The second approximate method that we are going to study to solve the Schrödinger equation, in two-electron systems, is the perturbation method. This method compares the system with no solution (perturbed system) with another, for which an analytical solution is available (unperturbed system). The difference between the two Hamiltonians is called a disturbance. For the error to be low, the disturbance must be as small as possible. That is, we must choose the unperturbed system that has a Hamiltonian as similar as possible to the perturbed system. Let be a time-independent Hamiltonian, $\hat{H}$, whose equation, $\hat{H}\phi_n=E_n\psi_n$, has no analytical solution (perturbed system).
Let $\hat{H}^0$ be the Hamiltonian of a system whose Schrödinger equation, $\hat{H}^{(0)}\phi_n^{(0)}=E_n^{(0)}\psi_n ^{(0)}$, we know how to solve, and it is only slightly different from $\hat{H}$ (unperturbed system). According to the perturbation method, the energy of the perturbed system is given by: $E_n=E_n^{(0)}+E_n^{(1)}$, where $E_n^{(1)}$ is the first-order correction. order in energy. \begin{equation} E_n^{(1)}=\int\psi_n^{\ast (0)}\hat{H'}\psi_n^{(0)}dq=\left\langle \psi_n^{( 0)}\left|\hat{H'}\right|\psi_n^{(0)}\right\rangle \end{equation} Where, $\hat{H'}$, is the disturbance. $\hat{H'}=\hat{H}-\hat{H}^{{0}}$ The wave function of the disturbed system will be given by: $\psi_n=\psi_n^{(0)}+ \psi_n^{(1)}$, where $\psi_n^{(1)}$ is the first order correction in the wave function. \begin{equation} \psi_m^{(1)}=\sum_{m\neq n}\frac{\int\psi_m^{\ast (0)}\hat{H'}\psi_n^{(0) }dq}{E_m^{(0)}-E_n^{(0)}} \end{equation} In the same way that the second order correction for the wave function and energy can be calculated. \begin{equation} E_n=E_n^{(0)}+E_n^{(1)}+E_n^{(2)} \end{equation} \begin{equation} \psi_n=\psi_n^{(0) }+\psi_n^{(1)}+\psi_n^{(2)} \end{equation} Where, the second order correction in energy is given by: \begin{equation} E_n^{(2) }=\int\psi_n^{\ast (0)}\hat{H'}\psi_n^{(1)}dq. \end{equation}