Most of our theoretical knowledge of molecular dynamics is derived from the time-
dependent Born-Oppenheimer approximation. The approximation is necessary because the
time-dependent Schrodinger equation for a molecule is not practical to solve, even numeri-
The accuracy of the Born-Oppenheimer approximation is based on the disparity be-
tween the masses of electrons and nuclei. The natural parameter governing the approxima-
tion is e, where e is the ratio of the mass of an electron to the average of the masses of the
nuclei in the system. In real molecules, e is small because the protons and neutrons that
make up nuclei are 1835 times more massive than electrons.
In the standard time-dependent Born-Oppenheimer approximation the electrons and
nuclei are treated separately, but in a way that respects the coupling of their motions. The
physical intuition that underlies the approximation is the following: Electrons move much
more rapidly than nuclei and quickly adjust their state to compensate for the relatively slow
nuclear motion. At any particular time, the electrons are approximately in a bound state as
though the nuclei were at fixed classical positions. This is the adiabatic approximation for
the electrons. The nuclear motion is well described by semiclassical approximations because
such approximations accurately describe the motion of particles of large mass. The nuclear
motion depends on the electronic motion because the effective potential energy function
that governs the nuclear evolution is the sum of the quantum energy of the electrons and the
nuclear-nuclear repulsion potential. Thus, the electronic evolution depends adiabatically on
the configuration of the nuclei, and the electrons produce an effective potential in which the
nuclei move semiclassically.
This physical intuition provides the groundwork for a mathematically rigorous asymp-
totic expansion for the quantum propagation of molecular systems [9,12,15,20]. The re-
sults that we require about this standard time-dependent Born-Oppenheimer expansion
are discussed in Section 4 of this paper. Section 3 contains the preliminary adiabatic and
semiclassical results that are required for Section 4.
Traditional Born-Oppenheimer approximations are valid only under the basic assump-
Received by the editor August 6, 1992.
Supported in part by National Science Foundation Grant DMS-9001635.