When doing an unrestricted calculation (doesn't matter the method, UHF, UMP2, UDFT or whatsoever) you are basically treating the two spin populations in an independent way, so that you loose the symmetry in the "doubly occupied" molecular orbitals that are below the Fermi surface.

So to the two spin populations are associated two sets of molecular orbitals alpha and beta that can be singly occupied or empty.

This symmetry breaking can in general introduce a small (or huge) spin contamination in your wave function, and is usually necessary to recover the property of size-consistency for example in the description of the dissociation processes.

This symmetry breaking is avoided in the restricted calculations (RHF RMP2 RDFT), for which both spin populations are represented with the same molecular orbitals, doubly occupied or empty (Physically this is more correct).

If you have unpaired orbitals you could try a restricted open shell calculation RODFT, in which for the double occupied orbitals the spin up and down electrons are assumed to be on identical orbitals, and the unpaired ones are treated independently.

This different behavior of the restricted and unrestricted calculations can lead to differences in the Kohn-Sham orbital energies, between the two spin populations.

Also the total energies change, because the Unrestricted calculations have more variational freedom then the restricted ones.

Unfortunately there is no "better or worse".

You must remember that despite the use that is sometimes made of the HOMO-LUMO energy differences, the KS orbital energies are not clearly defined physical quantities (unlike the energy of the HOMO orbital that is related to the ionization energy for Koopmans' theorem), and their meaning is strictly related only to the Kohn-Sham representation of non-interacting particles.

Their differences do not represent the excitation energies or the electronic gaps!!!!!!

Also for the ordering of states, you can read http://pubs.acs.org/doi/abs/10.1021/jp101758y

or search some articles on Time dependent DFT if you want to evaluate excitation energies.

I hope I haven't confused you.

Gook luck for you work!

Generally, at the restricted level of the theory, HOMO and LUMO are still the highest occupied and lowest unoccupied MOs. If the total spin of your system is zero (M=1, singlet), both alpha and beta MOs will have the same value. Otherwise, you must look at the energies of both alpha and beta HOMOs and the higher value will be the system HOMO. The same holds for the LUMOs, where the minor value will be the system LUMO. It could happens that your system has an alpha HOMO and beta LUMO, or vice-versa. The analysis become complicated if you are dealing with molecules that have open d and/or open f .

When doing an unrestricted calculation (doesn't matter the method, UHF, UMP2, UDFT or whatsoever) you are basically treating the two spin populations in an independent way, so that you loose the symmetry in the "doubly occupied" molecular orbitals that are below the Fermi surface.

So to the two spin populations are associated two sets of molecular orbitals alpha and beta that can be singly occupied or empty.

This symmetry breaking can in general introduce a small (or huge) spin contamination in your wave function, and is usually necessary to recover the property of size-consistency for example in the description of the dissociation processes.

This symmetry breaking is avoided in the restricted calculations (RHF RMP2 RDFT), for which both spin populations are represented with the same molecular orbitals, doubly occupied or empty (Physically this is more correct).

If you have unpaired orbitals you could try a restricted open shell calculation RODFT, in which for the double occupied orbitals the spin up and down electrons are assumed to be on identical orbitals, and the unpaired ones are treated independently.

This different behavior of the restricted and unrestricted calculations can lead to differences in the Kohn-Sham orbital energies, between the two spin populations.

Also the total energies change, because the Unrestricted calculations have more variational freedom then the restricted ones.

Unfortunately there is no "better or worse".

You must remember that despite the use that is sometimes made of the HOMO-LUMO energy differences, the KS orbital energies are not clearly defined physical quantities (unlike the energy of the HOMO orbital that is related to the ionization energy for Koopmans' theorem), and their meaning is strictly related only to the Kohn-Sham representation of non-interacting particles.

Their differences do not represent the excitation energies or the electronic gaps!!!!!!

Also for the ordering of states, you can read http://pubs.acs.org/doi/abs/10.1021/jp101758y

or search some articles on Time dependent DFT if you want to evaluate excitation energies.

I hope I haven't confused you.

Gook luck for you work!

Both MO's are equally important. With unrestricted calculation, HOMO will be a SOMO(singly occupied molecular orbital). Endorsing Dr. Alves's explanation, i would like to add that depiction of both MO's separately for alpha and beta spins are necessary for a detailed MO analysis.

You may use natural orbitals, for example, for visualization. Natural orbitals calculation is selected by Gaussian keyword pop=naturalorbitals, or if you use GAMESS software by $SCF UHFNOS=.T. $END. As far as I know, orbitals are usually used for visualization purpose, and thus natural orbitals in case of spin-polarized unrestricted solution are better, than alpha or beta set of canonical MOs. Even if you are interested in one-electron orbital energies (ionization potential or electron affinity) the canonical MOs are good only if they come from pure Hartree-Fock calculation. According to Koopmans theorem Hartree-Fock eigenvalues approximates IPs and EAs. DFT orbitals eigenvalues are not this case.

I hope that this is answer to your question.

When you do calculation with different functionals, you can see that the occupied orbital energy are very similar, when the virtual can be significantly different.

This is because the virtual orbitals are not included in the variational space. In another words, they are not used to construct the density.

So if Nalfa>Nbeta, take the alfa orbitals, in this way restricted density will be from optimized MO. If you take beta, the HOMO=SOMO in restricted calculation will be from virtual set of unrestricted.

Hello Sarah Padam!

I hope Mr. Mattel Barborini response has useful for you.

I has arose the similar question in Gaussian forums. Here, I have post their precious response from Gaussian HELPdesk to your attention.

The definition of HOMO-LUMO gap for open shell systems is not as trivially defined as for a closed shell system. There the energy of the HOMO and LUMO are unique and as an estimate of the excitation energy the difference is as well.

For open shell systems you have a couple of potential orbital energies which could be imagined and none of them are obviously related to the excitation. 1) You could have an excitation from a singly occupied orbital to a vacant orbital, and either an alpha or beta could be chosen. 2) You could have excitation from a doubly occupied orbital to a vacant orbital and again you have the option of excitation of an alpha or beta orbital.

So if your real goal is to look at electronic excitation a better approach would be to run a CIsingles or TD-DFT calculation. This will not give you a computational approach for the lowest few roots. You can look at the eigen vector to get a better look at which singly substituted determinants are most important. You may even get lucky and have one of them be very dominant which would let you compare the various HOMO/LUMO differences.

If your goal is just to get a quick and dirty value for estimating excitation you may be out of luck. If this is a single study doing the excitation energy is not that bad.

Does this help?

I grateful to Dr D. J. Fox for this response

Note that ROHF-RODFT methods produce Coopmans IP/EA eigenvalues for MO only in special case of so called coupling coefficients. See details in Plakhutin BN, Davidson ER. Canonical form of the Hartree-Fock orbitals in open-shell systems. J Chem Phys. 2014 Vol. 7 140(1):014102. doi: 10.1063/1.4849615. I don't know what input Gaussian requires to manage coupling coefficients to obtain canonical form ROHF. GAMESS provides such control via $SCF COUPLE = ... $END deck.

In The Name Of GOD

Salaam

The spin function is introduced in Slater determinant such that N/2 of particles have 1/2-spin and half f them have -1/2-spin. When the multiplicity i.e., 2S+1=1 or S=0.0, there is no difference between spin-up and spin-down electrons in eigenvalues, wave function, and so on. But when the multiplicity S is not equal to zero, spin-up differs from spin-down. In this case, you should report both of them and can not ignore one of them.

Best Regards

H.Simchi

Dear all,

I essentially agree with the latest comment by Hamidreza Simchi - without however any reference to Islam and to the Saint Coran.

One should pay the greatest attention in this discussion on the meaning of singlet "open-shell" wave functions. For many researchers (eg Houk), this implies a spin-polarized solution, with spin-alpha and spin-beta orbitals localized in different parts of the molecules. For many other researchers (eg Hans Lichka), singlet open-shell wave functions refer to strong deviations from 0 and 2 of natural orbital occupation numbers in multi-reference (eg CASSCF) calculations. Obviously, these two definitions are not equivalent, hence a lot of disputes in the scientific literature on for instance the electronic structure of graphene nanoflakes, which are essentially semantic in origin.

In this context, it is worth reminding that for singlet states, spin-densities must be identically zero at everypoint in space. Symmetry-breakings of spin-densities otherwize imply a spin contamination (i.e. deviation of the expectation value of the S^2 operator), which in some cases may take absolutely catastrophic proportions. It is well-known that strong spin-contampinations indicate a too approximate treatment of electronic correlation, in particular static correlation (multi-reference effects). I recommend everyone to have a look at works we published in  J. Chem. Phys. 135 (2011), 104704, Chem. Phys. Lett. 553 (2012), 6.,   J. Mol. Mod., 417 (2013), 17-25 for detailed discussions of these issues.When electron pairs dissociate in biradical (or multiradical) systems (as for instance in dissociating H2) one cannot , with single-reference (HF or DFT) treatments, recover both the exact (symmetry-adapted) wave function and the exact energy. This is the essence of the famous "symmetry-dilemma" by Per-Olov Löwdin. Unrestricted SR treatments give the right dissociation energies for the wrong symmetries, whereas restricted SR treatments give wrong dissociation energies for the right wave function. To get everything exact, one has to go to the full-CI limit. See also: J. Chem. Phys., 131 (2009), 224321, a work which among others prove that correct treatments of correlation restore the correct symmetries in symmetry-breaking situations.

With best regards,

                   Michael S. Deleuze,

Prize Winner 2005 of the Royal Academy 

of Belgium for the Sciences and the Arts.