Abstract
Machine learning techniques, specifically gradient-enhanced Kriging (GEK), have been implemented for molecular geometry optimization. GEK-based optimization has many advantages compared to conventional—step-restricted second-order truncated expansion—molecular optimization methods. In particular, the surrogate model given by GEK can have multiple stationary points, will smoothly converge to the exact model as the number of sample points increases, and contains an explicit expression for the expected error of the model function at an arbitrary point. Machine learning is, however, associated with abundance of data, contrary to the situation desired for efficient geometry optimizations. In this paper, we demonstrate how the GEK procedure can be utilized in a fashion such that in the presence of few data points, the surrogate surface will in a robust way guide the optimization to a minimum of a potential energy surface. In this respect, the GEK procedure will be used to mimic the behavior of a conventional second-order scheme but retaining the flexibility of the superior machine learning approach. Moreover, the expected error will be used in the optimizations to facilitate restricted-variance optimizations. A procedure which relates the eigenvalues of the approximate guessed Hessian with the individual characteristic lengths, used in the GEK model, reduces the number of empirical parameters to optimize to two: the value of the trend function and the maximum allowed variance. These parameters are determined using the extended Baker (e-Baker) and part of the Baker transition-state (Baker-TS) test suites as a training set. The so-created optimization procedure is tested using the e-Baker, full Baker-TS, and S22 test suites, at the density functional theory and second-order Møller–Plesset levels of approximation. The results show that the new method is generally of similar or better performance than a state-of-the-art conventional method, even for cases where no significant improvement was expected.
Highlights
The optimization of molecular structures is instrumental for the computational chemistry procedure to establish the fundamental thermodynamics of a chemical process the reaction enthalpy and the activation energy
We will present an alternative to the usual approach in computational chemistry the standard surrogate model of restricted-step second-order Taylor expansion approximations[1−3] in combination with approximative second derivatives[4] and a Hessian-update method, for example, the BFGS5−11 and MSP12−14 approaches used for minimum and transition-state optimizations, respectively
In analyzing the character and robustness of the GEKsupported molecular geometry optimization, three different test suites were employed, the extended Baker (e-Baker), Baker-TS, and S22 test suites results are listed in Tables 1, 2, and 3
Summary
The optimization of molecular structures is instrumental for the computational chemistry procedure to establish the fundamental thermodynamics of a chemical process the reaction enthalpy and the activation energy. The zerothorder understanding of the dynamics of a chemical reaction is based on the optimization of equilibrium structures, transition states, reaction pathways, constrained optimization on the ground-state potential energy surface, and so forth. Optimization, unconstrained or constrained, on ground- and excited-state potential energy surfaces is the essence in our extraction of a qualitative understanding and a quantitative prediction of the nature of a chemical process. For this reason, various efforts to make optimization procedures as robust and efficient as possible are of fundamental importance to computational chemistry. We will present an alternative to the usual approach in computational chemistry the standard surrogate model of restricted-step second-order Taylor expansion approximations[1−3] in combination with approximative second derivatives[4] and a Hessian-update method, for example, the BFGS5−11 and MSP12−14 approaches used for minimum and transition-state optimizations, respectively
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