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C-H···O Hydrogen Bonding. The Prototypical MethaneFormaldehyde System: A Critical Assessment Kevin B. Moore, Keyarash Sadeghian, C. David Sherrill, Christian Ochsenfeld, and Henry F. Schaefer J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.7b00753 • Publication Date (Web): 17 Oct 2017 Downloaded from http://pubs.acs.org on October 18, 2017

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Journal of Chemical Theory and Computation

C−H· · · O Hydrogen Bonding. The Prototypical Methane-Formaldehyde System: A Critical Assessment Kevin B. Moore III,† Keyarash Sadeghian,‡ C. David Sherrill,¶ Christian Ochsenfeld,‡ and Henry F. Schaefer III∗,† †Center for Computational Quantum Chemistry, University of Georgia, Athens, GA, 30602, USA ‡Department of Chemistry, Ludwig-Maximilians University (LMU), Munich D-81377, Germany ¶Center for Computational Molecular Science and Technology, School of Chemistry and Biochemistry, School of Computational Science and Engineering, Georgia Institute of Technology, Atlanta, GA, 30332, USA E-mail: [email protected]

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Abstract Distinguishing the functionality of C−H· · · O hydrogen bonds (HBs) remains challenging, bas their properties are difficult to reliably quantify. Herein, we present a study of the model methane-formaldehyde complex (MFC). Six stationary points on the MFC potential energy surface (PES) were obtained at the CCSD(T)/ANO2 level. The CCSDT(Q)/CBS interaction energies of the conformers range from only −1.12 to −0.33 kcal mol−1 , denoting a very flat PES. Notably, only the lowest energy stationary point (MFC1) corresponds to a genuine minimum, whereas all other stationary points — including the previously studied ideal case of ae (C−H· · · O) = 180◦ — exhibit some degree-of-freedom that leads to MFC1. Despite the flat PES, we clearly see that the HB properties of MFC1 align with those of the prototypical water dimer O−H· · · O HB. Each HB property generally becomes less prominent in the higher energy conformers. Only the MFC1 conformer prominently exhibits (1) elongated C−H donor bonds; (2) attractive C−H· · · O−C interactions; (3) n(O)→σ ∗ (C−H) hyperconjugation; (4) critical points in the electron density from Bader’s method and from the noncovalent interactions method; (5) positively charged donor hydrogen; and (6) downfield NMR chemical shifts and nonzero 2 J(CM –HM · · · OF ) coupling constants. Based on this research, some issues merit further study. The flat PES hinders reliable determinations of the HB-induced shifts of the C−H stretches; a similarly difficult challenge for experiment. The role of charge-transfer in HBs remains an intriguing open question, although our BLW and NBO computations suggest it is relevant to the C−H· · · O HB geometries. These issues notwithstanding, the prominence of the HB properties in MFC1 serves as clear evidence that the MFC is predominantly bound by a C−H· · · O HB.

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Introduction

1.1

Motivation

As early as 1937, Glasstone 1 attributed the increased polarization of ether- and acetonehaloform mixtures to hydrogen bonds (HBs) involving weakly polarized C−H donors. While our understanding of HBs has advanced substantially since then, assessing the full functionality of C−H· · · O HBs remains a challenging prospect in many studies. 2–4 Even the prototypical water dimer HB endures ongoing investigation. 5,6 The 2011 IUPAC redefinition of hydrogen bonding 7 no longer limits HBs be "considered as an electrostatic interaction." Unfortunately, like the 1960 definition of Pimentel and McClellan, 8 it only enumerates HB properties that could be used as "evidence of bond formation." The level of evidence necessary for affirmation of HBs still depends on the opinions of individual researchers. Szalewicz and co-workers claimed that one conformer of the methane-water complex is not bound by a C−H· · · O HB, because the complex "cannot be explained in terms of (longrange) electrostatic interactions but must be described in terms of a (delicate) balance of all components of the interaction energy." 9 This constraint seems inappropriate, as they later showed that the potential energy surfaces of several O−H· · · O hydrogen bonded dimers were better reproduced by dispersion, rather than electrostatics. 10 Such incongruous distinctions hinders the ability to intuit when C−H· · · O HBs are present and chemically meaningful. Analysis of C−H· · · O HBs often relies on examining the geometry of neighboring C−H groups and oxygen atoms, especially in more complex systems. 11–13 We used such an approach to suggest a potential dynamical role for a C−H· · · O HB in Streptococcus pneumoniae hyaluronate lyase. 14 In general however, Chamberlain and Bowie 15 have emphasized the tenuous nature of such approaches. More holistic approaches combining theory and experiment — exemplified by research from Trievel and co-workers 16–18 on C−H· · · O HBs in lysine methyltransferases — necessitates a thorough understanding of HB behavior.

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1.2

Properties of Hydrogen Bonds

The attractive nature of C−H· · · O contacts is well-established, albeit with energies of only 0.5–1.0 kcal mol−1 in most cases. 19–22 Hence, distinguishing the impacts of weak C−H· · · O HBs from dispersion is difficult. Angular distributions of C−H· · · O contacts in neutron 23 and X-ray 24 structures show a greater propensity for linearity than C−H· · · H contacts. 24 SAPT studies of several C−H donors 25,26 show that bending the HB angle increases the magnitudes of the attractive components; however, this is largely nullified by increases to repulsive exchange. Block-localized wavefunction (BLW) energy decompositions show that charge-transfer may also mediate the HB angle. 27 The C· · · O distance is generally be required to be less than the sum of the carbon and oxygen van der Waals (vdW) radii. 28 although the usefulness of this criteria is questionable, 29 and alternate metrics have been considered. 30 HBs are generally described as dipole-dipole interactions. For less polar C−H donors however, dispersion is important, as is negative hyperconjugation between the n(O) and σ ∗ (C−H) orbitals. This n(O)→σ ∗ (C−H) interaction should yield a weakening (lengthening) of the donor C−H bond, 31 and red-shift of the corresponding C−H antisymmetric stretch in accordance with Badger’s rule. 32 Traditional HBs almost uniformly exhibit "signature of H-bonding." 33 However, experimental 34 and theoretical 22,35 studies have noted blue-shifted vibrational frequencies and contracted C−H bonds from C−H· · · O HBs. No succinct model explains the unusual C−H· · · O HB-induced shifts. 36 Rather, several competing effects influence the X−H stretch. Red shifts arise from hyperconjugation and attractive electrostatic interactions, meanwhile Pauli repulsion tends increases the bond length and frequency. 37–39 Shortened X−H bonds may also increase dispersion. In the absence of hyperconjugation, the X−H bond may increase the s-character, shortening its length in accordance with Bent’s rule. 40 Jemmis et al. 41,42 related the frequency shift to the balance of inter- and intramolecular charge-transfer. The blue-shift may also stem from the antiparallel orientation of the induced-dipole derivatives of the donor X−H bond and electric field of the Y−Z acceptor. 43,44 The n(O)→σ ∗ (C−H) charge-transfer interaction also represents an elusive covalent na4


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ture of the HB. 45,46 Wang et al. 47 computed water dimer "hydrogen bonding orbitals" comprised of combinations of donor and acceptor orbitals. Isaacs et al. 48 measured ice X-ray scattering profiles which they argue are indicate covalent HB regions; however, this result has been contested by Romero et al. 49 Indirect spin-spin coupling constants (n J) involving HB nuclei have also been suggested as a sign of covalency, 50 although this notion is contested by the theoretical results of Del Bene et al. 51 and experimental results of Ledbetter et al. 52 C−H· · · O HB n J constants in Protein G 53 have been attributed to competing hyperconjugation and steric interactions by Weinhold, and Markley, and co-workers. 54,55 As to the NMR chemical shieldings of HBs, the donor hydrogen becomes deshielded as it loses electron density from HB formation, causing a downfield chemical shift. 56 The difficulty involved with reliably parsing C−H· · · O HB charge distributions is evident from the failures of classic schemes (e.g. Löwdin and Hirschfeld) to predict intuitive atomic charges. 57,58 The most popular topological measure may be Bader’s quantum theory of atoms-in-molecules (QTAIM). 59 Koch and Popelier 60 proposed eight criteria for affirming a C−H· · · O HB by QTAIM. While there is a positive correlation between BCP densities and bond strengths, the QTAIM framework has been scrutinized for predicting unusual H· · · H BCPs in several species. 61–63 This issue has inspired debate over whether BCPs are indicative of "stabilizing" bonding interactions. 64 From the opposite viewpoint, Lane et al. 65 have argued that the absence of an H· · · O BCP in 1,2-ethanediol should not preclude a HB. Rather, elimination of the reduced density gradient in the H· · · O region affirms HB formation. 66 Quantifying the HB properties described above allows for examining the extent to which interactions. 67 Kar and Scheiner 68 found that increasingly longwe chains of H2 CO and HFCO yield shorter C−H· · · O distances, blue-shifted C−H stretching frequencies, and larger NMR chemical shifts. Each property is diminished when a high-dielectric implicit solvent model is used. In contrast, Qingzhong et al. 69 found that the C−H· · · O HB properties of the DMSO−H2 O complex increase as more water molecules are explicitly included.

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1.3

Exemplar Studies of C−H· · · O Hydrogen Bonds

The above disagreements notwithstanding, C−H· · · O HBs have been shown to influence structure and function since the seminal study of Taylor and Kennard. 70 More recently, Calabrese et al. 71 resolved a microwave structure of the acrylnitrile-water complex with a C−H· · · O HB. Jones et al. 72 argued that generally unfavorable cis-planar OCCF arrangements in α-difluoroamides observed in X-ray structures are stabilized by C−H· · · O HBs. Vargas et al. 73 predicted that the lowest energy structures of N -N -N ′ -N ′ -tetramethylsuccinamide employ C−H· · · O HBs by performing a conformational scan with the MM3 force field, followed by refinement at the MP2 level. Takahashi et al. 74,75 argued that the anomeric effect in oxanes may in part stem from C−H· · · O HBs, as the more stable axial conformations are predicted to have shorter C−H· · · O contacts, and more polarized C−H donors. Bandyopadhyay et al. 76 used IR spectroscopy and computations to show that keto-enol tautomerization of β-cyclohexanedione is facilitated by dimerization via four C−H· · · O HBs. Khorief Nacereddine et al. 77 predicted that functionalizing ethylene with CO2 Me groups facilitates the ortho/endo pathway of [3+2] cycloaddition of nitrones, rather than the generally preferred para/exo pathway. The impact of a N−C−H· · · O−C HB on the stereoselectivity of the Houk-List transition state for proline-mediated aldol addition is more ambiguous. The presence of this HB is supported by the noncovalent interactions (NCI) method; 78 however, Bakr and Sherrill 79 used a new functional-group variant of SAPT to argue that this interaction contributed negligible electrostatic stabilization because of the partial negative charge of the nitrogen atom. Similarly, while the enantioselective oxazaborolidine Diels-Alder catalysts Corey and co-workers 80,81 are thought to employ a C−H· · · O HB, Paddon-Row, Houk, and co-workers et al. 82,83 have questioned whether this HB is needed for ester dienophiles.

1.4

The Methane-Formaldehyde Complex

We see that while C−H· · · O HBs are a capable mediating force, further research is needed to clarify their basic properties. As such, we investigated the methane-formaldehyde complex 6


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(MFC) using the highest-level theoretical methods available for a comprehensive suite of HB metrics. The MFC appears to have only been examined in five theoretical studies. 22,56,84–86 Our research will improve the C−H· · · O HB picture illustrated by these studies. The earliest discussion of a C−H· · · O HB in the MFC comes from a broad theoretical survey of HB complexes by Peter Kollman et al., 84 wherein they predict an MFC interaction energy of −0.4 kcal mol−1 using the RHF/4-31G energies of CH3 H· · · NH3 and FH· · · OCH2 . Notably, they predicted that charge-transfer correlates very poorly with the binding energy, in contrast to the BLW predictions of Shaik, Mo, and co-workers. 27 Novoa et al. 85 located a C−H· · · O BCP in the MFC for HB angles ranging from 90–180◦ via QTAIM. They discussed a MFC conformer with an idealized 180◦ C−H· · · O angle that exhibits a C−H· · · O BCP and a BSSE-corrected MP2/aug-cc-pVDZ interaction energy of −0.52 kcal mol−1 . They also located a C−H· · · C BCP within a higher energy conformer (−0.21 kcal mol−1 ) where the C−O is oriented towards the methane carbon, and along the methane C3v axis; breaking this high symmetry resulted in three separate C−H· · · O BCPs. Gu, Kar, and Scheiner 22 computed several HB properties of this conformer. They predicted a −0.46 kcal mol−1 interaction energy at the BSSE-corrected MP2/aug-cc-pVDZ level. With respect to the donor C−H bond, the MFC exhibited a contracted bond length and blue-shifted C−H antisymmetric stretch frequency. Complexation was found to increase (decreass) the NPA charge of the donor methane hydrogen (acceptor formaldehyde oxygen), and that there is a net charge-transfer towards methane. Morokuma decomposition of the RHF/6-31G* energy predicts little electrostatic stabilization in this conformer. Lastly, it was predicted that all HB properties are enhanced with increasing halogenation of methane. Gu, Kar, and Scheiner 56 later computed the NMR chemical shifts for the same conformer. At the MP2(GIAO)/6-311+G** level, the σ(1 HM ) shifted downfield 1.19 ppm and the σ(17 OF ) shifted upfield 0.38 ppm relative to the monomers. Facelli et al. 86 predicted a linear correlation between the repulsive interaction energies and RHF(GIAO)/6-31G** σ(17 OF ) shifts, which they attributed to electron density being pushed away from oxygen.

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As the MFC likely involves a very flat potential energy surface, there are questions in regards to the veracity of previously computed energies, geometries and vibrational frequencies. More generally, the choice of previous studies 22,56,85 to model the MFC C−H· · · O HB with an idealized 180◦ C−H· · · O angle may lead to erroneous predictions of HB properties, as there may be more lower energy structures with different values for such properties. As such, we investigate how the properties of the C−H· · · O HB in the MFC change with energy and geometry, and how they compare to the O−H· · · O HB properties of the water dimer. In this way, we can establish the presence and significance of a C−H· · · O HB in this species.

2 2.1

Theoretical Methods Geometries and Vibrational Frequencies

Equilibrium geometries and harmonic vibrational frequencies (ωe ) were obtained using coupledcluster theory with single, double, and perturbative triple excitations [CCSD(T)], 87–90 as implemented in CFOUR. 91,92 The atomic natural orbital (ANO) basis sets 93,94 were used for these computations. Since the MFC is described by a flat potential energy surface, we converged the SCF densities, CC amplitudes, and Lambda coefficients to 10−10 , and the RMS force to 10−10 Eh a−1 0 . Anharmonic contributions to the vibrational frequencies (δν) were determined with second-order vibrational perturbation theory (VPT2). 95 A semi-diagonal quartic force field was constructed by finite differences of CCSD(T)/ANO1 analytic second derivatives with CFOUR. Fundamental frequencies are then computed as ν = ωe + δν. The CCSD(T)/ANO2 optimized structures were used all energy and property computations.

2.2

Energetics

The focal point approach of Allen and co-workers. 96–99 was used to determine MFC interaction energies including the effects of geometric relaxation, computed as ∆Eint,gr = Egr [MFC] – (Egr [methane] + Egr [formaldehyde]). Herein, each Egr value was determined by performing 8


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a series of single-point energies at the HF, MP2, CCSD, CCSD(T), CCSDT, and CCSDT(Q) levels of theory with the Dunning 100 aug-cc-pVXZ (X = T, Q, 5) basis sets. The energies were extrapolated with the formulae of Feller 101 and Helgaker. 102 Using this approach, we obtain a CCSDT(Q)/CBS ∆Eint,gr value for which we can assess convergence towards the CBS and full configuration-interaction (full CI) limits. The CCSD(T) energies used for these extrapolation were computed using Molpro 2010.1, 103 whereas the CCSDT(Q) energies were computed with the NCC module of CFOUR. 91,92 To account for approximations used to compute the above energy, corrections were appended. A core-correlation correction (∆CORE ) — computed as the difference between allelectron and frozen-core CCSD(T)/aug-cc-pCVTZ 104 energies — accounts for not correlating the carbon and oxygen 1s-electrons. An RHF/ANO2 diagonal Born-Oppenheimer correction (∆DBOC ) 105,106 accounts for the clamped-nuclei approximation. A scalar relativistic correction (∆REL ) at first order in perturbation theory. 107,108 was obtained using CCSD(T)/augcc-pCVTZ wavefunctions. We computed ∆CORE with Molpro 2010.1, and ∆DBOC and ∆REL with CFOUR. All energies were computed to within 10−10 Eh . The above-described CCSDT(Q)/CBS energy (∆Eint,gr ) is equivalent to the negative of the equilibrium dissociation energy (–De ). We use the ∆Eint,gr notation so that signs of these energies are conceptually consistent with our symmetry-adapted perturbation theory (SAPT) interaction energies (∆Eint ) described below. Here, there is no gr subscript, as SAPT does not account for geometric relaxation of the separated monomers. Our SAPT ∆Eint and CCSDT(Q)/CBS ∆Eint,gr values do not include a correction for basis-set superposition error, 109 as the former utilizes a dimer-centered basis set, and the latter very large basis sets. The electrostatic, induction, dispersion, and exchange components of ∆Eint were determined using the SAPT0 formulation 110,111 implemented in Psi4 1.0. 112 The specific interaction energy of the donor C−H and acceptor C−O moieties was computed with the functional-group (F) partition 113 of the SAPT0 energy. The charge-transfer energy was then computed as the difference between the induction energies from the dimer basis and the

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monomer basis. 114 The aug-cc-pVTZ basis set describes all atoms in these computations.

2.3

Orbital Interactions

The occupations, hybridization, and interactions of significant orbitals were investigated using natural bonding orbital (NBO) theory. 115 Intermolecular hyperconjugation was com2

puted as the second-order energy [E(2) = qi Fǫ(i,j) ] for delocalizing electrons from a donor j −ǫi NBO (i) into an acceptor NBO (j), where F (i, j) is the NBO Fock matrix element, and qi and ǫi are the occupancy and energy of NBO i, respectively. 116 The contribution of these interactions to the total interaction energy was determined by the natural energy decomposition. 117 To see how the favorable n(O)→σ ∗ (C−H) hyperconjugation is counterbalanced by n(O)/σ(C−H) exchange-repulsion, we used Badenhoop and Weinhold’s natural steric analysis. 118 These computations were performed with the NBO 6.0 package 119 integrated with GAMESS 2015. 120,121 RHF/aug-cc-pVTZ orbitals used for all NBO analyses. The role of orbital interactions in C−H· · · O HB geometries was assessed with the blocklocalized wavefunctions (BLW) method. 122 Herein, the Kohn-Sham Fock matrix was block localized into two orbital subspaces: the first block contained the formaldehyde oxygen porbital, and the second block contained all the remaining orbitals of the complex. This partitioning prevents the construction of KS-MOs which incorporate both the formaldehyde p and methane C/H atomic orbitals, creating a diabatic state where HB charge-transfer has been eliminated; other intermolecular charge-transfer interactions are maintained. We compared the DFT and BLW-DFT optimized structures to see how differences in the C−H· · · O geometries are affected by the lack of HB charge-transfer. For this analysis, we used the M06, PBE0, and B3LYP-D3 functionals with the cc-pVTZ basis set. These computations were performed using a version of GAMESS 2013 provided by Professor Yirong Mo.

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2.4

Electron Topology

Shifts in the atomic charges (∆q) were determined via population analysis of the RHF/augcc-pVTZ orbitals. In this study, we used natural population analysis (NPA) 123 from NBO 6.0, QTAIM charges from AIMALL 16.10.31, 124 and intrinsic atomic orbital (IAO) 125 charges from MRCC 2015. 126,127 Bader’s quantum theory of atoms-in-molecules (QTAIM) was used to determine the topology of intermolecular bond critical points (BCPs), and check whether they comport with the C−H· · · O HB criteria of Koch and Popelier. 60 Due to the aforementioned issues with QTAIM, 65 we also employed the Noncovalent Interactions (NCI) 66,128 method, which involves locating regions of electron density (ρ) where the reduced density gradient (s =

|∇ρ| 1 ) 2(3π 2 )1/3 ρ4/3

goes to zero for low values of ρ. These regions correspond

to a BCP for a non-covalent interaction. Furthermore, the sign of the Laplacian eigenvalue (λ2 ) in the electron density indicates if the interactions are bonding (λ2 < 0) or nonbonding (λ2 > 0). 65 Both of these methods use an M06/aug-cc-pVTZ density. The QTAIM analysis was performed with AIMALL 16.10.31, 124 and NCI analysis with NCIPlot 3.0. 128

2.5

Magnetic Properties

The NMR chemical shieldings were computed for the the MFC and separated monomers using gauge-including atomic orbitals (GIAOs) 129–131 at the CCSD(T)/aug-cc-pVTZ level. 132 NMR chemical shifts (∆σ) were determined by comparing the MFC isotropic shieldings to those for separated methane and formaldehyde. The Fermi-contact, spin-orbit, diamagnetic spin-dipole, and paramagnetic spin-dipole components of the indirect spin-spin coupling constants (n J) were then computed. To minimize errors (such as those from triplet instabilities), we computed each component tensor at the all-electron CCSD/aug-cc-pVTZ level, with unrelaxed orbitals and an unrestricted HF reference. 133,134 Our discussion emphasizes constants associated with the C−H· · · O HB: 1 J(HM · · · OF ), 2 J(CM –HM · · · OF ), and 3 J(CM – HM · · · OF =CF ). All of the magnetic properties were computed with CFOUR.

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3

Results and discussion

3.1 3.1.1

Conformer Search MFC Stationary Points

We explored unique C−H· · · O HB configurations via six stationary points on the methaneformaldehyde complex (MFC) PES. Figure 1 depicts the conformers in ascending order of their interaction energies (∆Eint,gr ). Each conformer is labeled MFCn, where n denotes their place in this order, and n = 1 corresponds to the lowest energy conformer. MFC1 (−1.12 kcal mol−1 ), MFC2 (−1.03 kcal mol−1 ), and MFC3 (−0.85 kcal mol−1 ) were obtained by enforcing Cs symmetry. MFC4 (−0.62 kcal mol−1 ) was obtained by enforcing a linear C−H· · · O angle. MFC5 (−0.38 kcal mol−1 ) was constrained to C2v symmetry with two equivalent C−H· · · O distances. Lastly, MFC6 (−0.33 kcal mol−1 ) was constrained to have three equivalent C−H· · · O distances with the C−O oriented along a methane C3 axis. Each stationary point is characterized by their CCSD(T)/ANO2 Hessian index (HI ). Only MFC1 is a genuine minimum (HI = 0) on the PES, whereas the other conformers correspond to saddle-points with one energy-minimizing degree-of-freedom that leads to MFC1. MFC2 (HI = 1) and MFC3 (HI = 1) exhibit 34i and 74i cm−1 a′′ modes, respectively, which correspond to a C3 rotation about an in-plane methane C−H bond. MFC4 (HI = 2) has 14i cm−1 a′ and 5i cm−1 a′′ modes for rotating the C−O, and methane and formaldehyde hydrogens, respectively. MFC5 (HI = 2) contains 69i cm−1 b1 and 15i cm−1 b2 modes for rotating the methane hydrogens and C−O, respectively. MFC6 (HI = 3) has 37i cm−1 a′′ , 12i cm−1 a′′ , and 38i cm−1 a′ modes that correspond to rotation of the C−O and methane C−H off the methane C3 axis. Optimization along the symmetry-breaking imaginary modes of MFC4–MFC6 did not yield C 1 stationary points. This suggests that a C−H· · · O HB may predominate such that it yields the lowest energy structure of this complex.

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Table 1: Shifts of the X−H donor bond length [∆re (X−H); in Å)], (2) harmonic antisymmetric X−H stretch (∆ωe ; in cm−1 ), and (3) re (X−H) quadratic force constant (∆ke ). ∆re (X−H) ∆ωe This Research MFC6 MFC5 MFC4 MFC3 MFC2 MFC1 Water Dimer Previous Research 22 MFC4

∆ke

−0.0003 −0.0004 −0.0003 +0.0001 +0.0005 +0.0005 +0.0061

+4 +6 +8 +1 0 0 −28

+0.82 +1.42 +0.99 +0.26 −0.38 −0.41 −21.18

−0.0002

+7

-

distance (∼3.1 Å) and wider angle (∼105◦ ). The intermolecular C· · · O distance (in Å) of MFC1 (3.599), MFC4 (3.724), and MFC6 (3.422) shows that there is no correspondence between re (C· · · O) and ∆Eint,gr . These results show that C−H· · · O HBs are undeniably energetically beneficial; however, the HB distances and angles should be considered alongside the overall molecular orientations to predict their potential influence. 3.2.2

HB-induced Shifts of Bond Lengths and Vibrational Frequencies

Formation of the MFC breaks the Td symmetry of the methane C−H bond lengths and antisymmetric C−H stretch degeneracy. Table 1 reports the CCSD(T)/ANO2 HB-induced shift of the donor X−H bond length (∆re ), harmonic antisymmetric X−H stretch (∆ωe ), and re (X−H) quadratic force constant (∆ke ) for the MFC and water dimer. The directionality of ∆re , ∆ke , and ∆ωe for the water dimer and MFC4–MFC6 conformers align with the intuitive Badger’s rule; 32 however, MFC4–MFC6 exhibits a C−H bond contraction and blue-shift of the antisymmetric stretch frequency. Gu, Kar, and Scheiner 22 predicted a comparable 0.0002 Å C−H bond contraction and 7 cm−1 blue-shifted frequency for MFC4. This relationship begins to break down for MFC1 and MFC2, where the C−H bond length red-shifts by 0.0005 Å and force constant decreases by ∼0.4, but there is no observed shift in the frequency. These discrepancies are worsened with the CCSD(T)/ANO1 method: 15



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it predicts a 0.0003 Å C−H elongation and 3 cm−1 blue-shift. Similarly, Pandey 142 predicted an elongated N−H bond and concomitant blue-shifted frequency for an N−H· · · N HB in pyrrole· · · N2 complex. Notably, MFC1 and MFC4 exhibit qualitative differences in the relationship of ∆re and ∆ωe /∆ke , despite a similar energy and C−H· · · O geometry. Instead, similar ∆re -∆ωe / relationships exist among the side-on and direct conformers, respectively. 3.2.3

Anharmonic Treatments

The fundamental frequencies may be readily obtained from VPT2, which reproduces the experimental intramolecular fundamental frequencies of the water dimer 143–147 with 0.2% error (Supplementary Information). Unfortunately, similar VPT2 treatments of MFC1 are hindered by its flat PES, as indicated by its unusually large CCSD(T)/ANO1 diagonal quartic force-constants (Supplementary Information), in particular the constant for the lowest frequency intermolecular mode (ν21 ). Including all of the modes in the MFC1 VPT2 treatment, we predict an 8 cm−1 blue-shift of the antisymmetric stretch involving the C−H donor (ν1 ) with respect to methane; however, the veracity of this prediction is very tenuous, as the anharmonic correction to ν21 yields a −7 cm−1 fundamental frequency for this mode. Likely we are introducing error into the ν1 fundamental by allowing it to couple to the intermolecular modes. Hence, we computed anharmonic contributions by three additional approaches, each of which removes some number of the potentially problematic intermolecular modes: (1) ν21 , as its quartic force constant is very high; (2) the a′′ intermolecular modes (ν19 –ν21 ), as symmetry requirements weaken the coupling; and (3) all intermolecular modes (ν12 –ν14 , ν19 –ν21 ). This analysis is comparable to VCI treatments of methane-water conducted by the Bowman group, 148,149 who determined the C−H stretch fundamentals without explicit treatment of intermolecular modes. Fundamentals predicted from each approach are given in Table 2. The removal of more intermolecular modes increasingly red-shifts ν1 : ∆ν1 changes from +8 cm−1 (all modes), to −7 cm−1 (ν12 –ν14 and ν19 –ν21 removed). However, this is not a reliable affirmation of ∆ν1 , as its value varies so greatly across VPT2 treatments. 16


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Table 2: Harmonic (ωe ) and VPT2-determined fundamental frequencies (ν) fpr the C−H stretches of the lowest energy MFC conformer (MFC1), in cm−1 . ν[n] denotes the modes included in this analysis. The C−H stretch of the C−H HB donor is in bold. Mode ν1 (a′ ) asym. str. ν2 (a′ ) asym. str. ν3 (a′ ) sym. str. ν15 (a′′ ) asym. str. methane asym. str.

ωe 3158 3152 3028 3147 3158

ν[All] 3023 3011 2908 3008 3015

ν[1–20] 3020 3003 2907 3000

ν[1–18] 3016 3004 2906 2985

ν[1–11,15–18] 3008 2999 2904 2994

More reliable elucidation of the vibrational spectrum of the MFC necessitates a level of analysis beyond the scope of this work. The potential for only one MFC minimum-energy structure makes an experimental investigation of the vibrational spectrum an intriguing prospect; however, there are complicating factors. The structures and imaginary modes of MFC2 and MFC3 indicate that they are low-lying rotameric transition states which connect identical MFC1 minima. At the CCSD(T)/ANO2 level, the barriers to rotating MFC1 into MFC2 and MFC3 are 27 cm−1 and 100 cm−1 , respectively. The ν21 (ZPVE = 23 cm−1 ) and ν20 (ZPVE = 37 cm−1 ) intermolecular modes of MFC1 correspond to rotations that transform MFC1 into MFC2 and MFC3, respectively. Hence, the ZPVE of ν21 is nearly the exact amount needed to ascend the rotation barrier to MFC2. 150

3.3 3.3.1

Energetics Assessment of CCSDT(Q)/CBS Energies

Table 3 shows that the CCSDT(Q)/CBS energy (∆Eint,gr ) for MFC1 is converged to within 0.05 kcal mol−1 of the correlation and CBS limits, and that each auxiliary correction is below 0.02 kcal mol−1 . Similar convergence was obtained for MFC2–MFC6, as shown in the Supplementary Information. The water dimer D0 is also an encouraging benchmark: a sum of the CCSDT(Q)/CBS De (equivalent to –∆Eint,gr ) and anharmonic CCSD(T)/ANO ∆ZPVE gives a D0 of 3.12 kcal mol−1 , in great agreement with the 3.16 kcal mol−1 experimental value. 151 Hence, the ∆Eint,gr values in Figure 1 should be considered reliable.

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Table 3: The CCSDT(Q)/CBS MFC1 interaction energy [∆Eint,gr ]. Bracketed values obtained by additivity or extrapolation. Note that the other MFC conformers and water dimer show a similar level of energetic convergence. aug-cc-pVDZ aug-cc-pVTZ aug-cc-pVQZ aug-cc-pV5Z CBS LIMIT

3.3.2

RHF +δ MP2 +δ SD +δ SD(T) +δ SDT +δ SDT(Q) +0.08 −1.62 +0.18 −0.23 +0.00 −0.01 +0.33 −1.58 +0.18 −0.23 [+0.00] [−0.01] +0.37 −1.51 +0.20 −0.24 [+0.00] [−0.01] +0.38 −1.48 +0.19 −0.23 [+0.00] [−0.01] [+0.39] [−1.45] [+0.19] [−0.23] [+0.00] [−0.01] ∆Eint,gr = ∆ECBS + ∆CORE + ∆DBOC + ∆REL ∆Eint,gr = −1.11 − 0.0180 − 0.0047 + 0.0002 kcal mol−1 ∆Eint,gr = −1.13 kcal mol−1

NET −1.60 [−1.30] [−1.19] [−1.15] [−1.11]

SAPT Energy Decompositions

The physical origins of these MFC interaction energies are explored by the SAPT0/aug-ccpVTZ energy decompositions given in Figure 3. The SAPT0 ∆Eint values reliably reproduce the CCSDT(Q)/CBS ∆Eint,gr values. The choice to model the C−H· · · O interaction with the (C−H)M and (C−O)F fragments is not compulsory, but we feel this accounts for intermolecular orientation, which we know is significant. Practically, it avoids appreciable errors from fragmenting across multiple bonds. Note that we discuss "reduced" F-SAPT0 energies, whereby the fragment interaction energies include contributions from the link bonds connecting the fragments, which are assigned using the ratio of atomic charges. 113 As the total ∆Eint increases (proceeding from MFC1 to MFC6), we see that the dispersion energy increases from −1.54 kcal mol−1 in MFC1 to −0.74 kcal mol−1 in MFC6. Despite this, dispersion remains the most attractive component in each conformer, making up 53% and 90% of the total attractive energy in MFC1 and MFC6, respectively. Attractive dispersion is greatest in the side-on conformers (MFC1–MFC3), which orient the C−H· · · O HB donors along formaldehyde, facilitating greater surface area contact, and thus greater vdW interaction. In contrast, the orientation and intermolecular distance of the direct conformers (MFC4–MFC6) limits the vdW contact. The small intermolecular distances of MFC1 and MFC2 strengthens the short-range dispersion relative to MFC3. This viewpoint is buttressed by the C−H· · · O−C dispersion energies, which increases from

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MFC4 (0.61 kcal mol−1 ). Consider that these C−H· · · O HB geometries certainly strengthen the OFδ− · · · HMδ+ interaction; however, since the C−H bonds are not very polar, this interaction is weak. The repulsive OFδ− · · · CMδ− and CFδ+ · · · HMδ+ interactions lessen the overall attractiveness of the C−H· · · O−C electrostatic interaction. The side-on orientation and/or shorter intermolecular distance of MFC1 and MFC2 may ameliorate the issue, because they have a decreased C· · · C distance, and enhanced CFδ+ · · · CMδ− attraction. The increased distances and/or direct orientation of MFC3–MFC6 are less favorable for this interaction. In fact, for each conformer, electrostatic stabilization largely derives from the interaction of the formaldehyde C−O with the remaining methane hydrogens (Supplementary Information). The induction component changes little with increasing ∆Eint , only rising from −0.33 kcal mol−1 in MFC1 to −0.14 kcal mol−1 in MFC6. Moreover, induction essentially comprises a similar percentage of the total attractive energy of each conformer, as it only increases from 12% in MFC1 to 16% in MFC6. More attractive induction therefore correlates with favorable HB geometries of MFC1, MFC2, and MFC4, rather than intermolecular orientation. These total induction energy trends are exactly mirrored by the C−H· · · O−C IndMF and IndFM energies (note: IndMF denotes the induction of methane due to formaldehyde). The C−H· · · O−C IndMF energy comprises ∼85% of the total induction energy in each conformer. The contribution from all other induction interactions are almost negligible. The magnitude of exchange-repulsion decreases gradually from 1.69 kcal mol−1 in MFC1 to 0.42 kcal mol−1 in MFC6. Exchange in MFC1, MFC2, and MFC4 is such that no attractive component can singularly compensate for the larger steric repulsion. The C−H donor of MFC1, MFC2, and MFC4 is in close contact with formaldehyde, engendering significant repulsion. Indeed, the C−H· · · O−C exchange energies affirm that this contact is the primary source of exchange-repulsion, comprising ∼70% of the total exchange energy. Conformer MFC4 is a notable extreme, where the C−H· · · O−C exchange energy makes up 98% of the total exchange energy. There is also an appreciable contribution to exchange from the C−H· · · FH interaction in MFC1, MFC2, and MFC3 that derives from the side-on

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orientation. Since MFC3, MFC5, and MFC6 have multiple C−H donors, the C−O· · · MH interaction also constitutes an important portion of the exchange energy. SAPT clarifies the energetic role of the C−H· · · O HB: short HB distances and more linear HB angles produce C−H· · · O−C interactions with sizable attractive dispersion and induction components, but also sizable repulsive exchange. This trend has been computed previously. 25,26 The side-on orientation generally enhances the magnitudes of each component, in particular the C−H· · · O−C electrostatic interaction, which appears quite important. Increasingly attractive C−H· · · O interactions are clearly energetically beneficial to the MFC as a whole. The use of MFC4 as the C−H· · · O HB model in previous literature 22,56,85 is striking since F-SAPT0 describes the C−H· · · O−C interaction of this conformer as being very repulsive, apparently due to unfavorable OFδ− · · · CMδ− and CFδ+ · · · HMδ+ interactions. Our SAPT0 interaction energy for MFC4 is marginally lower than the MP2/aug-ccpVDZ interaction energies from Gu, Kar, and Scheiner 22 and Novoa et al., 85 respectively. Similarly, our interaction energy for MFC6 is not too far from the value predicted by Novoa et al. 85 More importantly, comparing the MFC4 and MFC6 interaction energies to that for MFC1 shows that previously computed MFC interaction energies 22,85 directly underestimate binding by ∼50%, and hence, the C−H· · · O HB strength. Notably, our MFC1 interaction energy is equivalent to a previous interaction energy for CH2 FH· · · OCH2 , 22 an intuitively stronger HB because of the inductive effects of the fluorine. Finishing this discussion, if we presume that MFC1 is the lowest energy conformer, and add the CCSD(T)/ANO2 ∆ZPVE to our ∆Eint,gr , we arrive at an estimate of the MFC D0 of 0.36 kcal mol−1 .

3.4 3.4.1

Orbital Interactions Donor-Acceptor Orbital Interactions

Figure 4 depicts intermolecular orbital interactions with a second-order perturbation energy [E(2)] greater than 0.10 kcal mol−1 . The strongest interactions are instances of negative hyperconjugation, whereby a formaldehyde oxygen lone-pair orbital [n(O)F ] donates elec21




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itive hyperconjugation, whereby a C−H bonding orbital of either formaldehyde [σ(C−H)F ] or methane [σ(C−H)M ] donates electron density to an antibonding orbital of the methane C−H [σ ∗ (C−H)M ] or formaldehyde C−O [σ ∗ (C−O)F ], respectively. This σ(C−H)F →σ ∗ (C−H)M interaction is observed for MFC1 (0.17 kcal mol−1 ) and MFC3 (0.21 kcal mol−1 ), while MFC4 exhibits a σ(C−H)M →σ ∗ (C−O)F interaction (0.18 kcal mol−1 ). Unlike the water dimer orbital interactions, which are dominated by a 7.67 kcal mol−1 n(O)→σ ∗ (O−H) interaction, interactions involving Rydberg orbitals are surprisingly significant in the MFC. Several of these interactions are even competitive with the intuitive interactions. The Rydberg interactions with the largest E(2) value for each MFC conformer are MFC1: σ(C−O)F →RY (H)M of 0.26 kcal mol−1 ; MFC2: σ(C−O)F →RY (H)F of 0.30 kcal mol−1 ; MFC3: σ(C−H)M →RY (H)M of 0.12 kcal mol−1 ; MFC4: σ(C−O)F →RY (H)M of 0.22 kcal mol−1 ; MFC5: σ(C−O)F →RY (C)M of 0.08 kcal mol−1 ; and MFC6: σ(C−O)F →RY (C)M of 0.21 kcal mol−1 . Note that the RY (H)M and RY (C)M orbital is centered on the donor hydrogen and carbon of the C−H· · · O HB, respectively. Hence, the Rydberg interactions listed above enhance the C−H· · · O HB charge transfer picture illustrated in Figure 4. 3.4.2

Relationship between E(2) and ∆Eint,gr

The E(2) values are greatest in MFC1, MFC2, and MFC4, which have more favorable HB geometries. Hence, E(2) trends similarly with the SAPT induction energies (Figure 3). Earlier, we noted that SAPT ascribes induction primarily to the (C−H)←(C−O) interaction (IndFM). This description clearly aligns with the NBO interaction picture (Figure 4), which predicts that the strongest interactions correspond to n(O)→σ ∗ (C−H). We may also highlight that a simple C3 rotation of MFC1 to MFC2 effectively eliminates the complementary σ(C−H)F →σ ∗ (C−H)M interaction. To compensate, MFC2 has a more linear C−H· · · O HB angle to strengthen the n(O)F →σ ∗ (C−H)M interaction. This suggests that a C−H· · · O HB may be strengthened in response to losing other favorable interactions.

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3.4.3

Steric Interactions of Orbitals

Natural steric analysis shows that attractive hyperconjugation is strongly counterbalanced by intermolecular orbital repulsion between the σ(C−H)M and multiple n(O)F orbitals. Additionally, for the significant steric interactions involving the σ(C−H)M and σ(C−H)F orbitals in the three side-on conformers. The energies are given in the Supplementary Information, but we state generally here that for each conformer, they are comparable to the corresponding hyperconjugation energies in Figure 4. For instance, the greatest n(O)F /σ(C−H)M repulsion occurs in conformers MFC1, MFC2, and MFC4. This correspondence is not surprising, as facilitating overlap of the n(O)F and σ ∗ (C−H)M orbitals for hyperconjugation brings the occupied n(O)F and σ(C−H)M orbitals into close contact. The NBO steric energies are in excellent alignment with the SAPT0 exchange energies. 3.4.4

Relationship between E(2) and HB-induced Geometries and Frequencies

Hyperconjugation into σ ∗ (C−H) orbitals centered on the C−H donor(s) of MFC1 and MFC2 increases the orbital occupation from 0.0002 e− (in the methane monomer) to ∼0.0014

e− ; this orbital only has an occupation of

∼0.0003

e− in the other conformers.

The C−H donors of MFC1, MFC2, and MFC4 are hybridized differently than in MFC3, MFC5, and MFC6: the former (latter) conformers carbon atoms have s-character and p-character of 25.5% (25.0%) and 74.2% (74.6%), respectively; methane is hybridized similarly to the latter conformers. This increased hyperconjugation and s-character balance in ambiguous manner to determine the HB-induced shift of the C−H donor lengths and stretch frequencies. In particular, it is unclear why the MFC4 C−H donor (Table 1) blue-shifts similarly to MFC5 and MFC6, which have less hyperconjugation and more s-character. 3.4.5

Charge-Transfer from NBO and Block-localized Wavefunction Methods

The NBO and SAPT methodologies differ strongly with regard to their predictions of charge transfer. SAPT predicts charge-transfer energies that are only 2–4% of the total attractive 24


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energy (Figure 3) in each conformer. In complete contrast, the natural energy decomposition analysis (NEDA) predicts charge transfer to be predominant, ranging from 39–54%. A similar discrepancy is also seen in the water dimer (see Supplementary Information). Certainly, the charge-transfer share will be overestimated since NEDA lacks any description of dispersion; however, Stone 152 asserts that the NEDA charge-transfer values are inherently susceptible to basis-set superposition error by construction. A more expansive examination of non-unique charge-transfer energies goes beyond the scope of this work. The impact of charge-transfer to the C−H· · · O HB geometries suggested by BLW 25–27 adds additional uncertainty. Herein, the significance of HB charge-transfer is evinced by the differences between the BLW-DFT and DFT optimized H· · · O distances [∆re (H· · · O)], X−H· · · O angles [∆ae (X-H· · · O)], and X−H donor lengths [∆re (X−H)] (provided in the Supplementary Information).. The X−H bond of MFC1 and the water dimer contract by 0.0003 and 0.005 Å, respectively, indicating less density in antibonding type orbitals. These values align with the re (X−H) shifts in Table 1. The MFC1 and water dimer H· · · O distances increase by 0.12 and 0.16 Å, respectively: a remarkable similarity considering the difference in the two HB strengths. Meanwhile, the differences in the HB angle are much more profound, as the MFC1 and water dimer ∆ae (X−H· · · O) values are −12.3◦ and −33.1◦ , respectively. Clearly, despite the small energetic contributions of charge-transfer, the HB geometries are sensitive to this interaction. This appears particularly true for the HB angle, where the strength of the interaction heavily mediates the linearity.

3.5 3.5.1

Electron Topology QTAIM Bond Critical Point Analysis

Each MFC conformer contains at least one intermolecular bond path, although the atoms connected by these paths differ slightly (Figure 5). A C−H· · · O bond critical point (BCP) −3 −3 is observed in MFC1 (0.007 ea−3 0 ), MFC2 (0.007 ea0 ), and MFC4 (0.005 ea0 ), while −3 an H−C· · · O BCP is observed in MFC5 (0.003 ea−3 0 ) and MFC6 (0.003 ea0 ), and an

25



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H−C· · · C BCP is observed in MFC3 (0.005 ea−3 0 ). C−H· · · O bond-path connectivity evidently depends on a proper C−H· · · O contact distance and angle. Regardless, the magnitudes of the BCP densities and ∆Eint,gr grow simultaneously, suggesting stronger HBs in the lower energy conformers. 59 As expected, the MFC BCP densities well below the 0.025 ea−3 0 BCP density for stronger O−H· · · O HB of the water dimer. Our predicted C−H· · · O BCP 85 density of MFC4 is similar to the 0.003 ea−3 MFC1 also 0 MP2 density from Novoa et al.

includes the now infamous H· · · H BCP; 61–63 this feature is discussed later. Comparison of Figure 4 and Figure 5 shows that intermolecular BCPs correspond with NBO interactions — BCP density and E(2) trend similarly with ∆Eint,gr . This congruity may be rationalized as orbital interactions giving rise to a resonance structure with an intermolecular covalent bond. 153 A C−H· · · O BCP is only observed in MFC1, MFC2, and MFC4, which experience n(O)→σ ∗ (C−H) hyperconjugation into the C−H bond, with hydrogen pointing at formaldehyde. In contrast, MFC6 exhibits n(O)→σ ∗ (C−H) hyperconjugation with the σ ∗ (C−H) accepting density through the lobe adjacent to the carbon, explaining the H−C· · · O BCP. Only, the σ(C−O)F →RY (C)M and σ(C−H)F →σ ∗ (C−H)M interactions of MFC5 and MFC3 may yield their respective H−C· · · O and H−C· · · C BCPs. The existence of a C−H· · · O BCP with a reasonable density and positive Laplacian would generally serve as sufficient electron-topological evidence for a C−H· · · O HB in MFC1, MFC2, and MFC4. However, the C−H· · · O HB in MFC1, MFC2, and MFC4 also satisfy the additional criteria specified by Koch and Popelier, 60 in that the HB-donated hydrogen atom’s charge and energy increases, and dipole moment and volume decreases. Meanwhile, MFC3, MFC5, and MFC6 lack even a C−H· · · O BCP, seemingly implying the absence of an HB. 154 Novoa et al. 85 noted that breaking the symmetry in MFC6 yields three C−H· · · O bond paths. Unfortunately, we could not reproduce this result, even for three different C 1 structures generated from perturbing our CCSD(T)/ANO2 structure (see Supplementary Information). It is unclear if these differences relate to the theoretical method or kind of perturbation used to augment the MFC6 structure. Therefore, it is ambiguous to what

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3.5.3

Surfaces of Interaction from NCI

Figure 6 depicts a green surface indicating weak (low ρ), attractive interactions in each conformer, as determined by NCI. 65,66 The attractive NCI surface extends over most intermolecular surface contact, including the donor C−H and acceptor C−O moieties. Hence, NCI may be predicting a C−H· · · O HB where QTAIM does not. There are no repulsive regions in the NCI surface, despite SAPT predicting significant total and C−H· · · O−C exchange energies for MFC1–MFC3, and MFC4–MFC6, respectively. The attractive interactions predicted by SAPT, NBO, and BLW apparently overcomes repulsion at all points of contact. The NCI surfaces denote a similar level of attraction in each conformer. As such, these surfaces alone can not readily distinguish C−H· · · O HBs from pure dispersion interactions. This notion aligns with the share of dispersion in the C−H· · · O−C interaction predicted by SAPT. For the reasons stated above, the relationship between these qualitative NCI surfaces and the MFC ∆Eint,gr values (or C−H· · · O HB strength) is ambiguous. However, for the MFC, the reduced density gradient plots (Figure 6) clearly delineate of the side-on conformers (MFC1–MFC3) from the direct conformers (MFC4–MFC6). The former conformers are represented by MFC1, and the latter by MFC4. The number of points in these NCI plots directly correlates with ∆Eint,gr : the lower energy conformers exhibit much more points, and have three peaks for which s ≈ 0. In sharp contrast, the NCI plots of the three higher energy conformers are much more sparse, and only have one peak that where s approaches zero. The density and Laplacian λ2 of each QTAIM CP (Figure 5) corresponds to a peak in the NCI reduced density gradient (s) plots (Figure 6). Even the infamous H· · · H BCP and RCP of MFC1 are seen in the NCI plots. In contrast, NCI predicts CPs in MFC2 and MFC3 that are not predicted by QTAIM. It is possible that these CPs undergo mutual annihilation in the QTAIM analysis, explaining their absence from Figure 5. More generally, the ability of NCI to detect interactions beyond the pairwise QTAIM bond paths may be helpful for understanding the ambiguous absence of C−H· · · O bond paths in MFC3, MFC5, and MFC6. As discussed previously, even traditional O−H· · · O HBs may be missed. 65 28


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Table 4: HB-induced shifts of the atomic populations (∆q, in e− ) of the X−H HB donors of the MFC conformers. MFC1 MFC2 MFC3 MFC4 MFC5 MFC6

HM CM HM CM HM CM HM CM HM CM HM CM

IAO +0.014 −0.006 +0.013 −0.004 +0.003 −0.001 +0.018 −0.004 +0.006 +0.001 +0.001 +0.005

NPA +0.018 −0.011 +0.017 −0.009 +0.004 −0.004 +0.021 −0.006 +0.005 +0.002 +0.001 +0.006

QTAIM +0.028 −0.017 +0.027 −0.013 +0.008 −0.010 +0.030 −0.008 +0.009 −0.002 +0.003 +0.002

In alignment with the IndMF energies predicted by SAPT, most of the HB-induced charge redistribution occurs within the methane monomer. The positive ∆q for the donor hydrogen atom shows that the charge from hyperconjugation (Figure 4) is ultimately redistributed to the other non-donor C−H bonds in each conformer (i.e. the carbon and hydrogen atoms become more negative), with symmetry dictating which bonds gain charge. These C−H bonds generally lengthen somewhat, possibly relating to repulsion of the increasingly negative carbon and hydrogen atoms, as the σ ∗ orbitals of these bonds are not heavily populated. The comparatively smaller-in-magnitude ∆q values of the C−O moiety consistently show the oxygen becoming more negative, as this atom gains charge from vicinal C−H bonds while concurrently losing electron density by σ(C−H)F →σ ∗ (C−H)M hyperconjugation. Some charge redistribution stems from charge-transfer towards the methane monomer, as seen by summing the ∆q values for all five atoms of the monomer. The greatest net chargetransfer of about 0.003 e− in MFC1 and MFC2, 0.002 e− in MFC4, and 0.001 e− or less in the other conformers. These electron density shifts nicely align with the E(2) values in Figure 4, which showed that MFC1, MFC2, and MFC4 have the greatest hyperconjugation. We note that our intramolecular and intermolecular ∆q values for MFC4 are comparable to the corresponding values from Gu, Kar, and Scheiner. 22 30


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3.6 3.6.1

Magnetic Properties NMR Chemical Shifts

Figure 7 illustrates the RHF-CCSD(T)(GIAO)/aug-cc-pVTZ NMR chemical shifts (∆σ) 129–132 and UHF-AE-CCSD/aug-cc-pVTZ indirect spin-spin coupling constants (n J) 133,134 for the HB donors (13 CM , 1 HM ) and acceptors (13 CF ,

17

OF ). The ∆σ values do not trend directly

with ∆Eint,gr . Rather, the ∆σ(13 CM , 1 HM ) values align with the ∆q values in Table 4; each are greatest in MFC1, MFC2, and MFC4. The similar HB geometries of these conformers induce a decrease (increase) of the hydrogen (carbon) electron density and concomitant downfield (upfield) shift. Unlike previously discussed HB properties, the +0.70 ppm ∆σ(1 HM ) value for MFC1 is not the largest observed for the MFC conformers. The +1.19 ppm value for MFC4 — in agreement with the previous value 56 — is greatest. Even the MFC5 ∆σ(1 HM ) value is not much less than MFC1, despite its poor measures for other computed HB metrics. These results suggest that while the ∆σ (1 HM ) can be signatures of a C−H· · · O HB, correlation of the magnitude of ∆σ (1 HM ) to HB strength seems more tenuous. It is immediately unclear whether the above NMR shifts are perturbed by ring current effects originating from the C−O π bond. 157 Using NMR computations, Wannere and Schleyer 158 for multiply bonded systems downplay the presence of impactful ring currents. To investigate this effect, we computed ∆σ values for a model Cs methane-water complex (depicted alongside the C−O "shielding cone" in the Supplementary Information), which has the same C−H· · · O HB geometry as MFC1. These results should simulate the C−H· · · O HB-induced ∆σ values of MFC1, absent of any perturbations from the C−O ring current. Compared to the MFC1 ∆σ(1 HM ) values in Figure 7, none of the methane protons are significantly different. Hence, there is no evidence of a perturbation by a ring current.

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the 1 J(HM · · · OF ) constant however, JSO and JSD are not negligible: they contribute up to 0.09 and 0.46 Hz (ignoring sign), respectively. For MFC1, MFC2, and MFC4 the magnitude of the JSO contribution to 1 J(HM · · · OF ) — in particular, the paramagnetic JSO part — dwarfs the JFC contribution. The increased impact from spin-orbit interactions on certain n

J constants has been attributed to covalent character of HBs. 160 Certainly, these three

conformers exhibited the most n(O)→σ ∗ (C−H) hyperconjugation (Figure 4). Of course, Del Bene 51 has highlighted the tenuous connection between covalency and n J constants.

4

Conclusions and Outlook

This research involves a comprehensive, theoretical assessment of C−H· · · O hydrogen bonding using the methane-formaldehyde complex (MFC) as a case study. In contrast to previous studies, which study HB properties across several hydrogen bonded species, we offer a unique approach: we analyze how the HB properties evolve across stationary points of increasing energy for one species. Optimized geometries and harmonic vibrational frequencies of six stationary points on the MFC potential energy surface (PES) — each with a unique C−H· · · O HB geometry — have been obtained at the CCSD(T)/ANO2 level. Highly accurate CCSDT(Q)/CBS interaction energies (∆Eint,gr ) provide a reliable energetic ordering of the conformers, labeled as MFCn, where n = 1 corresponds to the lowest energy conformer. The ∆Eint,gr values (equivalent to the negative of De in the spectroscopic literature) range from −1.12 kcal mol−1 (for MFC1) to −0.33 kcal mol−1 (for MFC6). These energies illustrate the immensely flat nature of the PES. Furthermore, the CCSD(T)/ANO2 Hessian indices show that only the lowest-energy stationary point (MFC1) corresponds to a genuine minimum on the PES; all of the other points have vibrational degrees-of-freedom that lower the energy in the direction of MFC1. Certainly, the immensely flat nature of the PES leaves open the possibility of locating a new conformer that is lower in energy than MFC1; however, no result from this study suggests that such a stationary point exists. This result is

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very important, since the three lowest energy conformers (MFC1–MFC3) have never been identified. Gu, Kar, and Scheiner 22 and Novoa et al. 85 discussed the fourth-highest (MFC4) and sixth-highest (MFC6) energy conformers. The energy of MFC4 is significant, as it contains the generally ideal linear C−H· · · O angle, and hence would intuitively serve as a good C−H· · · O HB model; however, its HB properties sometimes qualitatively differ from those of the minimum-energy MFC1. Despite the very flat PES, relationships between the HB properties and ∆Eint,gr can be established. In particular, examining the properties for all six conformers allows us to determine if the HB properties are more closely aligned to the C−H· · · O geometry or intermolecular orientation. We find that nearly HB property is most prominent in the minimum-energy MFC1; i.e., its properties most closely aligns with those of the prototypical water dimer O−H· · · O HB. Only the MFC1 conformer prominently exhibits (1) elongated C−H donor bond lengths; (2) attractive C−H−O−C interactions from F-SAPT0; (3) n(O)→σ ∗ (C−H) hyperconjugation; (4) greater bond critical point (BCP) critical point densities and attractive NCI plots; (5) charge distribution giving positive donor hydrogen atoms; and (6) downfield NMR chemical shifts and nonzero 2 J(CM –HM · · · OF ) coupling constants. Depending on how parsimonious the researcher is, different properties may be used to preclude HB formation for particular conformers. For instance, we see that (1) the C−H· · · O−C electrostatic interaction is repulsive in MFC3—MFC6, and (2) there is no n(O)→σ ∗ (C−H) hyperconjugation or C−H· · · O BCPs in MFC3, MFC5, and MFC6. As such, the satisfaction of the litany of HB criteria explored in this study provides unambiguous identification of C−H· · · O hydrogen bonding in MFC1. Moreover, since MFC1 is the lowest energy structure, we conclude that this C−H· · · O HB is the predominant force binding this complex. Therefore, this complex may be the simplest closed-shell dimer for which a C−H· · · O HB dictates the structure of the lowest energy conformer. In contrast, the heavily studied methane-water complex has a C−H· · · O bound conformer that is only a local minimum. 9,148,161–166 Presuming MFC1 is the lowest energy conformer, we estimate an MFC

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D0 of 0.36 kcal mol−1 . This energy should be an instructive reference since the donor C−H bond(s) are not polarized by geminal or vicinal groups. Some other interesting questions regarding the MFC remain. In this research, we were unable to unambiguously resolve the directionality of the HB-induced bond-length and frequency shifts; the latter proved particularly difficult. At the CCSDT(T)/ANO2 harmonic level, we see an increase of the methane C−H donor bond length, whereas the corresponding vibrational frequency of the antisymmetric stretch exhibits no shift. Unfortunately, the MFC PES prevents a reliable application of vibrational perturbation theory for obtaining fundamental frequencies of the C−H stretches. In addition, we find that intermolecular vibrations provide easy access to the higher energy MFC2 and MFC3, even at low temperatures. Hence, this problem clearly represents an enormous challenge to both theoretical and experimental vibrational spectroscopy. Definitive assignments of the HB-induced bond-length and frequency shifts in the MFC would be instrumental in understanding this elusive characteristic of HBs. While our NBO and BLW results suggests the significance of charge-transfer to the C−H· · · O geometries, SAPT predicts a nearly negligible charge-transfer energy. Hence, the significance of charge-transfer to weak HBs remains generally unclear. We find qualitative agreement between our predictions for MFC4 and MFC6 to those from lower level treatments reported in previous studies, 22,56,85 However, we have discussed at length how improper considerations of intermolecular orientation — which could arise from inadequate theoretical treatments — can yield quantitatively different HB properties. In this vein, we demonstrated that two common molecular mechanics force fields, GAFF and MM3, yield poor treatments of C−H· · · O hydrogen bonding. Even while using our MFC1 CCSD(T)/ANO2 structure as a guess. To avoid artifactual predictions of C−H· · · O HBs, MM force fields likely require higher level parameterizations. All of the above considerations motivate an urgent call for high-resolution spectroscopic studies combined with sophisticated analysis, of the methane-formaldehyde system: the prototypical C−H· · · O hydrogen bond.

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Acknowledgement This research was supported by the U.S. Department of Energy, Office of Basic Energy Science, Computational and Theoretical Chemistry Program Grant DE-SC0015517. KBM and HFS acknowledge the hospitality of the Ludwig-Maximilians University, where this research began. HFS is grateful to the Alexander von Humboldt Foundation for an AvH Fellowship.

Supporting Information Available Optimized geometries, vibrational frequencies, energetics, NBO, BLW, QTAIM, NCI, charges, and magnetic properties of each species are provided. This material is available free of charge via the Internet at http://pubs.acs.org/.

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