  We present a discussion of the consequences in perturbation theory when an
exact eigenfunctions and eigenvalues to to the zeroth order Hamiltonian $H_0$
cannot be found. Since the usual approximations such as projecting the
wavefunction on to a finite basis set and restricting the particle interaction
is a way of constructing an approximate zeroth order Hamiltonian $H_0'$ we will
here argue that the exact eigenfunctions and eigenvalues are always found for
$H_0'$. We will show that as long as the perturbative expansion does not depend
on any intrinsic properties of $H_0$ but only on knowing the exact
eigenfunctions and eigenvalues then any perturbative statement, such as origin
independence intensities, will be true for any $H_0'$ provided that $H_0'$ has
a spectrum. We will use this to show that the origin independence for the
intensities is trivially fulfilled in the velocity gauge but also can be
fulfilled exactly in the length gauge if an appropriate $H_0$ is chosen.
Finally a small numerically demonstration of the origin dependence of the terms
for the second-order intensities in both the length and velocity gauge is
undertaking to numerically illustrate the theoretical statements.
