Euler angles, generalized

For the interaction between a nonlinear molecule and an atom, one can place the coordinate system at the centre of mass of the molecule so that the PES is a fiinction of tlie three spherical polar coordinates needed to specify the location of the atom. If the molecule is linear, V does not depend on <() and the PES is a fiinction of only two variables. In the general case of two nonlinear molecules, the interaction energy depends on the distance between the centres of mass, and five of the six Euler angles needed to specify the relative orientation of the molecular axes with respect to the global or space-fixed coordinate axes. 

If D is taken as a traceless tensor, Tr(/)) = Da = 0, there remain only two independent components for D (neglecting the three Euler angles for orientation in a general coordinate system). Usually, these are the parameters D and E, for the axial and rhombic contribution to the ZFS ... 

In the most general case of a completely anisotropic diffusion tensor, six parameters have to be determined for the rotational diffusion tensor three principal values and three Euler angles. This determination requires an optimization search in a six-dimensional space, which could be a significantly more CPU-demanding procedure than that for an axially symmetric tensor. Possible efficient approaches to this problem suggested recently include a simulated annealing procedure [54] and a two-step procedure [55]. 

From this expression one has to remove the center-of-mass motion. The corresponding Hamiltonian can then be rewritten in terms of the intrinsic variables and the Euler angles. The general form of the Hamiltonian operator is, in the Eckart frame,1... 

The antisymmetric tensor is generally not observable in NMR experiments and is therefore ignored. The symmetric tensor is now diagonalized by a suitable coordinate transformation to orient into the principal axis system (PAS). After diagonalization there are still six independent parameters, the three principal components of the tensor and three Euler angles that specify the PAS in the molecular frame. 

A general rotation R( o n) in ft3 requires the specification of three independent parameters which can be chosen in various ways. The natural and familiar way is to specify the angle of rotation and the direction of the unit vector n. (The normalization condition on n means that there are only three independent parameters.) A second parameterization R(a b) introduced above involves the Cayley-Klein parameters a, b. A third common parameterization is in terms of the three Euler angles a, (3, and 7 (see Section 11.7). Yet another parameterization using the quaternion or Euler-Rodrigues parameters will be introduced in Chapter 12. 

Fig. 3.9. Experimental (dotted lines) and best-model (solid lines) generalized IRSE spectra J ij of a (1120) ZnO thin film (d = 1455 5 nm) on (1102) sapphire [43,71]. Spectra are shifted for clarity. Vertical dashed lines indicate the ZnO phonon mode frequencies. Spectra in (a,d,g), (b,e,h), and (c,f,i) belong to different sample azimuth angles, respectively. The best-model values of the Euler angle 0znO, Sapphire, which describe the c-axis inclination with respect to the sample normal are 89.0° 1.0° and 54.9° 0.8°, respectively. Reprinted with permission from [71]...

Figure 5.3. The Euler angles , LX OX" = 0 and LY Oy = x-...

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... 

In general, the position of a nonlinear rigid polyatomic molecule relative to a global axis system (fixed in space) can be defined by six coordinates three to define the position of the center of mass and three (usually the Euler angles a, p, y) to define the orientation of its local or molecule-fixed axis system relative to the global axis system. The local axis system is defined by the bonds within the molecule and should be chosen to reflect any symmetry because this simplifies the model potential. For example, a water molecule with C21, symme-... 

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... 

The reciprocality condition in Eq. (120) can be used to deduce some general properties of the measuring vectors associated with the shape coordinates and Euler angles. For example, because the measuring vector e 1 = u, is the same for any nucleus a, it follows that... 

Fig. 2. Reference frames and sets of Euler angles required for the description of rotational diffusion and sample spinning in a general NMR experiment. The magnetic tensor system (top, left) is characterized by a diagonal chemical shift tensor <7. The sample system (centre, left) is defined by the axis z of sample spinning while the laboratory system is determined by the direction of the external magnetic field. For a transformation from the magnetic tensor system into the sample system, a full set of three Euler angles 3>, 0, and W is needed. For the transformation from the sample system into the laboratory system, only two Euler angles a and /3 are required as the external magnetic field is assumed to be of rotational symmetry.

Consider two rigid molecules A and B, both of arbitrary shape. Let ft = (R, Q) = (R, 0, 4>) be the vector pointing from the center of mass of A to the center of mass of B. The coordinates of ft are measured with respect to a space-fixed frame. Let the orientation of molecule A be described by the Euler angles = (a, P, y ), which are the angles associated with an (active) rotation of the molecule from an initial position in which a reference frame fixed on A is parallel to the space-fixed frame, to its present position. Similarly, the orientation of B is determined by the Euler angles b — ( b Pb> Vb)- The interaction energy between A and B is most generally... 

As an Interlude between sections, it is useful to use Eq. (7) to obtain, at once, the general expressions for the angular momenta canonically conjugate to the Euler angles, as well as their relations with the corresponding cartesian components in the body frame. It is straightforward to obtain... 

Let us suppose that the liquid system is described by a MFPKE in N + 1 rigid bodies (the solute, or body 1 and N rotational solvent modes or bodies ), each characterized by inertia and friction tensors I and a set of Euler angles ft , and an angular momentum vector L (n = 1,..., N -I-1) plus K fields, each defined by a generalized mass tensor and friction tensor and a position vector and the conjugate linear momentum k = 1,..., K). The time evolution of the joint conditional probability x", L , P° 11, X, L, P, t) (where ft, X, etc. stand for the collection of Euler angles, field coordinates etc.) for the system is governed by the multivariate Fokker-Planck-Kramers equation... 

By analogy with our treatment of translation symmetry, we aim to derive an operator /f - which, when applied to a waveflmction, describes the effect of a general symmetry operation that causes the change in the Euler angles given in (equation A1.4.117). Because of the analogy between (equation A1.4.120) and the definition of f in (equation Al.4.97). we can repeat the arguments expressed in (equation A1 4.98). 

The orientation of a general anisotropic molecule (not necessarily a rigid rod) is given by the Euler angles O (a, fS, y) which specify the orientation of the molecular body-fixed axes with respect to the space-fixed axes (see Fig. 7.4.1).9 What is required is the conditional probability distribution KS(Q, t Do) which specifies the probability distribution for the molecule to have an orientation O at time t given thaPit had an orientation Qo at time 0. This conditional distribution must satisfy the initial condition... 

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