Some new energy formulas for atoms and molecules

An exact formula for the nonrelativistic ground-state energy of an atom or molecule is derived: $$\mathrm{E=}\frac{1}{2}{\displaystyle \sum _{\mathrm{A}}^{}}{Z}_{\mathrm{A}}^2{\left(\frac{{\mathrm{\partial \; \phi }}_{\mathrm{A}}}{{\mathrm{\partial \; r}}_{\mathrm{A}}}\right)}_{r_{\mathrm{A}}\text{=0}}-\frac{1}{2}{\displaystyle \sum _{\mathrm{A}}^{}}{\displaystyle {\int }_{\mathrm{Z\prime }}^{{Z_{\mathrm{A}}}_{\mathrm{A}\text{=0}}}}{\mathrm{Z\prime }}_{\mathrm{A}}^2{\left[\frac{\partial }{{\mathrm{\partial Z\prime }}_{\mathrm{A}}}{\left(\frac{{\mathrm{\partial \phi }}_{\mathrm{A}}}{{\mathrm{\partial r}}_{\mathrm{A}}}\right)}_{r_{\mathrm{A}}\text{=0}}\right]}_N{\mathrm{dZ\prime }}_{\mathrm{A}}.$$ The function φ_A is defined for each nucleus A by φ_A = r_A V/Z_A, where Z_A is the nuclear charge and V is the total electrostatic potential produced by the electrons and nuclei at a point r_A measured from nucleus A. φ_A represents the screening of nucleus A by the electrons and other nuclei. The integration must be carried out over an isoelectronic path, the number of electrons being equal to the actual number N in the atom or molecule being considered. Equation (I) gives the molecular energy as a sum of atomiclike terms. It is shown that atomic energies can be calculated with remarkable accuracy with the equation $$\mathrm{E=}\frac{3}{7}{Z}^2{\mathrm{(\partial \; \phi \; /\; \partial \; r)}}_{\text{r=0}},$$ using Hartree-Fock values for (∂φ/∂r)_r=0. Equation (II) is identical in form with a relationship in the Thomas-Fermi theory of atoms. These equations emphasize the dependence of atomic and molecular energies upon the electrostatic potentials at the nuclei. This is further discussed and illustrated. A semiempirical formula for the energy of an atom as a power series in Z^1/3 is also presented.