Matta, Boyd. The quantum theory of atoms in molecules

462 17 Relationships between QTAIM and the Decomposition of the Interaction Energy

Fig. 17.6 Relationship between the Gb=Vb ratio and the ratio of the interaction energy components (delocalization to electrostatic). The meanings of the symbol are as for Fig. 17.3.

ization and exchange energies are responsible for the stability of the systems containing them. One can see from Fig. 17.5 that the importance of electrostatic interaction increases with decreasing H H distance but not to the same extent as the delocalization energy. In other words, the ratio of the delocalization and electrostatic energy terms should correspond to the strength of hydrogen bonding. Such a correlation was found for the formamide dimer and its fluorine derivatives [52]. It has also been found [53] that when this ratio is @0.45 the hydrogen bonding interaction is covalent or partly covalent, as is apparent from the negative sign of the Laplacian of the electron density at the H Y BCP or at least the negative value of Hb. This ratio, similar to the Gb=Vb ratio, is thus a measure of the strength of the hydrogen bond and of the covalence of the interaction. The relationship between both ratios is depicted in Fig. 17.6 for a homogeneous sample of bonding interactions (empty squares correspond to related systems with OaH O and NaH O hydrogen bonds assisted by p-electron delocalization, the full square denotes the [FHF] system, and crosses correspond to other complexes, for example the water dimer). The linear correlation coe cient for the first class of systems is 0.988. Figure 17.6 also shows the regions of covalence, partial covalence, and of the weaker interactions for which both the Laplacian and Hb values are positive.

Figure 17.6 also shows that there is a correlation between topological data derived from the QTAIM and the energy data obtained from decomposition of the interaction energy. Hence the question arises of whether the topological data correlate with the energy components. The electronic potential energy density at the BCP correlates with the H Y distance for hydrogen bonded systems, because it was found for OaH O interactions that EHB A 12 Vb [54]. It follows that the high values of hydrogen bond energy and of the modulus of Vb are indicative of cova-

17.4 Relationships between the Topological and Energy Properties of Hydrogen Bonds 463

Fig. 17.7 Relationship between the exchange energy (kcal mol 1) and the potential energy density at the H Y BCP (Vb in au). Circles denote CAHB-DHBs and squares denote OH O and NH O.

lence because if jVbj > 2Gb then ‘2rb < 0. A negative value of the Laplacian indicates a concentration of electronic charge in the inter-atomic region.

Figure 17.7 shows the correlation between the exchange energy component and Vb. Two groups are considered:

1.OH O, NaH O, and OaH N (formamide and its derivatives, simple carboxylic acids); and

2.charge-assisted DHBs systems (H2OHþ HBeH complex and related species).

A strong correlation within each of these groups can be seen in the figure. This is connected with the strong exponential correlation between the exchange energy and the proton acceptor distance [55]. The exchange energy, similar to Vb, correlates well with the binding energy and with the H Y distance, the latter often reflects the strength of hydrogen bonding which (at the limit) is sometimes referred to as the ‘‘short strong hydrogen bond’’ (SSHB) [56].

Figure 17.8 shows the relationship between the delocalization energy and Vb, where again two groups of complexes are considered. The figure shows there is a strong correlation between the delocalization energy and Vb for both groups, as is revealed by the high regression coe cients.

Figure 17.9 shows the relationship between the electrostatic interaction energy term and Vb. It is very interesting that for the sample of systems in which there is strong p-electron delocalization (NH O and OH O hydrogen bonds in carboxylic acids and formamides) there is a linear relationship between both the topological and energy data. The situation is dramatically di erent for chargeassisted dihydrogen bonds for which interactions are very strong and H H distances are short. In such circumstances there is no correlation. The delocalization and exchange terms correlate with Vb for very strong interactions whereas the electrostatic term does not. This indicates that the electrostatic interaction, as op-

464 17 Relationships between QTAIM and the Decomposition of the Interaction Energy

Fig. 17.8 Relationship between the delocalization energy component (kcal mol 1) and the potential energy density at the H Y BCP (Vb in au). Circles denote CAHB-DHBs and squares denote RAHBs.

Fig. 17.9 Relationship between the electrostatic energy component (kcal mol 1) and the potential energy density at the H Y BCP (Vb in au) Circles denote CAHB-DHBs and squares denote RAHBs.

posed to the delocalization and the exchange terms, is not the ‘‘driving’’ interaction for very strong hydrogen bonds.

17.5

Various Other Interactions Related to Hydrogen Bonds

17.5.1

HB p Interactions

Figure 17.3 shows the range of dihydrogen bonds, from very strong chargeassisted DHBs to very weak bonds bordering with van der Waals interactions. Interactions such as p Hþ p in C2H2 Hþ C2H2 and C2H4 Hþ C2H2 complexes [57], which are related to hydrogen bonding, are also included in the fig-

17.5 Various Other Interactions Related to Hydrogen Bonds 465

Fig. 17.10 The Hþ p interaction for acetylene.

ure. For both complexes the proton is more firmly bound to one of two available p-electron systems. For C2H2 Hþ C2H2, the proton is closer to one of the acetylene molecules (Fig. 17.10); for C2H4 Hþ C2H2 the proton is closer to C2H4. In both complexes C2H2 Hþ and C2H4 Hþ can therefore be treated as protondonating systems. For the two short Hþ p contacts, the corresponding Lapla-

cians ð‘2rHþ pÞ are negative, indicating that the proton is covalently bonded to

the p-electron system. For the longer Hþ p contacts, the values of ‘2rHþ p are positive but the corresponding Hb values are negative, indicative of partly cova-

lent nature. Figure 17.3 shows that the Vb=Gb ratios for such complexes are extremely high, despite the longer Hþ p distances, both falling outside the main trend of the relationship.

It has also been found that the delocalization interaction energy is important in these complexes. For the C2H2 Hþ C2H2 complex the electrostatic, exchange, delocalization, and correlation energy terms are 24.5, 46.9, 33.2, and8.4 kcal mol 1, respectively. For the C2H4 Hþ C2H2 complex these interaction energy terms are 13.1, 19.9, 11.2 and 5.3 kcal mol 1, respectively.

Studies on p Hþ p bond complexes indicate these interactions can be regarded as very strong partly covalent hydrogen bonds. The binding energies for these complexes are 19.1 and 9.7 kcal mol 1 if the C2H2 Hþ and C2H4 Hþ moieties, respectively, are the proton donors to acetylene.

Summarizing, p Hþ p bond complexes are characterized by a relatively highVb=Gb and EDELðRÞ=EELð1Þ values which are unexpectedly high if the H Y distance is taken into account (for other hydrogen bonds with the similar H Y distances, these ratios are much smaller). It seems this is a common feature of the interactions between protons and p-electrons. For the T-shaped FH C2H2 complex and for the acetylene dimer, in both of which p-electrons are the proton acceptor, the H Y distance (where Y is the middle of a CcC triple bond) is equal to

46617 Relationships between QTAIM and the Decomposition of the Interaction Energy

2.186 and 2.697 A˚ , respectively, whereas the corresponding ratio EDELðRÞ=EELð1Þ is 0.45 and 0.25, respectively, and the Vb=Gb ratio is equal to 0.68 and 0.79, respectively.

17.5.2

Hydride Bonds

DHBs may be regarded as protic–hydric interactions, because the protic XaHþd bond contains a hydrogen atom carrying a positive charge whereas the hydric hydrogen ( dHaY) acts as the proton acceptor [58]. Rozas et al. [58] suggested that X dHaY systems are also possible in, for example, LiaH Li and BeaH Be. These authors named this interaction ‘‘inverse hydrogen bonding’’. It has also been suggested this interaction is called the ‘‘hydride bond’’ [59], because the term ‘‘inverse’’ is usually reserved for so-called blue-shift hydrogen bonds [60]. This type of interaction (BaH Naþ) has been observed in an experimental crystal structure [61]. The hydride bonding in the BeH2 Liþ, BeH2 Naþ, and BeH2 Mg2þ linear complexes, and the variation–perturbation partitioning of interaction energy, have recently been investigated by use of the QTAIM [59]. The binding energies calculated at the MP2/aug-cc-pVQZ level of theory for these systems are 13.3, 11.7, and 59.7 kcal mol 1, respectively. The delocalization energy term is the most important attractive term in these complexes. Similar results were obtained for agostic bonds, delocalization being responsible for stabilization of the systems. It is worth noting that the agostic bonds in CH4 LiNH2, CH4 NaNH2, and CH4 Naþ have weak-to-moderate binding energies ranging from 2 to 6 kcal mol 1 [59].

Figure 17.11 shows a classification of the interactions discussed. The arrows indicate electron transfer. One may expect that agostic bonding is a special kind of hydride bonding, because in the former the CaH (or SiaH) bond acts as a Lewis base and metal centre acts as a Lewis acid.

The direction of electronic charge transfer for numerous complexes bound by a variety of interactions (Fig. 17.11) has been analyzed at di erent levels of approximation. In one study [62] the CHelpG scheme was used to calculate the atomic charges. For the conventional OaH O hydrogen bond in the trans-linear water dimer, the transfer of electronic charge from acceptor to the donor amounts to 32 millielectrons (me) at the MP2/aug-cc-pVTZ//MP2/aug-cc-pVDZ level. More significant charge transfer of 205 me (from BeH2 to the hydronium ion) is observed in the strong charge-assisted dihydrogen bond of the H2OHþ HBeH complex at the MP2/aug-cc-pVDZ level. For hydride bonding in the HBeH Liþ system electron transfer to Liþ cation amounts to 41 me at the MP2/aug-cc-pVQZ level and for CH4 LiNH2, CH4 NaNH2, and CH4 Naþ transfer to the metal is 41, 31, and 52 me, respectively, at the MP2/aug-cc-pVDZ level. Only for the strongest (HBeH Mg2þ) interaction is Hb negative (at the MP2/aug-cc-pVTZ level) – Hb has positive values for the other complexes. This means that hydride bonds, even when very strong, cannot be classified as a covalent interaction.

17.6 Summary 467

Fig. 17.11 Classification of hydride bonds.

17.6 Summary

A wide range of interactions can be classified as hydrogen bonding. The proton acceptor distance (H Y) is an approximate measure of the strength of such interactions. The consensus in the literature is that short H Y distances imply covalent hydrogen bonding which turns gradually for longer distances into an essentially electrostatic interaction [18, 38, 39], the latter characterization is consistent with Pauling’s definition of the hydrogen bond [7]. This also agrees with the QTAIM, because for short H Y distances the electron density at the BCP is high, similar to that of covalent and polar bonds, and occasionally the Laplacian and/(or at least) Hb is negative. It has also been found that the Vb=Gb ratio increases with decreasing H Y distance. A similar relationship has been observed for the ratio of the delocalization and electrostatic interaction energy terms; this ratio increases for shorter proton–acceptor distances, indicative of greater importance of the delocalization energy for shorter distances. The latter relationship is valid for hydrogen bonds and sub-class of these interactions –

468 17 Relationships between QTAIM and the Decomposition of the Interaction Energy

dihydrogen bonds. In other words this is unique for Brønsted acid–Brønsted base interactions. For hydride bonds the delocalization interaction energy is the dominant attractive term. Topological data clearly show, however, that such interactions, despite of the significant binding energies, are not covalent, because the values of the Laplacian at the BCPs of Lewis acid–Lewis base contacts are positive. One may, therefore, expect that the increased covalence of hydrogen bonding connected with the increasing importance of the delocalization energy is the unique feature of this interaction.

Finally, DHBs are found to be a subclass of typical hydrogen bonds, ranging from very weak van der Waals-like interactions to very strong bordering on covalent bonds, as is also observed for hydrogen bonds.

References

1	C. F. Matta, J. Herna´ndez-Truillo,	13	S. J. Grabowski, W. A. Sokalski,
	T.-H. Tang, R. F. W. Bader, Chem.		J. Leszczynski, J. Phys. Chem. A 2005,
	Eur. J. 2003, 9, 1940–1951.		109, 4331–4341.
2	G. A. Je rey, W. Saenger, ‘‘Hydrogen	14	D. Cremer, E. Kraka, Angew. Chem.
	Bonding in Biological Structures’’,		Int. Ed. Engl. 1984, 23, 627–628.
	Springer, Berlin 1991.	15	M. Domagała, S. J. Grabowski, J. Phys.
3	G. A. Je rey, ‘‘An Introduction to		Chem. A 2005, 109, 5683–5688.
	Hydrogen Bonding’’, Oxford	16	O. Knop, K. N. Rankin, R. J. Boyd,
	University Press, New York, 1997.		J. Phys. Chem. A 2003, 107, 272–284.
4	G. R. Desiraju, T. Steiner, ‘‘The weak	17	K. N. Robertson, O. Knop, T. S.
	hydrogen bond in structural chemistry		Cameron, Can. J. Chem. 2003, 81,
	and biology’’, Oxford University		727–743.
	Press, Inc., New York, 1999.	18	G. R. Desiraju, Acc. Chem. Res. 2002,
5	S. Scheiner, ‘‘Hydrogen Bonding: A		35, 565–573.
	Theoretical Perspective’’, Oxford	19	R. Parthasarathi, V. Subramanian,
	University Press: New York, 1997.		N. Sathyamurthy, J. Phys. Chem. A
6	R. F. W. Bader, ‘‘Atoms in Molecules.		2006, 110, 3349–3351.
	A Quantum Theory’’, Oxford	20	S. J. Grabowski, A. J. Sadlej, W. A.
	University Press, New York, 1990.		Sokalski, J. Leszczynski, Chem. Phys.
7	L. Pauling, ‘‘The Nature of the		2006, 327, 151–158.
	Chemical Bond’’, Ithaca, NY: Cornell	21	S. Wojtulewski, S. J. Grabowski,
	University Press, Ithaca, New York,		J. Mol. Struct. 2002, 605, 235–240.
	third edition, 1960.	22	G. Pimentel, A. McClellan, ‘‘The
8	R. F. W. Bader, Chem. Rev. 1991, 91,		hydrogen bond’’, Freeman, San
	893–928.		Francisco, 1960.
9	M. S. Gordon, J. H. Jensen, Acc.	23	D. J. Suttor, J. Chem. Soc. 1963,
	Chem. Res. 1996, 29, 536–543.		1105–1110.
10	I. Rozas, I. Alkorta, J. Elguero, J. Am.	24	R. Taylor, O. Kennard, J. Am. Chem.
	Chem. Soc. 2000, 122, 11154–11161.		Soc. 1982, 104, 5063–5070.
11	T. K. Ghanty, V. N. Staroverov, P. R.	25	M. A. Viswamitra, R. Radhakrishnan,
	Koren, E. R. Davidson, J. Am. Chem.		J. Bandekar, G. R. Desiraju, J. Am.
	Soc. 2000, 122, 1210–1214.		Chem. Soc. 1993, 115, 4868–4869.
12	E. D. Isacs, A. Shukla, P. M.	26	J. M. A. Robinson, B. M. Kariuki, R. J.
	Platzman, D. R. Hamann, B.		Gough, K. D. M. Harris, D. Philp,
	Barbiellini, C. A. Tulk, Phys. Rev. Lett.		J. Solid State Chem. 1997, 134,
	1999, 82, 600–603.		203–206.

			References	469
	S. Harder, Chem. Eur. J. 1999, 5,
27		46	X. Fradera, M. A. Austen, R. F. W.
	1852–1861.		Bader, J. Phys. Chem. A 1999, 103,
28	S. J. Grabowski, J. Chem. Res. 1996,		304–314.
	534–535.	47	J. Poater, X. Fradera, M. Sola`,
29	M. Nishio, M. Hirota, Y. Umezawa,		M. Duran, S. Simon, Chem. Phys.
	‘‘The CH/p Interaction, Evidence,		Lett. 2003, 369, 248–255.
	Nature, and Consequences’’, Wiley–	48	S. F. Boys, F. Bernardi, Mol. Phys.
	VCH, 1998.		1970, 19, 553–561.
30	T. B. Richardson, S. de Gala, R. H.	49	A. Kaczmarek, A. J. Sadlej,
	Crabtree, J. Am. Chem. Soc. 1995, 117,		J. Leszczynski, Mol. Phys. 2006, 104,
	12875–12876.		395–407.
31	H. Cybulski, M. Pecul, J. Sadlej, T.	50	M. C. Daza, J. A. Dobado, J. M.
	Helgaker, J. Chem. Phys. 2003, 119,		Molina, J. Chem. Phys. 1999, 110,
	5094–5104.		11806–11813.
32	I. Rozas, I. Alkorta, J. Elguero, Chem.	51	D. Hugas, S. Simon, M. Duran,
	Phys. Lett. 1997, 275, 423–428.		Struct. Chem. 2005, 16, 257–263.
33	S. J. Grabowski, J. Phys. Chem. A	52	S. J. Grabowski, W. A. Sokalski,
	2000, 104, 5551–5557.		J. Leszczynski, J. Phys. Chem. A 2006,
34	S. J. Grabowski, J. Phys. Chem. A		110, 4772–4779.
	2001, 105, 10739–10746.	53	S. J. Grabowski, W. A. Sokalski,
35	S. J. Grabowski, T. L. Robinson,		E. Dyguda, J. Leszczynski, J. Phys.
	J. Leszczynski, Chem. Phys. Lett. 2004,		Chem. B 2006, 110, 6444–6446.
	386, 44–48.	54	E. Espinosa, E. Molins, C. Lecomte,
36	S. J. Grabowski, W. A. Sokalski,		Chem. Phys. Lett. 1998, 285, 170–
	J. Leszczynski, J. Phys. Chem. A 2004,		173.
	108, 5823–5830.	55	S. J. Grabowski, W. A. Sokalski,
37	E. Espinosa, I. Alkorta, J. Elguero,		J. Phys. Org. Chem. 2005, 18,
	E. Molins, J. Chem. Phys. 2002, 117,		779–784.
	5529–5542.	56	A. S. Mildvan, M. A. Massiah, T. K.
38	G. Gilli, F. Bellucci, V. Ferretti,		Harris, G. T. Marks, D. H. T.
	V. Bertolasi, J. Am. Chem. Soc. 1989,		Harrison, C. Viragh, P. M. Reddy,
	111, 1023–1028.		and I. M. Kovach, J. Mol. Struct. 2002,
39	P. Gilli, V. Bertolasi, V. Ferretti,		615, 163–175.
	G. Gilli, J. Am. Chem. Soc. 1994, 116,	57	S. J. Grabowski, W. A. Sokalski,
	909–915.		J. Leszczynski, J. Phys. Chem. A, 2004,
40	G. Gilli, P. Gilli, J. Mol. Struct. 2000,		108, 1806–1812.
	552, 1–15.	58	I. Rozas, I. Alkorta, J. Elguero,
41	H. Umeyama, K. Morokuma,		J. Phys. Chem. A 1997, 101, 4236–
	S. Yamabe, J. Am. Chem. Soc. 1977,		4244.
	99, 330–343.	59	S. J. Grabowski, W. A. Sokalski,
42	G. Chałasin´ski, M. M. Szcze˛s´niak,		J. Leszczynski, Chem. Phys. Lett. 2006,
	Chem. Rev. 1994, 94, 1723–1765.		422, 334–339.
43	W. A. Sokalski, S. Roszak, K. Pecul,	60	P. Hobza and Z. Havlas, Chem. Rev.
	Chem. Phys. Lett. 1988, 153, 153–159.		2000, 100, 4253–4264.
44	W. A. Sokalski, S. Roszak, J. Mol.	61	F. A. Cotton, J. H. Matonic, C. A.
	Struct. (Theochem) 1991, 234,		Murillo, J. Am. Chem. Soc. 1998, 120,
	387–400.		6047–6052.
45	S. M. Cybulski, G. Chałasin´ski,	62	C. M. Breneman, K. B. Wiberg,
	R. Moszyn´ski, J. Chem. Phys. 1990,		J. Comp. Chem. 1990, 11, 361–
	92, 4357–4363.		373.

Part V

Application to Biological Sciences and Drug Design