Z−1-expansion calculation of two-particle correlations in atoms

Abstract

A calculation of the expectation value of the mass-polarization operator $〈{\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}}}_{1}\ifmmode\cdot\else\textperiodcentered\fi{}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}}}_{2}〉$ for the two-electron atom is made using the ${Z}^{\ensuremath{-}1}$ perturbation expansion in order to determine the usefulness of this approach in calculating two-particle correlations in atoms. Evaluation of the first-order contribution to $〈{\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}}}_{1}\ifmmode\cdot\else\textperiodcentered\fi{}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}}}_{2}〉$ yields a value in good agreement both with an earlier value determined by another method and with the value which may be deduced from calculations using very accurate wave functions. An interesting feature of the calculation is the finding that the summations are dominated by contributions from matrix elements which involve continuum states, with more than 85% of the total coming from these contributions. The second-order contribution is then obtained by demonstrating that both first- and second-order contributions are included in the first-order formula if accurate uncorrelated wave functions for the ground state are used in the matrix elements, rather than the bare hydrogenic orbitals. After observing that most of the greater accuracy of such wave functions arises from inclusion of electronic screening, the first-order calculation is repeated using simple screened hydrogenic orbitals in the matrix elements. The values of $〈{\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}}}_{1}\ifmmode\cdot\else\textperiodcentered\fi{}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}}}_{2}〉$ obtained by means of this simple calculation for various $Z$ values agree quite well with those calculated by Hart and Herzberg using accurate 20-term wave functions. The conclusion which is drawn from this calculation is that this method appears to be relatively simple and quite promising for calculating certain types of correlation effects, such as those resulting from double excitation or ionization of valence or inner-shell electrons. As examples of this, qualitative estimates are made of the behavior of the double photoionization cross sections of the remainder of the He isoelectronic sequence ($Z\ensuremath{\ge}3$) and of other closed-shell atoms.