Rumer

Thus W. Heitler and G. Rumer, Nachr. Ges. Wiss. Gottingen, Math, physik. Klasse, 7, 277 (1930), in a paper on the quantum mechanics of polyatomic molecules, discuss only the second structure. 

Starting from the discovery by Rumer that the eigenfunctions corresponding to different distributions of valence bonds in a molecule can be represented by plane diagrams which provide information regarding their mutual linear... 

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)]. 

When discussing the SC models for the electronic mechanisms of the Diels-Alder reaction between butadiene and ethene [11], and of the disrotatory electrocyclic ringopening of cyclohexadiene, we shall make use of the Rumer spin basis, which allows a straightforward translation of our results into a form that resembles traditional VB ideas. 

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

The Rumer spin basis is nonorthogonal, which creates ambiguities in the definition of numerical estimates for the relative importance of the individual . . [see 0qq in Eq. (2)]. In this paper, we shall use the Chirgwin-Coulson expressions [15] (it is assumed... 

Figure 6. Composition of the active space spin-coupling pattern / oo in Eq. (2)]from the SC wavefunction for the disrotatory ring-opening of cyclohexadiene along the CASSCF(6,6) IRC, expressed in terms of Chirgwin-Coulson weights Pq [seeEq. (4)] in the Rumer basis [seeEq. (3)].

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... 

These Rumer tableaux are based upon the following diagrams ... 

Ts gives the same funetion. This function may be written in terms of Rumer tableaux, also, and we obtain... 

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. 

If Rumer tableaux are used for ft, these may in many cases be put in a one-to-one relation with classical bonding diagrams used by chemists. 

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