User:Blinn21/sandbox

= Natural Resonance Theory (NRT) = In computational chemistry, Natural Resonance Theory (NRT) is an iterative, variational functional embedded into the Natural Bond Orbital (NBO) program, commonly ran in Gaussian, GAMESS, ORCA, Ampac and other software packages. NRT was developed in 1997 by Frank A. Weinhold and Eric D. Glending, 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. 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 found in ab initio applications, including ...

= History = During the 1930's, Professor Linus Pauling and postdoctoral researcher George Wheland applied quantum-mechanical formulism 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:

$$\Psi=\sum_{\alpha} c_\alpha\Psi_\alpha$$

where cα and Ψα denote the weight and wavefunction for a Lewis structure α. Their formulism assumes that localized valence bond wavefunctions are mutually orthogonal.

$$\langle\Psi_\alpha|\Psi_\beta\rangle=\delta_{\alpha_\beta}$$

While this assumption ensures that the sum of the weights of the resonance structures describing the molecule is one, it creates difficulties in computing cα. The Pauling-Wheland formulism 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. Glending and Weinhold sought to create a new formulism, 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 (i.e., matrices) for localized structures.

$$\Gamma=\sum_{\alpha}\omega_\alpha\Gamma_\alpha$$

where the sum of all weights, ωα, is one:

$$\omega_\alpha\geq0 $$ and $$\sum_{\alpha}\omega_\alpha=1$$

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.

$$\delta_W=\underset{\{\omega_\alpha\}}{min}\|\Gamma-\sum_{\alpha}\omega_\alpha\Gamma_\alpha\|$$

To evaluate a single resonance structure, δref, the absolute difference between a single term expansion and the true density operator, approximated as the leading reference structure, can be taken.

$$\delta_{r_{}e_{}f}=\|\Gamma-\Gamma_{r_{}e_{}f}\|$$

Now, the extent to which each reference structure represents the true structure may be evaluated as the "fractional improvement", fw.

$$f_W= \frac{\delta_{r_{}e_{}f}-\delta_W}{\delta_{r_{}e_{}f}} $$

From this equation, it is evident that as fw approaches one and δw approaches zero, δref becomes a better representation of the true structure.

= Theory =

Generation of resonance structures and their density matrices:
From a given wavefunction, Ψ, a list of optimal NBOs for a Lewis-type wavefunction are generated along with a list of non-Lewis NBOs (e.g., incorporating some antibonding interactions). When these latter orbitals have nonzero value, there is "delocalization" (i.e., deviation from the ideal Lewis-type wavefunction). From this, NRT generates a "delocalization list" from deviation from the parent structure and describes a series of alternative structures reflecting the delocalization. A threshold for the number of generated resonance structures can be set by controlling the desired energetic maximum (NRTTHR threshold). The NBOs for a resonance structure formula can then be, subsequently, calculated from the CHOOSE option. Operationally, there are three ways in which alternative resonance structures may be generated: (1) from the LEWIS option, considering the Wiberg bond indices; (2) from the delocalization list; (3) specified by the user.

Below is an example of how NRT may generate a list of resonance structures.

(1) Given an input wavefunction, NRT creates a list of reference Lewis structures. The LEWIS option tests each structure and rejects those that do not conform to the Lewis bonding theory (i.e., those that do not fulfill the octet rule, pose unreasonable formal charges, etc.).

(2) The PARENT and CHOOSE operations determine the optimal set of NBOs corresponding to a specific resonance structure. Additionally, CHOOSE is able to eliminate identical resonance structures.

(3) A user may then call SELECT to select the structure that best matches to the true molecular structure. This option may also show other structures within a defined energy threshold NRTTHR, deviating from optimal Lewis density.

(4) Two other operations, CONDNS and KEKULE, are ran to remove redundant ionic structures and append structures related by bond shifts, respectively.

(5) Lastly, SECRES is called to calculate the NBOs and density matrices of each resonance structure.

Generation of resonance weights:
To compute the variational error, δw, NRT offers the following optimization methods: the steepest descent algorithms BFGS and POWELL and a "simulated annealing method" ANNEAL and MULTI. Most commonly, the NRT program computes an initial guess of the resonance weights by the following relation:

$$\overset{\backsim}{\underset{\alpha}{\omega}}\propto e^-3\rho_\alpha$$

where the weight is proportional to the exponential of the non-Lewis density, ρ, of structure α. Then the BFGS and POWELL steepest descent methods optimize for the nearest local minimum.

In contrast, the ANNEAL option finds the global maximum of the fractional improvement, fw, and performs a controlled, iterative random walk across the fw surface. This method is more computationally expensive than the BFGS and POWELL steepest descent methods.

After optimization, SUPPL evaluates the weight of each resonance structure and modifies the list of resonance structures by either retaining or adding resonance structures of high weight and deleting or excluding those of low weight. It continues this process until either convergence is achieved or oscillation occurs.

= Implications for Chemistry =

= Applications =

= Limitations =

= See Also = Natural Bond Orbital

Phosphaethynolate

= References =

= External Links = Frank Weinhold

https://www2.chem.wisc.edu/users/weinhold

https://scholar.google.com/citations?user=47IzzwYAAAAJ&hl=en

Eric D. Glending

https://www.indstate.edu/cas/chem_phys/eric-d-glendening

https://scholar.google.com/citations?user=iRjJ1Y0AAAAJ&hl=en