In computational chemistry, natural resonance theory (NRT) is an iterative, variational functional embedded into the natural bond orbital (NBO) program, commonly run in Gaussian, GAMESS, ORCA, Ampac and other software packages. NRT was developed in 1997 by Frank A. Weinhold and Eric D. Glendening, chemistry professors at University of Wisconsin-Madison and Indiana State University, respectively. Given a list of NBOs for an idealized natural Lewis structure, the NRT functional creates a list of Lewis resonance structures and calculates the resonance weights of each contributing resonance structure. Structural and chemical properties, such as bond order, valency, and bond polarity, may be calculated from resonance weights. Specifically, bond orders may be divided into their covalent and ionic contributions, while valency is the sum of bond orders of a given atom. This aims to provide quantitative results that agree with qualitative notions of chemical resonance. In contrast to the "wavefunction resonance theory" (i.e., the superposition of wavefunctions), NRT uses the density matrix resonance theory, performing a superposition of density matrices to realize resonance. NRT has applications in ab initio calculations, including calculating the bond orders of intra- and intermolecular interactions and the resonance weights of radical isomers.
History During the 1930s, Professor Linus Pauling and postdoctoral researcher George Wheland applied quantum-mechanical formalism to calculate the resonance energy of organic molecules. To do this, they estimated the structure and properties of molecules described by more than one Lewis structure as a linear combination of all Lewis structures:
Ψ A
i = ∑ κ a i
κ Ψ a
κ {\displaystyle \Psi {_{A}{}_{i}}=\sum _{\kappa }a{_{i}{}_{\kappa }}\Psi {_{a}{}_{\kappa }}}
where aiκ and Ψaκ denote the weight and single-electron eigenfunction from the wavefunction for a Lewis structure κ, respectively. Their formalism assumes that localized valence bond wavefunctions are mutually orthogonal.
⟨ Ψ α | Ψ β ⟩ = δ α β {\displaystyle \langle \Psi _{\alpha }|\Psi _{\beta }\rangle =\delta _{\alpha _{\beta }}}
While this assumption ensures that the sum of the weights of the resonance structures describing the molecule is one, it creates difficulties in computing aiκ. The Pauling-Wheland formalism also assumes that cross-terms from density matrix multiplication may be neglected. This facilitates the averaging of chemical properties, but, like the first assumption, is not true for actual wavefunctions. Additionally, in the case of polar bonding, these assumptions necessitate the generation of ionic resonance structures that often overlap with covalent structures. In other words, superfluous resonance structures are calculated for polar molecules. Overall, the Pauling-Wheland formulation of resonance theory was unsuitable for quantitative purposes. Glendening and Weinhold sought to create a new formalism, within their ab initio NBO program, that would provide an accurate quantitative measure of resonance theory, matching chemical intuition. To do this, instead of evaluating a linear combination of wavefunctions, they express a linear combination of density operators, Γ, (i.e., matrices) for localized structures, where the sum of all weights, ωα, is one.
Γ = ∑ α ω α Γ α {\displaystyle \Gamma =\sum _{\alpha }\omega _{\alpha }\Gamma _{\alpha }} where ω α ≥ 0 {\displaystyle \omega _{\alpha }\geq 0} and ∑ α ω α = 1 {\displaystyle \sum _{\alpha }\omega _{\alpha }=1}
In the context of NBO, the true density operator Γ represents the NBOs of an idealized natural Lewis structure. Once NRT has generated a set of density operators, Γα, for localized resonance structures, α, a least-squares variational functional is employed to quantify the resonance weights of each structure. It does this by measuring the variational error, δw, of the linear combination of resonance structures to the true density operator Γ.
δ W = m i n { ω α } ‖ Γ − ∑ α ω α Γ α ‖ {\displaystyle \delta _{W}={\underset {\{\omega _{\alpha }\}}{min}}\|\Gamma -\sum _{\alpha }\omega _{\alpha }\Gamma _{\alpha }\|}
… excerpt ends here. Continue reading the full article.
