Symmetry axis, rotation about

First, it is apparent that reflection through the xz plane, indicated by transforms H into H". More precisely, we could say that H and H" are interchanged by reflection. Because the z-axis contains a C2 rotation axis, rotation about the z-axis of the molecule by 180° will take H into H" and H" into H, but with the "halves" of each interchanged with respect to the yz plane. The same result would follow from reflection through the xz plane followed by reflection through the yz plane. Therefore, we can represent this series of symmetry operations in the following way ... 

Improper rotation axis. Rotation about an improper axis is analogous to rotation about a proper synunetry axis, except that upon completion of the rotation operation, the molecule is mirror reflected through a symmetry plane perpendicular to the improper rotation axis. These axes and their associated rotation/reflection operations are usually abbreviated X , where n is the order of the axis as defined above for proper rotational axes. Note that an axis is equivalent to a a plane of symmetry, since the initial rotation operation simply returns every atom to its original location. Note also that the presence of an X2 axis (or indeed any X axis of even order n) implies that for every atom at a position (x,y,z) that is not the origin, there will be an identical atom at position (—x,—y,—z) the origin in such a system is called a point of inversion , since one may regard every atom as having an identical... 

There are several forms of rotational symmetry. The simplest is cyclic symmetry, involving rotation about a single axis (Fig. 4—24a). If subunits can be superimposed by rotation about a single axis, the protein has a symmetry defined by convention as Gn (C for cyclic, n for the number of subunits related by the axis). The axis itself is described as an w-fold rotational axis. The a/3 protomers of hemoglobin (Fig. 4-23) are related by C2 symmetry. A somewhat more complicated rotational symmetry is dihedral symmetry, in which a twofold rotational axis intersects an w-fold axis at right angles. The symmetry is defined as DTO (Fig. 4—24b). A protein with dihedral symmetry has 2n protomers. 

There are three symmetry operations each involving a symmetry element rotation about a simple axis of symmetry (C ), reflection through a plane of symmetry (a), and inversion through a center of symmetry (i). More rigorously, symmetry operations may be described under two headings Cn and Sn. The latter is rotation... 

FIGURE 21.2 Three types of symmetry operation rotation about an n-fold axis, reflection in a plane, and inversion through a point. 

Chemists have adopted from mathematicians a way of classifying molecules according to their symmetry. The classification is based on the idea that if a molecule is sufficiently symmetrical, then an action can be found that will leave the molecule looking the same as it did when you started. You have already met two such actions inversion through a centre of symmetry and rotation about the molecular axis of a diatomic molecule. Actions such as this are called symmetry operations, and a molecule can be placed in a category according to how many such operations you can perform and still leave the molecule looking the same. 

Magnetic resonance (ESR and NMR) studies have provided additional details about the orientational motion of the water molecules adsorbed by vermiculites. ESR spectra of Cu-vermiculite and NMR spectra of Mg- and Na-vermiculite indicate clearly that the primary solvation shells of the cations on the two-layer hydrate are octahedral complexes with a preferred orientation relative to the siloxane surface. For Cu-vermiculite, the symmetry axis through the solvation complex, Cu(H20)6 , makes an angle of about 45 with the siloxane surface on Na-vermiculite the axis through Na(H20)6 makes an angle of 65°. The value of Tc, the correlation time for the rotation of Na(H20)6 around its symmetry axis, is about 10 s at 298 K. This value is four orders of magnitude larger than Tc for a solvation complex around a monovalent cation in aqueous solution. Not quite as disparate are T2 for Na(H20)6, equal to 100 ps at 298 K, and t2 for a monovalent solvation complex in dilute aqueous solution, equal to about 5 ps at the same temperature. These data show that the siloxane surface retards the orientational motion of the water molecules. 

As a first example, let s find the symmetry elements of the linear molecule HCN. In addition to the identity operation, there are two sets of symmetry elements rotation about the z axis, which is an infinite-fold Coo axis, and reflection through any of an infinite number of v mirror planes (Fig. 6.3). 

The representations of other point groups label the MOs of all nonlinear molecules. For linear molecule wavefunctions, we considered only four symmetry operations rotation about the z axis, inversion, reflection in the xy horizontal plane, and reflection in the vertical planes. Remember that when these operators act on the wavefunction, they may change if/ but not if/. The same principle remains true when we move on to polyatomic molecules, now with other possible symmetry elements. The symmetry properties of the orbital are denoted by the representation used to label the orbital. The Uj MOs of the Cjy molecule F2O, for example, have electronic wavefunctions that are antisymmetric with respect to reflection in either of the two vertical mirror planes (Fig. 6.10). 

A C—C covalent molecular orbital possess m-symmetry as well as C2-axis of symmetry because rotation about its mid-point brings same o-orbital. An antibonding sigma orbital < ) is antisymmetric w.r.t mirror plane as well as C2 axis. 

Atoms have complete spherical synnnetry, and the angidar momentum states can be considered as different synnnetry classes of that spherical symmetry. The nuclear framework of a molecule has a much lower synnnetry. Synnnetry operations for the molecule are transfonnations such as rotations about an axis, reflection in a plane, or inversion tlnough a point at the centre of the molecule, which leave the molecule in an equivalent configuration. Every molecule has one such operation, the identity operation, which just leaves the molecule alone. Many molecules have one or more additional operations. The set of operations for a molecule fonn a mathematical group, and the methods of group theory provide a way to classify electronic and vibrational states according to whatever symmetry does exist. That classification leads to selection rules for transitions between those states. A complete discussion of the methods is beyond the scope of this chapter, but we will consider a few illustrative examples. Additional details will also be found in section A 1.4 on molecular symmetry. 

Another one-dimensional representation of the group ean be obtained by taking rotation about the Z-axis (the C3 axis) as the objeet on whieh the symmetry operations aet ... 

The cyclobutene-butadiene interconversion can serve as an example of the reasoning employed in construction of an orbital correlation diagram. For this reaction, the four n orbitals of butadiene are converted smoothly into the two n and two a orbitals of the ground state of cyclobutene. The analysis is done as shown in Fig. 11.3. The n orbitals of butadiene are ip2, 3, and ij/. For cyclobutene, the four orbitals are a, iz, a, and n. Each of the orbitals is classified with respect to the symmetiy elements that are maintained in the course of the transformation. The relevant symmetry features depend on the structure of the reacting system. The most common elements of symmetiy to be considered are planes of symmetiy and rotation axes. An orbital is classified as symmetric (5) if it is unchanged by reflection in a plane of symmetiy or by rotation about an axis of symmetiy. If the orbital changes sign (phase) at each lobe as a result of the symmetry operation, it is called antisymmetric (A). Proper MOs must be either symmetric or antisymmetric. If an orbital is not sufficiently symmetric to be either S or A, it must be adapted by eombination with other orbitals to meet this requirement. 

C , rotation of the molecule about a symmetry axis through an angle of 360°/n n is called the order of the rotation (twofold, threefold, etc.) ... 

SHG has been used to study electrode surface symmetry and order using an approach known as SH rotational anisotropy. A single-crystal electrode is rotated about its surface normal and the modulation of the SH intensity is measured as the angle (9) between the plane of incidence and a given crystal axis or direction. Figure 27.34 shows in situ SHG results for an Au(ll 1) electrode in 0.1 M NaC104 + 0.002 M NaBr, using a p-polarized beam. The results indicate the presence of two distinct onefold... 

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