Spin: restricted vs. unrestricted
UNRESTRICTED
Specifies that spin-α and spin-β MOs may be spatially different and may have different occupation numbers. The default (absence of the key) is spin-restricted. The key has no argument. In the case of Spin-Orbit coupling it means that Kramer's symmetry does not have to be satisfied, in which case the key unrestricted should be used in combination with the key noncollinear or collinear.
The unrestricted mode roughly doubles the computational effort. The actual numbers of spin-α and spin-β electrons respectively are controlled by the keys charge and occupations.Not e carefully, that using only the keyword unrestricted, without either Charge or Occupations (or both) would not result in any spin polarization. This implies that you would effectively perform a spin-restricted calculation, but with increased computational effort. Therefore, the program will check that in an unrestricted calculation at least one of the keys Charge and Occupations is applied.
The unrestricted feature is equivalent with, in ab-initio terminology, (Spin-)Unrestricted-Hartree-Fock (uhf); the N-particle wavefunction is a single determinant and not necessarily an eigenfunction of the spin operator S2.
A restricted calculation implies that the (spatial) orbitals and the occupation numbers are identical for spin-α and spin-β.
The Fock operator, both in an unrestricted and in a restricted run, commutes with the spin operator Sz, but not (unless accidentally) with S2. The obtained one-determinant wave function may for instance be a mixture of a singlet and a triplet state.
In an unrestricted calculation the expectation value of S2 is now computed in ADF (note 29 in [98]). The implementation of an evaluation of S2 is not quite trivial. DFT is essentially a one-particle formalism, so the S-operator for the n-particle system has to be written out in single-particle operators [99]. The equations used in ADF to calculate the expectation value of S2 can be found in Szabo and Ostlund [100]. Note that the so called exact value (Sexact)2, which is printed in the ADF output, is defined as (Sexact)2 = (|Na- Nb|/2)(|Na- Nb|/2+1), where Na and Nb are the number of spin-α and spin-β electrons, respectively. The expectation value of S2 is not calculated in a Spin-Orbit coupled calculation.
Molecules that have been calculated using the unrestricted formalism cannot be employed as fragments. ADF will abort when you attach the TAPE21 result file from an unrestricted calculation as a fragment file.
A fair approximation to a computation with unrestricted fragments can be achieved with the key fragoccupations. See also the Examples.
