In the Born-Oppenheimer approximation, the positions of the nuclei are taken to be fixed so that the internuclear distances are constant. This is a sensible approximation because the massive nuclei are essentially immobile in comparison with light electrons. We first choose geometry (with fixed internuclear distances) for a molecule and solve the Schrödinger equation for that geometry. We then change the geometry slightly and solve the equation again. This continues until we find an optimum geometry with the lowest energy. It should be noted that without the Born-Oppenheimer approximation we would lack the concept of a potential energy surface: The PES is the surface defined by Eel over all possible nuclear coordinates. We would further lack the concepts of equilibrium and transition state geometries, since these are defined as critical points on the PES; instead we would be reduced to discussing high-probability regions of the nuclear wavefunctions.

Another approximation (the independent electron approximation) commonly made is that the wavefunction, R, can be written as a product of one-electron functions. The one-electron wavefunctions are called molecular orbitals - the molecular equivalent of atomic orbitals. Each molecular orbital then is expressed as a combination of the atomic orbitals from the atoms that make up the molecule. For example, the simplest molecular orbital function for the H2 molecule is written as c11s1 + c21s2, where 1si is a hydrogen 1s atomic orbital function and ci is a parameter. This method is called LCAO-MO theory for Linear Combination of Atomic Orbitals - Molecular Orbital Theory.

As an example, the two lowest energy molecular orbitals of the H2 molecule are shown here. These can be thought of as combinations of 1s orbitals from the two hydrogen atoms. The molecular orbital on the left is made by two H atom 1s orbitals combining constructively. This is a bonding MO because it helps hold the molecule together. The molecular orbital on the right is due to the destructive interference of the two 1s orbitals and is said to be antibonding.

Although molecular orbitals are written as combinations of atomic orbitals from the atoms in the molecule, molecular orbitals are not atomic orbitals.They are analogous to atomic orbitals, but instead of being defined for atoms, molecular orbitals are characteristic of the molecule as a whole.

Because of the large number of particles in a molecule (benzene, for instance, has 12 nuclei and 78 electrons) computer programs are used to do the calculations necessary for the solution of the Schrödinger equation. These calculations involve an enormous number of difficult integrals for large molecules. Ab Initio computational methods solve all of these integrals without approximation. Ab Initio methods are the most reliable for small and medium-sized molecules, but are prohibitively time-consuming for large molecules (20 atoms or so for PCs; around 100 atoms if workstations are available).

For larger molecules, semiempirical methods have been developed which ignore or approximate some of the integrals used in Ab Initio methods. To compensate for neglecting the integrals, the semiempirical methods introduce parameters based on molecular data.

Semiempirical molecular orbital theory methods have been developed which ignore or approximate some of the integrals used in the solution of the Schrödinger equation. To compensate for neglecting the integrals, the semiempirical methods introduce parameters based on molecular data. The most reliable semiempirical methods (PM3 and AM1) are used for predicting heats of formation, ground state geometries, and ionization potentials. The ZINDO method is also commonly used, which does a good job predicting the visible-UV bands for molecules containing hydrogen and first or second period elements.