Electron Correlation in Molecular Systems: Meaning & Types

[Karrar]

I wonder, what the physical meaning to electron correlation in molecular systems? and. Is there many types of it?

Thanks for interest colleagues

 

[cgk]

The term "Electron correlation" is used to refer to any electronic many-body effect which cannot be described by Hartree-Fock theory. That means, which requires a wave function more complicated than a single Determinant (or configuration state function) to be described adequately.

There are two qualitatively different kinds of electron correlation: static correlation and dynamic correlation.

Static correlation refers to situations in which multiple determinants are required to cover the coarse electronic structure of a state. This happens if there is only a very small energy gap between the ground state of the molecule and other states. For example, if you calculate a potential energy curve of a diatomic molecule until bond breaking, you have to deal with different configurations describing different electronic states at the same time, because these become quasi-degenerate. Static correlations deals with only few, but very important determinants. MCSCF is used to handle such correlations.

Dynamical correlation refers to capturing the effect of the instantaneous electron repulsion, mainly between opposite-spin electrons (because these don't have Pauli repulsion in HF). The Hartree-Fock wave function is rather restricted in form, and does not allow electrons to get out of each other's way. This is rectified by throwing in lots and lots of determinants with very small weight each. Dynamical correlation is required to get the energetics of a system right, but not for the coarse electronic structure. Methods dealing with dynamic correlation are MP2, CCSD(T) (on HF reference), MRCI (on MCSCF reference) or Kohn-Sham-DFT (instead of HF, dynamic correlation handled implicitly in an approximate way).

 

[alxm]

I gave the basics of what it is in a https://www.physicsforums.com/showpost.php?p=2907921&postcount=5", is simply the difference between the Hartree-Fock energy and the true energy.

But to address the question of physical meaning; it's generally described as the effect caused by electrons avoiding each other in their motion. Something I (hopefully) managed to illustrate in my previous post, which assumes a Hartree-Fock approach. (by assuming the electron-electron interaction can be described as a potential dependent only on the wavefunction of the other electron. I.e. a mean-field approach. )

Note that HF by construction obeys antisymmetry, so two electrons of the same spin still have a zero probability of being in the same location at the same time, so in this sense their motion is correlated. So the electrons of the same spin do still 'avoid' each other to some extent, it's merely the Coulomb repulsion correlation that's not taken into account. (The pair-density has a 'Fermi hole' but not a 'Coulomb hole') As cgk describes, there's also static correlation, which is related to the single-determinant nature of the Hartree-Fock approximation. This is a bit more relevant to MCSCF and the methods which build directly on Hartree-Fock.

So 'correlation', while usually physically described as the electrons "avoiding each other" doesn't actually include all the dynamical effects (since Pauli repulsion is taken into account), and it also includes some non-dynamical effects caused by the single-determinant HF description. But the bulk of correlation energy is still the coulomb-correlation of motion.

In DFT, correlation is "exchange-correlation", because the Kohn-Sham approach does not describe either. In other words, all dynamical effects are excluded. So the "functional" (within the KS methodology) only refers to this part. The exchange-correlation functionals are usually separated into a sum of exchange and correlation parts (although AFAIK, there is no rigorous theoretical justification for this. It does seem to work fairly well though). In principle at least, static correlation has no meaning in DFT, because if the exact density functional were known, a single determinant description would be exact.

 

[Karrar]

that's so good descreption cgk and alxm thanks so much .

but is HF theory fully neglected the electron correlation , in same time take correlation of average field that created by other electrons , is that Paradox ?

 

[raul_l]

I had the same question about electronic correlation and found the answers of cgk and alxm very helpful. If I understand correctly the original paper of Kohn and Hohenberg*, the exchange and correlation energy is given in terms of single and two particle density matrices
$$E_{xc} [n] = \frac{1}{2} \int {\frac{C_2(\mathbf r, \mathbf r')}{| \mathbf r - \mathbf r' |} d \mathbf r d \mathbf r'} = \frac{1}{2} \int {\frac{n_2(\mathbf r, \mathbf r';\mathbf r, \mathbf r') - n_1(\mathbf r, \mathbf r)n_1(\mathbf r', \mathbf r')}{| \mathbf r - \mathbf r' |} d \mathbf r d \mathbf r'},$$
where
$$n_2(\mathbf r, \mathbf r';\mathbf r, \mathbf r') = \langle \Psi | \hat\psi^{\dagger}(\mathbf r)\hat\psi^{\dagger}(\mathbf r')\hat\psi(\mathbf r')\hat\psi(\mathbf r) | \Psi \rangle$$
and
$$n_1(\mathbf r, \mathbf r) = n(\mathbf r).$$
Do I understand correctly that without exchange and correlation the n_2 term would be a simple product of single particle densities which would lead to Exc = 0?

*P. Hohenberg, W. Kohn, Phys. Rev. 136, B864 (1964)

 

[bsmile]

I am not an expert on this. Just try to share with you my understanding. As you noted in the definition of E_xc, the subtracted contribution n_1(r) n_1(r') is exactly the Hartree term, which I believe HK has isolated to be an independent term being easy to treat. So the exchange-correlation term must exist, coming out formally by definition. If it were not there, then yes, only Hartree term is present, Fock, viz exact exchange, and correlation terms would vanish, which is however actually physically not correct.