The Kullback-Leibler divergence between two (discrete) probability distributions is defined as $ D_{KL}(P\|Q) = \sum_i p_i \log \frac{p_i}{q_i}, $ where $p_i$ is the probability that $P$ assigns to the event $i$, and $q_i$ is the probability assigned by $Q$.
I know that the quantity $D_{KL}(P\|Q) + D_{KL}(Q\|P)$ (symmetrised Kullback-Leibler divergence) is sometimes used, because it is symmetric and thus behaves more like a distance between the two distributions. But does anyone know of a case where the quantity $ \sum_i (p_i-q_i) \log \frac{p_i}{q_i} $ is used, and whether it has a standard name? I ask because it came up in some statistical mechanics work I'm doing and I want to know if it has an interpretation in terms of information theory, or any particularly interesting known properties.