Classification and Normal Forms for Avoided Crossings of Quantum Mechanical Energy Levels

When using the Born–Oppenheimer approximation for molecular systems, one encounters a quantum mechanical Hamiltonian for the electrons that depends on several parameters that describe the positions of the nuclei. As these parameters are varied, the spectrum of the electron Hamiltonian may vary. In particular, discrete eigenvalues may approach very close to one another at “avoided crossings” of the electronic energy levels. We give a definition of an avoided crossing and classify generic avoided crossings of minimal multiplicity eigenvalues. There are six distinct types that depend on the dimension of the nuclear configuration space and on the symmetries of the electron Hamiltonian function. ∗ Supported in part by the National Science Foundation under Grant number DMS–9403401. In various situations, one encounters quantum mechanical Hamiltonians h(X) that depend on parameters X ∈ IR. For example, in the Born–Oppenheimer approximation of molecular physics, the electron Hamiltonian h(X) depends on the nuclear configuration parameters X. In such situations, the spectrum of h(X) may depend on X in a complicated way. In this paper we classify and study the local structure of “avoided crossings” of discrete eigenvalues of quantum mechanical Hamiltonian functions. These occur at values of the parameters X where two discrete eigenvalues EA(X) and EB(X) of h(X) approach very close to one another, but remain a positive distance apart. Avoided crossings are of interest because they may dramatically affect the physics of the situation. For example, in the time–dependent Born–Oppenheimer approximation the adiabatic approximation for the electrons can break down at an avoided crossing [ 3, 4 ], and this breakdown can provide a mechanism for certain chemical reactions to occur. There is no universal definition of what one means by an avoided crossing. In this paper we define one to be a “detuned crossing,” i.e., a situation where an actual eigenvalue crossing has been dismantled by a perturbation. We assume the Hamiltonian depends on the nuclear parameters X and an additional “detuning” parameter δ, such that h(X, 0) has a crossing, but that h(X, δ) does not for small δ > 0. Our precise definition is the following: Definition Suppose h(X, δ) is a family of self–adjoint operators with a fixed domain D in a Hilbert Space H, for X ∈ Ω and δ ∈ [0, α), where Ω is an open subset of IR. Suppose that the resolvent of h(X, δ) is a C function of X and δ as an operator from H to D. Suppose h(X, δ) has two eigenvalues EA(X, δ) and EB(X, δ) that depend continuously on X and δ and are isolated from the rest of the spectrum of h(X, δ). Assume Γ = {X : EA(X, 0) = EB(X, 0) } is a single point or non-empty connected proper submanifold of Ω, but that for all X ∈ Ω, EA(X, δ) 6= EB(X, δ) when δ > 0. Then we say h(X, δ) has an avoided crossing on Γ. In this definition, we have allowed the possibility that Γ is a manifold. The added generality has essentially no cost, and manifolds can arise in applications because of symmetries of some systems. For example, in the absence of external fields, electron Hamiltonians in molecular systems undergo similarity transformations as nuclear configurations are translated or rotated. The invariance of the eigenvalues under these transformations forces avoided 1 crossings to occur on manifolds of positive dimension. The motivation for this paper is the study of molecular propagation through avoided crossings [ 3, 4 ]. In that situation, the direction of propagation of the nuclei through an avoided crossing defines a special direction in the nuclear configuration space. Generically that direction has a non-trivial component in the hyperplane perpendicular to Γ at any particular point. Throughout the paper, we choose the X1 coordinate direction to be aligned with that component. The particular normal forms we obtain depend on having this distinguished X1 direction. Our classification depends on the codimension of Γ and the symmetries of h(X, δ). The codimension of Γ ⊂ IR is n −m, where m is the dimension of Γ, i.e., it is the minimum number of parameters that must be altered to move a generic point of IR near Γ onto Γ. Every Hamiltonian function h(X, δ) has a symmetry group G, which is the set of all (X, δ)– independent unitary and anti-unitary operators that commute with h(X, δ). An anti-unitary operator is complex conjugation composed with a unitary operator, and such operators have the feature of reversing time. In this paper, we consider only avoided crossings of energy levels EA(X, δ) and EB(X, δ) that are generic and have the minimal multiplicity allowed by the symmetry group. If G contains no anti-unitary operators, then each discrete energy level E(X, δ) of h(X, δ) is associated with a irreducible representation of G. In this case, the minimal multiplicity allowed is 1. If G contains anti-unitary operators, then each discrete energy level E(X, δ) of h(X, δ) is associated with a irreducible corepresentation of G [ 5, 7 ]. In this case, the unitary elements of G form a subgroup H of index 2, and each irreducible corepresentation belongs to one of three types [ 5, 7 ]. A corepresentation U of G is of Type I if its restriction UH to H is an irreducible representation of H. In this case, minimal multiplicity energy levels of h(X, δ) again have multiplicity 1. A corepresentation U of G is of Type II if UH is a direct sum of two equivalent irreducible representations of H, i.e., UH = D⊕D. Furthermore, for any anti-unitary K ∈ G, the corepresentation U can be cast in the form U(h) = ( D(h) 0 0 D(h) ) , U(K) = ( 0 −K K 0 ) , and U(Kh) = U(K)U(h), for all h ∈ H. Here K is an anti-unitary operator that satisfies K = −D(K) and KD(KhK)K = D(h) for all h ∈ H. In this case, minimal multi2 plicity energy levels have multiplicity 2. A corepresentation U of G is of Type III if UH is a direct sum of two inequivalent irreducible representations of H, i.e., UH = D ⊕ C. Furthermore, for any anti-unitary K ∈ G, the corepresentation U can be cast in the form U(h) = ( D(h) 0 0 C(h) )