Electron correlation has always been this intangible thing to me. You can talk about correlation energy, about multi-configurational wavefunctions, about differences between dynamic and static correlation. But what is actually going on? Let's tackle the problem with statistics! This is what we did in our new paper Statistical analysis of electronic excitation processes: Spatial location, compactness, charge transfer, and electron-hole correlation in J. Comp. Chem.
How do you quantify correlation? With a correlation coefficient! All you need is a function in two variables, then you can compute its covariance and normalize it by the standard deviations. A logical choice for such a function would be the 2-body density and given enough time and/or people doing it for me, I will look at that. For now we chose the 1-particle transition density matrix (1TDM) between the ground and excited state. This function describes the electron and hole quasi-particles in the exciton picture (see this post). And, among other things, we can compute the correlation coefficient between the electron and hole.
A good way to understand this new tool is in the case of symmetric dimers. Because of the symmetry all orbitals and states in such a system are delocalized over the whole system and no net charge transfer can be seen. But it is clear that the charge transfer states do not disappear: They are just arranged in symmetric linear combinations yielding the charge resonance states. On the other hand the local excitations are arranged in excitonic resonance states. Applying the new tool to this type of system shows that the difference is a correlation effect: Positive correlation yields bound excitonic states while negative correlation represents charge resonance
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