On the Unusual Synclinal Conformations of Hexafluorobutadiene and

Article pubs.acs.org/JPCA

Cite This: J. Phys. Chem. A XXXX, XXX, XXX−XXX

On the Unusual Synclinal Conformations of Hexafluorobutadiene and Structurally Similar Molecules Dale A. Braden* 19460 Wilderness Drive, West Linn, Oregon 97068, United States

Chérif F. Matta* Department of Chemistry and Physics, Mount Saint Vincent University, Halifax, Nova Scotia B3M 2J6, Canada S Supporting Information *

ABSTRACT: An explanation is presented for the unusual conformations of some molecules that contain the CC−CC core, namely, butadienes, biphenyls, and styrenes. Small substituents often induce a synclinal conformation, which brings the substituents into close proximity, and sometimes, there is no anticlinal minimum at all. This would not be predicted from steric repulsion arguments nor would it be expected that atoms that are nonbonded in a Lewis structure would approach closer than the sum of their van der Waals radii. Atomic energies calculated according to the quantum theory of atoms in molecules (QTAIM) do not show a consistent pattern for these structurally similar molecules, nor are intersubstituent bond paths consistently found, nor favorable diatomic interaction energies calculated using the interacting quantum atoms (IQA) scheme. Instead, the synclinal conformations are found to be driven by the attraction energy of the electron distribution of the carbon atoms and the nuclei of the molecule.

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INTRODUCTION Unexpected Conformations. The conformations of many organic molecules seem to support the idea that atoms that are not bonded to each other tend to repel each other as a function of distance: the closer the atoms, the greater the “steric repulsion” between them. Molecules whose conformations appear to violate this steric force model are therefore of interest, and their existence indicates that the model is not generally applicable. Tables 1 and 2 contain examples of substituted butadienes and biphenyls, respectively, which adopt synclinal conformations that bring the substituents close together, even though anticlinal conformations would be expected in the steric force model. Sometimes, the synclinal conformation is the only one observed experimentally, even in the gas phase. In such cases, postulating another force to overcome the assumed steric repulsion, such as one associated with “hyperconjugation,” still does not explain why the anticlinal conformation is not observed. The fact that synclinal conformations are observed with a variety of substituents suggests that the substituents themselves may not be responsible, and it may be that the carbon skeleton is the dominant factor. To investigate this problem, five butadienes (1−3, 4a, 4b) from Table 1, two biphenyls (5, 6) from Table 2, and two styrenes (7, 8) were selected for study, and their structures are shown in Figure 1. These molecules share the CC−CC core, in which multiply bonded carbon units are conjoined by a single bond. No structural data was © XXXX American Chemical Society

found in the literature for styrenes 7 or 8, but calculations (see below) show the same synclinal conformational preferences as for the butadienes and biphenyls. The calculated results may thus serve as predictions of what may be observed experimentally for 7 and 8. At room temperature, butadiene (1) exists in two conformations, anti and syn, labels that describe the relative orientation of the two ethylene units across the central single bond.40 The anti conformer is planar, with a CCCC dihedral angle of 180°, while the syn conformer is nonplanar, with a CCCC dihedral angle of 35.5°.1 In the literature, structures with a dihedral angle between 0 and 90° are sometimes labeled s-cisoid or skew s-cis or gauche, while structures with a dihedral angle between 90 and 180° are described as s-transoid or skew s-trans. These labels are discouraged by IUPAC in favor of the labels syn and anti, respectively.41 The more specific labels synclinal and anticlinal may be used to describe structures whose dihedral angles fall into the subranges of 30−90° and 90−150°, respectively. The IUPAC terms are used in this study. Synclinal conformations of butadienes are critical for the success of Diels−Alder reactions, because the mechanism is believed to require a diene in a synclinal conformation in order to react with the dienophile to form the desired cyclic Received: March 5, 2018 Revised: April 11, 2018

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DOI: 10.1021/acs.jpca.8b02157 J. Phys. Chem. A XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry A Table 1. CC−CC Dihedral Angles (Degrees) of Substituted 1,3-Butadienesa molecule butadiene (1) (E,E)-1,2,3,4-F (3) 2-vinyl 1,1,2,3,4,4-F (2) 1,1,4,4-phenyl-2,3-COOH 1,1,4,4-phenyl-2,3-CH2Br 1,1,4,4-phenyl-2,3COOCH3 1,1,3-CH3 2,3-Cl (Z,Z)-1,4-Cl 1,1,2,3,4,4-CH3 (E,Z)-1,2,3,4-CH3 (Z,Z)-1,2,3,4-CH3 (4a) 2,3-divinyl 2-CH3 (E,E)-1,2,3,4-Cl 1-NO2-2-NHPh-1,3,4,4-Cl 1,1,2,3,4,4-Cl 1,1,4,4-CH3-2,3-CH2OH 1,1,2,3,4,4-CN (E,E)-1,2,3,4-CH3 (4b)

dihedral angleb c

35.5, 180 (36.8, 180.0) (47.0, 180.0c) 39.3, ∼180d 47.4 (57.3) 48.1 48.2 49.2

c

∼50 52.3,c 180 60,c 180 60 65.7e 66.7e (60.8) 71.7f 73.5,c 180 76.6 78.8 89 104.8 140 153.4e (48.9,c 180.0)

method

reference

Raman

1

ED ED XRD XRD XRD

this work 2 3 4 4 4

NMR ED ED ED ED ED ED ED ED XRD ED XRD XRD ED

5 6 7 8 9 9 10 11 12 13 12 14 15 9

Table 2. Interannular Dihedral Angles (Degrees) for 2,2′Disubstituted Biphenylsa molecule biphenyl 2,2′-OH-3,3′-CH3O5,5′-Ar 2,2′-NH2 2,2′-F (5) 2,2′-Cl 2-NO2-2′diacetylamino 2,2′-CH3-4,4′-NH3+ 2,2′,4,4′-NH2 2,2′-Cl-4,4′-NH2 2-acetamido-2′diacetylamino 2,2′-CH3 (6)

a

A dihedral angle of 0° corresponds to a synplanar geometry. Note that the substitution pattern is consistently described here assuming a butadiene core, for ease of comparison. XRD = X-ray diffraction; ED = electron diffraction; NMR = nuclear magnetic resonance. bValues in parentheses are from the optimized geometries using density-fitted MP2(fc)/aug-cc-pVTZ in this study. cMinor conformer. dTwo butadiene skeletons exist in this molecule. One was described as anti, with nearly C2h local symmetry, while the other was found to have a synclinal conformation with a dihedral angle of 39.3°. eIn the original paper, a dihedral angle of 0° refers to an antiplanar geometry. The dihedral angles reported here consistently use a value of 0° for the synplanar geometry. fThis molecule is called [4]dendralene, but it can also be considered as 2,3-divinylbutadiene, in which case, the dihedral angle across the central CC bond is 71.7°, as reported here.

dihedral angleb

method

reference

44.4 50.6

ED XRD

16 17

58.2 60, 58, 51, 130 (55.2, 132.1) 70−75 67.1

XRD ED, XRD, NMR XRD, ED XRD

18 19, 20

70.6 72 72 72.2

XRD XRD XRD XRD

23 24 25 26

70−78 (85.0)

27−29

33 34 35 36 31 37 38 39

2,2′-Br 2,2′-CH2Br 2,2′-I 2,2′-NO2

75 76.3 79 72−80

2,2′-Br-4,4′-Ar 2,2′-COCl 2,2′-CH3-4,4′-NH2 2,2′-NO2-4,4′-Cl 2,2′-COOCH3 2,2′-NHCOCH3 2,2′-Br3,3′,4,4′,5,5′,6,6′-F 2,2′-COOH

80.1 83.1 86 87.2 89.8 91.6 104.1c

UV, NMR, PE ED XRD ED dipole moment XRD XRD XRD ED XRD XRD XRD

106.3, 119.3d

XRD

21 22

30 31 30 32

a

XRD = X-ray diffraction; ED = electron diffraction; NMR = nuclear magnetic resonance; UV = ultraviolet spectroscopy; PE = photoelectron spectroscopy. bCalculated values are in parentheses, from optimized geometries using df-MP2(fc)/aug-cc-pVTZ in this study. c The interannular dihedral angle is reported as 75.9°, but since the Br atoms are stated to be in a trans relationship, the complementary angle of 104.1° is reported here. dTwo different molecular structures in the unit cell.

product.42 In 1, however, the synclinal conformation comprises only about 2−5% of the gas at room temperature,1,43 because the antiplanar conformation is more stable. The preponderance of the antiplanar conformer seems to support the notion of steric repulsion between the H5 and H5′ atoms in the syn conformer (see Figure 1), so if all six hydrogen atoms in 1 were replaced with fluorine atoms, one would expect the steric clashes between the F5 and F5′ atoms to become quite pronounced, presumably eliminating the syn conformer altogether. Contrary to this expectation, hexafluorobutadiene (2) adopts a synclinal conformation with a CCCC dihedral angle of about 47°, and in fact, there is no experimental evidence for any anti minimum.3 Calculations show that the antiplanar structure of 2 is actually a transition state,44,45 and it has two close F···F distances of about 2.67 Å (F3···F3′ and F5··· F5′), well under the sum of the van der Waals radii for fluorine (2 × 1.47 = 2.94 Å). According to the steric model for predicting molecular conformation, steric forces of repulsion between these fluorine atoms in the transition state structure of 2 should induce a rotation of the CCCC dihedral angle so as to increase the distance between them, leading to an anticlinal geometry that would be expected to be the global minimum for this molecule. If the CCCC dihedral angle in our optimized

Figure 1. Structures of the molecules examined in this study.

geometry of the antiplanar structure of 2 is reduced from 180 to 135°, then the two pairs of F···F distances increase to 2.96 Å, at which point steric repulsions, if they exist, should be negligible. Yet, no anticlinal minimum has been found for this molecule either in spectroscopic experiments or in ab initio calculations employing basis sets that include diffuse functions.45 However, if one fluorine atom on each terminal carbon is replaced by a hydrogen atom, so as to form (E,E)-1,2,3,4-tetrafluorobutadiene (3), then, an antiplanar minimum appears; yet, it possesses the same F···F distances of 2.67 Å as in the planar transition state for 2 according to the calculations as discussed B


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The Journal of Physical Chemistry A

ethylene group, in a trans relationship, should cause the molecule to adopt a synclinal conformation (see Figure 1, structures 7 and 8). The calculations in this study show that synclinal conformations are indeed preferred for these two molecules, bringing the substituents into close proximity. Prior Studies. Bingham explained the conformations of conjugated organic molecules using the concept of conjugative destabilization.56 The idea is that electron delocalization is actually unfavorable when lone pairs of electrons on substituents interact with the occupied π orbitals of the conjugated carbon framework. This destabilization causes the molecule not only to deviate from planarity but also to favor syn conformations over anti conformations, because electronic delocalization in syn conformations is less than in anti conformations. The lone pair orbitals of electronegative substituents like fluorine are sufficiently low in energy to interact with the filled π orbitals of antiplanar butadiene, for example, rather than with the unfilled LUMO. The interaction creates two new molecular orbitals, one stabilized relative to the interacting orbitals and one destabilized. Because the degree of destabilization is always greater than the degree of stabilization, and because both new orbitals are filled, the net result is a destabilization of the antiplanar geometry. The molecule changes to a syn conformation to reduce this particular interaction. In contrast, the antiplanar conformation of the pentadienyl anion, which may be viewed as butadiene with a methyl anion substituent, is due to the lone pair of electrons on the methyl anion substituent interacting with the LUMO of the butadiene moiety, rather than with the HOMO, so that delocalization is favorable, and the antiplanar conformation is stabilized. However, Bingham’s model is not quantitative nor does it explain why 4a adopts a synclinal conformation, while 4b is anticlinal (Table 1 and Figure 1). These two isomers differ only in the relative placements of the methyl substituents on the terminal carbons of the butadiene core. A more general and quantitative explanation is sought here, one which does not rely on an assumed physicality of orbitals. Dixon and Smart explained the synclinal preference of 2 as being mainly due to minimizing the repulsion energy of the C− F bond dipoles.57 However, the usefulness of a bond dipole is questionable. The charge distribution in molecules consists of highly localized positive charges inside a highly delocalized envelope of negative charge. Such a nonuniform distribution of charge can only be approximated by point charges, dipoles, or quadrupoles when considered from a long distance away.58,59 Explaining a molecular conformation with the bond dipole model would require a point of reference lying within the molecular charge distribution, and in this situation, modeling pairs of bonded atoms as if they were point dipoles seems oversimplified. At close range, the interaction energy of two atomic systems can only be determined by explicitly calculating all of the particle−particle interactions, which is in fact what the Coulomb operators in the Hamiltonian of the Schrödinger equation require. In 1997, one of the authors (D.A.B.) participated in a resonance Raman study of 2.60 The spectroscopic information was supplemented by calculations of the equilibrium geometry and of the CCCC dihedral energy profile. There was no evidence in the spectroscopic or computational data for an anticlinal conformer, although a dihedral scan with HF/631+G* showed a shoulder in the potential energy curve at a dihedral angle of about 150°. The resonance Raman data were

below. These observations require an explanation that goes beyond the traditional model based on steric forces. Hexachlorobutadiene also has a single conformational minimum with a CCCC dihedral angle of about 83.5°.12 The electronegativity of fluorine or chlorine substituents is not responsible for the synclinal conformations of these molecules, however, because methyl substituents cause the same behavior. Thus, (Z,Z)-3,4-dimethylhexa-2,4-diene (4a), which may be regarded as butadiene substituted by a methyl group on each carbon, adopts a synclinal conformation, in which the CC− CC dihedral angle is 66.7°.9 Replacing the remaining two hydrogens on the terminal carbon atoms with methyl groups gives what one might call hexamethylbutadiene, which would be more properly named 2,3,4,5-tetramethylhexa-2,4-diene. The core CC−CC dihedral angle in this molecule, as determined by electron diffraction, is about 60°.8 Table 1 shows that a synclinal conformation in these molecules is quite common, and in many cases, no anticlinal conformer is observed. An interesting exception is 4b, the (E,E) stereoisomer of 4a. This molecule is found experimentally in the anticlinal conformation.9 A good model of molecular conformation should be able to explain the different conformational preferences of these two molecules. An unsubstituted biphenyl is nonplanar, with a CCCC dihedral angle of about 44°.16 In the synclinal conformation of 2,2′-difluorobiphenyl (5), the CCCC dihedral angle is about 55°, and the fluorine atoms are only 2.75 Å apart in the optimized MP2(fc)/aug-cc-pVTZ geometry from this study. For 5, both synclinal and anticlinal conformations of this molecule have been found experimentally20 and computationally,46 but for 2,2′-dimethylbiphenyl (6), only the synclinal structure is observed, with a dihedral angle of 70−78° as determined by various spectroscopic methods27−29 or 85.0° in the optimized MP2(fc)/aug-cc-pVTZ geometry. It is commonly believed that conjugation of the aromatic ring systems favors their coplanarity, while steric repulsion between the ortho substituents forces the two rings to twist relative to one another.47 If we follow this reasoning when analyzing 6, then we would expect two local minima, one anticlinal and one synclinal. But no anticlinal conformer has been found for 6. For the unsubstituted biphenyl, an analysis of the calculated atomic energies has shown that the twisted conformation is not due to steric repulsion between the ortho hydrogen atoms but to the significant lowering of the energies of the two junction carbon atoms.48 One significant chemical consequence of the conformation of 2,2′-disubstituted biphenyls is their atropisomerism.49 They can sometimes be chirally resolved,50 though additional substituents are needed to lock the conformation sufficiently to enable them to be separated at room temperature.51,52 Another noteworthy consequence is that the toxicity of polychlorinated biphenyls (PCBs) has been linked to the ease of rotation about the central single bond.53 Control of the dihedral angle in biphenyls is also important for their use in liquid crystals54 and in electrically conductive polymers.55 Understanding what controls the conformations of these molecules may lead to improving their design for various applications. Styrenes consist of an ethylene moiety joined by a single bond to a phenyl ring. Because they share the same CC− CC core as butadienes and biphenyls, one might expect certain substituted styrenes to exist in synclinal conformations. In particular, three fluorine atoms or methyl groups placed at the 2-position on the phenyl ring and on each carbon of the C


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the particular representation of the density, which can be a set of discrete gridpoints in space or analytic functions. According to QTAIM, for an equilibrium geometry, the energy of an atom in a molecule is a sum of energies that correspond to the terms in the Hamiltonian of the Schrödinger equation. The energy is obtained by integration over the atomic basin bounded by the zero-flux surface in the gradient of the electron density68,72

consistent with a nonplanar geometry. Explanations for the conformation based on the electronegativity of fluorine or on bond dipoles were ruled out. It was pointed out that steric forces, if present, would be just as reduced in an anticlinal minimum as in a synclinal minimum. More recently, Cheong et al. used natural steric analysis (NSA) and natural resonance theory (NRT) to study 2 and other fluorinated butadienes.61 They optimized the geometry of 2 with MP2 and the 6-311G(d,p) basis set and found minima for both synclinal and anticlinal conformers, the synclinal structure being lower in energy by approximately 1.0 kcal/mol. Their explanation for the conformation of 2 was that steric repulsion between the fluorine atoms, as quantified by NSA, is less in the synclinal conformer than in the anticlinal conformer, or in the antiplanar conformer, which was also found to be a transition state. Their finding of an anticlinal minimum was due to the lack of diffuse functions in the basis set, which was confirmed here by reoptimizing the anticlinal geometry after adding the usual diffuse functions to the basis set (i.e., 6311+G(d,p)), which resulted in the synclinal geometry. Karpfen had already shown that diffuse functions are essential for obtaining accurate energies and geometries for this molecule and for perfluoro-1,3,5-hexatriene.45 NSA always finds steric interactions in molecules to some degree, because it associates them with the energy change that results from orbital orthogonalization, and the reference orbitals used in NSA (“pre-NLMOs”) are not orthogonal.62 However, orbital orthogonalization is a mathematical operation, not a physical process. Using a wave function that has not been antisymmetrized as a reference ensures that any energy based upon it is unphysical, because the reference state does not correspond to anything that can be prepared, even in principle. If steric forces really do exist in the equilibrium conformations of molecules, and if such forces do in fact arise from the orthogonalization of orbitals, then maintaining the equilibrium conformation of a molecule would require continuously orthogonalizing the orbitals so as to sustain the steric force between the nonbonded atoms. The fact that NSA requires a nonorthogonal set of reference orbitals implies that if an orthogonal basis could be used for building a wave function, the steric energies would all be zero. Another problem with orbitalbased methods in general is that there are an infinite number of orbital sets which one can use to build a molecular wave function, so the choice of any one set is arbitrary.63 Quantum Mechanical Approach. If one accepts that the Schrödinger equation contains a complete description of a molecular system, then one realizes that the forces that are often invoked to explain what is observed in molecular structures, such as van der Waals forces, steric forces, dispersion forces, crystal packing forces, or “generalized intermolecular forces,”64 must all have their origin in one or more of the terms in the Hamiltonian of the Schrödinger equation. All of the “types” of bonding that chemists have defined, from ionic to dative to covalent, and including hydrogen bonding, halogen bonding, agostic bonding, pi-stacking, and charge-transfer bonding, must all originate in the electron−nuclear Coulomb term,65−67 as this is the only term that lowers the total energy to create a bound state. This is the view taken here, so the conformations of the molecules in this study are analyzed using the quantum theory of atoms in molecules (QTAIM).68−70 Unlike NSA, QTAIM analyzes the electron density rather than orbitals. Even experimental electron densities can be subjected to QTAIM analysis,71 because the model is not dependent on

E(Ω) = −T (Ω) = T (Ω) + Ven(Ω) + Vee(Ω) + Vnn(Ω) (1)

T is the kinetic energy of the electrons within the atomic basin denoted by Ω, Ven is the interaction energy of the atom’s electrons with all of the nuclei in the molecule, Vee is the repulsion energy of the atom’s electrons with themselves plus half of the repulsion energy of the atom’s electrons with the electrons of all other atoms in the molecule, and Vnn is the contribution the atom makes to the nuclear repulsion energy of the molecule. For a nonequilibrium geometry, there are forces on the nuclei, and the virial of these forces may be combined with Vnn to fully account for the energy.68,72 However, eq 1 may be used even for nonequilibrium geometries, as long as all bond distances are optimized, and only angular coordinates are constrained. In this case, the virials of the forces on the nuclei cancel such that the virial ratio (−V = 2T) for the molecule is maintained.73,74 Although there are no explicitly atomic terms in the Hamiltonian, chemists nonetheless consider atoms as the fundamental building blocks of matter. QTAIM recognizes this in its use of the zero-flux surfaces to define the boundaries of atoms in molecules so that the molecular energy and other properties can be meaningfully and quantitatively decomposed into atomic contributions. In QTAIM, the total energy of a molecule is stored in its atoms, not in bonds, and in fact, there are no interaction energy terms between atoms. This is a fundamentally different way of looking at a molecule compared to the traditional model, in which a molecule is composed of atoms and bonds, with the chemical energy being stored in the bonds. The lack of atomic interaction terms in QTAIM is perceived as a problem by some researchers, and it was one of the motivations for developing the interacting quantum atoms (IQA) scheme.75−78 IQA uses the same definition of atoms in molecules as QTAIM (i.e., bounded by zero-flux surfaces in the electron density gradient), but IQA partitions the molecular energy into atomic self-energies and diatomic interaction energies. The atomic self-energy contains all of the electronic kinetic energy within the atomic basin as well as the Coulomb interaction energies among the electrons and the nucleus in that basin E IQA (Ω) = T (Ω) + Ven(Ω) + Vee(Ω)

(2)

The diatomic interaction energy contains the remaining energy terms, all of which express the Coulombic interactions between the particles in two different atomic basins, Ω and Ω′ E IQA (Ω, Ω′) = Ven(Ω, Ω′) + Vne(Ω, Ω′) + Vee(Ω, Ω′) + Vnn(Ω, Ω′)

(3)

Eqs 2 and 3 may be used for either equilibrium structures or nonequilibrium structures. In this study, QTAIM was used to calculate the atomic energies in 1−8 in geometry scans of the central CCCC dihedral angle. The use of QTAIM in this manner has been described in detail in ref 74. Additionally, the IQA scheme was D


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Table 3. Relative Conformational Energies (ΔEsyn−anti), CCCC Dihedral Angles, Substituent Atoms Connected by Bond Pathsa, and Internuclear Distances between the Bonded Atomsb molecule

ΔEsyn−anti (kcal/mol)

dihedral angle (deg)

intersubstituent bond paths

internuclear distance (Å)

1 2 3 4b 5 6 7

2.8

36.8, 180.0 57.3 47.0, 180.0 48.9, 180.0 55.2, 132.1 85.0 44.8, 133.6

anti: F3···F5′, F3′···F5 syn: F5···F5′; anti: F3···F5′, F3′···F5 syn,anti: H2C3H···HC4H2 (2 pairs) syn: F···F H2CH···HCH2 syn: F···F

2.67, 2.67 syn: 2.82; anti: 2.67, 2.67 syn: 2.00, 2.00; anti: 1.93, 1.93 2.75 2.34 2.69

−1.4 1.6 −0.4 −0.3

a

See Figure 1 for atom numbering. bFully optimized geometries calculated with df-MP2(fc)/aug-cc-pVTZ. Molecules 4a and 8 are omitted from the table, because they have only one conformer and no intersubstituent bond paths.

Table 4. IQA Atomic Self-Energy Differences (ΔEsyn−anti, kcal/mol) and Atomic Charge Differences (Δqsyn−anti, Electrons) between the Optimized syn and anti Conformations of 1 and 2 for Selected Atoms and IQA Diatomic Interaction Energies (Esyn, Eanti, kcal/mol) between Selected Pairs of Atoms in 1 and 2a 1

a

2

1

2

atoms

ΔEsyn−anti

Δqsyn−anti

ΔEsyn−anti

Δqsyn−anti

atoms

Esyn

Eanti

Esyn

Eanti

C1 C2 R3 R5

0.1 0.8 0.2 0.9

−0.008 0.005 0.000 0.005

11.8 0.3 −6.8 −0.9

0.032 −0.024 −0.001 −0.006

C1−C1′ C1 = C2 C1-R3 C2-R5 R3-R3′ R5-R5′ R3-R5′

−179.3 −248.0 −145.6 −145.4 0.3 −1.0

−183.1 −246.9 −144.5 −146.0

−60.9 −52.1 −395.9 −507.6 56.2 52.3

−72.9 −46.0 −398.5 −510.5

−0.7

53.2

Atom numbering is given in Figure 1.

a disruption of the primary connectivity. All of the above calculations were performed using Psi4.84 The electron densities from the above electronic structure calculations were analyzed with AIMAll.85 AIMAll implements QTAIM to locate the boundaries (zero-flux surfaces in the gradient of the electron density) of the atoms in the molecules and to integrate the energy and other properties within those boundaries. For post-HF wave functions such as MP2, AIMAll uses the Müller approximation to the two-electron density matrix, expressing it in terms of natural orbitals of the oneelectron density matrix, to calculate properties such as Vee.86 The FCHKWriter() method in Psi4 was used to write the basis set information, density matrix, and other data to an input file for AIMAll. In order for AIMAll to recognize that the density matrix was from an MP2 calculation, it was necessary to manually edit the input files so that the heading “Total DFMP2 Density” was changed to “Total MP2 Density”. Because numerical integrations of this nature involve some error, AIMAll performs various checks to ensure that the integrated values are sufficiently accurate for interpretation. The sum of integrated atomic charges should be within 0.002 of the actual molecular charge. The integrated Laplacian of the density over each atomic basin as well as the sum of integrated atomic Laplacians, should be less than 0.002 au. The default automatic integration grid selection in AIMAll was usually sufficient to achieve these criteria. When problematic atoms were encountered, the integrations on those atoms were rerun using larger grids until the error criteria were met. AIMAll calculates atomic energies using both Bader’s virial QTAIM scheme and the IQA scheme. The atomic energies may be scaled using the calculated molecular virial ratio so that the sum of atomic energies is as close as possible to the calculated molecular energy. However, since Psi4 does not print

used to calculate the atomic self-energy and diatomic interaction energies of the two equilibrium conformers of 1 and of the equilibrium and transition state structures of 2. QTAIM has improved our understanding of the actual energetics that control molecular conformation. Recent examples of the application of QTAIM to problems of this nature include studies of the 1,3 diaxial interactions in cyclohexanes79 and of the anomeric80−82 and “hockey sticks” effects.83

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COMPUTATIONAL METHODS The geometries of all molecular structures were optimized using density-fitted second-order Møller−Plesset perturbation theory with the frozen-core approximation, together with the cc-pVTZ basis set augmented with diffuse functions on all atoms (df-MP2(fc)/aug-cc-pVTZ). Although all of the equilibrium butadiene and biphenyl structures possess at least C2 symmetry, symmetry was not enforced in any of the calculations. To simplify the presentation of the data in the figures in this article, the energies of symmetrically equivalent atoms were averaged together. Vibrational frequency calculations were performed on the antiplanar conformers of 2 and 3. The antiplanar conformer of 2 was confirmed to be a firstorder saddle point, while the antiplanar conformer of 3 was found to be a minimum-energy structure. The dihedral angle formed by the core CC−CC atoms in 1−8 was scanned in 10° increments, and all other geometric degrees of freedom were allowed to relax. In the phenyl rings of the biphenyl and styrene molecules, the dihedral angle was defined such that the carbon atoms that bear the F or CH3 substituents were chosen to correspond to the terminal carbons of the CC−CC butadiene core. For most of the molecules, the full 0−180° range of the dihedral coordinate was not scanned, in order to avoid forcing substituent atoms into close proximity and risking E


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Figure 2. Relative atomic energies vs CC−CC dihedral angle for molecules 1−4. For each molecule, the energy changes are relative to the minimum-energy scan point for the synclinal conformer in the total energy profile shown in (d). (a) Relative energies of the C1 junction atoms. (b) Relative energies of the C2 terminal atoms. (c) Summed relative energies of the C1 and C2 atoms. (d) Relative molecular energies.

diatomic interaction energies according to eqs 2 and 3 for some of the atoms in 1 and 2. The results are shown in Table 4. The IQA diatomic interaction energies for the two H3···H5′ pairs in the antiplanar conformation of 1 are stabilizing. The H5···H5′ interaction stabilizes the synclinal conformation of 1, while the vicinal H3···H3′ interaction is destabilizing. Taken in isolation, such data might be used to explain the preference for the antiplanar conformation of 1 over its synclinal conformation. However, when compared to the IQA data for 2, we see that such an explanation is not general. The F3···F3′ and F5···F5′ interaction energies in the synclinal conformation of 2 are both destabilizing, yet this is the only observed conformation for this molecule. Comparison of the data in Tables 3 and 4 show that a favorable IQA diatomic interaction energy does not imply the existence of a bond path nor does the presence of a bond path imply a favorable diatomic interaction energy. Another inconsistent result of the IQA scheme was that when 2 was modeled with Hartree−Fock theory (HF/aug-cc-pVTZ//HF/ aug-cc-pVTZ), the diatomic interaction energy between the junction carbon atoms was found to be highly unfavorable (+57.1 kcal/mol). If the diatomic interaction energy in the IQA scheme is to be interpreted as a measure of bonding, then the conclusion from this datum would be that in 2 there is no bonding between the ethylenic units, even though at the same level of theory the molecule is in fact bound, and there is in fact

out the value of the virial ratio, the unscaled atomic energies were used to construct the data plots in this study.

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RESULTS AND DISCUSSION Bond Paths and Diatomic Interactions. Since the synclinal conformations of the molecules in this study would be most easily explained by the presence of direct bonding interactions between the substituent atoms, a search was made for intersubstituent bond paths. According to QTAIM, a bond path is the physical manifestation of bonding in the electron density, which results from an interpolarization of the atomic electron clouds by the two nuclei that draws the electron density into a line of maximum density connecting the two nuclei, forming an observable and unambiguous indicator of an associative and stabilizing interaction between the atoms.66 Table 3 shows the intersubstituent bond paths found for all optimized stationary points. Surprisingly, the anticlinal conformation of 2 is stabilized by two F···F bond paths, yet this structure is a transition state. There are no F···F bond paths in the synclinal, equilibrium conformation of 2. Since intersubstituent bond paths were not found in the synclinal conformations of 2, 4a, or 8, they do not explain the unusual structures of these molecules. The possibility of favorable Coulombic interactions between substituent atoms was next examined by calculating the IQA F


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Figure 3. Relative Ven energies vs CC−CC dihedral angle.88 For each molecule, the energy changes are relative to the minimum-energy scan point for the synclinal conformer in the total energy profile shown in Figure 2d. (a) Relative Ven energies for the core CC−CC atoms in 1−4. (b) Relative Ven energies for the core CC−CC atoms in 5−8. (c) Relative molecular Ven energies for 1−4. (d) Relative molecular Ven energies for 5−8.

QTAIM Atomic Energies. In Bader’s formulation of QTAIM, the molecular energy is exhaustively partitioned only into atomic components. This partitioning is guided by the presence of the zero-flux surfaces in the electron density, because these features define the regions in which each nucleus acts as an attractor. The atoms are treated as open systems, meaning that as a molecule undergoes internal changes or responds to its environment, its electronic charge is redistributed throughout its atomic basins, with consequent changes to the atomic energies. Thus, by examining the atomic energy changes as a function of conformational angle, it may be possible to ascertain why a particular molecular conformation is preferred.74 Since the molecules here share the CC−CC core, whose torsional angle determines the conformation, the atomic energies for the junction carbons, C1, and the terminal carbons, C2 (see Figure 1 for the atom numbering), in 1−4 were calculated according to eq 1, and the results are shown in Figure 2a,b. These two plots show that the behavior of the energies of the junction carbons tends to be opposed to that of the terminal carbons, suggesting that there is a compensating effect by the terminal atoms on the energy of the junction atoms as the molecule undergoes internal rotation about the central C1−C1′ bond. Figure 2a,b also shows that the effect of

a bond path connecting the junction carbon atoms. Although some researchers have treated the diatomic interaction energies in the IQA scheme as establishing bonding interactions even in the absence of a bond path,87 it seems erroneous to do so, because the atomic self-energies may also be significantly affected by changes in the molecular structure or environment. In fact, the developers of IQA warned that this can happen if there is a large change in the atomic surface, in the amount of charge within the atomic basin, or in the structure of the charge within the basin.75 It is not clear, however, how large these changes would have to be in order to significantly affect the atomic self-energy. The values of ΔEsyn−anti in Table 4 show that even a conformational change of a molecule can affect the atomic self-energies just as much as the diatomic interaction energies. Comparing the equilibrium and transition state structures of 2, the self-energies of the junction carbon atoms differ by 11.8 kcal/mol, while the diatomic interaction energy between them differs by essentially the same amount (−60.9 − (−72.9) = 12.0 kcal/mol). The use of the IQA diatomic interaction energy as a bonding indicator seems risky, because too much of the energy due to an interaction may in fact be assigned to the atomic self-energies by the partitioning scheme. G


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the use of Ven in eq 1 is that it preserves the virial relationship between the potential and kinetic energies (−V = 2T) within each atomic basin for molecules at equilibrium. It is also faster to calculate than Vne, because the latter requires repeated use of the full molecular grid, on which the electron density is evaluated. Nevertheless, when discussing the interaction of electrons and nuclei within a system, one ought to be able to use either Vne or Ven with equal validity. To show this, Vne was calculated for the atoms of the CC−CC core in 1−4a, and plots of ΔVne vs dihedral angle were constructed. The plots of ΔVne are qualitatively similar to those of ΔVen shown in Figure 3a, all showing minima for synclinal geometries (see Supporting Information).

perfluorination reverses the behaviors of the C1 and C2 energies. Thus, in 1, the C1 atoms are most stabilized for synclinal conformations, and the C2 atoms are most stabilized in synplanar and antiplanar conformations, while in 2, the C1 atoms are most stable in the planar structures, and the C2 atoms are strongly stabilized in structures with nearly orthogonal dispositions of the two ethylenic units. Because of the opposing behaviors of the C1 and C2 atoms, their energies were combined and are shown in Figure 2c. The resulting curves are now qualitatively more similar to the full molecular potentials plotted in Figure 2d, but they still do not consistently show synclinal minima. The combined C1 and C2 atomic energy profiles for 1 and 2 do show very shallow minima for synclinal geometries, but the same plots for 3, 4a, and 4b show a clear preference for orthogonal orientations of the two ethylenic units. The atomic energies of the core carbon atoms in 1−4 do not show a consistent pattern and so do not appear to offer a general explanation for the observed conformational preferences. To gain more insight, the Ven term in eq 1 was examined, because it is the only one which lowers the total energy. It alone is associated with all bonding and otherwise favorable interactions.88 The sum of ΔVen values for the C1 and C2 atoms in 1−4 is shown in Figure 3a as a function of dihedral angle. These ΔVen profiles for the core carbon atoms all have minima for synclinal structures. The profiles for 1 and 4b, which are the only molecules in this study that favor antiplanar conformations, stand out as being relatively flatter for dihedral angles larger than 90°, compared to the other molecules. This indicates that in 1 and 4b, the effects of the two repulsive Coulomb terms in eq 1, Vee and Vnn, are enough to shift the minimum in the overall molecular potential from a synclinal preference to an antiplanar preference. Figure 3b contains the ΔVen profiles for the remaining molecules in this study, each of which also shows a single minimum for a synclinal conformation.89 For nearly all molecules, the minimum in ΔVen is within 10° of the synclinal minimum in the total molecular potential shown in Figure 2d (the exception is 2, for which the discrepancy is 17.2°; see the Supporting Information for the graph of the full molecular energy vs dihedral angle for 5−8). The ΔVen profiles in Figure 3a,b represent the total interaction energy between the electronic distribution in the common CC−CC core atoms and the molecule’s nuclear charges. These data show that the conformation of each of these molecules is driven by this interaction. That the four carbon atoms of the common core structure exert the greatest influence on the conformation may be discerned by comparing Figure 3a,b with Figure 3c,d, respectively, in which the ΔVen values summed over all atoms in each molecule are shown. For each molecule, the total attraction energy between the nuclei and the electron density is still greatest for a synclinal structure, but when the contribution to Ven from all atoms is considered, then the minimum in ΔVen is somewhat further away from the synclinal minimum in the full potential energy (the discrepancies for 3, 4a, and 7 increase to more than 10°, and the discrepancy for 2 increases to 27.2°). In eq 1, Ven is the interaction of an atom’s electrons with all nuclei in the molecule. One might instead consider Vne, which is the interaction of an atom’s nucleus with the electron density of the molecule. Although Ven is not equal to Vne for an atom in a molecule, the sum of atomic Ven energies is equal to the sum of atomic Vne energies, since both fully account for the attraction between all nuclei and all electrons. The reason for

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CONCLUSIONS The CC−CC skeleton shared by the molecules in this study behaves as a functional group, maintaining a striking conformational behavior as its substituents are changed, regardless of whether they are electronegative or not. No consistent evidence for repulsions or attractions or bond paths between substituents was found. It was necessary to consider the strictly associative, binding component of the energy, namely, Ven, between the electrons in the core CC−CC atoms and the nuclei of the molecule, in order to obtain a consistent prediction of the observed synclinal conformations. Although one cannot predict conformation from a consideration of Ven alone, it is worth remembering that the interaction of a nucleus with the electron density of other atoms is always favorable, even in the absence of a bond path. Thus, Ven is responsible not only for the bonding in molecules, but it also influences molecular conformation. It offers an explanation of why 1 and 4b possess stable synclinal conformations even though the antiplanar conformations are lower in energy. Accordingly, Ven for the CC−CC core is expected to be a significant factor in inducing the synclinal conformations of the molecules in Tables 1 and 2. Calculations on perfluoro-1,3,5-hexatriene led Dixon and Smart to pred ict sy nclinal st ructures for poly(difluoroacetylene).57 Salzner calculated the properties of some oligomer models of polydifluoroacetylene (up to eight carbon atoms) and likewise found that the oligomers favor synclinal geometries when optimized with MP2.90 Salzner also noted that dianions of decacyanooctatetraene (an oligomer of polydicyanoacetylene) are known from experimental studies to form synclinal helices. Based on the results presented above, it seems reasonable to suppose that the coiling of such polymers is due to the total Ven energy of the carbon skeleton, rather than to the electronegativity of the fluorine or cyano substituents. Whereas QTAIM may lead one to divide the total energy into favorable (Ven) and unfavorable (T, Vnn, and Vee) terms, Csizmadia et al. divided it into electronic (Eelec = T + Ven + Vee) and nuclear (Enuc = Vnn) terms and used the competition between the two to explain molecular conformation.91 They noted that Eelec was always opposite in sign to Enuc and showed that the maxima and minima in a torsional energy profile indicated where control of the total energy switched from one term to the other. Note that Eelec is opposite in sign to Enuc, because Eelec contains Ven, which is the only negative-valued term in the Hamiltonian, so a torsional energy profile can also be understood as a competition between Ven and the unfavorable terms, T, Vnn, and Vee. Explanations of molecular structure in terms of attractive or repulsive interactions between particular pairs of atoms appear H


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to be too simplistic for the molecules examined here. The traditional paradigm in which energy is assigned to bonds or to diatomic interactions can only be correct to a degree, because the energy of a molecule depends on all of its atoms. In some molecules, the total Ven may happen to be dominated by the interaction of the particles in two particular atomic basins, but one cannot always expect this to be true. What was found in this study is that molecular conformation can sometimes be controlled by a connected group of atoms, and the critical energy component for showing this is the same one that is responsible for bonding, namely, Ven.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpca.8b02157. Cartesian coordinates for all equilibrium structures and for the transition state structure of 2; plots of total molecular energy vs CC−CC dihedral angle for 5− 8; plots of ΔVne vs CC−CC dihedral angle for 1− 4a (PDF)

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (D.A.B.) *E-mail: [email protected] (C.F.M.) ORCID

Dale A. Braden: 0000-0002-8354-2846 Chérif F. Matta: 0000-0001-8397-5353 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS No sources of funding were used for this work. REFERENCES

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On the Unusual Synclinal Conformations of Hexafluorobutadiene and

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