Matta, Boyd. The quantum theory of atoms in molecules

422 15 Aromaticity Analysis by Means of the Quantum Theory of Atoms in Molecules

Clusters and Solids (Ed.: D. E. Ellis), Kluwer, Dordrecht, 1995, p. 1–46; E. J. Baerends, O. V. Gritsenko, J. Phys. Chem. A 1997, 101, 5383–5403; M. A. Buijse, E. J. Baerends, Mol. Phys. 2002, 100, 401–421.

21X. Fradera, M. A. Austen, R. F. W. Bader, J. Phys. Chem. A 1999, 103, 304–314.

22R. F. W. Bader, A. Streitwieser, A. Neuhaus, K. E. Laidig, P. Speers,

J.Am. Chem. Soc. 1996, 118, 4959–4965.

23R. L. Fulton, S. T. Mixon, J. Phys. Chem. 1993, 97, 7530–7534.

24E. Matito, M. Sola`, P. Salvador, M. Duran, Faraday Discuss. 2006, in press.

25F. Sondheimer, Pure Appl. Chem.

1964, 7, 363–388.

26P. v. R. Schleyer, H. Jiao, Pure Appl. Chem. 1996, 68, 209–218.

27C. F. Matta, J. Herna´ndez-Trujillo,

J.Phys. Chem. A 2003, 107, 7496– 7504; C. F. Matta, J. Herna´ndezTrujillo, J. Phys. Chem. A 2005, 109, 10798.

28P. Bultinck, M. Rafat, R. Ponec, B. van Gheluwe, R. Carbo´-Dorca, P. Popelier, J. Phys. Chem. A 2006, 110, 7642–7648.

29J. Poater, X. Fradera, M. Duran, M. Sola`, Chem. Eur. J. 2003, 9, 400– 406.

30C. W. Bird, Tetrahedron 1985, 41, 1409–1414.

31E. Matito, M. Duran, M. Sola`, J. Chem. Phys. 2005, 122, 014109.

32J. M. Matxain, L. A. Eriksson, J. M. Mercero, X. Lo´pez, J. M. Ugalde, J. Poater, E. Matito, M. Sola`, 2006, to be submitted.

33E. Matito, J. Poater, M. Duran, M. Sola`, J. Mol. Struct. (Theochem) 2005,

727, 165–171.

34I. Mayer, P. Salvador, Chem. Phys. Lett. 2004, 383, 368–375.

35E. Matito, J. Poater, M. Sola`, M. Duran, P. Salvador, J. Phys. Chem. A 2005, 109, 9904–9910; E. Matito, P. Salvador, M. Duran, M. Sola`, J. Phys. Chem. A 2006, 110, 5108–5113.

36L. T. Scott, J. S. Siegel, Tetrahedron 2001, 57, ix.

37M. Bu¨hl, A. Hirsch, Chem. Rev. 2001, 101, 1153–1183.

38R. C. Haddon, L. F. Schneemeyer, J. V. Waszczak, S. H. Glarum, R. Tycko, G. Dabbagh, A. R. Kortan, A. J. Muller,

A.M. Mujse, M. J. Rosseinsky, S. M. Zahurak, A. V. Makhija, F. A. Thiel,

K.Raghavachari, E. Cockayne, V. Elser, Nature 1991, 350, 46–47; T. M. Krygowski, A. Ciesielski, J. Chem. Inf. Comp. Sci. 1995, 35, 1001–1003.

39P. Fowler, Nature 1991, 350, 20–21;

R.C. Haddon, Science 1993, 261, 1545–1550.

40J. Poater, X. Fradera, M. Duran, M. Sola`, Chem. Eur. J. 2003, 9, 1113– 1122.

41J. Poater, X. Fradera, M. Duran, M. Sola`, in Fullerenes: The Exciting World of Nanocages and Nanotubes (Ed.: P. V. Kamat, K. M. Kadish, and D. Guldi), The Electrochemical Society Inc., Pennington, 2002, p. 707–719.

42E. Clar, The Aromatic Sextet, Wiley, New York, 1972.

43R. Taylor, J. P. Hare, A. K. AbdulSada, H. W. Kroto, J. Chem. Soc., Chem. Commun. 1990, 1423–1424; J. Mestres, M. Duran, M. Sola`, J. Phys. Chem. 1996, 100, 7449–7454; T. Akasaka, E. Mitsuhida, W. Ando, K. Kobayashi, S. Nagase, J. Am. Chem. Soc. 1994, 116, 2627–2628; A. L. Balch, J. W. Lee, M. M. Olmstead,

Angew. Chem. Int. Ed. Engl. 1992, 31, 1356–1358; C. Bellavia-Lund, F. Wudl, J. Am. Chem. Soc. 1997, 119, 9937–9937; M. S. Meier, G. W. Wang,

R.C. Haddon, C. P. Brock, M. A. Lloyd, J. P. Selegue, J. Am. Chem. Soc. 1998, 120, 2337–2342; C.

Thilgen, A. Herrmann, F. Diederich,

Angew. Chem. Int. Ed. Engl. 1997, 36, 2269–2280.

44Z. Chen, R. B. King, Chem. Rev. 2005, 105, 3613–3642.

45L. P. Hammett, Chem. Rev. 1935, 17, 125–136.

46T. M. Krygowski, K. Ejsmont, B. T. Stepie´n, M. K. Cyran´ski, J. Poater, M. Sola`, J. Org. Chem. 2004, 69, 6634– 6640.

47C. Hansch, A. Leo, R. W. Taft, Chem. Rev. 1991, 91, 165–195.

			References	423
48	J. Poater, I. Garcı´a-Cruz, F. Illas, M.	53	G. Portella, J. Poater, M. Sola´, J. Phys.
	Sola`, Phys. Chem. Chem. Phys. 2004,		Org. Chem. 2005, 18, 785–791.
	6, 314–318.	54	G. Portella, J. Poater, J. M. Bofill, P.
49	J. Poater, M. Sodupe, J. Bertran,		Alemany, M. Sola´, J. Org. Chem. 2005,
	M. Sola`, Mol. Phys. 2005, 103, 163–		70, 2509–2521.
	173.	55	K. N. Houk, J. Gonzalez, Y. Li, Acc.
50	M. K. Cyran´ski, M. Gilski, M.		Chem. Res. 1995, 28, 81–90; R.
	Jaskolski, T. M. Krygowski, J. Org.		Herges, H. Jiao, P. v. R. Schleyer,
	Chem. 2003, 68, 8607–8613; V. G. S.		Angew. Chem. Int. Ed. Engl. 1994, 33,
	Box, F. Jean-Mary, J. Mol. Mod. 2001,		1376.
	7, 334.	56	H. Jiao, P. v. R. Schleyer, J. Phys. Org.
51	E. Hu¨ckel, Z. Physik 1931, 70,		Chem. 1998, 11, 655–662.
	104–186; E. Hu¨ckel, Z. Physik 1931,	57	M. Manoharan, F. De Proft, P.
	72, 310–337; E. Hu¨ckel, Z. Physik		Geerlings, J. Org. Chem. 2000, 65,
	1932, 76, 628–648; E. Hu¨ckel, Z.		7971–7976; M. Manoharan, F. De
	Elektrochemie 1937, 43, 752–788,		Proft, P. Geerlings, J. Org. Chem.
	827–849.		2000, 65, 6132–6137; M. Manoharan,
52	M. Randic´, Chem. Rev. 2003, 103,		F. De Proft, P. Geerlings, J. Chem.
	3449–3605.		Soc., Perkin Trans. 2 2000, 1767–1773.

425

16

Topological Properties of the Electron Distribution in Hydrogen-bonded Systems

Ignasi Mata, Ibon Alkorta, Enrique Espinosa, Elies Molins,

and Jose´ Elguero

16.1 Introduction

The hydrogen bond (HB) is the most important weak interaction found in nature. It is responsible for the three dimensional shape of biopolymers (proteins and nucleic acids) and for the structure of water, in both the liquid and solid phases. Life processes extensively use the making and breaking of HBs as part of concatenated reactions involving huge amounts of biomolecules. In addition, it has profound implications in the mode of action of drugs and in molecular packing, recognition, and crystal engineering [1].

In the literature, the moieties involved in HB interaction are usually identified as hydrogen donor (or electron acceptor) and hydrogen acceptor (or electron donor). In this chapter the donor (D) and acceptor (A) terms are schematically represented as DaH A.

The H A interaction distance d(H A) varies from 1.2 to 2.5 A˚ (or up to 3.0 A˚ , depending of the criteria used). This is not true for covalent bonds, for which the range is much smaller. Dependence of hydrogen bond properties on the internuclear distance can, therefore, be clearly observed for hydrogen bonds (HBs), as shown by the DaH bond distance, which seems to depend on d(H A). The mutual dependence of bond distances on both sides of the hydrogen atom can easily be understood in terms of the bond-order model proposed by Pauling, which assumes a total valence equal to 1 for the hydrogen atom involved in the HB interaction [2–5].

Similar dependencies have been found for the electron-density properties from application of QTAIM methodology to the hydrogen bond. Topological analysis of rðrÞ was initially used to identify the presence of HB interactions. Thus, characterization of CaH O hydrogen bonds has been used to generalize a set of criteria to establish the presence of hydrogen-bonding interactions based on the QTAIM theory [6]. These criteria have been applied to the study of other hydrogen bonds apart from CaH O interactions, and a further extension has been

426 16 Topological Properties of the Electron Distribution in Hydrogen-bonded Systems

developed for dihydrogen bonds (DaH HaM) [7]. More recently, topological analysis has revealed the dependence of electron density properties on H A distance and the interdependences between these properties.

This chapter covers theoretical and experimental studies that analyze the dependence of the topological and energy properties of the electron distribution in HB systems either on the H A distance or on other properties of rðrÞ. As a particular point of interest, Section 16.6 discusses attempts which have been made to analyze the rðrÞ properties in a comprehensive range of interatomic distances, from covalent to weak HB interactions. In general, the theoretical studies considered here have been undertaken in two di erent manners. The first treats the complexes by following a reaction coordinate, where the HB distance between the two DaH and A moieties involved is artificially modified. The second approach assembles information from complexes that have di erent internuclear H A distances at their equilibrium geometries. The former method has the advantage of reducing secondary e ects that arise in the latter as a result of the different electronegativities of the D and Y groups in the DaH AaY systems. It should be noted, however, that, in the former instance, the observed dependencies do not necessarily follow the same quantitative behavior as systems in equilibrium. The second method can, moreover, be used with experimental electron densities in crystals. Occasionally both approaches have been combined in the same study, filling the gaps in the data for complexes in equilibrium with calculated magnitudes that have been obtained by making use of reaction coordinates.

16.2

Topological Properties of the Hydrogen Bond

16.2.1

Topological Properties at the Bond Critical Point (BCP)

Within the QTAIM theory a bond between two atoms is characterized by a line of maximum electron density (the bond path) that connects both nuclei and intersects the zero-flux surface of the electron density gradient field ‘rðrÞ at a topological ð3; 1Þ point, called the bond critical point (BCP).

Some particular properties of the electron distribution at the BCP are:

1.rb is the smallest value of the electron density along the bond path; and

2.it corresponds to the maximum of rðrÞ at the interatomic surface (IAS) between both atoms.

The uniqueness of this point has been used to define some characteristics of the whole bond (for example bond distance, degree of covalence, interaction energy, etc.) in terms of a number of rðrÞ properties at BCP. The most studied topo-

16.2 Topological Properties of the Hydrogen Bond 427

logical properties at the BCP are the electron density ðrbÞ, its Laplacian ð‘2rbÞ and the eigenvalues of the Hessian matrix (l1, l2 and l3), the latter indicating the three main curvatures of rðrÞ at the BCP. If rðrÞ at the BCP has a saddle distribution, two of the eigenvalues have negative values and correspond to the curvatures that are perpendicular to the bond path and the third is positive and represents the curvature of the rðrÞ distribution along the bond path. By convention, the negative curvatures are l1 and l2 ðl1 < l2 < 0Þ, and the positive one is l3. The Laplacian of the electron density ‘2rðrÞ, which is defined as the sum of the three eigenvalues of the Hessian, l1, l2 and l3, provides information about either the charge concentration ð‘2rðrÞ < 0Þ or the charge depletion ð‘2rðrÞ > 0Þ of the electron distribution. For hydrogen bonds, rb is usually small and ‘2rb > 0, both being characteristic magnitudes of closed-shell interactions. (See also Chapter 1 of this book).

The first publications in which topological analysis of rðrÞ was used to characterize hydrogen-bonded systems included theoretical calculations [HF/6-31G(d,p)] of the complexes formed between nitrile derivatives and hydrogen halides (HF and HCl) [8, 9]. In these studies the rb magnitudes calculated for each hydrogen halide were linearly correlated with the H A distance. A similar linear correlation was also observed for ‘2rb [9]. The small range of HB distances (0.28 and 0.36 A˚ for XCN HF and XCN HCl complexes, respectively) induced researchers to postulate linear relationships rather than other kinds of dependence. Similar conclusions were also reached by other authors when they considered a small range of distances. When the range expanded, however, the curvature of the rb and ‘2rb data distributions became evident, as shown by a theoretical study of carbenes and silylenes as HB acceptors [10] and experimental analysis of H O interactions [11].

The first study describing the nonlinear behavior of the rb magnitude with the bond distance does not correspond to the analysis of a weak interaction but to the SaS bond, which can expand by 0.5 A˚ ð1:70 < d(SaS) < 2:25 A˚ ) [12]. These results clearly showed that a double-logarithm function, which can be expressed as an equivalent power function, enables better fitting of the data distribution than a linear regression (Fig. 16.1). In the same way, the experimentally derived rb magnitudes of the CO bonds in citrinin were better fitted by a simple exponential function than by a linear function [13].

In experimental analysis of XaH O (X ¼ C, N, O) hydrogen bond interactions in crystals [11], rb, ‘2rb, and the curvatures were exponentially dependent on intermolecular distance (Fig. 16.2). For the same set of HB interactions, an exponential relationship was also found between the sum of the negative curvatures l1 þ l2 and the positive curvature l3 [14]. These dependencies have been also derived in theoretical studies of H F hydrogen-bonding interactions [15]. It is particularly worthy of note that the fitting data found for rb in experimental H O and in the theoretically calculated pure closed-shell H F interactions {rb ¼ 65(27) exp[ 3.2(2)d(H O)]} and {rb ¼ 63(10) exp[ 3.552(7)d(H F)]} are very similar.

428 16 Topological Properties of the Electron Distribution in Hydrogen-bonded Systems

Fig. 16.1 Relationship between d(SaS) distance (A˚ ) and rb (a.u.). The fitted curve corresponds to d(SaS) ¼ 1:229r 0:269. (Values taken from Ref. [12]).

Fig. 16.2 Relationship between l3 (e A˚ 5) and H O distance (A˚ ) data for experimental XaH O (X ¼ C, N, O) hydrogen-bonding interactions in crystals. (Values taken from Ref. [14]). The equation of the fitted curve is: l3 ¼ 0:65ð13Þ 103 exp½2:6ð1ÞdðH OÞ&.

16.2 Topological Properties of the Hydrogen Bond 429

16.2.2

Integrated Properties

Within a molecule, each atomic basin W is defined as the space region covered by all the gradient paths of rðrÞ ending at its nuclear position. These atomic regions are delimited by zero-flux surfaces of ‘rðrÞ which do not belong to the atoms and represent their topological borders. From QTAIM methodology, atomic basins can be regarded as quantum subsystems of the molecular quantum system. Thus, the magnitude of a given molecular property P can be obtained by using the integration within the basins W, which provides the atomic contributions PW to the molecular property P ðP ¼ SWPWÞ. Several integrated properties have been analyzed in theoretical studies of HBs, for example charge, energy, volume, and atomic dipole polarizability. On HB formation, the hydrogen atom involved in the interaction becomes more positively charged, energetically destabilized, and its volume and dipole moment decrease. These changes were first observed for weak CaH O hydrogen bonds by comparing the integrated properties of the complexes with those of the free monomers [6]. They have been included as four of the eight criteria used to establish hydrogen-bonding interactions from topological analysis of rðrÞ. When an HB is formed, electrons are typically promoted from the lone pair of the acceptor to the s antibonding orbital of DaH (except for blue shift hydrogen bonds, in which other orbitals of the donor molecule are involved [16–18]), weakening the covalent bond and producing changes in the energy, charge, and dipole moment of the hydrogen atomic basin that are intrinsically related to each other. Indeed, as shown for hydrogen bonds in dimers of tetrahydroimidazo[4,5-d]imidazole derivatives [19], there is a clear correlation between the variations of the net charge and the energy of the hydrogen atom on HB formation (Fig. 16.3).

The shrinking of the hydrogen atom volume is explained by the proximity of the HB acceptor and the concomitant overlap of their electron clouds. Exceptions are observed for very weak HB complexes, however – volume increments have sometimes been observed for this kind of atom [6, 20, 21]. According to the topological properties it must be noticed that, for these examples, the interaction should disappear if there are small changes in the geometry.

The literature contains few studies dealing with the dependence of these properties on intermolecular distance. For complexes in the equilibrium geometry formed with dihydrogen bonds it has been observed that changes in energy, charge, and volume of the protic hydrogen tend to decrease smoothly as the H H distance increases, following exponential relationships that depend on the donor moiety [21]. The behavior of these integrated magnitudes along reaction coordinates depends on the complex, as shown by the qualitatively di erent dependencies with the A H distance for (FH)2 and (H2O)2 (Fig. 16.4) [22]. Thus, for (FH)2 the energy of the hydrogen atom increases regularly as the distance is reduced, up to a local maximum that is observed at a distance slightly larger than the equilibrium distance Req. Similar behavior is also observed for the integrated charge of the hydrogen atom in the H F interaction of (FH)2, but here the max-

430 16 Topological Properties of the Electron Distribution in Hydrogen-bonded Systems

Fig. 16.3 Relationship between the change in of the net charge (e) and the change in energy (kJ mol 1) for all the hydrogen atoms of tetra- hydroimidazo[4,5-d]imidazole on dimer formation. (Values taken from Ref. [19]). The fitted linear equation is: DCharge ¼ 0:603ð4Þ 10 3 DEnergy 0:05ð22Þ 10 3.

imum occurs at the equilibrium distance (Fig. 16.4). These features are not observed for (H2O)2, a result that is explained by the incipient formation of a covalent bond in (FH)2 that should be able to stabilize the hydrogen in (FH)2, whereas for (H2O)2 this process would be hindered by stronger repulsion, leading to the more important variation observed for the integrated properties of its involved hydrogen atom (notice the di erent scale of Figs 16.4a and 16.4b).

Fig. 16.4 Relationship between the integrated charge (e) on hydrogen atoms involved in H F (a) and H O (b) hydrogen-bonding interactions and the interaction distances (A˚ ) for the (FH)2 and (OH2)2 dimers. (Values taken from Ref. [22]).

16.3 Energy Properties at the Bond Critical Point 431

Fig. 16.5 Variation of the net molecular charge (e) with intermolecular C C distance (A˚ ) in the formamide/formic acid complex (values taken from Ref. [23]). Req represents the C C distance corresponding to the equilibrium geometry of the complex.

The increase of the hydrogen net positive charge in the H A interaction is accompanied by a decrease in the net charge of the donor molecule, indicative of the well-known electron transfer from the acceptor molecule that reproduces the expected acid–base behavior. In general, this electron transfer tends to be less important as the interatomic distance increases, being almost negligible for weaker interactions. In complexes involving a single hydrogen bond the charge transfer increases regularly as the molecules approach [22] whereas for complexes involving several hydrogen bonds more complex behavior is observed. This is true for the formamide–formic acid complex [23], which is stabilized by two HB interactions (NaH O and OaH O, the former being only observed for C C distances shorter than 5.0 A˚ ), and where both molecules behave as donor and acceptor. Despite the intricate dependence of charge transfer between molecules (Fig. 16.5), the strongest OaH O interaction causes electron transfer from the formamide toward the formic acid in all the interaction range of distances. This transfer becomes more important as the intermolecular C C distance shrinks to approximately 3.7 A˚ , which corresponds to a shorter distance than the equilibrium geometry.

16.3

Energy Properties at the Bond Critical Point (BCP)

The total electron energy density at a given point of space, HðrÞ, is defined as the sum of the kinetic GðrÞ and the potential VðrÞ contributions. The last two are