Wave functions allyl system

Stabilizing resonances also occur in other systems. Some well-known ones are the allyl radical and square cyclobutadiene. It has been shown that in these cases, the ground-state wave function is constructed from the out-of-phase combination of the two components [24,30]. In Section HI, it is shown that this is also a necessary result of Pauli s principle and the permutational symmetry of the polyelectronic wave function When the number of electron pairs exchanged in a two-state system is even, the ground state is the out-of-phase combination [28]. Three electrons may be considered as two electron pairs, one of which is half-populated. When both electron pahs are fully populated, an antiaromatic system arises ("Section HI). 

The way this function represents the system is strongly influenced by the dynamics of the problem, as well as the flexibility allowed. If we were to find the set of three orbitals and value of a minimizing W, we obtain essentially the SCVB wave function. What this looks like depends significantly on the potential energy function. If we are treating the n system of the allyl radical, where all three orbitals are nearly degenerate, we obtain one sort of answer. If, on the other hand, we treat a deep narrow potential like the Li atom, we would obtain two orbitals close to one another and like the traditional s orbital of self-consistent-field (SCF) theory. The third would resemble the 2s orbital, of course. 

The Allyl Radical The case of allyl radical in Fig. 7.3, is the simplest example of a iT-system where the VB states can be generated by use of symmetry considerations (21). Allyl radical has two VB structures that are labeled in Fig. 7.3a as Kr and K/, where K denotes a Kekule structure (meaning a structure with maximum pairing), while the subscripts signify the location of the double bond on the right- and left-hand sides, respectively. The corresponding wave functions are given in Equation 7.1a and 7.1b, where normalization constants are dropped ... 

Fig. 12.7 Illustration of the rc-electron interaction of with Ni [3]. (Left) Contour plots of 7t orbitals from DFT calculations for in the gas phase and adsorbed on Ni(lOO). Solid and dashed lines indicate different phases of the wave function. Right) Schematic illustration of the k orbital interactions in the allylic configuration of the N -Ni adsorption system in terms of the atomic N2p and Ni34 orbitals. Reprinted with permission from ref [3]. Copyright 2004 Elsevier...

In the following, the Htickel reference wave functions for the cation, radical and anion will be noted, W and W, respectively (see Fig. 13.4). For each system, three Lewis structures are considered the left (Pl) and tight (Wr) stmctures, whereby the positive/negative charge or radical is located on the left and on the right of the molecule skeleton, respectively. In the third stmcture (Pc), the positive/negative charge or radical is located on the central carbon atom, whereas the electrons on opposite carbons are covalently paired (see Fig. 13.3). The mesomerism scheme for the allyl radical is represented in Fig. 13.5, as an example. 

In OCC and BCC theories, the wave function has the form of a standard coupled-cluster wave function with vanishing singles amplitudes. However, unlike standard coupled-cluster theory, where the orbitals are determined in a separate optimization of the reference state HF), the OCC and BCC orbitals are determined simultaneously with the optimization of the cluster amplitudes, making them more suitable for the description of correlation, in a manner reminiscent of MCSCF theory. In practice, the differences between the standard coupled-cluster wave functions and the BCC and OCC wave functions are small, except in systems characterized by Hartree-Fock singlet instabilities such as the allyl radical in Section 10.10.6. In such cases, the Harlree—Fock instability makes the standard approach unsuitable - the BCC and OCC models, by contrast, suffer from no such instabilities. 

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