Body-fixed frame

The electronic Hamiltonian and the comesponding eigenfunctions and eigenvalues are independent of the orientation of the nuclear body-fixed frame with respect to the space-fixed one, and hence depend only on m. The index i in Eq. (9) can span both discrete and continuous values. The q ) form... 

Let be a body-fixed frame IX, whose axes are the principal axes of... 

Applications of the theory described in Section III.A.2 to malonaldehyde with use of the high level ab initio quantum chemical methods are reported below [94,95]. The first necessary step is to define 21 internal coordinates of this nine-atom molecule. The nine atoms are numerated as shown in Fig. 12 and the Cartesian coordinates x, in the body-fixed frame of reference (BF) i where n= 1,2,... 9 numerates the atoms are introduced. This BF frame is defined by the two conditions. First, the origin is put at the center of mass of the molecule. 

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... 

Molecular multipole components are best described in the molecular (i.e., body-fixed) frame x, y, z. For example, a dipole when aligned with the z-axis is characterized by a single number, the strength p of the dipole. The other two independent components can then simply be expressed in terms of Euler angles or, in this case, of azimuthal and polar angle, q> and 3, between molecular (x, y, z) and laboratory-fixed frame (X, Y, Z). 

For molecules with inversion symmetry, like H2, the expansion parameter X must be even, Eqs. 4.15 through 4.17. (It also must be non-negative.) In order to relate the expansion coefficient Axl to the Cartesian dipole components calculated in a body-fixed frame, we choose the unit separation vector, R, to be parallel to the z-axis, hence M = 0, Yw = [(2L + 1)/4k] /2, and... 

Following the quenching process further, the system is described best in the body-fixed frame, at least for the smaller internuclear distances. Then no orbital angular momentum can be transferred to the relative internuclear motion lz = 0, and the question arises as to how the electronic orbital angular momentum Lz = 1 of the 3p II) state of Na should be disposed of since the final state has to be 3s 2). The obvious solution is to change the orientation of the molecular angular momentum j, that is, to induce a Aiz= 1 transition in the molecule and thus maintain a constant... 

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... 

This implies that three electrons are in the two pn orbitals with lobes perpendicular to the internuclear O-H axis which serves as the 2-axis of the body-fixed frame of the diatom (not to be confused with the body-fixed system of the triatomic molecule). Following Andresen et al. (1984) we assume that two of the pit electrons are paired and one is unpaired, the latter determining the open-shell character of the OH radical. For a more refined analysis see Alexander and Dagdigian (1984). 

Fig. 1.7. Two-dimensional Api2-0i2 correlation map for the three-body Coulomb explosion of CS + at a field intensity of 0.2PW/cm2. A logarithmic intensity scale is used to emphasize weak features. The solid curve represents the trajectory obtained from classical mechanical calculations for the sequential explosion pathway. The open circles represent the values for five rotational angles,

Here, Aftot is the total mass of the nuclei. Note that when eliminating the motion of the center-of-mass we arbitrarily eliminated the first nuclear coordinate (we could eliminate any single coordinate y). Equation (1-13) can further be transformed to the body-fixed frame. When applying the so-called Eckart conditions35 one would get the standard Watson s Hamiltonian describing the nuclear motions in molecules36. 

In this contribution, within the asymptotic approach, we have elaborated the basis for the calculation either of adiabatic channel potentials (diagonalization of the full Hamiltonian in a body-fixed frame at given interfragment distances) or of axially-nonadiabatic channel potentials (diagonalization of the full Hamiltonian in a space-fixed frame at given interfragment distances). As a by-product, we have compared our asymptotic PES on different levels of approximations with available local ab initio data. In subsequent work, we envisage the calculation of low temperature rate constants for complex-formation of the title reactions. 

D] a, P, y) depends on the Euler angles that rotate the spaced-fixed frame on to the body-fixed frame and are eigenfuetions of. The body-fixed internal angular momentum funetion in Eq. (21) is given by... 

Euler Angles. In the three-dimensional space, the rotor R can be parameterized by three scalar parameters. The Euler angles , 0, and y are popular among physicists and astronomers. The laboratory-fixed frame m,U2,u3 can be made to coincide with the body-fixed frame ii, . ip. ir( by three successive rotations... 

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