Editing Hartree–Fock method (section)

== Mathematical formulation ==

=== Derivation ===
According to the [[Slater–Condon rules]], the energy expectation value of the [[Molecular Hamiltonian#Clamped nucleus Hamiltonian|molecular electronic Hamiltonian]] <math>\hat{H}^e</math> for a [[Slater determinant]] is

: <math display="inline">\begin{aligned} E[\psi^{HF}] &= \left\langle\psi^{HF}|\hat{H}^e|\psi^{HF}\right\rangle \\
&=  \sum_{i=1}^N \int\text{d}\mathbf{x}_i \, \phi_i^*(\mathbf{x}_i) \hat{h}(\mathbf{x}_i) \phi_i(\mathbf{x}_i) \\ 
&+ \frac{1}{2} \sum_{i=1}^N\sum_{j=1}^N \int \mathrm{d}\mathbf{x}_i \int \text{d}\mathbf{x}_j \phi_i^*(\mathbf{x}_i)\phi_j^*(\mathbf{x}_j) \frac{1}{|\mathbf{x}_i-\mathbf{x}_j|}\phi_i(\mathbf{x}_i)\phi_j(\mathbf{x}_j) \\ 
&- \frac{1}{2} \sum_{i=1}^N\sum_{j=1}^N \int \text{d}\mathbf{x}_i \int \text{d}\mathbf{x}_j\phi_i^*(\mathbf{x}_i)\phi_j^*(\mathbf{x}_j) \frac{1}{|\mathbf{x}_i-\mathbf{x}_j|}\phi_i(\mathbf{x}_j)\phi_j(\mathbf{x}_i)  \end{aligned}
 </math>

where <math>\hat{h}</math> is the one electron operator including electronic kinetic energy and electron-nucleus Coulombic interaction, and
: <math>\begin{aligned}
 \psi^{HF} = \psi(\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_N) =
  \frac{1}{\sqrt{N!}}
  \begin{vmatrix} \phi_1(\mathbf{x}_1) & \phi_2(\mathbf{x}_1) & \cdots & \phi_N(\mathbf{x}_1) \\
                      \phi_1(\mathbf{x}_2) & \phi_2(\mathbf{x}_2) & \cdots & \phi_N(\mathbf{x}_2) \\
                      \vdots & \vdots & \ddots & \vdots \\
                      \phi_1(\mathbf{x}_N) & \phi_2(\mathbf{x}_N) & \cdots & \phi_N(\mathbf{x}_N)
  \end{vmatrix}.
\end{aligned}</math>

To derive the Hartree-Fock equation we minimize the energy functional for N electrons with orthonormal constraints.

: <math>\delta E[\phi_k^*(x_k)] = \delta \left\langle\psi^{HF}|\hat{H}^e|\psi^{HF}\right\rangle - \delta\left[\sum_{i=1}^N \sum_{j=1}^N \lambda_{ij} \left( \left\langle\phi_i, \phi_j\right\rangle - \delta_{ij}\right)\right] \stackrel{!}{=}\, 0,</math>

We choose a basis set <math>\phi_i(x_i)</math> in which the [[Lagrange multiplier]] matrix <math>\lambda_{ij}</math> becomes diagonal, i.e. <math>\lambda_{ij} = \epsilon_i \delta_{ij}</math>. Performing the [[Functional derivative|variation]], we obtain

: <math>\begin{aligned}
\delta E[\phi_k^*(x_k)] &= \sum_{i=1}^N \int\text{d}\mathbf{x}_i \, \hat{h}(\mathbf{x}_i) \phi_i(\mathbf{x}_i) \delta(\mathbf{x}_i -\mathbf{x}_k) \delta_{ik}\\ &+ \sum_{i=1}^N\sum_{j=1}^N \int \mathrm{d}\mathbf{x}_i \int \text{d}\mathbf{x}_j\phi_j^*(\mathbf{x}_j) \frac{1}{|\mathbf{x}_i-\mathbf{x}_j|}\phi_i(\mathbf{x}_i)\phi_j(\mathbf{x}_j) \delta(\mathbf{x}_i-\mathbf{x}_k)\delta_{ik}\\ 
&- \sum_{i=1}^N\sum_{j=1}^N \int \text{d}\mathbf{x}_i \int \text{d}\mathbf{x}_j\phi_j^*(\mathbf{x}_j) \frac{1}{|\mathbf{x}_i-\mathbf{x}_j|}\phi_i(\mathbf{x}_j)\phi_j(\mathbf{x}_i) \delta(\mathbf{x}_i-\mathbf{x}_k)\delta_{ik}\\
 &- \sum_{i=1}^N \epsilon_i \int \text{d}\mathbf{x}_i \, \phi_i(\mathbf{x}_i)  \delta(\mathbf{x}_i-\mathbf{x}_k)\delta_{ik}\\
&= \hat{h}(\mathbf{x}_k) \phi_k(\mathbf{x}_k)\\ 
&+ \sum_{j=1}^N  \int \text{d}\mathbf{x}_j\phi_j^*(\mathbf{x}_j) \frac{1}{|\mathbf{x}_k-\mathbf{x}_j|}\phi_k(\mathbf{x}_k)\phi_j(\mathbf{x}_j)\\ 
&- \sum_{j=1}^N  \int \text{d}\mathbf{x}_j\phi_j^*(\mathbf{x}_j) \frac{1}{|\mathbf{x}_k-\mathbf{x}_j|}\phi_k(\mathbf{x}_j)\phi_j(\mathbf{x}_k)\\ 
&- \epsilon_k \phi_k(\mathbf{x}_k)=0. \\
\end{aligned}</math>

The factor 1/2 before the double integrals in the molecular Hamiltonian drops out  due to symmetry and the product rule. We may define the [[Fock matrix|Fock operator]] to rewrite the equation

: <math>\hat{F}(\mathbf{x}_k)\phi_k(\mathbf{x}_k) \equiv \left[ \hat{h}(\mathbf{x}_k) + \hat{J}(\mathbf{x}_k) - \hat{K}(\mathbf{x}_k) \right]\phi_k(\mathbf{x}_k) = \epsilon_k \phi_k(\mathbf{x}_k),</math>

where the [[Coulomb operator]] <math>\hat{J}(\mathbf{x}_k)</math> and the [[exchange operator]] <math>\hat{K}(\mathbf{x}_k)</math> are defined as follows

: <math>\begin{aligned}
\hat{J}(\mathbf{x_k}) &\equiv \sum_{j=1}^N \int \mathrm{d}\mathbf{x}_j \frac{\phi_j^*(\mathbf{x}_j) \phi_j(\mathbf{x}_j)}{|\mathbf{x}_k-\mathbf{x}_j|}= \sum_{j=1}^N \int \mathrm{d}\mathbf{x}_j \frac{\rho(\mathbf{x}_j)}{|\mathbf{x}_k-\mathbf{x}_j|},\\
\hat{K}(\mathbf{x_k})\phi_{k}(\mathbf{x}_k) &\equiv  \sum_{j=1}^N  \phi_{j}(\mathbf{x}_k) \int \text{d}\mathbf{x}_j \frac{\phi_j^*(\mathbf{x}_j) \phi_k(\mathbf{x}_j)}{|\mathbf{x}_k-\mathbf{x}_j|}.\\
\end{aligned}</math>

The exchange operator has no classical analogue and can only be defined as an integral operator.

The solution <math>\phi_k</math> and <math>\epsilon_k</math> are called molecular orbital and orbital energy respectively.

Although Hartree-Fock equation appears in the form of a eigenvalue problem, the Fock operator itself depends on <math>\phi</math> and must be solved by a different technique.

===Total energy=== 
The optimal total energy <math> E_{HF} </math> can be written in terms of molecular orbitals.

:<math> E_{HF} = \sum_{i=1}^{N} \hat h_{ii} + \sum_{i=1}^{N} \sum_{j=1}^{N/2} [2\hat J_{ij} - \hat K_{ij}] + V_{\text{nucl}} </math>

<math>\hat J_{ij}</math> and <math>\hat K_{ij}</math> are matrix elements of the Coulomb and exchange operators respectively, and <math>V_{\text{nucl}}</math> is the total electrostatic repulsion between all the nuclei in the molecule.

The total energy is not equal to the sum of orbital energies.

If the atom or molecule is [[closed shell]], the total energy according to the Hartree-Fock method is
: <math>E_{HF} = 2 \sum_{i=1}^{N/2} \hat h_{ii} + \sum_{i=1}^{N/2} \sum_{j=1}^{N/2} [2\hat J_{ij} - \hat K_{ij}] + V_{\text{nucl}}.</math><ref name= Levine>Levine, Ira N. (1991). Quantum Chemistry (4th ed.). Englewood Cliffs, New Jersey: Prentice Hall. p. 402-3. {{ISBN|0-205-12770-3}}.</ref>

=== Linear combination of atomic orbitals ===
{{Main|Basis set (chemistry)}}

Typically, in modern Hartree–Fock calculations, the one-electron wave functions are approximated by a [[linear combination of atomic orbitals]].  These atomic orbitals are called [[Slater-type orbital]]s.  Furthermore, it is very common for the "atomic orbitals" in use to actually be composed of a linear combination of one or more [[Gaussian orbital|Gaussian-type orbitals]], rather than Slater-type orbitals, in the interests of saving large amounts of computation time.

Various [[basis set (chemistry)|basis sets]] are used in practice, most of which are composed of Gaussian functions.  In some applications, an orthogonalization method such as the [[Gram–Schmidt process]] is performed in order to produce a set of orthogonal basis functions.  This can in principle save computational time when the computer is solving the [[Roothaan equations|Roothaan–Hall equations]] by converting the [[overlap matrix]] effectively to an [[identity matrix]]. However, in most modern computer programs for molecular Hartree–Fock calculations this procedure is not followed due to the high numerical cost of orthogonalization and the advent of more efficient, often sparse, algorithms for solving the [[generalized eigenvalue problem]], of which the [[Roothaan equations|Roothaan–Hall equations]] are an example.