Multireference methods – a review (part 1)

Over the years, developing or benchmarking DFT functionals has been a hot topic, for example D. Truhlar paper on M06 has reached over 9000 citations, or this paper of D. Jacquemin on (just!) TD-DFT benchmark of organic molecules receives more than 500 citations. This leads to a zoo of DFT functionals, and it’s growing wild like weed.

How about ab initio methods? With the development of computer science, i.e. faster computers and smarter algorithms, many “novel” ab initio methods are being developed by physicists and chemists. In this post I create a list of multireference methods that I would like to try, or to include them all in my dream paper.

Generally, a multireference method is divided into two parts: First, a multi-configurational self-consistent field (MCSCF) is used to account for static correlation. On top of the MCSCF wavefunction, configuration interaction (CI), or perturbation method (PT) or coupled-cluster method (CC) can be applied to account for dynamic correlation. Voilà, we obtain a “near-exact” wavefunction.

In this post, I focus first on the first part: multi-configurational self-consistent field (MCSCF) to account for static correlation.

Traditionally, the wavefunction is expanded as a linear combination of Slater determinants. However, MCSCF is not a black-box method since one must choose which and how many determinants should be included in the wavefunction.


(i) I’m not an expert in the field, so please correct me if I’m wrong. (ii) This list is not exhaustive.

  • CASSCF pioneered by B. Roos
    • CASSCF, complete active space SCF, aka pain in the *** (NSFW). This is the first efficient approach of MCSCF. In general CASSCF = Full CI in a restricted configuration space. CASSCF is a “black-box” method, you just have to choose the so-called “active orbitals” that can describe your chemical problem. Thousands of computing hours have been wasted since the active space is not what you want. And thousands of bad(!) papers have been published using nonsense active spaces.
    • Nearly all quantum software can do CASSCF, even the you-know-who (YNW) software.
    • Limitation: legend say CASSCF is limited to around 16 electrons in 16 active orbitals.
    • Fun fact: Recently I saw a paper of T. Martinez ( a pioneer of applying GPU for quantum chemistry). By introducing an empirical value(!) into CASSCF, he invented a so-called α-CASSCF method that can approximate CASPT2. C’mon, give me a break!


  • RASSCF by PA. Malmqvist
    • restricted active space SCF, aka father of  pain in the ***. RASSCF adds another layer of complexity to CASSCF. Now you have to choose subspaces RAS1, RAS2, and RAS3.
    • Notable software: Molcas, Molpro.
    • Limitation: I saw somewhere, someone did RASSCF with an active space of 40(?) active orbitals.
    • Fun fact: who is using RASSCF, please raise your hand. I count less than 10.


  • GASSCF by L. Gagliardi
    • generalized active space SCF, aka grandfather of  pain in the ***. Now you can choose more subspaces instead of 3 in RASSCF.
    • Notable software: Molcas.
    • Limitation: L. Gagliardi did a calculation of Cr2 with nearly 50 active orbitals. Impressive!
    • Fun fact: who is using GASSCF, please raise your hand. I count less than 5.


  • DMRG by S. White
    • Density matrix renormalization group. DMRG is a hot topic recently since people claim that it can solve a problem with 100(!) active orbitals (but for H100 system, who wants to do H100?). DMRG approximates CASSCF in a very smart and unique way, but introduces some extra “parameters” that can significantly control the accuracy of your calculation. In conclusion: still less headache than RASSCF and GASSCF.
    • Notable software: there are quite a lot of free software that can do quantum chemistry DMRG such as QCMaquis by M. Reiher, Block by G. Chan, or CheMPS2 by S. Wouters. These software has been interfaced in some free (psi4, Orca) and commercial codes (Molpro).
    • Limitation: 100 active orbitals (for the hydrogen chain H100), 50(?) active orbitals for general molecules. This is exciting!
    • Fun fact: there is a very strong competition between different groups: who has the best DMRG software, who can do large calculations, who can produce better results. It’s good for science! We need more drama.


  • FCIQMC pioneered by A. Alavi
    • Semi-stochastic full configuration interaction quantum Monte Carlo. FCIQMC is also a hot topic. This is another smart way of approximating CASSCF using quantum Monte Carlo. Moreover, one can use thousands (or millions?) of CPU core to drastically speed-up a FCIQMC calculation. Last year, I was lucky to talk with one of the pioneer of FCIQMC and he told me that FCIQMC can do hundreds of active orbitals. Really impressive!
    • Notable software: NECI by A. Alavi. This code is already interfaced with Molpro.
    • Limitation: FCIQMC can do hundreds of active orbitals but for just small molecules, i.e. small number of active electrons. FCIQMC is not better than DMRG. Moreover, who has thousands of CPU core?
    • Fun fact: playing games in Casino de Monte-Carlo is fun!


  • 2-RDM-driven CASSCF by A. Eugene DePrince III
    • variational two-electron reduced-density-matrix (2-RDM)-driven complete active space self consistent field. This is another smart approximation of CASSCF. Since this methods scales only polynomially with system size, v-2RDM-CASSCF can be a competitor(?) of DMRG and FCIQMC.
    • Notable software: DePrince wrote a plugin for psi4. As far as I know, this is the only code.
    • Limitation: 50 active orbitals, as I can see in this paper.
    • Fun fact: in the paper, DePrince wrote “the chemistry community has resisted adopting v2RDM-based approaches to the static correlation problem”. The chemistry community is a jerk.


  • More to come? Please let me know.



More and more smarter algorithms to approximate CASSCF are coming. I really hope that soon, quantitative results (let’s say within 2 kcal/mol) for large molecule can be systematically obtained (at least in gas phase). DFT, watch out!


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