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arXiv:1808.00215 · Physical review. B./Physical review. B

Adiabatic generalized gradient approximation kernel in time-dependent density functional theory

Nisha Singh, Peter D. T. A. Elliott, Tashi Nautiyal, J. K. Dewhurst, Sangeeta Sharma

Published Jan 25, 2019• 34 citations
arXivHTMLPDF

Abstract

A complete understanding of a material requires both knowledge of the excited states as well as of the ground state. In particular, the low energy excitations are of utmost importance while studying the electronic, magnetic, dynamical, and thermodynamical properties of the material. Time-dependent density functional theory (TDDFT), within the linear regime, is a successful ab initio method to assess the electronic charge and spin excitations. However, it requires an approximation to the exchange-correlation (XC) kernel which encapsulates the effect of electron-electron interactions in the many-body system. In this work we derive and implement the spin-polarized XC kernel for semilocal approximation, the so-called adiabatic generalized gradient approximation (AGGA). This kernel has a quadratic dependence on the wave vector $\mathbf{q}$ of the perturbation, however the impact of this on the electron energy loss spectra (EELS) is small. We show that the AGGA generally worsens the spin-excitation spectra by overestimating the magnon energies and suppressing the intensity of spin waves.

DOI
10.1103/physrevb.99.035151

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A complete understanding of a material requires both knowledge of the excited states as well as of the ground state. In particular, the low energy excitations are of utmost importance while studying the electronic, magnetic, dynamical, and thermodynamical properties of the material. Time-dependent density functional theory (TDDFT), within the linear regime, is a successful ab initio method to assess the electronic charge and spin excitations. However, it requires an approximation to the exchange-correlation (XC) kernel which encapsulates the effect of electron-electron interactions in the many-body system. In this work we derive and implement the spin-polarized XC kernel for semilocal approximation, the so-called adiabatic generalized gradient approximation (AGGA). This kernel has a quadratic dependence on the wave vector q\mathbf{q}q of the perturbation, however the impact of this on the electron energy loss spectra (EELS) is small. We show that the AGGA generally worsens the spin-excitation spectra by overestimating the magnon energies and suppressing the intensity of spin waves.
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.
Abstract We formulate a rigorous method for calculating a nonadiabatic (frequency-dependent) exchange-correlation (XC) kernel appropriate for accurate description of both equilibrium and nonequilibrium properties of strongly correlated systems within the time-dependent density functional theory (TDDFT) via the charge susceptibility, which is in turn obtained from dynamical mean field theory (DMFT) based on the effective multi-orbital Hubbard model. Application to the simple case of the one-orbital Hubbard model already shows the importance of the nonadiabatic kernel as it leads to significant modification of the excitation spectrum—shifting the (adiabatic) peak and disclosing another that is reminiscent of the solution from DMFT. The impact of dynamical effects, naturally included through the nonadiabaticity of the XC kernel, becomes even more transparent in our consideration of the nonequilibrium charge-density response of a multi-orbital perovskite, YTiO 3 , to a perturbation by a femtosecond (fs) laser pulse. These initial results indicate that electron–electron correlations and nonadiabatic features may significantly affect the spectrum and nonequilibrium properties of strongly correlated systems. We also propose an algorithm for extension of the approach to non-linear response. The transparency and computational efficiency of this non-adiabatic TDDFT+DMFT approach opens the door to examination of the spectra and response of multi-orbital systems with many nonequivalent atoms—bulk material, films and nanostructures.
In time-dependent density-functional theory, an important quantity is the exchange-correlation kernel whose knowledge allows for the calculations of the excitation spectrum of electronic systems. In most applications, the exchange-correlation kernel is approximated by the adiabatic local-density approximation, ignoring the frequency dependence and nonlocality. To gain insight into the nature of the exact exchange-correlation kernel, we have considered a two-dimensional one-band Hubbard model. The calculated exact exchange-correlation kernel reveals a number of striking features. It has a weak energy dependence up to the main excitation energy. On the other hand, the exact kernel shows a strong energy dependence in energy regions, where there are many-body excitations that are not contained in the local-density approximation, even though these excitations have small weight. It is found that a static approximation can well reproduce the main excitation peaks, but the satellite structures arising from many-body interaction are unlikely to be accounted for with a static approximation.
The generator coordinate (GC) method is a variational approach to the quantum many-body problem in which interacting many-body wave functions are constructed as superpositions of (generally nonorthogonal) eigenstates of auxiliary Hamiltonians containing a deformation parameter. This paper presents a time-dependent extension of the GC method as a new approach to improve existing approximations of the exchange-correlation (XC) potential in time-dependent density-functional theory (TDDFT). The time-dependent GC method is shown to be a conceptually and computationally simple tool to build memory effects into any existing adiabatic XC potential. As an illustration, the method is applied to driven parametric oscillations of two interacting electrons in a harmonic potential (Hooke's atom). It is demonstrated that a proper choice of time-dependent generator coordinates in conjunction with the adiabatic local-density approximation reproduces the exact linear and nonlinear two-electron dynamics quite accurately, including features associated with double excitations that cannot be captured by TDDFT in the adiabatic approximation.
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.
We propose a nonadiabatic time-dependent spin-density functional theory (TDSDFT) approach for studying single-electron excited states and the ultrafast response of systems with strong electron correlations. The correlation part of the nonadiabatic exchange-correlation (XC) kernel is constructed by using exact results for the Hubbard model of strongly correlated electrons. We demonstrate that the corresponding nonadiabatic XC kernel reproduces the main features of the spectrum of the Hubbard dimer and the 2D, 3D and infinite-dimensional Hubbard models, some of which are impossible to obtain within the adiabatic approach. The formalism may be applied for ab initio examination of strongly correlated electron systems in- and out-of-equilibrium within the TDSDFT, extending it beyond the metallic and semiconductor structures with plasmons, excitons and other excitations.