremathω){f}_{x}^{h}(q,\ensuremath{\omega})fxh​(q,\ensuremathω) 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.
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).
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.
By minimizing, in the L2{L}_{2}L2​ norm, the difference between the left- and the right-hand sides of the time-dependent Schr\"odinger equation, the variational principle of McLachlan (McLVP) [A. McLachlan, Mol. Phys. 8, 39 (1964)] provides a powerful tool for the generation of equations of motion. If the trial wave function is the Slater determinant, McLVP produces a temporally and spatially nonlocal exchange potential [V. U. Nazarov, Phys. Rev. B 87, 165125 (2013)]. We study the performance of the corresponding wave-vector and frequency-dependent exchange kernel ${f}_{x}^{h}(q,\ensuremath{\omega})$ of the homogeneous electron liquid. While the McLVP-based ${f}_{x}^{h}(q,\ensuremath{\omega})$ lacks correlations by construction, we find that it accurately accounts for exchange, reproducing features in the quantum Monte Carlo data, which the known constraint-based kernels miss. We argue that the complementary use of the McLVP- and the constraint-based exchange-correlation kernels will enhance the performance of the linear-response time-dependent density functional theory of the electron liquid.
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.
Time-dependent orbital-free density functional theory (DFT) is an efficient method for calculating the dynamic properties of large-scale quantum systems due to the low computational cost compared to standard time-dependent DFT. We formalize this method by mapping the real system of interacting fermions onto a fictitious system of noninteracting bosons. The dynamic Pauli potential and associated kernel emerge as key ingredients of time-dependent orbital-free DFT. Using the uniform electron gas as a model system, we derive an approximate frequency-dependent Pauli kernel. Pilot calculations suggest that space nonlocality is a key feature for this kernel. Nonlocal terms arise already in the second-order expansion with respect to unitless frequency and reciprocal space variable ($\frac{\ensuremath{\omega}}{q\phantom{\rule{0.16em}{0ex}}{k}_{F}}$ and q2kF\frac{q}{2\phantom{\rule{0.16em}{0ex}}{k}_{F}}2kF​q​, respectively). Given the encouraging performance of the proposed kernel, we expect it will lead to more accurate orbital-free DFT simulations of nanoscale systems out of equilibrium. Additionally, the proposed path to formulate nonadiabatic Pauli kernels presents several avenues for further improvements which can be exploited in future work to improve the results.
For the calculation of neutral excitations, time-dependent density functional theory (TDDFT) is an exact reformulation of the many-body time-dependent Schrödinger equation, based on knowledge of the density instead of the many-body wavefunction. The density can be determined in an efficient scheme by solving one-particle non-interacting Schrödinger equations—the Kohn–Sham equations. The complication of the problem is hidden in the—unknown—time-dependent exchange and correlation potential that appears in the Kohn–Sham equations and for which it is essential to find good approximations. Many approximations have been suggested and tested for finite systems, where even the very simple adiabatic local-density approximation (ALDA) has often proved to be successful. In the case of solids, ALDA fails to reproduce optical absorption spectra, which are instead well described by solving the Bethe–Salpeter equation of many-body perturbation theory (MBPT). On the other hand, ALDA can lead to excellent results for loss functions (at vanishing and finite momentum transfer). In view of this and thanks to recent successful developments of improved linear-response kernels derived from MBPT, TDDFT is today considered a promising alternative to MBPT for the calculation of electronic spectra, even for solids. After reviewing the fundamentals of TDDFT within linear response, we discuss different approaches and a variety of applications to extended systems.