SPACE-AND BODY-FIXED COORDINATE SYSTEMS

A planetoid (or molecule) moves through empty space, we observe it from our (inertial ) space ship. Tb carry out observations of the planetoid (molecule), we have to install some equipment in our space ship and to fix a Cartesian coordinate system on it. This will enable us to describe the planetoid whatever happens to it. This is the Space-Fixed Coordinate System (SFCS), its orientation with respect to distant stars does not change in time. 

If the molecule does not interact with anything, then with respect to the SFCS (see Chapter 2) 

An observer on another space ship (also inertial) will see the same phenomena in exactly the same way, the energy, momentum and angular momentum will also be invariant, but in general they will be different from what was measured in the first space ship. 

Let us introduce the vectors r, = (Xi,yi,zi) into the SFCS showing (from the origin of the coordinate stem) the particles, from which our molecule is composed (i.e. the electrons and the nuclei), / = 1,2. Af. Then, using the SFCS, we write the Hamiltonian of the system, the operators of the mechanical quantities we are interested in, we calculate all the wave functions we need, compare with spectra measured in the SFCS, etc. 

One day, however, we may feel that we do not like the SFCS, because to describe the molecule we use too many variables. Of course, this is not a sin, but only a 

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Fig. 6.5. Space- and Body-Fixed Cooidinate Systems (SFCS and BFCS). (a) SFCS is a Cartesian coordinate system arbitrarily chosen in space (left). The origin of the BFCS is located in the centre of mass of the molecule (right). The centre of mass is shown by the vector Rcm front the SFCS. The nuclei of the atoms are indicated by vectors ri,r2,rj... from the BFCS. Fig. (b) shows what happens to the velocity of atom a, when the system is rotating with the angular velocity given as vector ta. In such a case the atom acquires additional velocity ta x r . Fig. (c) shows that if the molecule vibrates, then atomic positions ta differ from the equilibrium positions a by the displacements...

Euler angles relating space-fixed and body-fixed coordinate systems. 

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,... 

It ensues from the property (11) that it is sufficient to define (r R) and n(r)> only within the domain of internal nuclear coordinates R. The replacement of R by R = Rj>, where Rj = Xj,Yj,Zj>, which results in the removal of three degrees of freedom (two for linear molecules), corresponds to adopting a rotating ("body-fixed") coordinate system in place of the fixed ("space-fixed") one. Various definitions of the former coordinate system are possible, the most natural involving the requirement that the... 

More recently KUBPERMANN, SCHATZ and BAER /77c/ developed a method for an accurate treatment of complanar collisions of an atom A with a diatomic molecule BC This method was then extended by SCHATZ and KUPPERMANN /77d/ to atom-diatom collisions in a three-dimensional physical space, making use of the "tumbling-decoupling approximation /41b/ In this treatment the collision is conveniently described in a body-fixed coordinate system. Together with the quantum number v of BC-vibration, two quantum numbers j and J are introduced for the rotation of BC-molecule and the overall rotation of the system ABC, respectively. Two corresponding qiiantum numbers m and n. are associated with the projections of the BC-angular momentum... 

In principle there are two arbitrary ingredients in this prescription. One must define rotational angles and a bend angle. The rotational angles imply a transformation from a space fixed coordinate system to a body fixed coordinate system. There are an infinity of such transformations and one must choose one arbitrarily. Similarly the bend angle is defined arbitrarily. Of course, if a posteriori we find that indeed there is a good separation of time scales then this arbitrariness is irrelevant from a practical point of view. 

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... 

Let us consider a space fixed Cartesian coordinate i stem (SFCS, see Appendix 1 on p. 971), and vector Rcm indicating the centre of mass of a molecule composed of M atoms, Fig. 6.5. Let us construct a Cartesian coordinate i stem (Body-Fixed Coordinate System, BFCS) with the origin in the centre of mass and the axes parallel to those of the SFCS (the third possibility in Appendix I). 

The function koAf is the total wave function written in the centre-of-mass coordinate system (a special body-fixed coordinate system, see Appendix I), in which the total angular momentum operators and Jz are now defined. The three operators H,J and Jz commute in any space-fixed or body-fixed coordinate system (including the centre-of-mass coordinate system), and therefore the corresponding physical quantities (energy and angular momentum) have exact values. In this particular coordinate system p = pcM = 0. We may say, therefore, that... 

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. 

We now turn from transformations and operations in spin space to the more familiar three-dimensional real Euclidean space, which describes the structure of atoms and molecules. This space is usually referred to as R3. Molecules and atoms are finite systems, unlike solids and surfaces. Their Hamiltonians are most conveniently expressed in a body-fixed coordinate system where translational motion has been separated out, and so we will not consider translations in this treatment. We will also restrict our attention to operations that do not deform the molecule, and we will therefore be dealing with transformations that preserve length. 

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).

Fig. 1. Definition of body-fixed and space-fixed coordinate systems. Refer to Table 1 and to the text for a detailed explanation...

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