Cartesian components representation

Our task here is to determine whether any of the three Cartesian components is nonzero. Since, in DAh the z vector transforms according to the AZu representation and a and y jointly according to the representation, we need to know whether either of the direct products, Blg x AZu x BZu or BiK x Eu x BZu contains the Aljt representation. It is a simple matter to show that the first one is equal to AlK while the second is equal to Eg. Thus, the <5— S transition is electric-dipole allowed with z polarization and forbidden for radiation with its electric vector in the xy plane. 

In our description of spin reorientational relaxation processes, tensorial quantities are used for which it is necessary to know the transformation properties concerning rotation. A clear and compact formulation is obtained by replacing the cartesian components with a representation in terms of irreducible spherical components. It is known that any representation of the group of rotations can be developed into a sum of irreducible rqpre-sentations D of dimension 2/ +1. If for the description of general rotation R(U) we use the Euler angles Q = (a, p, y), this rotation will be defined by... 

Cartesian component notation offers an extremely convenient shorthand representation for vectors, tensors, and vector calculus operations. In this formalism, we represent vectors or tensors in terms of their typical components. For example, we can represent a vector A in terms of its typical component Ait where the index i has possible values 1, 2, or 3. Hence we represent the position vector x as x, and the (vector) gradient operator V as Note that there is nothing special about the letter that is chosen to represent the index. We could equally well write x . x/ . or x , as long as we remember that, whatever letter we choose, its possible values are 1, 2, or 3. The second-order identity tensor I is represented by its components %, defined to be equal to 1 when i = j and to be equal to 0 if z j. The third-order alternating tensor e is represented by its components ,< , defined as... 

In some cases two or more sets of cartesian components (say - y, xy or xz, yz in D3) belong to the same multidimensional irreducible representation so that a linear combination of them should be considered. Hence, the symmetry descent technique is applicable. 

An alternative form for specifying tensor principal values that is sometimes used in the EPRLL-family programs is the spherical representation. In terms of the Cartesian components of a tensor M, namely, M, My, and M, its spherical components are defined as... 

Due to the appearance of the position operator, this is called the dipole oscillator strength in the length representation. One can consider the oscillator strength as the trace of a tensor of cartesian components... 

Note that Equation [51] represents formally the tensorial relationship, while Equation [52] expresses this relation by the (Cartesian) components of P and E and some coefficients whereas Equation [53] states this relation by using Einstein s summation convention. Here, the coefficients are the components of the electric susceptibility tensor which is a tensor of rank 2. The tensor % is an example of what is usually called a property tensor or matter tensor. Strictly speaking, property tensors describe physical properties of the static crystal which belong to the totally symmetric irreducible representation of the relevant point group. Properties, however, that depend on vibrations of the crystal lattice are described by tensors which belong to the different irreducible representations. The corresponding tensors are then often designated as tensorial covariants. 

Here the entries TW, , , denote the Cartesian components of the tensor of rank [m] where i = 1, 2, 3 has to be taken into account. Equations [56] to [58] describe the transformation properties of an arbitrary tensor of rank [m] with respect to the orthogonal group 0(3, R), respectively. The matrix group Ml ] is the w-fold tensor representation of 0(3, R), where M g) is a real 3-dimensional matrix representation of g s 0(3, R), which implies that MM defines a real 3 -dimensional 0(3, R)-represen-tation. Moreover, note that in Equation [59] Einstein s summation convention has been used. Finally, one has to distinguish carefully between polar and axial tensors of rank [m] since their inherent transformation properties with respect to 0(3, R),... 

Under special conditions, discussed at the end of Chapter 13, it is possible for a mode to propagate with its electric field virtually parallel to either the x- or y-axis in the waveguide cross-section, assuming these are the principal axes. In such cases the displacement vector is parallel to the electric field, and, using the field representation of Eq. (30-4), the cartesian components of the transverse field, e, satisfy the transverse component of the vector wave equation of Eq. (30-31)... 

The symmetry selection rule may also be mqiressed in alternative ways. An infrared transition is not forbidden only in the case where the direct product of the presentations of the two interacting states Fy,xFy. coincides with the representation of at least one of the dipole moment Cartesian components. For a fundamental transition = 1, Vk = 0) the above requirement concerns the irreducible representation of die excited level (V). The selection rule for such transitions is simply... 

If this definition of is extended to the foil soft subspace, by letting a,b = 1in Eq. (A.42), then all elements of for which aoib corresponds to a component of the central position may be shown to vanish. Correspondingly, in the Cartesian representation, it is easily confirmed that... 

Exercise 11.7 For each nonnegative integer , decompose the representation of S U (F) on 0 mro a Cartesian sum of its irreducible components. Conclude that this representation is reducible. Is there a meaningful physical consequence or interpretation of this reducibility ... 

In the discussion of light polarization so far the Cartesian basis and spherical basis have been considered. Because the linear polarization might be tilted with respect to the (ex, e -basis, a third basis system has to be introduced against which such a tilted polarization state can be measured via its non-vanishing components. This coordinate system is called (e e and its axes are rotated by +45° with respect to the previous ones. This leads to a third representation of the arbitrary vector b ... 

Here the dot indicates contraction over Cartesian coordinates, tK is a component of the vector tjf(r) = (r — rK)/ATrsf r — rK 3, and M is the 3 x 3 block, corresponding to the Tfth and 7th atoms, of the so-called relay matrix, which gives an atomic representation of the molecular polarizability. The supermatrix M has dimension 3Na x 3Na, and is defined as ... 

The first step in the symmetry determination of the dynamic properties is the selection of the appropriate basis. Appropriate here means the correct representation of the changes in the properties examined. In the investigation of molecular vibrations (Chapter 5), either Cartesian displacement vectors or internal coordinate vectors are used. In the description of the molecular electronic structure (Chapter 6), the angular components of the atomic orbitals are frequently used... 

Fig. 2.11. Two-dimensional representations of the structural ensemble observed in the unfolding trajectories, which were mapped onto the two largest principal components in the 17-dimensional segmental Q-coordinates (a) and in the hyperspace of the Cartesian coordinates (b) [25]. Reproduced with permission from [25]...

Figure 2.7-6 A Assignment of the Cartesian coordinate axes and the symmetry operations of a planar molecule of point group C2,.. B Character table, 1 symbol of the point group after Schoen-flies 2 international notation of the point group 3 symmetry species (irreducible representations) 4 symmetry operations 5 characters of the symmetry operations in the symmetry species +1 means symmetric, -1 antisymmetric 6 x, y, z assignment of the normal coordinates of the translations, direction of the change of the dipole moment by the infrared active vibrations, R, Ry, and R stand for rotations about the axes specified in the subscript 7 x, xy,. .. assign the Raman active species by the change of the components of the tensor of polarizability, aw, (Xxy,. ...

For a molecule that has little or no symmetry, it is usually correct to assume that all its vibrational modes are both IR and Raman active. However, when the molecule has considerable symmetry, it is not always easy to picture whether the molecular dipole moment and polarizability will change during the vibration, especially for large and complex molecules. Fortunately, we can easily solve this problem by resorting to simple symmetry selection mles. The molecular vibration is active in IR absorption if it belongs to the same representation as at least one of the dipole moment components fjix, iiy, jj z) or, since the dipole moment is a vector, as one of the Cartesian coordinates (x, y, z). In contrast, the molecular vibration is active in Raman scattering if it belongs to the same representation as at least one of the polarizability components, etc.) or, since the polarizability is a tensor, as... 

The quantities relevant to the rotationally averaged situation of randomly oriented species in solution or the gas phase must necessarily be invariants of the rotational symmetry. Accordingly, they must transform under the irreducible representations of the rotation group in three dimensions (without inversion), R3, just like the angular momentum functions of an atom. The polarisability, po, is a second-rank cartesian tensor and gives rise to three irreducible tensors (5J), (o), a(i),o(2), corresponding in rotational behaviour to the spherical harmonics, with / = 0,1,2 respectively. The components W, - / < m < /, of the irreducible tensors are given below. 

The next example illustrates the different representation of normal and wavelet-transformed RDF descriptors. The first 50 training compounds were selected from a set of 100 benzene derivatives. The remaining 50 compounds were used for the test set. Compounds were encoded as low-pass filtered D20 Cartesian RDF descriptors, each of a length of 64 components, and were divided linearly into eight classes of mean molecular polarizability between 10 and 26 AT... 

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