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

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?

## 2 Answers

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.

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

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)

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