Fitting elephants in the density functionals zoo: Statistical criteria for the evaluation of density functional theory methods as a suitable replacement for counting parameters

Abstract

Counting parameters has become customary in the density functional theory community as a way to infer the transferability of popular approximations to the exchangecorrelation functionals. Recent work in data science, however, has demonstrated that the number of parameters of a fitted model is not related to the complexity of the model itself, nor to its eventual overfitting. Using similar arguments, here, we show that it is possible to represent every modern exchange-correlation functional approximations using just one single parameter. This procedure proves the futility of the number of parameters as a measure of transferability. To counteract this shortcoming, we introduce and analyze the performance of three statistical criteria for the evaluation of the transferability of exchange-correlation functionals. The three criteria are called Akaike information criterion, Vapnik-Chervonenkis criterion, and cross-validation criterion and are used in a preliminary assessment to rank 60 exchange-correlation functional approximations using the ASCDB database of chemical data.

Accuracy Meets Interpretability for Computational Spectroscopy by Means of Hybrid and Double-Hybrid Functionals

“…Quantum chemical calculations at the DFT level were carried out using some hybrid and double-hybrid density functional approximations of the last generation, which are considered the best performing and transferable according to a very recent benchmark (Peveratti, 2020). Among hybrid functionals, the PW6B95 meta exchange-correlation functional proposed by Zhao and Truhlar (2005) and the ωB97 family of long-range corrected functionals introduced by Chai and Head-Gordon (Chai and Head-Gordon, 2008a,b), namely ωB97, ωB97X and ωB97X-D, were considered in conjunction with aug-cc-pVDZ (Dunning, 1989;Kendall et al, 1992;Woon and Dunning, 1993) and jul-cc-pVDZ (Papajak et al, 2011) double-ζ basis sets.…”

Extrapolating DFT Towards the Complete Basis Set Limit: Lessons from the PBE Family of Functionals

“…The exchange and correlation functionals of Perdew, Burke, and Ernzerhof (PBE) 13 are used, as they represent a "non-empirical" DFA, even though it could be argued parameter-counting is a questionable metric of empiricity. 14 The generalization of the results obtained from this family of PBE-like functionals is discussed below.…”

Extrapolating DFT Toward the Complete Basis Set Limit: Lessons from the PBE Family of Functionals

“…The parameter sets are chosen to cover the range used by most of the common single and double hybrids. The exchange and correlation functionals of Perdew, Burke, and Ernzerhof (PBE) 17 are used, as they represent a "nonempirical" DFA, even though it is arguable whether parameter counting is a reliable metric of empiricity 18 or whether such nonempiricity improves the quality of the functional. 19 The generalization of the results obtained from this family of PBE-like functionals is discussed below.…”