# 8.3: Hartree-Fock Equations are Solved by the Self-Consistent Field Method

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The Hartree method is used to approximate the wavefunction and the energy of a quantum multi-electron system in a stationary state. This approximation assumes that the exact \(N\)-body wavefunction of the system can be approximated by a product of single-electron wavefucntions. By invoking the variational method, one can derive a set of \(N\)-coupled equations for the N spin orbitals. A solution of these equations yields the Hartree wavefunction and energy of the system. It is one step better than the "Ignorance is Bliss" approach, discussed previously, but still far from a good approximation.

## The Unsolvable System

The generic multielectron atom including terms for the additional electrons with a general charge \(Z\); e.g.

\[V_{\text{nuclear-electron}}(r_1) = -\dfrac {Z}{\left\vert\mathbf{r} - \mathbf{R}\right\vert} \label{8.3.1}\]

in atomic units with \(\left\vert\mathbf{r} - \mathbf{R}\right\vert\) is the distance between the electron and the nucleus, The Hamiltonian must also have terms for electron-electron repulsion (also in atomic units)

\[V_{\text{electron-electron}}(r_{12}) = \dfrac {1}{\left\vert \mathbf{r} - \mathbf{r}^{\prime} \right\vert} \label{8.3.2}\]

with \(\left\vert \mathbf{r} - \mathbf{r}^{\prime} \right\vert\) is the distance between electron 1 and electron 2. So the proper multi-electron Hamiltonian can be constructed

\[\hat {H} (r_1, r_2, ... r_n) = -\dfrac {\hbar ^2}{2m_e} \sum _i \nabla ^2_i + \sum _i V_{\text{nuclear-electron}} (r_i) + \sum _{i \ne j} V_{\text{electron-electron}} (r_{ij}) \label{8.3.3}\]

Given what we have learned from the previous quantum mechanical systems we’ve studied, we predict that exact solutions to the multi-electron Schrödinger equation would consist of a family of multi-electron wavefunctions, each with an associated energy eigenvalue. These wavefunctions and energies would describe the ground and excited states of the multi-electron atom, just as the hydrogen wavefunctions and their associated energies describe the ground and excited states of the hydrogen atom. We would predict quantum numbers to be involved, as well.

The fact that electrons interact through their electron-electron repulsion (final term in Equation \(\ref{8.3.3}\)) means that an exact wavefunction for a multi-electron system would be a single function that depends **simultaneously **upon the coordinates of all the electrons; i.e., a multi-electron wavefunction:

\[|\psi (r_1, r_2, \cdots r_i) \rangle \label{8.3.4}\]

Unfortunately, the electron-electron repulsion terms make it impossible to find an exact solution to the Schrödinger equation for many-electron atoms.

## The Hartree Approximation

The method for finding best possible one-electron wavefunctions that was published by Douglas Hartree in 1948 and improved two years later by Vladimir Fock. For the Schrödinger equation to be analytically solvable, the variables must be separable - the variables are the coordinates of the electrons. To separate the variables in a way that retains information about electron-electron interactions, the electron-electron term (Equation \(\ref{8.3.1}\)) must be approximated so it depends only on the coordinates of one electron. Such an approximate Hamiltonian can account for the interaction of the electrons in **an average way**. The exact one-electron eigenfunctions of this approximate Hamiltonian then can be found by solving the Schrödinger equation. These functions are the best possible **one-electron functions**.

The Hartree approximation starts by invoking an initial *ansatz* that the multi-electron wavefunction in Equation \(\ref{8.3.4}\) can be expanded as a product of single-electron wavefunctions (i.e., orbitals)

\[ | \psi(\mathbf{r}_1,\mathbf{r}_2, \ldots, \mathbf{r}_N) \rangle \approx \psi_{1}(\mathbf{r}_1)\psi_{2}(\mathbf{r}_2) \ldots \psi_{N}(\mathbf{r}_N) \label{2.3}\]

from which it follows that the electrons are *independent*, and interact only via the mean-field Coulomb potential. This yields one-electron Schrödinger equations of the form

\[-\dfrac{\hbar^{2}}{2m} \nabla^{2}\psi_{i}(\mathbf{r}) + V(\mathbf{r})\psi_{i}(\mathbf{r}) = \epsilon_{i}\psi_{i}(\mathbf{r}) \label{2.4}\]

or

\[H_e(r) \psi_{i}(\mathbf{r}) = \epsilon_{i}\psi_{i}(\mathbf{r}) \label{2.4B}\]

where \(V(r)\) is the potential in which the electron moves; this includes both the nuclear-electron interaction

\[V_{nucleus}(\mathbf{r}) = -Ze^{2}\sum_{R} \dfrac{1}{\left\vert\mathbf{r} - \mathbf{R}\right\vert} \label{2.5}\]

and the mean field arising from the \(N-1\) other electrons. We smear the other electrons out into a smooth negative charge density \(\rho(\mathbf{r}')\) leading to a potential of the form

\[V_{electron}(\mathbf{r}) = -e\int d\mathbf{r}^{\prime} \rho(\mathbf{r}^{\prime}) \dfrac{1}{\left\vert \mathbf{r} - \mathbf{r}^{\prime} \right\vert} \label{2.6}\]

where

\[\rho(\mathbf{r}) = \sum_{i}^{\text{occupied}}\vert\psi(\mathbf{r})\vert^{2}.\]

The sum over runs over all occupied states; i.e., only the states of electrons that exist in the atom. The wavefunctions that from this approach with the Hamiltonian \(H_e(r)\) involve possess three kinds of energies discussed below.

Three Energies within the Hartree Approximation

**Kinetic Energy:**The Kinetic energy of the electron has an average value is computed by taking the expectation value of the kinetic energy operator \[\dfrac{- \hbar^2}{2m} \nabla^2\] with respect to any particular solution \(\phi_j(r)\) to the Schrödinger equation: \[KE = \langle\phi_j| \dfrac{- \hbar^2}{2m} \nabla^2 |\phi_j\rangle \]**Nuclear-Electron Coulombic Attraction Energy:**Coulombic attraction energy with the nucleus of charge \(Z\): \[\langle\phi_j| \dfrac{-Z e^2}{\left\vert\mathbf{r} - \mathbf{R}\right\vert} |\phi_j\rangle\]**Electron-Electron Coulombic Repulsion Energy:**Coulomb repulsion energies with all of the \(N-1\) other electrons, which are assumed to occupy other atomic orbitals denoted \(\phi_K\), with this energy computed as \[\sum_{j\neq k} \langle\phi_j(r) \phi_k(r’) |\dfrac{e^2}{|r-r’|} | \phi_j(r) \phi_k(r’)\rangle.\label{8.3.8}\] The Dirac notation \(\langle\phi_j(r) \phi_k(r’) |\dfrac{e^2}{|r-r’|} | \phi_j(r) \phi_k(r’)\rangle\) is used to represent the two-electron (six-dimensional) Coulomb integral \[J_{j,k} = \int |\phi_j(r)|^2 |\phi_k(r’)|^2 \dfrac{e^2}{r-r'} dr dr’ \label{8.3.9}\] that describes the Coulomb repulsion between the charge density \(|\phi_j(r)|^2\) for the electron in \(\phi_j\) and the charge density \(|\phi_k(r’)|^2\) for the electron in \(\phi_k\). Of course, the sum over \(k\) must be limited to exclude \(k=j\) to avoid counting a “self-interaction” of the electron in orbital \(\phi_j\) with itself.

The total energy \(\epsilon_j\) of the orbital \(\phi_j\), is the sum of the above three contributions:

\[\epsilon_J = \langle\phi_j| \dfrac{- \hbar^2}{2m} \nabla^2 |\phi_j\rangle + \langle\phi_j| \dfrac{-Z e^2}{\left\vert\mathbf{r} - \mathbf{R}\right\vert} |\phi_j\rangle + \sum_{j\neq k} \langle\phi_j(r) \phi_k(r’) |\dfrac{e^2}{|r-r’|} | \phi_j(r) \phi_k(r’)\rangle.\label{8.3.10}\]

This treatment of the electrons and their orbitals is referred to as the Hartree-level of theory.