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arXiv:2408.16294 · Physical review. B./Physical review. B

Exchange kernel f x h ( q , ω ) of electron liquid from the variational principle of McLachlan

V. U. Nazarov, Vyacheslav M. Silkin

Published Sep 26, 2024• 2 citations
arXivHTMLPDF

Abstract

By minimizing, in the ${L}_{2}$ 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.

DOI
10.1103/physrevb.110.104310

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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 fxh(q,\ensuremathω){f}_{x}^{h}(q,\ensuremath{\omega})fxh​(q,\ensuremathω) of the homogeneous electron liquid. While the McLVP-based fxh(q,\ensuremathω){f}_{x}^{h}(q,\ensuremath{\omega})fxh​(q,\ensuremathω) 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.
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 fxh{f}_{x}^{h}fxh​ promising for applications.
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.
The key quantity in the formulation of linear-response theory within time-dependent density-functional theory is the so-called exchange-correlation (xc) kernel. This chapter discusses the properties of the xc kernel. In general, it is a complex and frequency-dependent object, and a functional of the ground-state density. Several exact properties such as the zero-force theorem, static and high-frequency limit, and the long-range behaviour are discussed. The xc kernel can be related to a time-dependent variational principle provided one pays attention to certain time-dependent boundary conditions to ensure causality. The adiabatic approximation of the xc kernel is discussed. Finally, the third section of the chapter reviews the xc kernel of the homogeneous electron liquid. Many of its exact properties are covered, such as the high- and low-frequency limit and the long-wavevector limit. Three widely used parametrizations are presented and the analytical behaviour in the complex frequency plane is investigated.
A framework for variational time-dependent current-density-functional theory beyond linear response and adiabatic approximations is given. The method allows incorporation of quantum memory into exchange-correlation functionals which respect Vignale’s “Galilean invariance” (GI) principle (Phys. Rev. Lett. 74, 3233 (1995)). The free variables of the theory are the electron velocity-field and density. Gauge-invariant time-dependent (TD) Kohn-Sham equations are derived. The action functionals allow memory effects and are expressed in terms Lagrangian quantities, to ensure GI. General closed-form expressions for the corresponding scalar and vector potentials are derived and applied within a specific functional, based on the homogeneous electron gas (HEG). The scalar potential depends on the future velocity field while the vector potential depends on both future and past velocity fields. These future dependencies naturally arise when non-local temporal effects are allowed and do not violate physical causality.
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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).