Notation, group theoretical

Configuration Mulliken Notation Group Theoretical Label... 

Point group symmetry, notation and representations, and the group theoretical condition for when an integral is zero. 

Dirac bra-ket notation, permutational symmetry, group theoretical properties, 672-674... 

Most of the stereochemical arrangements found in these complexes are based oil the octahedron or on the tetrahedron, and may be derived from these configurations by a suitable distortion. Both basic systems have cubic symmetry and d-orbital energies are split into a triplet and a doublet. The group-theoretical notation will be used for their description. ... 

The representations here are labelled by the group theoretical notation [13] A/, A2, By, B2, E. The first four are one dimensional, while the representation E is two dimensional. For 3, some representations are contained twice and the situation is slightly more complicated. In condensed matter physics, it has become customary to label the representations with the letter T [14]. When both positive and negative parity states are considered also the parity label is added, r and V. The conversion between the two notations is Aj — Ti, A2 — T2, Bi — r3, B2 —> T4, and E —> Ts. 

At the theoretical level of CCSD(T)/6-311+G(2d). The group-theoretic notation of the vibrational spectra will be made clear in Chapters 6 and 7. 

C=N, C=C , N=0 , all have essentially the same electronic configuration as N=N shown in Table 23.3, but without the g and u character in the case of the hetero-nuclear species. The extension to nonlinear molecules requires the introduction of a sophisticated way to describe the symmetry of the orbitals the group theoretical notation for the symmetry types is used. 

Fig. 3.30 Porphyrin HOMOs b], >2 and LUMOs Cj and Cj showing their original (Hiickel) group theoretical notation and their modified notation as used on the four-orbital model. The original coefficients are proportional to the size of the circles. SoUd or dashed circles indicate sign. Symmetry nodes are drawn in heavy lines.

Group-theoretical labels for the eigenstates may be written in several forms, the most common usage being A or B for singlets, E for doublets, and T for triplets (Hamermesh, 1962) in which subscripts, primes, etc. are used in addition to the letters to distinguish- different irreducible representations. Another notation due to Bethe (1929) is Fj, where j runs over the various representations of the group (Koster et al., 1963). 

Flurry, R. L. Jr., Symmetry Groups, Prentice-Hall, Englewood Cliffs, New Jersey, 1980. An excellent introduction to chemical applications of group theory. Our text assumes famil-arity with only the most elementary group theoretical ideas and notations. 

Excited methyl torsional state (group theoretical notation). 

The caption to this figure makes use of group theoretical notation. The reader who is unfamiliar with this notation should at least read Appendix 3, which includes an outline of it, at this point.. 

The group-theoretical approach is based on the fact that, if equation (5) contains products of internal (symmetry) coordinate displacements Sa(oi = /, y, k,...), then the product SaSfi - Sco and that obtained by any permutation of the indices are indistinguishable. Thus, the n-member products transform according to the permutation (symmetric) group Sn or in another, perhaps more appropriate, notation where... 

To answer this question, let us first suppose that a given set of spin eigenfunctions carries a representation, which we for denote by D , such that (with standard group-theoretical notation) (4.2.1) becomesj ... 

The notation of the torsion-inversion-rotational Hamiltonian [07Ily] is based on the group-theoretical high-barrier tunnelling formalism developed for this molecule in [870ha],... 

The notations shown in the parentheses after the group theoretical notations denote the transformation properties. For example, (0) and E ) transform in exactly the same way as the 6 and e d-orbitals, respectively. In terms of the wave-functions shown in Eqs. (23) and (26), the individual terms on the righthand sides of Eq. (27) are expressed as follows ... 

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