Exploring Nature and Predicting Strength of Hydrogen Bonds: A Correlation Analysis Between Atoms-in-Molecules Descriptors, Binding Energies, and Energy Components of Symmetry-Adapted Perturbation Theory
探索自然和预测氢键强度：分子原子描述符、结合能和对称适应扰动理论的能量成分之间的相关性分析
首次发表：2019 年 9 月 13 日 https://doi.org/10.1002/jcc.26068 引用次数：810
华南理工大学开放 URL
Abstract 摘要
This work studies the underlying nature of H-bonds (HBs) of different types and strengths and tries to predict binding energies (BEs) based on the properties derived from wave function analysis. A total of 42 HB complexes constructed from 28 neutral and 14 charged monomers were considered. This set was designed to sample a wide range of HB strengths to obtain a complete view about HBs. BEs were derived with the accurate coupled cluster singles and doubles with perturbative triples correction (CCSD(T))(T) method and the physical components of the BE were investigated by symmetry-adapted perturbation theory (SAPT). Quantum theory of atoms-in-molecules (QTAIM) descriptors and other HB indices were calculated based on high-quality density functional theory wave functions. We propose a new and rigorous classification of H-bonds (HBs) based on the SAPT decomposition. Neutral complexes are either classified as “very weak” HBs with a BE ≥ −2.5 kcal/mol that are mainly dominated by both dispersion and electrostatic interactions or as “weak-to-medium” HBs with a BE varying between −2.5 and −14.0 kcal/mol that are only dominated by electrostatic interactions. On the other hand, charged complexes are divided into “medium” HBs with a BE in the range of −11.0 to −15.0 kcal/mol, which are mainly dominated by electrostatic interactions, or into “strong” HBs whose BE is more negative than −15.0 kcal/mol, which are mainly dominated by electrostatic together with induction interactions. Among various explored correlations between BEs and wave function-based HB descriptors, a fairly satisfactory correlation was found for the electron density at the bond critical point (BCP;
ρBCP) of HBs. The fitted equation for neutral complexes is BE/kcal/mol = − 223.08 × ρBCP/a. u. + 0.7423 with a mean absolute percentage error (MAPE) of 14.7%, while that for charged complexes is BE/kcal/mol = − 332.34 × ρBCP/a. u. − 1.0661 with a MAPE of 10.0%. In practice, these equations may be used for a quick estimation of HB BEs, for example, for intramolecular HBs or large HB networks in biomolecules. © 2019 Wiley Periodicals, Inc.
这项工作研究了不同类型和强度的 H-键的本质，并尝试根据从波函数分析中得出的属性来预测结合能。考虑了由 28 个中性和 14 个带电单体构建的 42 个 H-键复合物。这个集合旨在通过获取广泛的 H-键强度视图来采样广泛的 H-键强度，以获得完整的 H-键视图。结合能使用准确的耦合簇单重和双重修正三重态（CCSD(T)）方法得出，并通过对称适配扰动理论（SAPT）研究了结合能的物理成分。基于高质量的密度泛函理论波函数，根据量子原子分子理论（QTAIM）描述符和其他 H-键指数计算了量子原子分子理论（QTAIM）描述符和其他 H-键指数。我们基于 SAPT 分解提出了一种新的和严格 H-键分类方法。中性复合物要么被分类为“非常弱”的 H-键，结合能≥-2.5 kcal/mol，主要由范德华力和静电相互作用主导，要么被分类为“弱到中等”的 H-键，结合能在-2.5 至-14.0 kcal/mol 之间，仅由静电相互作用主导。 另一方面，电荷复合体被分为“中等”HBs，其 BE 在-11.0 至-15.0 kcal/mol 的范围内，主要由静电相互作用主导，或者分为“强”HBs，其 BE 比-15.0 kcal/mol 更负，主要由静电和诱导相互作用共同主导。在探索 BE 与基于波函数的 HB 描述符之间的各种相关性中，发现 HBs 的键临界点（BCP；ρ BCP ）的电子密度与 BE 之间存在相当满意的相关性。对于中性复合体，拟合方程为 BE/kcal/mol = − 223.08 × ρ BCP /a. u. + 0.7423，平均绝对百分比误差（MAPE）为 14.7%，而对于电荷复合体，方程为 BE/kcal/mol = − 332.34 × ρ BCP /a. u. − 1.0661，MAPE 为 10.0%。实际上，这些方程可以用于快速估算 HB BE，例如，分子内 HB 或生物分子中的大型 HB 网络。版权所有 © 2019 Wiley Periodicals, Inc.
Introduction 简介
While the hydrogen bond (HB) was first discovered more than 100 years ago, it is still the topic of vital researches.1 Such a long-lasting interest is due to the eminent significance of HB in enzymatic catalysis,2, 3 proton transfer reactions,4 arrangement of molecules in crystals,5, 6 and also its highly important role in the life processes.7, 8 According to IUPAC definition,9 a typical hydrogen bond can be depicted as X─H⋯Y–Z and defined as “an attractive interaction between a hydrogen atom from a molecule or a molecular fragment X–H in which X is more electronegative than H, and an atom or a group of atoms in the same or a different molecule, in which there is evidence of bond formation.” The nature and strength of HB interaction are particularly sensitive to the character of the X, which is known as HB donor (D), and that of the Y, which is usually referred to as HB acceptor (A).
氢键（HB）的发现可以追溯到 100 多年前，但它仍然是至关重要的研究主题。这种持久的兴趣归因于氢键在酶催化、质子转移反应、晶体中分子排列以及生命过程中的重要性。根据国际纯粹与应用化学联合会（IUPAC）的定义，典型的氢键可以表示为 X─H⋯Y–Z，并定义为“分子或分子片段 X–H 中的氢原子与同一分子或不同分子中的原子或原子团之间的吸引性相互作用，其中 X 的电负性大于 H，且存在形成键的证据。”氢键相互作用的性质和强度特别敏感于 X 的特性，通常称为氢键供体（D），而 Y 的特性通常被称为氢键受体（A）。
HB interactions could be characterized by various aspects, among which, binding energy (BE) is one of the most important quantities. The BE, as used herein, is equivalent with “interaction energy” between the two monomers in complex geometry, that is, no structural relaxation of the monomers is taken into account. Quantum chemistry has become a routine and reliable way of estimating BE for various kinds of HBs.10-12 For example, the coupled cluster of singles and doubles with perturbative triples correction, such as CCSD(T),13 is able to calculate very accurate BE values with mean absolute deviation (MAE) of chemical accuracy (1 kcal/mol) or even better.10, 14 In addition, the symmetry-adapted perturbation theory (SAPT)12, 15-17 is not only able to evaluate BE but also capable of decomposing the BE into various physical components to facilitate the understanding of the nature of the interactions.
HB 相互作用可以由多个方面来描述，其中，结合能（BE）是最重要量之一。这里的 BE 等同于复合几何中两个单体之间的“相互作用能”，即不考虑单体的结构松弛。量子化学已经成为估算各种 HB 的 BE 的常规且可靠方法。10-12 例如，单重和双重耦合簇修正三重态，如 CCSD(T)13，能够以化学精度（1 kcal/mol）或甚至更好，计算出非常准确的 BE 值。此外，适应对称性的扰动理论（SAPT）12, 15-17 不仅能够评估 BE，还能够将 BE 分解为各种物理成分，以促进对相互作用本质的理解。
Aside from BEs, there are numerous analysis methods in the field of wave function analysis that are able to characterize weak interactions, including HBs. One of the most well-known theoretical tools to reveal inter- and intramolecular interactions together with their strengths is Bader's quantum theory of atoms in molecules (QTAIM).18 Within QTAIM, the electron density as well as other real space functions such as energy density and the Laplacian of electron density at the so-called bond critical point (BCP) of the HB interaction of interest can be comprehensively analyzed.19-25 For fully characterizing HB interactions, many other indices or descriptors have also been proposed, for instance, the core-valence bifurcation (CVB) index26 defined based on the electron localization function (ELF)27, 28 and the ΔΔV
n index defined by the molecular electrostatic potential (MEP) at nuclear positions.29
除了 BEs 之外，在波函数分析领域还有许多能够表征弱相互作用的分析方法，包括 HBs。揭示分子内外相互作用及其强度的最知名理论工具之一是 Bader 的分子中原子的量子理论（QTAIM）。18 在 QTAIM 中，可以全面分析电子密度以及能量密度等其他实空间函数，特别是对于所关注的 HB 相互作用的键临界点（BCP）处的电子密度拉普拉斯算子，可以进行综合分析。19-25 为了完全表征 HB 相互作用，还提出了许多其他指标或描述符，例如基于电子定位函数（ELF）定义的核心价分裂（CVB）指数 26，以及在核位置定义的分子静电势（MEP）处定义的ΔΔV
n 指数 29。
Despite existence of enormous amount of excellent research papers and reviews regarding the topic of HBs, we believe that there are still some points worthy of systematically exploring further. For example, is it possible to employ QTAIM descriptors or other HB indices to reliably predict BE of any kind or certain type of HBs? We note that some papers have already established a few useful relationships between QTAIM descriptors and BEs,23 such as the well-known BE ≈ VBCP/2 expression proposed by Espinosa and co-workers30; however, the quality of the reference BE values employed in their computational study is poor with respect to modern computational methodologies, and only very limited types of HBs have been considered, thus this relationship should be revisited. In addition, although there have been some works employing SAPT and QTAIM theories to attempt to shed light on nature of certain types of HB,19, 23 a comprehensive correlation analysis between the SAPT components and various QTAIM descriptors for a wide set of carefully curated HB systems is still lacking.
尽管关于 HBs 主题存在大量优秀的研究论文和综述，但我们认为仍有一些值得系统深入探索的点。例如，是否有可能利用 QTAIM 描述符或其他 HB 指标可靠地预测任何种类或特定类型的 HB 的 BE？我们注意到，一些论文已经建立了 QTAIM 描述符与 BE 之间的几条有用关系，23 例如 Espinosa 及其同事提出的广为人知的 BE ≈ V BCP /2 表达式 30；然而，在他们的计算研究中所使用的参考 BE 值的质量与现代计算方法相比较差，只考虑了非常有限类型的 HB，因此这种关系应该重新审视。此外，尽管已经有一些工作利用 SAPT 和 QTAIM 理论试图揭示某些类型 HB 的性质 19, 23，但对于广泛选择的精心挑选的 HB 系统，SAPT 组件与各种 QTAIM 描述符之间的全面相关性分析仍然缺乏。
In this work, we focus on revealing the underlying nature of HBs of different types and strengths and on finding the best correlation for predicting BE based on the properties that can be easily derived from wave function analysis. To realize this purpose, we will first employ the SAPT method to derive the physical components of the BEs to try to reveal the roles played by different physical sources in HB interactions and then perform a detailed correlation analysis between the easily reachable QTAIM topological parameters as well as some other HB indices and the BEs evaluated at very accurate CCSD(T) level. There is no doubt that such a relationship will be extremely useful in practical researches of HB systems. Finally, the covalency of HB interaction will be briefly examined based on correlation analysis between QTAIM descriptors and SAPT induction term. Although some of the findings and conclusions in the present work may already be explored, we believe such a study should be able to provide a more comprehensive view than ever for understanding characteristics of HB interactions.
在本工作中，我们专注于揭示不同类型和强度 HBs 的内在本质，并找到基于可以从波函数分析中轻松得出的属性来预测 BE 的最佳相关性。为了实现这一目标，我们首先将使用 SAPT 方法来推导 BE 的物理成分，试图揭示不同物理来源在 HB 相互作用中所扮演的角色，然后在非常准确的 CCSD(T)水平上对易于访问的 QTAIM 拓扑参数以及一些其他 HB 指标与 BE 之间进行详细的相关性分析。毫无疑问，这样的关系在 HB 系统的研究实践中将极为有用。最后，我们将基于 QTAIM 描述符与 SAPT 诱导项之间的相关性分析，简要探讨 HB 相互作用的共价性。尽管本工作中的一些发现和结论可能已经探索过，但我们相信这样的研究能够提供前所未有的全面理解 HB 相互作用的特性。
For conducting above-mentioned studies, we carefully selected 42 intermolecular hydrogen-bonded dimer complexes with a diverse set of monomers (Table 1). It is of great importance to note that using BE to characterize strength of HB interaction is reasonable just for sufficiently small interacting monomers. In other words, in the case of relatively large monomers, contribution of other parts of the monomers (particularly if these parts are polar or charged) will contribute prominently to the overall BE; as a result, the evaluated BE value cannot solely be pinpointed to the particular HB interaction. Taking this point into consideration, all of our selected monomers are reasonably small, so that secondary interaction (interaction between other parts of two interacting monomers) can safely be ignored and thus the BE only reflects the character of HB. The considered set covers a wide range of intermolecular HB interactions whose BE values vary from −0.6 to ca. −66.0 kcal/mol, both neutral and positively or negatively charged interacting monomers are included.
为了进行上述研究，我们精心挑选了 42 个不同单体组成的分子间氢键二聚复合物（表 1）。值得注意的是，使用 BE 来表征氢键相互作用的强度是合理的，仅适用于足够小的相互作用单体。换句话说，在相对较大的单体情况下，单体的其他部分（特别是如果这些部分是极性或带电的）将显著影响整体 BE；因此，评估的 BE 值不能仅归因于特定的氢键相互作用。考虑到这一点，我们所选择的所有单体都相当小，因此可以安全地忽略其他相互作用单体之间的次级相互作用，从而使得 BE 仅反映氢键的特性。考虑的范围涵盖了从-0.6 到约-66.0 kcal/mol 的广泛分子间氢键相互作用，既包括中性单体，也包括带正电或负电的相互作用单体。
表 1. 在 CCSD(T)/jul-cc-pVTZ 水平上计算的研究复合物的 BE，带有半对称修正（标记为 BE-1），以及在 SAPT2+(3)δMP2/aug-cc-pVTZ 水平上（标记为 BE-2）。
| Complex 复杂 | Structure 结构 | BE-1 | BE-2 | Complex 复杂 | Structure 结构 | BE-1 | BE-2 |
|---|---|---|---|---|---|---|---|
| 1 | H3CH⋯NCH | −0.60 -0.60 | −0.56 -0.56 | 22 | FH⋯SH2 | −5.06 -5.06 | −5.13 -5.13 |
| 2 | H3CH⋯OH2 | −0.62 -0.62 | −0.61 -0.61 | 23 | N3H⋯OH2 | −5.43 -5.43 | −5.56 -5.56 |
| 3 | MeCCH⋯OC 我 CCH⋯OC | −0.68 -0.68 | −0.64 -0.64 | 24 | HOH⋯NH3 OH⋯NH 3 | −6.41 -6.41 | −6.47 -6.47 |
| 4 | H3CH⋯NH3 抱歉，提供的字符串格式不明确，无法进行翻译。请提供清晰的文本内容以便进行翻译 | −0.71 -0.71 | −0.69 -0.69 | 25 | FH⋯N3H | −7.09 -7.09 | −7.98 -7.98 |
| 5 | HCCH⋯OC | −0.73 -0.73 | −0.70 -0.70 | 26 | N3H⋯NH3 | −7.54 -7.54 | −7.73 -7.73 |
| 6 | FCCH⋯OC | −0.78 -0.78 | −0.73 -0.73 | 27 | FH⋯OH2 | −8.89 -8.89 | −9.10 -9.10 |
| 7 | HOH⋯OC | −0.97 -0.97 | −0.98 -0.98 | 28 | FH⋯NH3 | −13.36 -13.36 | −13.65 -13.65 |
| 8 | HSH⋯SH2 抱歉，提供的文本似乎不完整或格式不正确，无法进行翻译。请提供完整且格式正确的文本 | −1.51 -1.51 | −1.57 -1.57 | 29 | [H2SH⋯FH]+ | −11.27 -11.27 | −11.62 -11.62 |
| 9 | HCCH⋯SH2 | −1.54 -1.54 | −1.54 -1.54 | 30 | [H3NH⋯FH]+ | −12.24 -12.24 | −12.49 -12.49 |
| 10 | HSH⋯N3H HSHN 3 H | −2.10 -2.10 | −2.42 -2.42 | 31 | [HOH⋯SH]− | −15.14 -15.14 | −15.36 -15.36 |
| 11 | HSH⋯OH2 | −2.58 | −2.66 | 32 | [H2NH⋯OH]− | −16.94 | −17.18 |
| 12 | HOH⋯SH2 | −2.78 | −2.82 | 33 | [H2NH⋯F]− | −17.34 | −17.74 |
| 13 | HCCH⋯OH2 | −2.83 | −2.85 | 34 | [H2OH⋯FH]+ | −19.29 | −19.68 |
| 14 | N3H⋯FH | −2.94 | −2.97 | 35 | [H3NH⋯OH2]+ | −20.91 | −21.25 |
| 15 | H2NH⋯NH3 | −3.03 | −3.02 | 36 | [H3NH⋯NH3]+ | −31.46 | −31.73 |
| 16 | N3H⋯SH2 | −3.16 | −3.29 | 37 | [HOH⋯F]− | −31.67 | −33.05 |
| 17 | HSH⋯NH3 | −3.34 | −3.43 | 38 | [HOH⋯OH]− | −37.17 | −39.12 |
| 18 | HCCH⋯NH3 | −3.56 | −3.57 | 39 | [H2OH⋯SH2]+ | −37.73 | −37.83 |
| 19 | HOH⋯N3H | −3.73 | −4.09 | 40 | [Cl⋯H⋯Cl]− | −42.70 | −43.98 |
| 20 | FH⋯FH | −4.52 | −4.56 | 41 | [H2O⋯H⋯OH2]+ | −52.74 | −53.86 |
| 21 | HOH⋯OH2 | −4.93 | −4.96 | 42 | [F⋯H⋯F]− | −65.47 | −69.38 |
- Values are given in kcal/mol.
值给出为千卡/摩尔。
Computational Details 计算细节
Geometries of all 42 hydrogen-bonded dimer complexes were fully optimized using the B3LYP exchange-correlation functional with Grimme's DFT-D3(BJ) empirical dispersion correction, abbreviated as B3LYP-D3(BJ).31 The ma-TZVPP basis set was adapted, which is the “minimally augmented” version of the def2-TZVPP basis set32, 33 for which s and p type diffuse basis functions are added to the non-hydrogen atoms. A further frequency calculation at the same level of theory was also performed at the optimized geometries to ensure that located stationary points do not have any imaginary frequency.
所有 42 个氢键二聚体复合体的几何形状均使用 B3LYP 交换相关功能与 Grimme 的 DFT-D3(BJ)经验性范德华修正相结合，简称 B3LYP-D3(BJ)进行了全面优化。31 使用了适应的 ma-TZVPP 基集，这是 def2-TZVPP 基集的“最小增强”版本，其中向非氢原子添加了 s 和 p 型扩散基函数。在同一理论水平下，还在优化的几何形状上进行了频率计算，以确保定位的稳定点没有任何虚频率。
For the optimized geometries, the BEs were evaluated through single-point energy calculations using the CCSD(T) method34 together with Truhlar's “calendar” basis set jul-cc-pVTZ,35 which is a reduced version of the well-known aug-cc-pVTZ basis36, 37 with removal of the less important diffuse functions on hydrogen atoms to reduce computational cost. The Boys and Bernardi's counterpoise (CP) technique38 was employed to correct the basis set superposition error (BSSE) problem. It has been demonstrated that when the aug-(or jul-)cc-pVTZ basis set is used in conjunction with CP correction, best results are obtained if only half of BSSE correction energy (
EBSSE) is taken into account.39 In consequence, our BEs were evaluated as BE(corrected) = BE(raw) + 0.5 × EBSSE. All above-mentioned computations were carried out using Gaussian 16 revision A.03.40 The deformation energy, which is related to the monomer geometry distortion due to the formation of complexes, was not taken into account in the BE evaluations, because BE is best to exhibit intrinsic binding strength of the monomers in a complex without this component. Note that all the monomers considered in present HB set are rather rigid; hence the deformation energy is essentially negligible.
优化几何结构后，通过使用 CCSD(T)方法 34 以及 Truhlar 的“日历”基组 jul-cc-pVTZ35 进行单点能量计算来评估 BE。jul-cc-pVTZ 是广为人知的 aug-cc-pVTZ 基组的简化版本，通过移除氢原子上较不重要的扩散函数来降低计算成本。Boys 和 Bernardi 的对称性校正（CP）技术 38 被用来修正基组叠加误差（BSSE）问题。已经证明，当使用 aug-(或 jul-)cc-pVTZ 基组与 CP 校正结合时，最好的结果是只考虑 BSSE 校正能（E BSSE ）的一半。39 因此，我们的 BE 被评估为 BE（校正）= BE（原始）+ 0.5 × E BSSE 。所有上述计算都在 Gaussian 16 修订版 A.03 中进行。40 在 BE 评估中没有考虑形成复合物时单体几何结构的变形能量，因为 BE 最好展示复合物中单体的内在结合强度，不包括这个成分。 请注意，当前 HB 设置中考虑的所有单体都非常刚性；因此变形能基本上可以忽略不计。
The high-order SAPT analyses were performed at the SAPT2 + (3)δMP2/aug-cc-pVTZ level of theory also on the optimized geometries. It has been denoted as “gold” level SAPT in reminiscence of the “gold standard CCSD(T)” and is able to generate highly accurate BE values.15 The PSI4 code41 was employed for the SAPT analyses. Since SAPT is essentially free of BSSE problem,10 no additional correction for BSSE is needed.
高级 SAPT 分析在理论水平为 SAPT2+(3)δMP2/aug-cc-pVTZ 上对优化的几何结构进行了执行。这被称为“黄金”级 SAPT，以“黄金标准 CCSD(T)”为灵感，能够生成高度准确的 BE 值。15 使用了 PSI4 代码进行 SAPT 分析。由于 SAPT 本质上不受 BSSE 问题的影响，因此不需要对 BSSE 进行额外修正。
The wave function-based quantities including QTAIM descriptors, ΔΔV
n
29 and CVB indices26 were all calculated based on the B3LYP-D3(BJ)/ma-TZVPP wave function at the optimized geometries by means of the Multiwfn version 3.6 code.42 Although it was pointed out that the quality of wave function of commonly used DFT functionals, including B3LYP, is somewhat poorer than that of post-HF ones such as MP2,43, 44 B3LYP is still one of the most employed functionals and B3LYP wave functions have been successfully adapted in numerous QTAIM studies45-47 providing chemically meaningful results48 even for noncovalent interactions.25, 49 Therefore, we chose B3LYP to generate wave function in the present investigation.
基于波函数的量，包括 QTAIM 描述符、ΔΔV
n 29 和 CVB 指数 26，都是在优化几何结构的基础上，使用 Multiwfn 版本 3.6 代码的 B3LYP-D3(BJ)/ma-TZVPP 波函数计算得出的。尽管指出，常用 DFT 功能，包括 B3LYP，的波函数质量略逊于后 HF 类型，如 MP2 的波函数，但 B3LYP 仍然是最常使用的功能之一，并且 B3LYP 波函数在众多 QTAIM 研究中已被成功应用，提供了化学意义明确的结果，即使对于非共价相互作用也是如此。因此，我们选择 B3LYP 来生成波函数进行当前的研究。
In the current study, we will provide linear regressions between many parameters describing the strength and nature of investigated HB interactions and, thus, using a quantity as a metric of error is unavoidable. The commonly employed quantities used to exhibit magnitude of error are mean squared error (MSE) and root mean squared error (RMSE); however both of which are evidently dependent on the absolute difference between the two set of data. It should be emphasized that the magnitude of the data involved in current study often spans a large range, for example, the magnitude of BEs of most neutral complexes and charged complexes are remarkably different, thus MSE and RMSE can hardly be employed as useful metrics in this work. Instead, we employ the mean absolute percentage error (MAPE), in which the aforementioned problem does not exist. MAPE is defined as 
, where Yact and Ypre are actually observed (calculated) data and predicted data, respectively, while n is the number of samples. In the current study, we use MAPE to portray the magnitude of error in the established linear regressions.
在当前的研究中，我们将提供许多描述 HB 相互作用强度和性质的参数之间的线性回归。因此，使用一个量作为误差的度量是不可避免的。通常用来展示误差大小的量是均方误差（MSE）和均方根误差（RMSE）；然而，两者都明显依赖于两组数据之间的绝对差异。应该强调的是，当前研究中涉及的数据范围往往很大，例如，大多数中性复合物和带电复合物的 BE 值差异显著，因此 MSE 和 RMSE 在本工作中几乎不能作为有用的度量。相反，我们采用平均绝对百分比误差（MAPE），其中不存在上述问题。MAPE 定义为 ，其中 Y act 和 Y pre 分别是实际（计算）数据和预测数据，n 是样本数量。在当前研究中，我们使用 MAPE 来描绘建立的线性回归中的误差大小。
Results and Discussion 结果与讨论
Evaluation of BEs BEs 的评估
In order to rigorously compare strength of the HB interactions, values of BE obtained at very accurate computational levels should first be at hand. Table 1 represents values of BEs calculated based on the CCSD(T)/jul-cc-pVTZ level (BE-1) including a half counterpoise correction and on the SAPT2 + (3)
δMP2/aug-cc-pVTZ level (BE-2) for the fully optimized 42 HB complexes. The set comprises 28 neutral complexes (1–28) and 14 charged ones (29–42). For each subset, the dimers are sorted in an increment order according to their values of BE-1. It is already well known that high-level SAPT method such as the SAPT2 + (3)
δMP2 is able to well reproduce the BE computed at very reliable CCSD(T) level for neutral HB complexes,15 the data in Table 1 also demonstrate that even for charged HB complexes, whose BEs are commonly much larger than those of the neutral ones, the SAPT2 + (3)
δMP2 result is still able to approximately reach the CCSD(T) quality. In addition, as shown in section S1 of the Supporting Information, linear correlation coefficient R2 between the BEs evaluated at the two different methods for both the neutral and charged sets is almost 1.0 with a very low MAPE of 3.0% and 1.4%, respectively. Therefore, using the physical components obtained from the SAPT2 + (3)
δMP2 approach to further discuss the nature of the HBs for all presently considered complexes should be reliable and meaningful.
为了严格比较 HB 相互作用的强度，首先应获得在非常准确计算水平上获得的 BE 值。表 1 基于 CCSD(T)/jul-cc-pVTZ 水平（BE-1）计算的包括半对称修正的 BE 值以及基于 SAPT2+(3)δMP2/aug-cc-pVTZ 水平（BE-2）对完全优化的 42 个 HB 复合体进行计算的 BE 值。该集合包括 28 个中性复合体（1-28）和 14 个带电复合体（29-42）。对于每个子集，根据其 BE-1 值对二聚体进行排序。已经很清楚，如 SAPT2+(3)δMP2 这样的高阶 SAPT 方法能够很好地重现 CCSD(T)水平上计算的中性 HB 复合体的 BE 值，15 表 1 中的数据也表明，即使对于 BE 通常远大于中性复合体的带电 HB 复合体，SAPT2+(3)δMP2 的结果仍然能够大致达到 CCSD(T)的质量。此外，如支持信息中的第 S1 节所示，对于中性和带电集合，使用两种方法评估的 BE 值之间的线性相关系数 R 2 几乎为 1.0，具有非常低的 MAPE 为 3.0%和 1。4% 分别。因此，使用 SAPT2 + (3)δMP2 方法获得的物理组件进一步讨论所有当前考虑的复合物中的 HBs 的性质应该是可靠和有意义的。
Although the actual complete basis set (CBS) limit is computationally inaccessible, it can approximately be approached using basis set extrapolation techniques employing sufficiently extended but still finite basis sets. The CBS extrapolation has been frequently adapted in estimation of extremely high quality of BE.10, 11 In the Supporting Information section S2, via a few selected complexes, we examined the potential improvement of the aug-cc-pV(T → Q)Z type of extrapolation on the BEs estimated at CCSD(T)/jul-cc-pVTZ level (Table 1). The data show that the improvement is marginal, therefore, the BE directly calculated using the relatively economical jul-cc-pVTZ basis set is already fully satisfactory and accurate enough for the present study. By means of regression analysis, in the Supporting Information section S2, we also showed that CCSD(T)/CBS data can be roughly estimated using a fitted equation based on the BEs produced by CCSD(T)/jul-cc-pVTZ calculation.
尽管实际的完整基组（CBS）极限在计算上不可达，但可以通过使用足够扩展但仍有限的基组集的外推技术来大致接近它。CBS 外推在估计极高质量的 BE 时经常被采用。10, 11 在补充信息部分 S2 中，我们通过几个选定的复合物，检查了 aug-cc-pV(T → Q)Z 类型外推对在 CCSD(T)/jul-cc-pVTZ 水平估计的 BE 的潜在改进（表 1）。数据显示，改进是微不足道的，因此，直接使用相对经济的 jul-cc-pVTZ 基组集计算的 BE 已经完全满足当前研究的需求。通过回归分析，在补充信息部分 S2 中，我们还展示了可以使用基于 CCSD(T)/jul-cc-pVTZ 计算产生的 BE 的拟合方程大致估计 CCSD(T)/CBS 数据。
Revealing dominating factors in the HB interactions via SAPT analysis
通过 SAPT 分析揭示 HB 相互作用中的主导因素
In the framework of SAPT approach, the BE can be decomposed into various physically meaningful components and can be expressed as BE = Eelst + Eexch + Eind + Edis,12, 15 by which the nature of a given HB interaction can be probed. The Eelst term mainly reflects the classical electrostatic interaction between the monomers. The Eexch term denotes the exchange-repulsion contribution caused by the overlap of monomer wave function as well as the antisymmetric requirements due to the fermionic behavior of the electrons in the dimer. The Eind term portrays the induction contribution, which comprises polarization as the response of each monomer to the electric field of the other one as well as charge transfer between two monomers. Note, however, that polarization and charge-transfer (CT) effects are not easily separable in the SAPT framework although recently some techniques have made the separation feasible.50 Finally, the Edis term is the dispersion contribution due to the Coulomb correlation between electrons in one monomer with those in another one. Within the formation of an HB, while electrostatic, induction, and dispersion terms contribute as attractive forces (displaying negative values), the exchange component behaves as a repulsive force (representing a positive value).
在 SAPT 方法的框架下，BE 可以分解为各种物理意义明确的组件，并可以表示为 BE = E elst + E exch + E ind + E dis ，12, 15，通过这种方式可以探究给定 HB 相互作用的性质。E elst 项主要反映了单体之间的经典静电相互作用。E exch 项表示由单体波函数重叠以及二聚体中电子的费米行为导致的交换排斥贡献。E ind 项描绘了诱导贡献，包括每个单体对另一个单体电场的响应以及两个单体之间的电荷转移。然而，需要注意的是，尽管最近有一些技术使得分离成为可能，但在 SAPT 框架下，极化和电荷转移（CT）效应并不容易分离。最后，E dis 项是由于一个单体中的电子与另一个单体中的电子之间的库仑相关性导致的散射贡献。 在 HB 的形成过程中，静电、感应和分散项作为吸引力（显示负值）贡献，而交换组件表现为排斥力（代表正值）。
In order to shed light on the electronic nature of the HB interactions and probe which factor(s) play the main stabilizing role, the different components in the SAPT-derived BEs were computed and summarized in Table 2. From the data in this table, it is quite obvious that the sum of attractive electrostatics, induction, and dispersion terms outweighs the repulsive exchange-repulsion contribution, leading to the negative values of BE-2 as presented in Table 1. On the other hand, if values given in Table 2 are converted into the contribution percentage for each attractive component, a more intuitive and deeper understanding could be obtained about the nature of the studied HB interactions. In this sense, the contribution percentage of each SAPT-derived attractive component was calculated for all of the studied complexes as 
, where
E
X signifies
Eelst,
Eind, or
Edis, the corresponding results are graphically portrayed in Figure 1. From Table 2 and more obviously Figure 1, one can find that electrostatic contribution always plays a major role, while the stabilization of the dimers is also effectively assisted by induction or dispersion interactions.
为了揭示 HB 相互作用的电子性质并探索主要稳定作用的因素，我们在表 2 中计算并总结了 SAPT 推导出的 BE 的不同组成部分。从表中的数据可以看出，吸引性静电、诱导和弥散项的总和明显超过排斥交换-排斥贡献，导致表 1 中呈现的 BE-2 为负值。另一方面，如果将表 2 中的值转换为每个吸引性组件的贡献百分比，可以获得对研究 HB 相互作用本质的更直观和深入的理解。在这种意义上，我们计算了所有研究复合物的每个 SAPT 推导出的吸引性组件的贡献百分比，表示为 ，其中 E
X 表示 E elst 、E ind 或 E dis ，对应的结果在图 1 中以图形方式展示。从表 2 和更明显地在图 1 中，我们可以发现静电贡献始终起主要作用，而二聚体的稳定也有效地得到了诱导或弥散相互作用的帮助。
表 2. 对复合体 1 至 42，在 SAPT2+(3)δMP2/aug-cc-pVTZ 水平计算的 BEs 的物理组件。
| Complex 复杂 | Eelst 对不起，提供的文本似乎不完整或格式错误，无法进行翻译。请提供完整且正确的文本以便进行翻译 | Eexch 对不起，提供的文本似乎不完整或格式错误，无法进行翻译。请提供完整且正确的文本以便进行翻译 | Eind 对不起，提供的文本似乎不完整或格式错误，无法进行翻译。请提供完整且正确的文本以便进行翻译 | Edis 对不起，提供的文本似乎不完整或格式错误，无法进行翻译。请提供完整且正确的文本以便进行翻译 | Complex 复杂 | Eelst 对不起，提供的文本似乎不完整或格式错误，无法进行翻译。请提供完整且正确的文本以便进行翻译 | Eexch 对不起，提供的文本似乎不完整或格式错误，无法进行翻译。请提供完整且正确的文本以便进行翻译 | Eind 对不起，提供的文本似乎不完整或格式错误，无法进行翻译。请提供完整且正确的文本以便进行翻译 | Edis |
|---|---|---|---|---|---|---|---|---|---|
| 1 | −0.51 -0.51 | 0.82 | −0.14 -0.14 | −0.73 -0.73 | 22 | −7.42 -7.42 | 9.95 | −4.89 -4.89 | −2.77 |
| 2 | −0.79 -0.79 | 1.25 | −0.22 -0.22 | −0.84 -0.84 | 23 | −8.59 -8.59 | 8.95 | −3.16 -3.16 | −2.76 |
| 3 | −0.54 -0.54 | 0.77 | −0.100 -0.100 | −0.77 -0.77 | 24 | −11.70 -11.70 | 12.63 | −4.16 -4.16 | −3.23 |
| 4 | −1.09 | 1.76 | −0.33 | −1.03 | 25 | −11.27 | 13.65 | −6.69 | −3.67 |
| 5 | −0.63 | 0.86 | −0.13 | −0.78 | 26 | −13.51 | 15.72 | −6.01 | −3.94 |
| 6 | −0.64 | 0.84 | −0.14 | −0.78 | 27 | −14.32 | 15.28 | −6.68 | −3.37 |
| 7 | −1.05 | 1.48 | −0.42 | −0.99 | 28 | −22.55 | 25.36 | −11.67 | −4.78 |
| 8 | −2.69 | 4.23 | −1.09 | −2.00 | 29 | −13.41 | 11.05 | −6.47 | −2.79 |
| 9 | −2.22 | 2.76 | −0.65 | −1.43 | 30 | −12.92 | 8.28 | −5.62 | −2.23 |
| 10 | −3.51 | 4.84 | −1.33 | −2.42 | 31 | −21.11 | 18.82 | −8.27 | −4.79 |
| 11 | −4.40 | 5.17 | −1.42 | −2.00 | 32 | −27.28 | 35.16 | −17.60 | −7.47 |
| 12 | −4.42 | 5.56 | −1.81 | −2.15 | 33 | −25.95 | 32.45 | −17.71 | −6.53 |
| 13 | −4.19 | 3.95 | −0.99 | −1.62 | 34 | −19.43 | 18.93 | −15.66 | −3.51 |
| 14 | −4.04 | 3.76 | −1.15 | −1.53 | 35 | −24.80 | 20.51 | −12.98 | −3.99 |
| 15 | −5.32 | 5.78 | −1.37 | −2.09 | 36 | −39.47 | 42.64 | −28.32 | −6.57 |
| 16 | −4.78 | 6.53 | −2.45 | −2.58 | 37 | −45.35 | 50.80 | −29.83 | −8.67 |
| 17 | −7.10 | 9.50 | −2.91 | −2.92 | 38 | −55.38 | 70.31 | −41.90 | −12.14 |
| 18 | −6.11 | 6.25 | −1.63 | −2.08 | 39 | −20.14 | 38.69 | −49.86 | −6.52 |
| 19 | −6.14 | 7.11 | −2.39 | −2.67 | 40 | −46.18 | 74.86 | −59.54 | −13.11 |
| 20 | −6.75 | 6.89 | −2.78 | −1.92 | 41 | −43.67 | 57.14 | −59.15 | −8.16 |
| 21 | −8.26 | 8.39 | −2.62 | −2.46 | 42 | −73.35 | 79.96 | −63.84 | −12.15 |
- Values are in kcal/mol.
值为每摩尔千卡。
SAPT 衍生的构造电静（Elst）、感应（Ind）和分散（Dis）相互作用在分子间氢键形成复合物 1-42 中的贡献百分比图。[彩色图可在 wileyonlinelibrary.com 上查看]
As portrayed in Figure 1, in the vast majority of the first 10 complexes, the main role within the HB formation is equally played by dispersion and electrostatic interactions and, meanwhile, the contribution of the induction interaction can safely be ignored. From complexes 11–31, as HBs become stronger, participation of dispersion interaction significantly decreases while that of induction interaction somewhat increases. Note, however, that in the complexes 22 and 25, both induction and electrostatic interactions are equally important, just as for the complexes 32–42, that comprise strong HB interactions. In some cases, such as complexes 39–41, the induction interaction even exceeds the electrostatic one.
如图 1 所示，在前 10 个复合体的绝大多数中，HB 形成中的主要作用由分散作用和静电相互作用共同承担，同时，诱导相互作用的贡献可以安全地忽略。从复合体 11 到 31，随着 HB 的增强，分散作用的参与显著减少，而诱导作用的参与略有增加。然而，值得注意的是，在复合体 22 和 25 中，诱导作用和静电作用同样重要，就像在由强 HB 相互作用组成的复合体 32 到 42 中一样。在某些情况下，如复合体 39 到 41 中，诱导作用甚至超过了静电作用。
The average contribution percentages of the SAPT-derived attractive components were calculated for the neutral and charged complexes and results are collected in Table 3. As shown, electrostatic is the main interaction in both neutral, with an average value of 52.2%, and charged complexes, with an average value of 50.6%, revealing that the importance of electrostatic interaction is universal. The average value of induction interactions for the charged complexes, 38.8%, is more than two times larger than that of neutral complexes, 19.1%. This observation clearly exhibits that induction interaction plays a much more significant role for charged HBs than the neutral HBs.
SAPT 衍生的吸引成分对中性和电荷复合物的平均贡献百分比进行了计算，结果收集在表 3 中。如图所示，静电作用是中性和电荷复合物中的主要相互作用，中性复合物的平均值为 52.2%，电荷复合物的平均值为 50.6%，这表明静电相互作用的重要性是普遍的。电荷复合物的诱导相互作用的平均值为 38.8%，是中性复合物的 19.1%的两倍多。这一观察清楚地表明，诱导相互作用在带电 HBs 中比中性 HBs 中扮演着更为重要的角色。
表 3. 本研究考虑的中性复合物和带电复合物中，通过 SAPT 推导出的构造性成分的平均贡献百分比。
| Type of complex 复合体类型 | Average of electrostatic contribution (%) 平均静电贡献（%） |
Average of induction contribution (%) 平均感应贡献率（%） |
Average of dispersion contribution (%) 离散贡献的平均值（%） |
|---|---|---|---|
| Neutral 中立 | 52.2 | 19.1 | 28.6 |
| Charged 被指控 | 50.6 | 38.8 | 10.6 |
It is instructive to further examine the variation of the SAPT terms with respect to the total SAPT-derived BEs, so that we can gain a deeper insight into the role played by different physical sources in HBs of different strengths, the plots exhibiting this point are given in Figure 2. Due to the very different characters of HB interaction in neutral and charged complexes, the map is, respectively plotted, for the two kinds of complexes. From Figure 2a, it can be seen that for neutral complexes, all the four SAPT terms have good linear relationships with the BE, thus the four fitted equations may be used to approximately estimate SAPT components in HB interactions without explicitly carrying out SAPT calculation. It is worth to note that when the HB strength is very low, for example, BE > −2.5 kcal/mol, the magnitude of Edis is larger than the magnitude of Eind. In contrast, since the slope of Eind (0.873) is by far steeper than that of Edis (0.329), showing that Eind varies faster with respect to the change of BE, for HBs of medium strength, the induction component has even notably higher contribution to BE than the dispersion component. A similar observation has also been reported in ref. 24, in which it was found that the ratio between dispersion term and delocalization term (which has analogous nature as the induction term in SAPT) is higher than 1.0 when electron density at the BCP (ρBCP) is very low, while the ratio is lower than 1.0 when the ρBCP is not quite small. As will be demonstrated later, the ρBCP mentioned here is a good measure of strength of HB interactions. For charged complexes, as shown in Figure 2b, the relationship between SAPT terms and BE is quite complicated. Except for the dispersion term, all other three SAPT terms fluctuate strongly with respect to change of BE; therefore, it is impossible to correlate any of these terms with the BE or directly roughly estimate their values according to BE by means of fitted equations as we have done for neutral HB systems. What we can conclude for the charged complexes from Figure 2b is that induction and electrostatic effects play comparable roles irrespective of the HB strength, and both of them are commonly much stronger than the dispersion effect.
进一步探讨 SAPT 项与总 SAPT 衍生 BE 之间的变化是有启发性的，这使我们能够更深入地了解不同强度 HB 中不同物理来源的作用。展示这一点的图表在图 2 中给出。由于中性复合物和带电复合物之间的 HB 相互作用的性质非常不同，因此分别绘制了两种类型的复合物的地图。从图 2a 可以看出，对于中性复合物，所有四个 SAPT 项都与 BE 有良好的线性关系，因此这四个拟合方程可以用来大致估计 HB 相互作用中的 SAPT 成分，而无需明确执行 SAPT 计算。值得注意的是，当 HB 强度非常低时，例如 BE > -2.5 kcal/mol，E dis 的大小大于 E ind 的大小。相比之下，由于 E ind 的斜率远大于 E dis 的斜率（0.873 远大于 0.329），表明 E ind 相对于 BE 的变化更快，对于中等强度的 HB，诱导成分甚至比分散成分对 BE 的贡献更高。 参考文献 24 中也有关于类似观察的报道，在其中发现，当 BCP 处的电子密度（ρ BCP ）非常低时，散射项与去局域化项（具有与 SAPT 中的诱导项类似性质的项）的比例高于 1.0，而当ρ BCP 不太小的时候，该比例低于 1.0。如后文将展示，这里提到的ρ BCP 是衡量氢键相互作用强度的好指标。对于带电复合物，如图 2b 所示，SAPT 项与 BE 之间的关系相当复杂。除了散射项外，其他三个 SAPT 项对 BE 的变化波动很大，因此不可能通过拟合方程根据 BE 来关联任何这些项或直接粗略估计它们的值，就像我们对中性氢键系统所做的那样。从图 2b 我们可以得出关于带电复合物的结论是，无论氢键强度如何，诱导作用和电荷效应扮演着同等重要的角色，并且两者通常都远比散射效应更强。
对于所有 28 个中性复合物（a）和 14 个带电复合物（b），在 SAPT2+(3)δMP2/aug-cc-pVTZ 水平上估计的 SAPT 术语的变化。由于图 2b 中的点分布相对分散，因此仅在图 2a 中给出了线性拟合。[彩色图可在 wileyonlinelibrary.com 上查看]
The BE and its physical components reported above only characterize HBs at equilibrium distance, while variation of the values versus the HB distance is able to fully display the contribution of various physical natures, resulting in a more complete understanding on how various factors stabilize HB complexes. In section S3 of the Supporting Information, variation of the SAPT interaction terms with respect to the molecular dimer separation is analyzed for three representative complexes, 2, 14, and 37, which are dispersion + electrostatics and pure electrostatics dominated neutral complexes as well as an electrostatics + induction dominated negatively charged complex, respectively. The figure in this section evidently displays that the dominating factor(s) at equilibrium separation also fully dominate the interaction over the entire interaction range, but induction interaction is attenuated faster than electrostatic interaction with increase in intermolecular spacing.
以上提到的 BE 及其物理组件仅描述了 HB 在平衡距离时的状态，而不同 HB 距离下值的变化则能完全展示各种物理性质的贡献，从而对各种因素如何稳定 HB 复合物提供了更全面的理解。在补充信息的第 S3 节中，分析了三个代表复合物（2、14 和 37）的 SAPT 相互作用项随分子二聚体分离的变化，这三个复合物分别是以范德华力和静电作用为主的中性复合物，以及以静电作用和诱导作用为主的负电荷复合物。本节的图表明显显示，在平衡分离时起主导作用的因素在整个相互作用范围内也完全主导了相互作用，但随着分子间距离的增加，诱导相互作用衰减速度比静电相互作用更快。
At the end of this section, it is worth to mention that SAPT or other kinds of energy decomposition methods have also been applied in many HB studies by other authors, interested readers are suggested to pay attention to these studies. For example, in the paper of the well-known S66 test set for measuring accuracy of weak interaction calculation,51 66 complexes including HB dimers were subjected to DFT-SAPT analysis, the ratio Edisp/Eelec was employed to determine the category of the interactions. Palusiak and coworkers characterized a series of charge-assisted hydrogen bonds (CAHBs) of N─H⋯Cl, N─H⋯Br, and P─H⋯Cl types by means of SAPT analysis.19 Grabowski and Lipkowski utilized the hybrid variational-perturbational approach to study many HB systems of X─H⋯π type and discussed the role played by different physical interactions.24
在本节结束时，值得一提的是，SAPT 或其他类型的能量分解方法也被其他作者在许多 HB 研究中应用。对有兴趣的读者，建议关注这些研究。例如，在用于衡量弱相互作用计算准确性的知名 S66 测试集中，对 51 个包含 HB 二聚体的 66 个复合物进行了 DFT-SAPT 分析，使用了 E disp /E elec 的比率来确定相互作用的类别。Palusiak 及其同事通过 SAPT 分析表征了一系列由 N─H⋯Cl、N─H⋯Br 和 P─H⋯Cl 类型组成的电荷辅助氢键（CAHBs）。Grabowski 和 Lipkowski 利用混合变分-扰动方法研究了多种 X─H⋯π类型的 HB 系统，并讨论了不同物理相互作用所扮演的角色。
New classification of H-bonds
新氢键分类
Classifying HBs facilitates researchers to recognize and identify HBs. However, obviously, the classification of HB is necessarily ambiguous and there does not exist a unique standard for classification. Currently, several HB classifications have already existed. The classification given by Kaplan52 argued that weak HB interactions represent a BE value between −0.5 and −4.0 kcal/mol, moderate HB interactions are distinguished with a BE value varying in the range −4.0 to −15.0 kcal/mol, and strong HB interactions have BE values fall in the range of −15.0 to −60.0 kcal/mol. Another classification has been proposed by Alkorta and co-workers53 indicating that HB interactions with a BE magnitude up to −5.0 kcal/mol portray a weak interaction, HB interactions whose BE magnitudes are larger than −5.0 kcal/mol but lower than −10.0 kcal/mol should be interpreted as medium interactions, and a BE magnitude greater than −10.0 kcal/mol characterizes a strong or a very strong HB interaction. While in the classification proposed by Grabowski,54 weak, medium, and strong HB interactions are characterized with the BE values fall within the range of −1.0 to −4.0, −4.0 to −15.0, and −15.0 to −40.0 kcal/mol, respectively. In addition, it is worth mentioning that the classification given by Rozas and co-workers not only considers BE but also takes wave function properties (energy density and Laplacian of electron density at BCP) into account.55
分类 HB 有助于研究人员识别和确定 HB。然而，显然，HB 的分类必然存在模糊性，并不存在唯一的分类标准。目前，已经存在几种 HB 分类。卡普兰给出的分类 52 认为，较弱的 HB 相互作用代表的 BE 值在-0.5 到-4.0 kcal/mol 之间，中等强度的 HB 相互作用的 BE 值在-4.0 到-15.0 kcal/mol 之间区分，而强烈的 HB 相互作用的 BE 值落在-15.0 到-60.0 kcal/mol 之间。Alkorta 及其同事提出的另一种分类 53 指出，BE 值达到-5.0 kcal/mol 的 HB 相互作用表示较弱的相互作用，BE 值大于-5.0 kcal/mol 但小于-10.0 kcal/mol 的 HB 相互作用应解释为中等相互作用，而 BE 值大于-10.0 kcal/mol 的 HB 相互作用则标志着强烈的或非常强烈的 HB 相互作用。而在格拉布斯基提出的分类中，较弱、中等和强烈的 HB 相互作用的 BE 值分别在-1.0 到-4.0、-4.0 到-15.0 和-15.0 到-40 之间。每摩尔为 0 千卡路里，分别。此外，值得注意的是，罗萨斯及其同事给出的分类不仅考虑了 BE，还考虑了波函数的性质（能量密度和 BCP 处电子密度的拉普拉斯算子）。 55
The highly physically meaningful SAPT2 + (3)
δMP2/aug-cc-pVTZ terms together with fairly accurate CCSD(T)/jul-cc-pVTZ BE values enable us to propose a new classification for the intermolecular HBs in a rigorous manner, see Table 4. As shown, neutral HB complexes are divided into “very weak” HBs whose BE magnitude is below 2.5 kcal/mol and “weak-to-medium” HBs, for which BE magnitude is larger than 2.5 kcal/mol but lower than 15.0 kcal/mol. The main sources of the HB binding for the former cases are dispersion and electrostatic interactions, while the latter ones are mainly driven by electrostatic attraction. On the other hand, charged HBs require their own classification and complexes are classified as “medium” HBs with a BE magnitude between 11.0 and 15.0 kcal/mol and “strong” HBs with a BE magnitude greater than 15.0 kcal/mol. The former ones are mainly dominated by electrostatic interaction while for the latter ones both electrostatic and induction interactions play a key role. Compared to existing classifications, a unique advantage of our new classification is that the correlation between class and major nature is explicitly given, therefore by comparing the value of BE for a certain HB interaction with BE values given in Table 4, one can quickly capture the physical component(s) that dominate the interaction.
高度物理意义的 SAPT2+(3)δMP2/aug-cc-pVTZ 项与相当准确的 CCSD(T)/jul-cc-pVTZ BE 值一起使用，使我们能够以严谨的方式提出一种新的分子间氢键分类，见表 4。如表所示，中性氢键复合物被分为“非常弱”的氢键，其 BE 值低于 2.5 千卡/摩尔，以及“弱到中等”的氢键，其 BE 值大于 2.5 千卡/摩尔但低于 15.0 千卡/摩尔。前者的氢键结合主要来源于色散和静电相互作用，而后者的主要是由静电吸引驱动。另一方面，带电氢键需要自己的分类，复合物被分类为“中等”氢键，其 BE 值在 11.0 至 15.0 千卡/摩尔之间，以及“强”氢键，其 BE 值大于 15.0 千卡/摩尔。前者的主导因素主要是静电相互作用，而后者则同时涉及静电和诱导相互作用的关键作用。 与现有分类相比，我们新分类的独特优势在于明确给出了类别与本质属性之间的关联，因此，通过将特定 HB 相互作用的 BE 值与表 4 中给出的 BE 值进行比较，可以快速捕捉主导该相互作用的物理成分。
表 4. 提议的分子间氢键分类
| Type of complex 复合体类型 | Strength 力量 | BE | Major nature 主要性质 |
|---|---|---|---|
| Neutral 中立 | Very weak 非常弱 | > −2.5 kcal/mol -2.5 千卡/摩尔 | Dispersion + Electrostatics 分散+静电学 |
| Weak to medium 虚弱到中等 | −2.5 to −14.0 kcal/mol -2.5 至-14.0 千卡/摩尔 | Electrostatics 静电学 | |
| Charged 被指控 | Medium 中等 | −11.0 to −15.0 kcal/mol -11.0 至 -15.0 千卡/摩尔 | Electrostatics 静电学 |
| Strong 强壮 | < −15.0 kcal/mol -15.0 千卡/摩尔 | Electrostatics + Induction 静电学+感应 |
Exploring correlation between various descriptors of HB and BEs
探索 HB 和 BEs 的各种描述符之间的相关性
In this section, we will study the correlation between various well-known HB descriptors and BE values estimated at CCSD(T)/jul-cc-pVTZ level (BE-1, Table 1), each of these descriptors has its own ability to characterize nature and strength of HB interactions. We focus on finding the best relationship that could be used for a reliably prediction of BEs so that its value can easily be estimated via a descriptor derived by a negligible cost approach. Moreover, such a valuable relationship would also allow to conveniently evaluate intramolecular HB BEs, which is often much more difficult to be estimated. Furthermore, when there are multiple HBs formed between two monomers, such a relationship could be used to individually estimate BE value of each HB interaction and thus revealing the main contribution to the binding if nonadditive effects are absent or marginal. In addition, correlation analysis between BE and HB descriptors enables us to understand characteristics of HBs from other perspectives rather than just from energetics consideration. All HB descriptors calculated using B3LYP-D3(BJ)/ma-TZVPP wave function are collected in Table 5. It may be surprising that why Table 5 does not include a column corresponding to the widely employed second-order perturbation interaction energy (E(2)) defined in the natural bond orbital (NBO) framework.56 This is because the E(2) energy essentially corresponds to charge transfer energy, while charge transfer is never the major source of most HBs, this point has already been largely implicitly reflected in the percentage contribution of induction term in Figure 1. In fact, E(2) even failed to present meaningful charge transfer energy for HB interaction, a recent theoretical study by Stone57 employing both NBO and SAPT(DFT) approaches has nicely demonstrated this point. As a consequence, explanation of HBs based on the E(2) perturbation analysis is omitted in the present study. In addition, the HB distance, namely the H⋯Y distance in X─H⋯Y contact, is evidently closely related to strength of HBs. However, this seemingly important geometric parameter was also not taken into our regression analyses, because elements in different rows have remarkably different van der Waals radii, which heavily influence the magnitude of interaction distance. Consequently, given that our complexes contain X atoms in both second and third rows, it is in principle impossible to well correlate H⋯Y distances with BEs.
在本节中，我们将研究各种广为人知的 HB 描述符与在 CCSD(T)/jul-cc-pVTZ 水平（BE-1，表 1）估算的 BE 值之间的相关性。每个这些描述符都有其自己的能力来表征 HB 相互作用的自然和强度。我们专注于寻找可用于可靠预测 BE 的最佳关系，这样其值可以通过几乎无成本的方法通过描述符来估算。此外，这样的宝贵关系还可以方便地评估分子内 HB BEs，这通常更难估算。此外，当两个单体之间形成多个 HB 时，可以使用此关系单独估算每个 HB 相互作用的 BE 值，从而揭示结合的主要贡献，如果不存在或边际的非叠加效应。此外，BE 与 HB 描述符之间的相关性分析使我们能够从能量考虑之外的其他角度理解 HB 的特性。使用 B3LYP-D3(BJ)/ma-TZVPP 波函数计算的所有 HB 描述符收集在表 5 中。 可能令人惊讶的是，为什么表格 5 中不包含对应于自然分子轨道（NBO）框架中广泛使用的二次扰动相互作用能（E (2) ）的列。这是因为 E (2) 能量本质上对应于电荷转移能，而电荷转移从来都不是大多数氢键的主要来源，这一点已经在图 1 中诱导项贡献百分比的大量隐含反映中得到了体现。实际上，E (2) 甚至未能呈现有意义的电荷转移能，对于氢键相互作用，这一点最近由 Stone 通过 NBO 和 SAPT(DFT)方法进行的理论研究已经很好地证明了这一点。因此，基于 E (2) 扰动分析对氢键的解释在当前的研究中被省略了。此外，氢键的距离，即 X─H⋯Y 接触中的 H⋯Y 距离，显然与氢键的强度密切相关。然而，这个看似重要的几何参数也被没有纳入我们的回归分析中，因为不同行的元素具有显著不同的范德华半径，这严重影响了相互作用距离的大小。 因此，由于我们的复合物在第二和第三行中都包含 X 个原子，原则上无法很好地将 H⋯Y 距离与 BEs 相关联。
表 5. 使用 B3LYP-D3(BJ)/ma-TZVPP 方法在氢键的 BCP 处计算了研究复合物的 QTAIM 拓扑描述符和其他氢键指数。
| Complex 复杂 | ρ(r) (a.u) | H(r) (a.u.) | 
|
∇2ρ(r) (a.u.) | 
|
V(r)/2 (kcal/mol) V(r)/2 (千卡/摩尔) | CVB (a.u.) CVB（a.u.） | ΔΔV n (a.u.) |
|---|---|---|---|---|---|---|---|---|
| 1 | 0.0047 | 0.0010 | 0.2127 | 0.0163 | 0.677 | −0.66 -0.66 | 0.072 | 0.0214 |
| 2 | 0.0064 | 0.0012 | 0.1875 | 0.0227 | 0.727 | −1.00 -1.00 | 0.067 | 0.0203 |
| 3 | 0.0054 | 0.0015 | 0.2778 | 0.0242 | 0.666 | −0.94 -0.94 | 0.085 | 0.0092 |
| 4 | 0.0069 | 0.0009 | 0.1304 | 0.0198 | 0.750 | −0.94 -0.94 | 0.056 | 0.0240 |
| 5 | 0.0059 | 0.0016 | 0.2712 | 0.0261 | 0.673 | −1.03 -1.03 | 0.086 | 0.0136 |
| 6 | 0.0059 | 0.0016 | 0.2712 | 0.0261 | 0.673 | −1.03 -1.03 | 0.081 | 0.0146 |
| 7 | 0.0088 | 0.0020 | 0.2273 | 0.0393 | 0.743 | −1.82 -1.82 | 0.071 | 0.0195 |
| 8 | 0.0113 | 0.0008 | 0.0708 | 0.0267 | 0.847 | −1.57 -1.57 | 0.019 | 0.0214 |
| 9 | 0.0087 | 0.0009 | 0.1034 | 0.0229 | 0.812 | −1.22 -1.22 | 0.056 | 0.0205 |
| 10 | 0.0146 | 0.0012 | 0.0822 | 0.0431 | 0.875 | −2.63 -2.63 | 0.026 | 0.0237 |
| 11 | 0.0161 | 0.0016 | 0.0994 | 0.0555 | 0.869 | −3.32 -3.32 | 0.034 | 0.0368 |
| 12 | 0.0157 | 0.0001 | 0.0064 | 0.0359 | 0.988 | −2.73 -2.73 | 0.004 | 0.0346 |
| 13 | 0.0144 | 0.0020 | 0.1389 | 0.0546 | 0.827 | −3.01 -3.01 | 0.058 | 0.0397 |
| 14 | 0.0157 | 0.0025 | 0.1592 | 0.0681 | 0.827 | −3.76 -3.76 | 0.052 | 0.0445 |
| 15 | 0.0159 | 0.0011 | 0.0692 | 0.0473 | 0.895 | −3.00 -3.00 | 0.024 | 0.0395 |
| 16 | 0.0167 | 0.0002 | 0.0120 | 0.0367 | 0.978 | −2.76 -2.76 | −0.011 -0.011 | 0.0399 |
| 17 | 0.0213 | 0.0001 | 0.0047 | 0.0523 | 0.992 | −4.02 -4.02 | −0.025 -0.025 | 0.0447 |
| 18 | 0.0165 | 0.0010 | 0.0606 | 0.0467 | 0.905 | −3.01 -3.01 | 0.028 | 0.0437 |
| 19 | 0.0221 | −0.0006 -0.0006 | −0.0271 -0.0271 | 0.0598 | 1.039 | −5.05 -5.05 | 0.001 | 0.0413 |
| 20 | 0.0281 | −0.0023 -0.0023 | −0.0819 -0.0819 | 0.0916 | 1.091 | −6.63 -6.63 | 0.014 | 0.0722 |
| 21 | 0.0259 | −0.0013 -0.0013 | −0.0502 -0.0502 | 0.0767 | 1.068 | −6.87 -6.87 | 0.002 | 0.0595 |
| 22 | 0.0274 | −0.0040 -0.0040 | −0.1460 -0.1460 | 0.0392 | 1.297 | −5.62 -5.62 | −0.101 -0.101 | 0.0685 |
| 23 | 0.0258 | −0.0005 -0.0005 | −0.0194 -0.0194 | 0.0807 | 1.024 | −6.62 -6.62 | 0.002 | 0.0618 |
| 24 | 0.0305 | −0.0035 -0.0035 | −0.1148 -0.1148 | 0.0643 | 1.179 | −7.22 -7.22 | −0.060 -0.060 | 0.0716 |
| 25 | 0.0399 | −0.0087 -0.0087 | −0.2180 -0.2180 | 0.0680 | 1.338 | −10.79 -10.79 | −0.110 -0.110 | 0.0778 |
| 26 | 0.0337 | −0.0039 -0.0039 | −0.1157 -0.1157 | 0.0677 | 1.187 | −7.75 -7.75 | −0.092 -0.092 | 0.0712 |
| 27 | 0.0450 | −0.0111 -0.0111 | −0.2467 -0.2467 | 0.0889 | 1.332 | −13.96 -13.96 | −0.092 -0.092 | 0.1010 |
| 28 | 0.0585 | −0.0201 -0.0201 | −0.3436 -0.3436 | 0.0524 | 1.605 | −16.72 -16.72 | −0.240 -0.240 | 0.1235 |
| 29 | 0.0351 | −0.0032 -0.0032 | −0.0912 -0.0912 | 0.1121 | 1.106 | −10.82 -10.82 | −0.009 -0.009 | 0.2181 |
| 30 | 0.0320 | −0.0014 -0.0014 | −0.0438 -0.0438 | 0.1194 | 1.045 | −10.22 -10.22 | 0.026 | 0.2289 |
| 31 | 0.0297 | −0.0041 -0.0041 | −0.1380 -0.1380 | 0.0407 | 1.288 | −5.74 -5.74 | −0.135 -0.135 | 0.2182 |
| 32 | 0.0538 | −0.0142 -0.0142 | −0.2639 -0.2639 | 0.0923 | 1.380 | −16.16 -16.16 | −0.153 -0.153 | 0.2469 |
| 33 | 0.0562 | −0.0148 -0.0148 | −0.2633 -0.2633 | 0.0127 | 1.318 | −19.23 -19.23 | −0.105 -0.105 | 0.2577 |
| 34 | 0.0662 | −0.0224 -0.0224 | −0.3384 -0.3384 | 0.1352 | 1.400 | −24.69 -24.69 | −0.121 -0.121 | 0.2991 |
| 35 | 0.0545 | −0.0137 -0.0137 | −0.2514 -0.2514 | 0.1085 | 1.338 | −17.13 -17.13 | −0.119 -0.119 | 0.2604 |
| 36 | 0.0863 | −0.0398 -0.0398 | −0.4612 -0.4612 | 0.0073 | 1.954 | −25.51 -25.51 | −0.432 -0.432 | 0.2951 |
| 37 | 0.0930 | −0.0458 -0.0458 | −0.4925 -0.4925 | 0.1094 | 1.625 | −37.33 -37.33 | −0.249 -0.249 | 0.3130 |
| 38 | 0.1164 | −0.0728 -0.0728 | −0.6254 -0.6254 | −0.0154 -0.0154 | 2.050 | −44.24 -44.24 | −0.451 -0.451 | 0.3128 |
| 39 | 0.0987 | −0.0578 -0.0578 | −0.5856 -0.5856 | −0.1318 -0.1318 | 3.325 | −25.99 -25.99 | −0.691 -0.691 | 0.3402 |
| 40 | 0.1243 | −0.0822 -0.0822 | −0.6613 -0.6613 | −0.1682 -0.1682 | 3.049 | −38.72 -38.72 | −0.742 -0.742 | 0.2833 |
| 41 | 0.1705 | −0.1939 -0.1939 | −1.1372 -1.1372 | −0.4438 -0.4438 | 3.340 | −86.88 -86.88 | −0.644 -0.644 | 0.3853 |
| 42 | 0.1783 | −0.2314 -0.2314 | −1.2978 -1.2978 | −0.4969 -0.4969 | 3.160 | −106.21 -106.21 | −0.578 -0.578 | 0.4131 |
In the QTAIM theory, the BCP is often viewed as the most representative point along the interaction path between two interacting atoms. The electron density,
ρ(r), is a quantity directly derived from wave function and experimentally observable; its value at the BCP has a central role in the QTAIM analysis framework. First, we examine the relationship between
ρ(r) at BCP, namely
ρBCP, and the calculated BE (i.e., BE-1 in Table 1, similarly hereinafter). As depicted in Figure 3a, the plot of BE against
ρBCP for the entire set leads to a satisfying linear correlation (R2 = 0.9716) but a large value of MAPE, 45.18%, is obviously unacceptable. As depicted in Figure 3b, once the BE is plotted versus
ρBCP for just neutral complexes, not only the linear dependency is slightly improved to 0.9732 but also the MAPE is significantly reduced to 14.69%. Figure 3c indicates that a similar situation exists if BE is plotted against
ρBCP for just charged complexes, while R2 remains basically unchanged, MAPE experiences a noticeable improvement to 10.03% compared with that of Figure 3a. These results, once again, emphasize that neutral complexes exhibit an electronically different behavior from charged ones and therefore dividing the entire set into neutral and charged complexes is completely reasonable and even indispensable. Using the linear regression equations given in Figures 3b and 3c, a satisfactory evaluation of BE from
ρBCP is made feasible in the case of neutral and charged complexes, respectively. It is also worthy to mention that the slope of regression equation in the case of charged complexes, −332.34, is conspicuously larger than that of neutral complexes, −223.08, implying that the change of BE is more sensitive to the change of
ρBCP for charged complexes compared with the neutral ones.
在 QTAIM 理论中，BCP 通常被视为两个相互作用原子之间相互作用路径上最具代表性的点。电子密度ρ(r)是一个直接从波函数和实验可观察量导出的量，在 QTAIM 分析框架中，BCP 处的ρ值起着核心作用。首先，我们检查 BCP 处的ρ(r)，即ρ BCP ，与计算得到的 BE（如表 1 中的 BE-1，以下同）之间的关系。如图 3a 所示，整个集合中 BE 与ρ BCP 的图呈现出令人满意的线性相关性（R 2 = 0.9716），但 MAPE 的值为 45.18%，显然是不可接受的。如图 3b 所示，当仅针对中性复合物绘制 BE 与ρ BCP 的关系图时，不仅线性依赖性略有改善至 0.9732，而且 MAPE 显著降低至 14.69%。图 3c 表明，如果仅针对带电复合物绘制 BE 与ρ BCP 的关系图，存在类似情况，尽管 R 2 基本保持不变，但与图 3a 相比，MAPE 经历了显著的改善，降至 10.03%。 这些结果再次强调，中性复合物表现出与带电复合物不同的电子行为，因此将整个集合分为中性和带电复合物是完全合理甚至必不可少的。使用图 3b 和 3c 给出的线性回归方程，在中性和带电复合物的情况下，从ρ BCP 评估 BE 是可行的。值得一提的是，带电复合物的回归方程斜率，-332.34，明显大于中性复合物的斜率，-223.08，这意味着对于带电复合物而言，BE 的变化对ρ BCP 的变化更为敏感，与中性复合物相比。
HB 在 BCP 处的 BE 与ρ(r)的回归图，a) 整体集合，b) 中性子集，c) 带电子集。[彩色图可在 wileyonlinelibrary.com 上查看]
It is well known that the larger value of
ρBCP, the stronger is the corresponding HB interaction. This traditional viewpoint is perfectly verified by our calculated data. A similar situation has also been confirmed by many other researchers, although the systems involved in their studies are different to ours. For example, ref. 24 showed that for C─H⋯Y and X─H⋯π-type intermolecular HBs, there is a significant negative correlation between
ρBCP and BE.
众所周知，ρ BCP 的较大值对应着更强的 HB 相互作用。这一传统观点通过我们的计算数据得到了完美验证。许多其他研究者也确认了类似的情况，尽管他们研究的系统与我们不同。例如，参考文献 24 显示，对于 C─H⋯Y 和 X─H⋯π型分子间 HB，ρ BCP 与 BE 之间存在显著的负相关关系。
The potential energy
V(r) is another important quantity in the QTAIM theory, it characterizes the potential energy of electrons at a given position
r.18 In 1998, Espinosa et al. showed that the BE can be approximately estimated as the half value of
V(r) at the BCP for HBs of X─O⋯H (X = C, N, O) type,30 namely BE ≈ VBCP/2. This equation is popular and has frequently been employed for estimating HB strengths.23, 25 We thus believe it is quite important to revisit this and look whether this relationship is really able to predict BE at an acceptable accuracy. In Figure 4, BEs are plotted against the values of
VBCP/2 in Table 5 for the studied complexes. Figure 4a represents a modest linear correlation (R2 = 0.8893) but a very large MAPE (98.48%), making this regression quite worthless. As shown in Figure 4b, the plot of the BE versus
VBCP/2 solely for the neutral complexes leads to a relatively satisfying linear correlation (R2 = 0.9482) and the MAPE is noticeably reduced to 23.36%. A similar situation exists in Figure 4c for the charged complexes, whose R2, 0.8679, is remarkably lower than that of the neutral complexes and also leads to a more satisfying MAPE value of 19.26%. As an important result, neither for the charged interacting monomers nor the neutral ones a sufficiently satisfying estimation of the BE can be achieved via Espinosa's expression. There should be two main reasons for the relatively poor prediction ability of Espinosa's expression: (1) the data quality of the BEs involved in Espinosa's study is low and (2) only very limited HB types were taken into the regression analysis in Espinosa's work. Given the fact that the BE can be predicted based on
ρBCP at a much better accuracy, using the BE ≈ VBCP/2 relationship for prediction of BEs is no longer recommendable.
在 QTAIM 理论中，势能 V(r)是另一个重要量，它描述了给定位置 r 处电子的势能。18 1998 年，埃斯皮诺萨等人表明，可以大约估计 BE 作为 BCP 处 HBs（X─O⋯H，X=C，N，O 类型）的 V(r)的一半值，即 BE ≈ V BCP /2。这个公式很受欢迎，经常被用来估计 HB 强度。23, 25 因此，我们相信重新审视这一点并查看这种关系是否真的能够在可接受的精度下预测 BE 是非常重要的。在图 4 中，BE 与表 5 中研究复合物的 V BCP /2 值进行了比较。图 4a 表示一个适度的线性相关（R 2 = 0.8893），但 MAPE（98.48%）非常大，使这个回归几乎毫无价值。如图 4b 所示，仅对于中性复合物的 BE 与 V BCP /2 的图，导致相对满意的线性相关（R 2 = 0.9482），MAPE 显著降低至 23.36%。对于带电复合物的情况如图 4c 所示，其 R 2 为 0，存在类似的情况。8679 的值明显低于中性复合物，并且导致 MAPE 值为 19.26%，这是一个显著的结果。对于带电相互作用单体和中性单体，Espinosa 的表达式都无法实现足够令人满意的 BE 估计。Espinosa 表达式预测能力相对较差有两个主要原因：（1）Espinosa 研究中涉及的 BE 数据质量低，（2）Espinosa 的工作中仅考虑了非常有限的 HB 类型进行回归分析。鉴于基于ρ BCP 可以以更高的准确性预测 BE 的事实，使用 BE ≈ V BCP /2 的关系来预测 BE 不再推荐。
回归图，显示 BE 与 E H − B = V BCP /2 的关系，分别为：a) 整体数据集，b) 中性子集，c) 带电子集。[彩色图可在 wileyonlinelibrary.com 上查看]
Next, we investigate whether good correlations could be found between other properties at the BCP and the BE. The electronic energy density
H(r) at the BCP position may behave as a useful index for characterizing the nature of HB interaction, as it has been involved in many QTAIM studies for HB systems23, 58 and other kinds of weakly interacting complexes.59, 60 It is expected that HBCP has a somewhat implicit correlation with HB strength, a stronger HB interaction should correspond to a more negative
HBCP. The 
corresponds to the energy density per electron at a given point, its value at the BCP is known as bond degree.58 The 
stands for the ratio of absolute potential energy density to Lagrangian kinetic energy density
G(r). As will be discussed later, this function at the BCP can be interpreted as a measure of covalent character in a given HB interaction.
接下来，我们探讨在 BCP 和 BE 之间是否存在其他性质的良好相关性。BCP 位置的电子能量密度 H(r)可能作为描述 HB 相互作用本质的有用指标，因为它在许多 QTAIM 研究中被用于 HB 系统 23, 58 和弱相互作用复合物的其他类型 59, 60。预期 H BCP 与 HB 强度之间存在某种隐含的相关性，更强的 HB 相互作用应对应更负的 H BCP 。 表示给定点的每个电子的能量密度，BCP 处的值被称为键度。58 表示绝对势能密度与拉格朗日动能密度 G(r)的比值。稍后将讨论，BCP 处的此函数可以解释为给定 HB 相互作用的共价性质的度量。
The regression plots of BEs against
H(r), 
, and 
at the BCP for the entire set are given in Figure 5. It can be seen that none of them correlates well with the BE, the distribution of data points is fairly scattered and MAPE values are rather large. However, as shown in Table 6, if respective fitting is performed for neutral and charged sets, some MAPE values could be significantly reduced; in particular, the MAPE of 
for the charged set is now merely 10.04%. Since none of the MAPE values are smaller than the one yielded by the linear fitting between BE versus
ρBCP (Fig. 3), there is little practical value of using any of these indices to empirically predict BE via the fitted equations.
整个数据集在 BCP 处，BEs 与 H(r)， 和 的回归图如图 5 所示。可以看出，它们与 BE 的相关性都不好，数据点的分布相当分散，MAPE 值也相当大。然而，如表 6 所示，如果分别对中性和带电集进行拟合，一些 MAPE 值可以显著降低；特别是带电集的 的 MAPE 现在仅为 10.04%。由于这些 MAPE 值中没有一个小于ρ BCP 与 BE 之间的线性拟合所得到的值（图 3），因此使用这些指标通过拟合方程来经验性预测 BE 几乎没有实际价值。
, and 
at the BCP for the entire set of complexes considered in this work. [Color figure can be viewed at wileyonlinelibrary.com]本工作中考虑的所有复合物集合在 BCP 处针对 H(r)，
和 对 BE 的回归图。[彩色图可在 wileyonlinelibrary.com 上查看]表 6. 对于研究的复合物，线性拟合各种 QTAIM 以及其它 HB 描述符与 BE 值之间的结果的 MAPE（%）的总结。
| Set 设置 | ρ(r) | H(r) | 
|
|
V(r)/2 | CVB | ΔΔV n |
|---|---|---|---|---|---|---|---|
| Entire 完整 | 45.18 | 181.09 | 134.67 | 56.43 | 98.48 | 51.20 | 56.42 |
| Neutral 中立 | 14.69 | 68.53 | 48.06 | 29.18 | 23.36 | 32.75 | 26.26 |
| Charged 被指控 | 10.03 | 19.34 | 10.04 | 16.44 | 19.26 | 16.05 | 15.28 |
The CVB and ΔΔV
n indices are defined beyond the QTAIM framework and both of them were demonstrated to be able to reliably reveal the strength of HB. The CVB index was first introduced by Fuster and Silvi26 based on the ELF.27, 28 For a D-H…A HB contact, this descriptor is expressed as
CVB = η(CV, D) − η(DH, A), in which the
η(CV, D) is the ELF value at the minimum point where the core and valence basins of D are separated from each other, while
η(DH, A) is the value of ELF at the minimum point where the valence basin of D-H moiety is separated from that of A. A graphical representation of CVB index is given in the Supporting Information Figure S4. Noteworthy, an excellent analysis about the correlation between CVB index and a variety of QTAIM descriptors has been given by Fuster and Grabowski for intramolecular O─H⋯O HBs, the reader is referred to ref. 23. The ΔΔV
n was proposed by Mohan and Suresh based on the molecular electrostatic potential (MEP).29 They studied a series of electrostatics dominated interacting systems including HBs, halogen bonds, and dihydrogen bond dimers and then defined a new parameter ΔΔV
n = (V
n − D′ − V
n − D) − (V
n − A′ − V
n − A) = ΔV
n − D − ΔV
n − A, where D and A stand for electron donor and electron acceptor atoms involved in the considered interactions, respectively.
ΔV
n − X denotes the MEP value at the nuclear position of atom X without nuclear charge contribution of this atom. In addition, the primed and unprimed terms stand for the corresponding quantity in the complex and isolated monomer, respectively. It was found that the ΔΔV
n values are nicely linearly correlated with BEs evaluated at the MP4(SDQ)/aug-cc-pVTZ level. Notice that the MP4(SDQ) method is not as accurate as the CCSD(T) employed in our present work.61
CVB 和ΔΔV
n 指数超出了 QTAIM 框架的定义，并且两者都证明了能够可靠地揭示氢键的强度。CVB 指数首先由 Fuster 和 Silvi 基于 ELF 引入，基于 D-H…A 氢键接触，此描述符表示为 CVB = η(CV, D) - η(DH, A)，其中η(CV, D)是 D 的核心和价电子盆地分离时的 ELF 值的最小点，而η(DH, A)是 D-H 基团的价电子盆地与 A 分离时的 ELF 值的最小点。CVB 指数的图形表示在补充信息图 S4 中给出。值得注意的是，Fuster 和 Grabowski 对分子内 O─H⋯O 氢键之间的 CVB 指数和各种 QTAIM 描述符的相关性进行了出色分析，读者可参阅参考文献 23。基于分子静电势（MEP）提出的ΔΔV
n 。 29 他们研究了一系列以静电学为主导的相互作用系统，包括 HBs、卤素键和二氢键，然后定义了一个新的参数 ΔΔV
n = (V
n − D′ − V
n − D ) − (V
n − A′ − V
n − A ) = ΔV
n − D − ΔV
n − A ，其中 D 和 A 分别代表参与考虑的相互作用的电子供体原子和电子受体原子。ΔV
n − X 表示在原子 X 的核位置处的 MEP 值，不包括该原子的核电荷贡献。此外，带撇和不带撇的术语分别代表在复合物和孤立单体中的相应量。发现ΔΔV
n 值与在 MP4(SDQ)/aug-cc-pVTZ 水平上评估的 BEs 之间有很好的线性相关性。请注意，MP4(SDQ)方法的准确性不如我们在当前工作中使用的 CCSD(T)方法。 61
The regression plots of BE against CVB index and ΔΔV
n for the entire set of complexes are given in Figure 6. From the scatter map, one can immediately find that both indices poorly correlate with the BE, especially in the region corresponding to medium and strong HBs. Not only the R2 but also the MAPE values are unsatisfactory. Individual regression analyses for the neutral and charged sets were also performed, the results are given in Table 6. It can be seen that the respective fittings indeed conspicuously reduce the MAPEs and thus greatly improve the accuracy of reproducibility. Unfortunately, for both CVB and ΔΔV
n indices, neither in the case of the neutral set nor in the case of the charged set, the value of MAPE is low enough to reach a reliable prediction of the BE based on these indices.
整个复合物集的 BE 对 CVB 指数和ΔΔV
n 的回归图在图 6 中给出。从散点图中，可以直接发现这两个指标与 BE 的相关性很差，尤其是在对应中等和强烈氢键的区域。不仅 R 2 的值，MAPE 的值也不令人满意。对中性和电荷集进行了单独的回归分析，结果在表 6 中给出。可以看出，相应的拟合确实明显降低了 MAPE 值，从而大大提高了可重复性的准确性。不幸的是，对于 CVB 和ΔΔV
n 指标，无论是中性集还是电荷集的情况，MAPE 的值都不足够低，无法基于这些指标可靠地预测 BE。
本工作中考虑的所有复合体的完整集合的回归图，针对 BE 与 CVB 指数以及ΔΔV n 。彩色图可以在 wileyonlinelibrary.com 上查看。
Table 6 summarizes the MAPE of all QTAIM or HB indices for various sets considered in present work. It can be immediately found that irrespective of the choice of the index, the charged set can always be evidently better represented by a linearly fitted equation than the neutral set, the reason may be that the magnitude of BEs of the charged set is generally larger than the neutral set. Therefore, achieving a lower MAPE is relatively easier, since MAPE is a quantity focusing on reflecting relative error. In addition, from Table 6, it can be seen that if the neutral and charged sets are not explicitly distinguished, the MAPE will be always too high and thus making the fitted equation useless for predicting BE. This observation also substantially implies that neutral HBs and charged HBs have essentially very different underlying natures.
表 6 总结了本工作中考虑的各种集合中所有 QTAIM 或 HB 指标的 MAPE。可以立即发现，无论选择哪个指标，带电集合总是可以通过线性拟合方程更明显地表示为中性集合，原因可能是带电集合的 BE 大小通常大于中性集合。因此，实现较低的 MAPE 相对更容易，因为 MAPE 关注反映相对误差。此外，从表 6 可以看出，如果没有明确区分中性和带电集合，MAPE 将总是太高，从而使拟合方程无法用于预测 BE。这一观察结果也明显表明，中性 HB 和带电 HB 本质上具有非常不同的内在性质。
Out of our expectations, although the electron density at the BCP is the simplest index considered in this work, it markedly outperforms all other indices in the correlation with BE. Its MAPEs for neutral and charged HB systems are only 14.69% and 10.03%, respectively; which, in our opinion, is acceptable, at least useful for using corresponding fitted equations to empirically estimate BE in practical studies when the requirement on accuracy is not quite high. Assume that there is a HB with a BE of −5.0 kcal/mol and the system is neutral, it is expected that the absolute error using our fitted equation will be merely 0.7 kcal/mol. This level of accuracy exceeds most low-level quantum chemical methods, for example, semi-empirical ones.
超出我们的预期，尽管在本工作中考虑的 BCP 的电子密度是最简单的指标，但它在与 BE 的相关性上显著优于所有其他指标。中性 HB 系统和带电 HB 系统的 MAPE 分别为 14.69%和 10.03%，在我们看来，这是可以接受的，至少在准确性要求不太高的实际研究中，使用相应的拟合方程来经验性地估算 BE 是有用的。假设有一个 BE 为-5.0 kcal/mol 的 HB 系统，并且系统是中性的，使用我们拟合的方程估计的绝对误差仅预计为 0.7 kcal/mol。这种精度超过了大多数低级量子化学方法，例如半经验方法。
On the covalent character of strong HB interaction: Correlation between QTAIM descriptors and SAPT induction term
强 HB 相互作用的共价特性：QTAIM 描述符与 SAPT 诱导项之间的相关性
It has been pointed out that the covalent character of strong HB interactions is significantly greater than that of other types of HB interactions22 and can be reflected in the magnitude of associated SAPT-derived induction term,
Eind.19 It is expected that the extent of covalent character of a given HB interaction can be correlated with the magnitude of corresponding induction interaction. In this sense, we will try to establish linear correlations between the
Eind and some QTAIM topological descriptors describing the magnitude of covalent character.
已指出，强氢键相互作用的共价特性显著高于其他类型的氢键相互作用 22，并可以通过相关 SAPT 衍生的诱导项的大小反映，E ind 。19 预期，给定氢键相互作用的共价特性的程度可以与相应的诱导相互作用的大小相关联。从这个意义上说，我们将尝试建立 E ind 和描述共价特性的大小的一些 QTAIM 拓扑描述符之间的线性关系。
In ref. 58, the 
ratio at the BCP is shown to be particularly useful for distinguishing the type of interactions. It is argued that 
, 
and 
correspond to closed-shell interaction, intermediate interaction, and covalent interaction, respectively. Hence, 
could be served as a measure of HB covalency and thus will be included in our regression analysis. In the same paper, the bond degree 
is proposed, which has been mentioned earlier. For covalent interactions, which is characterized by the condition
HBCP < 0, it is argued that the more negative the 
, the stronger will be the interaction. While for noncovalent interactions, which correspond to
HBCP > 0, more positive 
implies stronger interaction. Since 
is closely related to both covalent character and interaction strength, it is chosen as another quantity for regression analysis.
在参考 58 中，BCP 处的 比率被证明特别有用，用于区分交互类型。认为 、 和 分别对应于闭壳交互、中间交互和共价交互。因此， 可以作为 HB 共价性的度量，并且将被包括在我们的回归分析中。在同一篇论文中，提出了键度 ，之前已经提到过。对于共价交互，其特征是条件 H BCP < 0，认为 的负值越小，交互越强。而对于非共价交互，对应于 H BCP > 0， 的正值越大，交互越强。由于 与共价特性和交互强度密切相关，因此被选作另一个用于回归分析的量。
Regression plots of the
Eind in Table 2 versus the 
and 
at the BCP in Table 5 are sketched in Figure 7. For Figures 7a and 7c, we employed second-order polynomial fit because the fitting quality in this case is significantly better than the linear fitting. We first look at the case of neutral complexes. From Figures 7a and 7c, it can be seen that both 
and 
show satisfactory correlation with
Eind, their R2 reach as high as 0.971 and 0.959, respectively. The 
with MAPE = 14.46% has an evidently better ability for reproducing
Eind than 
, whose MAPE is notably larger (37.79%). This observation implies that the 
, which is often employed as a covalency metric, is indeed valuable and works for broad range of neutral HB systems.
图 7 中绘制了表 2 中 E ind 在表 5 中 BCP 处与 和 的回归图。对于图 7a 和 7c，我们采用了二次多项式拟合，因为在这种情况下，拟合质量明显优于线性拟合。我们首先关注中性复合物的情况。从图 7a 和 7c 可以看出， 和 均与 E ind 显示良好的相关性，它们的 R 值分别高达 0.971 和 0.959。具有 MAPE = 14.46%的 在重现 E ind 方面显然比 MAPE 为 37.79%的 更好。这一观察表明，通常作为共价性度量的 确实有价值，并适用于广泛的中性 HB 系统。
at BCP for (a) neutral and (b) charged complexes, and versus 
at BCP for (c) neutral and (d) charged complexes. [Color figure can be viewed at wileyonlinelibrary.com]BCP 点的 E ind 与
在（a）中性和（b）带电复合物之间的回归图，以及 BCP 点的 E ind 与 在（c）中性和（d）带电复合物之间的回归图。[彩色图可以在 wileyonlinelibrary.com 上查看]As mentioned earlier, the character of HB interaction for charged complexes is very different from that of the neutral ones, this point is also somewhat reflected in the scatter map between
Eind and 
, Figure 7b, as well as that between
Eind and 
, Figure 7d, for charged systems. Compared to Figures 7a and 7c, the most notable feature in Figures 7b and 7d is that the distribution of points is conspicuously more scattered and the regularity is not so strong, neither linear fitting nor second-order polynomial fit could correlate the 
or 
with
Eind quite satisfactorily, implying that the covalent character in the charged HB complexes is much more complex to identify and highly system dependent. One possible reason why the correlation between the Eind and the QTAIM descriptors degrades for charged complexes is that the Eind includes much stronger CT effect and polarization effect compared to the neutral systems, both are not unambiguously separable from the Eind according to the SAPT theory. If the CT part can be sufficiently excluded from the Eind, perhaps the correlation could be improved.
如前所述，带电复合物的 HB 相互作用的特性与中性的完全不同，这一点在图 7b 的 E ind 和 之间的散点图，以及图 7d 的 E ind 和 之间的散点图中也有所体现，对于带电系统而言。与图 7a 和 7c 相比，图 7b 和 7d 中最显著的特点是点的分布明显更为分散，规律性不强，无论是线性拟合还是二次多项式拟合都无法很好地将 或 与 E ind 相关联，这表明带电 HB 复合物中的共价特性要复杂得多，且高度依赖于系统。带电复合物中 E ind 与 QTAIM 描述符之间的相关性下降的一个可能原因是，E ind 包含了比中性系统更强的 CT 效应和极化效应，根据 SAPT 理论，两者都不清楚地从 E ind 中分离出来。如果能够充分排除 E ind 中的 CT 部分，也许相关性可以得到改善。
Conclusions 结论
在这项研究中，我们对 42 个不同的小分子间氢键二聚复合体进行了理论分析，这些复合体分为 28 个中性复合体和 14 个正负电荷相互作用的单体。在 B3LYP-D3(BJ)/ma-TZVPP 计算水平下对复合体进行了全面优化后，我们评估了它们的 BEs，在非常准确的 CCSD(T)/jul-cc-pVTZ 水平上包括 BSSE 校正以及 SAPT2 + (3)δMP2/aug-cc-pVTZ 水平，后者允许将 BE 分解为物理上可解释的成分，以揭示考虑的氢键相互作用的电子性质。此外，我们还计算了一些著名的 QTAIM 和其他与氢键相关的描述符。通过详细检查和相关性分析这些数据，我们可以提出许多吸引人的结论和有价值的研究发现：
- SAPT analysis fully reveals different nature of HBs of different types and strengths, finally allowing us to classify HB interactions in a more rigorous manner compared to already provided classifications, the key advantage of this new classification is that the magnitude of BE is directly correlated with its dominating physical components. Neutral complexes are classified as “very weak” HBs if its BE magnitude is <2.5 kcal/mol, which are mainly dominated by dispersion together with electrostatics interactions, or as “weak-to-medium” HBs with a BE magnitude varying between 2.5 and 14.0 kcal/mol, which are mostly dominated by electrostatics interaction. On the other hand, charged complexes require a separate classification and are divided into “medium” HBs with BE magnitude in the range of 11.0–15.0 kcal/mol, which are mainly dominated by electrostatics interaction, or into “strong” HBs whose BE magnitude is >15.0 kcal/mol, which are significantly contributed by both electrostatics and induction interactions.
SAPT 分析全面揭示了不同类型和强度的 HBs 的不同本质，最终使我们能够以比已提供的分类更严格的方式对 HB 相互作用进行分类。这种新分类的关键优势在于，BE 的大小直接与主导的物理成分相关。如果 BE 的大小小于 2.5 千卡/摩尔，则中性复合物被分类为“非常弱”的 HBs，主要由色散和静电相互作用共同主导。或者，如果 BE 的大小在 2.5 至 14.0 千卡/摩尔之间，则被分类为“弱至中等”的 HBs，主要由静电相互作用主导。另一方面，带电复合物需要单独分类，并分为“中等”的 HBs，其 BE 的大小在 11.0 至 15.0 千卡/摩尔之间，主要由静电相互作用主导，或者分为“强”的 HBs，其 BE 的大小大于 15.0 千卡/摩尔，这两种相互作用（静电和诱导）都对其有显著贡献。 - Correlation analyses were performed between various topological indices and the BE. The indices we considered include
ρ(r), ∇2ρ(r),
H(r),
, 
, and
V(r)/2 at the BCP defined by QTAIM theory, as well as two indices aiming for revealing HB strength, namely CVB index and ΔΔV
n. It is found that the
ρ(r) at the BCP, that is,
ρBCP, of our studied HBs showed the best linear dependency with the BE for both neutral and charged systems. The corresponding fitted equations are thus recommended for quickly and reliably predicting BE, namely BE/kcal/mol = − 223.08 × ρBCP/a. u. + 0.7423 for neutral complexes with a MAPE of 14.7%, and BE/kcal/mol = − 332.34 × ρBCP/a. u. − 1.0661 for charged complexes with a MAPE of 10.0%. Our test also showed that the popular
BE ≈ VBCP/2 relationship proposed by Espinosa and co-workers has an evidently larger error and thus cannot be recommended anymore. It should be noted that our equations were fitted based on B3LYP-D3(BJ)/ma-TZVP wave function, if the
ρ
BCP is computed at a level much poorer than the one we employed, the actual prediction error may be notably larger than our reported MAPE.
在各种拓扑指数与 BE 之间的相关性分析中，我们考虑了以下指数：由 QTAIM 理论定义的 BCP 处的ρ(r)，∇ 2 ρ(r)，H(r)， ， ，以及 V(r)/2，以及旨在揭示氢键强度的两个指数，即 CVB 指数和ΔΔV
n 。发现我们研究的氢键在 BCP 处的ρ(r)，即ρ BCP ，在中性和带电系统中都与 BE 表现出最佳的线性依赖关系。因此推荐相应的拟合方程用于快速可靠地预测 BE，对于中性复合物，方程为 BE/kcal/mol = − 223.08 × ρ BCP /a. u. + 0.7423，MAPE 为 14.7%；对于带电复合物，方程为 BE/kcal/mol = − 332.34 × ρ BCP /a. u. − 1.0661，MAPE 为 10.0%。我们的测试还表明，Espinosa 及其同事提出的流行 BE ≈ V BCP /2 关系明显存在更大的误差，因此不再推荐。值得注意的是，我们的方程是基于 B3LYP-D3(BJ)/ma-TZVP 波函数进行拟合的，如果ρ
BCP 的计算水平远低于我们所采用的水平，实际的预测误差可能远大于我们报告的 MAPE。 - The values of
and 
at the BCP are found to be correlated with the induction term derived by SAPT calculation, whose magnitude may be regarded as a meaningful metric of covalent character of a given HB interaction if excessive CT as expected for certain charged systems can be excluded. Our regression analysis validates the use of these two descriptors for measuring covalency of HBs, but only for neutral complexes.
在 BCP 处找到的 和 的值与 SAPT 计算得出的诱导项相关，该项的大小可以被视为衡量给定氢键相互作用共价性质的有意义度量，如果可以排除某些带电系统中预期的过量 CT。我们的回归分析验证了使用这两个描述符来测量氢键的共价性，但仅适用于中性复合物。
In addition, as a byproduct of our study, it is found that all SAPT energy components correlate well with BE for neutral HB complexes; therefore, a rough estimation of SAPT terms is fully possible by directly using our fitted equations (Fig. 2) based on the BE calculated in usual manner.
此外，作为我们研究的副产品，发现所有 SAPT 能量成分都与中性 HB 复合物的 BE 很好地相关；因此，通过直接使用我们根据常规方法计算的 BE 拟合方程（图 2）来大致估计 SAPT 项是完全可能的。
Acknowledgments 致谢
Saeedreza Emamian would like to present especial thanks to Dr Abdorreza Alavi Gharahbagh from Islamic Azad University of Shahrood for his effective advice regarding the regression analyses.
Saeedreza Emamian 特别感谢伊斯兰阿扎德大学沙赫罗德分校的阿布多雷扎·阿尔瓦伊·加拉赫巴赫博士，感谢他在回归分析方面的有效建议。
Conflicts of Interest 利益冲突
There are no conflicts to declare.
无需申报任何冲突。
