A Time–Dependent Born–Oppenheimer Approximation with Exponentially Small Error Estimates

We present the construction of an exponentially accurate time–dependent Born– Oppenheimer approximation for molecular quantum mechanics. We study molecular systems whose electron masses are held fixed and whose nuclear masses are proportional to ǫ−4, where ǫ is a small expansion parameter. By optimal truncation of an asymptotic expansion, we construct approximate solutions to the time–dependent Schrödinger equation that agree with exact normalized solutions up to errors whose norms are bounded by C exp ( −γ/ǫ ) , for some C and γ > 0. ∗Partially Supported by National Science Foundation Grant DMS–9703751.