What do the rows and columns of a Fock matrix represent?

Question

I am using the pyscf code, where the Fock matrix can be obtained by:

from pyscf import gto, scf
mol = gto.Mole()
mol.atom = geometry
mol.basis = '3-21G'
mol.build

mean_field = scf.RHF(mol)
mean_field.scf()

Fao = mean_field.get_fock()
print(Fao)

Where geometry can be set for a system of interest, and the basis set 3-21G can easily be changed.

I have realised I do not fully understand what the rows and columns of the Fock matrix actually represent, and I have read that this is in the atomic basis and not the molecular basis?

How does this relate to the molecular orbital coefficients obtained by: mo_coeff = mean_field.mo_coeff, having already understood that a realtionship can be obtained by the Roothan-Hall equations FC = SCe?

Answer 1

The short answer is: it is the matrix representation of the Fock operator in the given basis set, in this case, the atomic orbital (AO) basis. The Fock operator itself is a mean-field, independent particle approximation to the electronic Hamilton operator of the system (with other approximations beyond the scope of this Q&A).

The rows/columns (the matrix must be Hermitian, therefore it does not matter) thus refer to one basis function each. The equation $$ \mathbf{F}\mathbf{C} = \mathbf{S}\mathbf{C}\mathbf{\epsilon} $$ is a special eigenvalue problem, which can be similarity-transformed to an orthogonalized basis set to read $$ \mathbf{F'}\mathbf{C'} = \mathbf{C'}\mathbf{\epsilon} $$ which is a standard eigenvalue problem, solvable by diagonalization techniques. The $\mathbf{C}$ are the canonical molecular orbital (MO) coefficients for the non-orthogonal AO basis set.

Note that in the canonical MO basis set, the Fock matrix is diagonal by definition. Other choices for MO basis sets exist, such as localized MOs.

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Source: A Szabo, NS Ostlund: Modern Quantum Chemistry, Dover Publications, 1996.

Answer 2

Formally, the Fock matrix is the density matrix derivative of the Hartree-Fock or Kohn-Sham energy functional. The Fock matrix returned by PySCF is in the atomic-orbital basis, which is actually the same as the molecular basis. If the orbitals satisfy the self-consistent field equations, then in the molecular orbital basis the Fock matrix is diagonal.

See our open access overview on SCF calculations for more information: Molecules 25, 1218 (2020)