Assessment of two hybrid van der Waals density functionals for covalent and non-covalent binding of molecules

TL;DR

This study evaluates two hybrid van der Waals density functionals for their accuracy in modeling covalent and non-covalent molecular interactions, highlighting their strengths and limitations compared to existing methods.

Contribution

The paper introduces and assesses two new hybrid vdW-DFs incorporating 25% Fock exchange, providing insights into their performance for covalent and non-covalent binding properties.

Findings

Hybrid-vdW-DF-cx performs similarly to PBE0 for covalent properties.

Hybrid-vdW-DF2 shows larger deviations in covalent binding energies.

Hybrid-vdW-DF-cx excels in hydrogen-bonded systems, especially in non-covalent interactions.

Abstract

Two hybrid van der Waals density functionals (vdW-DFs) are constructed using 25%, Fock exchange with i) the consistent-exchange vdW-DF-cx functional and ii) with the vdW-DF2 functional. The ability to describe covalent and non-covalent binding properties of molecules are assessed. For properties related to covalent binding, atomization energies (G2-1 set), molecular reaction energies (G2RC set), as well as ionization energies (G21IP set) are benchmarked against experimental reference values. We find that hybrid-vdW-DF-cx yields results that are rather similar to those of the standard non-empirical hybrid PBE0 [JCP 110, 6158 (1996)]. Hybrid vdW-DF2 follows somewhat different trends, showing on average significantly larger deviations from the reference energies, with a MAD of 14.5 kcal/mol for the G2-1 set. Non-covalent binding properties of molecules are assessed using the S22 benchmark set of non-covalently bonded dimers and the X40 set of dimers of small halogenated molecules, using wavefunction-based quantum chemistry results for references. For the S22 set, hybrid-vdW-DF-cx performs better than standard vdW-DF-cx for the mostly hydrogen-bonded systems. Hybrid-vdW-DF2 offers a slight improvement over standard vdW-DF2. Similar trends are found for the X40 set, with hybrid-vdW-DF-cx performing particularly well for binding involving the strongly polar hydrogen halides, but poorly for systems with tiny binding energies. Our study of the X40 set reveals both the potential of mixing Fock exchange with vdW-DF, but also highlights shortcomings of the hybrids constructed here. The solid performance of hybrid-vdW-DF-cx for covalent-bonded systems, as well as the strengths and issues uncovered for non-covalently bonded systems, makes this study a good starting point for developing even more precise hybrid vdW-DFs.

Tables

Table 1. Table 1: Benchmark of the atomization energies in the G2-1 set. Goerigk and Grimme ( 2010 ) The table lists mean relative deviation (MRD), mean absolute relative deviation (MARD), mean deviation (MD), and MAD values of the functionals considered here relative to the experimental references.

Molecule	PBE	PBE0	DF2	DF2-0	DF-cx	DF-cx0
MRD [%]	3.63	-2.60	-0.82	-7.25	5.82	-1.40
MARD [%]	5.93	4.12	4.57	9.48	7.36	3.68
MD [kcal/mol]	5.82	-3.36	-2.34	-12.15	9.09	-1.48
MAD [kcal/mol]	7.52	4.99	6.60	14.47	9.73	4.90

Table 2. Table 2: Comparison of the performance of PBE0 and the hybrid van der Waals density functionals, vdW-DF2-0 and vdW-DF-cx0 for chemical reaction energies in the G2RC set. Goerigk and Grimme ( 2010 )

Molecule	PBE	PBE0	DF2	DF2-0	DF-cx	DF-cx0
MRD [%]	-15.8	5.3	-31.1	-11.4	-26.4	-8.6
MARD [%]	37.3	33.0	59.4	36.0	47.3	30.2
MD [kcal/mol]	0.73	-2.75	6.81	2.72	1.92	-1.09
MAD [kcal/mol]	5.63	6.06	10.01	5.76	6.14	4.77

Table 3. Table 3: Comparison of performance of PBE0 and the hybrid van der Waals density functionals vdW-DF2-0 and vdW-DF-cx0 for ionization potential values in the G21IP set. Goerigk and Grimme ( 2010 )

Molecule	PBE	PBE0	DF2	DF2-0	DF-cx	DF-cx0
MRD [%]	-0.35	-0.17	3.11	3.04	-0.62	0.17
MARD [%]	1.98	1.97	3.28	3.30	1.72	1.67
MD [kcal/mol]	-1.30	-0.91	7.38	7.05	-1.81	0.08
MAD [kcal/mol]	4.87	4.68	7.80	7.70	4.23	3.86

Table 4. Table 4: Summary of our benchmark of the X40 set binding energies.

	DF2	DF2-0	DF-cx	DF-cx0
involving methane
MAD[kcal/mol]	0.27	0.15	0.20	0.24
MADR[%]	39.3	20.1	25.7	29.9
involving benzene
MAD[kcal/mol]	0.11	0.12	0.25	0.45
MADR[%]	3.5	3.3	7.0	10.3
involving hydrogen halides
MAD[kcal/mol]	0.30	0.57	0.54	0.14
MADR[%]	3.0	5.9	5.1	1.6
other dimers
MAD[kcal/mol]	0.24	0.17	0.28	0.15
MADR[%]	10.3	6.0	10.4	6.2
Full set
MAD[kcal/mol]	0.22	0.21	0.32	0.24
MADR[%]	13.2	8.1	12.4	11.7

Full text

I Introduction

Hybrid functionals are widely used for computing molecular properties because they generally offer more accurate thermochemical and structural properties than non-hybrid semilocal functionals in density functional theory (DFT).Becke (1993); Stephens et al. (1994); Adamo and Barone (1999); Heyd, Scuseria, and Ernzerhof (2003, 2006); Yanai, Pew, and Handy (2004); Goerigk and Grimme (2011); Becke (2014) Through various procedures, hybrid functionals mix in a fraction of Fock exchange, introducing a Fock term in the exchange part of a Kohn-Sham density functional. Many semi-empirical or fitted hybrid functionals have been developed, with B3LYP being one of the most popular.Becke (2014, 1993); Stephens et al. (1994); Lee, Yang, and Parr (1988) One of the most successful hybrids, the PBE0 functional,Adamo and Barone (1999) one the other hand mixes in 25% of Fock exchange with the exchange of the PBE,Perdew, Burke, and Ernzerhof (1996); Burke (2012) which is based on arguments from many-particle theory.Perdew, Ernzerhof, and Burke (1996); Burke, Ernzerhof, and Perdew (1997); Ernzerhof, Perdew, and Burke (1997)

In many molecular complexes, London dispersion forces contribute significantly to the binding energy of the system. To accurately describe all binding properties of such system, it is desirable to include both non-local correlation and a Fock-exchange fraction within the same functional.Becke (2014) Non-local correlations are missing in both semi-local functionals in the generalized-gradient approximation (GGA) and in standard hybrid functionals, but they are included from the onset in the van der Waals density functional (vdW-DF) method.Dion et al. (2004); Thonhauser et al. (2007); Lee et al. (2010); Berland and Hyldgaard (2014); Hyldgaard, Berland, and Schröder (2014); Thonhauser et al. (2015); Berland et al. (2015); Román-Pérez and Soler (2009) vdW-DF has found a wide user base in physics and material science.Klimeš and Michaelides (2012); Berland et al. (2014, 2015) It is regularly used in the study of adsorption,Chakarova-Käck et al. (2006); Berland et al. (2011); Erhart, Hyldgaard, and Lindroth (2015); Liu et al. (2012); Carrasco et al. (2011) layered systems,Rydberg et al. (2003); Londero et al. (2012); Björkman (2014); Peelaers and de Walle (2014); Lindroth and Erhart (2016) molecular crystals,Ambrosch-Draxl et al. (2009); Berland and Hyldgaard (2010); Rangel et al. (2016) and more recently solids with covalent and metallic bonds.Klimeš, Bowler, and Michaelides (2011); Thonhauser et al. (2015); Gharaee, Erhart, and Hyldgaard (2017) However, assessments of how well vdW-DFs describe general properties of molecular systems are scarce, which is likely related to the fact that hybrid vdW-DFs have so far gone unexplored.

In this paper we define and assess the performance of two simple, unscreened hybrid van der Waals functionals, termed vdW-DF-cx0 and vdW-DF2-0, for molecular systems. A number of dispersion-corrected (DFT-D) hybrid functionals do already exist.Schwabe and Grimme (2007); Chai and Head-Gordon (2008); Grimme (2011); Grimme et al. (2010) Those descriptions rely on atom-centered pair-potential corrections to account the non-local correlation. In more advanced variants, the coefficients of these pair potentials can depend on for instance the electron density, as they are in DFT-TS and more recent extensionsTkatchenko and Scheffler (2009); Tkatchenko et al. (2012) and in associated hybrids. Marom et al. (2011) Hybrid variants of the Vydrov-Voorhis (VV) nonlocal density functional, Vydrov and Van Voorhis (2010) which borrows several features of vdW-DF, has also been developed. All of these schemes include one or several adjustable parameters that can be re-tuned when combined with a hybrid functional to achieve good performance for a set of reference systems. Here, we investigate the performance of two hybrid vdW-DFs where we deliberately avoid fitting to reference systems.

The vdW-DF framework is formally well anchored in many-body physics. Dion et al. (2004, 2005); Thonhauser et al. (2007); Hyldgaard, Berland, and Schröder (2014); Berland et al. (2015) However, different versions of the functional have been developed,Dion et al. (2004); Cooper (2010); Klimeš, Bowler, and Michaelides (2011); Hamada (2014) such as vdW-DF2Lee et al. (2010) and vdW-DF-cx,Berland and Hyldgaard (2014); Berland et al. (2014) which emphasize different limits of many-body theory. In this work we define two new vdW-DF hybrids, vdW-DF-cx0 and vdW-DF2-0, modifying vdW-DF-cx and vdW-DF2 to use 25% of Fock exchange, in analogy with the design of PBE0.Adamo and Barone (1999); Perdew, Ernzerhof, and Burke (1996); Burke, Ernzerhof, and Perdew (1997); Ernzerhof, Perdew, and Burke (1997) The 25% mixing fraction has been rationalizedBecke (1993); Perdew, Ernzerhof, and Burke (1996); Burke, Ernzerhof, and Perdew (1997); Ernzerhof, Perdew, and Burke (1997) in terms of a coupling-constant $\lambda$ integration in the adiabatic-connection formula (ACF).Gunnarsson and Lundqvist (1976); Langreth and Perdew (1977) In a mean-value evaluation of the ACF, using a fraction of Fock exchange takes care of the $\lambda\to 0$ limit, where the functional is exclusively represented by exchange. At the $\lambda\to 1$ end the properties of a system are dominated by plasmons,Nozières and Pines (1958) suggesting that functionals of the electron-gas tradition, like PBE, provide robust starting points for hybrid constructions, like PBE0.Perdew, Ernzerhof, and Burke (1996); Burke, Ernzerhof, and Perdew (1997) However, the same rationale also applies to the consistent-exchange vdW-DF-cx version, which rely on vdW-DF’s single plasmon-pole description for specifying the exchange and correlation, except in very inhomogeneous regions.Berland and Hyldgaard (2014); Berland et al. (2014) In fact, vdW-DF, and this version in particular, can be viewed as a series expansion of the effective response defined by a GGA-type exchange-correlation hole.Hyldgaard, Berland, and Schröder (2014); Berland et al. (2015)

Even if less strictly enforced, vdW-DF2 is also designed with consistency between the exchange and correlation in mind and can thus also be seen as plasmon based.Lee et al. (2010); Berland et al. (2014); Hyldgaard, Berland, and Schröder (2014); Berland et al. (2015) Whereas vdW-DF-cx has a good consistency between its rather soft exchange functional and the internal exchange that sets the strength of the non-local correlation term, vdW-DF2 relies on more rapidly increasing exchange enhancement factors in both the exchange and within the non-local correlation.Berland and Hyldgaard (2013); Schwinger (1981) vdW-DF-cx and vdW-DF2 can therefore be viewed as two different limits of viable vdW-inclusive density functionals. Thus by benchmarking these two functional for molecular systems, we gain insight into how to best construct an accurate hybrid vdW-DF.

II computational details

vdW-DF-cx0 and vdW-DF2-0 were implemented within version 6.0 of the quantum espresso package.Giannozzi et al. (2009) This version includes an implementation of spin-vdW-DF,Thonhauser et al. (2015) which is needed to correctly compute atomization energies. In all calculations, Martins-TroullierTroullier and Martins (1991) norm-conserving PBE pseudopotentials of the FHI-abinitFuc (1999) kind are employed. The energy cutoff is set to 80 Ry. Atomic coordinates were relaxed (to 0.0025 eV/Å) in all studies, except for the S22 set.Jurecka et al. (2006) Like in several earlier studies,Lee et al. (2010); Berland and Hyldgaard (2014) the internal molecular coordinates were kept fixed for the S22 set and the optimal separations were determined by calculating potential energy curves.

III Results

III.1 Covalent binding properties

A good hybrid vdW-DF should be able to describe both covalent and non-covalent binding accurately. To assess vdW-DF-cx0 and vdW-DF2-0 ability to describe covalent binding properties, we consider atomization energies (in the G2-1 subset) of molecules and chemical reaction energies (in the G2RC subset) of the G2 benchmark set.Goerigk and Grimme (2010) We furthermore benchmark molecular ionization potentials (the G2IP subset).Goerigk and Grimme (2010) We compare with experimental reference values, in the case of G2-1 and G2RC subsets back-corrected by wavefunction-based quantum chemistry results for zero-point energies.Goerigk and Grimme (2010)

Fig. 1 provides an overview of the mean absolute deviation (MAD) of the energies of the three sets, displaying results of vdW-DF-cx0, vdW-DF2-0 and their non-hybrid equivalents.111 Results of the G1 set (subset of G2-1) for the non-hybrid spin vdW-DF was presented in the supplementary material of Ref. Thonhauser et al., 2015. These calculations were based on norm-conserving pseudopotentials rather than ultrasoft pseudopotentials and were compared to the values of the G1G1set set rather than the updated reference values of the G2 sets.Goerigk and Grimme (2010) For reference, PBE and PBE0 results are also shown. The comparison shows that among the non-hybrid functionals, PBE performs best, except for atomization energies where vdW-DF2 performs better. Among the hybrids, vdW-DF-cx0 has the best overall performance, slightly better than that of PBE0. Both vdW-DF-cx0 and PBE0 perform better than the corresponding non-hybrid functionals. In contrast, vdW-DF2-0 does not necessarily perform better than vdW-DF2. To gain insight into these trends, we next consider the results of the three sets in more detail.

Table 1 gives an overview of our results for atomization energies, in terms of MAD, mean absolute relative deviation (MARD), mean deviation (MD), and mean relative deviation (MRD). vdW-DF2-0 is significantly less accurate than vdW-DF2, but vdW-DF-cx0 is significantly more accurate than vdW-DF-cx. Comparing the MRD values of a non-hybrid with a corresponding hybrid indicates that the atomization energies are on average about 10 % lower for the hybrid than for the non-hybrids. Since vdW-DF2 underestimates atomization energies on average by about 2 %, vdW-DF2-0 underestimates them by about 12 %. This systematic underestimation leads to large MAD and MARD values of vdW-DF2-0. vdW-DF-cx overestimates atomization energies by 9% on average, giving it the worst performance among the non-hybrid functionals. However, this overestimation is a good match with the reduction in atomization energies when introducing 25% Fock exchange, resulting in an MRD value of merely -1.48%. In turn, vdW-DF-cx0 ends up with the smallest MARD value (3.68 %) among the tested functionals, somewhat smaller than the PBE0 value 4.12 %.

For reaction energies, the hybrid vdW-DFs outperform the non-hybrids. While Table 2 shows that the MAD becomes somewhat smaller, the relative deviations are reduced significantly. For instance, the MARD value of 47% for vdW-DF-cx drops to 30% for vdW-DF-cx0. For vdW-DF2 and vdW-DF2-0, the corresponding numbers are 59% and 36%. Despite having poorer atomization energies, as discussed earlier, vdW-DF2-0 predicts reaction energies more accurately than vdW-DF2. This suggests a partial error cancellation between the atomization energies of both product and reactant.

Fig. 2 shows that the deviations grow quite slowly with the size of the reaction energies. In the figure, the reactions are ordered according to the size of their respective reaction energies (gray bars in the upper and middle panels, magnitude given by the right vertical axis). The comparison shows that the deviations of vdW-DF-cx0 are typically quite similar to those of PBE0, while overall being slightly more accurate. vdW-DF-cx also exhibits deviations that are fairly similar to those of PBE (as shown in panel 1 of Figure 2). In contrast, the deviations of vdW-DF2 and vdW-DF2-0 do not follow the same general trends as PBE, PBE0, vdW-DF-cx, and vdW-DF-cx0.

The bottom panel of figure 2 shows that the relative deviations of the different reaction energies. The relative deviations are typically less then 20% for the reactions with large reaction energies, as shown in the inset of the bottom panel, but they can be much larger for the cases with small reaction energies. In the worst cases, even the sign can be wrong, corresponding to relative deviations less than -1. The hybrid variants generally reduce the deviations of the reaction energies. As an example, the energy of the reaction \rm{CH}_{2}\rm{O}_{2}$$\rightarrow$$\rm{CO}_{2}+$\rm{H}_{2}$ (shown in the inset) is -2.0 kcal/mol. vdW-DF2 incorrectly predicts this to be an endothermic reaction with reaction energy of 6.1 kcal/mol which drops to 2.9 kcal/mol when going to vdW-DF2-0. vdW-DF-cx correctly predicts this reaction to be exothermic, but only barely so, with a reaction energy of -0.1 kcal/mol, vdW-DF-cx0 however gives a value of -2.3 kcal/mol which is only a $12\%$ overestimation of the reference energy. The improvements for the systems with small reaction energies are the main reason why the MARD value are significantly lower for the hybrids, while the reduction in MAD is more modest.

For ionization potentials, the comparison for the G21IP set, summarized in table 3 and Fig. 1, shows both the hybrids perform similar or slightly better. vdW-DF-cx and vdW-DF-cx0 perform better than vdW-DF2 and vdW-DF2-0.

The performance of vdW-DF-cx0 for covalent molecular binding properties makes this a promising vdW-DF hybrid functional candidate. vdW-DF2-0 is not as promising, clearly illustrated by its atomization energies. In a separate test, we find that the difference between the trends of vdW-DF2-0 and vdW-DF-cx0 for small-molecule atomization energies is predominantly due to the exchange component. For covalent interactions, the performance is rather insensitive to whether vdW-DF1 or vdW-DF2 correlation is employed for a fixed exchange functional. This is in contrast to the case for non-covalent interactions.Berland and Hyldgaard (2013)

III.2 Non-covalent molecular binding

Figure 3 shows the results for the S22 setJurecka et al. (2006) following the standard division into hybrid, dispersion, and mixed systems. vdW-DF2-0 has smaller MAD than vdW-DF2 for all the three cases. vdW-DF-cx0 on other hand has overall a somewhat larger MAD than vdW-DF-cx, increasing from 0.38 kcal/mol to 0.41 kcal/mol. Most striking, however, is the significantly worsened performance of vdW-DF-cx0 compared to vdW-DF-cx for purely dispersion-bonded systems — with MAD increasing from 0.19 to 0.63 kcal/mol —combined with a much improved performance for the hydrogen bonded systems, MAD decreasing from 0.62 to 0.27 kcal/mol. We also find that these poor results for dispersion bonded systems come from a significant overestimation of binding energies.

The improved performance for hydrogen bonded systems is linked to the fraction of Fock exchange, as the performance also improves corresponding for vdW-DF2 and PBE. In the latter case, MAD shrinks from 1.24 kcal/mol to 1.08 kcal/mol when going from PBE to PBE0. Nonetheless, these numbers are inferior to the vdW-DFs as dispersion forces generally contribute to the binding in all kinds of non-covalently bonded dimers, including systems that are mostly hydrogen bonded. We therefore omit PBE and PBE0 in further analysis of non-covalently bounded systems.

The clear overestimation of vdW-DF-cx0 for dispersion bounded systems shows that Fock exchange and vdW-DF-cx exchange behave quite differently for dispersion-bonded systems, as vdW-DF-cx itself has a rather low MAD. Thus the vdW-DF1 correlation (which is also used in vdW-DF-cx) may not be the optimal one for the exchange description used in vdW-DF-cx0. The low vdW-DF-cx0 MAD for hydrogen-bonded systems nonetheless indicates the promise of mixing a soft exchange like that of vdW-DF-cx with Fock exchange.

The mixed vdW-DF-cx0 performance for the S22 set motivates us to also consider the X40 setŘezáč, Riley, and Hobza (2012) of non-covalently bounded dimers involving at least one halogenated molecule. This set of dimers involve small molecules like methane, and its halogenated derivatives, an example being the methane-Cl2 dimer (reference binding energy of 0.69 kcal/mol) or bromobenzenezene-acetone dimer (2.48 kcal/mol. They also include the strongly polar hydrogen halides, such as the HF-methanol dimer (5.8 kcal/mol).

Figure 4 shows the deviation in binding energies (upper panel) and binding separation (lower panel) ordered by the magnitude of the reference energy. The bars in upper panel indicate the magnitude of the reference energies, with scale on the right vertical axis. The bar color indicates a sorting of the dimers into molecules involve methane (pink), benzene (green), hydrogen halide (orange), or none of these molecules (gray). We organize the X40 set into these subsystems for the following reasons: For all the dimers of methane and a "halogenated molecule", the binding energies are significantly overestimated and thus dominates the trends in the MRD. For benzene-"halogenated molecule" dimers, the aromatic $\pi$-orbitals could contribute to the binding, giving them a somewhat different binding nature. For the dimers involving hydrogen halides, the binding is dominated by the strong polarity of the halide. Table 4 summarizes the trends for the X40 set and these subsystems.

For the dimers involving methane, all the functionals overestimate binding energies, with overall somewhat better performance for the hybrids. The methane molecules in the X40 set are paired with another small molecule, either a diatomic halogen going from F2 to I2 (system 1, 5, 8 and 10), or halogenated methane derivatives (system 2, 3, 4, and 6). Related systems with low binding energies, such as the chloromethane-formaldehyde dimer (system 7), are also significantly overestimated. The overestimation is evident both in the table and by cases in which the data is outside either the first or second dashed curves indicating 10% and 30% deviation in the upper panel of Fig.4. vdW-DF-cx and vdW-DF-cx0 also significantly overestimate binding separations of the methane systems, on average by respectively 0.25 Å  and 0.17 Å, as with vdW-DF-cx for the methane dimer of the S22 set.Berland and Hyldgaard (2014) Issues for the smallest molecules like methane is a known shortcoming of vdW-DF-cx, which is consequence of using the Langreth-Vosko exchangeLangreth and Vosko (1987) form in a wide regime of reduced gradients $s$, making it overly repulsive for systems dominated by large $s$ values.Berland and Hyldgaard (2014) The separations of vdW-DF2 and vdW-DF2-0 are rather similar for all the systems and both are on average overestimated by about 0.05 Å. This result can be traced to the fact that for molecular dimers, PW86r exchange and Hartree-Fock exchange give quite similar binding curves.Murray, Lee, and Langreth (2009)

For the dimers involving benzene, vdW-DF2 and vdW-DF2-0 show excellent performance. vdW-DF-cx0 exhibits poorest performance, which arise from a poor description of the trifluorobenzenezene-benzene dimer and the hexafluorobenzenezene-benzene dimer. For the dimers involving hydrogen halides, which bind strongly, vdW-DF-cx0 exhibits a MADR of only 1.6%. Even if the binding separations are a bit underestimated, vdW-DF-cx0 is the only functional that does not result in a large deviation of the binding energy of the HCl-methylamine dimer. This is a case for which the binding energy shrinks once Foch exchange is mixed in.

Considering the remaining 20 dimers, both hybrids exhibit a clear improvement over the non-hybrid functionals with MARD dropping by about 40% for both vdW-DF-cx0 and vdW-DF2-0. These dimers involve a range of small organic and halogenated dimers and with varying binding energy and polarity. For the larger molecules, the overestimation of separations by both vdW-DF-cx and vdW-DF-cx0 turns into an underestimation in energies. vdW-DF-cx0 typically predicts a similar but somewhat more accurate separations than vdW-DF-cx.

Evaluating the entire X40 set, we find that hybrid variants do perform better than the non-hybrid functionals. Comparing the MAD and MARD values and Fig. 4 shows that vdW-DF2-0 mostly improve the performance over vdW-DF2 for the systems with small binding energy. vdW-DF-cx0 can worsen the binding energies for the systems with small binding energies, which is also linked to the poorer account of dispersion bounded systems for the S22 set. It can also improve the performance for systems with strong polar binding, which is linked to the good performance for hydrogen-bonded systems of the S22 set.

IV Discussion and outlook

We have tested the performance of two hybrid vdW-DFs, assessing both covalent and non-covalent molecular binding properties. For covalent molecular binding properties, we find that vdW-DF-cx0 performs very similar to, and even slightly better than PBE0, whereas vdW-DF2-0 is overall less accurate. This result indicates that the exchange of vdW-DF-cx0 is here a good match for the correlation component. In turn, this indicates that vdW-DF-cx can indeed, for these properties, serve the same role as PBE in mean-value evaluation of the adiabatic connection formula. Using hybrid functionals also improves the non-covalent interactions for systems where hydrogen bonding and other polar effects contribute significantly to the binding. However, vdW-DF-cx0 overestimates binding energies of purely dispersion-bonded systems of the S22 set.

Insight into a possible mechanism behind the finite vdW-DF-cx0 deviations in describing binding energies of dispersion bonded systems can be obtained by recalling the construction of vdW-DF-cx.Berland and Hyldgaard (2014) The vdW-DF-cx version rests on a consistent-exchange argument whereby exchange and correlation are brought in balance by effectively using the same plasmon-pole propagator to define both terms.Berland et al. (2014); Hyldgaard, Berland, and Schröder (2014) However, in going from vdW-DF-cx to vdW-DF-cx0, we only replaced the exchange contributing to the total energy, but did not update the internal exchange within the vdW-DF-cx correlation kernel and to some extent we thereby broke the internal consistency of the hybrid functional. Purely dispersion-bonded systems are precisely the regime where the parameterization of the internal functional is crucial for the total binding energies of the systems.Berland and Hyldgaard (2013) Further progress in constructing more accurate hybrid vdW-DF versions may therefore be achieved by ameliorating this broken consistency. Such an investigation is beyond the scope of the present paper.

Our study shows that hybrid vdW-DFs are promising general-purpose functionals for describing binding in molecules. vdW-DF-cx0 stands out as an excellent functional for describing covalent and hydrogen/halogen- bonding of molecules, but the simple mixing in of 25% Fock exchange does not preserve the good performance of vdW-DF-cx for typical dispersion bonded dimers nor does it resolve issues of vdW-DF-cx for binding between the smallest molecules. This study should therefore motivate further development of hybrid vdW functionals as well as tests of other classes of materials, both dense traditional materials, and sparse vdW materials. In particular materials where interactions compete and both the sparse- and the dense-matter properties must be described accurately at the same timeLangreth et al. (2009); Berland et al. (2014, 2015) are well suited testing grounds for hybrid vdW-DFs. An additional benefit of hybrid vdW-DFs not explored here is the potential for accurate computations of excited-state properties. It is practical and conceptual advantage to be able to compute the excited state properties of vdW bonded materials using the same hybrid vdW-DF version that was first used to relax the atomic coordinates.

Acknowledgment

The authors thank Zhenfei Liu for insightful discussions. Work in Sweden supported by the Swedish Research Council (VR), Chalmers Area-of-Advance Materials, and the Chalmers e-Science centre. KB acknowledges support from the Research Council of Norway (no. 250346). Work by JHL, TR, and JBN is supported by the Center for Gas Separations Relevant to Clean Energy Technologies, an Energy Frontier Research Center, funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Award DE-SC0001015. Portions of this work were performed at the Molecular Foundry, supported by the Office of Science, Office of Basic Energy Sciences, U.S. Department of Energy, under Contract DE-AC02-05CH11231. Computational resources were provided by DOE (LBNL Lawrencium and NERSC) and by the Swedish National Infrastructure for Computing (SNIC), under contract 2016-10-12, from the Chalmers Centre for computing, Science and Engineering (C3SE).