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Outline of a transition-state hydrogen-bond theory
Journal of Molecular Structure 790 (2006) 40–49 www.elsevier.com/locate/molstruc
Outline of a transition-state hydrogen-bond theory Paola Gilli *, Valerio Bertolasi, Loretta Pretto, Gastone Gilli Centro di Strutturistica Diffrattometrica and Dipartimento di Chimica, Universita` di Ferrara, Via L. Borsari 46, 44100 Ferrara, Italy Received 6 December 2005; received in revised form 18 January 2006; accepted 19 January 2006
Dedicated to Professor Dr.-Ing. Dr.h.c. Hartmut Fuess on the occasion of his 65th birthday
Abstract Though the H-bond is well characterized as a D–H/:A three-center-four-electron interaction, the formulation of a general H-bond theory has turned out to be a rather formidable problem because of the extreme variability of the bonds formed (for instance, O–H/O energies range from 0.1 to 31 kcal molK1). This paper surveys our previous contributions to the problem, including: (a) the H-bond chemical leitmotifs (CLs), showing that there are only four classes of strong H-bonds and one of moderately strong ones; (b) the PA/pKa equalization principle, showing that the four CLs forming strong H-bonds are actually molecular devices apt to equalize the acid–base properties (PA or pKa) of the H-bond donor and acceptor groups; (c) the driving variable of the H-bond strength, which remains so identified as the difference DpKaZpKAH(D–H)KpKBH(A–HC) or, alternatively, DPAZPA(DK)KPA(A); and, in particular, (d) the transition-state H-bond theory (TSHBT), which interprets the H-bond as a stationary point along the complete proton transfer pathway going from D–H/A to D/H–A via the D/H/A transition state. TSHBT is verified in connection with a series of seven 1-(X-phenylazo)-2-naphthols, a class of compounds forming a strong intramolecular resonance-assisted Hbond (RAHB), which is switched from N–H/O to N/H–O by the decreasing electron-withdrawing properties of the substituent X. The system is studied in terms of: (i) variable-temperature X-ray crystallography; (ii) DFT emulation of stationary points and full PT pathways; (iii) Marcus rateequilibrium analysis correlated with substituent LFER Hammett parameters. q 2006 Elsevier B.V. All rights reserved. Keywords: Hydrogen bond theory; Proton transfer; Solid-state tautomerism; Variable-temperature X-ray crystallography; Hydrogen bond chemical leitmotifs
1. Introduction 1.1. Chemical leitmotifs and PA/pKa equalization principle In the last 20 years, there has been a tremendous advancement of both experimental and computational methods used to study the hydrogen bond (H-bond) and, as a consequence, of the number of H-bonded systems investigated. The most important experimental improvements come from three sources: (i) neutron and X-ray diffraction at lowtemperatures, which allows more precise H-bond proton localization [1,2]; (ii) ion cyclotron resonance spectroscopy (ICRS) [3,4] and pulsed high-pressure mass spectrometry (PHPMS) [5,6], which provide accurate enthalpies of H-bonded complexes in the gas phase; (iii) crystallographic (CSD—Cambridge Structural Database) [7] and thermodynamic (NIST Chemistry WebBook) [8] databases, which * Corresponding author. Tel.: C39 0532 291141; fax: C39 0532 240709. E-mail address: [email protected] (P. Gilli).
0022-2860/$ - see front matter q 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.molstruc.2006.01.024
provide easy access to a complex and strongly intercorrelated literature. The systematic analysis of crystallographic and thermodynamic information leads to a new classification of the H-bond in terms of its strength, as derived from both H-bond energies or geometries. Roughly, H-bonds can be divided in three main groups [1]: (i) weak and dissymmetric H-bonds of electrostatic nature, which are by far the most numerous ones (more than 90%); (ii) strong and symmetric H-bonds, classifiable as threecenter-four-electron covalent bonds; (iii) moderate H-bonds, of intermediate nature. It is interesting that, when going to classify H-bonds according to their strength, we discover that all strong bonds correspond to a very small number of chemical schemes, that have been called the chemical leitmotifs (CLs) [9,10] and are summarized in Fig. 1. Strong and very strong bonds are exclusively associated with four CLs (CL #2: positive/negative charge-assisted; CL #3: negative charge-assisted; CL #4: positive charge-assisted H-bond; and CL #5: resonanceassisted or p-bond cooperative H-bond), which are generally indicated by the acronyms (G)CAHB, (K)CAHB, (C)CAHB and RAHB, respectively. CL #6 (PAHB, polarization-assisted
P. Gilli et al. / Journal of Molecular Structure 790 (2006) 40–49
41
Fig. 1. Summary of chemical patterns or H-bond chemical leitmotifs (CLs) able to transform weak H-bonds (CL #1: OHBs or ordinary H-bonds) into strong (CLs #25) or moderately strong (CL #6) H-bonds. The reason of this strengthening is reported in italic.
or s-bond cooperative H-bond) can only give rise to moderate H-bonds because of the scarce polarizability of s-bonds, while the leitmotiv #1 (OHBZordinary H-bond) collects the great majority of H-bonds known that, not being either chargeassisted or p- or s-bond cooperative, are weak and of electrostatic nature. This classification sets the purely taxonomic problem but does not tell us anything about the driving force able to transform weak, long, asymmetric and electrostatic H-bonds into strong, short, symmetric and covalent ones. In the past, there have been some important indications [3,5,11–16] that, for any specific D–H/:A bond, this driving force is to be sought for in the difference between the proton affinities (PAs) or related acid–base dissociation constants (pKa) of the H-bond donor (D–H) and acceptor (:A) moieties, that is DPAZ PA(DK)KPA(A) or DpKaZpKAH(D–H)KpKBH(A–HC). This hypothesis can be called the PA/pKa equalization principle and it has been already shown [17,18] that the reason why CLs #2–5 can produce such a dramatic H-bond strengthening is to be found in their ability of leveling out the normally very large differences between the PA or pKa values of the H-bond donor and acceptor moieties, so confirming the validity of the equalization rule for all cases of strong H-bond so far registered.
1.2. A new way of looking at the H-bond If the very driving force for H-bond strengthening lies in the PA/pKa equalization, it does not make much sense to study single spare R1–D–H/:A–R2 bonds but it seems much more reasonable and plausibly more conclusive to investigate sets of bonds formed by a same donor–acceptor D–H/A couple where the R1 and R2 substituents are changed in such a way to form a continuous series of DPA/DpKa values. In other words, any H-bond is much better studied as a point of minimum along the proton transfer (PT) pathway, which moves from D–H/A to D/H–A while DPA/DpKa is going to change. This is clearly the point of view normally adopted in the frame of the kinetic transition-state (or activated complex) theory [19–21] to treat any generic ACBC5ABCC reaction. Accordingly, any D–H/A H-bond can be considered as a D–H/A5D/H/A5D/H–A PT chemical reaction, which is bimolecular in both directions and proceeds via the D/H/A transition-state (TS) formation along a reaction pathway (the PT pathway) having two activation energies D‡E (the PT barriers in the two directions) and a reaction energy DEr (corresponding to the DPA or DpKa value) (Fig. 2).
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P. Gilli et al. / Journal of Molecular Structure 790 (2006) 40–49
investigated must belong to ‘a same coherent series’ [20,22], that is a series of compounds having a common intrinsic barrier and a formal reaction zone far removed from the substitution zone. 1.4. The choice of a suitable reaction: the ketohydrazone5azoenol system
Fig. 2. Any D–H/A bond can be considered as a stationary point of a bimolecular reaction pathway from D/H–A to D–H/A through the D/H/A transition state. D‡E1 and D‡E2 are the PT barriers in the two directions and DEr the reaction energy of the PT reaction, which also represents the PA (or pKa) difference between the H-bond donor and acceptor groups.
1.3. Criteria for choosing a suitable PT reaction Paradoxically, the most demanding operation is that of selecting a well-behaving PT reaction. Firstly, we must avoid very strong H-bonds because these are endowed with singlewell (SW) PT pathways (profiles 1 and 2 of Fig. 3(a)) and rather choose strong H-bonds because, for DPA or DpKa equal to zero, they can display the double-well (DW) symmetric (intrinsic) pathways shown by profiles 3 and 4 of Fig. 3(a). In this fortunate case, successive modifications produced by PA/pKa dissymmetrization will give rise to the complete set of PT profiles of Fig. 3(b). Accordingly, any well behaving R1–D–H/A–R2 PT system must have the following properties: (i) its intrinsic (i.e. symmetric) profile is double-well; (ii) changes of the electronic properties of one substituent (say R1) must be such to sweep the full range of properties from D–H/A to D/H–A; (iii) the substituent must be in such a position not to produce steric perturbation; (iv) all compounds
On the ground of previous studies [18,23,24] one of the H-bonded systems which appears to be more easily modulable by the effect of substituents is probably that connected with the N–H/O/N/H–O competition in ketohydrazone/azoenol derivatives. Simple ketohydrazones 1.I are known to form rather long N–H/O bonds with ˚ because the ketohydrazone N/O distances around 2.67 A tautomer is much more stable than the azoenol one in view of the higher PA of nitrogen with respect to oxygen. The second form, however, becomes the more stable after fusion of the H-bonded ring with a phenylene moiety (1.II) because the formation of the ketohydrazone tautomer would now require the loss of the large resonance energy of the aromatic ring. Accordingly, the azoenol 1.II is the form normally observed, with rather short N/O contact distances in the ˚ [25–28]. Fusion with a naphthalene ring, range 2.53–2.61 A having an intermediate resonance energy, leads to the most interesting situation 1.III of two roughly isoenergetic N–H/O and N/H–O tautomers, which can be tuned by changing the N-substituent (Scheme 1). Crystal data indicate a large prevalence of rather short N–H/O bonds ˚ ), [29] though two cases of N/ (2.50%d(N/O)%2.55 A H–O bond have also been reported for 1-(p-N,N-dimethyla˚ ) [24] minophenylazo)-2-naphthol (d(N/O)Z2.52–2.53 A ˚ ) [30]. and 1-(2-thioazolylazo)-2-naphthol (d(N/O)z2.56 A Quite recently, solid-state N–H/O5N/H–O dynamic disorder has been shown to occur by variable-temperature X-ray crystallography (VTXRC) in 1-(p-fluoro-phenylazo)-2naphthol and 1-(o-fluoro-phenylazo)-2-naphthol crystals [23] where energy differences of only 0.120–0.160 kcal molK1 between the two tautomers have been determined by van’t Hoff analysis of H-bond proton populations.
Fig. 3. (a) Possible shapes which may be adopted by a symmetric (intrinsic) PT reaction pathway (SW, single-well; DW, double-well); and (b) dissymmetrization of the symmetric (intrinsic) PT pathway 4 of (a) induced by increasing values of the reaction energy, DEr (sDW, symmetric DW; aDW, asymmetric DW; aSW, asymmetric SW).
P. Gilli et al. / Journal of Molecular Structure 790 (2006) 40–49 H N
O
H
N
N
O
N
N
H
N
O
H O
N
N
43
nearly invariant or randomly distributed populations, cannot follow any thermodynamic rule.
1.I ketohydrazone
1.II 1.IIIa
azoenol (azophenol)
1.IIIb
ketohydrazone (hydazonaphthone)
azoenol (azonaphthol)
X= p-Cl
All calculations were performed by DFT methods at the B3LYP/6-31CG(d,p)//B3LYP/6-31CG(d,p) level of theory using GAUSSIAN 98 [34] by (i) full geometry optimization of both N–H/O and N/H–O tautomers in the CS point group; (ii) TS location by the QST2 method [35]; and (iii) vibrational analysis for zero-point energy correction (ZPC) and check of the actual planarity of all molecules at their three stationary points.
X= p-F X= p-N(CH3)2
2.3. Data interpretation: the Marcus rate-equilibrium theory
Scheme 1.
X
X H N
H N
O
N
2.Ia
O
N
2.Ib
pNO2 mOM pH pCl pF pNM2
X= p-NO2 X= m-OCH3
pO
X= p-O
−
ketohydrazone (hydrazonaphthone)
2.2. Computationals
X= H
azoenol (azonaphthol)
Scheme 2.
In view of these considerations, we have decided to investigate the series of 1-(arylazo)-2-naphthols displayed in Scheme 2, where the phenyl substituents are chosen in such a way to change with continuity from electron-attracting to electron-donating properties [18]. The present paper surveys the results obtained in our laboratory for these compounds, whose structures have been investigated by VTXRC complemented by quantum-mechanical DFT emulation of their energies and geometries. Particular attention will be paid to the general interpretation of the results in the frame of the Marcus rate-equilibrium theory [31,32] and of the more general transition-state theory [19–21].
2. Methods 2.1. The VTXRC method VTXRC is a particularly useful tool to study H-bonds in potentially tautomeric systems. It is based on accurate crystal structure determinations at different temperatures (more often in the range 100–300 K) in association with full-matrix least squares refinement of the populations of the H-bonded proton (p and 1Kp), which are directly related to the equilibrium constant KZp/(1Kp) of the PT reaction. Further application of the van’t Hoff equation ln KZDS8/RKDH8/R(1/T) provides standard enthalpies (DH8) and entropies (DS8) of the tautomeric equilibrium. As far as we know, this method was firstly applied to H-bond studies by Destro in 1991 [33] and, beside providing the energy difference between the two minima of the DW PT potential, allows us to distinguish between static and dynamic disorder of the proton in tautomeric crystals, in the sense that only fast dynamic exchange at all the temperatures investigated guarantees the rapid reequilibration of the populations with the changing temperature which makes the van’t Hoff plot possible, at variance with the static disorder which, giving
The energetic and geometrical quantities evaluated by DFT methods have the following meaning. DE and DEZPC are the non-corrected and ZP-corrected energies of the stationary points relative to the TS chosen as zero. In terms of PT-reaction pathway (Fig. 4), they assume the meaning of the negative of the energy barriers, D‡E or D‡EZPC, for the PT process in the two directions, while the energy differences between the two minima, DEr or DEr,ZPC, that of reaction energies which, in the present series, are fairly similar before and after ZP correction because ZPC has the nearly constant value of 2.52(G11) kcal molK1 for all compounds investigated. RCZ[d(O–H)K d(N–H)] is the reaction coordinate for the PT process, while dZRCN–H/OKRCN/H–O is the total length of the PT pathway. It is advantageous to rescale RC to the relative reaction coordinate, rZRCKRCN/H–O (0%r%d), or to the fractional reaction coordinate, r/d (0%r/d%1). Results were interpreted in the frame of the Marcus rateequilibrium theory [21,22,31,32] where the PT pathway is represented by two harmonic oscillators which cross at the TS (where rZr‡) and have equations EN/H–OZ1/2kr2 and EN–H/O ZDErC1/2k(rKd)2. The theory makes use of the well-known
Fig. 4. Symbols for the energetic and geometric quantities computed by DFT methods and used for the subsequent Marcus analysis. D‡E and D‡EZPC are the non-corrected and ZP-corrected PT barriers in the two directions, DEr and DEr,ZPC the corresponding reaction energies, RC the reaction coordinate, defined as d(O–H)Kd(N–H), and d the total length of the PT pathway.
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P. Gilli et al. / Journal of Molecular Structure 790 (2006) 40–49
Marcus equations, that is D‡ E Z D‡ Eo C DEr =2 C ðDEr Þ2 =ð16D‡ Eo Þ
(1)
r ‡ =d Z DEr =ðkd 2 Þ C 1=2
(2)
k Z 8D‡ Eo =d 2
(3)
where D‡Eo is the intrinsic or symmetric barrier occurring when DErZ0. 3. Results 3.1. Analysis of VTXRC results Table 1 summarizes the crystallographic results obtained at the lowest temperature investigated, which is 100 K for the compounds studied in our laboratory (mOM, pCl, pF, pNM2) but 295 K for pNO2 and 213 K for pH which are taken from the literature [36,24]. The experimental structure of pOK is unknown and, for the sake of comparison, the DFT-emulated geometry is used. All molecules form remarkably strong ˚ . Data H-bonds with N/O distances in the range 2.52–2.56 A show that, with the progressive change of properties of the substituent X from electron-attracting to electron-donating (as measured by the mesomeric constant soR [37]), the H-bond changes from pure N–H/O to pure N/H–O through an intermediate region of dynamical (pCl and pF) or statical (pNM2) N–H/O5N/H–O tautomeric disorder. Two applications of the VTXRC method used for discriminating dynamic from static disorder are illustrated in Fig. 5. Fig. 5(a) shows the results obtained for the dynamically disordered 1-(p-fluorophenylazo)-2-naphthol (pF) [23] where the proton population ratio changes continuously from N–H/O:N/H–OZ0.64:0.36 at 100 K up to 0.54:0.46 at 295 K (see, for comparison, the difference Fourier maps computed in the H-bond region at these temperatures) and for which a reasonably linear van’t Hoff plot gives DH8ZK120(15) cal molK1 and DS8Z0.0(1) cal KK1 molK1. Conversely, Fig. 5(b) shows the different case of statical disorder in 1-(p-N,N-dimethylaminophenylazo)-2-naphthol
(pNM2) where the proton population ratio has a nearly constant value of 0.21:0.79 irrespectively of the temperature, as also indicated by the two difference Fourier maps shown. In summary, the intramolecular H-bond formed by the compounds investigated is seen to change significantly and continuously with the decreasing electronwithdrawing properties of the X substituent. pNO2, mOM and pH form ordered N–H/O bonds in agreement with the asymmetric SW profiles 7 or 8 of Fig. 3(b). pCl and pF form dynamically disordered N–H/O5N/H–O H-bonds with rather similar relative populations (z0.6:0.4) in a slightly asymmetrical (or nearly symmetrical) DW potential, as could be represented by the profile 5 of Fig. 3(b). A further increase of the electron donating properties of the substituent produces a large shift of the tautomeric equilibrium toward the N/H–O form in pNM2, whose populations become reversed with a ratio N–H/O:N/ H–O of 21:79 and the increased dissymmetrization of the DW PT pathway makes higher the PT barrier, so transforming the disorder from dynamic to static. This situation could be represented by the mirror image (i.e. reflected around the vertical zero line) of profile 6 of Fig. 3(b). We can only hypothesize from our DFT calculations that such a trend will continue in pOK, eventually producing the pure and ordered N/H–O bond in an asymmetric SW potential which is the mirror image of that found in pNO2. 3.2. DFT calculations and Marcus analysis Table 2 reports the DFT-computed values of the energy barriers corrected or less for the ZP energy, D‡EZPC and D‡E, reaction energies, DEr, and H-bond energies (DEHB: see below) for the compounds of Scheme 2. Complete list of all DFTcomputed energy and geometry data is available as Table 2 of Ref. [18]; for the aims of this paper, however, it is sufficient to discuss them in the graphical form illustrated in Fig. 6(a). The application of the Marcus method is seen to be successful in reducing the data of all individual molecules within the frame of a single reaction series having a common intrinsic barrier (black curve) and whose properties are continuously modulated by the substitution outside the formal reaction zone.
Table 1 Summary of the parameters derived from X-ray crystallography at the lowest temperature investigated Code
X
soR
T (K)
H-bond
d(N/O) ˚) (A
p(NH)%
pNO2 mOM pH pCl
p-NO2 m-OCH3 p-H p-Cl
0.17 – 0 K0.29
r.t. 100 213 100
2.56(1) 2.548(2) 2.553(1) 2.516(2)
100 100 100 69
pF
p-F
K0.40
100
2.535(2)
pNM2
p-N(CH3)2
K0.53
100
pOK
p-OK
K0.60
–
N–H/O N–H/O N–H/O N–H/O 5N/H–O N–H/O 5N/H–O N–H/O 5N/H–O N/H–O
p(OH)%
VTXRCPT profile
Order/ disorder
PT barrier
Ref.
0 0 0 31
aSW aSW aSW saDW
High High High Low
[36] [18] [24] [18]
64
36
zsDW
Low
[23]
2.534(2)
21
79
saDW
0
100
aSW
Medium– High High
[18]
(2.539)
Ordered Ordered Ordered Dynamically disordered Dynamically disordered Statically disordered Ordered (DFT computed)
[18]
Data for pOK are missing and those computed by DFT methods are used. PT profile: aSW, asymmetric single-well; saDW, slightly asymmetric double-well; zsDW, nearly symmetric double-well.
P. Gilli et al. / Journal of Molecular Structure 790 (2006) 40–49
45
Fig. 5. (a) Discrimination between static and dynamic disorder by VTXRC. Dynamic N–H/O5N/H–O proton disorder in pF as shown by the continuous proton population changes with the temperature and by the linearity of the van’t Hoff plot. Difference Fourier maps of pF at the temperatures of 100, 200 and 295 K, projected in the mean plane of the H-bonded ring, were computed after least-squares refinement carried out excluding the H-bond proton. Negative (dashed) and ˚ K3 intervals. Data from Ref. [23]. (b) Discrimination between static and dynamic disorder by VTXRC. Static N– positive (continuous) contours drawn at 0.04 e A H/O/N/H–O proton disorder in pNM2 as shown by the independence of proton populations from the experimental temperature. Difference Fourier maps of pNM2 at the temperatures of 100 and 295 K, projected in the mean plane of the H-bonded ring, were computed after least-squares refinement carried out excluding the H˚ K3 intervals. Data from Ref. [18]. bond proton. Negative (dashed) and positive (continuous) contours drawn at 0.04 e A
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P. Gilli et al. / Journal of Molecular Structure 790 (2006) 40–49
Table 2 ZP non-corrected, D‡E, and corrected, D‡EZPC, energy barriers, reaction energies, DEr, H-bond energies, DEHB, N/O contact distances, d(N/O), and relative positions of the TS, r‡/d, with respect to the total length, d, of the PT pathway as computed by DFT methods for the compounds of Scheme 2 Code
H-bond
D‡E (kcal molK1)
D‡EZPC (kcal molK1)
DEHB (kcal molK1)
d(N/O) ˚) (A
pNO2
N–H/O N/H–O DEr N–H/O N/H–O DEr N–H/O N/H–O DEr N–H/O N/H–O DEr N–H/O N/H–O DEr N–H/O N/H–O DEr N–H/O N/H–O DEr
3.83 2.23 K1.59 3.82 2.55 K1.26 3.69 2.58 K1.11 3.64 2.72 K0.91 3.33 2.93 K0.40 2.65 3.14 C0.49 1.44 2.78 C1.34
1.44 K0.12 K1.56 1.32 0.05 K1.27 0.99 0.11 K0.88 0.94 0.31 K0.63 0.65 0.36 K0.29 0.15 0.70 C0.55 K0.97 0.30 C1.28
K15.5 K13.9
2.555 2.529
K16.3 K15.0
2.557 2.534
K15.7 K14.6
2.555 2.535
K15.2 K14.3
2.550 2.536
K14.6 K14.2
2.546 2.538
K14.5 K15.0
2.541 2.541
K18.1 K19.4
2.527 2.539
mOM
pH
pCl
pF
pNM2
pO–
r‡/d
DFT-PT profile
0.438
aSW
0.451
aSW
0.458
aSW
0.468
saDW
0.477
zsDW
0.507
saDW
0.555
aSW
DEHB values are not ZP-corrected; corrected ones are smaller by 0.3–0.7 kcal molK1. PT profile: aSW, asymmetric single-well; saDW, slightly asymmetric doublewell; zsDW, nearly symmetric double-well.
The parameters of the intrinsic barrier found upon solving ˚ K2, harmonic the Marcus equations are: kZ15.0 kcal molK1 A ‡ K1 force constant; D EoZ3.08 kcal mol , intrinsic barrier; D‡E o,ZPCZ0.46 kcal mol K1, ZPC intrinsic barrier; dZ ˚ , total length of the PT pathway. The changes of the 1.283 A reaction energy, DEr, for the individual compounds are seen to induce parallel changes in the PT pathways (Fig. 6(a)) which are in agreement with the Leffler–Hammond rule [38,39], stating that: “the position of the TS is the more shifted towards the reagents (in this case N/H–O) the more the reaction is exoergonic (in the present case, the more DEr is negative)”. These progressive TS shifts are illustrated in Table 2 by the values of r‡/d, the relative positions of the TS with respect to the total length of the PT pathway. These results can be better represented by combining the information of the Marcus modeling with that coming from the DFT-computed H-bond energies, DEHB, calculated as energy differences between the closed, that is H-bonded, N/H–O form and the corresponding opened one obtained by 1808 rotation of the proton around the C–OH bond (Table 2). Fig. 6(b) shows what happens when these H-bond energies are added to the PT profiles of Fig. 6(a) (to notice that in the new figure some profiles, viz. pH, mOM and pF, are omitted to reduce the overcrowding without, however, loosing relevant information because of their intermediate positions among the other curves displayed). In this new representation the total energy of the system is now referred to a common hypothetical non-H-bonded reference state defined by setting equal to zero the energy of each open form. The horizontal lines mark the approximate vibrational level of the proton, estimated to have a nearly constant height of 2.52 kcal molK1
with respect to the DFT minima, as already indicated above. After applying this ZP correction, the energy heights of the stationary points of Fig. 6(b) are found to be in excellent agreement with and to account perfectly for the results of the diffraction experiments (Table 1). For example, in compounds forming pure N–H/O (pNO2) or N/H–O bonds (pOK) the vibrational level of the unstable form is higher than the PT barrier, while in the two tautomeric compounds (pCl and pNM2) both these levels lie slightly below it. This seems to indicate that the ZP-corrected PT barriers obtained by DFT methods at the present level of theory cannot be very far from the real ones. Finally, Table 2 and Fig. 6(b) show that the H-bond energies are very similar for all compounds investigated with the only exception of the pure N/H–O bond in pOK, a fact which will be further considered in the discussion.
4. Discussion and conclusions In three quite recent publications [17,18,23] we have tried to outline a novel method for the study of the H-bond in the perspective of the kinetic transition-state theory, a method which is based on the comparison of full H-bond PT profiles and is at variance with the traditional method of simply comparing ground-state properties for a number of H-bonds having different features. We have called this new method ‘transition-state H-bond theory’ [18] and, in this account, we have reviewed its application to the tautomeric N–H/O5N/H–O equilibrium in variously substituted 1(arylazo)-2-naphthols, and tried to show that it can provide a number of new ideas to help us to see the H-bond phenomenon
P. Gilli et al. / Journal of Molecular Structure 790 (2006) 40–49
(b)
(c)
Fig. 6. (a) Marcus modeling of the N/H–O/N–H/O PT reaction for all compounds of Table 2 (DFT-computed data not corrected for ZP). The two symmetric parabolas (in black) crossing at the intrinsic barrier D‡EoZ 3.08 kcal molK1 for r‡/dZ0.5 represent the intrinsic PT profile. Curves for the other compounds (in color) are shifted upwards or downwards by their respective reaction energies, DEr, and cross at r‡/d values larger or smaller than 0.5 with PT barriers D‡E higher or lower than D‡Eo, respectively. (b) Marcus modeling of (a) as modified by adding the values of the N/H–O bond energy, DEHB, to each individual curve. The black curve corresponds to the intrinsic H-bond with DErZ0 and D‡EZD‡Eo. Horizontal lines mark the approximate vibrational levels of the proton; full points indicate single-well and open symbols double-well H-bonds. Data from Ref. [18]. (For interpretation of the reference to colour in this legend, the reader is referred to the web version of this article.)
(d)
from a different point of view, ideas which can be shortly surveyed as follows: (a) The value of the intrinsic barrier found for this series of compounds is D‡EoZ3.08, which is reduced to D‡Eo,ZPCZ0.46 kcal molK1 by ZP correction. This value is rather small as could be expected for a D–H/A5D/H/A 5D/H–A reaction which proceeds via the D/H/A transition state, but where both reactants and products are pre-bound by the H-bond itself. This intrinsic barrier is low enough to allow H-bond dynamic disorder and therefore the formation of the socalled LBHBs [40–43] when DEr is close to zero. Any increase of DEr will make more and more dissymmetric the PT profile (Fig. 3(b)), causing a progressive increase of the PT barrier of the most stable tautomer which, in turn, originates the shift from dynamic to static disorder and, finally, the transition to the pure D–H/A or D/H–A bond having completely asymmetric SW profile (aSW). Though of not immediate application, this method of analysis seems to be the only one able to account for the complex behavior observed for these compounds in the solid state (Table 1). To
(e)
47
notice, finally, that the small value of D‡Eo,ZPCZ 0.46 kcal molK1 calculated for these compounds is such, anyway, to rule off the possibility of formation of the very short and strong SW H-bonds typical of some O/H/O RAHBs [17]. The harmonic constant of the parabolic expression EZ ˚ K2Z 1/2kr2 takes the value of kZ15.0 kcal molK1 A K1 K1 0.027 mdynZ0.027 N cm z75 cm . This force constant is some 250 times smaller than that associated with normal N–H or O–H stretchings (6–8 N cmK1) and can only be associated with the NH/O or N/HO stretching vibrations of the H-bond itself, in agreement with some early IR measurements which assign frequencies of 80– 250 cmK1 to these vibrations [44]. Accordingly, the properties of the N–H/O5N/H–O equilibrium can actually be described by two shallow nonbonded vibrations having such a small vibrational ˚ (the length of the PT constant, shifted by dZ1.283 A pathway) and intercrossing with an intrinsic PT barrier, D‡Eo,ZPC, as low as 0.46 kcal molK1. To notice that this barrier, having been calculated by MO quantum– mechanical methods, represents the best estimate of the true adiabatic barrier and does not require further corrections accounting for the TS resonance energy [45–48]. While the intrinsic profile is fully characterized by only two parameters (D‡Eo and k, or D‡Eo and d; see Eqs. (1– 3)), the reaction energy DEr is the independent variable acting as the thermodynamic driving force which modulates the shape of the PT pathway. The only point of it practically unaffected by DEr changes is the geometry of the TS, which displays nearly constant N/O ˚ associated with an almost complete distances of 2.38–2.40 A delocalization of the interleaving resonant fragment, suggesting that the TS H-bond can be interpreted as a nearly invariant three-center-four-electron covalent bond typical of the H-bonded system investigated [23,18]. Conversely, the DEr changes cause dramatic effects on the remaining parts of the PT pathway. While the intrinsic profile (DErZ0) has a perfectly symmetric DW potential, any increase (or decrease) of DEr splits the H-bond in two distinct bonds: (i) a less stable one which is shorter because closer to the TS and (ii) a more stable one which is longer because farther from the TS. This leads, in agreement with the Leffler–Hammond postulate [38,39], to the conclusion that the H-bond actually observed (the more stable) is always the longer in the tautomeric D–H/ A5D/H–A couple considered. The reaction energy DEr has the further physical meaning of being the difference of proton affinity, DPA, between the H-bond donor and acceptor groups. This PA difference cannot be either measured or calculated for intramolecular RAHBs, because of the large changes of p-delocalization occurring in the interleaving conjugated spacer (in this case the /HN–NaC–CaO/5/NaN–CaC–OH/ group) during H-bond formation, but can be tentatively evaluated making use of the extrathermodynamic LFER
48
P. Gilli et al. / Journal of Molecular Structure 790 (2006) 40–49
Fig. 7. Plot of the DFT-computed reaction energies, DEr, as a function of the mesomeric constant soR of the phenyl p-substituent [37]. Following Leffler and Grunwald [20], the two straight lines indicate that the two N–H/O/N/H–O and N/H–O/N–H/O reactions have different PT mechanisms. Full and open symbols refer to uncorrected and ZP-corrected DFT-computed data, respectively. Data from Ref. [18].
parameters of the changing substituent. In the present case, DEr is not found to correlate with the usual para and meta Hammett constants (sp and sm) but rather with the mesomeric constant, soR , most probably in view of the resonance-assisted nature of the H-bond treated. This correlation is shown in Fig. 7 for all the compounds of Scheme 2 except mOM, for which soR has no precise meaning. The correlation looks certainly impressive and such to support the idea that the driving force which modifies the PT profile is actually (all other factors remaining constant) the PA of the nitrogen as estimated by the soR value of the phenyl p-substituent. It appears, however, slightly different from what could have been expected, that is a unique straight line encompassing all compounds and not, as actually observed, two intercrossing lines with rather different slopes. Leffler and Grunwald, in their 1963 book ‘Rates and Equilibria of Organic Reactions’ [20] have considered the possibility of double-slope plots and interpreted them as characteristic of chemical reactions which change their mechanism beyond a certain value of s. This indicates that the two N–H/O/N/H–O and N/H–O/N–H/O reactions must have different features in the system presently considered, a fact that can be hypothetically accounted for in at least two non-mutually excluding ways: (i) The electronic effect of the p-phenyl substituent is felt in a different way by the N and by the O because 137°
3.Ia
H N 11 6° N
146° O
N N
Scheme 3.
H
O 6° 10
3.Ib
of the dissymmetric position of the phenyl; moreover, the effect felt by N appears to be larger when it behaves as acceptor (N/H–O) than when it acts as donor (N–H/O); (ii) It is well known that, for a same D/A distance, the energy of any D–H/A bond decreases steeply when the bond becomes less linear [49,50]. Accordingly, the fact that the pure N/H–O bond in pOK is calculated to be some 4.0 kcal molK1 stronger than any other N–H/O or N–H/O/N/ H–O bond of this series (19.4 against an average of 15.4 kcal molK1) can find a justification in the observation that the N–H–O angle is always more linear in the N/H–O (3.Ib) than in the N–H/O (3.Ia) tautomer (1468 against 1378) because of the different equilibrium values of the N–N–H (z1168) and C–O–H (z1068) angles (Scheme 3). These geometrical differences could then justify the change of mechanism occurring on the two sides of the intrinsic barrier (the horizontal line with DErZ0 in Fig. 7).
Acknowledgements We thank MIUR (Rome) for COFIN 2004 financial support in the frame of our project ‘Smart Hydrogen Bonds in Nature and in Functional Materials’. References [1] G.A. Jeffrey, An Introduction to Hydrogen Bonding, Oxford University Press, New York, 1997. [2] G.A. Jeffrey, W. Saenger, Hydrogen Bonding in Biological Structures, Springer, Berlin, 1991. [3] P. Kebarle, Annu. Rev. Phys. Chem. 28 (1977) 445. [4] T.B. McMahon, J.W. Larson, J. Am. Chem. Soc. 104 (1982) 5848. [5] M. Meot-Ner (Mautner), in: J.F. Liebman, A. Greenberg (Eds.), Molecular Structure and Energetics, vol. 4, VCH, Weinheim, 1987 (Chapter 3). [6] M. Meot-Ner (Mautner), Chem. Rev. 105 (2005) 213. [7] F.H. Allen, Acta Crystallogr., Sect. B 58 (2002) 380. [8] M. Meot-Ner (Mautner), S.G. Lias, Binding energies between ions and molecules and the thermochemistry of cluster ions, in: P.J. Linstrom, W.G. Mallard (Eds.), NIST Chemistry WebBook, NIST Standard Reference Database Number 69, National Institute of Standard and Technology, Gaithersburg, MD, June 2005 (http://webbook.nist.gov). [9] P. Gilli, V. Bertolasi, V. Ferretti, G. Gilli, J. Am. Chem. Soc. 116 (1994) 909. [10] G. Gilli, P. Gilli, J. Mol. Struct. 552 (2000) 1. [11] P.L. Huyskens, Th. Zeegers-Huyskens, J. Chim. Phys. Phys.–Chim. Biol. 61 (1964) 81. [12] Th. Zeegers-Huyskens, Chem. Phys. Lett. 129 (1986) 172. [13] Th. Zeegers-Huyskens, J. Mol. Struct. 177 (1988) 125. [14] Z. Malarski, M. Rospenk, L. Sobczyk, J. Phys. Chem. 86 (1982) 401. [15] B.S. Ault, G.G. Pimentel, J. Phys. Chem. 79 (1975) 615. [16] A.J. Barnes, J. Mol. Struct. 100 (1983) 259. [17] P. Gilli, V. Bertolasi, L. Pretto, V. Ferretti, G. Gilli, J. Am. Chem. Soc. 126 (2004) 3845. [18] P. Gilli, V. Bertolasi, L. Pretto, L. Antonov, G. Gilli, J. Am. Chem. Soc. 127 (2005) 4943. [19] S. Glasstone, K.J. Laidler, H. Eyring, The Theory of Rate Processes, McGraw Hill, New York, 1941.
P. Gilli et al. / Journal of Molecular Structure 790 (2006) 40–49 [20] J.E. Leffler, E. Grunwald, Rates and Equilibria of Organic Reactions, Wiley, New York, 1963. [21] S.S. Shaik, H.B. Schlegel, S. Wolfe, Theoretical Aspects of Physical Organic Chemistry. The SN2 Mechanism, Wiley, New York, 1992. [22] E. Grunwald, J. Am. Chem. Soc. 107 (1985) 125. [23] P. Gilli, V. Bertolasi, L. Pretto, A. Lycka, G. Gilli, J. Am. Chem. Soc. 124 (2002) 13554. [24] A.C. Olivieri, R.B. Wilson, I.C. Paul, D.Y. Curtin, J. Am. Chem. Soc. 111 (1989) 5525. [25] S. Isik, M. Aygun, H. Kocaokutgen, M.N. Tahir, O. Buyukgungor, A. Erdonmez, Acta Crystallogr., Sect. C 54 (1998) 859. [26] F. Jimenez-Cruz, G. Perez-Caballero, S. Hernandez-Ortega, M. RubioArroyo, Acta Crystallogr., Sect. C 56 (2000) 1028. [27] J. Pajak, M. Rospenk, R. Ramaekers, G. Maes, T. Glowiak, L. Sobczyk, Chem. Phys. 278 (2002) 89. [28] A. Filarowski, A. Koll, T. Glowiak, J. Chem. Soc., Perkin Trans. 2 (2002) 835. [29] P. Gilli, V. Bertolasi, V. Ferretti, G. Gilli, J. Am. Chem. Soc. 122 (2000) 10405. [30] M. Kurahashi, Bull. Chem. Soc. Jpn. 49 (1976) 2927. [31] R.A. Marcus, Discuss. Faraday Soc. 29 (1960) 21. [32] R.A. Marcus, J. Phys. Chem. 72 (1968) 891. [33] R. Destro, Chem. Phys. Lett. 181 (1991) 232. [34] M.J. Frisch, G.W. Trucks, H.B. Schlegel, G.E. Scuseria, M.A. Robb, J.R. Cheeseman, V.G. Zakrzewski, J.A. Montgomery Jr., R.E. Stratmann, J.C. Burant, S. Dapprich, J.M. Millam, A.D. Daniels, K.N. Kudin, M.C. Strain, O. Farkas, J. Tomasi, V. Barone, M. Cossi, R. Cammi, B. Mennucci, C. Pomelli, C. Adamo, S. Clifford, J. Ochterski, G.A. Petersson, P.Y. Ayala,
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Q. Cui, K. Morokuma, N. Rega, P. Salvador, J.J. Dannenberg, D.K. Malick, A.D. Rabuck, K. Raghavachari, J.B. Foresman, J. Cioslowski, J.V. Ortiz, A.G. Baboul, B.B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, I. Komaromi, R. Gomperts, R.L. Martin, D.J. Fox, T. Keith, M.A. AlLaham, C.Y. Peng, A. Nanayakkara, M. Challacombe, P.M.W. Gill, B. Johnson, W. Chen, M.W. Wong, J.L. Andres, C. Gonzalez, M. HeadGordon, E.S. Replogle, J.A. Pople, GAUSSIAN 98 (Revision A.11.3), Gaussian, Inc., Pittsburgh, PA, 2002. P.Y. Ayala, H.B. Schlegel, J. Chem. Phys. 107 (1997) 375. A. Whitaker, Z. Kristallogr. 152 (1980) 227. N.B. Chapman, J. Shorter (Eds.), Correlation Analysis in Chemistry, Plenum Press, New York, 1978 (Chapter 10). J.E. Leffler, Science 117 (1953) 340. G.S. Hammond, J. Am. Chem. Soc. 77 (1955) 334. W.W. Cleland, M.M. Kreevoy, Science 264 (1994) 1887. P.A. Frey, S.A. Whitt, J.B. Tobin, Science 264 (1994) 1927. T.K. Harris, A.S. Mildvan, Proteins 35 (1999) 275. P.A. Frey, Magn. Reson. Chem. 39 (2001) S190. S.N. Vinogradov, R.H. Linnel, Hydrogen Bonding, Van Nostrand Reinhold, New York, 1971. pp. 71–74. ˚ qvist, Faraday Discuss. 93 (1992) 225. A. Warshel, J.K. Hwang, J. A J.K. Hwang, G. King, S. Creighton, A. Warshel, J. Am. Chem. Soc. 110 (1988) 5297. C.N. Schutz, A. Warshel, J. Phys. Chem. B 108 (2004) 2066. ˚ qvist, A. Warshel, Chem. Rev. 93 (1993) 2523. J. A E.R. Lippincott, R. Schroeder, J. Chem. Phys. 23 (1955) 1099. R. Schroeder, E.R. Lippincott, J. Phys. Chem. 61 (1957) 921.
Outline of a transition-state hydrogen-bond theory Paola Gilli *, Valerio Bertolasi, Loretta Pretto, Gastone Gilli Centro di Strutturistica Diffrattometrica and Dipartimento di Chimica, Universita` di Ferrara, Via L. Borsari 46, 44100 Ferrara, Italy Received 6 December 2005; received in revised form 18 January 2006; accepted 19 January 2006
Dedicated to Professor Dr.-Ing. Dr.h.c. Hartmut Fuess on the occasion of his 65th birthday
Abstract Though the H-bond is well characterized as a D–H/:A three-center-four-electron interaction, the formulation of a general H-bond theory has turned out to be a rather formidable problem because of the extreme variability of the bonds formed (for instance, O–H/O energies range from 0.1 to 31 kcal molK1). This paper surveys our previous contributions to the problem, including: (a) the H-bond chemical leitmotifs (CLs), showing that there are only four classes of strong H-bonds and one of moderately strong ones; (b) the PA/pKa equalization principle, showing that the four CLs forming strong H-bonds are actually molecular devices apt to equalize the acid–base properties (PA or pKa) of the H-bond donor and acceptor groups; (c) the driving variable of the H-bond strength, which remains so identified as the difference DpKaZpKAH(D–H)KpKBH(A–HC) or, alternatively, DPAZPA(DK)KPA(A); and, in particular, (d) the transition-state H-bond theory (TSHBT), which interprets the H-bond as a stationary point along the complete proton transfer pathway going from D–H/A to D/H–A via the D/H/A transition state. TSHBT is verified in connection with a series of seven 1-(X-phenylazo)-2-naphthols, a class of compounds forming a strong intramolecular resonance-assisted Hbond (RAHB), which is switched from N–H/O to N/H–O by the decreasing electron-withdrawing properties of the substituent X. The system is studied in terms of: (i) variable-temperature X-ray crystallography; (ii) DFT emulation of stationary points and full PT pathways; (iii) Marcus rateequilibrium analysis correlated with substituent LFER Hammett parameters. q 2006 Elsevier B.V. All rights reserved. Keywords: Hydrogen bond theory; Proton transfer; Solid-state tautomerism; Variable-temperature X-ray crystallography; Hydrogen bond chemical leitmotifs
1. Introduction 1.1. Chemical leitmotifs and PA/pKa equalization principle In the last 20 years, there has been a tremendous advancement of both experimental and computational methods used to study the hydrogen bond (H-bond) and, as a consequence, of the number of H-bonded systems investigated. The most important experimental improvements come from three sources: (i) neutron and X-ray diffraction at lowtemperatures, which allows more precise H-bond proton localization [1,2]; (ii) ion cyclotron resonance spectroscopy (ICRS) [3,4] and pulsed high-pressure mass spectrometry (PHPMS) [5,6], which provide accurate enthalpies of H-bonded complexes in the gas phase; (iii) crystallographic (CSD—Cambridge Structural Database) [7] and thermodynamic (NIST Chemistry WebBook) [8] databases, which * Corresponding author. Tel.: C39 0532 291141; fax: C39 0532 240709. E-mail address: [email protected] (P. Gilli).
0022-2860/$ - see front matter q 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.molstruc.2006.01.024
provide easy access to a complex and strongly intercorrelated literature. The systematic analysis of crystallographic and thermodynamic information leads to a new classification of the H-bond in terms of its strength, as derived from both H-bond energies or geometries. Roughly, H-bonds can be divided in three main groups [1]: (i) weak and dissymmetric H-bonds of electrostatic nature, which are by far the most numerous ones (more than 90%); (ii) strong and symmetric H-bonds, classifiable as threecenter-four-electron covalent bonds; (iii) moderate H-bonds, of intermediate nature. It is interesting that, when going to classify H-bonds according to their strength, we discover that all strong bonds correspond to a very small number of chemical schemes, that have been called the chemical leitmotifs (CLs) [9,10] and are summarized in Fig. 1. Strong and very strong bonds are exclusively associated with four CLs (CL #2: positive/negative charge-assisted; CL #3: negative charge-assisted; CL #4: positive charge-assisted H-bond; and CL #5: resonanceassisted or p-bond cooperative H-bond), which are generally indicated by the acronyms (G)CAHB, (K)CAHB, (C)CAHB and RAHB, respectively. CL #6 (PAHB, polarization-assisted
P. Gilli et al. / Journal of Molecular Structure 790 (2006) 40–49
41
Fig. 1. Summary of chemical patterns or H-bond chemical leitmotifs (CLs) able to transform weak H-bonds (CL #1: OHBs or ordinary H-bonds) into strong (CLs #25) or moderately strong (CL #6) H-bonds. The reason of this strengthening is reported in italic.
or s-bond cooperative H-bond) can only give rise to moderate H-bonds because of the scarce polarizability of s-bonds, while the leitmotiv #1 (OHBZordinary H-bond) collects the great majority of H-bonds known that, not being either chargeassisted or p- or s-bond cooperative, are weak and of electrostatic nature. This classification sets the purely taxonomic problem but does not tell us anything about the driving force able to transform weak, long, asymmetric and electrostatic H-bonds into strong, short, symmetric and covalent ones. In the past, there have been some important indications [3,5,11–16] that, for any specific D–H/:A bond, this driving force is to be sought for in the difference between the proton affinities (PAs) or related acid–base dissociation constants (pKa) of the H-bond donor (D–H) and acceptor (:A) moieties, that is DPAZ PA(DK)KPA(A) or DpKaZpKAH(D–H)KpKBH(A–HC). This hypothesis can be called the PA/pKa equalization principle and it has been already shown [17,18] that the reason why CLs #2–5 can produce such a dramatic H-bond strengthening is to be found in their ability of leveling out the normally very large differences between the PA or pKa values of the H-bond donor and acceptor moieties, so confirming the validity of the equalization rule for all cases of strong H-bond so far registered.
1.2. A new way of looking at the H-bond If the very driving force for H-bond strengthening lies in the PA/pKa equalization, it does not make much sense to study single spare R1–D–H/:A–R2 bonds but it seems much more reasonable and plausibly more conclusive to investigate sets of bonds formed by a same donor–acceptor D–H/A couple where the R1 and R2 substituents are changed in such a way to form a continuous series of DPA/DpKa values. In other words, any H-bond is much better studied as a point of minimum along the proton transfer (PT) pathway, which moves from D–H/A to D/H–A while DPA/DpKa is going to change. This is clearly the point of view normally adopted in the frame of the kinetic transition-state (or activated complex) theory [19–21] to treat any generic ACBC5ABCC reaction. Accordingly, any D–H/A H-bond can be considered as a D–H/A5D/H/A5D/H–A PT chemical reaction, which is bimolecular in both directions and proceeds via the D/H/A transition-state (TS) formation along a reaction pathway (the PT pathway) having two activation energies D‡E (the PT barriers in the two directions) and a reaction energy DEr (corresponding to the DPA or DpKa value) (Fig. 2).
42
P. Gilli et al. / Journal of Molecular Structure 790 (2006) 40–49
investigated must belong to ‘a same coherent series’ [20,22], that is a series of compounds having a common intrinsic barrier and a formal reaction zone far removed from the substitution zone. 1.4. The choice of a suitable reaction: the ketohydrazone5azoenol system
Fig. 2. Any D–H/A bond can be considered as a stationary point of a bimolecular reaction pathway from D/H–A to D–H/A through the D/H/A transition state. D‡E1 and D‡E2 are the PT barriers in the two directions and DEr the reaction energy of the PT reaction, which also represents the PA (or pKa) difference between the H-bond donor and acceptor groups.
1.3. Criteria for choosing a suitable PT reaction Paradoxically, the most demanding operation is that of selecting a well-behaving PT reaction. Firstly, we must avoid very strong H-bonds because these are endowed with singlewell (SW) PT pathways (profiles 1 and 2 of Fig. 3(a)) and rather choose strong H-bonds because, for DPA or DpKa equal to zero, they can display the double-well (DW) symmetric (intrinsic) pathways shown by profiles 3 and 4 of Fig. 3(a). In this fortunate case, successive modifications produced by PA/pKa dissymmetrization will give rise to the complete set of PT profiles of Fig. 3(b). Accordingly, any well behaving R1–D–H/A–R2 PT system must have the following properties: (i) its intrinsic (i.e. symmetric) profile is double-well; (ii) changes of the electronic properties of one substituent (say R1) must be such to sweep the full range of properties from D–H/A to D/H–A; (iii) the substituent must be in such a position not to produce steric perturbation; (iv) all compounds
On the ground of previous studies [18,23,24] one of the H-bonded systems which appears to be more easily modulable by the effect of substituents is probably that connected with the N–H/O/N/H–O competition in ketohydrazone/azoenol derivatives. Simple ketohydrazones 1.I are known to form rather long N–H/O bonds with ˚ because the ketohydrazone N/O distances around 2.67 A tautomer is much more stable than the azoenol one in view of the higher PA of nitrogen with respect to oxygen. The second form, however, becomes the more stable after fusion of the H-bonded ring with a phenylene moiety (1.II) because the formation of the ketohydrazone tautomer would now require the loss of the large resonance energy of the aromatic ring. Accordingly, the azoenol 1.II is the form normally observed, with rather short N/O contact distances in the ˚ [25–28]. Fusion with a naphthalene ring, range 2.53–2.61 A having an intermediate resonance energy, leads to the most interesting situation 1.III of two roughly isoenergetic N–H/O and N/H–O tautomers, which can be tuned by changing the N-substituent (Scheme 1). Crystal data indicate a large prevalence of rather short N–H/O bonds ˚ ), [29] though two cases of N/ (2.50%d(N/O)%2.55 A H–O bond have also been reported for 1-(p-N,N-dimethyla˚ ) [24] minophenylazo)-2-naphthol (d(N/O)Z2.52–2.53 A ˚ ) [30]. and 1-(2-thioazolylazo)-2-naphthol (d(N/O)z2.56 A Quite recently, solid-state N–H/O5N/H–O dynamic disorder has been shown to occur by variable-temperature X-ray crystallography (VTXRC) in 1-(p-fluoro-phenylazo)-2naphthol and 1-(o-fluoro-phenylazo)-2-naphthol crystals [23] where energy differences of only 0.120–0.160 kcal molK1 between the two tautomers have been determined by van’t Hoff analysis of H-bond proton populations.
Fig. 3. (a) Possible shapes which may be adopted by a symmetric (intrinsic) PT reaction pathway (SW, single-well; DW, double-well); and (b) dissymmetrization of the symmetric (intrinsic) PT pathway 4 of (a) induced by increasing values of the reaction energy, DEr (sDW, symmetric DW; aDW, asymmetric DW; aSW, asymmetric SW).
P. Gilli et al. / Journal of Molecular Structure 790 (2006) 40–49 H N
O
H
N
N
O
N
N
H
N
O
H O
N
N
43
nearly invariant or randomly distributed populations, cannot follow any thermodynamic rule.
1.I ketohydrazone
1.II 1.IIIa
azoenol (azophenol)
1.IIIb
ketohydrazone (hydazonaphthone)
azoenol (azonaphthol)
X= p-Cl
All calculations were performed by DFT methods at the B3LYP/6-31CG(d,p)//B3LYP/6-31CG(d,p) level of theory using GAUSSIAN 98 [34] by (i) full geometry optimization of both N–H/O and N/H–O tautomers in the CS point group; (ii) TS location by the QST2 method [35]; and (iii) vibrational analysis for zero-point energy correction (ZPC) and check of the actual planarity of all molecules at their three stationary points.
X= p-F X= p-N(CH3)2
2.3. Data interpretation: the Marcus rate-equilibrium theory
Scheme 1.
X
X H N
H N
O
N
2.Ia
O
N
2.Ib
pNO2 mOM pH pCl pF pNM2
X= p-NO2 X= m-OCH3
pO
X= p-O
−
ketohydrazone (hydrazonaphthone)
2.2. Computationals
X= H
azoenol (azonaphthol)
Scheme 2.
In view of these considerations, we have decided to investigate the series of 1-(arylazo)-2-naphthols displayed in Scheme 2, where the phenyl substituents are chosen in such a way to change with continuity from electron-attracting to electron-donating properties [18]. The present paper surveys the results obtained in our laboratory for these compounds, whose structures have been investigated by VTXRC complemented by quantum-mechanical DFT emulation of their energies and geometries. Particular attention will be paid to the general interpretation of the results in the frame of the Marcus rate-equilibrium theory [31,32] and of the more general transition-state theory [19–21].
2. Methods 2.1. The VTXRC method VTXRC is a particularly useful tool to study H-bonds in potentially tautomeric systems. It is based on accurate crystal structure determinations at different temperatures (more often in the range 100–300 K) in association with full-matrix least squares refinement of the populations of the H-bonded proton (p and 1Kp), which are directly related to the equilibrium constant KZp/(1Kp) of the PT reaction. Further application of the van’t Hoff equation ln KZDS8/RKDH8/R(1/T) provides standard enthalpies (DH8) and entropies (DS8) of the tautomeric equilibrium. As far as we know, this method was firstly applied to H-bond studies by Destro in 1991 [33] and, beside providing the energy difference between the two minima of the DW PT potential, allows us to distinguish between static and dynamic disorder of the proton in tautomeric crystals, in the sense that only fast dynamic exchange at all the temperatures investigated guarantees the rapid reequilibration of the populations with the changing temperature which makes the van’t Hoff plot possible, at variance with the static disorder which, giving
The energetic and geometrical quantities evaluated by DFT methods have the following meaning. DE and DEZPC are the non-corrected and ZP-corrected energies of the stationary points relative to the TS chosen as zero. In terms of PT-reaction pathway (Fig. 4), they assume the meaning of the negative of the energy barriers, D‡E or D‡EZPC, for the PT process in the two directions, while the energy differences between the two minima, DEr or DEr,ZPC, that of reaction energies which, in the present series, are fairly similar before and after ZP correction because ZPC has the nearly constant value of 2.52(G11) kcal molK1 for all compounds investigated. RCZ[d(O–H)K d(N–H)] is the reaction coordinate for the PT process, while dZRCN–H/OKRCN/H–O is the total length of the PT pathway. It is advantageous to rescale RC to the relative reaction coordinate, rZRCKRCN/H–O (0%r%d), or to the fractional reaction coordinate, r/d (0%r/d%1). Results were interpreted in the frame of the Marcus rateequilibrium theory [21,22,31,32] where the PT pathway is represented by two harmonic oscillators which cross at the TS (where rZr‡) and have equations EN/H–OZ1/2kr2 and EN–H/O ZDErC1/2k(rKd)2. The theory makes use of the well-known
Fig. 4. Symbols for the energetic and geometric quantities computed by DFT methods and used for the subsequent Marcus analysis. D‡E and D‡EZPC are the non-corrected and ZP-corrected PT barriers in the two directions, DEr and DEr,ZPC the corresponding reaction energies, RC the reaction coordinate, defined as d(O–H)Kd(N–H), and d the total length of the PT pathway.
44
P. Gilli et al. / Journal of Molecular Structure 790 (2006) 40–49
Marcus equations, that is D‡ E Z D‡ Eo C DEr =2 C ðDEr Þ2 =ð16D‡ Eo Þ
(1)
r ‡ =d Z DEr =ðkd 2 Þ C 1=2
(2)
k Z 8D‡ Eo =d 2
(3)
where D‡Eo is the intrinsic or symmetric barrier occurring when DErZ0. 3. Results 3.1. Analysis of VTXRC results Table 1 summarizes the crystallographic results obtained at the lowest temperature investigated, which is 100 K for the compounds studied in our laboratory (mOM, pCl, pF, pNM2) but 295 K for pNO2 and 213 K for pH which are taken from the literature [36,24]. The experimental structure of pOK is unknown and, for the sake of comparison, the DFT-emulated geometry is used. All molecules form remarkably strong ˚ . Data H-bonds with N/O distances in the range 2.52–2.56 A show that, with the progressive change of properties of the substituent X from electron-attracting to electron-donating (as measured by the mesomeric constant soR [37]), the H-bond changes from pure N–H/O to pure N/H–O through an intermediate region of dynamical (pCl and pF) or statical (pNM2) N–H/O5N/H–O tautomeric disorder. Two applications of the VTXRC method used for discriminating dynamic from static disorder are illustrated in Fig. 5. Fig. 5(a) shows the results obtained for the dynamically disordered 1-(p-fluorophenylazo)-2-naphthol (pF) [23] where the proton population ratio changes continuously from N–H/O:N/H–OZ0.64:0.36 at 100 K up to 0.54:0.46 at 295 K (see, for comparison, the difference Fourier maps computed in the H-bond region at these temperatures) and for which a reasonably linear van’t Hoff plot gives DH8ZK120(15) cal molK1 and DS8Z0.0(1) cal KK1 molK1. Conversely, Fig. 5(b) shows the different case of statical disorder in 1-(p-N,N-dimethylaminophenylazo)-2-naphthol
(pNM2) where the proton population ratio has a nearly constant value of 0.21:0.79 irrespectively of the temperature, as also indicated by the two difference Fourier maps shown. In summary, the intramolecular H-bond formed by the compounds investigated is seen to change significantly and continuously with the decreasing electronwithdrawing properties of the X substituent. pNO2, mOM and pH form ordered N–H/O bonds in agreement with the asymmetric SW profiles 7 or 8 of Fig. 3(b). pCl and pF form dynamically disordered N–H/O5N/H–O H-bonds with rather similar relative populations (z0.6:0.4) in a slightly asymmetrical (or nearly symmetrical) DW potential, as could be represented by the profile 5 of Fig. 3(b). A further increase of the electron donating properties of the substituent produces a large shift of the tautomeric equilibrium toward the N/H–O form in pNM2, whose populations become reversed with a ratio N–H/O:N/ H–O of 21:79 and the increased dissymmetrization of the DW PT pathway makes higher the PT barrier, so transforming the disorder from dynamic to static. This situation could be represented by the mirror image (i.e. reflected around the vertical zero line) of profile 6 of Fig. 3(b). We can only hypothesize from our DFT calculations that such a trend will continue in pOK, eventually producing the pure and ordered N/H–O bond in an asymmetric SW potential which is the mirror image of that found in pNO2. 3.2. DFT calculations and Marcus analysis Table 2 reports the DFT-computed values of the energy barriers corrected or less for the ZP energy, D‡EZPC and D‡E, reaction energies, DEr, and H-bond energies (DEHB: see below) for the compounds of Scheme 2. Complete list of all DFTcomputed energy and geometry data is available as Table 2 of Ref. [18]; for the aims of this paper, however, it is sufficient to discuss them in the graphical form illustrated in Fig. 6(a). The application of the Marcus method is seen to be successful in reducing the data of all individual molecules within the frame of a single reaction series having a common intrinsic barrier (black curve) and whose properties are continuously modulated by the substitution outside the formal reaction zone.
Table 1 Summary of the parameters derived from X-ray crystallography at the lowest temperature investigated Code
X
soR
T (K)
H-bond
d(N/O) ˚) (A
p(NH)%
pNO2 mOM pH pCl
p-NO2 m-OCH3 p-H p-Cl
0.17 – 0 K0.29
r.t. 100 213 100
2.56(1) 2.548(2) 2.553(1) 2.516(2)
100 100 100 69
pF
p-F
K0.40
100
2.535(2)
pNM2
p-N(CH3)2
K0.53
100
pOK
p-OK
K0.60
–
N–H/O N–H/O N–H/O N–H/O 5N/H–O N–H/O 5N/H–O N–H/O 5N/H–O N/H–O
p(OH)%
VTXRCPT profile
Order/ disorder
PT barrier
Ref.
0 0 0 31
aSW aSW aSW saDW
High High High Low
[36] [18] [24] [18]
64
36
zsDW
Low
[23]
2.534(2)
21
79
saDW
0
100
aSW
Medium– High High
[18]
(2.539)
Ordered Ordered Ordered Dynamically disordered Dynamically disordered Statically disordered Ordered (DFT computed)
[18]
Data for pOK are missing and those computed by DFT methods are used. PT profile: aSW, asymmetric single-well; saDW, slightly asymmetric double-well; zsDW, nearly symmetric double-well.
P. Gilli et al. / Journal of Molecular Structure 790 (2006) 40–49
45
Fig. 5. (a) Discrimination between static and dynamic disorder by VTXRC. Dynamic N–H/O5N/H–O proton disorder in pF as shown by the continuous proton population changes with the temperature and by the linearity of the van’t Hoff plot. Difference Fourier maps of pF at the temperatures of 100, 200 and 295 K, projected in the mean plane of the H-bonded ring, were computed after least-squares refinement carried out excluding the H-bond proton. Negative (dashed) and ˚ K3 intervals. Data from Ref. [23]. (b) Discrimination between static and dynamic disorder by VTXRC. Static N– positive (continuous) contours drawn at 0.04 e A H/O/N/H–O proton disorder in pNM2 as shown by the independence of proton populations from the experimental temperature. Difference Fourier maps of pNM2 at the temperatures of 100 and 295 K, projected in the mean plane of the H-bonded ring, were computed after least-squares refinement carried out excluding the H˚ K3 intervals. Data from Ref. [18]. bond proton. Negative (dashed) and positive (continuous) contours drawn at 0.04 e A
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P. Gilli et al. / Journal of Molecular Structure 790 (2006) 40–49
Table 2 ZP non-corrected, D‡E, and corrected, D‡EZPC, energy barriers, reaction energies, DEr, H-bond energies, DEHB, N/O contact distances, d(N/O), and relative positions of the TS, r‡/d, with respect to the total length, d, of the PT pathway as computed by DFT methods for the compounds of Scheme 2 Code
H-bond
D‡E (kcal molK1)
D‡EZPC (kcal molK1)
DEHB (kcal molK1)
d(N/O) ˚) (A
pNO2
N–H/O N/H–O DEr N–H/O N/H–O DEr N–H/O N/H–O DEr N–H/O N/H–O DEr N–H/O N/H–O DEr N–H/O N/H–O DEr N–H/O N/H–O DEr
3.83 2.23 K1.59 3.82 2.55 K1.26 3.69 2.58 K1.11 3.64 2.72 K0.91 3.33 2.93 K0.40 2.65 3.14 C0.49 1.44 2.78 C1.34
1.44 K0.12 K1.56 1.32 0.05 K1.27 0.99 0.11 K0.88 0.94 0.31 K0.63 0.65 0.36 K0.29 0.15 0.70 C0.55 K0.97 0.30 C1.28
K15.5 K13.9
2.555 2.529
K16.3 K15.0
2.557 2.534
K15.7 K14.6
2.555 2.535
K15.2 K14.3
2.550 2.536
K14.6 K14.2
2.546 2.538
K14.5 K15.0
2.541 2.541
K18.1 K19.4
2.527 2.539
mOM
pH
pCl
pF
pNM2
pO–
r‡/d
DFT-PT profile
0.438
aSW
0.451
aSW
0.458
aSW
0.468
saDW
0.477
zsDW
0.507
saDW
0.555
aSW
DEHB values are not ZP-corrected; corrected ones are smaller by 0.3–0.7 kcal molK1. PT profile: aSW, asymmetric single-well; saDW, slightly asymmetric doublewell; zsDW, nearly symmetric double-well.
The parameters of the intrinsic barrier found upon solving ˚ K2, harmonic the Marcus equations are: kZ15.0 kcal molK1 A ‡ K1 force constant; D EoZ3.08 kcal mol , intrinsic barrier; D‡E o,ZPCZ0.46 kcal mol K1, ZPC intrinsic barrier; dZ ˚ , total length of the PT pathway. The changes of the 1.283 A reaction energy, DEr, for the individual compounds are seen to induce parallel changes in the PT pathways (Fig. 6(a)) which are in agreement with the Leffler–Hammond rule [38,39], stating that: “the position of the TS is the more shifted towards the reagents (in this case N/H–O) the more the reaction is exoergonic (in the present case, the more DEr is negative)”. These progressive TS shifts are illustrated in Table 2 by the values of r‡/d, the relative positions of the TS with respect to the total length of the PT pathway. These results can be better represented by combining the information of the Marcus modeling with that coming from the DFT-computed H-bond energies, DEHB, calculated as energy differences between the closed, that is H-bonded, N/H–O form and the corresponding opened one obtained by 1808 rotation of the proton around the C–OH bond (Table 2). Fig. 6(b) shows what happens when these H-bond energies are added to the PT profiles of Fig. 6(a) (to notice that in the new figure some profiles, viz. pH, mOM and pF, are omitted to reduce the overcrowding without, however, loosing relevant information because of their intermediate positions among the other curves displayed). In this new representation the total energy of the system is now referred to a common hypothetical non-H-bonded reference state defined by setting equal to zero the energy of each open form. The horizontal lines mark the approximate vibrational level of the proton, estimated to have a nearly constant height of 2.52 kcal molK1
with respect to the DFT minima, as already indicated above. After applying this ZP correction, the energy heights of the stationary points of Fig. 6(b) are found to be in excellent agreement with and to account perfectly for the results of the diffraction experiments (Table 1). For example, in compounds forming pure N–H/O (pNO2) or N/H–O bonds (pOK) the vibrational level of the unstable form is higher than the PT barrier, while in the two tautomeric compounds (pCl and pNM2) both these levels lie slightly below it. This seems to indicate that the ZP-corrected PT barriers obtained by DFT methods at the present level of theory cannot be very far from the real ones. Finally, Table 2 and Fig. 6(b) show that the H-bond energies are very similar for all compounds investigated with the only exception of the pure N/H–O bond in pOK, a fact which will be further considered in the discussion.
4. Discussion and conclusions In three quite recent publications [17,18,23] we have tried to outline a novel method for the study of the H-bond in the perspective of the kinetic transition-state theory, a method which is based on the comparison of full H-bond PT profiles and is at variance with the traditional method of simply comparing ground-state properties for a number of H-bonds having different features. We have called this new method ‘transition-state H-bond theory’ [18] and, in this account, we have reviewed its application to the tautomeric N–H/O5N/H–O equilibrium in variously substituted 1(arylazo)-2-naphthols, and tried to show that it can provide a number of new ideas to help us to see the H-bond phenomenon
P. Gilli et al. / Journal of Molecular Structure 790 (2006) 40–49
(b)
(c)
Fig. 6. (a) Marcus modeling of the N/H–O/N–H/O PT reaction for all compounds of Table 2 (DFT-computed data not corrected for ZP). The two symmetric parabolas (in black) crossing at the intrinsic barrier D‡EoZ 3.08 kcal molK1 for r‡/dZ0.5 represent the intrinsic PT profile. Curves for the other compounds (in color) are shifted upwards or downwards by their respective reaction energies, DEr, and cross at r‡/d values larger or smaller than 0.5 with PT barriers D‡E higher or lower than D‡Eo, respectively. (b) Marcus modeling of (a) as modified by adding the values of the N/H–O bond energy, DEHB, to each individual curve. The black curve corresponds to the intrinsic H-bond with DErZ0 and D‡EZD‡Eo. Horizontal lines mark the approximate vibrational levels of the proton; full points indicate single-well and open symbols double-well H-bonds. Data from Ref. [18]. (For interpretation of the reference to colour in this legend, the reader is referred to the web version of this article.)
(d)
from a different point of view, ideas which can be shortly surveyed as follows: (a) The value of the intrinsic barrier found for this series of compounds is D‡EoZ3.08, which is reduced to D‡Eo,ZPCZ0.46 kcal molK1 by ZP correction. This value is rather small as could be expected for a D–H/A5D/H/A 5D/H–A reaction which proceeds via the D/H/A transition state, but where both reactants and products are pre-bound by the H-bond itself. This intrinsic barrier is low enough to allow H-bond dynamic disorder and therefore the formation of the socalled LBHBs [40–43] when DEr is close to zero. Any increase of DEr will make more and more dissymmetric the PT profile (Fig. 3(b)), causing a progressive increase of the PT barrier of the most stable tautomer which, in turn, originates the shift from dynamic to static disorder and, finally, the transition to the pure D–H/A or D/H–A bond having completely asymmetric SW profile (aSW). Though of not immediate application, this method of analysis seems to be the only one able to account for the complex behavior observed for these compounds in the solid state (Table 1). To
(e)
47
notice, finally, that the small value of D‡Eo,ZPCZ 0.46 kcal molK1 calculated for these compounds is such, anyway, to rule off the possibility of formation of the very short and strong SW H-bonds typical of some O/H/O RAHBs [17]. The harmonic constant of the parabolic expression EZ ˚ K2Z 1/2kr2 takes the value of kZ15.0 kcal molK1 A K1 K1 0.027 mdynZ0.027 N cm z75 cm . This force constant is some 250 times smaller than that associated with normal N–H or O–H stretchings (6–8 N cmK1) and can only be associated with the NH/O or N/HO stretching vibrations of the H-bond itself, in agreement with some early IR measurements which assign frequencies of 80– 250 cmK1 to these vibrations [44]. Accordingly, the properties of the N–H/O5N/H–O equilibrium can actually be described by two shallow nonbonded vibrations having such a small vibrational ˚ (the length of the PT constant, shifted by dZ1.283 A pathway) and intercrossing with an intrinsic PT barrier, D‡Eo,ZPC, as low as 0.46 kcal molK1. To notice that this barrier, having been calculated by MO quantum– mechanical methods, represents the best estimate of the true adiabatic barrier and does not require further corrections accounting for the TS resonance energy [45–48]. While the intrinsic profile is fully characterized by only two parameters (D‡Eo and k, or D‡Eo and d; see Eqs. (1– 3)), the reaction energy DEr is the independent variable acting as the thermodynamic driving force which modulates the shape of the PT pathway. The only point of it practically unaffected by DEr changes is the geometry of the TS, which displays nearly constant N/O ˚ associated with an almost complete distances of 2.38–2.40 A delocalization of the interleaving resonant fragment, suggesting that the TS H-bond can be interpreted as a nearly invariant three-center-four-electron covalent bond typical of the H-bonded system investigated [23,18]. Conversely, the DEr changes cause dramatic effects on the remaining parts of the PT pathway. While the intrinsic profile (DErZ0) has a perfectly symmetric DW potential, any increase (or decrease) of DEr splits the H-bond in two distinct bonds: (i) a less stable one which is shorter because closer to the TS and (ii) a more stable one which is longer because farther from the TS. This leads, in agreement with the Leffler–Hammond postulate [38,39], to the conclusion that the H-bond actually observed (the more stable) is always the longer in the tautomeric D–H/ A5D/H–A couple considered. The reaction energy DEr has the further physical meaning of being the difference of proton affinity, DPA, between the H-bond donor and acceptor groups. This PA difference cannot be either measured or calculated for intramolecular RAHBs, because of the large changes of p-delocalization occurring in the interleaving conjugated spacer (in this case the /HN–NaC–CaO/5/NaN–CaC–OH/ group) during H-bond formation, but can be tentatively evaluated making use of the extrathermodynamic LFER
48
P. Gilli et al. / Journal of Molecular Structure 790 (2006) 40–49
Fig. 7. Plot of the DFT-computed reaction energies, DEr, as a function of the mesomeric constant soR of the phenyl p-substituent [37]. Following Leffler and Grunwald [20], the two straight lines indicate that the two N–H/O/N/H–O and N/H–O/N–H/O reactions have different PT mechanisms. Full and open symbols refer to uncorrected and ZP-corrected DFT-computed data, respectively. Data from Ref. [18].
parameters of the changing substituent. In the present case, DEr is not found to correlate with the usual para and meta Hammett constants (sp and sm) but rather with the mesomeric constant, soR , most probably in view of the resonance-assisted nature of the H-bond treated. This correlation is shown in Fig. 7 for all the compounds of Scheme 2 except mOM, for which soR has no precise meaning. The correlation looks certainly impressive and such to support the idea that the driving force which modifies the PT profile is actually (all other factors remaining constant) the PA of the nitrogen as estimated by the soR value of the phenyl p-substituent. It appears, however, slightly different from what could have been expected, that is a unique straight line encompassing all compounds and not, as actually observed, two intercrossing lines with rather different slopes. Leffler and Grunwald, in their 1963 book ‘Rates and Equilibria of Organic Reactions’ [20] have considered the possibility of double-slope plots and interpreted them as characteristic of chemical reactions which change their mechanism beyond a certain value of s. This indicates that the two N–H/O/N/H–O and N/H–O/N–H/O reactions must have different features in the system presently considered, a fact that can be hypothetically accounted for in at least two non-mutually excluding ways: (i) The electronic effect of the p-phenyl substituent is felt in a different way by the N and by the O because 137°
3.Ia
H N 11 6° N
146° O
N N
Scheme 3.
H
O 6° 10
3.Ib
of the dissymmetric position of the phenyl; moreover, the effect felt by N appears to be larger when it behaves as acceptor (N/H–O) than when it acts as donor (N–H/O); (ii) It is well known that, for a same D/A distance, the energy of any D–H/A bond decreases steeply when the bond becomes less linear [49,50]. Accordingly, the fact that the pure N/H–O bond in pOK is calculated to be some 4.0 kcal molK1 stronger than any other N–H/O or N–H/O/N/ H–O bond of this series (19.4 against an average of 15.4 kcal molK1) can find a justification in the observation that the N–H–O angle is always more linear in the N/H–O (3.Ib) than in the N–H/O (3.Ia) tautomer (1468 against 1378) because of the different equilibrium values of the N–N–H (z1168) and C–O–H (z1068) angles (Scheme 3). These geometrical differences could then justify the change of mechanism occurring on the two sides of the intrinsic barrier (the horizontal line with DErZ0 in Fig. 7).
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