Angular momentum nuclear rotational

For absorption by a diatomic molecule to produce an autoionizing state, the selection rule for the quantum number J associated with total angular momentum is AJ = 0, 1 with A J = 0 not allowed for transitions between two Z states. " In the process of autoionization there is a strong tendency for the angular momentum of rotation of the nuclear framework to remain unchanged. Thus there is usually little or no change in rotational... 

Electron orbital angular momentum Electron spin angular momentum Resultant of orbital and spin momenta Molecular rotational angular momentum Nuclear spin angular momentum... 

Initially, we neglect tenns depending on the electron spin and the nuclear spin / in the molecular Hamiltonian //. In this approximation, we can take the total angular momentum to be N(see (equation Al.4.1)) which results from the rotational motion of the nuclei and the orbital motion of the electrons. The components of. m the (X, Y, Z) axis system are given by ... 

The quantum numbers tliat are appropriate to describe tire vibrational levels of a quasilinear complex such as Ar-HCl are tluis tire monomer vibrational quantum number v, an intennolecular stretching quantum number n and two quantum numbers j and K to describe tire hindered rotational motion. For more rigid complexes, it becomes appropriate to replace j and K witli nonnal-mode vibrational quantum numbers, tliough tliere is an awkw ard intennediate regime in which neitlier description is satisfactory see [3] for a discussion of tire transition between tire two cases. In addition, tliere is always a quantum number J for tire total angular momentum (excluding nuclear spin). The total parity (symmetry under space-fixed inversion of all coordinates) is also a conserved quantity tliat is spectroscopically important. 

The permutational symmetry of the rotational wave function is determined by the rotational angular momentum J, which is the resultant of the electronic spin S, elecbonic orbital L, and nuclear orbital N angular momenta. We will now examine the permutational symmetry of the rotational wave functions. Two important remarks should first be made. The first refers to the 7 = 0 rotational... 

When the molecule is not in a S state there is an interaction between the rotation of the molecule and S and/or L, and the details of coupling the angular momenta are involved. Most nonsinglet molecules with electronic orbital angular momentum A = 0 obey Hund s case (b) coupling. In Case (b), the electronic orbital angular momentum combines with the nuclear orbital angular... 

Figure 5.12 shows the J= — 0 transition of the linear molecule cyanodiacetylene (H—C=C—C=C—C=N) observed in emission in Sagittarius B2 (Figure 5.4 shows part of the absorption spectrum in the laboratory). The three hyperfine components into which the transition is split are due to interaction between the rotational angular momentum and the nuclear spin of the nucleus for which 1= 1 (see Table 1.3). The vertical scale is a measure of the change of the temperature of the antenna due to the received signal. 

We consider a nuclear wave function describing collisions of type A + BC(n) AC(n ) + B, where n = vj, k are the vibrational v and rotational j quantum numbers of the reagents (with k the projection of j on the reagent velocity vector of the reagents), and n = v, f, k are similarly defined for the products. The wave function is expanded in the terms of the total angular momentum eigenfunctions t X) [63], and takes the form [57-61]... 

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. 

Closed shell molecules in the rotational ground state have no net magnetic dipole moment m apart from nuclear magnetic dipole moments. However, when molecules rotate with angular momentum /, they acquire a net rotational magnetic moment... 

As a consequence of the transformation, the equation of motion depends on three extra coordinates which describe the orientation in space of the rotating local system. Furthermore, there are additional terms in the Hamiltonian which represent uncoupled momenta of the nuclear and electronic motion and moment of inertia of the molecule. In general, the Hamiltonian has a structure which allows for separation of electronic and vibrational motions. The separation of rotations however is not obvious. Following the standard scheme of the various contributions to the energy, one may assume that the momentum and angular momentum of internal motions vanish. Thus, the Hamiltonian is simplified to the following form. 

The total vibrational energy is a sum of energies of 3N-6 distinct harmonic oscillators. Indeed, 3N-6 is the final number of coordinates in Equation 8. Namely, the number of 3N+3 coordinates of the initial equation has been reduced by three through the elimination of internal rotations. Furthermore, the equation of nuclear motion (mainly its potential) has to be invariant under rotations and translations of a molecule as a whole (which is equivalent to the momentum and angular momentum preservation laws). The latter requirement leads to a further reduction of the number of coordinates by six (five in the case of linear molecules for which there are only two possible rotations). 

The first line in this expression describes the rotational structure with color spin-doubling and the hyperflne interaction of the effective electron spin S with the nuclear spin I. B is the rotational constant, J is the electron-rotational angular momentum, A is the o -doubling constant. The second line describes the interaction of the molecule with the external fields B and E, (A is the unit vector directed from the heavy nucleus to the light one). The last line corresponds to the P-odd electromagnetic interaction of the electrons with the anapole moment of the nucleus described by the constant /ca [40], P,T-odd interaction of the electron EDM de with the interamolecular field, and P,T-odd scalar interactions of the electrons with the heavy nucleus [90]. 

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