1. Introduction

The success of density functional theory (DFT) as the method of choice for the calculation of the electronic structure of molecules is undeniable, and is intertwined with the development of improved approximations for the description of the exchange–correlation functional (xc functional, or just simply functional) \cite{becke_perspective:_2014}.  Such success is reflected by an indiscriminate proliferation of approximations, calling for a rugged safari across the “zoo of functionals” \cite{goerigk_look_2017,goerigk_trip_2019,jacobsen_directions_2017} to select an appropriate one \cite{pacchioni_first_2015}
Two philosophies are at odds in the world of functional development: the first one originated mainly within the chemistry community from the pioneering work of Becke \cite{becke_hartreefock_1983,becke_density-functional_1997}, which took the approach of using flexible parametrized mathematical forms that are fitted to chemical data, exact constraints, or a mix of both. The second philosophy originated primarily within the physics community from the ground-breaking work of Perdew \cite{langreth_gradient_1979,langreth_theory_1980,perdew_accurate_1985}, which expanded the knowledge and application of exact conditions and advocated for DFT to remain a purely ab initio method. These two philosophies have been largely constructive with each other, sharing ideas, providing criticisms, and validating results \cite{becke_perspective:_2014,yu_perspective:_2016,mardirossian_thirty_2017,goerigk_trip_2019}. A frequent question used to navigate the zoo of density functionals—perhaps guided by the famous John von Neumann's quote: “with four parameters I can fit an elephant, and with five I can make him wiggle his trunk” \cite{dyson_meeting_2004}—is: “how many parameters does this functional have?”. This question, in fact, underlies the more fundamental assumption that the number of parameters is a reliable criterion to evaluate the transferability of the results—but is it really? As pointed out in several occasions \cite{chan_extensive_2000,peverati_quest_2014,yu_perspective:_2016,civalleri_choosing_2012}, counting the number of parameters is not always as straightforward as it might initially appear, especially for functionals that are not directly fitted to data. In fact, there is no such thing as a truly parameter-free or “zero-parameter” xc functional approximation, since even functionals that are usually considered as such have mathematical forms that contain parameters that are then determined based on theoretical arguments. Since the true functional is still unknown, and potentially unknowable,\cite{schuch_computational_2009} it seems clear that every xc functional approximation must contain an empirical element \cite{yu_perspective:_2016}.
Instead of counting fitted parameters in “parametrized functionals” and compare them to hidden parameters in “zero-parameter” functionals, the first portion of this article explores the somehow opposite scenario where every functional—regardless of its development philosophy—is represented using a simple function containing one single parameter, as presented in section \ref{816226}.  This new representation is a direct adaptation of the recent works of Piantadosi \cite{piantadosi_one_2018} and Boué \cite{boue_real_2019}, where any distribution of points in any dimension is represented by a well-behaved scalar function with a single real-valued parameter. In other words, quoting Piantadosi’s paper title: “One parameter is always enough”, even for xc functionals. The result of this procedure is that every single functional on the first three rungs of Perdew and Schmidt’s “Jacob's ladder” \cite{perdew_jacobs_2001} (corresponding to LDA, GGA, and meta-GGA approximations) can be represented by just one single parameter. Famous “zero-parameter” functionals, such as PBE \cite{perdew_generalized_1996} and SCAN \cite{sun_strongly_2015}, as well as popular “parametrized functionals”, such as the Minnesota family \cite{zhao_exchange-correlation_2005,zhao_design_2006,zhao_new_2006,zhao_density_2006,zhao_m06_2008,zhao_exploring_2008,peverati_improving_2011,peverati_m11-l:_2012,peverati_screened-exchange_2012,peverati_improved_2012,yu_mn15:_2016,yu_mn15-l:_2016}, are all defined by one number. 
Having proven the inadequacy of the “number of parameters” as a measure of transferability of xc functionals, the focus of this article shifts to develop a set of statistical criteria that can be appropriately used for this task. Since the exact functional is still unknown, these criteria must rely on statistical analysis of data across as many different chemical and physical properties as possible. Luckily, several benchmark results with hundreds of functionals are already available in the literature \cite{goerigk_general_2010,goerigk_efficient_2011,goerigk_look_2017,yu_perspective:_2016,mardirossian_thirty_2017,santra_minimally_2019,martin_empirical_nodate}, but their analysis is not unequivocal, and might even produce contrasting recommendations. This is because the large number of data in these studies can be in principle sliced and grouped into any number of ad hoc subsets, that can then be used to statistically validate pretty much any hypothesis. Recent work from the Author’s lab has introduced a new unbiased subdivision of some of the most popular DFT databases generated without human intervention by means of data-science algorithms \cite{morgante_statistically_2019}. Interestingly enough, concepts that can be derived using simple chemical intuition have been also recovered by a posteriori analysis of the machine-generated groups. This reassuring fact validates the chemical-intuition–based approach that was used by DFT developers to group and analyze the data, but the data-science approach offer several other advantages nonetheless. One of this advantages is demonstrated in Section \ref{898228}, where  the unbiased subsets are used as the basis for three new statistical criteria obtained adapting the Akaike information criterion (AIC), the Vapnik–Chervonenkis criterion (VCC), and a new cross-validation criterion (CVC) to the DFT results. Preliminary rankings of 60 popular xc functionals are also presented and briefly discussed.