Total angular momentum basis

The total angular momentum basis is thus computationally more efficient, even for collision problems in external fields. There is a price to pay for this. The expressions for the matrix elements of the collision Hamiltonian for open-shell molecules in external fields become quite cumbersome in the total angular momentum basis. Consider, for example, the operator giving the interaction of an open-shell molecule in a 51 electronic state with an external magnetic field. In the uncoupled basis (8.43), the matrix of this operator is diagonal with the matrix elements equal to Mg, where is the projection of S on the magnetic field axis. In order to evaluate the matrix elements of this operator in the coupled basis, we must represent the operator 5 by spherical tensor of rank 1 (Sj = fl theorem [5]... 

The parity-adapted BF total angular momentum basis fimctions Yj (R, f, r2) in... 

The TD wavefunction satisfying the Schrodinger equation ih d/dt) F(f) = // (/,) can be expanded in a basis set whose elements are the product of the translational basis of R, vibrational wavefunctions for r, r2, and the body-fixed (BF) total angular momentum eigenfunctions as41... 

To illustrate the first point concerning a spectator bond for the abstraction reaction, Fig. 17 shows the total reaction probability for the abstraction reaction as a function of the translational energy for total angular momentum J = 0 on the YZCL2 PES with the H20 reactant in the ground rovibrational state [the (00)(0) state in the local mode notation], where the uncleaved bond OHb is treated in various ways. Using a limited number of one or five vibrational basis functions, VBF(OHb) = 1 or 5, means that the OHb bond is unreactive, a spectator. The abstraction reaction probability... 

If in the non-BO calculation one chooses a basis set of eigenfunctions of the operator representing the square of the total angular momentum and the... 

Nonadiabatic Variational Energies for 15 States of the H2 Molecule with Zero Total Angular Momentum (the Ground Rotational States) Obtained with 3000 Basis Functions for Each State and Expectation Values of the Internuclear Distance and the Square of the Internuclear... 

For a total angular momentum J, the Hamiltonian of Eq. (8) is represented in a set of basis functions consisting of linear combinations of... 

The angular momenta appearing in Eq. (8.43) can all be coupled to form a total angular momentum and the direct product states (8.43) can be transformed to the total angular momenrnm basis by a unitary transformation... 

The eigenvalues and eigenfunctions of the orbital angular momentum operators can also be derived solely on the basis their commutation relations. This derivability is particularly attractive because the spin operators and the total angular momentum obey the same commutation relations. 

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