We have investigated Hund’s spin-multiplicity rule for the second and third row atoms (C, N, O, Si, P, and S) and the methylene molecule (CH₂) by means of diffusion Monte Carlo method and complete active space self-consistent field method, respectively. It is found that Hund’s rule is interpreted to be ascribed to a lowering in the electron-nucleus attractive Coulomb interaction energy which is realized at the cost of increasing the electron-electron repulsive Coulomb interaction energy as well as the kinetic energy. We have also studied correlation in the hydrogen molecule H₂. Correlation in H₂ gives an increase of the electron density distribution n(r) in the left and right anti-binding regions, a reduction of n(r) in the binding region, and an increase in the equilibrium internuclear separation. The importance of the virial theorem is stressed in the evaluation of correlation effects on both Hund’s rule and chemical bonding in H₂.