Uncoupled product basis

The matrix elements (8.35) in the uncoupled space-fixed basis can be most easily evaluated if all interaction operators are represented as uncoupled products of spherical tensors, with each tensor defined in the space-fixed coordinate system. Since the Hamiltonian is always a scalar operator, we can write any interaction in the Hamiltonian as a sum... 

Since the uncoupled states )i Wy(i)) and y2W (2)) h form a complete basis, the (direct) product basis built from them is a complete basis for the Hilbert space representation of the total eigenstate JMj) in accord with Eq. (4.13). However, in this product basis is not diagonal. We may expand the coupled state Mj) into the product basis of the uncoupled states... 

It would appear that identical particle pemuitation groups are not of help in providing distinguishing syimnetry labels on molecular energy levels as are the other groups we have considered. However, they do provide very usefiil restrictions on the way we can build up the complete molecular wavefiinction from basis fiinctions. Molecular wavefiinctions are usually built up from basis fiinctions that are products of electronic and nuclear parts. Each of these parts is fiirther built up from products of separate uncoupled coordinate (or orbital) and spin basis fiinctions. Wlien we combine these separate fiinctions, the final overall product states must confonn to the pemuitation syimnetry mles that we stated above. This leads to restrictions in the way that we can combine the uncoupled basis fiinctions. 

The main conclusion of this section is that the matrix elements of all terms in the collision Hamiltonian in the fully uncoupled space-fixed representation can be reduced to simple products of integrals of the type (8.46). Such matrix elements are very easy to evaluate numerically. The fiilly uncoupled representation is therefore very convenient for the development of the coupled channel codes for collision problems involving open-shell molecules with many angular momenta that need to be accounted for. The price for simplicity is a very large number of basis states that need to be included in the expansion of the eigenstates of the full Hamiltonian to achieve full basis set convergence (see Section 8.3.4). 

In order to appreciate the size of the basis sets required for fully converged calculations, consider the interaction of the simplest radical, a molecule in a electronic state, with He. The helium atom, being structureless, does not contribute any angular momentum states to the coupled channel basis. If the molecule is treated as a rigid rotor and the hyperfine structure of the molecule is ignored, the uncoupled basis for the collision problem is comprised of the direct products NMf ) SMg) lnii), where N = is the quantum number... 

We have assumed that no angular momentum contribution assists. Then the basis set of spin functions consists of the uncoupled set Sa,Msa) Sb, Msb), or the coupled set SA,SB,S,Ms), its size is N = (2Sa + l)(2Ss + 1). Additionally, the orbital angular momentum can be added and then the basis set becomes a direct product of all orbital and spin functions. In a special case, spin delocalisation (double exchange) operates. 

These states can be expressed in terms of (direct) products of uncoupled basis states I/1/U-1/2M2) tiy direct application of (3.7), giving, for example. 

The uncoupled basis is formed by the determinants la o l, a, pa and The result of acting with the isotropic part of Hamiltonian on these determinants can directly be written down with the help of Eqs. 3.79, but the anisotropic part requires a little more work. Based on the relations given in Eqs. 1.16a and 1.20a, the following is easily derived for the products of one-electron operators... 

Asymptotically, when the separation between the collision partners is so large that the electrostatic interaction potential is negligibly small, the wavefunction of the combined system is most conveniently expanded in an uncoupled basis, namely a product of the wavefunction of the open-shell atom, iLSjn >, multiplied by the wavefunction which describes the relative orbital motion or the two collision partners, I Cm 2> Here j=L+S is the total ai ar momentum of the open-shell atom (e,g. j=l for Ip atoms, j=0, 1, or 2 for atoms) and mj and mQ denote the projections of j and C, respectively, along the space-fixed Z-axis. For collisions involving atoms... 

We consider the coherence evolution of uncoupled spins 7=1/2 during the pulsed gradient Hahn spin echo sequence schematically shown in Fig. 2a. A suitable basis for the treatment is the spherical product operator formalism. Explanations, definitions, and rules of the spin operator formahsm needed in this context can be found in Ref [2]. Times just before and immediately after RF (radio frequency) and field gradient pulses will be indicated by minus and plus signs, respectively. 

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