*From*: Billy McCann <thebillywayne]|[gmail.com>*Subject*: CCL: Measuring Instantaneous Correlation of Individual Orbitals*Date*: Sun, 31 May 2015 11:16:09 -0400

Sent to CCL by: Billy McCann [thebillywayne{=}gmail.com] Greetings All. This is a subject I've been considering for a while, but it seems I haven't a) found a way to express the problem to myself so that it becomes more clear to me and b) come across literature that deals with my line of questioning. If anyone can offer insight into this, it would be very much appreciated. As a background, I have some training in chemical physics, but am far from expert. So please bear with me if I expose my ignorance. :) I'd like to frame the discussion within the wavefunction interpretation of QM and canonical Hatree-Fock atomic orbitals and LCAO-MO level of theory. I'd like to, for now, leave aside density functional theory because I don't have much experience or insight into the nature of the exchange-correlation operators; I can't seem to get a systematic understanding of that particular operator in its various formulations. And it's this correlation energy which I'm curious about. That the operator contains both exchange, correlation, plus a correction to the kinetic energies of the Kohn-Sham orbitals confounds me even more when trying to understand it, not even mentioning double-hybrid DFA's. I know that brilliant scientists have worked on various density functional approximations, and I do not at all want to belittle their work. DFA is a great tools for physicists and chemists. Now, on to my questions. Regarding instantaneous, dynamical electron correlation, I understand that there are many ab initio methods which begin at the Hatree-Fock approximation, starting with a Slater determinant expanded to various numbers of basis functions, and then account for dynamical electron correlation in different ways, typically, from what I can understand, by the admixture of electronic states wherein n number of electrons have been promoted to higher energy orbitals. If I understand correctly, all methods begin from the HF approximation and correct for dynamical correlation by making a linear combination of Slater determinants by different methods. (Perhaps the electron propagator method and the use of Dyson orbitals represents an alternative approach that doesn't combine Slater determinants, but I'm unsure. I've read Ortiz's review and let's just say it's a little out of my depth. ;)) All of these methods measure the correlation energy of the entire system in question, i.e. the atom or molecule in question. But what I'm wondering about is the correlation energy of a *single* atomic or molecular orbital. Is it that comparing the HF orbital energy to, say, a corresponding orbital resulting from a CCSD(T) calculation would yield such an energy? I've pondered this question, but I've read others who say that this isn't entirely the case because HF does indeed account for some small degree of electron correlation, but only in an averaged way. (I think I remember reading this in Cramer's text.) Perhaps MC-SCF may provide such an answer, by measuring the coefficients of each determinant? So my question is two-fold: 1. How can the dynamical electron correlation energy of a single atomic or molecular orbital be measured? Can it even be done? 2. Is it possible to make a generalized statement such as, "Core electrons experience a greater degree of correlation because they are surrounded by more electrons," or "Valence electrons experience a greater degree of electron correlation because they are bound more loosely to the system, allowing their wavefunctions to fluctuate more freely,"? I'd appreciate any insight that anyone has or any references to the literature or textbooks. Also, if someone would like to reframe this question in terms of non-canonicalized HF orbitals or from a NAO/NBO viewpoint, that would be great as well. I hope I haven't embarrassed myself. Thanks for your attention, Billy Wayne -- Billy Wayne McCann, Ph.D. http://bwayne.sdf.org irc://irc.freenode.net:bwayne "There is nothing new under the sun." ~ Solomon