Angular momentum coupling diatomics

In our own work on both diatomic and polyatomic molecules, we have found it valuable to have a summary of the most important results from irreducible spherical tensor algebra, particularly those relating to the evaluation of matrix elements in various angular momentum coupling schemes. We now provide a summary of those results detailed derivations are, of course, to be found in the main body of the text. 

See [15,16] for the alternative diabatic representations which correspond to the alternative Himd s angular momentum coupling cases in the spectroscopy and dynamics of a diatomic system. 

Complexes formed from atoms and linear polyatomic molecules are very similar to atom-diatom systems the coupled equations are identical, and the same angular momentum coupling schemes apply. The only added degree of complexity is that perpendicular transitions of the polyatomic monomer are possible, and these introduce an extra quantum number (/ or k) for the monomer vibrational angular momentum. Such states are analogous to those arising from A > 0 states of a symmetric top monomer, as discussed below. 

Franck-Condon factors. The most common angular momentum coupling cases are discussed, and rotational fine structure in electronic transitions (cf. Fig. 4.3) is rationalized for heteronuclear and homonuclear diatomics using Herzberg diagrams. 

Angular momentum coupling is particularly transparent for this example, as there is no electronic (spin) angular momentum for reactants and products in equation (81), One just combines two diatomic rotors with j and 72 to a channel angular momentum j in equation (84) ... 

This is, first of all, due to the manifold of rotational and vibrational levels within each electronic state and furthermore to a larger variety of angular momentum coupling, such as spin-rotation interaction, A-type doubling, fine and hyperfine structure. In addition different kinds of perturbations may further increase the line density and the complexity of the spectrum. Even for small molecules, such as diatomic or triatomic molecules, the spacings between rotational lines of an electronic transition may become much smaller than the Doppler-width. This implies that single rotational lines often cannot be resolved with "classical" Doppler-limited techniques. 

For diatomic molecules, there is coupling of spin and orbital angular momenta by a coupling scheme that is similar to the Russell-Saunders procedure described for atoms. When the electrons are in a specific molecular orbital, they have the same orbital angular momentum as designated by the m value. As in the case of atoms, the m value depends on the type of orbital. When the internuclear axis is the z-axis, the orbitals that form a bonds (which are symmetric around the internuclear axis) are the s, pz, and dzi orbitals. Those which form 7r bonds are the px, p, dlz, and dyi orbitals. The cip-y2 an(i dxy can overlap in a "sideways" fashion with one stacked above the other, and the bond would be a 8 bond. For these types of molecular orbitals, the corresponding m values are... 

Let the diatomic (or linear) molecule satisfy the conditions for the Hund s case (a) coupling scheme (see Section 1.2, Fig. 1.3(a)), where the electronic orbital and spin angular momenta are coupled with the internuclear axis. The magnetic moment which is directed along the internuclear axis and corresponds to the projection Cl of the total angular momentum J upon the internuclear axis, has the value... 

The coordinate system used in the close-coupling method is the space-fixed frame. For simplicity we consider the atom-diatom scattering. The wave function iM(.R,r,R) for an atom-rigid rotor system corresponding to the total energy E, total angular momentum J, and its projection M on the space-fixed z axis can be written as an expansion,... 

This term describes the rotational Zeeman effect, that is, the coupling between the external field and the magnetic moment of the rotating nuclei. We note that there is no corresponding vibrational contribution since R a k is zero. The physical reason for this lack is that it is not possible to generate vibrational angular momentum in a diatomic molecule because it possesses only one, non-degenerate, vibrational mode. 

To take a specific example, let us consider P = S, the electron spin angular momentum for a diatomic molecule in a Hund s case (a) coupling scheme where the basis functions are simple products of orbital, rotational and spin functions. Using standard... 

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