Controlling Achiral and Chiral Properties with an Electric Field: A Next-Generation QTAIM Interpretation

: We used the recently introduced stress tensor trajectory U σ space construction within the framework of next-generation quantum theory of atoms in molecules (NG-QTAIM) for a chirality investigation of alanine when subjected to a non-structurally distorting electric field. The resultant sliding of the axial-bond critical point ( BCP ) responded significantly, up to twice as much, in the presence of the applied electric field in comparison to its absence. The bond flexing, a measure of bond strain, was always lower by up to a factor of four in the presence of the electric field, de-pending on its direction and magnitude. An achiral character of up to 7% was found for alanine in the presence of the applied electric field. The achiral character was entirely absent in the presence of the lowest value of the applied electric field. Future applications, including molecular devices using left and right circularly polarized laser pulses, are briefly discussed.


Introduction
Chirality is widely observed in nature and is definable as the geometric property of a molecule, where its mirror image is non-superimposable. Enantiomers are the left-and right-handed forms of chiral molecules that possess identical chemical properties when using scalar chemical measures but may demonstrate strong enantiomeric preference during chemical reactions. The development of reliable methods to discern enantiomers has received considerable attention in recent years [1].
Some of the current authors recently used NG-QTAIM to quantify a chirality-helicity measure [2], which is an association between molecular chirality and helical characteristics known as the chirality-helicity equivalence, first described by Wang [3], and is consistent with photoexcitation circular dichroism experiments [4]. Wang hypothesized that the origin of the helical characteristic was not attributable to steric hindrance or molecular geometries alone but required insight derived from the electronic structure. Recently, the interdependence of steric and electronic factors was discovered to be more complex [5] than discernable from the molecular geometries associated with the helical electronic transitions of spiro-conjugated molecules [6,7]. Enantiomers of isolated molecules or molecules during reactions can, however, be distinguished using the vector-based chemical measures that are used in next-generation QTAIM (NG-QTAIM) [8].
Previously [2], we established the presence of a distinct helical-shaped stress tensor trajectory Tσ(s) for lactic acid and alanine. If the value of the chirality Cσ > 0, and the value is larger for the CCW than for the CW directions of the torsion angle θ, a preference is indicated for Sσ compared to Rσ stereoisomers in Uσ space; see the Theoretical Background and Computational Details section. The Sσ and Rσ chirality assignments for lactic acid and alanine were in full agreement with the Cahn-Ingold-Prelog (CIP) priority rules [9]. Later, we developed the NG-QTAIM interpretation of chirality by quantifying the chirality-helicity equivalence by formulating the chirality-helicity function Chelicity [7,8,10]. This previous work, however, was limited by the use of only a single dihedral angle to construct Tσ(s), which consequently did not fully sample the bonding environment of the chiral carbon (C1). This limitation has been recently addressed using the new spanning Tσ(s) [11] for (achiral) singly halogen (F, Cl or Br) substituted and (chiral) doubly halogen substituted ethane. We also used the spanning Tσ(s) to construct the chirality-helicity function Chelicity to demonstrate the NG-QTAIM interpretation of achiral behavior: Chelicity = 0 comprised an equal and cancelling mix of Sσ and Rσ stereoisomer contributions in Uσ space. This was also discovered for ethane [12]. We found that within NG-QTAIM, formally achiral molecules may contain chiral contributions but these summed to zero when using the spanning form of the Tσ(s).
An enhanced ability of tracking changes to the chirality and the ability to distinguish enantiomers during the course of a reaction when subjected to an electric (E) field is vital to a wide range of sciences. For instance, Shaik et al. recently investigated an E field for use as a 'smart reagent' for the control of the reactivity and structure for chemical catalysis in a range of reactions [13].
Recently, we examined formally achiral glycine subjected to an E field [14] using NG-QTAIM. This work followed on from investigations performed by Wolk et al. on the application of an E field to induce stereoisomers via symmetry breaking changes to the length of the C-H bonds [15]. Again, this previous study on glycine was unable to provide the mix of Sσ and Rσ stereoisomer contributions in Uσ space for each of the Sa and Ra geometric stereoisomers due to the use of only a single dihedral angle to construct Tσ(s).
The construction of the spanning Tσ(s) enables the determination of the degree of mixing of Sσ and Rσ stereoisomers that may occur when alanine is subjected to an E field, as shown in Scheme 1. A goal of this investigation was to understand the consequences of any mixing of the Sσ and Rσ stereoisomers that may occur when alanine is subjected to a non-structurally distorting E field, since Sσ and Rσ mixing indicates the presence of an achiral character. Another goal was to test the effect of the E fields on the chirali-ty-helicity function Chelicity of alanine, the NG-QTAIM interpretation of chirality, along the C3→O10 BCP bond path and the C3←O10 BCP bond path attached to the chiral carbon (C1), as shown in Scheme 1. We used the E fields: ±25 × 10 −4 a.u., ±50 × 10 −4 a.u., and ±100 × 10 −4 a.u., which are experimentally accessible, using a scanning tunneling microscope (STM).

Theoretical Background
The background of QTAIM and next-generation QTAIM (NG-QTAIM), including the procedure used to generate the stress tensor trajectories Tσ(s), is provided in the Supplementary Materials S1. The ellipticity, ε, quantifies the relative accumulation of the electronic charge density ρ(rb) distribution in the two directions perpendicular to the bond path at a bond critical point (BCP) with position rb. For values of the ellipticity ε > 0 and where 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, and the ellipticity is defined as ε = λ1/λ2 − 1. Note that λ1 and λ2 both possess negative signs, where λ1 ≤ λ2 < λ3 and λ3 > 0.
We used the Bader's formulation of the quantum stress tensor σ(r) [16] to quantify the mechanics of the forces that act on the electron density distribution in open systems defined by: where γ(r,r′) is the one-body density matrix: , , , … , * ′, , … , The stress tensor is any quantity σ(r) that satisfies Equation (2) since any divergence-free tensor can be added to the stress tensor while satisfying this definition [16][17][18]. Bader's formulation of the stress tensor [16], Equation (1), is used in the AIMAll QTAIM package [19] and in this investigation due to the superior performance of the stress tensor compared with the Hessian of ρ(r) for more clearly distinguishing the Ra and Sa stereoisomers of lactic acid and alanine [20]. Earlier, we demonstrated that the most and least preferred directions for bond displacement correspond to the most and least preferred direction of ρ(r) displacement, respectively, namely the e1σ and e2σ eigenvectors, respectively, of the stress tensor [8].
The chirality Cσ is quantified by the bond torsion direction CCW vs. CW, where the largest magnitude stress tensor eigenvalue (λ1σ) is associated with e1σ. The stress tensor σ(r) eigenvector e1σ corresponds to the direction in which the electrons at the C1-C2 BCP are subject to the most compressive forces. Therefore, e1σ corresponds to the direction along which the C1-C2 BCP electrons will be most displaced when the C1-C2 BCP is subjected to torsion [21]. The chirality Cσ for each dihedral angle is defined as the difference in the maximum projections: the dot product of the stress tensor e1σ eigenvector and the BCP displacement dr of the Tσ(s) values between the CCW and CW torsion θ is defined as: The bond flexing Fσ, defined as: The bond flexing Fσ, see Equation (4), provides a measure of the 'flexing-strain' of a bond path for each dihedral angle, which is particularly of use when a molecule is subjected to an E field.
The bond axiality Aσ for each dihedral angle provides a measure of the chiral asymmetry, defined as: The bond axiality Aσ, see Equation (5), quantifies the direction of axial displacement of the bond critical point (BCP) in response to the bond torsion (CCW vs. CW), i.e., the sliding of the BCP along the bond path [22]. The (±) sign of the chirality Cσ, see Equation (3). bond flexing Fσ, and bond axiality Aσ determine the prevalence of the Sσ (Cσ > 0, Fσ > 0, Aσ > 0) or Rσ (Cσ < 0, Fσ < 0, Aσ < 0) character, as shown in Table 1. For formally achiral molecules, we may define an additional null-chirality assignment Qσ (≈ 0 chiral character) that occurs for the ethane molecule [12] and singly halogen substituted ethane [11]. The ± sign, however, is not used with the assignment Qσ as it is for the Sσ and Rσ assignments since Cσ = 0, Fσ = 0, and Aσ = 0 in this case. We include all the contributions to the Uσ space chirality from the 'chiral' center C1 of alanine. This is undertaken by constructing all nine torsion C1-C2 BCP Tσ(s) that use dihedral angles that include the C1 atom, see Scheme 1. We refer to this process of using all nine torsion C1-C2 BCP Tσ(s) as the so-called spanning Uσ space chirality construction. The result of this process is a complete set of alanine Uσ space isomers, with possible chirality assignments Qσ, Sσ, or Rσ. The linear sum of the individual components of the symmetry inequivalent Uσ space distortion sets ∑{Cσ,Fσ,Aσ} is calculated.
The chirality-helicity function Chelicity (=Cσ|Aσ|) summed over each of the dihedral angles used to construct the Tσ(s) is required to determine whether a molecule is formally achiral. We tabulate Chelicity in the absence of an applied E field and refer to this as ChelicityE = CσE|AσE, which is the product of the ratio of the chirality CσE = ∑Cσ/∑Cσ|E = 0 and the ratio of the bond axiality AσE = ∑Aσ/∑Aσ|E=0, see Table 2. Ethane, for instance, comprises Chelicity values: Qσ (=0), Sσ (=+0.0003), and Rσ (=−0.0003), which sum to give ∑Chelicity = 0. The chirality Cσ is formed from the e1σ•dr (bond twist) BCP shift in the plane perpendicular to e3σ (the bond path). The axiality Aσ is formed from the axial BCP sliding e3σ•dr (bond axiality) [22], where the BCP sliding is the shift of the BCP position along the containing bond path due to changes in the bonded inter-nuclear separations. In this investigation, the presence of a mix of Sσ and Rσ chirality assignments for the components of the Sa or Ra geometric stereoisomers of alanine are referred to as mixed chirality Cσ in Uσ space and are determined by a value of Cσmixing = ∑Sσ{Cσ}/|∑Rσ{Cσ}|} > 0. Earlier, an equal mixing of the Sσ and Rσ chirality was found for the formally achiral ethane [12] and corresponds to the maximum value possible Cσmixing = 1. We consider the degree of mixing of the chirality Cσmixing = ∑Sσ{Cσ}/|∑Rσ{Cσ}|, the bond flexing Fσmixing = ∑Sσ{Fσ}/|∑Rσ{Fσ}|, and bond axiality Aσmixing = ∑Sσ{Aσ}/|∑Rσ{Aσ}|, where all are bounded by the limits [0, 1]. Values of Aσmixing = 0 would occur in instances of insignificant torsional C1-C2 BCP bond path curvature since the eigenvector e3σ, which is directed along the bond path (r), is always perpendicular to the plane defined by the eigenvectors e1σ and e2σ. The presence of non-zero bond path curvature is determined by a non-zero difference of the bond path length (BPL) and the geometric separation (GBL) of the pair of bonded nuclei, as shown in the Supplementary Materials S2.

Computational Details
The alanine molecular geometry was initially optimized with the 'verytight' convergence criteria with the B3LYP/cc-pVQZ level of DFT theory using Gaussian 09.E01 [23] and employing an 'ultrafine' integration grid. The wavefunctions were converged to less than 10 −10 RMS change in the density matrix and less than 10 −8 maximum change in the density matrix. All the subsequent E field optimization, torsion, and single-point steps then used identical convergence criteria. An iterative process is used to create the E-field-induced isomers. This was undertaken by directing an E field parallel (+E field) or anti-parallel (-E field) to the C3-O10 BCP bond path, see Scheme 2.
Each of the Sa and Ra geometric stereoisomers were subjected to an iterative process consisting of two steps: Step (I): a molecule alignment step: the alpha C3 atom was fixed at the origin of the coordinate frame, whereas the selected C3-O10 bond was aligned along a reference axis with the positive direction of the axis from C3 to O10 and the C3 atom consistently aligned in the same plane. Step (II): a constrained optimization step with the selected E field directed along the reference axis. The sign convention by default G09 is for the E field relative to the reference axis used. This two-step process was repeated ten times to ensure the consistency of the E field application direction and the required bond (C3-O10) direction. The resulting molecular geometries were subsequently used in the torsion calculations, where the C3-O10 bond lengths were constrained to their E-field-optimized values. The alanine molecule was subjected to E fields = ±25 × 10 −4 a.u., ±50 × 10 −4 a.u., and ±100 × 10 −4 a.u. before the molecule was torsioned to construct the trajectories Tσ(s) from the series of rotational isomers −180.0° ≤ θ ≤ +180.0° for the torsional C1-C2 BCP of alanine. The direction of torsion is defined as CCW (−180.0° ≤ θ ≤ 0.0°) or CW (0.0° ≤ θ ≤ +180.0°) from a decrease or an increase in the dihedral angle, respectively, see Single-point calculations were undertaken on each torsion scan geometry where the SCF iterations converged to less than 10 −10 RMS change in the density matrix and less than 10 −8 maximum change in the density matrix to yield the final wavefunctions for the QTAIM and stress tensor σ(r), which was conducted using the AIMAll [19] and QuantVec [24] suite on each wavefunction, obtained in the previous step. All molecular graphs were confirmed to be free of non-nuclear attractor (NNA) critical points.

Results and Discussions
In this investigation, the scalar distance measures: geometric bond length (GBL) and bond path length (BPL), were not sufficient to quantify any chiral effects with or without the presence of an applied non-structurally distorting E field = ±25 × 10 −4 a.u., ±50 × 10 −4 a.u., and ±100 × 10 −4 a.u. and are supplied in the Supplementary Materials S2. None of the torsional C1-C2 BCP bond path curvatures were significantly non-zero, see Supplementary Materials S2. The other scalar measures for alanine without and with an E field are supplied in the Supplementary Materials S3.
The stress tensor trajectories Tσ(s) for the torsion C1-C2 BCP demonstrate the helical form that appears to be characteristic of chiral molecules, as shown in Figures 1-4, where the same axis scales are used throughout. This finding is consistent with the helical-shaped Tσ(s) previously observed for the alanine C1-C2 BCP Tσ(s), which were obtained using only a single dihedral angle [9].
The total Sσ and Rσ chirality assignments in Uσ space correspond to the Sa and Ra geometric stereoisomers, respectively, of alanine, fully consistent with the CIP priority rules, as shown in Table 1 Table  1. Without an applied E field, a degree of mixing of the Sσ and Rσ Uσ space chirality Cσ of the stereoisomers is apparent from the non-zero value of Cσmixing.
The values of Cσmixing = 0 indicate there was no mixing of the Sσ and Rσ chirality Cσ stereoisomers for E field values of −25.0 × 10 −4 a.u. and +25.0 × 10 −4 a.u., in contrast to the behavior in the absence of an applied E field. Therefore, the effect of the E field = ±25.0 × 10 −4 a.u. is to render the Sa and Ra geometric stereoisomers as comprising pure Sσ and pure Rσ chirality Cσ components, respectively. The values of Cσmixing are larger by at least a factor of two for values of the E field = −50.0 × 10 −4 a.u and −100.0x10 −4 a.u than for the oppositely directed E field, i.e., for E field = +50.0 × 10 −4 a.u and +100.0 × 10 −4 a.u. This indicates that Cσmixing is enhanced for the C3←O10 BCP bond path direction and reduced for the oppositely directed C3→O10 BCP bond path.
The value of Fσmixing = 0 in the absence of an E field; however, the Fσmixing values are rather significant with the application of an E field. The presence of non-zero Fσmixing for the applied E fields can explain the reduction in the magnitude of the bond flexing ∑Fσ, i.e., an increase in the bond stiffness, indicated by the values of FσE < 1 in Table 2. Fσmixing remains unaffected by the E field direction, in contrast to Cσmixing.
The values of Aσmixing = 0 for all values of the applied E field, which corresponds to a complete lack of mixing of the Sσ and Rσ bond axiality Aσ components and is due to a lack of significant torsional C1-C2 BCP bond path curvature; see the Theoretical Background.
We observe that the application of a -E field consistently reduces the values of CσE compared with a +E field. The CσE values increase with the magnitude of the +E field, with CσE = 1.0456 for +E field = +100.0 × 10 −4 a.u and only just exceeding that of alanine in the absence of an E field. The effect of the E field on the bond flexing FσE is very similar for the +E field and -E field in that both cause a significant decrease. The ±E field causes the bond axiality values AσE > 1.0 in all cases and AσE exceeds 2.0 for +E field = +100.0 × 10 −4 a.u.
The chirality-helicity values ChelicityE all increase in the presence of the applied ±E field due to the induction of increases in AσE, where the increase is greater for the +E field values. The values of AσE indicate a very significant enhancing interaction with the ±E field compared with CσE.

Conclusions
In this NG-QTAIM investigation, we determined the effects of a non-structurally distorting E field on the Sa and Ra geometric stereoisomers of the alanine molecule using the recently introduced spanning Tσ(s). We discovered that the values of the chirality-helicity ChelicityE, the complete NG-QTAIM quantification of chirality, can be manipulated with an applied E field whilst reducing the structural strain in the form of the ratio of the bond flexing FσE. We found that in the presence of the E field, a mix of Sσ and Rσ components, in all but one case, resulted in a reduction in the chirality ∑Cσ and bond flexing ∑Fσ but for none of the bond axiality values Aσ. The Uσ space Sσ and Rσ chirality stereoisomers were found to be in agreement with the CIP naming schemes for all ±E field values.
In all cases, the applied E field resulted in the ratio of the bond axiality values AσE > 1 responding strongly due to the increased ease of the resultant C1-C2 BCP sliding. The complete absence of Sσ and Rσ bond axiality Aσmixing mixing for all values of the applied ±E field was due to a lack of significant torsional C1-C2 BCP bond path curvature. The lack of Aσmixing mixing for all values of the applied E field resulted in an increase in the chirality-helicity Chelicity compared to the absence of the E field. Therefore, the large effect on the ratio of the AσE values due to the ±E field demonstrates that it is essential to account for ∑Aσ for a complete understanding of the manipulation of chirality.
Values of Cσmixing > 0 were found to indicate the presence of an achiral character in the range 2% to 7%, both in the absence and presence of the applied E field. A single instance of purely chiral alanine was found for an applied E field = ±25 × 10 −4 a.u., corresponding to a value of Cσmixing = 0, thus indicating a complete lack of achiral character. The quantification of mixing of the Sσ and Rσ components determined by Cσmixing is an additional feature provided by NG-QTAIM that enables the presence of achiral behaviors to be accounted for.
The magnitude of the ratio of the bond flexing FσE decreased in the presence of all values of the applied E field. This finding indicates the significance of monitoring the E field magnitude and direction to minimize the bond flexing ∑Fσ to achieve less destructive, i.e., bond flexing FσE, manipulation of the chirality ∑Cσ. A useful effect of tracking the changes in the Uσ space distortion sets ∑{Cσ,Fσ,Aσ} was provided by the ability to choose ±E field values that increased or decreased ChelicityE whilst reducing the unwanted outcome of bond flexing ∑Fσ that occurred for E field = +100.0 × 10 −4 a.u.
Future investigations using NG-QTAIM could include manipulation of the components of the Uσ space distortion sets ∑{Cσ,Fσ,Aσ} using a pair of left and right circularly-polarized laser pulses. This has the benefit of not requiring the geometric C1-C2 BCP torsions that are used to create the spanning stress tensor trajectories Tσ(s). The application of non-ionizing ultra-fast laser irradiation that would be fast enough to avoid disrupting atomic positions would enable non-destructive manipulation, i.e., increasing or decreasing the ratio of the chirality CσE and bond axiality AσE whilst monitoring the degree of bond flexing FσE. NG-QTAIM could consequently open up a wide scientific field for chiral solid-state and molecular systems to track, control, and quantify the chirality of a wide range of molecular devices, including photochromic switches [25] and azobenzene chiroptical switches [26]. In addition, the design of chiral-optical molecular rotary motors [27] that use synthetic controllable chiral light for ultrafast imaging of chiral dynamics in gases [28] could be investigated with NG-QTAIM.