CCL:G: Understanding restricted vs unrestricted calculations

 Sent to CCL by: Igors Mihailovs [igors.mihailovs0/a\gmail.com]
 Hello Ashika,
 UNrestricted calculations are used for systems with odd number of
 electrons (free radicals, in particular radical cations and anions
 that occur upon ionization of a molecule / electron capture by it) and
 also for systems that have even number of electrons but unpaired:
 these are biradicals, e. g., oxygen molecule, or, most commonly, these
 are electronically excited species - singlet and triplet states. All
 this correspond to spin multiplicity of specie beig higher than 1.
 By default, Gaussian 09 sets calculation type to be unrestricted only
 if spin multiplicity is higher than one. However, the initial guess
 still has alpha and beta states coinciding in energy, so the
 optimization might not break the spin symmetry by its own, and the
 resulting wavefuction will be something like a "saddle point" between
 two unrestricted functions. Spin symmetry means that there is no
 biradical character of the full wavefunction of Your system, e.g.,
 that alpha and beta electrons are distributed equally – in fact,
 sharing spatial parts of their wavefunctions, as in restricted
 calculation. To check if Your unrestricted wavefunction is really
 unrestricted, You ought to run the same calculation with "Stable=Opt"
 keyword. This requires a bit more resources than plain single point,
 as it have to diagonalize configuration interaction (CI) matrices.
 Personally I appeared at questioning about this when I noticed that
 <S**2> values for closed-shell molecules in my unrestricted
 calculations (not only Kohn–Sham, but even Hartree–Fock) were
 surprisingly equal to 0.000, with no spin contamination. The last one
 is "the beast" of unrestricted calculations: those wavefunctions that
 we obtain by solving Schroedinger equation are not neccessarily
 eigenfunctions of spin-squared operator, S**2, but in reality they
 should be; so, in fact unrestricted wavefunction contain some spurious
 contribution from energetically higher-laying states with greater spin
 than we have set up in the calculation parameters (electronically
 "more excited" states). This means that we have somewhat incorrect
 wavefunction and, as a concequence, somewhat incorrect energy. Spin
 contamination is usually relatively high for Hartree–Fock and
 correlated methods (MPn, etc.) based upon it, but particularily less
 for unrestricted Kohn–Sham DFT calculations.
 There is one more alternative: restricted open-shell calculations
 (RO). Here, all spatial shells that in the reality contain two
 electrons (usually inner shells) are forced to stay together, as in
 restricted calculation, whereas those containing only one electron are
 computed as they are. This method should give more reasonable results,
 but it requires more computational resources, because we now have not
 only one Slater determinant as in unrestricted calculation, double
 that of restricted one, but several (multiple) determinants, a bit
 like in CI.
 Hope this will be useful,
 Igors Mihailovs (engineer)
 Institute of Solid State Physics
 University of Latvia
 2015-04-08 17:23 GMT+03:00, ashika torikora
 ashika.torikora(_)gmail.com <owner-chemistry!=!ccl.net>:
 > Hello everyone,
 >
 > I am trying to understand the difference between restricted and
 > unrestricted calculations. So far I have somewhat understood the basic
 > difference (in restricted calculations consider the calculation is
 > restricted to having two electrons per occupied orbital whereas for
 > unrestricted calculations there are two complete sets of orbitals, one for
 > the alpha electrons and one for the beta electrons).
 > My question is more of a practical one.
 > When should one use an unrestricted calculation?
 > When is restricted or unrestricted used by default in gaussian(09) and when
 > should one specify it?
 >
 > All answers welcome,
 > ( ´ ▽ ` )ﾉ
 >