We discuss the multi-configuration
time-dependent Hartree (MCTDH) method for the approximation
of the time-dependent Schrödinger equation in quantum molecular dynamics.
This method approximates the high-dimensional nuclear
wave function by a linear combination of products of functions depending
only on a single degree of freedom. The
equations of motion, obtained via the Dirac-Frenkel
time-dependent variational principle,
consist of a coupled system of low-dimensional
nonlinear partial differential equations and ordinary differential equations.
We show that, with a smooth and bounded potential, the MCTDH equations
are well-posed and retain high-order Sobolev regularity globally in time, that is,
as long as the density matrices appearing in the method formulation remain invertible.
In particular, the solutions are regular enough to ensure
local quasi-optimality of the approximation and to admit an efficient numerical treatment.