# Why is no dynamical correlation recovered in the CASSCF method?

Configuration interaction is the only certain way to get a calculation to converge toward the exact solution of the many-body Schrödinger equation (within the approximations used for the framework of the method). In such calculations, the wave function is expanded in all possible electron configurations, and will therefore often take too long to be practical. For full CI, all dynamical correlation is recovered.

In truncated CI, part of the dynamical correlation is recovered. The wave function expansion is, for example, limited to include only single and double excitations (CISD).

In a CASSCF calculation, a subset of all orbitals is chosen, and the wave function is expanded in all possible electron configurations within this subset of orbitals. However, here it is said that no dynamical correlation is recovered. This is stated so often I suspect it is common knowledge for computational chemists, but I can provide references if someone want them.

It is my understanding that the CI method recovers dynamical correlation in allowing electrons to be moved into virtual orbitals (and hence "react" to the instantaneous coulombic repulsion from other electrons). In allowing for higher a number of excitations, a larger portion of the dynamical correlation is recovered (CISDT > CISD).

So, why is no dynamical correlation recovered in the CASSCF method? Imagine a good set of orbitals are chosen for the active space, such that no important configurations are left outside of the wave function expansion, surely dynamical correlation is described? What if we extend this to the generalized actice space (GAS) formulation. In developing GAS, the original authors aimed to remove the "deadwood" of configuration state functions; that is, exclude CSF's with near-zero coefficients. So now a larger active space can be used, and hence a larger number of excitations can be used. Why is not more dynamical correlation recovered?

If indeed CASSCF recovers no dynamical correlation, a CASSCF calculation should give the same energy as a HF calculation. Is this the case?

• You can reach full-ci by including everything inside the active space. It is somewhat a convention that people call the correlation energy captured by CASSCF as non-dynamical correlation, when the active space is chosen correctly. – Rodriguez May 7 '16 at 18:22
• This convention is confusing to me. – Yoda May 7 '16 at 18:43
• Try to be a big name in quantum chemistry community and let the majority accept your convention :) – Rodriguez May 7 '16 at 18:56
• Give me 15 years and we'll see :) – Yoda May 7 '16 at 18:59

The difference between dynamical and static correlation has always been ill-defined, and there is no clear-cut delineation between them.

Correlation simply means that the probability of finding an electron somewhere at position $a$ and another electron somewhere at position $b$ is not simply the product of the two probabilities.

(In that respect, even Hartree-Fock has some correlation since it forbids two same spin electrons occupying the same state (Fermi correlation).)

Correlation was originally thoguht of as the instantaneous repulsion of electrons. This was dynamic correlation. The origin of the difference between static and dynamical correlation came from studies of bond dissociation in molecules. As you pull a bond apart, the electrons should be further apart, and correlation energy should decrease. However, the opposite was found, and in many cases the correlation energy increases. Therefore, there must be some sort of "other" correlation occurring. This was static correlation.

This is because there are additional (near-)degenerate configurations that contribute strongly to the nature of the wave function during dissociation. You can think of this as a manifestation of entanglement, of which Hartree-Fock, being single determinant, fails miserably.

In short, static correlation came to describe situations where the underlying Hartree-Fock reference was a qualitative and quantitative failure.

Dynamic correlation, on the other hand, was used to describe the cases where the underlying Hartree-Fock reference has the correct qualitative behavior, but was not quantitatively accurate.

CASSCF does a better job at calculating static correlation because most of the important near degenerate configurations are multi-electron excitations within a small valence space. Thus it generally captures the configurations necessary to describe bond breaking, etc. with a well-chosen active space.

Single reference coupled cluster, does better at capturing dynamic correlation because it is usually truncated to few-pair excitations, like the double excitations of CCSD, but cover the whole virtual space.

For high levels of coupled cluster, e.g. CCSDTQ, the distinction between static and dynamic correlation really breaks down, and some high level coupled cluster calculations actually do quite a good job at otherwise "multireference problems." Multireference coupled cluster --- combining the best of CASSCF and CC --- is an active area of research today.

• A paraphrase of something I read somewhere: "When it comes down to it, static correlation is just dynamic correlation where the correlated states have started to closely approach degeneracy." This seems consistent with what you wrote, but ... is it at all correct? – hBy2Py May 8 '16 at 3:11
• I would say that's true if the correlated states are near the state of interest. – pentavalentcarbon May 8 '16 at 18:08
• So "fermi correlation" is basically the same as the electron exchange? "The movement one depends on the movement of the other" – Yoda May 9 '16 at 8:26