The Potential Energy Surface

The complete mathematical description of a molecule, including both quantum mechanical and relativistic effects, is a formidable problem, due to the small scales and large velocities. However, for the purposes of this discussion, these intricacies are ignored and the focus is on general concepts, because molecular mechanics and dynamics are based on empirical data that implicitly incorporate all the relativistic and quantum effects. Since there is no complete relativistic quantum mechanical theory suitable for the description of molecules, this discussion starts with the nonrelativistic Schrödinger description:

Eq. 2-1:

where H is the Hamiltonian for the system,

is the wavefunction, and E is the energy. In general,

is a function of the coordinates of the nuclei (R) and of the electrons (r). Although it is quite general, this equation is too complex for any practical use, so approximations are made. Noting that the electrons are several thousand times lighter than the nuclei and therefore move much faster, Born and Oppenheimer (1927) proposed what is known as the Born-Oppenheimer approximation: the motion of the electrons can be decoupled from that of the nuclei, giving two separate equations. The first equation describes the electronic motion:

Eq. 2-2:

and depends only parametrically on the positions of the nuclei. Note that this equation defines an energy E(R), which is a function of only the coordinates of the nuclei. This energy is usually called the potential energy surface.

The second equation then describes the motion of the nuclei on this potential energy surface E(R):

Eq. 2-3:

The direct solution of Eq. 2-2 is the province of ab initio quantum chemical codes such as Gaussian, Cadpac, Hondo, GAMESS, DMol, and Turbomole. Semiempirical codes such as Zindo, MNDO, MINDO, MOPAC, and AMPAC also solve Eq. 2-2, but they approximate many of the integrals needed with empirically fit functions. The common feature of these programs, though, is that they solve for the electronic wavefunction and energy as a function of nuclear coordinates.

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Describe-System access Empirical Fit of the Surface