New techniques for evaluating parity-conserving and parity-violating contact interactions

The accurate evaluation of expectation values such as I₁=〈ψ|δ(→r₁)|ψ〉 and I₁₂=〈ψ|δ(→r₁-→r₂)|ψ〉, where ${ψ}={ψ}({{{→}}{r}}₁, {{{→}}{r}}₂, {}{{{→}}{r}}N)$ is an eigenfunction of a Hamiltonian $H$ is of interest for a variety of problems in atomic physics. Transformations are found to new forms ${I}₁^{{'}}$ and ${I}₁₂^{{'}}$, 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 γ₅δ(→r₁), which arise from the parity-violating part of the neutral-current interaction and are important in the calculation of parity mixing in atoms.