Open broken spin symmetry

Quantum chemical studies of cyclizations of enediynes and enyneallenes have been reviewed.180 The intermediates are computationally tractable as a result of the unrestricted broken spin symmetry (UBS) approach using GGA functionals for the description of open-shell biradicals. The intermediacy of biradicals in Cope-type rearrangements, to which the Bergman and Myers-Saito reactions belong, are shown to be predictable using a very simple rule biradicals are likely to be intermediates if they are stabilized either by allyl resonance or by aromaticity. 

All the early applications, as well as the vast majority of many-body calculations performed to this day, are single-reference in character. They start from an appropriate single determinant, usually (but not necessarily) Hartree-Fock, and include correlation by finite-order perturbation or infinite-order summation of certain perturbation terms (the CC approach). The starting determinant may be closed-shell or open-shell the latter leads to contaminated spin states and occasionally to broken spatial symmetry [8], but acceptable results are obtained in most cases. States involving degeneracy or quasidegeneracy, where a single determinant... 

Here, we should mention that there exists an extensive discussion in the literature on the capabilities of spin-DFT regarding, for instance, the question whether the Kohn-Sham spin density has to be equal to the spin density of the fully interacting system of electrons (and in the case of open-shell singlet broken-symmetry (BS) determinants (see below) for binuclear transition-metal clusters this is certainly not the case see Ref. (33) for a more detailed discussion). But the situation is much more subtle and one may basically set up the variational procedure in a Kohn-Sham framework such that the spin density of the Kohn-Sham system of noninteracting fermions represents the true spin density. However, the frame of this review is not sufficient to present all details on this matter (34,35). 

All electron calculations were carried out with the DFT program suite Turbomole (152,153). The clusters were treated as open-shell systems in the unrestricted Kohn-Sham framework. For the calculations we used the Becke-Perdew exchange-correlation functional dubbed BP86 (154,155) and the hybrid B3LYP functional (156,157). For BP86 we invoked the resolution-of-the-iden-tity (RI) approximation as implemented in Turbomole. For all atoms included in our models we employed Ahlrichs valence triple-C TZVP basis set with polarization functions on all atoms (158). If not noted otherwise, initial guess orbitals were obtained by extended Hiickel theory. Local spin analyses were performed with our local Turbomole version, where either Lowdin (131) or Mulliken (132) pseudo-projection operators were employed. Broken-symmetry determinants were obtained with our restrained optimization tool (136). Pictures of molecular structures were created with Pymol (159). 

The strong nuclear force is insensitive to the distinction between neutrons and protons. These can be treated as alternative states of a single particle called a nucleon, differing in isotopic spin or isospin. It is found, for example, that the nuclei and He have similar energy-level spectra. Isospin is, however, only an approximate symmetry. It is broken by electromagnetic interactions since protons have electric charge, while neutrons do not. Broken symmetry is a central theme in fundamental physics. An open question is how our universe evolved to break the symmetry between matter and antimatter, so that it is now dominated by matter. 

Recently, a combination of closed-shell singlet, open-shell broken symmetry singlet and spin triplet DFT calculations was used to probe the reaction coordinate for nitrite reduction catalyzed by mARC. Regarding an oxyl radical transfer mechanism (Figure 2.36), the computed reaction barrier (AH was found to be 14 kcal moF. This reaction is approximately thermoneutral since Mo(v) dioxo species are very unstable toward proton-ation. However, protonation of the equatorial oxo is expected to lead to an overall increase in the exergonic nature of this reaction (Figure 2.36). However, if the 0 oxygen of N02 that is coordinated to the Mo(iv) ion is... 

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