Broken symmetry determinant

Technical Issues Optimization of Broken-Symmetry Determinants 213 Studies on Open-Shell Polynuclear Transition-Metal Clusters 216... 

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). 

One the other hand, the expectation value of 0gs is neither close to zero (singlet) nor to two (triplet), but rather somewhere in between. It is therefore a logical step to approximate the broken symmetry determinant as a linear combination of the spin-restricted singlet and triplet states [17] ... 

The broken symmetry determinant is written as a linear combination of the singlet... 

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). 

For our SCF calculation the way to obtain such a solution is to optimize the orbitals not for the energy of determinant 7.6, but for the average energy of both determinants. This is termed the imposition of symmetry and equivalence restrictions. It involves imposing a constraint on a variational calculation, and consequently the symmetry and equivalence restricted solution will have an energy no lower than the broken symmetry solution it will usually have a higher energy. We may note that in a UHF calculation we impose neither spin nor spatial symmetry and equivalence restrictions — the -terms restricted and unrestricted were first used in exactly this context of whether to impose symmetry constraints on the wave function. 

Lindgren et al. studied the excited state and phosphorescence of platinum(II) acetylides (5.9) with the DFT/B3LYP method [103], The calculated UV/Vis absorption spectra revealed that the orientation of the phenyl rings relative to the P-Pt-P axis has a strong correlation with the intensity of the absorption band. The broken symmetry phenomenon was also confirmed in the lowest triplet excited state, which leads to a OC-Ph bond on one side and a C=C=C bond on the opposing side. Quadratic response calculations of spin-orbit coupling showed that the intensity of the phosphorescence of these complexes arises mainly from the a—>n type T- l) transitions localized at the OC-Pt-P fragment, but not from the delocalized 7t—type transitions. The quadratic response calculations also reproduced an increase of xR from PEj to PE2 (5.9), which was also determined experimentally. 

Figure 2. As in Figure 1, but for the broken symmetry state. Iron "a" is the unique iron, and "b" labels the equivalent pair. The three orbitals marked with an asterisk show the g and u spindown orbitals on the "b" irons (whose energy difference determines B), and the spin-up orbital mostly localized on iron "a". The plot corresponds to the state with the final electron in the lower a orbital marked with an asterisk. 

Deuterium diffusion was studied in VDX, where 0.4 < x < 0.6, by means of 2H NMR measurements.599 1H and 51V spin-lattice relaxation times for TaV2Hx, where x < 0.18, were consistent with two co-existing proton-jump processes.600 3H, 2H and 51V spin-lattice relaxation times were also determined for NbVCrHo.3, NbVCrD0.3g and NbV 14Cro.6Ho.6 in the temperature range 11-424 K.601 A DFT/broken symmetry approach has been used to study exchange... 

The concept of anisotropic fluctuations leads to a simple explanation of the epitaxial relationships observed in the transition from one ordered phase to another (Schulz et al., 1994 Koppi et al., 1994). Because of the broken symmetry in a periodically ordered structure, the dominant fluctuation modes reside at specific locations in the reciprocal space, determined by the symmetry of the reciprocal lattice of the initial ordered structure. This spatial relationship is maintained when the new structure—dictated by the most unstable fluctuation modes—grows out of the initial structure, giving rise to the observed epitaxy. 

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