How accurate are the Minnesota density functionals for non-covalent interactions, isomerization energies, thermochemistry, and barrier heights involving molecules composed of main-group elements? Narbe Mardirossiany and Martin Head-Gordon∗,y yKenneth S. Pitzer Center for Theoretical Chemistry, Department of Chemistry, University of California, Berkeley, California 94720, USA zChemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA E-mail: [email protected] Abstract ten hybrid Minnesota functionals). Similarly, the main strength of the local Minnesota den- The 14 Minnesota density functionals pub- sity functionals is that the best ones provide lished between the years 2005 and early 2016 very good performance for thermochemistry are benchmarked on a comprehensive database (e.g. MN15-L), barrier heights (e.g. MN12- of 4986 data points (84 datasets) involving L), and systems heavily characterized by self- molecules composed of main-group elements. interaction error (e.g. MN12-L and MN15-L), The database includes non-covalent interac- while the main weakness is that none of them tions, isomerization energies, thermochemistry, are state-of-the-art for the full spectrum of non- and barrier heights, as well as equilibrium bond covalent interactions and isomerization energies lengths and equilibrium binding energies of (although M06-L is clearly the best from the non-covalent dimers. Additionally, the sensi- four local Minnesota functionals). As an over- tivity of the Minnesota density functionals to all guide, M06-2X and MN15 are perhaps the the choice of basis set and integration grid is most broadly useful hybrid Minnesota function- explored for both non-covalent interactions and als, while M06-L and MN15-L are perhaps the thermochemistry. most broadly useful local Minnesota function- Overall, the main strength of the hybrid als, although each has different strengths and Minnesota density functionals is that the best weaknesses. ones provide very good performance for ther- mochemistry (e.g. M06-2X), barrier heights (e.g. M08-HX, M08-SO, MN15), and systems 1 Introduction heavily characterized by self-interaction error (e.g. M06-2X, M08-HX, M08-SO, MN15), while Density functional theory (DFT) provides an the main weakness is that none of them are in principle exact approach to the problem 1 state-of-the-art for the full spectrum of non- of electronic structure theory. While the ex- covalent interactions and isomerization energies act exchange-correlation functional is unknown, (although M06-2X is recommended from the and approximate functionals continue to face 1 challenging problems,2,3 tremendous progress A second reason is more subtle. More recent has been made in developing practical function- and more sophisticated Minnesota functional als that offer good accuracy for many types of design (for instance the introduction of meta- chemical problems at moderate computational NGAs) has been accompanied by the addition cost. While functionals from the 1990s such as of many more empirical parameters. For the BLYP,4,5 B3LYP,4{6 PBE,7 and PBE08 remain case of local functionals for example, one can widely used (with a combined total of more compare the 34 parameters of M06-L against than 200,000 citations), there has been a prolif- the 44 of M11-L against the 58 of MN12-L and eration of new functionals in the past decade. MN15-L. The same trend can be seen for the Specifically, since 2005, Truhlar and cowork- hybrid functionals as well, increasing from 29 ers have published 14 highly-parameterized lo- (M06-2X) to 47 (M08-HX) to 59 (MN15) pa- cal (M06-L,9 M11-L,10 MN12-L,11 and MN15- rameters. Large increases in the number of pa- L12), global hybrid (M05,13 M05-2X,14 M06,15 rameters can lead to overfitting, which reduces M06-2X,15 M06-HF,16 M08-HX,17 M08-SO,17 predictive power.23 Furthermore, published as- and MN1518), and range-separated hybrid sessments of new Minnesota functionals upon (M1119 and MN12-SX20) density functionals introduction have largely been for datasets on with kinetic energy density dependence. Of which the parameters of each new functional these 14 functionals, 10 have been of the meta- were trained.10{12,19,20 For example, M11-L was generalized gradient approximation (meta- exclusively validated on its own training set GGA) variety, while 4 (MN12-SX, MN15, upon publication, while MN15-L was tested on MN12-L, and MN15-L) have been of the meta- a set of only 56 data points, mostly consist- nonseparable gradient approximation (meta- ing of solid state properties. Success on train- NGA) variety. Table 1 contains a compact ing datasets only confirms the flexibility of the summary of the features of these 14 functionals, functional form to fit the training data, but which are often designated as the \Minnesota" does not indicate accuracy when transferred to functionals. independent problems. Collectively, the Minnesota functionals are Finally, the inclusion of novel datasets in the perhaps the most widely adopted of recently- training set of a semi-empirical density func- developed functionals, with a combined total of tional (such as those containing transition met- more than 10,000 citations across the 12 pub- als, solid-state properties, or multi-reference lished papers. However, since there are now bond energies) can potentially extend its range 14 of them, the question naturally arises as to of applicability (a clearly desirable attribute). which is the best choice for chemical applica- However, it also has the potential of having a tions? That is the main question this work detrimental effect on the accuracy of a func- seeks to address. The answer may not be the tional for conventional interactions. With the obvious one, which is to use the most recent aim of achieving broader accuracy, the train- and most sophisticated functional that is com- ing sets for the most recent Minnesota func- putationally affordable for the task at hand. tionals have been expanded in this direction, One reason is that the failures of semi-empirical with the hope that a large improvement in these density functionals vary for different classes of traditionally unexplored areas (e.g. multi- problems. For systems where strong correla- reference transition-metal bond energies) comes tion (SC) effects are important (multi-reference at the cost of minor loss in accuracy for typical character: primarily a defect of the correlation datasets (e.g. closed-shell non-covalent interac- functional), exact exchange leads to larger er- tions). rors.21 For systems where self-interaction er- For reasons such as those given above, it is ror (SIE) is in play (odd electrons or charge- common to perform project-specific validation transfer excited states: primarily a defect of the calculations using a variety of different func- exchange functional), more exact exchange typ- tionals in order to make a final choice for ap- ically reduces errors.22 plied DFT studies. This is a pragmatic strat- 2 egy, but has the limitation that the conclusions sented in this section in order to summarize the are usually non-transferable, because one reli- evolution of the various exchange and correla- able density functional is virtually never uni- tion functional forms that have been utilized formly better than another. In order to draw since 2005. For further discussion of the Min- more general conclusions about the relative nesota functionals, the reader may wish to con- virtues of the Minnesota functionals, it is es- sult a comprehensive review by Peverati and sential to gather as much high quality test data Truhlar.24 together in one place as possible, and compare The exchange component of the first Min- all 14 functionals against these reference val- nesota functional, M05, used the PBE exchange ues on an equal, unbiased footing. With results functional7 as its foundation and enhanced it of statistical significance in hand, more general with a 12-term power series inhomogeneity cor- conclusions can be drawn. That is the goal of rection factor (ICF) in Becke's τ-dependent this paper. variable,25 w. On the other hand, the corre- After a qualitative description of the 14 Min- lation component of M05 was similar to that of nesota functionals, details regarding the calcu- B9723 with two differences: the same-spin cor- lations, as well as the 84 datasets used for the relation functional utilized Becke's τ-dependent assessment, are given. In brief, the assessment self-correlation correction (SCC) factor26 and database comprises nearly 5000 data points both power series had five terms instead of that have been culled from the benchmarking three. Furthermore, the two nonlinear corre- activities of numerous groups, including Hobza, lation parameters (γc;ss and γc;os) were reopti- Sherrill, Herbert, Grimme, Karton, and Martin. mized and all three uniform electron gas (UEG) The reference data is typically estimated to be limits were satisfied, resulting in a total of 22 at least 10 times more accurate than the very fitted parameters (20 linear and 2 nonlinear), best available density functionals, so that ro- including the linear exact exchange mixing pa- bust and meaningful conclusions can be drawn. rameter (28%). The construction of M05-2X The data points are divided into several cate- was identical to that of M05, except the exact gories: non-covalent interactions, isomerization exchange mixing parameter of M05 was doubled energies, thermochemistry, barrier heights, and and fixed (56%) and the two nonlinear correla- potential energy curves for non-covalent inter- tion parameters were borrowed from M05, for a actions. Within certain categories, the data total of 19 fitted parameters. points are further subdivided into easy cases With M06-L, Zhao and Truhlar set out to de- versus difficult cases, where the latter should velop an efficient, local meta-GGA functional. reflect a strong presence of one of the two afore- The functional form of M06-L was identical to mentioned remaining shortcomings in modern that of an unhybridized M05 or M05-2X, with density functionals { either strong correlation one exception: all three components (exchange, or self-interaction error.