Original file was Chapter5.tex - RUA:...

This is a previous version of the article published in Theoretical Chemistry Accounts. 2015, 134: 1602. doi:10.1007/s00214-014-1602-6

Non parametrized functionals with empirical

dispersion corrections: a happy match?

Diane Bousquet1, Eric Brémond1, Juan C. Sancho-García2, Ilaria Ciofini1 and Carlo Adamo1,3*

1Laboratoire d’Electrochimie, Chimie des Interfaces et Modélisation pour l’Energie, CNRS

UMR-7575, Chimie ParisTech, 11 rue P. et M. Curie, F-75231 Paris Cedex 05, France.2Departamento de Química-Física, Universidad de Alicante, E-03080 Alicante, Spain 3Institut Universitaire de France, 103 Boulevard Saint Michel, F-75005 Paris, France.

Abstract

The performances of two parametrized functionals (namely B3LYP and B2PYLP) have been

compared to those of two non parametrized functionals (PBE0 and PBE0-DH) on a relatively

large benchmark set when three different types of dispersion corrections are applied (namely

the D2, D3 and D3(BJ) models). Globally the MAD computed using non parametrized

functionals decreases when adding dispersion terms although the accuracy not necessarily

increases with the complexity of the model of dispersion correction used. In particular, the

D2 correction is found to improve the performances of both PBE0 and PBE0-DH, while no

systematic improvement is observed going from D2 to D3 or D3(BJ) corrections. Indeed

when including dispersion, the number of sets for which PBE0-DH is the best performing

functional decreases at the benefit of B2PLYP.

Overall our results clearly show that inclusion of dispersion corrections is more beneficial to

parametrized DH functionals than for non parametrized ones. The same conclusions globally

hold for the corresponding global hybrids, showing that the marriage between non

parametrized functionals and empirical corrections may be a difficult deal.

*corresponding author: [email protected]

http://dx.doi.org/10.1007/s00214-014-1602-6

1. Introduction

In the field of Density Functional Theory (DFT), the development of more accurate

approximate forms of exchange correlation functionals for the description of a wider variety

of properties, ranging from thermochemistry to photophysics, still remains a major research

topic. In the last few years, many functionals belonging to the higher rung of the Perdew

ladder1 have indeed been developed. Among these, global or range-separated hybrid

functionals, which are characterized by a dependence on occupied Kohn-Sham (KS) orbitals

due to the inclusion of a fraction of Hartree-Fock (HF) like exchange, such as for instance

PBE02, B3LYP3, CAM-B3LYP4 or LC-wPBE5-7, normally present a good performances to

computational time ratio, and for this reason they have been intensively applied to model the

reactivity and properties of many chemical systems during the last years8-11. More recently,

following an approach firstly proposed by Truhlar12, different double-hybrid (DH) functionals

have been developed including also a dependency on the virtual KS orbitals. This latter

dependence is related to the presence of a second-order perturbative Moller-Plesset (MP2)

contribution to the correlation energy13,14. Since that pioneering work, a significant number of

DHs have been developed, the most famous family of parametrized DHs being the one

introduced by Grimme and including functionals such as B2PLYP15,16 and mPW2-PLYP17.

Following the same parametrization philosophy, other double-hybrids have been developed

by Martin, such as B2GP-PLYP18, B2T-PLYP and B2K-PLYP19, demonstrating to be

particularly accurate on an impressing number of properties. Furthermore, the inclusion of a

spin component or spin-opposite scaling of the MP2 contribution was recently proven to

ameliorate the performances such in the case of DSD-BLYP20, DSD-PBEB8621 or the latest

Grimme’s functional (PWPB9522).

In this overall context a DH, the so-called PBE0-DH23-25, was recently developed by Adamo

and coworkers, following the same parameter free approach used for the development of the

parent and well performing global hybrid PBE026-32. This functional actually includes the

Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional in a double hybrid

formulation without the inclusion of any empirical parameters since both the fraction of HF-

like exchange and MP2 correlation contribution were fixed making use of theoretical

constrains and physical considerations33 without resorting on fitting over different

experimental sets.

In a recent paper34, the performances of this double hybrid were tested on the extended

benchmark set GMTKN30, developed by Goerigk and Grimme35,36 and compared to those of

B2PLYP and of their parent Global Hybrids (GHs), PBE0 and B3LYP, in order to compare

2

two DHs functionals possessing the same level of complexity and representatives of the

parametrized (B2PYLP) and non parametrized (PBE0-DH) development philosophies. In

particular, many recent papers of Grimme and collaborators clearly showed that explicit

inclusion of dispersion corrections strongly ameliorate the overall performances of

functionals not only for properties directly related to dispersion, such as for instance

structures of systems of biological relevance, but also for properties that could appear, at first

sight, hardly affected by London interactions37-39.

The development of corrections able to correctly deal with London forces, has indeed become

a flourishing field in DFT. Clearly, the inclusion of a fraction of MP2 correlation in global

hybrids represents a good starting point to correctly estimate non covalent interactions, such

as van der Waals ones. Indeed, a systematic improvement of functionals’ performances going

from global to corresponding double hybrid functionals was observed for benchmark sets

dominated by dispersion and non covalent interactions36. The better performances of DH with

respect to GH is indeed related to the different long range behavior of the correlation energy

terms in the two families of functionals: an exponential decrease exponentially with the

interatomic distance in the case of GH versus the presence of a long range term in DH, due

the inclusion of an MP2 correlation fraction. However, according to Grimme works22,40, the

contribution of the MP2 correlation term is not sufficient to compensate the DFT repulsive

part, so that even at DH level, dispersion forces are not correctly described making the

addition of an empirical correction term necessary for the correct description dispersion

interactions41.

To this end, different empirical corrections, containing parameters fitted on relevant

benchmark sets, were developed and tested in conjunction of a wide variety of parametrized

DHs showing excellent results. Nonetheless, up to now none of these corrections were tested

when combined to a non-parametrized DH such as PBE0-DH. In order to test if the

systematic and sizable improvement of DH performances by inclusion of empirical

dispersion corrections observed for parametrized DHs also holds in the case of PBE0-DH, the

inclusion of three different empirical dispersion corrections, namely the so-called –D238, D339

and D3(BJ)42-45 ones, in the PBE0-DH functional form will be considered in this work. The

performances of the so-obtained dispersion corrected DH will be compared to those of an

empirical DH of analogous complexity (B2PLYP), also dispersion corrected, in order to

basically show if non-empirical DH and empirical dispersion corrections really make a happy

match.

3

The paper is structured as follows: after a brief recall of the dispersion corrections that will be

applied and a description of the fitted parameters that will be used (Section 2), the

computational details will be given in Section 3. Next, the performances of dispersion

corrected PBE0-DH and B2PYLP will be compared over the GMTKN30 benchmark set

(Section 4) and some general conclusions will be drawn in Section 5.

2. Dispersions interactions: the models and the parameters

In order to deal with inter- and intra-molecular dispersion contributions in the DFT

framework, a set of corrections to Kohn-Sham energies where recently introduced by

Grimme, namely the DFT-D137, DFT-D238 and, more recently, the DFT-D339 models, these

can be viewed as a kind of hierarchy of increasing complexity arising from the approximate

yet accurate modeling of pair-wise long-range interactions between weakly interacting

densities or fragments.

Basically, the total energy of a system can be written as the sum of the Kohn-Sham energy

and of an energy correction for dispersion interactions as follows:

EDFT−Dx=EKS−DFT+ EdispDx

(

1

)

where EdispDx is an atom pair wise correction term expressed as:

EdispDx =∑

k=2

x

Edisp(k)

(

2

)

and where Edisp(k) (for k>1) represents respectively the two body (k=2) and three-body (k=3)

dispersion correction terms. Truncated expression of (2) to the first order correction (x=2)

leads to the so-called DFT-D2 correction, while the DFT-D3 model is obtained truncating for

x=3. The leading term is indeed the two-body one, expressed as:

Edisp(2) =−∑

n=3

Norder

s2 n ∑i=1

N n−1

∑j=i+1

Nn C2 nij

r ij2 n f dmp ,2n ( Rij )

(

3

)

where Nn represent the number of atoms in the system, n represent the order (here limited to

3) Rij'=|Ri

'−R j'|is the internuclear distance between atomsi and j and C2n

ij is the dispersion

coefficient for a pair of atoms ij, calculated as the geometrical average of the dispersion

coefficient of the atoms pair:

4

C2nij =√C2 n

i C2 nj

(

4

)

In equation (3), f dmp ,2n is a damping function introduced in order to avoid as much as possible

the double counting of electronic correlation at intermediate distances. f dmp actually defines

the range of in which the correction plays a role, and, in the case of the DFT-D2 model is

expressed as:

f dmp ,2 n ( rij)=1

1+exp[−d ( r ij

r iVdW +r j

VdW −1)](

5

)

Starting from the DFT-D2 model, the DFT-D3 one was developed allowing to obtain a better

accuracy for the description of all systems, including particularly metallic and heavy atoms

containing systems39, while reducing concomitantly the empiricism contained in the model. In

this latter model, the dispersion coefficients specific to each atoms pair and the cutoff radii

are explicitly calculated, taking into account the chemical environment too. On the other hand

the three body terms have been proven to yield negligible effects or even a deterioration of

the results when included for small sized systems, most probably related to an overestimation

of the three body terms in the strong density regions39. Therefore these terms are often

neglected, although they may play a role, as in the case of the Grimme S12L subset46 for

which three-body corrections are proved to play an important role. In this line it is worth

mentioning the recent work of Tkatchenko and collaborators dealing with the effect of many

body corrections47.

Overall, the dispersion correction term at DFT-D3 level (Edisp(3) ) is thus very similar to DFT-D2

one, except that all the terms in n should be taken into account. The scaling factors s2n (eq. 3)

are adjusted in the case of n<3, to allow a correct asymptotic behavior, which is only the case

when C6ij is exact (s6=1). For functionals that already take into account a part of the long

range correlation, a correct asymptotic behavior will only be reproduced when using s6 ≠1,

which will be the case for the studied double hybrids PBE0-DH and B2PLYP. Indeed, after

intensive testing, Grimme found out that the inclusion of terms higher in n−¿order than 8 can

generate instability while not upgrading the performances significantly39. Thus, as in the case

of DFT-D2, only s6 and s8 terms will be included. On the other hand, the DFT-D3 zero

damping function used is slightly different from the one of the DFT-D2 model, as proposed

by Chai and Head Gordon48:

5

f dmp ,2n ( rij)=1

1+d ( rij

sr ,2n+R0ij )

−α 2 n

(

6

)

where sr ,n is the scaling factor dependent on the cutoff radii (R0ij). Grimme38 proposed to fix

sr ,8 to a value of unity, needing thus to only optimize the sr ,6 value. Regarding the α n

parameters, they were adjusted manually to fit some physical constrains39, being α 6=¿ 14 and

α 8=α6+2=¿ 16.

At last, the DFT-D3 model was also tested using the damping function presented by Becke

and Johnson, leading to the so-called DFT-D3(BJ). In this variant of the DFT-D3 dispersion

model, the dispersion energy is expressed as:

EdispDFT− D3(BJ )=− ∑

n=6,8 , …sn ∑

i=1

Nn−1

∑j=i+1

N n C 2nij

rij2 n+ f BJ ,2n(Rij

0)2 n

(

7

)

with the corresponding damping function defined as f BJ , 2n(R¿¿ij0)=a1 Rij0+a2¿. Because of

the form of the dumping funtion, there is no need to define cutoff radii. The parameters a1

and a2 are fitted parameters, and Rij0 is defined as:

Rij0=√C8

ij

C6ij

(

8

)

6

In order to use the three above mentioned models in conjunction with the PBE0-DH

functional, some parameters were needed to be defined. All coefficients used in this work are

listed in Table 1, along with the corresponding values obtained from literature when

available35,40,49-51. In the case of the DFT-D2 model, being d fixed to 20, the only parameter

needed, that is the s6coefficient51, was obtained via a fitting procedure on the S2249 subset.

For the DFT-D3 model, for which the d value is fixed to 6, three parameters need to be fitted,

and the procedure described by Grimme39 was followed. This latter consists in fixing the s6

parameter to one in order to ensure a correct asymptotic behavior for global hybrid

functionals (ie PBE0 and B3LYP). The determination of s6 for double hybrids is done based

on a system of three rare gas dimers (Ne2, Ar2 et Kr2). By applying this method to PBE0-DH,

a value of 0.84 is obtained for s6 . Then, the other parameters are determined by minimizing

the Mean Absolute Deviation (MAD) with a non linear least square method on the S130 set.

This set is composed of the following subsets: S2239,49,50 S22+, PCONF52, SCONF53,

ACONF54, CYCONF55, ADIM656, RG657-60. Note that S22+ contains the same complexes as

the S22 set, only considering systems at larger internuclear distances. The total MAD is then

calculated, using the MAD of RG6 weighted by 20. The mapping of the MAD evolution and

its dependence on the s8 and sr ,6 parameters, can be found in Figure 1. This procedure allows

us to find s8 and sr ,6 values of 0.784 and 1.394, respectively as reported in Table 1. Regarding

the optimization of the DFT-D3(BJ) parameters, the procedure followed is strictly the same

applied for the DFT-D3 model, and allow obtaining s8 = 0.095, a1 = 0.0 and a2 = 6.102.

Inspection of the parameters obtained for the different GHs (B3LYP and PBE0) and DHs

(B2PYLP and PBE0-DH) allows to drawn some general comments. First, in the case of the

DFT-D2 model, the s6 values are substantially different for B3LYP and PBE0; the latter

being significantly smaller while PBE0-DH and B2PYLP shows significantly similar

parameters, the latter being only slightly larger. On the other hand, in the case of DFT-D3

models, PBE0-DH always possesses a larger s6 and a smaller s8. Interestingly, for PBE0-DH

in the case of D3(BJ) model for the same s6 value two sets of parameters providing very

similar MAEs on the fitting subset (i.e. the S130 one) have been found. The one (hereafter

referred to as PBE0-DH-D3(BJ)) providing the smallest MAE (s6= 0.840; a1=0.0 ; s8= 0.095

and a2= 6.102, Table 1) will be used throughout in the paper, and the results obtained using

the second set parameters, referred to as PBE0-DH-D3(BJ)2 (s6= 0.840; a1=0.345 ;s8= 0.747

and a2= 5.276, Table 1) and providing a MAE of only 0.003 kcal/mol larger on the S130, will

be discussed in section 4.

7

Interestingly, these two sets of parameters closely resemble to the D3(BJ) parameters

obtained for DSD-BLYP and PWPB95 functionals (both having a null a1 value) and for the

B2PLYP one, clearly pointing out how the search of global minima over the MAE surface

can be quite tricky and that parameter of the same range can be obtained independently of the

functional used.

3. Computational details

The PBE0-DH functional has been implemented in a local version of the Gaussian program61,

so that all the standard features, including analytical derivatives for geometry optimization,

harmonic frequencies and properties, are also available. In analogy with previous work (see

for instance references 2, 36, 62, 63), the performances of the PBE0, B3LYP, B2PLYP and

PBE0-DH functionals have been tested on the GMTKN30 database using both Gaussian and

Orca64 Version 2.9.1-RELEASE program packages.

To easily compare the obtained results with the values presented by Grimme, we used in

particular the large Ahlrichs’ type quadruple-ζ basis set def2-QZVP,65 which is expected to

provide converged results even for DHs. Indeed, the use of a quadruple-ζ base is sufficient to

reach the convergence limit,51,66 even if DH approaches could have a significantly larger

dependency on basis set then corresponding non-DH forms, due to the presence of the MP2

component. Regarding the calculations of electron affinities, and of binding energies of water

(subsets G21EA67 and WATER2768), diffuse s and p functions were taken from the Dunning

aug-cc-pVQZ,69 obtaining the aug-def2-QZVP basis, which was recommended by Grimme

for those particular subsets.

The PCONF52, WATER2768, IDISP70, ISOL2271, ADIM656, BHPERI72 and BSR3673-77 were

computed making use of the RI-MP278-80 approach as implemented in Orca. Available data

for B3LYP, B2PYLP, PBE0, B3LYP-D3, B2PYLP-D3 and PBE0 were retrived from the

Grimme website81.

8

4. Results and discussion

Being mostly interested in the effect of dispersion correction on non parametrized

functionals, firstly the global behavior of PBE0-Dx and PBE0-DH-Dx functionals will be

compared to that computed in absence of dispersion correction on the whole GMTKN3035

set. Next, the performances of the different functionals will be analyzed over different groups

of subsets, defined in a previous paper36 and based on the dominant properties they are aimed

to benchmark, in order to see which type of interaction are mostly affected by the inclusion of

dispersion. In particular, the GMTKN30 datasets will be classed in subset probing

thermochemistry (decomposition and atomization energies), adiabatic processes, reaction

energies and barriers heights, conformers and isomers relative energies. On the other hand the

DC982-89 subset, which includes nine reactions known to be difficult to be treated at DFT

level, and SIE1133 subset, dealing with test cases for self interaction error, will be discussed

separately. The MADs computed on each GMTKN30 subset are reported in Supporting

Information.

4.1 Dispersion corrected global (PBE0) and double hybrid (PBE0-DH) benchmarked over

the GMTKN30 set

The overall performances of PBE0 and PBE0-DH over the GMTKN30 set with and without

inclusion of dispersion corrections are summarized in Figure 2 by their associated Mean

Absolute Deviation (MAD), together with the results obtained at B3LYP and B2PLYP level.

Globally speaking, the MAD computed with both PBE0 and PBE0-DH decreases when

adding dispersion terms, although not necessarily after increasing the complexity of the

correction used. In particular, if the D2 correction sizably ameliorates the performances of

both PBE0 and PBE0-DH, the systematic improvement observed going from -D2 to -D3 (or -

D3(BJ)) corrections in the case of B3LYP is here not observed, in analogy to what is

computed in the case of B2PLYP.

On the other hand, both in the case of the non-parametrized and parametrized families of

hybrids, the effects of dispersion correction terms are quantitatively larger on global hybrids

with respect to double hybrids so that the computed MAD difference between DH and

corresponding GH decreases by addition of dispersion. For instance, the difference of MADs

between PBE0 and its corresponding double hybrid (PBE0-DH) is smaller when adding

dispersion corrections, and even if PBE0-DH still performs better than PBE0, the difference

in MAD decreases from 0.29 kcal/mol when dispersion is not included to 0.08, 0.19 or 0.15

kcal/mol when dispersion is included at the D2, D3 and D3(BJ) level, respectively.

9

Globally speaking, this is also what is observed in the case of the parameterized family of

functionals (BLYP and B2PLYP) although the overall MAD difference between GH and DH

is always larger than 1 kcal/mol. Furthermore, while the largest decrease in MAD is

computed when applying the D3 type corrections in the case of B3LYP, for PBE0 the largest

effect is found when using the D2 corrections. Indeed, both for PBE0-DH and PBE0, the

more complex D3 corrections seem to add a negligible accuracy.

The difference in performances of the two families of functionals (i.e. the PBE-based and

BLYP-based) could be related to the dissimilar behavior of the exchange component in the

limit of large reduced density gradient. Indeed, it has been shown that the Becke's exchange

functional (B88)90 is repulsive in dispersion-bound complexes so that simple van der Waals

systems are precited to be unbounded91,92. This is not the case for the PBE exchange, which is

more actractive93. It is therefore likely that BLYP based functionals benefit more from the

always attractive, DFT-D type corrections. More insigths on this point could be obtained by

analyzing the performances of “cross” functionals, such as BPBE-DH or PBE-LYP-DH.

However, these unusual exchange and correlation functioals’ combinations are of scarse

theoretical interest, since the PBE contributions were defined to match each-other94.

Furthermore, a correct definition of these functionals requires the full optimizations of the

coefficients ruling the different contributions to the functionals, so that the differences

between the two orginal DH functionals (PBE0-DH and B2PLYP) and their derivatives could

be not clearly ascribed to a single effect, making the obtained results difficult to interpret. In

this intricate context, we believe that this point does not deserve further explorations and

prefer to rest on the above mentioned qualitative interpretation of the exchange behavior.

Analyzing in more details the results obtained over the GMTKN30 database, with and

without dispersion corrections for PBE0 and PBE0-DH, one can notice that, in absence of

dispersion the PBE0-DH functional is better than its parent PBE0 one for 20 over 31 subsets.

Adding the dispersion corrections computed using either the -D2 or the -D3 models, PBE0-

DH is still better than PBE0 for 20 subsets only. Indeed, even if the number of subsets where

PBE0-DH outperforms PBE0 does not change, this is only fortunate, because the concerned

sets are not the same. For example, the O3ADD694 and ALK639 subsets, for which PBE0

provides better results in absence of correction, are better described by PBE0-DH when

adding dispersion terms. Analogously, the IDISP70 subset presenting lower MADs for PBE0-

DH is actually better described at PBE0 level when adding the -D2 correction. Generally, the

MADs over the different subsets vary in consistent way for the two functionals when adding -

10

D2 or -D3 dispersion corrections with the only exception of the O3ADD6, AL2X95,

ISOL2271, DC9, ALK6, and IDISP subsets.

On the other hand, regarding the results obtained when adding dispersion corrections at DFT-

D3(BJ) level, a significant deterioration of the performances of PBE0-DH functional with

respect to the parent PBE0 one is found. Indeed, PBE0-DH-D3(BJ) outperforms PBE0-

D3(BJ) only for 15 subsets over 31.

For PBE0 and PBE0-DH, and for all subsets concerning thermochemistry, a deterioration of

the performances can be observed when adding dispersion terms, except for the BSR3672-77

subset, which shows a reduction of the MAD of 6.48 kcal/mol (for PBE0-D2) and 4.55

kcal/mol (for PBE0-DH-D2). For adiabatic processes, the inclusion of dispersion does not

play an important role while the change in accuracy when including dispersion corrections

for subsets probing reactions energies is quite inhomogeneous.

Indeed, for some subsets, such as BH76, BH76RC, the inclusion of dispersion does not

change the overall behavior of PBE0 and PBE0-DH while for other subsets, O3ADD694,

G2RC96, NBPRC97,98 , DARC95, ALK639 et AL2X95, results are sizably worst or better, such as

for the BHPERI72 or RSE4399,100 subset. On the other hand, and as it was expected, the

inclusion of dispersion corrections allows for a significant amelioration of the results and a

systematic decrease of the MADs for all subsets probing dispersion and relative energies of

conformers.

4.2 Decomposition and atomization energies

Figure 3 reports the MAD computed for PBE0, B3LYP, PBE0-DH, B2PYLP (both in

presence and in absence of different dispersion corrections) for all four subsets concerning

thermochemistry, that are: the MB08-165101, containing the decomposition energies of

artificial molecules, the W4-0818, including atomization energies of 99 small artificial

molecules, the W4-08woMR, obtained by removing multireference cases of the previous

subset, and the BSR3673-77 which includes bond separations reactions. Those four sets present

large MAD but, since the reference data are large, the MADs obtained are, on a relative scale,

not so large. Interestingly, a reduction of MADs by inclusion of dispersion terms is observed.

For B2PLYP and PBE0, the smallest MADs are found using the -D2 correction, with a

difference of 1.44 kcal/mol and 1.47 kcal/mol, respectively, with respect to non corrected

values. For B3LYP functional, a systematic decrease of the MAD is observed following the

dispersion correction model used, with a reduction of the MAD with respect to the non-

dispersion corrected results going from 0.23 kcal/mol (DFT-D2), 1.82 kcal/mol (DFT-D3)

11

and 2.84 kcal/mol for the DFT-D3(BJ) model. Basically, it seems that B3LYP benefits most

of corrections of the D3 form. The same trend is indeed found for PBE0-DH although the

overall amelioration of performances is quantitatively smaller (MADs difference between

PBE0-DH results and PBE0-DH-D3(BJ) of only 1.17 kcal/mol).

Overall, the results obtained show that the parameterized family of functionals, already

providing the best results in absence of dispersion corrections in the case of B2PYLP, is the

one which benefits the most (in the case of the B3LYP functional) of the inclusion of

dispersion corrections especially of the -D3 type. Indeed, besides global MADs computed for

all the different subsets, it is interesting to analyze the effects of the inclusion of dispersion

for each subset, since, as previously mentioned, the behavior is rather inhomogeneous.

Indeed for the MB08-165, a significant decrease in errors is computed for B2PLYP with

MADs going from 5.12 kcal/mol (B2PLYP) to 3.67 kcal/mol (B2PLYP-D3(BJ)). For

B3LYP, however, a reduction of MAD is only obtained when using the -D3 or -D3(BJ)

corrections, with differences of 1.8 kcal/mol and 2.64 kcal/mol, respectively, with respect to

the B3LYP MADs. For the PBE0 family, inclusion of dispersion only leads to an increase of

the errors: the most important deviations are observed for the DFT-D2 model, with an

increase in MAD with respect to the non dispersion-corrected values of 0.66 (PBE0-D2) and

0.54 kcal/mol for PBE0-DH-D2.

For the W4-08 subset on the other hand, only small variations are observed among inclusion

of dispersion independently of the functional used. The most important improvements are

observed using -D3(BJ) dispersion model, with reduction of the MADs of 0.87 kcal/mol

(B3LYP), 0.59 kcal/mol (B2PLYP), 0.07 kcal/mol (PBE0) and 0.45 kcal/mol (PBE0-DH). A

similar behavior is found for the W4-08 woMR subsets, with the best improvement on MAD

obtained also applying the -D3(BJ) model but with very small overall effects ranging from

0.01 kcal/mol (PBE0) to 0.46 kcal/mol ( B3LYP).

On the other hand, the BSR36 subset, probing bond separation reactions, seems to be the

most affected by the inclusion of dispersion corrections. The minimal MADs are computed

making use of the -D2 model, with extremely low MAD values of 0.76 kcal/mol (B2PLYP),

1.60 kcal/mol (PBE0) and 1.14 kcal/mol (PBE0-DH). The most important reduction in MAD

observed after introduction of dispersion terms is of 6.48 kcal/mol (PBE0-D2), whereas a

decrease of 4.55 and 4.52 kcal/mol is computed for PBE0-DH-D2 and B2PLYP-D2

respectively. As noticed above, the B3LYP functional presents a behavior slightly different

from the other functionals with an increase in accuracy obtained only starting from -D3 type

dispersion models, the best results being obtained at DFT-D3(BJ) level with an associated

12

MAD of 3.84 kcal/mol, representing thus a decrease of 7.41 kcal/mol with respect to the

value computed without any dispersion corrections.

4.3 Adiabatic processes

Three different subsets, namely the G21IP, G21EA67 and the PA102-104 ones, probing adiabatic

ionization potentials, electron and proton affinities, are used to estimate the performances of

the functionals in the prediction of adiabatic processes. When looking at Figure 4, which

reports the MADs for these subsets, we can observe that the performances of all the

considered functionals are practically unaffected by the inclusion of dispersion. Indeed,

adding dispersion correction implies a systematic but almost negligible increase of the MAD

for B3LYP, B2PLYP and PBE0-DH, whereas the best value for PBE0 is obtained for DFT-

D3 model but with a reduction of MAD with respect to the pure PBE0 functional of only 0.02

kcal/mol.

Analyzing the subsets, the G21IP one, including 36 adiabatic ionization potentials of atoms

and small molecules, non parameterized functionals are clearly not influenced by dispersion

terms with difference in MAD with respect to pure functionals smaller than 0.03 kcal/mol.

For the G21EA subset, including 25 adiabatic electron affinities, the behavior is similar for

the two double hybrids, presenting a maximum enhancement of 0.03 kcal/mol. The

dependence on dispersion for hybrids is slightly larger, since the MAD computed with

B3LYP varies from 1.81 (for DFT and DFT-D3) to 2.44 kcal/mol for DFT-D2 and DFT-

D3(BJ).

Analogously, for the PA set, which includes 12 adiabatic proton affinities of four conjugated

polyenes104 and eight small molecules102-103, B2PLYP and B3LYP results are sizably better

than PBE0-DH and PBE0 ones, and dispersion corrections do not change significantly the

functional behavior, actually very slightly deteriorating the MADs.

Globally, a dispersion correction for non parameterized functionals seems to be useless for

the description of adiabatic processes.

4.4 Barriers heights and reaction energies

The Figure 5 shows the MADs computed for all subsets discussed in this section. Generally

speaking, if the parametrized functionals benefit from the inclusion of dispersion terms, with

a decrease of the MADs from 5.75 kcal/mol (B3LYP) to 3.16 kcal/mol (B3LYP-D3(BJ)) for

example, this global trend is not observed for the PBE0 family. For this family indeed, an

increase of the MADs is systematically found upon inclusion of dispersion corrections with a

13

maximal deterioration of performances of 0.71 kcal/mol for PBE0-D2 and 1.47 kcal/mol for

PBE0-DH-D3(BJ).

Considering the performances obtained for barrier heights on two subsets, the BHPERI54 and

the BH76105,106, as for adiabatic processes, no amelioration in the MADs regarding the BH76

subset can be found when adding dispersion corrections. Better results are indeed obtained

for the non dispersion corrected functionals, and for example, the MAD of PBE0 increases

from 4.12 to 4.56 kcal/mol when adding the -D2 correction. The same behavior is observed

for all functionals, the largest increase being obtained for B3LYP-D3(BJ) with a

augmentation of 0.76 kcal/mol.

Globally, the dispersion correction allows a nuanced amelioration of the results. Indeed, if

PBE0-DH-D3 gives smaller MADs than PBE0-DH-D2 (improvement of 0.31 kcal/mol),

functional which improves the results shown by PBE0-DH of 0.42 kcal/mol, this tendency is

not followed by the others functionals. The two global hybrids are not getting better when

adding dispersion using the DFT-D2 model, and accurate performances are only found

starting from DFT-D3 correction models, with a MAD of 2.79 kcal/mol for B3LYP.

B2PLYP-D2, on the contrary shows smaller values of MADs than B2PLYP-D3. Overall,

double hybrids are better than their hybrids parents for barrier heights, but on the BHPERI

set, the differences become smaller and smaller when adding the dispersion, to finally obtain

similar MAD for all functionals : from 1.29 kcal/mol (B2PLYP-D3(BJ) to 1.94 for PBE0-

D3(BJ), for example.

For the corresponding reaction energies (included in the BH76RC) the computed MADs are

overall smaller, the largest MAD being 2.59 kcal/mol at PBE0-D3(BJ) level. Including

dispersion, the PBE0 family shows a slight increase of the MADs, with difference in MAD

with respect to non-dispersion corrected functional between 0.03 kcal/mol (PBE0-D3) and

0.08 kcal/mol (PBE0-DH-D2), whereas B2PLYP results are practically insensitive and

BL3YP ones are slightly enhanced by dispersion corrections.

Also in the case of the O3ADD694 set, including the reactions of addition of ozone on ethane

or ethyne, the computed MADs are found to increase upon inclusion of dispersion correction

terms. One interesting example is given by the behavior of the B3LYP functional, with the

best MAD of 1.97 kcal/mol, whose accuracy is worsen to 3.12 kcal/mol when using the

D3(BJ) correction. Indeed, the B3LYP family, with or without correction, is still

outperforming the PBE0 one on the O3ADD6 subset.

For the G2RC subset, the behavior of all functionals is almost identical in presence or

absence of dispersion interactions, while large MADs are computed when considering 14

14

Diels-Alder reactions, contained in the DARC95 set. The PBE0 functional presents in this case

the best behavior with the smallest MAD (3.13 kcal/mol) computed at PBE0-D3 level, while

PBE0-D2 and PBE0-D3(BJ) model give MADs larger than 4.0 kcal/mol. This behavior is not

followed by the PBE0-DH functional since in this case the inclusion of dispersion terms is

systematically deteriorating the MADs with an increase in MADs of 2.27 and 3.84 at PBE0-

DH-D3 and PBE0-DH-D3(BJ) level, respectively. B2PLYP results, on the other hand, are

enhanced by the inclusion of dispersion terms, the best performances being found at

B2PLYP-D2 level (4.23 kcal/mol). B3LYP benefits the most of the inclusion of dispersion

and it significantly increases its performances as the dispersion model included becomes

more complex, to reach a MAD of 7.4 kcal/mol for B3LYP-D3(BJ) which represents a half

of the MAD computed at pure B3LYP level (15.41 kcal/mol).

The performances of the functionals are not significantly modified by inclusion of dispersion

in the case of the RSE3499,100 subset, and the best results are still obtained for at double

hybrids level with MAD of 0.91 kcal/mol for B2PLYP and 0.86 kcal/mol for PBE0-DH

including dispersion at -D2 level. At this level, the MAD of PBE0 is also reduced of 0.37

kcal/mol. Regarding H2 activation, (NBPRC subset97,98), the B3LYP family benefits most

from the dispersion, with MADs of 1.86 kcal/mol for B3LYP-D3(BJ) and 1.20 kcal/mol for

B2PLYP-D3(BJ) whereas for the PBE0 family, results are getting worst by inclusion of

dispersion. The same trends are also observed for the ALK6 subset: the PBE0 family shows

an increase of MAD of 2.39 kcal/mol for D2 and 1.76 kcal/mol for D3 corrections using the

PBE0-DH functional. On the other hand, using the DFT-D3 model allows a decrease of 2.34

kcal/mol for the B2PLYP functional, which was not the case of the inclusion of the D2 model

leading, for B2PYLP, an increase of 2.79 kcal/mol.

Finally, also in the case of the AL2X95 subset, containing the binding energies of seven

dimers of AlX3 (X=H, CH3, F, Cl and Br), only the B3LPY family benefits from the inclusion

of dispersion corrections although overall the PBE0-DH functional without inclusion of

dispersion is still providing the smallest MAD.

4.5 Conformers and isomers

In the case of isomer relative energies, probed by the ISOL2271 and ISO34107 sets, the

addition of dispersion correction terms seems beneficial especially in the case of the B3LYP

family. Nonetheless, even the inclusion of dispersion does not significantly enhance the

results of the PBE0 family of functional, these latter are still overall better than corresponding

parametrized ones.

15

More in details, the ISOL22 subsets presents more important MADs than the ISO34 subset.

B2PLYP, PBE0 and PBE0-DH are more sensitive to dispersion computed using the DFT-D2

model, whereas B3LYP shows better performances with the DFT-D3 and D3(BJ) models. As

in the general case for isomers, the PBE0 family performs better than the B3LYP one in the

case of this set. The tendency is reversed for ISO34 subset, since B2PLYP is the best

performing functional (MAD of 1.08 kcal/mol for B2PLYP-D2), the PBE0 family also

presenting also better results for the DFT-D2 model, and all functionals being actually pretty

accurate when using dispersion corrections allowing to obtain MAD always smaller than 1.74

kcal/mol.

The PCONF,52 ACONF,54 SCONF53 and CYCONF55 sets all include relative energies of

conformers. The Figure 6 shows the change in the mean absolute deviation as function of the

dispersion correction used. Conformers energies are supposed to be dominated by dispersion

interactions, and this assumption can be verified by inspection of the results obtained using

the -D2 model: the MADs decrease with respect to the non corrected values of 0.66 (PBE0),

0.51 (PBE0-DH) and 0.56 (B2PLYP) practically reducing to half the errors. Furthermore,

more complex dispersion models give more accurate results, although the relative decrease in

MAD becomes less important, and so that for all functional but B3LYP the DFT-D2 model

seems to provide sufficiently good MADs. Only in the case of B3LYP, indeed the inclusion

of the –D3(BJ) correction seem to provide a sizable improvement (of 1.27 kcal/mol).

Globally, for all these subsets, double hybrids always outperform the corresponding hybrids

and the PCONF, showing the largest MADs, it is also the one that benefits the most of the

inclusion of dispersion. Again for this subset important improvement of the performances of

all functionals are observed by inclusion of dispersion at D2 level, with MAD decreasing of

1.55 (B2PLYP), 2.09 (PBE0) and 1.47 kcal/mol (PBE0-DH) with respect to the non-

corrected values.

Also in the case of the other three subsets (ACONF, SCONF, and CYCONF), dispersion

terms globally improve the MADs, leading to extremely accurate results: for example the

largest MAD for ACONF at D3(BJ) level is only 0.07 kcal/mol obtained using B2PLYP.

4.6 Dispersion and non covalent interactions

The performances of the functionals to describe dispersion and non covalent interactions

were tested using the six following sets: ADIM656, S2239,49,50, HEAVY2839 and RG657-61,

WATER2768 and IDISP70. The corresponding global MADs obtained are collected in Figure

7. Globally, double hybrids functionals present better performances than corresponding

16

hybrids even after inclusion of dispersion terms. These latter particularly enhance the

accuracy of the B3LYP family especially when introduced at -D3 and -D3(BJ) level for

B3LYP functional. Large effects are indeed computed already at -D2 level for PBE0, PBE0-

DH and B2PLYP: the change in MAD going from pure to -D2 corrected functional is indeed

larger than the one observed going from DFT-D2 to -D3 models. For example, the MAD of

PBE0-DH decreases from 2.39 to 1.55 kcal/mol when using the -D2 correction, while a

further decrease of 0.11 kcal/mol is computed going from the -D2 to the -D3(BJ) correction

models.

Considering the different subsets, the S22 and ADIM6 ones show exactly the same behavior

with best results obtained at DFT-D3(BJ) level, although very close, except for B3LYP

functionals, to errors obtained at DFT-D2 level. For these two sets, the MADs are indeed

very low, in the 0.11 kcal/mol (PBE0, ADIM6) to 0.48 kcal/mol (PBE0, S22) range.

Dispersion corrections over these two subsets also tend to level off the difference between

double and corresponding global hybrids accuracy. The same global trends are also valid for

the RG6 and HEAVY28, which indeed show very low MADs.

On the other hand, it is difficult to define global trends for the WATER27 subset. Indeed, for

the WATER27 subset, larger MADs are computed when including dispersion correction,

with the notable exception of the B3LYP functional which presents some improvement in

accuracy for all dispersion models tested. A slight enhancement of the accuracy is also found

for B2PLYP-D3 (0.29 kcal/mol).

Finally, concerning the IDISP subset, the behavior is less erratic, and the same patterns

described for the ADIM and S22 subsets can be found : an improvement of the results for

DFT-D2 models, with an impressive decrease in MAD, especially for global hybrids, with

respect to the non corrected functionals of 6.88 (B2PLYP), 9.32 (PBE0) and 4.0 kcal/mol

(PBE0-DH). The smallest MADs, of 1.27 kcal/mol, are indeed obtained at PBE0-D2 level of

theory.

4.7 Self interaction error

The performances of the four functionals over the SIE1134 subset, probing issues related to

self-interaction error, are not enhanced by the inclusion of dispersion corrections. Indeed, a

systematic increase of the MADs as a function of the inclusion of dispersion terms can be

observed. The variations in MADs are comprised between 0.02 kcal/mol (when going from

PBE0 to PBE0-D3) and 0.86 kcal/mol (when going from B3LYP to B3LYP-D3). However,

17

also when considering dispersion corrections double hybrids outperform their parent global

hybrids.

4.8 Nine difficult cases for DFT: the DC9 subset

The DC982-89 set contains nine reactions, not sharing any common features, considered as

particularly difficult cases for DFT approaches.

Globally, an improvement of the accuracy of the functionals due to the inclusion of

dispersion corrections can only be found for four reactions out of nine. Analyzing each

reaction in more details, it can be noticed that the reaction energy associated to the 2-

pyridone -> 2-hydropyridine reaction82, of only -1.0 kcal/mol, is actually very difficult to be

correctly reproduced. Only PBE0-DH gives a qualitatively and quantitatively correct result

predicting a reaction energy of -0.70 kcal/mol. Introduction of dispersion corrections globally

deteriorates the results and reaction energies of -0.48 kcal/mol (PBE0-DH-D2) and -0.55

kcal/mol (PBE0-DH-D3) are found. No change upon addition of dispersion is observed for

four reactions: the energy difference between hepta-1,2,3,5,6 hexaene and hepta-1,3,5 tryine

isomers80, the decomposition89 of S8 to diatomic S2, isomerization85 of (CH)12 and

cycloaddition between ethene and diazomethane87. A decrease of the MADs is observed for

the dimerization of tetramethyl-ethene to octamethyl-cyclobutane, especially in the case of

B2PLYP and PBE0 : 2.67 kcal/mol for PBE0-DH-D3, 1.51 kcal/mol for PBE0-D3, and 5.85

kcal/mol for B2PLYP-D3. Regarding the isomerization of C20 cage to its bowl isomer83, the

relative stability (of -13.3 kcal/mol) of the isomers is affected by large errors, independently

of the functional and of the dispersion corrections used. Only B2PLYP are improved by a –

D2 dispersion term allowing a reduction of the MAD of 5.03 kcal/mol. The same behavior is

found for the isomerization reaction of carbo-[3]-oxacarbon86, where the B3LYP family

provides relatively small errors and the dispersion further reduces the MAD computed with

B2PLYP: B2PLYP-D2 (1.63 kcal/mol) and B2PLYP-D3 (1.92 kcal/mol). The same holds for

the decomposition of the Be4 cluster into beryllium atoms88 with MADs of 6.87 (B3LYP-D3)

and 3.7 kcal/mol (B2PLYP-D3).

5.Effect of the parameterization of D3(BJ) model: the PBE0-DH case

As previously discussed in section 2, two possible set of D3(BJ) parameters can be defined

providing a very similar MAD on the S130 set for the PBE0-DH functional. Since the

difference in MAD obtained using the D3(BJ) parameterization proving the minimal MAD

and the D3(BJ)2 is very tiny (only 0.003 kcal/mol) and since the latter parameters are much

18

closer to those of Grimme for B2PLYP (refer to Table 1), we also decided to investigate the

performances of the PBE0-DH-D3(BJ)(2) dispersion model on all different GMTKN30

subsets. Detailed results are reported in Supporting Information.

Overall, the use of the PBE0-DH-D3(BJ)(2) model allows to decrease the global MAD on the

GMTKN30 database from 3.54 (computed using PBE0-DH-D3(BJ) model) to 3.49 kcal/mol

(as reported in Figure 2), that is a slight amelioration of the performances.

Most of the subsets are not affected by the change in parametrization going from the D3(BJ)

to the D3(BJ)(2) model, except for ALK6, DARC, DC9 and AL2X ones.

Indeed, considering the sets probing thermochemistry or conformers and isomers stability, a

slight increase is even found (from 5.74 to 5.60 kcal/mol for the W4-08 subset, for instance)

while the change in parametrization practically do not affect the accuracy in the prediction of

the adiabatic processes. On the other hand, subsets proving non covalent and dispersion

interactions present an overall slight improvement of the MAD for five sets out of six.

Regarding the four sets presenting largest amelioration of performances with respect to the

PBE0-DH-D3(BJ) parametrization, a decrease of 0.85, 1.18, 1.68 and 4.44 kcal/mol in MAD

is computed for the DARC, DC9, AL2X and ALK6 subsets, respectively, the latter subset

thus being the one mostly affected by the change in parametrization.

6. Conclusions

The overall performances of the functionals studied upon addition of dispersion corrections

are summarized in Figure 2 and 9.

As discussed above, globally the MAD computed using non parametrized functionals

decreases when adding dispersion terms although not necessarily increasing with the

complexity of the model of correction used. In particular, the D2 correction is found to

improve the performances of both PBE0 and PBE0-DH, but the systematic improvement

observed going from D2 to D3 (or D3(BJ)) corrections in the case of B3LYP is not observed.

On the other hand, both in the case of the non-parametrized and parametrized families of

hybrids, the effects of dispersion corrections terms are quantitatively larger on global hybrids

with respect to double hybrids so that the computed MAD difference between DH and

corresponding GH decreases by addition of dispersion terms. The Figure 9 shows how double

hybrids perform better for 28 sets out of the 31 studied: 13 times for PBE0-DH and 15 in the

case of B2PLYP. Adding dispersion does not change this ratio, since 27 sets out of 31 present

better MADs for double hybrids corrected with D2, and 29 out of 31 with D3 dispersion

correction terms.

19

Indeed when including dispersion, the number of sets for which PBE0-DH is the best

performing functional decreases at the benefit of B2PLYP: from 15 sets to 22 sets for

B2PLYP-D3. This data clearly shows that inclusion of dispersion is clearly more beneficial in

the case of the parametrized DH functionals than for non parametrized ones, independently of

the BJ parametrization used. The same conclusion globally holds for the corresponding

global hybrids, showing that the marriage between non parametrized functionals and

empirical corrections may be a difficult deal.

Supporting Information Available: Computed MADs over the different subsets.

Acknowledgments

This work was funded by the ANR agency under the project DinfDFT ANR 2010 BLANC n.

0425.

20

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Table 1: Empirical coefficients of the different models of dispersion for each functional considered in this paper.

DFT-D2 DFT-D3 DFT-D3(BJ)s6 s6 sr ,6 s8 s6 a1 s8 a2

B3LYP 1.05a 1.000 c 1.261 c 1.703 c 1.000 d 0.398 d 1.989 d 4.421 d

B2PLYP 0.55 a 0.640 c 1.427 c 1.022 c 0.640 d 0.307 d 0.915 d 5.057 d

PBE0 0.67 a 1.000 c 1.287 c 0.928 c 1.000 d 0.415 d 1.218 d 4.859 d

PBE0-DH 0.47 b 0.840 1.394 0.748 0.840 0.000 0.095 6.1020.840 e 0.345 e 0.747 e 5.276e

a Reference [49,50], b Reference [51], c Reference [36] ], d Reference [40]. e The model developed using this set of parameters will be referred to as PBE0-DH-D3(BJ)(2).

26

Figure captions

Figure 1: MAD (in kcal/mol) computed with the PBE0-DH functional for the S130 set, when

s8 and sr ,6 are varying, at fixed s6=¿0.84.

Figure 2: MAD (in kcal/mol) computed with the PBE0, PBE0-DH, B3LYP and B2PLYP

functionals over the GMTKN30 set including the dispersion corrections at different level.


functionals for sets concerning decomposition and atomization energies, including dispersion

corrections with different models.


functionals for sets concerning adiabatic processes, including dispersion corrections with

different models.


functionals for sets regarding barriers heights and reactions energies, including dispersion



functionals for sets regarding isomers (left) and conformers (right) stability, including

dispersion corrections with different models.


functionals for sets regarding dispersion and non covalent interactions, including dispersion



functionals for the sets DC9 (left) and SIE11(right), including dispersion corrections with

different models.

Figure 9: Count of best (up) and worst (low) MAD obtained for PBE0, PBE0-DH, B3LYP

and B2PLYP functionals, including dispersion at different level, over all subsets of the

GMTKN30 database.

27

Figure 1

28

Figure 2

29

Figure 3

30

Figure 4

31

Figure 5

32

Figure 6

33

Figure 7

34

Figure 8

35

Figure 9

36

SIE11

BH76

BSR36ACONF

HEAVY28

MB08-165

RSE34

DARC

BHPERIISO34

G2RC

DC9

G21EAALK6

WATER 27O3ADD6

AL2X

ISOL22

S22

IDISP

HEAVY 28

RG6

Table of contents

The performances of the B3LYP and B2PYLP functionals have been compared to those of PBE0 and PBE0-DH when three different types of dispersion corrections are applied (D2, D3 and D3(BJ)).

37

CYCONF

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