What is The integrals of the LCAO method?

For two real atomic orbitals $\varphi_A$, $\varphi_B$ and the energy operator $\hat H$ of an electron in the molecule:

• the overlap integral $S = \int\varphi_A\varphi_B\,dV$ measures how much the two orbitals occupy the same region ($S = 1$ for identical, superimposed orbitals);
• the Coulomb integral $\alpha_A = \int\varphi_A\hat H\varphi_A\,dV$ is the energy of an electron in $\varphi_A$ within the molecule, close to the atomic orbital energy;
• the resonance integral $\beta = \int\varphi_A\hat H\varphi_B\,dV$, negative, measures the coupling of the two orbitals.

The energies $E$ of the molecular orbitals are the roots of the secular determinant