Atomic and ionic energy expressions obtained by scaling both the nuclear and electronic charges

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The differential Hellmann–Feynman and virial theorems were used in the scaling of both electronic and nuclear charges of atomic systems. We have obtained expressions for the energy, the inverse electron-nucleus distance, and the inverse electron–electron distance as series expansions in powers of 1/Z. The expansion coefficients depend on the number of electrons and the particular state considered. We show that the first two terms of the energy expansion can be exactly calculated in terms of hydrogen wave functions. Using a three-term-energy expression and fitting the third coefficient we obtain an agreement better than 0.5% with the experimental results of all 190 atomic structures with 2≤Z≤20 and 2≤n≤Z. Based on these calculations we propose a semiempirical energy expression which agrees within 2% with the same experimental results and with the formula of Politzer and Parr for atoms.