Focal Point Evaluation of Energies for Tautomers and Isomers for 3-hydroxy-2-butenamide: Evaluation of Competing Internal Hydrogen Bonds of Types -OH…O=, -OH…N, -NH…O=, and CH…X (X=O and N)

The title compound is a small molecule with many structural variations; it can illustrate a variety of internal hydrogen bonds, among other noncovalent interactions. Here we examine structures displaying hydrogen bonding between carbonyl oxygen and hydroxyl H; between carbonyl oxygen and amino H; hydroxyl H and amino N; hydroxyl O and amino H. We also consider H-bonding in its tautomer 2-oxopropanamide. By extrapolation algorithms applied to Hartree-Fock and correlation energies as estimated in HF, MP2, and CCSD calculations using the cc-pVNZ correlation-consistent basis sets (N = 2, 3, and 4) we obtain reliable relative energies of the isomeric forms. Assuming that such energy differences may be attributed to the presence of the various types of hydrogen bonding, we attempt to infer relative strengths of types of H-bonding. The Atoms in Molecules theory of Bader and the Local Vibrational Modes analysis of Cremer and Kraka are applied to this task. Hydrogen bonds are ranked by relative strength as measured by local stretching force constants, with the stronger =O…HO- > NH…O= > -OH…N well separated from a cluster > NH…O= ≈ >NH…OH ≈ CH…O= of comparable and intermediate strength. Weaker but still significant interactions are of type CH…N which is stronger than CH…OH.


Introduction
The concept of hydrogen bonding has evolved considerably over its century of history [1,2]. The anomalous properties of water prompted the first suspicion of an attractive interaction between a hydrogen atom in one water molecule with the oxygen of another (HOH . . . OH 2 ). Similar phenomena involving N and F were soon recognized as well. H bonding has been recognized as a primarily electrostatic phenomenon expressed by its effect on vibrational frequencies (most often red-shifting the OH stretch), molecular structure (short X . . . HY distances, where X and Y are electronegative atoms F, O, N etc.), and characteristic NMR parameters (owing to charge shifts) to mention the most prominent. The concept of H bonding has been expanded from its original context so to include a number of surprising interactions, involving atoms other than oxygen, fluorine, and nitrogen [3,4]; "strong" H-bonding [5]; and "anomalous" (blue-shifting) H bonding [6]. Further expansion of the concept also has been recognized [7].
The computational modeling of hydrogen bonding was reviewed in 1997 [8] and 1999 [9], in 2006 [10], and in 2009 [11]. The definition of hydrogen bonding has been formalized by IUPAC [12]. IUPAC's criteria for recognition of H bonding in a structure X-H . . . YZ include: origins of bonding (largely electrostatic, but also including contributions from charge transfer and dispersion); polarization of the XH bond; geometry of the structure

Results
In the following sections we define the systems under study, describe the extrapolation techniques by which we obtain accurate relative energies, and lay out results of AIM analy-sis of their charge distributions. Complementary information on non-covalent interactions is provided by graphical representation of the reduced density gradient. We interpret the vibrational spectra for all species, and recover the force constants for significant local modes by the LMODEA algorithms. Those force constants are found to be useful indices of relative intramolecular bond strength. All these analyses are described in detail in the section on methods and software.

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.

Results
In the following sections we define the systems under study, describe the extrapolation techniques by which we obtain accurate relative energies, and lay out results of AIM analysis of their charge distributions. Complementary information on non-covalent interactions is provided by graphical representation of the reduced density gradient. We interpret the vibrational spectra for all species, and recover the force constants for significant local modes by the LMODEA algorithms. Those force constants are found to be useful indices of relative intramolecular bond strength. All these analyses are described in detail in the section on methods and software.

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. All other systems TWO to ELEVEN (shown in  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). All other systems TWO to ELEVEN (shown in  have an unsaturated link and a hydroxyl group at either the methyl-substituted C (CH 3 -C-OH) or at the aminosubstituted C group (NH 2 -C-OH). The most stable of all species, TWO, has an evident OH . . . O = hydrogen bond. Rotating the CH 3 -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)  Beginning with THREE, exchanging methyl and hydroxyl produces SIX. A subsequent rotation of the amide produces SEVEN (Figure 3).  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).

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  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).

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= 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  (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).

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. 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.

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. n = 0.54 ρ 0.32 (1)  The positive values of the Laplacian ( . . 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.
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. ure 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. 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. 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.

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.

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, NH 2 , and CH 3 group modes. See Table 3.  None  TWO  3198s  3164a  3084b  246  1356  THREE  3202s  3155a  3074b  178  1385  FOUR  3198a  3143s  3063b  226  1347  FIVE  3203s  3162a  3084b  226  1317  SIX  3213s  3135a  3062b  226  1348, 1267  SEVEN  3213s  3132a  3063b  193  1351, 1273  EIGHT  3202s  3155a  3074b  178  1467  NINE  3205s  3175a  3089b  207  1435  TEN  3214a  3132s  3063b  194  1273  ELEVEN  3204s  3161a  3079b  242  1367 1 Predominantly C=O. 2 Predominantly C=C. NH 2 stretches are either antisymmetric (a) or symmetric (s) combinations of local NH motions 4 CH stretches descend in symmetry from the C 3v limit; the A 1 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 NH 2 , 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 . . . NH 2 interaction in FIVE, and that OH is not strongly engaged in hydrogen bonding in any other system. NH 2 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 ∆ν between asymmetric and symmetric stretching frequencies correspond to NH participation in H bonding for (especially) system ELEVEN (∆ν = 308) and to a lesser extent for system ONE (∆ν = 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 (∆ν=151), which has an NH . . . OH interaction.
The NH stretching modes in EIGHT (∆ν = 120) and NINE (∆ν = 119) have minimal differences. In both cases the NH 2 group is isolated from H bonding. The remaining systems have splitting ranging from 132 to 151. TWO (∆ν = 147) and SIX (∆ν = 144) have comparable splitting; for each, NH 2 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 ∆ν = 138) is unique.
CH stretches in the methyl group: Methyl CH stretches in a C 3v environment include the A 1 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 A 1 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 . . . NH 2 , 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].

Local Mode Analysis
The Local Mode Analysis allowed identification of force constants for several diatomic stretching motions for all species ONE through ELEVEN. These included the two C=O stretches for ONE (Table 4) and the single C=O stretches for TWO through ELEVEN (Table 5). CO and OH stretches were also defined for TWO through ELEVEN. The three methyl CH stretches and the two NH stretches were chosen for all species. For systems which can plausibly be assigned hydrogen bonding structures, the X . . . HY stretches were included in the analysis. Species TWO, THREE, and SIX have both local NH force constants above 7.2 and both local frequencies above 3600. These systems do not involve the NH 2 group in H bonding. ONE and ELEVEN engage one NH bond in hydrogen bonding which is reflected in one low NH stretching frequency (3564 and 3212). FOUR is a puzzling exception, with nearidentical NH force contants. TEN shows both NH force constants of 6.8 and frequencies near 3500, and FIVE has its two modes with force constants near 7.1 and frequencies near 3579. EIGHT and NINE have one local mode with low frequency (below 3600) with the other mode higher than 3600. These values are consistent with an interaction between substituents OH and NH 2 on an unsaturated carbon.
Yannacone et al. [18] established an empirical power law relation between density at a bond critical point and local force constants, of the form We fit the computed local force constants for TWO and ELEVEN to this form, finding A = 0.740 and B = 0.934, and inferred the remaining values reported in Table 4

Discussion
We have described the intramolecular interactions of conformations of acetoacetamide and isomers of its tautomer 2-oxopropanolamine by accurate ab intio calculations of relative energies and analysis of their density by AIM and NCI theories. Recasting the vibrational modes by the Cremer-Kraka local modes analysis allow reliable assignments of relative strengths of hydrogen bonds XH . . . Y with (X, Y) including (O, O), (O, N), (N, C), and (O, C) ( Figure 8). Rank ordering of bond strength by measures bond order n, electron density ρ and potential energy density V, and also local force constants k produce an agreed sequence of bond strengths. In specific, ordering according to a bond energy BE estimate based on the potential energy density at the bond critical point agrees with ranking based on local force constants.
quence of H bond strength begins with the strongest interactions, found in TWO > ELEVEN > FIVE which have kLOCAL above 0.200. The central cluster includes ONE, FOUR, and SIX with NH…O=, >NH…OH, and CH…O= interactions. The weakest interactions (in descending order) are CH…N, CH…N, and CH…OH. It is interesting to see that the strongest CH…X interaction (with X a carbonyl oxygen) is comparable to NH…O interactions.

Discussion
We have described the intramolecular interactions of conformations of acetoacetamide and isomers of its tautomer 2-oxopropanolamine by accurate ab intio calculations of relative energies and analysis of their density by AIM and NCI theories. Recasting the vibrational modes by the Cremer-Kraka local modes analysis allow reliable assignments of relative strengths of hydrogen bonds XH…Y with (X, Y) including (O, O), (O, N), (N, C), and (O, C) (Figure 8). Rank ordering of bond strength by measures bond order n, electron density  and potential energy density V, and also local force constants k produce an agreed sequence of bond strengths. In specific, ordering according to a bond energy BE estimate based on the potential energy density at the bond critical point agrees with ranking based on local force constants. All measures agree that the H bond strength decreases in magnitude in the order shown in Figure 9, which shows the values of the local force constants for H bonds. All measures agree that the H bond strength decreases in magnitude in the order shown in Figure 9, which shows the values of the local force constants for H bonds. We expect that the analysis can be extended to many molecules of biological importance and to the description of other kinds of noncovalent interactions. We expect that the analysis can be extended to many molecules of biological importance and to the description of other kinds of noncovalent interactions.

Materials and Methods
We employed extrapolation techniques to arrive at accurate relative energies. The Atoms in Molecules analysis of the computed density, and its extension to the reduced gradient description of non-covalent interactions allowed one perspective on the relative strength of our system's hydrogen bonds. An alternative view is provided by analysis of the vibrational spectrum and its canonical (normal) modes. Recovery of local vibrational modes and their associated frequencies allows more direct discussion properties of the hydrogen bonds.

Extrapolation Techniques
A well-established method for obtaining highly accurate energies (including energy differences among conformational isomers) is the extrapolation of energies obtained in a sequence of more and more flexible basis sets [35]. Such techniques depend on the behavior of energy values obtained in a sequence of basis sets embracing a series of values of the angular momentum. The Dunning basis sets of general form cc-pVLZ, with maximum angular momentum L = 2, 3, 4, . . . are most often used [36,37]. The Hartree-Fock energy is known to scale exponentially, while the correlation energy follows a L −3 inverse-cubic scaling for sufficiently large L [38]. Simple schemes based on this behavior are often usefully accurate. We employ the formulas shown below.
Here L denotes the maximum angular momentum in the extrapolation, so if we use cc-pVQZ as the largest basis, E HF (L) and E CORR (L) refer to the energies obtained with that basis, L − 1 to cc-pVTZ, and L − 2 to cc-pVDZ.
A very recent refinement in the extrapolation of the correlation energy has been reported by Lesiuk and Jeziorski [39] which employs E CORR for L and L − 1. Defining the constant a by a = L 4 (E CORR (L) − E CORR (L − 1)) The limit of the correlation energy is We found that the two-point Zeta-extrapolation lowered the final estimate of total electronic energy by 15 millihartrees. However, the Zeta-extrapolation had only a very small impact (<0.2 kJ/mol.) on relative energies obtained with the simpler scheme.
We do not employ any correction intended to overcome basis set superposition error, since it appears that the complete basis set limit of energies is only very slightly altered by its inclusion [35]. The fact that we are studying intramolecular effects may further discount its significance.
Extrapolation is also an important part of well-established thermochemical schemes, which include the CBS series [33,40], the Gn series [41], and more demanding schemes such as Wn [42,43] and HEAT [44]. These methods include empirical corrections lacking in our calculations. To complement our extrapolations we performed G4 calculations, finding that the G4 estimates of relative electronic energies of species in question were in disagreement with our values by no more than 1-2 kJ/mol.

Atoms in Molecules
The atoms in Molecules (AIM) theory of Bader [13,27,45] is constituted of a description of the molecular charge density ρ, its gradient and Hessian, and related quantities including kinetic and potential energy densities. Extreme values of the density demark significant regions in the molecular charge distribution. Points with ∇ρ = 0 may be described as quantum atoms, bond critical points (BCPs), ring critical points (RCPs) and cage critical points (CCPs) depending on the sign structure of the set of eigenvalues of the Hessian matrix of the density at those points. A locus of points with zero density gradient connecting atom centers through a BCP is termed a bond path. These paths very often correspond to intuitive ideas of chemical bonds. However serious complications are often encountered in the chemical interpretation of bond paths and critical points for weak interactions, as mentioned already [22][23][24][25][26][27][28].
The AIM analysis was applied to H bonding from the beginning. Koch and Popelier [46] developed criteria for judging H-bonding in AIM context. They include: 1.
Topology: There should appear a bond path between atoms considered to be linked by a H bond. This path should contain a BCP.

2.
Density: At this point the density should have a "reasonable" value, about an order of magnitude smaller than the values for covalent bonds or ca 0.01 atomic units, with a range up to 0.035 atomic units. There is a correlation between the BCP density and intermolecular attraction.

3.
Laplacian: ∇ 2 ρ must be positive, with values in an approximate range 0.02 to 0.10 atomic units. This sign suggests a depletion of charge at the BCP, and is characteristic of an interaction between closed shells [13,21]. 4.
H atom charge: q(H) involved in a hydrogen bond XH . . . Y should be substantially smaller than q(H) for H atoms not participating in a H bond but with the same X atom in XH. Here q(H) is the charge in the basin containing the nuclear attractor for H atom.
Further criteria include Atomic interpenetration; Destabilization of the H atom; Diminished atomic polarizability for the H atom; and Decreased volume of the H atom. In this work we confine our attention to the first four criteria, as has been the practice in most studies of AIM-based description of intramolecular H bonding in conformational isomers.
The delocalization index δ(A, B), in contrast to many of the parameters of AIM theory, refers to the two-electron density matrix, and is a measure of electrons shared between two basins A and B. A perfectly covalent bond would have a value for δ(A, B) near the integer associated with a Lewis structure, but an ionic component to bonding reduces the value of δ(A, B) [47,48]. The parameter has been used to discuss the strengths of intermolecular hydrogen bonds [48].

Non-Covalent Interactions and the Reduced Density Gradient
The measure s(r) for non-covalent interaction (NCI) was introduced by Johnson et al. [49,50] The reduced (dimensionless) density gradient s (also called RDG) is defined as The RDG has been applied to the description of H-bond strength [51]. A useful description of the RDG, its graphical characterization, and its interpretation is given by Contreras-Garcia, et al. [52].
Two NCI graphic realizations are useful. A two-dimensional plot of s vs. the density (given the sign of the second eigenvalue of the Hessian H) will produce a broad sweep from one extreme at small density but large s through an extreme and large s and small density and then to a second extreme of small density and small s. (Figure 6) This is characteristic of an exponentially-decaying density. Superimposed on this sweep may be spikes in s extending to values near zero at modest values of density. The high-density (but low s) regions near nuclei are far to the right or left of the range of densities shown. The lowdensity region is located primarily in Cartesian space close to the density tails for which s is large. However, s can be small precisely where the density is disturbed by non-covalent interactions, such as van der Waals/dispersion, electrostatic forces, and hydrogen bonds. Figure 10 is the plot for TWO, the most stable species, which contains a strong -OH . . . O= hydrogen bond with its corresponding critical point, and a closed ring marked by a ring critical point. Two such spikes are shown, at about +0.015 and −0.025 signed density units. Generally spikes fall into three types: (i) negative values of the signed density indicative of attractive interactions, such as dipole-dipole or H-bonding, (ii) positive signed density indicating non-bonding interactions, such as electrostatic or steric repulsion in the ring/cage, and (iii) values near zero indicating very weak interactions, such as van der Waals interaction.  Figure 11 shows the isosurface obtained by mapping data from Figure 10 into threedimensional space. The blue region shows an attractive interaction (according to the reduced density gradient criterion) at the Bond Critical Point along the Path O…H. At a Ring Critical Point a green isosurface encloses a region of repulsive non-covalent interaction.   . The NCI isosurfaces for ONE make evident the repulsive noncovalent interaction at the RCP interior to the ring (green coded) and the attractive noncovalent interaction at the BCP for the hydrogen bond (blue).

Canonical Vibrational Spectra and Local Mode Analysis
Computed vibrational frequencies are generally recovered as eigenvalues of the matrix of second derivatives of the energy with respect to mass-weighted displacement coordinates, in the formulation developed by Wilson and coworkers [53]. The associated eigenvectors, normal modes, are generally delocalized. In the study of the strength of hydrogen bonding discussion is eased by the introduction of local force constants without reference to atomic masses. A means of recovery of local force constants from vibrational data was developed by Cremer and coworkers [54]. An overall review is provided by Kraka, Zou, and Tao [55]. The hydrogen bond, among many other kinds of noncovalent interaction has been analyzed [56][57][58].

Software
Gaussian 09 [59] and 16 [60] produced optimized structures and the extrapolation sequence of RHF, MP2, and coupled-cluster energies. Atoms-in-Molecules results were obtained with AIMALL [61]. NCIPLOT was obtained by download from the Contreras-Garcia group [62] and LMODEA software for the local modes analysis [63] was provided by the Kraka group.  . The NCI isosurfaces for ONE make evident the repulsive noncovalent interaction at the RCP interior to the ring (green coded) and the attractive noncovalent interaction at the BCP for the hydrogen bond (blue).

Canonical Vibrational Spectra and Local Mode Analysis
Computed vibrational frequencies are generally recovered as eigenvalues of the matrix of second derivatives of the energy with respect to mass-weighted displacement coordinates, in the formulation developed by Wilson and coworkers [53]. The associated eigenvectors, normal modes, are generally delocalized. In the study of the strength of hydrogen bonding discussion is eased by the introduction of local force constants without reference to atomic masses. A means of recovery of local force constants from vibrational data was developed by Cremer and coworkers [54]. An overall review is provided by Kraka, Zou, and Tao [55]. The hydrogen bond, among many other kinds of noncovalent interaction has been analyzed [56][57][58].

Software
Gaussian 09 [59] and 16 [60] produced optimized structures and the extrapolation sequence of RHF, MP2, and coupled-cluster energies. Atoms-in-Molecules results were obtained with AIMALL [61]. NCIPLOT was obtained by download from the Contreras-Garcia group [62] and LMODEA software for the local modes analysis [63] was provided by the Kraka group.