4.1. Born-Oppenheimer approximation
Recall that the atomic orbital of an electron in an atom and the molecular orbital of an electron in a molecule are time-independent stationary states. In Section 2.4, Postulate 4 – Time-independent stationary states, we introduced the time-independent Schrödinger equation:
where 
is the non-relative Hamiltonian operator obtained by quantizing the classical energy in Hamilton form (first quantization), and it represents the total energy (
) of all its particles; 
electrons and 
nuclei. For a molecular system, the electric charge of two nuclei 
and 
are 
and 
with masses 
and 
. The position of the particles in the molecule is determined by using a laboratory (LAB) frame coordinate system, as shown in Figure 4.2, where the origin of the coordinate system is outside the molecule. The origin of the coordinate system can be placed anywhere in free space.
Figure 4.2 – LAB frame coordinates
The Hamiltonian...