The Potential Energy of Reactive Systems

Abstract

The accurate theoretical study of any chemical reaction requires the solution of a many-body problem which involves the motion of all nuclei and electrons constituting the interacting atoms and molecules. We denote by x the set of nuclear coordinates and by z the set of electron coordinates. From the point of view of quantum mechanics, the system is described quite generally by a wave function Ψ(x,z,t) which depends on all these coordinatei and on time t. This function is a solution of the time-dependent Schrödinger equation

$$\widehat{{\text{H}}}\psi = i\hbar \frac{{\partial \psi }}{{\partial t}}$$

((1.I))

where the Hamiltonian

$$\widehat{{\text{H}}} = {{\widehat{{\text{T}}}}_{x}} + {{\widehat{{\text{T}}}}_{z}} + U\left( {x,z} \right)$$

((1a.I))

includes the kinetic energy operator of nuclei

$${{\hat{T}}_{x}} = - \frac{{{{\hbar }^{2}}}}{2}\sum\limits_{i} {\left| {\frac{1}{{{{m}_{i}}}}} \right|} \frac{{{{\partial }^{2}}}}{{\partial x_{i}^{2}}}$$

m_i being the corresponding nuclear masses, the kinetic energy operator of electrons

$${{\widehat{{\text{T}}}}_{x}} = - \frac{{{{\hbar }^{2}}}}{{2{{m}_{o}}}}\sum\limits_{k} {\left| {\frac{{{{\partial }^{2}}}}{{\partial {{z}_{k}}}}} \right|}$$

m_o being the electron mass, and the classical potential energy U(x,z) of the system of nuclei and electrons, which for an isolated system does not depend explicitly on time.