The Roothaan–Hall Equations

In the LCAO (Linear Combination of Atomic Orbitals) approximation, molecular orbitals are expanded in terms of atomic orbitals:

$${\varphi }_i=\sum _{\mu }{C}_{\mu i}{\chi }_{\mu }$$

This leads to the Roothaan–Hall matrix equations:

$$\mathbf{FC}=\mathbf{SC}\mathbf{\epsilon }$$

where:

• $\mathbf{F}$ is the Fock matrix
• $\mathbf{C}$ is the MO coefficient matrix
• $\mathbf{S}$ is the overlap matrix
• $\mathbf{\epsilon }$ is the diagonal matrix of orbital energies

Derivation

The HF energy for a single determinant is:

$$E=\sum _i⟨i|\hat{h}|i⟩+\frac{1}{2}\sum _{i,j}\left[⟨ij|ij⟩-⟨ij|ji⟩\right]$$

Minimizing $E$ with respect to orbital rotations subject to orthonormality constraints $⟨{\varphi }_i|{\varphi }_j⟩={\delta }_{ij}$ leads to the eigenvalue problem above.

Solution Procedure

The Roothaan–Hall equations are solved iteratively (SCF procedure):

1. Initial guess for $\mathbf{C}$ (core Hamiltonian, Hückel, SAD, or read from checkpoint)
2. Build density matrix: $P_{\mu \nu }=2{\sum }_{i(\text{occ})}{C}_{\mu i}{C}_{\nu i}$
3. Build Fock matrix using current density
4. Solve $\mathbf{FC}=\mathbf{SC}\mathbf{\epsilon }$ for new $\mathbf{C}$
5. Check convergence (density change, energy change, gradient norm)
6. Repeat from step 2 if not converged

Symmetric Orthogonalization

To solve the generalized eigenvalue problem, we transform to an orthonormal basis:

$$\mathbf{S}=\mathbf{Us}{\mathbf{U}}^{†}$$

$$\mathbf{X}=\mathbf{U}{\mathbf{s}}^{-1/2}{\mathbf{U}}^{†}$$

Then ${\mathbf{F}}^{\prime }={\mathbf{X}}^{†}\mathbf{FX}$ is diagonalized:

$${\mathbf{F}}^{\prime }{\mathbf{C}}^{\prime }={\mathbf{C}}^{\prime }\mathbf{\epsilon }$$

And the MO coefficients in the original basis are $\mathbf{C}=\mathbf{X}{\mathbf{C}}^{\prime }$.

Symmetry Adaptation

When point-group symmetry is enabled, the basis functions are symmetry-adapted, block-diagonalizing both $\mathbf{F}$ and $\mathbf{S}$ by irreducible representation. Each block is diagonalized separately, producing canonical-like orbital energies within each irrep. This is essential for post-HF calculations (MP2, CCSD(T)) which require symmetry-labeled orbitals.