*2.1. Systems under Study*

In the following enumeration of structures under study, actual reaction paths are not described; there is instead a series of logical connections between related species.

We begin with the tautomer ONE (acetoacetamide) which has a saturated link and two carbonyl groups. Species ONE-A and ONE-B obtained by rotation of the amido and acyl groups revert spontaneously to species ONE as shown in Figure 1, and are not assigned a relative energy.

**Figure 1.** Diketone variant ONE with NH ... O hydrogen bonding. Species ONE-A and ONE-B (which lacks the NH ... O = H bond) proceed spontaneously upon optimization to ONE, which lies about 4.0 kJ/mol above the most stable isomer TWO.

All other systems TWO to ELEVEN (shown in Figures 2–5) have an unsaturated link and a hydroxyl group at either the methyl-substituted C (CH3-C-OH) or at the aminosubstituted C group (NH2-C-OH). The most stable of all species, TWO, has an evident OH ... O = hydrogen bond. Rotating the CH3-C-O-H dihedral forms THREE, which lacks that hydrogen bond (Figure 2). Simple torsion of THREE around the CC-amide bond produces an unstable form (unlabeled) which upon optimization can establish either a NH donor -OH acceptor H bond (FOUR) or an N acceptor OH-donor H bond (FIVE).

**Figure 2.** Enolic form THREE can assume forms FOUR with NH...O and FIVE with OH...N H-bonding.

**Figure 3.** SIX, which may exhibit CH...O= hydrogen bonding, can form SEVEN by a rotation of the amide group. SEVEN may contain CH...N hydrogen bonding.

**Figure 4.** A high energy isomer EIGHT with no hydrogen bonding can achieve OH ... O = hydrogen bonding; the resultant species rearranges spontaneously to stable TWO. By rotation of the acyl fragment EIGHT can produce NINE which may have CH ... OH hydrogen bonding. Swapping OH and NH2 produces TEN which may contain a CH . . . NH2 interaction.

**Figure 5.** Swapping amino and hydroxyl groups in EIGHT produces a form (unlabeled) which spontaneously forms ELEVEN which has NH . . . O= hydrogen bonding.

Beginning with THREE, exchanging methyl and hydroxyl produces SIX. A subsequent rotation of the amide produces SEVEN (Figure 3).

We turn to structures with an unsaturated link and one carbonyl group at the methylsubstituted C. The system EIGHT which lacks H bonding occupies a high-energy relative

minimum (Figure 4). If the OH group is positioned to donate a H bond to the carbonyl oxygen, the system spontaneously rearranged to species TWO. Rotating the acyl group produces species NINE. Exchange of amino and hydroxyl groups in NINE produces TEN. NINE and TEN may be stabilized by CH . . . OH and CH . . . N interactions.

On the other hand, exchange of amino and hydroxyl groups in EIGHT produces a species lacking H bonding (unlabeled). Upon optimization it rearranges spontaneously to ELEVEN which is stabilized by > NH . . . O= H bonding (Figure 5).

#### *2.2. Extrapolation of Accurate Relative Energies*

The numerical values (kJ/mol) in Figures 1–5 are the result of the extrapolations described in the methods section. Estimates of electronic energies from the G4 thermochemical scheme agree within 1–2 kJ/mol with our extrapolated values. Detailed tables of energies obtained with RHF, MP2, and CCSD Hamiltonians in Dunning basis sets cc-pVNZ with N = 2, 3, and 4 are provided in Supplemental Information. A broad overview is set forth in Table 1.



There is a rough clustering of H bonding types. Species ONE and TWO with NH ... O= and OH ... O= hydrogen bonding are most stable, while systems THREE and EIGHT with little or no H bonding are relatively unstable. However systems with presumably much weaker CH ... O or CH ... N interactions (SIX, SEVEN, and TEN) are not entirely separated from systems incorporating XH ... Y (X, Y = O and N) which are generally thought to be stronger.

It appears that the relative stability, a global property for each of the molecules in question, is not simply explained as a consequence only of differences in local H bonding. Our task, to describe the hydrogen bonds, needs more local analysis.

#### *2.3. Atoms in Molecules (AIM) Characterization of Interactions*

The Quantum Theory of Atoms-in-Molecules, which defines local properties of the charge distribution at significant points, has been widely employed to characterize bonding of many kinds. (See further discussion in the Methods and Software section below.) Table 2 collects the electron density and its Laplacian at bond critical points, with the associated kinetic energy density G and the potential energy V. The total charge *Q(H)* in the basin containing H atom and the delocalization index *δ(H, B)* are shown as well. The delocalization index is a measure of the number of electrons shared between two basins, and is related to the extent of covalent bonding and, indirectly, to bond strength. Kraka and co-workers [34] define an empirical bond order *n* derived from the density at a BCP.

$$m = 0.54 \,\,\rho^{0.32} \tag{1}$$


**Table 2.** Topological data for critical points associated with non-covalent interaction in species ONE–ELEVEN.

*ρ* = density; <sup>∇</sup><sup>2</sup>*ρ* = Laplacian of density; **G** = Kinetic energy density; **V** = potential energy density; **Q(H)** = integral of charge in H atom basin. Order = Kraka definition of bond order and DI = delocalization index for H atom and its partner in the species' putative hydrogen bond. Analysis by AIMALL of density computed with ωB97XD/cc-pVTZ.

The positive values of the Laplacian (Table 2) indicate that the interaction is between closed shells. The values of bond order indicate that the strongest H bond is the link OH ... O of TWO at 0.221, while the bond orders descend from 0.191 to 0.180 to 0.164 to 0.158 for ELEVEN, FIVE, ONE, and FOUR. These all have O to N hydrogen bonds. CH ... X interactions are found in SIX (0.155), TEN (0.142), SEVEN (0.138), and NINE (0.129). OH seems to be a weaker H-bond acceptor than carbonyl =O. We do not assign a bond order to THREE and EIGHT, despite the presence of appreciable density at BCPs between O atoms. See further discussion on THREE and EIGHT below and in the section on non-covalent interactions.

An energy density diagnostic adopted by Cremer and Kraka [34] identifies interactions as mainly electrostatic (if H = G + V > 0) or mainly covalent (if H < 0). [35–40] By this criterion the only covalent interactions are for TWO (-OH ... O=), FIVE (-OH ... NH) and ELEVEN (=O ... HN). These also have the shortest X ... Y distances (2.538, 2.631, and 2.617 Å). ONE, which has an =O ... HN- interaction has crossed over to be predominantly electrostatic. This may be attributed to its greater O ... N distance, 2.789 Å, since the electrostatic interaction is of longer range than the covalent interaction which depends on orbital overlap.

H > O for the systems with no plausible H bonding (THREE and EIGHT) and those for which CH ... X hydrogen bonding is conceivable (SIX, SEVEN, NINE, and TEN). It is notable that ONE and FOUR with =O ... HN and HO ... HN interactions are to be considered electrostatic according to the H diagnostic.

Several images representative of interactions as characterized by AIM appear in Figure 6 The complete set is to be found in Supplemental Information.

The diagrams display the "bond paths" (solid and dashed lines) and the bond (or line) critical points (green) for each species, and the ring critical points as well (red). The paths and BCPs close a ring; the location of the ring critical point is related to the strength of the ring closure. In the weaker ring closing interactions, the RCP approaches the BCP, as in THREE and EIGHT. For CH ... X interactions, the RCP is further removed from the BCP, and for stronger hydrogen bonds the separation is even greater.

**Figure 6.** Atoms in Molecules analysis of ωB97XD/cc-pVTZ density. Ring CPs in red, BCPs in green. Laplacian values at BCPs are shown. (**a**) Species ONE, with NH ... O= hydrogen bonding; (**b**) Species TWO, with OH ... O= hydrogen bonding; (**c**) Species THREE with repulsive noncovalent interaction; (**d**) Species FOUR, with NH . . . OH.

Our emphasis has been on the BCPs associated with hydrogen bonding and the properties of density at those points. What are we to make of THREE and EIGHT, with O ... O paths and a substantial positive Laplacian considerable density at the (path) critical point? In common with the other systems, the Laplacians show that the interaction depletes density at the CP, as is characteristic of interactions between closed shells. To deal with such cases we turn to the reduced density gradient, which diagnoses non-covalent interactions.

#### *2.4. Non-Covalent Interaction (NCI) Characterization of Interactions*

Isosurfaces for the reduced density gradient characterizing the noncovalent interactions in all species are shown in Figure 7. The color coding identifies the O ... O interaction in THREE and EIGHT (entirely green) as repulsive, despite the substantial density at the BCPs. In cases TWO and ELEVEN the portion of the NCI isosurface enclosing the BCP is blue (attractive), and the portion enclosing the RCP is green. The NCI enclosing surfaces for weakly interacting CH ... O and CH ... N systems are of more subtly varying hue. All these qualitative observations comport with our understanding of the relative strength of the hydrogen bonding.

**Figure 7.** Panels display the isosurfaces for the noncovalent interactions in species TWO through ELEVEN. Species ONE is discussed in detail in the Methods and Software section below. Color coding identify regions of repulsive interaction as green and the attractive regions as blue.

#### *2.5. Expression of Hydrogen Bonding in MP2/cc-pVTZ Computed Harmonic Vibrations*

Hydrogen bonding is often expressed in the vibrational spectrum. Here we discuss the canonical frequencies which correspond to normal modes. These are in principle

delocalized combinations of local modes, but in some cases the local modes are well isolated. These include OH, NH2, and CH3 group modes. See Table 3.

**Table 3.** anonical harmonic frequencies (in cm<sup>−</sup>1) for normal modes of all systems, computed by MP2/cc-pVTZ.


1 Predominantly C=O. 2 Predominantly C=C. NH2 stretches are either antisymmetric (a) or symmetric (s) combinations of local NH motions 4 CH stretches descend in symmetry from the C3v limit; the A1 mode can be recognized as "breathing" breathing (b) and components of the E pair are the symmetric (s) and antisymmetric (a) modes. 5 The C-C-C bend serves as a surrogate for the hydrogen bond stretch.

**OH Mode:** The reference OH stretches–uninfluenced by H bonding–fall in the set (THREE, SEVEN, and ELEVEN) and have values 3836, 3850, and 3842 respectively. EIGHT and TEN have OH modes at 3807, which may be associated with coupling of OH at a carbon also bearing NH2, both being uninvolved in intramolecular hydrogen bonding. The most drastically shifted OH stretches are for TWO to 3087 and FIVE to 3571, suggesting that the OH ... O= interaction in TWO is stronger than the OH ... NH2 interaction in FIVE, and that OH is not strongly engaged in hydrogen bonding in any other system.

**NH2 Modes:** NH stretching modes for most systems cluster in the range 3720–3780 for the asymmetric combination and 3600–3630 for the symmetric combination. High values for the differences Δν between asymmetric and symmetric stretching frequencies correspond to NH participation in H bonding for (especially) system ELEVEN ( Δν = 308) and to a lesser extent for system ONE ( Δν = 159). ONE and ELEVEN have the H-bonding structure (NH . . . O=) in common, but the larger shift in ELEVEN is easily attributed to the shorter N ... O distance in ELEVEN (2.617 Å) than is found for ONE (2.789 Å). The next largest difference is for FOUR ( Δν=151), which has an NH . . . OH interaction.

The NH stretching modes in EIGHT ( Δν = 120) and NINE ( Δν = 119) have minimal differences. In both cases the NH2 group is isolated from H bonding. The remaining systems have splitting ranging from 132 to 151. TWO ( Δν = 147) and SIX ( Δν = 144) have comparable splitting; for each, NH2 is attached to a carbonyl carbon. FIVE (132), SEVEN (135), and TEN (136) fall in a narrow range. In each of these, N is a hydrogen bond acceptor. THREE (with Δν = 138) is unique.

**CH stretches in the methyl group:** Methyl CH stretches in a C3v environment include the A1 all-in-phase "breathing" mode and an E set of out-of-phase motions. In this low symmetry setting we can identify motions corresponding to those in high symmetry. The mode analogous to the A1 breathing is lowest in frequency, ranging from 3060 to 3090 for species ONE through ELEVEN. The former E stretching combinations, which split into in-phase and out-of-phase modes in the low symmetry environment, appear in ranges 3130–3180 and 3060–3090 respectively. There seems to be no pattern in these values indicating whether the methyl group can participate in CH . . . X interaction.

**CCC bend:** For THREE and EIGHT, for which no H bonding is recognizable and a repulsive NCI zone lies between the C=O and OH oxygens, the C-C=C bending modes have the lowest a frequency, 178. For other systems the C-C=C bend is our rough surrogate for hydrogen bond stretching. The highest bending mode frequencies are found for ELEVEN (242), SEVEN (247), TWO (246) and ONE (242). These have NH ... O=, CH ... NH2, OH ... O=, and weakened NH ... O= interactions respectively. Mixing with the methyl internal rotation sometimes makes the isolation of the CCC bend difficult; strong coupling is evident for SEVEN and other species with CH . . . X interaction.

Coupling of local modes within canonical normal modes complicates the interpretation of hydrogen bonding by inspection of vibrations. For example, the C=O and C=C stretching modes are strongly coupled in FOUR, but are weakly coupled in NINE and TEN. Furthermore, the COH bend is often strongly coupled to the CO stretch. as well. We expect that extraction of local modes from the canonical normal modes will simplify the discussion of H bonding and vibrations. This is accomplished by the Local Mode Analysis [41,42].
