Pauling’s Conceptions of Hybridization and Resonance in Modern Quantum Chemistry

We employ the tools of natural bond orbital (NBO) and natural resonance theory (NRT) analysis to demonstrate the robustness, consistency, and accuracy with which Linus Pauling’s qualitative conceptions of directional hybridization and resonance delocalization are manifested in all known variants of modern computational quantum chemistry methodology.


Introduction
The present authors proudly claim direct line of descent in the academic family tree of Linus Pauling. Senior author FW was an academic grandson (through Doctorvater E. B. Wilson, Jr. at Harvard University, 1963-1967), a faculty colleague (at Stanford University, 1974-1976, and a student of Pauling (in the 1975 Special Topics course on the valence bond theory of nuclear structure). Junior author EDG's Ph.D studies with FW on chemical bonding [1][2][3] and resonance theory [4][5][6] at UW-Madison (1985-1991 were largely based on classic works of Pauling and Wilson [7,8] and conducted under their watchful eyes in photographic portraits that overlooked both the Theoretical Chemistry Institute (TCI) Lecture Room and FW's office.
In the quarter-century following the first applications of quantum theory to chemical bonding [9,10], the powerful influence of Pauling's valence bond (VB) formulation of hybridization [11,12] and resonance [13,14] theory could hardly be overstated. However, this influence waned as the rival molecular orbital (MO) formulation [15][16][17] achieved efficient numerical implementation [18][19][20][21][22][23] in the 1960s. Traditional VB theory was further weakened when Norbeck and Gallup [24] demonstrated that a strictly ab initio evaluation of the VB wavefunction for benzene gave results that were variationally inferior to MO theory and contradicted many semi-empirical VB assumptions of the time. Some limitations of the original VB formulation were removed in the self-consistent generalized GVB formulation of Goddard and co-workers [25,26] (and the related spin-coupled SCGVB variant [27]). However, the self-consistent orbital mixings tend to obscure interpretation of final GVB numerical results in terms of the VB-type initial guess. As density functional theoretic (DFT) and other MO-based methodologies advanced [28], VB-based methods were reduced to a niche role in quantum chemistry.
It is important to recognize that the validity of Pauling-type hybridization and resonance concepts is essentially independent of whether VB/GVB-type wavefunctions are computationally competitive. Pauling's inspiration to "hybridize" free-atom sphericalharmonics to achieve improved bonding orbitals and compact wavefunctions was intended to rationalize the empirically known directionality of atomic valency (e.g., the tetrahedral decades. Pauling's hybridization and resonance conceptions thereby seem to gain increasing theoretical support as the accuracy and applicability of modern quantum chemistry methods continue to improve.

Computational Methods
The present overview involves comparisons of many computational levels that are commonly identified in the arcane "method/basis" acronyms of modern computational quantum chemistry (see [28] for additional explanations and original references). In addition to RHF (restricted Hartree-Fock), the employed methods include B3LYP (Becke 3-parameter, Lee-Yang-Parr correlation functional variant of DFT theory), SCGVB, CAS (complete active space self-consistent-field), MP2 (2nd-order Møller-Plesset), and CCSD (coupled-cluster with single and double excitations). The basis set was chosen uniformly as "aVTZ" (Dunning-type augmented correlation-consistent valence triple zeta), but many other basis sets of higher or lower quality could be expected to give qualitatively similar numerical results. Geometries were optimized at the B3LYP/aVTZ or MP2/aVTZ level, as detailed below. Transition state searches and intrinsic reaction coordinate (IRC) calculations were performed at the B3LYP/aVTZ level. All calculations were completed with Gaussian-16 [72] except for single-point energy evaluations at the SCGVB and CAS levels [73,74], which were completed using Molpro [75][76][77]. Further numerical details of optimizations, IRC evaluation, and NRT keyword settings are described in Supplementary Materials.

Directional Hybridization
Hybridization of atomic orbitals is a central concept in modern chemical bonding theory. As described by Pauling [11] and Slater [12], the mixing of valence s and p orbitals at a tetrahedral carbon atom facilitates electron-pair bonding by forming four equivalent hybrids that are directed toward the vertices of the regular tetrahedron. More generally, valence orbitals of any main group atom can undergo hybridization in a molecular environment to give a set of four directed hybrids (i = 1-4) of sp λ i character, where p θ i is a valence p orbital aligned with direction θ i and the hybridization parameter λ i can range from 0 (pure s) to ∞ (pure p). We assume here that mixing is limited to orbitals of s and p symmetry only, which is typical for normal-valent main group atoms (where d-character in these hybrids is generally less than 0.2%). Conservation of valence sand p-character requires that where 1/(1 + λ i ) and λ i /(1 + λ i ), respectively, represent the fractional sand p-character of the ith hybrid and the summations run over all four hybrids. These conservation expressions are only satisfied for a mutually orthogonal set of atomic hybrids. Before illustrating hybridization in NBO analysis, let us briefly review the procedure that yields the "natural hybrid orbitals" (NHOs). NBO analysis begins with the firstorder reduced density matrix Γ for any N-electron wavefunction ψ (1,2, . . . ,N). This matrix has elements for atom-centered basis functions {χ k } and density (integral-) operatorΓ(1|1 ), We assume here that the density matrix is represented in an orthogonal basis. If the basis functions are instead non-orthogonal, as is usually the case, the density is first transformed to an orthogonal "natural atomic orbital" (NAO) representation, the details of which are described elsewhere [78]. NBO analysis then seeks the set of localized oneand two-center orbitals, the natural bond orbitals (NBOs), that best represent the electron density. The NHOs are the atomic components of these NBOs.
NBOs are obtained from eigenvectors of one-and two-center blocks of the density matrix. The NBO search procedure initially searches one-center blocks, selecting all eigenvectors having occupancies (eigenvalues) that exceed threshold (initially 1.90e, the "occupancy threshold"). These vectors are identified as atomic core and lone pair orbitals, and the density associated with these functions is projected from the density matrix. The procedure next searches two-center blocks of the projected density matrix, selecting eigenvectors that again have occupancies exceeding threshold. These two-center vectors are generally non-orthogonal, and those vectors that overlap considerably (squared-overlap exceeding 0.70) at common centers are eliminated. The remaining vectors are orthogonalized using an occupancy-weighted symmetry orthogonalization procedure [78]. This yields the set of orthogonal, two-center orbitals (the bonds), each A-B bonding orbital represented as a linear combination of atomic bonding hybrids, h A , h B , with polarization coefficients c A , c B . The set of NHOs includes all one-center NBOs and all hybrids, h A , h B , of the two-center NBOs, along with extra-valence Rydberg functions that complete the span of the basis set. The one-and two-center NBOs together often account for over 99.9% of the calculated electron density. Figure 1 shows representative bonding hybrids for the central atoms of CH 4 , SiH 4 , and GeH 4 . The orbitals depicted in this figure are "pre-orthogonal" because although they are orthogonal to all other hybrids on the central atom, each can strongly overlap the 1 s orbital of the adjacent H atom to which the hybrid is directed.
for atom-centered basis functions { } and density (integral-) operator Γ 1|1 , We assume here that the density matrix is represented in an orthogonal basis basis functions are instead non-orthogonal, as is usually the case, the density is first formed to an orthogonal "natural atomic orbital" (NAO) representation, the det which are described elsewhere [78]. NBO analysis then seeks the set of localized on two-center orbitals, the natural bond orbitals (NBOs), that best represent the electro sity. The NHOs are the atomic components of these NBOs.
NBOs are obtained from eigenvectors of one-and two-center blocks of the d matrix. The NBO search procedure initially searches one-center blocks, selecting all vectors having occupancies (eigenvalues) that exceed threshold (initially 1.90e, the pancy threshold"). These vectors are identified as atomic core and lone pair orbital the density associated with these functions is projected from the density matrix. Th cedure next searches two-center blocks of the projected density matrix, selecting vectors that again have occupancies exceeding threshold. These two-center vecto generally non-orthogonal, and those vectors that overlap considerably (squared-o exceeding 0.70) at common centers are eliminated. The remaining vectors are ort nalized using an occupancy-weighted symmetry orthogonalization procedure [78 yields the set of orthogonal, two-center orbitals (the bonds), each A-B bonding orb Ω ℎ ℎ represented as a linear combination of atomic bonding hybrids, hA, hB, with polari coefficients cA, cB. The set of NHOs includes all one-center NBOs and all hybrids, hA the two-center NBOs, along with extra-valence Rydberg functions that complete th of the basis set. The one-and two-center NBOs together often account for over 99 the calculated electron density. Figure 1 shows representative bonding hybrids for the central atoms of CH4 and GeH4. The orbitals depicted in this figure are "pre-orthogonal" because althoug are orthogonal to all other hybrids on the central atom, each can strongly overlap orbital of the adjacent H atom to which the hybrid is directed. NHO character is found to be largely independent of the ab initio or density tional method employed, as illustrated for the 15 main group hydrides of Table 1. character of the bonding hybrids is reported for a range of computational method densities calculated at the single-determinantal uncorrelated (RHF), multi-determi correlated (SCGVB, CAS), and single-reference correlated (MP2, CCSD) levels, and density functional theory (B3LYP), all at fixed MP2/aVTZ optimized geometries. Fo hydride, the p-character varies weakly across the series of densities. Even for HBr, NHO character is found to be largely independent of the ab initio or density functional method employed, as illustrated for the 15 main group hydrides of Table 1. The p-character of the bonding hybrids is reported for a range of computational methods, for densities calculated at the single-determinantal uncorrelated (RHF), multi-determinantal correlated (SCGVB, CAS), and single-reference correlated (MP2, CCSD) levels, and with density functional theory (B3LYP), all at fixed MP2/aVTZ optimized geometries. For each hydride, the p-character varies weakly across the series of densities. Even for HBr, which exhibits the largest λ variation (from 6.82 at the RHF level to 7.81 for SCGVB), the percent p-character changes by only 1.4% (from 87.2% to 88.6%). Note specifically that the SCGVB hybrid descriptors of Table 1 are generally in line with the near-Pauling results that are found both at higher and lower computational levels, contrary to the conclusions of [57,58]. Thus, the NBO user can be confident that the hybrid description offered at one level of theory will be largely consistent with that obtained using nearly any other level, particularly for densities from correlated or density functional calculations.  Figure 2 shows the character of the X-H bonds, including bond polarization (c X 2 ) and hybridization (λ) of the main group atom. As the electronegativity of X increases, the bond increasingly polarizes and the bonding hybrid gains p-character, as anticipated by Bent's rule [79,80]. The Group 13 hydrides (XH 3 , X = B, Al, Ga) have trigonal planar geometries so that the central atoms are essentially sp 2 hybridized (67% p), as confirmed by the NHOs. Similarly, the Group 14 hydrides (XH 4 , X = C, Si, Ge) are tetrahedral with sp 3 -like hybrids (75% p), consistent with Pauling's original inferences from molecular symmetry. [57,58]. Thus, the NBO user can be confident that the hybrid description offered at on level of theory will be largely consistent with that obtained using nearly any other leve particularly for densities from correlated or density functional calculations.  Figure 2 shows the character of the X-H bonds, including bond polarization (cX 2 ) an hybridization (λ) of the main group atom. As the electronegativity of X increases, the bon increasingly polarizes and the bonding hybrid gains p-character, as anticipated by Bent rule [79,80]. The Group 13 hydrides (XH3, X = B, Al, Ga) have trigonal planar geometri so that the central atoms are essentially sp 2 hybridized (67% p), as confirmed by the NHO Similarly, the Group 14 hydrides (XH4, X = C, Si, Ge) are tetrahedral with sp 3 -like hybrid (75% p), consistent with Pauling's original inferences from molecular symmetry.  The Group 17 hydrides reveal the highest p-character, with λ = 4.10 for HF, λ = 6.17 for HCl, and λ = 7.43 for HBr. Elevated p-character arises as s-character shifts from the bonding hybrid into a lone pair, thereby stabilizing the molecule. To illustrate, consider HF. The F atom has four valence orbitals, including the bonding hybrid and three lone pairs. Two of the lone pairs are π-type orbitals (unhybridized 2p) directed along vectors that are orthogonal to the line-of-centers. The third lone pair is σ-type, directed along the line-of-centers but away from the H atom. The latter orbital is sp 0.24 hybridized (80.4% s), so by conservation of hybrid character, only 19.6% s-character is available for the bonding hybrid (sp 4.10 ). The lone pair is essentially doubly occupied (1.982e), whereas the bonding hybrid has an occupancy (1.553e) considerably less than two electrons. HF is therefore stabilized because the higher occupancy lone pair has enhanced s-character, leaving limited s-character for the bonding hybrid. The bonding hybrids for the Group 15 and 16 hydrides have similarly elevated p-character-s-character concentrates in a lone pair of the central atom [79]. More general vertical (size-dependent) aspects of Bent's rule are discussed elsewhere [81].
Group 15 and 16 hydrides may exhibit some bond bending if the central atom hybrids deviate from the X-H line-of-centers. In contrast, there is no bending in the Group 13 (XH 3 , D 3h ), 14 (XH 4 , T d ), and 17 (HX, C ∞v ) hydrides because symmetry requires alignment with the line-of-centers. Table 2 compares the MP2 optimized bond angles of the Group 15 and 16 hydrides with two measures of inter-hybrid angle. The first of these, α = cos −1 (1/λ), is the angle between a pair of sp λ -hybridized orbitals, equivalent to the angle between the p θ i orbitals [cf Equation (1)] for the hybrid pair. This measure assumes no contribution from polarization (d, f, etc.) functions. A second measure, β, is the angle subtended by the line segment that connects the points of maximum amplitude for the pair of bonding hybrids. We see in Table 2 that the inter-hybrid angles are consistently several degrees larger than the inter-nuclear bond angle. That the α angles are particularly large is not surprising because this measure ignores polarization effects. The β angles are somewhat smaller than α because d-character (typically approximately 0.2% of the hybrid) allows for the polarization of the hybrids, thereby shifting the amplitude maxima to somewhat more acute angles. We find that the β values are usually in fairly good accord with the optimized bond angles, except in cyclic species with appreciable ring-strain (e.g., cyclopropane). All the foregoing results are qualitatively consistent with the intuitions that animated Pauling's original conception of hybridization, long before the availability of respectably accurate wavefunctions by current standards. Accordingly, these hybrid intuitions continue to warrant central focus in chemical pedagogy, contrary to the conclusions expressed by Grushow [45].

Resonance Delocalization
As mentioned above, Pauling's original formulation of the theory of resonance in chemistry [13] was grounded in mesomerism concepts [36][37][38][39][40][41][42][43] that could be rationalized and broadly extended in the abstract language and mathematical constructs of quantum mechanics. However, Pauling's powerful resonance-based intuitions were largely honed by encyclopedic familiarity with available chemical structural data, rather than thenavailable VB formulations (later shown to be significantly flawed [24]). As recounted by Eisenberg [82], Pauling's celebrated discovery of the α-helix was inspired by folding a cut-out paper model of a protein chain with resonance-enforced planarity at each amide group, a crude "analog device" to effectively bypass numerical VB-based modeling. In the present section, we employ NRT analysis to re-examine Pauling's resonance-type concepts of amide structure and reactivity in the framework of modern quantum-chemical computations for a simple amide tautomerization reaction.
In the NRT formulation [4], resonance weightings {w α } are obtained from convex-type (w α ≥ 0, ∑α w α = 1) expansion of the electron density operator, with weightings chosen to optimally approximate the full quantum-chemical density operatorΓ QC for the chosen ab initio or density functional calculation. The corresponding resonance-typeΓ NRT expansion is expressed as a weighted sum of localized density operatorsΓ α , one operator for each idealized localized bonding pattern α contributing to the resonance hybrid. Resonance weights, w α , are variationally optimized subject to normalization and positivity constraints An efficient and parallelized implementation of NRT is available in NBO 7.0 [69]. In addition to reporting the details of the resonance hybrid (weights and structures), NRT calculates "natural bond orders" AB is the integer bond order of the A-B atom pair of resonance structure α. We illustrate application of NRT by considering formamide (F)-formimidic acid (FA) tautomerization ( Figure 3). Formamide is the simplest naturally occurring molecule that features the N-C=O peptide bond. Its conversion to formimidic acid is catalyzed by solvent molecules or by another formamide molecule, but we only examine here the uncatalyzed intramolecular reaction that proceeds via a simple 1,3-proton transfer mechanism.
idealized localized bonding pattern α contributing weights, wα, are variationally optimized subject t straints 1; by minimizing the Frobenius norm min An efficient and parallelized implementation o addition to reporting the details of the resonance h calculates "natural bond orders" where is the integer bond order of the A-B ato We illustrate application of NRT by considering tautomerization (Figure 3). Formamide is the simpl features the N-C=O peptide bond. Its conversion to vent molecules or by another formamide molecule, lyzed intramolecular reaction that proceeds via a si  Figure 4 shows the B3LYP/aVTZ energy profile is 12.3 kcal/mol less stable than formamide and is kcal/mol barrier. The barrier is consistent with the Hazra and Chakraborty [83] at the MP2/6-311++G **   NRT analysis of formamide yields the resonance hybrid of well represented by just four resonance structures that collectivel resonance expansion. The molecule is sufficiently delocalized th nance form (F1, the "natural Lewis structure") only contributes hybrid. The leading secondary structure, the charge-transfer form strong π-type resonance as electron density from the N lone pair, πCO * antibond. An image of this donor-acceptor interaction in Ta orbital overlap between the C and N atoms (on the right) that lend bond character to the CN bond, while electron transfer into the πCO acts to reduce CO double-bond character. These effects on bond or the mixing of the F2 structure into the resonance hybrid. Pertu Kohn-Sham matrix suggests that the nN →πCO * interaction alone s about 62 kcal/mol. Two additional interactions, both involving σ electrons from an O lone pair are somewhat weaker (at 26.2 and 2 in smaller, although still significant, contributions to the resonanc tures F3 and F4.  NRT analysis of formamide yields the resonance hybrid of Table 3. Formamide is well represented by just four resonance structures that collectively describe 99.3% of the resonance expansion. The molecule is sufficiently delocalized that the dominant resonance form (F1, the "natural Lewis structure") only contributes 43.8% of the resonance hybrid. The leading secondary structure, the charge-transfer form F2 at 33.7%, stems from strong π-type resonance as electron density from the N lone pair, n N , delocalizes into the π CO * antibond. An image of this donor-acceptor interaction in Table 3 shows significant orbital overlap between the C and N atoms (on the right) that lends considerable double-bond character to the CN bond, while electron transfer into the π CO * antibond (on the left) acts to reduce CO double-bond character. These effects on bond order are consistent with the mixing of the F2 structure into the resonance hybrid. Perturbative analysis of the Kohn-Sham matrix suggests that the n N → π CO * interaction alone stabilizes formamide by about 62 kcal/mol. Two additional interactions, both involving σ-type delocalization of electrons from an O lone pair are somewhat weaker (at 26.2 and 22.8 kcal/mol) and result in smaller, although still significant, contributions to the resonance expansion from structures F3 and F4. Table 4 shows the corresponding analysis for formimidic acid. Like formamide, formimidic acid is fairly well described by four resonance structures, with weights totaling 98.2%. The Lewis structure (FA1) dominates the resonance expansion at 64.2%, and the leading charge-transfer form (FA2) at 20.2% arises from π-type delocalization of an O lone pair, n O , into the π CN * antibond. This charge-transfer interaction is stabilizing by about 40.4 kcal/mol. Two weaker σ-type delocalizations, involving the N lone pair and NH bond, contribute about 14% of the resonance hybrid. The resonance expansion clearly suggests that resonance delocalization effects are somewhat weaker in formimidic acid than in formamide, which probably accounts for the greater stability (by~12 kcal/mol) of the latter tautomer.
Natural bond orders for formamide and formimidic acid are shown in Figure 5, along with the optimized bond lengths. These bond orders are weighted-averages of the integer bond orders for the structures F1-F4 of Table 3 and FA1-FA4 of Table 4, respectively. Ignoring the proton transfer, the principal geometry changes during tautomerization are the lengthening of the CO bond (by 0.135 Å) and shortening of the CN bond (by 0.097 Å).
These changes result from the loss of CO double-bond character (bond order decreasing from 1.877 to 1.160) and gain of CN double-bond character (increasing from 1.223 to 1.948) as the resonance description morphs from mostly F1 to predominantly FA1.
We examined tautomerization more fully by performing NRT analysis at geometries across the reaction profile of Figure 4. To simplify the analysis, we limited the NRT expansion to only four resonance forms, including the two dominant structures of formamide (F1 and F2) and the two dominant structures of formimidic acid (FA1 and FA2) [84]. These four structures alone constitute the minimal set required to simultaneously describe bond breaking/formation during proton transfer and resonance effects in the π system (neglecting weaker σ-type resonance contributions). Figure 6 shows the dependence of the resonance weights on reaction coordinate. F → FA conversion begins with formamide electron density described by F1 and F2 in roughly 80%:20% proportion. As proton transfer begins π resonance strengthens as the F2 contribution increases. Note that F2 has the same N=C-O bonding pattern as the product Lewis structure FA1, although the latter only begins to contribute importantly to the resonance hybrid within close proximity to the transition state (IRC = 0). The transition state is strongly delocalized with nearly equal contributions (~28%) from F1, F2, and FA1. When the reaction is complete, the formimidic acid is roughly 90% FA1 and 10% FA2.  NRT analysis of formamide yields the resonance hybrid of Table 3. Formamide is well represented by just four resonance structures that collectively describe 99.3% of the resonance expansion. The molecule is sufficiently delocalized that the dominant resonance form (F1, the "natural Lewis structure") only contributes 43.8% of the resonance hybrid. The leading secondary structure, the charge-transfer form F2 at 33.7%, stems from strong π-type resonance as electron density from the N lone pair, nN, delocalizes into the πCO * antibond. An image of this donor-acceptor interaction in Table 3 shows significant orbital overlap between the C and N atoms (on the right) that lends considerable doublebond character to the CN bond, while electron transfer into the πCO * antibond (on the left) acts to reduce CO double-bond character. These effects on bond order are consistent with the mixing of the F2 structure into the resonance hybrid. Perturbative analysis of the Kohn-Sham matrix suggests that the nN →πCO * interaction alone stabilizes formamide by about 62 kcal/mol. Two additional interactions, both involving σ-type delocalization of electrons from an O lone pair are somewhat weaker (at 26.2 and 22.8 kcal/mol) and result in smaller, although still significant, contributions to the resonance expansion from structures F3 and F4.  Figure 4. B3LYP/aVTZ energy profile for the tautomerization of formamide (F) to formimidic acid (FA). IRC = 0 corresponds to the transition state.
NRT analysis of formamide yields the resonance hybrid of Table 3. Formamide is well represented by just four resonance structures that collectively describe 99.3% of the resonance expansion. The molecule is sufficiently delocalized that the dominant resonance form (F1, the "natural Lewis structure") only contributes 43.8% of the resonance hybrid. The leading secondary structure, the charge-transfer form F2 at 33.7%, stems from strong π-type resonance as electron density from the N lone pair, nN, delocalizes into the πCO * antibond. An image of this donor-acceptor interaction in Table 3 shows significant orbital overlap between the C and N atoms (on the right) that lends considerable doublebond character to the CN bond, while electron transfer into the πCO * antibond (on the left) acts to reduce CO double-bond character. These effects on bond order are consistent with the mixing of the F2 structure into the resonance hybrid. Perturbative analysis of the Kohn-Sham matrix suggests that the nN →πCO * interaction alone stabilizes formamide by about 62 kcal/mol. Two additional interactions, both involving σ-type delocalization of electrons from an O lone pair are somewhat weaker (at 26.2 and 22.8 kcal/mol) and result in smaller, although still significant, contributions to the resonance expansion from structures F3 and F4.  NRT analysis of formamide yields the resonance hybrid of Table 3. Formamide is well represented by just four resonance structures that collectively describe 99.3% of the resonance expansion. The molecule is sufficiently delocalized that the dominant resonance form (F1, the "natural Lewis structure") only contributes 43.8% of the resonance hybrid. The leading secondary structure, the charge-transfer form F2 at 33.7%, stems from strong π-type resonance as electron density from the N lone pair, nN, delocalizes into the πCO * antibond. An image of this donor-acceptor interaction in Table 3 shows significant orbital overlap between the C and N atoms (on the right) that lends considerable doublebond character to the CN bond, while electron transfer into the πCO * antibond (on the left) acts to reduce CO double-bond character. These effects on bond order are consistent with the mixing of the F2 structure into the resonance hybrid. Perturbative analysis of the Kohn-Sham matrix suggests that the nN →πCO * interaction alone stabilizes formamide by about 62 kcal/mol. Two additional interactions, both involving σ-type delocalization of electrons from an O lone pair are somewhat weaker (at 26.2 and 22.8 kcal/mol) and result in smaller, although still significant, contributions to the resonance expansion from structures F3 and F4.   Table 4 shows the corresponding analysis for formimidic acid. Like formamide, formimidic acid is fairly well described by four resonance structures, with weights totaling 98.2%. The Lewis structure (FA1) dominates the resonance expansion at 64.2%, and the leading charge-transfer form (FA2) at 20.2% arises from π-type delocalization of an O  Table 4 shows the corresponding analysis for formimidic acid. Like formamide, formimidic acid is fairly well described by four resonance structures, with weights totaling 98.2%. The Lewis structure (FA1) dominates the resonance expansion at 64.2%, and the leading charge-transfer form (FA2) at 20.2% arises from π-type delocalization of an O  Table 4 shows the corresponding analysis for formimidic acid. Like formamide, formimidic acid is fairly well described by four resonance structures, with weights totaling 98.2%. The Lewis structure (FA1) dominates the resonance expansion at 64.2%, and the leading charge-transfer form (FA2) at 20.2% arises from π-type delocalization of an O the leading charge-transfer form (FA2) at 20.2% arises from π-type delocalization of an O lone pair, nO, into the πCN * antibond. This charge-transfer interaction is stabilizing by about 40.4 kcal/mol. Two weaker σ-type delocalizations, involving the N lone pair and NH bond, contribute about 14% of the resonance hybrid. The resonance expansion clearly suggests that resonance delocalization effects are somewhat weaker in formimidic acid than in formamide, which probably accounts for the greater stability (by ~12 kcal/mol) of the latter tautomer. ing 98.2%. The Lewis structure (FA1) dominates the resonance expansion at 64.2%, and the leading charge-transfer form (FA2) at 20.2% arises from π-type delocalization of an O lone pair, nO, into the πCN * antibond. This charge-transfer interaction is stabilizing by about 40.4 kcal/mol. Two weaker σ-type delocalizations, involving the N lone pair and NH bond, contribute about 14% of the resonance hybrid. The resonance expansion clearly suggests that resonance delocalization effects are somewhat weaker in formimidic acid than in formamide, which probably accounts for the greater stability (by ~12 kcal/mol) of the latter tautomer. ing 98.2%. The Lewis structure (FA1) dominates the resonance expansion at 64.2%, and the leading charge-transfer form (FA2) at 20.2% arises from π-type delocalization of an O lone pair, nO, into the πCN * antibond. This charge-transfer interaction is stabilizing by about 40.4 kcal/mol. Two weaker σ-type delocalizations, involving the N lone pair and NH bond, contribute about 14% of the resonance hybrid. The resonance expansion clearly suggests that resonance delocalization effects are somewhat weaker in formimidic acid than in formamide, which probably accounts for the greater stability (by ~12 kcal/mol) of the latter tautomer. the leading charge-transfer form (FA2) at 20.2% arises from π-type delocalization of an O lone pair, nO, into the πCN * antibond. This charge-transfer interaction is stabilizing by about 40.4 kcal/mol. Two weaker σ-type delocalizations, involving the N lone pair and NH bond, contribute about 14% of the resonance hybrid. The resonance expansion clearly suggests that resonance delocalization effects are somewhat weaker in formimidic acid than in formamide, which probably accounts for the greater stability (by ~12 kcal/mol) of the latter tautomer. ing 98.2%. The Lewis structure (FA1) dominates the resonance expansion at 64.2%, and the leading charge-transfer form (FA2) at 20.2% arises from π-type delocalization of an O lone pair, nO, into the πCN * antibond. This charge-transfer interaction is stabilizing by about 40.4 kcal/mol. Two weaker σ-type delocalizations, involving the N lone pair and NH bond, contribute about 14% of the resonance hybrid. The resonance expansion clearly suggests that resonance delocalization effects are somewhat weaker in formimidic acid than in formamide, which probably accounts for the greater stability (by ~12 kcal/mol) of the latter tautomer. Natural bond orders for formamide and formimidic acid are shown in Figure 5, along with the optimized bond lengths. These bond orders are weighted-averages of the integer bond orders for the structures F1-F4 of Table 3 and FA1-FA4 of Table 4, respectively. Ignoring the proton transfer, the principal geometry changes during tautomerization are the lengthening of the CO bond (by 0.135 Å) and shortening of the CN bond (by 0.097 Å). These changes result from the loss of CO double-bond character (bond order decreasing from 1.877 to 1.160) and gain of CN double-bond character (increasing from 1.223 to 1.948) as the resonance description morphs from mostly F1 to predominantly FA1.  Natural bond orders for formamide and formimidic acid are shown in Figure 5, along with the optimized bond lengths. These bond orders are weighted-averages of the integer bond orders for the structures F1-F4 of Table 3 and FA1-FA4 of Table 4, respectively. Ignoring the proton transfer, the principal geometry changes during tautomerization are the lengthening of the CO bond (by 0.135 Å) and shortening of the CN bond (by 0.097 Å). These changes result from the loss of CO double-bond character (bond order decreasing from 1.877 to 1.160) and gain of CN double-bond character (increasing from 1.223 to 1.948) as the resonance description morphs from mostly F1 to predominantly FA1.  Natural bond orders for formamide and formimidic acid are sh with the optimized bond lengths. These bond orders are weighted-a bond orders for the structures F1-F4 of Table 3 and FA1-FA4 of Ignoring the proton transfer, the principal geometry changes durin the lengthening of the CO bond (by 0.135 Å) and shortening of the C These changes result from the loss of CO double-bond character (b from 1.877 to 1.160) and gain of CN double-bond character (increasin as the resonance description morphs from mostly F1 to predominan We examined tautomerization more fully by performing NRT across the reaction profile of Figure 4. To simplify the analysis, w F → FA conversion begins with formamide electron dens in roughly 80%:20% proportion. As proton transfer begins π re F2 contribution increases. Note that F2 has the same N=C-O bo uct Lewis structure FA1, although the latter only begins to con resonance hybrid within close proximity to the transition stat state is strongly delocalized with nearly equal contributions (~ When the reaction is complete, the formimidic acid is roughly Figure 7 shows the correlation of natural bond order with across the IRC. The correlations reveal slight S-shaped curvat near-perfect linear correlations around each integer (single dom (two-state bond-shift [85]) bond order with connecting curva slightly different slopes of different bond types, but their esse by the robust |χ| 2 coefficients. Such correlations strongly sup associations of NRT bond orders with experimentally measur with well-known empirical relationships connecting a variety ing bond lengths [86][87][88], bond energies [89][90][91][92], IR vibratio NMR spin-spin coupling constants [95].  Figure 7 shows the correlation of natural bond order with bond length for geometries across the IRC. The correlations reveal slight S-shaped curvatures, or more specifically, near-perfect linear correlations around each integer (single dominant NLS) or half-integer (two-state bond-shift [85]) bond order with connecting curvatures to accommodate the slightly different slopes of different bond types, but their essential linearity is suggested by the robust |χ| 2 coefficients. Such correlations strongly support the useful predictive associations of NRT bond orders with experimentally measurable quantities, consistent with well-known empirical relationships connecting a variety of bond properties, including bond lengths [86][87][88], bond energies [89][90][91][92], IR vibration frequencies [93,94], and NMR spin-spin coupling constants [95]. F → FA conversion begins with formamide electron density described by F1 and F2 in roughly 80%:20% proportion. As proton transfer begins π resonance strengthens as the F2 contribution increases. Note that F2 has the same N=C-O bonding pattern as the product Lewis structure FA1, although the latter only begins to contribute importantly to the resonance hybrid within close proximity to the transition state (IRC = 0). The transition state is strongly delocalized with nearly equal contributions (~28%) from F1, F2, and FA1. When the reaction is complete, the formimidic acid is roughly 90% FA1 and 10% FA2. Figure 7 shows the correlation of natural bond order with bond length for geometries across the IRC. The correlations reveal slight S-shaped curvatures, or more specifically, near-perfect linear correlations around each integer (single dominant NLS) or half-integer (two-state bond-shift [85]) bond order with connecting curvatures to accommodate the slightly different slopes of different bond types, but their essential linearity is suggested by the robust |χ| 2 coefficients. Such correlations strongly support the useful predictive associations of NRT bond orders with experimentally measurable quantities, consistent with well-known empirical relationships connecting a variety of bond properties, including bond lengths [86][87][88], bond energies [89][90][91][92], IR vibration frequencies [93,94], and NMR spin-spin coupling constants [95].  Variations in the NRT weights for the four resonance structures across the IRC, as well as concomitant changes in natural bond orders, are entirely consistent with the electronpushing, curly-arrow representation that the bench chemist would use to depict the reaction mechanism ( Figure 8). Red arrows correspond to the bond/lone pair rearrangement associated with proton migration, and blue arrows represent the change in π electron distribution of the peptide bond. 1, 26, x FOR PEER REVIEW reaction mechanism (Figure 8). Red arrows corr ment associated with proton migration, and blu tron distribution of the peptide bond. With this simple example, we have shown t ily obtaining compact and chemical intuitive des tivity that are fully consistent with the prescient ling, Robinson, Ingold, and other bonding pione chanical era. Similar to the hybridization r B3LYP/aVTZ results are fully representative of th quantum chemical wavefunctions at any reasona

Summary and Conclusions
Contrary to skepticism that is sometimes e present results confirm the essential correctness a and resonance concepts, as consistently found i from the best currently available quantum che quantitative accuracy of the wavefunction tends erful intuitions, developed long before numeric equation became routinely available.
In closing this tribute, it may be appropriate With this simple example, we have shown that NRT analysis provides a tool for easily obtaining compact and chemical intuitive descriptors of molecular structure and reactivity that are fully consistent with the prescient mesomerism/resonance insights of Pauling, Robinson, Ingold, and other bonding pioneers, dating back to the pre-quantum mechanical era. Similar to the hybridization results presented above, the present B3LYP/aVTZ results are fully representative of those obtained from numerically complex quantum chemical wavefunctions at any reasonably current computational level.

Summary and Conclusions
Contrary to skepticism that is sometimes expressed [45,54,58], we believe that the present results confirm the essential correctness and usefulness of Pauling's hybridization and resonance concepts, as consistently found in NBO/NRT analysis of wavefunctions from the best currently available quantum chemical methods. If anything, improved quantitative accuracy of the wavefunction tends to enhance admiration of Pauling's powerful intuitions, developed long before numerically reliable solutions of Schrödinger's equation became routinely available.
In closing this tribute, it may be appropriate to relate that E. Bright Wilson considered John von Neumann and Linus Pauling to be the only two authentic geniuses he ever met. Elite company indeed! Supplementary Materials: Supplementary Materials are available online, including (i) Gaussian input files for all optimized geometries and the formamide-formimidic acid transition state, (ii) all IRC geometries, and (iii) a sample Gaussian input file, with NBO/NRT input for one of the IRC geometries.
Author Contributions: F.W. conceived this study; E.D.G. performed the calculations; E.D.G. and F.W. analyzed the results; F.W. and E.D.G. wrote the paper. All authors have read and agreed to the published version of the manuscript.
Funding: Support for computational facilities at ISU was provided by the College of Arts and Sciences and Office of Information Technology. Support for computational facilities at UW-Madison was provided in part by National Science Foundation Grant CHE-0840494.

Data Availability Statement:
The data presented in this study are available in Supplementary Materials.

Conflicts of Interest:
The authors declare no conflict of interest.
Sample Availability: Not applicable.