CCL: Measuring Instantaneous Correlation of Individual Orbitals

 Sent to CCL by: JC Womack [jw5533]=[my.bristol.ac.uk]
 These are interesting thoughts!
 You asked for some literature references. I would suggest looking at
 this excellent review:
 Hättig, C., Klopper, W., Köhn, A. & Tew, D. P. Explicitly
 Correlated
 Electrons in Molecules. Chem. Rev. 112, 4–74 (2012).
 http://pubs.acs.org/doi/abs/10.1021/cr200168z
 The review is about explicitly correlated electronic structure methods,
 but the introduction gives some useful insights into the nature of
 electron correlation.
 On 31/05/15 16:16, Billy McCann thebillywayne-,-gmail.com wrote:
 >
 > 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>
 >
 --
 James C. Womack
 PhD research student
 Centre for Computational Chemistry
 School of Chemistry
 University of Bristol
 BRISTOL BS8 1TS
 Email: jw5533-x-my.bristol.ac.uk
 Web: http://jcwomack.uk