Identity representation

Allenes, spiranes (c) Identical representation of the same compound. 

Consequently generates the identical representation T1 of the point group and we say that the ground state vibrational wavefunction is totally symmetric. 

In stage 2, we have an identical representation of matches for all streams to that of stage 1. At the right end of stage 2 we allow for coolers for HI and H2, while in the left end of stage 1 we allow for heaters for Cl and C2. The simplified HEN superstructure for this example is shown in Figure 8.30. 

Following ref. 8, let us expand the total energy loss scattering cross section in a correlational series. The terms of this series represent contributions to the cross section by isolated many-particle fluctuational complexes of different ranges. Using the identical representation for /g,iv(e),... 

Theorem 1. A doubly transitive permutation representation of a group G over the complex field is the sum of the identical representation and an absolutely irreducible representation [13],... 

The symmetry properties of spontaneous strains are most conveniently understood by referring to the irreducible representations and basis functions for the point group of the high symmetry phase of a crystal of interest. These are given in Table 2 for the point group Almmm as an example. Basis functions x + y ) and are associated with the identity representation and are equivalent to (ei + ei) and C3 respectively. This is the same as saying that both strains are consistent with Almmm symmetry e = ei). The shear strain e - ei) is equivalent to the basis function (x - y ) which is associated with the Big representation, the shear strain e is equivalent to xy (B2g) and shear strains e and e to xz, yz respectively (Eg). The combinations (ci + 62) and (ci - ei) are referred to as symmetry-adapted strains because they have the form of specific basis functions of the... 

The coupling terms and elastic energy terms must comply with standard symmetry rules, which require that each term in the free energy expansion transforms (in the group theoretical sense) as the identity representation, Lidentity- The order parameter, by definition, transforms as the active representation, Ractive- The rules for coupling terms are then as follows ... 

It can be seen that the PCA and NLM plots are quite similar. This is to be expected when the first two principal components explain a high percentage of the variation in the data (73% in this case). The main difference between the plots is on the left-hand side where the H, F, OH and NH2 are less closely clustered in the NLM than in the PCA plot. Similar clustering is apparent in the cluster analysis and KM. In fact all four approaches give similar, but not identical, representations of similarity in this example. 

Figure 6. A ngular dependence of apparent structure factor S(K) for PCS Al-Dl in water (0) C = 0J2 mg/mL fVl C = 0.46 mg/mL C ) C = 0.71 mg/mL (O) C -- 1.1 mg/mL and (X) C = 2.2 mg/mL ( ) values at small scattering vector calculated using Equation 5. Identical representation for S(K) is obtained by assuming P(K) for PCS to be that of a star polymer with polydisperse rays fsee Equation 6).

Invariance under the point-group operations requires that the crystal-field Hamiltonian contain only operators that transform as the identity representation of the point group. These operators are easy to determine in general, since, for all the point groups except the cubic groups (T, Tj, T, O, and Oh), all group operators may be constructed from the following operators (Leavitt, 1980) ... 

The amount that any specific resonance structure representation actually contributes to the overall hybrid will depend on the presumed energy of that contributor. Generally, the larger the number of identical representations of the lowest-energy contributor, the more stable the structure. 

The Hyperfine Tab (Fig. 15) is only valid if the atom containing one or more unpaired electrons (the electron spin is greater than zero) has isotopes wifli a nonzero nuclear spin. The interaction can be turned ofEon with a middle mouse click on the red/blue tick on the hyperfine tab. Identical representations (isotropic, axial, and orthorhombic) to those for the electron Zeeman interaction are available for the hyperfine interaction. The hyperfine coupling constants correspond to the isotope of the particular atom that has the largest isotopic abundance. For randomly orientated solutions, molecules containing a single unpaired electron with monoclinic or triclinic symmetries, the Euler angles, a, P, and y will be nonzero, a corresponds to a rotation about the z (gP axis, p a rotation about the new axis, and y a rotation about the new axis. 

Let us note some features of this diagram. To begin, it has a familiar air to it. We have seen an identical representation of operating line and equilibrium curve in the operating diagram dealing with a countercurrent staged... 

The five structures in the top row are considered identical representations of the same molecule, whereas the five structures in the bottom row are considered five different resonance structures. 

Rotational symmetry axis = an axis around which the molecule as a whole can be rotated (some distance less than 360°) to result in an identical representation. 

Rotating Object 3 degrees (niunber less than 360) results in an identical representation. 

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