Calculated Nuclear Magnetic Resonance Parameters for Multiproton

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Calculated Nuclear Magnetic Resonance Parameters for MultiprotonExchange and Nonbonded-Hydrogen Rotation Processes in Cyclic Water Clusters Hubert Cybulski and Joanna Sadlej* Faculty of Chemistry, University of Warsaw, Pasteura 1, 02-093 Warsaw, Poland

bS Supporting Information ABSTRACT: In this work we report, for the first time, calculations of nuclear magnetic resonance parameters for the processes of multiproton-exchange and nonbonded-proton rotations in small, cyclic water clusters. The simultaneous proton exchange induces a large decrease in the oxygen shielding constants in both clusters, with a mean value of -52.6 ppm for the water trimer and -50.1 ppm for the water tetramer. The 1(h) JOH coupling constant between an oxygen nucleus and exchanging proton decreases (in absolute value) along the path, changes sign, finally reaching a value of 5-7 Hz. The changes in the NMR parameters induced by the nonbonded proton rotations are smaller. The calculated dependencies of the intermolecular spin-spin coupling constants upon rotation reveal that the largest changes are expected for the couplings transmitted through the hydrogen bond between the rotating and neighboring molecule which acts as a proton donor. The symmetry-adapted perturbation theory (SAPT) interaction energy calculations for each dimer forming the water trimer have allowed us to relate a strength of interactions within pairs of water molecules with coupling constant values. The predicted maximal values of the interaction-energy terms (energetically unfavorable orientations of the constituent dimers) along paths correlate with the extremal values of the spin-spin coupling constants, which mostly correspond to the maximal couplings along pathways.

I. INTRODUCTION Long-range proton transfer constitutes a key step not only in water but also in many biological reactions.1,2 In most of these processes, water plays an important role in the transfer of protons. Cyclic water clusters are convenient models to study proton-exchange phenomenon, since they approach the description of the environment of an “infinite” molecular chain. They are optimal to investigate concerted proton transfer avoiding the effects of ionic contributions in ionic water clusters. Because of their size, the most frequently investigated systems are the water trimer and tetramer. The literature data for the simplest cyclic (H2O)3 cluster consists of numerous experimental and theoretical studies3-9 such data for the water tetramer is less abundant.10-15 The cyclic structures for both clusters, in which the monomers simultaneously act as proton donors and proton acceptors, have been identified as global minima.16 Several theoretical approaches have been proposed to model and understand the environmental effects in liquid water on the nuclear magnetic resonace (NMR) parameters. Representative clusters of water molecules taken from molecular dynamics17,18 or the Car-Parrinello method simulations19-21 were used to calculate the 1H and 17O shielding constants, and the results seem to be promising. The calculated oxygen liquid shift is in qualitative agreement with experiment, although the results depend, e.g., on the r 2010 American Chemical Society

chosen interatomic potential, method of averaging molecular properties, or on the cluster size. Besides, supermolecular calculations for small water clusters have also been performed.22-24 The enhancement of the cooperative effects with cluster size and changes in ligand environment results in decreasing 17O shielding constants. A very large shift of -76.2 ppm has been predicted for the oxygen nucleus of the four-coordinated, central water molecule in the largest n = 17 cluster.24 The indirect nuclear spin-spin coupling constants in water are less widely investigated than the shielding constants. Only a few studies have been devoted to the calculations of spin-spin coupling constants for small water clusters.25,26 In the aforementioned studies, the influence of a proton transfer on the calculated NMR parameters was completely neglected. The methods based on molecular dynamics or Car-Parrinello method, where randomly chosen snapshots of molecules extracted from liquid water simulations were used to predict molecular properties, or rigid water clusters from firstprinciples computational studies account for the deformation Special Issue: Victoria Buch Memorial Received: August 11, 2010 Revised: August 31, 2010 Published: September 22, 2010 5774

dx.doi.org/10.1021/jp107595n | J. Phys. Chem. A 2011, 115, 5774–5784

The Journal of Physical Chemistry A of water molecules mainly due to stretching of the O-H bonds or changes in the H-O-H angle induced by interactions with surrounding molecules. Numerous pathways of proton rearrangement in cyclic clusters may be described5,10,15,27-31 however, in present work we employ two simple models of proton-transfer processes. First, concerted proton transfer, involving all of the inner ring protons in small cyclic water clusters,29-32 which may mimic the phenomenon of proton transfer expected to take place in liquid water. Second, we analyze energetically more accessible effect of nonbondedproton rotations in the water trimer. One of these rotational processes follows the lowest energy path, corresponding to the flipping of one of the two free O-H bonds which are on the same side of the ring plane across this plane.5 The object of this paper is two-fold. The first goal is to investigate, for the first time, the NMR parameters in systems with simultaneous exchange of three or four protons that allows us to estimate the influence of such process on nuclear shielding constants and spin-spin coupling constants. This is also complementary to and extends our previous studies on the calculations of the NMR parameters for double proton exchange pathways in the formamide-formic acid and formamide-formamidine complexes.33 The second purpose is to analyze in more details effects of frequent low-barrier processes on the observed shielding constants and coupling constants, using models of nonbonded-proton rotations, and try to correlate them with structural and energetic parameters.

II. COMPUTATIONAL DETAILS A. Geometry Optimization. The structures of the cyclic water trimer and tetramer were optimized by means of the frozencore Møller-Plesset (MP2) perturbation theory with the aug-ccpVTZ basis set.34,35 In the geometry optimization procedure, no counterpoise corrections were made for the basis-set superposition error (BSSE). The intrinsic reaction-coordinate (IRC) calculations were performed starting from a transition-state structure in both forward and reverse directions of the vibrational mode in order to follow the multiproton exchange and nonbonded-hydrogen rotation energy paths. In the location of the IRC the Gonzalez-Schlegel second-order method has been employed. Geometry optimization and IRC calculations were carried out using GAMESS program package.36,37 B. Calculation of the NMR Parameters. The indirect nuclear spin-spin coupling constants and nuclear shielding constants were calculated employing the density functional theory (DFT) method with the hybrid three-parameter Becke-Lee-YangParr (B3LYP) functional.38,39 B3LYP is the functional the most intensively and widely tested for this purpose and its usefulness to reproduce the sign and the order of magnitude of the intra- and intermolecular spin-spin coupling constants is well-known.40-42 Couplings transmitted through a hydrogen bond have been studied using approaches based on DFT and their results are quite promising.43-46 Spin-spin coupling constants and nuclear shielding constants have been calculated with Huzinaga’s Huz-IVsu4 basis set modified by van W€ullen47,48 derived from Huz-IV basis set by decontracting s orbitals and adding 4 tight orbitals using the geometrical progression for the exponents49,50 (10s3p1d/ 10s3p1d for H and 15s7p3d1f/15s7p3d1f for O). This basis set yields good results for the spin-spin coupling constants in isolated molecules.40,41,49

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The calculations have been performed with Dalton 2.051 program. The NMR parameters were calculated for the 1H and 17 O nuclei. No vibrational or temperature corrections were accounted for. All four nonrelativistic contributions to the indirect spin-spin coupling constants were calculated: the Fermi-contact (FC) term, the spin-dipole (SD) term, the paramagnetic spin-orbit (PSO) term, and the diamagnetic spin-orbit (DSO) term. C. Symmetry-Adapted Perturbation Theory. In this work we have employed the following symmetry-adapted perturbation theory (SAPT) approximation to the intermolecular interaction energy, roughly equivalent to the supermolecular second-order many-body perturbation-theory (MBPT2) calculation, namely :52,53 ESAPT2 int ð12Þ

ð1Þ

ð22Þ

t ESAPT2 ¼ EHF int int þ Eelst, r þ εexch ð2Þ þ Eind ð22Þ

ð20Þ

ð20Þ

þ t Eexch-ind þ Edisp þ Eexch-disp

ð1Þ

with the interaction energy calculated at the Hartree-Fock level defined as ð10Þ

ð10Þ

ð20Þ

ð20Þ

HF EHF int ¼ Eelst þ Eexch þ Eind, r þ Eexch-ind, r þ δEint

ð2Þ

(10) E(10) elst is the classical (Coulombic) electrostatic energy, Eexch is the exchange term that results from the antisymmetrization (symmetry adaptation) of the wave function, E(20) ind, r denotes the induction (with response) energy, E(20) exch-ind, r is the second-order exchange-induction (with response) energy term, E(20) disp is the dispersion energy, and E(20) exch-disp denotes the exchange-dispersion contribution. δEHF int collects all third- and higher-order induction and exchange-induction terms. For more details, see refs 52 and 53. In order to simplify the analysis of the SAPT decomposition, we have collected the electrostatic, exchange and induction terms into their respective summary terms

ð10Þ

ð12Þ

Esum elst ¼ Eelst þ Eelst, r ð10Þ

ð20Þ

ð1Þ

ð3Þ ð22Þ

ð20Þ

t Esum exch ¼ Eexch þ Eexch-ind, r þ εexch ð2Þ þ Eexch-ind þ Eexch-disp

ð4Þ ð20Þ

ð22Þ

t Esum ind ¼ Eind, r þ Eind

ð5Þ

The interaction energy components were calculated using the SAPT2008 program.54

III. RESULTS OF CALCULATIONS A. Structures and Energetics of the Multiproton-Exchange and Hydrogen-Rotation Paths. The structures of the cyclic water

clusters are of different symmetry. We employed, partially after ref 6, the following notation for water trimer structures: the C1 global minimum [uud] (see Figure 1a), the C3 local minimum [uuu], the Cs multiproton exchange transition state (TS) [mpt] (see Figure 1b), two C1 nonbonded-proton rotation TSs: [udp] (the [uud] f [udd] minima transition, between two equivalent minima, see Figure 1c), and [uup] (the [uud] f [uuu] minima transition, see Figure 1d). The structures of the water tetramer along the proton-exchange paths are of S4 symmetry, inlcuding the global minimum [udud] (see Figure 1e), whereas in the multiproton exchange TS [mpt] is of higher D2 symmetry (see Figure 1f). The symmetry of the multiproton exchange TSs implies that the relevant pathways are symmetric and the structures (however, not necessarily the parameters) in both forward and back 5775

dx.doi.org/10.1021/jp107595n |J. Phys. Chem. A 2011, 115, 5774–5784

The Journal of Physical Chemistry A

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Figure 1. Structures of the global minima and transition states of cyclic water clusters optimized at the MP2/aug-cc-pVTZ level (see text for description): (a) the C1 global minimum [uud] of (H2O)3, (b) the Cs [mpt] TS of (H2O)3, (c) the [udp] TS of (H2O)3, (d) the [uup] TS of (H2O)3, (e) the S4 global minimum [udud] of (H2O)4, and (f) the D2 [mpt] TS of (H2O)4

directions are the mirror reflections. The rotating hydrogens in the water trimer (H9 and H8, for the [uud] f [udd] and [uud] f [uuu] minima transition, respectively; see Figure 1, panels c and d) are indicated with black arrows and the terminal positions of the hydrogens are represented by broken-line contours. These arrows also show the direction in which the largest magnitude component of the imaginary normal mode is positive and this direction is represented by a positive intrinsic reaction coordinate. The nonbonded-proton rotation pathways are clearly nonsymmetric. The found structures of cyclic water clusters’ minima agree with previous studies: the water trimer6,29 and water tetramer.32,55 The energy profiles depicted in Figure 2a present steep and quite high energy barriers for simultaneous multiprotonexchange processes. Such a barrier is of 110.3 kJ/mol for the water trimer. This barrier is lower (94.6 kJ/mol) in the water tetramer. The calculated barriers agree well with previous studies: 107.19 kJ/ mol for the water trimer (MP2/cc-pVTZ, ref 29) and 97.45 kJ/mol for the water tetramer (MP2/6-311þþG(3pd,3df), ref 32 and 55). The predicted energy barriers for nonbonded-proton rotations are much smaller (see Figure 2b): 1.17 kJ/mol for the [uud] f [udd] minima transition and 3.34 kJ/mol for the [uud] f [uuu] minima transition (0.19 kJ/mol in the reversed direction). They are the same as calculated in ref 6. B. NMR Parameters. The calculated nuclear shielding constants are presented in Table 1, whereas the selected calculated intra- and intermolecular spin-spin coupling constants, nJXY and nh JXY, respectively, are collected in Table 2. We employed, after

ref 56, the notation nhJXY for the indirect intermolecular coupling constant between nuclei X and Y, where the number n in superscript denotes the number of bonds through which the intermolecular coupling is formally transmitted (including the intermolecular bond). The parameters for the multiproton-exchange and nonbondedproton rotation pathways are discussed separately since they differ in magnitude. 1. Multiproton-Exchange Pathways. The nuclei in the clusters we can divide into three groups: a group of the oxygen nuclei and two groups of protons: involved in hydrogen bonding (the inner rings) and nonbonded (dangling) hydrogens (cf. Figure 1, panels b and f). The nuclei are identical within each group (because of the symmetry) in the water tetramer. Despite the lower symmetry in the water trimer the calculated parameters are also practically the same within these three groups of nuclei and to ease the analysis in this work we shall use their average values. a. Nuclear Shielding Constants. The dependencies of the isotropic oxygen nuclear shielding constants, σO, on the multiproton-exchange path are shown in Figure 3a. One can notice a strong deshielding (defined as a difference between the values of σO in the transition-state and minimal geometries) for the oxygen nuclei with a mean value of -52.6 ppm for the water trimer and -50.1 ppm for the water tetramer (cf. Table 1). The shape of the graph in Figure 3a reveals that the dependence is mostly linear while a rapid decrease of the oxygen shielding constant appears in the proximity of a [mpt] TS. This becomes 5776



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Figure 2. Relative energy barriers, in kJ/mol, for (a) the multiproton exchange and (b) the nonbonded-proton rotation processes in the cyclic water trimer and tetramer.

Table 1. Isotropic (σ) and Anisotropic (Δσ) Nuclear Shielding Constants [ppm] for the Water Trimer and Tetramer Calculated at the DFT(B3LYP)/HuzIVsu4 Level minimum

TS

minimum

(H2O)3

(H2O)4

(H2O)3

(H2O)4

307.93

300.97

255.37

250.87

σH-ring

26.61

24.53

11.34

11.01

σH-dang ΔσO

30.89 62.42

30.79 52.36

30.37 58.44

30.21 35.15

ΔσH-ring

28.22

33.41

47.03

50.06

ΔσH-dang

19.77

18.98

16.52

15.68

σO

Table 2. Spin-Spin Coupling Constants [Hz] for the Water Trimer and Tetramer Calculated at the DFT(B3LYP)/HuzIVsu4 Level

more evident while the shielding anisotropy, ΔσO, is concerned (see inset in Figure 3a) A similar pattern exhibits the proton isotropic shielding constant (σH) of exchanging protons, Figure 3b. The average deshielding for the proton forming the inner ring (σH-ring) is -15.3 ppm for the water trimer. Such deshielding for the water tetramer is -13.5 ppm. The changes in proton shielding anisotropy, ΔσH, are in opposite direction to the changes in the isotropic shielding (see inset in Figure 3a). In a [mpt] TS, the anisotropy increases in comparison to its value in a minimal geometry. The average deshielding for the dangling hydrogens (σH-dang) is small and is ca. -0.5 ppm in the water trimer and -0.6 ppm in the water tetramer; as one can expect, the changes in anisotropy are also small. b. Indirect Nuclear Spin-Spin Coupling Constants. There is only one intramolecular spin-spin coupling constant, 1JOH, between an oxygen atom and dangling hydrogen. This coupling in the geometries of global minima has a value of ca. -82 Hz increasing irregularly to ca. -84 Hz for a [mpt] TS structure (see Table 2). Small changes are related to small changes in geometry, especially in the internuclear separation. As it was shown in numerous studies,25,26,50 the main contribution arises from FC term and this pattern is valid along the whole pathways. The remaining couplings either change from intra- to intermolecular along the multiproton exchange path (for that reason we have employed the notation n(h)JXY) or they have intermolecular character within the whole pathway (nhJXY). The behavior

minimum

(back)

1 JOH 1(h) JOH 2(h) JHH 2h JOO

TS

(forward)

(H2O)3

(H2O)4

(H2O)3

(H2O)4

(H2O)3

(H2O)4

-81.79

-82.62

-83.31

-84.08

-81.79

-82.62

-82.28 -7.74

-83.38 -7.83

-19.38 -2.15

-22.83 -2.47

5.44 -0.35

7.51 -0.15

2.62

3.89

12.17

15.69

2.62

3.89

of the one-bond coupling constant between an oxygen nucleus and exchanging proton, 1(h)JOH, is shown in Figure 4a. In minimal geometry (in which the hydrogen is bonded to the parent oxygen atom) this coupling has a slightly higher (absolute) value than 1JOH of dangling hydrogens but decreases (also in absolute value) along the multiproton exchange path and changes sign becoming the intermolecular 1hJOH coupling. When the proton is transferred 1(h) JOH has a value of 5-7 Hz and a rapid decrease appears in the proximity of a [mpt] TS (Figure 4a). The total value and behavior of this coupling are also determined by FC term, the three remaining terms are rather small. Concerning the reduced FC terms and reduced coupling constants, such smooth change of sign for 1(h) KXH along proton-exchange path (a negative 1hKXH to a positive 1 KXH) has been predicted57 and calculated for the model systems of the formamide-formic acid and formamide-formamidine dimers.33 Let us now discuss the behavior of the two-bond protonproton coupling constants between dangling and exchanging protons. 2(h)JHH, depicted in Figure 4b presents a similar pattern as for the 1(h)JOH coupling. 2(h)JHH decreases (in absolute value) from -8 Hz to ca. 0 Hz and a rapid drop is in the vicinity of a [mpt] TS. The substantial contributions to 2(h)JHH come from FC and two spin-orbit terms;25,26,50 however, it is the FC term which influences the total value of the coupling. For shorter internuclear distances it dominates the total coupling, while for longer distances (passing a TS) two substantial spin-orbit of nearly equal values and opposite signs cancel each other making the smaller FC term still the most important. 5777



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Figure 3. Oxygen and proton shielding constants [ppm] for the multiproton exchange paths in the water trimer and tetramer: (a) isotropic, σO; inset: anisotropic, ΔσO; (b) isotropic, σH; inset: anisotropic, ΔσH.

Another interesting coupling is between two oxygen nuclei of neighboring water molecules, 2hJOO (see Figure 4c). This coupling with a value of 3-4 Hz in minimal geometries increases to 12-16 Hz in a [mpt] transition-state geometry. Such increase/ decrease is nearly linear with a steep growth in the proximity of the TS. This coupling is also dominated by FC term for all internuclear distances along the multiproton exchange path; the DSO term, which is of second importance, constitutes less than 15% of FC contribution. 2. Nonbonded-Proton Rotation Pathways. The calculated NMR properties for both the [uud] f [udd] (C1 f C1) and [uud] f [uuu] (C1 f C3) rotation pathways shall be presented jointly to visualized the possible similarities and discrepancies. a. Nuclear Shielding Constants. The behavior of the isotropic oxygen nuclear shielding constants along the rotational pathways is depicted in Figure 5a. One can easily recognize that the shielding-constant dependency for each oxygen nucleus exhibits a maximum (the nucleus is shielded) in the proximity of a TS. The largest increase in the oxygen shielding constant is predicted for the rotating water molecule for both pathways and the changes are of similar magnitude (for instance, an increase of 4.7 ppm is observed for the [uud] f [udd] minima transition). The smallest changes characterize the water molecule which is a proton acceptor for the rotating molecule: the σO2 dependence for the [uud] f [udd] minima transition is flat and σO may slightly decrease along the path. The trend in the changes of the anisotropy (ΔσO, inset in Figure 5a) follows the behavior of the isotropic σO; however, it may have other direction: the σO2 dependence for the [uud] f [udd] minima transition reveals a maximum, while the ΔσO2 exhibits a minium along the path. An interesting pattern is observed for the isotropic proton shielding constants shown in Figure 5b. The greatest changes are observed not for the rotating protons but for the protons of neighboring molecule involved in hydrogen bonding connecting both molecules (the H8 and H4 hydrogens in Figures 1c and 1d, respectively). The corresponding dependencies present maxima (the nuclei are shielded) around the TSs and σH may increase by 0.8 Hz (σH8 for the [uud] f [udd] minima transition). The remaining proton shielding constants behave more irregularly; however, such changes are relatively small - the second important

change (σH5 for the [uud] f [uuu] minima transition) does not exceed 0.25 Hz (cf. Figure 5b). The changes in the anisotropy depicted in inset in Figure 5b are small and do not cross a value of 2.0 Hz with the largest changes corresponding to the shielding constants of the aforementioned protons of the hydrogen bonds connecting the rotating water and neighboring, proton-donating molecule. Substantial changes are also observed for the rotating protons. b. Indirect Nuclear Spin-Spin Coupling Constants. As in the previous section, first we present the changes in the intramolecular spin-spin coupling constants upon nonbonded proton rotation. Then we report the corresponding changes of the intermolecular couplings. Among the intramolecular 1JOH spin-spin coupling constants, as expected, the largest changes are calculated for the rotating water molecules (see Figure 6a). Nevertheless, the changes in the coupling between rotating (dangling) protons and oxygen nuclei are slightly larger than those involving the proton of hydrogen bonding. The coupling constants dependencies exhibit pronounced minima along the paths: 1JOH may decrease (increase in absolute value) by ca. 5.1 Hz (the [uud] f [uuu] minima transition). The changes in remaining 1JOH coupling constants are much smaller and the predicted curves (not shown Figure 6a) are relatively flat. Similarly, the intramolecular 2JHH coupling constant between protons in a rotating water molecule experiences the most substantial changes, reaching a maximum in the proximity of a TS connected with an increase by 1.3 Hz (the [uud] f [uuu] minima transition). The changes in 2JHH of two other water molecules are substantially smaller. The intermolecular spin-spin coupling constants are also affected by rotation of a water molecule. The 1hJOH coupling constant (shown in Figure 6b) between the rotating-molecule oxygen nuclei and neighboring water-molecule proton (the H8 and H4 hydrogens in Figure 1, panels c and d, respectively) may increase by ca. 1.6 Hz (a maximum on the pathway), whereas the changes for the proton of rotating molecule and the oxygen of a neighboring molecule are approximately three times smaller and in opposite direction. 1hJOH between nonrotating water molecules changes slightly and irregularly along the path. The 2hJOO dependencies (depicted in Figure 6c) reveal that the most 5778



Figure 4. Indirect spin-spin coupling constants [Hz] for the multiproton exchange paths in the water trimer and tetramer: (a) one-bond oxygen-proton coupling constants, 1(h)JOH, (b) two-bond protonproton coupling constants, 2(h)JHH, and (c) intermolecular two-bond oxygen-oxygen coupling constants, 2hJOO.

pronounced maxima on the curves correspond to the coupling between the oxygen of the rotating molecule and the oxygen of the

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neighboring molecule which donates a proton into hydrogen bonding (the O1 oxygens in Figure 1, panels c and d). The range of the changes depends on the process direction and the greatest value of 2.5 Hz is predicted for the [uud] f [uuu] minima transition. The least sensitive 2hJOO couplings are those between the oxygen nuclei of nonrotating water molecules. The 2hJHH involving the dangling protons and the protons of inner ring are interesting, too. Two of them are small and the changes along the rotational path are small, while the third, between the rotating proton and the proton of the neighboring molecule forming a hydrogen bond between these two molecules experiences substantial changes upon proton rotation with a minimum around a TS (see Figure 1a in the Supporting Information (SI)). The maximal absolute value of 2hJHH is ca. 1.7 Hz. The changes of 2hJHH between the proton of the inner ring are noticeable; however, they are small since the coupling are also small and do not exceed 0.16 Hz. The most of the 3hJOH coupling constants change monotonically on the pathway, the exceptions are the couplings between the rotating proton and the oxygen of the neighboring molecule which donates a proton into hydrogen bonding (the O1 and H9, O1 and H8 nuclei pairs in Figure 1, panels c and d, respectively). The 3hJO1-H8 coupling constant for the [uud] f [uuu] minima transition increases in absolute value by ca. 2.0 Hz (see Figure 1b in the SI). The changes of 3hJOH between the oxygen and the proton of the inner ring are noticeable but small and do not cross 0.2 Hz. C. SAPT Decomposition along Hydrogen-Rotation Paths. The analysis of the calculated NMR parameters for the rotational pathways clearly exhibits the complexity of interactions in the water trimer and indicates an existence of preferred paths for coupling transmission. To gain more insight, we have performed the SAPT calculations for each pair (three dimers) of water molecules along both rotational pathways. The SAPT interaction energy decomposition for the global minimum of the water dimer can be found elsewhere.58 The calculated SAPT2 energy (cf. eq 1) agrees thoroughly with the MP2 interaction energy. The dependence of the SAPT2 decomposition and its four contributions (vide supra), namely electrostatic, exchange, induction, and dispersion energies, on the rotational pathways is depicted in Figure 7. A comparison of Figures 2b and 7a proves that the minimum on the potentialenergy surface of the system does not necessarily correspond to the optimal mutual orientations of the constituents what results in the increasing and decreasing interaction energy for the dimers along the rotational pathways. The SAPT2 graphs (see Figure 7a) show that the largest changes appear in the interaction energy between a rotating molecule and each of its neighbors. They depend on whether the neighboring water molecule acts as a proton acceptor or proton donor to the rotating molecule: the former being monotonical, while the latter exhibit flat maxima in the proximity of the TSs. The corresponding changes for the remaining pair of water molecules are small and the interaction energy is almost constant along the pathway. The analysis of the SAPT terms (Figure 7b-e) reveals that, regarding the individual contributions, the differences are even more pronounced and the changes for the rotating molecule-proton donor pair for each pathway are twice as for the rotating molecule-proton acceptor pair, with an extremum in the vicinity of the corresponding TS. The pattern in the graphs depicted in Figure 7b-e is similar for each of the contributions. 5779



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Figure 5. Oxygen and proton shielding constants [ppm] for the nonbonded-proton rotation paths in the water trimer: (a) isotropic, σO; inset: anisotropic, ΔσO; (b) isotropic, σH; inset: anisotropic, ΔσH.

IV. DISCUSSION In this section we shall discuss two issues. First, we intent to interpret the calculated changes in the oxygen and proton nuclear shielding constants upon the multiproton exchange in our models of water clusters. Second, we try to point out that the intermolecular interactions within the cluster greatly influence observed molecular properties. The gas-to-liquid NMR experimental chemical shifts for water are known to be -36.1 ppm for oxygen59 (using new data of 323.6 ppm in gas and 287.5 ppm in liquid phase59) and -4.26 ppm for hydrogen.60,61 Previous ab initio calculations of the shielding constants for small water clusters22-24 have predicted an increase with a size of the cluster, gas-to-liquid (or, more precisely, monomer-to-cluster) chemical shift. It yielded a value of -76.2 ppm for the central oxygen atom surrounded by two hydration shells in the largest water cluster, (H2O)17,24 whereas the largest chemical shift for the proton shielding constant with a value of -10.2 ppm was found for the longest O-H bond. The data for the nuclear spin-spin coupling constants is much modest and consists of the experimental estimate of 1JOH in gaseous water,62 -79 Hz (close to the value of -78.70 Hz measured for cyclohexane-d12 solution), and in liquid water,63 89.8 Hz, yielding the gas-to-liquid shift of ca. -10 Hz. The measurements of 2JHD in nitromethane-d3 at 297 K yield a value of -1.127 Hz (this value renormalized to 2JHH is -7.342 Hz),64 and to our knowledge there is no experimental value of 2JHD in liquid water reported. The range of the calculated complexationinduced changes in 1JOH showed that a simple model of small water clusters is adequate to reproduce the sign and approximated magnitude of the gas-to-liquid shift in 1JOH.25,26 The predicted changes for the 2JHH coupling constants (9.08 Hz in the water monomer) cover a range between -2.9 and 3.0 Hz.25 However, the previously employed model of rigid water clusters does not account for the proton exchange effect. In this study, we have considered a case of concerted multiproton exchange in the water trimer and tetramer which may mimic simultaneous proton exchange process expected to take place in liquid water. The calculations presented in section III.B.1 have shown that the proton exchange process induces a decrease by -52.6 and 50.1 ppm in the isotropic shielding constant of the oxygen nuclei

in the water trimer and tetramer, respectively. A summation of this effect along with the difference between the shielding constants in the water trimer or tetramer and the water monomer (-17.3 and -24.3 ppm, respectively) yields a large deshielding on the oxygen nuclei with values of -69.9 and -74.4 ppm. The corresponding decreases in the proton shieldings upon proton exchange process (cf. section III.B.1) are -15.3 and -13.5 ppm. The calculated proton shifts defined as a difference of the shieldings in the water trimer or tetramer and the water monomer are -4.6 and -6.7 ppm (the protons of the inner ring). A sum of these two effects gives rise to the values of -19.9 and 20.2 ppm. For consistency, the values calculated at the DFT(B3LYP)/HuzIVsu4 level in the MP2/aug-cc-pVTZ geometry have been used for the water monomer. The changes in 1JOH involving nonbonded protons are ca. -2 Hz, thus, increase (in absolute) the value of the earlier estimated changes (from -5.9 to -7.0 Hz) upon cluster formation for dangling hydrogen in the water trimer and tetramer.24 The influence of the proton exchange on the remaining 1JOH as well as on the 2JHH couplings is more difficult to estimate since they smoothly change their character from intra- to intermolecular. The results contained in section III.B.2, concerning the nonbonded-proton rotations, clearly indicates that the strength of an interaction and coupling in a cluster depends on the orientation of the molecules; however, their reciprocal relationships are quite complex since the optimal structure of the cluster (an energy minimum) does not necessarily correspond to the maximal interaction (coupling) among the constituent molecules. The changes in the total energy of the water trimer upon proton rotation (see Figure 2b) reveal that the minima on the potential-energy surface of the system do not relate to the simultaneous minimal interaction energies for all of the constituent dimers (cf. Figure 7a) and, that is, the most favorable mutual orientation of all the water molecules. As one may expect, the largest changes along the rotational pathways correspond to the interaction energy between a rotating molecule and its neighbors, while the analogous changes for a nonrotating pair are small. Nevertheless, the pattern in the interaction-energy changes depends on whether the rotating molecule is a proton donor to the neighboring one or vice versa. In the latter case, the plotted graphs exhibit a distinct extrema along the 5780



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considered (cf. Figures 7b-e). The origin of such difference may be sought in the orientation of the electron lone pair of the oxygen nucleus of the rotating molecule: the movement of the hydrogen and, thus, of the whole molecule as well causes the rotation of the lone pair involved in the hydrogen bonding. This results in changes in the hydrogen bond strength, influencing the observed molecular properties. While the calculated changes in the oxygen shielding constants can only discriminate among the rotating and nonrotating water molecules (see Figure 5a), the greatest changes in the proton shielding constants correspond to the proton involved in the hydrogen bonding between the rotating molecule and neighboring proton-donor water molecule (cf. Figure 5b). The predicted intermolecular spin-spin coupling constants also prove that the most substantial changes correspond to the couplings transmitted directly through the hydrogen bonding involving the rotating molecule as a proton acceptor. As the most illustrative example of such tendency may serve a comparison of Figures 6c and 7b-e where a similarity in the patterns of changes is apparent. On the other hand, the maximal values of the interaction-energy terms signify energetically unfavorable orientations of the constituent dimers, while the maxima in the oxygen-oxygen spin-spin coupling constants correspond to the maximal coupling along pathways. Previously, theoretical studies for different systems showed that the intermolecular spin-spin coupling constants monotonically decrease with the distance, unlike the interaction energy, and evidently the minimal interaction energy does not correspond to the maximal coupling (e.g., a simplest case of the helium dimer65). Current work proves that not only a distance but also a mutual orientation between molecules, especially molecules with lone electron pairs, may influence the values of observed NMR parameters. Such discrimination between a proton-donor and protonacceptor effect have been already observed. In our previous calculations of the shielding constants for small water clusters,24 a significantly larger deshielding in the oxygen nuclei was observed in (H2O)12 and (H2O)17, when molecules of proton-donor type had been added to a single, central water molecule, in comparison with an analogous addition of proton-accepting molecules.

Figure 6. Indirect spin-spin coupling constants [Hz] for the nonbonded-proton rotation paths in the water trimer: (a) one-bond oxygenproton coupling constants, 1(h)JOH, (b) intermolecular one-bond oxygenproton coupling constants, 1hJOH, and (c) intermolecular two-bond oxygen-oxygen coupling constants, 2hJOO.

proton-rotation paths, as shown in Figure 7a, which become more pronounced while the individual contributions are

V. SUMMARY AND CONCLUSIONS In this study, the cyclic water trimer and tetramer have been employed as convenient models to explore the influence of the processes of simultaneous multiproton exchange and nonbonded proton rotation on the NMR parameters. The predicted energy profiles for the processes are of different magnitude: the barriers of the multiproton exchange are 110.3 and 94.6 kJ/mol in the water trimer and tetramer, respectively, whereas the nonbonded-proton rotations characterize energies smaller than 3.5 kJ/mol. The simultaneous proton exchange induces a large decrease in the oxygen shielding constants in both clusters, with a mean value of -52.6 ppm for the water trimer and -50.1 ppm for the water tetramer. A strong deshielding is also calculated for the protons forming an inner ring (-15.3 and -13.5 ppm for the water trimer and tetramer, respectively), while the nonbonded protons experience small changes in the proton shielding constants. A further correction of this values by an addition of the cluster formation effect, defined as a difference between the shielding constants in the water cluster and the water monomer, shows that the theoretical gas-to-liquid chemical shift is overestimated in comparison with experimental data. That proves once again that such models of water clusters are 5781



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Figure 7. SAPT decomposition along the nonbonded-proton rotation pathways in the water trimer: (a) the total SAPT2 interaction energy, ESAPT2 , int sum sum (20) (b) the summary electrostatic term, Esum elst , (c) the summary exchange term, Eexch, (d) the summary induction, Eind , and (e) the dispersion term, Edisp .

still too simple to reproduce correctly the network of molecules in liquid water. The changes in the NMR parameters upon the nonbonded proton rotations are smaller. The largest increases in the oxygen shielding constants are predicted for the rotating water molecule for both pathways and the changes are of similar magnitude (they do not exceed 5 ppm). The greatest changes for isotropic proton shielding constants are calculated for the hydrogen of the intermolecular bonding between the rotating water molecule and its neighbor acting as a proton donor. These shieldings may increase by 0.8 Hz. The intermolecular spin-spin coupling constants are also affected by rotation of a water molecule. The calculated dependencies reveal that the largest changes are expected

for the couplings transmitted through the hydrogen bonding between the rotating molecule and neighboring molecule which acts as a proton donor. The SAPT interaction energy calculations for each dimer forming the water trimer have allowed us to relate a strength of interactions within pairs of water molecules with spin-spin coupling constants values. The largest changes along the rotational pathways correspond to the interaction energy between a rotating molecule and its neighbors and the pattern in the interaction-energy changes depends on whether the rotating molecule is a proton donor to the neighboring one or vice versa. The predicted maximal values of the interaction-energy terms (energetically unfavorable orientations of the constituent dimers) 5782


The Journal of Physical Chemistry A along paths correlate with the extremal values of the spin-spin coupling constants, which mostly correspond to the maximal couplings along pathways. Such effect may be related to the orientation of the electron lone pair of the oxygen nucleus of the rotating molecule.

’ ASSOCIATED CONTENT

bS

Supporting Information. Indirect intermolecular spinspin coupling constants [Hz] for the nonbonded-proton rotation paths in the water trimer: (a) two-bond proton-proton coupling constants, 2hJHH, and (b) three-bond oxygen-proton coupling constants, 3hJOH. This material is available free of charge via the Internet at http://pubs.acs.org.

’ AUTHOR INFORMATION Corresponding Author

* E-mail: [email protected].

’ ACKNOWLEDGMENT This paper is dedicated to the memory of Professor Victoria Buch, a brilliant scientist and brave woman, in deep appreciation for her long-lasting friendship and stimulating collaboration over the years. The computational part of this work was done using the computer cluster at the Computing Centre of Faculty of Chemistry, University of Warsaw. The authors also acknowledge the computational Grant G18-4 from the Interdisciplinary Centre of Mathematical and Computer Modelling (ICM) of the University of Warsaw. ’ REFERENCES (1) Jeffrey, G. A.; Saenger, W. Hydrogen Bonding in Biological Structures; Springer-Verlag: Berlin, 1991. (2) Bountis, T. Proton Transfer in Hydrogen-Bonded Systems; Plenum Press: New York, 1992. (3) Liu, K.; Loeser, J. G.; Elrod, M. J.; Host, B. C.; Rzepiela, J. A.; Pugliano, N.; Saykally, R. J. J. Am. Chem. Soc. 1994, 116, 3507. (4) Pugliano, N.; Saykally, R. J. Science 1992, 257, 1937. (5) Sch€utz, M.; B€urgi, T.; Leutwyler, S.; B€urgi, H. B. J. Chem. Phys. 1993, 99 (7), 5228. (6) Anderson, J. A.; Crager, K.; Fedoroff, L.; Tschumper, G. S. J. Chem. Phys. 2004, 121, 11023. (7) Chazasinski, G.; Szcze) sniak, M. M.; Cieplak, P.; Scheiner, S. J. Chem. Phys. 1991, 94, 2873. (8) Fowler, J. E.; Schaefer, H. F., III J. Am. Chem. Soc. 1995, 117, 446. (9) Del Bene, J. E.; Pople, J. A. J. Chem. Phys. 1971, 55, 2296. (10) Wales, D. J.; Walsh, T. R. J. Chem. Phys. 1997, 106, 7193. (11) Lin, W.; Han, J.-X.; Takahashi, L. K.; Harker, H. A.; Keutsch, F. N.; Saykally, R. J. J. Chem. Phys. 2008, 128, No. 094302. (12) Perez, J. F.; Hadad, C. Z.; Restrepo, A. Int. J. Quantum Chem. 2008, 108, 1653. (13) Xantheas, S. S.; Burnham, C. J.; Harrison, R. J. J. Chem. Phys. 2002, 116, 1493. (14) Cruzan, J. D.; Braly, L. B.; Liu, K.; Brown, M. G.; Loeser, J. G.; Saykally, R. J. Science 1996, 271, 59. (15) Sch€utz, M.; Klopper, W.; L€uthi, H.-P.; Leutwyler, S. J. Chem. Phys. 1995, 103, 6114. (16) Xantheas, S. S.; Dunning, T. H., Jr. J. Chem. Phys. 1993, 99, 8774. (17) Malkin, V. G.; Malkina, O. L.; Steinebrunner, G.; Huber, H. Chem.—Eur. J. 1996, 2, 452. (18) Kongsted, J.; Nielsen, C. B.; Mikkelsen, K. V.; Christiansen, O.; Ruud, K. J. Chem. Phys. 2007, 126, No. 034510.

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Calculated Nuclear Magnetic Resonance Parameters for Multiproton

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