Symmetry elements, molecular

For a nucleus sharing all the molecular symmetry elements (e.g., the metal nucleus in a mononuclear complex), the hyperfine matrix is subject to the same... 

It should be noted that the displacement coordinates have been labeled and then combined into the -type symmetry coordinates in a particular way. The angles have been indexed so that 0, is opposite to rh This assures that 0, and the change therein, A0 are related to the molecular symmetry elements in the same way as are r, and Ar,. Then, when the SALCs are written, A0, and Ar, occupy corresponding positions in the expressions. Unless this is done the symmetry factorization will not work out. 

A final feature of importance in 13C NMR spectra is the notion of equivalency. Because some type of decoupling is normally done, either broad band or off resonance, magnetic equivalency is not an issue in 13C NMR, but chemical equivalence remains an issue. If two carbon atoms share the same chemical environment, then of course they will have the same chemical shift. Thus it is important to recognize local or molecular symmetry elements. In a previous... 

With the above symmetry considerations one can explain the unexpected trend in the tetrasubstituted TEE series by the molecular symmetry elements without knowing the wavefunctions themselves. It was assumed that the dipolar term dominates and that the negative and two-photon term are of secondary importance. As all the nonlinearities of the tetrasubstituted TEEs are positive the negative term can assumed to be small. 

For the analysis of these sigmatropic reactions, correlation diagrams are not relevant since it is only the transition state and not the reactants or products which may possess molecular symmetry elements. 

Performing a symmetry operation on a molecule gives a nuclear configuration physically indistinguishable from the original one. Hence the center of mass must have the same position in space before and after a symmetry operation. For the operation the only points that do not move are those on the C axis. Therefore a C symmetry axis must pass through the molecular center of mass. Similarly, a center of symmetry must coincide with the center of mass a plane of symmetry and an 5 axis of symmetry must pass through the center of mass. The center of mass is the common intersection of all the molecular symmetry elements. 

In their discussion of sigmatropic rearrangements, Woodward and Hoffmann state For the analysis of these reactions correlation diagrams are not relevant since it is only the transition state and not the reactants or products which may possess molecular symmetry elements. [1, p. 114] This is something of an overstatement. For example, they could hardly have meant it to apply to 1,5-hexadiene, which is no less symmetrical than any transition state that can be assumed for its Cope rearrangement. 

Correlation diagram method is not suitable for the analysis of sigmatropic-rearrangements because only transition state but not reactants or products possess molecular symmetry elements. Methods for the analysis of this type of reactions are discussed in forthcomming discussion. 

Fe(CN)g] (ferricyanide ion) symmetry elements, 85 [Fe(CN)g] (ferrocyanide ion) molecular orbitals, 270ff FlCiiN (cyanopentaacetylene) interstellar, 120... 

CC and CH bond orbitals but also for the CTL, ami CH3 group orbitals. If the local symmetry elements are preserved in the full molecule, the 7r (or a) local orbitals can combine to give v (or o) molecular orbitals. The reader should, therefore, not be surprised to find, for instance, tt type molecular orbitals in cyclopropane which are delocalized over the CH2 groups. 

Recognize molecular symmetry planes and axes. Even approximate, or local symmetry elements may be useful. One should not step back just because, formally, the molecule has no symmetry elements. A methyl and an ethyl substituent, or chlorine and bromine substituent, can be equated. Substituents that disrupt the molecular symmetry but have trivial electronic requirements may be deleted. 

Vibrations of the symmetry class Ai are totally symmetrical, that means all symmetry elements are conserved during the vibrational motion of the atoms. Vibrations of type B are anti-symmetrical with respect to the principal axis. The species of symmetry E are symmetrical with respect to the two in-plane molecular C2 axes and, therefore, two-fold degenerate. In consequence, the free molecule should have 11 observable vibrations. From the character table of the point group 04a the activity of the vibrations is as follows modes of Ai, E2, and 3 symmetry are Raman active, modes of B2 and El are infrared active, and Bi modes are inactive in the free molecule therefore, the number of observable vibrations is reduced to 10. 

A computer program for the theoretical determination of electric polarizabilities and hyperpolarizabilitieshas been implemented at the ab initio level using a computational scheme based on CHF perturbation theory [7-11]. Zero-order SCF, and first-and second-order CHF equations are solved to obtain the corresponding perturbed wavefunctions and density matrices, exploiting the entire molecular symmetry to reduce the number of matrix element which are to be stored in, and processed by, computer. Then a /j, and iap-iS tensors are evaluated. This method has been applied to evaluate the second hyperpolarizability of benzene using extended basis sets of Gaussian functions, see Sec. VI. 

It should be noted that the trace of a matrix that represents a given geo] operation is equal to 2 cos y 1, the choice of signs is appropriate to or improper operations. Furthermore, it should be noted that the aim direction of rotation has no effect on the value of the trace, as a inverse sense corresponds only to a change in sign of the element sin y. TE se operations and their matrix representations will be employed in the following chapter, where the theory of groups is applied to the analysis of molecular symmetry. 

---