LESSON NOTES · 03

Roothaan equations and a reproducible SCF cycle

Prerequisites: HF equations; nonorthogonal finite bases

Prerequisite lessons: Hartree–Fock from constrained variation · Variational principle and finite bases

1. Specify the reference and basis

Use a real spatial AO basis and restricted closed-shell HF, two electrons per occupied orbital. Atomic basis functions are generally nonorthogonal. Report basis, charge, multiplicity, reference type, thresholds and final residuals. Basis error and ansatz error are distinct.

\[ \phi_i=\sum_\mu C_{\mu i}\chi_\mu,\quad S_{\mu\nu}=\langle\chi_\mu|\chi_\nu\rangle,\quad C^TSC=I,\qquad FC=SC\epsilon. \]

2. Define the AO density and Fock matrix

Projection of \(f\phi_i=\varepsilon_i\phi_i\) gives the generalized matrix equation. Now switch explicitly to chemists’ real-AO integral notation. The density includes a factor two for closed-shell spatial occupancy. Do not transplant spin-orbital formulas without converting conventions.

\[ \begin{aligned}(\mu\nu|\lambda\sigma)&=\iint\chi_\mu(r_1)\chi_\nu(r_1)r_{12}^{-1}\chi_\lambda(r_2)\chi_\sigma(r_2)dr_1dr_2,\\P_{\mu\nu}&=2\sum_{i\in occ}C_{\mu i}C_{\nu i},\\F_{\mu\nu}&=h_{\mu\nu}+\sum_{\lambda\sigma}P_{\lambda\sigma}[(\mu\nu|\lambda\sigma)-\tfrac12(\mu\lambda|\nu\sigma)],\\E_{RHF}&=\tfrac12\sum_{\mu\nu}P_{\mu\nu}(h_{\mu\nu}+F_{\mu\nu})+V_{NN},\quad N=\operatorname{Tr}(PS).\end{aligned} \]

3. A reproducible iteration

1. Compute overlap, one-electron and repulsion integrals; document the threshold used to remove near-dependent basis directions.
2. In the retained subspace form \(X=S^{-1/2}\) and an initial \(P\).
3. Build \(F[P]\), diagonalize \(X^TFX\), transform \(C=XC\prime\), occupy orbitals and form \(P_{new}\).
4. Test energy, density change and \(FPS-SPF\); mix or use DIIS if needed, then repeat.
5. Verify electron count, orthonormality and relevant stability. Solving once is insufficient because \(F\) depends on occupied coefficients.

Teaching schematic drawn from the equations or algorithm steps in this lesson; not measured data.

Worked generalized eigenproblem

A toy two-function one-electron matrix, not molecular benchmark data, has diagonal \(\alpha\), off-diagonal \(\beta\), and overlap \(s\) with \(|s|<1\). Generalized normalization fixes the symmetric/antisymmetric eigenvectors. For \(\alpha=-1\), \(\beta=-0.2\), \(s=0.1\) atomic units, \(E_+=-1.090909\), \(E_-=-0.888889\). Ignoring \(S\) instead gives \(-1.2,-0.8\), the answer to a different problem.

\[ H=\begin{pmatrix}\alpha&\beta\\\beta&\alpha\end{pmatrix},\quad S=\begin{pmatrix}1&s\\s&1\end{pmatrix},\quad c_\pm=\frac{(1,\pm1)^T}{\sqrt{2(1\pm s)}},\quad E_\pm=\frac{\alpha\pm\beta}{1\pm s}. \]

Check yourself and limitations

Is \(\operatorname{Tr}P=N\) generally?

Answer and reasoning

No, use \(\operatorname{Tr}(PS)\).

Does a larger basis fix missing correlation?

Answer and reasoning

No: it improves representation within the same ansatz.

Is energy change alone enough?

Answer and reasoning

No, examine the commutator residual and density; small energy changes can hide poor stationarity.

References and further reading

• Florida State University: HF-SCF tutorial

Derivations and numerical examples here are original teaching constructions, not copied passages or reported research data.

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