The Cis-Effect Explained Using Next-Generation QTAIM

We used next-generation QTAIM (NG-QTAIM) to explain the cis-effect for two families of molecules: C2X2 (X = H, F, Cl) and N2X2 (X = H, F, Cl). We explained why the cis-effect is the exception rather than the rule. This was undertaken by tracking the motion of the bond critical point (BCP) of the stress tensor trajectories Tσ(s) used to sample the Uσ-space cis- and trans-characteristics. The Tσ(s) were constructed by subjecting the C1-C2 BCP and N1-N2 BCP to torsions ± θ and summing all possible Tσ(s) from the bonding environment. During this process, care was taken to fully account for multi-reference effects. We associated bond-bending and bond-twisting components of the Tσ(s) with cis- and trans-characteristics, respectively, based on the relative ease of motion of the electronic charge density ρ(rb). Qualitative agreement is found with existing experimental data and predictions are made where experimental data is not available.

Earlier, some of the current authors provided a scalar physics-inspired coupling mechanism explaining the cis-effect in terms of electronic and nuclear degrees of freedom for three families of molecules, including halogen-substituted ethene and diazenes [18]. We undertook a static investigation of the properties of the central X = X or N = N bond paths and found that those of the cis-isomers were more bent: the difference between the length of the bond path and the internuclear separation was up to 1.5% larger in the cis-isomers than in the corresponding trans-isomers. In our earlier contribution, we therefore concluded that the physical origin of the cis-effect was associated with greater bond-path bending. This earlier work, however, only provided correlations of the bond-path bending with the energy and did not explain why the cis-effect is the exception rather than the rule. In this work, the physical basis of the cis-effect will be provided in terms of the least and most preferred directions of electronic charge density motion.
Recently, some of the current authors used an electronic charge-density-based analysis to investigate steric effects within the formulation of next-generation Quantum Theory of Atoms in Molecules (NG-QTAIM) [19]. We found that the presence of chiral contributions suggested that steric effects, rather than hyperconjugation, explained the staggered geometry of ethane [20]. This recent work on steric effects relates to the current investigation on cis-effects, since in both cases we subject the central C = C or N = N bond to a torsion to probe either steric or cis-effects. Low/high values of the NG-QTAIM interpretation of chirality (C σ ) were associated with low/high steric effects due to the absence/presence of an asymmetry. The chirality C σ [21] was earlier used to redefine a related quantity for cumulenes, the bond-twist T σ [22].
In this investigation, we will use NG-QTAIM to explain why the cis-effect was previously found, in our scalar investigation, to be associated with bond bending [18]. This will be undertaken by subjecting the axial bonds, C1-C2 and N1-N2, to a torsion θ to sample the directional response of the electronic charge density ρ(r b ) at the bond cross-section. This will provide a better understanding of the greater (topological) stability of the cis-isomer over the trans-isomer in these halogen-containing species; see Scheme 1. Recently, some of the current authors used an electronic charge-density-based analysis to investigate steric effects within the formulation of next-generation Quantum Theory of Atoms in Molecules (NG-QTAIM) [19]. We found that the presence of chiral contributions suggested that steric effects, rather than hyperconjugation, explained the staggered geometry of ethane [20]. This recent work on steric effects relates to the current investigation on cis-effects, since in both cases we subject the central C = C or N = N bond to a torsion to probe either steric or cis-effects. Low/high values of the NG-QTAIM interpretation of chirality (Cσ) were associated with low/high steric effects due to the absence/presence of an asymmetry. The chirality Cσ [21] was earlier used to redefine a related quantity for cumulenes, the bond-twist Tσ [22].
In this investigation, we will use NG-QTAIM to explain why the cis-effect was previously found, in our scalar investigation, to be associated with bond bending [18]. This will be undertaken by subjecting the axial bonds, C1-C2 and N1-N2, to a torsion  to sample the directional response of the electronic charge density ρ(rb) at the bond cross-section. This will provide a better understanding of the greater (topological) stability of the cisisomer over the trans-isomer in these halogen-containing species; see Scheme 1.
We use Bader's formulation of the quantum stress tensor σ(r) [30] to characterize the forces on the electron density distribution in open systems that is defined by: where γ(r,r′) is the one-body density matrix, (r, r ′ ) = ∫ (r, r 2 , … , r ) * (r′, r 2 , … , r )dr 2 ⋯ dr (2) The stress tensor is then any quantity σ(r) that can satisfy equation (2): any divergence-free tensor can be added [30][31][32]. Bader's formulation of the stress tensor σ(r), equation (1), is a standard option in the AIMAll QTAIM package [33]. Earlier Bader's formulation of σ(r) demonstrated superior performance compared with the Hessian of ρ(r) for distinguishing the Sa-and Ra-geometric stereoisomers of lactic acid [34] and therefore will be used in this investigation.
In this investigation, we include the entire bonding environment, including all contributions to the Uσ-space cis-and trans-characteristics, by considering the C1-C2 BCP Tσ(s)
We use Bader's formulation of the quantum stress tensor σ(r) [30] to characterize the forces on the electron density distribution in open systems that is defined by: where γ(r,r ) is the one-body density matrix, The stress tensor is then any quantity σ(r) that can satisfy equation (2): any divergencefree tensor can be added [30][31][32]. Bader's formulation of the stress tensor σ(r), equation (1), is a standard option in the AIMAll QTAIM package [33]. Earlier Bader's formulation of σ(r) demonstrated superior performance compared with the Hessian of ρ(r) for distinguishing the S a -and R a -geometric stereoisomers of lactic acid [34] and therefore will be used in this investigation.
In this investigation, we include the entire bonding environment, including all contributions to the U σ -space cisand trans-characteristics, by considering the C1-C2 BCP T σ (s) of the asymmetric, i.e., 'reference' carbon atom (C1) or the N1-N2 BCP T σ (s) of the nitrogen atom (N1); see Scheme 1 and the Computational Details section. The C1-C2 BCP T σ (s) and N1-N2 BCP T σ (s) are created by subjecting these BCP bond paths to a set of torsions θ; see the Computational Details section.
The bond-twist T σ is the difference in the maximum projections, the dot product of the stress tensor e 1σ eigenvector and the BCP displacement dr, of the T σ (s) values between the counter-clockwise (CCW) and clockwise (CW) torsion θ.
Equation (3) for the bond-twist T σ quantifies the bond torsion BCP-induced bond twist for the CCW vs. CW direction, where the largest magnitude stress tensor eigenvalue (λ 1σ ) is associated with e 1σ ; see Figures 1 and 2. The eigenvector e 1σ corresponds to the direction along which electrons at the BCP are subject to the most compressive forces. Therefore, e 1σ corresponds to the direction along which the BCP electrons will be displaced most readily when the BCP is subjected to a torsion [35]. Higher values of the bond twist T σ correspond to greater asymmetry, and therefore to a dominance of the transcompared with the cis-isomer in U σ space. This reflects the structural symmetry, with respect to the positioning of the halogen substituents, of the transrather than the cis-isomer.  Table 1. Notice the markers at 30° intervals.   -figures (a-c), respectively; see the caption of Figure 1 and Table 2.   Figure 1 and Table 2. Conversely, the eigenvector e 2σ corresponds to the direction along which the electrons at the BCP are subject to the least compressive forces. Therefore, e 2σ corresponds to the direction along which the BCP electrons will be least readily displaced when the BCP undergoes a torsion distortion. The bond-flexing F σ associated with e 2σ is defined as: The bond-flexing F σ is calculated from the torsion BCP bond flexing defined by equation (4); see Figures 1 and 2. Equation (4) provides a U σ -space measure of the 'flexing-strain' or bond bending that a bond path is under in the cisor trans-isomer configurations. This is consistent with greater 'flexing-strain' or bond bending that previously correlated with a greater presence of the cis-effect [18]. Higher values of F σ correspond to the dominance of the ciscompared with the trans-isomer in U σ space, because bond bending reflects the symmetry of the cis-isomer rather than the trans-isomer, with respect to the positioning of the halogen substituents.
The bond-axiality A σ is part of the U σ -space distortion set ∑{Tσ,Fσ,Aσ}, which provides a measure of the chiral asymmetry. It is defined as: 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. We will, however, not use A σ as it does not comprise the bond cross-section, but provide it in the Supplementary Materials S5 and S6. Instead, we will use the so-called U σ -space bond cross-section set ∑{Tσ,Fσ} developed for cisand trans-isomers.
The (+/−) sign of the bond-twist T σ and bond-flexing F σ determines the S σ (T σ > 0, F σ > 0) or R σ (T σ < 0, F σ < 0) character; see Tables 1 and 2. The bond cross-section set ∑{Tσ,Fσ} is related to the cross-section of a BCP bond path that is quantified by the λ 1σ and λ 2σ eigenvalues associated with the e 1σ and e 2σ eigenvectors, respectively. Note, the e 1σ and e 2σ eigenvectors are the directions along which the BCP electrons are displaced most readily and least readily, respectively, when the BCP is subject to a torsional distortion. The trans-isomer is dominant in U σ space if the magnitude of the bond-twist T σ value is larger for the transthan for the cis-isomer. Conversely, dominance of the cis-isomer is determined by the presence of a larger magnitude bondflexing F σ value for the cis-isomer compared with the trans-isomer; see Tables 1 and 2.

Computational Details
The electronic wavefunction for molecular structures incorporating a chemically conventional double bond is usually well-represented by a single-reference wavefunction in the 'eclipsed' configurations 0 • (cis) or 180 • (trans). It is also well-known that as the dihedral angle across the double bond deviates from the 'eclipsed' configurations, the nature of the wavefunction changes, becoming fully multi-reference in nature at the twisted 90 • 'staggered' configuration. The multi-reference character is determined for ethene using the frequently used T1 measure [36], where values of T1 > 0.02 indicate that a singlereference description is inadequate. For this reason, in all of this work, for both the C 2 X 2 (X = H, F, Cl) and N 2 X 2 (X = H, F, Cl) molecules, we use a multi-reference CAS-SCF(2,2) method [37,38], using Slater determinants for the active space, implemented in Gaussian G09.E01 [39] with symmetry disabled, an 'ultrafine' integration grid and convergence criteria of 'VeryTight' geometry convergence and an SCF convergence criterion of 10 −12 . The cc-pVTZ triple-zeta basis set was used during geometry optimization and the dihedral coordinate scan constrained geometry optimization process. The magnitude of the dihedral angle scan steps was 1 • . Additionally, the second atom used in each sequence defining the dihedral scan angle, C1 and N1, respectively, for the ethene and diazene derivatives, was constrained to be fixed at the origin of the Cartesian spatial coordinates. All initial 'eclipsed' (cis-and trans-) optimized molecular geometries were generated (and checked to be energy minima with no imaginary vibrational frequencies) with these settings; see Supplementary Materials S2 for the optimized structures and tabulated experimental data. The final single-point wavefunctions and densities for each structure produced during the dihedral scans were calculated, as recommended for accurate NG-QTAIM properties [40], using a quadruple-zeta basis set (cc-pVQZ).
The direction of torsion is determined to be CCW (0.0 • ≤ θ ≤ +90.0 • ) or CW (−90.0 • ≤ θ ≤ 0.0 • ) from an increase or a decrease in the dihedral angle, respectively. An exception is made for N 2 Cl 2 where the respective angular limits used were −80 • and +80 • . These latter limits are chosen due to the well-known destabilizing interactions between the nitrogen lone pairs and the relatively weak N-Cl bonds [41,42], which cause dissociation of the molecule into N 2 and Cl 2 when a larger twist is applied: we observed and confirmed this dissociation for dihedral twists > 80 • .
QTAIM and stress tensor analysis were then performed on each single-point wavefunction obtained in the previous step with the AIMAll [33] and QuantVec [43] suite. In addition, all molecular graphs were confirmed to be free of non-nuclear attractor critical points.

Results and Discussions
The scalar distance measures geometric bond length (GBL) and bond path length (BPL) used in this investigation on ethene, doubly substituted ethene and diazene are insufficient to quantify the presence of the cis-effect and are provided in the Supplementary Materials S3. The variation in the (scalar) relative energy ∆E for ethene, doubly substituted ethene and diazene molecules do not provide any insights either into the cis-effect for these molecules and are provided in the Supplementary Materials S4. The intermediate and the complete C1-C2 BCP U σ -space distortion sets are provided in the Supplementary Materials S5 and S6, respectively.
The sum of the bond-cross-section sets ∑{Tσ,Fσ} of the C1-C2 BCP T σ (s) was calculated for all four possible isomers of the formally achiral molecules ethene and doubly substituted ethane, C 2 X 2 (X = H, F, Cl). The results for the molecular graph of pure ethene are provided as a control to enable a better understanding of the effect of the halogen atom substitutions; see Table 1 and Scheme 1. The corresponding results for the formally achiral diazenes N 2 X 2 (X = H, F, Cl) comprising a single isomer N1-N2 BCP T σ (s) are presented in Table 2. The magnitude of the values of the bond cross-section set ∑{Tσ,Fσ} increases with atomic weight, as is demonstrated for F 2 and Cl 2 substitution of ethene; see Table 1. This relationship between the magnitude of the ∑{Tσ,Fσ} values and halogen substituent also occurred in a recent investigation of singly and doubly halogen-substituted ethane [44]. This dependency of {T σ ,F σ } on the atomic weight of the substituent, however, does not occur for the diazenes.
The magnitude of the bond-twist T σ is significantly smaller for the ciscompared with the trans-isomer for C 2 X 2 (X = F, Cl) and slightly smaller for N 2 X 2 (X = Cl). The magnitude of the bond-flexing F σ is significantly larger for the ciscompared with the trans-isomer for C 2 X 2 (X = F, Cl) and N 2 X 2 (X = F, Cl).
These results for the bond cross-section set ∑{Tσ,Fσ} are consistent with the presence of the cis-effect and therefore indicate the occurrence of the cis-effect in U σ space for C 2 X 2 (X = F, Cl) and N 2 X 2 (X = F, Cl). The very large component of the bond-twist T σ for N 2 X 2 (X = H) indicates a complete lack of the cis-effect and a dominance of the trans-isomer in U σ space for this molecule.
All of the investigated molecular graphs comprised a significant degree of chiral character as indicated by the magnitudes of the bond-twist T σ and bond-flexing F σ , particularly for C 2 X 2 (X = F, Cl).

Conclusions
In this investigation, NG-QTAIM was used to determine the presence or absence of the cis-effect for the C 2 X 2 (X = H, F, Cl) and N 2 X 2 (X = H, F, Cl) molecules. Qualitative agreement with experimental data for differences in the energies of the cisand trans-isomers was found.
The molecules of this investigation are formally achiral according to the Cahn-Ingold-Prelog (CIP) priority rules [45], but all comprise at least a degree of chiral character in U σ space, on the basis of the magnitude of the T σ values, with C 2 X 2 (X = Cl) displaying a very significant degree of chirality. This finding reflects the conventional understanding that steric effects are among the reasons for the differences between the relative energetic stabilities of cisand trans-isomers, consistent with our previous association of chiral character in Uσ space for the steric effects for ethane [20].
We found that both C 2 X 2 (X = F, Cl) and N 2 X 2 (X = F, Cl) display the cis-effect. This includes the prediction of a cis-effect in N 2 X 2 (X = Cl), for which no experimental data on the cis-isomer and trans-isomer energy difference are available. The cis-effect is determined on the basis of the much larger values of the bond-flexing Fσ for the ciscompared with the trans-isomer.
We provided a physical explanation as to why the cis-effect is the exception rather than the rule, by defining a dominant bond-flexing F σ component of the bond cross-section set {T σ ,F σ } as characterizing the cis-effect. This is on the basis that it is more difficult to bend (F σ ) than to twist (T σ ) the C1-C2 BCP bond path and N1-N2 BCP bond path. This difference in the difficulty of performing bond-bending (F σ ) and bond-twisting (T σ ) distortions is explained by their construction, using the least preferred e 2σ and most preferred e 1σ eigenvectors, respectively, that determine the relative ease of motion of the electronic charge density ρ(r b ).
Suggestions for future work include the exploration of the newly discovered NG-QTAIM bond cross-section set {T σ ,F σ } for cisand trans-isomers, which could be undertaken by manipulating the cisand trans-isomer character in U σ space with laser irradiation. We make this suggestion since NG-QTAIM chirality has already been found to be reversed by the application of an electric field [46]. Reversing the cisand trans-isomer character in U σ space is possible with laser irradiation that is fast enough to avoid disrupting atomic positions. Such a reversal in U σ space could result in the cisand trans-geometric isomers comprising transand cis-isomer assignments in U σ space, respectively.