Dynamic versus static electronic correlation

In quantum chemistry, when a nomenclature in which one distinguishes between “static” and “dynamic” correlation is used, “correlation” referrers to all the deficiencies of the Hartree-Fock (HF) single-determinantal approach. For instance, the the *correlation energy* is defined as the difference between the exact (non-relativistic) energy and the HF energy (calculated with a complete basis),

$E_{\mathrm{corr}} = E_{\mathrm{exact}} - E_{\mathrm{HF}}$

Now, what are the deficiencies of a Hartree-Fock approach?

First, electrons in this model do not *instantaneously* interact with each other, as they do in reality, but rather each and every electron interacts with the average, or mean, field created by all other electrons. Classically speaking, each electron moves in a way so that it avoids locations in a close proximity to the instantaneous positions of all other electrons. And the failure of the HF model to correctly reproduce such motion of electrons is the first source of $E_{\mathrm{corr}}$ . This type of correlation is called *dynamic correlation* since it is directly related to electron dynamics.

Secondly, the wave function in the HF model is a single Slater determinant, which might be a rather poor representation of a many-electron system’s state: in certain cases an electronic state can be well described only by a linear combination of more than one (nearly-)degenerate Slater determinants. This is the second reason why $E_{\mathrm{HF}}$ may differ from $E_{\mathrm{exact}}$ and the corresponding type of correlation is called static, or nondynamic, since it ist not related to electron dynamics.

Interestingly, both static and dynamic correlation effects can be taken into account by “mixing in” more Slater determinants $\Phi_i$ to the Hartree-Fock one $\Phi_0$ ,

$\Psi_{e}(\vec{r}_{e}) = c_0 \Phi_0 + \sum_i c_i \Phi_i$

Here, if $c_0$ is assumed to be close to $1$ and a large number of excited determinants $\Phi_i$ are added each of which is assumed to give only a small contribution, then the method primarily treats dynamic correlation. And if, on the other hand, it is assumed that there are just a few excited determinants $\Phi_i$ with weights close to that of the reference determinant $\Phi_0$ , then the method primarily treats static correlation.

An example of a method that recovers primarily dynamic correlation is Møller–Plesset perturbation theory (MP $n$ ), while multi-configurational self-consistent field (MCSCF) method primarily takes account of static correlation. Note the word “primarily” here and above. It is almost impossible in principle to keep dynamic and static correlation effects separated since they both arise from the very same physical interaction. Thus, methods that typically cover dynamical correlation effects include at high order also some of the non-dynamical correlation effects and vice versa.

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