Molecular frame of reference

Another important dynamic propaty to consider is the water molecule s reorientation. Given a vector u that is fixed in the molecular frame of reference, one is interested in the time-dependent quantities... 

The distribution of bonding, or valence, electrons is the largest contributor to the electric field gradient. In a molecular frame of reference, we can define three orthogonal components of the electric field gradient, Vxx, Vyy, and Vzz, where Vxx + Vyy + Vzz = 0 and, by convention, VXX < V l < VZZ. From these electric field gradient components, we define two parameters ... 

With this reservation, the Hamiltonian appropriate to a molecule with electronic spin quantised in the molecular frame of reference becomes... 

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... 

Considerable information on molecular interactions and molecular structure can be obtained from studies of ordered materials and samples here the degree of averaging between laboratory and molecular frames of reference is either wholly removed or greatly reduced. There is now a net degree of molecular order in the laboratory frame of reference. As in the case of probe fluorescence [see Equation (11.18)], this intrinsic distribution is conveniently expressed as an expansion in spherical harmonics. 

In Eq. (338) y is already written in the molecular frame of reference in which it is diagonalized, as illustrated in Fig. 76, with the principal axes 1, 2, and 3. Because the C2 axis perpendicular to the tilt plane has to appear in the property (Neumann principle - we use it here even for a dynamic parameter), this has to be the direction for y2. As for the other directions, there are no compelling arguments, but a natural choice for a second principal axis is along the director. We take this as the 3 direction. The remaining axis 1 is then in the tilt plane, perpendicular to n. The 3 axis is special because, along n, the rotation is supposed to be characterized by a very low viscosity. In other words, we assume that the eigenvalue is very small, i.e., yi[Pg.1633]

Both tensors are described in a different frame of reference the tensor is given in the laboratory frame of reference based on the plane of incidence, whereas the hyperpolarisability tensor in the molecular frame of reference defined by the symmetry of the molecule. An Euler transform relates both, leading to rather lengthy expressions between the corresponding tensor components, which are best handled and manipulated with the aid of a symbolic programming language such as Maple or Mathematica. 

In order to calculate the dipole correlation function, let us consider an ensemble of N identical rigid molecules, each possessing a dipole moment m. To describe the orientation of a molecule in space, two coordinate systems are introduced. The laboratory frame of reference (XYZ), which we will call LF, is traditionally defined as having the Z-axis in the direction of the probing electric field. The molecular frame of reference fixed within the molecule (xyz), which we shall refer to as mF, usually has axes chosen along the principal axes of the moment of inertia tensor (or any other molecular tensor). The orientation of the molecule is then given by the orientation of mF with respect to LF, which is determined by a set of Eulerian angles Q = a, jS, (see Fig. 4.4). The molecular dynam-... 

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