Carbohydrate–aromatic p interactions: a test of density functionals

and the DFT-D methodw

Rajesh K. Raju, Anitha Ramraj, Ian H. Hillier,* Mark A. Vincent

and Neil A. Burton

Received 19th December 2008, Accepted 28th January 2009

First published as an Advance Article on the web 2nd March 2009

DOI: 10.1039/b822877a

The performance of a number of computational approaches based upon density functional theory

(DFT) for the accurate description of carbohydrate–p interactions is described. A database

containing interaction energies of a small number of representative complexes, computed at a

high ab initio level, is described, and is used to judge 18 different density functionals including the

M05 and M06 families as well as the DFT method augmented with empirical dispersive

corrections (DFT-D). The DFT-D method and the M06 functionals are found to perform

particularly well, whilst traditional functionals such as B3LYP perform poorly. The interaction

energies for 23 sugar–aromatic complexes calculated by the DFT-D method are compared with

the values from the 18 functionals. Again, the M06 class of functional is found to be superior.

Introduction

Weak interactions, particularly hydrogen bonds and dispersive

interactions, are central to many substrate–protein interactions,1

and as such their accurate description is needed to understand

structure–function relationships of proteins and to assist in the

associated problem of rational drug design.2 Thus, there is

much current interest in searching for the most appropriate

quantum mechanical model for quantifying these interactions

to a high level of accuracy. To achieve this using ab initio

methods requires methods beyond MP2, such as at the

CCSD(T) level, as well as the use of large basis sets, and basis

set superposition (BSSE) corrections.3 Calculations at this

level are still quite computationally demanding, but have been

used to establish databases containing complexes which

exemplify the varied range of such non-covalent interactions

found in biological systems. A variety of methods including

semiempirical Hartree Fock4 and DFT models,5 as well as

DFT-based approaches using different functionals,6–11 have

been explored in attempts to accurately predict these inter-

actions. Although the inadequacies of a number of popular

functionals such as B3LYP in describing non-covalent inter-

actions, particularly p–p stacking, have been noted, some

functionals, such as BH&H,12 and in particular the newer

meta and hybrid meta functionals of Zhao and Truhlar,6–8 are

considerably more successful. In addition, a particularly fruit-

ful approach13,14 has been to augment traditional DFT and

semiempirical15 methods with an empirical atom–atom

dispersive correction having the usual R�6 form. These various

methods have been used to study, in particular, hydrogen-

bonding and p–p stacking interactions. There have been far

fewer studies directed at the accurate modelling of those

interactions responsible for sugar–protein recognition, which

are important, for example, in the case of various amyloid-

forming proteins such as those associated with Alzheimer and

Creutzfeldt–Jacob diseases.16

We have recently shown17 that a density functional model

using a BLYP functional, when augmented with an empirical

dispersive term (DFT-D), is successful in describing a range of

complexes involving simple monosaccharides and analogues

of phenylalanine, tyrosine and tryptophan, when judged

against a very limited number of high level calculations. We

find that the important interactions involve the aliphatic C–H

and O–H groups which point toward the aromatic p system,

and that there can be a delicate balance between these

two classes of interaction. Thus, although crystal structures

display mainly C–H–p interactions, we find that for binary

fucose–toluene and a-methyl glucose–toluene complexes, the

most stable structures involve O–H–p interactions, which are

reflected in the infrared shift of the corresponding O–H

stretching frequency, our calculations being in good quantita-

tive agreement with experiment.18

It is thus important to find the most appropriate ways of

carrying out these calculations and this is the focus of the present

paper. We first construct a small database of high level calcula-

tions of monosaccharide–aromatic complexes, and then use these

data to assess the accuracy of a variety of DFT calculations,

particularly the newM05 andM06 families of functionals, as well

as the DFT-D approach. We then compare these DFT methods

to describe a larger set of monosaccharide–aromatic complexes.

In view of the importance of the aliphatic C–H and O–H–pinteractions in such complexes, and indications that such inter-

actions are somewhat stronger with tryptophan than with

phenylalanine,17 we have also studied the prototype complexes

of water and methane with benzene and 3-methylindole.

School of Chemistry, University of Manchester, Oxford Road,M13 9PL Manchester, UK. E-mail: Ian.Hillier@manchester.ac.ukw Electronic supplementary information (ESI) available: Optimizedstructures of fucose–toluene complexes at DFT-D/TZV2D level(Fig. S1); optimized structures of glucose–3-methylindole and–p-hydroxytoluene complexes (Fig. S2); coordinates of the structuresfor which ‘benchmark’ calculations were carried out (Table S1);interaction energies for full dataset, for 18 functionals, DFT-D andMP2, the MUE values compared to the DFT-D values are also given(Table S2). See DOI: 10.1039/b822877a

This journal is �c the Owner Societies 2009 Phys. Chem. Chem. Phys., 2009, 11, 3411–3416 | 3411

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Computational details

A number of benchmark studies were carried out employing

the strategy of Hobza and co-workers.3 Here the MP2 energies

of the complex and monomers are extrapolated to the basis set

limit by the use of the aug-cc-pVDZ and aug-cc-pVTZ basis

sets giving the MP2/CBS energy. This value is then corrected

for higher order correlation effects obtained by computing

MP2 and CCSD(T) energies using a modified 6-31G** basis

followed by employing the difference between these energies to

obtain the correction. The modified 6-31G** basis used more

diffuse polarization functions than are standard, in order to

describe the long-range interactions.19,20 Basis set super-

position errors (BSSE) are also taken into account in the

calculation of the interaction energy. We note that in their

study of the benzene dimer, Sinnokrot and Sherrill20 found

that both the standard aug-cc-pVTZ and the larger modified

basis, aug-cc-pVQZ, gave a similar potential energy surface

when BSSE is taken into account, giving us confidence in our

basis set extrapolation procedure.

We have previously found14 that DFT-D calculations can

describe the extensive range of structures in the full JSCH-

2005 database3 with a mean unsigned error (MUE) of less than

1 kcal mol�1. Recently,17 we have shown that for a limited

number of structures, a similar accuracy is found for mono-

saccharide sugar complexes. Briefly, in the DFT-D method the

dispersion corrected total energy is given by

EDFT-D = EKS-DFT + Edisp (1)

where EKS-DFT is the normal self-consistent Kohn–Sham

energy, and Edisp is an empirical term involving pair-wise

dispersive interactions

Edisp = �s6P

i

Pj4i(C

ij6/R

6ij)fdmp(Rij) (2)

Here, the summation is over all atom pairs, Cij6 is the disper-

sion coefficient for the pair of atoms i and j (calculated from

the atomic C6 coefficients), s6 is a scaling factor that depends

on the density functional used and Rij is the interatomic

distance between atoms i and j. A damping function is used

in order to avoid near singularities for small distances. This

function is given by

fdmp(Rij) = 1/(1 + exp(�a(Rij/R0 � 1))) (3)

where R0 is the sum of atomic van der Waals radii and a is a

parameter determining the steepness of the damping function.

In order to obtain the composite dispersion coefficients Cij6, a

simple average of the form

Cij6 = 2Ci

6Cj6/(C

i6 + Cj

6) (4)

is used. We here use the BLYP functional together with the

values for the C6, R0, s6, and a parameters taken from the

original parameterization of the DFT-D method.13 Following

Grimme,13 we do not consider basis set superposition errors

(BSSE), as we use a basis of triple zeta quality for the valence

orbitals, complemented with two sets of polarization functions

(TZV2D).21

We have carried out calculations using our implementation

of the DFT-D method, as well as for a range of pure func-

tionals. We have used a number of functionals in everyday use

such as B3LYP, BLYP and BH&H, together with the four

hybrid meta GGA functionals, MPW1B95, MPWB1K,

PW6B95, and PWB6K, developed by Zhao and Truhlar,

and the five new meta and hybrid meta functionals (M05,

M05-2X, M06, M06-2X and M06-L) also developed by the

same group. In all we have used 18 functionals as well as the

MP2 method. Those functionals not available in Gaussian

release (G03)22 were implemented by us in this code. All

calculations employed a TZV2D basis and were not corrected

for BSSE.

Computational results

Complexes of water and methane with benzene and

3-methylindole

We first discuss the results for the complexes of water and

methane with benzene. For the water–benzene complex

we have investigated three possible structures in which one

or both hydrogen atoms point towards the benzene ring. Two

are of C2v symmetry and one of Cs symmetry (Fig. 1). At the

DFT-D level it was possible to locate only the symmetric C2v

structures, and a number of stationary structures were found

to have one or more imaginary frequencies for various func-

tionals (Table 1). A similar situation was found for the

hydrogen sulfide–benzene complex studied with a variety of

functionals.9 The DFT-D structure of the water–benzene

complex (Fig. 1f) has a ‘benchmark’ interaction energy

of �3.35 kcal mol�1 (Table 2), close to the complete basis

set limit value of �3.33 kcal mol�1 of Min et al.23 for a Cs

structure. All the nine functionals of Zhao and Truhlar, as well

as the MP2 method, give interaction energies within

1 kcal mol�1 of the benchmark value, most being within

0.2 kcal mol�1 of this value. Of these M05, M06-L, MPWB1K

and PW6B95 can be singled out, all giving interaction energies

within 0.1 kcal mol�1 of the benchmark value. The three

structures we have studied are very close in energy, all methods

Fig. 1 Structures of methane–benzene (a–d) and water–benzene (e–g)

complexes.

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giving corresponding interaction energies which differ by less

than 0.1 kcal mol�1. To investigate the energetics of the

different structures further we have optimized both the C2v

and Cs structures at the MP4(SDQ)/TZV2D level, followed by

evaluation of their energies at the CCSD(T)/TZV2D level.

This procedure gave the Cs structure (Fig. 1g) to be lower in

energy by 0.05 kcal mol�1 than the C2v structure (Fig. 1f).

For the methane–benzene complex we have studied four

possible C3v structures, which have either three or a single

hydrogen atom pointing towards the benzene ring (Table 3,

Fig. 1). We find that the two structures having a single hydrogen

interacting with the benzene ring are preferred, with these two

structures, (c) and (d), having essentially the same energy. Our

benchmark value of �1.52 kcal mol�1, again for the DFT-D

Table 1 Interaction energy (I.E) (kcal mol�1) for benzene–water complex (Fig. 1), and distance between the centroid of benzene ring and centre ofmass of water molecule (A). The number of imaginary frequencies (I.F.) is also given. Blanks indicate that a structure could not be located

C2V (e) C2V (f) Cs (g)

I.E I.F. Distance I.E I.F. Distance I.E I.F. Distance

MP2 �4.17 2 3.22 �4.17 1 3.21 �4.21 1 3.24DFT-D �3.76 0 3.26 �3.77 1 3.25 — — —M05 �3.39 1 3.36 �3.40 1 3.36 �3.41 0 3.37M05-2X �4.09 0 3.17 �4.09 0 3.17 �4.13 2 3.20M06 �3.41 2 3.21 �3.43 2 3.20 �3.47 0 3.28M06-2X �4.40 0 3.13 �4.41 0 3.12 �4.41 1 3.13M06-L �3.26 0 3.16 �3.28 0 3.16 �3.19 1 3.21MPWB1K �3.23 0 3.20 �3.22 0 3.19 �3.26 0 3.24MPW1B95 �3.13 0 3.21 �3.13 2 3.24 �3.16 1 3.27PW6B95 �3.33 0 3.21 �3.35 2 3.26 �3.35 1 3.29PWB6K �3.74 0 3.21 �3.74 0 3.19 �3.79 0 3.23

Table 2 Interaction energies (kcal mol�1) at ‘benchmark’, DFT-D and MP2 levels, and for 18 functionals, for database of 7 DFT-D structures.The MUE values with respect to the CCSD(T) values are given

Complex ‘Benchmark’ DFT-D BLYP B3LYP B97-2 B98 BMK BH and H BH and HLYP PBE VSXC

II �6.81 �6.52 2.09 0.55 0.61 �1.28 �2.37 �7.65 �0.80 �1.61 �27.02I �9.21 �9.28 �0.36 �1.92 �1.58 �3.78 �4.59 �11.05 �3.28 �4.33 �31.88V �7.69 �7.97 0.44 �0.87 �0.61 �2.65 �3.16 �9.04 �1.98 �3.28 �28.10III �5.92 �5.14 2.07 0.67 0.61 �0.99 �1.72 �6.27 �0.57 �1.12 �23.42Benzene–H2O �3.35 �3.76 �1.33 �1.91 �1.85 �2.70 �2.56 �5.44 �2.44 �2.96 �10.26Benzene–CH4 �1.52 �0.97 0.93 0.44 0.36 �0.28 0.21 �1.98 0.02 �0.32 �5.28IV �7.42 �7.30 1.91 0.35 0.48 �1.58 �2.35 �8.23 �1.01 �2.00 �29.35MUE — 0.36 6.81 5.60 5.71 4.10 3.62 1.11 4.55 3.76 16.20

Complex MPW1B95 MPWB1K PWB6K PW6B95 MP2 M05 M05-2X M06 M06-2X M06-LII �2.99 �3.41 �4.41 �3.33 �8.94 �2.99 �5.46 �6.26 �6.39 �6.06I �5.09 �5.46 �6.62 �5.51 �11.61 �5.81 �8.23 �9.21 �8.93 �8.72V �3.77 �4.04 �5.17 �4.24 �10.02 �4.53 �6.64 �7.41 �7.48 �7.11III �2.59 �2.97 �3.89 �2.93 �7.73 �2.55 �4.77 �5.21 �5.60 �5.14Benzene–H2O �3.26 �3.38 �3.90 �3.44 �4.18 �3.41 �4.18 �3.56 �4.38 �3.29Benzene–CH4 �0.52 �0.64 �1.10 �0.72 �1.87 �0.81 �1.19 �0.91 �1.27 �0.80IV �3.1 �3.5 �4.6 �3.5 �10.0 �3.5 �6.00 �6.80 �6.80 �6.60MUE 2.94 2.66 1.90 2.63 1.78 2.64 1.01 0.43 0.45 0.60

Table 3 Interaction energy (I.E) (kcal mol�1) for benzene–methane complex (Fig. 1), and the distance between the centroid of benzene ring andcarbon atom of methane (A). The number of imaginary frequencies (I.F.) is also given. Blanks indicate that the structure could not be located

C3V (a) C3V (b) C3V (c) C3V (d)

I.E I.F. Distance I.E I.F. Distance I.E I.F. Distance I.E I.F. Distance

MP2 �1.62 2 3.58 �1.62 2 3.57 �1.90 0 3.68 �1.90 0 3.68DFT-D �0.71 2 3.67 �0.71 2 3.67 �0.97 0 3.82 �0.97 0 3.81M05 �0.66 2 3.80 �0.69 0 3.78 �0.94 1 3.95 �0.89 0 3.95M05-2X — — — �0.93 1 3.65 �1.15 2 3.77 �1.15 2 3.76M06 �0.62 6 3.59 �0.79 0 3.63 �0.88 2 3.83 �0.88 2 3.83M06-2X — — — �1.01 2 3.47 �1.34 3 3.63 �1.33 3 3.63M06-L — — — �0.85 1 3.65 �0.78 2 3.75 �0.81 1 3.75MPWB1K �0.37 0 3.90 �0.37 2 3.90 �0.61 0 3.79 �0.61 0 3.79MPW1B95 �0.34 1 3.90 �0.28 0 3.79 �0.50 0 3.81 �0.50 0 3.80PW6B95 — — — �0.54 0 3.78 — — — �0.80 1 3.98PWB6K — — — — — — �1.07 0 3.77 — — —

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structure (Fig. 1d), is very close to the value of Hobza et al.3 Other

estimates of the interaction energy are slightly less than this value,

at �1.4324 and �1.45 kcal mol�1.25 All of the nine functionals, as

well as the DFT-D method, give a value up to 1 kcal mol�1 less

than these values, with both M05-2X and M06-2X being in error

by less than 0.5 kcal mol�1, and the DFT-D value, 0.6 kcal mol�1,

being too small.

Methane–indole complexes have been studied at the

CCSD(T) level by Ringer et al.25 They have located a number

of structures, the lowest energy configuration, with an inter-

action energy of �2.08 kcal mol�1, having one hydrogen atom

pointing to each of the two rings. At the DFT-D level, we

find such a methane–3-methylindole structure to be the most

stable, with an interaction energy of �1.69 kcal mol�1

(Table 4). For the water–methylindole complex our p-complex

at the DFT-D level has one hydrogen atom pointing

towards each of the two rings, with an interaction energy of

�5.31 kcal mol�1, in contrast to the structure given by the

B3LYP functional in which water binds to a single ring.26 We

have also studied the more stable structure having the water

molecule hydrogen bonded to the indole nitrogen atom, but

do not discuss these results further. We have carried out

benchmark ab initio calculations of the interaction energy

of the water–methylindole and methane–methylindole

complexes using the DFT-D structures, yielding values

of �4.83 and �2.36 kcal mol�1, respectively. We have again

studied the performance of the nine functionals of Zhao

and Truhlar, the interaction energies for the corresponding

optimized structures are given in Table 4. The values for the

methane–indole complex are all greater by B0.5 kcal mol�1

than the corresponding ones for the methane–benzene

complex (Table 3), but for both complexes the trends in the

values for different functionals are very similar. A similar

picture emerges for the water–methylindole complex

(Table 4), where the trend is close to that for the water–

benzene complex (Table 1), although the interaction energies

are again greater, now by 1–2 kcal mol�1. All values

are within 0.5 kcal mol�1 of the benchmark value

(�4.83 kcal mol�1), except for the M05-2X and M06-2X

values, which are noticeably larger. The performance of

the MPW1B95, MPWB1K, PW6B95 and PWB6K functionals

in describing the complexes of water with benzene and methyl-

indole has been investigated by Zhao et al.,27 who find similar

interaction energies to the ones reported here.

Sugar–aromatic complexes

Benchmark studies

We have previously17 studied the fucose–toluene system at the

DFT-D level and have located ten minimum energy structures

which are close in energy, involving the three low energy con-

formers of fucose which differ in the orientation of O–H1 or

O–H2, which in turn leads to different interactions with the

p-system of toluene. We have described these structures in

Fig. S1 of ref. 17, and use the same labelling, 1–10, in this paper.

For convenience, we show these structures again in Fig. S1 (ESIw)of the present paper. In these structures, the aromatic group

interacts with either the upper or the lower face of the sugar. The

most stable group of complexes (8, 9, 10) involves the interaction

with one O–H and one C–H group of the upper face of the sugar.

The next group of complexes, (1, 2, 3, 5, 6), is higher in energy by

1–3 kcal mol�1, and involve the interaction of only C–H groups

of the lower face of the sugar. We have also studied 11 complexes

of glucose with 3-methylindole and p-hydroxytoluene (Fig. S2 of

ref. 17), which we have optimized at the DFT-D level, initial

structures being based upon crystal structures. Here, we number

these structures 11–21, and they are again shown in Fig. S2 (ESIw)of the present paper. Again, these structures can be characterized

in terms of the interaction of the C–H and O–H groups of the

sugar with the aromatic ring. In addition there is the possibility of

OH� � �N hydrogen bonding in the case of the indole systems.

We have chosen five complexes (I–V) shown in Fig. 2,

together with the water–benzene and methane–benzene com-

plexes, to be members of our small database, and have

evaluated the corresponding interaction energies at the high

ab initio level previously outlined, using structures optimized

at the DFT-D level. The coordinates of these seven structures

are given in Table S1 (ESIw). Structures I and II are 8 and 5,

respectively, from Fig. S1 (ESIw), and involve the interaction

of toluene with the upper and lower faces, respectively, of

fucose. Complex II has three C–H–p interactions, whilst

complex I has one C–H–p and one O–H–p interaction. The

p-hydroxytoluene–glucose complex IV is structure 17 from

Fig. S2 (ESIw), and also has three C–H–p interactions, and an

interaction energy close to that of II. We have chosen two

other sugar–aromatic complexes for our small database, a

galactose–benzene complex (III), which also has three C–H–pinteractions, and a a-methyl glucose–toluene complex (V)

which has two O–H–p interactions and a single C–H–pinteraction, with an interaction energy 1–2 kcal mol�1 less

than that of (I). Thus, in this small database we included five

quite diverse structures with interaction energies in the

range �5.9 to �9.2 kcal mol�1. It is to be expected that the

accurate prediction of these values, both absolute and relative,

will be a quite severe test of the various DFT methods.

Our benchmark consists of these five sugar–aromatic com-

plexes as well as the benzene–water and benzene–methane

complexes previously discussed. We have not included the

water–3-methylindole and methane–3-methylindole complexes

since these are similar to the corresponding benzene com-

plexes. We have evaluated the interaction energies of these

seven complexes at the DFT-D optimal geometry using the

DFT-D method, 18 functionals and MP2. These energies and

Table 4 Methane–3-methylindole and water–3-methylindole inter-action energies (kcal mol�1) using different functionals

Methane–3-methylindole Water–3-methylindole

DFT-D �1.69 �5.31M05 �1.26 �4.60M05-2X �1.89 �5.86M06 �2.01 �5.16M06-2X �2.49 �6.21M06-L �1.88 �4.91MPW1B95 �0.87 �4.44MPWB1K �0.95 �4.56PW6B95 �1.16 �4.63PWB6K �1.50 �5.12MP2 �3.25 �6.42

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the corresponding MUE values are given in Table 2. Of the

computational schemes used we see that the DFT-D method

performs the best, with a MUE of 0.36 kcal mol�1. The

traditional functionals, with the exception of BH&H, perform

particularly poorly. Of the Zhao and Truhlar functionals

tested, the newer M06 family performs much better than the

older M05 functionals, with MUE values only slightly greater

than for the DFT-D method.

Density functional studies of 23 sugar–aromatic complexes

We have studied the performance of the 18 functionals in

predicting the interaction energies of the 21 sugar–aromatic

complexes previously reported,17 together with structures

III and V. Since we have benchmark values for only five of

these complexes, we judge the interaction energies against the

corresponding DFT-D values, which we have found to repro-

duce those energies of our benchmark to better than

0.5 kcal mol�1. The interaction energies of these 23 complexes,

using a variety of functionals, are shown in Table S2 (ESIw),

the corresponding MUE values being shown in Table 5, which

also include the water–benzene and methane–benzene inter-

action energies. The inadequacy of the traditional functionals is

evident, all of which, except VSXC and BH&H, underestimate

the interaction energies by 3–7 kcal mol�1 whilst VSXC over-

estimates the interaction energy by a much larger amount, and

BH&H is again more successful, overestimating the inter-

action by only 1.5 kcal mol�1. The four hybrid meta GGA

functionals, MPW1B95, MPWB1K, PWB6K and PW6B95,

are generally more successful than the previous functionals,

apart from BH&H, and underestimate the interaction energies

by between 2.9 and 4.0 kcal mol�1, with PWB6K being the

more accurate. Of the two newer M05 functionals investigated,

M05-2X is the better performer, giving interaction energies

within B1 kcal mol�1 of the DFT-D values, whilst the

corresponding value for M05 is B3 kcal mol�1. We find the

M06 group of new functionals to perform even better, with all

three giving average interaction energies within 0.7 kcal mol�1

of the DFT-D values, with M06 and M06-2X performing

the best.

In these studies of the sugar–aromatic complexes using

different functionals, we have employed the DFT-D optimized

structures. As a check of this procedure we have optimized the

structures of the fucose–toluene complexes (3, 8) and evalu-

ated the corresponding interaction energies. We find values

of (�5.42, �5.82 kcal mol�1) for the M05 functional, and

values of (�7.72, �8.86 kcal mol�1) for the M06 functional.

These are close to the corresponding values obtained with the

DFT-D optimized structures of �5.32 and �5.81 kcal mol�1

for M05, and �7.98 and �9.21 kcal mol�1 for M06 (Table S2,

ESIw). We note, for comparison, that the DFT-D interaction

energies are �8.11 and �9.28 kcal mol�1 for 3 and 8,

respectively.

Conclusions

We have here shown that a small database of complexes

involving the interaction of –OH and –CH groups with

Fig. 2 DFT-D structures used in ‘benchmark’ calculations.

Table 5 Mean unsigned error (MUE, kcal mol�1) of interactionenergies for full dataset (25 structures, Table S2, ESIw), for 18functionals and MP2, compared to DFT-D values

Functional MUE

BLYP 8.12B3LYP 6.64B97-2 6.83B98 4.97BMK 4.12BH and H 1.48BH and HLYP 5.35PBE 4.57VSXC 18.62MPW1B95 3.62MPWB1K 3.24PWB6K 2.23PW6B95 3.26MP2 2.44M05 3.24M05-2X 1.00M06 0.40M06-2X 0.36M06-L 0.73

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aromatic p systems can be used to judge the performance of

both the DFT-D scheme and of a range of traditional and

newer density functionals. The studies we have reported here

reinforce previous findings concerning the application of DFT

methods to study non-covalent interactions of biological

importance. Thus, the DFT-D method is here particularly

successful with a MUE value of less than 1 kcal mol�1, a

similar value being found previously for a range of different

interactions.14 The poor performance of traditional func-

tionals is again evident, with the newer set of functionals of

Zhao and Truhlar being definitely superior. However, it is only

the latest M06 family which here gives MUE values less than

1 kcal mol�1, and is thus competitive with the DFT-D

approach.

Acknowledgements

We thank Dr Y Zhao for advice on the M0X functionals.

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