The adiabatic approximation I. The physical background of the Born-Handy ansatz

There are two current ways to evaluate the adiabatic corrections to a potential energy hypersurface: firstly a rigorous way in which one starts with the separation of the centre-of-mass motion and where one gets rather complicated expressions in terms of relative coordinates in a molecule-fixed frame, and secondly a pragmatic way, recently advocated by Handy et al., where one starts from a one-term Born ansatz in the laboratory system and arrives at a rather simple expression in terms of nuclear coordinates. In this note it is shown that the latter simple expression can also be obtained rigorously, that is by separating the centre-of-mass motion first. Why the pragmatic approach leads to a correct result, and why the specification of a molecule-fixed frame is not necessary for the evaluation of the adiabatic corrections, are analysed.