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Oxford University Press eBooks

The time-dependent Kohn–Sham scheme

Carsten A. Ullrich

Published Dec 22, 2011• 1 citation
Publisher

Abstract

This chapter introduces the time-dependent Kohn-Sham scheme. The first section shows how an interacting time-dependent many-body system can be replaced by an auxiliary noninteracting system which reproduces the exact time-dependent density. This noninteracting system is called the Kohn-Sham system. It features an effective single-particle potential, consisting of external, Hartree, and exchange-correlation potential. The Kohn-Sham formalism is then generalized to spin-dependent systems. The exchange-correlation potential needs to be approximated in practice. Of particular importance is the adiabatic approximation, which uses any static exchange-correlation potential and evaluates it with time-dependent densities. The last two sections deal with practical issues related to the numerical implementation of the time-dependent Kohn-Sham scheme, namely, the meaning of self-consistency and unitary time propagation. The Crank-Nicholson algorithm is introduced, and important practical concepts such as the predictor-corrector scheme and absorbing boundary conditions are discussed.

DOI
10.1093/acprof:oso/9780199563029.003.0004

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The Journal of Chemical Physics•Aug 10, 2004•600 citations
This chapter introduces the time-dependent Kohn-Sham scheme. The first section shows how an interacting time-dependent many-body system can be replaced by an auxiliary noninteracting system which reproduces the exact time-dependent density. This noninteracting system is called the Kohn-Sham system. It features an effective single-particle potential, consisting of external, Hartree, and exchange-correlation potential. The Kohn-Sham formalism is then generalized to spin-dependent systems. The exchange-correlation potential needs to be approximated in practice. Of particular importance is the adiabatic approximation, which uses any static exchange-correlation potential and evaluates it with time-dependent densities. The last two sections deal with practical issues related to the numerical implementation of the time-dependent Kohn-Sham scheme, namely, the meaning of self-consistency and unitary time propagation. The Crank-Nicholson algorithm is introduced, and important practical concepts such as the predictor-corrector scheme and absorbing boundary conditions are discussed.
The exact static and time-dependent Kohn-Sham (KS) exchange-correlation potential is extremely challenging to approximate as it is a local multiplicative potential that depends on the electron density everywhere in the system. The KS approach can be generalized by allowing part of the potential to be spatially nonlocal. We take this nonlocal part to be that of unrestricted Hartree-Fock theory. The additional local correlation potential in principle ensures that the single-particle density exactly equals the many-body density. In our case, the local correlation potential is predominantly nearsighted in its dependence on the density and hence an (adiabatic) local-density approximation to this potential yields accurate ground-state properties and real-time densities for one-dimensional test systems.
Exact Time-Dependent Kohn-Sham Potential for Interacting Few-Body SystemsTime-dependent density functional theory enables practical simulations of the dynamic many-electron systems, but one of the biggest obstacles to reliable application is the quality of the approximate potential.It is often difficult to determine whether ever-more sophisticated approximations properly include new physics, as there exist few benchmark exact potentials.Towards this ends, we have developed and tested a scheme to extract the exact (non-adiabatic) time-dependent Kohn-Sham potential for few body systems.We will present some examples on 1D model systems.The approach is general and can be used to back engineer high-level quantum mechanical simulations to gain insight into TDDFT on a broad scale.
The Kohn-Sham approach to time-dependent density-functional theory (TDDFT) can be formulated, in principle, exactly by invoking the force-balance equation for the density, which leads to an explicit expression for the exchange-correlation potential as an implicit density functional. It is shown that this suggests a reformulation of TDDFT in terms of the second time derivative of the density, rather than the density itself. The result is a time-local Kohn-Sham scheme of second-order in time whose causal structure is more transparent than that of the usual Kohn-Sham formalism. The scheme can be used to construct new approximations at the exchange-only level and beyond, and it offers a straightforward definition of the exact adiabatic approximation.
In the present thesis, a rigorous mathematical formulation of time-dependent density-functional theory for lattice systems is derived, serving a formally exact approach towards solving time-dependent many-particle Schrödinger problems. After introducing the necessary mathematical foundations, a one-to-one mapping from external potentials onto electronic densities obtained by solving the Schrödinger equation is introduced. Its properties and the related Banach spaces of scalar potentials and electronic densities are discussed in detail. It allows to precisely formulate the Kohn-Sham Iteration scheme, mapping a given many-particle to an effective single-particle Schrödinger problem, both generating the exact same density. Employing the Banach fixed point theorem, the Kohn-Sham Iteration scheme is proven to be convergent depending on the chosen initial density.
The State--Specific Kohn--Sham Density Functional Theory [arXiv:physics/0506037] is used to derive the Kohn-Sham exchange-correlation potential $\vxc$ and exchange-correlation energy $\Eco$ as explicit functionals of vsv_svs​ and ϕ1ϕ_1ϕ1​, where vsv_svs​ is the local, one-body potential from the Kohn--Sham equations, and ϕ1ϕ_1ϕ1​ is the spinless one-particle density matrix from the Kohn--Sham noninteracting state, say $|ϕ_1\ran$. In other words, $|ϕ_1\ran$ is the ground state eigenfunction of the noninteracting Schrödinger equation with the one-body potential vsv_svs​. For simplicity, we only consider noninteracting states that are closed-shell states and interacting states that are nondegenerate, singlet ground-states.
In this paper we address the problem of the numerical integration of the time-dependent Schrodinger equation i partial differential (t)phi=Hphi. In particular, we are concerned with the important case where H is the self-consistent Kohn-Sham Hamiltonian that stems from time-dependent functional theory. As the Kohn-Sham potential depends parametrically on the time-dependent density, H is in general time dependent, even in the absence of an external time-dependent field. The present analysis also holds for the description of the excited state dynamics of a many-electron system under the influence of arbitrary external time-dependent electromagnetic fields. Our discussion is separated in two parts: (i) First, we look at several algorithms to approximate exp(A), where A is a time-independent operator [e.g., A=-iDeltatH(tau) for some given time tau]. In particular, polynomial expansions, projection in Krylov subspaces, and split-operator methods are investigated. (ii) We then discuss different approximations for the time-evolution operator, such as the midpoint and implicit rules, and Magnus expansions. Split-operator techniques can also be modified to approximate the full time-dependent propagator. As the Hamiltonian is time dependent, problem (ii) is not equivalent to (i). All these techniques have been implemented and tested in our computer code OCTOPUS, but can be of general use in other frameworks and implementations.