# Variational method: Hydrogen atom ground state in STO-3G basis expansion

I computed numerically the ground state energy of hydrogen by variational procedure, firstly expanding the wave function over the s-wave basis set STO-3G $$\psi(r)=\sum_{i=0}^3 C_{i} e^{-\alpha_ir^2}$$ and then solving the generalized eigenvalue problem (in matrix notation) $$\begin{equation} \textbf{HC}=E\textbf{SC} \end{equation}$$ As output I have all 3x3 matrices. $$E$$ is a diagonal 3x3 matrix and my eigenvalue is the smallest of the three on the diagonal. My question is: what do the other two elements in the $$E$$ diagonal represent?

• The eigenvalues you get are the energies of your orbitals. These energies don't necessarily have any physical meaning except for the fact that they can be approximately related to the true ionization energy by Koopmans' theorem. – jheindel Apr 30 '19 at 23:20
• In the case of Hydrogen I get -0.49501 H and other two positive values. These positive values are energies of which orbitals? – Liuuuuk May 1 '19 at 12:22
• @Liuuuuk Orbitals are, by definition, bound states, hence have negative energy. The positive eigenvalues hardly represent anything of chemical significance. Try a larger basis set so that it can approximate both 1s and 2s orbitals, and you should get an eigenvalue close to -0.125. – lisyarus May 4 '19 at 14:19