Semiempirical Electron Correlation in the Carbon Atom

An empirical method of including the correlation of the two valence electrons has been applied to the carbon atom, which has a ${\mathrm{}\left(1s\right)\mathrm{}}^2{\mathrm{}\left(2s\right)\mathrm{}}^2{\mathrm{}\left(2p\right)\mathrm{}}^2$ ground electronic configuration. The carbon atom was chosen as a test of the method because of the penetrating nature of the $2p$ orbital. The atom was taken as a two-electron system with each electron moving in some kind of effective potential. The choice of this effective potential determines the type of raidal function to use as well as the screening function of one of the valence electrons on the other. For each choice of effective potential, a correlation factor ($1+c{r}_{12}$) is inserted into the wave function of each of the multiplet members obtained as Clebsch-Gordan combinations of one-electron orbitals, and the values of $c$ are determined by the variation method. It is found that the ratio of multiplet spacings is very sensitive to the type of effective potential used and that the Hartree-Fock average of the configuration calculation gives the best results. In this calculation, the use of the correlated wave function gives a value for the ratio of the multiplet spacings of 1.34, compared with 1.43 obtained from the unrestricted Hartree-Fock calculation and 1.13 from experiment.