Fock matrices and SCF procedure

Density matrix

For a given set of molecular orbital (MO) coefficients, the density matrix becomes equal to

\[ D_{\gamma\delta} = \sum_{i=1}^N C_{\gamma i}^* C_{\delta i} \]

where the summation over occupied orbitals include \(\alpha\)- as well as \(\beta\)-spin orbitals. For a closed shell system, these two sets of MO coefficients are equal.

Fock matrix

A density matrix defines a Fock matrix according to

\[ F_{\alpha\beta} = h_{\alpha\beta} + \sum_{\gamma\delta} D_{\gamma\delta} \left[ \langle \alpha \gamma | \beta \delta \rangle - \langle \alpha \gamma | \delta \beta \rangle \right] \]

introducing the one-electron Hamiltonian as well as the Coulomb and exchange two-electron integrals.

Hartree–Fock equation

The Hartree–Fock (HF) equation in the atomic orbital (AO) basis takes the form

\[ \mathbf{F}\mathbf{C} = \mathbf{S} \mathbf{C} \boldsymbol{\varepsilon} \]

The Fock matrix depends on the MO coefficients through the density matrix so this equation needs to be solved iteratively in an SCF procedure. The HF equation corresponds to a matrix eigenvalue problem in a non-orthogonal basis. The overlaps between basis functions are collected in matrix \(\mathbf{S}\). In this exercise, we will go through the steps required to solve the HF equation for the eigenvalues (or orbital energies) collected on the diagonal of matrix \(\boldsymbol{\varepsilon}\) and the associated eigenvectors (or MO coefficients). This exercise will use the results from the associated problem in the problem set.

Exercises

1. Perform an SCF optimization to obtain the Hartree–Fock wave function of water as described here.

2. Diagonalize the overlap matrix, \(\mathbf{S}\), with use of NumPy routine linalg.eigh. The access to the overlap matrix and other SCF tensors is described in here. Check that all eigenvalues are positive and, with use of NumPy routine matmul for matrix multiplication, verify that \(\mathbf{U}^\dagger \mathbf{U} = \mathbf{I}\), where \( \mathbf{U}\) collects the eigenvectors of \(\mathbf{S}\) as columns.

3. Determine the transformation matrix \(\mathbf{X} = \mathbf{S}^{-1/2}\).

4. Determine the transformed Fock matrix \(\mathbf{F}'\).

5. Solve the transformed Hartree–Fock equation \(\mathbf{F}'\mathbf{C}' = \mathbf{C}' \boldsymbol{\varepsilon}\) for eigenvalues \(\boldsymbol{\varepsilon}\) and eigenvectors \(\mathbf{C}'\).

6. Determine \(\mathbf{C}\) from \(\mathbf{C}'\).

7. Determine the density matrix as expressed in terms of MO coefficients in the equation above and check the SCF convergence by comparing to the SCF tensor object.