A detailed understanding of reactive molecular systems would support the optimization of chemical processes through directed manipulations of promoting and interfering factors during the course of a reaction. For this purpose, we need to uncover the kinetic principles of a general (complex and noisy) chemical reaction network from a quantum chemical perspective. Free-energy differences are fundamental for the kinetic modeling of chemical systems at thermal equilibrium since they serve to derive rates of elementary reactions. The set of all rates associated with a reaction network can be represented by a system of coupled differential equations, and their numerical integration yields particle concentrations for every molecular species as a function of time.
Possible reaction mechanism for the first steps of the formose reaction (a) and a network representation thereof (b). J. Proppe, T. Husch, G. N. Simm, M. Reiher, Faraday Discuss. 2016, 195, 497 – Published by The Royal Society of Chemistry.
Due to the approximations involved in actual quantum chemical models, free-energy predictions are generally plagued by systematic errors, which can be translated into statistical uncertainties. These uncertainties propagate through the differential equations such that predictions of time-dependent concentrations are also uncertain to some degree, usually in an unpredictable way. Therefore, the kinetic information to be extracted is more or less noisy, which may render reliable predictions of product distributions or reaction mechanisms impossible. During the Faraday Discussions on Reaction Rate Theory in 2016, we presented a general workflow to arrive at uncertainty-equipped free-energy differences on the basis of quantum chemical calculations. We showed that the propagation of free-energy uncertainties can lead to striking variances in reactant half-lives. For a standard deviation of about 3 kcal/mol in the kinetically most relevant activation free energy, we found a maximum shift in the reactant half-life of almost 23 orders of magnitude.
Time-dependent concentrations of the chemical species 1–6 of the mechanism proposed above. Red and black concentration profiles result from ensemble-averaged and sampled activation free energies, respectively. J. Proppe, T. Husch, G. N. Simm, M. Reiher, Faraday Discuss. 2016, 195, 497 – Published by The Royal Society of Chemistry.
Recently, we introduced KiNetX, a fully automated meta-algorithm for the kinetic simulation and analysis of complex and noisy reaction networks with rigorous uncertainty control. It is designed to cope with method-inherent errors in quantum chemical calculations on elementary reactions. For a rigorous analysis of the KiNetX algorithm, we developed a random generator of artificial reaction networks, AutoNetGen, which encodes chemical logic into their underlying graph structure. AutoNetGen allows us to consider a vast number of distinct chemical scenarios which is necessary to investigate the reliability and efficiency of KiNetX in a statistical context. Our results reveal that reliable mechanism deduction from noisy chemical reaction networks is feasible through the combination of first-principles calculations, kinetic-modeling techniques, and rigorous statistical methods.
Property predictions by computers become increasingly important in all fields of chemistry. For this purpose, determining the predictive accuracy of routine quantum chemical calculations is an essential but arduous task that requires a sophisticated statistical analysis. It is common practice to account for predictive accuracy by average error statistics, which are generally unrepresentative of the actual prediction uncertainty (the virtual analog to measurement uncertainty). We developed a generic workflow for the rigorous calibration of computational results to arrive at reliable prediction uncertainties for arbitrary physicochemical properties. A compelling feature of this workflow is its availability as a fully automated implementation named reBoot (a portmanteau of resampling and Bootstrapping).
Prediction uncertainty (R632) of the 57Fe Mössbauer isomer shift as a function of the exact-exchange admixture c3 for common density functionals. The black curve corresponds to B3LYP (default: c3 = 0.2). Hybrid density functionals with c3 about 0.2 reveal lowest prediction uncertainty. Reprinted with permission from J. Proppe, M. Reiher, J. Chem. Theory Comput. 2017, 13, 3297. Copyright 2017 American Chemical Society.
The single steps of our calibration workflow were presented in detail at the example of Mössbauer isomer shift prediction based on quantum chemical data (electron densities). A wide range of distinct and established density functionals was employed to determine the most predictive model and, hence, to support structure elucidation in (bio)inorganic research. Our reliable approach for estimating virtual error bars allows us to determine the probability that two different molecular iron compounds can be distinguished on the basis of their Mössbauer isomer shifts.
Parameter-induced uncertainty of the D3 dispersion energy as a function of the molecular size. Except for rare gases, the uncertainty in the dispersion energy uncertainty is an approximately linear function of the molecular size. Reprinted with permission from T. Weymuth, J. Proppe, M. Reiher, J. Chem. Theory Comput. 2018, 14, 2480. Copyright 2018 American Chemical Society.
Our calibration workflow has recently been extended to nonlinear prediction models, and has been assessed in detail at the example of Grimme’s semiclassical D3 approach for the calculation of London dispersion energies. A rigorous analysis of error accumulation arising from different parameterizations of the D3 model reveals a monotonically increasing deviation in the dispersion energy with increasing molecular size. We demonstrated this issue at the prominent example of the C60 buckycatcher, and recommend to always determine the parametric uncertainty associated with dispersion corrections for which we provide the BootD3 software tool.