Variational estimates of iriq for q=-1,1,and 2 for atoms with Z from 2 to 18

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The utility of variational principles for the estimation of the expectation values of certain operators, so far established only for helium, has been tested for atoms with Z from 2 through 18 for the operators r̄qiriq for q=-1,1,2 using simple atomic wave functions. The trial wave function Ψt is taken to be a Hartree product of hydrogenic orbitals with different effective charges fixed through phenomenological approaches such as Slater's empirical screening rules. This form for Ψt is particularly useful for these exploratory calculations since all steps in the variational method can actually be carried out analytically. Results for r̄-1 and r̄ are obtained through neon, while for the more interesting case of r̄2 they are extended up through argon. The results for r̄2 are generally the least accurate and are especially sensitive to the choice of effective charges, since they depend almost entirely on the outermost electrons whereas the Ψt used represents an average description of the atom. The variational estimates of r̄2 represent a substantial improvement over the zeroth-order estimates and agree to within about 3 to 10% with the Hartree-Fock results. The simplicity and relative accuracy of these calculations generally support the utility of the variational principle and suggest future calculations with a better Ψt as a starting point. © 1974 The American Physical Society.

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Physical Review A

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