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Configuration interaction (CISD and FCI)

Modules: :py:mod:`pyscf.ci`, :py:mod:`pyscf.fci`, :py:mod:`pyscf.pbc.ci`

PySCF has separate molecular implementations for configuration interaction singles and doubles (CISD) (:mod:`ci`) and full configuration interaction (FCI) (:mod:`fci`). The functionalities of the CISD implementation are similar to the functionalities of MP2 (:ref:`user_mp2`) and CCSD (:ref:`user_cc`) in PySCF.

Introduction

Configuration interaction (see e.g. :cite:`Knowles1984,SzaOst2012ci,Helgaker2000ci`) is a post-Hartree--Fock method, diagonalising the many-electron Hamiltonian matrix. FCI includes all Slater determinants of appropriate symmetry in the eigenvector basis. CISD includes only those that differ from the Hartree--Fock determinant by at most two occupied spinorbitals.

Configuration interaction singles and doubles (CISD)

A simple example (see :source:`examples/ci/00-simple_cisd.py`) of running a restricted and an unrestricted CISD calculation is

.. literalinclude:: ../../examples/ci/00-simple_cisd.py

which outputs

converged SCF energy = -99.9873974403488
E(RCISD) = -100.196182975762  E_corr = -0.2087855354132202
RCISD correlation energy -0.2087855354132202

namely, the (restricted) Hartree--Fock energy, the RCISD energy and their difference, the RCISD correlation energy, as well as

converged SCF energy = -99.9873974403486  <S^2> = 5.3734794e-13  2S+1 = 1
E(UCISD) = -100.196182975774  E_corr = -0.2087855354254317
UCISD correlation energy -0.20878553542543174

namely, the (unrestricted) Hartree--Fock energy, the UCISD energy and their difference, the UCISD correlation energy.

Note

mycc = mf.CISD().run()

could have been replaced by

myci = pyscf.ci.CISD(mf)
myci.kernel()

or

myci = pyscf.ci.CISD(mf).run()

for the same result.

Note

Density fitting is not implemented for CISD.

Spin symmetry

The CISD module in PySCF supports a number of reference wavefunctions with broken spin symmetry. In particular, CISD can be performed with a spin-restricted, spin-unrestricted, and general (spin-mixed) Hartree-Fock solution, leading to the RCISD, UCISD, and GCISD methods. The small example above shows this for RCISD and UCISD.

The module-level ci.CISD(mf) constructor can infer the correct method based on the level of symmetry-breaking in the mean-field argument. For more explicit control or inspection, the respective classes and functions can be found in cisd.py (restricted), ucisd.py (unrestricted), and gcisd.py (general).

For example, a spin-unrestricted calculation on triplet oxygen can be performed as follows:

from pyscf import gto, scf, ci
mol = gto.M(
    atom = 'O 0 0 0; O 0 0 1.2',  # in Angstrom
    basis = 'ccpvdz',
    spin = 2
)
mf = scf.HF(mol).run() # this is UHF
myci = ci.CISD(mf).run() # this is UCISD
print('UCISD total energy = ', myci.e_tot)

Properties

A number of properties are available at the CISD level.

Unrelaxed 1- and 2-electron reduced density matrices can be calculated. They are returned in the MO basis:

dm1 = myci.make_rdm1()
dm2 = myci.make_rdm2()

Analytical nuclear gradients can be calculated (see :cite:`Osamura1987,Yamaguchi2011` and references therein):

mygrad = myci.nuc_grad_method().run()

Frozen orbitals

By default, CISD calculations in PySCF correlate all electrons in all available orbitals. To freeze the lowest-energy core orbitals, use the frozen keyword argument:

myci = ci.CISD(mf, frozen=2).run()

To freeze occupied and/or unoccupied orbitals with finer control, a list of 0-based orbital indices can be provided as the frozen keyword argument:

# freeze 2 core orbitals
myci = ci.CISD(mf, frozen=[0,1]).run()
# freeze 2 core orbitals and 3 unoccupied orbitals
myci = ci.CISD(mf, frozen=[0,1,16,17,18]).run()

Wavefunction overlap

The following example shows how to evaluate the overlap of two different CISD wavefunctions.

.. literalinclude:: ../../examples/ci/32-wfn_overlap.py

Full configuration interaction (FCI)

A simple example (see :source:`examples/fci/00-simple-fci.py`) of running a restricted and an unrestricted FCI calculation is:

.. literalinclude:: ../../examples/fci/00-simple_fci.py

The second and third FCI calculation in the example show two different ways of running unrestricted FCI calculations.

Setting Spin and Symmetry of Wavefunction

The following example discusses how to set the spin for an FCI calculations:

.. literalinclude:: ../../examples/fci/10-spin.py

and the next example how to set symmetry:

.. literalinclude:: ../../examples/fci/13-wfn_symmetry.py

Properties

Reduced density matrices can be evaluated, see the following example:

.. literalinclude:: ../../examples/fci/14-density_matrix.py