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Symmetry displacements

Consider a force constant as a real scalar quantity that can conveniently be denoted K ". Such an object would appear multiplying symmetry displacements that jure respectively of species (a, i), (/ , j), (7, k),..., for example. First, we observe that unless a / (8)7. .. contains the tofally symmetric irrep, all such force constants will vanish. Proceeding further, we can operate on K with G, and (using the fact that our irrep matrices are real here) obtain... [Pg.158]

Another series of tetranuclear compounds has been basically obtained by adding a ferrocenyl moiety to the tricobalt methylidyne cluster Co3(p3-CHXCO)9 [79, 80]. Fig. 31, which refers to Co3(C (C5H4)-Fe(C5H5) (CO)9, illustrates the main structural features of this class of compounds [80]. With respect to the symmetrical assembly of Co3(CH)(CO)9 [81], the presence of the sterically encumbering ferrocenyl substituent induces deformation from C3v symmetry displacing the capping-carbon atom towards one cobalt vertex. [Pg.135]

Moreover, in transforming dj to symmetry displacements the actual values of djiitf) are important only for the totally symmetric displacements since they cancel out for all the others. [Pg.31]

The out-of-plane distortion of a five-membered ring can always be described in terms of only two coordinates, since, of the nine (3x5-6) independent coordinates, seven (2x5-3) may be chosen in an arbitrary plane, the mean plane of the ring. These two coordinates can be chosen as a pair of symmetry displacement coordinates of a regular pentagon that transform together as the doubly degenerate representation of Ds. ... [Pg.39]

Fig. 13.27. Correlation of the symmetry displacement coordinates S, (horizontal) and Sj (vertical) mapping the expansion of the coordination state at Si from tetra- to pentacoordination... Fig. 13.27. Correlation of the symmetry displacement coordinates S, (horizontal) and Sj (vertical) mapping the expansion of the coordination state at Si from tetra- to pentacoordination...
From the last relationship the energetic symmetry displacement is noted, to the alpha branch, as an exponential fimction of the term which represents the effect of dynamic localization manifested by the standing waves this way, the quantum dynamic localization of the phases between the directions (on DS branches) is realized such that the photons pass from the field associated with the beta branch of the DS to the one associated with the alpha branch of DS, in order to keep the statistic equilibrium. [Pg.613]

In the following Subsections, it will be shown how the HF theorem becomes a powerful tool when applied to low-symmetry displacement patterns. [Pg.244]

Fig. 2. The axj and ejj carbon-carbon Cartesian symmetry displacements of benzene. The radial and tangential displacements and y, respectively, are positive at each atom in the direction indicated in and its clockwise rotated counterparts. The analogous angle bending and bond stretching displacements have been pictured in Fig. 3. A counterclockwise rotation of the molecule by 60 about the z -axis replaces by - (l/2) -j + (V3/2) 2, and by - V3j2)Sl ia — (1/2) 2j SC = ). [Such a physical rotation mathematically... Fig. 2. The axj and ejj carbon-carbon Cartesian symmetry displacements of benzene. The radial and tangential displacements and y, respectively, are positive at each atom in the direction indicated in and its clockwise rotated counterparts. The analogous angle bending and bond stretching displacements have been pictured in Fig. 3. A counterclockwise rotation of the molecule by 60 about the z -axis replaces by - (l/2) -j + (V3/2) 2, and by - V3j2)Sl ia — (1/2) 2j SC = ). [Such a physical rotation mathematically...
Fig. 3. The aj, and e g carbon-carbon internal symmetry displacements of benzene. A counterclockwise rotation of the molecule by 60° about the z-axis replaces by —(1I2)S — (V3j2)S, and Si.j by [V3j2)S g — (l/2)S i , (k = 6, 8). (Please note the error in reference 12c the counterclockwise rotation of the molecule there used was 60° not 120° as stated. Also, in Fig. 1 of I2c the signs in both 0, and 0,.0 should be reversed.) The relationship connecting the Cartesian symmetry coordinates of Fig. 2 to the internal symmetry coordinates above is readily established by means of the vector addition of the appropriate displacement diagrams. This procedure yields 3i = Sj, 2( )= (1/8)V2/3 X (3S,(j)-b V3S,( )), and = (1I8)V2I3 x (S,(J) - - 3 /3Ss( ). Fig. 3. The aj, and e g carbon-carbon internal symmetry displacements of benzene. A counterclockwise rotation of the molecule by 60° about the z-axis replaces by —(1I2)S — (V3j2)S, and Si.j by [V3j2)S g — (l/2)S i , (k = 6, 8). (Please note the error in reference 12c the counterclockwise rotation of the molecule there used was 60° not 120° as stated. Also, in Fig. 1 of I2c the signs in both 0, and 0,.0 should be reversed.) The relationship connecting the Cartesian symmetry coordinates of Fig. 2 to the internal symmetry coordinates above is readily established by means of the vector addition of the appropriate displacement diagrams. This procedure yields 3i = Sj, 2( )= (1/8)V2/3 X (3S,(j)-b V3S,( )), and = (1I8)V2I3 x (S,(J) - - 3 /3Ss( ).
Fig. 9. A general displacement vector of an octahedral complex. Such vectors may be uniquely expressed as a linear combination of the symmetry displacements (e.g., see Fig. 6), which in turn may be expressed as linear combinations of the normal coordinates. Fig. 9. A general displacement vector of an octahedral complex. Such vectors may be uniquely expressed as a linear combination of the symmetry displacements (e.g., see Fig. 6), which in turn may be expressed as linear combinations of the normal coordinates.
Piezoelectric response is related to ionic displacement dielectric response. In a heteropolar (partially ionic) material that lacks a center of inversion symmetry, displacement of atoms of one polarity with respect to atoms of another polarity results in a change in shape of the material. A relationship between shape and applied electric field is termed a piezoelectric response. When the unit cell of the lattice includes inversion symmetry such a displacement moves charge but does not change the shape. Consequently, such materials are not piezoelectric. An example of how a material can lack an inversion center is found in all zincblende-structure materials. In these materials, a cation and anion lie at opposite ends of each bond and the structure is not symmetric around this bond. Furthermore, all bond pairs are... [Pg.51]


See other pages where Symmetry displacements is mentioned: [Pg.171]    [Pg.11]    [Pg.4]    [Pg.161]    [Pg.31]    [Pg.31]    [Pg.45]    [Pg.219]    [Pg.220]    [Pg.314]    [Pg.590]    [Pg.159]    [Pg.244]    [Pg.255]    [Pg.179]    [Pg.319]   
See also in sourсe #XX -- [ Pg.255 ]




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Internal displacement coordinate symmetry coordinates

Low-symmetry nuclear displacement

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