An expression for the total energies of molecules E was derived using the power series for the one-electron density matrix (DM) P obtained previously in the basis of bond orbitals (BOs) [Gineityte, V. J Mol Struct (Theochem) 1995, 343, 183] and the well-known relation between the energy E and the DM P [E = Trace(PH), where H is the Hamiltonian matrix]. Inasmuch as the power series for the DM P is based on a certain generalization of the usual Rayleigh-SchrΓΆdinger perturbation theory (RSPT), namely, on the so-called non-commutative RSPT [Gineityte, V. Int J Quantum Chem 1998, 68, 119], the new expression for total energies of molecules proved to be a generalization of the well-known Dewar formula obtained using the usual RSPT in the framework of the HΓΌckel model. The generalization consists of passing to the case of zero-order resonance parameters between pairs of bonding BOs (BBOs) and/or of antibonding BOs (ABOs). Comparative analysis of the Dewar formula and of its generalized version was carried out, and some new interpretations of the former are suggested. In particular, the negative second-order correction within the Dewar energy is shown to be made up of a difference between the stabilization energy due to the formation of bond orders between BBOs and ABOs and the destabilization energy describing the intramolecular charge transfer. The case of alkanes, in general, and a model system of two interacting C-C(C-H) bonds, in particular, are discussed as examples.