Pauling-Wheland model

Beyond the question of similarity in general predictions for a system s ground state, there are questions concerning excited-state spectra. But results in this area seem to be more meager, perhaps because results for either type of model are more meager. Moreover, since the MO-model space includes the ionic structures while the simple covalent Pauling-Wheland model does not, there arises a natural limitation, in the absence of extension to include these excited structures. 

The number of Kekule stmctures (Kekule structure count, KSC) of a conjugated hydrocarbon gives information on the thermodynamic stability and chemical reactivity and is used for the computation of various structural indices for ben-zenoid hydrocarbons " the Pauling bond order, PBO the stability index SI, SI = KSC the r-electron energy, Kekul6 structures are considered in valence-bond resonance-theoretical models such as the Pauling-Wheland model, the... 

Pauling- Wheland Valence-Bond Model 3.1 First Orthogonalization... 

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]... 

One approach to a complete solution of the Pauling-Wheland resonance-theoretic model is to treat it as a quantum-chemical configuration-interaction problem, now defined on a space of reduced size (corresponding only to those VB structures which are Kekule structures). However this space still has a size that often increases exponentially with the size of G. Thence one (ultimately) wishes further simplifications in dealing with the Hamiltonian and overlap matrixes H and S. 

A second approach to solving the Hemdon-Simpson model is to make a wave-function Ansatz which then leads to an upper bound to the ground state. The simplest Ansatz (of Pauling Wheland [12]) is a uniform sum over all Kekule structures,... 

To reasonably limit the focus here it the survey is primarily of many-body solution techniques as applied to a particular VB model, the covalent-space Pauling-Wheland VB model, represented by the Heisenberg spin Hamiltonian... 

A variety of VB models arise, with possibly the most natural hierarchy [22] indicated in fig. 1. The hierarchy of models occur in the column on the left and corresponding methods of solution are indicated on the right (though in some cases the methods have been little explored to date and are then identified with a "question mark") - the abbreviation Cl refers to "configuration interaction". The Pauling-Wheland... 

Both the simple VB and MO models for organic it-networks are quantum- mechanical models explicitly expressible in terms of their molecular graphs. For the neutral molecule the (Pauling-Wheland) VB model assigns one 71-electron (spin up or down) to each (carbon) center, so that for N sites the (2 -dimensional) model space is sparmed by products of N different electron spins. Then the simple VB Hamiltonian may be written as... 

The VB work of Oosterhoff et al [43] is relevant also to the neutral case but makes use of the non-orthogonalized VB model. Epiotis [44] also deals with the general case, possibly utilizing "anti-orthogonalized" AOs. Basically these workers note (beyond the exchange permutations) the importance of the cyclic permutations around the cycle, such as typically are discarded in the lowest order derivations to the Pauling-Wheland VB model. The inclusion of such terms is crucial most especially for 4-cycles - and such corrected models for quantitative work are available [33]. 

Wheland and Pauling (1959) tried to explain the inductive effect in terms of ar-electron theory by varying the ax and ySxY parameters for nearest-neighbour atoms, then for next-nearest-neighbour atoms and so on. But, as many authors have also pointed out, it is always easy to introduce yet more parameters into a simple model, obtain agreement with an experimental finding and then claim that the model represents some kind of absolute truth. 

The Yamanouchi-Kotani basis is best suited if we want to solve the Heisenberg problem in the complete spin space. However, the number of spins that can be handled this way, soon reaches an end due to the rapid growth of the spin space dimension f(S,N). Even with the present day computers, the maximum number of spins that can be treated clusters around N = 30. For larger values of N one must resort to approximate treatments, one of which, as described hereafter, is based on the idea of resonating valence bonds (RVB) coming from the classical VB model developed by Pauling and Wheland back in the early 1930 s [37, 51]. In essence,... 

---