Angular rotational momentum vector

Figure 5.5 The rotational angular momentum vector P for (a) a linear molecule and (b) the prolate symmetric rotor CH3I where is the component along the a axis...

It depends on both the angle a of the angular momentum vector rotation and other Euler angles F and q, which determine the molecule s axis shift. Besides, the angle F is also the azimuth of the change in angular momentum A/ = J(t + 0) — J(t — 0), which is the result of collision. 

In addition to CO(v = 0—2,7) populations, Houston and Kable recorded CO Doppler profiles to measure the translational energy release, and the vector correlation between the recoil velocity vector and the angular momentum vector of CO. Together, these data paint a compelling picture that two pathways to CH4 + CO are operative. The rotationally hot CO population (85% of total CO)... 

After the separation of the kinetic energy operator due to the center-of-mass motion from the Hamiltonian, the Hamiltonian describes the internal motions of electrons and nuclei in the system. These in the BO approximation can be separated into the vibrational and rotational motions of the nuclear frame of the molecule and the electronic motion that only parametrically depends on the instantenous positions of the nuclei. When the BO approximation is removed, the electronic and nuclear motions become coupled and the only good quantum numbers, which can be used to quantize the stationary states of the system, are the principle quantum number, the quantum number quantizing the square of the total (nuclear and electronic) squared angular momentum, and the quantum number quantizing the projection of the total angular momentum vector on a selected direction (usually the z axis). The separation of different rotational states is an important feamre that can considerably simplify the calculations. 

Equation (5.2) is a combination of the two two-dimensional Hamiltonians (2.39) and (3.15) which describe the vibrational and rotational excitations of BC separately. The Jacobi coordinates R, r, and 7 are defined in Figures 2.1 and 3.1 and P and p denote the linear momenta corresponding to R and r, respectively, j is the classical angular momentum vector of BC and 1 stands for the classical orbital angular momentum vector describing the rotation of A with respect to BC. For zero total angular momentum J=j+l = 0we have 1 = — j and the Hamilton function reduces to... 

The vector of the electromagnetic field defines a well specified direction in the laboratory frame relative to which all other vectors relevant in photodissociation can be measured. This includes the transition dipole moment, fi, the recoil velocity of the fragments, v, and the angular momentum vector of the products, j. Vector correlations in photodissociation contain a wealth of information about the symmetry of the excited electronic state as well as the dynamics of the fragmentation. Section 11.4 gives a short introduction. Finally, we elucidate in Section 11.5 the correlation between the rotational excitation of the products if the parent molecule breaks up into two diatomic fragments. 

Fig. 11.11. Schematic illustration of orientation and alignment of the fragment rotational angular momentum vector in terms of the distribution P(mj) where mj is the projection or magnetic quantum number, i.e., the eigenvalue of the 2-component of j.

A further interesting result concerns spin component conservation. Each rotational level of this 2S state has two spin components, one (termed F ) with the spin and rotational angular momentum vectors parallel, and the other (F2) having them antiparallel. Excitation of an F component results in transfer primarily to F- components of other rotational levels, and similarly for F2 excitation. In a collision, the OH would as soon exchange several hundred cm" of energy as flip its spin around at no energy cost. Similar results were also observed in the fluorescence scans of the NH molecule (16). 

When molecules arrive at the state with rotation quantum number J" from the state J in the process of spontaneous radiation at rate (see Fig. 3.14), a photon possessing unit spin is emitted in an arbitrary direction. Let us assume that the angular momentum carried away by the emitted photon is small, as compared with both 3 [ and Jr. This means that the angular momentum vector of each separate molecule does not change its value and does not turn in space as a result of the spontaneous transition. Consequently, the angular momenta distribution pji(0,ip) is... 

It has the same form as the cartesian components and the solution, = ke tmf describes rotation about the polar axis in terms of the orbital angular momentum vector LZ1 specified by the eigenvalue equation... 

Figure 2.15 Classical circular orbits of p-electrons with angular momentum Hm in the direction of z. The orbit (shown shaded) rotates with the perpendicular angular momentum vector to sweep out the surfaces shown on the right.

Fig. 1.11. Angular distributions of the fragment ions for the two-body Coulomb explosion of CH3CN2+ through the an = 0, bn = l,cn = 2 pathways derived from a thin slice (5pz = 16 x 103amum/s) of the momentum vector distribution at pz 0. The results from the least-squares fit [32] in which the effect of molecular rotation is taken into account are shown for comparison (solid lines). The fit was improved for the n = 2 pathways (c) when the effect of fragment ejection along the tilted direction with respect to the molecular axis is included (dashed lines)...

The optimization procedure yields a set of coefficients a,-. Of considerable interiasff is the question of whether these coefficients merely define a new vector that is simply a vector in a rotated coordinate system. If so, this would indicate that the optimqifl solution corresponds to a simple classical reorientation of the di atomic-moleciilee angular momentum vector. Examination of the optimal results [238] indicate " this is not the case. That is, control is the result of quantum interference effects ... 

Orientational hindrance does not occur only for molecules with angular momentum vectors parallel to the surface, i.e. cartwheel rotating molecules. The PES topography also depends strongly on the azimuthal orientation of the molecular bond. For example, in the H2/Cu(l 11) system the lowest dissociation barrier occurs at the bridge site when the H atoms are directed towards adjacent 3-fold hollow sites, but the barrier is substantially higher if the molecule is rotated 90° so that the atoms are directed towards atop sites. In this case, there is also substantial orientational hindrance of dissociation for helicopter states (those with angular momentum perpendicular to the surface). 

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