General Theory

The orbital basis used throughout this chapter is 6-31+G*. This is small enough to allow the study of large molecules, yet also has enough complexity to contain the general features required of a basis set [100]. The G2 geometries are the MP2/6-31G* geometries, and zero-point vibrational corrections are calculated from Hartree-Fock harmonic frequencies. From the original G2 set the two excited states, N₂⁺ and SH₂⁺, have been removed. The atomization energy and proton affinity of H₂ and the ionization potentials of inert gas atoms have been added. Also the electron affinity of the H atom has been excluded, as the 6-31+G* basis set does not place a diffuse orbital on hydrogen.

The quality of a functional is judged by the root-mean-square (RMS) deviation of the computed results from the experimental values for the 129 data points. This requires the evaluation of self-consistent energies on 150 atoms and molecules. The RMS deviation is minimized with respect to parameters included in the functionals. Linear combinations of different functionals and combinations of the same functional with different parameters are considered. This optimization can be carried out in two ways. In the first, termed `internal' optimization, the full functional is written as a linear combination of component functionals
$$\begin{align}E_{xc} = \int \sum_{i} c_{i} f_{i}(\rho_{\alpha},\rho_{\beta},\nabla \rho_{\alpha},\nabla \rho_{\beta})\, d{\bf r} \end{align}$$
with adjustable coefficients c_i. This functional is then used to calculate the 150 self-consistent energies, leading to an RMS deviation from experiment, which is minimized with respect to the c_i. The Hartree-Fock exchange can be included by adding another term to this sum with an additional c coefficient. This is the conventional mixing method introduced by Becke [119].

The second method used is termed `external' optimization. Here the 150 energies are calculated for each of the component functionals
$$\begin{align}E_{xc}^{i} = \int f_{i}(\rho_{\alpha},\rho_{\beta},\nabla \rho_{\alpha},\nabla \rho_{\beta})\, d{\bf r}. \end{align}$$
Note that, if there are n functionals, this will require the 150n single point Kohn-Sham calculations. The energies for each functional can be arranged as n vectors ${\bf E}^{i}$ and then combined, using coefficients c<sub>i</sub>, to give a single set of 150 energies as a vector
$$\begin{align}{\bf E}({\bf c}) = \sum_{i}^{n} c_{i} {\bf E}^{i}. \end{align}$$
This mixture is, in effect, a `linear combination of model chemistries' defined by the coefficients c_i. As before, the RMS deviation from experiment is minimized with respect to the c_i.

While the second type could be used as a model, the internal optimization is clearly preferable as only one calculation is required (compared with n calculations). However optimization by the `external' method is much faster as it only involves quadratic minimization of the coefficients. Thus the external optimization can be used first as a pointer to worthwhile candidates for internal optimization.