# 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?

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.