Rumer spin functions

S lines are then drawn between pairs of the first N points, additional lines are drawn between the remaining 2S points and the pole none of the lines are allowed to intersect. It can be shown" that the number of distinct extended Rumer diagrams which can be drawn in this way is Jg, and the associated Rumer spin eigenfunctions, in which the singlet pairs contain electrons with numbers connected by the first i V — >S lines, while the remaining electrons (with numbers connected to the pole) are placed within a spin functions, are linearly independent. The extended Rumer diagrams and the associated Rumer spin functions for N = 5,5 = (/, = 5) are shown in Fig. 1. ... 

Fig. 1 Extended Rumer diagrams and Rumer spin functions for N...

In all type A calculations the active-space spin function was found to be dominated by the two Kekule-type Rumer spin functions, with weights between 38-40% each. All of this is consistent with the traditional VB notions of aromaticity. 

N form singlets is identical to the Rumer spin function... 

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]. 

Each of the five Rumer spin eigenfunctions for a six-electron singlet represents a product of three singlet two-electron spin functions ... 

For a system consisting of six electrons with a net spin of zero, = 5. For our present purposes, it is probably most useful to consider the modes of spin coupling in terms of the traditional basis of Rumer functions used in classical VB theory. For a discussion of different spin functions, and of the relationships between them, see... 

Now we are concerned with how the VB model (18) can be solved for a given conjugated system. In fact, the model Hamiltonian (18) actually acts on the space of pure spin functions, either of the Weyl-Rumer (WR) form [34] or the simple product of one-electron spin functions. The matrix element between any two WR functions can be obtained by using Pauling s graphical rules [4], while the matrix element between two simple spin products is easily available using the following expression... 

It is perhaps worth remarking that had we chosen, instead of the standard basis, the Rumer basis of spin functions, then the five VB singlet covalent functions are just the two well-known Kekule Structures and the three Dewar structures. 

The Rumer basis turns out to be particularly useful for interpreting the total spin functions for aromatic systems. In the case of Af=6 and S=0, there are just five linearly independent modes of spin coupling [29], which may be represented as in Figure 1, in which an arrow i—>j signifies a factor in the total spin function of 2 (a(j)P(/)-a(/)p(0). The similarity to Kekule and para-bonded structures for benzene is obvious. 

In the Rumer basis [29] (see Figure 1), the total spin function corresponds to weights of 40.6% each for the two Kekule structures (RirR4) and of 6.3% each for the three para-bonded ( Dewar ) structures These values are very close... 

Rumer basis, spin functions, 199 Spin-Dipolar (SD) operator, 251 method, 322 Wave function stability, 76... 

The Rumer spin basis represents a set offg linearly independent spin functions, in which N—2S electrons (/ii,- form singlet pairs, and the remaining 2S... 

Two popular modem VB approaches, GVB-PP-SO and SC, use fully-variational wavefunctions including a single orbital product. In the GVB-PP-SO wavefunction this single orbital product is combined with a single perfect-pairing (PP) spin function which in the Rumer spin 3-4,. .. , (JV-2S-1)-(N-2S)) ... 

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). 

The changes in the shapes of the orbitals are accompanied by a re-coupling of the electron spins. For this reaction, it proves most convenient to express the total active-space spin function oo in the Rumer basis. As shown in Fig. 5(b), the two Kekule-like functions (1-2, 3-4, 5-6) and (1-6, 2-3, 4-5) are dominant over the... 

---