Variational-splitting time integration of the multi-configuration time-dependent Hartree-Fock equations in electron dynamics

Abstract

We discuss the numerical approximation of the solution to the multi-configuration time-dependent Hartree-Fock (MCTDHF) equations in quantum dynamics. The MCTDHF method approximates the high-dimensional wave function of the time-dependent electronic Schrödinger equation by an antisymmetric linear combination of products of functions depending only on three-dimensional spatial coordinates. The equations of motion, obtained via the Dirac-Frenkel time-dependent variational principle, consist of a coupled system of three-dimensional nonlinear partial differential equations and ordinary differential equations. We investigate the convergence properties of a time integrator based on a splitting of the Hamiltonian directly in the variational principle. First-order convergence in the H 1 Sobolev norm and second-order convergence in the L 2 norm are established under a solution regularity of H 2 . As a prerequisite, we show that the MCTDHF equations have a solution in this Sobolev space if the initial data has such regularity.

Error Analysis of Trigonometric Integrators for Semilinear Wave Equations

“…With the second-order Peano kernel K 2 (σ) = 1 2 σ(σ − 1) of the trapezoidal rule we further get Proof. As in the proof of Proposition 3.6, we start from (24). Using (14b) with β = 0, we estimate as in that proof…”

Multiconfiguration time-dependent Hartree-Fock treatment of electronic and nuclear dynamics in diatomic molecules

“…(13) and (8) has been the source of several theoretical and numerical studies over the past years. Splitting of the orbital equation by separating the onebody, stiff kinetic energy terms from the two-body, local, nonstiff potential terms has received considerable attention [51][52][53], but we do not pursue this avenue here.…”

Functionally-fitted energy-preserving integrators for Poisson systems

“…By the two-stage arguments used in [9,10,26,32,33], the proof of Theorem 3.3 will be divided into two parts. We will first show the proof of the lower-order error bounds in higher-order Sobolev spaces (i.e., −1 ≤ α ≤ 0) in Section 4, and then present the proof of the higher-order error bounds in lower-order Sobolev spaces (i.e., 0 < α ≤ 1) in Section 5.…”