Serber basis

Table 4 Accumulated weights in the Serber basis, expressed as percentages Sjj is the total spin of electrons i andj.

An example of a basis of this kind is the SC x x... (x basis in which pairs of electron spins are first coupled to form singlets or triplets, the pairs then being coupled to form the desired resultant S. This is, of course, the natural basis to use when constructing pair wavefunctions (Section 4), and will be referred to as the Serber basis since it was first used by him in VB theory.12-13 It should be noted that in this basis the matrices VS(P) representing simple pair interchanges P/t-i/, even) are all diagonal,... 

The use of the Serber basis for spin functions combined with the strong orthogonality condition (100) ensures that separated pair functions (98)... 

Spin-coupled calculations at the idealized Dm geometry of cyclooctatetraene reveal a description dominated by triplet coupling of pairs of electrons [12], as anticipated earlier. Expressing the total spin function in the Serber basis [29], we find that the mode made up only of triplet-coupled pairs is responsible for 75% of the total. We find that the n orbitals for this antiaromatic system (see Figure 9) adopt localized forms that resemble closely those shown in Figure 4 for benzene, rather than the antipair representation shown for cyclobutadiene in Figure 6. 

Finally it is worth mentioning the Serber basis of spin functions in which pairs of electrons, 1 and 2, 3 and 4,...,etc., are coupled first to singlets or triplets, these pairs subsequently being coupled to one another to produce the required resultant spin. This differs from the Rumer basis in that triplet spin functions are used for the pairs as well as singlets, and the final set of spin functions is orthogonal. The construction of spin eigenfunctions is discussed in detail in the book by Pauncz. ... 

The Serber basis of spin functions is particularly useful here. 

An important result which arises out of our extensive use of different bases of spin functions, is the great utility of the little-known Serber basis. This set of spin functions is constructed by considering pairs of electrons (1, 2), (3,4),..., ((V — 1, N) in a similar manner to that of Rumer. The pairs of spins are then coupled to form either a singlet (5 = 0) or triplet (5=1) spin, which are subsequently coupled successively together to form the final spin. A particular function in this basis is identified by the quantum numbers... 

Projected spin functions have recently been reintroduced by Friis-Jensen and Rettrup (see also Refs. 12, 13, and 18). These spin eigenfunctions are linearly independent, but are nonorthogonal. They have been introduced into some versions of the spin-coupled codes and, using modem technology, provide a useful shortening of execution times, sometimes by as much as a factor of four. The main drawback of the projected spin functions is their lack of physical interpretability. However, the resulting SC coefficients can always be transformed into a more familiar representation such as the Kotani, Rumer, or Serber basis. Used in this way, the projected spin functions provide a useful addition to the SC codes. 

In addition, however, we must consider the spin functions, of which there are 14. This example illustrates very clearly some of the features of different spin functions. Table 1 reports spin coupling coefficients, using the Serber basis, with the orbitals ordered as ( i, 02>. ( 4), (s)- The notation of column 2 has been explained in Section 3. 

For (a), of course, the choice of spin basis may be very important for highlighting different features of the spin coupling, with our most common choices being the Rumer, Kotani, Serber, or projected spin function bases. Transformation between these (complete) bases is, in any case, very straightforward [36,66,67]. 

A complete set of spin eigenfunctions, e.g. oo i l = 1, 2,. .., 5) in the case of a six-electron singlet, can be constructed by means of one of several available algorithms. The most commonly used ones are those due to Kotani, Rumer and Serber [13]. Once the set of optimized values of the coefficients detining a spin-coupling pattern is available [see in EQ- (2)], it can be transformed easily [14] to a different spin basis, or to a modified set reflecting a change to the order in which the active orbitals appear in the SC wavefunction [see Eq. (1)]. 

SC theory does not assume any orthogonality between the orbitals ij/ which, just as in the GVB-PP-SO case, are expanded in the AO basis for the whole molecule Xp P 1,2,..., M. The use of the full spin space and the absence of orthogonality requirements allow the SC wavefunction to accommodate resonance which is particularly easy to identify if 0 sm is expressed within the Rumer spin basis. In addition to the Rumer spin basis, the SC approach makes use of the Kotani spin basis, as well as of the less common Serber spin basis. When analysing the nature of the overall spin function in the SC wavefunction (3.9), it is often convenient to switch between different spin bases. The transformations between the representations of 5M in the Kotani, Rumer and Serber spin bases can be carried out in a straightforward manner with the use of a specialised code for symbolic generation and manipulation of spin eigenfunctions (SPINS, see ref. 51). 

However, as shown in ref. 92, if the orbitals in benzene are ordered as in eqn (4.2) and the active-space spin-coupling pattern is expressed in the Serber spin basis, it becomes very similar to that at the TS for the [1,5]-H shift in (Z)-l,3-pentadiene. An additional symmetry-constrained SC calculation on benzene produced an antipair solution which is just about 1 mhartree above the well-known unconstrained solution with localised orbitals. The orbitals from this antipair solution are shown in Fig. 9 (the orbitals from the standard solution are visually indistinguishable from i/ i). 

Semi-classical picture. The first results of high energy machine experiments with nucleons were immediately interpreted by Serber [4] on the basis of a very simple picture it continues as the basis for our physical understanding of many high energy experiments. It is difficult to improve on Berber s statements so several direct quotations from his paper follows ... 

The method based on the Serber-Wilson condition is especially suited to the treatment of reactor systems which are spherically symmetric. We will develop the basic ideas involved in this method by analyzing, on the basis of the one-velocity model, the simple case of a spherical core surrounded by an infinite reflector. However, the method is generally applicable to finite systems, to multiregion configurations, and to multigroup (energy) calculations. These problems will not be discussed here. In order to demonstrate the effect on the criticality requirements of the... 

Of course for equilibrium intemuclear distances, HCN is better described in terms of electron-pair bonds, i.e., in the Serber or Rumer basis of spin functions. This brings us to the problem of transforming from one set of spin functions to another. In practical terms the problem is as follows. A calculation has been carried out using, e.g., the Kotani basis (which may be the most convenient) and we have... 

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