  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.
