Diabatic states via a diabatic Hamiltonian

A new class of diabatic states for atom-atom and atom-molecule collisions is obtained by formulating a diabatic Hamiltonian. The true Hamiltonian describing the collision system is decomposed by means of topologically defined projection operators into a diabatic Hamiltonian plus an interaction term. A pseudosymmetry operator, which commutes with the diabatic Hamiltonian, is constructed from these same projection operators. The diabatic states are simultaneous eigenstates of the diabatic Hamiltonian and the pseudosymmetry operator. They cross and maintain their character through the crossing region. Such level crossings do not constitute a violation of the noncrossing rule, because the diabatic states involved have different pseudosymmetry. The diabatic states are coupled by the interaction term of the Hamiltonian. A set of coupled equations of motion for the amplitudes of the diabatic states is derived from the time-dependent Schrödinger equation. The (${\mathrm{HeH}}_2$ ${)}^{+}$ triatomic molecular ion is considered as an example. A pseudoreflection operator is constructed from the projection operators, which reduces to the usual reflection operator in those geometries for which the potential energy has true reflection symmetry. Pseudoinversion symmetry is also considered for this molecular system, treating the ${\mathrm{H}}_2$ at small separations as a near He. In addition, a new configuration constant of the motion is constructed for multielectron systems which is not an extension of a conventional symmetry. The diabatic states obtained in this work predict a recently observed excitation of He to an n=2 state in ${\mathrm{H}}_2^{+}$ on He collisions.