Rumer basis

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)].

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).

Similar descriptions emerge for the other five-membered rings we have studied, and occupation numbers in the Rumer basis are collected in Table 3. The main difference is in the delocalization of the second orbital stemming from the heteroatom ([Pg.52]

A point of some confusion is that there are different representations of the Pauling-Wheland VB model. In fact, Pauling and Wheland [1] did not represent it in the form of Eq. (3.1.10), but rather they presented it as a matrix on the Rumer basis (mentioned in Sect. 4.2 here). The appearance of (3.1.10) may be further modified through the use of the Dirac identity [25]... 

Projecting the Molecular Orbital Configuration Interaction Wave Function Onto the Rumer Basis of Valence Bond Structures... 

As shown above in Scheme 4.2, the Rumer basis for butadiene is made of the VB structures 6 and 7. From Section 3.1.4, the VB function corresponding to a given bonding scheme is the one that involves singlet coupling between the AOs that are paired in this scheme. This, however, can be done in two ways. In the first way, the AOs are kept in the same order in the various determinants, and the determinants display all the possible 2x2 spin permutations between the orbitals that are singlet coupled. This is the convention used in Equation 4.16, which is similar to Equation 3.9. In this case, the determinant has a positive or... 

The projection of flMO on the Rumer basis consists of combining Equation 4.19 with the final expression of fiMO in terms of VB structures, Equation 4.20 ... 

A difference between the qualitative VB theory, discussed in Chapter 3, and the spin-Hamiltonian VB theory is that the basic constituent of the latter theory is the AO-based determinant, without any a priori bias for a given electronic coupling into bond pairs like those used in the Rumer basis set of VB structures. The bond coupling results from the diagonalization of the Hamiltonian matrix in the space of the determinant basis set. The theory is restricted to determinants having one electron per AO. This restriction does not mean, however, that the ionic structures are neglected since their effect is effectively included in the parameters of the theory. Nevertheless, since ionicity is introduced only in an effective manner, the treatment does not yield electronic states that are ionic in nature, and excludes molecules bearing lone pairs. Another simplification is the zero-differential overlap approximation, between the AOs. 

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 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... 

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. ... 

Occupation numbers of the spin functions in the Rumer basis". 

The spin functions used by the program are the set of Rumer ° (or bonded) functions. Besides highlighting the formation or disintegration of covalent bonds, this basis has the advantage that the structures in it are very easily decomposed into Slater determinants. The coefficients of the individual determinants are just + 1. The Rumer basis of spin functions is not orthogonal, and a linearly independent set is systematically generated using... 

TABLE I. Weights of the f =5 different VB structures in 4 vb, with the spin hinction expressed in the Rumer basis. 

The Rumer basis of spin functions has found extensive use in organic chemistry in the description of the mechanisms of a great variety of organic reactions such as aromatic electrophilic and nucleophilic attack and a host of addition reactions. Even after 40 or more years of development of molecular orbital methods, this mode of description obstinately remains a major part of theoretical organic chemistry. 

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... 

If we transform these spin-coupling coefficients to the Rumer basis, we find that the perfectly-paired spin function now contributes 80% to the total spin function. The remainder is made up almost entirely from structures containing a single H-H pairing and only two C-H bonds. There are four such structures which, at the equilibrium geometry of CH4, are all symmetrically equivalent. The occurrence of this type of spin pairing with significant contributions is 20%. 

Table 2 Spin-coupling Coefficients in the Rumer Basis...

RMP2 method, 132 ROMP method, 132 Roothaan-Hall equations, 65 Rumer basis, spin functions, 199 Runge-Kutta (RK) integration method, 344... 

---