FR13
Molecular electrons and the breaking of a bond
A shared orbital describes the hydrogen bond well near its usual length. Stretch the same bond, and the electrons need a different way to stay apart. Start with one electron around one nucleus, build the simplest molecular calculation, then follow exactly where its assumptions stop working.
Pull two hydrogen atoms apart
At the start, the electrons gain by spreading between the nuclei. Far apart, one electron should stay with each atom. A useful molecular theory must describe both situations and the continuous change between them. Hydrogen is small enough that we can compare an approximate answer with the best possible answer in exactly the same basis.
One shared doubly occupied orbital
At infinite separation, half the weight has one electron on each atom; half has both on the same atom.
A correlated singlet at separation
Mix bonding and antibonding pairs with opposite signs. The double-occupancy terms cancel.
- One electron on each atom
- Both electrons on one atom
The first two chapters build orbitals and matrix elements. Chapters three to six follow self-consistency, spin symmetry, configuration mixing and perturbation theory. The final three explain what DFT changes, how to compare entire energy curves, and how an exact condition on fractional spin exposes a difficulty in approximate functionals.
L1 · 1
Choosing an orbital for one hydrogen electron
Use the hydrogen atom to understand probability, trial orbitals and the variational energy before combining two atomic functions.
L1 · 2
Turning a molecule into a small matrix problem
Fix the nuclei, choose STO-3G functions and identify the overlap, one-electron and repulsion integrals that define the calculation.
L2 · 3
Making the density and its field agree
Derive the restricted Hartree–Fock field, follow SCF residuals and see why orbital energies cannot simply be added into a total energy.
L2 · 4
Why a shared orbital struggles when the bond stretches
Expand the restricted hydrogen state at separation, then see how UHF lowers its energy and introduces spin contamination.
L3 · 5
Mixing configurations to keep the electrons apart
Build the small FCI problem, derive the correlated singlet and use natural occupations to follow the growth of static correlation.
L3 · 6
When a small correlation correction becomes too large
Derive a second-order energy correction and connect shrinking orbital gaps to the failure of restricted MP2 for a stretched bond.
L3 · 7
What density functional theory changes
Build the Kohn–Sham energy expression, distinguish exact DFT from PBE and compare a versioned restricted calculation with same-basis FCI.
L3 · 8
An accurate energy can still give the wrong bond force
Compare hydrogen energy curves, force errors and natural occupations, then separate electronic-method error from the limits of a small basis.
L4 · 9
A fractional-spin test for a broken bond
Read the exact constancy condition from Cohen, Mori-Sánchez and Yang, then reproduce a bounded one-orbital hydrogen calculation.
Follow the calculation in three experiments
- Watch a density and its self-consistent field agree
- Stretch a bond and compare RHF, UHF and FCI
- Compare energy errors, forces and occupations along one geometry set
Earlier models and the calculation used here
The Morse wavepacket experiment describes nuclear motion on a prescribed empirical potential. The XPS experiment relates a synthetic spectrum to chemical shifts. This course adds finite-basis electronic calculations with fixed nuclei. Its orbitals and densities are calculated distributions, and the reference energies are numerical calculations, not experimental measurements.
Sources and calculation scope
- Sandbox Physics, FR13 reference inputs, raw numerical records and reproduction guide (2026)
Original generated data: H2/STO-3G method energies, forces and occupations; independent Psi4 checks; basis and DFT-grid checks; fixed-orbital fractional-spin data. The guide links scripts, raw outputs, runtime versions and data licenses.
- A. E. DePrince III, The Hartree–Fock Self-Consistent Field Procedure
Author tutorial: electronic Hamiltonian, variational principle, density matrices, SCF, basis convergence and spin-symmetry breaking. The browser calculation specializes these equations to two hydrogen STO-3G functions.
- Psi4 manual, HF: Hartree–Fock Theory
Theory and numerical methods sections: restricted and unrestricted references, generalized eigenproblem, total energy and convergence. The reproducible reference manifest records the installed version and settings.
- Psi4 manual, CI: Configuration Interaction
CI expansion and full configuration interaction within a chosen orbital space. Here FCI is exact only for the stated finite-basis Hamiltonian.
- C. Møller and M. S. Plesset, Note on an Approximation Treatment for Many-Electron Systems (1934)
Original perturbation partition. The two-orbital worked example uses ordinary restricted second-order perturbation theory with conventional integrals.
- P. Hohenberg and W. Kohn, Inhomogeneous Electron Gas (1964)
Ground-state density and the exact density variational principle. This existence result does not provide an approximate functional.
- W. Kohn and L. J. Sham, Self-Consistent Equations Including Exchange and Correlation Effects (1965)
Auxiliary noninteracting orbitals and the separation of kinetic, electrostatic and exchange-correlation energies.
- J. P. Perdew, K. Burke and M. Ernzerhof, Generalized Gradient Approximation Made Simple (1996)
Published PBE generalized-gradient functional, with the 1997 erratum. The method-error experiment selects PBE; its saved numerical record fixes the library, grid and spin restriction.
- A. J. Cohen, P. Mori-Sánchez and W. Yang, Fractional spins and static correlation error in density functional theory (2008)
Locked preprint v1, 12 May 2008; published as J. Chem. Phys. 129, 121104, DOI 10.1063/1.2987202. The final chapter studies the exact constancy condition and a newly computed finite-basis illustration of the hydrogen-atom figure.
- Basis Set Exchange, STO-3G basis data
Hydrogen STO-3G contraction: one contracted s function per atom, made from three Gaussian primitives. Basis coefficients and normalization are fixed in the calculation manifest.