Matrix Eulerian rotation

Show also that the overall Eulerian rotation matrix E = AB C is given by... 

PROBLEM 2.4.21. If the Eulerian matrix E given by Eq. (2.4.60) must equal the rotation matrix R determined in Problem 2.4.20 above ... 

It can be shown that det C = 1 (Problem 2.13.1). The norm of C, or trace of C, or sum of its diagonal terms, is 2 + 2y. Since det C = 1, we can consider the Lorentz transformation matrix X like the four-dimensional analog of the Eulerian rotation in 3-space. We now seek quantities that are "covariant with the Lorentz transformation"—that is, are "relativistically correct". We next define in this new four-space a few essential quantities ... 

The local frame for rigid molecules, once chosen, will always be clearly defined. The necessary transformation of the separation vector from the laboratory to the local frame is usually accomplished by multiplication by the rotation matrix of the central molecule. The construction of this rotation matrix is usually a straightforward task. In fact, it will already be available in any program that describes molecular orientations in terms of quaternions (coordinate-transformed eulerian angles) [3,24]. 

Let us consider the electron-vibrational matrix element. As is usually done, we consider two coordinate systems, the origins of which are located at the center of mass of the molecule. The first coordinate system is fixed in space, while the second system (the rotational one) is fixed to the molecule. For describing the orientation of the rotational system with respect to the fixed frame we use the Eulerian angles 6 = a, / , y. In the Born-Oppenheimer approximation, the motion of nuclei may be decomposed into the vibrations of the nuclei about their equilibrium position and the rotation of the molecule as a whole. Accordingly, the wave function of the nuclei X (R) is presented as a product of the vibrational wave function A V(Q) and the rotational wave function... 

As is well known, the 3x3 matrix Oy can be diagonalized by an appropriate orthogonal coordinate transformation (rotational transformation), provided it is a symmetric matrix generally it is considered to be symmetric because of its physical meaning. If the principal-axes frame of o, where o is expressed by a diagonal matrix, is transformed to the laboratory frame by a rotational transformation R(o, /3, y) which is defined by three Eulerian angles a, /3 and y, then the representations of o in both frames are related to each other by the equation = (5)... 

Figure 12. Orientation of a solute molecule in the liquid crystal matrix. U, V, W is a space fixed axes system. The optical axis L of the homogeneously oriented liquid crystal is assumed to be parallel to W. The solute molecule fixed coordinate system x, y, z is rotated by the Eulerian angles

Starting from the Coulomb Hamiltonian, to arrive at a Hamiltonian which had an angular part of this form, a neutral and natural choice would seem to be to choose the Eulerian angles to define an orthogonal matrix C that diagonalises the nuclear inertia tensor. This yields moments of inertia and puts the Hamiltonian in principal axis form, just the form appropriate to describe a rigid rotator in classical mechanics. The principal axis approach to the molecule was attempted in classical mechanics by Eckart [Eckart, 1934] shortly before the approach [Eckart, 1935] that we have referred to in 2. It was tried too, almost simultaneously, by Hirschfelder... 

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