Symmetry axis

In summary, the moleeular orbitals of a linear moleeule ean be labeled by their m quantum number, whieh plays the same role as the point group labels did for non-linear polyatomie moleeules, and whieh gives the eigenvalue of the angular momentum of the orbital about the moleeule s symmetry axis. Beeause the kinetie energy part of the... 

For symmetric top species, Pave hes along the symmetry axis of the molecule, so the orientation of Pave can again be described in terms of 0 and (j), the angles used to locate the orientation of the molecule s symmetry axis relative to the lab-fixed coordinate system. As a result, the El integral again can be decomposed into three pieces ... 

In a symmetric top molecule such as NH3, if the transition dipole lies along the molecule s symmetry axis, only k = 0 contributes. Such vibrations preserve the molecule s symmetry relative to this symmetry axis (e.g. the totally symmetric N-H stretching mode in NH3). The additional selection rule AK = 0... 

If the transition dipole lies perpendicular to the symmetry axis, only k = 1 contribute. In this case, one finds... 

The n ==> n transition thus involves ground Ai) and exeited Ai) states whose direet produet (Ai x Ai) is of Ai symmetry. This transition thus requires that the eleetrie dipole operator possess a eomponent of Ai symmetry. A glanee at the C2v point group s eharaeter table shows that the moleeular z-axis is of A symmetry. Thus, if the light s eleetrie field has a non-zero eomponent along the C2 symmetry axis (the moleeule s z-axis), the n ==> 71 transition is predieted to be allowed. Light polarized along either of the moleeule s other two axes eannot induee this transition. 

For non-linear molecules of the spherical or symmetric top variety, pf j(Rg) (or dpf j/dRa) may be aligned along or perdendicular to a symmetry axis of the molecule. The selection rules that result are... 

The dipole moment )T lies along the molecule s C3 symmetry axis. Let the electric field of... 

Microwave spectroscopy (Stark effect). d, the probable approximate angle between the direction of fj, and the symmetry (or near symmetry) axis of the ring through atom N-1, in the molecular plane, has the value 69°. 

Palindromic base pairs that are most frequent at the two ends are green, and the pseudo-twofold symmetry axis is Indicated by a red dot. 

The protein-DNA interactions have been analyzed in detail at high resolution in the complex between the 434 repressor fragment and the ORl containing 20mer DNA. A pseudo-twofold symmetry axis relates the halves of this complex. The symmetry is not exact since the nucleotide sequence of the DNA is slightly different in each half (see Table 8.2). However, the interactions between one protein subunit and one half of the DNA are very similar to those between the second subunit and the other half of the DNA since most of the bases that interact with the protein are identical in both halves. Details of the interaction are very similar to those in the complex with the palindromic synthetic 14mer of DNA shown in Figures 8.14 and 8.15. The base pairs at one end of the DNA, 1-14, 2-13, etc. are called base pairs 1, 2, etc. 

The structurally similar L and M subunits are related by a pseudo-twofold symmetry axis through the core, between the helices of the four-helix bundle motif. The photosynthetic pigments are bound to these subunits, most of them to the transmembrane helices, and they are also related by the same twofold symmetry axis (Figure 12.15). The pigments are arranged so that they form two possible pathways for electron transfer across the membrane, one on each side of the symmetry axis. 

This symmetry is important in bringing the two chlorophyll molecules of the "special pair" into close contact, giving them their unique function in initiating electron transfer. They are bound in a hydrophobic pocket close to the symmetry axis between the D and E transmembrane a helices of both... 

Figure 12.15 Schematic arrangement of the photosynthetic pigments in the reaction center of Rhodopseudomonas viridis. The twofold symmetry axis that relates the L and the M subunits is aligned vertically in the plane of the paper. Electron transfer proceeds preferentially along the branch to the right. The periplasmic side of the membrane is near the top, and the cytoplasmic side is near the bottom of the structure. (From B. Furugren, courtesy of the Royal Swedish Academy of Science.)...

This pair of chlorophyll molecules, which as we shall see accepts photons and thereby excites electrons, is close to the membrane surface on the periplasmic side. At the other side of the membrane the symmetry axis passes through the Fe atom. The remaining pigments are symmetrically arranged on each side of the symmetry axis (Figure 12.15). Two bacteriochlorophyll molecules, the accessory chlorophylls, make hydrophobic contacts with the special pair of chlorophylls on one side and with the pheophytin molecules on the other side. Both the accessory chlorophyll molecules and the pheophytin molecules are bound between transmembrane helices from both subunits in pockets lined by hydrophobic residues from the transmembrane helices (Figure 12.16). 

Any symmetric object is built up from smaller pieces that are identical and that are related to each other by symmetry. An icosahedron can therefore be divided into a number of smaller identical pieces called symmetry-related units. Protein subunits are asymmetric objects hence, a symmetry axis cannot pass through them. The minimum number of protein subunits that can form a virus shell with icosahedral symmetry is therefore equal to... 

Figure 16.4 The division of the surface of an icosahedron into asymmetric units, (a) One triangular face is divided into three asymmetric units into which an object is placed. These are related by the threefold symmetry axis.

Subunits VP2 and VP3 from different pentamers alternate around the threefold symmetry axes like subunits B and C in the plant viruses (Figure 16.12b). Since VP2 and VP3 are quite different polypeptide chains, they cannot be related to each other by strict symmetry, or even by quasi-symmetry in the original sense of the word. To a first approximation, however, they are related by a quasi-sixfold symmetry axis, since the folded structures of the cores of the subunits are very similar. 

A view down the fivefold symmetry axis of the icosahedtal structure (a) shows that the central capsomer is pentameric in shape and surrounded by five other capsomers as expected. The view down the pseudosixfold axis (h) shows, however, that the central capsomer is pentameric in shape and not hexameric as required for a T = 7 structure. (Adapted from 1. Rayment et al., Nature 295 110-115, 1982, hy copyright permission of Macmillan Magazines Limited.)... 

As distinct from ihe ideal connection of Dunlap, we now describe the series of nanotubule knees (9 ,0)-(5m,5 ), with n an integer. We call this series the perfectly graphitizahle carbon nanotuhules because the difference of diameter between the two connected segments of each knee is constant for all knees of the series (Fig. 4). The two straight tubules connected to form the = 1 knee of that series are directly related to Cfio, the most perfect fullerene[15], as shown by the fact that the (9,0) tubule can be closed by 1/2 Qo cut at the equatorial plane perpendicular to its threefold rotation symmetry axis, while the (5,5) tubule can be closed by 1/2 Qo cut at the equatorial plane perpendicular to its fivefold rotation symmetry axis [Fig. 5(a)]. 

For a removal attempt a molecule is selected irrespective of its orientation. To enhance the efficiency of addition attempts in cases where the system possesses a high degree of orientational order, the orientation of the molecule to be added is selected in a biased way from a distribution function. For a system of linear molecules this distribution, say, g u n ), depends on the unit vector u parallel to the molecule s symmetry axis (the so-called microscopic director [70,71]) and on the macroscopic director h which is a measure of the average orientation in the entire sample [72]. The distribution g can be chosen in various ways, depending on the physical nature of the fluid (see below). However, g u n ) must be normalized to one [73,74]. In other words, an addition is attempted with a preferred orientation of the molecule determined by the macroscopic director n of the entire simulation cell. The position of the center of mass of the molecule is again chosen randomly. According to the principle of detailed balance the probability for a realization of an addition attempt is given by [73]... 

In some studies of pure N2 solids more realistic potentials have been used by considering electrostatic interaction in addition. The electrostatic interaction sites are positive charges of q = cj2 = 0.313e at distances 1.044 A away from the molecular center-of-mass on the molecular symmetry axis and negative charges = /4 = —0.313 e at distances 0.874 A respectively [143-145]. 

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