The Hartree-Fock approximation and beyond

With the Born-Oppenheimer approximation the general problem of the electron and nuclei is separated. The first problem awaiting solution now is the Schrödinger equation of the interacting electrons moving in the potential of the momentary configuration of the nuclei
\begin{equation*}
\left(\hat{T}_{\mathrm{e}}+\hat{V}_{\mathrm{ee}}
+\hat{V}_{\mathrm{ek}}(\{\mathbf{R}_I\}_{N_\mathrm{k}})\right)
\,\Phi_n(\{\mathbf{r}_i\}_{N_\mathrm{e}};\{\mathbf{R}_I\}_{N_\mathrm{k}})
=E_n(\{\mathbf{R}_I\}_{N_\mathrm{k}})\,
\Phi_n(\{\mathbf{r}_i\}_{N_\mathrm{e}};\{\mathbf{R}_I\}_{N_\mathrm{k}})
\end{equation*}
The potential of the electron-nuclei interaction $\hat{V}_{\mathrm{ek}}$ can be cast into the form
$\hat{V}_{\mathrm{ek}}(\{\mathbf{r}_i\}_{N_\mathrm{e}},\{\mathbf{R}_I\}_{N_\mathrm{k}})=\sum_i V_\mathrm{ext}(\mathbf{r}_i)$, where the single particle potential $ V_\mathrm{ext}(\mathbf{r})$ in our context is the Coulomb potential of all nuclei with positions $\{\mathbf{R}_I\}$. In the following, however, $V_\mathrm{ext}(\mathbf{r})$ is not limited to this context and can be any single particle potential. Since no explicit reference to the coordinates of the nuclei will be made in the following, the above electron wave function $\Phi_n$ is written as a function of the electron coordinates only, i.e. $\Phi_n(\{\mathbf{r}_i\}_{N_\mathrm{e}})$.

Already for small atoms and molecules, not to mention solids, the Schrödinger equation cannot be solved, neither analytically nor numerically. For the purpose of assessing $\Psi_0$, therefore approximations to the wave function become necessary in one or the other form. The focus in the following will lie on the outline of the concepts for the discussion of the methods of theoretical spectroscopy, which also offer a complementary approach to this problem. The primary interest at the moment is to describe the electron ground state.

The idea of the Hartree-Fock method for this purpose is to construct an approximation of the many electron wave function from single particle wave functions. This approximate wave function must obey the Pauli principle. It is given in terms of the slater determinant
\begin{equation}
\Phi_{\mathrm{HF}}=\frac{1}{N_\mathrm{e}!}\det \bar{\phi}_{i,\sigma_j}(\mathbf{r}_j)
\qquad\mathrm{with}\quad j=1,\ldots,N_\mathrm{e}\quad.
\end{equation}
The single particle spin orbitals $\bar{\phi}_{i}(\mathbf{r})=\phi_i(\mathbf{r})\,\alpha_i(\sigma)$ are required to be orthonormal wave functions, i.e. $\langle\bar{\phi}_i(\mathbf{r})|\bar{\phi}_j(\mathbf{r})\rangle=\delta_{i j}$, such that $\Phi_{\mathrm{HF}}$ is normalized. Here $\alpha_i(\sigma)$ is the spin eigenfunction with eigenvalue $\bar{\sigma}_i$. The equation for the $\phi_i(\mathbf{r})$ is obtained from the Ritz principle. The variation is executed with respect to the complex conjugate of the independent single particle orbitals $\delta\phi^*_j(\mathbf{r})$ under the constraint $\langle \bar{\phi}_j(\mathbf{r})|\bar{\phi}_j(\mathbf{r})\rangle=1$. This leads to an equation of the form
\begin{equation}
\frac{\delta}{\delta\phi_j^*}\,\left({\langle\Phi_\mathrm{HF}|\hat{H}_\mathrm{e}|\Phi_\mathrm{HF}\rangle}
-\sum_i \varepsilon_i\langle\phi_i|\phi_i\rangle\right)=0\quad,
\end{equation}
where the $\varepsilon_i$ are the Lagrange multipliers to account for the constraint. In order to execute this variation, the expectation value of the Hamltonian $\hat{H}_\mathrm{e}$, namely the ground state energy $E_{\mathrm{HF}}$, has to be evaluated. It is given by
\begin{align}
E_\mathrm{HF}=\sum_{i}&\int d^3r \phi_i^*(\mathbf{r})\,
\left(-\frac{1}{2}\nabla^2+V_{\mathrm{ext}}(\mathbf{r})\right)\,
\phi_i(\mathbf{r}) \\
&+\frac{1}{2} \sum_{i\neq j}\iint d^3r\,d^3r'
\frac{\phi^*_i(\mathbf{r})\,\phi^*_j(\mathbf{r}')\phi_i(\mathbf{r})\,\phi_j(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|}\\
&-\frac{1}{2} \sum_{{i\neq j}} \delta_{\bar{\sigma}_j,\bar{\sigma}_j}\iint d^3r\,d^3r'
\frac{\phi^*_i(\mathbf{r})\,\phi^*_j(\mathbf{r}')\phi_j(\mathbf{r})\,\phi_i(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|}\quad,
\end{align}
where $\bar{\sigma}_i$ denotes the spin eigenvalue associated with the spin orbital $\alpha_i(\sigma)$. The above expression for the energy can be brought into a more conventional form. Firstly note that in the constraint $i\neq j$ in the summation can be dropped as the additional term is the same in both sums, therefore cancels out. The second simplification is accomplished then with the help of the electron density $n(\mathbf{r})$. It is given by
\begin{equation}
n(\mathbf{r})=\sum_i |\phi_i(\mathbf{r})|^2\quad.
\end{equation}
The second term, after dropping the constraint, now can be expressed with the help of $n(\mathbf{r})$. This energy term is the Hartree energy $E_\mathrm{H}$
\begin{equation}
E_\mathrm{H}=\frac{1}{2} \sum_{i,j}\iint d^3r\,d^3r'
\frac{|\phi_i(\mathbf{r})|^2\,|\phi_j(\mathbf{r}')|^2}{|\mathbf{r}-\mathbf{r}'|}=\frac{1}{2}\iint d^3r\,d^3r'\,
\frac{n(\mathbf{r})\,n(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|}
\end{equation}
The Hartree energy thus corresponds to the classical electrostatic energy of the charge density $n(\mathbf{r})$. It contains a self-interaction of the charge density at $\mathbf{r}$ with itself, which is canceled by the following exchange energy term. This last sum in the above expression for $E_\mathrm{HF}$ solely arises from the Pauli principle as a consequence of the quantum mechanical nature of the indistinguishable electrons
\begin{equation}
E_{\mathrm{X}}=-\frac{1}{2}\sum_{i\,j}\delta_{\bar{\sigma}_1,\bar{\sigma}_j}
\iint d^3r\,d^3r'\,\frac{\phi^*_i(\mathbf{r})\,\phi^*_j(\mathbf{r}')\phi_j(\mathbf{r})
\,\phi_i(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|}
\end{equation}

The Hartree-Fock equations for the orbitals $\phi_i$ are then obtained from the variation of $E_{\mathrm{HF}}$ respecting the constraint of orthonormality
\begin{equation}
\left(\vphantom{\int}-\frac{1}{2}\nabla^2+V_{\mathrm{ext}}(\mathbf{r})+V_\mathrm{H}(\mathbf{r})\right)\phi_i(\mathbf{r})
+\int d^3r'\,\frac{\phi^*_j(\mathbf{r}')\phi_j(\mathbf{r})\,\phi_i(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|}=\varepsilon_i\,\phi_i(\mathbf{r})\quad,
\end{equation}
where the $\varepsilon_{i}$ are Lagrange multipliers to ensure the orthonormality constraint of the single particle orbitals. The Hartree potential $V_{\mathrm{H}}$ arises from the Hartree energy $E_{\mathrm{H}}$
\begin{equation}
V_\mathrm{H}(\mathbf{r})=\int d^3r' \frac{n(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|}
\end{equation}
and although it is written here like a typical potential it depends on the electron density, which is calculated from all orbitals $\phi_i$. The last sum gives rise to the so called exchange potential. The exchange potential operator $\hat{V}_\mathrm{X}$ is an integral operator
\begin{equation}
\hat{V}_\mathrm{X}\,\phi_i(\mathbf{r})=-\sum_j \delta_{\bar{\sigma}_i,\bar{\sigma}_j}
\int d^3r' \frac{\phi^*_j(\mathbf{r}')\phi_j(\mathbf{r})}{|\mathbf{r}-\mathbf{r}'|}\,\phi_i(\mathbf{r}')
\end{equation}
$\hat{V}_\mathrm{X}$ is a non-local operator as $\phi_i$ appears under the integral together with $\phi_j^*$, whereas $\phi_j$ remains outside the intergration. Hence the exchange potential for each orbital $\phi_i$ is different. $V_\mathrm{H}$, in contrast, despite being an integral operator too, is the same for all orbitals and which is expressed by writing it in the form $V_\mathrm{H}(\mathbf{r})$.

Solving the Hartree-Fock equations is more complicated than solving the Schrödinger equation of a single-particle problem. The reason is its non-linearity in the orbitals $\phi_i$, or, in other words, the fact that the expressions for the potential operators $\hat{V}_{\mathrm{H}}$ and $\hat{V}_{\mathrm{X}}$ contain the orbitals $\phi_i$ themselves. A practicle scheme illustrated in Fig. 1.2 is the iterative solution of the equations for fixed approximations to the potentials $\hat{V}_{\mathrm{H}}$ and $\hat{V}_{\mathrm{X}}$ improved in each step. At the start of this iterative procedure an approximation of the initial density and orbitals $\phi^{\tau=0}$ is used for evaluating the first approximation $\hat{V}^{\tau=1}_{\mathrm{H}}$ and $\hat{V}^{\tau=1}_{\mathrm{X}}$, where $\tau$ is an iteration index. In the $\tau$-th. iteration one first solves the Hartree-Fock equations using the potential operators $\hat{V}^{\tau}_{\mathrm{H}}$ and $\hat{V}^{\tau}_{\mathrm{X}}$ of the previous iteration. From the new orbitals $\phi^{\tau}_i$ one evaluates the potential operators to generate a new approximation $\hat{V}^{\tau+1}_{\mathrm{H}}$ and $\hat{V}^{\tau+1}_{\mathrm{X}}$. The equations are solved when the sequence of potentials and orbitals (within a certain tolerance) have converged to final values.

Fig. 1.2: Illustration of the method of the self-consistent field for solving iteratively the Hartree-Fock equations. The step highlighted in orange is the actual step being executed. Follow the scheme until a solution is obtained.

So far the $\varepsilon_i$ are mathematical objects only and were introduced as Lagrange multipliers to ensure the orthonormality of the orbital $\phi_i$. Allthough the similarity of the Hartree-Fock equation with a Schrödinger equation might suggest this, an interpretation in terms of a single particle eigenenergies is not permitted on a strict basis. The following considerations will illustrate this.

First of all, it is interesting to compare the Hartree-Fock groundstate energy $E_\mathrm{HF}$ with that of a single particle system with single particle eigenvalues $\varepsilon_i$. For this purpose $E_\mathrm{HF}$ is rewritten. First the expectation value $\langle\phi_i|\frac{1}{2}\nabla^2+V_\mathrm{ext}|\phi_i\rangle$ is expressed with help of the Hartree-Fock equation
\begin{align}
\int d^3r\
\phi_i^*(\mathbf{r})\left(\vphantom{\int}-\frac{1}{2}\nabla^2+V_\mathrm{ext}\right)
\phi_i(\mathbf{r}) =\varepsilon_i&-\iint
d^3r\,d^3r'\,\frac{|\phi_i(\mathbf{r})|^2\,n(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|}
\\ &+\sum_j \delta_{\bar{\sigma}_i,\bar{\sigma}_j} \iint d^3r\,d^3r'
\frac{\phi^*_i(\mathbf{r})\phi^*_j(\mathbf{r}')\phi_j(\mathbf{r})\,\phi_i(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|}\quad.
\end{align}
Substituting this into the expression for $E_\mathrm{HF}$ yields
\begin{align}
E_\mathrm{HF}&=\sum_{i} \varepsilon_i
-\frac{1}{2} \sum_{i,j}\iint d^3r\,d^3r'
\frac{\phi^*_i(\mathbf{r})\,\phi^*_j(\mathbf{r}')\phi_i(\mathbf{r})\,\phi_j(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|}\\
&\hphantom{-\frac{1}{2} \sum_{{i,j}}}
+\frac{1}{2} \sum_{{i,j}} \delta_{\bar{\sigma}_j,\bar{\sigma}_j}\iint d^3r\,d^3r'
\frac{\phi^*_i(\mathbf{r})\,\phi^*_j(\mathbf{r}')\phi_j(\mathbf{r})\,\phi_i(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|}\\
&=\sum_i \varepsilon_i-E_\mathrm{H}-E_\mathrm{X}
\end{align}
The Hartree-Fock ground state energy unlike the ground state energy of a single particle system is not only given by the sum over the occupied single particle eigenvalues $\varepsilon_i$. In addition it contains two terms that arise from the mutual interaction of the electrons. Now consider the ionization of the $N_\mathrm{e}$-electron system by removing the particle with the highest value of $\varepsilon_i$. As a consequence the slater determinante will contain $N_\mathrm{e}-1$ single particle orbitals accept the orbital $\phi_i$. The ionization energy $I$ of the system is obtained from $E_\mathrm{HF}(N_\mathrm{e})-E_\mathrm{HF}(N_\mathrm{e}-1)$. Under the assumption that the number of electrons is large, one may approximate the orbitals of the ionized system by the remaining $\phi_j$ orbitals. With this approximation one obtains the ionization energy of an electron in the $i$th. quantum state by
\[
I=E_\mathrm{HF}(N_\mathrm{e}-1_i)-E_\mathrm{HF}(N_\mathrm{e})=-\varepsilon_i
\]
Thus, under the above assumption, the eigenvalue $\varepsilon_i$ can be identified with the ionization energy of the system. This statement is known as Kopmann's theorem.

Problems of the Hartree-Fock approximation

• In metallic systems a logarithmic divergence of the
  density of states appears at the Fermi level.
• In semiconductors or insulators the band gap turns out to be too large.
• In molecules the HOMO-LUMO gap often is overestimated

Direct improvements of the Hartree-Fock approximation follow its spirit and address a refinement of the many-electron wave function $\Phi$. The central idea is to add to the solutions $\phi_i$, $i\leq N_\mathrm{e}$, of the Hartree-Fock equation orbitals $\phi_k$ with $k>N_\mathrm{e}$. These latter states are obtained by solving the Hartree-Fock equation with the density $n(\mathbf{r})$ given by the orbitals $\phi_i$ with $i\leq N_\mathrm{e}$ and the summation of the exchange term still restricted to the same set of orbitals. This set of orbitals forms a complete basis of the single particle Hilbert space.

With this complete set of single particle orbitals one expands $\Phi$
\begin{equation}
\Phi(\{\mathbf{r}_j\})=\sum_{\nu_1,\ldots\nu_{N_\mathrm{e}}} a_{\nu_1,\ldots,\nu_{N_\mathrm{e}}}
\phi_{\nu_1}(\mathbf{r}_1)\cdot\phi_{\nu_2}(\mathbf{r}_2)\cdots
\phi_{\nu_{N_\mathrm{e}-1}}(\mathbf{r}_{N_\mathrm{e}-1})\cdot\phi_{\nu_{N_\mathrm{e}}}(\mathbf{r}_{N_\mathrm{e}})
\end{equation}
here for each coordinate $\mathbf{r}_i$ the summation contains all possible orbitals $\phi_{\nu_i}(\mathbf{r}_i)$ with the quantum number $\nu_i$. The $a_{\nu_1,\ldots\nu_{N_\mathrm{e}}}$ are the expansion coefficients. They are determind such that the Pauli principle is fullfilled, i.e. that the sign of $\Phi(\{\mathbf{r}_j\})$ changes when the coordinates $\mathbf{r}_i$ and $\mathbf{r}_j$ are exchanged. This implies, that the coefficients with identical indices $\nu_i$ and $\nu_j$ must vanish. Therefore one can write $\Phi(\{\mathbf{r}_j\})$ as a linear combination of slater determinants
\begin{equation}
\Phi(\{\mathbf{r}_j\})=\sum_{\nu_1<\ldots<\nu_{N_\mathrm{e}}}
f_{\nu_1,\ldots\nu_{N_\mathrm{e}}} \frac{1}{\sqrt{N_\mathrm{e}!}}
\mathrm{det}\,\phi_{\nu_i}(\mathbf{r}_j)
\end{equation}
The summation thus contains the slater determinant of the ground state $\Phi_{\mathrm{HF}}$, determinants where one of the orbitals in $\Phi_{\mathrm{HF}}$ is replaced by an unoccupied orbitals, determinants where two of the orbitals $\Phi_{\mathrm{HF}}$ are replaced by unoccupied orbitals and so forth. Each of the unoccupied orbitals is associated with an $\varepsilon_{\nu_i}$ that is larger than any of the $\varepsilon_{\nu_i}$ of the orbitals $\phi_{\nu_j}$ included in $\Phi_{\mathrm{HF}}$. This can be rationalized as if one excites a number of electrons from the ground state into the unoccupied states as sketched in Fig. 1.3.

Fig. 1.3: Schematic visualisation of the expansion of $\Phi(\{\mathbf{r}_{i}\})$ in terms of the Hartree-Fock ground state and slater determinats of excited states. Single particle orbitals corresponding to the lowest $\varepsilon_{i}$ (denoted by black lines) form the slater determinat of the ground state. In the second set of determinants one ground state orbital is replaced by an unoccupied orbital (unoccupied levels are represented by gray lines). The third set contains to unoccupied orbitals and so forth. The inclusion of the orbitals indicated by up and down arrows representing the electron spin.

The coefficients $f_{\nu_1,\ldots\nu_{N_\mathrm{e}}}$ are determined now by the application of the Ritz principle to the energy expectation value $\langle \Phi|\hat{H}_\mathrm{e}|\Phi\rangle$. This expectation value can be expressed in the form
\begin{equation}
\langle \Phi|\hat{H}_\mathrm{e}|\Phi\rangle=
\sum_{\nu_1<\ldots<\nu_{N_\mathrm{e}}}\sum_{\mu_1<\ldots<\mu_{N_\mathrm{e}}}f^{*}_{\nu_1,\ldots,\nu_{N_\mathrm{e}}}
f_{\mu_1,\ldots,\mu_{N_\mathrm{e}}}\frac{1}{N_\mathrm{e}!}
\langle \mathrm{det}\,
\phi_{\nu_i}(\mathbf{r}_j)|\hat{H}_\mathrm{e}|\mathrm{det}\,\phi_{\mu_i}(\mathbf{r}_j)\rangle
\end{equation}
The summation contains an infinite number of terms unless the basis set is finite. And even then the number of matrix elements one has to include is typically large. Thus in practise not all excited states can be considered and approximations have to be made. It is interesting to note, that the matrix elements among determinants corresponding to single excitations vanish, thus the smallest set of slater determinants contains the ground state and double excitations and the next larger set includes also the single excitations.

At least, in principle, one can describe the exact ground state with this procedure. In this ground state slater determinants of the Hartree-Fock ground state and excited states are correlated via the coefficients $f_{\nu_1,\ldots,\nu_N}$. Therefore the energy difference between the exact ground state energy $E_\mathrm{exact}$ and the Hartree-Fock ground state is called correlation energy
\begin{equation}
E_\mathrm{C}=E_\mathrm{exact}-E_\mathrm{HF}
\end{equation}
The concepts of exchange and correlation of identical particles is central to the physics of a many electron system and will be used in the subsequent discussion of methods of theoretical spectroscopy.

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The Rayleigh-Ritz principle

The energy eigenvalue of each eigenstate $\Psi_n$ of the Hamiltonian $\hat{H}$ or more specifically $\hat{H}_\mathrm{e}$ or $\hat{H}_\mathrm{k}$ is equal to the expectation value
\begin{equation}
E_n=\frac{\langle\Psi_n|\hat{H}|\Psi_n\rangle}{\langle\Psi_n|\Psi_n\rangle}
\end{equation}
For any quantum state $\Psi$ of the system's Hilbert space, the expectation value of the energy $E$ is a functional of the wave function $\Psi$
\begin{equation}
E[\Psi]=\frac{\langle\Psi|\hat{H}|\Psi\rangle}{\langle\Psi|\Psi\rangle}
\end{equation}
The ground state $\Psi_0$ is associated per definition with the lowest energy, $E_0\lt E_{n\neq0}$. Therefore the variation of the functional with respect to $\Psi$ or its complex conjugate $\Psi^*$ has to be stationary for the ground state, i.e. $\delta E[\Psi]/\delta\Psi=0$ or $\delta E/\delta\Psi^*=0$. This variation yields the Schrödinger equation
\begin{equation}
\frac{\delta E[\Psi_0]}{\delta\Psi_0^*}=\frac{\hat{H}\Psi_0
-\langle \Psi_0|\hat{H}|\Psi_0\rangle \Psi_0}{\langle\Psi_0|\Psi_0\rangle}
=0\qquad\mathrm{and\ hence}\qquad \hat{H}\Psi_0=E_0\,\Psi_0\quad.
\end{equation}
Consider now a small deviation $\delta\Psi$ from the ground state wave function $\Psi_0$. Here $\delta\Psi$ -- without loss of generality -- shall be orthogonal to $\Psi_0$, i.e. $\langle\Psi_0|\delta\Psi\rangle=0$.
The energy expectation value of this state $\Psi_0+\delta\Psi$ shall be expanded in $\delta\Psi$
\begin{align}
E[\Psi_0+\delta\Psi]&=\frac{\langle\Psi_0+
\delta\Psi |\hat{H}|\Psi_0+\delta\Psi\rangle}{\langle\Psi_0+\delta\Psi|\Psi_0+\delta\Psi\rangle}\\
&=\frac{\langle\Psi_0|\hat{H}|\Psi_0\rangle+\langle\Psi_0|\hat{H}|\delta\Psi\rangle
+\langle\delta\Psi|\hat{H}|\Psi_0\rangle
+\langle\delta\Psi_0|\hat{H}|\delta\Psi_0\rangle}{\langle\Psi_0|\Psi_0\rangle+\langle\Psi_0|\delta\Psi\rangle
+\langle\delta\Psi|\Psi_0\rangle+\langle\delta\Psi|\delta\Psi\rangle}\\
&=\frac{E_0\,\langle\Psi_0|\Psi_0\rangle+\langle\delta\Psi |\hat{H}|\delta\Psi\rangle}{\langle\Psi_0|\Psi_0\rangle+\langle\delta\Psi|\delta\Psi\rangle}=E_0+o(\delta\Psi^2)
\end{align}
In the last line the fact that $\hat{H}\Psi_0=E_0\,\Psi_0$ and $\langle\Psi_0|\delta\Psi\rangle=0$ were used. Thus, for an approximation $\Psi_0+\delta\Psi$ to the ground state wave function $\Psi_0$ the error in the ground state energy will be quadratic in the deviation $\delta\Psi$. In particular, the approximation value $E[\Psi_0+\delta\Psi]$ for the ground state energy will always be larger than the true value $E_0$.

The Rayleigh-Ritz principle provides therefore a strategy to find approximations to the ground state wave function in terms of an energy minimization. Among two distinct approximations the one with the lower energy expectation value is the better approximation. The error in the approximation for the ground state energy converges with second order of the deviation $\delta\Psi$ from the ground state wave function.

For excited states the Functional $E[\Psi_n+\delta\Psi]$ is only stationary. Also the error is quadratic in $\delta\Psi$: Nevertheless the convergence of $E[\Psi_0+\delta\Psi]$ to $E_n$ is not monotonic as for the ground state.

One important application of this principle for the purpose of numerical calculations is the approximation of the infinite Hilbert space by a vector space spanned by a finite basis of normalized wave
functions $\{\bar{\Psi}_n\}$. The approximate wave function $\bar{\Psi}$ is thus expanded in terms of the basis functions and coefficients $a_n$
\begin{equation}
\bar{\Psi}=\sum_n a_n\,\bar{\Psi}_n
\end{equation}
Application of the Rayleigh-Ritz principle to the finite vector space yields the generalized eigenvalue problem
\begin{equation}
\sum_m \langle \bar{\Psi}_n|\hat{H}|\bar{\Psi}_m\rangle\,a_m
=E\,\sum_m \langle \bar{\Psi}_n|\bar{\Psi}_m\rangle\,a_m\quad.
\end{equation}

Another application is the simplification of the many particle wave function in terms of a generic form. The adiabatic approximation for the separation of electronic and nuclear degrees of freedom, for instance, can be seen in this light. Also Hartree-Fock theory and theories beyond it are based on approximations regarding the functional form of the wave function, namely they express $\Psi$ in terms of single particle wave functions as detailed in the following.

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