Page 272 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

10

Molecular structure

A molecule is composed of positively charged nuclei surrounded by electrons.
The stability of a molecule is due to a balance among the mutual repulsions of
nuclear pairs, attractions of nuclear±electron pairs, and repulsions of electron
pairs as modi®ed by the interactions of their spins. Both the nuclei and the
electrons are in constant motion relative to the center of mass of the molecule.
However, the nuclear masses are much greater than the electronic mass and, as
a result, the nuclei move much more slowly than the electrons. Thus, the basic
molecular structure is a stable framework of nuclei undergoing rotational and
vibrational motions surrounded by a cloud of electrons described by the
electronic probability density.
In this chapter we present in detail the separation of the nuclear and
electronic motions, the nuclear motion within a molecule, and the coupling
between nuclear and electronic motion.

10.1 Nuclear structure and motion
We consider a molecule with Ù nuclei, each with atomic number Z á and mass
M á (á ˆ 1, 2, ... , Ù), and N electrons, each of mass m e . We denote by Q the
set of all nuclear coordinates and by r the set of all electronic coordinates. The
positions of the nuclei and electrons are speci®ed relative to an external set of
coordinate axes which are ®xed in space.
^
The Hamiltonian operator H for this system of Ù ‡ N particles may be
written in the form

^ ^ ^
H ˆ T Q ‡ V Q ‡ H e (10:1)
^
where T Q is the kinetic energy operator for the nuclei

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