Symmetrically complete sets

Symmetrically Complete Sets. It is easiest to construct each internal symmetry coordinate out of equivalent internal coordinates, i.e., internal coordinates which arc exchanged by the symmetry operations of the molecules, such as the four CII bond extensions in CH4. Then the construction of the symmetry coordinates for a given molecule breaks down into separate problems for the different equivalent sets. Further, it is very desirable to utilize symmetrically complete sets i.e., sets containing all the coordinates resulting from the application of the symmetry operations of the molecule to an arbitrarily chosen coordinate. Thus the six IICH bond angles in CH4 form such a set and should all bo used, even though only five are independent. This use of redundant coordinates will introduce zero roots into the secular equation, but they are most... 

Since F(Q) is symmetric (antisymmetric), it may be expanded in terms of a complete set of symmetric (antisymmetric) wave functions v(Q) (we omit the subscript S, A)... 

Then the moment induced by the electric vector of the incident light is parallel to that vector resulting in complete polarization of the scattered radiation. The A lg i>(CO) mode of the hexacarbonyls provides a pertinent example08. Suppose we have a set of coupled vibrators, equidistant from some origin. Then it must be possible to express the basis functions for the vibrations in terms of spherical harmonics, for the former are orthogonal and the latter comprise a complete set. The polarization of a totally symmetric vibration will be determined by its overlap with the spherically symmetrical term which may be taken as r2 = x2 + y1 + z2. Because of the orthogo-... 

A complete set of symmetry elements is known to be intrinsic only to the sphere. However, objects may be referred to as symmetrical if they retain certain elements of symmetry. The term asymmetrical should be applied only to those objects where not a single element of symmetry is present... 

By equation 74, we see that [ ], depending on the form of the wavefunction, and by equation 75 that ) can be linearly expanded (hence linearly varied) in some set of N functions. This is directly analogous to expanding the asymmetric top rotational wavefunctions in a complete set of symmetric top rotational wavefunctions. The task is to minimize E subject to the single constraint that the wavefunction remain normalized, or... 

Exercise. For one-step processes W is a tridiagonal matrix. With the aid of (3.8) a similarity transformation can be constructed which makes the matrix symmetric, as in (V.6.15). Prove in this way that any finite one-step process has a complete set of eigenfunctions, and that its autocorrelation function consists of a sum of exponentials.510... 

To perform PCA, we need the complete set of eigenvectors, V, and eigenvalues, D, that diagonalize the square, symmetric variance-covariance matrix, Z, where D is the diagonal matrix of eigenvalues. 

It should be noted that for non-isothermal case (and also for isothermal case with autocatalytic kinetics) the local equation may have multiple solutions. When this occurs, the averaged model obtained by the L-S method captures the complete set of solutions of the full CDR equations only within the region of convergence of the local equation. For example, for the wall-catalyzed non-isothermal reaction case, we have shown that the averaged two-mode model can capture only the three azimuthally symmetric solutions of the full CDR equation. The latter has three symmetric solutions (of which two are stable) as... 

The NMR spectrum of pyridazine shows two symmetrical quartets of A2X2 type and t values and coupling constants are as follows Te = 0.76, T4 = T5 = 2.45 t/3 4 = /5,e = 4.9 (5.05 ),. Ig 5 = 7 4,6 = 2.0, Jg g = 3.5 (1.4 ), J4 5 = 8.4. The C satellites in natural abundance in the liquid state have been examined in order to obtain a complete set of proton magnetic resonance parameters for pyridazine. It has been found that chemical shifts are strongly concentration-dependent in a sense opposite to that normally found with aromatic compounds. Coupling constants are, however, virtually invariant. From the NMR spectrum, a deshielding effect on the hydrogen atoms at positions 3 and 6 is revealed. ... 

An ED study requires a complete set of intensity data, measured to a much higher Bragg angle (sin 9IX > 1.1 A , sometimes up to 1.77A ) than is required for an lAM study, and with much more accuracy and precision. It is necessary to measure all symmetrical equivalents of each reflection, carefully address all possible sources of systematic errors, especially extinction and TDS. With a point detector, the data collection required many weeks or even months, but with an area detector (especially using SR) this time has been reduced to a few days, without any loss in precision. There is an intrinsic difficulty to distinguish between the ED distribution of a stationary atom and an apparent smearing of its ED from thermal motion, therefore the data must be collected at as low a temperature as possible. ... 

The complete set of equations, relating fluxes and forces can be written in a matrix form and is called the set of phenomenological equations. It was shown by Onsager [1] that the matrix of the phenomenological proportionality constants is symmetrical ... 

As established above, the interaction between a pair of symmetry elements (or symmetry operations) results in another symmetry element (or symmetry operation). The former may be new or already present within a given combination of symmetrically equivalent objects. If no new symmetry element(s) appear, and when interactions between all pairs of the existing ones are examined, the generation of all symmetry elements is completed. The complete set of symmetry elements is called a symmetry group. 

Inasmuch as the known solutions of the wave equation for a symmetrical-top molecule form a complete set of orthogonal functions (discussed in the preceding section), we can expand the wave function p in terms of them, writing... 

The singularities in the liquid crystals cause the deformation of the director field of liquid crystals and thus affect the symmetry of liquid crystals. This idea provides an approach to analyze the characteristics of the defects. The order vectors (or scalars, or tensors) of various liquid crystals are not the same. The director n is the order vector of the nematic liquid crystals, but the order for the cholesteric liquid crystals is a symmetric matrix, i.e., a tensor. Because the order vector space is thus a topological one, any configuration of the director field of liquid crystals is thus represented by a point in the order vector space. The order vector space (designated by M) is associated with the symmetry of liquid crystals. The topologically equivalent defects in liquid crystals constitutes the homotopy class. The complete set of homotopy classes constitutes a homotopy group, denoted Hr(M). r is the dimension of the sub-space surrounding a defect, which is related to the dimension of the defect (point, line or wall) d, and the dimension of the liquid crystal sample d by... 

When more than one anharmonic interaction term couples near-degenerate, zero-order levels, a simple vector orthogonalization technique can be used to generate a complete set of the dynamically important (i.e., approximately conserved) polyad quantum numbers (Fried and Ezra, 1987 Kellman, 1990). For example, in acetylene, HC = CH, where the ratios of normal mode frequencies u2 W3 W4 W5 are approximately 5 3 5 1 1, modes 1, 2, and 3 are stretching modes (respectively symmetric CH stretch, and CC stretch, and antisymmetric CH stretch), modes 4 and 5 are bending modes (trans-bend and cis-bend), each polyad is labeled by 3 polyad quantum numbers,... 

Experimental data are available only for the symmetric stretching and bending vibrations, Vi(A,) and V2(Ai). The antisymmetric stretching vibration Va(Bi) has not been observed. Complete sets of Vj (or of harmonic frequencies cOj) were obtained from strictly theoretical calculations for PHg in both its electronic ground state X Bi and its observed excited state A Ai and also for the isotopic species PDg. 

Complete sets of the fundamental vibrations Vi (symmetric stretching), V2 (bending), and V3 (antisymmetric stretching), all in cm", were obtained from IR absorption bands of MPH2 (M = K, Rb, Cs) in KBr pellets (and assigned to PHg by comparison with known vibrational frequencies of PH3, NH3, NH2, H2O, and H2S) [25] (see table on the next page). 

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