Symmetry, reduced

In general, CijU is a 9 x 9 tensor with 81 terms, but symmetry reduces this considerably. Thus, for the cubic crystal system, it has only three terms (Cmi, Cm2, and C4444) and for an isotropic material only two terms remain B = bulk modulus and G = shear modulus. A further simplification is that the bulk modulus, B for the cubic system is given by (Cmi + 2Ci2i2)/3, and the two shear moduli are C44 and (Cmi - Ci2i2)/2. 

A more complicated example, in which the presence of symmetry reduces the problem of creating eight chiral centres to only four, is the case of (+)-a-onocerine (8), the synthesis of which [15] was performed by coupling of the two enantiomerically pure C 5 units (7, X = COOH), with the correct stereochemistry, in order to prevent the formation of the unnatural meso form d,l + l,d) (Scheme 4.2). 

The splitting of the free-ion term in octahedral symmetry Oh symmetry) reduces the degeneracy of the five d orbitals. Three orbitals have energy lower than the other two. This means that if the orbitals are populated by one electron, three degenerate states are possible, according to the three possible positions for the electron in the low-energy levels (T symmetry) ... 

That this comparison seems to have validity (neglecting the legitimated analogy between bis-JT-allylnickel and symmetry-reduced benzene) may be atMboted to the fact that open-chain TT-systems may display some residual aromaticity (see Ref. ). Metal complexes with C3 symmetry must have three identical ligands (S) in the same state. When we change only one l and (S versus L) symmetry is reduced. The structural consequence is the change to a T- or Y-shaped moiety (see Fig. 1 in Scheme 2.5-4 and similar discussions in Refs. Within thcK arran ... 

Single crystals of /S-A1203 are essentially two dimensional conductors. The conducting plane has hexagonal symmetry (honeycomb lattice). This characteristic feature made -alumina a useful model substance for testing atomistic transport theory, for example with the aid of computer simulations. Low dimensionality and high symmetry reduce the computing time of the simulations considerably (e.g., for the calculation of correlation factors of solid solutions). 

One particular advantage of Slater determinants constructed from orthonormsd spin-orbitals is that matrix elements between determinants over operators such as H sure very simple. Only three distinct cases arise, as is well known and treated elsewhere. It is perhaps not surprising that the simplest matrix element formulas should be obtained from the treatment that exploits symmetry the least, as only the fermion antisymmetry has been accounted for in the determinants. As more symmetry is introduced, the formulas become more complicated. On the other hand, the symmetry reduces the dimension of the problem more and more, because selection rules eliminate more terms. We consider here the spin adaptation of Slater determinants. 

The equation of continuity for an incompressible liquid, and with spherical symmetry, reduces to... 

By introducing the thienyl ring instead of the phenyl ring for the DDAA(cro55)-TEE the symmetry in the (x,y)-plane is completely broken and all terms in the numerator of Eq.(81) occur. The dipolar contribution is not symmetry reduced and therefore DDAA(cross)-TEE exhibits the largest nonlinearity (Table 4). 

Although symmetry reduces the number of independent force constants, they still grow rapidly with the order of anharmonicity. For example, for a CH3X molecule of C.. symmetry there are 12 independent quadratic constants, 38 cubic constants, and 102 quartic constants. 

Figure 6-40. The symmetry-reducing vibrational mode of eg symmetry for an octahedron.

For an AX3 planar molecule the symmetry is D3h. The puckering mode (Figure 7-5b) of A2 symmetry reduces it to C3v. In the C3v point group, the symmetry of this vibration is A . 

As in the ABC case, the basis functions divide into four sets according to fz with 1,3, 3, and 1 functions in each set. However, of the three functions in the set with fz = % or — V2 two are symmetric and one antisymmetric. Hence each of the two 3X3 blocks of the secular equation factors into a 2 X 2 block and 1X1 block. Algebraic solutions are thus possible. Furthermore, the presence of symmetry reduces the number of allowed transitions from 15 to 9, because no transitions are allowed between states of different symmetry. (One of the nine is of extremely low intensity and is not observed.) Thus the A2B system provides a good example of the importance of symmetry in determining the structure of NMR spectra. 

The transformation of the one-electron integrals is computationally inexpensive and easily accomplished without point group symmetry, this transformation can be performed as two half-transformations, each of which requires a multiplication of the one-electron integral matrix by the SCF coefficient matrix, for a total of 2n3 multiplications. Spatial symmetry reduces this cost because the one-electron integral and SCF coefficient matrices are block diagonal according to irreducible representation (irrep), and the transformation can be carried out an irrep at a time (cf. Figure 1). Note that it would also be... 

Oy is a phase factor that is not determined by symmetry. It is a free parameter and it is significant only for an independent set of magnetic atoms (one orbit) with respect to another one. The component. Ay. is a phase factor determined by symmetry as shown in Equation (55). The sign of (Ay> changes for —k. In the general case Sy is a complex vector with six components. These six components per magnetic orbit constitute the parameters that have to be refined from the diffraction data. Symmetry reduces the number of free parameters per orbit to be refined. Notice that we have adopted a different phase convention than that used in ref. 22. 

Figure 2.41. The H spectrum of ammonium nitrate in DMSO (a) before and (b) after acidification. In (b) the NHJ ion dominates and the induced symmetry reduces the quadrupolar relaxation rate and reveals the one-bond coupling constant (51 Hz).

Within the Trigonal model, this Tt-symmetry reduces to trigonal. The extended Hermann-Mauguin symbols, depending on which of the two maximal jion-isomorphic subgroups is preserved (either P(3)lm or P 6)2m, become P 1 1 1 (3) 2lm 2 m 2 m and P 22 2 (6) mmm. 

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