Directional property symmetrical

Orbitals are described by specifying their size shape and directional properties Spherically symmetrical ones such as shown m Figure 1 1 are called y orbitals The let ter s IS preceded by the principal quantum number n n = 2 3 etc ) which speci ties the shell and is related to the energy of the orbital An electron m a Is orbital is likely to be found closer to the nucleus is lower m energy and is more strongly held than an electron m a 2s orbital... 

The concepts of directed valence and orbital hybridization were developed by Linus Pauling soon after the description of the hydrogen molecule by the valence bond theory. These concepts were applied to an issue of specific concern to organic chemistry, the tetrahedral orientation of the bonds to tetracoordinate carbon. Pauling reasoned that because covalent bonds require mutual overlap of orbitals, stronger bonds would result from better overlap. Orbitals that possess directional properties, such as p orbitals, should therefore be more effective than spherically symmetric 5 orbitals. 

Recall that the character of an operation is equal to the number of vectors unshifted by that operation. Previously, for AH molecules, in determining 7h, the Is orbitals on the hydrogen atoms are spherically symmetric. In the present case, however, the 2p orbitals have directional properties. Take [Pg.222]

Regarding the property symmetric , antisymmetric , or degenerate with respect to all. symmetry operations, vibrations can be classified according to symmetry species. Each symmetiy species possesses certain spectroscopic characteristics, like forbidden in IR and Raman spectra , or IR-active with dipole moment change in. v-direction , or modulates the xy component of the polarizability tensor . They are given in character tables (Figure 2.7-6), Sec. 7. 

In solids, the chemical shift is a directional property and varies with the crystal s orientation in the magnetic field. This orientation dependence, the chemical shift anisotropy (CSA), can be described by a symmetric second-rank tensor that can be diagonalized and reduced to three principal values 5n, 822,833, where the isotropic chemical shift, 8i, is the average of these values 8i = l/3-(8n + >22 + 833). For a single crystal, or any particular nucleus in a powdered sample, the frequency varies according to ... 

The directional properties of the naphthalene absorption system pointed to interpenetrating allowed and vibrationally induced bands, the former long-axis and the latter short-axis polarized. There did, however, remain a difficulty. Given the different polarizations the separation between the true and false origin(s) must be the frequency of a nontotally symmetrical vibration. The observed hot band separation however was close to a ground state Raman-active 512 cm-1 vibration, known with certainty to be totally symmetrical. 

In models employing simple pair potentials (Morse, Lennard-Jones, Buckingham) only the direct interaction between two atoms is considered. These potentials are radially symmetric and ignore the directional property of the interatomic bond. They make the best use for molecules. One may estimate the total energy of a solid by the use of pair potentials though they involve no further cohesive term. 

Figure 3.15 shows contour representations of the three 7p orbitals. These are seen to have different orientations in space they have directional properties. In fact, s orbitals are the only ones that are spherically symmetrical. The shapes of orbitals are involved in molecular geometry and help to determine the shapes of molecules. The shapes of d and/orbitals are more complex and varied than those of p orbitals and are not discussed in this book. 

Each such nonual mode can be assigned a synuuetry in the point group of the molecule. The wavefrmctions for non-degenerate modes have the following simple synuuetry properties the wavefrmctions with an odd vibrational quantum number v. have the same synuuetry as their nonual mode 2the ones with an even v. are totally symmetric. The synuuetry of the total vibrational wavefrmction (Q) is tlien the direct product of the synuuetries of its constituent nonual coordinate frmctions (p, (2,). In particular, the lowest vibrational state. 

The important point to note from this Example is that in a non-symmetrical laminate the behaviour is very complex. It can be seen that the effect of a simple uniaxial stress, or, is to produce strains and curvatures in all directions. This has relevance in a number of polymer processing situations because unbalanced cooling (for example) can result in layers which have different properties, across a moulding wall thickness. This is effectively a composite laminate structure which is likely to be non-symmetrical and complex behaviour can be expected when loading is applied. 

For laminates that are symmetric in both geometry and material properties about the middle surface, the general stiffness equations. Equation (4.24), simplify considerably. That symmetry has the form such that for each pair of equal-thickness laminae (1) both laminae are of the same material properties and principal material direction orientations, i.e., both laminae have the same (Qjjlk and (2) if one lamina is a certain distance above the middle surface, then the other lamina is the same distance below the middle surface. A single layer that straddles the middle surface can be considered a pair of half-thickness laminae that satisfies the symmetry requirement (note that such a lamina is inherently symmetric about the middle surface). ... 

Because of the analytical complications involving the stiffnesses Ai6, A26, D g, and D26, a laminate is sometimes desired that does not have these stiffnesses. Laminates can be made with orthotropic layers that have principal material directions aligned with the laminate axes. If the thicknesses, locations, and material properties of the laminae are symmetric about the middle surface of the laminate, there is no coupling between bending and extension. A general example is shown in Table 4-2. Note that the material property symmetry requires equal [Q j], of the two layers that are placed at the same distance above and below the middle surface. Thus, both the orthotropic material properties, [Qjjlk. of the layers and the angle of the principal material directions to the laminate axes (i.e., the orientation of each layer) must be identical. 

Antisymmetry of a laminate requires (1) symmetry about the middle surface of geometry (i.e., consider a pair of equal-thickness laminae, one some distance above the middle surface and the other the same distance below the middle surface), but (2i some kind of a reversal or mirror image of the material properties [Qjjlk- In fact, the orthotropic material properties [Qjj], are symmetric, but the orientations of the laminae principal material directions are not symmetric about the middle surface. Those orientations are reversed from 0° to 90° (or vice versa) or from + a to - a (a mirror image about the laminate x-axis). Because the [Qjj]k are not symmetric, bending-extension coupling exists. 

Consider an angle-ply laminate composed of orthotropic laminae that are symmetrically arranged about the middle surface as shown in Figure 4-48. Because of the symmetry of both material properties and geometry, there is no coupling between bending and extension. That is, the laminate in Figure 4-48 can be subjected to and will only extend in the x-direction and contract in the y- and z-directions, but will not bend. 

As described above, the ground state vibrational wavefunction is totally symmetric for most common molecules. Therefore, the product, -(1)0 must at least contain a totally symmetric component. The direct product of two irreducible representations contains the totally symmetric representation only if the two irreducible representations are identical. Therefore, transitions can occur from a symmetrical initial state only to those states that have the same symmetry properties as the transition operator, 0. 

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