The Hamiltonian is the total energy operator for a system, and is written as the sum of the kinetic
energy of all the components of the system and the internal potential energy. In an atom or
molecule, comprised of positive nuclei and negative electrons, the potential energy is simply that
due to the coulombic interactions present. Thus for the kinetic energy in a system of M nuclei and
N electrons:
And for the potential energy:
Since 
,
Within the Born-Oppenheimer approximation, we assume the nuclei are held fixed while the electrons
move really fast around them. (note: 
.) In this case, nuclear motion and
electronic motion are seperated. The last two terms can be removed from the total hamiltonian to
give the electronic hamiltonian, 
, since 
, and 
.
The nuclear motion is handled in a rotational/vibrational analysis. We will be working within the
B-O approximation, so realizing
we completly define the problem. Solving the electronic Schrödinger equation using this will give
the electronic structure of a molecular system at a fixed nuclear geometry.