:orphan:

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.


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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.


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    <div class="sphx-glr-thumbcontainer" tooltip="Compares QuantumHarmonicOscillator&#x27;s perfectly evenly spaced levels against MorseOscillator&#x27;s exact (anharmonic) levels at the same force constant (~chemistrykit.quantum.systems.harmonic_oscillator.compare_harmonic_vs_morse), using HCl-like parameters, and shows the harmonic-oscillator ground-state wavefunction.">

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  .. image:: /api/gallery/quantum/harmonic_oscillator/images/thumb/sphx_glr_plot_01_harmonic_vs_morse_thumb.png
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  :doc:`/api/gallery/quantum/harmonic_oscillator/plot_01_harmonic_vs_morse`

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      <div class="sphx-glr-thumbnail-title">Harmonic vs. Morse vibrational levels: anharmonicity toward dissociation</div>
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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.


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    <div class="sphx-glr-thumbcontainer" tooltip="Hund and Mulliken&#x27;s molecular-orbital theory builds one-electron orbitals spread over the whole molecule as a linear combination of atomic orbitals, \psi_\pm=c_A\chi_A\pm c_B\chi_B. H2PlusVariational does this for H2+ with one Gaussian orbital per proton: diagonalizing the 2x2 problem gives the in-phase bonding orbital (equal coefficients, energy below the isolated-atom level) and the out-of-phase antibonding orbital (opposite coefficients, energy above it). Following both orbital energies from short bond lengths out to separated atoms gives Mulliken&#x27;s correlation diagram for the simplest molecule.">

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  .. image:: /api/gallery/quantum/hartree_fock/images/thumb/sphx_glr_plot_01_lcao_molecular_orbitals_thumb.png
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  :doc:`/api/gallery/quantum/hartree_fock/plot_01_lcao_molecular_orbitals`

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      <div class="sphx-glr-thumbnail-title">Hund-Mulliken LCAO molecular orbitals of H2+: bonding and antibonding</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Walter Heitler and Fritz London&#x27;s 1927 quantum-mechanical treatment of H2 -- the paper usually credited with founding quantum chemistry as a field -- showed for the first time why two hydrogen atoms bind: the symmetric (in-phase) combination of their two atomic orbitals builds up electron density in the region between the nuclei, lowering the energy below that of the separated atoms, while the antisymmetric (out-of-phase) combination depletes density there and raises the energy instead. Their original calculation used two-electron integrals this package does not implement (chemistrykit.quantum.utils.basis_sets stops at the one-electron integrals H2PlusVariational needs for H2+), but the same qualitative physics -- constructive vs. destructive interference of atomic orbitals along the bond axis -- is exactly what H2+&#x27;s own bonding and antibonding molecular orbitals show, using the genuine LCAO coefficients solve computes, not an assumed equal-weight combination.">

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  .. image:: /api/gallery/quantum/hartree_fock/images/thumb/sphx_glr_plot_02_bonding_antibonding_density_thumb.png
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  :doc:`/api/gallery/quantum/hartree_fock/plot_02_bonding_antibonding_density`

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      <div class="sphx-glr-thumbnail-title">Heitler-London's insight: bonding vs. antibonding electron density</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Born and Oppenheimer separated the fast electrons from the slow nuclei: first solve the electronic problem with the nuclei clamped at a fixed separation R, then let the nuclei move on the resulting potential energy curve U(R)=E_{el}(R)+e^2/4\pi\varepsilon_0R. Here each point of the curve is a separate clamped-nuclei calculation with H2PlusVariational (``bond_length`` is a fixed input, never a dynamical variable). The curvature at the minimum then gives a nuclear vibrational frequency, and because the electronic curve does not depend on nuclear mass, H2+ and D2+ share the same curve but have different zero-point energies.">

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  .. image:: /api/gallery/quantum/hartree_fock/images/thumb/sphx_glr_plot_03_born_oppenheimer_potential_curve_thumb.png
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  :doc:`/api/gallery/quantum/hartree_fock/plot_03_born_oppenheimer_potential_curve`

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      <div class="sphx-glr-thumbnail-title">Born-Oppenheimer clamped-nuclei potential curve of H2+</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="S. F. Boys (1950) proposed building molecular wavefunctions from Gaussian functions e^{-\alpha r^2} instead of the physically natural Slater-type e^{-\zeta r}. The payoff is the Gaussian product theorem: the product of two Gaussians on different centres is a single Gaussian on a point between them, so every multi-centre integral reduces to a closed form. This example checks the theorem numerically, checks the closed-form overlap integral (~chemistrykit.quantum.utils.basis_sets.overlap_integral) and the Boys function (~chemistrykit.quantum.utils.basis_sets.boys_f0) against brute-force quadrature, and shows the price paid: one Gaussian has no cusp at the nucleus, while a contraction of three (STO-3G) comes close to the Slater 1s orbital.">

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  .. image:: /api/gallery/quantum/hartree_fock/images/thumb/sphx_glr_plot_04_boys_gaussian_orbitals_thumb.png
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  :doc:`/api/gallery/quantum/hartree_fock/plot_04_boys_gaussian_orbitals`

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      <div class="sphx-glr-thumbnail-title">Boys's Gaussian-type orbitals and the Gaussian product theorem</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Roothaan and Hall turned the molecular orbital problem into matrix algebra: expand each orbital in a finite, non-orthogonal basis and the variational condition becomes the generalized eigenvalue problem HC=SCE, solved here by solve_secular_equation. For the one-electron H2+ the Fock matrix is just the core Hamiltonian, so the equation is solved exactly once in each basis. The example shows that the overlap matrix S matters, that Lowdin&#x27;s symmetric orthogonalization S^{-1/2}HS^{-1/2} gives the same answer, and that enlarging the basis lowers the energy toward the exact H2+ value, as the variational principle requires.">

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  .. image:: /api/gallery/quantum/hartree_fock/images/thumb/sphx_glr_plot_05_roothaan_hall_secular_equation_thumb.png
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  :doc:`/api/gallery/quantum/hartree_fock/plot_05_roothaan_hall_secular_equation`

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      <div class="sphx-glr-thumbnail-title">Roothaan-Hall equations: solving HC = SCE in a finite basis</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="G. N. Lewis&#x27;s 1916 covalent bond is a pair of electrons shared between two atoms. In molecular-orbital language that pair is two electrons of opposite spin in the same bonding orbital. Using the two orbitals of a hydrogen-like diatomic from H2PlusVariational (at the H2 bond length, electron-electron repulsion neglected), fill them with 1 to 4 electrons: the bond order (n_{bonding}-n_{antibonding})/2 is 1 exactly when a single Lewis pair sits in the bonding orbital, and adding a second pair to the antibonding orbital cancels the bond (the He2 case).">

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  :doc:`/api/gallery/quantum/hartree_fock/plot_06_lewis_shared_electron_pair`

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      <div class="sphx-glr-thumbnail-title">Lewis's shared electron pair: two opposite-spin electrons in one bonding orbital</div>
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Variational helium atom
-------------------------

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


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    <div class="sphx-glr-thumbcontainer" tooltip="Helium was the first test of quantum mechanics beyond one electron. Kellner (1927) and Hylleraas (1928-1929) attacked it with the variational method: take both electrons in hydrogen-like 1s orbitals but let the nuclear charge they &quot;see&quot; be an adjustable \zeta. The energy E(\zeta)=(\zeta^2-2Z\zeta+\tfrac{5}{8}\zeta)E_h (~chemistrykit.quantum.systems.helium.helium_like_variational_energy) is lowest at \zeta=Z-5/16=27/16 for helium (~chemistrykit.quantum.systems.helium.optimize_helium_like_effective_charge): each electron shields the other from 5/16 of a nuclear charge. The result, -2.848\,E_h, is within 2% of the exact -2.9037\,E_h, which Hylleraas approached by adding the interelectronic distance to the trial function.">

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  :doc:`/api/gallery/quantum/helium/plot_01_hylleraas_variational_helium`

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      <div class="sphx-glr-thumbnail-title">Kellner and Hylleraas: the variational helium atom and its screened charge</div>
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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.


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    <div class="sphx-glr-thumbcontainer" tooltip="Erich Huckel&#x27;s 1931 pi-electron theory reduces a planar conjugated ring to a small matrix eigenvalue problem: one 2p_z orbital per carbon, Coulomb integral \alpha on the diagonal, and resonance integral \beta only between bonded neighbours (~chemistrykit.quantum.systems.huckel.HuckelSystem). Filling the resulting levels two electrons at a time shows why rings with 4n+2 pi electrons (benzene, 6) form a closed shell while rings with 4n (cyclobutadiene, 4) leave two electrons unpaired in a degenerate pair. is_aromatic_by_huckel_rule checks the rule against the computed degeneracies, not only by counting electrons.">

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  .. image:: /api/gallery/quantum/huckel/images/thumb/sphx_glr_plot_01_huckel_aromaticity_rule_thumb.png
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  :doc:`/api/gallery/quantum/huckel/plot_01_huckel_aromaticity_rule`

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      <div class="sphx-glr-thumbnail-title">Huckel's 4n+2 aromaticity rule from the computed pi spectrum</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Linus Pauling&#x27;s The Nature of the Chemical Bond made &quot;resonance energy&quot; the chemist&#x27;s measure of the extra stability a molecule gains because its true structure is a superposition of several Lewis structures (benzene&#x27;s two Kekule structures foremost). In Huckel theory that stabilization is the delocalization energy (~chemistrykit.quantum.systems.huckel.HuckelSystem.delocalization_energy): the computed pi energy minus that of the same electrons held in isolated, ethylene-like double bonds, 2(\alpha+\beta) per bond.">

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  .. image:: /api/gallery/quantum/huckel/images/thumb/sphx_glr_plot_02_pauling_resonance_energy_thumb.png
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  :doc:`/api/gallery/quantum/huckel/plot_02_pauling_resonance_energy`

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      <div class="sphx-glr-thumbnail-title">Pauling's resonance energy as Huckel delocalization energy</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Coulson (1939) wrote the Huckel eigenvalues of a linear chain of n atoms in closed form, E_k=\alpha+2\beta\cos(k\pi/(n+1)), and Frost and Musulin (1953) turned the ring result E_k=\alpha+2\beta\cos(2\pi k/n) into a drawing: put a regular n-gon, one vertex pointing down, inside a circle of radius 2|\beta|; the heights of the vertices are the orbital energies. Both closed forms (~chemistrykit.quantum.systems.huckel.linear_polyene_eigenvalues, cyclic_polyene_eigenvalues) are checked here against numerical diagonalization.">

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  :doc:`/api/gallery/quantum/huckel/plot_03_coulson_frost_closed_forms`

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      <div class="sphx-glr-thumbnail-title">Coulson's polyene formula and the Frost circle</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Kenichi Fukui, Teijiro Yonezawa and Haruo Shingu (1952) proposed that an aromatic hydrocarbon&#x27;s site of electrophilic substitution is set by one orbital alone, the highest occupied molecular orbital: the attacking electrophile goes where the HOMO density 2c_r^2 is largest. For naphthalene the Huckel HOMO has coefficient 0.425 on the alpha carbons (1, 4, 5, 8), 0.263 on the beta carbons (2, 3, 6, 7), and zero on the bridgeheads, so frontier_electron_density predicts alpha substitution, which is what nitration and halogenation mostly give.">

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  :doc:`/api/gallery/quantum/huckel/plot_04_fukui_frontier_orbitals`

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      <div class="sphx-glr-thumbnail-title">Fukui's frontier orbitals: why naphthalene reacts at the alpha position</div>
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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.


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    <div class="sphx-glr-thumbcontainer" tooltip="Schrodinger&#x27;s first 1926 paper solved his wave equation for the hydrogen atom and recovered the Bohr energies E_n\propto-1/n^2 without any quantization postulate. HydrogenLikeAtom reproduces the textbook 13.6 eV hydrogen ground-state energy exactly, and its radial wavefunctions (built from the associated Laguerre polynomials of scipy.special) are independently checked here by numerically integrating them to a normalized probability of 1.">

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  :doc:`/api/gallery/quantum/hydrogenlike/plot_01_hydrogen_orbitals`

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      <div class="sphx-glr-thumbnail-title">Schrodinger's hydrogen atom: radial wavefunctions and orbital energies</div>
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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.


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    <div class="sphx-glr-thumbcontainer" tooltip="Plots the first few 1D particle-in-a-box wavefunctions (~chemistrykit.quantum.systems.particle_in_box.ParticleInBox1D), each offset by its own energy level, then checks the textbook 3D cubic-box degeneracy pattern (~chemistrykit.quantum.systems.particle_in_box.ParticleInBox3D). These are the simplest exact solutions of Schrodinger&#x27;s 1926 time-independent equation -\frac{\hbar^2}{2m}\nabla^2\psi=E\psi inside hard walls: quantized energies E_n=n^2h^2/8mL^2 appear from the boundary conditions alone, with no quantization postulate.">

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  :doc:`/api/gallery/quantum/particle_in_box/plot_01_particle_in_box`

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      <div class="sphx-glr-thumbnail-title">Solving Schrodinger's equation for a particle in a box</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Hans Kuhn (1949) treated the pi electrons of a linear conjugated dye as free electrons in a one-dimensional box as long as the conjugated chain. With N pi electrons filling the box levels two at a time, the lowest absorption is the HOMO to LUMO jump, \Delta E=\frac{h^2}{8mL^2}(N+1), so a longer chain (more electrons and a longer box) absorbs at longer wavelength. conjugated_dye_absorption_wavelength implements this on top of ParticleInBox1D.">

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  .. image:: /api/gallery/quantum/particle_in_box/images/thumb/sphx_glr_plot_02_kuhn_free_electron_dye_thumb.png
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  :doc:`/api/gallery/quantum/particle_in_box/plot_02_kuhn_free_electron_dye`

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      <div class="sphx-glr-thumbnail-title">Kuhn's free-electron model of conjugated dye color</div>
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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.


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    <div class="sphx-glr-thumbcontainer" tooltip="Schrodinger&#x27;s third 1926 paper introduced his perturbation theory; here it is applied to a harmonic oscillator with an added quartic term. Checks quartic_perturbation_first_order_correction&#x27;s first-order Rayleigh-Schrodinger estimate against exact numerical diagonalization of the full anharmonic Hamiltonian (~chemistrykit.quantum.systems.perturbation.anharmonic_energy_levels, built in a truncated harmonic-oscillator basis via position_operator_matrix), and confirms that a cubic perturbation gives exactly zero first-order correction by parity.">

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  :doc:`/api/gallery/quantum/perturbation/plot_01_anharmonic_perturbation`

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      <div class="sphx-glr-thumbnail-title">Rayleigh-Schrodinger perturbation theory vs. exact diagonalization for the quartic oscillator</div>
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Rigid rotor
------------

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


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    <div class="sphx-glr-thumbcontainer" tooltip="Dennison (1926) solved Schrodinger&#x27;s equation for a rotating diatomic molecule, giving levels E_J=J(J+1)\hbar^2/2I. Builds an HCl-like RigidRotor from its atomic masses and bond length (~chemistrykit.quantum.systems.rigid_rotor.RigidRotor.from_diatomic), plots its energy-level diagram (each level&#x27;s height showing its (2J+1)-fold degeneracy), and shows that the J\to J+1 microwave absorption lines are evenly spaced by 2B.">

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  .. image:: /api/gallery/quantum/rigid_rotor/images/thumb/sphx_glr_plot_01_rigid_rotor_thumb.png
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  :doc:`/api/gallery/quantum/rigid_rotor/plot_01_rigid_rotor`

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      <div class="sphx-glr-thumbnail-title">Dennison's rigid rotor: rotational levels and the microwave spectrum</div>
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.. toctree::
   :hidden:
   :includehidden:


   /api/gallery/quantum/harmonic_oscillator/index.rst
   /api/gallery/quantum/hartree_fock/index.rst
   /api/gallery/quantum/helium/index.rst
   /api/gallery/quantum/huckel/index.rst
   /api/gallery/quantum/hydrogenlike/index.rst
   /api/gallery/quantum/particle_in_box/index.rst
   /api/gallery/quantum/perturbation/index.rst
   /api/gallery/quantum/rigid_rotor/index.rst



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    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_