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This is a tutorial on how to implement a simple program to solve the Hartree-Fock with SCF iterations.

We start from step in the algorithm described in Szabo and Ostlund, 3.4.6, p. 146

• Obtain a guess at the density matrix.
• Calculate the exchange and coulomb matrices from the density matrix and the two-electron repulsion integrals.
• Add exchange and coulomb matrices to the core-Hamiltonian to obtain the Fock matrix.
• Diagonalize the Fock matrix.
• Select the occupied orbitals and calculate the new density matrix.
• Compute the energy
• Compute the errors and check convergence
  • If converged, return the energy
  • If not converged, return to second step with new density matrix

You can first implement it for hydrogen molecule by running the following code:

    mol = "h2"
    r   = 1.0
    inp = f"{mol}-{r:.4f}"
    e = main(inp) # H2 molecule with bond length 1.0

then try some more complicated molecules, such as HeH+

    mol = "heh+"
    r   = 1.0
    inp = f"{mol}-{r:.4f}"
    e = main(inp) # HeH+ molecule with bond length 1.0

and H2O

    mol = "h2o"
    r   = 1.0
    inp = f"{mol}-{r:.4f}"
    e = main(inp) # water molecule with bond length 1.0

After implementing the SCF procedure, we can use it to compute the potential energy surface of the $\mathrm{H}_2$ molecule. We can do this by varying the internuclear distance and computing the energy at each point.

    for r in numpy.linspace(0.5, 2.5, 21):
        inp = f"h2-{r:.4f}"
        e = main(inp)
        print(f"H2: r={r: 6.4f}, e={e: 12.8f}")

You may also try other molecules, such as $\mathrm{HeH}^+$ and $\mathrm{H_2O}$.

Write a new function solve_uhf to implement the unrestricted Hartree-Fock method. See what's the difference between the restricted and unrestricted Hartree-Fock potential energy surfaces. (Note that to obtain a reasonable dissociation potential energy surface, you may need to play some tricks to get the broken spin-symmetry UHF solution.)

• numpy
• scipy
• If you wish to use ./data/gen_data.py to generate the integrals, pyscf is also required.

• Szabo and Ostlund, Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory, Dover Publications, New York, 1996
• Helgaker, Jørgensen, and Olsen, Molecular Electronic-Structure Theory, Wiley, New York, 2000