Euler rotation matrix

Here and are vectors of the Cartesian coordinates and velocities selected above and R(ft x) s the Euler rotation matrix. The angles 6. , x are chosen randomly according to... 

Other than linear molecules. If molecules of symmetry other than axial are considered, it is not possible to describe their orientation by an azimuthal and polar angle, Euler angles, Q = a pi, y and Wigner rotation matrices are then needed as Eq. 4.8 suggests. In that case, besides the set of parameters X, X2, A, L that has been used for linear molecules, two new parameters, u, with i = 1,2, occur that enter through the rotation matrices. These must be chosen so that the dipole moment is invariant under any rotation belonging to the molecular symmetry group. The rotation matrix is expressed as a linear combination of such... 

The macroscopic property observed in sum-frequency experiments, Xs . is a sum of the molecular hyperpolarizabilities, over all vibrational modes and all of the molecules at the interface, which takes into account the orientation of each molecule. Orientational information is obtained from the experimental spectra through consideration of the relationship between the observed Cartesian components of the macroscopic second-order susceptibility Xuk, the corresponding spectroscopically active components of the molecular hyperpolarizability, This is accomplished through an Euler angle rotation of the molecular axis system into the laboratory axis system as defined through the use of the rotational matrix iiuK imn- The general expression for the transformation from a molecular-fixed axis system to a laboratory-fixed system is... 

P indicates that the components are those for the principal axis system (PAS) of the tensor. The terms o(QpL(f)) are Wigner rotation matrix elements. They are functions of the set of Euler angles, Qpf (f), which relates the PAS of the chemical shift to the laboratory frame. Due to MAS, these angles are time dependent. A full treatment of the orientation dependence of the chemical shift requires the transformation between several different reference frames. 

A disadvantage of the Euler angle approach is that the rotation matrix contains a total of six trigonometric functions (sine and cosine for each of the three Euler angles). These trigonometric functions are computationally expensive to calculate. An alternative is to use quaternions. A quaternion is a four-dimensional vector such that its components sum to 1 0 + 1 + <72 + = 1- quaternion components are related to the Euler angles as follows ... 

The Euler angle rotation matrix can then be written... 

J. B. Marion, Classical Dynamics of Particles and Systems. Academic Press, New York, 1965. In that reference, the second Euler rotation is made about the i rather than y axis the rotation matrix correspondingly differs from ours. 

The reorientation of the molecule is described within the molecular frames of reference for both the a and b states for the forward and reverse moves, respectively, as illustrated in Figure 7. The rotation matrix R, corresponding to the Euler angles (A, 0 (i ), is... 

To develop an integrator for the complete system (including external forces), we invoke the splitting technique, solving successively for the translational motion from the center of mass equations, integrating the torques due to the rotational interactions, then solving the Euler equations, and recovering the rotation matrix from the linear system... 

There are several ways to handle mathematically this orientation g. One can define as a rotation matrix or specify a crystal plane (hkl) and a crystal direction [uvw] which are parallel to (A, B) and A, respectively. In the texture analysis field the most used description for g consists of the set of 3 angles, the Euler angles. The coincidence of the two co-ordinate systems is then achieved by three rotations, which is illustrated in the stereographic projection in Fig. 7. The notation g denotes then in fact three variables. 

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