Molecular propagation through crossings and avoided crossings of electron energy levels

The time–dependent Born–Oppenheimer approximation describes the quantum mechanical motion of molecular systems. This approximation fails if a wavepacket propagates through an electron energy level crossing or “avoided crossing.” We discuss the various types of crossings and avoided crossings and describe what happens when molecular systems propagate through them. It is not practical to solve the time–dependent Schrödinger equation for molecular systems, but one can obtain useful information by using the time–dependent Born– Oppenheimer approximation. Although it is extremely useful, this approximation fails under certain circumstances. The simplest failures occur at crossings and avoided crossings of the electron energy levels. In this note we briefly summarize the results of [2, 3, 4, 5, 6]. These papers classify crossings and avoided crossings and analyze what happens when molecular systems propagate through them. ∗Partially Supported by National Science Foundation Grant DMS–9703751. The standard Born–Oppenheimer approximation exploits the smallness of the parameter ǫ, where ǫ is the ratio of the mass of an electron to the average of the masses of the nuclei in the molecular system. The largest value of ǫ that occurs in a real molecule is approximately 0.15. The time–dependent Schrödinger equation for a molecule can be written in the form i ǫ ∂ψ ∂t = − ǫ 2 ∆X ψ + h(X)ψ, (1) where h(X) is a family of self-adjoint operators on the electron Hilbert space Hel that depends paramterically on the nuclear configuration variable X ∈ IR. The variable X describes the positions of all the nuclei, and the dimension n is 3k, where k is the number of nuclei. The Born–Oppenheimer approximation [1] provides an algorithm for approximately solving this equation for small values of ǫ. We briefly describe this algorithm in the simplest situation where the electron state is non-degenerate. The details can be found, e.g., in [1] or [3]. The first step is to solve the eigenvalue problem h(X) Φ(X) = E(X) Φ(X) for the electron Hamiltonian h(X) for each X. We assume E(X) ∈ IR and Φ(X) ∈ Hel are chosen to depend continuously on X. In elementary situations, E(X) is a simple, isolated eigenvalue of h(X) for each X. The second step is to solve for the semiclassical motion of the nuclei with the electron energy level E(·) playing the role of an effective potential. To do this, we solve Hamilton’s equations