One way to simplify the Schroedinger equation is to assume that the nuclei do not move, and hence have no kinetic energy. This is called the Born-Oppenheimer approximation. It leads to an electronic Hamiltonian:

The term for nuclear-nuclear repulsions (omitted above) becomes a constant, and must be added to the electronic energy to produce the total energy for the system:

We still have not simplified things enough for us poor, mathematically challenged chemists. The next step is to replace the many-electron wavefunction by a product of one-electron wavefunctions.

• The simplest such replacement, called the Hartree-Fock (or single determinant) wavefunction, involves a single determinant of products of one-electron functions, termed spin orbitals.

• Each spin orbital is written as a product of a space part, Y, describing the location of a single electron, and one of two possible (and opposite) spin parts, a or b.

• The space part is essentially a molecular orbital, which can be occupied by only two electrons, which must be of opposite spin.

The differential equations that arise from the Hartree-Fock approximation can be solved numerically. However, one further approximation is useful.

• This is the LCAO approximation, which suggests that the one-electron wavefunctions for many-electron molecules should look much like the wavefunctions for the hydrogen atom.

• That is, we consider molecular wavefunctions to be composed of atomic wavefunctions, combined linearly:
  

• Here the f are the atomic wavefunctions, called the basis functions, and the c are the coefficients of those basis functions in the linear combination.

• Thus, the acronym, LCAO, represents the Linear Combination of Atomic Orbitals.