The accurate evaluation of expectation values such as $I_1=〈\psi \left|\delta \right({\overrightarrow{\mathrm{r}}}_1\left)\right|\psi 〉$ and $I_{12}=〈\psi \left|\delta \right({\overrightarrow{\mathrm{r}}}_1-{\overrightarrow{\mathrm{r}}}_2\left)\right|\psi 〉$, where $\psi =\psi ({\overrightarrow{\mathrm{r}}}_1, {\overrightarrow{\mathrm{r}}}_2, \dots {\overrightarrow{\mathrm{r}}}_N)$ is an eigenfunction of a Hamiltonian $H$ is of interest for a variety of problems in atomic physics. Transformations are found to new forms $I_1^{\prime }$ and $I_{12}^{\prime }$, which are likely to give considerably more accurate values when, as is usually the case, only approximate wave functions are available. A successful test of the method is presented for the case of electron-electron and electron-nucleus contact interactions in helium. We give some identities which may be similarly useful in the evaluation of off-diagonal matrix elements of relativistic operators such as ${\gamma }_5\delta \left({\overrightarrow{\mathrm{r}}}_1\right)$, which arise from the parity-violating part of the neutral-current interaction and are important in the calculation of parity mixing in atoms.

• Received 5 June 1978

DOI:https://doi.org/10.1103/PhysRevA.18.2399