Body-fixed coordinates

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 form of the scattering wavefunction has been represented in a very general manner, specifically, as a product of a function in body-fixed coordinates the parity-adapted total angular momenrnm... 

After making these adjustments to allow for the fact that the analysis line cannot be located in the region of space where the centrifugal coupling in the body-fixed coordinates is negligible, and also for the fact that the analysis of Ref. 75 did not account for the long-range analytic form of the spherical Bessel functions, the space-fixed S matrix of Eq. (4.47) must be transformed back to the body-fixed axes and Eq. (4.46) must be used to compute the state-to-state differential cross sections [136,160]. 

For an atom-diatom collision the initial wavepacket, in body-fixed coordinates, is defined as [47,133]... 

No one wants to work with 18 -y symbols. There are two work-arounds for the numerical implementation of the coupled channel theory in the total angular basis. First - as suggested by Tscherbul and Dalgarno [21] - one can use a basis of angular momentum states defined in the body-fixed coordinate frame. For example, for two molecules in a S electronic state with nonzero electron spin, the eigenstates of the full Hamiltonian can be written as... 

These combinations represent the case when the body fixed coordinate frame has its axes parallel to the cube faces [374]. Since only the linear combinations just mentioned may appear in the expansion of the dipole moment, relations between the B components of the dipole moment which differ in their o values are seen seen to be [141]... 

The intramolecular structure factor Iin,ra>M given in body-fixed coordinates (M) can be written in terms of atomic scattering factors and the atomic distance distribution, yielding... 

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... 

Let us apply the above general formalism to two simple examples that are central to this book chapter, namely that of a bulk fluid and a fluid confined to a slit-pore (see Sections 1.3.2 and 1.3.3). In both cases, we take as the reference system a rectangular prism of volume Vo = SxoSyoSzo, where a body-fixed coordinate system is employed such that the faces of the prism coincide with the planes x = ,Sxo/2, y = Syo/2, and = .Szo/2. If the rmstrained system is exposed to an infinitesimally small compressional or shear strain, Vo —> 1/ = SxSy z- This implies that a mass element originally at a point To in the unstrained. system changes position to a point r in the strained system. 

Hamiltonian and Complex-Coordinate Coupled-Channel Formulation in Body-Fixed Coordinates (23). In the BF fr jne, the Hamiltonian is identical to equation 3 except that R and r are expressed relative to the unprlmed axes of figure 1 and the angular momentum operator of the rotation of R (i.e. X) is written as X = J-j. The operator (J-J)2 may be expressed as... 

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. 

Now consider the case where 7)/ is a tensor in the laboratory-fixed coordinate frame. Then the spherical components in Eq. (7.C.13) are also in the laboratory frame, and we denote this by writing these elements as Tf (L). The elements Tf (L) that appear in Eq. (7.C.13) can be related through Eq. (7.C.8) to the spherical components in the molecule or body-fixed coordinate system... 

To remove the operator dependence of the C , while maintaining the simple form of H given in Eq. (19), we reexpress the rotation operators in terms of the raising and lowering operators for two coupled, degenerate harmonic oscillators. Following Schwinger (75), in a body-fixed coordinate system, the rotation operators are redefined... 

Let us start with the Hamiltonian H of a diatomic molecule, which is given in the body-fixed coordinate system as (the mass polarization and relativistic effects are disregarded)... 

For a complete treatment of a laser-driven molecule, one must solve the many-body, multidimensional time-dependent Schrodinger equation (TDSE). This represents a tremendous task and direct wavepacket simulations of nuclear and electronic motions under an intense laser pulse is presently restricted to a few bodies (at most three or four) and/or to a model of low dimensionality [27]. For a more general treatment, an approximate separation of variables between electrons (fast subsystem) and nuclei (slow subsystem) is customarily made, in the spirit of the BO approximation. To lay out the ideas underlying this approximation as adapted to field-driven molecular dynamics, we will consider from now on a molecule consisting of Nn nuclei (labeled a, p,...) and Ne electrons (labeled /, j,...), with position vectors Ro, and r respectively, defined in the center of mass (rotating) body-fixed coordinate system, in a classical field E(f) of the form Eof t) cos cot). The full semiclassical length gauge Hamiltonian is written, for a system of electrons and nuclei, as [4]... 

The translation motion of the whole system of interacting particles can be described by the motion of its center-of-mass in respect to a body-fixed coordinate system This will be a free (inertial) motion with a constant velocity as far as the collision complex can be considered as an isolated system Such is approximately the situation during a collision in a dilute gas where, because of the large intermolecular distances, the interactions of the collision complex with the other molecules may be neglected. As is known from classical mechanics, the free center-of-mass motion can be completely separated from the internal motions, which can then be described in a coordinate system having its origin in the center-of-mass. In quantum mechanics a similar separation is possible by a product representation of the wave function... 

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