Multiple Determinant Wavefunctions

The main deficiency of HF theory is the inadequate treatment of the correlation between motions of electrons. No account is made for the correlation between electrons of opposite spin. Electrons of the same spin are partially, but not completely, correlated. This leads to a HF energy above the exact non-relativistic value. This difference is defined to be the correlation energy [30]:
$$\begin{align}E_{corr} = E_{exact} - E_{HF}. \end{align}$$
Unfortunately, the energy change in a reactive chemical processes is often of the same magnitude as the correlation energy, and the correlation energy can change markedly for many chemical processes, especially those where the number of electron pairs change. Hence, HF theory performs well for isodesmic reactions and for locating equilibrium structures (bond lengths usually to within 0.01Å and angles with $1^{\circ}$) [31]. Vibrational frequencies are usually within 10%. For relative energies, however, more accurate calculations are often required.

The are a number of techniques that seek to improve on the HF wavefunction. The method of choice depends very much on the characteristics of the problem. Some of the desirable features of a method involving electron correlation are [32]:

1.

it should be well defined, giving a continuous potential surface and a unique energy for any nuclear configuration.

2.

it should be ``size consistent'': the energy of a sum of non-interacting fragments should be exactly the sum of separate calculations on the fragments.

3.

it should be exact when applied to a two-electron system.

4.

it should be efficient, computational cost scaling slowly with system size.

5.

it should be accurate enough to be an adequate approximation to the exact result.

6.

it should be variational: that is, the energy is an upper bound to the exact result.

Unfortunately, no current method satisfies all of the above criteria!