The enhancement of X–H⋯π hydrogen bond by cooperativity effects – Ab initio and QTAIM calculations

Chemical Physics 355 (2009) 169–176

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The enhancement of X–H  p hydrogen bond by cooperativity effects – Ab initio and QTAIM calculations Sławomir J. Grabowski a,b,*, Jerzy Leszczynski a a b

Computational Center for Molecular Structure and Interactions, Department of Chemistry, Jackson State University, Jackson, MS 39217, USA Department of Chemistry, University of Łódz´, 90-236 Łódz´, ul.Pomorska 149/153, Poland

a r t i c l e

i n f o

Article history: Received 23 October 2008 Accepted 9 December 2008 Available online 24 December 2008 Keywords: Hydrogen bonding Cooperativity effect Quantum Theory of ‘Atoms in Molecules’ H  p interaction

a b s t r a c t The cooperativity effects for C2H2  (HF)n and C2H4  (HF)n (n = 1–4) complexes are analyzed using the results of MP2/6-311++G(d,p) calculations. It has been revealed that F–H  p and F–H  F hydrogen bonds exist for these complexes and those interactions are enhanced if the number of HF molecules increases. It is shown that cooperativity effect causes the shortening of H  F and H  p distances, simultaneously the electron density and its Laplacian at the corresponding bond critical point (BCP) increase. There is also the greater charge transfer corresponding to p(C@C) ? r*(F–H), p(C„C) ? r*(F–H) and n(F) ? r*(F–H) interactions. One notices the greater elongation of H–F bonds within complexes if the number of HF molecules increases. The various correlations were found between geometrical, energetic and topological parameters. There are unique bond paths of the complexes analyzed that connect the hydrogen attractors with the BCPs of C@C and C„C bonds of Lewis base sub-systems. Ó 2008 Elsevier B.V. All rights reserved.

1. Introduction There are different explanations of the existence of very strong hydrogen bonding interactions [1–4]. For example, the so-called electrostatic-covalent hydrogen bond (ECHB) model was proposed by Gilli and co-workers [5,6]. They stated that hydrogen bonding possesses the double nature since it may be treated as an electrostatic interaction from one side and the covalent one from the other side. The term covalency is usually attributed to strong hydrogen bonds while weak and medium in strength hydrogen bonds are rather considered as electrostatic interactions [7]. In order to follow such classification one can ask which hydrogen bonds are the strong ones and hence possess characteristics of covalent bonds. According to the ECHB model [6] these are the charge assisted hydrogen bonds (CAHBs). If negatively charged they are designated as ()CAHBs, if positively charged the (+)CAHB term is applied. The H5 Oþ 2 system is an example where the positively charged assisted O–H  O hydrogen bond exists. This interaction should be practically designated as O  H  O (or O–H–O) since for the energetic minimum the proton is situated in the middle of O  O distance or nearly so [8] and both the H  O interactions are equivalent. Hence one can not classify them as interactions of closed-shell systems and they also do not possess the characteristics typical

* Corresponding author. Address: Department of Chemistry, University of Łódz´, 90-236 Łódz´, ul.Pomorska 149/153, Poland. Tel.: +48 42 635 57 37; fax: +48 42 665 57 71. E-mail address: [email protected] (S.J. Grabowski). 0301-0104/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2008.12.011

for covalent bonds. The similar situation was found for ()CAHB hydrogen bonding revealed for the [FHF] ion [9–11]. The latter interaction was considered early by Pauling as an example of the covalent character of hydrogen bonding [2]. There is also the third class of very strong hydrogen bonds according to the ECHB model [6]. These are the so-called resonance assisted hydrogen bonds (RAHBs). It is worth mentioning that intramolecular O–H  O hydrogen bonds existing in malonaldehyde and its derivatives are examples of such kind of interactions which have been very often analyzed [12–15]. For those molecular systems the conjugated double and single covalent bonds exist where there is p-electron delocalization leading to the equalization of bonds within a pseudo-ring and consequently to the enhancement of the strength of hydrogen bonding. However, it was found that such equalization could be also the result of external agents interacting with the systems with intramolecular RAHB [16]. For example, Lewis acids may cause the additional elongation of carbonyl group involved in hydrogen bonding and the charge redistribution within the system what leads to the equalization of pseudo-ring bonds. Interestingly, for the latter systems the weakening of hydrogen bond is observed as a result of the additional interaction with external Lewis acid. Thus the covalent bonds’ equalization is not always related to the enhancement of hydrogen bonding. The other assumptions of the RAHB model were also criticized [17–20]. Anyway, it seems that the p-electron delocalization connected with the closure of the pseudo-ring enhances [12] hydrogen bonding in the ring in spite of the fact that some of the statements of RAHB model seem to be controversial.

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It was proven that the term ‘‘covalency” is strongly related to the delocalization term of the interaction energy [21]. There are short proton-acceptor distances for very strong hydrogen bonds and in extreme cases those distances are close to the lengths of typical covalent bonds. The binding energy for the intermolecular H-bond linked system is high (if modulus of that energy is considered), the Bader QTAIM (Quantum Theory of ‘Atoms in Molecules’) theory [22,23] applied for short and strong hydrogen bonds indicates that for the proton–acceptor interactions there is a negative value of Laplacian of electron density at the corresponding bond critical point (BCP) or at least there is the negative value of the total electron energy density at this BCP [24]. In the other words there are numerous characteristics of very strong hydrogen bonds which may be classified as being covalent in nature. It is worth mentioning that the RAHBs’ systems are sometimes characterized as those ones where so-called p-cooperativity exists, on the other hand there are r-cooperativity effects which may also enhance the strength of hydrogen bonding [4]. For the first kind of cooperativity there is the p-electron delocalization, for the second kind of cooperativity r-bonds are connected to each other within the chain or a cycle. The chain of species connected by OH bonds is an example [4]. Very often the cooperativity effect is defined as the enhancement of hydrogen bond if the additional species interacts either with the proton donor or with the proton acceptor forming the next hydrogen bonding [25–27]. For example, H3N  HF and H3N  HF  HF complexes were investigated using the experimental microwave and ab initio techniques [28]. It was found that the addition of the second HF molecule to the H3N  HF complex leads to the contraction of the N  H hydrogen bond. Such a contraction amounts to 0.21(6) Å. In other words the cooperativity effect exists in cluster systems where a monomer participates concertedly as a donor and as an acceptor. The cooperativity effects were often studied both experimentally and theoretically [29]. As an example, cooperativity in C–H  O and OH  O hydrogen bonds can be considered [30]. The authors found that in the case of O–H  O interactions this effect enhances the H-bond strength and also leads to the elongation and the red-shift of the corresponding stretching mode of the proton donating bond. In the case of C–H  O interactions the blue-shifting was found for H-bonds of (H2CO)n and (HFCO)n aggregates. However, the increase of the shift to the blue for the latter C–H  O interactions is not related to the number of n-mers. One can mention the numerous studies on cooperativity effects; such effects in two-dimensional cyclic networks containing three-centered hydrogen bond interactions [31], in intramolecular bifurcated hydrogen bonds [32], in amide hydrogen bonding chains [33], in (HCN)n homo and (HCN)n   HF heterochains [34,35]. Cooperativity effect for different kinds of hydrogen bond was also analyzed. For the CH  F interaction in HCF3  (HF)n complexes an enhancement of blue-shift is observed as a result of cooperativity [36]; the latter observation is opposite to the other one mentioned above and concerning C–H  O interactions. This means that generally the cooperativity is not dependent to blue-shift. The H2C@O  (H–F)n (n up to 10) clusters were also studied and the decomposition of the interaction energy was performed [37]. It was revealed that for the larger number of hydrogen fluoride molecules there are the shorter and stronger (F)H  O(@CH2) interactions and the ratio of delocalization and electrostatic attractive interaction energy terms increases. Since the delocalization energy is often attributed to covalency thus the covalent character may manifest for the greater number of HF molecules. It is worth mentioning that many-body interactions often play important role in stabilizing complexes containing the greater number of moieties and connected by hydrogen bonding networks. For example, it was proofed that purely two-body pairwise additive potentials re-

sult in errors exceeding 20% for clusters containing more than five water molecules [38]. The similar cooperativity effects were investigated for halogen bonds existing in H2C@O  (Cl–F)n aggregates [39]. In the other words the cooperativity may exist for different kinds of hydrogen bonds and also for the other type of interactions (for example halogen bonding). Among different examples pH-bonded interactions analyzed at MP2/6-311++(2d,2p) level of approximation have been recently studied [40]. The latter studies concern O–H  p hydrogen bonds since O–H bonds of water molecules act as the proton donating bonds and p-electrons of ethene, propene and trans-2-butene as the proton acceptor. To our knowledge this study of DuPré and Yappert [40] is the first one where the comprehensive investigation of the enhancement of H  p interaction as an effect of cooperativity was performed. In this study also the other O–H  O hydrogen bonds were characterized, for example, the C2H4  HOH complex with O–H  p hydrogen bond formed when the next water molecule was added. Thus the NBO analysis was also carried out to characterize the intermolecular p(C@C) ? r*(O–H) chargetransfer interactions as well as the n(O) ? r*(O–H) ones [40]. Very recently the B3LYP/6-311++G(d,p) calculations for the C2H2  HF and C2H2  (HF)2 complexes were carried out to study their geometries, the nature of F–H  p and F–H  F interactions as well as to perform the infrared spectrum and QTAIM analyses [41]. The aim of the present study is to analyze the cooperativity phenomenon for the similar systems using the results of ab initio calculations as well as to reveal the characteristics of critical points derived from the QTAIM (Quantum Theory of ‘Atoms in Molecules’) theory [22,23]. The C2H2  (HF)n and C2H4  (HF)n complexes are considered here since for such species both the enhancement of p  H(F) interaction as well as of F–H  F hydrogen bond as an effect of cooperativity may be analyzed. This may be also considered for the latter systems if the cooperativity effect is strong enough to enhance the F–H  p hydrogen bonding to possess the characteristics of the covalent interaction. 2. Computational details The calculations have been performed with the Gaussian03 set of codes [42]. The complexes of C2H2  (HF)n and C2H4  (HF)n (n = 1, 2, 3, 4) were optimized at the MP2/6-311++G(d,p) level of approximation and all systems are in energetic minima since no imaginary frequencies were found. These complexes were chosen to analyze cooperativity effect influence on F–H  p and F–H  F hydrogen bonds. Fig. 1 presents selected examples of those complexes investigated here, these are: C2H2  HF, C2H2  (HF)4, C2H4  HF and C2H4  (HF)4. The binding energy for the analyzed complexes has been computed according to the supermolecular approach [43] as the difference between the total energy of the complex and the energies of monomers. It has been corrected for the basis set superposition error (BSSE) using the counterpoise method [44]. It is worth mentioning that in this study ‘monomers’ are classified in the following way: since the H  p interaction and the influence of the number of HF molecules attached on the strength of that interaction are mainly analyzed thus C2H2 or C2H4 are treated as one unit and the remaining HF molecules as the other monomer within the supermolecular approach. Such an approach is in line with the following expression [45,46]:

EHBm ¼ Etotal  E1;2;...;m  Emþ1;mþ2;...;n ;

ð1Þ

EHBm is the mth hydrogen bonding within the n-element system. Thus there are two sub-systems which are connected through mth hydrogen bonding and the strength of the latter interaction is

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171

using the AIM2000 program [50]. The natural bonds orbital (NBO) analysis [51,52] was also carried out to characterize the following intermolecular charge-transfer interactions: p(C@C) ? r*(F–H), and p(C„C) ? r*(F–H) related to the F–H  p hydrogen bonding, as well as n(F) ? r*(F–H) component of the F–H  F interaction. 3. Results and discussion 3.1. Characteristics of H  p interactions

Fig. 1. The selected complexes analyzed here: C2H2  HF, C2H2  (HF)4, C2H4  HF and C2H4  (HF)4.

expressed in the same way as the binding energy is usually expressed. For the systems analyzed here the mth hydrogen bonding exists between the first element (acetylene or ethylene) and the first hydrogen fluoride molecule. The CHelpG scheme [47] implemented within the Gaussian03 package was applied to calculate the atomic charges. The CHelpG procedure produces charges fitted to the electrostatic molecular potential (EMP) using a grid based method. Its application based on EMP expectation values provides much better estimates of intermolecular charge transfer than the other approaches – any arbitrary population analysis, where the corresponding relative error values were doubled reaching 50% [48]. The QTAIM theory of Bader was also applied here to find the critical points [22,49] and to analyze them in terms of electron densities and their Laplacians. The properties of BCPs were also studied in terms of the local energy density at BCP (HC) and its components (the local kinetic energy density GC and the local potential energy density VC). The QTAIM calculations were performed

Table 1 displays the selected characteristics of the systems analyzed here. The binding energies calculated according to the supermolecular approach described briefly in the previous section are presented. One can note that the binding energy (correctly its modulus) increases for sub-samples containing ethylene or acetylene molecules if the number of hydrogen fluoride molecules increases. For C2H2  HF complex the binding energy amounts to 3.18 kcal/ mol. It increases by 78% (to 5.53 kcal/mol) if the additional HF molecule is inserted into the complex. The next HF molecule inserted results in formation of the C2H2  (HF)3 complex and the binding energy increases by 38%, related to the C2H2  (HF)2 system. The addition of the next HF molecule causes the increase of the binding energy by 12%. Similarly, in the case of C2H4 acting as the Lewis base two first corresponding energy increases amount to 53% and 26% respectively, however for the largest C2H4  (HF)4 complex the binding energy decreases (Table 1). This may be the result of the other effects opposing the systematic increase of the binding energy with the increasing number of HF molecules. If n is equal to 3 or 4 thus the Lewis bases act also as the proton donors (Lewis acids) for C–H  F weak hydrogen bonds. This may be the other reason of the non-systematic energetic changes for the complexes with ethylene. It was found that many-body effects play important role for water complexes, they may exceed even 20% for pentamer systems [38]. It was shown earlier [37] that for H2CO  (HF)n linear complexes of the C2v symmetry the binding energy increases systematically and such an increase is meaningless for n = 8, 9. For fully optimized H2CO  (HF)n complexes the increase of the binding energy is greater after inserting additional HF molecules but the systems are not linear. For the complexes analyzed here the full optimization was performed. There are two T-shaped complexes: C2H4  HF and C2H2  HF of the C2v symmetry (see Fig. 1) for which this symmetry is broken if n increases. Fig. 2 presents the molecular graph of C2H4  (HF)3 complex where bond paths and critical points are presented, one can observe the bond path of the (F)H  p interaction, the bond paths corresponding to H–F shared and F  H closed-shell interactions as well as those of the mentioned above (C)H  F interactions. Two bond paths were detected for the latter interactions (Fig. 2). It is worth to mention that for the (F)H  p interaction there is the bond path connecting the critical point (the attractor) of hydrogen atom and the bond critical point (BCP) corresponding to C@C bond of ethylene. For such a path of H  p interaction there is the corresponding BCP (see Fig. 2). The bond path (BP) represents an atomic interaction line (AIL) of the molecular system being in minimum and it usually connects two attractors [53]. Hence the specific kind of the H  p bond path is a characteristic of hydrogen bonds where p-electrons act as the proton acceptor. Fig. 3 presents the relief map of the electron density of the C2H2  (HF)4 complex. One can see the pseudo-ring created by H  p, H–F, F  H(F) and F  H(C) interactions. Table 1 presents the other parameters which satisfactorily follow the changes of the binding energy. One can note from this table that if the number of HF molecules increases also the binding energy increases (is ‘more negative’; except of C2H4  (HF)4 complex). In addition, it is followed by the lengthening of the CC bond

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Table 1 P The characteristics of H  p interactions, binding energy (Ebin, in kcal/mol), the C„C or C@C bond length (r, in Å), the sum of net atomic charges of acetylene or ethylene ( q, in au), the H  p distance (R, in Å), the electron density at H  p BCP (qC, in au), the Laplacian of the electron density (r2qC, au), GC and VC values (in au). Complex

Ebin

r

P q

R

qC

r2qC

GC

VC

C2H2  HF C2H2  (HF)2 C2H2  (HF)3 C2H2  (HF)4 C2H4  HF C2H4  (HF)2 C2H4  (HF)3 C2H4  (HF)4

3.18 5.53 7.65 8.55 3.35 5.13 6.45 5.37

1.2177 1.2185 1.2192 1.2194 1.3423 1.3435 1.3445 1.3442

0.0680 0.0885 0.1112 0.1303 0.0072 0.1007 0.1141 0.0928

2.0098 1.8558 1.7885 1.7608 1.9876 1.8286 1.7567 1.8009

0.0167 0.0220 0.0262 0.0282 0.0165 0.0227 0.0263 0.0239

0.0505 0.0601 0.0647 0.0664 0.0439 0.0536 0.0562 0.0539

0.0098 0.0131 0.0153 0.0164 0.0086 0.0119 0.0135 0.0124

0.0071 0.0112 0.0144 0.0162 0.0062 0.0103 0.0130 0.0113

Fig. 2. The molecular graph of the C2H4  (HF)3 complex, bond paths, attractors (big circles) and critical points (small circles) are indicated (bond critical points and ring critical points).

of ethylene or acetylene as a result of complexation and the greater electron charge transfer from the Lewis base to the complex of HF molecules. For the greater n the H  p distances are shorter (these are the distances between the H-atom and the BCP of CC bond– approximately the mid-point of CC bond). This is reflected in the increase of the positive charge (the sum of Chelp charges is taken into account in Table 1) of the Lewis base molecule. Table 1 also includes the QTAIM parameters, electron density at H  p BCP, its Laplacian and the energetic parameters: the total electron energy density at BCP (HC) and its components – the kinetic and potential energies (GC and VC). The brief insight into results of this table indicates that they all are interrelated, at least within the sub-samples concerning two different proton acceptors: C2H2 and C2H4. Fig. 4 presents the linear dependencies between the H  p distance and Laplacian of the electron density at the corresponding BCP. Since four systems are included within each of sub-samples thus the correlation analysis is not performed and the figure indicates only the tendencies. For the same H  p distances but for different proton acceptors the stronger interactions occur for acetylene since the electron density at BCP may be treated as a measure of the strength of an interaction. Similarly for the same number of HF molecules the greater binding energy (its modulus) occur for the complex with acetylene than for the complex with ethylene (Table 1), only complexes with one HF molecule present the reverse tendency. The latter observation is a consequence and is in line with the well known fact that acetylene proton affinity is greater than that of ethylene [54]. It is also worth to be noted that the characteristics of H  p interactions indicate that they are at least medium in strength if

laplacian of the electron density at BCP (in au)

0.07 C2H2…(HF)4 0.065

C2H2…(HF)3 C2H2…(HF)2

0.06 C2H4…(HF)3 0.055 C2H4…(HF)2

C2H2…HF

C2H4…(HF)4 0.05

C2H4…HF

0.045

0.04 1.75

1.8

1.85

1.9

1.95

2

2.05

H...pi distance (in Å)

Fig. 3. The relief map of the C2H2  (HF)4 complex.

Fig. 4. The linear correlations between the H  p distance (in Å) and the Laplacian of the electron density at the corresponding BCP (in au), two sub-samples differing in the Lewis base, acetylene (empty circles) or ethylene (full circles) are considered.

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compared to the other typical hydrogen bonds. For example for the O–H  O hydrogen bonding of the trans-linear water dimer the binding energy calculated at MP2/6-311++G(d,p) level of approximation is equal to 4.5 kcal/mol while the electron density at H  O BCP amounts to 0.023 au [55]. For H  p interactions analyzed here the electron density at the corresponding BCP is within the range 0.017–0.028 au (Table 1). The natural bond orbitals (NBO) [52] calculations were also performed here. According to the NBO approach the hydrogen bonding formation is possible mostly owing to the nB ! rAH interaction. This interaction is connected with the maximum nB ! rAH overlap within A–H  B hydrogen bonded system. nB designates the lone electron pair of the proton acceptor, rAH is an antibonding orbital of the proton donating bond. This charge transfer leads to the weakening and lengthening of the A–H bond and to the redshifted of mAH stretching frequency. The nB ! rAH interaction may be estimated by second order perturbation theory (Eq. (2))

DEðnB ! rAH Þ ¼ 2hnB jFjrAH i=ðeðrAH Þ  eðnB ÞÞ;

ð2Þ

hnB jFjr is the Fock matrix element while ðeðr  eðnB ÞÞ is the orbital energy difference. There are typical F–H  F hydrogen bonds for the systems analyzed here and containing at least two hydrogen fluoride molecules. For such interactions the related expression has the following form:  AH i

ð3Þ

However for F–H  p interactions the slightly modified expression should be taken into account since the corresponding energy term is connected with the charge transfer from pCC-bond orbital to the rAH (exactly rFH ) antibonding orbital:

DEðpCC ! rFH Þ ¼ 2hpCC jFjrFH ji=ðeðrFH Þ  eðpCC ÞÞ

ð4Þ

Table 2 presents the energies (DE) expressed by Eqs. (3) and (4) for the complexes analyzed in this study. One notices the enhancement of the H  p interaction as a result of cooperativity what is reflected by all data collected in Table 1. It is visible that the DE ðpCC ! rFH Þ energy is also related to all parameters corresponding to the H  p interaction. Fig. 5 presents one of the examples – the exponential relationship between H  p distance and DE ðpCC ! rFH Þ energy with the correlation coefficient close to the unity. It is worth mentioning that both sub-samples are taken into account together with-

Table 2 The HF bond length (r, in Å) and the corresponding topological parameters (in au); the DE energy corresponding to the transfer of charge (in kcal/mol)  pCC ! rAH or nB ! rAH is also included.

a

C2H2  HF C2H2  (HF)2a C2H2  (HF)2 C2H2  (HF)3a C2H2  (HF)3 C2H2  (HF)3 C2H2  (HF)4a C2H2  (HF)4 C2H2  (HF)4 C2H2  (HF)4 C2H4  HFa C2H4  (HF)2a C2H4  (HF)2 C2H4  (HF)3a C2H4  (HF)3 C2H4  (HF)3 C2H4  (HF)4a C2H4  (HF)4 C2H4  (HF)4 C2H4  (HF)4 a

13 11 9

y = 20957e

-4.1742x

2

R = 0.9918

7 5 3 1.75

1.8

1.85

1.9

1.95

2

2.05

H...pi distance (in Å) Fig. 5. The exponential relationship between the H  p distance (in Å) and the charge transfer energy expressed by Eq. (4) (in kcal/mol), two sub-samples differing in the Lewis base, acetylene (empty circles) or ethylene (full circles) are presented however the correlation concerns all species.

 AH Þ

DEðnF ! rFH Þ ¼ 2hnF jFjrFH i=ðeðrFH Þ  eðnF ÞÞ

System

energy (kcal/mol)

15

R

qC

r2qC

GC

VC

DE

0.9231 0.9316 0.9257 0.9374 0.9335 0.9293 0.9400 0.9366 0.9355 0.9311 0.9242 0.9332 0.9244 0.9402 0.9335 0.9281 0.9399 0.9383 0.9361 0.9309

0.3648 0.3512 0.3599 0.3417 0.3468 0.3537 0.3372 0.3411 0.3434 0.3502 0.3631 0.3483 0.362 0.3381 0.3474 0.3561 0.3367 0.3391 0.3436 0.3512

2.7749 2.6589 2.7715 2.5606 2.6638 2.7183 2.513 2.6133 2.6302 2.6875 2.7556 2.6192 2.7862 2.5106 2.6652 2.7339 2.526 2.5887 2.6264 2.6877

0.0866 0.0831 0.0821 0.0826 0.0792 0.0811 0.0825 0.0784 0.0791 0.0807 0.0865 0.084 0.0834 0.0831 0.0796 0.082 0.08 0.0784 0.0795 0.0814

0.8669 0.8309 0.8571 0.8053 0.8242 0.8417 0.7933 0.8102 0.8159 0.8333 0.8619 0.8228 0.8633 0.7938 0.8256 0.8475 0.7916 0.804 0.8156 0.8347

4.81 8.29 8.4 11.99 14.07 11.53 13.83 16.6 16.14 13.4 5.33 10.21 8.69 13.7 13.95 11.02 11.7 17.52 16.42 13.43

pCC ! rAH , for the remaining systems that energy corresponds to nB ! rAH .

in this correlation, i.e. the systems containing C2H4 and C2H2 Lewis bases. For complexes containing at least two HF molecules, the closest neighbors of C2H4 and C2H2 molecules participate in two kinds of interactions. These are pCC ! rFH and nF ! rFH . In spite of this fact the energy expressed by Eq. (4) correlates well with H  p distance and the other parameters of Table 1. It was pointed out that the energy expressed by Eq. (2) is the most important term describing and stimulating the nature of hydrogen bonding interaction [52]. 3.2. F–H and F  H interactions It was mentioned in the previous section that Table 2 contains the energies related to the pCC ! rFH and nF ! rFH electron charge transfer energies. This table also presents the corresponding values of H–F bond lengths. This is well known that the greater lengthening of the AH proton donating bond is connected with the greater energy value expressed by Eq. (2). Fig. 6 presents the relationship between the H–F bond length and the energy connected with the charge transfer from the proton acceptor to the proton donating

18

energy (kcal/mol)

16 14 R2 = 0.9778 12 10 R2 = 0.962 8 6 4 0.92

0.93

0.94

HF bond length (in Å) Fig. 6. The linear correlations between the HF bond length (in Å) and the charge transfer energy expressed by Eq. (3) (in kcal/mol), two sub-samples differing in the type of interaction leading to H–F bond elongation are considered, one of subsamples concerns the HF molecules directly interacting with C2H4 or C2H2 molecules (empty squares), the other sub-sample concerns F–H  F hydrogen bonds (full squares).

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Table 3 The F  H distance (R, in Å) and the corresponding C2H2  (HF)2 topological parameters (in au). System

R

qC

r2qC

GC

VC

C2H2  (HF)2 C2H2  (HF)3 C2H2  (HF)3 C2H2  (HF)4 C2H2  (HF)4 C2H2  (HF)4 C2H4  (HF)2 C2H4  (HF)3 C2H4  (HF)3 C2H4  (HF)4 C2H4  (HF)4 C2H4  (HF)4

1.8227 1.6964 1.7471 1.6471 1.6583 1.7077 1.7915 1.6929 1.7635 1.6313 1.6640 1.7202

0.0266 0.035 0.0314 0.0386 0.0382 0.0341 0.0271 0.0356 0.0307 0.0403 0.0387 0.0342

0.1081 0.1418 0.1277 0.1578 0.1543 0.1394 0.1175 0.1435 0.1234 0.1632 0.1520 0.1350

0.0256 0.0350 0.0309 0.0394 0.0386 0.0341 0.0271 0.0356 0.0299 0.0412 0.0386 0.0335

0.0242 0.0345 0.0299 0.0394 0.0387 0.0334 0.0249 0.0353 0.0290 0.0416 0.0392 0.0333

0.37 electron density at BCP (in au)

bond. Two sub-samples are considered since there are two mentioned above charge transfers. One of sub-samples relates to the HF molecules directly interacting with C2H4 or C2H2 molecules, i.e. connected through H  p contacts. The corresponding energies are expressed by Eq. (4) and the results are designated by empty squares (Fig. 6). The sub-sample of nF ! rFH interactions of F– H  F hydrogen bonds where the energies are expressed by Eq. (3) is designated by full squares. One can see fine linear correlations for both sub-samples. Additionally, it is evident that the same value of the charge transfer energy causes the greater elongation of H–F bond length for H  p interaction than for the F  H interaction (Table 2). This elongation increases with the increase of the number of HF molecules. It seems that the upper limit of H–F bond length is 0.94 Å or nearly so. Table 2 shows that for C2H2  (HF)3 complex the HF bond length of the molecule directly interacting with acetylene amounts to 0.937 Å while for the C2H2  (HF)4 complex the bond length of the corresponding HF molecule is equal to 0.940 Å. For C2H4  (HF)3 and C2H4  (HF)4 complexes the bond length of HF molecule involved in H  p interaction is practically the same – 0.940 Å. The mentioned above upper limit of 0.940 Å concerns cooperativity effects of H  p interactions, for example for H2CO  (HF)3 complex calculated at MP2/6-311++G(d,p) level of approximation the H–F bond length of the molecule involved in F–H  O hydrogen bonding and enhanced by cooperativity amounts to 0.956 Å [37]. Tables 2 and 3 present the topological parameters representing the characteristics of BCP of H–F bond and F  H contact, respectively. All Laplacian values for H–F shared interactions are negative and for all F  H interactions these values are positive. However it was pointed out that HC value (the total electron energy density) may be negative for partially covalent interactions, for example, for strong hydrogen bonds [56–59]. One can observe (Table 3) for complexes with the greater number of HF molecules (n = 3, 4) that |VC| > GC thus HC < 0 or HC  0. This means that such stronger F  H interactions are enhanced by cooperativity effect. It is also well known that QTAIM parameters correlate very often with the geometrical and energetic characteristics [60–63]. The distance–electron density dependence at the corresponding BCP is mostly known. Fig. 7 displays the relationship between the H–F bond length (and F  H distance) and the electron density at the corresponding BCP; these are fine linear correlations. For the broader range of interactions (broader range of distances) starting from closed-shell interactions through transition states and ending on the shared covalent bonds, the distance–electron density correlation is an exponential one. Fig. 7a and b presents chosen ranges of F  H distance and H-F bond length thus both correlations are linear. The broad range of various F  H interactions was analyzed before and the non-linear dependence was revealed for the whole sample of interactions [64]. For such broad range there is no the

0.365 0.36 0.355 0.35

2

R = 0.994 0.345 0.34 0.335 0.92

0.925

0.93

0.935

0.94

0.945

1.8

1.85

HF bond length (in Å) 0.044 electron density at BCP (in au)

174

0.04

0.036

0.032 2

R = 0.9836

0.028

0.024 1.6

1.65

1.7

1.75

F...H distance (in Å) Fig. 7. The linear correlation between the H–F bond length (figure a, in Å) or H  F distance (figure b, in Å) and the electron density at the corresponding BCP (in au).

sharp border between H–F covalent bonds and H  F closed shell interactions if the QTAIM parameters are considered, the only conventional difference between shared and closed shell interactions is expressed by the sign of Laplacian of the electron density at H–F (H  F) BCP or the sign of HC value. For the complexes analyzed here these interactions are well separated since the longest H–F bond reaches 0.940 Å while the shortest H  F distance is equal to 1.631 Å, however for the shortest H  F distances detected in this study the HC values are negative informing of the partially covalent character of such interactions. 3.3. Infrared spectrum for C2H2  (HF)n and C2H2  (HF)n complexes The interrelations between the cooperativity effect and the infrared spectrum of the multi-molecular systems were analyzed before [31,33,36,65,66]. For example, cooperativity effects were investigated for formamide chains [33] where the formamide molecules are connected through N–H  O hydrogen bonds. For formamide monomer the wave number for N–H stretch mode is equal to 3608 cm1, for the dimer there are two such modes (3549 and 3604 cm1), for decamer there are ten N–H stretching modes ranging from 3337 to 3603 cm1. Hence the red shifts are observed for all formamide molecules involved in hydrogen bonds, the greater shifts are observed if there is the greater number of formamide molecules. Each of these modes may be attributed to the N–H bond involved in any N–H  O interaction, this is the approximate attribution since these are coupled vibrations of N–H bonds. The similar situation is observed here for C2H2  (HF)n and C2H2  (HF)n complexes, Table 4 presents the N–H vibrational frequencies: wave numbers and the corresponding intensities are

S.J. Grabowski, J. Leszczynski / Chemical Physics 355 (2009) 169–176

175

Table 4 The vibrational frequencies for H–F stretching modes, wave numbers (cm1) and the corresponding intensities (km/mol) as well as H–F bond lengths (in Å) are given.

H-atom attractor and the bond critical point of CC bond of the Lewis base.

Moiety

Wave number

Intensity

Bond length

Acknowledgments

HF C2H2  HFa C2H2  (HF)2a C2H2  (HF)2 C2H2  (HF)3a C2H2  (HF)3 C2H2  (HF)3 C2H2  (HF)4a C2H2  (HF)4 C2H2  (HF)4 C2H2  (HF)4 C2H4  HFa C2H4  (HF)2a C2H4  (HF)2 C2H4  (HF)3a C2H4  (HF)3 C2H4  (HF)3 C2H4  (HF)4a C2H4  (HF)4 C2H4  (HF)4 C2H4  (HF)4

4197.12 4042.66 3871.18 4019.8 3736.54 3867.31 3955.97 3668.07 3777.98 3854.38 3920.33 4016.62 3828.32 4041.15 3677.09 3863.69 3973.27 3662.39 3767.67 3836.97 3918.83

141.52 593.47 760.26 437.71 1119.93 750.63 522.91 1445.08 1297.08 595.5 480.09 653.9 1163.72 439.38 1291.79 614.35 590.34 1529.18 963.72 557.14 694.43

0.9167 0.9231 0.9316 0.9257 0.9374 0.9335 0.9293 0.94 0.9366 0.9355 0.9311 0.9242 0.9332 0.9244 0.9402 0.9335 0.9281 0.9399 0.9383 0.9361 0.9309

a HF molecules directly interacting with C2H2 or C2H4 molecules (involved in F– H  p interactions).

included. It is worth mentioning that a given mHF stretching frequency is associated with the collective motion of different H– F bonds. The wave numbers are not scaled since they are given to show tendencies. The greatest red-shifts are observed for HF molecules involved in F–H  p interactions, the next molecules of n-membered H–F chains are characterized by lower red-shifts. Similarly there are the decreases of corresponding intensities. The C2H4  (HF)4 complex is a slight exception (Table 4) since the third in the chain HF molecule is characterized by the lower intensity than the fourth one. There is the correlation between the HF bond length and the shift of the H–F stretching frequency being the result of complexation since the second order polynomial dependence between these parameters was found (R2 = 0.973). It is worth mentioning that for the species analyzed here the additional frequencies are observed which are connected with the H  p and H  F interactions corresponding to the F–H  p and F–H  F hydrogen bonds, respectively. The wave numbers for such frequencies are situated in the range 121–287 cm1 and the range of the corresponding intensities is characterized by low values (0.8–20 km/mol), The similar values were observed earlier by Oliveira and co-workers for C2H2  HF and C2H2  (HF)2 complexes analyzed by DFT results [41] as well as for C2H4O  2HF complex [66].

4. Summary The cooperativity effect was investigated here and its influence on the F–H  p hydrogen bonds was analyzed. It is revealed that, similarly as for typical O–H  O hydrogen bonds, cooperativity enhances the H  p interaction. If the number (n) of HF molecules increases in C2H2  (HF)n and C2H4  (HF)n complexes thus all descriptors of the strength of interaction indicate that the hydrogen bonds in these complexes are stronger (except for the C2H4  (HF)4 complex where the decrease of the binding energy is observed if compared with the C2H4  (HF)3 complex). The numerous correlations between geometric, topological (QTAIM), energetic and spectroscopic parameters are found. It is interesting that for H  p interaction there is the unique bond path connecting

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