Coupled product basis

Edmonds classic text on the theory of angular momentum is recommended for further studies on this subject. Our presentation was very brief in many respects. For instance, we have neither introduced ladder operators to explicitly constmct the eigenfunctions of angular momentum eigenvalue equations nor have we deduced the expression for total angular momentum eigenstates in the coupled product basis. Edmonds book fills all these gaps. 

For a coupled spin system, the matrix of the Liouvillian must be calculated in the basis set for the spin system. Usually this is a simple product basis, often called product operators, since the vectors in Liouville space are spm operators. The matrix elements can be calculated in various ways. The Liouvillian is the conmuitator with the Hamiltonian, so matrix elements can be calculated from the commutation rules of spin operators. Alternatively, the angular momentum properties of Liouville space can be used. In either case, the chemical shift temis are easily calculated, but the coupling temis (since they are products of operators) are more complex. In section B2.4.2.7. the Liouville matrix for the single-quantum transitions for an AB spin system is presented. 

Diazo Coupling Reactions. Alkylphenols undergo a coupling reaction with dia2onium salts which is the basis for the preparation of a class of uv light stabilizers for polymers. The interaction of orxv i -nitrobenzenediazonium chloride with 2,4-di-/ r2 -butylphenol results in an azo-coupled product (30). Reduction of the nitro group followed by m situ cyclization affords the benzottiazole (31) (19). 

The performance of a pigment which is produced by mixed coupling does not always equal the sum of the characteristics of the individually coupled products. A new crystal modification, for instance, may lead to an unpredictable deviation in the application properties. On a cost-performance basis, the best approach is to add to the reaction mixture traces of a costly coupling component which is known to confer excellent properties on the product, in an attempt to promote a more favorable crystal modification. The resulting mixed product is expected to assume the modification of the trace coupling component and thus to improve its commercial performance without representing a financial strain [7]. 

Racemic aldehyde 2 under similar reaction conditions yielded two diastereo-meric pinacols the configuration of all the products was assigned by X-ray diffraction. On the basis of these results a mechanistic scheme could be proposed to explain the stereochemical course of the coupling reactions [19]. It is assumed that the reaction proceeds via an intermediate titanaoxirane by insertion of a molecule of aldehyde 2 into the Ti—C bond - a proposal that has been put forward recently on the basis of DFT calculations [20]. The pinacol-type coupling products could be employed as bidentate P,P chelate ligands to Mo(CO)4 fragments however analogous experiments carried out with the olefinic derivatives 18 were unsuccessful [19]. 

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... 

When a copper anode is used instead of platinum, the resulting chlorosilane is subsequently reduced to a Si—Si coupling product in a one-pot reaction (equation 49)57. Interestingly, when a mixture of hydrosilane and chlorosilane is electrolyzed using a copper anode and a platinum cathode, a Si—Si coupling product is obtained in 64% yield on the basis of the sum of both reagents used. Thus, the paired electrolysis of hydrosilane on the anode and chlorosilane on the cathode proceeds to give disilane (equation 50)57. 

Propylene oxide represents a very attractive epoxide monomer for copolymerization with C02, as polypropylene carbonate) is industrially valuable. The low glass transition temperature (Tg) of 313 K, the sharp and clean decomposition above 473 K, and biodegradability of this copolymer are the reasons for its attracting interest in several applications. On a similar basis, H NMR spectroscopy is useful for assessing the coupling products resulting from the reaction of PO and C02 (Figure 8.21). 

The calculation method presented here also provides a possible extension of the single spin vector model. This extension is performed in two steps first to weakly coupled spin systems, then to strongly coupled ones. In the first case, the introduction of the well-known product basis functions and their coherences is sufficient while in the latter one the solution is not so trivial. The crucial point is the interpretation of the linear transformation between the basis functions and the eigenfunctions (or coherences) during the detection and exchange processes. These two processes can be described by the population changes of single quantum... 

When the glycopyranosyl radical adopts a B2 5 conformation then the preference for axial coupling products is readily understood on the basis of well-known steric interactions. Approach of a reagent from the P-face is significantly sterically hindered, particularly by the substituent at C(3), whereas attack from the a-face is less so. If ease of approach to position-2 of the norbornyl system [145] is taken as a model for ease of approach to C(l) of a pyranose ring in the B2 5 conformation, then the favoured approach to the carbohydrate corresponds to the favoured exo-approach to the norbornyl system. If the glycopyranosyl radical adopts the 4H3 conformation, then axial reaction is favoured by the least motion effects set out in Scheme 13. 

We now apply the concepts developed in the preceding section to the system of just two nuclei, first considering the case in which there is no spin coupling between them. We digress from our usual notation to call the two spins A and B, rather than A and X, because we later wish to use some of the present results in treating the coupled AB system. The four product basis functions are given in Eq. 6.1. We now compute the matrix elements needed for the secular determinant. Because there are four basis functions, the determinant is 4 X 4 in size, with 16 matrix elements. Many of these will turn out to be zero. For <3CU we have, from... 

For the sake of completeness, it should be mentioned that at variance with the corresponding silyl enolates, the oxidation of titanium bis(enolates) with a variety of oxidants does not show any diastereoselectivity in the formation of the enolate coupling product . On the basis of crossover experiments, it has been shown that the C—C bond formation occurs via an intramolecular route in the case of the silyl derivatives and intermolecularly in the case of the titanium derivatives. 

The effective spin state of the radical pairs [RuLs " MV" ]) may be represented in the direct product basis of two doublet, whereby the MV" radical is represented by a normal spin doublet (a and P), but the RuL3 " moiety by a strongly spin-orbit coupled Kramers doublet (a and P ). The lowest Kramers doublet of a Ru(III) complex of D3 symmetry may be expressed in the general form... 

The condition of Eq. (179) is fulfilled if the matrix elements of both operators A and B are either purely real or purely imaginary. For example, the effective Hamiltonians for characteristic zero-quantum coupling topologies with coupling tensor types /, P, L, or O (see Section V.B) are symmetric in the usual product basis. Because the matrix representations of the operators /, and /y are either real-valued (a =x or z) or purely imaginary (a = y) in this basis, the condition of Eq. (179) if fulfilled. In a system consisting of N spins 1/2, the operators and / have the same norm Tr /, = Tr /y = 2 and... 

In contrast, for the acetyl-substituted compound, 91, where intersystem crossing from the excited singlet state to T, is expected to be efficient, homolytic cleavage to triplet radical pairs was proposed on the basis of formation of the out-of-cage coupling product and by CIDEP [131] and QDNP [133] studies. Finally, 92 reacted from S on direct irradiation again to give the phosphonate as the major product, but from Ti on sensitization with benzophenone [131]. 

The triple sum is over all nearest-neighbor combinations, with modes 1 and 6 coupled. For example the term Q QhQi is included, but QiQ2Q is not, because mode 5 is not a nearest neighbor to mode 2. The calculations are carried out using a direct product basis... 

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