Bond alternation parameter

Fig. 9.11 Spatially extended soliton (f = 7) on a polyacetylene chain centred on the 30th carbon atom, (a) variation in bond alternation parameter, (b) the electronic wavefunction and (c) bonding structure.

Figure 1.10. Definition of the bond-alternation parameter in equation (1.3). Odd and even lattice sites after marking a zero. (Reprinted with permission from ref 22)...

Hiickel calculations of Siegel cyclohexatriene with inclusion of /3-parameters between the bridge jr orbital and the jr orbitals of the benzene moiety change the bond orders and predict the direction of bond alternation observed in the molecule. Heil bronner, E. Personal communication to S.S. 

We start with some biographical notes on Erich Huckel, in the context of which we also mention the merits of Otto Schmidt, the inventor of the free-electron model. The basic assumptions behind the HMO (Huckel Molecular Orbital) model are discussed, and those aspects of this model are reviewed that make it still a powerful tool in Theoretical Chemistry. We ask whether HMO should be regarded as semiempirical or parameter-free. We present closed solutions for special classes of molecules, review the important concept of alternant hydrocarbons and point out how useful perturbation theory within the HMO model is. We then come to bond alternation and the question whether the pi or the sigma bonds are responsible for bond delocalization in benzene and related molecules. Mobius hydrocarbons and diamagnetic ring currents are other topics. We come to optimistic conclusions as to the further role of the HMO model, not as an approximation for the solution of the Schrodinger equation, but as a way towards the understanding of some aspects of the Chemical Bond. 

In Ref. 31 it has been shown that long linear chains, as seen from the point of view of VB theory, show a long-range order, measured by the parameter p in Eq. (23), leading to a ground-state instability to bond alternation. From Sec. 5.3 also this is expected to occur. 

Table 19 The bond-length alternation parameter (in A) for the push-pull oligomers of Fig. 8. The phase shows whether the calculations were performed in vacuum or in the water phase, and HF and MP2 mark Hartree-Fock calculations without (HF) or with (MP2) the inclusion of correlation effects via Moller-Plesset perturbation theory. All results are from ref. 70... 

A variety of other molecular properties can be found, including details of the spatial distribution of segments, the probability of gauche-conformations, bond order parameters (which display alternations between odd and even chain elements, counted from the head group) and tilt behaviour 1). 

Deng and Yang [40] tabulate expressions for the HMOs that are appropriate to the case where single and double carbon-carbon bonds are equivalent (in the sense that their respective resonance integrals fis and are equal). However, the theory is sufficient to allow easy extension to a picture in which Pd- To this end, the authors introduce a parameter a = Ps which requires that = a — 2. In an earlier work [41], we introduced a similar parameter to account for this bond alternation , as it is often termed, r = Pd/Ps- Thus, the two treatments are related by... 

In particular, the description of bond length alternation parameters in B3LYP-optimized geometries is believed to be the main factor resulting in overestimated electronic delocalization. Subsequently, bathochromic shifts were obtained for the calculated excited state energies. In contrast, calculations based on... 

The opposite conclusion that correlation increases the dimerization reached in PPP valence bond (VB) theory [36] holds only in comparison to Hiickel theory which is not the semiempirical equivalent to ab initio HF. In fact Hiickel theory would correspond to diagonalization of the one-electron part of the ab initio Fock matrix rather than to HF itself although in Hiickel theory correlation may appear implicitely via the parameters. And indeed, if one considers only the one-electron part of the ab initio Fock matrix (core diagonalization) in a 6-31G basis set the equidistant geometry for trans-butadiene is favored. Inclusion of electron-electron interactions on one-particle level (HF), which corresponds to PPP theory yields an increase of the bond-alternation beyond its experimental value. The inclusion of Jt-n correlations (CCD-level) decreases this value again to the experimental one [16]. 

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