CCL: Measuring Instantaneous Correlation of Individual Orbitals

 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