A semiclassical correction for quantum mechanical energy levels

摘要

We propose a semiclassical method for correcting molecular energy levels obtained from a quantum mechanical variational calculation. A variational calculation gives the energy level (i.e., eigenvalue) as the expectation value of the molecular Hamiltonian 〈φ ? φ〉, where φ〉 is the trial wave function. The true (i.e., exact) eigenvalue E can thus be expressed as this variational result plus a correction, i.e., E= 〈φ ?φ〉 +ΔE, the correction being due to the lack of exactness of the trial wave function. A formally exact expression for ΔE is usually given (via L?wdin partitioning methodology) in terms of the Greens function of the Hamiltonian projected onto the orthogonal complement of φ〉. Formal treatment of this expression (using Brillouin-Wigner perturbation theory to infinite order) leads to an expression for ΔE that involves matrix elements of the Greens function for the unprojected, i.e., full molecular Hamiltonian, which can then be approximated semiclassically. (Specifically, the Greens function is expressed as the Fourier transform of the quantum mechanical time evolution operator, e~(-i?t/H), which in turn is approximated by using an initial value representation of semiclassical theory.) Calculations for several test problems (a one dimensional quartic potential, and vibrational energy levels of H_2 O and H_2 CO) clearly support our proposition that the error in the total eigenvalue E arises solely due to the semiclassical error in approximating ΔE, which is usually a small fraction of the total energy E itself.