Abstract In imaginary-time evolution, the same technique, Crank–Nicolson’s method, as for the analysis of real-time evolution is employed. For the verification using the model oscillators such as quantum well, Morse, and harmonic potentials, practical density functional theory (DFT) calculations are performed for hydrogen atoms and molecules. The convergence is confirmed to be faster under appropriate initial and boundary conditions. The results are consistent with the analytical solutions and those of other DFT calculations. Imaginary-time evolution is quite convenient to subsequent real-time real-space time-dependent DFT calculation. Although the operators are different, similar evolution is adopted. This technique is expected to unify into practical electronic state calculations for the self-consistent electronic structure and system’s response due to perturbation because calculations are in the same framework as the time-dependent equation.
Abstract In imaginary-time evolution, the same technique, Crank–Nicolson’s method, as for the analysis of real-time evolution is employed. For the verification using the model oscillators such as quantum well, Morse, and harmonic potentials, practical density functional theory (DFT) calculations are performed for hydrogen atoms and molecules. The convergence is confirmed to be faster under appropriate initial and boundary conditions. The results are consistent with the analytical solutions and those of other DFT calculations. Imaginary-time evolution is quite convenient to subsequent real-time real-space time-dependent DFT calculation. Although the operators are different, similar evolution is adopted. This technique is expected to unify into practical electronic state calculations for the self-consistent electronic structure and system’s response due to perturbation because calculations are in the same framework as the time-dependent equation.
In this paper, we propose an imaginary-time evolution method to obtain self-consistent static electronic states according to real-time TDDFT calculations. This method provides very good convergence accuracy and stability with respect to the ground state, as well as by a large number of computational steps due to obtain all states to evolve simultaneously. Also, we can get a diffusion equation from this method as same as the real-time TDDFT and can probably unify these time-evolution calculations in the same framework which is used in electronic structure calculations. In this study, we show that this method is effective in practical electronic state calculations. We will show several improvements and calculations with the computational process using the imaginary time evolution method in combination with the conventional DFT algorithm for practical use. The results of calculations of systems such as hydrogen atoms, hydrogen molecules and ethylene can show the computational efficiency and accuracy of this method, which probably improve traditional self-consistent electronic structure calculations for practical systems.
Since the early days of quantum mechanics, the wave function theory has proven to be very successful in describing many different quantum processes and phenomena. However, in many problems of quantum chemistry and solid-state physics, where the dimensionality of the systems studied is relatively high, ab initio calculations of the structure of atoms, molecules, clusters, and crystals, and their interactions are very often prohibitive. Hence, alternative formulations based on the direct use of the probability density, gathered under what is generally known as the density matrix theory [1], were also developed since the very beginning of the new mechanics. The independent electron approximation or Thomas-Fermi model, and the Hartree and Hartree-Fock approaches are former statistical models developed in that direction [2]. These models can be considered direct predecessors of the more recent density functional theory (DFT) [3], whose principles were established by Hohenberg, Kohn, and Sham [4,5] in the mid-1960s. According to this theory, the fundamental physical information about a many-body system is provided by single-particle densities in a three-dimensional space, which are obtained variationally within a time-independent framework. When compared with other previous formalisms, DFT presents two clear advantages: (1) it is able to treat many-body problems in a sufficiently accurate way and (2) it is computationally simple. This explains why it is one of the most widely used theories to deal with electronic structure-the electronic ground-state energy as a function of the position of the atomic nuclei determines molecular structures and solids, providing at the same time the forces acting on the atomic nuclei when they are not at their equilibrium positions. At present, DFT is used routinely to solve many problems in gas phase and condensed matter. Furthermore, it has made possible the development of accurate molecular dynamics schemes in which the forces are evaluated quantum mechanically ‘‘on the fly.’’ Nonetheless, DFT is a fundamental tool provided the systems studied are relatively large; for small systems, standard methods based on the use of the wave function render quite accurate results [6]. Moreover, it is also worth stressing that all practical applications of DFT rely on essentially uncontrolled approximations [7] (e.g., the local density approximation [4,5], the local spin-density approximation, or generalized gradient approximations [8]), and therefore the validity of DFT is conditioned to its capability to provide fairly good values of the experimental data.
The problem of calculating real-time correlation functions is formulated in terms of an imaginary-time partial differential equation. The latter is solved analytically for the perturbed harmonic oscillator and compared with the known exact result. The first order approximation for the short-time propagator is derived and used for numerical solution of the equation by a Monte Carlo integration. In general, the method provides a reformulation of the dynamic sign problem, and is applicable to any two-time correlation function including single-particle, density-density, current-current, spin-spin, and others. The prospects of extending the technique onto multi-dimensional problems are discussed.
We explore the usefulness of a quantum fluid dynamics (QFD) approach for quantitative electronic structure calculations of many-electron systems. By combining QFD and density functional theory, a single time-dependent nonlinear QFD equation can be derived. The equation is further transformed into a diffusion-type form by an imaginary-time evolution method, whose asymptotic solution reaches a global minimum and the many-body ground-state wavefunction. The time-dependent generalized pseudospectral method is extended to solve the diffusion equation in spherical coordinates, allowing optimal and nonuniform spatial discretization and accurate and efficient solution of the diffusion function in space and time. The procedure is applied to the study of electronic energies, densities and other ground-state properties of noble gas atoms (He, Ne, Ar, Kr, Xe). The results are in good agreement with other best available values. The method offers a conceptually appealing and computationally practical procedure for the treatment of many-electron systems beyond the Hartree-Fock level.
We discuss a new analytical approach to real-time evolution in quantum many-body systems. Our approach extends the framework of continuous unitary transformations such that it amounts to a novel solution method for the Heisenberg equations of motion for an operator. It is our purpose to illustrate the accuracy of this approach by studying dissipative quantum systems on all timescales. In particular, we obtain results for non-equilibrium correlation functions for general initial conditions. We illustrate our ideas for the exactly solvable dissipative oscillator and, as a non-trivial model, for the dissipative two-state system.