I will take the indices $1$ and $2$ to correspond to the two states involved in the (Raman) transition of interest; these will be the $0$ and $1$ vibrational (or perhaps rotational) levels of the electronic ground state. Indices $3$ and above will correspond to (quoting HPB) “translational motion on higher electronic states”.

[*Note: I can see that assigning these indices to higher electronic states will conveniently keep the energy separations for transitions to greater than that of the $1 \leftrightarrow 2$ transitions, but it’s not so clear to me why I can ignore all the higher vibrational (or rotational) states.*]{}

Since I am interested only in the Raman transition, I specialise the above equations for the coefficients $c_n$, calculating $c_1$ and $c_2$ only, and assuming that the $d_{12} = \left< 1 \right| d \left| \right>$ dipole moment is zero. This means we will only be including transitions between indices $1$ and $2$ that [*go via one of the higher states*]{} $j \ge 3$, since we still allow $d_{1j}, ~d_{2j} \neq 0 ~~ \forall j \ge 3$. Further, I solve for the coefficients for the higher states in terms of $c_1$ and $c_2$, in an adiabatic approximation justified when $c_1$ 