Orbital angular momentum total

S, L, and J are the total spin angular momentum, total orbital angular momentum, and total angular momentum, respectively. In a Ln -doped insulator, the coordination of anions aroimd the Ln cation produces an electrostatic field having a symmetry lower than spherical. As a result, the 2J + l)-fold degeneracy of the multiplets in the free ion case is partially or completely... 

Ion f electron Angular momentum Total orbital Term associated... 

Fig. 2-10.—The interaction of Bpin angular momentum and orbital angular momentum to form the total angular momentum for the states 2Z>j and 2D .

Each electronic state is defined by three angular momenta total orbital angular momentum vector L, total spin angular momentum vector S, and total angular momentum vector J. These vectors have the following definitions ... 

Total orbital angular momentum Total spin angular momentum... 

J,S,L total angular momentum, total spin, total orbital momentum, respectively Xo temperature-independent susceptibility... 

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 ... 

There are significant differences between tliese two types of reactions as far as how they are treated experimentally and theoretically. Photodissociation typically involves excitation to an excited electronic state, whereas bimolecular reactions often occur on the ground-state potential energy surface for a reaction. In addition, the initial conditions are very different. In bimolecular collisions one has no control over the reactant orbital angular momentum (impact parameter), whereas m photodissociation one can start with cold molecules with total angular momentum 0. Nonetheless, many theoretical constructs and experimental methods can be applied to both types of reactions, and from the point of view of this chapter their similarities are more important than their differences. 

The simplest case arises when the electronic motion can be considered in temis of just one electron for example, in hydrogen or alkali metal atoms. That electron will have various values of orbital angular momentum described by a quantum number /. It also has a spin angular momentum described by a spin quantum number s of d, and a total angular momentum which is the vector sum of orbital and spin parts with... 

The simplest case is a transition in a linear molecule. In this case there is no orbital or spin angular momentum. The total angular momentum, represented by tire quantum number J, is entirely rotational angular momentum. The rotational energy levels of each state approximately fit a simple fomuila ... 

For high rotational levels, or for a moleeule like OFI, for whieh the spin-orbit splitting is small, even for low J, the pattern of rotational/fme-stnieture levels approaehes the Flund s ease (b) limit. In this situation, it is not meaningful to speak of the projeetion quantum number Rather, we first eonsider the rotational angular momentum N exelusive of the eleetron spin. This is then eoupled with the spin to yield levels with total angular momentum J = N + dand A - d. As before, there are two nearly degenerate pairs of levels assoeiated... 

One current limitation of orbital-free DFT is that since only the total density is calculated, there is no way to identify contributions from electronic states of a certain angular momentum character /. This identification is exploited in non-local pseudopotentials so that electrons of different / character see different potentials, considerably improving the quality of these pseudopotentials. The orbital-free metliods thus are limited to local pseudopotentials, connecting the quality of their results to the quality of tlie available local potentials. Good local pseudopotentials are available for the alkali metals, the alkaline earth metals and aluminium [100. 101] and methods exist for obtaining them for other atoms (see section VI.2 of [97]). 

P, Jy, and J , are the components of the total orbital angular momentum J of the nuclei in the IX frame. The Euler angles a%, b, cx appear only in the P, P and P angular momentum operators. Since the results of their operation on Wigner rotation functions are known, we do not need then explicit expressions in temis of the partial derivatives of those Euler angles. 

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