Matta, Boyd. The quantum theory of atoms in molecules

453

17

Relationships between QTAIM and the Decomposition of the Interaction Energy – Comparison of Di erent Kinds of Hydrogen Bond

Sławomir J. Grabowski

17.1 Introduction

Intramolecular and intermolecular hydrogen-bonding interactions are the subject of intense interest because of their role in many physical, chemical, and biological processes, for example they a ect the arrangement of molecules in crystals, the behavior of liquids and gases, and determine properties of materials such acidity, basicity, and susceptibility to electrophilic or nucleophilic substitution, etc. All of these e ects of hydrogen bonding are rooted in the underlying electron density distribution, because ‘‘matter is a distribution of charge in real space, of pointlike nuclei embedded in the di use density of electronic charge’’ [1]. The quantum theory of atoms in molecules (QTAIM) is a powerful tool that can be used for analysis of inter-atomic interactions, as is apparent from this book.

Analysis of hydrogen bonds as a specific class of weak interactions is often performed in physicochemical studies [2–5]. QTAIM [6] uses novel descriptors, for example the properties of the electron density at the bond critical point (BCP), which enable one to gain deeper insight into the nature of the chemical bond as reviewed in this book (the properties of BCP and their interpretation are reviewed in Chapter 1).

In a typical hydrogen-bonding interaction, generally symbolized XaH Y, where XaH designates the proton donating bond and Y is an acceptor centre, the bond critical points of both the XaH and of the H Y bonding interactions provide crucial information characterizing the bonding. The bond properties determined at the XaH and H Y BCPs include:

the electron densities at the BCPs (rXH, rH Y);

their Laplacians (‘2rXH, ‘2rH Y); and

the energy properties of both bond critical points, usually designated Hb, Gb, and Vb (Chapter 1).

In other words, QTAIM puts at ones disposal an entire set of topological properties for characterizing both bonding interactions of all types and traditional

454 17 Relationships between QTAIM and the Decomposition of the Interaction Energy

Fig. 17.1 The hydrogen bond in the acetylene–water complex.

geometrical properties. Figure 17.1 shows an example of a hydrogen-bonded system, the acetylene–water complex, with a CaH proton-donating bond and the oxygen atom of water as an acceptor, in which the topological characteristics of these interactions may be considered in addition to the XaH (CaH) and H Y (H O) bond lengths.

One of the early definitions of hydrogen bonding is that of Pauling [7] who stated, ‘‘under certain conditions an atom of hydrogen is attracted by rather strong forces to two atoms, instead of only one, so that it may be considered to be acting as a bond between them. This is called the hydrogen bond.’’ QTAIM enables characterization of a wide diversity of hydrogen bonding interactions (XaH Y) by use of the above-mentioned topological descriptors, irrespective of the nature of X and Y. ‘‘A hydrogen bond, which includes the van der Waals complexes, is defined to be one in which a hydrogen atom is bound to the acid fragment by a shared interaction, rðrcÞ large and ‘2rðrcÞ < 0, and to the base by a closed-shell interaction, rðrcÞ small and ‘2rðrcÞ > 0’’ [8].

The wide diversity of hydrogen bonds is the subject of di erent classification schemes. In several monographs classification is achieved on the basis of the hydrogen-bond energies – weak hydrogen bonds (1–4 kcal mol 1); medium (4– 15 kcal mol 1), and strong (15–40 kcal mol 1) [3, 4]. It is worth mentioning that the hydrogen bond energy ðEHBÞ is often identified with the binding energy [9], which is usually computed as the di erence between the total energy of the complex ðEABÞ and the energies of the isolated monomers (EA and EB). EAB assumes negative values for stable complexes.

EHB ¼ EAB ðEA þ EBÞ

ð1Þ

For simplicity, the absolute values of binding energies are denoted jEHBj in the text and EHB values are given in figures. In this chapter the hydrogen bond energy is the energy of the H Y interaction within the XaH Y system and the binding energy refers to the interacting system as a whole, as defined in Eq. (1). The terminal parts of molecules sometimes contribute substantially to binding energies. In this chapter, both terms are equivalent because:

1.interactions between terminal moieties are negligible in most systems (with a few exceptions which are described in detail); and

2.all reported hydrogen bond energies are calculated in accordance with Eq. (1).

17.1 Introduction 455

Rozas et al. [10] have proposed a classification of hydrogen-bonding interactions based on energy and topological data. For weak hydrogen bonds with hydro-

gen bond energy jEHBj < 12 kcal mol 1, ‘2rH Y > 0 and Hb > 0. For hydrogen bonds of medium strength 12 kcal mol 1 < jEHBj < 24 kcal mol 1, ‘2rH Y > 0 and Hb < 0. For strong hydrogen bonds jEHBj > 24 kcal mol 1, ‘2rH Y < 0 and Hb < 0 (where Hb is the electron energy density at H Y BCP). This classification

shows that weak hydrogen bonds eventually merge with (weaker) van der Waals interactions whereas strong hydrogen bonds merge, at the other end of the continuum, with covalent and polar bonds. ‘2rAB < 0 is indicative of covalent A–B bonds and ‘2rH Y < 0 is also characteristic of very strong hydrogen bonds; this means that such hydrogen bonds have the characteristics of covalent bonds. (The covalence of strong hydrogen bonds is discussed elsewhere [11–13].) A negative value of Hb is often regarded as su cient reason to classify an interaction as covalent [14]. It is, in contrast, di cult to classify an interaction as hydrogen bond if it is very weak, conditions under which hydrogen-bonding criteria are not universally accepted [15]. These very weak interactions may be regarded as van der Waals attractive interactions. One can state there is a continuous transition from covalent bonds to hydrogen bonds and, finally, to van der Waals interactions [16, 17]. For this reason, Desiraju has suggested that hydrogen bonding is an ‘‘interaction without borders’’ [18].

Figure 17.2 shows the relationship between H Y distance and hydrogen bond energy; black circles correspond to ‘2rb < 0, grey circles to ‘2rb > 0 and Hb < 0 and white circles to ‘2rb > 0 and Hb > 0. It is apparent that energies and H Y distances correspond approximately to the topological properties. Hydrogen bonds are strong if the H Y distance is less than ca. 1.2 A˚ and weak if the distance is greater than ca. 1.8 A˚ . Figure 17.2 was obtained on the basis of the di erent types of hydrogen bond discussed below.

Fig. 17.2 The relationship between H Y distance (A˚ ) and the binding energy (kcal mol 1).

456 17 Relationships between QTAIM and the Decomposition of the Interaction Energy

Parthasarathi et al. [19] recently proposed yet another classification based on the relationship between H Y distance and the electron density at the corresponding BCP, covering the regions from covalence to van der Waals interactions. The topological properties at the BCP can enable better characterization of a particular hydrogen bonding interaction than the binding energy, because the properties determined at the BCP are specific to the particular H Y interaction whereas the binding energy, as already mentioned, more often than not includes, in addition to the contribution of the hydrogen bond of interest, contributions from interactions between other parts of molecules, which can sometimes be significant [20]. The BCP properties single out the characteristic of the H Y interaction of interest from the rest of the system. Having said that, however, one must also recognize that the bond path and BCP attributed to H Y are a ected by the electron density of the whole system. An example is represented by the T-shaped configuration of the LiCcCLi HF complex, in which an FaH p hydrogen bond path is present. The binding energy calculated for that system at the MP2/6- 311þþG(d,p) level of theory [21] amounts to 15.4 kcal mol 1 whereas such energy for the T-shaped HCCH HF complex is 3.1 kcal mol 1. The principal contribution to the energy for the former system seems to be Li F electrostatic interactions.

17.2

Diversity of Hydrogen-bonding Interactions

Pauling argued that hydrogen bonds may only be formed by electronegative atoms (the X and Y atoms of XaH Y) and stressed that the hydrogen bond is electrostatic in nature [7]. Despite Pauling’s statements, early studies indicated that even atoms of very low electronegativity, for example carbon, can act as hydrogen donors in a hydrogen bond and that CaH Y hydrogen bonds are possible [22]. Such CaH Y hydrogen bonds were first found in crystal structures [23]; later their existence was proved by use of refined statistical methods [24]. It has been pointed out that carbon atoms may also act as proton acceptors and that XaH C hydrogen bonds [25] and even CaH C [26–28], are possible.

Interestingly, p-electrons can also act as proton acceptors, i.e. as Lewis bases. CaH p hydrogen bonds are often responsible for the arrangement of molecules in crystals [29]. These unconventional hydrogen bonds are usually very weak.

The dihydrogen bond (DHB) is a special type of hydrogen-bonding interactions. The DHB may be denoted XaHþd dHaY, because of the presence of a proton-donating XaH bond, as for a typical hydrogen bond, but, unlike a typical hydrogen bond, the proton acceptor is a second hydrogen atom bearing excess negative charge [30]. Such interactions were extensively studied in the last decade and were found to have characteristics very similar to those of typical hydrogen bonds [31]. Dihydrogen bonding is extremely important in (bio)chemistry because, for example, it occurs in the preliminary stages of the release of hydrogen gas in several biochemical processes [32].

17.2 Diversity of Hydrogen-bonding Interactions 457

An important characteristic of DHBs is that they are relatively strong compared with typical hydrogen bonds – the binding energy of a dihydrogen bonded complex very often exceeds 10 kcal mol 1 [33, 34]. For example, the binding energy for an FH HLi model system is 11.9 kcal mol 1 at the QCISD(T)/6- 311þþG(d,p) level of approximation [33]. Very strong DHBs [13, 35], for example those in the H2OHþ HBeH complex and other, related, systems have binding energies in the range 20–30 kcal mol 1. At the other end of the scale, very weak DHBs are also known which border van der Waals interactions [36]. In other words, DHBs cover a broad spectrum of interactions from very weak (van der Waals-like) to very strong (covalent-like) [17] in the same way as conventional hydrogen bonds.

It is crucial to know the nature of a DHB interaction, because H H contacts are very common, for example in crystal structures of organic compounds. Such contacts may be stabilized by DHBs, by van der Waals interactions, and by hydrogen–hydrogen interactions, which have recently been detected and characterized [1].

Figure 17.3 shows the relationship between proton acceptor distance andVb=Gb ratio. This relationship for H F interactions – covalent-shared and

Fig. 17.3 Relationship between proton– acceptor distance and Vb=Gb ratio. Filled circles denote dihydrogen bonds. Open circles denote species with s-electrons as proton acceptors (two species are covered in the figure because the H Y distance andVb=Gb ratio are close to each other). The filled square denotes [FHF] . Open squares denote so-called ‘‘resonance-assisted

hydrogen bonds’’, for example OH O, NH O, and OH N. Filled triangles denote p Hþ p interactions. Open triangles denote complexes with p-electrons as a proton acceptor (e.g. the T-shaped acetylene dimer and the T-shaped FH C2H2 complex). Crosses denote other hydrogen bonded systems; among these is the trans-linear water dimer.

458 17 Relationships between QTAIM and the Decomposition of the Interaction Energy

non-covalent-closed-shell interactions – has been investigated [37]. The dependence presented here (Fig. 17.3) is mainly that for DHB systems (filled circles), but other types of hydrogen bond are also included. The ‘‘charge-assisted hydrogen bond’’ (filled square) is represented by [FHF] and there are also so-called ‘‘resonance-assisted hydrogen bonds’’ [38] and, finally, typical hydrogen bonds such as that in the trans-linear water dimer (see caption of Fig. 17.3).

From Fig. 17.3 it is apparent that Vb=Gb increases if the H Y distance (where Y denotes the proton acceptor) decreases. The (nonlinear) correlation is particularly strong for dihydrogen-bonded systems. Changes of Vb=Gb have the same trends as for other hydrogen-bonded interactions.

Because the electronic potential and kinetic energy density at the BCP (Vb and Gb, respectively) are negative and positive, respectively, everywhere the decrease in the H Y distance results in an increase of VC=GC. This last ratio may be treated as a measure of the covalence of a chemical bonding interaction. When the ratio is greater than 2, the corresponding Laplacian at the H Y BCP is negative ð1=4‘2rb ¼ 2Gb þ VbÞ, and when the ratio is between 1 and 2 the Laplacian is positive and Hb ðHb ¼ Vb þ GbÞ is negative. Finally, when this ratio is less than 1, both values, i.e. the Laplacian and Hb, are positive. These three previously mentioned regions correspond, respectively, to covalent interactions, partially covalent interactions, and noncovalent attractive interactions, for example moderate and weak hydrogen bonds and van der Waals interactions.

Figure 17.4 shows the relationship between H H distance and the Laplacian of the electron density at the corresponding BCP. The region of negative values of the Laplacian corresponding to very strong covalent interactions is clearly appar-

Fig. 17.4 The relationship between the H H distance (in A˚ ) and the Laplacian of the electron density at the bond critical point (in au) for dihydrogen bonded systems.

17.3 The Decomposition of the Interaction Energy 459

ent, as also is the region of positive Laplacian for the interactions of closed-shell systems; this is in agreement with Fig. 17.3.

17.3

The Decomposition of the Interaction Energy

It is claimed in di erent definitions of hydrogen bonding that it is an electrostatic interaction [2–4]. It is also stated, occasionally, that hydrogen bonding has a covalent nature especially when it is strong or very strong. According to the electrostatic–covalent hydrogen bond (ECHB) model [39, 40] weak and moderate hydrogen bonds are mostly electrostatic. When the proton acceptor distance decreases, the strength of the hydrogen bond increases and so does its covalence and the electrostatic nature becomes less important.

Desiraju [18] claims that hydrogen bonding is a conglomerate of electrostatic (acid/base), polarization (hard/soft), van der Waals (dispersion/repulsion), and covalent (charge-transfer) interactions. These energy components can be obtained from an interaction energy-decomposition scheme. The approach proposed by Morokuma [41] can be used to decompose the binding energy. The Hartree– Fock energy is expressed as:

DESCF ¼ ES þ PL þ EX þ CT þ MIX

ð2Þ

where ES is the electrostatic interaction energy, PL the polarization interaction energy, defined as the energy of the distortion of the monomers’ charge distributions, and CT the energy of charge transfer between the monomers (both CT and PL interaction energy terms are referred to changes of the electronic charge distribution as a result of complexation). These terms are most often attractive for stable complexes whereas EX, the exchange energy resulting from antisymmetrization of the wave function, is usually not. MIX is the energy di erence between the SCF interaction energy and these four components. Although the correlation energy (CORR) is not included in the SCF binding energy ðDESCFÞ, it can be calculated as the di erence between the energy when correlation is taken into account ðDEÞ and the SCF energy.

DE ¼ DESCF þ CORR

ð3Þ

The dispersion energy, often attributed to the van der Waals interaction, is the most important attractive energy component within the electron correlation energy. The dispersion energy originates from mutual polarization of the electron charge distribution of interacting monomers. In other words, it is the interaction of instantaneous multipoles [42].

Results from the decomposition scheme based on the variation–perturbation approach [43, 44] are presented here. The components of the interaction energy are obtained within the MP2 method for this decomposition scheme (designa-

460 17 Relationships between QTAIM and the Decomposition of the Interaction Energy

tions di erent from those of the Morokuma scheme are used here to distinguish both approaches):

DEMP2 ¼ EELð1Þ þ EEXð1Þ þ EDEL HF þ ECORRð2Þ ð4Þ

where EELð1Þ is the first-order electrostatic term describing the Coulombic interaction of static charge distributions of interacting molecules, EEXð1Þ is the first-order exchange component resulting from the Pauli exclusion principle, and EDEL HF and ECORR correspond to higher-order delocalization and correlation terms. The delocalization term represents the e ect of mutual deformation of the electron density as a result of complexation [45]. In other words, changes of the electron distribution within connected monomers a ect both interacting species and electron transfer between them. Hence this term approximately contains CT and PL interaction energies of the Morokuma and Kitaura approach.

The delocalization interaction energy term can be related to the delocalization index (DI) [46], because the latter is a quantitative measure of the sharing of electrons between two interacting species (because it measures the number of electrons pairs delocalized between two atoms in the absence of significant charge transfer between them). The sharing of electrons between the hydrogen atom and the proton acceptor Y is usually larger the stronger the hydrogen bond [47].

According to the variation–perturbation approach (Eq. 4) the starting wave functions of the subsystems are obtained by using a dimer-centered basis set (DCBS) [48], significantly reducing the basis set superposition error (BSSE) for the total interaction energy and its components. It has been shown that this approach enables removal of the BSSE contributions to the Heitler–London firstorder interaction energy term, EHL [49]:

EHL ¼ EELð1Þ þ EEXð1Þ ð5Þ

It is apparent that decomposition schemes can be useful for gaining insight into the nature of the interactions of interest and, perhaps, for obtaining a definition for the term ‘‘covalence’’.

17.4

Relationships between the Topological and Energy Properties of Hydrogen Bonds

Di erent interactions are	analyzed	in this	section in	accordance with
the variation–perturbation	approach	based on	calculations	at the MP2/6-

311þþG(d,p) level of approximation. Full-geometry optimizations were performed for all complexes considered and in this step no BSSE correction was applied to the Born–Oppenheimer (BO) energy surface. In recent studies, geometry optimizations have been performed on the BO surface corrected for BSSE. For weakly interacting monomers, some reports indicate that the di erences between the corrected and uncorrected BO surfaces are negligible [50] whereas

17.4 Relationships between the Topological and Energy Properties of Hydrogen Bonds 461

other studies indicate that such di erences can be significant [51]. For all results presented here BO energy surface was not corrected for BSSE, however, for such optimized geometry the BSSE was reduced significantly by the applied decomposition scheme described in the previous section.

Decomposition of the interaction energy yields insight on the character of the interaction. For example, for the trans-linear dimer of water the electrostatic component is 8.8 kcal mol 1, the exchange interaction energy term is 6.8 kcal mol 1, and the remaining delocalization and correlation terms are 2.2 and 0.3 kcal mol 1, respectively. For the H2OHþ HBeH complex already discussed, decomposition of interaction energy is as follows. The electrostatic, exchange, delocalization, and correlation terms are 12.0, 19.9, 24.4, and 4.2 kcal mol 1, respectively. It is apparent that for the water dimer the electrostatic interaction energy is the most important attractive term. For the much stronger dihydrogen bond in the second example the delocalization is twice as large (if one considers the modulus) than the electrostatic term. In other words one may expect for strong short hydrogen bonds that the most important attractive term is delocalization whereas for typical moderate and weak hydrogen bonds the electrostatic contribution dominates. The greater importance of delocalization is, hence, attributed to covalence. For very strong DHBs, as in the H2OHþ HBeH complex and related species, a strong correlation is found between both the exchange energy and the delocalization on one hand and the H H distance on the other. The other interaction energy terms do not correlate with this ‘‘intermolecular’’ distance (H H distance). Figure 17.5 shows these relations for very strong DHBs. For very strong hydrogen bonds, which are covalent in nature, the delocal-

Fig. 17.5 Correlations between H H distance (A˚ ) and the interaction energy components (kcal mol 1). Empty squares show the exchange energy, full squares the delocalization energy, full circles the electrostatic energy, and empty triangles the correlation energy.