Dynamic Behavior of Hydrogen Bonds from Pure Closed Shell to

Article pubs.acs.org/JPCA

Dynamic Behavior of Hydrogen Bonds from Pure Closed Shell to Shared Shell Interaction Regions Elucidated by AIM Dual Functional Analysis Satoko Hayashi, Kohei Matsuiwa, Masayuki Kitamoto, and Waro Nakanishi* Department of Material Science and Chemistry, Faculty of Systems Engineering, Wakayama University, 930 Sakaedani, Wakayama 640-8510, Japan S Supporting Information *

ABSTRACT: The dynamic behavior of hydrogen bonds (HBs) was clarified for the wide range of interactions applying AIM dual functional analysis. Plots of Hb(rc) versus Hb(rc) − Vb(rc)/2 are analyzed in the polar (R, θ) representation, where Hb(rc) and Vb(rc) are total electron and potential energy densities at bond critical points, respectively, for the fully optimized structures. Data of the fully optimized structure and four perturbed ones around it are plotted for each interaction, which give a specific curve. The curve is analyzed by (θp, κp): θp corresponds to the tangent line from the y-direction and κp is the curvature. Whereas (R, θ) correspond to the static nature, (θp, κp) represent the dynamic nature of interactions. Indeed, HBs can be classified only by one parameter of θ, but θp supplies more information necessary for better understanding of HBs. Although H2Se-*-HSeH and H3N-*-HNH2 show the nature of pure CS (closed shell) of the vdW-type, H2S-*-HSH and H2O-*-HOH contain the nature of pure CS other than the vdW-type (HB-typical). The regular CS nature is observed for B-*-HF (B = HF, H2Se, H2S, H2O, and H2CO). The HF-*HF interaction is described as HB-typical, whereas others are by CTMC-type. The nature of H3N-*-HX (X = F, Cl, Br) is regular CS of the CTTBP-type. HBs in charged species, such as [HOH-*-OH]− and [H2O-*-H-*-OH2]+, show the weak covalent nature of SS (shard shell). The dynamic behavior of HBs helps us to understand HBs in more detail, in addition to the static behavior.

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bonds and interactions.35−39 Electron densities at bond critical points (BCPs: rc, *) of (ω, σ) = (3, −1)33 (ρb(rc)) are strongly related to the binding energies14a,40−45 and bond orders.46 The sign of Laplacian ρb(rc) (∇2ρb(rc)) indicates that ρb(rc) is depleted or concentrated with respect to its surroundings, because ∇2ρb(rc) is the second derivative of ρb(rc). ρb(rc) is locally concentrated relative to the average distribution around BCP if ∇2ρb(rc) < 0, but it is depleted when ∇2ρb(rc) > 0. Total electron energy densities at BCPs (Hb(rc)) are the sum of kinetic energy densities (Gb(rc)) and potential energy densities (Vb(rc)) at BCPs. Electrons at BCPs are stabilized when Hb(rc) < 0; therefore, interactions exhibit the covalent nature in this region, whereas they exhibit no covalency if Hb(rc) > 0 due to the destabilization of electrons at BCPs under the conditions. The positive sign of Hb(rc) changes to negative as interactions

INTRODUCTION Hydrogen bonds (HBs) are fundamentally important by their ability of the molecular association due to the stabilization of the system in energy. The direction-control through the formation of HBs plays a crucial role in all fields of chemical and biological sciences.1−32 Although weak HBs contain the van der Waals (vdW) nature, strong ones show that of weak covalent bonds (Cov-w). This variety of HB enriches chemical and biological sciences. HBs control various chemical processes depending on the strength. That the duplex DNA structure once opens and then closes in active proliferation at around room temperature is a typical example. It is inevitable to evaluate, classify, and understand the nature of HBs for better understanding of chemical processes controlled by HBs. The understanding of HBs has been much growing, recently, through the analysis.7−11,31 Here, we discuss the dynamic nature of HBs elucidated by the AIM dual functional analysis, in addition to the static nature. AIM (atoms-in-molecules method), proposed by Bader,33,34 enables us to analyze, evaluate, and classify the nature of chemical © XXXX American Chemical Society

Received: October 6, 2012 Revised: January 24, 2013

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dx.doi.org/10.1021/jp3098928 | J. Phys. Chem. A XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry A

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Scheme 1. Classification of Interactions by AIM Functions, Where Hb(rc) − Vb(rc)/2 = (ℏ2/8m)∇2ρb(rc)

Scheme 2. Requirements for Data to Appear in a Certain Quadrant in the Plots of Hb(rc) versus Hb(rc) − Vb(rc)/2, Where Hb(rc) − Vb(rc)/2 = (ℏ2/8m)∇2ρb(rc)

the classification shown in Scheme 1, because (ℏ2/8m)∇2ρb(rc) = Hb(rc) − Vb(rc)/2 (eq 2), of which signs necessary for the classification are the same with each other. The classification by the signs of ∇2ρb(rc) and Hb(rc) can be achieved only by one parameter of θ, as shown in Scheme 1 (cf. Figure 1).49,50 We proposed the AIM dual functional analysis of weak to strong interactions by plotting Hb(rc) versus Hb(rc) − Vb(rc)/2,49 after the proposal of Hb(rc) versus ∇2ρb(rc).48 Two treatments are essentially the same with each other, because Hb(rc) − Vb(rc)/2 = (ℏ2/8m)∇2ρb(rc) (eq 2). The treatments incorporate the classification shown in Scheme 1.49,50 The former is obtained through reducing the x-axis of the latter by 1/8 in atomic units according to eq 2. Scheme 2 shows the requirements for data to appear in certain quadrant in the proposed plots of Hb(rc) versus Hb(rc) − Vb(rc)/2. Data appear in first quadrant if they are pure CS, in the fourth quadrant for the regular CS, and those of SS drop in the third quadrant. No data appear in the second one. The results are essentially the same as the classification shown in Scheme 1. Both axes in Scheme 2 are given in energy unit; therefore, distances on the (x, y) = (Hb(rc) − Vb(rc)/2, Hb(rc)) plane can be expressed in energy units. In our treatment, data at BCPs of perturbed structures for interactions in question around the fully optimized structures are also employed for the plots, in addition to those of the fully optimized ones. The interaction distances in question are suitably fixed longer and shorter than those of fully optimized structures.48−50 Plots for weak to strong interactions show spiral streams. Though data for fully optimized structures form an averaged stream in the plots, as a whole, those

become stronger than pure closed-shell (CS) interactions obtaining some covalency. Therefore, Hb(rc) must be a more appropriate measure for weak interactions on the energy basis.33,34,47−51 Equations 1, 2, and 2′ represent the relations between Hb(rc), Gb(rc), Vb(rc), and ∇2ρb(rc). Hb(rc) = G b(rc) + Vb(rc)

(1)

(ℏ2 /8m)∇2 ρb (rc) = Hb(rc) − Vb(rc)/2

(2)

(ℏ2 /8m)∇2 ρb (rc) = G b(rc) + Vb(rc)/2

(2′)

Chemical bonds and interactions are classified by the signs of ∇2ρb(rc) and Hb(rc). Scheme 1 summarizes the classification. Interactions in the region of ∇2ρb(rc) < 0 are called shared-shell (SS) interactions and they are CS interactions for ∇2ρb(rc) > 0. Hb(rc) must be negative when ∇2ρb(rc) < 0, because Hb(rc) are larger than (ℏ2/8m)∇2ρb(rc) by Vb(rc)/2 where Vb(rc) are negative at all BCPs (eq 2). Consequently, ∇2ρb(rc) < 0 (and Hb(rc) < 0) for the SS interactions. The CS interactions are especially called pure CS interactions for Hb(rc) > 0 (and ∇2ρb(rc) > 0), because electrons at BCPs are depleted and destabilized under the conditions.33 Electrons in the intermediate region between SS and pure CS, which belong to CS, are locally depleted but stabilized at BCPs, because ∇2ρb(rc) > 0 but Hb(rc) < 0.42 The redistribution of ρb(rc) occurs between those electronic states in this region. We will call the interactions in this intermediate region regular CS, when it is necessary to distinguish from pure CS. ∇2ρb(rc) can be replaced by Hb(rc) − Vb(rc)/2 in B



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reason for the interactions classified only by θ, which are usually done by Hb(rc) and ∇2ρb(rc), as shown in Scheme 1.48−50

for perturbed structures do local streams, together with those of fully optimized ones. How can the perturbed structures be generated, necessary to evaluate the dynamic behavior of interactions? We proposed to employ the motion at the zero-point energy,52 which is given by normal coordinates of internal vibrations (NIV). NIV in the Gaussian output are used to generate the perturbed structures.52,53 The magnitude for the displacement of an interatomic distance in question is amplified to Δr = wao (w = ±0.05 and ±0.1 and ao being Bohr radius) (cf. eq 9), because it is necessary to adjust the magnitude to apply NIV for the AIM dual functional analysis. This method is called NIV. The motion in NIV is closely related to the quantified adiabatic process, because it is originated from the motion at the zero-point energy of molecules and adducts. We often employ a primitive method to generate the perturbed structures, which is called POM (the partial optimization method).49,50 The perturbed structures are obtained by POM with the length of an interaction in question being fixed suitably (cf. eq 9). A perturbed structure by POM must exist on the potential energy surface; therefore, POM should be closely related to the thermal process. We also proposed the concept of “dynamic nature of interaction”, which originates from the perturbed structures in addition to the fully optimized structures. The behavior of interactions analyzed with the data of fully optimized structures corresponds to the static nature, whereas the dynamic nature for an interaction arises from the data of the perturbed structures with the fully optimized structure. The dynamic nature is evaluated at the fully optimized structure, as if it were extrapolated. We consider this method to be well-suited to clarify the dynamic nature of HBs of wide range from pure CS to SS interaction regions, together with static one on the basis of AIM dual functional analysis. Survey of AIM Dual Functional Analysis. Figure 1 and eqs 3−7 explain the method to analyze the plots of Hb(rc) versus

R = (x 2 + y 2 )1/2

(3)

θ = 90° − tan−1(y/x)

(4)

θp = 90° − tan−1(dy/dx)

(5)

κ p = |d2y/dx 2| /[1 + (dy/dx)2 ]3/2

(6)

k = Vb(rc)/G b(rc)

(7)

where (x, y) = (Hb(rc) − Vb(rc)/2, Hb(rc)). Each plot for an interaction shows a specific curve, as shown in Figure 1, which must provide important information about the interaction. The curve is expressed by (θp, κp): θp defined by eq 5 corresponds to the tangent line measured from the y-direction and κp is the curvature of the plot at BCP, which is given by eq 6. Equation 7 defines k, which is equal to Vb(rc)/Gb(rc) and serves as a nice parameter to analyze weak to strong interactions. Whereas (R, θ) correspond to the static nature, (θp, κp) represent the dynamic nature of interactions. Methodological Details in Calculations. Equation 8 explains the method to generate perturbed structures with NIV.52 The kth perturbed structures in question (Skw) are obtained by adding the normal coordinates of the kth internal vibration (Nk) printed out in the frequency analysis to the standard orientation of a fully optimized structure (So) in the matrix representation.56 The coefficient f kw in eq 8 controls the difference in structures between Skw and So: f kw is determined to satisfy eq 9 for r in question in the kth perturbed structures, where r and ro are the interaction distances in the perturbed and fully optimized structures, respectively.57 Therefore, r in Skw must be longer and shorter than ro in So by 0.05ao and 0.1ao. Nk of five digits are used to predict Skw, although only two digits are usually printed out.58 Skw = So + fkw ·Nk

(8)

r = ro + wao (w = (0), ±0.05, and ±0.1; ao = 0.52918 Å)

(9)

The perturbed structures are also generated with POM, where a molecule or an adduct is (partially) optimized with an interaction in question (r) being fixed to satisfy eq 9. Therefore, r in the perturbed structures must be fixed longer and shorter than ro by 0.05ao and 0.1ao, with other structural parameters being at the minimum values. Molecules and adducts were optimized with the 6-311++G(3df,3pd) basis sets of the Gaussian 03 program.59 The Møller− Plesset second-order energy correlation (MP2) level was applied to the calculations.60 BSSE is not considered.61 The HB adducts were usually optimized assuming the suitable symmetry at the initial stage. However, the symmetry sometimes reduced to lower one containing C1 in the optimization process. The complexes were also optimized assuming C1 from the initial stage when the optimization seemed not well converged at the higher symmetry. Frequency analysis was performed on all optimized structures. Each optimized structure was confirmed by all positive frequencies of it. AIM functions were calculated with the Gaussian 03 program59 and analyzed by the AIM2000 program.62 Each plot of Hb(rc) versus Hb(rc) − Vb(rc)/2 for an interaction gave an curve with the data of five points (w = 0, ±0.05,

Figure 1. Polar (R, θ) coordinate representation of Hb(rc) versus Hb(rc) − Vb(rc)/2. A regression curve is also shown, which connects the data at BCPs for perturbed and fully optimized structures.

Hb(rc) − Vb(rc)/2 in the AIM dual functional analysis. Data of fully optimized structures in the plots are analyzed employing the polar (R, θ) coordinate and those for perturbed structures with a fully optimized one are by the (θp, κp) parameters, as shown in Figure 1.49,50,52,53 R in (R, θ) is defined by eq 3, which is given in the energy unit. R corresponds to the energy for an interaction at BCP relative to that at the origin, where no interactions must be detected at the origin (Hb(rc) = Hb(rc) − Vb(rc)/2 = 0).49,50,52,54 θ defined by eq 4 controls the spiral stream of the plot, which is measured from the y-axis. The acceptable range of θ is limited to 45.0° < θ < 206.6°.49,50,55 The range is further divided into 45.0° < θ < 90.0° for pure CS, 90.0° < θ < 180.0° for regular CS, and 180.0° < θ < 206.6° for SS interactions (Scheme 1). This is the C



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and ±0.1 in eq 9). The curve was analyzed using an auxiliary regression curve assuming a cubic function,63 as shown in eq 10. The R2 values (square of correlation coefficients) were usually very good (>0.99999). Equations 11 and 12 show the first and second derivatives of eq 10, respectively, necessary to evaluate θp and κp according to eqs 5 and 6, where (x, y) = (Hb(rc) − Vb(rc)/2, Hb(rc)).50

Table 1. Energies Evaluated for the Formation of Hydrogen Bonded Adducts at the MP2 Level,a Together with the Optimized Distances and Symmetry

y = ao + a1x + a 2x 2 + a3x 3 (R2: square of correlation coefficient)

(10)

y′ = a1 + 2a 2x + 3a3x 2

(11)

y″ = 2a 2 + 6a3x

(12)

Various neutral (1−23) and charged (24−27) hydrogen bonded species were optimized at the MP2 level,60 together with the components, as shown in Table 1 (see also Table S1 of the Supporting Information). The θp and κp values are evaluated for 1−27 with NIV and POM, where X-*-Y denotes an atomic interaction lines33,48 with BCP by an asterisk between X and Y and X--*--Y will be used if it is necessary to emphasize the weakness of an interaction, although tentative. The method to evaluate θ p and κ p is already established. 48−50,52,64 The outline of the treatment will be explained and the results are discussed exemplified by H−F (C∞h), H2O (C2v), HF---H−F (4: C1, very close to Cs), and H2O---H−OH (8: Cs), prior to the detailed discussion on 1−27.

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RESULTS AND DISCUSSION Optimized Structures of Hydrogen Bonded Species and the Stabilities. Table 1 collects the optimized HB distances in question (ro) and the distances relative to the sum of the vdW radii65 (Δr = ro(X, Y) − ΣrvdW), together with the relative energies from the components [ΔE = E(HB adduct) − E(components)] and the symmetry of the adducts for various neutral (1−23) and charged (24−27) hydrogen bonded species optimized at the MP2 level.60,61 The relative energies in Table 1 are well correlated to those evaluated with the aug-cc-pVDZ basis sets at the MP2 Level, reported earlier.64 How do ΔE correlate to Δr in HBs? ΔE are plotted versus Δr, separately by the neutral and charged species. Figure 2 shows the plots. The plots are analyzed assuming the linear correlation (y = ax + b (R2: square of correlation coefficient)). Equations 13 and 14 show the correlations for the neutral and charged species, respectively. Although the correlation for the latter was very good, that for the former was not so. The plot for the neutral species seems better to be analyzed assuming the quadratic function (y = ax2 + bx + c (R2: square of correlation coefficient)). Equation 15 shows the correlation. ΔE = 34.17Δr + 2.07

ΔE = 52.99Δr + 47.73

(R2 = 0.882)

(R2 = 0.997)

ΔE = −31.49(Δr )2 − 5.32Δr − 8.18

species (X-*-Y)

ro(X,Y) (Å)

Δrb (Å)

ΔEc,d (kJ mol−1)

symmetry

H2Se--*--HSeH (1) H2S--*--HSH (2) H3N--*--HNH2 (3) H2O-*-HOH (4) H3N-*-HOH (5) HBr--*--HBr (6) HCl--*--HCl (7) HF-*-HF (8) H2Se-*-HBr (9) H2Se-*-HCl (10) H2Se-*-HF (11) H2S-*-HBr (12) H2S-*-HCl (13) H2S-*-HF (14) H2O-*-HBr (15) H2O-*-HCl (16) H2O-*-HF (17) H2CO-*-HBr (18) H2CO-*-HCl (19) H2CO-*-HF (20) H3N-*-HBr (21) H3N-*-HCl (22) H3N-*-HF (23) H2O-*-HNH3+ (24) H3N-*-HNH3+ (25) [HOH-*-OH]− (26) [H2O-*-H-*-OH2]+ (27)

2.9375 2.7998 2.2845 1.9427 1.9585 2.7078 2.5109 1.8196 2.6194 2.5726 2.4078 2.4919 2.4410 2.2719 1.9304 1.8727 1.7054 1.8868 1.8477 1.7069 1.7176 1.7459 1.6837 1.6328 1.5611 1.3429 1.1941

−0.1625 −0.2002 −0.4655 −0.7773 −0.7915 −0.3422 −0.4391 −0.8504 −0.4806 −0.5274 −0.6922 −0.5081 −0.5590 −0.7281 −0.7896 −0.8473 −1.0146 −0.8332 −0.8723 −1.0131 −1.0324 −1.0041 −1.0663 −0.7615 −1.1889 −1.3771 −1.8559

−7.6 −8.7 −13.8 −22.2 −28.2 −8.3 −10.0 −20.7 −13.9 −15.5 −21.3 −14.6 −16.6 −23.2 −20.7 −24.7 −38.4 −22.4 −25.9 −36.3 −33.7 −38.0 −54.8 −87.3 −112.6 −119.9 −145.8

Cs Cs Cs Cs C1 Cs Cs C1e Cs Cs Cs Cs Cs Cs Cs Cs Cs Cs Cs Cs C3v C3v C3v Cs C3 C1 C2

With the 6-311++G(3df,3pd) basis set. bΔr = ro(X,Y) − ΣrvdW (vdW radii of Bondi being employed; see ref 65). cΔE = E(HB adduct) − E(components)]. dTotal energies of E(HB adduct) and E(components) are given in the Table S1 of the Supporting Information. eVery close to Cs. a

(13) (14)

(R2 = 0.931)

Figure 2. Plots of ΔE versus Δr for various HBs. Data for neutral and charged species are plotted separately. Those for H3N-*-HF are deviated from the correlation.

(15)

Data for H3N-*-HF deviated somewhat from the correlation for the neutral species. Whereas the magnitudes of ΔE are larger than 87 kJ mol−1 for the charged species, the values are less than 39 kJ mol−1 for the neutral species, except for H3N-*-HF. The

magnitude for H3N-*-HF of 55 kJ mol−1 is just the intermediate between the two groups and each group is analyzed by the D



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different correlation. As shown in Figure 2, the ΔE values seem to deviate downward of the correlation for the neutral species if Δr values become smaller than −1.000 Å. The values are less than −1.000 Å for H2O-*-HF, H2CO-*-HF, H3N-*-HBr, H3N-*-HCl, and H3N-*-HF. The Δr values of the HF adducts and the H3N adducts are less than −1.000 Å. Consequently, Δr of H3N-*-HF must be strongly controlled by the two factors. The contribution from the (Δr)2 term also operates to decrease ΔE. It must be larger than that expected from eq 15 for H3N-*-HF. However, the reason for the deviation of H3N-*-HF must be more complex. The higher terms would operate to decrease ΔE of H3N-*-HF. The contribution from the (Δr)2 term, together with the higher terms, seems larger than that expected from eq 15 to decrease ΔE for H3N-*-HF with Δr = −1.066 Å. This must be one of reasons for the deviation. The ΔE values of B-*-HX (B = H2Se, H2S, H2CO, and H3N; X = F, Cl, and Br) are plotted versus those of H2O-*-HX (X = F, Cl, and Br), where the values of common X are compared. Figure 3 shows the plots. The plot of ΔE(H2O-*-HX) versus

ΔE(H3N‐*‐HX) = 1.204ΔE(H 2O‐*‐HX) − 8.58 (X = F, Cl, and Br: R2 = 0.9996)

(19)

ΔE(Table 1): B‐*‐HBr (ΔE = −8.3 kJ mol−1 for B = H 2O) ≫ B‐*‐HCl ( −24.7 for B = H 2O) ≫ B‐*‐HF ( −38.4 for B = H 2O) (B = H 2Se, H 2S, H 2CO, H 2O, and H3N)

(20)

ΔE(Table 1): HF‐*‐HF (ΔE = −20.7 kJ mol−1) > H 2Se‐*‐HF ( −21.3) > H 2S‐*‐HF ( −23.2) ≫ H 2CO‐*‐HF ( −36.3) > H 2O‐*‐HF ( −38.4) ≫ H3N‐*‐HF ( −54.8)

(21)

ΔE(Table 1): HCl‐*‐HCl (ΔE = −10.0 kJ mol−1) > H 2Se‐*‐HCl (− 15.5) > H 2S‐*‐HCl ( − 16.6) > H 2O‐*‐HCl ( −24.7) > H 2CO‐*‐HCl ( − 25.9) ≫ H3N‐*‐HCl ( −38.0)

(22)

ΔE(Table 1): HBr‐*‐HBr(ΔE = −8.3 kJ mol−1) > H 2Se‐*‐HBr (− 13.9) ≥ H 2S‐*‐HBr (− 14.6) > H 2O‐*‐HBr (− 20.7) > H 2CO‐*‐HBr (− 22.4) ≫ H3N‐*‐HBr (− 33.7)

(23)

ΔE(Table 1): H 2Se‐*‐HSeH ( −7.6 kJ mol−1) ≥ HBr‐*‐HBr ( −8.3 kJ mol−1) ≥ H 2S‐*‐HSH ( −8.7) > HCl‐*‐HCl ( −10.0 kJ mol−1) > H3N‐*‐HNH 2 ( −13.8) > HF‐*‐HF ( −20.7 kJ mol−1) > H 2O‐*‐HOH (− 22.2) Figure 3. Plots of ΔE for B-*-HX (B = H2Se, H2S, H2CO, and H3N) versus ΔE for H2O-*-HX with X = F, Cl, and Br.

(24)

Basicity (−ΔH °): HF (484.0 kJ mol−1) < HCl (556.9) < HBr (584.2) < H 2O (691.0) < H 2S (705.0)

ΔE(H2O-*-HX) is also shown in Figure 3 by the dotted line, for convenience of comparison. The correlations are very good, which are exhibited in eqs 16−19. The results show that ΔE of B-*-HX (B = H2Se, H2S, H2CO, H2O, and H3N; X = F, Cl, and Br) are well correlated with each other, if ΔE of common X are compared. The magnitudes of ΔE become larger in the order of H2Se < H2S ≪ H2O < H2CO ≪ H3N for X = F, although the order changes partially to H2CO < H2O when X = Cl and Br are compared.

< MeCOMe (812.0) < H3N (853.6) Acidity ( −ΔH °): H3N ( −1691.6 kJ mol−1)

< H 2O ( −1634.7) < HF ( −1554.0) < H 2S (− 1469.0) < HCl ( −1395.0) < HBr (− 1353.0)

(16)

ΔE(H 2S‐*‐HX) = 0.485ΔE(H 2O‐*‐HX) − 4.57 (X = F, Cl, and Br: R2 = 0.9999)

(17)

ΔE(H 2CO‐*‐HX) = 0.781ΔE(H 2O‐*‐HX) − 6.39 (X = F, Cl, and Br: R2 = 0.9991)

(26)

The very good correlations shown in eqs 16−19 remind us that the common mechanisms are operating mainly in the formation of the HB adducts. The order of ΔE for B-*-HX (B = H2Se > H2S ≫ H2O ≈ H2CO ≫ H3N; X = F, Cl, and Br) shown by eqs 20−23 are essentially explained by the basicity measured in gas phase (eq 25),66 where the basicity for H2CO is estimated to be somewhat smaller than 812 kJ mol−1. Indeed, the order of ΔE for H2O and H2S is not reproduced by the basicity, but the values are very close with each other in eq 25. However, the contribution of HX in B-*-HX to ΔE cannot be explained by the acidity in gas phase. The results may show that the mechanism other than the acid−base type interaction operates to stabilize the HB adducts. The interaction with some covalency would be

ΔE(H 2Se‐*‐HX) = 0.420ΔE(H 2O‐*‐HX) − 5.17 (X = F, Cl, and Br: R2 = 0.99999)

(25)

(18) E



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Figure 4. Some internal vibrations illustrated for H−F (C∞h), H2O (C2v: ∠HOH = 104.04°), HF---H−F (8) (C1, very close to Cs: ∠FHF = 174.06°), and H2O---H−OH (4) (Cs: ∠OHO = 172.98°). The largest displacement of the interaction in each vibration is also shown by r1, r2, and/or r3. Arrows in some νk are drawn using −Nk, in place of Nk, for better description.

Table 2. Functions and Parameters for Some HB Interactions Evaluated with NIV and POMa,b adduct: method

interaction with BCP

frequency (cm−1)

NIV-1 (N1)

F-*-H

4176.4

NIV-1 (N1) NIV-2 (N5)

F2--*--H1 H1-*-F1 F2--*--H1 H2-*-F2 F2--*--H1 F2--*--H1 H1-*-F1 F2--*--H1 H1-*-F1 F2--*--H1

166.9 4044.5

NIV-1A (N2) NIV-1B (N3) POM-1

O-*-H1 O-*-H1 O-*-H1 O-*-H2

3896.6 3989.0

NIV-1 (N4) NIV-2 (N9)

O2--*--H1 O1-*-H1 O2--*--H1 O2--*--H1 O1-*-H1 O2--*--H1

188.1 3767.0

NIV-3 (N6) POM-1 POM-2 POM-3

POM-1 POM-2

4132.1

kfc,f

θpd (deg)

H-*-F (C∞h) 10.88 204.6 H2-*-F2--*--H1-*-F1 (8: C1) 0.08 128.5 10.20 204.1 172.8 10.64 204.7 164.5 128.3 204.7 169.8 204.7 147.3 H1-*-O-*-H2 (C2v) 9.22 204.6 10.14 204.6 204.6 207.3 H2O2--*--H1-*-O1H (4: Cs) 0.04 116.7 8.80 204.5 162.5 123.8 204.6 157.6

κpe (au−1) 0.03 103.57 0.02 109.18 0.02 122.82 108.51 0.02 79.58 0.02 78.50 0.05 0.05 0.05 1.58 158.13 0.04 308.94 159.13 0.04 195.16

w2/w1

character of freq

f

F-*-H

(1.000)g (1.000) −0.948 (1.000) 0.154 (1.000)h (1.000) −1.105 (1.000) −0.254

F2--*--H1 H1-*-F1

1.000 −1.000 (1.000) −0.025

νA1: O-*-H1 νB2: O-*-H1 O-*-H1i

(1.000)j (1.000) −0.021 (1.000)k (1.000) −1.125

νA′: O2--*--H1 νA′: O1-*-H1

H1-*-F1 F2--*--H1i H1-*-F1i H1-*-F1i

O2--*--H1i O1-*-H1i

a The 6-311+G(3df,3pd) basis sets being employed at the MP2 level. bMajor interactions are shown by plain text and minor ones by italic. cForce constant correspond to the frequency. dDefined by eq 10. eDefined by eq 11. fNot defined. gw2/w1 = −0.001 for H1-*-F1 and 0.013 for H2-*-F2. hw2/ w1 = −0.017 for H1-*-F1 and 0.004 for H2-*-F2. iFixed interaction in POM. jw2/w1 = −0.020 for O1-*-H1. kw2/w1 = −0.019 for O1-*-H1.

Behavior of θp and κp in H2O and HF Dimers. Figure 4 draws the fully optimized structures with some internal vibrations for H2O--*--H-*-OH (4: Cs) and H2−F2--*--H1−F1 (8: C1, very close to Cs), together with H2O (C2v) and H−F (C∞h). Before discussion of 4, 8 will be discussed, because 8 is simpler. There are six internal vibrations in 8, which are three

the candidate for the mechanism. The order in the HA-*-HA type shown by eq 24 should be explained on the viewpoints of both basicity and acidity, given by eqs 25 and 26. The values of ρb(rc), Vb(rc), Gb(rc), and R are also well correlated to Δr. The results are displayed in the Figures S1 and S2 of the Supporting Information. F



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stretching vibrations ν1 (N1: A) = 166.9 cm−1, ν5 (N5: A) = 4044.5 cm−1, and ν6 (N6: A) = 4132.1 cm−1 with three angular deformations ν2 (N2: A) = 241.0 cm−1, ν3 (N3: A) = 549.8 cm−1, and ν4 (N4: A) = 614.9 cm−1. N1, N5, and N6 will be employed to generate the perturbed structures of 8 with NIV, because angular deformations affect the dynamic behavior of interactions a little.52 In the case of 4, the stretching vibrations ν4 (N4: A′) = 188.1 cm−1 and ν9 (N9: A′) = 3767.0 cm−1 are employed to generate the perturbed structures of 4 with NIV. The modes of stretching vibrations are summarized in Figure 4. Table 2 collects some interactions in question with BCPs, the frequencies located on the interactions, and the corresponding force constants of the frequencies. The f1 value of 0.08209 satisfies w = 0.100 in eq 9 (cf. eq 8) for N1 of 8, where the displacement of F2---H1 is largest. In this case, wao = 0.052918 Å (r = ro + wao) for F2---H1, whereas w′ao = −0.000066 Å (r′ = ro′ + w′ao) for H1−F1 and w″ao = 0.000669 Å (r″ = ro″ + w″ao) for H2−F2. Consequently, w′/w = −0.001 for H1−F1 and w″/w (usually denoted by w′/w) = 0.013 for H2−F2, when w = 0.100 for F2---H1. The F2---H1 interaction is called major with respect to N1 and others are minor. The magnitude of w′/w is larger for H2−F2 than that for H1−F1, which may show that H2−F2 is more easily influenced from the motion of F2---H1 than the case of H1−F1, although H1−F1 constructs the linear F2---H1−F1 interaction in 8 but H2−F2 is almost perpendicular to it. Does this really mean that F2---H1 is more easily influenced from the motion of H2−F2 than that of H1−F1? Similar calculations were also performed with NIV of N5 and N6. Figure 5 shows the plots of Hb(rc) versus Hb(rc) − Vb(rc)/2

F2---H1 versus H1−F1 may show that weaker interaction is much more easily influenced by the surroundings if the orientation is suitable. Weak interactions such as F2---H1 in 8 could be called soft, because it is easily affected, whereas strong interactions such as H1−F1 and H2−F2 should be hard. The length of curved segments for F2---H1 shown in Figure 5 decreases in the order of NIV (N1) > NIV (N5) > NIV (N6), which support the above discussion. The length must be closely related to the magnitude of w′/w. The results with POM are similarly understood and support above discussion, although the signs of w′/w should be carefully examined. Table 2 summarizes the results for 4 and 8, together with H−F and H2O, analyzed by applying AIM dual functional analysis with NIV and POM. θp and κp Values Evaluated with NIV and POM for Various HBs. The dynamic behavior of various HBs is investigated employing AIM dual functional analysis with NIV and POM. Table 3 summarizes the θp and κp values for the neutral species, together with the ν and kf values, which correspond to the major HB interactions. The θp and κp values calculated with NIV are denoted by θp:NIV and κp:NIV, respectively, and those with POM by θp:POM and κp:POM, respectively. θp:NIV are very close to θp:POM for the usual cases, so are κp:NIV to κp:POM, as pointed out recently.53 Table 3 also collects the Hb(rc) − Vb(rc)/2, Hb(rc), k (=(Vb(rc)/Gb(rc)), R, and θ values at BCPs of the fully optimized structures. Table 4 collects those for the charged species for the data of both A-*-H and B-*-H in B-*-H-*-A. The A-*-H interaction in B-*-H-*-A becomes weaker as the strength of B-*-H increases. Such phenomena can be typically observed in charged species. For example, both O-*-H interactions in [H2O-*H-*-OH2]+ are equivalent due to the C2 symmetry, although it could be produced from H2O and H3O+. Figure 6 shows the plots of Hb(rc) versus Hb(rc) − Vb(rc)/2 for HBs of the fully optimized structures shown in Tables 3 and 4, together with the perturbed ones, although data are limited to the weaker HB interactions in 24A, 25A, 26A, and 27 for the plots. Survey of HBs Elucidated by AIM Dual Functional Analysis. The acceptable range of θ is limited to 45.0° < θ < 206.6°.49,50,55 The range is subdivided into 45.0° < θ < 90.0° for pure CS (Hb(rc) > 0; Hb(rc) − Vb(rc)/2 (=(ℏ2/8m)∇2ρb(rc)) > 0), 90.0° < θ < 180.0° for regular CS (Hb(rc) < 0; Hb(rc) − Vb(rc)/2 > 0), and 180.0° < θ < 206.6° for SS interactions (Hb(rc) < 0; Hb(rc) − Vb(rc)/2 < 0). The θ values become larger in the order shown by eq 27 for B-*HX (B = HF, H2Se, H2S, H2O, H2CO, and H3N), if X = F are compared. The values increase in the order shown by eq 28 for B = H2Se, H2S, H2O, and H2CO in B-*-HX (X = F, Cl, and Br), but not for H3N (eq 29). The θ values for the dimer of HCl, H3N, HBr, H2Se, H2S, H2O, and HF increase in the order shown by eq 30. The θ values in eq 30 are less than 90°; therefore, they are classified as the pure CS interactions, except for HF-*-HF. The value of HF-*-HF (θ = 90.8°) is larger than 90°, consequently, the interactions contain some covalent nature and are classified as regular CS.

Figure 5. Plots of Hb(rc) versus Hb(rc) − Vb(rc)/2 for F2--*--H1 in 8. Perturbed structures are generated by NIV with N1, N5, and N6, of which F2--*--H1 interactions correspond to the major, minor, and minor, respectively.

for F2---H1 in 8 with NIV of N1, N5, and N6. The F2---H1 interaction under the conditions corresponds to major, minor, and minor, respectively. As shown in Table 2, w′/w = −0.948 for F2---H1 versus H1−F1 (NIV with N5) and w′/w = 0.154 for F2---H1 versus H2−F2 (NIV with N6). The results seem well explained by assuming that F2---H1 is more easily affected from the motion of H1−F1 than that of H2−F2, irrespective of the results predicted with NIV of N1. The unexpectedly large w′/w value of −0.948 for

HF‐*‐HF (θ = 90.8°) < H 2Se‐*‐HF (104.3°) < H 2S‐*‐HF (108.5°) < H 2O‐*‐HF (124.0°) < H 2CO‐*‐HF (142.2°) < H3N‐*‐HF (156.4°) (27) G



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Table 3. AIM Functions and Parameters Evaluated for Hydrogen Bonds (HB) Evaluated by the AIM Dual Functional Analysis with NIV at the MP2 Levela species (X-*-Y) H2Se-*-HSeH (1) H2S-*-HSH (2) H3N-*-HNH2 (3) H2O-*-HOH (4) H3N-*-HOH (5) HBr-*-HBr (6) HCl-*-HCl (7) HF-*-HF (8) H2Se-*-HBr (9) H2Se-*-HCl (10) H2Se-*-HF (11) H2S-*-HBr (12) H2S-*-HCl (13) H2S-*-HF (14) H2O-*-HBr (15) H2O-*-HCl (16) H2O-*-HF (17) H2CO-*-HBr (18) H2CO-*-HCl (19) H2CO-*-HF (20) H3N-*-HBr (21) H3N-*-HCl (22) H3N-*-HF (23) species (X-*-Y) H2Se-*-HSeH (1) H2S-*-HSH (2) H3N-*-HNH2 (3) H2O-*-HOH (4) H3N-*-HOH (5) HBr-*-HBr (6) HCl-*-HCl (7) HF-*-HF (8) H2Se-*-HBr (9) H2Se-*-HCl (10) H2Se-*-HF (11) H2S-*-HBr (12) H2S-*-HCl (13) H2S-*-HF (14) H2O-*-HBr (15) H2O-*-HCl (16) H2O-*-HF (17) H2CO-*-HBr (18) H2CO-*-HCl (19) H2CO-*-HF (20) H3N-*-HBr (21) H3N-*-HCl (22) H3N-*-HF (23)

ρb(rc)b,c (au) 0.0090 0.0102 0.0146 0.0244 0.0291 0.0113 0.0127 0.0250 0.0159 0.0168 0.0212 0.0175 0.0186 0.0238 0.0274 0.0303 0.0417 0.0321 0.0430 0.0431 0.0563 0.0513 0.0547 νf (cm−1) kfg (h) 41.8 69.1 141.2 188.1 200.2 48.8 76.4 166.9 57.9 79.3 123.0 77.5 98.2 145.7 119.6 150.1 229.9 152.0 176.0 246.7 148.1 186.8 227.0

0.016 0.009 0.036 0.043 0.050 0.028 0.021 0.081 0.044 0.057 0.101 0.028 0.042 0.096 0.034 0.048 0.079 0.070 0.115 0.267 0.059 0.105 0.241

c∇2ρb(rc)b,c,d (au) 0.0026 0.0032 0.0059 0.0106 0.0094 0.0038 0.0049 0.0125 0.0040 0.0044 0.0051 0.0047 0.0051 0.0061 0.0103 0.0091 0.0131 0.0108 0.0115 0.0127 0.0069 0.0080 0.0085 θp:NIV (deg)

Hb(rc)b,c (au)

kb,c,e

θ (deg)

R (au)

κp:NIV (au−1)

0.0006 0.0008 0.0016 0.0005 −0.0020 0.0010 0.0015 −0.0002 0.0002 0.0001 −0.0013 −0.0001 −0.0002 −0.0020 −0.0006 −0.0128 −0.0089 −0.0022 −0.0032 −0.0099 −0.0189 −0.0155 −0.0195 θp:POM (deg)

−0.858 −0.861 −0.844 −0.976 −1.096 −0.853 −0.828 −1.007 −0.978 −0.989 −1.113 −1.010 −1.024 −1.143 −1.028 −1.413 −1.252 −1.147 −1.122 −1.279 −1.579 −1.492 −1.533 κp:POM (au−1)

0.0027 0.0033 0.0062 0.0107 0.0096 0.0039 0.0052 0.0125 0.0040 0.0044 0.0052 0.0047 0.0051 0.0064 0.0103 0.0114 0.0158 0.0111 0.0119 0.0161 0.0201 0.0174 0.0213 classification

76.0 76.3 74.9 87.3 102.1 75.6 73.6 90.8 87.6 88.7 104.3 91.1 92.8 108.5 93.2 98.9 124.0 101.6 105.9 142.2 160.0 152.7 156.4 character

195.81 262.47 158.75 158.13 82.97 258.47 267.32 103.33 479.46 423.08 144.24 300.62 259.88 117.08 168.36 103.85 7.08 139.90 91.78 5.67 6.34 9.33 1.83

88.0 91.8 85.4 123.8 157.5 91.4 95.0 128.3 130.0 137.4 164.5 133.9 140.6 165.1 138.1 149.9 166.1 154.5 160.4 170.0 190.3 186.9 181.9

201.96 231.77 168.34 159.13 88.06 269.39 294.83 108.51 498.34 437.80 150.54 317.25 273.70 121.16 185.77 119.96 8.54 135.36 91.68 7.96 8.06 11.67 5.41

p-CS p-CS p-CS p-CS r-CS p-CS p-CS r-CS p-CS p-CS r-CS r-CS r-CS r-CS r-CS r-CS r-CS r-CS r-CS r-CS r-CS r-CS r-CS

vdW-type HB-typical vdW-type HB-typical CTMC-type HB-typical HB-typical HB-typical HB-typical HB-typical CTMC-type HB-typical HB-typical CTMC-type HB-typical HB-typical CTMC-type CTMC-type CTMC-type CTMC-type CTTBP-type CTTBP-type CTTBP-type

88.3 93.3 86.6 116.7 158.6 91.2 94.8 128.5 129.9 137.1 163.9 134.3 140.7 164.6 140.7 152.0 167.6 152.8 158.6 168.3 189.8 186.2 180.6

a The 6-311++G(3df,3pd) basis sets being employed at the MP2 level. bCorresponding to the optimized structure. cGiven values corresponding to the static nature of interactions. dc∇2ρb(rc) = Hb(rc) − Vb(rc)/2 where c = ℏ2/8m. ek = Vb(rc)/Gb(rc). fInternal vibrational frequency corresponding to the interaction. gForce constant corresponding to the frequency. hIn mdyn Å−1.

B‐*‐HBr (θ = 93.2° for B = H 2O)

HCl‐*‐HCl (θ = 73.6°) ≤ H3N‐*‐HNH 2 (74.9°)

< B‐*‐HCl (98.9° for B = H 2O)

≤ HBr‐*‐HBr (75.6°) ≤ H 2Se‐*‐HSeH (76.0°)

< B‐*‐HF (124.0° for B = H 2O)

≤ H 2S‐*‐HSH (76.3°) < H 2O‐*‐HOH (87.3°)

(B = H 2Se, H 2S, H 2O, and H 2CO)

< HF‐*‐HF (90.8°)

(28)

The order for θ shown by eq 27 is well explained by that for ΔE given by eq 21. The order of H2O-*-HF and H2CO-*-HF is not the same, but the difference in ΔE seems small in eq 27. The order is also reproduced by the basicity determined in gas phase,

H3N‐*‐HCl (θ = 152.7°) < H3N‐*‐HF (156.4°) < H3N‐*‐HBr (160.0°)

(30)

(29) H



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Table 4. AIM Functions and Parameters Evaluated for Hydrogen Bonds (HB) Evaluated by the AIM Dual Functional Analysis with NIV at the MP2 Levela species (X-*-Y) +

[H3NH-*-OH2] (24A) [H3NH-*-NH3]+ (25A) [HO-*-HOH]− (26A) [H2O-*-H-*-OH2]+ (27) [HOH-*-OH]− (26B)j [H3N-*-HNH3]+ (25B)k [H3N-*-HOH2]+ (24B)l species (X-*-Y) [H3NH-*-OH2]+ (24A) [H3N-*-HNH3]+ (25A) [HOH-*-OH]− (26A) [H2O-*-H-*-OH2]+ (27)i [HO-*-HOH]− (26B) [H3NH-*-NH3]+ (25B) [H3N-*-HOH2]+ (24B)

ρb(rc)b,c (au) 0.0524 0.0806 0.1192 0.1703 0.2214 0.2496 0.3050 νf (cm−1) kfg (h) 304.5 317.1 313.0 899.8 1203.4 1852.4 2905.3

0.239 0.200 0.137 0.502 1.091 2.366 5.348

c∇2ρb(rc)b,c,d (au)

Hb(rc)b,c (au)

0.0143 0.0030 −0.0041 −0.0684 −0.1443 −0.1661 −0.2522 θp:NIV (deg) κp:NIV (au−1) 168.9 193.0 200.2 206.2 206.0 206.1 205.7

2.35 4.25 1.58 0.07 0.02 90°. Notations of CTMC-type and CTTBP-type are also employed to show the character of molecular complex (MC) formation through charge transfer (CT) and that of trigonal bipyramidal adduct (TBP) formation through CT, respectively. The charge of the species is also shown for the classification. As a result, He--*--HF, H2O-*-HOH, HF-*-HF, H3N-*-HCl, and [H2O-*-H-*-OH2]+ are denoted as (null: p-CS; vdW-type), (null: p-CS; HB-typical), (null: r-CS; CTMC-type), (null: r-CS; CTTBP-type), and (+1: SS; Cov-w), respectively. The nature of various HBs are similarly characterized and classified, of which range spans over pure CS of vdW-type to SS of Cov-w region.

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REFERENCES

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ASSOCIATED CONTENT

S Supporting Information *

Total energies of E(HB adduct) and E(components) in Table S1. Plots of ρb(rc), Vb(rc), Gb(rc), and R versus Δr for various HBs. Plot of the reported ΔE values14a versus those in Table 1 for the BHs of the CS interactions. The full-optimized structures given by Cartesian coordinates for examined molecules and adducts. This material is available free of charge via the Internet at http:// pubs.acs.org.

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Article

AUTHOR INFORMATION

Corresponding Author

*Tel: +81 73 457 8252. Fax: +81 73 457 8253. E-mail: nakanisi@ sys.wakayama-u.ac.jp. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was partially supported by a Grant-in-Aid for Scientific Research (Nos. 20550042, 21550046, and 23350019) from the Ministry of Education, Culture, Sports, Science, and Technology, Japan. The support of the Wakayama University Original Research Support Project Grant and the Wakayama University Graduate School Project Research Grant is also acknowledged. K



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Anthraquinone and 9-Methoxyanthracene Systems. Chem.Eur. J. 2007, 13, 255−268. (48) Nakanishi, W.; Hayashi, S.; Narahara, K. Atoms-in-Molecules Dual Parameter Analysis of Weak to Strong Interactions: Behaviors of Electronic Energy Densities versus Laplacian of Electron Densities at Bond Critical Points. J. Phys. Chem. A 2008, 112, 13593−13599. (49) Nakanishi, W.; Hayashi, S.; Narahara, K. Polar Coordinate Representation of Hb(rc) versus (ℏ2/8m)∇2ρb(rc) at BCP in AIM Analysis: Classification and Evaluation of Weak to Strong Interactions. J. Phys. Chem. A 2009, 113, 10050−10057. (50) Nakanishi, W.; Hayashi, S. Atoms-in-Molecules Dual Functional Analysis of Weak to Strong Interactions. Curr. Org. Chem. 2010, 14, 181−197. (51) Potts, K. T.; Kane, J. Bridgehead Nitrogen Heterocycles. VIII. Dimroth Rearrangement of 3H-1,2,4-Thiadiazolopyrimidines. J. Org. Chem. 1974, 39, 3783−3785. (52) Nakanishi, W.; Hayashi, S. Dynamic Behaviors of Interactions: Application of Normal Coordinates of Internal Vibrations to AIM Dual Functional Analysis. J. Phys. Chem. A 2010, 114, 7423−7430. (53) Nakanishi, W.; Hayashi, S.; Matsuiwa, K.; Kitamoto, M. Applications of Normal Coordinates of Internal Vibrations to Generate Perturbed Structures: Dynamic Behavior of Weak to Strong Interactions Elucidated by Atoms-in-Molecules Dual Functional Analysis. Bull. Chem. Soc. Jpn. 2012, 85, 1293−1305. (54) The origin of the plot corresponds to those without any interactions at BCPs, where Vb(rc) = Gb(rc) = 0. (55) For the plot of Hb(rc) versus Hb(rc) − Vb(rc)/2, tan θ = Hb(rc)/ (Hb(rc) − Vb(rc)/2) = (Gb(rc) + Vb(rc))/(Gb(rc) + Vb(rc)/2) = {1 + Vb(rc)/Gb(rc)}/{1 + Vb(rc)/[Gb(rc)/2]}, which converges to 1 at the weak limit of an interaction with Vb(rc)/Gb(rc) = 0. θ = 45.0° is derived from tan θ = 1 with Hb(rc) > 0 and (Hb(rc) − Vb(rc)/2) > 0. Similarly, tan θ = (Gb(rc)/Vb(rc) + 1)/(Gb(rc)/Vb(rc) + 1/2), which converges to 2 at the strong limit with Gb(rc)/Vb(rc) = 0. θ = 206.6° is derived from tan θ = 2 with Hb(rc) < 0 and Hb(rc) − Vb(rc)/2 < 0. (56) For the m × n matrix representation, m corresponds to the number of atoms and n (=3) to the x, y, and z components of the space. (57) Once the values of w = (0), ±0.1, and ±0.2 in r = ro + wao were employed for the perturbed structures in POM in refs 49 and 50, because the bond orders become 2/3 and 3/2 times larger at w = +0.2 and −0.2 relative to the original values at w = 0, respectively. However, it seems better to employ the perturbed structures as close as possible to the fully optimized ones in NIV. The perturbed structures closer to the fully optimized one will reduce the errors in the AIM functions of the perturbed structures generated by NIV and/or POM. Therefore, w = (0), ±0.05, and ±0.1 for r = ro + wao are employed for the analysis in this paper. Similarly, it is recommended that w = (0), ±0.025, and ±0.05 for θs = θso + wbo are applied to the perturbed structures when the effect from the angular deformations is discussed, because ±0.1bo (≈±5.73° with bo = 180°/π ≈ 57.30°) would be too large as the perturbations in angles. (58) It is achieved by changing the corresponding parameters in Gaussian 03 from the default values to print out the normal coordinates of five digits for the purpose. (59) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Montgomery, J. A., Jr.; Vreven, T.; Kudin, K. N.; Burant, J. C.; Millam, J. M.; Iyengar, S. S.; Tomasi, J.; Barone, V.; Mennucci, B.; Cossi, M.; Scalmani, G.; Rega, N.; Petersson, G. A.; Nakatsuji, H.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Klene, M.; Li, X.; Knox, J. E.; Hratchian, H. P.; Cross, J. B.; Bakken, V.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.; Pomelli, C.; Ochterski, J. W.; Ayala, P. Y.; Morokuma, K.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.; Zakrzewski, V. G.; Dapprich, S.; Daniels, A. D.; Strain, M. C.; Farkas, O.; Malick, D. K.; Rabuck, A. D.; Raghavachari, K.; Foresman, J. B.; Ortiz, J. V.; Cui, Q.; Baboul, A. G.; Clifford, S.; Cioslowski, J.; Stefanov, B. B.; Liu, G.; Liashenko, A.; Piskorz, P.; Komaromi, I.; Martin, R. L.; Fox, D. J.; Keith, T.; Al-Laham, M. A.; Peng, C. Y.; Nanayakkara, A.; Challacombe, M.; Gill, P. M. W.;

Johnson, B.; Chen, W.; Wong, M. W.; Gonzalez, C.; and Pople, J. A. Gaussian 03, revision D.02; Gaussian, Inc.: Wallingford, CT, 2004. (60) Møller, C.; Plesset, M. S. Note on an Approximation Treatment for Many-Electron Systems. Phys. Rev. 1934, 46, 618−622. Gauss, J. Effects of Electron Correlation in the Calculation of Nuclear Magnetic Resonance Chemical Shifts. J. Chem. Phys. 1993, 99, 3629−3643. Gau ss, J. Accurate Calculation of NMR Chemical Shifts. Ber. Bunsen-Ges. Phys. Chem. 1995, 99, 1001−1008. (61) (a) Kestner, N. R. He−He Interaction in the SCF−MO Approximation. J. Chem. Phys. 1968, 48, 252−257. (b) Jansen, H. B.; Ros, P. Non-empirical Molecular Orbital Calculations on the Protonation of Carbon Monoxide. Chem. Phys. Lett. 1969, 3, 140− 143. (c) Liu, B.; McLean, A. D. Accurate Calculation of the Attractive Interaction of Two Ground State Helium atoms. J. Chem. Phys. 1973, 59, 4557−4558. (62) The AIM2000 program (Version 2.0) is employed to analyze and visualize atoms-in-molecules: Biegler-König, F. Calculation of Atomic Integration Data. J. Comput. Chem. 2000, 21, 1040−1048. See also ref 34g. (63) Indeed, fourth functions can be applied to the five points in a plot, but fourth functions may show an irregular behavior with five points in some cases. Therefore, a regression curve of cubic functions would be more suitable as applied in the text, because the calculation errors in the data will be processed suitably. (64) The reported ΔE values14a are plotted versus those in Table 1 for the BHs of the CS interactions. The plot is analyzed by assuming the linear correlation (y = ax + b (R2: square of correlation coefficient)). Equation R1 shows the correlation. Data for H2Se-*-HF deviated from the correlation. However, the data moved on the line after recalculation with the proposed method employing the Gaussian 03 program package, although some differences would exist in the procedures between the two calculations. The minus sign for a (correlation constant) must be the reflection from the signs of ΔE, which are negative in Table 1 but positive for the reported values. The plot is shown in Figure S3 of the Supporting Information.

(R2 = 0.994) (R1) (65) Bondi, A. van der Waals Volumes and Radii. J. Phys. Chem. 1964, 68, 441−451. (66) Hunter, E. P. L.; Lias, S. G. Evaluated Gas Phase Basicities and Proton Affinities of Molecules: An Update. J. Phys. Ref. Data 1998, 27, 413−656. Hunter, E. P. L.; Lias, S. G. Proton Affinity Data. Bartmess, L. E. Negative Ion Energetic Data. In NIST Chemistry WebBook; NIST Standard Reference Database Number 69; Mallard, W. G., Linstrom, P. J., Eds.; National Institute of Standards and Technology: Gaithersburg, MD, 2001; http://webbook.nist.gov. Kagakubenran (Handbook of Chemistry), 5th ed.; Iwasawa, Y., Ed.; Chemical Society of Japan, Marzen Co.: Tokyo, 2003. ΔE(ref 14a) = −1.074 × ΔE(Table 1) − 5.28

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Dynamic Behavior of Hydrogen Bonds from Pure Closed Shell to

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