Symmetry-inequivalent atoms

Both isomers lack symmetry elements and, consequently, the two three-coordinate phosphorus atoms are inequivalent. The chemical environments of the two P(III) centres are similar in both isomers, but the large value of /ab=263 Hz indicates a P-P bond as found in 3.11b. ... 

Fig. 5.4. Top view of all possible on-surface/subsurface site combinations at a total coverage 0 = 1 ML with one oxygen atom per (2 x 2) surface unit-cell located below the surface, i.e., corresponding to a fraction of 25% below the surface (see Fig. 5.1 for the explanation of the different sites). The metal atoms are drawn as big spheres (white for the surface layer, gray for second layer), and the oxygen atoms are drawn as small spheres (black for on-surface, gray for subsurface). Oxygen atoms in tetra-I I sites below the first layer metal atoms are invisible in this plot and are schematically indicated by small white circles. Symmetry inequivalent occupation of the same kind of on- and subsurface sites are denoted with (a) and (b), respectively...

In general, only small molecules, usually diatomics, have been studied with four-component methods. Often, correlation effects have not yet been taken into account. Those larger molecules, which have also been studied to some extent, exhibit high symmetry like Oh or 7d consisting of only two symmetry-inequivalent atoms. Therefore, hydrides, oxides and halides are by far the most extensively studied molecules. 

Efficient use of symmetry can greatly speed up localized-orbital density-functional-exchange-and-correlation calculations. The local potential of density functional theory makes this process simpler than it is in Hartree-Fock-based methods. The greatest efficiency can be achieved by using non-Abelian point-group symmetry. Such groups have multidimensional irreducible representations. Only one member of each such representation need be used in the calculation. However efficient localized-orbital evaluation of the chosen matrix element requires the sum of the magnitude squared of the components of all the members on one of the symmetry inequivalent atoms, based on Eq. 13. 

The two axial substituents are symmetry equivalent since they change places when the molecule is rotated 180" about one of the twofold axes, the three equatorial substituents are symmetry equivalent since they change places by rotation about the threefold or twofold symmetry axes. Since axial and equatorial atoms cannot be exchanged by any symmetry operation, they are symmetry inequivalent. 

Linking Results for Symmetry-Inequivalent Sets of Atoms... 

Carbon. - Pulsed HFEPR and ENDOR measurements at 95 GHz have been reported for a frozen solution of fully C-enriched Cm molecules in their photoexcited triplet state. The triplet state was populated by irradiation at 532 nm using the second harmonic of a Nd YAG laser. The shape of the electron spin echo detected HFEPR spectrum showed that the g factor was anisotropic. There are four types of symmetry-inequivalent C atom in the triplet state and the ENDOR measurements showed that the electron spin density was largely on the equator atoms (3.8%) and on atoms of type 3 (1.1%). The calculated triplet wave function was found to reproduce the large spin density on the equator atoms. 

The "unfolded-drum" or "ladder" compound 2 has crystallographic symmetry. This corresponds to the idealized molecular symmetry and, therefore, there are three chemically inequivalent types of Sn atoms in the molecule, although all are hexacoordinated. The oxygen atoms in the open form can be subdivided into two types, as in the case of the drum molecule tricoordinate framework oxygen atoms and the dicoordinate oxygen atoms of the bridging carboxylate ligands. 

Any 6- or 12-coordinate ion in the graph is initially assumed to have the site symmetry m3m (Oh) if all the ligands are equivalent in the bond graph. If they are not all equivalent, then one must choose a lower site symmetry that is compatible with this inequivalence. Similarly an ion with four equivalent ligands is assumed to be tetrahedrally coordinated with site symmetry 43m (T ). The constraints 1-3 above are then examined to ensure that all have been satisfied. If they are, then one can look in Appendix 2 to find a matching space group using the procedure described below. If they are not, the symmetry of one or more atoms must be lowered until all the constraints are satisfied. 

While the supercell approach works well for localized systems, it is typically necessary to consider a very large supercell. This results in a plane-wave basis replicating not only the relevant electronic states but also vacuum regions imposed by the supercell. A much more efficient method to implement for investigating the electronic structures of localized systems is to use real space methods such as the recursion methods [27] and the moments methods [28], These methods do not require symmetry and their cost grows linearly with the number of inequivalent atoms being considered. For these reasons, real space methods are very useful for a description of the electronic properties of complex systems, for which the usual k-space methods are either inapplicable or extremely costly. 

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