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:mod:`mcscf` --- Multi-configurational self-consistent field

The :mod:`mcscf` implements orbital optimization for MCSCF and CASSCF. 1-step (combined orbital and wavefunction optimization) and 2-step algorithms (alternating orbital and wavefunction optimization) are available. Different kinds of active space solvers can be used with this module.

For example, a simple CASCI calculation can be run as:

import pyscf

mol = pyscf.M(
    atom = 'O 0 0 0; O 0 0 1.2',
    basis = 'ccpvdz',
    spin = 2)

myhf = mol.RHF().run()

# 6 orbitals, 8 electrons
mycas = myhf.CASCI(6, 8).run()

and a simple CASSCF can be run as:

import pyscf

 mol = pyscf.M(
     atom = 'O 0 0 0; O 0 0 1.2',
     basis = 'ccpvdz',
     spin = 2)

 myhf = mol.RHF().run()

 # 6 orbitals, 8 electrons
 mycas = myhf.CASSCF(6, 8).run()

The CASSCF orbital optimization is general and can be combined with many different solvers, such as DMRG and selected CI solvers. Optimized orbitals are stored in the attribute :attr:`mycas.mo_coeff`.

Examples

CASSCF active space solver

.. automodule:: pyscf.mcscf

DMRG solver

FCIQMC solver

State-average FCI solver

State-average with mixed solver

Symmetry broken

Initial guess

Canonical orbitals

Orbital canonicalization are controlled by parameters :attr:`mc.canonicalization` and :attr:`mc.natorb` (assuming the MCSCF object is mc). The order of orbitals are affected by the parameter :attr:`mc.sorting_mo_energy`.

  • canonicalization: This flag canonicalizes orbitals in core and external space using general Fock matrix.
  • natorb: Transforms active orbitals using 1-particle density matrices.
  • sorting_mo_energy: Sort orbitals based on the diagonal elements of the general Fock matrix.
  • Enabling natorb or sorting_mo_energy may slightly change the total energy of DMRG solver or selected CI solver.

General Fock matrix is defined as

\mathbf{F} &= \mathbf{h}_{core} + \mathbf{J} - \mathbf{K} \\
J_{pq} &= \sum_{rs} (pq|rs) \gamma_{sr} \\
K_{pq} &= \sum_{qr} (pq|rs) \gamma_{qr} \\

\gamma is the total density matrix which includes the doubly occupied core density matrix and correlated density matrix in active space.

If :attr:`mc.canonicalization` is enabled, CASCI/CASSCF will call the :func:`mc.canonicalize` function to diagonalize orbitals in core space and external space. Orbitals in active space are not changed if merely setting :attr:`mc.canonicalization`. In the attribute :attr:`mc.mo_energy`, eigenvalues of general Fock matrix for core and external subspaces are stored in the corresponding sub-sectors. The sector of active space in :attr:`mc.mo_energy` stores the expectation value of general Fock \langle \phi|F|\phi\rangle. By default, :attr:`mc.canonicalization` is enabled because the canonicalized MCSCF orbitals can simplify the implementations of MRPT (NEVPT2) methods.

:attr:`mc.natorb` controls whether the CASCI/CASSCF active space orbitals are transformed to natural orbitals w.r.t. the correlated density matrix. When this parameter is enabled, the natural orbitals will be stored in the active part of the attribute :attr:`mc.mo_coeff` and the CI coefficients :attr:`mc.ci` (if applicable) will be transformed accordingly. By default :attr:`mc.natorb` is disabled because natural orbitals may not be favored by total energy for an arbitrary CI solver. We make this default value to ensure that a CASCI calculation followed by a CASSCF calculation (e.g. DMRG-CASSCF then DMRG-CASCI) produces results same to the CASSCF results. The CASCI calculation may produce different The value of :attr:`mc.natorb` does not affect (the default) FCI solver. But this is not true for external large active space solvers such as DMRG, selected CI methods. It is recommended to disable :attr:`mc.natorb` in these calculations.

Following presents what the :attr:`mc.mo_coeff` would be like for different combinations of :attr:`mc.canonicalization` and :attr:`mc.natorb` in a CASCI calculation:

  • mc.canonicalization = False and mc.natorb = False:

The resultant orbitals :attr:`mc.mo_coeff` are identical to the input orbitals. If the CASCI was initialized with a RHF calculation, :attr:`mc.mo_coeff` points to RHF orbitals.

  • mc.canonicalization = True and mc.natorb = False:

Core part and external part of :attr:`mc.mo_coeff` are canonicalized orbitals, which diagonalize the core and external blocks of general Fock matrix. The orbitals in active space are identical to the active orbitals in the input.

  • mc.canonicalization = False and mc.natorb = True

Core and external part of :attr:`mc.mo_coeff` are identical to the core and external part of the input orbitals. Active space orbitals are transformed to the natural orbitals of the correlated density matrix.

  • mc.canonicalization = True and mc.natorb = True

:attr:`mc.mo_coeff` are completely different to the input orbitals.

Please note that elements of :attr:`mc.mo_energy` may not be sorted ascendantly. Parameter :attr:`mc.sorting_mo_energy` can affect the ordering of MCSCF orbitals when :attr:`mc.canonicalization` or :attr:`mc.natorb` is enabled.

By default, canonical orbitals in the core and external space are sorted by the orbital energies (from low to high) and the natural orbitals in the active space are sorted by natural occupations (from large to small). If point group symmetry is enabled in the calculation, canonical orbitals are sorted within each symmetry sector only (rather than the entire core or external space). Irreducible representation labels (can be accessed via :attr:`mc.mo_coeff.orbsym`) are assigned to orbitals in the initial guess and they will not be changed in the MCSCF optimization and the subsequent canonicalization procedure. Setting :attr:`mc.sorting_mo_energy` (though not recommended) can force the orbitals to be sorted against energy (or occupations in active space) regardless whether the point group symmetry is used.

In certain scenario, you may want to enable both :attr:`mc.natorb` and :attr:`mc.sorting_mo_energy`. examples/dmrg/31-cr2_scan/cr2-scan.py provides one example that needs both parameters. In that example, the dissociation curve of Cr dimer was scanned using heat-bath selected-CI method in which the active space of selected-CI-CASSCF was gradually enlarged in a series of CASSCF calculations. Since the selected-CI algorithm depends on the initial single determinant, the orbital ordering does matter to the final CASSCF results. :attr:`mc.natorb` and :attr:`mc.sorting_mo_energy` have to be enabled to make sure that the each selected-CI starts from the similar initial reference at each point on the dissociation curve. Without these settings, the differences in the orbital ordering can lead to discontinuous potential energy curve.

Program reference

CASCI

.. automodule:: pyscf.mcscf.casci
   :members:


.. automodule:: pyscf.mcscf.casci_symm
   :members:


.. automodule:: pyscf.mcscf.ucasci
   :members:

CASSCF

.. automodule:: pyscf.mcscf.mc1step
   :members:


.. automodule:: pyscf.mcscf.mc1step_symm
   :members:


.. automodule:: pyscf.mcscf.umc1step
   :members:


.. automodule:: pyscf.mcscf.mc_ao2mo
   :members:


.. automodule:: pyscf.mcscf.umc_ao2mo
   :members:

addons

.. automodule:: pyscf.mcscf.addons
   :members: