Some new energy formulas for atoms and molecules

Abstract

An exact formula for the nonrelativistic ground-state energy of an atom or molecule is derived: E=12 ∑ A ZA2(∂ φA∂ rA)rA=0−12 ∑ A ∫ Z′ZAA=0 Z′A2[∂∂Z′A(∂φA∂rA)rA=0]NdZ′A. The function φA is defined for each nucleus A by φA = rA V/ZA, where ZA is the nuclear charge and V is the total electrostatic potential produced by the electrons and nuclei at a point rA measured from nucleus A. φA 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 E=37Z2(∂ φ / ∂ r)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 Z1/3 is also presented.