Euler angles/space

We consider rotations of the molecule about space-fixed axes in the active picture. Such a rotation causes the (x, y, z) axis system to rotate so that the Euler angles change... 

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. 

Figure 1. The space-fixed (ATZ) and body-fixed (xyz) frames. Any rotation of the coordinate system XYZ) to (xyz) may be performed by three successive rotations, denoted by the Euler angles (a, 3, y), about the coordinate axes as follows a) rotation about the Z axis through an angle a(0 < a < 2n), (b) rotation about the new yi axis through an angle P(0 < P < 7i), (c) rotation about the new zi axis through an angle y(0 Y < 2n). The relative orientations of the initial and final coordinate axes are shown in panel (d).

As discussed in Section II. A, the adiabatic electronic wave functions, a and / 1,ad depend on the nuclear coordinates R> only through the subset (which in the triatomic case consists of a nuclear coordinate hyperradius p and a set of two internal hyperangles this permits one to relate the 6D vector W(1)ad(Rx) to another one w(1 ad(q J that is 3D. For a triatomic system, let aIX = (a1 -. blk, crx) be the Euler angles that rotate the space-fixed Cartesian frame into the body-fixed principal axis of inertia frame IX, and let be the 6D gradient vector in this rotated frame. The relation between the space-fixed VRi and is given by... 

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]. 

Figure 3.1 Bond and intrinsic variables for triatomic molecules (a) and the Euler angles characterizing the orientation in space of the molecule (b).

As discussed in Ref. [1], we describe the rotation of the molecule by means of a molecule-fixed axis system xyz defined in terms of Eckart and Sayvetz conditions (see Ref. [1] and references therein). The orientation of the xyz axis system relative to the XYZ system is defined by the three standard Euler angles (6, (j), %) [1]. To simplify equation (4), we must first express the space-fixed dipole moment components (p,x> Mz) in this equation in terms of the components (p. py, p along the molecule-fixed axes. This transformation is most easily done by rewriting the dipole moment components in terms of so-called irreducible spherical tensor operators. In the notation in Ref. [3], the space-fixed irreducible tensor operators are... 

If we know the components of the electronically averaged dipole moment (fix, Py, Pz) in the space-fixed axis system XYZ, and we know the values of the Euler angles (d, f, x) that define the orientation of the xyz axes relative to the XYZ axes (see, for example, Ref. [3]), we can compute the xyz components (p, ff, pf) of the dipole moment from the relations... 

Any rotation in 3D space can be composed of successive rotations through the three Euler angles (see Figure 11) ... 

The first and the third of the Euler angles, rotations about z axes. The corresponding transformation matrices in 3D coordinate space read... 

The Van der Waals constants Cn(a)A, (oB, R) depend on the Euler angles (oA and (oB specifying the orientation of the monomers in an arbitrary space-fixed frame, and on the polar angles R = (fi, a) determining the orientation of the intermolecular axis (R is assumed to join the monomer centers of mass) with respect to the same space-fixed frame. 

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