Chirality-Helicity Equivalence in the S and R Stereoisomers: A

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Chirality-Helicity Equivalence in the S and R Stereoisomers: A Theoretical Insight Tianlv Xu, Jia Hui Li, Roya Momen, Wei Jie Huang, Steven Robert Kirk, Yasuteru Shigeta, and Samantha Jenkins J. Am. Chem. Soc., Just Accepted Manuscript • DOI: 10.1021/jacs.9b00823 • Publication Date (Web): 13 Mar 2019 Downloaded from http://pubs.acs.org on March 13, 2019

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Journal of the American Chemical Society

Chirality-Helicity Equivalence in the S and R Stereoisomers: A Theoretical Insight Tianlv Xu1, Jia Hui Li1, Roya Momen1, Wei Jie Huang1, Steven Robert Kirk1*, Yasuteru Shigeta2 and Samantha Jenkins1*

1Key

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 2Center

for Computational Sciences, University of Tsukuba, Tsukuba 305-8577, Japan

email: [email protected] email: [email protected]

Abstract We located the unknown chirality-helicity equivalence in molecules with a chiral center and as a consequence the degeneracy of the S and R stereoisomers of lactic acid was lifted. Agreement was found with the naming schemes of S and R stereoisomers from optical experiments. This was made possible by the construction of the stress tensor trajectories in a non-Cartesian space defined by the variation of the position of the torsional bond critical point upon a structural change, along the torsion angle, θ, involving a chiral carbon atom. This was undertaken by applying a torsion θ, -180.0° ≤θ ≤+180.0° corresponding to clockwise and counter-clockwise directions. We explain why scalar measures can at best only partially lift the degeneracy of the S and R stereoisomers, as opposed to vector-based measures that can fully lift the degeneracy. We explained the consequences for stereochemistry in terms of the ability to determine the chirality of industrially relevant reaction products.

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Introduction Different stereoisomers of a compound may have substantially different biological effects: examples include the lactate dehydrogenase molecule that is stereospecific to the S stereoisomer of lactic acid1. In nature, usually only one stereoisomer of chiral biological compounds is present. For drug synthesis, both stereoisomers may be present2 in a racemic mixture, creating a need for efficient separation of the stereoisomers3, which is an active field of research4–7. A particularly important field is the enantioselective synthesis of chiral compounds8,9, for instance the enantioselective hydrogenation of enamides for the synthesis of natural or unnatural α-amino acids, where the choice of the chiral ligand for the metal catalyst is critical10. The use of trial-and-error and high throughput screening is expensive both terms of time taken and materials cost11. The ability to determine the S or R assignments of the products would therefore be very valuable and unprecedented.

(a)

(b)

Scheme 1: The molecular graphs of the S- and R-stereoisomers of lactic acid with the torsional C1-C2 bond critical point (BCP) indicated by the undecorated green sphere, with the undecorated red sphere indicating the location of the ring critical point (RCP), are shown in sub-figures (a) and (b) respectively. The clockwise (CW) direction of torsion θ is indicated by dashed black lines surrounding the torsional C1-C2 BCP. The chiral center is located at the carbon atom C1.

Conventionally, a chiral molecule has at least a chiral center, and the relative configuration (R/S) of the chiral molecule is determined by the Cahn–Ingold–Prelog (CIP) priority rules without any calculation

12,13.

Some optical isomers of coordination compounds used as stereoselectivity catalysts however, cannot be assigned from CIP rules. Therefore, the quantitative method for determining the relative configurations of the chiral molecules is significant. As far back as 1821 Fresnel14 first proposed unknown helical characteristics of stereoisomers as the origin of chirality. Evidence for this has come from optical experiments that reveal different refractive indices for right (R) and left (S) circularly polarized light, thus exhibiting a form of optical activity15 known as circular dichroism. This optical activity is in agreement with theories of optical activity that correlate the inherent helical identities with direction of rotation of the circularly polarized light16–18. More recently, D. Z. Wang proposed a ‘helix theory for molecular chirality and chiral interaction’ and noted that the evidence for the helical character from an examination of molecular geometries had proved elusive19. In his treatise, Wang proposed that the unknown chirality-helicity equivalence would be discovered in future ACS Paragon Plus Environment

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by examination of the molecular electronic properties and not be solely attributed to steric hindrance. In this investigation, we will pursue Wang’s suggestion by undertaking an in-depth investigation of the molecular properties of the commonly occurring S and R stereoisomers of lactic acid due to the existence of an asymmetrical carbon chiral center (C1), indicated in Scheme 1. We therefore need a measure that can capture the asymmetry of the chiral center C1. Very recent groundbreaking experiments by Beaulieu, Comby

et al.20 revealing the helical motion of bound electrons provide further evidence that the total

electronic charge density distribution will be useful in the search for the chiral-helicity equivalence. There have been no theoretical considerations of helicity and the relationship with chirality prior to this investigation that include specific examples, apart from the conjectures and predictions by D. Z. Wang. The symmetry21 needs to be broken for helical character to appear: this is undertaken by constructing a series of rotational isomers for the S and R stereoisomers to sample the effects of steric hindrance on the molecular electronic properties in the form of the total electronic charge density distribution ρ(r). To extract information revealing the chirality-helicity equivalence, we here adopt the quantum theory of atoms in molecules (QTAIM) 22 . We suggest that previous sole use of scalar, as opposed to vector-based directional, chemical topological measures is the reason why the relationship between chirality and helicity has yet to be found satisfactorily. Within the QTAIM framework, the cross-section of a bond or bond critical point (BCP), see the next Section and Supplementary Materials S1, can be quantified as possessing a non-circular distribution of ρ(r). This asymmetry in the distribution of ρ(r) is already well known and can be used to quantify the asymmetry that arises as a consequence of the chiral center C1. It may not, however, fully explain the chirality-helicity equivalence. In this investigation, we will search for the unknown chirality-helicity equivalence and the consequences for differentiating the S and R stereoisomers from the other physical quantities evaluated by QTAIM. 2. Theoretical Background and Computational Details The QTAIM and stress tensor BCP properties; ellipticity ε, total local energy density H(rb), stiffness S and stress tensor stiffness, Sσ and the stress tensor trajectories Tσ(s) The physical quantities treated here may be unfamiliar to the readership, and therefore in this section we will briefly explain them and their significance within the context of QTAIM. We also provide a set of complete mathematical expressions and explanations in the Supplementary Materials S1. We use QTAIM and the stress tensor analysis23–29 which exploit higher derivatives of ρ(r), in effect acting as a ‘magnifying lens’ on the ρ(r). We use QTAIM to identify critical points in the total electronic charge density distribution ρ(r) where the gradient vector field ∇ρ(r) = 0. These critical points can further be divided into four topologically stable types according to the set of ordered eigenvalues λ1 < λ2 < λ3, with the corresponding set of eigenvectors (e1, e2, e3), of the Hessian matrix of the electronic charge density, ρ(r), which is defined as the matrix of partial second derivatives with respect to the spatial coordinates, ∇:∇ρ(r). Critical points are labeled using the notation (R, ω) where R is the rank of the Hessian matrix, i.e., the number of distinct ACS Paragon Plus Environment


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non-zero eigenvalues, and ω is the signature (the algebraic sum of the signs of the eigenvalues); the (3, -3) [nuclear critical point (NCP), a local maximum generally corresponding to a nuclear location], (3, -1) and (3, 1) [saddle points, called bond critical points (BCP) and ring critical points (RCP), respectively] and (3, 3) [the cage critical points (CCP)]. In the limit that the forces on the nuclei become vanishingly small, an atomic interaction line30 , the line passing through a BCP and terminating on two nuclear attractors along which the charge density ρ(r) is locally maximal with respect to nearby lines, becomes a bond-path, although not necessarily a chemical bond31. The complete set of critical points together with the bond-paths of a molecule or cluster is referred to as the molecular graph. In this section, we focus on the change of physical quantities as a result of the torsion of the asymmetric (chiral) carbon atom, C1, see Scheme 1. The eigenvalues and eigenvectors obtained from the QTAIM analysis provide measures used to define and characterize the nature of the unknown helicity, see Supplementary Materials S1. The first measure is the ellipticity, ε, that quantifies the relative accumulation of ρ(rb) in the two directions perpendicular to the bond-path at a BCP with position rb. For values of the ellipticity ε > 0, the shortest and longest axes of the elliptical distribution of ρ(rb) are associated with the λ1 and λ2 eigenvalues respectively. The second measure is the bond-path stiffness, S = λ2/λ3, as a measure of the rigidity of the bond-path32. The third measure is the total local energy density H(rb) at rb33,34. A value of H(rb) < 0 for the closed-shell interaction, ∇2ρ(rb) > 0, indicates a BCP with a degree of covalent character, and conversely H(rb) > 0 reveals a lack of significant covalent character for the closed-shell BCP. The fourth measure is directional, i.e., vector-based: the stress tensor trajectory, Tσ(s), where s indicates a reaction coordinate, mapped onto a space, Uσ, which is a central (or major) quantity to distinguish two stereoisomers. The Tσ(s) comprised of a series of contiguous points and is a 3-D vector path that displays the effect of the structural change, i.e., the bond torsion, see Supplementary Materials S1. We depict all four of these measures that attempt to determine the unknown helicity as a function of the torsion angle θ involving a chiral carbon (C1) in lactic acid and extract the indicator for characterizing the chirality from the electronic charge density distribution rather than from optical spectra and/or simple molecular geometry, see Figures 1- 3 and Table 1. The first step of the computational protocol is to perform a constrained scan of the potential energy surface (E), see Figure 1(a) and Scheme 1. The scan was performed with a constrained (Z-matrix) geometry optimization performed at all steps with all coordinates free to vary except for the torsion coordinate θ, see Scheme 1, where θ was defined by the dihedral angle C3-C1-C2-H8 (for the S stereoisomer) and C3-C1-C2-H7 (for the R stereoisomer), in the range -180.0° ≤θ ≤+180.0° with 1.0° intervals. These torsions correspond to the Cartesian clockwise (CW) and Cartesian counter-clockwise (CCW) directions of the torsion θ, applied to the S and R stereoisomers. For further analysis, we first focused on the physical quantities measured at the bond-critical point of the C1-C2 bond (hereafter referred to as the C1-C2 BCP). We chose the C1-C2 bond attached to the chiral center C1 because it is free to rotate: the other choices C1-C3 and C1-O4 are unsuitable due to the applied torsion θ being hindered by the H12--O4 bond, see ACS Paragon Plus Environment

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Figure 1(a) and the Supplementary Materials S2. For a torsion performed about a non-chiral atom (e.g. C3) the variation of the relative energy ∆E for the S and R rotational isomers is distinguishable, see Figure S2(c) of the Supplementary Materials S2 corresponding to the torsion about the C3-O11 BCP. The B3LYP DFT functional was used with the 6-311G(2d,3p) basis set and computations performed using Gaussian 09E0135. In the calculations, the following tight convergence thresholds (stated here in atomic units) were used: SCF energy 10-6, density matrix 10-8, maximum force 4.5x10-4, RMS force 3x10-4. A built-in pruned ’Ultrafine’ DFT integration grid with 99 radial shells and 950 angular points/shell convergence criteria was also used. Subsequent single point energies for each step in the potential energy surface were evaluated using the same theory level, convergence criteria and integration grids. QTAIM and stress tensor analysis was performed with the AIMAll36 suite on each wave function obtained in the previous step. All molecular graphs were additionally confirmed to be free of non-nuclear attractor critical points.

(a) (b) Figure 1. The variation of the relative energy ∆E and the bond-path lengths (BPL) of the C1-C2 BCP with torsion θ of the rotational isomers and S and R stereoisomers of lactic acid with the CW (-180.0º ≤ θ ≤ 0.0º) and CCW (0.0º ≤ θ ≤ 180.0º ) are presented in sub-figures (a) and (b) respectively, see Scheme 1.

3. Results and discussions A comparison of the S and R stereoisomers of lactic acid using the QTAIM and stress tensor bond critical

point (BCP) properties In this work, only the relaxed geometries, i.e., for a given θ, where θ = 0.0º will be referred to as the S and R stereoisomers. All the remaining isomers will be discussed as rotational isomers of the S and R stereoisomers, see Scheme 1 and Figure 1(a). The relative energy ∆E plots for the series of rotational isomers of the S and R stereoisomers of lactic acid are indistinguishable, for both the Cartesian clockwise (CW) and counter-clockwise (CCW) directions of θ, since the energies are almost degenerate, see Figure 1(a). The corresponding plots for the variation of the bond-path length (BPL) that relates to the λ3 eigenvalue are also indistinguishable, see Figure 1(b). The “eclipsing-effect” commonly associated with alkanes37 is apparent for θ, where θ = ±60.0º and θ = ±180.0º are located at the maxima. ACS Paragon Plus Environment


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(b)

(c)

(d)

Figure 2. The variation of the local total energy density H(rb), stiffness S, stress tensor stiffness Sσ and ellipticity ε with θ for the S and R stereoisomers and rotational isomers of lactic acid with the CW and CCW torsions about the C1-C2 BCP are presented in sub-figures (a-d) respectively, for further details see Scheme 1 and Figure 1.


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The corresponding variations of the total local energy density H(rb), see Figure 2(a) are also indistinguishable and follow the same form as the relative energy plots ∆E, see Figure 1(a). The variations for both the stiffness S and the stress tensor stiffness Sσ show very slight differences between the S and R stereoisomers, see Figure 2(b) and Figure 2(c), respectively. For the stress tensor stiffness Sσ we notice in particular that slight differences occur at θ = 0.0º because the C1-C2 BCP is not located at the geometric mid-point of the torsion C1-C2 BCP bond-path, see the Supplementary Materials S1 for an explanation. The plots of the ellipticity ε values of the rotational isomers of the S and R stereoisomers for θ from -180.0º ≤ θ ≤ +180.0º are mirror images of each other see Figure 2(d), consistent with the geometries of the molecular graphs of the S and R stereoisomers, see Scheme 1. For rotational isomers of the S stereoisomer, i.e., -180.0º ≤ θ ≤ 0.0º about the C1-C2 BCP for the CW torsion θ the ellipticity ε values are greater than or equal to those of the rotational isomers of R in the CW direction. As a consequence of the mirror symmetry, the ellipticity ε values of the rotational isomers of the R stereoisomer for the CCW torsion θ, i.e., 0.0º ≤ θ ≤ 180.0º are correspondingly greater than or equal to those the rotational isomers of S. We see significant differences in the ellipticity ε values of the rotational isomers of S and R stereoisomers for values of θ, except for the degenerate points at θ = 0.0º, ±60.0º, ±120.0º and ±180.0º that correspond to the extrema in the relative energy ∆E. The separation of the ellipticity ε values of the rotational isomers of S and R stereoisomers is due to the inherent asymmetries in the distributions of the λ1 and λ2 eigenvalues of the torsional C1-C2 BCP due to the C1 atom being the chiral center. We provide two new bond-path measures that also partially lift the degeneracy, i.e., with respect to the e1 and e2 eigenvectors, but not the e3 eigenvector. The first is the bond-path set B = {p,q,r}, a vector-based interpretation of the chemical bond with three strands p-, q- and r-, constructed from the three QTAIM eigenvectors {e1, e2, e3}, see the Supplementary Materials S3-S8 for all theoretical deviations, procedures to generate the results and surrounding discussion. The p- and q-paths are twisted about the r-path, i.e., the bond-path, for the torsion θ, -180.0° (CW) ≤ θ ≤ +180.0° (CCW) and due to their dependence on r, the pand q-paths cannot be used to distinguish the S and R stereoisomers at θ values corresponding to the extrema in ∆E. The second bond-path measure provided is the ellipticity ε profile, which was able to lift the degeneracy for the same limited set of θ values, see the Supplementary Materials S7 and Supplementary Materials S8. We will therefore now examine the directional measure required for chirality and to search for preferences of the S and R stereoisomers in the form of the associated e1 and e2 eigenvector properties using the stress tensor

trajectories,

Tσ(s),

see


Figure

3.


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(a)

(b) Figure 3. The helical stress tensor trajectories Tσ(s) of the S and R stereoisomers (θ = 0.0°) and rotational isomers of lactic acid for the Cartesian torsions CW (θ = -30.0°, -60.0°, -90.0°, -120.0°, -150.0°, -180.0°) and CCW (θ = 30.0°, 60.0°, 90.0°, 120.0°, 150.0°, 180.0°) of the torsional C1-C2 BCP in Cartesian space are presented in sub-figures (a) and (b) respectively. The maximum projections Tσ(s)max = {t1,max = (e1σ∙dr)max, t2,max = (e2σ∙dr)max, t3,max = (e3σ∙dr)max}, where dr are the BCP shift vectors. The chirality assignments of the Tσ(s) helix screw axes: CWH and CCWH may not coincide with the Cartesian CW and CCW directions of torsion θ shown in Scheme 1 that are only appropriate for Figure 1 and Figure 2. The subscript ‘H’ indicates the left- or right-handed screw axes of the Tσ(s) helices.


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The trajectories Tσ(s) of the S and R stereoisomers (θ = 0.0°) are constructed from the series of rotational isomers (θ = ±30.0°, ±60.0°, ±90.0°, ±120.0°, ±150.0°, ±180.0°) for the torsional C1-C2 BCP. The Tσ(s) forms

helices in Uσ space with repeating motifs at intervals of 120.0°, consistent with the relative energy, ∆E, bond-path length (BPL) and total local energy density H(rb) values, see Figure 1, Figure 2(a) and Figure 3. The chirality assignments of the Tσ(s) helix screw axes: CWH (right-handed) and CCWH (left handed) may not coincide with the Cartesian CW and CCW directions of torsion θ: the subscript ‘H’ is used to make the distinction from the Cartesian CW and CCW directions of torsion θ shown in Scheme 1. The stress tensor trajectories, Tσ(s), are vector quantities constructed from the S and R stereoisomers: it was observed that both stereoisomers and rotational isomers for the torsion C1-C2 BCP are unique, because the Tσ(s) occupy separate regions of the stress tensor trajectory space, Uσ, for both the applied Cartesian (CW and CCW) torsions, see Figure 3. The reason for the uniqueness of the Tσ(s), and therefore lack of overlap, e.g., at a value of θ = 0.0º of the rotational isomers is due to the lifting of the degeneracy with respect to the e3σ eigenvectors. The lifting of the degeneracy of the Tσ(s) is visible as the separation parallel to the t3 axis, e.g., for the S and R stereoisomers at θ = 0.0º, see Figure 3(a) and Figure 3(b). The maximum projections Tσ(s)max = {t1,max, t2,max, t3,max} corresponding to stress tensor trajectory space lengths , Lσ, and real space lengths, l, are all scalars, see Table 1 and the Supplementary Materials S4, respectively. As a consequence, these scalar measures cannot preserve the helical character of the Tσ(s) and instead retain the mirror symmetry of the λ2 and λ1 eigenvalues that comprise the ellipticity ε. The scalar t1,max and t2,max define values of the maximum projections Tσ(s)max and map out approximate ‘long and short axes’ dimensions of the helices of the Tσ(s) trajectories seen in Figure 3 and Table 1. In addition, the t1 and t2 components also define the maximum extent associated with the most and least preferred directions of charge density ρ(r) accumulation respectively. The t3 component is associated with the motion of the torsional C1-C2 BCP along ±e3σ associated with λ3 that defines the bond-path. We observed that t1,max > t2,max for both the CW and CCW directions of torsion θ for the S and R rotational isomers. The presence of the Tσ(s) in the form of the helix indicates the presence of translation as well as torsion of the C1-C2 BCP. This was demonstrated by the presence of values of t3,max > 0 in all cases. The stress tensor trajectory lengths Lσ and real space lengths l, despite being scalars, can be used to distinguish the rotational isomers of S and R stereoisomers everywhere except for θ = 0.0°, ±60.0°, ±120.0° and ±180.0° that correspond to extrema in ∆E. This is due to the construction of Lσ and l from the inherently asymmetrical λ1 and λ2 eigenvalues and the lack of dependency on λ3. The application of the Cartesian CW vs. CCW, however, results in differences in the e1 and e2 eigenvectors but not e3 and therefore lifts the degeneracy of the λ1 and λ2 eigenvalues for the rotational isomers and stereoisomers of S and R but not the associated λ3 eigenvalue, which does not remove the mirror symmetry. The same result is true for the corresponding Cartesian CCW vs. CW torsional C1-C2 BCP of the rotational isomers and stereoisomers of S and R. The mirror symmetry is not present for the helix geometry of the Tσ(s) due to the mapping from each ±dr shift vector in real space to a point in Uσ space that lifts the degeneracy with respect to the e3σ eigenvectors. ACS Paragon Plus Environment


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The Tσ(s)max components of the torsional C1-C2 BCP corresponding to the S and R rotational isomers display mirror image symmetry with respect to the CWH and CCWH directions. This is consistent with values of the scalar lengths Lσ and l, which are provided in the Supplementary Materials S4. We see that the magnitudes of the t1,max component of the S stereoisomer are significantly (4%) larger for the CCWH (left-handed) direction than the CWH (right-handed) direction. As a consequence of the mirror symmetry, the converse is true for R where the t1,max component possesses significantly (4%) larger magnitude in the CWH direction compared with CCWH. Therefore, the CCWH direction is most preferred for the helices of the Tσ(s) corresponding to the S stereoisomer compared with the less preferred CWH direction. The converse is true for the R stereoisomer where the CWH direction is preferred over the CCWH. These results are consistent with optical experiment results used to determine the S (CCWH, left-handed) and R (CWH, right-handed) naming assignments. The same analysis was also performed on the S and R stereoisomers and rotational isomers of the chiral amino acid Alanine, where the magnitudes of the t1,max component of the S stereoisomer are larger (by 5.3%) for the CCWH direction than the CWH direction, also in agreement with the optical experiment results, see the Supplementary Materials S9. We also performed the same analysis on the achiral amino acid Glycine, where no helical geometry in the trajectory Tσ(s) was observed, due to the achiral nature of the molecule, see the Supplementary Materials S10. This finding is consistent with recently published work on the achiral ethene molecule subjected to CW and CCW torsions where the no helical character was found to be present for the Tσ(s)25 .


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Table 1. The maximum stress tensor projections {t1max, t2max, t3max} for the S and R stereoisomers for the torsional C1-C2 BCP corresponding are shown highlighted in a bold font. The chirality assignments of the left- and right-handed screw axes of the Tσ(s) helices are denoted as CCWH and CWH respectively, the subscript “H” refers these helices as opposed to the Cartesian torsion CCW and CW shown in Scheme 1.

{t1max, t2max, t3max} S CWH BCP C1-C2 C1-C3 C1-O4 C1-H5 C2-H7 C2-H8 C2-H9 C3-O10 C3-O11 H6-O4 H12-O4 H12-O11

R CCWH

{1.900, 0.971, 0.880} {1.978, 1.077, 0.883} {0.226, 0.515, 0.281} {0.217, 0.507, 0.281} {2.950, 1.990, 0.627} {2.780, 2.127, 0.567} {1.788, 4.270, 0.200} {1.708, 4.218, 0.181} {40.50, 7.380, 18.83} {40.86, 7.251, 19.69} {44.60, 8.900, 20.65} {44.67, 8.099, 21.62} {40.50, 7.573, 24.88} {41.25, 10.90, 22.40} {6.290, 0.695, 0.686} {7.078, 0.789, 0.758} {1.201, 8.420, 0.935} {1.306, 9.349, 1.030} {22.50, 6.380, 7.720} {23.56, 6.485, 7.870} {19.17, 6.430, 1.962} {21.03, 7.191, 1.937} {30.10, 2.800, 2.170} {33.76, 3.045, 2.246}

CWH

CCWH

{1.976, 1.066, 0.884} {1.899, 0.967, 0.881} {0.212, 0.506, 0.282} {0.229, 0.517, 0.280} {2.760, 2.140, 0.555} {2.962, 1.980, 0.624} {1.704, 4.215, 0.185} {1.783, 4.259, 0.206} {44.88, 7.977, 21.75} {44.88, 8.764, 20.65} {40.66, 7.447, 19.85} {40.36, 7.467, 18.85} {41.14, 10.90, 22.49} {40.65, 7.545, 24.72} {6.958, 0.764, 0.724} {6.424, 0.674, 0.672} {1.264, 9.179, 0.990} {1.233, 8.583, 0.915} {23.52, 6.531, 7.909} {22.81, 6.374, 7.677} {20.59, 6.987, 1.951} {19.43, 6.515, 2.010} {33.13, 2.930, 2.135} {30.74, 2.672, 2.136}



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Conclusions In this investigation of stereochemistry, we located the unknown chirality-helicity equivalence present in chiral molecules required to explain the chemical origins of differences in behaviors of the S and R stereoisomers of lactic acid and alanine. We find agreement with the prediction made by David Z. Wang19 that the molecular electronic properties and not solely steric hindrance would be the source of the unknown chirality-helicity equivalence. This finding is consistent with the recent groundbreaking experimental work that uses the helical motion of bound electrons instead of the traditional magnetic field effects20. The symmetry breaking that enabled the chirality-helicity equivalence to be located was made possible by the construction of the stress tensor trajectories Tσ(s). Helical Tσ(s) in Uσ space were created from a series of rotational isomers for the S and R stereoisomers of lactic acid by the use of an applied Cartesian torsion θ, -180.0° (CW) ≤ θ ≤ +180.0° (CCW). We examined the variation with θ of the scalar measures by QTAIM: relative energy, ∆E, bond-path length (BPL), local total energy density H(rb), stiffness S, stress tensor stiffness Sσ, and the ellipticity ε. We determined that the scalar measures that depend on the bond-path (∆E and H(rb)) or depend on the λ3 eigenvalue (S and Sσ) cannot be used to distinguish either the stereoisomers or rotational isomers of the S and R stereoisomers of lactic acid. The ellipticity ε does not contain a dependency on the λ3 eigenvalue and can, therefore, distinguish the rotational isomers of the S and R stereoisomers. The ellipticity ε, however, cannot distinguish the S and R stereoisomers, i.e., at θ values of θ = 0.0º or the rotational isomers corresponding to the extrema in the relative energy ∆E values for θ = ±60.0°, ±120.0° and ±180.0°. So far all chemical explanations of chirality are incomplete and stated only in terms of steric effects such as the eclipsing effect. The reason for this failure, we suggest, is due to the sole use of scalar measures. No scalar measure will be able to capture the directional character of chirality and therefore can never locate the unknown chirality-helicity equivalence. This demonstrated the need to quantify the directional behavior of all three of the associated e1σ, e2σ and e3σ eigenvectors in the form of the vector stress tensor trajectories Tσ(s) as the source of the unknown chirality-helicity equivalence. We found, however, that by providing a measure of asymmetry or ‘symmetry-breaking’ by varying the torsion angle θ, we observe that the scalar measures of ellipticity ε, Tσ(s)max, the stress tensor trajectory lengths Lσ and the real space lengths l were able to distinguish the rotational isomers of S and R everywhere except for θ = 0.0°, ±60.0°, ±120.0° and ±180.0. We have, therefore, outlined an approach to obtain the correct assignment of S and R stereoisomers in agreement with optical experiment i.e. S (CCWH, left-handed) and R (CWH, right-handed). Verification was provided by the same analysis performed on the stereoisomers and rotational isomers S and R of the amino acid Alanine. In this article, we offered the methodology to determine the relative configuration of chiral molecules and applied to measure the chirality quantitatively. Here however, we only applied our methodology for characterizing the molecule itself. Since our methodology can be used to characterize the chirality during the chemical reaction processes, in the future, we will apply it to characterize the heterotopicity in the ACS Paragon Plus Environment

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addition/substitution of a chemical group to a planar molecule quantitatively in relation to the energetically favorable reaction pathway. This extension can be used to explain the reactivity towards both prochiral planes and helps one to understand the origin of the (homo-) chirality in the enantio-selective synthesis using catalysis. In practice, this approach could be used to straightforwardly determine the chirality of industrially relevant reaction products, e.g., as outlined in the introduction for the Rh-catalyzed hydrogenation of enamides. These structures would be obtained from standard quantum mechanical simulation methods. This approach, therefore, removes the requirement for the trial and error screening methods currently used for such reactions. Plans for future work include the investigation of chirality in molecules with more than one chiral atom. Supporting Information: 1. Supplementary Materials S1. QTAIM and stress tensor bond theoretical background. 2. Supplementary Materials S2. The variation of the relative energy ∆E with torsion θ. 3. Supplementary Materials S3. The procedure to generate the trajectory length Lσ and implementation details of the calculation of the eigenvector-following path lengths H and H*. 4. Supplementary Materials S4. Tables of the stress tensor Uσ space trajectory lengths Lσ and real space trajectory lengths l in atomic units (a.u.) of the S and R stereoisomers. 5. Supplementary Materials S5. The variation of bond-path properties of the torsional C1-C2 BCP. 6. Supplementary Materials S6. The combined plots of bond-path properties of the torsion C1-C2 BCP. 7. Supplementary Materials S7. Magnified (x10) versions of the bond-path set B{p, q, r} for the C1-C2 BCP. 8. Supplementary Materials S8. Variations of the ellipticity ε with the distance along bond-path (r) of the torsion C1-C2 BCP corresponding to the relative energy ∆E extrema. 9. Supplementary Materials S9. The QTAIM and stress tensor properties of the S and R stereoisomers of Alanine. 10. Supplementary Materials S10. The QTAIM and stress tensor properties of Glycine. Acknowledgements The National Natural Science Foundation of China is gratefully acknowledged, project approval number: 21673071. The One Hundred Talents Foundation of Hunan Province is also gratefully acknowledged for the support of S.J. and S.R.K. References (1) (2)

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Chirality-Helicity Equivalence in the S and R Stereoisomers: A

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