# Estimation of exponentially small overlap
integrals in the framework of phase space distributions

## Alexey V. Sergeev and Bilha Segev

*Department of Chemistry, Ben-Gurion University of the Negev*

POB 653, Beer-Sheva 84105, Israel

The integral $\int d\vec{q} \psi_0(\vec{q}) \psi_1(\vec{q})$ where
$\psi_0(\vec{q})$ is the ground state eigenfunction of a
Hamiltonian $H_0$, and $\psi_1(\vec{q})$ is the eigenfunction of
the highly excited state of a Hamiltonian $H_1$ corresponding to
some energy $E$ is estimated by replacing the eigenfunctions by
their Wigner functions, and by analyzing the phase space integral.
In the quasiclassical limit only a vicinity of one point where
$H_1(\vec{q},\vec{p})=E$ and where the Wigner function
$\rho_1(\vec{q},\vec{p})$ is maximal contributes to the integral
[1, 2]. Results are illustrated for Morse and Poescl - Teller
oscillators.

1. Segev B. and Heller E. J. Phase-space derivation of propensity
rules for energy transfer processes between Born-Oppenheimer
surfaces. J. Chem. Phys. 112, 4004 (2000).

2. Sergeev A. V. and Segev B. Most probable path in phase space
for a radiationless transition in a molecule. J. Phys. A: Math.
Gen. (2002).

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Designed by A. Sergeev.