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.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_quantum_hartree_fock_plot_08_sto3g_minimal_basis.py>`
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.. _sphx_glr_api_gallery_quantum_hartree_fock_plot_08_sto3g_minimal_basis.py:


Hehre, Stewart and Pople's STO-3G basis: three Gaussians for one Slater orbital
===============================================================================

A Slater 1s orbital :math:`e^{-\zeta r}` has the right cusp and tail but
awkward integrals; a Gaussian has easy integrals but the wrong shape.
STO-3G (:func:`~chemistrykit.quantum.utils.basis_sets.sto3g_1s`) fixes
three Gaussian exponents and coefficients by a least-squares fit to the
Slater function once, for :math:`\zeta=1`. Every other exponent follows by
scaling the Gaussian exponents by :math:`\zeta^2`. Below, the STO-3G
contraction for hydrogen in molecules (:math:`\zeta=1.24`) is compared
with its Slater target and with a single best-fit Gaussian. The error
concentrates at the nucleus and in the tail, but the region that matters
for bonding is reproduced well.

.. GENERATED FROM PYTHON SOURCE LINES 18-55

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.quantum import RestrictedHartreeFock
    from chemistrykit.quantum.systems.helium import HARTREE_ENERGY
    from chemistrykit.quantum.utils.basis_sets import BOHR_RADIUS, sto3g_1s

    zeta = 1.24
    r = np.linspace(0.0, 4.0, 400)  # bohr
    slater = (zeta**3 / np.pi) ** 0.5 * np.exp(-zeta * r)

    phi = sto3g_1s(zeta, [0.0, 0.0, 0.0])
    alphas = np.array([p.alpha for p in phi.primitives]) * BOHR_RADIUS**2  # back to bohr^-2
    norms = (2 * alphas / np.pi) ** 0.75
    sto3g = sum(d * n * np.exp(-a * r**2) for d, n, a in zip(phi.coefficients, norms, alphas, strict=True))

    alpha_1g = 0.270950 * zeta**2  # STO-1G exponent (Szabo & Ostlund, eq. 3.221)
    sto1g = (2 * alpha_1g / np.pi) ** 0.75 * np.exp(-alpha_1g * r**2)

    fig, axes = plt.subplots(1, 2, figsize=(12, 4.8))
    axes[0].plot(r, slater, color="black", linewidth=2, label=r"Slater 1s, $\zeta=1.24$")
    axes[0].plot(r, sto3g, "--", color="darkorange", label="STO-3G")
    axes[0].plot(r, sto1g, ":", color="steelblue", label="STO-1G (one Gaussian)")
    axes[0].set_xlabel("r (bohr)")
    axes[0].set_ylabel(r"$\phi(r)$ (bohr$^{-3/2}$)")
    axes[0].set_title("Fitting a Slater orbital with Gaussians")
    axes[0].legend()

    axes[1].plot(r, r**2 * (sto3g - slater), color="darkorange", label="STO-3G")
    axes[1].plot(r, r**2 * (sto1g - slater), color="steelblue", label="STO-1G")
    axes[1].axhline(0.0, color="gray", linewidth=0.8)
    axes[1].set_xlabel("r (bohr)")
    axes[1].set_ylabel(r"$r^2\,[\phi_{fit}-\phi_{Slater}]$")
    axes[1].set_title("Radially weighted fitting error")
    axes[1].legend()
    fig.tight_layout()




.. image-sg:: /api/gallery/quantum/hartree_fock/images/sphx_glr_plot_08_sto3g_minimal_basis_001.png
   :alt: Fitting a Slater orbital with Gaussians, Radially weighted fitting error
   :srcset: /api/gallery/quantum/hartree_fock/images/sphx_glr_plot_08_sto3g_minimal_basis_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 56-58

The same fitted contraction, rescaled for helium (:math:`\zeta=2.0925`),
reproduces Szabo & Ostlund's minimal-basis results:

.. GENERATED FROM PYTHON SOURCE LINES 58-63

.. code-block:: Python


    for name, rhf, reference in (("H2", RestrictedHartreeFock.h2(), -1.1167), ("HeH+", RestrictedHartreeFock.heh_plus(), -2.8607)):
        print(f"{name:5s} STO-3G RHF energy: {rhf.scf().total_energy / HARTREE_ENERGY:.4f} hartree (Szabo & Ostlund: {reference})")

    plt.show()




.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    H2    STO-3G RHF energy: -1.1167 hartree (Szabo & Ostlund: -1.1167)
    HeH+  STO-3G RHF energy: -2.8607 hartree (Szabo & Ostlund: -2.8607)





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