Intrinsic Dynamic Nature of Neutral Hydrogen Bonds Elucidated with QTAIM Dual Functional Analysis: Role of the Compliance Force Constants and QTAIM-DFA Parameters in Stability.

The dynamic and static nature of various neutral hydrogen bonds (nHBs) is elucidated with quantum theory of atoms-in-molecules dual functional analysis (QTAIM-DFA). The perturbed structures generated by using the coordinates derived from the compliance force constants (Cij ) of internal vibrations are employed for QTAIM-DFA. The method is called CIV. The dynamic nature of CIV is described as the "intrinsic dynamic nature", as the coordinates are invariant to the choice of the coordinate system. nHBs are, for example, predicted to be van der Waals (H2Se-✶-HSeH; ✶=bond critical point), t-HBnc (typical-HBs with no covalency: HI-✶-HI), t-HBwc (t-HBs with covalency: H2C=O-✶-HI), CT-MC [molecular complex formation through charge transfer (CT): H2C=O-✶-HF], and CT-TBP (trigonal bipyramidal adduct formation through CT: H3N-✶-HI) in nature. The results with CIV were the same as those with POM in the calculation errors, for which the perturbed structures were generated by partial optimization, and the interaction distances in question were fixed suitably in POM. The highly excellent applicability of CIV for QTAIM-DFA was demonstrated for the various nHBs, as well as for the standard interactions previously reported. The stability of the HBs, evaluated by ΔE, is well correlated with Cij (ΔE×Cij =constant value of -165.64), and the QTAIM parameters, although a few deviations were detected.

Introduction

Hydrogen bonds (HBs) are fundamentally important because of their ability to form molecular associations, which stabilizes a system in terms of energy; the direction of the interacting three atoms in B⋅⋅⋅H−Y (see below) is controlled through the formation of a HB that is almost a linear asymmetric σ bond (3c–4e: three‐center, four‐electron bond).1, 2, 3, 4, 5, 6 Weak HBs can be considered to be van der Waals (vdW) interactions, whereas strong HBs tend to be more covalent (Cov) in nature. The formation of HBs plays a crucial role in all fields of chemical and biological sciences. HBs control various chemical processes depending on their strength. It is imperative to clarify the nature of HBs for better understanding of chemical processes controlled by HBs.7, 8, 9, 10 We previously reported the dynamic and static nature of HBs in the neutral and charged forms by applying the quantum theory of atoms‐in‐molecules dual functional analysis (QTAIM‐DFA).11, 12, 13, 14, 15, 16 Perturbed structures were employed for QTAIM‐DFA to clarify the dynamic behavior of the interactions. The perturbed structures were generated by using the normal coordinates of the (best‐fitted) internal vibrations and/or by partial optimization, for which the interaction distances in question were suitably fixed. The methods are called NIV14, 15, 16 and POM,11, 12, 13 respectively. Neutral HBs (nHBs) are predicted to be vdW to CT‐TBP [trigonal bipyramidal adduct formation through charge transfer (CT)] in nature, whereas charged HBs are typically Cov in nature.17

The QTAIM approach, introduced by Bader,18, 19 enables us to analyze the nature of chemical bonds and interactions.20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31 Many QTAIM investigations have been reported, mainly by theoretical researchers, and experimental scientists have recently used QTAIM to explain their results by considering chemical bonding and interactions.20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39 A bond critical point (BCP, ✶)40 is an important concept in QTAIM, for which ρ() (charge density) reaches a minimum along the interatomic (bond) path, whereas it is a maximum on the interatomic surface separating the atomic basins. The ρ() at the BCP is described by ρ b( c) and so are other QTAIM functions, such as the total electron energy densities H b( c), potential energy densities V b( c), and kinetic energy densities G b( c) at the BCPs. A chemical bond or interaction between A and B is denoted by A−B, which corresponds to a bond path (BP) in QTAIM. We will use A−✶−B for a BP, in which the asterisk emphasizes the existence of a BCP in A−B.18, 19, 40 Equations (1), (2), (2′) represent the relations between G b( c), V b( c), H b( c), and ∇2 ρ b( c); H b( c) must be negative if ∇2 ρ b( c) <0, as confirmed by Equation (2) with negative V b( c) at all BCPs.

QTAIM‐DFA was recently formulated on the basis of QTAIM.41, 42, 43, 44, 45, 46, 47, 48 The H b( c) parameters are plotted versus H b( c)−V b( c)/2 [=(ħ2/8 m)∇2 ρ b( c)] at the BCPs in QTAIM‐DFA. In our treatment, data from the perturbed structures around the fully optimized structures are employed, in addition to those from the fully optimized ones. Data from the fully optimized structures are analyzed by using the polar coordinate (R, θ) representation, which corresponds to the static natures of the interactions.12, 13, 14, 15, 16 Each interaction plot, which contains data from both the perturbed structures and the fully optimized ones, includes a specific curve that provides important information about the interaction. This plot is expressed by (θ p, κ p), for which θ p corresponds to the tangent line of the plot and κ p is the curvature. The dynamic nature of the interactions has been proposed on the basis of (θ p, κ p).12, 13, 14, 15, 16 Namely, the signs of the first and second derivatives of H b( c) and H b( c)−V b( c)/2 [=(ħ2/8 m)∇2 ρ b( c)] are also employed to characterize the interactions in QTAIM‐DFA, whereas the signs of H b( c) and H b( c)−V b( c)/2 are employed for classification in QTAIM. We call (R, θ) and (θ p, κ p) the QTAIM‐DFA parameters, which are drawn in Figure 2, exemplified by H3N−✶−HI (26: C 3). QTAIM‐DFA is applied to standard interactions, and rough criteria that distinguish the interaction in question from others are obtained. QTAIM‐DFA and the criteria are explained in the Supporting Information by using Schemes S1 and S2, Figures S1, and Equations (S1)–(S7). The basic concept of the QTAIM approach is also discussed. QTAIM‐DFA has excellent potential for evaluating, classifying, characterizing, and understanding weak to strong interactions according to a unified form.12, 13, 14, 15, 16

Figure 2

Plots of H b( c) versus H b( c)−V b( c)/2 for 1–29, for which data from the perturbed structures generated with CIV were employed, in addition to the data from the optimized structures. Definitions of (R, θ) and (θ p, κ p) are illustrated, as exemplified by H3N−✶−HI (26: C 3).

However, the predicted dynamic nature would somewhat change depending on the perturbed structures employed. Very recently, we proposed a new method to generate the perturbed structures, other than NIV and POM, for QTAIM‐DFA.49 The method employs the coordinates corresponding to the compliance force constants (C) for the internal vibrations.50, 51, 52, 53, 54 We call the method CIV.49 The compliance force constants (C)50, 51, 52, 53, 55 are defined as the partial second derivatives of the potential energy due to an external force, as shown in Equation (3), for which i and j refer to internal coordinates, and the force constants f and f correspond to i and j, respectively. The value in Equation (3) corresponds to a lower numerical value (i) of a compliance constant representing a stronger bond (j), that is, the compliance constants measure the flexibility (or compliance) of a particular bond. The applications of CIV to the closed‐shell (CS) interactions are substantially more effective than those to the shared‐shell (SS) interactions in QTAIM‐DFA.49

The very high applicability of CIV is demonstrated to generate the perturbed structures for QTAIM‐DFA. The dynamic nature of the interactions based on the perturbed structures with CIV is described as the “intrinsic dynamic nature of interactions”, as the coordinates corresponding to C are invariant to the choice of the coordinate system. The results with CIV are the same as those with POM in terms of the calculation errors. However, CIV has been applied only to the typical interactions of a limited number of HBs, and the default in NIV seems large for HBs.49 The establishment of QTAIM‐DFA on the firm basis of employing the perturbed structures with CIV for the wide range of nHBs is another purpose of this paper. The neutral HBs in the species examined in this work are denoted by B−✶−HY (1–29), containing HI adducts, which are found in Table 1.

Table 1

QTAIM functions and QTAIM‐DFA parameters evaluated for the neutral hydrogen bonds (nHBs) in 1–29 by applying the QTAIM dual functional analysis by employing the perturbed structures generated with CIV, NIV, and POM.[a,b]

Species (X−✶−Y)	c∇2 ρ b(r c)[c] [au]	H b(r c) [au]	k b(r c)[d]	R [au]	θ [°]	Cij [e] [mDyn−1 Å3]	θ p:CIV [°]	κ p:CIV [au−1]
H2Se−✶−HSeH (1)	0.0026	0.0006	−0.858	0.0027	76.0 [f]	23.4	88.1	194
H2S−✶−HSH (2)	0.0032	0.0008	−0.861	0.0033	76.3	19.2	91.8	229
H3N−✶−HNH2 (3)	0.0059	0.0016	−0.844	0.0062	74.9	12.1	87.5	188
H2O−✶−HOH (4)	0.0106	0.0005	−0.976	0.0107	87.3	6.4	123.7	159
H3N−✶−HOH (5)	0.0094	−0.0020	−1.096	0.0096	102.1 [f]	5.5	157.3	87.3
HI−✶−HI (6)	0.0034	0.0004	−0.945	0.0034	84.1	13.8	102.6	304
HBr−✶−HBr (7)	0.0038	0.0010	−0.853	0.0039	75.6	17.9	91.4	269
HCl−✶−HCl (8)	0.0049	0.0015	−0.828	0.0052	73.6	16.1	95.0	294
HF−✶−HF (9)	0.0125	−0.0002	−1.007	0.0125	90.8	5.9	128.2	107
H2Se−✶−HI (10)	0.0040	0.0001	−0.986	0.0040	88.5	12.7	126.5 [f]	464
H2Se−✶−HBr (11)	0.0040	0.0002	−0.978	0.0040	87.6	13.0	130.1 [f]	488
H2Se−✶−HCl (12)	0.0044	0.0001	−0.989	0.0044	88.7	11.2	137.3 [f]	431
H2Se−✶−HF (13)	0.0051	−0.0013	−1.113	0.0053	104.3 [f]	7.3	164.5	146
H2S−✶−HI (14)	0.0043	0.0001	−0.991	0.0043	89.0	13.4	124.5	334
H2S−✶−HBr (15)	0.0047	−0.0001	−1.010	0.0047	91.1	12.0	133.9	309
H2S−✶−HCl (16)	0.0051	−0.0002	−1.024	0.0051	92.8	10.3	140.5	269
H2S−✶−HF (17)	0.0061	−0.0020	−1.143	0.0064	108.5 [f]	6.6	165.1	120
H2O−✶−HI (18)	0.0091	0.0009	−0.949	0.0091	84.5	10.1	113.6	217
H2O−✶−HBr (19)	0.0103	−0.0006	−1.028	0.0103	93.2	8.2	138.6	182
H2O−✶−HCl (20)	0.0112	−0.0018	−1.072	0.0114	98.9	6.4	149.9	116
H2O−✶−HF (21)	0.0131	−0.0089	−1.252	0.0158	124.0	3.4	166.1	6.9
H2C=O−✶−HI (22)	0.0102	−0.0009	−1.044	0.0103	95.3	9.7	139.7	216
H2C=O−✶−HBr (23)	0.0108	−0.0022	−1.093	0.0111	101.6 [f]	8.2	154.4	138
H2C=O−✶−HCl (24)	0.0115	−0.0032	−1.122	0.0119	105.9 [f]	6.6	160.4	92.0
H2C=O−✶−HF (25)	0.0127	−0.0099	−1.279	0.0161	127.8	3.5	170.1	6.7
H3N−✶−HI (26)	0.0050	−0.0268	−1.728	0.0272	169.4	19.8	194.1	4.2
H3N−✶−HBr (27)	0.0069	−0.0189	−1.579	0.0201	160.0	7.9	190.3	6.4
H3N−✶−HCl (28)	0.0080	−0.0155	−1.492	0.0174	152.7	5.5	186.9	9.3
H3N−✶−HF (29)	0.0085	−0.0195	−1.533	0.0213	156.4	2.8	182.0	2.8

[a] The functions and parameters were evaluated at the BCPs of the nHBs in the fully optimized structures. [b] With MP2/6–311++G(3df,3pd), except for I, for which calculations were performed with (7433111/743111/7411/2+1s1p1d1f) from the Sapporo Basis Set Factory, which is called MP2/BSS‐A. [c] c∇2 ρ b( c)=H b( c)−V b( c)/2, for which c=ħ2/8 m. [d] k b( c)=V b( c)/G b( c). [e] Defined in Equation (3). [f] Minor values that do not satisfy the characterization from the major ones are shown in italics. [g] Internal vibrational frequency corresponding to the interaction. [h] Force constant corresponding to the frequency. [i] From the components. [j] The nature of r‐CS/CT‐MC was predicted with NIV. [k] On the borderline area between r‐CS/t‐HBwc and r‐CS/CT‐MC if evaluated with CIV and POM.

Herein, we present the results of investigations on the “intrinsic dynamic nature of nHBs”, together with the static nature in B−✶−HY (1–29). To elucidate the nature, QTAIM‐DFA is applied to B−✶−HY (1–29) by employing the perturbed structures generated with CIV. Each HB interaction in 1–29 is classified and characterized by employing the criteria obtained on the basis of the standard interactions as a reference. The applicability of CIV to QTAIM‐DFA is also established in the nHBs of 1–29, for which the QTAIM‐DFA parameters elucidated by using CIV are compared with those elucidated by using NIV and POM. As a result, a firm basis for QTAIM‐DFA by employing the perturbed structures generated with CIV is established over the wide range of the nHBs in 1–29. The stability of 1–29 is discussed by examining the relations between the stability and the C, (R, θ), and (θ p, κ p) values.

Computational Details

The 6‐311++G(3df,3pd) basis sets of the Gaussian 09 programs56 were employed for the calculations of 1–29, together with the basis set of the 7433111/743111/7411/2+1s1p1d1f type for I, as implemented in the Sapporo Basis Set Factory.57 The basis set system is called BSS‐A. All calculations were performed under nonrelativistic conditions. The Møller–Plesset second‐order energy correlation (MP2) level58, 59, 60 was applied to the calculations (MP2/BSS‐A). The optimized structures were confirmed by frequency analysis. To obtain the perturbed structures with POM, the optimized structures were further (partially) optimized by employing Z matrices. The distances in question in the perturbed structures (r) were fixed to satisfy Equation (4), in which r o is the distance in the fully optimized structure with a o of Bohr radius (0.52918 Å). The method to generate the perturbed structures with NIV is explained by Equation (5). A k‐th perturbed structure in question (S ) is generated by the addition of the normal coordinates of the k‐th internal vibration (N ) to the standard orientation of a fully optimized structure (S o) in the matrix representation. The coefficient f in Equation (5) controls the structural difference between S and S o: f is determined to satisfy Equation (4) for r. The selected motion must be most effectively localized on the interaction in question among the zero‐point internal vibrations in NIV. Equations (4) and (6) are similarly applied to generate the perturbed structures with CIV, for which C is the coordinates corresponding to C in Equation (3).50, 51, 52, 53, 55 The N and C values of five digits are used to predict S and S , respectively:

QTAIM functions are calculated by using the same basis sets at the MP2 level as in the optimizations (MP2/BSS‐A), unless otherwise noted, and are analyzed with the AIM2000 program.18, 61 H b( c) are plotted versus H b( c)−V b( c)/2 for the data of five data points of w=0, ±0.05, and ±0.1 in Equation (4) in QTAIM‐DFA. Each plot is analyzed by using a regression curve of the cubic function, shown in Equation (7), for which (x, y)=[H b( c)−V b( c)/2, H b( c)] (R c 2>0.99999 per usual).16

Results and Discussion

Optimized Structures of Neutral Hydrogen‐Bonded Species and Stability

Neutral HB species, B−HY [1–29: B=H2Se, H2S, H2O, H2CO, H3N, and HX (=HI, HBr, HCl, and HF); HY=HSeH, HSH, HOH, HNH2, and HX], were optimized with MP2/BSS‐A, although some were optimized in a previous study.17 The optimized B−H distances [r o(B, H)] are collected in Table S1, together with the sum of the vdW radii62 [Δr=r o(B, H)−Σr vdW(B, H)]. The symmetries are also given in Table S1. The energies for 1–29 on the energy surface (E) and the relative energies from the components (ΔE) [=E(HB)−E(components)] are collected in Table S1. The ΔE values of 1–29 are also shown in Table 1 for convenience of discussion. The ΔE values are plotted versus Δr for 1–29. The plot is shown in Figure S2, and the correlation is given in the figure.63

The ΔE values of B−✶−HX (B=H2Se, H2S, H2C=O, and H3N; HX=HF, HCl, HBr, and HI) are plotted versus those of H2O−✶−HX. The plot is shown in Figure S3, which also contains the plot of ΔE(H2O−✶−HX) versus ΔE(H2O−✶−HX) for convenience of comparison. The correlations are very good (Table 2, entries 1–4). The results show that the ΔE values of B−✶−HX (B=H2Se, H2S, H2C=O, H2O, and H3N) are well correlated with each other if the ΔE values of common HX are compared, although the ΔE value of H2Se−✶−HI seems somewhat smaller (more stable) than that predicted from the correlation for H2Se−✶−HX. The magnitudes of ΔE become larger in the order H2Se≤H2S≪H2O≤H2C=O≪H3N, although ΔE(H2O−✶−HF)<ΔE(H2C=O−✶−HF). The relations between ΔE in B−✶−HX are also confirmed in this work for HX=HI in addition to HX=HF, HCl, and HBr, although the E values are all evaluated under nonrelativistic conditions.

Table 2

Correlations in 1–29, evaluated with NIV, POM, and CIV, under the MP2/BSS‐A conditions.[a]

Entry	Correlation	a	b	R c 2	n
1	ΔE _H_2Se-HX vs. ΔE _H_2O-HX	0.371	−6.83	0.960	4
2	ΔE _H_2S-HX vs. ΔE _H_2O-HX	0.468	−5.13	0.997	4
3	ΔE _H_2CO-HX vs. ΔE _H_2O-HX	0.751	−7.39	0.996	4
4	ΔE _H_3N-HX vs. ΔE _H_2O-HX	1.200	−8.72	0.9997	4
5	θ p:NIV vs. θ p:CIV	0.988	1.71	0.994	29
6	θ p:NIV vs. θ p:CIV	0.992	1.02	0.999	27[b]
7	θ p:POM vs. θ p:CIV	1.001	−0.15	0.99997	29
8	κ p:NIV vs. κ p:CIV	0.980	−0.31	0.994	29
9	κ p:POM vs. κ p:CIV	1.009	−0.42	0.998	29
10	ΔE vs. R	−2012.0	−3.83	0.866	27[c]
11	ΔE vs. θ	−0.479	25.70	0.891	27[c]
12	θ p:CIV vs. θ	2.390	−86.95	0.957	23[d]
13	ΔE vs. θ p:CIV	−0.314	19.69	0.971	8[e]
14	ΔE vs. θ p:CIV	−0.219	14.01	0.957	8[f]
15	ΔE vs. θ p:CIV	−0.155	−0.05	0.838	6[g]
16	ΔE vs. θ p:CIV	1.994	−414.66	0.898	4[h]

[a] Analyzed by assuming the linear correlation y=ax+b (R c 2: square of correlation coefficient). [b] For 1–29, except for 4 and 18. [c] For 1–29, except for 26 and 27. [d] For 1–29, except for 21, 24, and 25–29. [e] For 1–9, except for 3. [f] For 10–17. [g] For 18–25, except for 21 and 25. [h] For 24–29.

After clarifying the basic behavior in Δr and ΔE, molecular graphs with contour plots of ρ() are examined before detailed discussion of the nature of the nHBs in 1–29.

Molecular Graphs with Contour Plots for B−✶−HX

Figure 1 illustrates molecular graphs for B−✶−HI (B=H2Se, H2S, H2O, H2C=O, H3N, and HI) containing the contour plots of ρ(). All of the BCPs expected for B−✶−HI are clearly detected. They seem to be well located at three‐dimensional saddle points of ρ(). The molecular graphs of 1–29, other than B−✶−HI, were similarly drawn, and although they are not shown, they are very close to those of B−✶−HI.

Figure 1

Molecular graphs, with contour plots of ρ(r) for a) HI−✶−HI (6), b) H2Se−✶−HI (10), c) H2S−✶−HI (14), d) H2O−✶−HI (18), e) H2C=O−✶−HI (22), and f) H3N−✶−HI (26).

Survey of the B−✶−HY Interactions

The HB interactions seem straight for B−✶−HX on the basis of the BPs, as shown in Figure 1. To examine the linearity of the BPs, further, the lengths of the BPs (r BP) in question are collected in Table S2 for 1–29, together with the corresponding straight‐line distances (R SL). The differences between them (Δr BP=r BP−R SL) are less than 0.04 Å. Consequently, the BPs for all B−✶−HY of 1–29 can be described by straight lines.

QTAIM functions were calculated for B−✶−HY (1–29) at the BCPs. Table 1 collects the H b( c)−V b( c)/2 [=(ħ2/8 m)∇2 ρ b( c)] and H b( c) values, and the ρ b( c) values are given in Table S3, whereas some were reported previously.17 The H b( c) values are plotted versus H b( c)−V b( c)/2 for the data shown in Table 1 together with those from the perturbed structures generated with CIV. Figure 2 shows the plots. The plots appear in the region of H b( c)−V b( c)/2>0; therefore, the HBs are all classified by CS interactions. The CS interactions will be further classified by the signs of H b( c). They are specifically called pure CS (p‐CS) interactions if they appear in the region of H b( c)>0, whereas they will be regular CS (r‐CS) interactions for H b( c)<0. The behavior of the nHBs in 1–29 will be discussed in detail after evaluations of the QTAIM‐DFA parameters (see Table 1).

QTAIM‐DFA Parameters of (R, θ) and (θ p, κ p) for Neutral HBs in 1–29 Evaluated with POM, NIV, and CIV

The QTAIM‐DFA parameters of (R, θ) and (θ p, κ p) were obtained by analyzing the plots of H b( c) versus H b( c)−V b( c)/2 according to Equations (S3)–(S6). The (θ p, κ p) values evaluated by employing the perturbed structures generated with CIV, POM, and NIV are denoted by (θ p:CIV, κ p:CIV), (θ p:POM, κ p:POM), and (θ p:NIV, κ p:NIV), respectively. The (θ p:CIV, κ p:CIV) values can be obtained if the plots shown in Figure 2 are analyzed. Table 1 collects the QTAIM‐DFA parameters for 1–29. Table 1 contains the C values for the nHBs in 1–29 together with the frequencies correlated to the NIVs employed to generate the perturbed structures and the force constants (k f).

Behavior of θ p:CIV, θ p:POM, and θ p:NIV Together with That of κ p:CIV, κ p:POM, and κ p:NIV

Figure 3 a shows the plot of θ p:NIV versus θ p:CIV, which gives very good correlation. The correlation is shown in entry 5 of Table 2 (see also Figure 3 a). The magnitudes of the differences between θ p:NIV and θ p:CIV (Δθ p:NIV−CIV=θ p:NIV−θ p:CIV) are less than 2° for most of the interactions. The magnitudes of Δθ p:NIV−CIV are larger than 2.0° for H2O−✶−HOH (Δθ p:NIV−CIV=−7.0°),49 H2O−✶−HI (9.3°), H2O−✶−HBr (2.1°), and H2O−✶−HCl (2.1°). Large deviations are detected for H2O−✶−HX (HX=HOH and HI). The selected internal vibration for H2O−✶−HX could not be located effectively on O−✶−H by mixing with some other vibrational modes in the same symmetry,49 although the selected mode is the best fit for the O−✶−H interaction. The correlation for the plot is much improved (Table 2, entry 6; see also Figure 3 a) if the data for H2O−✶−HX (X=HOH and HI) are omitted from the correlation. On the other hand, excellent correlation is obtained if θ p:POM is plotted versus θ p:CIV, as shown in Figure 3 b (for the correlation, also see Table 2, entry 7). The magnitudes of Δθ p:POM−CIV are equal to or less than 0.1° for all HB adducts examined, except for H3N−✶−HOH (Δθ p:POM−CIV=0.2°), H2S−✶−HI (−0.2°), H2O−✶−HI (−0.7°), and H2O−✶−HBr (−0.5°). The results must be a reflection of the fact that the perturbed structures generated with POM and CIV are very similar.49 The results demonstrate the excellent applicability of CIV to generate the perturbed structures also for the nHB species in QTAIM‐DFA.

Figure 3

Plots of a) θ p:NIV versus θ p:CIV and b) θ p:POM versus θ p:CIV.

Figure 4 a, b shows the plots of p:NIV versus p:CIV and p:POM versus p:CIV. The correlations are given in entries 8 and 9 of Table 2 (see also Figure 4 a, b, respectively). The correlations seem very good, although substantial deviations are observed in the plots. The magnitudes of Δκ p:NIV−CIV are larger than 10 au−1 for H2S−✶−HSH (Δκ p:NIV−CIV=34 au−1), H3N−✶−HNH2 (−29 au−1), HBr−✶−HBr (−10 au−1), HCl−✶−HCl (−27 au−1), H2O−✶−HI (10 au−1), H2O−✶−HBr (−14 au−1), and H2O−✶−HCl (−12 au−1), together with magnitudes of 5 to 10 au−1 for HI−✶−HI (−8.7 au−1), H2Se−✶−HI (−9.3 au−1), H2Se−✶−HBr (−8.5 au−1), H2Se−✶−HCl (−7.6 au−1), H2S−✶−HI (−9.2 au−1), H2S−✶− HBr (−8.7 au−1), and H2S−✶−HCl (−9.1 au−1). In the case of Δκ p:POM−CIV, the magnitudes are less than 5 au−1 for most cases. The values are larger than 10 au−1 for H3N−✶−HNH2 (Δκ p:POM−CIV=−37 au−1), H2Se−✶−HBr (10 au−1), and H2C=O−✶−HI (−14 au−1), together with magnitudes of 5 to 10 au−1 for H2Se−✶−HSeH (8.0 au−1), H2Se−✶−HCl (7.1 au−1), H2S−✶−HI (6.2 au−1), and H2S−✶−HBr (8.0 au−1). The magnitudes for Δκ p:NIV−CIV seem very large at first glance. However, the very large values of κ p would be responsible for the large magnitudes of Δκ p as a whole. The magnitudes of Δκ p:POM−CIV seem to be much improved relative to the case of Δκ p:NIV−CIV; however, there are some severe deviations, such as H3N−✶−HNH2 (−37 au−1).

Figure 4

Plots of a) κ p:NIV versus κ p:CIV and b) κ p:POM versus κ p:CIV.

The correlation of θ p:POM versus θ p:CIV is much better than that of κ p:POM versus κ p:CIV (see Table 2, entries 7 and 9). This observation seems curious at first glance, as the same perturbed structures are employed to evaluate θ p and κ p in QTAIM‐DFA. The differences may be mainly attributable to the much more complex route to evaluate κ p (=[d2 y/dx 2]/[1+(dy/dx)2]3/2) relative to the case of θ p (=90°−tan−1 (dy/dx)). The θ p and κ p values are evaluated by using the common regression curve shown in Equation (7), as pointed out in a previous paper.49 The small differences in the QTAIM functions based on the perturbed structures generated with CIV and POM will be magnified in the second derivatives of the regression curves used to evaluate κ p.

As discussed above, the θ p:POM values can be recognized to be the same as the θ p:CIV values in terms of the calculation errors as a whole, although the Δθ p:POM−CIV values of −0.7° for H2O−✶−HI and −0.5° for H2O−✶−HBr seem slightly larger than the calculation errors. There must be a reason for these results, but we did not examine this point further. Larger magnitudes of Δκ p:POM−CIV are usually detected if κ p is very large. However, the results will not damage the excellent reliability in the characterization of the nHBs, as the κ p values are not used to characterize the interactions. Namely, the excellent applicability of CIV to generate the perturbed structures for QTAIM‐DFA is also well established for the various nHBs, as discussed above.

Nature of Neutral HBs Evaluated with the (θ, θ p) Values

The nature of the wide range of HBs for neutral forms 1–29 is now classified and characterized on the basis of the θ and θ p:CIV values obtained in this work. It is instructive to survey the criteria shown in Scheme S2 before a detailed discussion. The θ values classify interactions, whereas the θ p values predict the character of these interactions. The criteria tell us that 45°<θ<180° [0c)−V b( c)/2] for the CS interactions and 180°<θ<206.6° [H b( c)−V b( c)/2<0] for the SS interactions.12, 13, 14, 15, 16 The CS interactions can be subdivided into 45°<θ<90° [H b( c)>0] for the pure CS interactions (p‐CS) and 90°<θ<180° [H b( c)<0] for the regular CS interactions (r‐CS). In the p‐CS region of 45°<θ<90°, the character of interactions will be of the vdW type for 45°<θ p<90° (45°<θ<75°), whereas it will be of the typical hydrogen bond (t‐HB) type with no covalency (t‐HBnc) for 90°<θ p<125° (75°<θ<90°), for which θ=75° and θ p=125° are tentatively given for θ p=90° and θ=90°, respectively. The CT interaction will appear in the r‐CS region of 90°<θ<180°. The t‐HB interactions with covalency (t‐HBwc) appear in the range of 125°<θ p<150° (90°<θ<115°), for which (θ, θ p)=(115°, 150°) are tentatively given as the borderline between the t‐HBwc and CT‐MC (interactions in the molecular complex formation through CT) natures. The borderline in the interactions between CT‐MC and CT‐TBP is defined by (θ, θ p)=(150, 180°), for which θ=150° is tentatively given corresponding to θ p=180°. As a result, CT‐MC and CT‐TBP will appear in the ranges of 150°<θ p<180° (115°<θ<150°) and 180°<θ p<190° (150°<θ<180°), respectively. Namely, the (θ, θ p) values of (75°, 90°), (90°, 125°), (115°, 150°), (150°, 180°), and (180°,190°) correspond to the borderlines between the interactions for vdW/t‐HBnc, t‐HBnc/t‐HBwc, t‐HBwc/CT‐MC, CT‐MC/CT‐TBP, and CT‐TBP/Cov‐w (weak covalent bonds), respectively. The basic parameters of θ and θ p, described in bold, are superior to the tentatively given parameters in the classification and characterization of the interactions. The classical chemical bonds of the SS (180°<θ) will be strong if R>0.15 au (Cov‐s: strong covalent bonds), whereas they will be weak for R<0.15 au (Cov‐w), although SS interactions are not detected in the nHBs studied in this work.

The (θ, θ p:CIV) values of H2Se−✶−HSeH (1) are (76.0°, 88.1°), and therefore, it is classified by the p‐CS interaction and is characterized by its vdW nature (p‐CS/vdW). The θ p:CIV value of 88.1° should be superior to θ=76.0° (>75°) to predict the nature. The HB interaction in H3N−✶−HNH2 (3) is also predicted to be p‐CS/vdW in nature with (θ, θ p:CIV)=(74.9°, 87.5°) for the interaction. However, the HB interactions in 1 and 3 would be close to the borderline area between p‐CS/vdW and p‐CS/t‐HBnc judging from the (θ, θ p:CIV) values. HB interactions other than these two were similarly classified and characterized. The nHB interactions are predicted to have the p‐CS/t‐HBnc nature for H2S−✶−HSH (2), H2O−✶−HOH (4), HX−✶−HX [6 (HX=HI), 7 (HBr), and 8 (HCl)], H2Se−✶− HX [10 (HX=HI), 11 (HBr), and 12 (HCl)], H2S−✶−HI (14), and H2O−✶−HI (18). The r‐CS/t‐HBwc nature is predicted for HF−✶−HF (9), H2S−✶−HX [15 (HX=HBr) and 16 (HCl)], H2O−✶−HX [19 (HX=HBr) and 20 (HCl)], and H2C=O−✶−HI (22). On the other hand, the r‐CS/CT‐MC nature is predicted for H3N−✶−HOH (5), H2Se−✶−HF (13), H2S−✶−HF (17), H2O−✶−HF (21), and H2C=O−✶−HX [23 (HX=Br), 24 (HCl), and 25 (HF)], whereas the r‐CS/CT‐TBP nature is predicted for H3N−✶−HX [26 (HX=HI), 27 (HBr), 28 (HCl), and 29 (HF)]. The results are summarized in Table 1. The superior values of θ or θ p can be employed to predict the nature if either θ or θ p:CIV does not satisfy the categories to determine the nature. The characterization based on POM is the same as that based on CIV, and the characterization based on NIV is equal to that based on CIV and POM, except for 20 (H2O−✶−HCl). The nature of r‐CS/CT‐MC is predicted for 20 with NIV, whereas it is just borderline between r‐CS/t‐HBwc and r‐CS/CT‐MC if evaluated with CIV and POM. The results show that the HB interactions can also be characterized satisfactorily by employing θ p:NIV for most cases, irrespective of the substantial differences between θ p:NIV and θ p:CIV in some cases.

The predicted nature for B−✶−HX is summarized in Table 3, exemplified by the formation of B−✶−HX from B (=H2Se, H2S, H2O, H2C=O, and H3N) and HX (=HI, HBr, HCl, and HF). It enables us to visualize the roles of B and HX in the formation of B−✶−HX. The HB interactions are predicted to be stronger in the order shown in Equations (8), (9). The results shown in Table 3 and Equations (8) and (9) can be essentially explained on the basis of the results shown in Figure S3, although there are some differences, namely, the order shown in Equation (9) holds for B=H2Se, H2S, H2O, and H2C=O in BH−✶−HX, but it is reversed for B=H3N. The indirect B⋅⋅⋅(H)−X soft–soft interactions may affect the (θ, θ p:CIV) values in H2Se−✶−HI and H2S−✶−HI.

Table 3

The predicted nature of the nHBs in B−✶−HX with the (θ, θ p) values, for which B=H2Se, H2S, H2O, H2C=O, and H3N with HX=HI, HBr, HCl, and HF.[a–c]

HX	B=H2Se		H2S		H2O		H2C=O		H3N
	(θ [°], θ p [°])	nature	(θ [°], θ p [°])	nature	(θ [°], θ p [°])	nature	(θ [°], θ p [°])	nature	(θ [°], θ p [°])	nature
HI	(88.5, 126.5)	t‐HBnc	(89.0, 124.5)	t‐HBnc	(84.5, 113.5)	t‐HBnc	(95.3, 139.7)	t‐HBwc	(164.9, 194.1)	CT‐TBP
HBr	(87.6, 130.1)	t‐HBnc	(91.1, 133.9)	t‐HBwc	(93.2, 138.6)	t‐HBwc	(101.6, 154.6)	CT‐MC	(160.0, 190.3)	CT‐TBP
HCl	(88.7, 137.3)	t‐HBnc	(92.8, 140.5)	t‐HBwc	(98.9, 149.9 [d])	t‐HBwc	(105.9, 160.4)	CT‐MC	(152.7, 186.9)	CT‐TBP
HF	(104.3, 164.5)	CT‐MC	(108.5, 165.1)	CT‐MC	(124.0, 166.1)	CT‐MC	(127.8, 170.1)	CT‐MC	(156.4, 182.0)	CT‐TBP

[a] Evaluated with MP2/BSS‐A by employing the perturbed structures generated by using CIV. [b] Basic (superior) parameters are shown in bold. In the case of t‐HBwc, θ is basic if it is close to 90°, whereas θ p becomes basic if it is close to 150°. [c] Nonsuperior parameters are shown in italics if they do not satisfy the predicted nature on the basis of the superior parameters. [d] The value shows that the nature is borderline between t‐HBwc and CT‐MC.

The wide range of nHB interactions in 1–29 were satisfactorily classified and characterized by employing the perturbed structures generated with CIV in QTAIM‐DFA, resulting in the prediction of the reliable intrinsic dynamic nature of these interactions.

Behavior of Neutral HBs, Examined by the Parameters

What is the behavior of the nHBs in 1–29? The behavior was examined on the basis of the relation of ΔE with the compliance force constants (C) and the QTAIM (R, θ) and (θ p, κ p) parameters for 1–29. Figure 5 draws the plot of ΔE versus C. It seems that the plot can be well described by an inverse relationship, although data for H3N⋅⋅⋅HX [26 (X=I) and 27 (Br)] deviate from the correlation. Equation (10) shows the inverse relationship for 1–29, except for 26 and 27. No effort was made to get a best‐fit relationship; instead, the averaged value of ΔE×C (=−165.64) for 1–25, 28, and 29 was employed in Equation (10). The regression curve, given in Equation (10), is drawn in Figure 5 by a dotted line. The ΔE values for the nHBs seem to be well correlated to C if the data for 26 and 27 are omitted. As a result, the stability of B−✶−HY, denoted by ΔE, can be well described by the inverse nature of the compliance for the B−✶−HY interactions, evaluated by C, in 1–25, 28, and 29, according to Equation (3).

Figure 5

Plot of ΔE versus C for 1–29.

The ΔE values for 1–29 were next plotted versus R in (R, θ), which is drawn in Figure S4. The correlation is given in Table 2 (entry 10), although the data for 26 and 27 are again omitted from the correlation. The ΔE values in the nHBs of 1–25, 28, and 29 seem well correlated to R, for which both have the energy unit, although the ΔE values are on the energy surface, whereas the R values are at the BCPs on the BPs corresponding to the HBs. Figure 6 shows the plot of ΔE versus θ for 1–29. The correlation seems good if the data of 26 and 27 are omitted from the correlation. The correlation is given in Table 2 (entry 11) (see also Figure 6). Why does ΔE correlate rather well with θ for 1–25, 28, and 29? One may not expect such a correlation. However, it is of interest that the plot of θ versus R for 1–29 gives a good correlation, although this data is not shown (θ=3956.3R+64.73: R c 2=0.813). The correlation of ΔE versus R through θ versus R leads to the correlation of ΔE versus θ.

Figure 6

Plot of ΔE versus θ for 1–29.

The relations between ΔE and (θ p, κ p) are next discussed. Before a detailed discussion is given, the relation between θ p and θ is examined. The θ p values are plotted versus θ in Figure S5. A good correlation is obtained for 1–20 and 22–24, which is given in Table 2 (entry 12); the data for 21 (H2O⋅⋅⋅HF), 25 (H2CO⋅⋅⋅HF), and 26–29 (H3N−HX: X=I, Br, Cl, and F) are omitted from the correlation. Figure 7 illustrates the plot of ΔE versus θ p for 1–29. The plot was analyzed separately for four groups. Data for 1–9 (HA−✶−HA) belong to group A [G (A)]. A very good correlation is obtained for G (A), although data for 3 (H3N−✶−HNH2) deviate from the correlation. The correlation is shown in Table 2 (entry 13). The data for 10–17 (H2Se−✶−HX and H2S−✶−HX: X=I, Br, Cl and F) make up group B [G (B)]. Very good correlation is also obtained for G (B), which is shown in Table 2 (entry 14). The data for 18–25 (H2O−✶−HX and H2C=O−✶−HX: X=I, Br, Cl, and F) form group C [G (C)]. The correlation seems poorer than those for G (A) and G (B), and the data for 21 (H2O−✶−HF) and 25 (H2C=O−✶−HF) deviate from the correlation. The correlation is given in Table 2 (entry 15). Group D [G (D)] consists of 26–29 (H3N−✶−HX: X=I, Br, Cl, and F). A positive correlation constant is predicted for G (D) (a=1.99>0; see Table 2, entry 16), contrary to the cases of G (B) and G (C) with negative correlations (a<0; see Table 2, entries 13–15). As shown in Figure 7, ΔE correlates rather well with θ p as a whole, with a few deviations. The correlation of ΔE versus θ p should be a reflection of the correlation of ΔE versus θ through the correlation of θ p versus θ.

Figure 7

Plot of ΔE versus θ p for 1–29. Black dots for 1–9 belong to G (A), red triangles for 10–17 to G (B), blue squares for 18–25 to G (C), and green diamonds for 26–29 to G (D), although a few deviations are also included.

The plot of ΔE versus κ p for 1–29 is shown in Figure S6. A linear correlation is not detected between κ p and ΔE, as expected. The plot shows a characteristic shape. The κ p values are less than 10 au−1 if ΔE<−30 kJ mol−1, 80 au−1<κ p<300 au−1 for −30 kJ mol−1<ΔE<−7 kJ mol−1, and especially 270 au−1<κ p<490 au−1 if −17 kJ mol−1<ΔE<−10 kJ mol−1, except for κ p=188 au−1 with ΔE=−13.8 kJ mol−1 for 6 (HI−✶−HI). Very large values of κ p are observed around the borderline area for HBs between the vdW and t‐HBnc types.

The stability of the HBs in 1–29, evaluated by ΔE, is well explained on the basis of C, R, θ, and θ p. For the plots of ΔE versus C, the magnitudes of ΔE for 26 and 27 seem to be overestimated relative to those expected on the basis of the correlations for 1–25, 28, and 29. On the other hand, the magnitudes of ΔE for 26 and 27 would be underestimated relative to those expected from the correlations of ΔE versus R and θ for 1–25, 28, and 29. As shown in Figure 7, the plot for ΔE versus θ p could be recognized as a correlation as a whole, with deviation for G (D) (H3N−✶−HX) from the whole correlation for G (A)–G (C), if the correlation constants for the a values for the groups are compared.

Conclusions

Hydrogen bonds (HBs) are fundamentally important in all fields of chemical and biological sciences. Therefore, HBs have been variously investigated. However, it has been difficult to characterize the nature of HBs spread over the range of van der Waals (vdW) type for pure closed‐shell (CS) interactions to the covalent type of shared‐shell (SS) interactions. In this paper, HBs of the neutral form were characterized by applying quantum theory of atoms‐in‐molecules dual functional analysis (QTAIM‐DFA) by employing the perturbed structures generated with CIV. The neutral hydrogen bond (nHB) interactions were characterized on the basis of the static and dynamic behavior predicted with QTAIM‐DFA. The static nature arises from the data of the fully optimized structures, whereas the dynamic nature originates from the data of the perturbed structures around the fully optimized structures. The dynamic nature of the interactions could be described as the “intrinsic dynamic nature of interactions” if the perturbed structures were generated with CIV, as the coordinates corresponding to the compliance force constants (C), used in CIV, are invariant to the choice of the coordinate system. The method was applied to nHBs and the interactions were characterized. Some of them are as follows: nHBs in H2Se−✶−HSeH and H3N−✶−HNH2 were characterized by the p‐CS (pure CS)/vdW nature. The p‐CS/t‐HBnc (typical hydrogen bond with no covalency) nature was predicted for H2S−✶−HSH, H2O−✶−HOH, and HX−✶−HX (HX=HI, HBr, and HCl), whereas the r‐CS (regular CS)/t‐HBwc (typical‐HB interactions with covalency) nature was predicted for HF−✶−HF, H2S−✶−HX (HX=HBr and HCl), and H2O−✶−HX (HX=HBr and HCl). On the other hand, HBs in H2C=O−✶−HX (HX=HBr, HCl, and HF) were predicted to have the r‐CS/CT‐MC (interactions in the molecular complex formation through CT) nature, whereas the r‐CS/CT‐TBP (trigonal bipyramidal adduct formation through CT) nature was predicted for H3N−✶−HX (HX=HI, HBr, HCl, and HF). Characterization based on POM was the same as that based on CIV, and characterization based on NIV was equal to that based on CIV and POM, except for 20 (H2O−✶−HCl). The r‐CS/CT‐MC nature was predicted for 20 with NIV, whereas it was borderline between r‐CS/t‐HBwc and r‐CS/CT‐MC if evaluated with CIV and POM. The highly excellent applicability of CIV is well demonstrated in QTAIM‐DFA by applying the method to nHBs, in addition to the standard interactions in a previous paper.

Relations between ΔE and C or the QTAIM‐DFA parameters were examined, together with the reasons. A relation of ΔE×C=−165.64 was found for 1–25, 28, and 29. Namely, ΔE could be well described by the inverse nature of C. Similarly, the R, θ, and θ p values correlated linearly well with ΔE. The results showed that the values became larger as the stability of the HBs, described by ΔE, increased in the region examined, although there were a few deviations.

Conflict of interest

The authors declare no conflict of interest.

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