Body-fixed representation

The theory behind body-fixed representations and the associated angular momentum function expansions of the wavefunction (or wave packet) in terms of bases parameterized by the relevant constants of the motion and approximate constants of the motion is highly technical. Some pertinent results will simply be stated. The two good constants of the motion are total angular momentum, J, and parity, p = +1 or 1. An approximate constant of the motion is K, the body-fixed projection of total angular momentum on the body-fixed axis. For simplicity, we will restrict attention to the helicity-decoupled or centrifugal sudden (CS) approximation in which K can be assumed to be a constant of the motion. In terms of aU its components, and the iteration number k, the real wave packet is taken to be [21]... 

The most well-known and dramatic manifestation of an INR is the appearance of a narrow feature in the integral cross-section (ICS), cr(E) at total energy E = Er of width T. Obviously the resonance peak is closely related to the existence of the resonance pole in the S-matrix. Using the normal body-fixed representation for an A + BC v,j) — AB(v, j ) + C reaction, the ICS is related to the S-matrix by... 

In order to transform to the body-fixed representation, we will need to relate the angular functions Wj (R,r) to angular functions defined relative to the body-fixed axes [L., J,K,M,p)QjK ), where J,K,M,p) are the parity-adapted total angular momentum eigenfunctions of Eq. (4.5) and x(0) normalized associated Legendre polynomials of the body-fixed Jacobi angle]. 

M is the projection of J on the space-fixed z-axis, 0 its projection on the body-fixed z-axis, which is chosen here along the r vector. The D Ijq are Wigner matrices and are angular functions in the coupled BF representation. 

In this work we use an adiabatic electronic representation, and Jacobi nuclear coordinates are chosen r, the HE internuclear vector, and R, the vector joining the HE center-of-mass to the Li atom, in a body-fixed frame in which the three atoms lie on the a — body-fixed plane, with R being parallel to the body-fixed... 

The projection of the electronic orbital angular momentum is neglected in this adiabatic representation, and the parity of the electronic function under reflection through the x — z body-fixed plane, (Txz, is given by... 

This work introduced the concept of a vibronic R-matrix, defined on a hypersurface in the joint coordinate space of electrons and intemuclear coordinates. In considering the vibronic problem, it is assumed that a matrix representation of the Schrodinger equation for N+1 electrons has been partitioned to produce an equivalent set of multichannel one-electron equations coupled by a matrix array of nonlocal optical potential operators [270], In the body-fixed reference frame, partial wave functions in the separate channels have the form p(q xN)YL(0, radial channel orbital function i/(q r) and antisymmetrized in the electronic coordinates. Here 0 is a fixed-nuclei A-electron target state or pseudostate and Y] is a spherical harmonic function. Both and i r are parametric functions of the intemuclear coordinate q. It is assumed that the target states 0 for each value of q diagonalize the A-electron Hamiltonian matrix and are orthonormal. 

Assuming the IMU to be stationary with respect to the earth, the components of these vectors are measured by the accelerometers and gyros, respectively. Let us denote the body fixed orthonormal base vectors of the IMU by ef. For the sake of convenience, we choose as the ideal computational frame the local horizontal frame, denoting its base vectors by ef. Thus we have the following representations of the two vectors ... 

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... 

Figure 5.1. Schematic representation of a diatomic molecule, consisting of atoms A and B, in the body-fixed molecular frame where the origin is taken to be the center of mass and the z axis is along the principal axis of the molecule, za and zb denote the z coordinates of atoms A and B, respectively, and Zab the bond length.

Figure 5.13. Schematic representation of the three-site SPC/E water model. Indicated are the principal axes of the body-fixed reference frame (the x axis being normal to the molecular plane) and the coordinates of constituent atoms. The values of the parameters are zo = —0.0646 A, yu = 0.8165 A, zn = 0.5127 A, 9 = 54.74 °.

Here j is the total (rotation-plus-electronic) angular momentum of the molecule, and m and m are the projections of j on the laboratory and body-fixed axes, respectively. Using the total j angular momentum, rather than just the molecular rotation n, marks the use of a Hund s case a representation, rather than the Hund s case b that was implicit in the previous section [2]. 

Claim 4.1.1 A field t described by the Euler equations of motion of a rigid body fixed in the centre of mass is tangent to the orbits 0 of the adjoint representation in the Lie algebra so(3) and is Hamiltonian on these orbits (which are homeomorphic to the spheres S ). 

The potential matrix elements in the body-fixed representation are jKJ V j KJ) =... 

The potential matrix elements in the body-fixed representation have been given by Danby, but e too complicated to reproduce here. The oflF-diagonal matrix elements of the operator J — jY in equation (25) are the same as in the atom-diatom case, and are given by equation (9) with an additional factor of... 

To complete the definition of the truncated basis set, we consider the allowed values of A", the body-fixed projection quantum number. In principle K = 0,..., J for even J - P and 1,..., J for odd J A P- With a finite basis for the Jacobi angle, however, K can not exceed min(J, 2N>il — 1). We have found that for threaction probabilities considered in the present Chapter, convergence is reached with Kmax = 2, in accord with the basis set contraction results of Zhang [35]. This rapid convergence with respect to K ax facilitates exact calculations with very modest increases in CPU time as J increases, and is one of the many useful aspects of the body-fixed representation. 

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