Unifying Conceptual Density Functional and Valence Bond Theory: The Hardness–Softness Conundrum Associated with Protonation Reactions and Uncovering Complementary Reactivity Modes

In this study, we address the long-standing issue—arising prominently from conceptual density functional theory (CDFT)—of the relative importance of electrostatic, i.e., “hard–hard”, versus spin-pairing, i.e., “soft–soft”, interactions in determining regiochemical preferences. We do so from a valence bond (VB) perspective and demonstrate that VB theory readily enables a clear-cut resolution of both of these contributions to the bond formation/breaking process. Our calculations indicate that appropriate local reactivity descriptors can be used to gauge the magnitude of both interactions individually, e.g., Fukui functions or HOMO/LUMO orbitals for the spin-pairing/(frontier) orbital interactions and molecular electrostatic potentials (and/or partial charges) for the electrostatic interactions. In contrast to previous reports, we find that protonation reactions cannot generally be classified as either charge- or frontier orbital-controlled; instead, our results indicate that these two bonding contributions generally interplay in more subtle patterns, only giving the impression of a clear-cut dichotomy. Finally, we demonstrate that important covalent, i.e., spin pairing, reactivity modes can be missed when only a single spin-pairing/orbital interaction descriptor is considered. This study constitutes an important step in the unification of CDFT and VB theory.


S1. Comparison of the BOVB results with the DFVB results for a selected subset of considered systems
From Table S1 it is clear that the bonding situations are equally well described by BOVB and DFVB level-of-theory; the only quantity that differs significantly between both methods is the bonding energy. The latter finding is in line with expectations, since DFVB generally does a better job at recovering the full bonding energy than BOVB (see also Section S2; cf. Ref. 46 in the main text). Table S1. The weights of the HL and the main ionic structures (wHL and wion,1), the resonance energy (RE) and the spacing between the HL and main ionic state (DEHL-ion,1) at the optimal bonding distance and the adiabatic bonding energy obtained for [ H3N-H]      a The geometry of dissociated H3NOH was selected to evaluate the energetics in the asymptotic limit b The geometry of dissociated N-protonated pyridine was selected to evaluate the energetics in the asymptotic limit   values. This finding can also be connected to our model through realization that ! acts as a probe for the spacing between the HL and ionic structure in our model, i.e.
IH -IR (cf. Eq. 2 in the main text): since ! is a rather small quantity in most (saturated) organic compounds (relative to ! ) and IH is by definition a constant for all protonation processes, the magnitude of ! effectively acts as an indicator of the magnitude of IR. Hence, this quantity tells us something about the magnitude of the S8 spin-pairing interaction when a single elemental type is considered for the association site within a set of structurally related acids. This realization is further corroborated by the finding in other studies that the pKa correlates with EHOMO,R (another quantity connected to ! ), cf. Ref. 65.

Fukui function
Throughout the main text, we focus on the VB-inspired analogue of the Fukui function, i.e. the spin density of the (positively) charged compound, as a descriptor of the spin-pairing/orbital interaction. In the literature, this descriptor is often referred to as a so-called "Parr function".

S6. Spin densities of the ground-state and first excited state for [HSHNSH] + • in its optimal
and twisted geometry Figure S4. The spin density associated with the ground-state and first excited state of [HSHNSH] +• in its optimal geometry (DEexcitation = 11.7 kcal/mol) and the geometry in which the dihedral angle H-S-N-H has been twisted to maximize the overlap between the lone pairs (DEexcitation = 22.5 kcal/mol).

S8. Utility to scan for a minimum electrostatic potential value
As indicated in the main text, a small utility was written to scan for the minimum value of the electrostatic potential along a specific axis away from the atom center for pyridine. The utility takes a regular Gaussian .cube file as input. Because the pyridine molecule in our calculation was aligned with the xy-plane, the electrostatic potential for the C-sites could simply be scanned in the z-direction; the electrostatic potential for the in-plane lone pair associated with the N-site was scanned in the y-direction.
For the ortho-site, the minimum of the electrostatic potential (reached at a distance of 2.3 Å from the atomic center) amounted to -6.7 kcal/mol. For the meta-site, this quantity amounted to -5.1 kcal/mol and for the para-site to -3.9 kcal/mol (reached at 2.2 Å and 2.1 Å respectively). For the N-site, the minimum, amounting to -59.0 kcal/mol, was reached at a distance of 1.4 Å away from the atom center.

S9. Geometries and Energies
Units: • Coordinates are expressed in Å