Matta, Boyd. The quantum theory of atoms in molecules

442 16 Topological Properties of the Electron Distribution in Hydrogen-bonded Systems

involving the pairs H and X ¼ C, N, S confirmed the quality of this method for representing the whole interaction range with a single function not only for rb but for all the topological and energy properties of rðrÞ at BCP [5, 41, 42]. Most recently, and making use of the elegantly matching Morse-type function and the potential energy function defined in terms of Hb [31], the joint function has been used to fit Hb and ‘2rb data [43].

Before the joint function was proposed, the behavior of rb in the transition region between closed-shell and covalent interactions was studied for protic systems A H D [44]. By displacing the hydrogen atom along the reaction coordinates the behavior of rb between the states AaH D and A HaD was characterized. The results showed that rb at both critical points are related by:

	rbð1Þ	rbð2Þ		1		9
	rbð01Þ	þ rbð02Þ	¼		ð		Þ

where rb(01) and rb(02) are, respectively, the rb values for the free donors AaH and DaH, and rb(1) and rb(2) are the corresponding rb magnitudes in the complex. This expression, which can be derived from Eq. (7), shows that the rb magnitude of interactions involving hydrogen atoms is an additive property when it is expressed as a relative quantity without dimensions. Accordingly, the relationship between the rb magnitudes at both critical points in the complex is linear. Indeed, in theoretical analysis of the electron distribution in NaH N HBs, this linear relationship was revealed between the rb magnitudes corresponding to the covalent NaH bond and to the hydrogen bonding N H interaction [28]:

rbðNaHÞ ¼ 0:3506 1:302rbðN HÞ

ð10Þ

The same study showed that the relationship between the curvatures at both sides of the hydrogen atom is much more complex than is found for rb. Thus, good quality fitting could only be achieved for the ratios l3(NaH)/l3(N H) and l12(NaH)/l12(N H), where ½l12 ¼ ðl1 þ l2Þ=2&, but for neither l3(NaH) nor l12(NaH) as functions of l3(N H) or l12(N H). Considering logarithmic dependencies, these relationships are expressed as:

ln l				ðN		HÞ ¼ 1:237 1:939 ln½ l12ðN HÞ&
	l12			NaH
		12ð Þ
							0:1389 ln½ l12ðN HÞ&2	ð11Þ
and						HÞ
ln l			ðN				¼ 0:9033 þ 1:5434 lnfln½l3ðN HÞ&g	ð12Þ
	l3			NaH
		3ð				Þ

where r2 ¼ 0:998 and r2 ¼ 0:988 for Eqs (11) and (12), respectively.

16.6 Complete Interaction Range 443

Despite the di erent relationships of rb for closed-shell and for shared-shell interactions, it is di cult to recognize the range of distances where the transition between these two regions occurs, because of the smooth changes observed for this topological property. A similar situation is found for l12, as shown for the isolated H F system and for a set of 79 XaH FaY complexes (Fig. 16.12a) [15]. Considering the parallel curvature, l3, the change of dependence associated

Fig. 16.12 Relationships between (a) curvatures (e A˚ 5) and (b) local energy densities (kJ mol 1 a0 3) at BCP and H F distance (A˚ ) for the isolated H F interaction (white symbols) and for a set of 79 XaH FaY complexes (black symbols). The lines were obtained by fitting joint functions [15]. The inset focuses Gb data corresponding to the [FaH F]-system.

444 16 Topological Properties of the Electron Distribution in Hydrogen-bonded Systems

with the transition between both kind of interaction is revealed by a shoulder for the isolated H F pairwise interaction and by a broad peak for the set of XaH FaY complexes (Fig. 16.12a). The latter corresponds to calculated values for the [F H F] system along the reaction coordinates from [FaH F] to

[F HaF] . This proton-transfer			process is	energetically	characterized		by		Gb
(Fig. 16.12b),	which has a	local	maximum	at the distance d(F H)¼					˚
							1:120 A
and is very	˚	of the symmetrical proton			position [F		H		F]
	close to that

(d(F H) ¼ 1:138 A). The dependencies of the curvatures and the local energies on d(F H) are similar when comparing l3 to Gb and l1 to Vb for both data sets.

Of all the topological properties at BCP, the Laplacian ‘2rb is the only one with qualitatively di erent behavior in covalent and noncovalent regions. Thus, in the covalent region the negative magnitude of ‘2rb increases exponentially as the bond distance decreases, because of the larger charge concentration. For a typical hydrogen-bond region ‘2rb is positive and increases as the atoms approach, because the interaction remains of the closed-shell type, and, therefore, greater charge depletion of rðrÞ is observed.

Several studies involving reaction coordinates of the molecular approach covering positive and negative regions of ‘2r have shown the generic shape of this topological property along the interatomic distance (Fig. 16.13) [15, 26]. A maximum of ‘2rb is always observed in the transition region between typical hydrogen bonds and covalent interactions, and can be fitted by making use of a joint function (Eq. 8). A similar profile, also with a maximum of ‘2rb, has been ob-

Fig. 16.13 Dependence of ‘2rb (e A˚ 5) and Hb (kJ mol 1 a0 3) on the interatomic distance (A˚ ) for isolated H F systems [15].

16.6 Complete Interaction Range 445

served in transitions from closed-shell to shared-shell interactions involving atoms other than hydrogens, as for example LiF or CO interactions [45]. Further studies using a set of complexes at their equilibrium geometries, some of which have interaction distances in the transition region, confirmed the profile of the dependence of ‘2rb on internuclear distance [5, 15, 41, 42]. The dihydrogen bond has been also studied [46]; it has a similar ‘2rb profile, indicating that for very strong hydrogen bonds the interaction is covalent. Indeed, the negative magnitude of the Laplacian and results from energy decomposition analysis have shown that, in contrast with medium and weak hydrogen bonds, the H H electrostatic interaction is no longer the largest attractive term for the strong hydrogen bond.

The dependence of the total electron energy density Hb on the interaction distance has a profile similar to that of ‘2rb, also having a local maximum. Whereas for weak hydrogen bonds Hb is positive, because of the excess of kinetic energy Gb over potential energy Vb, for covalent bonds Hb is negative, because of the reverse situation, and becomes more negative as the atoms approach each other, leading to a greater charge concentration in the interatomic region. For intermediate examples of stronger hydrogen-bonding interactions, in which covalent features start to appear, Hb is found to be negative while ‘2rb is still positive. Positive and negative values of Hb, respectively, have been used as an alternative way of defining ionic and covalent bonds, because this quantity avoids the problems observed for ‘2rb with some covalent bonds [47]. Bonds containing very electronegative atoms (for example F2, CO, H2CO, and HCN) have positive values of ‘2rb whereas the corresponding values of Hb for these molecules are negative, as in other covalent bonds.

All the topological and energy properties of rðrÞ at BCP, except ‘2rb and Hb, change smoothly with distance, varying from closed-shell to shared-shell characteristics in a continuous way and without any sudden alteration that could be associated with a transition from hydrogen bonding to covalent interactions. This transition has been studied in strongly hydrogen-bonded complexes in which proton transfer occurs [37]. For these systems, however, neither ‘2rb nor Hb have maxima in the range of distances considered, because only strong hydrogen bonds and covalent interactions are found along the proton-transfer process and the ‘2rb and Hb data calculated along the reaction coordinates were fitted to a single exponential function plus an independent term.

A coe cient which could be used to monitor closed-shell or shared-shell character and the strength of the interaction would be useful for qualitative and quantitative characterization of bonds. The ratio Gb=rb was initially proposed as a classification criterion for distinguishing between di erent types of chemical bond [48]. Covalent interactions involve a large amount of electron density and, because the charge is locally concentrated, the kinetic energy density is expected to be comparatively small; the opposite situation is expected for closed-shell interactions, for example ionic or hydrogen bonding or van der Waals interactions. Thus, Gb=rb < 1 and Gb=rb > 1 have been proposed for classifying shared and

446 16 Topological Properties of the Electron Distribution in Hydrogen-bonded Systems

closed-shell interactions, respectively. In a theoretical study involving NaH N hydrogen bonds with internuclear distances up to 2.3 A˚ , however, the Gb=rb ratio never reached unity [28].

When assembling XaH and X H interactions the characteristic features shown by both ‘2rb and Hb have been used to divide the whole range of interaction distances into three regions (Fig. 16.13) [15, 49]. Region I corresponds to weak hydrogen bonds with interaction energies Ei < 12:0 kcal mol 1 and larger interaction distances. Within this range of distances ‘2rb > 0 and Hb > 0, and the interaction can be regarded as pure closed-shell type. The maximum of Hb occurs in this region, and the distance where Hb ¼ 0 marks the border between regions I and II. Region II is situated at intermediate distances and is characterized by ‘2rb > 0 and Hb < 0. This region is associated with a closed-shell interaction with some covalent character and contains from medium to strong hydrogen bonds with interaction energies typically in the range 12:0 < Ei < 24:0 kcal mol 1. The maximum of ‘2rb is observed in this region and is a direct consequence of the excess of Vb over Gb, starting the concentration of rðrÞ and, there-

fore, being at the origin of the loss of exponential behavior of ‘2rb in region I. The transition to region III occurs at ‘2rb ¼ 0, where the shortest distances and the strongest interaction energies (Ei > 24:0 kcal mol 1) are found. In this region, ‘2rb < 0 and Hb < 0, both falling to deep negative magnitudes with shortening of the interaction distance. Region III corresponds to strong HBs, low-barrier hydrogen bonds (LBHB), and covalent bonds. In accordance with the topological and energy properties of rðrÞ, all these interactions are of shared-shell type.

From the local form of the virial theorem and the definition of Hb, the three regions can also be defined by the magnitudes of the ratio jVbj=Gb. Thus, regions I, II, and III correspond to jVbj=Gb < 1, 1 < jVbj=Gb < 2 and jVbj=Gb > 2, respectively. This ratio enables identification and quantification of the closed-shell and shared-shell characteristics of interactions. In particular, this index has been used to classify metal–oxide bonded interactions [50] and to investigate the nature of metal–metal interactions [51].

Calculations, using the natural bond orbital (NBO) method, performed for the population of the bonding molecular orbitals for the isolated H F system, support the partition of the whole interaction range of distances into three regions (Fig. 16.14). Indeed, according to these data, the formation of the bonding molecular orbital starts (as observed from the first converged calculation of the orbital) when the hydrogen and the fluorine atoms approach at d(F H) A2:1 A˚ . This distance corresponds to the change in the behavior of the ratio jVbj=Gb (inset in Fig. 16.14), indicating a reorganization of rðrÞ associated with a more important increase of jVbj relative to Gb, and it occurs close to that corresponding to jVbj=Gb ¼ 1, defining the border between regions I and II. For shorter distances there is rapid filling of the bonding orbital, which extends along the narrow region II. The bonding orbital is almost filled at the border of regions II and III, where jVbj=Gb ¼ 2. Further shrinkage of the bonding distance has very small

16.6 Complete Interaction Range 447

Fig. 16.14 Dependences of the index jVbj=Gb (squares) and the number of electrons filling the bonding molecular orbital (triangles) on the interatomic distance (A˚ ) for the isolated H F system [15].

consequences on the bonding-orbital population, which remains within the values expected for a covalent bond.

The index jVbj=Gb and the bond-degree (BD), defined as Hb=rb, have been derived from analysis of the H F interaction with the objective of characterizing pairwise atom–atom interactions [15, 51]. BD, which can be interpreted as either the total pressure per electron density unit or the total energy per electron at BCP, has been defined as the softening degree (SD) for Hb > 0 (Section 16.4) and as the covalence degree (CD) for Hb < 0. For jVbj=Gb < 1 (i.e. for Hb > 0), the electron distribution is ideally depleted ð‘2rb > 0Þ according to the local form of the virial theorem. As the internuclear distance shortens, the ratio jVbj=Gb increases and the interaction becomes stronger, leading to more charge in the internuclear region (rb increases) and to a regular decrease of SD to zero at jVbj=Gb ¼ 1. At shorter distances, 1 < jVbj=Gb < 2 ðHb < 0Þ and, from Eq. (1), ‘2rb > 0. Despite the positive value of ‘2rb, the more important increase of jVb | in relation to Gb reduces the magnitude of ‘2rb to zero at jVbj=Gb ¼ 2. Then, for shorter interaction distances, an important increase of the negative magnitudes of both ‘2rb and Hb is observed, indicating that electrons are concentrated and the interaction has strong covalent character. As a consequence of the observed electron redistribution in the H F bonding molecular orbital, an initial amount of covalence (CD ¼ Hb=rb, for Hb < 0) appears in H F interactions for which ‘2rb > 0 and Hb < 0, and the negative magnitude of CD increases regularly with shortening of the distance from Hb ¼ 0 to very short geometries for covalent

448 16 Topological Properties of the Electron Distribution in Hydrogen-bonded Systems

Fig. 16.15 Relationship between ‘2rb (a.u.) and the intermolecular d(C C) distance (A˚ ) for the HCOOHaHCONH2 complex [23].

interactions ðjVbj=Gb > 2Þ. Thus, BD decreases continuously and regularly from positive to negative values as the interatomic distance shortens along the entire interaction range.

16.6.2

Perturbed Systems

The dependence of the topological and energy properties of rðrÞ at BCP throughout the complete range of interaction distances is qualitatively the same for all the A H interactions studied. The exact quantitative form of the corresponding profiles, which can be regarded as the signature of rðrÞ, depends on the environment around the A H interacting atoms, however. Thus, the profiles of ‘2rb and of Hb observed for di erent complexes that have been studied along reaction coordinates, appear displaced from one complex to another [23].

Accordingly, changes in the environment of the interaction should a ect the BCP properties. Further evidence of this is given by the behavior of ‘2rb in the two hydrogen bonds present in the formamide–formic acid complex (Fig. 16.15) [23]. For large intermolecular separations a single OaH O hydrogen bond is observed; a second NaH O hydrogen bond is formed when the molecules are close enough. While the NaH O interaction has the expected dependence on the reaction coordinate, a second local maximum and, consequently, a local minimum, is observed for OaH O. The minimum is reached at the largest distance where the BCP associated with the NaH O bond is observed, indicating that the OaH O hydrogen bond is a ected by the modification of its environment induced by formation and breaking of the NaH O interaction.

16.6 Complete Interaction Range 449

The e ect of external perturbations on the topological and energy properties at BCP has been studied by considering both addition and subtraction of one electron to and from the neutral F H system, leading to (F H)þ and (F H) [52]. Thus, both ionic states are regarded as extreme cases of a net charge perturbation induced on the system by the environment. The most stable of the three calculated systems is the neutral one, as shown by its deepest energy, and by its largest rb and ‘2rb values at the equilibrium configuration, which are indicative of greater stabilization energy and a larger and more concentrated quantity of charge in the interatomic region.

Comparing the dependence of the BCP properties on bond distance for the three systems within the range 0.8–3.0 A˚ enables observation of the e ect of perturbation on the electron distribution of the neutral system. Thus, taking into account the electron configuration of the (F H) neutral system ðð1sÞ2ð2sÞ2ð3sÞ2ð1pÞ2ð2pÞ2ð4sÞ0Þ, in (F H) the supplementary electron is added to the 4s antibonding orbital. As a consequence, the extra charge expands the bonding molecular orbital toward both nuclei, increasing the depletion of the electron distribution and removing part of the charge from the bonding region, as observed in the dependencies of rb and ‘2rb on bond distance, as shown in Fig. 16.16 for the latter. The presence of an additional electron also hinders formation of the bonding orbital, as seen by the highest value of the maximum of the Laplacian and its displacement toward shorter distances.

For (F H)þ, vertical ionization of one electron from the 2p nonbonding orbital, which is mainly localized around the fluorine, polarizes the bonding molecular orbital toward this atom, removing charge from the bonding region and, there-

Fig. 16.16	Relationship between ‘2rb (e A˚ 5) and the F H distance
(A˚ ) for F	H, (F	H)þ, and (F H) [52].

450 16 Topological Properties of the Electron Distribution in Hydrogen-bonded Systems

fore, destabilizing the bond. This molecule is formed from a neutral fluorine and a proton (it has no electron charge as counterpart), so the exchange interaction does not occur within this system, leading to negative ‘2rb values and, therefore, to a net local concentration of charge over almost the full interaction range of distances. As a consequence, no local maximum of ‘2rb appears in region II (Fig. 16.16).

16.7

Concluding Remarks

This chapter contains a summary of the relationships found, so far, for the topological and energy properties of rðrÞ in hydrogen-bonded systems, and their observed dependence on the interatomic H A distance. The long and interesting path which awaits new research workers in this field may enable explanation of macroscopic physicochemical properties in terms of microscopic quantities derived from electron density properties. Understanding these relationships is a challenge, and a major objective in this field.

Acknowledgments

This work was supported by the Spanish Ministerio de Ciencia y Tecnologı´a (Project No. BQU2003-01251) (IA and JE) and by the Pla de Recerca de Catalunya (Grant 2005SGR-452) (EM and IM). EE thanks Professor R. Guilard for supporting the development of a part of this work at the LIMSAG laboratory.

References

1	G. A. Je rey, An Introduction to	8	R. J. Boyd, S. C. Choi, Chem. Phys.
	Hydrogen Bonding, Oxford University		Lett. 1985, 120, 80–85.
	Press, 1997.	9	R. J. Boyd, S. C. Choi, Chem. Phys.
2	T. Steiner, W. Saenger, Acta		Lett. 1986, 129, 62–65.
	Crystallogr. B 1994, 50, 348–357.	10	I. Alkorta, J. Elguero, J. Phys. Chem.
3	T. Steiner, J. Chem. Soc., Chem.		1996, 100, 19367–19370.
	Commun. 1995, 1331–1332.	11	E. Espinosa, M. Souhassou, H.
4	M. Ramos, I. Alkorta, J. Elguero, N. S.		Lachekar, C. Lecomte, Acta Crystallogr.
	Golubev, G. S. Denisov, H. Benedict,		B 1999, 55, 563–572.
	H. H. Limbach, J. Phys. Chem. A	12	O. Knop, R. J. Boyd, S. C. Choi,
	1997, 101, 9791–9800.		J. Am. Chem. Soc. 1988, 110, 7299–
5	M. Sa´nchez, P. F. Provasi, G. A.		7301.
	Aucar, I. Alkorta, J. Elguero, J. Phys.	13	P. Roversi, M. Barzaghi, F. Merati, R.
	Chem. B 2005, 109, 18189–18194.		Destro, Can. J. Chem. 1996, 74, 1145–
6	U. Koch, P. L. A. Popelier, J. Phys.		1161.
	Chem. 1995, 99, 9747–9754.	14	E. Espinosa, C. Lecomte, E. Molins,
7	P. L. A. Popelier, J. Phys. Chem. A		Chem. Phys. Lett. 1999, 300, 745–
	1998, 102, 1873–1878.		748.

			References	451
15	E. Espinosa, I. Alkorta, J. Elguero, E.	35	M. E. Alikhani, F. Fuster, B. Silvi,
	Molins, J. Chem. Phys. 2002, 117,		Struct. Chem. 2005, 16, 203–210.
	5529–5542.	36	F. Fuster, B. Silvi, Theor. Chem. Acc.
16	P. Hobza, Z. Havlas, Chem. Rev. 2000,		2000, 104, 13–21.
	100, 4253–4264.	37	L. F. Pacios, O. Ga´lvez, P. C. Go´mez,
17	X. S. Li, L. Liu, H. B. Schlegel, J. Am.		J. Chem. Phys. 2005, 122, 214307.
	Chem. Soc. 2002, 124, 9639–9647.	38	I. Alkorta, I. Rozas, J. Elguero, Struct.
18	I. V. Alabugin, M. Manoharan, S.		Chem. 1998, 9, 243–247.
	Peabody, F. Weinhold, J. Am. Chem.	39	I. Alkorta, L. Barrios, I. Rozas, J.
	Soc. 2003, 125, 5973–5987.		Elguero, J. Mol. Struct. (Theochem)
19	I. Alkorta, O. Picazo, J. Elguero,		2000, 496, 131–137.
	J. Phys. Chem. A 2006, 110, 2259–	40	O. Knop, K. N. Rankin, R. J. Boyd,
	2268.		J. Phys. Chem. A 2001, 105, 6552–
20	I. Alkorta, I. Rozas, J. Elguero, Ber		6566.
	Bunsen Phys Chem 1998, 102,	41	O. Picazo, I. Alkorta, J. Elguero,
	429–435.		J. Org. Chem. 2003, 68, 7485–7489.
21	I. Alkorta, K. Zborowski, J. Elguero,	42	I. Alkorta, O. Picazo, J. Elguero,
	M. Solimannejad, J. Phys. Chem. A, in		Tetrahedron Asymmetry 2004, 15,
	press (DOI: 10.1021/jp061481x).		1391–1399.
22	O. Ga´lvez, P. C. Go´mez, L. F. Pacios,	43	P. M. Dominiak, A. Makal, P. R.
	J. Chem. Phys. 2001, 115, 11166–		Mallinson, K. Trzcinska, J. Eilmes, E.
	11184.		Grech, M. Chruszcz, W. Minor, K.
23	O. Ga´lvez, P. C. Go´mez, L. F. Pacios,		Wozniak, Chem. Eur. J. 2006, 12,
	J. Chem. Phys. 2003, 118, 4878–4895.		1941–1949.
24	E. Espinosa, E. Molins, C. Lecomte,	44	I. Alkorta, I. Rozas, J. Elguero, J. Mol.
	Chem. Phys. Lett. 1998, 285, 170–173.		Struct. (Theochem) 1998, 452, 227–
25	Y. A. Abramov, Acta Crystallogr. A		232.
	1997, 53, 264–272.	45	J. Hernandez-Trujillo, R. F. W. Bader,
26	O. Ga´lvez, P. C. Ga´mez, L. F. Pacios,		J. Phys. Chem. A 2000, 104, 1779–
	Chem. Phys. Lett. 2001, 337, 263–268.		1794.
27	E. Espinosa, I. Alkorta, I. Rozas, J.	46	S. J. Grabowski, W. A. Sokalski, J.
	Elguero, E. Molins, Chem. Phys. Lett.		Leszczynski, J. Phys. Chem. A 2005,
	2001, 336, 457–461.		109, 4331–4341.
28	O. Knop, K. N. Rankin, R. J. Boyd,	47	D. Cremer, E. Kraka, Croat. Chem.
	J. Phys. Chem. A 2003, 107, 272–284.		Acta 1984, 57, 1259.
29	R. Parthasarathi, V. Subramanian, N.	48	R. F. W. Bader, H. Essen, J. Chem.
	Sathyamurthy, J. Phys. Chem. A 2006,		Phys. 1984, 80, 1943–1960.
	110, 3349–3351.	49	I. Rozas, I. Alkorta, J. Elguero, J. Am.
30	G. V. Gibbs, D. F. Cox, K. M. Rosso,		Chem. Soc. 2000, 122, 11154–11161.
	J. Phys. Chem. A 2004, 108, 7643–	50	G. V. Gibbs, D. F. Cox, T. D.
	7645.		Crawford, K. M. Rosso, N. L. Ross,
31	E. Espinosa, E. Molins, J. Chem. Phys.		R. T. Downs, J. Chem. Phys. 2006,
	2000, 113, 5686–5694.		124, 847041–847048.
32	P. T. T. Wong, E. Whalley, J. Chem.	51	G. Gervasio, R. Bianchi, D.
	Phys. 1976, 64, 2359–2366.		Marabello, Chem. Phys. Lett. 2004,
33	C. Gatti, B. Silvi, F. Colonna, Chem.		387, 481–484.
	Phys. Lett. 1995, 247, 135–141.	52	E. Espinosa, I. Alkorta, I. Mata, E.
34	A. D. Becke, K. E. Edgecombe,		Molins, J. Phys. Chem. A 2005, 109,
	J. Chem. Phys. 1990, 92, 5397–5403.		6532–6539.