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Bulletin of the American Physical Society

Exact-Exchange Optimized Effective Potential and Memory Effect in Time-dependent Density Functional theory

Sheng-Lun Liao, Tak‐San Ho, HERSCHEL A. RABITZ, Shih‐I Chu

Published May 29, 2018• 0 citations
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Abstract

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.

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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.
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.
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.
Most applications of time-dependent density-functional theory (TDDFT) use the adiabatic local-density approximation (ALDA) for the dynamical exchange-correlation potential V(xc)(r,t). An exact (i.e., nonadiabatic) extension of the ground-state LDA into the dynamical regime leads to a V(xc)(r,t) with a memory, which causes the electron dynamics to become dissipative. To illustrate and explain this nonadiabatic behavior, this paper studies the dynamics of two interacting electrons on a two-dimensional quantum strip of finite size, comparing TDDFT within and beyond the ALDA with numerical solutions of the two-electron time-dependent Schrodinger equation. It is shown explicitly how dissipation arises through multiple particle-hole excitations, and how the nonadiabatic extension of the ALDA fails for finite systems but becomes correct in the thermodynamic limit.
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.
The exact time-dependent correlation energy in time-dependent density-functional theory is shown to sometimes become positive, which is impossible with most present TDDFT approximations. Both the correlation potential and energy can be strongly nonlocal in time. A new inequality is derived for the time-dependent exchange-correlation energy. The correlation energy appears to scale to a constant function of scaled time in the high-density limit. In the linear response regime, the correlation energy is shown to become purely adiabatic, but the correlation potential is generally nonadiabatic. The usefulness of the virial theorem as a test of numerical accuracy is demonstrated. All results are found or inspired by exact numerical solution of a simple model system (Hooke’s atom), and inversion of the corresponding Kohn–Sham equations.
We address and quantify the role of nonadiabaticity (``memory effects'') in the exchange-correlation (xc) functional of time-dependent density functional theory (TDDFT) for describing nonlinear dynamics of many-body systems. Time-dependent resonant processes are particularly challenging for available TDDFT approximations, due to their strong nonlinear and nonadiabatic character. None of the known approximate density functionals are able to cope with this class of problems in a satisfactory manner. In this work we look at the prototypical example of the resonant processes by considering Rabi oscillations within the exactly soluble two-site Hubbard model. We construct the exact adiabatic xc functional and show that (i) it does not reproduce correctly resonant Rabi dynamics, and (ii) there is a sizable nonadiabatic contribution to the exact xc potential, which turns out to be small only at the beginning and at the end of the Rabi cycle when the ground-state population is dominant. We then propose a ``two-level'' approximation for the time-dependent xc potential which can capture Rabi dynamics in the two-site problem. It works well both for resonant and for detuned Rabi oscillations and becomes essentially exact in the linear response regime.