The Calculation of Matrix Elements in Relativistic Quantum Mechanics

Employing a relativistic version of a hypervirial result, recurrence relations for arbitrary non-diagonal radial hydrogenic matrix elements have recently been obtained in Dirac relativistic quantum mechanics. In this contribution honoring Professor L\"owdin, we report on a new relation we have recently discovered between the matrix elements $<2| r^\lambda |1 >$ and $<2|\beta r^\lambda |1>$---where $\beta$ is a Dirac matrix and the numbers distiguish between different radial eigenstates--- that allow for a simplification and hence for a more convenient way of expressing the recurrence relations. We additionally derive another relation that can be employed for simplifying two center matrix element calculations in relativistic atomic or molecular calculations.