I'm a physical chemist and have been trying to work my way through some of the literature around vibrational anharmonicity and second order vibrational perturbation theory (VPT2).
Unfortunately, while chemistry is generally good at defining conventional notations for things, this is not the case with VPT2, which makes following the equations in different papers a bit of an adventure...
I have been writing a chapter of my thesis which covers vibrations in molecules and have been using a lot equations I have obtained from different sources, many of which include sums over multiple . I would like to make the notation more uniform, however. At the moment, I have several equations with sum notations appearing as follows: $$ \sum_{i} \sum_{j} \sum_{k} \sum_{l} g_{ijkl}q_iq_jq_kq_l $$ $$ \sum_{i,j,k,l = 1}\phi_{ijkl} Q_iQ_jQ_kQ_l $$ $$ \sum_{ijkl} q_iq_jq_kq_l $$ While I have been able to find out what the first example means, there are no resources I can find which explain what the second and third examples mean. To me, it looks like they are all mean the same thing, and I would therefore like to simplify all sums in this chapter using the third notation, but I'm not 100% certain. Considering some mathematical symbols look equivalent (i.e., are simplifications of more cumbersome equations) but are actually referring to slightly different operations, I am wary of changing the nomenclature without expert advice.
So, are all of these different notations equivalent (i.e., summing over all combinations of $i$, $j$, $k$, $l$, from 1 to a final number [usually $3N$]) or do all have slightly different meanings?
These notations are indeed the exact same. The second and third ones are better in the case where you impose conditions like "$(i,j,k,l), i \neq k, k \neq l+2$" and are generally preferred over the longer one besides manipulations like sum exchanges and the like.