Effect of Symmetry

In a unit cell the positions of many of the atoms are related to each other by symmetry relationships. Recognizing the presence of such symmetry elements is an important step in solving the structure of a crystal. The symmetry implies that the content of the unit cell can be grouped into several equivalent parts, and the structure of only one of these parts need be determined. Knowledge of the various symmetry elements present may also give valuable hints about the packing of atoms even before the structure analysis has begun. 

The various symmetry elements that can be present in a crystal with three-dimensional lattices are (1) rc-fold rotation axes, (2) ra-fold rotation-inversion axes, (3) mirror planes, (4) rc-fold screw axes, and (5) glide planes. The n-fold rotation 

To illustrate how the presence of symmetry elements gives rise to a systematic absence of certain hkl reflections, consider a crystal, for example, that possesses a two-fold screw axis passing through the origin and parallel to the a axis. The symmetry element implies that whenever there is an atom at (X, Y,Z), there is another atom of the same type at ( + X, — T, — Z). We now rewrite Equation (3.9) in terms of atomic positions (X -, Yj, Zj) rather than in terms of scattering length density distribution pu(r) as 

---

Reduction in cation symmetry (ideally to Cl) lowers the freezing point and markedly expands the range of room-temperature liquid salts. Table 3.1-4 shows the effect of symmetry for a series of [NR4]X salts, in which all the cations contain 20 carbon atoms in the allcyl substituents [44]. 

Eq. (22) have been derived from the variation principle alone (given the structure of H) they contain only the single model approximation of Eq. (9) the typically chemical idea that the electronic structure of a complex many-electron system can be (quantitatively as well as qualitatively) understood in terms of the interactions among conceptually identifiable separate electron groups. In the discussion of the exact solutions of the Schrodinger equation for simple systems the operators which commute with the relevant H ( symmetries ) play a central role. We therefore devote the next section to an examination of the effect of symmetry constraints on the solutions of (22). 

The minimal basis calculation on the hydrogen molecule is a well-worn but eminently suitable example for our purposes. It has a convenient symmetry element and orbital basis calculations can be carried through which are quantitatively acceptable and yet not prohibitively unwiedly to report. We give below variational calculations on the H2 molecule using the familiar simplest AO basis in the one-electron-group (MO) model and the electron-pair (VB) model. These calculations have been performed explicitly to investigate the effect of symmetry constraints . 

As an example, to construct the character table for the Oh symmetry group we could apply the symmetry operations of the ABg center over a particularly suitable set of basis functions the orbital wavefunctions s, p, d,... of atom (ion) A. These orbitals are real functions (linear combinations of the imaginary atomic functions) and the electron density probability can be spatially represented. In such a way, it is easy to understand the effect of symmetry transformations over these atomic functions. 

At elevated temperatures, ys falls off to near zero as a consequence of rotational excitation of the H2 molecules (which is here not accounted for), and the effect of the summation over many partial waves will further reduce the effects of symmetry. 

As a very persuasive illustration of the effectiveness of symmetry factorization in reducing a computational task that would be entirely impractical without a digital computer to one that is a straightforward pencil-and-paper operation, we shall again consider the naphthalene molecule. It has been shown in Section 7.1 that the secular equation for the n MOs is the 10 x 10 determinantal equation, 7.1-15, if the set of 10pn orbitals is used directly for constructing LCAO-MOs. 

Fig. 3.18 Effects of symmetry operations in C2l. symmetry rotation about the z axis (a) identity. E (b) rotation about the C2 axis, (c, d) reflection in p, planes.

The effect of symmetry is conveniently taken into account by... 

Figure 10.96. Hj (0,2)-(18,3) hyperfine, spin-rotation and symmetry-breaking energy level diagram, showing the six AF = AN transitions, (a) denotes the Fermi contact splitting, (b) is the spin-rotation splitting and (c) shows the effect of symmetry breaking.

Figure 6.9 Effect of symmetry and substituents on the stereochemistry of the resultant polypropylene. 6.32 has Cs symmetry. 6.33 is chiral, but the effect of Me is moderate. 6.34 is also chiral, and the effect of bulky Bu is more marked.

Two contributions from the Cooper group in 2006 focused on the delocalization of the triplet excitons in Pt-acetylides [86, 87], They investigated the effect of symmetry within the complexes by comparing chromophores with either one pendant acetylide or two symmetric acetylides trans disposed to each other. Similar to earlier... 

Figure 7-5. The effect of symmetry lowering on the reaction coordinate (a) Bending of a linear AX2 molecule [v2(ttw) - v2C4i) (b) Puckering of a planar AX3 molecule [v2 (A%) —> v2(/l ). ...

The effect of symmetry on the spectral intensity is significant. All the observed f f... 

The simplest demonstration of how symmetry fixes natural laws is by the effect of symmetries on the motion of non-relativistic classical particles. 

FIG. A5-2. The effects of symmetry operations on an arbitrary point, designated 0, thus generating sets of points. 

Effect of Symmetry on the Relationships Between the Stress and Strain... 

Wilson, A. J. C. The probability distribution of X-ray intensities. III. Effects of symmetry elements on zones and rows. Acta Cryst. 3, 258-261 (1950). 

In general, the more symmetrical compounds have higher melting points than do the related, less symmetrical compounds. This is illustrated in Table 2-1. The effect of symmetry on these somewhat extreme examples is quite profound. It may be noted that in each case the heat of fusion is lowest for the more symmetrical compound. This is easily understood for these cases in which polar effects are not operating. The energy factor which favors the solid state is the van der Waals attraction between molecules, which is stronger in the solid... 

---