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The substituent effects on the biphenyl H⋯H bonding interactions subjected to torsion
Accepted Manuscript Title: The Substituent Effects on the Biphenyl H—H Bonding Interactions Subjected to Torsion Author: Dong Jiajun Yuning Xu Tianlv Xu Roya Momen Steven R. Kirk Samantha Jenkins PII: DOI: Reference:
S0009-2614(16)30151-8 http://dx.doi.org/doi:10.1016/j.cplett.2016.03.042 CPLETT 33728
To appear in: Received date: Revised date: Accepted date:
28-2-2016 12-3-2016 16-3-2016
Please cite this article as: D. Jiajun, Y. Xu, T. Xu, R. Momen, S.R. Kirk, S. Jenkins, The Substituent Effects on the Biphenyl HmdashH Bonding Interactions Subjected to Torsion, Chem. Phys. Lett. (2016), http://dx.doi.org/10.1016/j.cplett.2016.03.042 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
The Substituent Effects on the Biphenyl H---H Bonding Interactions Subjected to Torsion Dong Jiajun, Yuning Xu, Tianlv Xu, Roya Momen, Steven R. Kirk and Samantha Jenkins* Key Laboratory of Chemical Biology and Traditional Chinese Medicine Research and Key Laboratory of Resource Fine-Processing and Advanced Materials of Hunan Province of MOE, College of Chemistry and Chemical Engineering, Hunan Normal University, Changsha Hunan 410081, China *email: [email protected]
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QTAIM and the stress tensor describe a torsion φ, 0.0° ≤ φ < 25.0°, of the C4-C7 torsional bond that affects the H---H bonding interactions linking the two phenyl rings of para-substituted biphenyl, C12H9X, X = N(CH3)2, NH2, CH3, CHO, CN, NO2 and C12H9-Y, Y = SiH3, ZnCl, COOCH3, SO2NH2, SO2OH, COCl, CB3. For all values of X and Y, shorter H---H bond-path lengths corresponded to higher values of both the stiffness and torsion φ. The onset of a phase transition-like behavior is found by the stress tensor stiffness. The atomic basins of the H---H interactions are affected by the para-substituent groups.
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Highlights QTAIM and stress tensor properties of the H---H interactions of biphenyl subjected to torsion φ depend on para-substituent groups. QTAIM and stress tensor stiffness S and S follow in line with physical intuition. A phase-transition-like behavior for the stress tensor stiffness S is found to vary with the parasubstituent groups.
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The H---H atomic basins are affected by the choice of para-substituent groups.
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The application of this investigation to molecular rotors is briefly discussed.
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*Graphical Abstract (pictogram) (for review)
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Recently, some of the current authors used QTAIM and the stress tensor topological approaches to add to the ongoing discussion about the biphenyl molecule [1]. We explained the effects of the torsion φ of the C-C bond linking the two phenyl rings of the biphenyl molecule on a bond-by-bond basis. Using the total local energy density H(rb) we demonstrated the favorable conditions for the formation of the controversial H---H bonding interactions for a planar biphenyl geometry. This bond-by-bond QTAIM analysis was found to be in agreement with an earlier alternative QTAIM atom-by-atom [2] approach that indicated that the H---H bonding interaction provided a locally stabilizing effect that is overwhelmed by the destabilizing role of the C-C bond. This was shown to lead to the global destabilization of the planar biphenyl conformation compared to the twisted global minimum. Additionally, the H---H bonding interaction in biphenyl has been recently related by some of the current authors to previous work by Benkert and collaborators [3] who suggested an incommensurate phase transition for 5° φ 21°. This was explained as corresponding to the gas/solid phase caused by the pure biphenyl molecule stiffening at approximately φ ≈ 5° and suggested that the lowering of the symmetry caused by the torsion process was inducing the phase transition. Our earlier result for the undecorated biphenyl that the symmetry is reduced at φ = 5.0° as detected from the stress tensor properties of the H---H bonding was consistent with the onset of the incommensurate phase transition being caused by symmetry lowering. It was also noticed that for φ = 22.5° the Poincaré-Hopf relationship which coincided approximately with the upper end of the incommensurate phase transition noticed by Benkert and collaborators. The controversial H---H bonding interactions continue to be investigated for a wide range of chemical environments [4]–[6] and in particular the bay-type H---H bonding interactions in cis-2-butene [7] and biphenyl [8]. The H---H BCPs are very tenuous interactions and as such they can be used as a sensitive probe to detect subtle differences in the chemical environment in the context of the QTAIM and stress tensor topological approaches. An important goal for this work therefore, is to determine the effect that the thirteen substituent groups have on the QTAIM and stress tensor properties of the H---H bonding interactions and H atomic basins subjected to a torsion φ of the C4-C7 torsional bond sufficient to rupture the H---H bonding interactions. In this work the torsion φ always refers to the C4-C7 torsional bond. Previously, it has been determined that the QTAIM formulation of the bond-path stiffness S produced unphysical results for the biphenyl ‘pivot’ C-C bond directly subjected to a torsion φ ≥ 40° beyond the position of the global energy minimum [1]. This was found to be in contrast with the stress tensor stiffness Sσ that produced results in line with expectations from physical intuition. In this work we can investigate the effect that the presence or otherwise of non-physical effects of the bond-path stiffness S and stress tensor stiffness Sσ of the H---H bonding interactions subjected to a torsion. This will include developing new tools suitable for the examination of very small changes in the topological features of the H---H BCPs and the H atomic basins comprising them. The analysis used will need to be able to be sensitive in particular to the changes that occur as a result of the addition of the para-substituted groups of C12H9X, X = N(CH3)2, NH2, CH3, CHO, CN, NO2, and C12H9Y, Y = SiH3, ZnCl, COOCH3, SO2NH2, SO2OH, COCl, CB3, see Figure 1. The reason for the partitioning of the groups into X and Y is because there are Hammett energies available for the X groups but not the Y groups. Figure 1. The representation of the molecular graphs of the para-substituted biphenyl that affected by the torsion φ of the C4-C7 BCP bond-path, C12H9X = X or Y, where X = N(CH3)2, NH2, CH3, CHO, CN, NO2, and Y = SiH3, ZnCl, COOCH3, SO2NH2, SO2OH, COCl, CB3. The C1-C2-C3C4-C5-C6 ring is the fixed ‘stator’ ring. The C7-C8-C9C10-C11-C12 ring, the ‘rotor’ is rotated so that the nuclei comprising the substituent groups move out of the plane in an anti-clockwise direction and the H20 nuclei moves into the plane of the C1-C2-C3-C4-C5-C6 ring.
We can regard the biphenyl molecule subjected to a torsion φ as a simple molecular machine, at least in the sense that the left hand ring is stationary, i.e. the ‘stator’, and the right hand right rotates, i.e. the ‘rotor’ and the C4-C7 bond-path is the ‘axle’, see Figure 1. The H---H interactions and other such weak interactions, will
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likely be frequently present and changing their connectivity in the functioning of the molecular machine. Recently H---H interactions have been found by the current authors in unpublished work based on light driven fluorene molecular rotary [9]. Therefore, quantifying the effect of the addition of a wide range of substituent groups on the H---H interactions seems to be a worthwhile pursuit. We will use QTAIM [10] to identify critical points in the total electronic charge density distribution ρ(r) by analyzing ρ(r). These critical points can further be divided into four types of topologically stable critical points according to the set of ordered eigenvalues λ1 < λ2 < λ3, with corresponding eigenvectors e1, e2, e3. of the Hessian matrix, i.e. the matrix of partial second derivatives of ρ(r) with respect to the components of r evaluated at these points, using classifiers of the form (rank, signature), where the rank is the number of distinct eigenvalues and the signature is the sum of their numerical signs. These are denoted as: (3,-3), which are local maxima usually corresponding to a nuclear position (NCP), but can be non-nuclear maxima (NNA) other critical points are denoted as (3,-1) and (3,+1), which correspond to bond critical point (BCP) and ring critical point (RCP), respectively and are both saddle points; finally (3,+3) correspond to cage critical point (CCP) which is a local minima. The numbers of the four different types of critical points are related by the fundamental theorem of topology for molecules and clusters: the Poincaré-Hopf relationship [11], described below: n–b+r–c=1
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The Poincaré-Hopf solution set {n, b, r, c} are the number of NCPs, BCPs, RCPs and CCPs, respectively and are valid in this work either for QTAIM or the stress tensor within the QTAIM partitioning scheme. In this investigation, we use the eigenvalues and eigenvectors of the Hessian matrix of the total electronic charge density ρ(r), at the bond critical points, BCPs. The pair of special gradient paths linking a BCP with two nuclei (NCPs) and along which ρ(rb), where the subscript ‘b’ denotes the value at the BCP and is a maximum with respect to any neighboring path is denoted as an atomic interaction line (AIL) [12]. In the limit that the forces on the nuclei become vanishingly small an AIL becomes a bond-path, although not necessarily a chemical bond [13]. The complete set of critical points together with the bond-paths of a molecule or cluster is referred to as the molecular graph. Recently, some of the current authors found that the solution set {n, b, r, c} of the stress tensor Poincaré-Hopf relationship is different to that of the QTAIM topologies for the pure biphenyl molecular graph undergoing torsion [14]. Therefore, the Poincaré-Hopf relationship for the stress tensor no longer always referred to the same types of critical points as does the QTAIM version. For example, the stress tensor the ‘b’ and ‘r’ terms of the Poincaré-Hopf relationship do not automatically correspond to features in the molecular graph topology corresponding to a BCP and an RCP respectively. In addition, for a given molecular graph the absence of both an RCP and a BCP would lead to the satisfaction of the Poincaré-Hopf relationship with an incomplete Poincaré-Hopf solution set. As a consequence of the possibility of the breakdown of the stress tensor Poincaré– Hopf relationship is that the definition of the stress tensor stiffness should be changed to Sσ = |λ1σ|/|λ3σ|, to take into account situations where the sign of the λ3σ eigenvalue is negative. The stress tensor stiffness, Sσ = |λ1σ|/|λ3σ| has been found as a good descriptor of the ‘resistance’ of the bond-path to the twist in line with physical intuition, as well as the eigenvector, e1σ indicating the direction of the π-bond[14] [15]. Within QTAIM the topological condition of zero flux ρ(r)∙n(r) = 0 of the total electronic charge density ρ(r) serves as the boundary condition of the atomic basins for the application of Schwinger's principle of stationary action in the definition of an open system. The atomic basins are bounded by these zero-flux surfaces referred to as interatomic surfaces (IAS), which in turn are defined by trajectories of ρ(r) ending at the BCP. A basinpath set is defined as a collection of such trajectories of ρ(r), where the trajectories originate at a chosen number of equi-spaced points placed on a small circle around a BCP and terminate based on a common criterion. The ρ(r) trajectory termination is determined by the total electronic charge density ρ(r) falling below some value, conventionally the threshold value for ρ(r) adopted in atomic integrations to denote the edge of an atomic basin, i.e. 10-4 au. Therefore, the basin-path set area is defined as the area of the surface swept out by the ρ(r) trajectories. Previously, for the bond-path of the fulvene molecular graph undergoing torsion a qualitatively judgment indicated this to be the case[15]. The reason for this was due to the visibly smaller basin-
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path set areas of the C-C atomic basins linking the carbon ring and the H2 group undergoing the e2 eigenvector defined torsion than for the e1 eigenvector defined torsion. It had been found that the direction defined by the eigenvector e2 and e1 of the BCP and bond-path undergoing a torsion φ indicated the most and least preferred sense of rotation respectively[16]. A recent investigation on biphenyl found that the basin-path set areas of the C atomic basins defined by the e2 eigenvector of the BCP undergoing a torsion φ was always greater than for the corresponding e1 eigenvector except at the planar position, φ = 0.0°. The converse was found to be true for the H atomic basins associated with the H---H BCPs [1]. The first step of the computational protocol is to perform a relaxed scan of the potential energy surface (PES) of the C4-C7 pivot-BCP. The PES was performed at 1.0° intervals except for 0.0° φ 10.0° where it was performed at 0.2° intervals with a geometry optimization performed at all steps, spanning the reaction coordinate φ; 0.0° φ < 25.0°, with tight convergence criteria at B3LYP/6-311G(2d,3p) with Gaussian 09B01[17]. Subsequent single point energies for each step in the PES were evaluated at the same theory level. Then the QTAIM and stress analysis was performed with the AIMAll [18] suite on each wave function obtained in the previous step.
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Figure 2. The variation of the H14---H17 BCP and H15---H20 BCP bond-path length (BPL) with the reaction coordinate 0.0° ≤ φ ≤ 10.0° for the substituent groups of biphenyl C12H9X, are shown in sub-figure (a) and (b) respectively. The corresponding values for C12H9Y are presented in sub-figures (c) and (d) respectively, see the caption of Figure 1. The results for the undecorated biphenyl, denoted by ‘H’ are added for comparison [1]. The variation of the BPL with the reaction coordinate φ until the H---H BCPs rupture is shown in Supplementary Materials S1.
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In contrast to the C4-C7 pivot-BCP, the variation stiffness S with the torsion φ for the H14---H17 BCPs is in accordance with physical intuition, the stiffness S decreases smoothly with the increase in the bond-path length, see Figure 3 and Figure S1(e) of the Supplementary Materials. The reason that the stiffness S does not fail for the H---H BCPs is because there is insufficient twisting of the bond-paths due the H14---H17 BCP and H15--H20 BCP rupturing at a low value of the torsion φ, 21.0° ≤ φ ≤ 23.0° respectively. We notice that the H15--H20 BCP ruptures for a lower value of torsion φ due to way the torsion was performed; the bond-path associated with the H15---H20 BCP for each of the substituted groups is always more stretched than the bondpath of the H14---H17 BCP. The bond-path lengths of H14---H17 BCP and H15---H20 BCP smoothly increase with the reaction coordinate φ until they rupture, see Figure 2.
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Despite the transient nature of the QTAIM H---H BCPs with respect to the torsion φ, the QTAIM PoincaréHopf relationship for the biphenyl molecular graph is satisfied throughout the torsion process. This is not always the case however, for the stress tensor topology and we will investigate the consequences of the breakdown of the Poincaré-Hopf relationship.
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Figure 3. The variation of the H14---H17 BCP and H15---H20 BCP stiffness S with the reaction coordinate 0.0° ≤ φ ≤ 10.0° for the substituent groups of biphenyl C12H9X, are shown in sub-figure (a) and (b) respectively. The corresponding values for C12H9Y are presented in sub-figures (c) and (d) respectively, see the captions of Figure 1 and Figure 2. The variation of the stiffness S with the reaction coordinate φ until the H---H BCPs rupture is shown in Supplementary Materials S2.
The QTAIM characteristic sets, {n, b, r, c}, always satisfy the Poincaré-Hopf relationship for all values of φ and all thirteen of the substitution groups. Conversely, the stress tensor characteristic set {nσ, bσ, rσ, cσ} violates the Poincaré-Hopf relationship for the H---H BCPs when the stress tensor eigenvalue 3 becomes negative. In this work we display the stress tensor stiffness Sσ = |λ1σ|/|λ3σ| because it is a physically intuitive way of showing phase transition-like behavior as a peak that coincides also with the change in sign of the stress tensor
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eigenvalue 3, see Figure 4. The lower bound for the failure of the stress tensor Poincaré-Hopf relationship is given by 2.6° φ 5.0°, see Figure 4(b) and Figure 4(a). The failure the stress tensor Poincaré-Hopf relationship is due to the lowering of symmetry of the H---H bond-paths with the increase in the reaction coordinate φ, where QTAIM RCPs always are classified as NNACPs for the stress tensor Poincaré-Hopf relationship for all values of φ. Again the values of torsion φ for the H14---H17 BCP are higher than for the H15---H20 BCP in accordance with the differences in the BPL, see Figure 4(a) and Figure 4(b) respectively. The lower bounds for the failure of the stress tensor Poincaré-Hopf relationship are 3.4° φ 5.0° and 2.6° φ 4.8° for the H14---H17 BCP and H15---H20 BCP respectively.
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Figure 4. The variation of the H14---H17 BCP and H15---H20 BCP stress tensor stiffness Sσ with the reaction coordinate 0.0° ≤ φ ≤ 10.0° for the substituent groups of biphenyl C12H9X, are shown in sub-figure (a) and (b) respectively. The corresponding values for C12H9Y are presented in sub-figures (c) and (d) respectively, see the captions of Figure 1 and Figure 2. The variation of the stress tensor stiffness Sσ with the reaction coordinate φ until the H---H BCPs rupture is shown in Supplementary Materials S3.
It can be seen that the magnitude of the basin-path set area of e2 > e1 for the H14---H17 basin-path set areas for both the H14 and H17 nuclei, except for the position φ = 0.0°, see Figure 5 and Supplementary Materials S4 respectively. The observation that except for the initial position φ = 0.0° that e2 > e1 for the H14 and H17 basinpath set areas indicates that the both the H14 and H17 atomic basins as well as the H14---H17 bond-paths are topologically unstable. This is because any torsion φ > 0.0° causes the H14 and H17 basin-path set area of e2 to exceed that of e1. It can be seen that the for the H17 atomic basin that is associated with the ‘rotor’ part of the molecular graph that the undecorated biphenyl, indicated by ‘H’ is the considerably more stable because than any of the substituent groups, as indicated by the lowest e2 basin-path set area, see Figure 5(b) and Figure 5(d) respectively.
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Figure 5. The variation of the basin-path set areas of the e2 eigenvectors of the H14 and H17 atomic basins comprising the bond-paths of the H14---H17 BCP with the reaction coordinate φ, 0.0° ≤ φ ≤ 3.0° for the substituent groups of biphenyl C12H9X, are shown in sub-figure (a) and (b) respectively. The corresponding values for C12H9Y are presented in subfigures (c) and (d) respectively, see the captions of Figure 1 and Figure 2. The variation of the basin-path set areas e1 with the reaction coordinate φ until the H---H BCPs rupture are shown in Supplementary Materials S4.
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To summarize, we examined the variation of the QTAIM and stress tensor properties of the H---H BCPs with the addition of the para-substituent groups C12H9X, X = N(CH3)2, NH2, CH3, CHO, CN, NO2, and C12H9Y, Y = SiH3, ZnCl, COOCH3, SO2NH2, SO2OH, COCl, CB3 as a function of the torsion φ of the C4-C7 BCP bondpath. Variation was found in the values of the bond-path lengths dependent on the substituent groups X and Y. The low value of the torsion φ, φ < 25.0° meant that the QTAIM could provide reliable values, i.e. physically intuitive, of the stiffness S as seen from the consistent variation with the BPL. The basin-path set area of e2 > e1 for both the H14 and H17 atomic basins demonstrate that the topologically instability of the H---H BCPs was increased for all choices of X and Y. In addition, the sensitivity of the H---H atomic basins to the choice of parasubstituent group was evident. Differences were apparent for all of the QTAIM and stress tensor properties between the ‘stator’ and ‘rotor’ portions of para-substituted biphenyl molecular graphs. We hypothesize that the addition of the substituent groups to biphenyl would broaden the range of the incommensurate phase transition, from 5° φ 21° in the undecorated biphenyl to 2.6° φ ≤ 23.0° in the decorated biphenyl. The basis of this broadening effect, we suggest, is a lowering of symmetry resulting in the change in sign of the λ3σ from positive to negative that induces a phase transition-like peak in the stress tensor stiffness Sσ. The lowering of the symmetry sufficient to change the sign of λ3σ is due to the stress tensor properties being calculated within the QTAIM partitioning and not a stress tensor partitioning
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Acknowledgements The One Hundred Talents Foundation of Hunan Province and the aid program for the Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province are gratefully acknowledged for the support of S.J. and S.R.K. The National Natural Science Foundation of China is also acknowledged, project approval number: 21273069.
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References [1] S. Jenkins, J. R. Maza, T. Xu, D. Jiajun, and S. R. Kirk, ‘Biphenyl: A stress tensor and vector-based perspective explored within the quantum theory of atoms in molecules’, Int. J. Quantum Chem., vol. 115, no. 23, pp. 1678–1690, 2015. [2] C. F. Matta, J. Hernández-Trujillo, T.-H. Tang, and R. F. W. Bader, ‘Hydrogen–Hydrogen Bonding: A Stabilizing Interaction in Molecules and Crystals’, Chem. – Eur. J., vol. 9, no. 9, pp. 1940–1951, May 2003. [3] C. Benkert, V. Heine, and E. H. Simmons, ‘The incommensurate phase transition of biphenyl’, J. Phys. C Solid State Phys., vol. 20, no. 22, pp. 3337–3354, Aug. 1987. [4] J. Poater, M. Solà, and F. M. Bickelhaupt, ‘Hydrogen–Hydrogen Bonding in Planar Biphenyl, Predicted by Atoms-In-Molecules Theory, Does Not Exist’, Chem. – Eur. J., vol. 12, no. 10, pp. 2889–2895, Mar. 2006. [5] C. F. Matta, ‘Hydrogen–Hydrogen Bonding: The Non-Electrostatic Limit of Closed-Shell Interaction Between Two Hydro’, in Hydrogen Bonding—New Insights, S. J. Grabowski, Ed. Springer Netherlands, 2006, pp. 337–375. [6] L. F. Pacios, ‘A theoretical study of the intramolecular interaction between proximal atoms in planar conformations of biphenyl and related systems’, Struct. Chem., vol. 18, no. 6, pp. 785–795, Aug. 2007. [7] F. Weinhold, P. von R. Schleyer, and W. C. McKee, ‘Bay-type H···H “bonding” in cis-2-butene and related species: QTAIM versus NBO description’, J. Comput. Chem., vol. 35, no. 20, pp. 1499–1508, Jul. 2014. [8] K. Eskandari and C. Van Alsenoy, ‘Hydrogen–hydrogen interaction in planar biphenyl: A theoretical study based on the interacting quantum atoms and Hirshfeld atomic energy partitioning methods’, J. Comput. Chem., vol. 35, no. 26, pp. 1883–1889, Oct. 2014. [9] A. Kazaryan, J. C. M. Kistemaker, L. V. Schäfer, W. R. Browne, B. L. Feringa, and M. Filatov, ‘Understanding the Dynamics Behind the Photoisomerization of a Light-Driven Fluorene Molecular Rotary Motor’, J. Phys. Chem. A, vol. 114, no. 15, pp. 5058–5067, Apr. 2010. [10] R. F. W. Bader, Atoms in Molecules: A Quantum Theory. Oxford University Press, USA, 1994. [11] S. Jenkins, Z. Liu, and S. R. Kirk, ‘A bond, ring and cage resolved Poincaré–Hopf relationship for isomerisation reaction pathways’, Mol. Phys., vol. 111, no. 20, pp. 3104–3116, Oct. 2013. [12] R. F. W. Bader, ‘A Bond Path: A Universal Indicator of Bonded Interactions’, J. Phys. Chem. A, vol. 102, no. 37, pp. 7314–7323, Sep. 1998. [13] R. F. W. Bader, ‘Bond Paths Are Not Chemical Bonds’, J. Phys. Chem. A, vol. 113, no. 38, pp. 10391– 10396, Sep. 2009. [14] S. Jenkins, J. R. Maza, T. Xu, D. Jiajun, and S. R. Kirk, ‘Biphenyl: A stress tensor and vector-based perspective explored within the quantum theory of atoms in molecules’, Int. J. Quantum Chem., vol. 115, no. 23, pp. 1678–1690, 2015. [15] S. Jenkins, L. Blancafort, S. R. Kirk, and M. J. Bearpark, ‘The response of the electronic structure to electronic excitation and double bond torsion in fulvene: a combined QTAIM, stress tensor and MO perspective’, Phys. Chem. Chem. Phys., Mar. 2014. [16] F. A. Figueredo, J. R. Maza, S. R. Kirk, and S. Jenkins, ‘Quantum topology phase diagrams for the cisand trans-isomers of the cyclic contryphan-Sm peptide’, Int. J. Quantum Chem., vol. 114, no. 24, pp. 1697–1706, 2014. [17] See the Supplementary Materials, M. Frisch, et al, Gaussian 09, Revision B.01. Wallingford, CT 06492 USA: Gaussian, Inc., 2009. [18] T. A. Keith, AIMAll. Overland Park KS, USA: TK Gristmill Software, 2012.
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S0009-2614(16)30151-8 http://dx.doi.org/doi:10.1016/j.cplett.2016.03.042 CPLETT 33728
To appear in: Received date: Revised date: Accepted date:
28-2-2016 12-3-2016 16-3-2016
Please cite this article as: D. Jiajun, Y. Xu, T. Xu, R. Momen, S.R. Kirk, S. Jenkins, The Substituent Effects on the Biphenyl HmdashH Bonding Interactions Subjected to Torsion, Chem. Phys. Lett. (2016), http://dx.doi.org/10.1016/j.cplett.2016.03.042 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
The Substituent Effects on the Biphenyl H---H Bonding Interactions Subjected to Torsion Dong Jiajun, Yuning Xu, Tianlv Xu, Roya Momen, Steven R. Kirk and Samantha Jenkins* Key Laboratory of Chemical Biology and Traditional Chinese Medicine Research and Key Laboratory of Resource Fine-Processing and Advanced Materials of Hunan Province of MOE, College of Chemistry and Chemical Engineering, Hunan Normal University, Changsha Hunan 410081, China *email: [email protected]
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QTAIM and the stress tensor describe a torsion φ, 0.0° ≤ φ < 25.0°, of the C4-C7 torsional bond that affects the H---H bonding interactions linking the two phenyl rings of para-substituted biphenyl, C12H9X, X = N(CH3)2, NH2, CH3, CHO, CN, NO2 and C12H9-Y, Y = SiH3, ZnCl, COOCH3, SO2NH2, SO2OH, COCl, CB3. For all values of X and Y, shorter H---H bond-path lengths corresponded to higher values of both the stiffness and torsion φ. The onset of a phase transition-like behavior is found by the stress tensor stiffness. The atomic basins of the H---H interactions are affected by the para-substituent groups.
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Highlights QTAIM and stress tensor properties of the H---H interactions of biphenyl subjected to torsion φ depend on para-substituent groups. QTAIM and stress tensor stiffness S and S follow in line with physical intuition. A phase-transition-like behavior for the stress tensor stiffness S is found to vary with the parasubstituent groups.
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The H---H atomic basins are affected by the choice of para-substituent groups.
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The application of this investigation to molecular rotors is briefly discussed.
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*Graphical Abstract (pictogram) (for review)
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Recently, some of the current authors used QTAIM and the stress tensor topological approaches to add to the ongoing discussion about the biphenyl molecule [1]. We explained the effects of the torsion φ of the C-C bond linking the two phenyl rings of the biphenyl molecule on a bond-by-bond basis. Using the total local energy density H(rb) we demonstrated the favorable conditions for the formation of the controversial H---H bonding interactions for a planar biphenyl geometry. This bond-by-bond QTAIM analysis was found to be in agreement with an earlier alternative QTAIM atom-by-atom [2] approach that indicated that the H---H bonding interaction provided a locally stabilizing effect that is overwhelmed by the destabilizing role of the C-C bond. This was shown to lead to the global destabilization of the planar biphenyl conformation compared to the twisted global minimum. Additionally, the H---H bonding interaction in biphenyl has been recently related by some of the current authors to previous work by Benkert and collaborators [3] who suggested an incommensurate phase transition for 5° φ 21°. This was explained as corresponding to the gas/solid phase caused by the pure biphenyl molecule stiffening at approximately φ ≈ 5° and suggested that the lowering of the symmetry caused by the torsion process was inducing the phase transition. Our earlier result for the undecorated biphenyl that the symmetry is reduced at φ = 5.0° as detected from the stress tensor properties of the H---H bonding was consistent with the onset of the incommensurate phase transition being caused by symmetry lowering. It was also noticed that for φ = 22.5° the Poincaré-Hopf relationship which coincided approximately with the upper end of the incommensurate phase transition noticed by Benkert and collaborators. The controversial H---H bonding interactions continue to be investigated for a wide range of chemical environments [4]–[6] and in particular the bay-type H---H bonding interactions in cis-2-butene [7] and biphenyl [8]. The H---H BCPs are very tenuous interactions and as such they can be used as a sensitive probe to detect subtle differences in the chemical environment in the context of the QTAIM and stress tensor topological approaches. An important goal for this work therefore, is to determine the effect that the thirteen substituent groups have on the QTAIM and stress tensor properties of the H---H bonding interactions and H atomic basins subjected to a torsion φ of the C4-C7 torsional bond sufficient to rupture the H---H bonding interactions. In this work the torsion φ always refers to the C4-C7 torsional bond. Previously, it has been determined that the QTAIM formulation of the bond-path stiffness S produced unphysical results for the biphenyl ‘pivot’ C-C bond directly subjected to a torsion φ ≥ 40° beyond the position of the global energy minimum [1]. This was found to be in contrast with the stress tensor stiffness Sσ that produced results in line with expectations from physical intuition. In this work we can investigate the effect that the presence or otherwise of non-physical effects of the bond-path stiffness S and stress tensor stiffness Sσ of the H---H bonding interactions subjected to a torsion. This will include developing new tools suitable for the examination of very small changes in the topological features of the H---H BCPs and the H atomic basins comprising them. The analysis used will need to be able to be sensitive in particular to the changes that occur as a result of the addition of the para-substituted groups of C12H9X, X = N(CH3)2, NH2, CH3, CHO, CN, NO2, and C12H9Y, Y = SiH3, ZnCl, COOCH3, SO2NH2, SO2OH, COCl, CB3, see Figure 1. The reason for the partitioning of the groups into X and Y is because there are Hammett energies available for the X groups but not the Y groups. Figure 1. The representation of the molecular graphs of the para-substituted biphenyl that affected by the torsion φ of the C4-C7 BCP bond-path, C12H9X = X or Y, where X = N(CH3)2, NH2, CH3, CHO, CN, NO2, and Y = SiH3, ZnCl, COOCH3, SO2NH2, SO2OH, COCl, CB3. The C1-C2-C3C4-C5-C6 ring is the fixed ‘stator’ ring. The C7-C8-C9C10-C11-C12 ring, the ‘rotor’ is rotated so that the nuclei comprising the substituent groups move out of the plane in an anti-clockwise direction and the H20 nuclei moves into the plane of the C1-C2-C3-C4-C5-C6 ring.
We can regard the biphenyl molecule subjected to a torsion φ as a simple molecular machine, at least in the sense that the left hand ring is stationary, i.e. the ‘stator’, and the right hand right rotates, i.e. the ‘rotor’ and the C4-C7 bond-path is the ‘axle’, see Figure 1. The H---H interactions and other such weak interactions, will
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likely be frequently present and changing their connectivity in the functioning of the molecular machine. Recently H---H interactions have been found by the current authors in unpublished work based on light driven fluorene molecular rotary [9]. Therefore, quantifying the effect of the addition of a wide range of substituent groups on the H---H interactions seems to be a worthwhile pursuit. We will use QTAIM [10] to identify critical points in the total electronic charge density distribution ρ(r) by analyzing ρ(r). These critical points can further be divided into four types of topologically stable critical points according to the set of ordered eigenvalues λ1 < λ2 < λ3, with corresponding eigenvectors e1, e2, e3. of the Hessian matrix, i.e. the matrix of partial second derivatives of ρ(r) with respect to the components of r evaluated at these points, using classifiers of the form (rank, signature), where the rank is the number of distinct eigenvalues and the signature is the sum of their numerical signs. These are denoted as: (3,-3), which are local maxima usually corresponding to a nuclear position (NCP), but can be non-nuclear maxima (NNA) other critical points are denoted as (3,-1) and (3,+1), which correspond to bond critical point (BCP) and ring critical point (RCP), respectively and are both saddle points; finally (3,+3) correspond to cage critical point (CCP) which is a local minima. The numbers of the four different types of critical points are related by the fundamental theorem of topology for molecules and clusters: the Poincaré-Hopf relationship [11], described below: n–b+r–c=1
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The Poincaré-Hopf solution set {n, b, r, c} are the number of NCPs, BCPs, RCPs and CCPs, respectively and are valid in this work either for QTAIM or the stress tensor within the QTAIM partitioning scheme. In this investigation, we use the eigenvalues and eigenvectors of the Hessian matrix of the total electronic charge density ρ(r), at the bond critical points, BCPs. The pair of special gradient paths linking a BCP with two nuclei (NCPs) and along which ρ(rb), where the subscript ‘b’ denotes the value at the BCP and is a maximum with respect to any neighboring path is denoted as an atomic interaction line (AIL) [12]. In the limit that the forces on the nuclei become vanishingly small an AIL becomes a bond-path, although not necessarily a chemical bond [13]. The complete set of critical points together with the bond-paths of a molecule or cluster is referred to as the molecular graph. Recently, some of the current authors found that the solution set {n, b, r, c} of the stress tensor Poincaré-Hopf relationship is different to that of the QTAIM topologies for the pure biphenyl molecular graph undergoing torsion [14]. Therefore, the Poincaré-Hopf relationship for the stress tensor no longer always referred to the same types of critical points as does the QTAIM version. For example, the stress tensor the ‘b’ and ‘r’ terms of the Poincaré-Hopf relationship do not automatically correspond to features in the molecular graph topology corresponding to a BCP and an RCP respectively. In addition, for a given molecular graph the absence of both an RCP and a BCP would lead to the satisfaction of the Poincaré-Hopf relationship with an incomplete Poincaré-Hopf solution set. As a consequence of the possibility of the breakdown of the stress tensor Poincaré– Hopf relationship is that the definition of the stress tensor stiffness should be changed to Sσ = |λ1σ|/|λ3σ|, to take into account situations where the sign of the λ3σ eigenvalue is negative. The stress tensor stiffness, Sσ = |λ1σ|/|λ3σ| has been found as a good descriptor of the ‘resistance’ of the bond-path to the twist in line with physical intuition, as well as the eigenvector, e1σ indicating the direction of the π-bond[14] [15]. Within QTAIM the topological condition of zero flux ρ(r)∙n(r) = 0 of the total electronic charge density ρ(r) serves as the boundary condition of the atomic basins for the application of Schwinger's principle of stationary action in the definition of an open system. The atomic basins are bounded by these zero-flux surfaces referred to as interatomic surfaces (IAS), which in turn are defined by trajectories of ρ(r) ending at the BCP. A basinpath set is defined as a collection of such trajectories of ρ(r), where the trajectories originate at a chosen number of equi-spaced points placed on a small circle around a BCP and terminate based on a common criterion. The ρ(r) trajectory termination is determined by the total electronic charge density ρ(r) falling below some value, conventionally the threshold value for ρ(r) adopted in atomic integrations to denote the edge of an atomic basin, i.e. 10-4 au. Therefore, the basin-path set area is defined as the area of the surface swept out by the ρ(r) trajectories. Previously, for the bond-path of the fulvene molecular graph undergoing torsion a qualitatively judgment indicated this to be the case[15]. The reason for this was due to the visibly smaller basin-
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path set areas of the C-C atomic basins linking the carbon ring and the H2 group undergoing the e2 eigenvector defined torsion than for the e1 eigenvector defined torsion. It had been found that the direction defined by the eigenvector e2 and e1 of the BCP and bond-path undergoing a torsion φ indicated the most and least preferred sense of rotation respectively[16]. A recent investigation on biphenyl found that the basin-path set areas of the C atomic basins defined by the e2 eigenvector of the BCP undergoing a torsion φ was always greater than for the corresponding e1 eigenvector except at the planar position, φ = 0.0°. The converse was found to be true for the H atomic basins associated with the H---H BCPs [1]. The first step of the computational protocol is to perform a relaxed scan of the potential energy surface (PES) of the C4-C7 pivot-BCP. The PES was performed at 1.0° intervals except for 0.0° φ 10.0° where it was performed at 0.2° intervals with a geometry optimization performed at all steps, spanning the reaction coordinate φ; 0.0° φ < 25.0°, with tight convergence criteria at B3LYP/6-311G(2d,3p) with Gaussian 09B01[17]. Subsequent single point energies for each step in the PES were evaluated at the same theory level. Then the QTAIM and stress analysis was performed with the AIMAll [18] suite on each wave function obtained in the previous step.
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Figure 2. The variation of the H14---H17 BCP and H15---H20 BCP bond-path length (BPL) with the reaction coordinate 0.0° ≤ φ ≤ 10.0° for the substituent groups of biphenyl C12H9X, are shown in sub-figure (a) and (b) respectively. The corresponding values for C12H9Y are presented in sub-figures (c) and (d) respectively, see the caption of Figure 1. The results for the undecorated biphenyl, denoted by ‘H’ are added for comparison [1]. The variation of the BPL with the reaction coordinate φ until the H---H BCPs rupture is shown in Supplementary Materials S1.
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In contrast to the C4-C7 pivot-BCP, the variation stiffness S with the torsion φ for the H14---H17 BCPs is in accordance with physical intuition, the stiffness S decreases smoothly with the increase in the bond-path length, see Figure 3 and Figure S1(e) of the Supplementary Materials. The reason that the stiffness S does not fail for the H---H BCPs is because there is insufficient twisting of the bond-paths due the H14---H17 BCP and H15--H20 BCP rupturing at a low value of the torsion φ, 21.0° ≤ φ ≤ 23.0° respectively. We notice that the H15--H20 BCP ruptures for a lower value of torsion φ due to way the torsion was performed; the bond-path associated with the H15---H20 BCP for each of the substituted groups is always more stretched than the bondpath of the H14---H17 BCP. The bond-path lengths of H14---H17 BCP and H15---H20 BCP smoothly increase with the reaction coordinate φ until they rupture, see Figure 2.
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Despite the transient nature of the QTAIM H---H BCPs with respect to the torsion φ, the QTAIM PoincaréHopf relationship for the biphenyl molecular graph is satisfied throughout the torsion process. This is not always the case however, for the stress tensor topology and we will investigate the consequences of the breakdown of the Poincaré-Hopf relationship.
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Figure 3. The variation of the H14---H17 BCP and H15---H20 BCP stiffness S with the reaction coordinate 0.0° ≤ φ ≤ 10.0° for the substituent groups of biphenyl C12H9X, are shown in sub-figure (a) and (b) respectively. The corresponding values for C12H9Y are presented in sub-figures (c) and (d) respectively, see the captions of Figure 1 and Figure 2. The variation of the stiffness S with the reaction coordinate φ until the H---H BCPs rupture is shown in Supplementary Materials S2.
The QTAIM characteristic sets, {n, b, r, c}, always satisfy the Poincaré-Hopf relationship for all values of φ and all thirteen of the substitution groups. Conversely, the stress tensor characteristic set {nσ, bσ, rσ, cσ} violates the Poincaré-Hopf relationship for the H---H BCPs when the stress tensor eigenvalue 3 becomes negative. In this work we display the stress tensor stiffness Sσ = |λ1σ|/|λ3σ| because it is a physically intuitive way of showing phase transition-like behavior as a peak that coincides also with the change in sign of the stress tensor
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eigenvalue 3, see Figure 4. The lower bound for the failure of the stress tensor Poincaré-Hopf relationship is given by 2.6° φ 5.0°, see Figure 4(b) and Figure 4(a). The failure the stress tensor Poincaré-Hopf relationship is due to the lowering of symmetry of the H---H bond-paths with the increase in the reaction coordinate φ, where QTAIM RCPs always are classified as NNACPs for the stress tensor Poincaré-Hopf relationship for all values of φ. Again the values of torsion φ for the H14---H17 BCP are higher than for the H15---H20 BCP in accordance with the differences in the BPL, see Figure 4(a) and Figure 4(b) respectively. The lower bounds for the failure of the stress tensor Poincaré-Hopf relationship are 3.4° φ 5.0° and 2.6° φ 4.8° for the H14---H17 BCP and H15---H20 BCP respectively.
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Figure 4. The variation of the H14---H17 BCP and H15---H20 BCP stress tensor stiffness Sσ with the reaction coordinate 0.0° ≤ φ ≤ 10.0° for the substituent groups of biphenyl C12H9X, are shown in sub-figure (a) and (b) respectively. The corresponding values for C12H9Y are presented in sub-figures (c) and (d) respectively, see the captions of Figure 1 and Figure 2. The variation of the stress tensor stiffness Sσ with the reaction coordinate φ until the H---H BCPs rupture is shown in Supplementary Materials S3.
It can be seen that the magnitude of the basin-path set area of e2 > e1 for the H14---H17 basin-path set areas for both the H14 and H17 nuclei, except for the position φ = 0.0°, see Figure 5 and Supplementary Materials S4 respectively. The observation that except for the initial position φ = 0.0° that e2 > e1 for the H14 and H17 basinpath set areas indicates that the both the H14 and H17 atomic basins as well as the H14---H17 bond-paths are topologically unstable. This is because any torsion φ > 0.0° causes the H14 and H17 basin-path set area of e2 to exceed that of e1. It can be seen that the for the H17 atomic basin that is associated with the ‘rotor’ part of the molecular graph that the undecorated biphenyl, indicated by ‘H’ is the considerably more stable because than any of the substituent groups, as indicated by the lowest e2 basin-path set area, see Figure 5(b) and Figure 5(d) respectively.
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Figure 5. The variation of the basin-path set areas of the e2 eigenvectors of the H14 and H17 atomic basins comprising the bond-paths of the H14---H17 BCP with the reaction coordinate φ, 0.0° ≤ φ ≤ 3.0° for the substituent groups of biphenyl C12H9X, are shown in sub-figure (a) and (b) respectively. The corresponding values for C12H9Y are presented in subfigures (c) and (d) respectively, see the captions of Figure 1 and Figure 2. The variation of the basin-path set areas e1 with the reaction coordinate φ until the H---H BCPs rupture are shown in Supplementary Materials S4.
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To summarize, we examined the variation of the QTAIM and stress tensor properties of the H---H BCPs with the addition of the para-substituent groups C12H9X, X = N(CH3)2, NH2, CH3, CHO, CN, NO2, and C12H9Y, Y = SiH3, ZnCl, COOCH3, SO2NH2, SO2OH, COCl, CB3 as a function of the torsion φ of the C4-C7 BCP bondpath. Variation was found in the values of the bond-path lengths dependent on the substituent groups X and Y. The low value of the torsion φ, φ < 25.0° meant that the QTAIM could provide reliable values, i.e. physically intuitive, of the stiffness S as seen from the consistent variation with the BPL. The basin-path set area of e2 > e1 for both the H14 and H17 atomic basins demonstrate that the topologically instability of the H---H BCPs was increased for all choices of X and Y. In addition, the sensitivity of the H---H atomic basins to the choice of parasubstituent group was evident. Differences were apparent for all of the QTAIM and stress tensor properties between the ‘stator’ and ‘rotor’ portions of para-substituted biphenyl molecular graphs. We hypothesize that the addition of the substituent groups to biphenyl would broaden the range of the incommensurate phase transition, from 5° φ 21° in the undecorated biphenyl to 2.6° φ ≤ 23.0° in the decorated biphenyl. The basis of this broadening effect, we suggest, is a lowering of symmetry resulting in the change in sign of the λ3σ from positive to negative that induces a phase transition-like peak in the stress tensor stiffness Sσ. The lowering of the symmetry sufficient to change the sign of λ3σ is due to the stress tensor properties being calculated within the QTAIM partitioning and not a stress tensor partitioning
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Acknowledgements The One Hundred Talents Foundation of Hunan Province and the aid program for the Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province are gratefully acknowledged for the support of S.J. and S.R.K. The National Natural Science Foundation of China is also acknowledged, project approval number: 21273069.
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References [1] S. Jenkins, J. R. Maza, T. Xu, D. Jiajun, and S. R. Kirk, ‘Biphenyl: A stress tensor and vector-based perspective explored within the quantum theory of atoms in molecules’, Int. J. Quantum Chem., vol. 115, no. 23, pp. 1678–1690, 2015. [2] C. F. Matta, J. Hernández-Trujillo, T.-H. Tang, and R. F. W. Bader, ‘Hydrogen–Hydrogen Bonding: A Stabilizing Interaction in Molecules and Crystals’, Chem. – Eur. J., vol. 9, no. 9, pp. 1940–1951, May 2003. [3] C. Benkert, V. Heine, and E. H. Simmons, ‘The incommensurate phase transition of biphenyl’, J. Phys. C Solid State Phys., vol. 20, no. 22, pp. 3337–3354, Aug. 1987. [4] J. Poater, M. Solà, and F. M. Bickelhaupt, ‘Hydrogen–Hydrogen Bonding in Planar Biphenyl, Predicted by Atoms-In-Molecules Theory, Does Not Exist’, Chem. – Eur. J., vol. 12, no. 10, pp. 2889–2895, Mar. 2006. [5] C. F. Matta, ‘Hydrogen–Hydrogen Bonding: The Non-Electrostatic Limit of Closed-Shell Interaction Between Two Hydro’, in Hydrogen Bonding—New Insights, S. J. Grabowski, Ed. Springer Netherlands, 2006, pp. 337–375. [6] L. F. Pacios, ‘A theoretical study of the intramolecular interaction between proximal atoms in planar conformations of biphenyl and related systems’, Struct. Chem., vol. 18, no. 6, pp. 785–795, Aug. 2007. [7] F. Weinhold, P. von R. Schleyer, and W. C. McKee, ‘Bay-type H···H “bonding” in cis-2-butene and related species: QTAIM versus NBO description’, J. Comput. Chem., vol. 35, no. 20, pp. 1499–1508, Jul. 2014. [8] K. Eskandari and C. Van Alsenoy, ‘Hydrogen–hydrogen interaction in planar biphenyl: A theoretical study based on the interacting quantum atoms and Hirshfeld atomic energy partitioning methods’, J. Comput. Chem., vol. 35, no. 26, pp. 1883–1889, Oct. 2014. [9] A. Kazaryan, J. C. M. Kistemaker, L. V. Schäfer, W. R. Browne, B. L. Feringa, and M. Filatov, ‘Understanding the Dynamics Behind the Photoisomerization of a Light-Driven Fluorene Molecular Rotary Motor’, J. Phys. Chem. A, vol. 114, no. 15, pp. 5058–5067, Apr. 2010. [10] R. F. W. Bader, Atoms in Molecules: A Quantum Theory. Oxford University Press, USA, 1994. [11] S. Jenkins, Z. Liu, and S. R. Kirk, ‘A bond, ring and cage resolved Poincaré–Hopf relationship for isomerisation reaction pathways’, Mol. Phys., vol. 111, no. 20, pp. 3104–3116, Oct. 2013. [12] R. F. W. Bader, ‘A Bond Path: A Universal Indicator of Bonded Interactions’, J. Phys. Chem. A, vol. 102, no. 37, pp. 7314–7323, Sep. 1998. [13] R. F. W. Bader, ‘Bond Paths Are Not Chemical Bonds’, J. Phys. Chem. A, vol. 113, no. 38, pp. 10391– 10396, Sep. 2009. [14] S. Jenkins, J. R. Maza, T. Xu, D. Jiajun, and S. R. Kirk, ‘Biphenyl: A stress tensor and vector-based perspective explored within the quantum theory of atoms in molecules’, Int. J. Quantum Chem., vol. 115, no. 23, pp. 1678–1690, 2015. [15] S. Jenkins, L. Blancafort, S. R. Kirk, and M. J. Bearpark, ‘The response of the electronic structure to electronic excitation and double bond torsion in fulvene: a combined QTAIM, stress tensor and MO perspective’, Phys. Chem. Chem. Phys., Mar. 2014. [16] F. A. Figueredo, J. R. Maza, S. R. Kirk, and S. Jenkins, ‘Quantum topology phase diagrams for the cisand trans-isomers of the cyclic contryphan-Sm peptide’, Int. J. Quantum Chem., vol. 114, no. 24, pp. 1697–1706, 2014. [17] See the Supplementary Materials, M. Frisch, et al, Gaussian 09, Revision B.01. Wallingford, CT 06492 USA: Gaussian, Inc., 2009. [18] T. A. Keith, AIMAll. Overland Park KS, USA: TK Gristmill Software, 2012.
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