Variational method
Variational method is one way of finding approximations to the ground state, and some excited states. This allows for calculating approximate wavefunctions such as molecular orbitals. The basis for this method is the variational principle.
The method consists of choosing a “trial wavefunction” depending on one or more parameters, and finding the values of these parameters for which the expectation value of the energy is the lowest possible. The wavefunction obtained by fixing the parameters to such values is then an approximation to the ground state wavefunction, and the expectation value of the energy in that state is an upper bound to the ground state energy.
$ \global\def\psit{\psi_\text{trial}} $ $ \global\def\Egs{E_\text{gs}} $
\[\boxed{ \brktop{\psit}{ \hat{\hamiltonian} }{\psit}\ge \Egs }\]
Since the wavefunction is an element of Hilbert space, it can be represented as a linear combination of the eigenfunctions of the Hamiltonian.
\[ \ket{\psit} = \sum_n \ket{\psi_n}\brkt{\psi_n}{\psit} \]
Therefore, the expectation value of the Hamiltonian is:
\[ \begin{align*} \expct{\hat{\hamiltonian}} &= \sum_n \abs{\brkt{\psi_n}{\psit}}^2 E_n \nl &= \sum_n \abs{\brkt{\psi_n}{\psit}}^2 (E_n-\Egs) + \Egs \nl &\ge \Egs \end{align*} \]
Also, we can compute an upper bound to the energy level of the first excited state similarly. If we fully know the ground state wavefunction $\psi_\text{gs}$ and have the trial wavefunction orthogonal to it, i.e. $\brkt{\psi_\text{gs}}{\psit}=0$, we get:
\[ \expct{\hat{\hamiltonian}} \ge E_\text{fe} \]
Trivially,
\[ \begin{gather*} \begin{cases} \brkt{\psi_\text{gs}}{\psit}=0 \nl \brkt{\psi_\text{fe}}{\psit}=0 \end{cases} \Rightarrow \expct{\hat{\hamiltonian}}\ge E_\text{se} \nl \vdots \end{gather*} \]