The Nature of the Chemical Bond. V. The Quantum-Mechanical Calculation of the Resonance Energy of Benzene and Naphthalene and the Hydrocarbon Free Radicals

Abstract

The secular equations corresponding to the five canonical structures for benzene and the forty-two for naphthalene, considered as six and ten-electron systems, respectively, are set up and solved with certain simplifying assumptions, leading to energy values differing by 1.1055α and 2.0153α, respectively, from those corresponding to unexcited (Kekulé-type) structures, α being a single exchange integral involving neighboring carbon atoms. Equating these values to the empirical values of the resonance energy, α is found to be about — 1.5 v.e. It is pointed out that the dissociation of certain substituted ethanes into free radicals is due not to weakness of the carbon-carbon bond in the ethane but to the stabilization of the free radicals resulting from resonance among the structures in which the unpaired electron is located on the methyl carbon and those in which it is on other atoms (ortho, para, etc., to the methyl carbon). The secular equations for a number of such radicals have been solved, neglecting excited structures. The experimentally determined heat of formation of hexaphenylethane from triphenylmethyl, 0.5 v.e., when equated to the calculated value C–C+2.2156α, with C–C = 3.65 v.e., leads to α = — 1.4 v.e. The calculated tendencies towards dissociation are in satisfactory agreement with observation, such features as the smaller dissociating power of β-naphthyl than of α-naphthyl and of biphenylene than of diphenyl being accounted for, so that resonance among the structures considered may be accepted as the principal effect causing the stability of the hydrocarbon free radicals.

A comparison of approaches to estimate the resonance energy

“…However, employing a realistic basis set and ''as long as the atomic characteristics are well retained in the basis functions'', the BLW will generate stable results [13]. We infer that the resonance energy definition using the BLW approach [11,13] does not seem comparable with the Pauling and Wheland concept [9] and should be actually referred to as delocalisation energy.…”

“…In any case, the hypothetical molecules are only accessible computationally by restricting the variational space in which the wave function is expanded or by the use of fixed (non-optimised) orbitals. Pauling and Wheland [9] comment on the Pauling and Sherman [10] approach in their paper: ''the resonance energy calculated by Pauling and Sherman is not strictly comparable with that found by the present method''.…”

“…Many approaches for its calculation have been suggested in the literature, both empirical [1][2][3] and employing Ab Initio quantum chemistry [4][5][6][7][8]. Historically, Pauling and Wheland [9] have defined the resonance energy of an aromatic compound as the difference in energy between the resonating multi-structure valence bond wave function and the lowest contributing structure. This definition has been widely used by VB calculations [5][6][7][8].…”

1,3,5-Cyclohexatriene captured in computro; the importance of resonance

“…The calculation of the seemingly trivial defined concept of the resonance energy is hampered by the arbitrariness in the choice of orbitals. Depending on the orbital optimization model chosen, we find resonance energies of )25.37 (local), ) 19.82 (delocal) and )44.13 (breathing) kcal/mol, using the Pauling definition [2]. The last one employs the most extended wavefunction with the lowest total energy, and currently represents the best result.…”

“…The resonance energy as defined by Pauling [2] can be calculated in a Valence Bond (VB) calculation as the energy difference between the total energy and the energy of the most stable structure in the same calculation (thus with the same orbitals). The energy difference between 1,3,5-cyclohexatriene with the same geometry as benzene, but lacking resonance, and benzene gives the vertical resonance energy (VRE) [3].…”