9 Hartree–Fock Theory

9.1 Spin Functions

The spin variable $\sigma_i$ can take two values:

\[ \sigma_i = +1 \quad \text{(spin up)} \]

\[ \sigma_i = -1 \quad \text{(spin down)} \]

Introduce two orthonormal spin functions:

$\alpha(\sigma)$ : spin up
$\beta(\sigma)$ : spin down

Thus, the spin–orbital can be written as a product of spatial and spin parts:

\[ \chi_i(\mathbf{r}_i, \sigma_i) = \psi_i(\mathbf{r}_i)\,\alpha(\sigma_i) \quad \text{if } \sigma_i = +1 \]

\[ \chi_i(\mathbf{r}_i, \sigma_i) = \psi_i(\mathbf{r}_i)\,\beta(\sigma_i) \quad \text{if } \sigma_i = -1 \]

where

\[ \alpha(1) = 1, \quad \alpha(-1) = 0 \]

\[ \beta(1) = 0, \quad \beta(-1) = 1 \]

The orthonormality conditions on the spin functions are:

\[ \int |\alpha(\sigma)|^2\, d\sigma = \sum_{\sigma=\pm 1} |\alpha(\sigma)|^2 = 1 \]

\[ \int \alpha(\sigma)\beta(\sigma)\, d\sigma = \sum_{\sigma=\pm 1} \alpha(\sigma)\beta(\sigma) = 0 \]

9.2 Slater Determinant Ansatz

Hartree–Fock assumes the wavefunction is a single Slater determinant of spin-orbitals $\{\chi_p(\mathbf{x})\}$:

\[ \Psi(x_1, \ldots, x_N) = \frac{1}{\sqrt{N!}} \begin{vmatrix} \chi_1(x_1) & \cdots & \chi_N(x_1) \\ \vdots & \ddots & \vdots \\ \chi_1(x_N) & \cdots & \chi_N(x_N) \end{vmatrix} \]

where $x \equiv (\mathbf{r}, \sigma)$, $x_1 \equiv (\mathbf{r}_1, \sigma_1)$, $x_2 \equiv (\mathbf{r}_2, \sigma_2)$, and so on. Thus,

\[ \Psi = \frac{1}{\sqrt{N!}} \sum_P (-1)^p\, P\, \chi_1(x_1)\chi_2(x_2)\cdots\chi_N(x_N) \]

Here, $P$ is the permutation operator and $p$ is the number of interchanges making up this permutation.

9.3 Expectation Value of the Energy

The expectation value of the energy can be written as

\[ E = \langle \Psi | H | \Psi \rangle \]

where the Hamiltonian is

\[ H = -\sum_i \frac{1}{2}\nabla_i^2 - \sum_i\sum_\ell \frac{Z_l}{|\mathbf{r}_i - \mathbf{R}_\ell|} + \frac{1}{2}\sum_{\substack{ij \\ i \neq j}} \frac{1}{|\mathbf{r}_i - \mathbf{r}_j|} \]

Thus,

\[ E = \langle \Psi | H | \Psi \rangle = \int \Psi^* H\Psi\, d\tau \]

where

\[ d\tau = dx_1\, dx_2 \cdots dx_N \]

Then,

\[ E = \frac{1}{\sqrt{N!}} \int \Psi^* H \sum_P (-1)^p P\,\chi_1(x_1)\cdots\chi_N(x_N)\, d\tau \]

\[ = \frac{1}{\sqrt{N!}} \sum_P (-1)^p \int (P^{-1}\Psi^*)\,H\,\chi_1(x_1)\cdots\chi_N(x_N)\, d\tau \qquad [\text{as } PH = H] \]

\[ = \frac{1}{\sqrt{N!}} \sum_P \int \Psi^* H\,\chi_1(x_1)\cdots\chi_N(x_N)\, d\tau \]

Note that all $x_i$ are integrated out in the above equation and the integral is independent of any $x_i$. Thus, the operation of $P$ over the integral gives the same number and, therefore, the sum over all the permutations gives just $N!$ times the integral.

Thus, we get the final simplified expression for the energy. From the previous result,

\[ E = \sqrt{N!} \int \Psi^* H\,\chi_1(x_1)\cdots\chi_N(x_N)\, d\tau \]

Expanding explicitly:

\[ \begin{aligned} E = \sum_P \int &\, (-1)^p P\chi_1^*(x_1)\cdots\chi_N^*(x_N) \\[4pt] &\times \left[ -\sum_i \frac{1}{2}\nabla_i^2 - \sum_{i\ell} \frac{Z_\ell}{|\mathbf{r}_i - \mathbf{R}_\ell|} + \frac{1}{2}\sum_{\substack{ij \\ i\neq j}} \frac{1}{|\mathbf{r}_i - \mathbf{r}_j|} \right] \\[4pt] &\times \chi_1(x_1)\cdots\chi_N(x_N)\, d\tau \end{aligned} \]

Carrying out the permutations and integrations gives

\[ \begin{aligned} E ={}& \sum_i \int \chi_i^*(x_i) \left[ -\frac{1}{2}\nabla_i^2 - \sum_{i\ell} \frac{Z}{|\mathbf{r}_i - \mathbf{R}_\ell|} \right] \chi_i(x_i)\, dx_i \\[6pt] &+ \frac{1}{2}\sum_{i \neq j} \iint \frac{ \left[ \chi_i^*(x_i)\chi_j^*(x_j) - \chi_j^*(x_i)\chi_i^*(x_j) \right] }{ |\mathbf{r}_i - \mathbf{r}_j| } \\[6pt] &\qquad\qquad \times\, \chi_i(x_i)\chi_j(x_j)\, dx_i\, dx_j \end{aligned} \]

Rewriting in compact form:

\[ \begin{aligned} E &= \sum_i \int \chi_i^*(x_i) \left[ -\frac{1}{2}\nabla_i^2 - \sum_{i\ell} \frac{Z}{|\mathbf{r}_i - \mathbf{R}_\ell|} \right] \chi_i(x_i)\, dx_i \\[6pt] &+ \frac{1}{2}\sum_{i \neq j} \iint \frac{|\chi_i(x_i)|^2\,|\chi_j(x_j)|^2}{|\mathbf{r}_i - \mathbf{r}_j|}\, dx_i\, dx_j \\[6pt] &- \frac{1}{2}\sum_{i \neq j} \iint \frac{\chi_j^*(x_i)\,\chi_i^*(x_j)\,\chi_i(x_i)\,\chi_j(x_j)}{|\mathbf{r}_i - \mathbf{r}_j|}\, dx_i\, dx_j \end{aligned} \]

9.4 Exchange Energy

The above expression differs from the Hartree expression for $E$ due to the presence of the last term, known as the exchange energy, because it arises due to the exchange of electronic coordinates. We shall now show that this term is finite only for parallel spins.

In the first two terms, the integration over spin variables is easily performed, and the result is that $x_i$ is replaced by $\mathbf{r}_i$. Let us consider the last term of the equation. Assuming $\psi_i$ and $\psi_j$ have opposite spins, we have

\[ \begin{aligned} &\sum_{i,j} \sum_{i \neq j} \iint \frac{1}{|\mathbf{r}_i - \mathbf{r}_j|} \, \psi_i^*(\mathbf{r}_j)\,\alpha(\sigma_j)\, \psi_j^*(\mathbf{r}_i)\,\beta(\sigma_i) \\[6pt] &\qquad\qquad \times\, \psi_i(\mathbf{r}_i)\,\alpha(\sigma_i)\, \psi_j(\mathbf{r}_j)\,\beta(\sigma_j) \, d\mathbf{r}_i\, d\mathbf{r}_j\, d\sigma_i\, d\sigma_j = 0 \end{aligned} \]

Therefore, the last term gives a finite contribution only if $\psi_i$ and $\psi_j$ have the same spins. Thus, the exchange energy can be written as

\[ \begin{aligned} &-\frac{1}{2} \sum_{\substack{i \neq j \\ (\parallel\,\text{spins})}} \iint \frac{ \psi_i^*(\mathbf{r}_j)\, \psi_j^*(\mathbf{r}_i)\, \psi_i(\mathbf{r}_i)\, \psi_j(\mathbf{r}_j) }{ |\mathbf{r}_i - \mathbf{r}_j| } \, d\mathbf{r}_i\, d\mathbf{r}_j \\[8pt] &\qquad = -\frac{1}{2} \sum_{\substack{i \neq j \\ (\parallel\,\text{spins})}} \iint \frac{ \psi_i^*(\mathbf{r})\, \psi_j^*(\mathbf{r}')\, \psi_i(\mathbf{r}')\, \psi_j(\mathbf{r}) }{ |\mathbf{r} - \mathbf{r}'| } \, d\mathbf{r}\, d\mathbf{r}' \end{aligned} \]

9.5 Hartree–Fock Equation

Let us define $v_I(\mathbf{r})$ as

\[ v_I(\mathbf{r}) = -\sum_\ell \frac{Z}{|\mathbf{r} - \mathbf{R}_\ell|} \]

$E$ can be written as

\[ \begin{aligned} E &= \sum_i \int \psi_i^*(\mathbf{r}) \left[ -\frac{1}{2}\nabla^2 + v_I(\mathbf{r}) \right] \psi_i(\mathbf{r})\, d\mathbf{r} \\[6pt] &+ \frac{1}{2}\sum_{i \neq j} \iint \frac{|\psi_i(\mathbf{r})|^2\,|\psi_j(\mathbf{r}')|^2}{|\mathbf{r} - \mathbf{r}'|}\, d\mathbf{r}\, d\mathbf{r}' \\[6pt] &- \frac{1}{2}\sum_{\substack{i \neq j \\ (\parallel\,\text{spins})}} \iint \frac{\psi_i^*(\mathbf{r})\,\psi_j^*(\mathbf{r}')\,\psi_i(\mathbf{r}')\,\psi_j(\mathbf{r})}{|\mathbf{r} - \mathbf{r}'|}\, d\mathbf{r}\, d\mathbf{r}' \end{aligned} \]

Note that the $i = j$ contribution cancels out in the last two terms; therefore, we shall drop this restriction.

Now, as in the Hartree case, minimize $E$, subject to the constraint that

\[ \int \psi_i^*(\mathbf{r})\,\psi_i(\mathbf{r})\, d\mathbf{r} = 1 \]

that is, minimize

\[ E - \sum_i \varepsilon_i \left[\int \psi_i^*(\mathbf{r})\,\psi_i(\mathbf{r})\, d\mathbf{r} - 1\right] \]

Similar to Hartree’s case, this gives

\[ \frac{\delta E}{\delta \psi_i^*(\mathbf{r})} - \varepsilon_i\,\psi_i(\mathbf{r}) = 0 \]

\[ \begin{aligned} & \left[ -\frac{1}{2}\nabla^2 + v_I(\mathbf{r}) \right] \psi_i(\mathbf{r}) \\[6pt] &\quad + \sum_j \int \frac{|\psi_j(\mathbf{r}')|^2} {|\mathbf{r} - \mathbf{r}'|} \, d\mathbf{r}' \, \psi_i(\mathbf{r}) \\[6pt] &\quad - \sum_{\substack{j \\ (\text{spin } j\, =\, \text{spin } i)}} \int \frac{ \psi_j^*(\mathbf{r}')\,\psi_i(\mathbf{r}') }{ |\mathbf{r} - \mathbf{r}'| } \, d\mathbf{r}' \, \psi_j(\mathbf{r}) \\[6pt] &= \varepsilon_i\,\psi_i(\mathbf{r}) \end{aligned} \]

\[ \boxed{ \left[ -\frac{1}{2}\nabla^2 + v_I(\mathbf{r}) + v_H(\mathbf{r}) \right]\psi_i(\mathbf{r}) - \int \sum_{\substack{j \neq i \\ (\text{spin } j\, =\, \text{spin } i)}} \frac{\psi_j^*(\mathbf{r}')\,\psi_i(\mathbf{r}')}{|\mathbf{r} - \mathbf{r}'|}\, d\mathbf{r}'\, \psi_j(\mathbf{r}) = \varepsilon_i\,\psi_i(\mathbf{r}) } \]

which is known as the Hartree–Fock equation.

In the above equation $v_H(\mathbf{r})$ is the Hartree potential. The H–F method introduces an improvement over Hartree energy, and it is due to the appearance of exchange energy, which is caused by taking the antisymmetry of the wave function into account. This exchange interaction is a real quantum mechanical effect and must appear in any one-electron model.

The H–F method is a mean-field theory, that is, the many-body problem is reduced to a one-electron problem in which the electron moves in an average field generated by other electrons. It ignores Coulomb correlations that arise because electrons repel each other and move in such a way as to avoid each other. However, the H–F method does include some correlations between parallel spin electrons, which give rise to the exchange interaction. The Coulomb correlations actually reduce the exchange interaction between electrons with parallel spins, so that the H–F equation greatly overestimates the importance of the exchange interaction. This equation describes compact systems such as atoms well but it predicts totally incorrect behavior for the conduction electrons in solids.

--- title: "Hartree–Fock Theory" format: html: toc: true toc-depth: 3 toc-title: "Contents" number-sections: true theme: light: cosmo dark: darkly theme-toggle: true highlight-style: github code-fold: false html-math-method: mathjax smooth-scroll: true include-in-header: text: | <style> body { font-size: 1.05rem; line-height: 1.75; } h2 { margin-top: 2.5rem; padding-bottom: 0.4rem; border-bottom: 2px solid var(--bs-border-color); } h3 { margin-top: 1.8rem; color: var(--bs-body-color); } h4 { color: var(--bs-secondary-color); font-style: italic; } .boxed-eq { background: var(--bs-tertiary-bg); border-left: 4px solid #4c6ef5; padding: 0.8rem 1.2rem; border-radius: 4px; margin: 1rem 0; } </style> --- ## Spin Functions {#sec-11-spin-functions} The spin variable $\sigma_i$ can take two values: $$ \sigma_i = +1 \quad \text{(spin up)} $$ $$ \sigma_i = -1 \quad \text{(spin down)} $$ Introduce two orthonormal spin functions: - $\alpha(\sigma)$ : spin up - $\beta(\sigma)$ : spin down Thus, the spin–orbital can be written as a product of spatial and spin parts: $$ \chi_i(\mathbf{r}_i, \sigma_i) = \psi_i(\mathbf{r}_i)\,\alpha(\sigma_i) \quad \text{if } \sigma_i = +1 $$ $$ \chi_i(\mathbf{r}_i, \sigma_i) = \psi_i(\mathbf{r}_i)\,\beta(\sigma_i) \quad \text{if } \sigma_i = -1 $$ where $$ \alpha(1) = 1, \quad \alpha(-1) = 0 $$ $$ \beta(1) = 0, \quad \beta(-1) = 1 $$ The orthonormality conditions on the spin functions are: $$ \int |\alpha(\sigma)|^2\, d\sigma = \sum_{\sigma=\pm 1} |\alpha(\sigma)|^2 = 1 $$ $$ \int \alpha(\sigma)\beta(\sigma)\, d\sigma = \sum_{\sigma=\pm 1} \alpha(\sigma)\beta(\sigma) = 0 $$ --- ## Slater Determinant Ansatz {#sec-11-slater-determinant} Hartree–Fock assumes the wavefunction is a single Slater determinant of spin-orbitals $\{\chi_p(\mathbf{x})\}$: $$ \Psi(x_1, \ldots, x_N) = \frac{1}{\sqrt{N!}} \begin{vmatrix} \chi_1(x_1) & \cdots & \chi_N(x_1) \\ \vdots & \ddots & \vdots \\ \chi_1(x_N) & \cdots & \chi_N(x_N) \end{vmatrix} $$ where $x \equiv (\mathbf{r}, \sigma)$, $x_1 \equiv (\mathbf{r}_1, \sigma_1)$, $x_2 \equiv (\mathbf{r}_2, \sigma_2)$, and so on. Thus, $$ \Psi = \frac{1}{\sqrt{N!}} \sum_P (-1)^p\, P\, \chi_1(x_1)\chi_2(x_2)\cdots\chi_N(x_N) $$ Here, $P$ is the permutation operator and $p$ is the number of interchanges making up this permutation. --- ## Expectation Value of the Energy {#sec-11-expectation-energy} The expectation value of the energy can be written as $$ E = \langle \Psi | H | \Psi \rangle $$ where the Hamiltonian is $$ H = -\sum_i \frac{1}{2}\nabla_i^2 - \sum_i\sum_\ell \frac{Z_l}{|\mathbf{r}_i - \mathbf{R}_\ell|} + \frac{1}{2}\sum_{\substack{ij \\ i \neq j}} \frac{1}{|\mathbf{r}_i - \mathbf{r}_j|} $$ Thus, $$ E = \langle \Psi | H | \Psi \rangle = \int \Psi^* H\Psi\, d\tau $$ where $$ d\tau = dx_1\, dx_2 \cdots dx_N $$ Then, $$ E = \frac{1}{\sqrt{N!}} \int \Psi^* H \sum_P (-1)^p P\,\chi_1(x_1)\cdots\chi_N(x_N)\, d\tau $$ $$ = \frac{1}{\sqrt{N!}} \sum_P (-1)^p \int (P^{-1}\Psi^*)\,H\,\chi_1(x_1)\cdots\chi_N(x_N)\, d\tau \qquad [\text{as } PH = H] $$ $$ = \frac{1}{\sqrt{N!}} \sum_P \int \Psi^* H\,\chi_1(x_1)\cdots\chi_N(x_N)\, d\tau $$ Note that all $x_i$ are integrated out in the above equation and the integral is independent of any $x_i$. Thus, the operation of $P$ over the integral gives the same number and, therefore, the sum over all the permutations gives just $N!$ times the integral. Thus, we get the final simplified expression for the energy. From the previous result, $$ E = \sqrt{N!} \int \Psi^* H\,\chi_1(x_1)\cdots\chi_N(x_N)\, d\tau $$ Expanding explicitly: $$ \begin{aligned} E = \sum_P \int &\, (-1)^p P\chi_1^*(x_1)\cdots\chi_N^*(x_N) \\[4pt] &\times \left[ -\sum_i \frac{1}{2}\nabla_i^2 - \sum_{i\ell} \frac{Z_\ell}{|\mathbf{r}_i - \mathbf{R}_\ell|} + \frac{1}{2}\sum_{\substack{ij \\ i\neq j}} \frac{1}{|\mathbf{r}_i - \mathbf{r}_j|} \right] \\[4pt] &\times \chi_1(x_1)\cdots\chi_N(x_N)\, d\tau \end{aligned} $$ Carrying out the permutations and integrations gives $$ \begin{aligned} E ={}& \sum_i \int \chi_i^*(x_i) \left[ -\frac{1}{2}\nabla_i^2 - \sum_{i\ell} \frac{Z}{|\mathbf{r}_i - \mathbf{R}_\ell|} \right] \chi_i(x_i)\, dx_i \\[6pt] &+ \frac{1}{2}\sum_{i \neq j} \iint \frac{ \left[ \chi_i^*(x_i)\chi_j^*(x_j) - \chi_j^*(x_i)\chi_i^*(x_j) \right] }{ |\mathbf{r}_i - \mathbf{r}_j| } \\[6pt] &\qquad\qquad \times\, \chi_i(x_i)\chi_j(x_j)\, dx_i\, dx_j \end{aligned} $$ Rewriting in compact form: $$ \begin{aligned} E &= \sum_i \int \chi_i^*(x_i) \left[ -\frac{1}{2}\nabla_i^2 - \sum_{i\ell} \frac{Z}{|\mathbf{r}_i - \mathbf{R}_\ell|} \right] \chi_i(x_i)\, dx_i \\[6pt] &+ \frac{1}{2}\sum_{i \neq j} \iint \frac{|\chi_i(x_i)|^2\,|\chi_j(x_j)|^2}{|\mathbf{r}_i - \mathbf{r}_j|}\, dx_i\, dx_j \\[6pt] &- \frac{1}{2}\sum_{i \neq j} \iint \frac{\chi_j^*(x_i)\,\chi_i^*(x_j)\,\chi_i(x_i)\,\chi_j(x_j)}{|\mathbf{r}_i - \mathbf{r}_j|}\, dx_i\, dx_j \end{aligned} $$ --- ## Exchange Energy {#sec-11-exchange-energy} The above expression differs from the Hartree expression for $E$ due to the presence of the last term, known as the **exchange energy**, because it arises due to the exchange of electronic coordinates. We shall now show that this term is finite only for parallel spins. In the first two terms, the integration over spin variables is easily performed, and the result is that $x_i$ is replaced by $\mathbf{r}_i$. Let us consider the last term of the equation. Assuming $\psi_i$ and $\psi_j$ have opposite spins, we have $$ \begin{aligned} &\sum_{i,j} \sum_{i \neq j} \iint \frac{1}{|\mathbf{r}_i - \mathbf{r}_j|} \, \psi_i^*(\mathbf{r}_j)\,\alpha(\sigma_j)\, \psi_j^*(\mathbf{r}_i)\,\beta(\sigma_i) \\[6pt] &\qquad\qquad \times\, \psi_i(\mathbf{r}_i)\,\alpha(\sigma_i)\, \psi_j(\mathbf{r}_j)\,\beta(\sigma_j) \, d\mathbf{r}_i\, d\mathbf{r}_j\, d\sigma_i\, d\sigma_j = 0 \end{aligned} $$ Therefore, the last term gives a finite contribution only if $\psi_i$ and $\psi_j$ have the same spins. Thus, the exchange energy can be written as $$ \begin{aligned} &-\frac{1}{2} \sum_{\substack{i \neq j \\ (\parallel\,\text{spins})}} \iint \frac{ \psi_i^*(\mathbf{r}_j)\, \psi_j^*(\mathbf{r}_i)\, \psi_i(\mathbf{r}_i)\, \psi_j(\mathbf{r}_j) }{ |\mathbf{r}_i - \mathbf{r}_j| } \, d\mathbf{r}_i\, d\mathbf{r}_j \\[8pt] &\qquad = -\frac{1}{2} \sum_{\substack{i \neq j \\ (\parallel\,\text{spins})}} \iint \frac{ \psi_i^*(\mathbf{r})\, \psi_j^*(\mathbf{r}')\, \psi_i(\mathbf{r}')\, \psi_j(\mathbf{r}) }{ |\mathbf{r} - \mathbf{r}'| } \, d\mathbf{r}\, d\mathbf{r}' \end{aligned} $$ --- ## Hartree–Fock Equation {#sec-11-hf-equation} Let us define $v_I(\mathbf{r})$ as $$ v_I(\mathbf{r}) = -\sum_\ell \frac{Z}{|\mathbf{r} - \mathbf{R}_\ell|} $$ $E$ can be written as $$ \begin{aligned} E &= \sum_i \int \psi_i^*(\mathbf{r}) \left[ -\frac{1}{2}\nabla^2 + v_I(\mathbf{r}) \right] \psi_i(\mathbf{r})\, d\mathbf{r} \\[6pt] &+ \frac{1}{2}\sum_{i \neq j} \iint \frac{|\psi_i(\mathbf{r})|^2\,|\psi_j(\mathbf{r}')|^2}{|\mathbf{r} - \mathbf{r}'|}\, d\mathbf{r}\, d\mathbf{r}' \\[6pt] &- \frac{1}{2}\sum_{\substack{i \neq j \\ (\parallel\,\text{spins})}} \iint \frac{\psi_i^*(\mathbf{r})\,\psi_j^*(\mathbf{r}')\,\psi_i(\mathbf{r}')\,\psi_j(\mathbf{r})}{|\mathbf{r} - \mathbf{r}'|}\, d\mathbf{r}\, d\mathbf{r}' \end{aligned} $$ Note that the $i = j$ contribution cancels out in the last two terms; therefore, we shall drop this restriction. Now, as in the Hartree case, minimize $E$, subject to the constraint that $$ \int \psi_i^*(\mathbf{r})\,\psi_i(\mathbf{r})\, d\mathbf{r} = 1 $$ that is, minimize $$ E - \sum_i \varepsilon_i \left[\int \psi_i^*(\mathbf{r})\,\psi_i(\mathbf{r})\, d\mathbf{r} - 1\right] $$ Similar to Hartree's case, this gives $$ \frac{\delta E}{\delta \psi_i^*(\mathbf{r})} - \varepsilon_i\,\psi_i(\mathbf{r}) = 0 $$ or $$ \begin{aligned} & \left[ -\frac{1}{2}\nabla^2 + v_I(\mathbf{r}) \right] \psi_i(\mathbf{r}) \\[6pt] &\quad + \sum_j \int \frac{|\psi_j(\mathbf{r}')|^2} {|\mathbf{r} - \mathbf{r}'|} \, d\mathbf{r}' \, \psi_i(\mathbf{r}) \\[6pt] &\quad - \sum_{\substack{j \\ (\text{spin } j\, =\, \text{spin } i)}} \int \frac{ \psi_j^*(\mathbf{r}')\,\psi_i(\mathbf{r}') }{ |\mathbf{r} - \mathbf{r}'| } \, d\mathbf{r}' \, \psi_j(\mathbf{r}) \\[6pt] &= \varepsilon_i\,\psi_i(\mathbf{r}) \end{aligned} $$ or $$ \boxed{ \left[ -\frac{1}{2}\nabla^2 + v_I(\mathbf{r}) + v_H(\mathbf{r}) \right]\psi_i(\mathbf{r}) - \int \sum_{\substack{j \neq i \\ (\text{spin } j\, =\, \text{spin } i)}} \frac{\psi_j^*(\mathbf{r}')\,\psi_i(\mathbf{r}')}{|\mathbf{r} - \mathbf{r}'|}\, d\mathbf{r}'\, \psi_j(\mathbf{r}) = \varepsilon_i\,\psi_i(\mathbf{r}) } $$ which is known as the **Hartree–Fock equation**. In the above equation $v_H(\mathbf{r})$ is the Hartree potential. The H–F method introduces an improvement over Hartree energy, and it is due to the appearance of exchange energy, which is caused by taking the antisymmetry of the wave function into account. This exchange interaction is a real quantum mechanical effect and must appear in any one-electron model. The H–F method is a mean-field theory, that is, the many-body problem is reduced to a one-electron problem in which the electron moves in an average field generated by other electrons. It ignores Coulomb correlations that arise because electrons repel each other and move in such a way as to avoid each other. However, the H–F method does include some correlations between parallel spin electrons, which give rise to the exchange interaction. The Coulomb correlations actually reduce the exchange interaction between electrons with parallel spins, so that the H–F equation greatly overestimates the importance of the exchange interaction. This equation describes compact systems such as atoms well but it predicts totally incorrect behavior for the conduction electrons in solids. ---