Multi-configuration self-consistent field (MCSCF)

Modules: mcscf

Introduction

Multiconfigurational self-consistent field (MCSCF) methods go beyond the single-determinant Hartree-Fock (HF) method by allowing the wave function to become a linear combination of multiple determinants. While the configurations i.e. determinants can in principle be chosen in an arbitrary number of ways, PySCF focuses on the complete active space (CAS) family of methods, where the set of electron configurations is defined in terms of a given set of active orbitals, also known as the “active space”. The CAS method generates all possible electron configurations that can be formed from the set of the active orbitals, and is therefore equivalent to an FCI procedure on a subset of the molecular orbitals; please see Configuration interaction (CISD and FCI) for a discussion on the FCI method. The use of MCSCF methods is crucial for reliable modeling of systems that exhibit nearly degenerate orbitals i.e. static correlation, such as transition metal complexes. For a general discussion of MCSCF methods, we direct the reader to References [1] and [2], and for specific details about PySCF’s implementation of CASSCF see [3].

The MCSCF module has two main flavors: CASCI and CASSCF. In CASCI, the wave function is written as a linear combination of Slater determinants, and the expansion coefficients are solved in a variational procedure. The orbitals are fixed in a CASCI calculation. In contrast, in a CASSCF calculation, the orbitals are relaxed to minimize the resulting CASCI energy. This means performing multiple CASCI steps followed by updates to the molecular orbitals, in analogy to Hartree-Fock calculations which also relies on iterative algorithms to minimize the HF energy with respect to the orbitals.

Minimal examples for each type of calculations are shown below.

CASCI

from pyscf import gto, mp, mcscf
mol = gto.M(
    atom = 'O 0 0 0; O 0 0 1.2',
    basis = 'ccpvdz',
    spin = 2)
myhf = mol.RHF().run()
# Use MP2 natural orbitals to define the active space for the single-point CAS-CI calculation
mymp = mp.UMP2(myhf).run()

noons, natorbs = mcscf.addons.make_natural_orbitals(mymp)
ncas, nelecas = (6,8)
mycas = mcscf.CASCI(myhf, ncas, nelecas)
mycas.kernel(natorbs)

CASSCF

import pyscf
mol = pyscf.M(
    atom = 'O 0 0 0; O 0 0 1.2',
    basis = 'ccpvdz',
    spin = 2)
myhf = mol.RHF().run()
ncas, nelecas = (6,(5,3))
 # We can also run CAS calculations starting from the Hartree-Fock orbitals.
 mycas = myhf.CASSCF(ncas, nelecas).run()

Like many other modules in PySCF, mcscf works with density-fitting (Density fitting (DF)) and x2c (Scalar relativistic correction). It can also be used in its unrestricted formulation, please see examples/mcscf/60-uhf_based_ucasscf.py for an example. An important feature of mcscf is that it can interface with external CI solver such as DMRG, FCIQMC, or selected CI methods, see the external projects for more details. You can even use MP2 or CC methods as solvers! Check out /examples/cc/42-as_casci_fcisolver.py for a working example.

Picking an Active Space

In general, selecting an active space can be cumbersome and PySCF offers severals ways to facilitate this process. There are several strategies and they all contain two main components:

  • Specify the number of electrons and orbitals in the active space.

  • (Optional) Specify the which orbitals are “active”

Note

We always advise users to visualize their chosen active orbitals before starting large/expensive calculations. This involves dumping the MO coefficients to a molden file (see example examples/tools/02-molden.py) and visualizing with your chosen program. While there are many great softwares available to visualize orbitals, JMol is one of the easiest to use and is recommended for less experienced users.

Below is a list of several general strategies one could employ to pick active space orbitals:

  1. (Default) Specifying no additional information.

This is the most minimal strategy for selecting an active space and chooses orbitals (and electrons) around the Fermi level that match the number of orbitals and electrons specified by the user. In most circumstances, this is not an ideal strategy and will lead to poor convergence or none at all.

For example:

ncas, nelecas = (6,8)
mycas = myhf.CASSCF(ncas, nelecas)

Note

Hartree-Fock orbitals are often poor for systems with significant static correlation. In such cases, orbitals from density functional calculations often yield better starting points for CAS calculations.

  1. Specifying the active space orbitals as a list of molecular orbital (MO) indices. This is often useful combined with a visual analysis of localized orbitals (see the section on localized orbitals).

The user can select the MO orbital indices “manually” and pass them to the sort_mo function. NB! The orbitals are numbered from 1, not 0. See examples/mcscf/10-define_cas_space.py and examples/mcscf/34-init_guess_localization.py for more details.

mycas = mcscf.CASSCF(myhf, 4, 4)
# Note sort_mo by default take the 1-based orbital indices.
mo = mycas.sort_mo([5,6,8,9])
mycas.kernel(mo)

3) Specifying the number of orbitals in each symmetry group. This strategy is often helpful when the system has a high degree of symmetry, or when there is no physically evident best choise for the active orbitals.

mycas = mcscf.CASSCF(mf, 12, 12)
ncore = {'A1g':5, 'A1u':5}
ncas = {'A1g':2, 'A1u':2,'E1ux':1, 'E1uy':1, 'E1gx':1, 'E1gy':1,
            'E2ux':1, 'E2uy':1, 'E2gx':1, 'E2gy':1}
mo = mcscf.sort_mo_by_irrep(mycas, mf.mo_coeff, ncas, ncore)
mycas.kernel(mo)

A similar approach where we specify the number of electrons in each irreducible representation is also possible by setting mycas.fcisolver.irrep_nelec.

mycas = mcscf.CASSCF(myhf, 8, 8)
mycas.fcisolver.irrep_nelec = {"A1g": (2, 1), "A1u": (1, 1), "E1ux": (1, 1), "E1uy": (1, 0)}

Note

This strategy is often combined with calculations at a lower level of theory. For instance, MP2 or CISD natural orbitals and their occupation numbers can be used determine the suitable active space in each symmetry block. Natural orbitals with occupations close to 2 are strongly occupied, and can be frozen in the CAS calculation. Natural orbitals with small occupation numbers can likewise be omitted from the active space.

4) Use automated strategies (avas and dmet_cas) to pick an active space based on AO orbitals you’re targeting. For more details, see examples/mcscf/43-avas.py and examples/mcscf/43-dmet_cas.py.

from pyscf.mcscf import avas
ao_labels = ['Fe 3d', 'Fe 4d', 'C 2pz']
ncas, nelecas, orbs = avas.avas(mf, ao_labels)
mycas = mcscf.CASSCF(mf, ncas, nelecas)
from pyscf.mcscf import dmet_cas
ao_labels = ['Fe 3d', 'Fe 4d', 'C 2pz']
ncas, nelecas, mo = dmet_cas.guess_cas(mf, mf.make_rdm1(), ao_labels)
mycas = mcscf.CASSCF(mf, ncas, nelecas)
mycas.kernel(mo)

Frozen-orbital MCSCF

Orbitals can be frozen in the orbital optimization to e.g. reduce the computational effort of CASSCF calculations. The orbitals will remain fixed throughout the optimization.

Users can specify the number of lowest orbitals to freeze:

mycas = mcscf.CASSCF(myhf, 6, 8)
mycas.frozen = 2
mycas.kernel()

Users can also specify a list of orbital indices (0-based). These may be occupied, virtual, or active orbitals.

See examples/mcscf/19-frozen_core.py for a complete example.

Note

The frozen keyword of the CASSCF optimizer should not be confused with the frozen keyword of the FCI solver, which controls the number of orbitals that are constrained to be doubly occupied.

State-Averaged Calculations

When your system has states that are close-by in energy, their ordering may change during the orbital optimization and result in non-convergence of the CASSCF optimization. In such cases it is often helpful to optimize the orbitals for a state average. The state_average_ function (note the hanging underscore) is a member function of CASCI/CASSCF objects and takes the weights of the states as input. The weights can be any normalized and non-negative array of values, but equal weights are typically used. See Section 12.7.2 in Ref. [1] for more details.

n_states = 5
weights = np.ones(n_states)/n_states
mycas = mcscf.CASSCF(mf, 4, 4).state_average_(weights)

See examples/mcscf/15-state_average.py for a complete example.

By default, only a single spin and/or point group symmetry is targeted, but it is possible to target a mixture of both:

weights = [.5, .5]
solver1 = fci.direct_spin1_symm.FCI(mol)
solver1.wfnsym= 'A1'
solver1.spin = 0
solver2 = fci.direct_spin1_symm.FCI(mol)
solver2.wfnsym= 'A2'
solver2.spin = 2

mycas = mcscf.CASSCF(mf, 4, 4)
mcscf.state_average_mix_(mycas, [solver1, solver2], weights)
mycas.kernel()

See examples/mcscf/41-state_average.py for a complete example.

Job Control

Optimization Settings

For CASSCF calculations, users may want to modify several of the convergence thresholds such as the energy (conv_tol), the orbital gradient (conv_tol_grad), and the maximum number of MCSCF iterations (max_cycle_macro).

mycas = mcscf.CASSCF(mf, 6, 6)
mycas.conv_tol = 1e-12
mycas.conv_tol_grad = 1e-6
mycas.max_cycle_macro = 25
mycas.kernel()

Initial Guess

Initial guess orbitals for the CASSCF calculation (starting orbitals) may be passed to the kernel member function of an MCSCF object.

mycas = mcscf.CASSCF(myhf, 8, 8)
mycas.kernel(my_custom_mos)

CI coefficients from a previous calculation can also be passed as an initial guess to expedite the calculation:

Examples:

Restarting

Warning

When running large calculations, it’s always recommended that you specify a checkpoint file for your calculation.

mycas.chkfile = "casscf.chk"

Much like scf, if a job is interrupted, users can restart the MCSCF calculations using checkpoint files from a crashed calculation.

from pyscf.lib import chkfile
old_chk_file = "old_casscf.chk"
mycas = mcscf.CASSCF(scf.RHF(mol), 6, 6)
mycas.chkfile = "restarted_casscf.chk"
mo = chkfile.load(old_chk_file, 'mcscf/mo_coeff')
mycas.kernel(mo)

See examples/mcscf/13-restart.py for a complete example.

Restarting calculations can be also be useful when using results from a smaller active space to speed up calculations on a larger one.

Observables and Properties

Wave Function Analysis

The analyze member functions of MCSCF objects prints many useful properties to stdout when the verbosity is >=4.

  1. Natural orbital occupancies

  2. Natural orbital AO expansions

  3. Overlap between canonical MCSCF orbitals and the initial guess orbitals.

  4. Analysis of the CI coefficients, i.e. the leading configurations and their weights

  5. AO populations

  6. Atomic populations

  7. AO spin densities (if applicable)

  8. Atomic spin densities (if applicable)

mycas = myhf.CASCI(6, 8).run()
mycas.verbose = 4
mycas.analyze()

Natural Orbitals

The CAS wave function is invariant under orbital rotation within the inactive core, active, and inactive virtual spaces. The inactive core and virtual orbitals are by default canonicalized, i.e., they are transformed by block-diagonalizing the generalized Fock matrix within the inactive core and virtual subblocks. This also yields energy eigenvalues for the inactive core and virtual orbitals. In contrast, no energy or electron occupation can be assigned to the active orbitals, strictly speaking. The user may request the active orbitals to be transformed to the so-called natural representation, in which the reduced one-body density matrix is diagonal, and the formal natural orbital occupation numbers can be found on its diagonal.

Warning

When mycas.natorb is set, the natural orbitals may NOT be sorted by the active space occupancy.

mycas = mcscf.CASSCF(myhf, 6, 8)
mycas.natorb = True
mycas.kernel()

References

1(1,2)

Trygve Helgaker, Poul Jorgensen, and Jeppe Olsen. Molecular Electronic-Structure Theory. Wiley, 2013. ISBN 9781118531471.

2

Esqc lectures 2019. URL: http://www.esqc.org/?page=lectures.

3

Qiming Sun, Jun Yang, and Garnet Kin-lic Lic Chan. A general second order complete active space self-consistent-field solver for large-scale systems. Chemical Physics Letters, 683:291–299, 2017. arXiv:1610.08394, doi:10.1016/j.cplett.2017.03.004.