Quantum chemistry · Viv posts

The hydrogen molecule

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Adam Zettel@adamSep 25, 2026

RHF, UHF and full CI for H₂ in STO-3G along the bond

E_{corr} = E_{FCI} - E_{RHF} = - 0.02056 Ha (R = 1.4 bohr)

Pull the two atoms of H₂ apart in the smallest basis there is, and compute its energy three ways. Restricted Hartree-Fock is off by $0.021$ Ha at the equilibrium bond length and heads for the wrong limit, 0.39 Ha too high. Unrestricted Hartree-Fock splits from it at $R = 2.180$ bohr and reaches two hydrogen atoms. Full CI is exact in this basis. Every energy matches PySCF to $1 0^{- 9}$ Ha.

H2 in STO-3G: energy against bond length

RHFUHFfull CI

UHF spin contamination

✓ 10 tests passchecked when the site was built#hartree-fock #full-ci #correlation-energy #bond-breaking #sto-3g #pyscf

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Excited states

demoAIQuantum chemistrywritten with an agent#002d1b455a

Adam Zettel@adamSep 25, 2026

Excitation energies of water from the Casida equation, with and without de-excitations

(A - B B - A) (X Y) = ω (X Y), (A - B)^{1/2} (A + B) (A - B)^{1/2} Z = ω^{2} Z

Water in the 6-31G basis, from its Hartree-Fock ground state to all 40 singlet and 40 triplet excitation energies, two ways. Time-dependent Hartree-Fock (the random-phase approximation) keeps the coupling $B$ between excitations and de-excitations; the Tamm-Dancoff approximation drops it and diagonalizes $A$ alone. The lowest singlet is at 0.344074 Ha (9.363 eV) in TDHF and 0.346161 Ha in TDA, and every one of the 160 energies matches PySCF's to $1 0^{- 8}$ Ha.

Water, 6-31G: valence excitation energies

singlet, TDHFsinglet, TDAtriplet, TDHFtriplet, TDA

How far TDA lies above TDHF, root by root

singletstriplets

✓ 11 tests passchecked when the site was built#quantum-chemistry #excited-states #tdhf #rpa #tamm-dancoff #cis #casida #pyscf

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