I have read some other posts explaining what multi-reference and multi-configuration are with regards to the wave-function (such as What exactly is meant by 'multi-configurational' and 'multireference'?).

However, what does it mean for a molecule specifically to exhibit high multi-reference character? Does it just mean the results from calculations will be strongly dependent on if the level of theory incorporates multi-reference?

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    $\begingroup$ There are molecules, which are qualitatively treated wrong when only using one configuration (for example in the form of a Slater Determinant). This is often also called strong correlation, or dynamic correlation. This happens when you have partially occupied (near-)degenerate orbitals (like in transition metal complexes for example). $\endgroup$ – Martin - マーチン Aug 26 '19 at 13:05
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    $\begingroup$ @Martin-マーチン , I think that maybe is not very clear that you mean by "dynamic correlation". For clarification, those systems that are treated wrong when only using one configuration suffer of static correlation and have multi-reference character. $\endgroup$ – user1420303 Aug 26 '19 at 21:01
  • $\begingroup$ @user1420303 yes indeed, I misspelt. It should have been static correlation. If you have the time, maybe you could add an answer. I will unfortunately not be able to. $\endgroup$ – Martin - マーチン Aug 26 '19 at 21:19

When doing a CI calculation based on Hartree-Fock orbitals, then usually the HF configuration has a very high weight in the CI vector (>90%). The next largest weight is then very small (for example only 1%), and there is a very large number of those very small configurations. Such cases are called weakly or dynamically correlated. Because of the large weight, it is reasonable to optimize the orbitals for the leading configuration only and then run some truncated CI calculations (typically CCSD) on top.

If there are multiple configurations with similar (or same) weight, then it is no longer reasonable to optimize the orbitals for just a single configuration, because this will introduce a bias towards this configuration. The number of these configurations is rather small. Additionally, those configurations are not necessarily related by Single or Double excitations, so CISD may not include all. However, they are commonly generated by an active space of just a few orbitals. This is called strong or static correlation, and is typically found in systems with partially occupied close-degenerate orbitals, e.g. d orbitals in transition metals.

In FCI this orbital bias would not matter, because the basis is (numerically) complete. But in truncations like CISD or CCSD this leads to a qualitative error. Some important configurations are poorly represented (due to the bias), other are completely missing (due to the truncation).

A CASSCF calculation resolves the bias by considering all important configurations of similar weight (by chosing an appropriate active space) and balancing the orbitals to all of the considered configurations. Since only some configurations are required for static correlation, it is often feasible to optimize both (CI and orbital coefficients) simultaneously.

The term multi-reference then refers to a calculation where all (or most) of the CASSCF configurations are used to generate excitations from.

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