# Most probable path in phase space for a radiationless transition in a molecule

*Alexey V. Sergeev and Bilha Segev*

## Abstract

We study a radiationless transition in a polyatomic molecule when
the electronic energy of an excited electronic state is
transferred to the vibrational degrees of freedom of the nuclei,
and when some nuclear coordinates change abruptly. This jump
between the donor energy surface to the acceptor one gives the
initial conditions for the subsequent dynamics on the acceptor
surface, and the partition of energy between competing accepting
modes. In the Wigner representation, the physical problem of
recognizing the accepting modes for a radiationless vibronic
relaxation reduces to the mathematical problem of finding the
maximum of a function of many variables under a constraint. The
function is the initial Wigner function of the nuclei and the constraint is
energy conservation. In a harmonic approximation for the potential surfaces, the
problem is equivalent to finding the distance from a given point
to a multidimensional ellipsoid. This geometrical problem is
solved in closed form. For nonharmonic potentials, the
optimization problem is solved perturbatively.

Text of the paper: PDF format (with figures), TeX file.

5 separate figure files: 1, 2, 3, 4, 5.

On-line constraint minimization of the function *W*(*x*,*y*)

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Designed by A. Sergeev.