*4.1. 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-pV*L*Z, 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*−<sup>3</sup> inverse-cubic scaling for sufficiently large *L* [38]. Simple schemes based on this behavior are often usefully accurate. We employ the formulas shown below.

$$E\_{HF}(\infty) = \frac{\left(E\_{HF}(L-2) - E\_{HF}(L-1)\right)^2}{\left(E\_{HF}(L-2) - 2E\_{HF}(L-1) - E\_{HF}(L)\right)}\tag{3}$$

$$E\_{CORR}(\infty) = \frac{\left(L^3 E\_{CORR}(L) - \left(L - 1\right)^3 E\_{CORR}(L - 1)\right)}{\left(4L^3 - \left(L - 1\right)^3\right)}\tag{4}$$

Here *L* denotes the maximum angular momentum in the extrapolation, so if we use cc-pVQZ as the largest basis, *EHF*(*L*) and *ECORR(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 *ECORR* for *L* and *L* − 1. Defining the constant *a* by

$$a = L^4(E\_{CORR}(L) - E\_{CORR}(L-1))\tag{5}$$

The limit of the correlation energy is

$$E(\infty) = E\_{CORR}(L) + a \left[ \frac{\pi^4}{90} - \sum\_{l=1}^{L} l^{-4} \right] \tag{6}$$

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.

## *4.2. 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 [22–28].

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:


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].

#### *4.3. 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

$$s(r) = \frac{|\nabla \rho(r)|}{2 \left(3\pi^2\right)^{1/3} \rho(r) r^{4/3}}\tag{7}$$

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 10.** Reduced density gradient *s* vs. the density signed by the second eigenvalue of the Hessian for species ONE. A positive sign indicates a repulsive non-covalent interaction (often found in the center of a closed ring), and a negative value denotes attraction (H bonding in this case).

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.

**Figure 11.** 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).

#### *4.4. 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 [56–58].
