Group theoretical analysis

The pair symmetry classifications in the C4V group are given on the right and only the last pair function has Bi symmetry. We are hence restricted to three possible singlet pair functions of JBi symmetry of which one is likely to make a dominant contribution to the condensate state function. 

This table has been previously published in part by Sawatzky et al [44] 

Energetically, a condensate dominated by a2, b2 pairs is unlikely since the a2 single particle state is essentially a non-bonding p-combinadon of 0(2p) orbitals and is expected to lie well below the Fermi level. Similar conclusions can be made about all other non-bonding or weakly bonding orbitals. Therefore aibi and e-pairs seem more likely. 

Other things being equal, for repulsive electronic correlations the ground state of a2, bi pairs is likely to be the triplet state, 3Bi, by Hund s rule. This principle energetically favours the most antisymmetrical space function and symmetrical spin function for a pair by minimising the short range Coulomb repulsion between electrons and is found to be widely applicable [40] in molecular systems. 

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In general, the number of phonon branches for a carbon nanotube is very large, since every nanotube has 6N vibrational degrees of freedom. The symmetry types of the phonon branches for a general chiral nanotube are obtained using a standard group theoretical analysis [194]... 

BalK82 Balasubramanian, K. Topological and group theoretical analysis of dynamics nuclear magnetic resonance spectroscopy. J. Phys. Chem. 86 (1982) 4668-4674. 

Kneubuhl37,38 has given a detailed group theoretical analysis of symmetry restrictions on the orientations of g- and hyperfine matrix principal axes. His results are summarized in Table 4.9. 

Vibrational spectroscopy has been widely applied in the study of LDHs [161,162] but a somewhat confusing variety of spectral data and interpretations have appeared in the literature, hi this section, we focus on the information that can be obtained regarding the structure of the interlayer anions. The unperturbed carbonate ion has point symmetry Dsh. Group theoretical analysis predicts four normal modes the vi symmetric stretch of Aj symmetry at 1063 cm the V2 out of plane bend of A 2 symmetry at 880 cm the V3 asymmetric stretch of E symmetry at 1415 cm , and the V4 in plane bend of E symmetry at 680 cm [22]. The V2 mode is IR active only, the vi mode is Raman active only, whilst the two E modes are both IR and Ra-... 

Infrared Spectrum and Group Theoretical Analysis of the Vibrational Modes of Carbonyl Sulfide 160... 

Wilhin a group theoretical analysis, this is normally accomplished with projection operators. For a discussion of (his method, see the group iheory texts listed in Footnote I of Chapter 3. 

The ZOA is based more on expediency than on it being a good approximation in fact, the value of, S is about 0.2 0.3 (rather than zero) for carbon 2pz orbitals on adjacent atoms. When 5 is not joined to r, it is much more reasonable. Nevertheless, it is customary to use the ZOA at this level of approximation since it yields normalization constants without performing any calculations. One should remark that it affects only the NJ, the ratio of the coefficients being given by the group theoretical analysis. Using the ZOA,... 

Worlton, T. G. and Warren, J. L. (1972) Group theoretical analysis of lattice vibrations. Comp. Phys. Commun. 3, 88-117. 

The group-theoretical analysis of the several p shell configurations is given below ... 

Systems containing more than one identical particles are invariant under the interchange of these particles. The permutations form a symmetry group. If these particles have several degrees of freedom, the group theoretical analysis is essential to extract symmetry properties of the permissible physical states. Examples include Bose-Einstein, Fermi-Dirac, Maxwell-Boltzmann statistics, Pauli exclusion principle, etc. 

Special attention is given to the localized vibrations with the symmetries Ai and E. According to a group-theoretic analysis, they are active in the EA spectra of ZnO Ni crystals. The frequencies of such localized vibrations induced by nickel impurities in ZnO crystals are listed in Table 4. 

The Ih symmetry of the 60-atoms molecule allows 2Ag and SH modes to be Raman active and 4T modes to be IR active. The four IR active modes are at 1430, 1185, 580 and 528 cm , respectively. The most important Raman modes are at 1469 (tangential bond alternation or pinch mode, Ag), 495 (radial breathing mode, T ) and 271 (squashing mode, Hg) cm , respectively. In the low temperatur phase degenerate modes split from a crystal field and Davidov interaction. Good reviews on the group theoretical analysis and on the line positions are given in (Dresselhaus et al., 1992 Matus and Kuzmany, 1993). 

Group-Theoretical Analysis of Jahn-Teller Systems... 

This study deals with group-theoretical analysis of JT systems, especially with the prediction of the symmetries of the structures caused by JTE. Two alternative treatments based on JT active coordinates and on the step-by-step splitting degenerate electronic states are explained and their results are compared within several examples. Despite producing equal results for some low-dimensional groups, both treatments have their advantages and shortages. 

Howard CJ, Stokes HT (1998) Group-theoretical analysis of octahedral tilting in perovskites. Acta CiystallogrB54 782-789... 

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