What exactly is meant by 'multi-configurational' and 'multireference'?

Question

Some topics here have touched on this before (see 1, 2, 3), but I haven't found a clear definition yet.

I would like to know what exact property of the wave function these terms refer to. It would also be helpful to have a clear definition of 'reference' and 'configuration'. I'll try to explain below where my problems are in clearly understanding/defining these terms:

Starting with Hartree-Fock, it is obvious that the wave function in HF is both single-reference and 'single-configurational': there is only one Slater determinant.

Going to configuration interaction methods, the wave function now becomes a linear combination of several Slater determinants. The additional Slater determinants are excitations of the ground state determinant: virtual orbitals from HF are taken and replace previously occupied orbitals in the determinant. However, these orbitals still have the same coefficients as in HF - only the coefficients in front of the Slater determinants are optimized for the linear combination. If I'm correct, 'configuration' here refers to one particular Slater determinant - a method is, therefore 'multi-configurational' if the wave function that is used has two or more (different) Slater determinants, correct? That also means that none of the CI methods (be it CIS, CISD,..., or Full CI) are multireference methods?

Continuing with CASSCF, this method is basically Full CI limited to the chosen active space of orbitals. It is therefore multi-configurational. At the same time, it is also often referred to as being 'multireference'. The only difference to CI, however, seems to be the optimization of the coefficients in the Slater determinants themselves, hence, this must be the defining criterion for 'multireference'? What does 'reference' here refer to?

Now there is also multireference-CI. From the above definition, I would expect this to be a form of CI where I also optimize the orbitals, but that does not seem to be the case. The Wikipedia article on MRCI starts with:

In quantum chemistry, the multireference configuration interaction (MRCI) method consists of a configuration interaction expansion of the eigenstates of the electronic molecular Hamiltonian in a set of Slater determinants which correspond to excitations of the ground state electronic configuration but also of some excited states. The Slater determinants from which the excitations are performed are called reference determinants.

This is confusing to me: Is 'excitation of the ground state electronic configuration' vs. 'excited states' referring to the optimization of the Slater determinants themselves? Or does 'excited state' refer to configurations with different total spin? That would be a different definition of 'reference', but then CASSCF would only be a multireference method if it uses the corresponding SA-CSFs, regardless whether the Slater determinants are optimized or not?

Answer 1

Your problem seems to be with the terminology used in CI methods, so let me go through the different terms you mentioned:

• A configuration is a certain occupation of (molecular) orbitals. Mathematically configurations can be represented in 2 ways. The first one is the Slater Determinant (SD), an anti-symmetrized product of spin-orbitals. Slater Determinants, however, are not eigenfunctions of the spin operator $\hat S^2$, but the electronic wave function needs to fulfill this requirement. Therefore one constructs spin-adapted Configuration State Functions (CSF) as certain linear-combinations of SDs. Using CSFs instead of SDs usually makes a calculation more stable. "Configuration" is a more general term, which does not explicitly say whether an SD or CSF is considered.

• Multi-configurational just means the method considers more than one configuration.

• Reference means we have a designated configuration from which the excitations are generated. Single-reference thus means we only have one such configurations (usually the HF configuration), e.g. in CISD or CCSD. Multi-reference means we have more than one configurations to generate excitations from.

So multi-configurational just means we have many configurations, while single-/multireference says something about how those configurations are selected/generated.

How does single-/multireference apply to the different methods?

• In HF we do not generate any excitations, therefore this concept does not really apply here. But I think one could still call it single-reference and single-configurational.

• CISD and CCSD are the typical examples of single-reference methods. Nothing special here.

• In FCI and CASCI we don't restrict the excitations to certain degrees (Single, Double, etc.), instead, we just take all of them. The concept of having one (or more) reference configurations is a possible perspective, but it is not really necessary. FCI can be viewed as having one reference configuration and taking all excitation degrees (up to the number of electrons available), which would make it single-reference. But from another perspective, we can argue that every MRCI wave function is just a truncation of the corresponding FCI wave function. So if FCI covers everything and more than MRCI does, would it not be multi-reference as well? Again, I would just say the concept does not apply here.

• MCSCF (CASSCF) is a combination of CI (CASCI) and SCF, where SCF means we optimize the orbitals as well. This is not done in CI(SD...), CC(SD...), MRCI, etc. Other than that, the above arguments apply in the same way to the CI space as before.

Multi-Reference CI and CASSCF

As argued above, personally I would not consider CASSCF to be multi-reference, as the concept does not really apply. But I can see why people would consider it as such.

One usually does CASSCF because single-reference methods fall short of describing the wave function correctly, in a qualitative way. Such systems are then called strongly correlated and CASSCF can treat that strong correlation (also called static correlation) which is missing in single-reference calculations. In turn, however, CASSCF is missing that kind of correlation single-reference method can treat well, called weak or dynamic correlation.

Multi-reference is now the approach to combine both, strong and weak correlation, by doing first a CASSCF, and then using those configurations as a reference space, to generate all the excitations from.

Orbital optimization, however, is usually not feasible for dynamic correlation, because dynamic correlation usually means to have a lot of configurations and therefore huge CI space. With static correlation there are much fewer configurations, so doing orbital optimization additional to CI optimization is feasible and yields qualitative improvements of the orbitals. Therefore this is done in CASSCF, but not in MRCI.

"Excited states" in MRCI

What is meant here by "excited states" are actually "excited configurations" (with respect to the HF configuration). If you, for example, choose a CAS as your reference space, then all configurations, except for the HF configuration itself, are excited configurations.

Please note, that "excited state" is not a good choice of words here since it is actually something different: a state is a result of a CI calculation and is physically observable. On the other hand, a configuration is a basis function you put in a CI calculation and therefore is more of an abstract mathematical tool. However, you can approximate a state as a configuration to some extent, this is for example done in HF.

Answer 2

I would like to add to the above description. The reference states involve the spin function. For two electrons there are four spin functions, aa, (ab + ba)/sqrt 2, and bb for the triple state and (ab - ba)/sqrt 2 for the single state. (a is spin up and b is spin down.) For three electrons the spin functions are aaa, (aab + aba + baa)/sqrt 3, (abb + bab + bba)/sqrt 3, bbb for the quartet state. There are two spin functions for the three electron S sub z = 1/2, S = 1/2 state: (-2aab + baa + aba)/sqrt 6, and (aba - baa)/sqrt 2. Including one of these spin functions in the wave function is a single reference wave function. Including both of these spin functions is a multireference wave function. For four electrons S = 0, S sub z = 0 state there are two spin functions: (ab - ba)(ab - ba)/2, and (2aabb +2bbaa - abab - baba - abba - baab)/sqrt 12. The first of these is the spin function used in the single determinant Hartree - Fock wave function. Both of these spin function are used for multireference situations. Ruedenberg discusses the spin function in some of his MCSCF papers, such as Ruedenberg and Sundberg, in Quantum Science, J.-L. Calais et al., Springer, 1976, pp 505 -515.