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arXiv:1209.2949 · Physical Review A

Analytic density functionals with initial-state dependence and memory

Michael Ruggenthaler, S. E. B. Nielsen, Robert van Leeuwen

Published Aug 21, 2013• 22 citations
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Abstract

We analytically construct the wave function that, for a given initial state, produces a prescribed density for a quantum ring with two noninteracting particles in a singlet state. In this case the initial state is completely determined by the initial density, the initial time derivative of the density and a single integer that characterizes the (angular) momentum of the system. We then give an exact analytic expression for the exchange-correlation potential that relates two noninteracting systems with different initial states. This is used to demonstrate how the Kohn-Sham procedure predicts the density of a reference system without the need of solving the reference system's Schr\"odinger equation. We further numerically construct the exchange-correlation potential for an analytically solvable system of two electrons on a quantum ring with a squared cosine two-body interaction. For the same case we derive an explicit analytic expression for the exchange-correlation kernel and analyze its frequency dependence (memory) in detail. We compare the result to simple adiabatic approximations and investigate the single-pole approximation. These approximations fail to describe the doubly excited states, but perform well in describing the singly excited states.

DOI
10.1103/physreva.88.022512

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We analytically construct the wave function that, for a given initial state, produces a prescribed density for a quantum ring with two noninteracting particles in a singlet state. In this case the initial state is completely determined by the initial density, the initial time derivative of the density and a single integer that characterizes the (angular) momentum of the system. We then give an exact analytic expression for the exchange-correlation potential that relates two noninteracting systems with different initial states. This is used to demonstrate how the Kohn-Sham procedure predicts the density of a reference system without the need of solving the reference system's Schr"odinger equation. We further numerically construct the exchange-correlation potential for an analytically solvable system of two electrons on a quantum ring with a squared cosine two-body interaction. For the same case we derive an explicit analytic expression for the exchange-correlation kernel and analyze its frequency dependence (memory) in detail. We compare the result to simple adiabatic approximations and investigate the single-pole approximation. These approximations fail to describe the doubly excited states, but perform well in describing the singly excited states.
The adiabatic approximation of time-dependent density-functional theory is studied in the context of nonlinear excitations of two-electron singlet systems. We compare the exact time evolution of these systems to the adiabatically exact one obtained from time-dependent Kohn-Sham calculations relying on the exact ground-state exchange-correlation potential. Thus, we can show under which conditions the adiabatic approximation breaks down and memory effects become important. The hydrodynamic formulation of quantum mechanics allows us to interpret these results and relate them to dissipative effects in the Kohn-Sham system. We show how the breakdown of the adiabatic approximation can be inferred from the rate of change of the ground-state noninteracting kinetic energy.
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.
We present a local control scheme to construct the external potential v that, for a given initial state, produces a prescribed time-dependent density in an interacting quantum many-body system. This numerical method is efficient and stable even for large and rapid density variations irrespective of the initial state and the interactions. It can at the same time be used to answer fundamental v -representability questions in density functional theory. In particular, in the absence of interactions, it allows us to construct the exact time-dependent Kohn-Sham potential for arbitrary initial states. We illustrate the method in a correlated one-dimensional two-electron system with different interactions, initial states and densities. For a Kohn-Sham system with a correlated initial state we demonstrate the interplay between memory and initial-state dependence as well as the failure of any adiabatic approximation.
The interacting and the Kohn-Sham static density-density response functions for different one-dimensional two-electron singlet systems are reconstructed numerically. From their inverse we obtain the exact static exchange-correlation kernel. This quantity represents the adiabatically exact approximation of the frequency-dependent exchange-correlation kernel that is crucial for time-dependent linear density-response theory. We investigate its performance for nonlocal perturbations and analyze its sum rule properties. We also compute the adiabatically exact transition energies that follow from the static kernel within linear-response theory.
When a system has evolved far from a ground-state, the adiabatic approximations commonly used in time-dependent density functional theory calculations completely fail in some applications, while giving qualitatively good predictions in others, and sometimes even quantitative predictions. It is not clearly understood why this is so, and developing practical approximations going beyond the adiabatic approximation remains a challenge. This paper explores three different lines of investigation. First, an expression for the exact time-dependent exchange-correlation potential suggests that the accuracy of an adiabatic approximation is intimately related to the deviation between the natural orbital occupation numbers of the physical system and those of the Kohn-Sham system, and we explore this on some exactly-solvable model systems. The exact expression further suggests a path to go beyond the adiabatic approximations, and in the second part we discuss a newly proposed class of memory-dependent approximations developed in this way. Finally, we derive a new expression for the exact exchange-correlation potential from a coupling-constant path integration.
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.