Variational-Splitting Time Integration of the Multi-Configuration Time-Dependent Hartree–Fock Equations in Electron Dynamics

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 H1 Sobolev norm and second-order convergence in the L2 norm are established under a solution regularity of H2. As a prerequisite, we show that the MCTDHF equations have a solution in this Sobolev space if the initial data has such regularity.