Pair Correlation Energies as Derived from the Analysis of Semiempirical Values of Correlation Energies of Atoms

Abstract

For the purpose of obtaining the semiempirical formulas for the pair correlation energy of various types of atoms and ions having the electron configurations 1s²2s^m2pⁿ(m=0, 1, 2; n=0, 1, ..., 6), semiempirical values of the correlation energies of such atoms are analyzed. It is pointed out that the correlation energy in the average of a configuration is more convenient than the correlation energy in each of the stationary states belonging to the configuration, for the purpose of this analysis. The correlation energy in the average of the configuration is shown to be equal to $E_{\mathrm{Av}}^{\mathrm{corr}},$ the average of the correlation energy in various states belonging to the configuration. The $E_{\mathrm{Av}}^{\mathrm{corr}}$ values are obtained for a wide variety of the 1s²2s^m2pⁿ atoms and regularities are found among these $E_{\mathrm{Av}}^{\mathrm{corr}}$ values. From the analysis of these regularities, it is found that the $E_{\mathrm{Av}}^{\mathrm{corr}}$ values of the 1s²2s^m2pⁿ atoms can be reproduced very well with an equation $$E_{\mathrm{Av}}^{\mathrm{corr}}{\mathrm{=\; C}}_1{\mathrm{+\; C}}_2{\mathrm{m\; +\; C}}_3{\mathrm{n\; +\; C}}_4{m}^2{\mathrm{+\; C}}_5{\mathrm{mn\; +\; C}}_6{n}^2{\mathrm{+\; C}}_7{m}^2{\mathrm{n\; +\; C}}_8{\mathrm{mn}}^2{\mathrm{+\; C}}_9{m}^2{n}^2{\mathrm{+\; C}}_{10}^m{\text{(m − 1)Z + C}}_{11}{\text{m(m − 1) nZ + C}}_{12}{\text{m (m − 1)Z}}^2$$ , where Z is the atomic number, along with the numerical values of the coefficients C_i's (i=1, 2,..., 12) determined from a least‐squares fitting of this equation to various empirical $E_{\mathrm{Av}}^{\mathrm{corr}}$ values. The concept of the average pair correlation energy of the configuration ε_kl is also proposed. Expressions for ε _kl's are given in terms of the pair correlation energies of various type in each of the states belonging to the configuration. It is shown that $E_{\mathrm{Av}}^{\mathrm{corr}}$ can be expressed in a simple form in terms of ε_kl's. From this expression and the semiempirical expression of $E_{\mathrm{Av}}^{\mathrm{corr}}$ mentioned above, a semiempirical formula for the average pair correlation energy is obtained for each type of electron pairs. It is found that these semiempirical formulas are in accord with the results of the Sinanoğlu theory for the ground and the excited states and of nonempirical calculations of pair correlation energies, rather than with those of a previous purely empirical analysis of the correlation energies by Allen, Clementi, and Gladney.