Diagram Rumer

For singlet systems the bonding patterns for Rumer diagrams are conventionally obtained by writing the symbols for the orbitals in a ring (shown here for six), and drawing all diagrams where all pairs of orbital symbols are connected by a line and no lines cross[2, 13]. 

Our treatment has been oriented towards using tableaux to represent functions rather than Rumer diagrams, and it will be convenient to continue. Thus, corresponding to the five canonical diagrams for a ring of six orbital symbols one can write... 

The Rumer diagrams for benzene actually mirror the real spatial symmetry, and thus the Kekule and Dewar structures emerge. 

In this case the S5munetry functions in terms of the Rumer tableau are fairly obvious, as can be seen by inspection of the Rumer diagrams added together in them. 

This happens when we consider the most important configuration using HLSP functions. The two Rumer diagrams are shown with dots to indicate the extra electron. 

We show again the traditional five covalent Rumer diagrams for six electrons and six orbitals in a singlet coupling and emphasize that the similarity between the ring of orbitals and the shape of the molecule considerably simplifies the understanding of the S5mimetry for benzene. 

We now consider naphthalene, which possesses 42 covalent Rumer diagrams. Many of these, however, will have long bonds between the two rings and are probably not very important. To the author s knowledge no systematic ab initio study has been made of this question. The molecule has D21, symmetry, and these 42 covalent functions are combined into only 16 4 symmetry functions. 

This term together with the first provides the two Rumer diagrams for the bonding scheme. That this has such a large coefficient indicates that electron pair bonds are not near perfect pairing. 

Table 2. Occupation numbers of the different spin couplings in the Rumer basis for six-membered heterocycles. The Rumer diagrams (Rf) are numbered as in Fig. 2, and the orbitals as in Fig. 1 (and as described in the text).

Let us take the ir-electronic system of benzene as an example, with six AOs (Xi — X6) and six electrons. The five possible Rumer diagrams that can be drawn are shown in Scheme 4.1. 

Of course, Schemes 4.1 and 4.2 focus only on the covalent structures. However, Rumer diagrams for ionic VB structures can be generated in the same manner. Negative and positive charges are first assigned to some specific atoms,... 

For naphthalene we examine the H and S matrices based upon the both the HLSP functions and the standard tableaux functions for the system. In both cases we include the non-ionic structures, only. This will give a picture of how the situation compares for the two sorts of basis functions. In both cases, of course, the dimensions of the matrices are 42x42, the number of non-ionic Rumer diagrams for a naphthalene structure. Some statistics concerning the commutator are shown in Table 6. It is clear that,... 

This produces a non-standard Rumer function as shown by the representation on the right hand side. These non-standard functions, which correspond to a Rumer diagram in which the lines cross, are not used in calculations. A standard set of functions can be selected using the following procedure the open shell orbitals are represented by a 1 if their number occurs in the left hand column of the Weyl tableau or a 2 if it appears in the right hand column. Hence for the example above... 

The phenomenon of four singly-occupied active-space orbitals (AO or MO) arises in many seemingly-unrelated molecular situations. The singlet spin (S = 0) Rumer diagrams for these orbitals indicate that there are two linearly-independent or canonical spin-pairings schemes. These spin-pairings are well-exemplified by the 7i-electrons of butadiene, for which two canonical Lewis VB structures with different 7i-electron spin pairings are those of 1 and 2. 

Rumer diagram for circulants C3H3 through C7H7. Only C6H6 is stabilized as an aromatic system (Huckel 4/7 + 2 rule). 

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...

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