Development of an exchange–correlation functional with uncertainty quantification capabilities for density functional theory

Abstract

This paper presents the development of a new exchange-correlation functional from the point of view of machine learning. Using atomization energies of solids and small molecules, we train a linear model for the exchange enhancement factor using a Bayesian approach which allows for the quantification of uncertainties in the predictions. A relevance vector machine is used to automatically select the most relevant terms of the model. We then test this model on atomization energies and also on bulk properties. The average model provides a mean absolute error of only 0.116 eV for the test points of the G2/97 set but a larger 0.314 eV for the test solids. In terms of bulk properties, the prediction for transition metals and monovalent semiconductors has a very low test error. However, as expected, predictions for types of materials not represented in the training set such as ionic solids show much larger errors.

mBEEF-vdW: Robust fitting of error estimation density functionals

“…16 For recent advancements in the development of meta-GGA functionals, both empirical and non-empirical, see Ref. 6,[17][18][19][20][21][22][23][24][25][26] The BEEF-vdW 9 functional was fitted within semilocal generalized gradient approximation (GGA) for exchange, which depends on the electronic density and its derivative. Its correlation was a fitted mixture of a Local Density Approximation (LDA), semi-local GGA, and non-local correlation of Rutgers-Chalmers approximation for van der Waals forces.…”

Optimization of an exchange-correlation density functional for water

“…The resulting Bayesian method for functional optimization shares many methodological characteristics of the BEEF method of Wellendorff et al [31][32][33] , as well as with that of Aldegunde et al 34 . Like them, it has the ability to quantify the errors due to the uncertainty of the resulting functional, although we will not discuss this ability in the present work.…”

Efficient and accurate evaluation of potential energy matrix elements for quantum dynamics using Gaussian process regression

“…For example, recent work has employed Gaussian process methods 65 to achieve machine-learning of accurate PESs describing both atom and molecular chemical systems; the resulting Gaussian approximation potential (GAP [69][70][71] ) or kriging method 64,[66][67][68] offers a route to performing molecular simulations on PESs which approximate ab initio electronic structure, albeit at a much lower computational cost. Moving away from PES interpolation, machine learning methods such as kernel ridge regression, Bayesian inference, and artificial neural networks, have found application in relating ab initio atomization energies to simple molecular descriptors such as atomic partial charges, 78 in learning optimized exchangecorrelation functionals for density functional theory (DFT) calculations, 79,80 and in learning accurate interatomic PESs for large-scale molecular simulations. 81 Within the same field of chemical dynamics, a particular interest of the present work, machine-learning in the form of a support vector machine has also found application in determination of optimal dividing surfaces for transition state theory calculations.…”