Examples#

This gallery walks through every public feature of chemistrykit.quantum: particle-in-a-box models (with the free-electron model of conjugated-dye color); the quantum harmonic oscillator compared against the exact Morse potential; the rigid rotor; hydrogen-like orbitals; Huckel molecular-orbital theory and its aromaticity rule; a minimal variational treatment of H2+; and Rayleigh-Schrodinger perturbation theory for the anharmonic oscillator.

Each script in this gallery is self-contained and can be run directly with python examples/quantum/<section>/<script>.py. Every script also carries an RST module docstring as its title/description and uses # %% markers to split narrative text from code, which is exactly what Sphinx-Gallery renders into the pages below – the script is the source of truth for what you see, not a copy of it.

Sections#

particle_in_box – 1D/3D particle-in-a-box energy levels and wavefunctions, and Kuhn’s free-electron model of a conjugated dye’s UV-Vis absorption wavelength.
harmonic_oscillator – the quantum harmonic oscillator’s evenly spaced vibrational levels, compared against the exact (anharmonic) Morse-potential levels for the same force constant.
rigid_rotor – rigid-rotor rotational energy levels, degeneracies, and the evenly spaced microwave absorption spectrum they predict.
hydrogenlike – hydrogen-like radial wavefunctions, radial distribution functions, and orbital energies for 1s/2s/2p/3d.
huckel – Huckel molecular-orbital theory for butadiene and benzene: building and diagonalizing the secular matrix, and checking Huckel’s 4n+2 aromaticity rule against the computed spectrum.
hartree_fock – a minimal 2-Gaussian LCAO variational treatment of H2+: solving HC=SCE and variationally optimizing the orbital exponent to improve on a naive guess.
perturbation – Rayleigh-Schrodinger perturbation theory for the quartic anharmonic oscillator, checked against exact numerical diagonalization in a truncated basis.

Harmonic oscillator vs. Morse potential#

The quantum harmonic oscillator’s perfectly evenly spaced vibrational levels, compared against the exact (anharmonic) Morse-potential levels for the same force constant – showing how real bonds’ level spacing shrinks toward dissociation.

Harmonic vs. Morse vibrational levels: anharmonicity toward dissociation

Minimal variational H2+ and the LCAO method#

The one-electron H2+ molecular ion in a small Gaussian basis: LCAO bonding and antibonding orbitals, bonding density, the Born-Oppenheimer potential curve, Gaussian-type orbital integrals, the Roothaan-Hall secular equation HC=SCE, and the shared electron pair.

Hund-Mulliken LCAO molecular orbitals of H2+: bonding and antibonding

Heitler-London’s insight: bonding vs. antibonding electron density

Heitler-London's insight: bonding vs. antibonding electron density

Born-Oppenheimer clamped-nuclei potential curve of H2+

Boys’s Gaussian-type orbitals and the Gaussian product theorem

Boys's Gaussian-type orbitals and the Gaussian product theorem

Roothaan-Hall equations: solving HC = SCE in a finite basis

Lewis’s shared electron pair: two opposite-spin electrons in one bonding orbital

Lewis's shared electron pair: two opposite-spin electrons in one bonding orbital

Variational helium atom#

The Kellner-Hylleraas effective-nuclear-charge treatment of helium and its two-electron isoelectronic ions: electron screening from the variational principle.

Kellner and Hylleraas: the variational helium atom and its screened charge

Huckel molecular-orbital theory#

Huckel pi-electron theory for conjugated molecules: the 4n+2 aromaticity rule checked against computed spectra, resonance (delocalization) energies, the Coulson and Frost-circle closed forms, and Fukui’s frontier-orbital reactivity index.

Huckel’s 4n+2 aromaticity rule from the computed pi spectrum

Huckel's 4n+2 aromaticity rule from the computed pi spectrum

Pauling’s resonance energy as Huckel delocalization energy

Pauling's resonance energy as Huckel delocalization energy

Coulson’s polyene formula and the Frost circle

Coulson's polyene formula and the Frost circle

Fukui’s frontier orbitals: why naphthalene reacts at the alpha position

Fukui's frontier orbitals: why naphthalene reacts at the alpha position

Hydrogen-like atoms#

Hydrogen-like radial wavefunctions, radial distribution functions, and orbital energies for 1s/2s/2p/3d, plus the exact ground-state ionization energy.

Schrodinger’s hydrogen atom: radial wavefunctions and orbital energies

Schrodinger's hydrogen atom: radial wavefunctions and orbital energies

Particle in a box#

1D/3D particle-in-a-box energy levels and wavefunctions, cubic-box degeneracy, and Kuhn’s free-electron model of a conjugated dye’s UV-Vis absorption wavelength.

Solving Schrodinger’s equation for a particle in a box

Solving Schrodinger's equation for a particle in a box

Kuhn’s free-electron model of conjugated dye color

Kuhn's free-electron model of conjugated dye color

Rayleigh-Schrodinger perturbation theory#

First-order perturbation theory for the quartic anharmonic oscillator, checked against exact numerical diagonalization of the full Hamiltonian in a truncated harmonic-oscillator basis.

Rayleigh-Schrodinger perturbation theory vs. exact diagonalization for the quartic oscillator

Rigid rotor#

Rigid-rotor rotational energy levels, their (2J+1) degeneracy, and the evenly spaced microwave absorption spectrum they predict for a diatomic molecule.

Dennison’s rigid rotor: rotational levels and the microwave spectrum

Dennison's rigid rotor: rotational levels and the microwave spectrum

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