MOLECULAR PROPAGATION THROUGH LEVEL CROSSINGS
3
The chemical physicists refer to such levels as being from different symmetry classes. To
leading order in e, these crossings do not affect the Born-Oppenheimer propagation. To
zeroth order in e, a Born-Oppenheimer wave packet associated with the level Ej± behaves as
though the Eg level were not there. However, as the wave packet moves through the cross-
ing, a correction term is produced that is first order in e. It is associated with the Eg level,
and once it is produced, it persists and propagates according to the Born-Oppenheimer ap-
proximation associated with the Eg level. In Section 5 we make precise statements of these
results and present detailed proofs.
The situation is dramatically different in the case of codimension 2, 3, and 5 crossings.
A Born-Oppenheimer wave packet originally associated with the Ej± level is split into two
nontrivial zeroth order components as it passes through the crossing. One of these compo-
nents is associated with the Ej± level and propagates according to the Born-Oppenheimer
approximation associated with the Ej± level. The other component is associated with the E$
level and propagates according to the Born-Oppenheimer approximation associated with the
Eg level. Precise statements and detailed proofs for the codimension 2, 3, and 5 crossings
are presented in Sections 6, 7, and 8, respectively.
The Hamiltonian for a molecular system with k nuclei and N K electrons has the
form
i= l J j=K+l J ij
Here Xj G H* denotes the position of the j t h particle, the mass of the j t h nucleus is Mj
(for 1 j K), the mass of the j
t h
electron is mj (for K + 1 j TV), and V^j is the
potential between particles i and j . For convenience we assume Mj 1 for 1 j K. We
set n = kl and let X = (Xi, X2, ... ,XK) G IR71 denote the nuclear configuration vector.
We decompose H(e) as
H(e) = -jAx+h(X). (1.2)
This defines the electronic Hamiltonian h(X) that depends parametrically on X.
Nowhere in this paper do we require the Hamiltonian to have the particular form (1.1).
We only require the Hamiltonian to be of the form (1.2).
Electron energy levels are the discrete eigenvalues E(X) of h(X). We assume through-
out the paper that h(X) satisfies the smoothness condition that the resolvent operator
(h(X) - i)~l be a Ck function of X for various values of A: 2. This forces the elec-
Previous Page Next Page