Intramolecular Hydrogen Bonds: the QTAIM and ELF Characteristics

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Intramolecular Hydrogen Bonds: the QTAIM and ELF Characteristics Franck Fuster† and Szawomir J. Grabowski*,‡,§ †

Laboratorie de Chimie Th
eorique (UMR-CNRS 7616), Universit
e Pierre et Marie Curie, 4 Place Jussieu 75252-Paris c
edex, France Kimika Fakultatea, Euskal Herriko Unibertsitatea and Donostia International Physics Center (DIPC), P.K. 1072, 200080 Donostia, Spain § IKERBASQUE, Basque Foundation for Science, 48011, Bilbao, Spain ‡

ABSTRACT: B3LYP/aug-cc-pVTZ calculations were performed on the species with intramolecular O
H 3 3 3 O hydrogen bonds. The Quantum Theory of Atoms in Molecules (QTAIM) and the Electron Localization Function (ELF) method were applied to analyze these interactions. Numerous relationships between ELF and QTAIM parameters were found. It is interesting that the CVB index based on the ELF method as well as the total electron energy density at the bond critical point of the proton
acceptor distance (Hbcp) may be treated as universal descriptors of the hydrogen bond strength, they are also useful to estimate the covalent character of this interaction. There are so-called resonance-assisted hydrogen bonds (RAHBs) among the species analyzed here. It was found that there are not any distinct differences between RAHBs and the other intramolecular hydrogen bonds.

’ INTRODUCTION There are numerous theoretical studies where various tools are applied for the analysis of inter- and intramolecular interactions. It seems that the Quantum Theory of
Atoms in Molecule
s (QTAIM),1 the topology of the electron localization function (hereafter designated as ELF or ELF method),2 the Natural Bond Orbitals theory (NBO)3 and various schemes of the decomposition of the interaction energy are among the most powerful tools which allow deepen the nature of interactions.4 The range of applications of these methods covers broad spectrum of chemical, physical, and biological processes, even if one restricts only to the mentioned above inter- and intramolecular interactions. Hence, the goal of this study is limited to the comparison of the QTAIM and ELF approaches applied to analyze the intramolecular O
H 3 3 3 O hydrogen bonds. Such a choice of specific systems where pseudorings exist being the result of the mentioned above hydrogen bonds5 may be treated as the starting point for the more extended analysis where a greater number of methods and other types of interactions are to be analyzed. Both methods, QTAIM and ELF, were often applied to analyze different types of interactions. For example, QTAIM was often used to characterize the hydrogen bond. One can mention numerous early studies; the set of complexes with HF and HCl molecules acting as the proton donors was early analyzed,6 the study on H-bonded complexes with the hydrogen fluoride acting as the proton donor where numerous topological characteristics and relationships were discussed,7 the other studies on conventional intermolecular8 and intramolecular hydrogen bonds9 and the study on X-H 3 3 3 π interactions.10 The C
H 3 3 3 O interactions were analyzed early with the use of r 2011 American Chemical Society

QTAIM,11 and the topological criteria for the existence of the hydrogen bond was proposed. It was proven that C
H 3 3 3 O interactions may be classified as hydrogen bonds.11 It is difficult to briefly mention numerous QTAIM studies on the hydrogen bond. Among the other ones there is the classification of hydrogen bonds in relation to their strength based on the QTAIM parameters;12 the inter-relations between QTAIM and the decomposition of the interaction energy;13 a recent study on a few H-bonded systems to show how the choice of the method and the basis set influence the calculated QTAIM parameters;14 and the application of QTAIM to analyze hydrogen bonds enhanced by π-electron delocalization15 or to characterize the covalency of hydrogen bond.16 The number of studies on hydrogen bonds where the ELF method is applied is significantly smaller. For example, hydrogen bonds from weak ones to very strong were analyzed by the ELF method17 and the distinct differences between weak, medium in strength and strong hydrogen bonds were found; the hydrogen bihalide anions were the subject of a topological study to compare the hydrogen bonds in the (X 3 3 3 H 3 3 3 X) complexes.18 The core
valence bifurcation index (CVBI) was defined to build up a scale for the hydrogen bond interactions.17 The further ELF analysis on the hydrogen bonds was performed later.19 However, in the latter study, the other set of hydrogen bonds was taken into account, also containing charge-assisted hydrogen bonds (CAHBs) in (FHF)
and C2H2 3 3 3 H+ 3 3 3 C2H2 complexes. For these systems, the monosynaptic basins Received: June 17, 2011 Revised: July 21, 2011 Published: July 22, 2011 10078

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The Journal of Physical Chemistry A V(H) are observed being the signature of very strong hydrogen bonds.20 It seems to be interesting if QTAIM and ELF characteristics are related to energetic and geometrical parameters and if they may be used as descriptors of the hydrogen bond strength. It is worth mentioning that there are few studies where both approaches, QTAIM and ELF, were compared. One can mention the analysis of grids of data for the determination of surfaces and volumes around maxima of the electron density21 or the study on hydrogen bonded systems where relationships between the characteristics of critical points found by ELF and by QTAIM are presented.22 The aim of this study is to check which ELF and QTAIM parameters are useful to categorize and especially to estimate the strength of hydrogen bonds. This is very important for the intramolecular hydrogen bonds where the lack of reference states does not allow to calculate the energy of this interaction. For the intermolecular hydrogen bond, the difference in energy between the complex and the monomers involved in the latter interaction is calculated. This is the binding energy usually identified with the H-bond energy. The latter attitude is rather justified because the hydrogen bond is the main interaction responsible for the stability of the complex, the other interactions are usually negligible.23 In the case of the intramolecular hydrogen bond, different approaches were proposed and discussed for the approximate estimation of its energy.24,25 However, there are meaningful differences between those approaches, thus, the problem of which of the methods provides the results corresponding to the intramolecular H-bond strength seems to be open. This is the reason why the systems containing intramolecular hydrogen bonds were chosen here for the deeper analysis. The so-called resonance assisted hydrogen bonds (RAHBs)26 are well represented within the sample chosen because for such systems the enhancement of the hydrogen bond strength is observed being the result of π-electron delocalization. There are controversies connected with the assumptions of RAHB model27 because it is claimed that the proximity of the proton donor and the proton acceptor is the result of the rigidity of the σ-framework and not of the π-electron delocalization. Hence, it is interesting to check if the QTAIM and ELF approaches allow to detect any meaningful differences between RAHBs and the other intramolecular hydrogen bonds.

’ COMPUTATIONAL DETAILS AND FEW DETAILS ON QTAIM AND ELF METHODS The calculations were performed using the Gaussian03 set of codes.28 The structures were optimized using the aug-cc-pVTZ Dunning’s correlated consistent basis set29 and the B3LYP functional.30 Frequency calculations were carried out to check that all the geometries correspond to energy minima. The Gaussian output wfn-files were used as inputs for AIM2000 program31 to perform the QTAIM analysis. The evaluation of the ELF function and of the basin properties has been carried out with the TopMoD program.32 The calculations connected with ELF analysis were then carried out in four steps: (i) the evaluation of the ELF function over a 3D grid; (ii) the identification of the various basins and assignment of the corresponding grid points; (iii) the location of the critical points of the ELF function; and (iv) the integration of the charge density over the basins. The electron localization function (ELF) introduced by Becke and Edgecombe33 is a local function that describes how much the

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Pauli repulsion is efficient at a given point of the molecular space. In the seminal paper of Becke and Edgecombe, ELF was derived from the Laplacian of the conditional probability. An alternative interpretation has been given by Savin and co-workers34 in terms of the local excess kinetic energy density due to the Pauli repulsion principle. This interpretation allows to generalize ELF to any wave function and in particular to the exact one. It was also proposed to use the gradient field of ELF to perform the topological analysis of the molecular space2 in the spirit of Bader’s theory of atoms in molecules (designated here as QTAIM: Quantum Theory of
Atoms in Molecules).1 To establish a scale for the weak and medium in strength intermolecular hydrogen bonds based on topological criteria the core
valence bifurcation index (CVBI) has been defined.17 It has been demonstrated that the value of ELF at the critical points between the V(D,H) and V(A) basins, ηvv(DH 3 3 3 A), on the one hand, and between V(D,H) and C(D), ηcv(DH), determine whether the complex can be considered or not as a single chemical species. This index has been applied successfully for a series of doubly hydrogen bonded complexes,35 the 2c
3e bonded complexes,36 and to interpret the “anomalous” strength of the intramolecular O
H 3 3 3 X hydrogen bond formed in cisortho-X substituted phenols.37 Moreover, a linear relationship between the CVBI and the complexation energy for a series of hydrogen bonds of different strength has been demonstrated.17,19 Additionally, CVBI is directly related to the bond energy estimated using an orbital approach, such as, the bond energy
bond order (BEBO) model.34 In the case of the intramolecular hydrogen bond where the acceptor and the donor are parts of the same molecule, this approach has been generalized by calculating the value of the ELF for which the core basins of the atoms are separated from the valence, ηcv(DH 3 3 3 A) and the ELF value at the (3,
1) critical point between the V(D,H) and V(A) basins, ηvv(DH 3 3 3 A). The core valence bifurcation index denoted hereafter by ϑ(DHA) is the difference of these two quantities, that is: ϑðDHAÞ ¼ ηcv ðDH 3 3 3 AÞ
ηvv ðDH 3 3 3 AÞ

ð1Þ

This index is negative when the first bifurcation creates two molecular-reducible domains, and this indicates a weak interaction between two moieties D
H and A. This interaction is physically interpreted as being mostly of the electrostatic nature. On the other hand, a positive CVBI is due to the core
valence separation, and it is the first bifurcation occurring in the reduction of the localization process. This is a moderate interaction where the main electrostatic nature of the hydrogen bond is preserved but there is also an additional covalent contribution due to the electronic delocalization between the V(D,H) and V(A) basins. Qualitatively, it is possible to define basin properties as the integral of a property density F(r) over the basin volume Ωi, which is particularly important for the discussion of the bonding of the basin populations N(Ωi) Z NðΩ FðrÞdr ð2Þ ̅ iÞ ¼ Ωi

and their variances Z Z 2 2 σ ðN; dr1 πðr1 , r2 Þdr2
½NðΩ ̅ Ωi Þ ¼ ̅ i Þ Ωi

Ωi

þ NðΩ ̅ iÞ 10079

ð3Þ

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Figure 1. Systems analyzed in this study; the designations used through the text and in tables are included.

where π(r1,r2) is the spinless pair function. The population of a protonated disynaptic basin, like V(D,H), may be either more or less than 2e
and it never exceeds 3e
. Typical values range from 1.3 to 2.5 e
. Moreover, the monosynaptic basins, like lone pairs, always occupy a large volume and their population might be larger than 2e
without noticeably increasing the Pauli repulsion within them. The relative fluctuation of the basin populations has been introduced by Bader in the case of atomic basins.38 Its generalization to localization basins σ2 ðN; ̅ Ωi Þ λðN; ̅ Ωi Þ ¼ NðΩ ̅ iÞ

ð4Þ

provides an indication of the delocalization within the Ωi basin. Usually a relative fluctuation larger than 0.45 betokens

delocalization.39 Because of the relative stability of the protonated basin population and to take into account the pressure generated by the V(A) basin over V(D,H) during the creation of hydrogen bond, the electron concentration concept was introduced ½NðΩ ̅ i Þ ¼

NðΩ ̅ iÞ Ωi

ð5Þ

In the case of QTAIM studies on intra- and intermolecular interactions, the properties of critical points are most often analyzed. Particularly, for the D
H 3 3 3 A hydrogen bond, these are the characteristics of the H 3 3 3 A bond critical point (in this study, H 3 3 3 O bond critical point); the electron density at H 3 3 3 A bcp, Fbcp, its laplacian, r2Fbcp, the total electron energy 10080

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density at bcp, Hbcp, and the components of the latter values, the potential electron energy density, Vbcp, and the kinetic electron energy density, Gbcp. There are relationships between topological parameters at critical point (expressed in atomic units, eq 6, virial equation, and eq 7). 0:25∇ Fcp ¼ 2Gcp þ Vcp 2

Hcp ¼ Vcp þ Gcp

Table 1. Geometrical Parameters (in Å) of O
H 3 3 3 O Hydrogen Bonds; the Designations of Species According to the Figure 1a species

O
H

H3 3 3O

O3 3 3O

1a

0.981

1.778

2.668

ð6Þ

1b

0.976

1.841

2.713

1c

0.981

1.768

2.658

ð7Þ

2

0.969

1.878

2.793

3 4

0.981 0.985

1.825 1.746

2.758 2.626

5

0.991

1.679

2.563

6

1.040

1.511

2.471

7a

1.001

1.673

2.570

7b

0.985

1.808

2.659

7c

0.990

1.744

2.616

7d

1.009

1.629

2.540

7e 7f

1.012 1.034

1.569 1.450

2.496 2.383

7g

1.003

1.649

2.532

8a

1.001

1.657

2.534

8b

0.998

1.669

2.539

9

0.993

1.702

2.565

Gcp is a positive value, whereas Vcp is a negative one. One can see that if the absolute value of the potential energy outweighs two times the kinetic energy then the laplacian is negative (eq 6). This implies the covalent character of interaction and it may concern covalent bonds as well as very strong hydrogen bonds.12,16 However, there are such interactions, especially hydrogen bonds, where the modulus of the potential energy only one time outweighs the kinetic energy; in such a case, the laplacian is positive but Hcp is negative (eq 7). Rozas et al. proposed the classification of hydrogen bonds.12 Weak and medium in strength hydrogen bonds show both positive r2Fcp and Hcp values. For strong H-bonds, r2Fcp is positive and Hcp is negative. For very strong hydrogen bonds, r2Fcp and Hcp values are negative. The electron density at H 3 3 3 A bcp is also an indicator of the hydrogen bond strength because numerous correlations between this value and the other descriptors of this interaction were found.40 This is the reason why it seems to be useful to compare the descriptors of hydrogen bond strength of QTAIM and ELF because in both methods we have found the hydrogen bond critical point (H 3 3 3 A bcp) and the ring critical point connected with the closing of the system owing to the intramolecular hydrogen bond formation (rcp). The latter critical points characterize topologically the existence of the hydrogen bond and they demonstrate the contribution of this interaction into the electron delocalization. These critical points (H 3 3 3 A bcp and rcp) were found for all systems considered here.

’ RESULTS AND DISCUSSION Different systems containing intramolecular hydrogen bonds are analyzed here (see Figure 1). Among these systems there are so-called resonance assisted hydrogen bonds (RAHBs).26 They are usually classified as strong hydrogen bonds. There are numerous controversies connected with the model of RAHB.25,27 In this study we do not relate to the correctness of the model or to those controversies. It is worth to mention that for RAHB systems the conjugated double and single bonds exist and this leads to the π-electron delocalization within the system and the equalization of single and double bonds. The enhancement of the hydrogen bonding strength is connected with such equalization according to the RAHB model.26 Such RAHB systems are well represented in the sample analyzed here. One of the systems taken into account, 1,4-butanediol, is characterized by the lack of π-electrons. It was chosen as the reference species because one can expect for this system the existence of the weak hydrogen bond, and that should be reflected in the geometrical and topological parameters. The other system is analyzed here for comparison (system 6: see tables and Figure 1); it may be classified as a proton sponge, for such moieties one can expect rather strong hydrogen bonds, similar to those of RAHBs. It is interesting to compare the RAHBs parameters with parameters

a

Bold species: such systems are usually classified as RAHBs.

of the other species to detect if there are any specific characteristics of each kind of interaction. Table 1 shows geometries of H-bridges. One can observe that generally, the shorter H 3 3 3 O distances occur for RAHBs and also for proton sponge (system 6). For the latter species, the shortest H 3 3 3 O distance is observed amounting to 1.511 Å. There is also the longest O
H bond for the latter kind of the hydrogen bond. The rough dependence between the protonacceptor distance and the strength of interaction was reported very often for hydrogen bonded systems.16,41 Also, the greater elongation of the proton donating bond is observed for stronger interactions. It means that RAHBs and the proton sponge are characterized by stronger interactions than the other hydrogen bonds analyzed in this study. For the sample considered here there is the linear correlation (R is equal to 0.964) between the O
H bond length and H 3 3 3 O distance. Such a correlation is observed in spite of the fact that the species analyzed are not strongly related. It was stated before that the elongation of the proton donating bond for hydrogen bonded systems should be normalized and related to the length of this bond not participating in any external interaction.42 However, such a reference is possible for the intermolecular hydrogen bonds and not for intramolecular ones. The shortest O
H bond, 0.969 Å, and the longest H 3 3 3 O distance of 1.878 Å occur for 1,4-butanediol. It means that, for this system, where there are not π-electrons, the hydrogen bond is the weakest one. Tables 2 and 3 present selected QTAIM and ELF parameters, respectively. The results collected in Table 2 are connected with the proton-acceptor (H 3 3 3 A) bond critical point (bcp). For the sample analyzed here, the oxygen atom is the proton acceptor center. It was found in numerous studies that the strength of interaction, particularly of the hydrogen bond, is expressed by the characteristics of H 3 3 3 A bcp.12,13 The increase of the strength of hydrogen bond is connected with the increase of the electron density at H 3 3 3 A bcp, with the increase of the kinetic energy 10081

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Table 2. QTAIM Parameters (in a.u.) Corresponding to the H 3 3 3 O bcp, the Electron Density at H 3 3 3 O bcp, Gbcp, its Laplacian, r2Gbcp, the Total Electron Energy Density at bcp, Hbcp, and the Components of the Latter Value; the Kinetic Electron Energy Density, Gbcp, the Potential Electron Energy Density, Vbcp; H 3 3 3 bcp is the H-Atom Radius (in Å)a species

a

Fbcp

r2Fbcp

Gbcp

Vbcp

Hbcp

H 3 3 3 bcp

1a

0.041

0.107

0.032


0.038


0.006

0.602

1b

0.035

0.104

0.028


0.031


0.002

0.640

Table 3. ELF Parameters (in a.u.) Corresponding to the ELF Value at the ELF bcp, η(bcp); ELF Value at the Density bcp, η(H 3 3 3 bcp); the Core Valence Bifurcation Index, CVBI; the V(H,O) Population (in electron), N; the V(H,O) Population Variance (in Electron), σ2; the Relative Fluctuation of the V(H,O) Basin Populations (in electron), λ; the Electron Concentration, [N]; the Cross Exchange Contribution V(H,O) into V(O), CECa species η(bcp) η(H 3 3 3 bcp)

CVBI

N

σ2

λ

[N]

CEC

1c

0.042

0.109

0.033


0.039


0.006

0.598

2 3

0.031 0.035

0.094 0.095

0.025 0.027


0.027
0.031


0.002
0.004

0.657 0.625

1c

0.163

0.590

0.024

4

0.045

0.110

0.035


0.043


0.008

0.583

2

0.114

0.648


0.032 1.71 0.84 0.49 0.061 0.04

1a 1b

0.163 0.128

0.593 0.628

0.018 1.81 0.87 0.48 0.066 0.05
0.009 1.80 0.86 0.48 0.062 0.05 1.80 0.87 0.48 0.066 0.05

5

0.052

0.118

0.041


0.053


0.012

0.549

3

0.133

0.615


0.003 1.80 0.86 0.48 0.065 0.05

6

0.080

0.092

0.056


0.090


0.034

0.449

4

0.182

0.574

0.047

1.80 0.87 0.48 0.066 0.06

7a

0.054

0.111

0.041


0.055


0.014

0.541

5

0.212

0.538

0.086

1.82 0.88 0.49 0.070 0.06

7b

0.038

0.108

0.031


0.034


0.004

0.620

6

0.366

0.429

0.230

2.00 0.95 0.48 0.081 0.09

7c

0.045

0.112

0.035


0.043


0.007

0.584

7d 7e

0.060 0.072

0.109 0.110

0.045 0.054


0.063
0.080


0.018
0.026

0.516 0.497

7a 7b

0.234 0.149

0.532 0.608

0.092 0.014

1.85 0.89 0.48 0.070 0.07 1.83 0.87 0.47 0.063 0.05

7c

0.182

0.574

0.047

1.82 0.87 0.48 0.066 0.06

7f

0.110

0.052

0.076


0.139


0.063

0.387

7d

0.269

0.506

0.139

1.89 0.91 0.48 0.073 0.07

7g

0.056

0.116

0.044


0.058


0.014

0.531

7e

0.315

0.460

0.174

1.87 0.82 0.48 0.079 0.08

8a

0.056

0.118

0.044


0.058


0.014

0.535

7f

0.472

0.359

0.323

1.84 0.94 0.51 0.086 0.10

8b

0.054

0.120

0.043


0.055


0.013

0.544

7g

0.237

0.520

0.108

1.85 0.89 0.48 0.071 0.07

9

0.050

0.118

0.040


0.050


0.019

0.560

8a

0.236

0.526

0.095

1.87 0.90 0.48 0.070 0.07

8b 9

0.222 0.206

0.535 0.548

0.087 0.068

1.85 0.89 0.48 0.069 0.07 1.84 0.88 0.48 0.067 0.06

The designations of species according to Figure 1.

electron density at bcp, the decrease of the potential energy (the increase of its modulus) and the decrease of the total electron energy density at bcp.16 The relation between the strength of interaction, and particularly, the H 3 3 3 A distance, and the laplacian, r2Fbcp, is more complicated; starting from the weakest hydrogen bonds the positive value of the laplacian increases with the increase of the strength of interaction. However, for very strong hydrogen bonds, it decreases and is even negative, like for the shared interactions, particularly covalent bonds.43 These changes of QTAIM parameters are also observed here if one compares the results of Tables 1 and 2. One can see that not only H 3 3 3 O distance may be treated as a descriptor of H-bond strength. For all interactions analyzed here, r2Fbcp is positive, which may mean that hydrogen bonds could not be classified as very strong ones. However, all Hbcp values are negative, which may indicate that such interactions are rather strong. For the lowest values of Fbcp, 0.035 a.u. and below (three cases, including 1,4-butanediol), Hbcp is very close to 0 and it is greater than
0.004 a.u. The linear correlation coefficient for the relation between Fbcp and Hbcp is equal to 0.987, while for the second order polynomial regression, the correlation coefficient is equal to 1.000. Table 2 also presents the QTAIM radius of the H-atom involved in the hydrogen bond interaction. The latter is the distance between the H-atom attractor and the H 3 3 3 O bcp. This value is smaller for the stronger interaction. This is in agreement with the conditions of the existence of hydrogen bond proposed by Koch and Popelier.11 It was stated there that the decrease of the H-atom volume and the penetration of H and A (acceptor) atoms are results of the hydrogen bond formation. This also means that the last effects are greater for the stronger hydrogen bonds.

a

The designations of species according to the Figure 1.

Table 3 presents the ELF characteristics, which also may be treated as descriptors of the hydrogen bond strength; ELF at H 3 3 3 A bcp, CVBI which is negative for weak hydrogen bonds, and increases with the increase of the strength of interaction. It is interesting that three negative values of CVBI are observed for the species where Hbcp is very close to 0, Fbcp values are the lowest ones and the H 3 3 3 O distances are greater than for the other hydrogen bonds. Table 3 also contains the ELF radius of the H-atom, the value corresponding to that one determined by QTAIM (and included in Table 2), population of V(D,H) basin, their variance, and the relative fluctuation. The cross exchange contribution is also included in Table 3. The population of V(D,H) basin never exceeded 2e, as expected: this low value is the consequence of the competition between the Pauli repulsion and the electron
nucleus potential. The electron
nucleus potential tends to concentrate the electron density in the neighborhood of the nuclei, as well as its local spherical symmetry tends to preserve the atomic shell structure. The minimization of the Pauli repulsion favors a partition into basins either large enough to enable same spin electrons to avoid one another or within which the integrated same spin pair density is low. The latter implies that the basin population does not noticeably exceed 2e. The delocalization of the V(H,D) basin is indicated by a rather high relative fluctuation (0.47
0.51). The ELF results of Table 3 lead to the conclusion that the electronic concentration and the cross exchange contribution grow with the strength of interaction, while the volume decreases. Moreover, the cross exchange contribution (CEC) correlates with the CVBI because the linear correlation coefficient for this relationship amounts to 0.977. The lowest values of CVBI and CEC are observed for the 10082

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Figure 2. Relationship between the ELF value at bcp (in a.u.) and the electron density at H 3 3 3 O bcp.

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Figure 4. Relationship between the CVBI and the electron density at H 3 3 3 O bcp (in a.u.).

Figure 3. Relationship between the ELF H 3 3 3 bcp distance and the QTAIM H 3 3 3 bcp distance (both in Å).

1,4-butanediol,
0.032 and 0.04, respectively. The highest values of the latter indices are observed for one of the RAHB systems (Figure 1, 7f), 0.323 and 0.10, respectively. Thus, one can observe correlations between the ELF characteristics and the geometrical and energetic parameters usually attributed to the hydrogen bond strength, similarly as it is observed for the QTAIM characteristics. However, numerous relationships were analyzed before, especially in studies where QTAIM was applied. Hence, they are not presented here. In this study we concentrate on the relations between the ELF method and QTAIM. Figure 2 presents the relationship between the ELF value at the H 3 3 3 O bcp and the electron density at H 3 3 3 O bcp. This is the linear correlation and the correlation coefficient, R, is equal to 0.996. Both values increase with the increase of the strength of interaction. Empty circles correspond to RAHBs (this designation is also applied for the remaining figures). Generally for RAHBs there are greater values of electron density, Fbcp, and ELF than for the other systems. This picture may not be so clear because the proton sponge included in the sample is also characterized by the strong interaction. However, this is the most important because there is no qualitative difference between RAHBs and the other intramolecular H-bonds, that all species are characterized by the same linear regression. It is worth mentioning that the position of bcp determined by QTAIM is not the same as that one determined by the ELF method

Figure 5. Exponential dependence between the electron density at the H 3 3 3 O bcp, Fbcp (in a.u.), and the H 3 3 3 O distance (in Å), as well as the dependence between the ELF calculated at bcp and this distance.

(see Tables 2 and 3). It is connected with the different physical meaning of both points. The QTAIM bond critical point corresponds to a minimum of the electron density in the normal direction of the bond which is a maximum in the orthogonal directions and its curvature is designated as (3,
1) in the QTAIM notation.1 The ELF bcp is connected with the electron localization function and not with the electron density. However, there is the correlation between both H 3 3 3 bcp distances determined by ELF and QTAIM approaches (Figure 3), the linear correlation coefficient is equal to 0.997. The increase of Fbcp and the ELF at bcp follows the shortening of H 3 3 3 O distance (see Tables 1
3), thus, it means that all these parameters are attributed to the strength of interaction. Figure 4 shows good linear correlation between Fbcp and CVBI (R = 0.992). This means that CVBI may be treated as a descriptor of the hydrogen bond strength; one can mention that Fbcp was analyzed in numerous studies and various correlations between this value and the binding energy for intermolecular hydrogen bonded systems were found.13,40 It is worth mentioning that the linear correlation coefficient between the ELF at bcp and CVBI is equal to 1.000. Figure 5 is in agreement with these findings, both 10083

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Table 4. H-Bond Energies (kcal/mol) Evaluated from Different Approaches Described in the Text As Well As in the Footnote of This Table; the Designations of Species According to Figure 1 system

Figure 6. Dependencies between the QTAIM parameters and CVBI; for Gbcp, the linear correlation is presented, while for Hbcp and Vbcp, the second order polynomial relationships are presented.

1a

2b

3c

4d

1a

3.28

10.14

9.35

12.02

1b

2.45

8.08

7.78

9.62

1c

3.45

10.52

9.74

12.32

2

2.10

7.09

5.58

8.43

3

2.60

8.58

6.45

9.69

4

3.77

11.38

11.12

13.50

5 6

5.12 9.40

14.48 26.57

14.38 21.02

16.65 28.22

7a

5.05

14.81

14.01

17.16

7b

2.77

9.10

9.70

10.82

7c

3.72

11.45

11.60

13.42

7d

6.03

17.34

15.83

19.70

7e

8.02

21.51

18.99

25.18

7f

13.05

33.08

30.25

43.63

7g 8a

5.62 5.45

16.16 15.66

16.35 16.24

18.18 18.17

8b

5.19

15.01

15.90

17.38

9

4.54

13.36

14.32

15.69

a

1. The Lippincott and Schroeder model, refs 41a, 41b, and 45; EHB = (
43.8 + 0.38AO
H 3 3 3 O) exp[
5.1(dO 3 3 3 O
2.4)]. b 2. Relation of Espinosa et al., ref 40c; EHB = [25300 exp(
3.6dH 3 3 3 O)]/4.14. c 3. Relation of Musin and Mariam, ref 46; EHB = 555400 exp(
4.12dO 3 3 3 O). d 4. Relation of Espinosa et al., ref 40c; EHB ≈ 0.5Vbcp; dO 3 3 3 O, dH 3 3 3 O, AO
H 3 3 3 O, are O 3 3 3 O distance, H 3 3 3 O distance, O
H 3 3 3 O angle, respectively.

Figure 7. Two linear correlations between the H 3 3 3 O distance and CVBI as well as between the H 3 3 3 bcp distance and CVBI (distances in Å).

ELF at bcp and Fbcp depend exponentially on the protonacceptor distance (the H 3 3 3 O distance for the sample analyzed here). Figure 6 presents dependencies between QTAIM parameters, Gbcp, Vbcp, and Hbcp (see eq 7) and CVBI. These are similar relationships as those found before for different samples of hydrogen bonded systems between the proton-acceptor distance or the electron density at H 3 3 3 A bcp and the QTAIM parameters, Gbcp, Vbcp, and Hbcp, presented in Figure 6.40c,44 There is the linear relationship between CVBI and Gbcp, while two other dependencies are fitted by the second order polynomials. Hbcp is negative for all systems analyzed here, which means that these are the strong interactions. CVBI is positive for almost all cases; in three cases, as it was mentioned earlier here, CVBI is negative. It was proposed earlier17 that medium in strength and weak H-bonds are characterized by negative CVBIs, while strong interactions are characterized by a positive value. The changes of Hbcp are in agreement

with the changes of CVBI values (there are only reverse signs of these parameters). For three cases characterized by a negative CVBI, the Hbcp value is very close to 0. Figure 7 presents the strong linear relationships between the H 3 3 3 A distance and CVBI on one hand and between the H 3 3 3 bcp (ELF) distance and CVBI on the other hand. The similar linear correlation is found if the QTAIM H 3 3 3 bcp distance is considered. The latter correlation is in line with the dependence between the ELF and QTAIM H 3 3 3 bcp radii presented in Figure 3. One can see that the QTAIM and ELF parameters may be treated as descriptors of the strength of interaction. However, these parameters are useful if the greater sample of species is analyzed and one compares the interactions within such a sample. In the other words, the values of the energy of interaction, especially H-bond energies are not known if only QTAIM and ELF approaches are applied. Especially, this is a serious problem for intramolecular hydrogen bonds where it is not possible to calculate these energies.24 It was mentioned earlier here that there are different approaches to evaluate the hydrogen bond energy for intramolecular interactions.24 Table 4 presents different approaches applied for the sample considered here. The hydrogen bond energy was determined from the Lippincott
Schroeder model,41a,b,45 from the relation proposed by Espinosa et al.40c and from the function proposed by Musin and Mariam.46 Espinosa and co-workers also found that H-bond energy is equal approximately to 0.5Vbcp,40c where the proton-acceptor bcp is considered. The corresponding values determined from the latter relation are also included in Table 4. One can see that sometimes there are meaningful differences between different approaches. 10084

dx.doi.org/10.1021/jp2056859 |J. Phys. Chem. A 2011, 115, 10078–10086

The Journal of Physical Chemistry A

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’ REFERENCES

Figure 8. Relationships between the H-bond energies (in kcal/mol) estimated from the different functions and CVBI. (1) Circles, H-bond energy determined from the Lippincott and Schroeder model [ref 45], y = 30.633x + 2.5233, R = 0.993; (2) squares, H-bond energy from the relation of Espinosa et al. [ref 40c], y = 74.868x + 8.4119, R = 0.996; (3) triangles, H-bond energy from the relation of Musin and Mariam [ref 46], y = 64.846x + 8.3795, R = 0.981; (4) rectangles, H-bond energy from the relation of Espinosa et al. [ref 40c] that EHB ≈
0.5Vbcp, y = 92.544x + 9.4564, R = 0.983.

Figure 8 presents the relationship between the H-bond energy calculated from the approaches described above and CVBI. It was shown in this study that CVBI is a good descriptor of the strength of hydrogen bonding. The linear correlations of Figure 8 are pretty good for the approaches applied here; even the second order polynomial or exponential correlations, not presented here, are slightly better. The energies calculated from different approaches (Figure 8) correlate well between themselves. However, absolute values sometimes differ significantly. Especially the values obtained from the model of Lippincott and Schroder are significantly different than those obtained from the other models.

’ SUMMARY Numerous correlations were found between geometrical, ELF, and QTAIM parameters for the sample of systems containing intramolecular O
H 3 3 3 O hydrogen bonds. So-called RAHB systems are well represented within this sample. These hydrogen bonds may be classified as the systems where the πelectron delocalization enhances the strength of the interaction. There is no meaningful difference between RAHBs and the other hydrogen bonds analyzed here. However, it may be concluded that RAHBs are often stronger than the other ones. Besides, it was shown that numerous QTAIM and ELF parameters may be useful as descriptors of the strength of H-bond interaction, especially CVB index and Hbcp seem to be useful. ’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

’ ACKNOWLEDGMENT Technical and human support provided by IZO-SGI SGIker (UPV/EHU, MICINN, GV/EJ, ESF) is gratefully acknowledged (S.J.G.).

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