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.