Rumer basis, spin functions

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

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

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]. 

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... 

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... 

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... 

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... 

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... 

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... 

Another basis of spin functions which has proved itself of great value in chemistry is that due to Rumer " and was much used in classical VB theory. It is specially suited to describing chemical bonds and hence it is applied almost exclusively to systems (or subsystems) with zero net spin. The functions in this basis are constructed by considering all distinct pairs of electrons 1, j and coupling the associated spins a-, Oj to singlets... 

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. 

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. 

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%. 

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