Imaginary-Time Time-Dependent Density Functional Theory and Its Application for Robust Convergence of Electronic States

Abstract

Reliable and robust convergence to the electronic ground state within density functional theory (DFT) Kohn-Sham (KS) calculations remains a thorny issue in many systems of interest. In such cases, charge sloshing can delay or completely hinder the convergence. Here, we use an approach based on transforming the time-dependent DFT equations to imaginary time, followed by imaginary-time evolution, as a reliable alternative to the self-consistent field (SCF) procedure for determining the KS ground state. We discuss the theoretical and technical aspects of this approach and show that the KS ground state should be expected to be the long-imaginary-time output of the evolution, independent of the exchange-correlation functional or the level of theory used to simulate the system. By maintaining self-consistency between the single-particle wavefunctions and the electronic density throughout the determination of the stationary state, our method avoids the typical difficulties encountered in SCF. To demonstrate dependability of our approach, we apply it to selected systems which struggle to converge with SCF schemes. In addition, through the van Leeuwen theorem, we affirm the physical meaningfulness of imaginary time TDDFT, justifying its use in certain topics of statistical mechanics such as in computing imaginary time path integrals.

Imaginary-time time-dependent density functional theory for periodic systems

“…Imaginary time propagation is an alternative method for calculating the DFT ground state, and may be useful for large metallic systems or any others where the standard SCF has difficulty converging to the correct ground state. 9 We have implemented it for periodic systems by modifying the open source package Quantum ESPRESSO. Our implementation uses a plane-wave basis with the options of multiple k-points, DFT+U, collinear or non-collinear, and norm-conserving or ultrasoft pseudo-potentials.…”

“…al. 9 adjusted the time-step on the fly by decreasing it when the total energy increased and increasing it otherwise. Our experience with this variable time-step was that in some cases it would constantly decrease to effectively zero, never resulting in monotonically decreasing energy.…”

“…al. 9 as a more reliable method for generating the KS ground state in various cases of atomic clusters where the SCF iteration approach has difficulty converging to the KS ground-state. In this approach the KS orbitals are propagated in imaginary time with the KS Hamiltonian, while their orthonormalization is maintained.…”

Crystal-momentum-resolved contributions to multiple plateaus of high-order harmonic generation from band-gap materials

“…The convergence is further checked against time propagation using the Arnoldi-Lanczos method [65] with adaptive Krylov subspaces. The initial conditions for the TDDFT calculations, i.e., the fieldfree ground-state KS orbitals, are found via imaginary time propagation with orthogonalization in each time step [64,66]. We note that as in previous studies [49,50,53,55], the considered laser interaction does not cause significant changes to the density; therefore we also perform time propagation with the KS potential frozen to its ground-state form, and find that the frozen KS approach captures basically the same HHG features as the dynamic KS approach.…”

Crystal-momentum-resolved contributions to multiple plateaus of high-order harmonic generation from band-gap materials

“…The convergence is further checked against time propagation using the Arnoldi-Lanczos method [65] with adaptive Krylov subspaces. The initial conditions for the TDDFT calculations, i.e., the field-free ground-state KS orbitals are found via imaginary time propagation with orthogonalization in each time step [64,66]. We note that as in previous studies [49,50,53,55], the considered laser interaction does not cause significant changes to the density; therefore we also perform time propagation with the KS potential frozen to its ground-state form, and find that the frozen KS approach captures basically the same HHG features as the dynamic KS approach.…”