Rotations, symmetry species

All molecules possess the identity element of symmetry, for which the symbol is / (some authors use E, but this may cause confusion with the E symmetry species see Section 4.3.2). The symmetry operation / consists of doing nothing to the molecule, so that it may seem too trivial to be of importance but it is a necessary element required by the mles of group theory. Since the C operation is a rotation by 2n radians, Ci = I and the symbol is not used. 

In the sixth column of the main body of the character table is indicated the symmetry species of translations (7) of the molecule along and rotations (R) about the cartesian axes. In Figure 4.14 vectors attached to the nuclei of H2O represent these motions which are assigned to symmetry species by their behaviour under the operations C2 and n (xz). Figure 4.14(a) shows that... 

In Table 6.6 the results for the point group are summarized and the translational and rotational degrees of freedom are subtracted to give, in the final column, the number of vibrations of each symmetry species. 

Whether the molecule is a prolate or an oblate asymmetric rotor, type A, B or C selection mles result in characteristic band shapes. These shapes, or contours, are particularly important in gas-phase infrared spectra of large asymmetric rotors, whose rotational lines are not resolved, for assigning symmetry species to observed fundamentals. 

In the theory of deuteron spin-lattice relaxation we apply a simple model to describe the relaxation of the magnetizations T and (A+E), for symmetry species of four coupled deuterons in CD4 free rotators. Expressions are derived for their direct relaxation rate via the intra and external quadrupole couplings. The jump motion between the equilibrium positions averages the relaxation rate within the same symmetry species. Spin conversion transitions couple the relaxation of T and (A+E). This mixing is included in the calculations by reapplying the simple model under somewhat different conditions. The results compare favorably with the experimental data for the zeolites HY, NaA and NaMordenite [6] and NaY presented here. Incoherent tunnelling is believed to dominate the relaxation process at lowest temperatures as soon as CD4 molecules become localized. 

Vibrations may be decomposed into three orthogonal components Ta (a = x, y, z) in three directions. These displacements have the same symmetry properties as cartesian coordinates. Likewise, any rotation may be decomposed into components Ra. The i.r. spanned by translations and rotations must clearly follow the appropriate symmetry type of the point-group character table. In quantum formalism, a transition will be allowed only if the symmetry product of the initial and final-state wave functions contains the symmetry species of the operator appropriate to the transition process. Definition of the symmetry product will be explained in terms of a simple example. 

Still another aspect of the Li and F valence orbitals is modified by ionic-bond formation. In an isolated ionic or neutral species, each NAO retains the characteristic angular shape of the pure s and p hydrogenic orbitals shown in Fig. 1.1, reflecting the full rotational symmetry of atoms. However, in the presence of another atom or ion this symmetry is broken, and the optimal valence orbitals acquire sp hybrid form (assumed for simplicity to include only valence s and p orbitals), as represented mathematically by... 

The calculated state energies, the transition moments, and the symmetry classification are given in Table 3. The symmetry species of the triplet functions is obtained by taking the direct product of irreducible representation of the space and the spin functions Fx, Fy, Fz, which transform as the rotations Rx, Ry, and Rz-... 

Similar analysis assigns a symmetry species to rotations about the three coordinate axes Rx, Ry and Rz- These may be visualized as directed loops around the axes. A symmetry operation that reverses the sense of the rotation is said to take the rotation into its negative. Thus in C2v, ERx = +Rx, C2RX = —Rx, and ayRx = —Rx and a yRx = +Rx- Remember that the molecular plane is the y-z plane. The character vector is (1 —1 —1 1) ... 

The symmetry species of translations and rotations are used so frequently that they are generally included in the character tables. 

The 3N degrees of freedom for nuclear motion are divided into 3 translational, 3 (or 2) rotational, and 3N-6 (or 3N-5) vibrational (degrees of freedom. (The translations and rotations are often called nongenuine vibrations.) The 9 irreducible representations in (9.104) include the 3 translations and the 3 rotations. To find the symmetry species of the 3 vibrations, we must find the symmetry species of the translations and rotations. 

There are two totally symmetric ( ,) normal modes and one b2 normal mode. (The convention is to use lowercase letters for the symmetry species of the normal modes.) The symmetry species of the normal modes have been found without solving the vibrational secular equation. Moreover, since there is only one b2 normal mode, the form of this vibration must be determined from symmetry considerations together with the requirement that the vibration have no translational or rotational energy associated with it. Thus (Fig 6.1), any bent XYX molecule has a b2 normal mode with the X atoms vibrating along the X—Y bonds and the Y atom vibrating in the plane of the molecule and perpendicular to the symmetry axis. On the other hand, there are two ax symmetry coordinates and the two ax normal vibrations are linear combinations of the ax symmetry coordinates, where the coefficients are dependent on the nuclear masses and the force constants. Thus the angles between the displacement vectors of the X atoms and the X—Y bonds for the ax modes of a bent XYX molecule vary from molecule to molecule. 

The overall molecular wave function is the product of vibrational, rotational, and electronic wave functions and can be classified according to one of the symmetry species of the molecular point group.10... 

We can now find the symmetry species of the rotations in HzO. Taking +1 times the matrices (9.105) and —1 times the matrices (9.106), we obtain rrot, which is seen to be reducible as follows ... 

The spherical-top Hamiltonian is Htot= P2/1I. The ellipsoid of inertia is a sphere. The spherical-top rotational wave functions can thus be classified according to the symmetry species of %h. (The angular-momentum operator P2 =P2 + Pf+P2 remains unchanged no matter how the abc axes are rotated.)... 

In connection to control in dynamics I would like to take here a general point of view in terms of symmetries (see Scheme 1) We would start with control of some symmetries in an initial state and follow their time dependence. This can be used as a test of fundamental symmetries, such as parity, P, time reversal symmetry, T, CP, and CPT, or else we can use the procedure to discover and analyze certain approximate symmetries of the molecular dynamics such as nuclear spin symmetry species [2], or certain structural vibrational, rotational symmetries [3]. 

Having the character table, we will now show how to determine the number of modes associated with each symmetry species and their activity. We will quote the relevant formulae [Bhagavantam and Venkatarayudu (75)] and apply them to the example of the polyethylene chain. The total number of normal modes (including translations and rotations) under a given species is given by... 

For reasons discussed later (Section II, E) the ground state of a stable organic molecule is almost always totally symmetrical and it is then possible, at least in principle, to determine the excited state symmetry species from the rotational structure of the electronic bands. The basis for the decision lies in a symmetry that exists between the conditions for the production and absorption of light waves. According... 

The rotational levels can be classified according to the symmetry species of the rotational wavefunction tfir. Since the moment of inertia about any axis is invariant to a rotation by 180° about one of the principal axes, ijir must either remain unchanged or change in sign only... 

The rotational analysis gives a comprehensive picture of the excited state structure. Ingold and King (1953) established that the dipole moment associated with the transition is oriented along the inertial c axis (species Au), that is, perpendicular to the plane of the excited trans-bent structure hence the excited state symmetry species must be 1AU, its singlet character being inferred from the intensity of the transition (/ 10-4). Innes (1954) showed that the intensity alternation in the rotational lines requires the axis of greatest inertia to coincide... 

Calculation of partition functions requires spectroscopic quantities for the rotational and vibrational partition functions. The quantities required are moments of inertia, rotational symmetry numbers and fundamental vibration frequencies for all normal modes of vibration. The translational terms require the mass of the molecule. All terms depend on temperature. Calculation of partition functions is routine for species for which a detailed spectroscopic analysis has been made. 

The wavefunctions V /, may be classified according to their symmetry properties. If we take the symmetry point group of this system to be C3V, there are three symmetry species in this group A (symmetric with respect to all operations of this group), A2 (symmetric with respect to the threefold rotations but antisymmetric with respect to the vertical symmetry planes), and E (a two-dimensional representation). 

Step 3. From 7 3/v subtract the symmetry species rtrans(= r -) for the translation of the molecule as a whole and TTot (based on Rx, Ry, and Rz) for the rotational motion to obtain the representation for molecular vibration, TTitv... 

Octahedral complex ML6 has Oh symmetry. However, for simplicity, we may work with the 0 point group, which has only rotations as its symmetry operations and five irreducible representations, A, A2,. ..,T2. The character table for this group is shown in Table 8.4.1. It is seen that the main difference between the Oh and 0 groups is that the former has inversion center i, while the latter does not. As a result, the Oh group has ten symmetry species Aig, Aiu, A2g, A2u, , T2g, r2u. 

One possible such mechanism for fixing a pattern is to have a phase transition. For example, if the pattern is in terms of a distribution of large molecules on the outer membrane surface, as in the Fucus-like models discussed here, then a membrane phase transition from a more liquid-like to a more crystal-like state of the membrane could essentially immobilize the membrane bound species and freeze in the pattern. In fact several hours after fertilization in Fucus the lability (rotatability) of the polar axis significantly decreases. Indeed this freezing of the Fucus patterning is not easily explained in terms of a Turing mechanism since the rotational symmetry of the Fucus egg, as discussed previously, implies that the electrical polarity is not stable (or more precisely is marginally stable) to polar axis rotation. 

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