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LESSON NOTES · 04

4. Roothaan equations and a reproducible SCF cycle

Position in the course: Lesson 4 of 10. Complete the preceding derivation and use the explained exercises to check understanding.

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.

Roothaan equations and a reproducible SCF cycle — schematic under the stated model assumptions

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

4. 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.

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

5. 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.

6. Residuals, acceleration and multiple solutions

Self-consistency means that the occupied subspace is invariant under the Fock matrix built from its own density. In an orthogonal basis, stationarity is measured by a commutator \([F,P]\). Returning to an AO basis gives the residual \(FPS-SPF\). It tests occupied-virtual mixing gradients; the density difference between iterations is a separate convergence indicator. Small energy changes alone can occur during a slow oscillation or near a flat direction.

DIIS combines previous Fock matrices using coefficients whose sum is one, minimizing an extrapolated residual norm. It is a convergence accelerator, not a physical averaging of different states. Early iterations with nearly dependent residuals can make the extrapolation ill conditioned. Damping, level shifts or a revised initial guess may help, but document the final undistorted convergence criteria rather than reporting only the stabilization settings.

Different guesses can converge to different stationary determinants. Compare physically relevant spin, charge and occupation patterns, and perform a stability check where appropriate. A converged solution can be a saddle point of the unrestricted variational space. Convergence is a numerical statement about a chosen equation, whereas stability asks how the energy changes under allowed orbital rotations.

Additional exercise: In an orthogonal basis with \(P=\operatorname{diag}(2,0)\) and \(F=\begin{pmatrix}a&b\\b&d\end{pmatrix}\), compute \(FP-PF=\begin{pmatrix}0&-2b\\2b&0\end{pmatrix}\). The residual vanishes only when the occupied and virtual directions do not mix, \(b=0\). Changing \(a\) and \(d\) alone does not eliminate an off-diagonal gradient. This toy calculation explains why eigenvalue or energy checks cannot replace residual checks.

7. References and further reading

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

Quantum mechanics · Density functional theory · Quantum Monte Carlo · Gaussian

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