Valence bond and molecular orbital descriptions of the three-electron bond

When two atomic orbitals are used to accommodate the electrons of the three-electron bond (or three-electron two-center bond), it is well-known that the valence bond (A . B = AB <----> AB) and molecular orbital (one antibonding + two bonding electrons) descriptions of this type of bond are equivalent, i.e. Psi = Psi(VB) = Psi(MO). With three atomic spin orbitals to accommodate the electrons of AB, and three additional atomic spin orbitals to accommodate the electrons of AB, it is deduced that a wave function of the form Psi = Psi(1)(VB) + Psi(2)(VB) = Psi(1)(MO) + Psi(2)(MO) may be constructed from each set of three atomic spin orbitals, for which the Psi(1) and Psi(2) are three-electron bond wave functions. The equivalence is illustrated via the results of some ab initio calculations for the ground states of H-2(-) and He-2(+). For H-2(-), the use of canonical double-zeta molecular orbitals constructed from 1s' and 1s'' atomic orbitals on each atomic center must lead to ionization of this anion to form H-2 when the exponents of the diffuse (1s'') orbital components of these molecular orbitals are energy-optimized.