Hydrogen-bonded complexes upon spatial confinement: structural and energetic aspects.

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Hydrogen-bonded complexes upon spatial confinement: structural and energetic aspects† Paweł Lipkowski,* Justyna Kozłowska, Agnieszka Roztoczyn ´ ska and Wojciech Bartkowiak In the present study we consider structural and energetic aspects of spatial confinement of the H-bonded systems. The model dimeric systems: HF HF, HCN HCN and HCN HCCH have been chosen for a case study. Two-dimensional harmonic oscillator potential, mimicking a cylindrical confinement, was applied in order to render the impact of orbital compression on the analyzed molecular complexes. The calculations have been performed employing the MP2 method as well as the Kohn–Sham formulation of density functional theory. In the latter case, two exchange–correlation potentials have been used, namely B3LYP and M06-2X. The geometries of studied complexes have been optimized (without any constraints) in the presence of the applied model confining potential. A thorough analysis of topological parameters

Received 23rd August 2013, Accepted 10th November 2013

characterizing hydrogen bonds upon orbital compression has been performed within the Quantum Theory of Atoms in Molecules (QTAIM). Furthermore, an energetic analysis performed for the confined

DOI: 10.1039/c3cp53583e

H-bonded complexes has shown a different trend in the interaction energy changes. Additionally, a variational–perturbational decomposition scheme was applied to study the interaction energy compo-

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nents in the presence of spatial confinement.

1. Introduction Hydrogen bonds (HBs) undoubtedly represent one of the major types of weak intermolecular interactions with a huge impact on the surrounding world.1–8 In principle, according to the IUPAC definition, ‘‘the hydrogen bond is an attractive interaction between a hydrogen atom from a molecule or a molecular fragment X–H in which X is more electronegative than H, and an atom or a group of atoms in the same or a different molecule, in which there is evidence of bond formation’’.9,10 The phenomenon of hydrogen bonding is primarily about the fast association and dissociation of molecules, depending on the ambient temperature.3,4,11 This ability is a direct consequence of the value of the hydrogen bond energy, which is located between the van der Waals interactions and covalent bonds. Therefore hydrogen bonds play a crucial role in a variety of chemical, physical and biochemical processes. These include for example the enzymatic catalysis,12,13 the formation of supramolecular architectures,6 crystal engineering8 as well as molecular recognition, which are important for many life processes.14–17 Additionally, linear complexes linked by hydrogen bonds can be found in many

Theoretical Chemistry Group, Institute of Physical and Theoretical Chemistry, ´skiego 27, Wrocław, Poland. Wrocław University of Technology, Wybrzez˙e Wyspian E-mail: [email protected] † Electronic supplementary information (ESI) available. See DOI: 10.1039/ c3cp53583e

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molecular materials, including biomaterials, which makes them attractive objects of interest from both experimental and theoretical points of view.18–22 What is more, they constitute very important structural elements that determine the topological, electrical and optical properties of these systems.23–25 A significant part of physical and chemical phenomena occurs in spatially confined environments, therefore, this problem is of great importance for many areas of science. However, despite the considerable amount of data, and due to the extremely complicated nature of this phenomenon, knowledge of the properties of compressed matter is still lacking. Basically, spatial restriction influences strongly the stability, electrical properties, absorption spectra, dielectric constant as well as reactivity of atoms and molecules, as has been clearly demonstrated both theoretically and experimentally.26–36 This is particularly true in the case of guest–host systems, of which endohedral complexes with nanotubes and fullerenes are excellent examples and can serve as archetypal benchmarks.37,38 The unique properties of carbon nanotubes, including an extraordinary thermal conductivity and electrical properties, make them thoroughly studied systems by physicists, chemists and material engineers as promising materials for potential applications in nanotechnology, nanoelectronics, optics and medicine.39,40 Due to their hollow monolithic structure carbon nanotubes and other porous materials are intensively investigated as a medium for storing high energy materials, e.g. hydrogen.41,42 Moreover, the nanoneedle shape and ease of functionalization

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of CNTs make them emerging carriers for drug delivery and cancer therapy.43–45 Except for several studies of molecules exposed to orbital compression, an understanding of properties of hydrogen bonds upon spatial confinement is still far from being satisfactory. A great majority of studies on this subject were primarily focused on molecular dynamics simulations for the water complexes embedded in carbon nanotubes.29,46–49 An interesting outcome of these studies is, inter alia, that the average number of H-bonds decreases along with increasing strength of confinement.49 On the other hand, in order to obtain a more complete picture of the outlined research problem it becomes necessary to resort to the calculations based on ab initio methods. Recently, ´ski and Sadlej have studied the influence of the external Jabłon pressure on the character of the improper hydrogen bond for the 3-methylacrolein molecule embedded inside model helium clusters of different shapes and volumes.50 A large blue shift has been found based on their calculations for the C–H stretching frequency while the C–H bond length was decreased. In yet ´ski and Sola analyzed the FH NCH another study, Jabłon complex between two helium atoms placed on both sides of the dimer.51 Performed calculations have shown a reduction of the hydrogen bond strength of the analyzed system due to the spatial confinement. Similar observations have been made for the water chain as well as for complexes such as Cl3CH NH3 and HNO HNO encapsulated in a carbon nanotube.52,53 On the other hand, encapsulation of the ClH NH3 complex inside a carbon nanotube leads to shortening of the length of the hydrogen bond.53 Furthermore, a study by Ramachandran and Sathyamurthy based on the stabilization energy values indicated that the hydrogen bonds between water molecules embedded in fullerene C60 will be broken.54 In contrast, the electron localization function (ELF) analysis performed for a series of bihalides embedded inside fullerenes of different diameter (Cn, n = 60, 70, 80, 90) clearly demonstrates that spatial restriction leads to the shortening and strengthening of the hydrogen bond.55 As suggested by the authors, beyond the electrostatic interactions the presence of additional effects might also be responsible for the enhancement of the HB strength in confined systems. Apart from theoretical investigations, there are also few experimental reports regarding the problem of spatially limited hydrogenbonded complexes.56–63 For instance, the results of Raman scattering measurements conducted for crystalline deuterated formic acid demonstrate that under the influence of high pressure the strength of deuterium bonds increased.57 Similarly, an increase of both number and strength of the hydrogen bond has been demonstrated for the H2S in various high-pressure experiments.58–60 In more recent studies of Ajami et al. hydrogen bonded carboxylic acid dimers enclosed in molecular capsules have been considered.61,62 On the basis of NMR spectroscopic analysis it has been found that the spatial restriction caused shortening of the hydrogen bond in investigated complexes.61 These findings, in turn, are not in line with the results of density functional calculations performed for the similar guest–host systems.64,65 It has been shown that the confinement effects lead to nonmonotonic changes or increase of the hydrogen


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bond length. However, it should be noted that the calculated dimerization energies for the confined complexes became smaller as compared to those obtained in the gas phase. Based on what has been already discussed, one cannot draw clear and general conclusions regarding the characteristics of hydrogen bonds in confined spaces. This is also due to the fact that in the presented works the effect of orbital compression was taken into account in various ways. For this reason, we wish to undertake a systematic analysis of the impact of the spatial confinement on the properties of selected hydrogen bonds. Basically there are many approximate models of spatial confinement suitable for such investigations.36,66–69 Among them model potentials have become a useful tool that enables the representation of general aspects of spatial restrictions, especially effects related to the valence exchange repulsion. This type of interaction appears as a result of the Pauli exclusion principle and increases rapidly, when the two subsystems approach each other. Several observations presented in the literature show that the model potentials quantitatively describe the confinement effects related to the interaction with a non-polarizable, electronically inert environment.28,36,68–75 In addition to valence repulsion other types of interactions (electrostatic, induction and dispersion) will also influence properties of the confined systems.76 Therefore, theoretical studies in which the effect of confinement is modeled by an exact representation of a chemical environment are also very valuable.77 However, in the present contribution, being the first part of a series of systematic studies, the model confining harmonic oscillator potential (HO) has been employed in order to describe the influence of spatial restriction on the energetic and topological parameters of selected linear H-bonded systems. It should be noted that such a methodology has been used previously for the description of confinement effects on the hydrogen-bonded [FHF] complex.78 In the above mentioned study, theoretical analysis indicated that the strength of the hydrogen bond increases along with increasing confinement strength. To the best of our knowledge, the present work is a first attempt to describe changes in geometrical and topological parameters as well as the interaction energy and its counterparts of hydrogen-bonded complexes under spatial restriction modeled by the harmonic confining potential.

2. Methodology The subject of the present study are closed-shell hydrogenbonded complexes, characterized by the decreasing strength of intermolecular interaction: HCN HCN, HF HF, HCN HCCH. In order to obtain changes of hydrogen bond properties, due to the presence of orbital compression, a two-dimensional model harmonic confining potential has been applied:67,69,70,73,78–80 Vconf = 12(x2 + y2)o2.

(1)

In the above equation o is related to the quadratic force constant and controls the strength of orbital compression. Such an approach on one hand mimics the confinement induced by

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Fig. 1 The molecular graphs of the investigated complexes: HF HF, HCN HCN, HCN HCCH. The big circles correspond to attractors attributed to the atoms while the small ones to bond critical points.

the nanotubelike environment, but on the other hand it allows to account for pure valence repulsion effects. In our studies we used multiple values of parameter o in the range of 0.0–0.8 au. Most of the calculations were carried out using the MP2 method, as well as employing density functional theory. In the latter case, B3LYP and M06-2X functionals were used. Calculations have been primarily carried out using the 6-311++G(2df,2pd) basis set with the aid of the Gaussian09 package.81 The analyzed complexes have been oriented along the z-axis in the Cartesian coordinate system, as has been shown in Fig. 1. The geometrical parameters of analyzed complexes have been optimized in vacuum as well as in the presence of spatial confinement without any constraints. In the latter case, the Broyden– Fletcher–Goldfarb–Shanno method was used.82 At this point it should be also noted that HO potential is included in the molecular Hamiltonian in the form of an one-electron operator and therefore acts only on the electrons in the entire volume of the molecule. On the other hand the relaxation of nuclei in this repulsive potential is only indirect and results from the electron density reorganization in the presence of spatial confinement. The energetic analysis was carried out in a supermolecular manner. In accordance with the supermolecular approach the interaction energy is defined as the difference between the total electron energy of the system and the sum of electron energies of subsystems (calculated at the same positions of nuclei R, as in the case of the interacting complex):76 DE = Eabc. . .(R) [Ea(R) + Eb(R) + Ec(R) + . . .].

(2)

So the obtained value of DE can be significantly overestimated, which is connected with the fact that calculations made with the basis sets of limited size lead to a non-physical effect known as a basis set superposition error, BSSE.76 This effect not only results in overestimating the total interaction energy value, but it can also cause a revaluation of other properties of the molecular system.83 It is therefore essential to eliminate basis set superposition error during calculations. In the simplest case, which is the correction for BSSE in two body systems, one employs the counterpoise correction scheme proposed by Boys and Bernardi.84 In this method energy of monomers is calculated in a basis set of the entire system (Oab): DE = Eab(Oab) [Ea(Oab) + Eb(Oab)].

(3)

It should be underlined that in the above equation energies of monomers have been calculated using the same geometry as in the interacting complex.


Moreover, the nature of the interaction for the selected systems has been analyzed based on a variational–perturbational scheme. All calculations in vacuum have been performed employing the modified version of the GAMESS US code,85–90 while in the presence of spatial confinement using the MOLPRO program.91 According to the variational–perturbational approach the total interaction energy (DEMP2) is partitioned into the Hartree–Fock (DEHF) and Coulomb electron correlation (DEMP2 corr ) components. The Hartree–Fock interaction energy term is decomposed into the electrostatic interaction of unperturbed monomer charge densities (e(10) el ), the corresponding exchange repulsion HF (DEHL ex ) and the charge-delocalization term (DEdel). The second MP2 order electron correlation correction term (DEcorr ) is decomposed into the second-order dispersion interaction (e(20) disp), the electron correlation correction to the Hartree–Fock components (e(12) el,r ) and the exchange-delocalization component (DE(2) ex-del) encompasses the remaining electron correlation corrections. Hence, the interaction energy may be expressed in the following way: DEMP2 = DEHF + DEMP2 corr ,

(4)

HL HF DEHF = e(10) el + DEex + DEdel,

(5)

(20) (12) (2) DEMP2 corr = edisp + eel,r + DEex-del.

(6)

where

The topological analysis of the electron density has been performed using the QTAIM method proposed by Bader.92 It is based on the fact that location and properties (including in particular the electron density and its Laplacian as well as density of the kinetic and potential energy) of the characteristic critical points allow to define the strength and nature of the studied interactions.92–99 The AIM calculations were performed using the AIMAll program100

3. Results and discussion Three H-bonded complexes, namely: HF HF, HCN HCN and HCN HCCH, have been chosen for a case study (Fig. 1). Due to their simplicity and versatility some of them have been intensively investigated as the model complexes for the cooperative effect analysis,101–103 topological and energetic property calculations94,104,105 as well as in the field of nonlinear optics.106–110 In the scope of present study the linearity of all of the investigated systems has been assumed. It should be noted that only for the HCN HCN and HCN HCCH complexes the linear structures have been experimentally observed.111,112 Additionally, in our calculations the equilibrium linear geometries of HCN HCN and HCN HCCH have been confirmed by the analytically determined vibrational spectra. On the other hand, it is well established that the HF molecules form the zig-zag chains in the solid state.113 However, the linearity of the HF dimer has been also adopted herein for the consistency of further discussion. Such an approach has been previously used in various theoretical studies in which properties of the linear HF HF complexes were considered.94,105,107–110


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Table 1 Intermolecular interaction energy components for the HF HF, HCN HCN, HCN HCCH complexes computed using the variational– perturbational scheme and the 6-311++G(2df,2pd) basis set. All data are given in kcal/mol

HF HF HCN HCN HCN HCCH

e(10) el

DEHL ex

DEHF del

e(12) el,r

e(20) disp

DE(2) ex-del

DEMP2

4.66 6.54 3.42

2.60 4.22 2.59

1.26 1.68 0.82

0.61 0.38 0.12

1.04 1.84 1.47

0.53 0.91 0.60

3.23 4.56 2.41

In order to get a deeper insight into the nature of the selected complexes as well as to distinguish differences among them, the decomposition of the interaction energy in vacuum, according to the variational–perturbational scheme, has been conducted. From the data reported in Table 1 one can notice that for all investigated complexes the electrostatic contribution to DE is the most significant. Nevertheless the contribution of the dispersion term to the interaction energy varies depending on the system and in HCN HCCH it is the most significant, whereas in the HF dimer it is the least important. According to the above considerations it is interesting to examine whether the influence of the spatial confinement on the H-bond properties is similar for all investigated systems. 3.1

Energetic and geometrical analysis

It is well established that in order to correctly predict the properties of the hydrogen bonded systems it is crucial to select an appropriate level of theoretical approximation. In particular, a method should account for the electron correlation and yet, in order to reliably describe the electron density between the proton donor and the acceptor, the basis set should be augmented with a set of diffuse and polarization functions.18 So far the MP2/6-311++G(d,p) methodology commonly used for such complexes proved to be successful.105,114 In the present study, in order to improve the accuracy of the results, all calculations have been carried out employing the 6-311++G(2df,2pd) basis set, which is also an often choice for the description of other noncovalent interactions.115,116 Moreover, due to the lack of systematic studies concerning the properties of H-bonded systems in the harmonic confining potential, some additional tests have been performed to confirm the reliability of the selected basis set. In doing so, we have compared the values of the interaction energies, calculated with and without the counterpoise correction scheme (eqn (3)), with these obtained for a series of Dunning correlation consistent basis sets aug-ccpVXZ, where X = D,Q,6 (cf. Fig. 2). The system characterized by the largest interaction energy, e.g. the HCN HCN complex, has been selected for test calculations. The external potential values (see eqn (1)) in the range 0.0–0.8 au were considered. The results presented in Fig. 2 are based on the geometry of the HCN dimer in vacuum, optimized at the MP2/6-311++G(2df,2pd) level of theory. Obtained data clearly indicate that the corrected and uncorrected results in general demonstrate a quite opposite tendency. It is connected with the fact that the basis set superposition error gradually increases with the increasing strength of the applied HO potential. Moreover, in the case of the basis


Fig. 2 Intermolecular interaction energy changes for the HCN HCN complex in the presence of the confining potential of cylindrical symmetry. All calculations have been performed using the MP2 method. The acronym AccX denotes the aug-cc-pVXZ basis set while the CP refers to the results obtained using the counterpoise correction procedure. Lines are drawn to guide the eye.

sets smaller in size than that of the sextuple-x basis set, the BSSE was found to be a few orders of magnitude larger than the estimated interaction energy. Only the calculations employing the aug-cc-pV6Z basis set and without the CP procedure reproduced the proper trend of changes in the interaction energy together with a relatively small value of the BSSE. Therefore, in order to obtain reliable values of DE of the HB systems in the spatial confinement represented by the HO potential, the elimination of the basis set superposition error is needed. Otherwise, preliminary tests have to be performed to select the basis set providing correct results. Furthermore, the data depicted in Fig. 2 demonstrate that the results obtained using the 6-311++G(2df,2pd) basis set and the CP procedure are in line with those obtained using Dunning’s correlation consistent basis sets. Thus, the selected Pople’s basis set is found to be adequate for the further analysis. As it was mentioned before, two computational strategies, namely MP2 and KS-DFT (using B3LYP and M06-2X functionals), have been used in the calculations in the present study. Unless otherwise indicated, all energetic and geometrical parameters are discussed based on the MP2 data. The results obtained using the remaining methods have been collected in the ESI† (Tables S1, S3, S5 and S7–S9). It is worth underscoring that all three methods provide qualitatively and quantitatively similar outcomes. This has been demonstrated on the basis of the interaction energy values for the HCN dimer in the presence of the confining potential (see Fig. 3). The data depicted in Fig. 3 have been calculated for the geometry optimized in vacuum (model ‘u’) as well as for the geometry relaxed for each o value (model ‘r’). As one can notice the quantitatively significant difference occurs only as a result of geometry relaxation and not because of the applied method. The interaction energy values, estimated as a function of the confinement strength using the supermolecular method, for all complexes in question are presented in Fig. 4. Additionally,


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Fig. 3 Intermolecular interaction energy changes for the HCN HCN complex in the presence of the confining potential of cylindrical symmetry. All calculations have been performed using the 6-311++G(2df,2pd) basis set. The acronym ‘u’ refers to the geometry optimized in vacuum while ‘r’ denotes the geometry relaxed for each o value. Lines are drawn to guide the eye.

Fig. 4 Intermolecular interaction energy changes for the HF HF, HCN HCN, HCN HCCH complexes in the presence of the confining potential of cylindrical symmetry. All calculations have been performed using the MP2/6-311++G(2df,2pd) method. The acronym ‘u’ refers to the geometry optimized in vacuum while ‘r’ denotes the geometry relaxed for each o value. Lines are drawn to guide the eye.

in an attempt to explain the observed changes in DE, the variational–perturbational scheme in the presence of the spatial confinement has been applied. The four leading interaction energy terms, contributing to DEMP2, have been analyzed: e(10) el , (20) (HF) e(10) , e and DE +D +D . The last term, which is a sum of ex disp del F W the charge-delocalization and two other small ‘‘zeroth-order exchange’’ contributions without any physical interpretation,117–119 has been calculated as DEHF e(10) e(10) (note ex el (HL) (10) 117 that DEex = eex + DF + DW). The results obtained for models ‘u’ and ‘r’ are illustrated in Fig. 5 and collected in Tables S10 and S11 in the ESI.† In the case of the HCN dimer (model ‘u’), for small o values, one can observe a slight increase in the stability of the complex followed by a noticeable decrease in the DE value for the increasing confinement strength (see Fig. 4).


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Such an effect is even more pronounced when the system is optimized in the presence of the external potential (model ‘r’). One should note that for the geometry relaxed in the harmonic confining potential e(10) ex increases gradually with the increasing value of o. Hence, at some point, this repulsive term dominates to a large extent the interaction energy of the HCN dimer in the spatial restriction. This in turn is reflected by the noticeable decrease in the interaction energy value (destabilization of the complex), which can be observed for o values greater than 0.20 au. It is due to the fact that structural relaxation leads to the shortening of the hydrogen bond distance (see Fig. 6). As follows from the theory of the intermolecular interactions the valance repulsion increases rapidly together with the decreasing distance between two subsystems.76 Therefore, the more prominent destabilization of the complex is caused by the rapid growth of the exchange energy term being a result of the H-bond length shortening. On the contrary, for model ‘u’, in which the H-bond length is constant, only a small decrease of the e(10) ex term is observed. Additionally, for both employed models, the electrostatic contribution to the interaction energy initially increases (is more stabilizing) and becomes less stabilizing from o values ca. 0.40 au. Quite a similar trend of changes in the interaction energy as well as in the case of the e(10) ex term is also observed for HF HF. However, this system became less stable for a much larger o value. This is due to the fact that in the case of interacting HF monomers the e(10) el term is more essential than in the case of the HCN HCN complex and increases in the whole range of o values. Moreover, the difference in the o value, for which the change in the interaction energy trends occurs, is quite significant between the vacuum based geometry and the relaxed one, as compared to the HCN dimer. In contrast, the picture of DE changes is completely different for the last of the investigated complexes (HCN HCCH). In this hydrogen–bonded system even for the smallest o value a decrease in the interaction energy has been found. This is connected with the fact that for this complex the e(10) term diminishes gradually in the whole range of o. The el remaining counterparts of the interaction energy considered herein behave similarly for all analyzed H-bonded dimers. In the case of model ‘r’ the DE(HF) del + DF + DW term increases with confinement strength, while for model ‘u’ it remains almost constant. Moreover, negligible changes in e(20) disp do not affect the interaction energy trend in the presence of spatial restriction. In the light of gathered information it appears that the influence of the confining potential on the interaction energy of H-bonded systems varies and strongly depends on the investigated complex. This is caused by the different character of changes in the electrostatic term for each of the analyzed systems. Furthermore, the careful analysis of the obtained results led to the subsequent observation that the tendency of e(10) el follows the interaction energy changes in the case of model ‘u’ for each of the analyzed HB systems. Referring in turn to the results of structural parameters, obtained during the optimization in the presence of HO potential, we have found that for all investigated systems the spatial confinement leads to the significant shortening of the


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Fig. 5 Changes in DE and the leading interaction energy components computed using the variational–perturbational scheme for the HCN HCN, HF HF, HCN HCCH complexes in the presence of spatial confinement. The data obtained within model ‘u’ are prestented on the left-hand side of the figure, while those for model ‘r’ are depicted on the right-hand side of the figure.

hydrogen bond length. In this regard, our findings are consistent with those from experimental studies.57,61 The changes in the HB distance upon increasing value of o are demonstrated in Fig. 6. As one can notice, the most pronounced changes in the HB length, up to 24.5% for o = 0.8 au, appear for the


HF HF system. On the other hand, for the remaining complexes this change is still noticeable but does not exceed 10.0%. Moreover, a similar behavior is found for all covalent bonds. In this case, the observed decrease is also substantial and approximately equals to 11.5–13.0% (see Tables S7–S9 in ESI†).


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Fig. 6 Intermolecular hydrogen bond length changes for the HF HF, HCN HCN, HCN HCCH complexes in the presence of the confining potential of cylindrical symmetry. All calculations have been performed using the MP2/6-311++G(2df,2pd) method. Lines are drawn to guide the eye.

This is in line with previously reported theoretical results, where the shortening of all covalent bonds was observed for the diatomic molecules,67,69,80 cyanoacetylene73 as well as 1,3-butadiene and diborane.120,121 Additionally, Fig. 7 presents the influence of the spatial confinement on the Dd X–H parameter which can be described as follows: Dd X–H = d(X–H)opt,complex d(X–H)opt,monomer.

(7)

In the above equation d(X–H)opt,complex refers to the bond length in the H-bond donor within the complex, and d(X–H)opt,monomer represents the corresponding value in the isolated proton donor. The geometry relaxation has been carried out for both the complex and the monomer for each o value. The elongation of the X–H proton donating bond due to the complexation may allow us to estimate the hydrogen bond strength.18 According to the presented results, in the case of HF and HCN dimers, the increasing value of Dd X–H is due to the fact that under the influence of the HO potential the decrease of the X–H length within the complex is smaller than in the case of the isolated proton donor. This in turn suggests that the confinement effects cause an increase of the HB strength in these complexes. The opposite tendency observed for the HCN HCCH system indicates weakening of the hydrogen bond strength. Therefore, it is worth mentioning that the obtained results demonstrate different behavior of investigated systems under the spatial restriction. This is consistent with the conclusions reached on the basis of the interaction energy analysis (cf. discussion for Fig. 4). 3.2

Fig. 7 The influence of the spatial confinement on the Dd X–H parameter. All calculations have been performed using the MP2/6-311++G(2df,2pd) method. Lines are drawn to guide the eye.

density (rHBCP), its Laplacian (r2rHBCP) as well as density of the kinetic (GHBCP) and potential (VHBCP) electron energy at the hydrogen bond critical point (HBCP). In the present study, due to the lack of analytic energy derivatives we have not attempted to determine electron density for the complexes in the external confining potential employing correlated wave functions. Instead, we analyze the electron density obtained using the Kohn–Sham formulation of density functional theory. The discussion of QTAIM parameters at HBCP has been carried out based on the results obtained using the M06-2X functional. Additionally, the qualitatively similar B3LYP data are collected in Tables S1–S6 in ESI.† In Fig. 8 the dependence of electron density at the hydrogen bond critical point on o is shown. For the sake of clarity, Fig. 8 (as well as Fig. 9) are depicted for o values in the range of 0.0 to 0.6 au. It follows from this plot that rHBCP gets larger upon increasing the spatial confinement. This result was quite expected, as a direct consequence of the shortening of the hydrogen bond in the presence of HO potential. However, it should not be overlooked that trends observed for relaxed and unrelaxed

QTAIM analysis

A topological analysis of the electron density based on the QTAIM method, which was proposed by Bader, is very useful in the description of noncovalent interactions, especially in the case of H-bond interactions.92,95,105,122 Location and properties of the characteristic bond critical points allow to define the strength and nature of the studied interactions. In this regard, of particular importance are parameters such as: the electron


Fig. 8 The electron density at the hydrogen bond critical point in the presence of the confining potential of cylindrical symmetry. All calculations have been performed using the MP2/6-311++G(2df,2pd) method. Lines are drawn to guide the eye.


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Fig. 9 The total electron energy density at the hydrogen bond critical point in the presence of the confining potential of cylindrical symmetry. All calculations have been performed using the MP2/6-311++G(2df,2pd) method. Lines are drawn to guide the eye.

geometries are different. In the case of vacuum-based geometry this relation is clearly linear (the coefficient of determination (R2) equals to 0.9912, 0.9891, 0.9839 for HF HF, HCN HCN and HCN HCCH respectively), while for relaxed one it is described by a second order polynomial (R2 equals to 0.977, 0.9966, 0.9981 for HF HF, HCN HCCH and HCN HCCH respectively). Therefore, it is evident that in order to describe properly the changes in the rHBCP in spatial confinement, it is necessary to carry out the geometry optimization. Moreover, on the basis of other properties i.e. total electron energy density (HHBCP, where HHBCP = GHBCP + VHBCP) and its relation with r2rHBCP one can distinguish H-bond interactions of different nature:92–99 strong, covalent HB: r2rHBCP o 0 and HHBCP o 0 medium, partially covalent HB r2rHBCP > 0 and HHBCP o 0 weak, closed shell HB r2rHBCP > 0 and HHBCP > 0. The results of our calculations demonstrate that r2rHBCP > 0 for the entire range of o values, which indicates the closed shell nature of HB in complexes under consideration (cf. Tables S1–S6 in ESI†). Nevertheless, as illustrated in Fig. 9, the change in sign (from positive to negative value) is observed for the total electron energy density. This in turn reflects the modification in the nature of analyzed hydrogen bonds from noncovalent to partially covalent. Noteworthy, such a change occurs for smaller o values in the case of relaxed geometry. What is more, the order in which change of the sign for HHBCP appears is directly connected with the strength of the H-bond in the case of analyzed systems.

4. Conclusions In this work, we report on the effect of confinement on the energetic, geometric as well as topological parameters of three hydrogen bonded systems (i.e. HF HF, HCN HCH, HCN HCCH). In order to assess the influence of spatial restriction


on the properties in question, the two-dimensional harmonic confining potential, mimicking a topology of nanotubelike carbon cages, has been used. Our results demonstrate that changes in the interaction energy trends, arising from the presence of the external potential, differ for each of the analyzed complexes and strongly depend on the strength of applied repulsive potential. In general, however, at some strength of the potential the interaction energy becomes less attractive for all investigated complexes. According to the results of energy decomposition obtained using the variational– perturbational scheme, the shape of the interaction energy curve is dominated by the electrostatic term. What is more, in the case of model ‘u’ e(10) reproduces the tendency of the interaction energy el changes for each of the analyzed HB systems. The remaining contributions to the interaction energy analyzed herein behave similarly for all investigated complexes in the spatial confinement. Likewise, the changes in the X–H proton donating bond, due to complexation, also demonstrate different behaviors of investigated systems in spatial confinement. The geometry relaxation in the presence of external repulsive potential leads to decrease of all covalent as well as hydrogen bonds of investigated systems. It is worth mentioning that the shortening of the hydrogen bond upon spatial confinement has been also reported previously in some experimental work. The performed QTAIM analysis indicates that the spatial confinement causes an increase in electron density at the hydrogen bond critical point. Moreover, it has been also found that the spatial restriction causes the modification of the nature of analyzed hydrogen bonds from noncovalent to partially covalent. Another important finding of this study, of significance for future theoretical considerations, is that the basis set superposition error is enhanced significantly in the presence of external harmonic confining potential. Finally, in our opinion results presented in this study constitute a very significant step toward the description of the hydrogen bond behavior under the influence of the orbital compression. However, it would be also interesting to confront our findings with other confinement models. This aspect is currently studied and will be considered in our forthcoming contribution.

Acknowledgements This work was supported by a statutory activity subsidy from the Polish Ministry of Science and Higher Education for the Faculty of Chemistry of Wrocław University of Technology and NCN grant no.: 2011/03/D/ST4/00744. The authors gratefully acknowledge Wrocław Center of Networking and Supercomputing for the generous allotment of computer time. This research was supported in part by PL-Grid Infrastructure. The authors would also like to thank Dr Josep Maria Luis for his computer program to perform geometry optimization using the BFGS method and Dr Robert Zalesńy for technical assistance.

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