Halogen and Hydrogen Bonding in Halogenabenzene/NH3 Complexes Compared Using Next-Generation QTAIM

Next-generation quantum theory of atoms in molecules (QTAIM) was used to investigate the competition between hydrogen bonding and halogen bonding for the recently proposed (Y = Br, I, At)/halogenabenzene/NH3 complex. Differences between using the SR-ZORA Hamiltonian and effective core potentials (ECPs) to account for relativistic effects with increased atomic mass demonstrated that next-generation QTAIM is a much more responsive tool than conventional QTAIM. Subtle details of the competition between halogen bonding and hydrogen bonding were observed, indicating a mixed chemical character shown in the 3-D paths constructed from the bond-path framework set B. In addition, the use of SR-ZORA reduced or entirely removed spurious features of B on the site of the halogen atoms.

1. Supplementary Materials S1. A discussion of the construction of the bond-path framework set B The reasons for the choice of the ellipticity ε as scaling factor. This was motivated by the fact that the scaled vector tip paths drop smoothly onto the bond-path, ensuring that the tip paths are always continuous. We previously discussed in detail the unsuitability of alternative scaling factors that included |λ1 -λ2| but this and other choices were not used because they lack the universal chemical interpretation of the ellipticity ε e.g. double-bond ε > 0.25 vs. single bond character ε ≈ 0.10. Also unsuitable choices for scaling factors, on the basis of not attaining zero, included either ratios involving the λ1 and λ2 eigenvalue or any inclusion of the λ3 eigenvalue. The λ3 eigenvalue was also found to unsuitable because it contains no information about the least (e1) and most (e2) preferred directions of the total charge density ρ(r) accumulation.
Discussion on the uniqueness of the H * and H. Because the scaling factor, εi is identical in equation (3a) and equation (3b) and H * and H are defined by the distances swept out by the e2 tip path points pi = ri + εie1,i and qi = ri + εie2,i respectively then H * = H will result in a linear bond-path r.
The bond-path framework set B = {p,q,r} should consider the bond-path to comprise the unique p-, q-and r-paths, swept out by the e1, e2 and e3, eigenvectors that form the eigenvector-following paths with lengths H * , H and BPL respectively. The p-and q-paths are unique even when the lengths of H * and H are the same or very similar because the p-and q-paths traverse different regions of space. Bond-paths r with non-zero bond-path curvature are more likely to occur for the equilibrium geometries of closed-shell BCPs than for shared-shell BCPs and will result in H * and H with different values. This is because the p-and q-paths will be different because of the greater distance travelled around the outside of a twisted bond-path r compared with the inside of the same twisted bond-path r. This is because within QTAIM the e1, e2 and e3, eigenvectors can only be defined to within to a factor of -1, i.e. (e1,-e1), (e2,-e2) and (e3,-e3) therefore there will be two possible tip-paths. The consequences of this (within QTAIM) calculation of the H * is that we dynamically update the sign convention to define H * as being the shorter of the two possible tip-paths because e1 is the least preferred direction of accumulation of ρ(r). A similar procedure is used for H except that we chose the longer of the two possible tip-paths because e2 is the most preferred direction of accumulation of ρ(r).
It should be noted that the direction of the p-and q-paths always remain orthogonal to each other since they are constructed from the e1 and e2 eigenvectors respectively. The ellipticity ε is used as a scaling factor in the construction of the p-and q-paths: The lengths of the p-and q-paths are defined as the eigenvector-following paths H * or H: Similar expressions to equations (S1a-S1b) and equations (S1c-S1d) can be constructed using the stress tensor ellipticity εσ = |λ2σ|/|λ1σ| -1; note the different numerator and denominator orderings compared with the ellipticity ε. In the limit of vanishing ellipticity ε = 0, for all steps i along the bond-path then H = BPL. The form of the constituent p-and q-paths along each bondpath (r) can be used to provide a 3-D interpretation of bonding to track precisely the mechanisms of bond evolution throughout the functioning of the switch i.e. the hydrogen transfer reaction.
Two paths (q and q') are associated with the e2 eigenvector because e2 = -e2 lie in the same plane for the same point on the bond-path (r), correspondingly there are two paths (p and p') associated with the e1 eigenvector, see Scheme S1(a). The q (equivalently qσ) is always defined to be longer than the q' (equivalently qσ') because it is constructed from the preferred direction (the e2 eigenvector). Conversely, the p (equivalently pσ) is always defined to be the shorter of the two paths associated with the e1 eigenvector. For very curved bond-paths, however, p may be shorter than r (the bond-path length), so we only chose p'; see Scheme S1(a).
Scheme S1(a). A sketch, not to scale, of the {p,p'} path-packets illustrating that for the highly curved bondpath (r) the p-path may be shorter than r-path.
Implementation details of the calculation of the eigenvector-following path lengths H and H * .
When the QTAIM eigenvectors of the Hessian of the charge density ρ(r) are evaluated at points along the bond-path, this is done by requesting them via a spawned process which runs the selected underlying QTAIM code, which then passes the results back to the analysis code. For some datasets, it occurs that, as this evaluation considers one point after another in sequence along the bond-path, the returned calculated e2 (correspondingly e1 is used to obtain H * ) eigenvectors can experience a 180-degree 'flip' at the 'current' bond-path point compared with those evaluated at both the 'previous' and 'next' bond-path points in the sequence. These 'flipped' e2 (or e1) eigenvectors, caused by the underlying details of the numerical implementation in the code that computed them, are perfectly valid, as these are defined to within a scale factor of -1 (i.e. inversion). The analysis code used in this work detects and re-inverts such temporary 'flips' in the e2 (or e1) eigenvectors to maintain consistency with the calculated e2 (or e1) eigenvectors at neighboring bond-path points, in the evaluation of eigenvector-following path lengths H and H * , see Scheme S1(b).
(a) (b) Scheme S1(b). The pale-blue line in sub-figure (a) represents the path, referred to as the eigenvectorfollowing path with length H * , swept out by the tips of the scaled e1 eigenvectors, shown in magenta, and defined by equation (1c). The red path in sub-figure (b) corresponds to H, constructed from the path swept out by the tips of the scaled e2 eigenvectors, shown in mid-blue and is defined by equation (1d). The paleblue and mid-blue arrows representing the e1 and e2 eigenvectors are scaled by the ellipticity ε respectively, where the vertical scales are exaggerated for visualization purposes. The green sphere indicates the position of a given BCP.

Supplementary Materials S2.
where the term in equation (2) is the Pauli matrix operator, is the momentum operator, is the speed of light, cis the speed of light, is the mass, V is the potential operator, = (-1) 1/2 , is the reduced Plank's constant, is the wave-function, and is time. The first term on the lefthand side of equation (2)  . (3) The first operator on the right-hand-side in equation (3) is the scalar relativistic part and the second operator on the right-hand-side is the spin-orbit operator part, which is often neglected.
This gives rise to a popular method for including relativity into electronic structure calculations, which is known as the scalar relativistic ZORA (SR-ZORA) equation: .