# What are the types of MCSCF?

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Let's be the first resource that explains the following in up to 3 paragraphs:

For FCIQMC-SCF, DMRG-SCF, and SHCI-SCF, instead of explaining what FCQIMC, DMRG and SHCI are, please instead say which programs implement these methods, and anything special that had to be done in order for these to work (for example SHCI-SCF can be done with variational SHCI or with SHCI-PT2, and the one with PT2 is not variational, so the authors had to do a little extra to make it work).

• FCIQMC, DMRG, SHCI and many others like ASCI and coupled-cluster methods aren't actually types of MCSCF, but approximations to CASSCF where one just uses a different FCI solver. RASSCF and GASSCF can also be thought of as CASSCF with a non-exact CAS solver, where the CAS is just split into subspaces. – Susi Lehtola Jul 29 '20 at 8:20

## LASSCF: Localized Active Space SCF

An approximation to or generalization of (depending on how you look at it) CASSCF. In CASSCF, the wave function consists of an antisymmetrized product of two factors defined in two non-overlapping sets of orbitals: a single determinant of occupied inactive orbitals, and a general correlated wave function describing the electrons occupying the active orbitals. In LASSCF, the wave function is a product of one single determinant and several general correlated wave function terms, defined in the Fock spaces of multiple, distinct, non-overlapping sets of active orbitals; usually, each active subspace is localized (hence the name) around a particular atom or cluster of atoms within a large molecule, and the difference active subspaces are presumed to interact with one another only weakly. The initialism "vLASSCF" corresponds to the variational extension in which all orbitals are optimized without any (direct or indirect) constraint and the Hellmann-Feynman theorem applies, and corresponds to a realization for ab initio chemical models of the cluster mean-field (cMF) method developed by Jiménez-Hoyos and Scuseria.

The purpose of LASSCF is similar to that of RASSCF and GASSCF, in that one purports to reduce the cost of an untenably expensive CASSCF calculation by splitting a large active space, described by a factorially large CI vector, into weakly-interacting subspace parts. However, the approximation (compared to CAS with the same active orbitals) inherent to a LAS wave function is more severe than in either RAS or GAS, and the corresponding savings in computational cost are likewise greater in principle. RAS and GAS limit the configurations the CI vector of the active space is allowed to explore in terms of excitations from a reference determinant, while LAS requires that the notional CI vector describing all active subspaces collectively must factorize into a product of subspace parts. This means that the total cost of solving the CI problems in LASSCF is linear with respect to the size of the molecule, provided the size of a single active subspace is held fixed.

The method is currently only implemented in mrh, an extension to PySCF.

• Is this the same as the work by evangelisti? Localized orbitals casscf? – Cody Aldaz Jul 21 '20 at 19:39

## RASSCF: Restricted Active Space SCF

In a Complete Active Space (CAS) calculation, one chooses a set of occupied/virtual orbitals (the active space) from an initial Slater determinant and forms additional configurations from all the possible rearrangements (hence, complete active space) of the electrons among those orbitals. As the name suggests, a RAS calculation defines an active space of orbitals, but restricts the type of rearrangements that are made. This is typically done by subdividing the active space in to three parts: R1, R2, R3.

R2 is similar to the complete active space and all possible rearrangements among these orbitals are used. R1 is a set of initially occupied orbitals which is restricted to have at most $$n$$ holes, where $$n$$ is some parameter set by the user. R3 is a set of initially virtual orbitals, restricted to have at most $$m$$ occupancies in the generated configurations. In the limit where $$n$$ and $$m$$ are the size of R1 and R3 respectively, the calculation becomes a CASSCF.

RASSCF is useful for situations where one would like to use CASSCF (e.g. degenerate ground states from SCF, modeling bond breaking), but where the system size is too large for CASSCF to be feasible.

Implementations:

• +1. Just added some programs that implement it. Looked for the original paper where it was first described, but might have to continue tomorrow since it's close to 1am! – Nike Dattani Jul 19 '20 at 4:48
• @Tyberius this answer also contains CASSCF. – Susi Lehtola Jul 19 '20 at 9:38
• @SusiLehtola true, I felt that an explanation of RASSCF only really made sense with the basics of CASSCF. However I feel I have given a bare bones enough explanation of CASSCF where a more detailed answer would still be useful. – Tyberius Jul 19 '20 at 13:31
• I didn't include CASSCF in the list, but if someone wants to answer in detail (for example finding and posting the first ever CASSCF paper or the first paper where the term came, or review paper on the topic, and records of how large a CAS has ever been used in a CASSCF, etc.) they would be welcome to. – Nike Dattani Jul 19 '20 at 17:14

## DOLO (Do Localized Orbital) CASSCF

CASSCF can capture static electron correlation (i.e. orbital degeneracies), and it is very important to select the active space which includes the degenerate orbitals. For example, in a bond breaking event the $$\sigma$$ and $$\sigma$$* orbital will become degenerate and should be in the active space. However, anyone familiar with CASSCF will know how difficult it is to get the active space that you want. Particularly, if you are trying to solve for orbitals which are not the frontier orbitals. CASSCF can have many solutions, and sometimes you have to rotate really weird orbitals into the active space region and even that is not guaranteed to work and you might ask,... why? The reason is orbital rotations which are involved in the SCF part mix the orbitals. So even if you start with something that you like, you aren't guaranteed to keep that.

So what can we do?

Use DOLO! In DOLO convergence is imposed with the condition that the orbitals remain “as similar as possible” to the initial orbital, i.e. it avoid total diagonalization of Fock or Density matrices. Therefore, nice localized orbitals can be used and they remain nice localized orbitals.

For some nice free pictures see these slides. Also the original reference: Direct generation of local orbitals for multireference treatment and subsequent uses for the calculation of the correlation energy, Daniel Maynau, Stefano Evangelisti, Nathalie Guihéry, Carmen J. Calzado and Jean-Paul Malrieu, J. Chem. Phys. 116, 10060 (2002). My favorite example, is the dissociation of a C-H bond in ethylene. Without DOLO these requires the full valence active space. But with DOLO the minimal (2,2) active space can be used.

Note, I haven't actually used DOLO. It can be found here, in the COST package from Toulouse. But why isn't this more popular?!