Exchange interaction is the conceptual foundation to the understanding of the covalent bond, ferromagnetism, and electron transport phenomenon. However, its evaluation has always been a formidable problem. We show that within the Hartree-Fock and muffin-tin potential approximations, using time-dependent perturbation theory, simple analytic expressions of the exchange interaction between two atomic states can be obtained. To assess the accuracy of the method, we evaluated the dissociation energies and the vibrational frequencies for homonuclear diatomic molecules built from atoms in the first two rows of the periodic table. The theoretical predictions agree well with experimental values and the accuracy matches that of typical first-principle computations.