Angular momenta total

Total orbital angular momentum Total spin angular momentum... 

Angular momenmm, linear momentum and energy are all subject to universal conservation laws such that in any interaction the total angular momentum, total linear momentum, and total energy, before and after the event, remain unchanged. 

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... 

The "favourable" behaviour is most interesting in that the strong alignment is not due to kinematic constraints. In a A + BC(J) -> AB(J ) + C reaction where 1 and are the reactant and product rotational angular momenta, conservation of total angular momentum, total requires that ... 

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

The Flamiltonian commutes widi the angular momentum operator as well as that for the square of the angular momentum I . The wavefiinctions above are also eigenfiinctions of these operators, with eigenvalues tndi li-zland It should be emphasized that the total angular momentum is L = //(/ + )/j,... 

We consider an isolated molecule in field-free space with Hamiltonian //. We let Pbe the total angular momentum operator of the molecule, that is... 

We hope that by now the reader has it finnly in mind that the way molecular symmetry is defined and used is based on energy invariance and not on considerations of the geometry of molecular equilibrium structures. Synnnetry defined in this way leads to the idea of consenntion. For example, the total angular momentum of an isolated molecule m field-free space is a conserved quantity (like the total energy) since there are no tenns in the Hamiltonian that can mix states having different values of F. This point is discussed fiirther in section Al.4.3.1 and section Al.4.3.2. 

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. 

In addition to affecting the number of active degrees of freedom, the fixed n also affects the iinimolecular tln-eshold E in). Since the total angular momentum j is a constant of motion and quantized according to... 

Regardless of the nature of the intramolecular dynamics of the reactant A, there are two constants of the motion in a nnimolecular reaction, i.e. the energy E and the total angular momentum j. The latter ensures the rotational quantum number J is fixed during the nnimolecular reaction and the quantum RRKM rate constant is specified as k E, J). 

Skokov S and Bowman J M 1999 Variation of the resonance width of HOCI (6uq ) with total angular momentum comparison between ab /M/o theory and experiment J. Chem. Phys. 110 9789-92... 

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]). 

Figure B3.4.6. Reaction probabilities for the initial-state-selected process H2(v = 0,J = 0)+OH(v,y = 0) — H2O+H, for zero total angular momentum. Taken from [75] with pennission.

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. 

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