Body-fixed axis system

The results simplify considerably if the body-fixed axis system is a principal axis system for the polarizability tensor as well as for the rotational diffusion tensor. In this case Ag = At = A5 = 0 in Eq. (7.5.27). Then the spectrum consists of only two Lorentzians. Many asymmetric diffusors do have enough symmetry to rigorously satisfy this condition-for instance, planar molecules with at least one two fold rotation axis in the molecular plane. Others may have these axes so close together that Ag = A4 = A5 = 0 and the spectrum effectively consists of only two Lorentzians. In any particular application, it must be kept in mind that the spectrum might very well be the five-Lorentzian form given by the Fourier transform of Eq. (7.5.25). 

Satchler" and YjK(0,(j)) is a spherical harmonic involving the angular coordinates of the diatomic molecule in the body-fixed axis system. Physically, the end-over-end angular momentum of the complex cannot have any body-fixed projection along R, so that the K quantum numbers appearing in the rotation matrix element and in the spherical harmonic must be the same. 

The coordinate system needed for an atom-nonlinear molecule complex is a straightforward generalisation of that for an atom-diatom complex. The body-fixed axis system is defined as before, with Euler angles (a,, 0) specifying the orientation of R. However, it is now necessary to define an axis system (x y z) fixed in the monomer the relationship of these axes to the body-fixed axes is specified by Euler angles (0,, x)- md 0 describe the orientation of the z axis, and x describes rotations about the 2 axis. 

There has been rather more work on the specific case of diatom-diatom systems, mostly aimed at understanding the spectrum of the HF dimer. The coordinate system used is again a generalisation of the atom-diatom coordinates, with angles (a,/ ) defining the body-fixed axis system and angles and ( 2 < 2) specifying the orientation of the two diatomic... 

The coordinate system needed for a trimeric system in this approximation is a straightforward generalisation of that for an atom-diatqm complex. The R vector is now defined as running from the HX centre of mass to the midpoint of the Ar2 pair, and forms the axis of a body-fixed axis system. An additional vector, p, runs between the two Ar atoms. The body-fixed x axis is perpendicular to R and coplanar with p. The relationship between the body-fixed and the space-fixed axes now requires three Euler angles Once again, the... 

On the left hand side is the space-fixed wavefunction. The Euler angles are the three angles required to rotate the space-fixed axes to the body-fixed axes. The wavepacket r, 7, t) is defined in the body-fixed axis system. 

We represent the four-atom problem in terms of diatom-diatom Jacobi coordinates R, the vector between the AB and CD centers of mass, and rj and r2, the AB and CD bond vectors. In a body-fixed coordinate system [19,20] with the z-axis chosen to R, only six coordinate variables need be considered, which we choose to be / , ri, and ra, the magnitudes of the Jacobi vectors, and the angles 01, 02, and (j). Here 0, denotes the usual polar angle of r, relative to the z-axis, and 4> is the difference between the azimuthal angles for ri and r2 (i.e., a torsion angle). 

The solution to this problem is to transform, or half-transform, the S matrix from the body-fixed to the space-fixed axis system then to use the known analytic properties of the spherical Bessel functions, which are the solutions to the potential-free scattering problem in the space-fixed axes and finally to transform back to the body-fixed axes and then to use Eq. (4.46) to calculate the differential cross section. 

The dipole operator d is a vector defined in the body-fixed frame of the molecule. Consequently, the transition dipole moment /a defined in (2.35) is a vector field with three components each depending — like the potential — on R, r, and 7. For a parallel transition the transition dipole lies in the plane defined by the three atoms and for a perpendicular transition it is perpendicular to this plane. Following Balint-Kurti and Shapiro, the projection of /z, which is normally calculated in the body-fixed coordinate system, on the space-fixed z-axis, which is assumed to be parallel to the polarization of the electric field, can be written as... 

Equations (1-124) and (1-133) are valid in an arbitrary space-fixed coordinate system. However, since the angular functions A A (a)A, coB, R) are invariant with respect to any frame rotation162, a specific choice of the coordinate system may considerably simplify Eq. (1-125). In particular, in the body-fixed coordinate system with the z axis along the vector R the polar angles R = (/ , a) are zero. Using the fact that (r = (0,0)) = 8Mfi 14S, one gets,... 

The orientation of a nonlinear molecule can be described by three Euler angles <, 0, j, because it takes two angles to describe the orientation of any body-fixed vector and takes one angle to describe the orientation of the body about that vector. The Euler angles relate the orientation of an orthonormal molecule-fixed axis system u), u 2, u to some standard orthonormal space-fixed frame u 1, u2,113 (see Fig. 1 and Eq. (A73) in Appendix A, Section 3.c). 

Equation (7.5.22) applies rigorously if the Z axis is a four fold or more symmetry axis of rotation. The more general result given in Eq. (7.5.19) holds if the polarizability tensor does not have cylindrical symmetry about the Z axis in the body-fixed coordinate system while the rotational diffusion tensor does. 

The absolute velocity and acceleration of G, are expressed in terms of the body-fixed coordinate system, b ba, b3, which is fixed to the pendulum and rotates about the b3 axis as before. Although it is equivalent to expressing within the inertial frame of reference, I, j, k, the body-fixed coordinate system, b bj, bj, uses fewer terms. The velocity and acceleration for G, are respectively as follows ... 

The orientation of each molecule may be specified by the three Euler angles a, and If we associate with a molecule a body-fixed coordinate system (to be denoted by primes), then the Euler angles locate the body-fixed system with respect to the space-fixed system. If the symmetry axis 0 is taken to be in the z direction, then... 

Rotating, xyz the molecule-fixed axis system. The conditions specifying how this system is attached to a nonrigid molecule are discussed in Chaps. 2 and 11. For a rigid body, these axes may be taken coincident with the principal axes of inertia. 

The programs (17,18) used to solve the ro-vibrational problems discussed here are based substantially on two theoretical developments the use of a flexible body-fixed coordinate system (19) and a two-step variational method for rotationally-excited states (12). The problem has been formulated as a generalised body-fixed Hamiltonian which allows internal coordinates and the orientation of the axis system to be input parameters in the programs. This Hamiltonian has the form ... 

The symmetiy axis is normally chosen as the quantization axis, z, in the rotating body-fixed coordinate system where the pure rotational states = E-g J, k,M)wQ degenerate in for AT = A > 0. Here, k = 0, 1, 2, J is the quantum number of the projection of the rotational angular momentum on the symmetry axis,M= 0, 1, 2,. .., J the one for the projection on the space-fixed quantization axis, Z. In a field-free environment, J, k, and M are good quantum numbers. There are only two rotational parameters to be determined in the rigid-rotor approximation A and B = C for a prolate top, and C and for an oblate one. 

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).

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