Narbe Mardirossian

Mapping the genome of meta-generalized gradient approximation density functionals: The search for B97M-V

A meta-generalized gradient approximation density functional paired with the VV10 nonlocal correlation functional is presented. The functional form is selected from more than 10(10) choices carved out of a functional space of almost 10(40) possibilities. Raw data come from training a vast number of candidate functional forms on a comprehensive training set of 1095 data points and testing the resulting fits on a comprehensive primary test set of 1153 data points. Functional forms are ranked based on their ability to reproduce the data in both the training and primary test sets with minimum empiricism, and filtered based on a set of physical constraints and an often-overlooked condition of satisfactory numerical precision with medium-sized integration grids. The resulting optimal functional form has 4 linear exchange parameters, 4 linear same-spin correlation parameters, and 4 linear opposite-spin correlation parameters, for a total of 12 fitted parameters. The final density functional, B97M-V, is further assessed on a secondary test set of 212 data points, applied to several large systems including the coronene dimer and water clusters, tested for the accurate prediction of intramolecular and intermolecular geometries, verified to have a readily attainable basis set limit, and checked for grid sensitivity. Compared to existing density functionals, B97M-V is remarkably accurate for non-bonded interactions and very satisfactory for thermochemical quantities such as atomization energies, but inherits the demonstrable limitations of existing local density functionals for barrier heights.

ωB97M-V: A combinatorially optimized, range-separated hybrid, meta-GGA density functional with VV10 nonlocal correlation

A combinatorially optimized, range-separated hybrid, meta-GGA density functional with VV10 nonlocal correlation is presented. The final 12-parameter functional form is selected from approximately 10 × 10(9) candidate fits that are trained on a training set of 870 data points and tested on a primary test set of 2964 data points. The resulting density functional, ωB97M-V, is further tested for transferability on a secondary test set of 1152 data points. For comparison, ωB97M-V is benchmarked against 11 leading density functionals including M06-2X, ωB97X-D, M08-HX, M11, ωM05-D, ωB97X-V, and MN15. Encouragingly, the overall performance of ωB97M-V on nearly 5000 data points clearly surpasses that of all of the tested density functionals. In order to facilitate the use of ωB97M-V, its basis set dependence and integration grid sensitivity are thoroughly assessed, and recommendations that take into account both efficiency and accuracy are provided.

How Accurate Are the Minnesota Density Functionals for Noncovalent Interactions, Isomerization Energies, Thermochemistry, and Barrier Heights Involving Molecules Composed of Main-Group Elements?

The 14 Minnesota density functionals published between the years 2005 and early 2016 are benchmarked on a comprehensive database of 4986 data points (84 data sets) involving molecules composed of main-group elements. The database includes noncovalent interactions, isomerization energies, thermochemistry, and barrier heights, as well as equilibrium bond lengths and equilibrium binding energies of noncovalent dimers. Additionally, the sensitivity of the Minnesota density functionals to the choice of basis set and integration grid is explored for both noncovalent interactions and thermochemistry. Overall, the main strength of the hybrid Minnesota density functionals is that the best ones provide very good performance for thermochemistry (e.g., M06-2X), barrier heights (e.g., M08-HX, M08-SO, MN15), and systems heavily characterized by self-interaction error (e.g., M06-2X, M08-HX, M08-SO, MN15), while the main weakness is that none of them are state-of-the-art for the full spectrum of noncovalent interactions and isomerization energies (although M06-2X is recommended from the 10 hybrid Minnesota functionals). Similarly, the main strength of the local Minnesota density functionals is that the best ones provide very good performance for thermochemistry (e.g., MN15-L), barrier heights (e.g., MN12-L), and systems heavily characterized by self-interaction error (e.g., MN12-L and MN15-L), while the main weakness is that none of them are state-of-the-art for the full spectrum of noncovalent interactions and isomerization energies (although M06-L is clearly the best from the four local Minnesota functionals). As an overall guide, M06-2X and MN15 are perhaps the most broadly useful hybrid Minnesota functionals, while M06-L and MN15-L are perhaps the most broadly useful local Minnesota functionals, although each has different strengths and weaknesses.

Thirty years of density functional theory in computational chemistry: an overview and extensive assessment of 200 density functionals

ABSTRACT In the past 30 years, Kohn–Sham density functional theory has emerged as the most popular electronic structure method in computational chemistry. To assess the ever-increasing number of approximate exchange-correlation functionals, this review benchmarks a total of 200 density functionals on a molecular database (MGCDB84) of nearly 5000 data points. The database employed, provided as Supplemental Data, is comprised of 84 data-sets and contains non-covalent interactions, isomerisation energies, thermochemistry, and barrier heights. In addition, the evolution of non-empirical and semi-empirical density functional design is reviewed, and guidelines are provided for the proper and effective use of density functionals. The most promising functional considered is ωB97M-V, a range-separated hybrid meta-GGA with VV10 nonlocal correlation, designed using a combinatorial approach. From the local GGAs, B97-D3, revPBE-D3, and BLYP-D3 are recommended, while from the local meta-GGAs, B97M-rV is the leading choice, followed by MS1-D3 and M06-L-D3. The best hybrid GGAs are ωB97X-V, ωB97X-D3, and ωB97X-D, while useful hybrid meta-GGAs (besides ωB97M-V) include ωM05-D, M06-2X-D3, and MN15. Ultimately, todays state-of-the-art functionals are close to achieving the level of accuracy desired for a broad range of chemical applications, and the principal remaining limitations are associated with systems that exhibit significant self-interaction/delocalisation errors and/or strong correlation effects.

The Performance of Density Functionals for Sulfate–Water Clusters

The performance of 24 density functionals, Hartree-Fock, and MP2 is assessed with respect to the CCSD(T)/CBS* energetics of 49 sulfate-water clusters with three to six water molecules. Included among the density functionals are GGA, meta-GGA, hybrid GGA, hybrid meta-GGA, and double hybrid density functionals, as well as the LDA. Three types of dispersion corrections (VV10, XDM, and -D) are tested in conjunction with these functionals. The 26 methods are compared using the relative and binding energies of the sulfate-water clusters as the main criteria. It was discovered that a majority of the tested density functionals are unable to simultaneously capture the physics necessary to describe both the relative and binding energies of the anionic solvation clusters. The three density functionals with the best overall performance are XYG3, ωB97X-2, and XYGJ-OS. The only other density functional that performs comparably to these three double hybrids is M11. A majority of the density functionals that contain a fraction of exact exchange tend to perform well only for the relative energies, while functionals lacking exact exchange generally perform poorly with respect to both criteria. However, the meta-GGA functional, M11-L, stands out due to its superior performance for the relative energies. While dispersion correction functionals cannot replace the accuracy provided by MP2 correlation, it is shown that the proper combination of a hybrid GGA functional (LC-ωPBE) with a dispersion correction functional (VV10) can lead to drastic improvements in the binding energies of the parent functional, while preserving its performance with respect to the relative energies. Ultimately, however, MP2 has the best overall performance out of the 26 benchmarked methods.

Characterizing and Understanding the Remarkably Slow Basis Set Convergence of Several Minnesota Density Functionals for Intermolecular Interaction Energies.

For a set of eight equilibrium intermolecular complexes, it is discovered that the basis set limit (BSL) cannot be reached by aug-cc-pV5Z for three of the Minnesota density functionals: M06-L, M06-HF, and M11-L. In addition, the M06 and M11 functionals exhibit substantial, but less severe, difficulties in reaching the BSL. By using successively finer grids, it is demonstrated that this issue is not related to the numerical integration of the exchange-correlation functional. In addition, it is shown that the difficulty in reaching the BSL is not a direct consequence of the structure of the augmented functions in Dunnings basis sets, since modified augmentation yields similar results. By using a very large custom basis set, the BSL appears to be reached for the HF dimer for all of the functionals. As a result, it is concluded that the difficulties faced by several of the Minnesota density functionals are related to an interplay between the form of these functionals and the structure of standard basis sets. It is speculated that the difficulty in reaching the basis set limit is related to the magnitude of the inhomogeneity correction factor (ICF) of the exchange functional. A simple modification of the M06-L exchange functional that systematically reduces the basis set superposition error (BSSE) for the HF dimer in the aug-cc-pVQZ basis set is presented, further supporting the speculation that the difficulty in reaching the BSL is caused by the magnitude of the exchange functional ICF. Finally, the BSSE is plotted with respect to the internuclear distance of the neon dimer for two of the examined functionals.

Use of the rVV10 Nonlocal Correlation Functional in the B97M-V Density Functional: Defining B97M-rV and Related Functionals

The VV10 and rVV10 nonlocal correlation functionals are consistently implemented and assessed, with the goal of determining if the rVV10 nonlocal correlation functional can replace the VV10 nonlocal correlation functional in the recently developed B97M-V density functional, to give the B97M-rV density functional. Along the way, four density functionals are simultaneously tested: VV10, rVV10, B97M-V, and B97M-rV. An initial assessment is carried out across the S22 data set, and the short-range damping variable, b, is varied for all four density functionals in order to determine the sensitivity of the functionals to the empirical parameter. The results of this test indicate that a value of b = 6 (fortuitously the same as that in B97M-V) is suitable for B97M-rV. The functionals are then compared across an extensive database of interaction energies, and it is demonstrated that B97M-rV either matches or outperforms B97M-V for all of the tests considered. Finally, the optimization of b across the S22 data set is extended to two range-separated hybrid density functionals, ωB97X-V and ωB97M-V, and a value of b = 6.2 is recommended for both ωB97X-rV and ωB97M-rV.

Self-consistent implementation of meta-GGA functionals for the ONETEP linear-scaling electronic structure package

Accurate and computationally efficient exchange-correlation functionals are critical to the successful application of linear-scaling density functional theory (DFT). Local and semi-local functionals of the density are naturally compatible with linear-scaling approaches, having a general form which assumes the locality of electronic interactions and which can be efficiently evaluated by numerical quadrature. Presently, the most sophisticated and flexible semi-local functionals are members of the meta-generalized-gradient approximation (meta-GGA) family, and depend upon the kinetic energy density, τ, in addition to the charge density and its gradient. In order to extend the theoretical and computational advantages of τ-dependent meta-GGA functionals to large-scale DFT calculations on thousands of atoms, we have implemented support for τ-dependent meta-GGA functionals in the ONETEP program. In this paper we lay out the theoretical innovations necessary to implement τ-dependent meta-GGA functionals within ONETEPs linear-scaling formalism. We present expressions for the gradient of the τ-dependent exchange-correlation energy, necessary for direct energy minimization. We also derive the forms of the τ-dependent exchange-correlation potential and kinetic energy density in terms of the strictly localized, self-consistently optimized orbitals used by ONETEP. To validate the numerical accuracy of our self-consistent meta-GGA implementation, we performed calculations using the B97M-V and PKZB meta-GGAs on a variety of small molecules. Using only a minimal basis set of self-consistently optimized local orbitals, we obtain energies in excellent agreement with large basis set calculations performed using other codes. Finally, to establish the linear-scaling computational cost and applicability of our approach to large-scale calculations, we present the outcome of self-consistent meta-GGA calculations on amyloid fibrils of increasing size, up to tens of thousands of atoms.

Survival of the most transferable at the top of Jacob’s ladder: Defining and testing the ωB97M(2) double hybrid density functional

A meta-generalized gradient approximation, range-separated double hybrid (DH) density functional with VV10 non-local correlation is presented. The final 14-parameter functional form is determined by screening trillions of candidate fits through a combination of best subset selection, forward stepwise selection, and random sample consensus (RANSAC) outlier detection. The MGCDB84 database of 4986 data points is employed in this work, containing a training set of 870 data points, a validation set of 2964 data points, and a test set of 1152 data points. Following an xDH approach, orbitals from the ωB97M-V density functional are used to compute the second-order perturbation theory correction. The resulting functional, ωB97M(2), is benchmarked against a variety of leading double hybrid density functionals, including B2PLYP-D3(BJ), B2GPPLYP-D3(BJ), ωB97X-2(TQZ), XYG3, PTPSS-D3(0), XYGJ-OS, DSD-PBEP86-D3(BJ), and DSD-PBEPBE-D3(BJ). Encouragingly, the overall performance of ωB97M(2) on nearly 5000 data points clearly surpasses that of all of the tested density functionals. As a Rung 5 density functional, ωB97M(2) completes our family of combinatorially optimized functionals, complementing B97M-V on Rung 3, and ωB97X-V and ωB97M-V on Rung 4. The results suggest that ωB97M(2) has the potential to serve as a powerful predictive tool for accurate and efficient electronic structure calculations of main-group chemistry.

Approaching the basis set limit for DFT calculations using an environment-adapted minimal basis with perturbation theory: Formulation, proof of concept, and a pilot implementation

Recently developed density functionals have good accuracy for both thermochemistry (TC) and non-covalent interactions (NC) if very large atomic orbital basis sets are used. To approach the basis set limit with potentially lower computational cost, a new self-consistent field (SCF) scheme is presented that employs minimal adaptive basis (MAB) functions. The MAB functions are optimized on each atomic site by minimizing a surrogate function. High accuracy is obtained by applying a perturbative correction (PC) to the MAB calculation, similar to dual basis approaches. Compared to exact SCF results, using this MAB-SCF (PC) approach with the same large target basis set produces <0.15 kcal/mol root-mean-square deviations for most of the tested TC datasets, and <0.1 kcal/mol for most of the NC datasets. The performance of density functionals near the basis set limit can be even better reproduced. With further improvement to its implementation, MAB-SCF (PC) is a promising lower-cost substitute for conventional large-basis calculations as a method to approach the basis set limit of modern density functionals.