# What are the references in multi-reference methods?

After reading What exactly is meant by 'multi-configurational' and 'multireference'? I still don't understand what the multiple "references" are. I understand that you can for example use a HF ansatz and exited states for multi-configurational methods. And that multi reference methods use more than one configuration to build exited states. But what are these configurations? There is only one HF Slater determinant, how do the other references look?

I would appreciate if you could answer the questions with some additional formulas instead of just describing it by words :)

In a comment on the question I previously asked in Chemistry Stack Exchange, I was told that these other determinants have just different occupied orbitals. But then they are excited states and should already be included in the normal multi-configurational ansatz (CISD for example).

So how does CISD and multi-reference CISD differ?

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I hope this basic example helps you to see the difference. I'll denote configurations by simple tuples where each number indicates the occupation of a spatial orbital, with increasing energy from left to right.

Let's take only the ground state determinant as reference and do CIS within a space of 4 spatial orbitals (yielding 8 spin orbitals in total) and two electrons. The additional configurations that we obtain when we look at all single excitations are the following:

(2 0 0 0) -> (1 1 0 0) | (1 0 1 0) | (1 0 0 1)


Now let's add a second configuration and do CIS on this new configuration. I'll omit configurations that have already been generated in the former step.

(0 1 1 0) -> (0 2 0 0) | (0 0 2 0) | (0 0 1 1)


You see that you get new determinants based on single excitations starting from this reference. This set of configurations differs from the set that you would have obtained by doing all single and double excitations based on the groundstate configuration. If we had done CISD we would have gotten one more determinant,

(0 0 0 2)


In this basic example the difference is small but as the active space and the number of electrons grows, the number of possible configurations "explodes" in size.

The same transfers to multireference CISD. By adding a reference and doing single and double excitations you will get additional configurations that aren't generated based on a single reference. It is also different from simply doing the next higher excitation scheme, as you can see in my CIS example.

You can also include determinants that correspond to higher excitations by starting from a doubly excited determinant and doing single excitations on top of it, if we had more electrons and a larger active space.

The multireference ansatz allows us thus to include some higher excited configurations without including the whole set of all higher excited determinants, which is often unfeasible due to the large number of possible combinations that are available given a larger active space. It also allows us to add higher excited configurations without the cost of doing the full higher excited scheme.

The diagonalization of the CI matrix based on the generated configurations yields your electronic states and eigenvalues. The electronic states are mathematically linear combinations of the determinants although many states often have a dominant contribution. This allows us to approximately identify a state with the determinant, that has the highest coefficient in the linear combination.

• Thanks a lot. So are multi-refernece methods good at picking the most important higher excited configurations? Why are they superior above including them randomly?
– M0M0
Aug 8 at 20:55
• Usually you don't pick them at random. Choosing the proper determinants is not easy and depends on the system. Aug 9 at 6:57
• I would add that if 2000 denotes the restricted HF configuration, the 0200, 0020 and 0002 configurations are likewise perfectly valid HF configurations. The problem with multireference systems is that the HF energies for these configurations may be very close to the 2000 configuration; the differences may be just fractions of an electron volt. The configurations and their cross-couplings are optimized simultaneously in multireference methods. Aug 9 at 14:29

Rather than starting with a single-determinant (single-reference) such as the Hartree-Fock determinant which is obtained by an HF-SCF calculation, multi-reference methods such as MRCI, MR-ACPF, MR-AQCC, and even MR-CC will start with a multi-configurational SCF (MCSCF) calcualtion. There's many types of these, and many can be found here: What are the types of MCSCF?

Perhaps the most common starting point for an MRCI calculation is to do a CASSCF calculation. For example you might choose 3 spatial orbitals and 3 electrons and form a CAS(3,3) calculation which can result in a wavefunction that is the sum of many determinants. Now rather than doing the post-SCF (i.e. CI, CC, etc.) calculation on top of only one reference determinant, you can have many of them.

Finally, based on what you wrote in your question, I want to emphasize strongly that multi-reference methods are not just for excited states, but also for ground states in cases where the most dominant determinant does not have a much stronger weight in the overall wavefunction compared to some other determinants, and this happens in many molecules such as $$\ce{Cr2}$$, in which the ground electronic state requires a multi-reference treatment.

• Thanks. When I wrote exited states, I meant "exited determinants". My question was probably not clear enough (or I asked the wrong question): My problem is that when I do a CAS calculation, my wavefunction will be described by several determinants (the hf and several exited ones) but if I do CISD I do basically the same. So I still don't see which determinants will be included in a multi-refernce CISD Ansatz that are not already included in the normal CISD Ansatz.
– M0M0
Jul 29 at 4:19
• Perhaps it is noteworthy that CASSCF(x,y) usually also 'optimizes the orbitals' of the active space. Aug 6 at 22:59
• Determinants are not the same as states. Also, the list of all singles and doubles from one reference, is going to be different from the list of singles and doubles from several references (especially if those references are very different from each other). Aug 9 at 15:41
• By the way I was surprised to see a downvote on this answer and that you got the critic badge shortly after. Was it you? Please carefully consider the part highlighted in yellow here: i.stack.imgur.com/ixGof.png Aug 9 at 15:42
• @NikeDattani Yes, that was me. Sorry, I misunderstood what downvotes are for. I just wanted to express that I like the other answer more (explaining with an example why the methods differ). The downvote should be gone now :)
– M0M0
Aug 11 at 16:48