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A significant part of quantum chemistry involves developing methods that go beyond the Hartree-Fock theory which treats electron-electron interactions through a mean-field approach. From what I understand much of the emphasis in this program of research is towards capturing as much of the correlation energy as possible. Interestingly, the definition of the correlation energy here, is the relative difference in energy calculated using the exact and mean-field approaches.

However, I find it hard to develop the link between correlation or entanglement between electrons to chemical bonding. For example, in this article, electron correlation is referred to as the "chemical glue". How does one make sense of correlation in motion of electrons being somehow related to bonding?

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  • $\begingroup$ @lex2763 mentioned in his answer, it could be technically wrong to call it "chemical glue". However, the question that remains is the link between bonding theories and modern electronic structure theories. $\endgroup$
    – Fracton
    Jan 26 at 1:02

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Coincidentially, I recently read Perdew's recent autobiographical article where he also mentions the concept of "chemical glue." His simple image is as follows:

The exchange–correlation energy is a relatively small part of the total energy, but it dominates the binding energy and has been called “nature’s glue.” In a nutshell, the electrons swerve to avoid one another (like shoppers in a crowded mall) because of the Pauli exclusion principle and the Coulomb repulsion between electrons. This lowers the total energy, and does so more in bonded systems than in separated atoms because bonded systems have more nearby electrons to be avoided. This “dance of the electrons” is the largest part of what binds atoms together.

Perdew is a DFT person, so he lumps together exchange and correlation contributions. Assuming exchange (i.e., the Pauli principle) is accounted for (as in Hartree-Fock theory), only correlation remains. Correlation lowers the energy of the system because electrons avoid each other (compared to mean field approximations). The more electrons there are, the more this effect will be in play (a one-electron system will have no correlation). Thus, at least part of chemical bonding (which brings electrons together in a small volume) arises from correlation.

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    $\begingroup$ That makes me wonder, if bonding theories are all but dead. With the advent of DFT and sophisticated wave function theories, the ideas of chemical bonding seem to be lost. Or am I wrong? Are there any recent developments in bonding theories? $\endgroup$
    – Fracton
    Jan 24 at 2:23
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Normally the correlation energy is defined to be:

$E_{corr}^{basis} = E_{FCI}^{basis} - E_{HF}^{basis}$

where the superscript basis denotes each calculated in a fixed basis set. The correlation energy must be $E_{corr}^{basis}\leq 0$. This is because the the full configuration interaction (FCI) wavefunction allows N electrons to be distributed in M spin orbitals (there are N choose M Slater determinants). Whereas, Hartree-Fock (HF) only allows electrons to fill the N lowest lying orbitals (a single Slater determinant). I.e. in a qualitative way the electrons can distribute themselves in more ways in FCI than HF. In more handwavey terms, the mathematics of FCI allow more arrangements of electrons such that they can position themselves in a better way such that the energy is lowered. As Kristof answer details this is how electrons 'avoid' each other.

In fact FCI includes the HF state and so by the variational principle $E_{FCI}^{basis}$ cannot be greater than $E_{HF}^{basis}$ hence why $E_{corr}^{basis} \leq 0$.

It turns out that for naturally occurring molecular systems it is energetically favorable to be bonded together - as seen in nature (usually at least). When we calculate the energy of separated atoms (i.e. split them apart to infinity) and vs at equilibrium bond length. The energy of the system together is lower (more stable) than dissociated and is why it costs energy to split molecular systems in a lab apart. I guess this is why electron correlation can be called "chemical glue", although I would avoid using the term in a computational chemistry paper as it isn't really correct to think of it as 'glue'. Yes it does cause things to bond, but this stems from the wavefunction of the system and has a more mathematical interpretation as discussed above. *Having said that, I think it is perfectly okay to use this phrase in a non-technical setting.

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  • $\begingroup$ Thanks for the answer! I know there's a lot of correlation energy that in quantum chemistry. However, the definition by itself doesn't seem to be rigourous. In general, there should be other ways to measure correlation in molecules? For example, von Neumann entropy is generally used as measure for entanglement in spin chains or model systems. $\endgroup$
    – Fracton
    Jan 26 at 0:58
  • $\begingroup$ (1/2) I believe the terms you are looking for are static and dynamical correlation - which are discussed at length in many quantum chemistry papers. My take is: dynamical correlation is caused by the wavefunction being described by a dominant single Slater determinant (but needs others to get the full energy) static correlation: is caused by degeneracies between different slater determinants and the reference determinant (if any) $\endgroup$
    – lex2763
    Jan 26 at 20:52
  • $\begingroup$ (2/2) So heuristically: systems with strong static correlation energy have wavefunctions that differ qualitatively from the reference Slater Determinant (or have coefficients that are nearly all equal) vs systems with strong dynamic correlation implies a wavefunction that includes a large number of excited determinants (all with comparable small occupations) $\endgroup$
    – lex2763
    Jan 26 at 20:55
  • $\begingroup$ (3/2 EXTRA) von Neumann entropy is used as a measure in these contexts. The following paper: doi.org/10.1007/s00214-012-1291-y gives a nice overview $\endgroup$
    – lex2763
    Jan 26 at 20:59
  • $\begingroup$ Point 1 and 2 seem to contradict each other. "dynamical correlation is caused by the wavefunction being described by a dominant single Slater determinant " and "strong dynamic correlation implies a wavefunction that includes a large number of excited determinants". These two don't add up somehow? $\endgroup$
    – Fracton
    Jan 27 at 9:13

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