HydrogenâHydrogen Bonds in Highly Branched Alkanes and in

Article pubs.acs.org/JPCA

Hydrogen−Hydrogen Bonds in Highly Branched Alkanes and in Alkane Complexes: A DFT, ab initio, QTAIM, and ELF Study Norberto K. V. Monteiro and Caio L. Firme* Institute of Chemistry, Federal University of Rio Grande do Norte, Natal, Rio Grande do Norte, Brazil, CEP 59078-970. S Supporting Information *

ABSTRACT: The hydrogen−hydrogen (H−H) bond or hydrogen−hydrogen bonding is formed by the interaction between a pair of identical or similar hydrogen atoms that are close to electrical neutrality and it yields a stabilizing contribution to the overall molecular energy. This work provides new, important information regarding hydrogen−hydrogen bonds. We report that stability of alkane complexes and boiling point of alkanes are directly related to H−H bond, which means that intermolecular interactions between alkane chains are directional H−H bond, not nondirectional induced dipole−induced dipole. Moreover, we show the existence of intramolecular H−H bonds in highly branched alkanes playing a secondary role in their increased stabilities in comparison with linear or less branched isomers. These results were accomplished by different approaches: density functional theory (DFT), ab initio, quantum theory of atoms in molecules (QTAIM), and electron localization function (ELF).

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INTRODUCTION

bond or hydrogen−hydrogen bonding, wherein a bond path links a pair of identical or similar hydrogen atoms that are close to electrical neutrality, opposing the dihydrogen bonding with positively and negatively charged hydrogens.5,9 It is demonstrated that these interactions are found to link hydrogen atoms bonded to both unsaturated and saturated carbon atoms in hydrocarbon molecules and contribute to molecular stabilization, where each H−H bond makes a stabilizing contribution of up to 10 kcal/mol to the energy of the molecule.9 These interactions are found between the ortho-hydrogen atoms in planar biphenyl, between the hydrogen atoms bonded to the C1−C4 carbon atoms in phenanthrene and other angular polybenzenoids, and between the methyl hydrogen atoms in the cyclobutadiene, tetrahedrane and indacene molecules corseted with tertiary-tetra-butyl groups.9 Some experimental10−14 and computational15−18 studies exemplify interactions between two hydrogen atoms, although not differentiating H−H bond from dihydrogen bond. Nonetheless, the interactions cited in these works occur between similar or identical hydrogen atoms close to electrical neutrality. Some studies used the term “non-bonding repulsive interaction” to describe the H−H bond.

Bader defined bond path as an atomic interaction line (AIL) which is mirrored by its corresponding virial path, where the potential energy density is maximally negative.1,2 Bader3,4 has stated that bond paths are indicative of bonded interactions, which encompass all kinds of chemical interactions. Bond paths or virial paths can be used to characterize four types of bonded interactions involving hydrogen atoms: dihydrogen bond, hydrogen bond, hydride bond, and hydrogen−hydrogen bond. The dihydrogen bond is a bonded interaction consisting of electrostatic interaction between two hydrogen atoms of opposite charges and its formation is restricted to systems that possess a hydridic hydrogen wherein transition metals and boron atom are typical elements that can interact with this hydridic hydrogen.5 On the other hand, the hydrogen bond involves only one hydrogen atom and it can be represented by X−H···Y interaction, where X and Y are electronegative atoms; X−H is named as a proton-donating bond, and Y is an acceptor center.6 An unconventional type of hydrogen bond is named hydride bond, also known as “inverse hydrogen bond”. In the hydride bond, a negatively charged hydrogen atom will be the electron donor for non-hydrogen acceptors. The hydride bond consists of a hydride placed between two electrophilic centers.7,8 Recently,9 a new kind of interaction involving hydrogen atoms has been discovered by means of the quantum theory of atoms in molecule (QTAIM): the hydrogen−hydrogen (H−H) © 2014 American Chemical Society

Received: January 6, 2014 Revised: February 8, 2014 Published: February 15, 2014 1730

dx.doi.org/10.1021/jp500131z | J. Phys. Chem. A 2014, 118, 1730−1740

The Journal of Physical Chemistry A

Article

The H−H bond in planar biphenyl was questioned.19 However, no other studied molecular system with hydrogen− hydrogen bond has been questioned to date. Bader20 argued that there is no repulsion in rotamers such as ethane or biphenyl, only decrease of nuclear attraction. In reply21 to Bader’s statements, an arbitrary set of MO’s was used (which is obviously not univocal22). Nonetheless, QTAIM is dependent upon the electron density which, in turn, arises from a variational calculation and it represents “the distribution that minimizes the energy for any geometry and thus the presence of bond paths represent spatial accumulation of density that lower the energy of the system”, as stated by Bader.20 Therefore, bond path is an universal indicator of bonded interactions.3 Accordingly, QTAIM has become a method that has already been successfully applied to understand conventional hydrogen bonds,23,24 C−H···O bonds,25 and dihydrogen bonds.26,27 In this work, we show that there are very good linear relations between H−H bond in alkane complexes and their stabilities and between H−H bond in alkane complexes and their boiling points. We also demonstrate that there are H−H bonds in highly branched alkanes, and we show that the H−H bonds play a secondary role in the stability of highly branched alkanes. All results in this work were obtained by means of different approaches: the topology of the virial field function based on QTAIM, the critical points of electron localization function (ELF), ab initio, and density functional theory.

H−H bond were used to locate the corresponding ELF BCP. All ELF BCPs were confirmed as an index 1 critical point since their corresponding ∇η[r(QTAIM BCP coordinates)] is 0.0000. Fuster and Grabowski24 obtained a 0.9935 coefficient of determination involving ELF H···BCP distance and QTAIM H···BCP distance, which reinforces our approach for finding ELF bond critical point of H−H bond from QTAIM coordinates. QTAIM and van der Waals Interactions. The QTAIM is an extension of quantum mechanics to an open system that is made up of atomic basins in a molecule, and it can be developed by deriving Heisenberg’s equations of motion from Schrödinger’s equation wherein any atomic property in a molecule can be calculated.43 The quantum theory of atoms in molecules was developed by Bader to study the electronic structure and especially the chemical bond by trespassing the limits of orbital theories when approaching the physics underlying atoms in a molecule.1,2 This quantum model is considered innovative in the study of chemical bonding and characterization of intra- or intermolecular interactions.6,44 According to the QTAIM quantummechanic concepts,45,46 the calculation of gradient of charge density,∇ρ, of the studied molecular system is performed through the electron density’s Hessian matrix in order to find critical points of the charge density.47 The critical points of the charge density are obtained where ∇ρ = 0. The bond critical point is a saddle point on an electron density curvature, being a minimum in the direction of the atomic interaction line and a maximum in the two directions perpendicular to it.48−50 The BCP’s give information about the character of different chemical bonds or chemical interactions, for example, metal− ligand interactions,48,51,52 covalent bonds,1,2,53 different kinds of hydrogen bonds,24,54,55 including dihydrogen bonds26,56,57 and the H−H bond.36,58 The Laplacian of the charge density (∇2ρ) is obtained from the sum of the three eigenvalues of the Hessian matrix (λ1, λ2 and λ3) of the charge density.1,2 The negative (∇2ρ < 0) or positive (∇2ρ > 0) sign of the Laplacian represents concentration and depletion of the charge density, respectively.59 The H−H bonds exhibit characteristics of closed-shell interaction since they have low value of charge density (ρb) ( 0, and the positive value for the total energy density Hb that is close to zero. The H−H bond presents BCP topological values similar to those from weak hydrogen bonds.60−62 All properties of matter are somewhat related to its corresponding density of electronic charge ρ(r), where one can associate its topology with delineating atoms and the bonding between them.9 In molecules, the nuclei of bonded atoms are linked by a line along which the electron density is a maximum with respect to any neighboring gradient path, and it is termed bond path.9 In truth, according to Bader, a bond path has to be mirrored by its corresponding virial path, otherwise it cannot be regarded as a bond path. In a virial path, the potential energy density is maximally stabilizing.63 When the molecular system is also in a stable electrostatic equilibrium, it ensures both a necessary and a sufficient condition for bonding.64 Bader also defined bond path as a measured consequence of the action of the density operator along the trajectories that originate in bond critical point and end in both interacting nuclei.65 The bond path is based on theorems of quantum mechanics that govern the interactions between atoms. Only two forces operate in a molecular system: Feynman and Ehrenfest, which act on the nuclei and electrons, respectively.

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COMPUTATIONAL DETAILS AND THEORETICAL METHODS The geometries of the studied species were optimized by using standard techniques.28 Vibrational modes of the optimized geometry were calculated in order to determine whether the resulting geometries are true minima or transition states. Calculations were performed at ωB97XD,29 G4,30 G4/CCSD(T),31 M062X,32 and B3LYP33,34 methods with the 6-311+ +G(d,p) basis set by using the Gaussian 09 package,35 including the electron density, which was further used for QTAIM calculations. The 6-311++G(d,p) basis set was applied because the inclusion of the diffuse components in the basis set is desired to describe properly the hydrogen−hydrogen bond.36 Among the density functionals accounting for dispersion, ωB97XD37,38 contains empirical dispersion terms and also long-range corrections, exhibiting a van der Waals correction. All topological information was calculated by means of AIM2000 software.39 The algorithm of AIM2000 for searching critical points is based on the Newton−Raphson method, which relies heavily on the chosen starting point.40 Every available alternative for increasing calculation accuracy and choosing different starting points was done. Integrations of the atomic basins were calculated in natural coordinates in default options of integration. All integrations yielded 10−3 to 10−4 order of magnitude for the Laplacian of the calculated atomic basin. Atomic energies were calculated from the atomic virial approach. All calculated bond paths are mirrored by their corresponding virial paths (where maximum negative potential energy exists between two atoms). According to atomic virial approach, the bond paths mirrored by virial paths are indicative of bonded interaction. The evaluation of the electron localization function (ELF) values, η(r), and topological information of bond critical point (BCP) have been carried out with the TopMoD program.41,42 For each alkane complex, the QTAIM coordinates of the bond critical point of a selected 1731



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They both are united in the virial theorem,66 which relates these forces to the energy of the molecule.67 This is the essential physics of all bonded interactions that, when combined with the properties of the bond critical point, provides all the features of a chemical bond. Many studies involving critical points and bond paths in van der Waals and weak interactions have been performed.68−70 The van der Waals interactions have been identified as directional bonded interactions because of the existence of bond paths in solid chloride and N4S4, which are important to understand the structure of these crystals. Hydrogen−hydrogen bond is also a directional interaction, different from induced dipole−induced dipole interaction ,which is not directional. Wolstenholme and Cameron71 showed that crystal structures of tetraphenylphosphonium squarate, bianthrone, and bis(benzophenone)azine contains a variety of C−Hδ+···+δH−C interactions, as well as a variety of C−H···O and C−H···Cπ interactions. This study established similarities and differences between these weak interactions, leading to a better understanding of H−H bonds. Slater has shown in his paper72 that there are no fundamental differences between van der Waals and covalent bonding, because they have the same quantum mechanism.73 According to Feynman,74 the origin of chemical bond is derived from electron density between nuclei, as well as the requirement for occurrence of a bond path. ELF. The electron localization function introduced by Becke and Edgecombe75 is measured from the Fermi hole curvature in terms of the local excess kinetic energy density due to Pauli repulsion. It has been shown that ELF is confined to the [1,0] interval, where the tendency to 1 means that parallel spins are highly improbable, (otherwise opposite spin pairs are highly probable), while the tendency to zero indicates a high probability of same spin pairs. This interpretation gives ELF function a deep physical meaning. The electron localization function provides a rigorous basis for the analysis of the wave function and electronic structure in molecules and crystals.76 The ELF topology has been widely applied for study of chemical bonding.77−81 The topological partitioning of ELF gradient field82,83 provides two types of basins of attractors that can be thought of as corresponding to atomic cores, bonds, and lone pairs: core basins surrounding nuclei with atomic number Z > 2 and labeled by C(A), where A is the atomic symbol of the element, and valence basins that are characterized by their synaptic order,84 which is the number of the atomic valence shells in which they participate. Therefore, there are monosynaptic, disynaptic, trisynaptic basins etc., labeled V(A), V(A, X), V(A, X, Y), etc., respectively. Monosynaptyc basins correspond to the lone pairs of the Lewis model. Disynaptyc and trisynaptyc basins correspond to two-center and three center bonds respectively. The ELF local function, η(r), provides the chemical information and ∇η(r) provides two types of points in real space: wandering points where ∇η(r) ≠ 0 and critical points where ∇η(r) = 0. The critical points are characterized by their index which is the number of positive eigenvalues from Hessian matrix of η(r). The topology of the ELF has been applied in the study of hydrogen bond.85−88 In FH···N2 complex, the value of ELF at the index 1 critical point between the V(F,H) and V(N) basins is 0.047, while for FH···NH3 the corresponding η(r) is 0.226. These values are attributed to weak and strong hydrogen bonds, respectively. In this work, the ELF was used as a complementary method for evaluating hydrogen−hydrogen

bond for alkane complexes from methane−methane until hexane−hexane.

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RESULTS AND DISCUSSION We analyzed the H−H bonds as intramolecular interactions in some C8H18 isomers and their contribution to molecular stabilization. Afterward, we analyzed the H−H bonds as intermolecular interactions in the case of alkane−alkane complexes and its relation with complex stability and boiling point. Octane Isomers. We investigated the possible relation between the amount of H−H bonds and branched alkane stabilization based on heat of combustion (ΔHcomb) of the corresponding alkanes. The isomer with the lowest ΔHcomb is the most stable. It is well-known that branched alkanes are more stable than their linear isomers.89 Among some C8H18 isomers (Table 1), the highly branched isomer, 2,2,3,3Table 1. Amount of Branching, Amount of H−H Bonds and Heats of Combustion (ΔHcomb) in Branched Alkanes octane isomersa

branching

H−H bonds

ΔHcomb (kcal/mol)

ΔΔHcomb (kcal/mol)

octane 3-methyl-heptane 3,3-dimethyl-hexane 2,2,3-trimethylpentane 2,2,3,3-tetramethylbutane

0 1 2 3

0 0 0 2

−1647.6 −1647.3 −1646.9 −1645.8

2.6 2.3 1.9 0.8

4

3

−1645.0

0

a

Optimized at ωB97XD/6-311++(d,p) level of theory.

tetramethylbutane, is the most stable and the linear isomer octane is the least stable. The heats of combustion were calculated at the ωB97xd/6-311++G(d,p) level, since this functional has significantly lower absolute errors when considering the noncovalent attractions and empirical dispersion terms.90 Figure 1 shows the virial graphs of some C8H18 isomers (octane, 3-methylheptane, 3−3-dimethylhexane, 2−2−3-trimethylpentane and 2−2−3−3-tetramethylbutane). The averaged value of charge density of the bond critical point, ρb, from the H−H bonds, is 0.0098 au. which is typical for weak hydrogen bonds and higher than that for van der Waals interactions.60 On going from octane, to 3-methyl-heptane, to 3,3-dimethylhexane, there is a unitary increase of methyl branching accompanying an increasing branching stabilization on C8H18 system (Table 1). However, no H−H bond was found in this set of alkanes. Then, other factors contribute for 3-methylheptane and 3,3-dimethyl-hexane stabilizations relative to octane, which is not the focus of this work. In literature, one may find different explanations for the alkane branching stabilization. Gronert’s work91 suggests that germinal steric interactions in linear alkane are relieved on branching. Opposing Gronert’s hypothesis, Wodrich and Schleyer92 attributed branching stabilization to 1,3-alkyl−alkyl stabilizing interaction (so-called protobranching). In a more recent work, Kennitz and collaborators93 attribute alkane branching stabilization to a greater electron delocalization in branched alkanes relative to linear alkanes. Nonetheless, when comparing 3,3-dimethyl-hexane, 2,2,3trimethyl-pentane, and 2,2,3,3-tetramethyl-butane, there is an increasing trend of intramolecular H−H bond and branching stabilization (Table 1). It was already been shown that these 1732



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Figure 1. Virial graphs of C8H18 isomers: (A) octane, (B) 3-methylheptane, (C) 3,3 dimethylhexane, (D) 2,2,3-trimethylpentane, and (E) 2,2,3,3tetramethylbutane. The geometry optimization of the molecules were performed at ωB97xd/6-311++G(d,p) level of theory.

interactions contribute to energy stabilization in hydrocarbons.9,36 The 2,2,3,3-tetramethylbutane is the most stable branched C8H18 isomer, and it has three H−H bonds, while the second most stable C8H18 isomer, 2,2,3-trimethylpentane, has two intermolecular H−H bonds. Moreover, in the series 3methyl-heptane to 3,3-dimethyl-hexane, the average stabilization, ΔΔHcomb, is 0.35 kcal/mol, while for the 3,3-dimethylhexane to 2,2,3,3-etramethyl-butane series, the average stabilization is 0.95 kcal/mol, which implies the influence of an additional stabilization factor when leaving the first series and going to the latter. Additional evidence on stabilization due to H−H bond of highly branched C8H18 isomer was given by comparing hydrogen atomic energy belonging or not to H−H bond. The atomic energy of hydrogen atoms belonging to H−H bond (H1 in Figure 1E) are on average 0.80 kcal/mol lower when compared to the hydrogen atoms not belonging to H−H bond (H2 in Figure 1E). Thus, for highly branched alkanes, H−H bond can be regarded as secondary stabilization factor. One possible reason for the nonexistence of H−H bond in octane, 3-methylheptane, and 3,3-dimethylhexane can be ascribed to the lack of necessary alignment involving C−H··· H-C bonds of vicinal or germinal methyl groups. Further details on this geometrical requirement are given in the following subsection. Alkane Complexes. In this section we evaluate the H−H bonds in linear bimolecular alkane complexes: methane−

methane, ethane−ethane, propane−propane, butane−butane, pentane−pentane, and hexane−hexane. These alkane complexes were obtained from two different optimization steps: first, the optimization of each single alkane molecule (the geometry of each single molecule is in Supporting Information Figure S1); second, the optimization of the whole bimolecular complex built from an initial geometry where each single optimized molecule was arbitrarily positioned. The optimizations were done in different levels of theory: ωB97XD/6-311+ +G(d,p), G4/CCSD(T)/6-311++G(d,p), B3LYP/6-311++G(d,p) (Figure S2, Supporting Information), and M062X/6311++G(d,p) (Figure S3, Supporting Information). Only optimized structures of alkane complexes obtained from G4/ CCSD(T)/6-311++G(d,p) and ωB97XD/6-311++G(d,p) are depicted in Figures 2 and 3, respectively. We emphasize that the single molecules in Figures 2 and 3 comes from optimized geometries, but both Figures contain randomized nonoptimized initial geometry (to the left) and optimized geometry (to the right) for the complex where individual molecules are optimized as above-mentioned. Both (columns) are shown for comparison reasons, which depict differences of some geometrical parameters after optimization of the whole complex affording better understanding of the H−H bond. In Figure 2, the CH···HC dihedral angles for methane− methane, ethane−ethane, propane−propane, and butane− butane from the G4/CCSD(T)/6-311++G(d,p) level of theory are depicted. In this case, geometry optimization proceeded 1733



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Figure 2. Randomized, nonoptimized, and optimized geometries of methane−methane, ethane−ethane, propone−propane and butane−butane complexes depicting their corresponding selected CH···HC dihedral angles, in degrees. The geometry optimization of the molecules were performed at the G4/CCSD(T)/6-311++G(d,p) level of theory. The dashed lines represent the selected dihedral angles.

different methods for optimization calculation. The optimization with ωB97XD functional tends to impose a conformation that increases the number of intramolecular H−H bonds in the alkane complex. This is confirmed when comparing the virial graphs of G4- and ωB97XD-optimized structures in Figures 4 and 5, respectively. Figure 4 shows the virial graphs of the optimized alkane complexes using the G4/CCSD(T)/6-311++G(d,p) level of theory. There is one H−H bond in the methane−methane complex and two H−H bonds in other alkane complexes. The values of charge density of BCP (ρb) and the Laplacian of ρb (∇2ρ) indicate that all H−H bond paths from alkane complexes exhibit the characteristics of closed-shell interactions. Figure 5 shows the virial graphs of the optimized alkane complexes using the ωB97XD/6-311++G(d,p) level of theory. There is an increasing number of H−H bonds as the number of carbon atoms increases in the alkane complex. The number of H−H bonds increases by a unit from methane−methane to butane−butane. From butane−butane to hexane−hexane, the increase is two units of H−H bond. In the ethane−ethane complex, there are also two C−H intramolecular bonds whose virial paths are curved, meaning less attractive and stable interactions, with a low value of ρb (about 8.5 × 10−6 at 6.7 ×

through the G4 method. The CCSD(T)/6-311++G(d,p) was used for self-consistent field and density matrix calculations from the G4-optimized geometry (Figure S4, Supporting Information). Due to infrastructure limitations, it was not possible to calculate pentane−pentane and hexane−hexane complexes from the G4/CCSD(T)/6-311++G(d,p) level of theory. From Figure 2, one can see that all CH···HC dihedral angles decrease from nonoptimized to optimized structures. Except for the ethane−ethane complex, this decrease is reasonably considerable: in methane−methane (from 17.53 to 0.34°), in propane−propane (from 30.51 to 8.02°), and in butane−butane (from 65.18 to 18.56°), which tends to point both hydrogen atoms in nearly the same direction. This is a clear indication that hydrogen atoms tend to form an intramolecular interaction. In Figure 3, selected H−H interatomic distances for methane−methane, ethane−ethane, propane−propane, butane−butane, pentane−pentane and hexane−hexane from ωB97XD/6-311++G(d,p) level of theory are depicted, where all molecular pairs are closer after optimization. Except for the methane−methane complex, the optimized structures of the alkane complexes shown in Figure 3 are notably different from those in Figure 2. This is a consequence in the usage of 1734



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Figure 3. Randomized nonoptimized initial geometry and optimized geometry of methane−methane, ethane−ethane, propone−propane, butane− butane, pentane−pentane, and hexane−hexane complexes depicting their corresponding selected H−H interatomic distances, in angstroms. The geometry optimization of the molecules were performed at the ωB97XD/6-311++G(d,p) level of theory. The dashed lines represent the selected interatomic distances.

10−3 au range values). Accordingly, Table 2 shows the number of H−H bonds in the alkane complexes calculated at ωB97XD/ 6-311++G(d,p) level of theory, along with the experimental values of the boiling points of the corresponding alkanes and the so-called complex stability, ΔG(complex‑2*isolated), derived from Gibbs energy of complex and 2-fold Gibbs energy of the corresponding isolated alkane, whose equation is: ΔG= Gcomplex − 2Gisolated.

We can see that as the number of H−H bonds increases from methane−methane to hexane−hexane, the ΔG(complex‑2*isolated) increases as well. This linear relation is plotted in Figure 6, with a reasonably good coefficient of determination. In another words, the hydrogen−hydrogen bonds influence the stability of alkane complexes. From Table 2 there is another linear relation: between alkanes’ boiling point and number of H−H bonds, which is 1735



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Figure 4. Virial graphs of Alkane complexes optimized at the G4/CCSD(T)/6-311++G(d,p) level of theory. Methane−methane (Met-Met), ethane−ethane (Eth-Eth), propane−propane (Pro-Pro), and butane−butane (But-But) along with the charge density, ρ(r), and Laplacian of the electron density, ∇2 ρ(r) at H−H bond critical, in au. The others complexes were not optimized due to lack of computational resources.

Figure 5. Virial graphs of Alkane complexes optimized at the ωB97XD/6-311++G(d,p) level of theory. Methane−methane (Met-Met), ethane− ethane (Eth-Eth), propane−propane (Pro-Pro), butane−butane (But-But), pentane−pentane (Pen-Pen), and hexane−hexane (Hex-Hex) complexes along with the charge density, ρ(r), and Laplacian of the electron density, ∇2 ρ(r) at H−H bond critical, in au.

Table 2. Number of H−H Bonds, Boiling Point, in °C, Gibbs Energy of Alkane Complexes (Gcomplex), Two-Fold Gibbs Energy of the Corresponding Isolated Alkane (2*Gisolated), and Complex Stability (ΔG(complex‑2*isolated)) of the Alkane Complexes complexes*

number of H−H bonds

boiling point (°C)

Gcomplex (kcal/mol)

2*Gisolated (kcal/mol)

ΔG(complex‑2*isolated) (kcal/mol)

Met-Met Eth-Eth Pro-Pro But-But Pen-Pen Hex-Hex

1 2 3 4 6 8

−162 −89 −42 0 36 69

−50848.0 −100184.2 −149522.3 −198860.6 −248201.1 −297540.0

−50847.8 −100182.2 −149520.2 −198858.2 −248196.5 −297534.5

−0.2 −2.0 −2.1 −2.3 −4.6 −5.4

*

Optimized at the ωB97XD/6-311++(d,p) level of theory.

plotted in Figure 7, whose coefficient of determination is 0.9052. The boiling point is directly dependent upon the magnitude of the intermolecular interactions which, in turn, is

directly related to the complex stability. Each H−H bond in these complexes contribute to decrease on complex electronic energy (Ecomplex) compared to the sum of electronic energy of 1736



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Figure 6. Plot of number of H−H bond paths versus ΔG (complex2*isolated), in kcal/mol, of selected alkanes at ωB97XD/6-311+ +G(d,p) level of theory.

Figure 8. Plot of energy, in hartrees, of a hydrogen atom belonging to the H−H bond (EH(H−H)) in the ethane−ethane complex versus the energy (E) in the ethane−ethane complex of selected steps in the optimization steps.

Figure 7. Plot of the number of H−H bond paths, from the ωB97XD/ 6-311++G(d,p) wave function, versus the boiling point, in °C, of selected alkanes. Figure 9. Plot of energy, in hartrees of a hydrogen atom not belonging to the H−H bond (EH) in the ethane−ethane complex versus the energy (E) in the ethane−ethane complex of selected steps in the optimization steps.

the two corresponding isolated alkanes (Eisolated), leading to an increased complex stability and boiling point. As a consequence, as the number of H−H bonds increase, the complex stability increases, which leads to an expected increase in the alkane boiling point. These trends are confirmed with the reasonably good coefficient of determination in Figure 7. It is well-known the correlation between boiling/melting point in linear alkanes with increasing number of carbon atoms.89,94 The intermolecular attractive forces acting in neutral species are known as van der Waals forces based on nondirectional induced dipole−induced dipole interactions. However, the results from this work strongly indicate that the intramolecular forces in alkane complexes are consequence of directional attractive H−H bonds. In order to demonstrate the stabilizing effect of the H−H bond, the atomic energies of hydrogen atoms belonging and not to the H−H bond were obtained from atomic virial theorem of QTAIM. Figures 8 and 9 show the energy of a hydrogen atom belonging (EH(H−H)) and not belonging (EH) to H−H bond, respectively, versus energy of the ethane−ethane complex in some selected optimization steps, which includes the first and the last steps of optimization plus five other points (steps 1, 22, 27, 33, 43, 49, and 52). Figure S5, in the Supporting Information, shows the plot of energy, E, in hartrees, versus optimization steps of ethane−ethane complex. The decrease of EH(H−H) of the hydrogen atom belonging to the H−H bond versus E (Figure 8) reveals the stabilizing effect of

this type of interaction in the ethane−ethane complex. On the other hand, there is no decrease in the overall EH of a hydrogen atom not belonging to the H−H bond in the ethane−ethane complex (Figure 9). Then, the decreasing trend in alkane complex energy is accompanied by the decreasing trend of the energy of hydrogen atoms from H−H bonds, while no overall change in hydrogen energy not belonging to H−H bond occurs when the alkane complex reaches the minimum in the potential energy surface. A complementary ELF analysis of H−H bond in alkane complexes (from methane−methane until hexane−hexane) was carried out. The coordinates of QTAIM BCP from a selected H−H bond in each alkane complex was used for locating the corresponding ELF BCP. The calculation of gradient of ELF function at QTAIM BCP coordinates confirmed the ELF BCP for all alkane complexes. These BCP’s are index 1 critical points, η(r), whose values are 0.010, 0.015, 0.021, 0.017, 0.026, and 0.025 for the methane−methane, ethane−ethane, propane−propane, butane−butane, pentane−pentane, and hexane−hexane complexes, respectively (Table 3). These η(r) values are nearly half the η(r) values for weak hydrogen bonds.88 Furthermore, the electron density and Laplacian of 1737



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Table 3. ELF Value, η(r), Electron Density, ρ(r), and Laplacian of the Electron Density, ∇2 ρ(r), at the Bond Critical Point of a Selected H−H Bond in Alkane Complexes complex

η(r)

ρ(r)

∇2 ρ(r)

methane−methane ethane−ethane propane−propane butane−butane pentane−pentane hexane−hexane

0.010 0.015 0.021 0.017 0.026 0.025

0.0032 0.0053 0.0051 0.0050 0.0067 0.0067

0.0093 0.0180 0.0137 0.0158 0.0193 0.0197

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CONCLUSIONS This work gives new insights and understanding about intramolecular and intermolecular interactions in alkanes. There are H−H bonds in highly branched C8H18 isomers, but not in linear or less branched alkanes. Therefore, the hydrogen−hydrogen bond plays a secondary role on the stability of branched alkanes, which partly accounts for the increased stability of highly branched alkanes when compared with linear or less branched isomers. We found very good linear relations between H−H bond in complex alkanes and their stabilities and between H−H bond in alkanes and their boiling points. For all studied molecular systems, the hydrogen−hydrogen bonds perform a stabilizing interaction. The hydrogen−hydrogen bond plays an important role on the boiling point of linear alkanes. Intermolecular interactions in alkane complexes are related to the H−H bond, not to nondirectional induced dipole−induced dipole. The electron localization function can also be used for identification of hydrogen−hydrogen bonds. The ELF topological approach enables a clear identification of hydrogen−hydrogen bonds in alkane complexes whose η(r) are smaller than that from a weak hydrogen bond. ASSOCIATED CONTENT

S Supporting Information *

Optimized geometries of alkane molecules; Virial graphs of Alkane complexes optimized at B3LYP/6-311++G(d,p), M062X/6-311++G(d,p) levels and G4 method; plot of energy optimization of ethane−ethane complex. This material is available free of charge via the Internet at http://pubs.acs.org.

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REFERENCES

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electron density values at these points have topological features of closed-shell interactions. Then, ELF is capable of identifying H−H bonds as well as QTAIM, indicating that each H−H bond in the alkane complexes is weaker than a weak hydrogen bond.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: caiofi[email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank, Fundaçaõ de Apoio à Pesquisa do Estado do Rio Grande do Norte (FAPERN), Coordenaçaõ de Ensino ́ Superior (CAPES), de Aperfeiçoamento de Pessoal de Nivel ́ and Conselho Nacional de Desenvolvimento Cientifico e Tecnológico (CNPq) for financial support. 1738



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