In a nutshell, a HOMO with a positive eigenvalue has no physical meaning. The eigenvalue of the HOMO represents the energy of a bonded electron, while the zero value represents the ionization limit (i.e. the energy of a free unbound electron). Accordingly in HF theory Koopmans theorem, which can be strictly applied only to closed shell systems, states that the first ionization energy can be evaluated as the module of the HOMO eigenvalue. While Koopmans theorem has been generalized to DFT (see for example J. Phys. Chem. A 2006, 110, 12005), the energy of a bound system remains inherently negative. Hence, most probably the HOMO and LUMO energy were incorrectly recorded and have been possibly swapped.

@Massimiliano Arca

I have tried to calculate the anionic state(n=1-6) of particular molecule. In 5 th state, I got negative energy gap in alpha spin and positive energy gap in beta spin. For the 6th state, I got positive energy gap value.

Please note that on varying the charge the total number of electrons changes, so the HOMO is not the same MO for all the anions considered. In particular, species with an odd number of electrons will have a number of alpha and beta electrons differing by one.

Massimiliano Arca

The difference in charges cause alpha and beta spin. But does it change the high value to HOMO and low value to LUMO. Wheather the energies given in software (gaussview) may be incorrectly recorded