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Challenging Dogmas: Hydrogen Bond Revisited Maxim Tafipolsky J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.6b04861 • Publication Date (Web): 14 Jun 2016 Downloaded from http://pubs.acs.org on June 17, 2016

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The Journal of Physical Chemistry

Challenging Dogmas: Hydrogen Bond Revisited

Maxim Tapolsky* June 10, 2016 Institut für Physikalische und Theoretische Chemie, Universität Würzburg, Campus Hubland Nord, Emil-Fischer-Strasse 42, D-97074 Würzburg, Germany

Corresponding Author * Dr. Maxim Tapolsky, Institut für Physikalische und Theoretische Chemie, Universität Würzburg, Campus Hubland Nord, Emil-Fischer-Strasse 42, D-97074 Würzburg, Germany

Telefone: +49 9313186983, E-mail: maxim.ta[email protected]

Abstract Hydrogen bond directionality in the water dimer is explained based on symmetry-adapted intermolecular perturbation theory which directly separates the intermolecular interaction energy into four physically interpretable components: electrostatics, exchange-repulsion, dispersion, and induction. Analysis of these four main contributions to the binding energy allows a deeper understanding of the dominant factors ruling the mutual arrangement of the two monomers. A preference for the linear conguration is shown to be due to a subtle interplay of all the four energy components. While the rst-order terms, electrostatic and exchange-repulsion, almost perfectly cancel each other near the equilibrium geometry of the dimer, the importance of the second and higher-order terms, induction and dispersion, becomes evident.

1

Introduction Understanding intermolecular interactions is probably one of the most intriguing but challenging task of computational

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Figure 1: Denition of the two angular variables for the linear water dimer. chemistry. The interactions between an electron-decient hydrogen atom (attached to a more electronegative atom, like oxygen or nitrogen) and an electron-rich atom (such as oxygen or nitrogen) are ubiquitous in nature, including liquid water, ice and DNA base pairing, to name a few. Water dimer serves naturally as a prototype for the so-called hydrogen bond, and has been (and still is) the subject of both theoretical 1,2 and experimental 3,4 studies. First ab initio and semiempirical calculations on a water dimer were carried out almost 50 years ago (beginning in the late 1960s). Recent highly accurate ab initio calculations culminated in the linear geometry (see Fig. 1) as the global minimum on the potential energy surface of the dimer with the binding energy of around 5 kcal/mol: 5.02(5), 5 5.02, 6 5.01, 7 5.02, 8 4.99(4), 9 (all values are at zero Kelvin and without the zero-point energy correction). To estimate the zero-point energy, the benchmark anharmonic vibrational frequencies from the work of Howard et al. 10 can be used.

Figure 1 near here

The question now arises: What is behind this gure of the weakly bound dimer? Or put it another way: What is the driving force for such an arrangement of the two water monomers? Previous attempts were mainly based on the energy decomposition analysis within the Hartree-Fock theory, 1114 thus ignoring the contribution of the electron correlation to both intra- and intermolecular terms. A more rigorous answer to this question can be readily obtained by utilizing Symmetry-Adapted intermolecular Perturbation Theory (SAPT) 15,16 which gives the interaction energy directly as a sum of four physically well-dened components: electrostatics, exchange-repulsion, dispersion, and induction. This technique, still underappreciated in the community, 17 is a very powerful tool to get deeper insight into the intermolecular forces operating between molecules. Recent developments and ecient implementations of SAPT 1820 enable highly accurate studies of intermolecular interactions at a level comparable to a state-of-the-art methods such as coupled-cluster theory, CCSD(T), but with a much reduced computational eorts. Moreover, it allows a deeper understanding of dierent contributions to the intermolecular interactions thus giving a more detailed picture in comparison with the supermolecular approach where only the total interaction energy is available. Despite the fact that only the total energy is a quantummechanical observable and all partitions into dierent components are ad hoc, the SAPT partitioning is physically sound and very useful for both interpretive purposes and force eld parameterizations. 2123 Below we shall analyze each of the four energy components calculated within SAPT(DFT) as a function of mutual


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arrangements of the two rigid water molecules. Two angular variables we are particularly interested in are dened in

= =

Figure 1. α is the angle between the line connecting two oxygen atoms and the H O H bisector of the acceptor water

=

molecule, whereas β is the angle between the O...O axis and the O H bond of the donor water molecule. These two angles are varied keeping the O...O distance between the two rigid monomers xed at 2.91 Å (see Table S1 of the Supporting Information). Upon either angle variation the other one is kept frozen at its approximate equilibrium value ( α ≈ 46◦ and β ≈ 3◦ ). 24 The focus of the present study is mainly on the relative importance of the four main contributions to the binding energy of water dimer. The crucial points concerning the choice of the monomer geometry, the level of electron correlation needed and the accuracy/completeness of the basis set used upon the binding energy of the water dimer have been discussed previously, see, for example, ref. 25

2

Methods

=

= =

º

The vibrationally averaged monomer geometry of water 24 (r(O H) = 0.9716 Å, ∠H O H = 104.69 ) was used and kept rigid in all dimer calculations. The dimer-centered basis sets, i.e. all atomic basis functions of both monomers were employed in both monomer and dimer calculations. Density-functional theory is used in combination with SAPT to describe intramolecular electron correlation eects (SAPT(DFT)) and all the monomer properties. To speed up the calculations, we employed the density-tting variant 26 of SAPT(DFT) as implemented in the MOLPRO 27,28 program package using the augmented correlation consistent aug-cc-pVQZ basis sets of Dunning 29 along with an appropriate combination of cc-pV(Q+1)Z JK and aug-cc-pVQZ MP2-tting basis sets of Weigend et al. 30,31 To produce accurate results, the asymptotic correction to the exchange-correlation potential was applied. We used the gradient-regulated asymptotic correction approach of Grüning et al., 32 to correct the PBE0 33,34 hybrid exchange and PW91 35 correlation functionals by employing the calculated ionization potentials of 0.4660 Hartree (exp: 0.4638 36 for water along with the HOMO energy of

=0.3339 Hartree obtained using PBE0 functional).

The corresponding shift parameters were approximated by the

dierence between the HOMO energy and the (negative) ionization potential of each monomer. A similar methodology was used previously by Leforestier et al. 37 and Fiethen et al. 38 Accurate evaluation of the dispersion contribution (and its exchange counterpart) is the most time-consuming and dicult part of the SAPT calculation. 39,40 This particular term usually requires the use of extended basis sets. 41,42 To check the accuracy of our SAPT results, the total intermolecular energies were obtained with the CCSD(T) method (only valence electrons were correlated) corrected for the basis set superposition error (BSSE) through the counterpoise correction, 43 and utilizing the aug-cc-pVQZ basis set 29 using the MOLPRO program package. 27,28 The deviations of the SAPT total interaction energies from their CCSD(T) counterparts were not exceeded 0.1 kcal/mol (see Figs. S1-S3 of the Supporting Information). In a recent study by Korona 44 it has been shown (see Table 4 there for the water dimer) that the PBE0 density functional is able to reproduce very closely the results from SAPT with intramonomer electron correlation described by coupled cluster theory limited to single and



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double excitations. To further validate the chosen level of theory, we performed the calculations of the binding energy of the water dimer at the geometry optimized by Lane at the CCSDTQ level of theory. 8 Our values of 4.93 kcal/mol (CCSD(T)/aug-cc-pVQZ, counterpoise corrected) and 4.83 kcal/mol (SAPT(DFT)) come very close to the CBS limit of 5.02 kcal/mol. 8 The total interaction energy calculated within the SAPT(DFT) framework, truncated at second order, is usually written as the sum of individual components containing the rst (electrostatic) and the second (induction and dispersion) order interaction terms accompanied by their respective exchange counterparts (the superscripts 1 or 2 in parentheses refer to the order of the perturbation correction):

(1)

(1)

(2)

(2)

(2)

(2)

Etot = Eelst + Eexch + Eind + Eexch−ind + Edisp + Eexch−disp + δ(HF)

(1)

As no higher than second-order terms are currently implemented in the program we used, third- and higher-order induction and exchange-induction contributions were estimated using the supermolecular approach at the Hartree-Fock level 45 (counterpoise corrected for basis set superposition errors) and added to the sum of rst- and second-order SAPT(DFT) energy contributions:

(1)

(1)

(2)

(2)

δ(HF) = Eint (HF) − Eelst (HF) − Eexch (HF) − Eind (HF) − Eexch−ind (HF)

(2)

where the last two terms on the right-hand side are evaluated at the coupled Hartree-Fock level. To get the dispersion energy, we combine two terms (dispersion and exchange-dispersion) together:

(2)

(2)

Edisp = Edisp + Eexch−disp

(3)

whereas the sum of induction, exchange-induction and higher-order ( δ(HF)) contributions is used to represent the induction energy:

(2)

(2)

Eind = Eind + Eexch−ind + δ(HF)

(4)

Based on the analysis of the third-order induction energy in SAPT, Patkowski et al. 46 have shown that the hybrid approach


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including δ(HF) for the polar molecules (like water) gives reliable results. For brevity, we will skip the superscript (1) for the rst-order electrostatic and exchange-repulsion terms (Eq. 1) in the Figures below.

3

Results and Discussion Analysis of the components of the binding energy based on SAPT suggests that the global minimum conguration

corresponding to the linear dimer is a consequence of a very subtle interplay between the two rst-order contributions: (1)

(1)

electrostatic (Eelst ) and exchange-repulsion ( Eexch ). Both of them have a pronounced angular dependence. While elec-

=

trostatics favors the geometrical arrangements away from α = 0 deg., where the O H bond of one monomer points to the lone pair of the oxygen atom of another one (see Fig. 1), exchange-repulsion acts in the opposite direction making such geometrical arrangements signicantly less favorable. These two contributions, however, cancel each other almost perfectly. This is clearly seen in Figure 2, where the angular dependence of all energy components is shown. The rstorder electrostatic energy is the result of the interaction between the unperturbed charge densities of the two monomers, thus violating the Pauli exclusion principle (with respect to the exchange of electrons between the interacting molecules). Therefore, at short and intermediate range, the rst-order exchange-repulsion contribution should be taken into account.

Figure 2 near here

A closer look at the separate contributions (see Fig. 2, top) reveals that a particular conguration, corresponding to an almost linear O-H · · · O arrangement (known as a ngerprint of the hydrogen bonding in water 47 ), is a result of the (2)

(2)

attractive second-order contributions ( Eind and Edisp ), being equally important, modulated by the sum of the rst-order (1)

(1)

terms (Eelst + Eexch ). As can be seen from Fig. 2 (top right), the weak dependence of all contributions (the sum of the rst-order terms, induction and dispersion), and hence the total energy, on the angle α indicates that the hydrogen acceptor molecule is quite mobile as noted long before. 48 The dependence of the SAPT components on the β angle is, on the other hand, more pronounced (see Fig. 2, bottom right). It can be readily seen that the second-order terms (especially induction) favor a close to linear arrangement of the two water molecules (where β ∼ 0 deg.). We can give a simple explanation: on getting closer, the electron-decient hydrogen atom of the donor molecule starts to signicantly polarize the electron-rich oxygen atom of the acceptor molecule. Hence, the polarizability (and not just its electronegativity!) of the electron-rich atom, especially its lone pair regions, along with the polarizing ability of the electron-decient hydrogen atom (a more deshielded proton could polarize better!) would be the important factors which inuence the binding energy. Again, the rst-order terms act in exactly opposite directions just nicely mirroring the behavior of each other (see Fig. 2, bottom left). If we now simultaneously change both angles (their precise values are given in Table S3 of the Supporting Information) so that to mimic a transition from the global minimum conguration (linear) to the bifurcated one shown in Fig. 3 (which is known to be a rst-order saddle point 6,49 ), the following picture emerges (see Fig. 4).



2

8 6 4 2 0 −2 −4 −6 −8

ESAPT (kcal/mol)

ESAPT (kcal/mol)

Eelst + Eexch Edisp Eind Etot

1

Eelst Eexch Edisp Eind Etot

0

−1 −2 −3 −4

−40

−20

0 20 α (deg)

40

−5

60

−40

−20

0 20 α (deg)

40

60

3

8 6 4 2 0 −2 −4 −6 −8

ESAPT (kcal/mol)

2

ESAPT (kcal/mol)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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1 0

−1 −2


−20

0

20 40 β (deg)

60

−3


−4 −5

80

−20

0

20 40 β (deg)

60

80

Figure 2: Angular dependence of the SAPT energy components (in kcal/mol) for the linear water dimer with the O...O distance xed at 2.91 Å. The vertical broken line marks the approximate equilibrium angle ( α ≈ 46◦ and β ≈ 3◦ ). Top: the value of β is set to the approximate equilibrium value. Bottom: the value of α is set to the approximate equilibrium value. See Figure 1 for the denition of the angles α and β .


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Figure 3: Two geometrical arrangements of the water dimer: linear (left) and bifurcated (right).

8

0

6


4 2


−1

ESAPT (kcal/mol)

ESAPT (kcal/mol)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


−2

0

−3

−2 −4

−4

−6 −8

0

1

2 3 pathway

4

−5

5

0

1

2 3 pathway

4

5

Figure 4: The change of the SAPT energy components (in kcal/mol) along the pathway from the linear conguration (marked as 0) to the bifurcated one (marked as 5). See Figure 3 for the geometries.

Figure 3 near here

Figure 4 near here

As in the two cases above, the sum of the rst-order terms contributes only insignicantly to the binding energy. Moreover, taken together, they favor slightly the bifurcated structure. It is the induction term that plays an important role in the energy lowering for the linear conguration. Indeed, as can be clearly seen from Fig. 4 (right), the induction curve along the pathway nicely mimics the total energy curve, which is quite remarkable! That a combination of intra- and intermolecular charge delocalization eects plays the critical role along the transition pathway has been shown recently by utilizing a quite dierent methodology based on an energy decomposition analysis for second-order MøllerPlesset perturbation theory based on absolutely localized molecular orbitals. 50 In addition to the variation of two angles, it is also interesting to analyze the various energy contributions as a function of the intermonomer separation. Figures 5 and 6, respectively, display the SAPT energy decomposition for both the linear and bifurcated dimers.




30 20 10 0

−10 −20 1.5


15 ESAPT (kcal/mol)

ESAPT (kcal/mol)

40

10 5 0 −5

−10 2

2.5 3 RO...H (Å)

3.5

4

1.5

2

2.5 3 RO...H (Å)

3.5

4

Figure 5: SAPT energy components (in kcal/mol) as a function of the intermonomer separation for the linear conguration. See Figure 3 for the geometries.

20

8


15 10 5 0

4 2 0

−2

−5

−10 2.5


6 ESAPT (kcal/mol)

ESAPT (kcal/mol)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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−4 3

3.5 RO...O (Å)

4

4.5

2.5

3

3.5 RO...O (Å)

4

4.5

Figure 6: SAPT energy components (in kcal/mol) as a function of the intermonomer separation for the bifurcated conguration. See Figure 3 for the geometries.

Figure 5 near here

Figure 6 near here

It can be seen that the exchange-repulsion term changes more steeply than the electrostatic contribution so that their sum becomes more and more repulsive as the intermonomer distance decreases. At the same time, the rst-order term near the minimum energy geometry is attractive albeit insignicantly. The water dimer is in this respect quite unique which has been pointed out recently by Misquitta 51 (see also a discussion in ref. 52 ). For many medium to large neutral dimers at their equilibrium intermolecular geometries, the rst-order term is usually strongly repulsive (see below). We also note that for the linear conguration the induction term becomes more attractive than the dispersion one (especially at shorter


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distances), whereas the opposite is true for the bifurcated dimer. We have also investigated the energy contributions for the ten stationary points on the potential energy surface of the water dimer taken from the literature. 53 For all these dimers, the sum of the rst-order electrostatic and exchange-repulsion contributions is close to zero thus making the sum of the induction and dispersion terms the important contributor to the total energy of the observed congurations. Interestingly, the dispersion-to-induction ratio is found to depend on a particular arrangement of the monomers in the dimer (see Table S2 in the Supporting Information). We should, therefore, stress that an accurate description of the second and higher-order contributions, both induction and dispersion (describing electron charge redistribution), is mandatory in order to explain geometrical preferences in the water dimer. It should be pointed out that, in the SAPT context, a further separation of mutual perturbation of interacting monomers into internal polarization and external charge transfer is not done (see a discussion in refs. 54,55 ). There is still some debate in the literature on the importance of the charge-transfer contribution in the water dimer. 5664 Even though the signicance of this particular term might be overemphasized in the past, recent developments of Misquitta, 65 based on SAPT formalism, can shed more light on this issue. It should also be mentioned that there is no unique method to perform such a separation and that two dierent methods can give drastically dierent values of the charge-transfer contribution to the interaction energy. Interestingly, in one of the rst attempts to understand the hydrogen-bonding phenomenon, Umeyama and Morokuma 12 based on the energy decomposition analysis developed by Kitaura and Morokuma 66,67 have emphasized the importance of the induction energy (divided by them into polarization and charge-transfer) to the binding energy of the water dimer.

In their work, however, the dispersion contribution was ignored completely despite the

important nding by Jeziorski and van Hemert 68 (essentially at the same time!) that the dispersion term should also play an important role (see also ref. 6971 and references therein). Utilizing the same geometry for the water dimer as used by Jeziorski and van Hemert 68 , we found that their value of -1.54 kcal/mol underestimates somewhat our value (-1.90 20 kcal/mol). Such a discrepancy is mainly a result of calculating the Edisp (HF) term only and not accounting for other 21 22 terms (Edisp and Edisp ) in that earlier estimation of the dispersion contribution.

=

Interestingly, the classical model based on the dipole-dipole interactions ( Edip−dip = (R2 µA µB 3(µA R)(µB

R))/(4π0 R5 )) of the two rigid water molecules at a center-of-mass distance

R

apart (with the molecular dipole, µ, of 1.87

D and ε0 =1) qualitatively reproduces only the α−dependence of our reference total energy (see Fig. 7, left), whereas incorrectly predicts the bifurcated conguration to be the lowest energy structure (Fig. 7, right), which is a well-known failure. 72

Figure 7 near here

It seems that such a preference for the bifurcated geometry is not a consequence of using rigid monomers. To prove that, we have calculated the electron densities at the MP2(full)/aug-cc-pVQZ level of theory using the optimized geometries of



2 Total Energy (kcal/mol)

ESAPT Edip−dip

−1 −2 −3 −4 −5

−40

−20

0 20 α (deg)

40

60

1

ESAPT Edip−dip

Total Energy (kcal/mol)

0 Total Energy (kcal/mol)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

0 −1 −2 −3 −4 −5

−20

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0

20 40 β (deg)

60

80

−3

−4

−5

ESAPT Edip−dip

0

1

2 3 pathway

4

5

Figure 7: Angular dependence of the total interaction energy from SAPT (in kcal/mol) against dipole-dipole interactions for the water dimer with the O...O distance xed at 2.91 Å. See Figure 1 for the denition of the angles α and β . The pathway denes the transition from the linear conguration (marked as 0) to the bifurcated one (marked as 5). See Figure 3 for the geometries. both dimers taken from Tschumper et al. 6 The dipole moments of the monomers ( µA and µB ) were obtained applying Bader topological analysis 73 to partition the electron charge density among the monomers in a dimer and integrating 74 the electron density over the atomic basins belonging to each monomer. According to the theory of atoms in molecules, the monomer dipole moment in a dimer is enhanced by around 15% (donor) and 10% (acceptor) for the linear and by around 18% (donor) and 4% (acceptor) for the bifurcated congurations as compared to the value of 1.87 D calculated for the isolated water molecule (at the MP2(full)/aug-cc-pVQZ level). Our results agree with those by Handley and Popelier, 75 where a similar approach was used, and with the ndings by Gregory et al., 76 and by Piquemal et al. 77 despite the fact that a completely dierent partitioning of the electron density was used in those studies, and with those by Batista et al. 78 . In the latter work, the inuence of the partitioning scheme on the water monomer multipole moments has been

=

=

analyzed. Even though the calculated energies ( Edip−dip ) for the linear ( 3.6 kcal/mol) and bifurcated ( 4.5 kcal/mol) congurations turned out to be slightly more negative than those calculated for the rigid monomers (see Fig. 7, right), the preference for the bifurcated geometry, as judged solely from the dipole-dipole interactions, is conserved. We should also point out that at the O...O distance near the equilibrium geometry ( ∼2.9 Å) the dipole-dipole interaction model is not even able to reproduce the rst-order electrostatic contribution either. Thus, some widely used simplied electrostatic models are not able to capture the right physics behind the intermolecular interactions and need to be revised. 79 To get an insight why the linear geometry is preferred, the analysis of various contributions calculated by SAPT(DFT) is particularly instructive. In Table 1 are compared the energy contributions for both dimers calculated using two dierent monomer geometries (vibrationally averaged and fully optimized).

Table 1 near here

We discuss the results obtained for the fully optimized geometries since the case with the vibrationally averaged monomers (1)

(1)

is qualitatively similar. We note that the sum of the rst-order terms ( Eelst + Eexch ) is more negative in the case of the


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Table 1: SAPT(DFT) energy decomposition for the two water dimers (in kcal/mol). (1) Eelst

Vib. av.a Opt.b

(1)

Eexch

=7.9558 =8.0091

7.8026 7.8661

= =

(2)

Edisp

=3.0168 =3.0184 = =

(2)

(2)

Eexch−disp

Eind

(2)

Eexch−ind

δ(HF)

Etot

linear 0.6078 3.4370 2.0497 0.9379 4.8874 0.6148 3.4753 2.1068 0.9155 4.8306 bifurcated 0.2942 1.0319 0.6847 0.1421 3.0542 0.3289 1.2104 0.8078 0.1664 3.3052 water monomer is used: r(O H) = 0.9716 Å,

Challenging Dogmas: Hydrogen Bond Revisited - The Journal of

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