Diagram branching

The classic approach to representing this result is via the branching diagram of Fig. 6.3. 

In general, n equivalent nuclei of spin I will produce an EPR multiplet consisting of (2nl + 1) equally spaced lines. For spins of I = Vi, the intensity ratios are given by the coefficients of the binomial expansion (a + b)n but for I > /2, the formula for intensities is much more complicated. For modest values of n it is easy to find the intensity ratios by sketching a branching diagram thus... 

Branching diagrams are often crucial to obtaining an assignment of a spectrum when a radical contains more than one set of equivalent nuclei. The spectrum of the naphthalene anion is shown in Figure 29.6 together with a diagram... 

The situation becomes more complicated for a system with four electrons, but similar arguments as in the three electron case can be applied. The total spin S in the four electron case can take three distinct values, namely, 5 = 0 (singlet), 1 (triplet), and 2 (quintet). According to the spin branching diagram... 

Figure 5 The spin branching diagram for 0 to 6 electrons (horizontal axis). The total spin quantum number is on the vertical axis. The numbers in the circles give the spin degeneracies.

The Yamanouchi-Kotani basis in the 77-electron --adapted spin space is closely related to the standard Young tableaux used in characterizing irreps of the symmetric group [50] and is conveniently represented by Van Vleck s branching diagram [18, 42]. To a basis function QfM we assign an array... 

Figure 1 Van-Vleck s branching diagram (A) and the reversed branching diagram (B) for S = 1, N = 8. Either 6 (full lines) or 8 (full and broken lines) spins are coupled in all ways allowed by the Yamanouchi-Kotani scheme. At vertices and arcs their weights (if different from 0) are shown.

A transposition (k — 1, k) which interchanges two adjacent elements is called an elementary transposition. When this elementary transposition acts on a branching diagram spin function QfM, it affects only the arcs corresponding to k — 1 and k, that is the path segments contained between... 

A major drawback of RVB calculations is the fact that RVB functions cannot be cast into a compact graphical form as it is possible in SGA using the branching diagram. Each RVB function has to be generated explicitly and special techniques (e.g. bit representation) have to be used... 

Figure 2 Two-segment (A) and three-segment (B) loop in the branching diagram...

Kotani branching diagram (Fig. 2), however, on the Serber branching diagram these states have no successors. 

A branching diagram that summarizes this method of assigning point groups was given in Figure 4-7 and more examples are given in Table 4-4. 

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