Hartree-Fock

The Hartree-Fock method is one of simplest ab initio methods where the wave function is modeled as a single Slater determinant of spin-orbitals

\[\Psi_\text{HF}(1,2,...,N) = \frac{1}{\sqrt{N!}} \left| \begin{array}{c c c c} \phi_1(1) & \phi_2(1) & ... & \phi_N(1) \\ \phi_1(2) & \phi_2(2) & ... & \phi_N(2) \\ \vdots & \vdots & \ddots & \vdots \\ \phi_1(N) & \phi_2(N) & ... & \phi_N(N) \\ \end{array}\right|\]

The spin-orbitals are constructed under the basis set approximation

\[\phi_i(\vec{r}) = C_{\mu i} \chi_\mu(\vec{r})\]

Note that sum over repeated indices is assumed.

where $\chi(\vec{r})$ are contracted Gaussian basis functions from pre-constructed basis set such as STO-3G, cc-pVDZ, ANO, etc.

These orbitals are constrained to be orthonormal to each other. Moreover, we choose to solve for the set of orbitals that diagonalize the Fock matrix

\[D^\alpha_{\mu \nu} = C^\alpha_{\mu i} C^\alpha_{\nu i} \\[2mm] F^\alpha_{\mu \nu} = H_{\mu \nu} + (D^\alpha_{\lambda\sigma} + D^\beta_{\lambda\sigma})(\mu \nu | \lambda \sigma) + D^\alpha_{\lambda\sigma}(\mu \nu | \lambda \sigma)\]

such that

\[C^\alpha_{\mu i} F^\alpha_{\mu\nu} C^\alpha_{\nu j} = \delta_{ij} \epsilon_i\]

These are the canonical Hartree-Fock orbitals.

Equations also need to be solved for $F^\beta$, but in the case of a restricted calculation, i.e. orbitals for both spins are taken to be the same, solving for $\beta$ will yield the same results as for $\alpha$.

Restricted Hartree-Fock (RHF)

Minimal example

using Fermi

@molecule {
    He 0.0 0.0 0.0
}

@set basis 3-21g
@energy rhf

This will run a RHF computation on Helium using the 3-21G basis set. Currently, Fermi does not support point group symmetry.

Output file

The first part of the output gives an overview of the input information

He    0.000000000000    0.000000000000    0.000000000000


Charge: 0   Multiplicity: 1   
Nuclear repulsion:    0.0000000000
 Number of AOs:                            2
 Number of Doubly Occupied Orbitals:       1
 Number of Virtual Spatial Orbitals:       1

First the molecule XYZ is print. Followed by charge and multiplicity. Those will be taken as 0 and 1 by default, but can be controlled using @set charge and @set multiplicity.

⚠️ RHF can only be used if the multiplicity is 1.

Next, we see the information about the iterations

 Iter.            E[RHF]         ΔE       Dᵣₘₛ        t     DIIS     damp
--------------------------------------------------------------------------------
    1     -2.8352184971  -2.835e+00   1.166e-01     0.78    false     4.71
    2     -2.8260289197   9.190e-03   2.885e-02     0.00    false     1.45
    3     -2.8157915919   1.024e-02   1.601e-02     0.00    false     0.00
    4     -2.8355956172  -1.980e-02   4.948e-02     0.18     true     0.00
    5     -2.8356798736  -8.426e-05   3.475e-03     0.00     true     0.00
    6     -2.8356798733   2.662e-10   8.346e-06     0.00     true     0.00
    7     -2.8356798736  -2.908e-10   6.418e-06     0.00     true     0.00
    8     -2.8356798736  -1.554e-14   4.527e-08     0.00     true     0.00
    9     -2.8356798728   8.546e-10   1.108e-05     0.14     true     0.00
    10    -2.8356798735  -7.070e-10   6.475e-06     0.00     true     0.00
    11    -2.8356798736  -1.477e-10   4.596e-06     0.00     true     0.00
    12    -2.8356798736  -8.882e-16   9.630e-09     0.00     true     0.00
    13    -2.8356798736   4.441e-16   1.687e-10     0.00     true     0.00
--------------------------------------------------------------------------------
 RHF done in  1.46s

Iterations are controlled using a few keywords. The convergence is achieved when

• The number of iterations reaches scf_max_iter

or

• $\Delta E$ is less than scf_e_conv and $D_{rms}$ is less than scf_max_rms

DIIS and damp are auxiliary strategies to reach convergency faster.

Finally, the RHF energy is listed along with orbital energies

    @Final RHF Energy          -2.835679873641 Eₕ

   • Orbitals Summary

    Orbital            Energy    Occupancy
          1     -0.9035715084       ↿⇂
          2      2.0817026436         

   ✔  SCF Equations converged 😄

RHF object

The computation returns a wave function object Fermi.HartreeFock.RHF which contains data useful for post-processing.

Fermi.HartreeFock.RHF

julia> @energy rhf

julia> Fermi.HartreeFock.RHF()