The odd-electron (one and three) bond involves resonance between two charge-shift related structures, in both MO and VB theory. A physically correct description of odd-electron bonding should reflect the instantaneous response of the orbital size and shape to the charge fluctuation inherent in the odd-electron bonding. VB theory contains this response and involves charge-fluctuation-adapted orbitals in the resonance structures, resulting in reliable bond energies. In contrast, due to its constraints, the Hartree-Fock theory fails to represent this crucial feature of the odd-electron bond and generates thereby poor bond energies. A nonempirical remedy for this Hartree-Fock bias is proposed. This is the "Uniform Mean-Field Hartree-Fock" (UMHF) procedure which is based on the simple unrestricted Hartree-Fock method, but involves orbital occupancy constraints and correction of the resonance energies by non-empirical factors. The UMHF approach is tested on three-electron- and one-electron-bonded molecules and is shown to yield bonding energies in satisfactory agreement with more sophisticated calculations (up to and beyond fourth order pf Moller-Plesset perturbation theory). The UMHF procedure is offered as a routine inexpensive tool for obtaining odd-electron bond energies for large molecules.