Rotational representations

The two Is orbitals are unaffected by the E (identity) operation, and hence the number 2 is written down in the representation. Rotation by any angle, (]), around the axis does not affect the orbitals hence the second 2 appears as the character of the two Is orbitals. The third 2 appears because the two orbitals are unaffected by reflexion in any of the infinite number of vertical planes which contain the molecular axis. The operation of inversion affects both orbitals in that they exchange places with each other, and so a zero is written down in the i column. Likewise an operation causes the orbitals to exchange places and a zero is written in that column. There are an infinite number of C2 axes passing through the inversion centre and these are perpendicular to the molecular axis. The associated operation of rotation through 180° around any C2 axis causes the Is orbitals to exchange places with each other, so there is a final zero to be placed in the representation. 

FIGURE 1.16 Isomers of cyclopropane-1,2-dicarboxylic acid, (a) is-l,2-Dicarboxycyclopropane (trans isomers) (b) Z-l,2-Dicarboxycyclopropane isomer, (meso isomer) (c) Molecular models of E- (trans-) 1,2-dimethylcyclo-propane shown in the tube representation. Rotation of the right hand structure about a vertical axis through the center of the cyclopropane will superimpose the two methyl groups. The methylene of the rotated structure will be in the back, rather than the front, and not superimposed, (d) A mirror plane through the methylene and the back carbon-carbon bond is a plane of symmetry. The two carboxyl groups appear not to reflect each other in the model shown but they can rotate freely and will reflect each other on an instantaneous basis. 

Fig. 16.4. A Tafel plot for the situation represented in Fig. 16.3 (in normal representation rotated 90° anticlockwise).

Figure 3. Dopant content as a function of current denisty, i (in mA/sq cm) same as figure 2 in two-dimensional representation rotation speed Omega (in rpm) is a parameter...

To Edmund Pendleton, Aug. 26, 1776 A brief defense of Tf s ideas for a draft constitution for Virginia, with special reference to representation, rotation in office, and constitutional interpretation - let the judge be a mere machine ... 

The 3D representation of the test object can be rotated by means of an ARCBALL interface. Clicking on the main client area will produce a circle which is actually the silhouette of a sphere. Dragging the mouse rotates the sphere, and the model moves aceordingly. An arc on the surface of the sphere is drawn for visual feedback of orientation additionally a set of coordinate axes in the bottom left comer provides further feedback. 

The exact position of reflectors within the weld volume is calculated by means of the known probe position plus weld geometry and transferred to a true-to-scale representation of the weld (top view and side view). Repeated scanning of the same zone only overwrites the stored indications in cases where they reach a higher echo amplitude. The scanning movement of the probe is recorded in the sketch at the top, however, only if the coupling is adequate and the probe is situated within the permissible rotation angle. 

We have described here one particular type of molecular synnnetry, rotational symmetry. On one hand, this example is complicated because the appropriate symmetry group, K (spatial), has infinitely many elements. On the other hand, it is simple because each irreducible representation of K (spatial) corresponds to a particular value of the quantum number F which is associated with a physically observable quantity, the angular momentum. Below we describe other types of molecular synnnetry, some of which give rise to finite synnnetry groups. 

Finally, we consider the complete molecular Hamiltonian which contains not only temis depending on the electron spin, but also temis depending on the nuclear spin / (see chapter 7 of [1]). This Hamiltonian conmiutes with the components of Pgiven in (equation Al.4,1). The diagonalization of the matrix representation of the complete molecular Hamiltonian proceeds as described in section Al.4,1.1. The theory of rotational synnnetry is an extensive subject and we have only scratched the surface here. A relatively new book, which is concemed with molecules, is by Zare [6] (see [7] for the solutions to all the problems in [6] and a list of the errors). This book describes, for example, the method for obtaining the fimctioiis ... 

The rotation-vibration-electronic energy levels of the PH3 molecule (neglecting nuclear spin) can be labelled with the irreducible representation labels of the group The character table of this group is given in table Al.4.10. 

This representation is slightly inconvenient since Ey and 2 in equation (Al.6.56) are explicitly time-dependent. For a monocln-omatic light field of frequency oi, we can transfonn to a frame of reference rotating at the frequency of the light field so that the vector j s a constant. To completely remove the time dependence... 

Figure Bl.15.7. Transient EPR. Bottom time-resolved EPR signal of the laser-flash-indueed triplet state of pentaeene in /j-terphenyl. BpO.085 mT. Top initially, the transient magnetization M is aligned along B z. In the presenee of a MW magnetie field B the magnetization preeesses about BJ x (rotating frame representation).

Figure Bl.15.10. FT EPR. (A) Evolution of the magnetization during an FT EPR experiment (rotating frame representation). (B) The COSY FT EPR experiment.

Figure Bl.15.11. Fomiation of electron spin echoes. (A) Magnetization of spin packets i,j, /rand / during a two-pulse experiment (rotating frame representation). (B) The pulse sequence used to produce a stimulated echo. In addition to this echo, which appears at r after the third pulse, all possible pairs of the tluee pulses produce primary echoes. These occur at times 2x, 2(x+T) and (x+2T).

Figure 2-51. a) The rotational barrier in amides can only be explained by VB representation using two resonance structures, b) RAMSES accounts for the (albeit partial) conjugation between the carbonyl double bond and the lone pair on the nitrogen atom. 

Thus, each stereochemical stnicttirc can be described and recognised with this rotational list if the structure is designated, c.g., in the STEREO block of the SMD format. The compact and extensible representation of the rotational list can include additional information, such as the name specification of the geometiy or whether the configuration is absohtte, relative, or racemic (Eigitre 2-73). 

The two structures in our example are identical and are rotated by only 1 20 h Clearly, rotation of a molecule docs not change its stereochemistry, Thus, the permutation descriptor of both representations should be (+ I). On this basis, we can define an equation where the number of transpositions is correlated with the permutation descriptors in an exponential term (Eq. (9)). 

In order to represent 3D molecular models it is necessary to supply structure files with 3D information (e.g., pdb, xyz, df, mol, etc.. If structures from a structure editor are used directly, the files do not normally include 3D data. Indusion of such data can be achieved only via 3D structure generators, force-field calculations, etc. 3D structures can then be represented in various display modes, e.g., wire frame, balls and sticks, space-filling (see Section 2.11). Proteins are visualized by various representations of helices, / -strains, or tertiary structures. An additional feature is the ability to color the atoms according to subunits, temperature, or chain types. During all such operations the molecule can be interactively moved, rotated, or zoomed by the user. 

Projeetors are used to find the SALC-AOs for these irredueible representations. Define C3 = 120 degree rotation, C3 = 240 degree rotation,... 

The rotational eigenfunctions and energy levels of a molecule for which all three principal moments of inertia are distinct (a so-called asymmetric top) can not easily be expressed in terms of the angular momentum eigenstates and the J, M, and K quantum numbers. However, given the three principal moments of inertia la, Ib, and Ic, a matrix representation of each of the three contributions to the rotational Hamiltonian... 

Symmetry tools are used to eombine these M objeets into M new objeets eaeh of whieh belongs to a speeifie symmetry of the point group. Beeause the hamiltonian (eleetronie in the m.o. ease and vibration/rotation in the latter ease) eommutes with the symmetry operations of the point group, the matrix representation of H within the symmetry adapted basis will be "bloek diagonal". That is, objeets of different symmetry will not interaet only interaetions among those of the same symmetry need be eonsidered. 

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 ... 

In writing these relations, we use the faet that refleetion reverses the sense of a rotation. The matrix representations eorresponding to this one-dimensional basis are ... 

The basic idea of symmetry analysis is that any basis of orbitals, displacements, rotations, etc. transforms either as one of the irreducible representations or as a direct sum (reducible) representation. Symmetry tools are used to first determine how the basis transforms under action of the symmetry operations. They are then used to decompose the resultant representations into their irreducible components. 

Before considering other concepts and group-theoretical machinery, it should once again be stressed that these same tools can be used in symmetry analysis of the translational, vibrational and rotational motions of a molecule. The twelve motions of NH3 (three translations, three rotations, six vibrations) can be described in terms of combinations of displacements of each of the four atoms in each of three (x,y,z) directions. Hence, unit vectors placed on each atom directed in the x, y, and z directions form a basis for action by the operations S of the point group. In the case of NH3, the characters of the resultant 12x12 representation matrices form a reducible representation... 

From the information on the right side of the C3v eharaeter table, translations of all four atoms in the z, x and y direetions transform as Ai(z) and E(x,y), respeetively, whereas rotations about the z(Rz), x(Rx), and y(Ry) axes transform as A2 and E. Henee, of the twelve motions, three translations have A and E symmetry and three rotations have A2 and E symmetry. This leaves six vibrations, of whieh two have A symmetry, none have A2 symmetry, and two (pairs) have E symmetry. We eould obtain symmetry-adapted vibrational and rotational bases by allowing symmetry projeetion operators of the irredueible representation symmetries to operate on various elementary eartesian (x,y,z) atomie displaeement veetors. Both Cotton and Wilson, Deeius and Cross show in detail how this is aeeomplished. 

---