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When I want to compare the energy between two different spin states of the same molecule, i.e. singlet-triplet-gap, do I better use unrestricted or restricted open formalism to compare the energies between both?

Further on, do I need to calculate the singlet state also with restricted open/unrestricted method? At least within Gaussian I did not recognize any energy difference between all of them. (Which sounds logic for me, as if the ground state is a singlet, also the open shell version will find that best version would be to have the same orbitals in alpha and beta.)

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To expand on user1420303's answer a bit:

When I want to compare the energy between two different spin states of the same molecule, i.e. singlet-triplet-gap, do I better use unrestricted or restricted open formalism to compare the energies between both?

It depends.

The unrestricted formalism will almost always give lower absolute electronic energies (i.e., closer to the "real" value) than restricted-open will, due to the greater variational flexibility introduced by the separate spin-up and spin-down orbital sets. So, from a purely energetic perspective, yes, unrestricted is likely preferred.

If you're interested in other properties, though, the spin contamination introduced by the unrestricted formalism may be problematic. Of course, in systems with sufficient static correlation (viz., a degenerate or near-degenerate ground state) to result in spin contamination (radicals, bond-breaking states, transition metals, etc.), single-reference methods like Hartree-Fock and DFT may not give reliable results anyways.

Further on, do I need to calculate the singlet state also with restricted open/unrestricted method?

It depends.

For most 'non-exotic' organic systems, there is no need to calculate the singlet state using an unrestricted method since, as you note, the results will often be indistinguishable. For systems with static correlation, though, if you still want to try to use a single-reference method, you will need to explore the unrestricted orbital space, as there may be a lower-energy solution where the alpha and beta orbital compositions appreciably differ. The primary method I'm aware of for such exploration is the "broken-symmetry" method. Two papers I know of that discuss the method, albeit not in extensive detail, are the following reviews by Frank Neese:

  • F. Neese. "Prediction of molecular properties and molecular spectroscopy with density functional theory: From fundamental theory to exchange-coupling." Coord Chem Rev 253: 526, 2009. doi:10.1016/j.ccr.2008.05.014

  • F. Neese. "A critical evaluation of DFT, including time-dependent DFT, applied to bioinorganic chemistry." J Biol Inorg Chem 11: 702, 2006. doi:10.1007/s00775-006-0138-1

The method is integrated into the current version of ORCA (see Section 5.9.10 of the v3.0.3 ORCA manual), and tips for running such calculations are available at the ORCA Input Library. Other software packages likely include this capability; search their manuals for more information.

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  • $\begingroup$ Is an unrestricted method really a single-reference method? Different orbitals for different spins appears to me as to have one determinant (ref) per spin. $\endgroup$ – Martin - マーチン Oct 13 '16 at 11:51
  • $\begingroup$ @Martin-マーチン Mathematically you're right, but I've never heard anyone call UHF 'multireference' before. It's still a mean-field, uncorrelated method... it just has greater variational flexibility as compared to RHF/ROHF. $\endgroup$ – hBy2Py Oct 13 '16 at 12:06
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    $\begingroup$ I'd call it a dirty, dirty method (Fifty shades of spin). It's not that I would go out and say UHF is multi-reference (although in BS that is exactly what it tries to mimic), but I wouldn't call it single-reference either. $\endgroup$ – Martin - マーチン Oct 13 '16 at 12:10
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    $\begingroup$ @Martin-マーチン How about a "dual coupled field" method? ;-) $\endgroup$ – hBy2Py Jun 19 '17 at 18:48
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1 It is better to use the unrestricted (U) formalism when the computational cost is feasible, because of course it does not impose thee extra restrictions that take place in RO. Although, RO was always suitable in my experience.

2 Normally you don't have to, e.i. in most of cases for a singlet state restricted treatment works good (giving the same results that U and RO). But you can not generalize it. Take a look at http://www.cup.uni-muenchen.de/ch/compchem/energy/stability.html

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  • $\begingroup$ Beat me to the punch. :-) $\endgroup$ – hBy2Py Apr 8 '16 at 13:07
  • $\begingroup$ @Brian, because you elaborate more, +1..., after reading your answer I am curious about the relationship between static correlation and spin contamination. $\endgroup$ – user1420303 Apr 8 '16 at 13:28
  • $\begingroup$ I never really found a good, concise exposition of it. I would start with the relevant sections of the Neese papers, and references therein. Section 5.1.7.6 of the ORCA v3.0.3 manual actually has a nice mini-example showing how BS-UHF can treat statically correlated systems. Beyond that, a literature search for "broken-symmetry" should turn up a lot. $\endgroup$ – hBy2Py Apr 8 '16 at 13:37

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