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

Home , Natural bond orbital, Peptide bond, Resonance (chemistry)

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

Eric D. Glendening 1 and Frank Weinhold 2,*

1 Department of Chemistry and Physics, Indiana State University, Terre Haute, IN 47809, USA; [email protected] 2 Theoretical Chemistry Institute and Department of Chemistry, University of Wisconsin-Madison, Madison, WI 53706, USA * Correspondence: [email protected]

Abstract: 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.

Keywords: chemical bonding; directed hybridization; resonance delocalization; mesomerism; natural bond orbitals; natural resonance theory

1. Introduction

Citation: Glendening, E.D.; Weinhold, F. The present authors proudly claim direct line of descent in the academic family tree Pauling’s Conceptions of Hybridization of Linus Pauling. Senior author FW was an academic grandson (through Doctorvater E. B. and Resonance in Modern Quantum Wilson, Jr. at Harvard University, 1963–1967), a faculty colleague (at Stanford University, Chemistry. Molecules 2021, 26, 4110. 1974–1976), and a student of Pauling (in the 1975 Special Topics course on the valence https://doi.org/10.3390/ bond theory of nuclear structure). Junior author EDG’s Ph.D studies with FW on chemical molecules26144110 bonding [1–3] and resonance theory [4–6] at UW-Madison (1985–1991) were largely based on classic works of Pauling and Wilson [7,8] and conducted under their watchful eyes Academic Editors: Maxim L. Kuznetsov, in photographic portraits that overlooked both the Theoretical Chemistry Institute (TCI) Carlo Gatti, David L. Cooper and Lecture Room and FW’s office. Miroslav Kohout 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 Received: 3 June 2021 hybridization [11,12] and resonance [13,14] theory could hardly be overstated. However, Accepted: 2 July 2021 this influence waned as the rival molecular orbital (MO) formulation [15–17] achieved Published: 6 July 2021 efficient numerical implementation [18–23] in the 1960s. Traditional VB theory was further weakened when Norbeck and Gallup [24] demonstrated that a strictly ab initio evaluation of Publisher’s Note: MDPI stays neutral the VB wavefunction for benzene gave results that were variationally inferior to MO theory with regard to jurisdictional claims in and contradicted many semi-empirical VB assumptions of the time. Some limitations of the published maps and institutional affil- original VB formulation were removed in the self-consistent generalized GVB formulation iations. 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 Copyright: © 2021 by the authors. to a niche role in quantum chemistry. Licensee MDPI, Basel, Switzerland. It is important to recognize that the validity of Pauling-type hybridization and reso- This article is an open access article nance concepts is essentially independent of whether VB/GVB-type wavefunctions are distributed under the terms and computationally competitive. Pauling’s inspiration to “hybridize” free-atom spherical- conditions of the Creative Commons Attribution (CC BY) license (https:// harmonics to achieve improved bonding orbitals and compact wavefunctions was intended creativecommons.org/licenses/by/ to rationalize the empirically known directionality of atomic valency (e.g., the tetrahedral 4.0/).

Molecules 2021, 26, 4110. https://doi.org/10.3390/molecules26144110 https://www.mdpi.com/journal/molecules Molecules 2021, 26, 4110 2 of 15

carbon atom of van’t Hoff and LeBel [29–31]), as demanded by early structural studies [32–34]. As shown by Coulson [35], such directional hybrids (linear combinations of directionless free-atom s,p,d, ... orbitals) can serve equally well as conceptual building blocks in MO and VB theory. Similarly, Pauling’s motivation to combine two (or more) Lewis-structural bonding patterns into a “resonance hybrid” was intended to rationalize the empirically known ambivalence of certain molecules (such as benzenoid species or practically any molecule containing allyl or amide groups) whose properties appear “intermediate” or “averaged” between the possible Lewis-structural bonding patterns that might be envisioned [36–43]. Pauling’s basic resonance concept was first expressed mathematically in terms of then-standard Heitler-London pair functions [9], which, in light of subsequent Norbeck-Gallup [24] and Coulson-Fischer [44] studies, can be recognized as a rather arbitrary and sub-optimal choice. In more recent times, basic precepts of hybridization and resonance theory have been questioned or criticized on various grounds. Specific technical criticisms of hybridization theory are often based on uncritical application of Koopmans’-type approximations to interpret photoionization spectroscopy or pedagogical preference for VSEPR-type ratio- nalizations of molecular structure [45] (but see contrarian views [46–51]). Early criticisms of resonance concepts were philosophically based on supposed conflicts with “realism” as perceived in dialectical materialism theory [52,53]. More specific technical criticisms of resonance (e.g., as a conceptual “unicorn” [54]) are often based on preferred use of energy decomposition analysis (EDA) methods that require a specific choice of “reference state” [55,56] for each interacting fragment (perforce eliminating resonance-type state-mixing in either fragment). However, more general questioning of hybridization and resonance concepts can be attributed to the complex mathematical forms of modern wavefunctions and density functionals that no longer allow chemists to easily “see” the hybridization and resonance features that appear explicitly in VB-based formulations. Ironically, even some advocates of modern SCGVB theory have expressed skepticism about Pauling’s hybridization concepts [57,58], because the final orbital shapes no longer resemble localized VB-inspired forms. Related attempts to obtain directed hybrids of localized chemical bonding in the MO/DFT framework by transforming canonical MOs to localized LMO form [59–63] are similarly frustrated, because the localization procedure can be chosen rather arbitrarily to yield a virtually unlimited variety of orbital energies and shapes, with no effect on the calculated total energy or other measurable properties of the system. All such conceptual dilemmas can be averted by adopting a uniform analysis of diverse wavefunctions in a common language of localized bonding constructs. For this purpose, we employ natural bond orbital (NBO) [64–66] and natural resonance theory (NRT) algorithms [67,68] that are implemented in a widely used program (currently NBO 7.0 [69,70]). Although NBO/NRT methods rest on mathematical foundations somewhat beyond those usually discussed in introductory quantum chemistry, the NBO 7.0 program is often integrated into the selfsame quantum chemistry program that generates the wavefunctions to be analyzed [71]. In other cases, the host quantum chemistry program that performs the wavefunction calculation is able to write the “wavefunction archive” (job.47) file that serves as input to a current NBO analysis program, either in stand-alone form or as included in other quantum chemistry program systems. The resulting NBO analysis allows consistent apples-to-apples comparisons of key bonding descriptors (such as atomic s,p,d, ... composition of NBO-based bonding hybrids, or the NRT bond orders between atoms) no matter how diverse the wavefunctions to be compared. In the present work, we employ a consistent protocol to obtain NBO/NRT descriptors of hybridization and resonance for prototype chemical species described at a wide variety of modern quantum chemistry levels, including GVB, DFT, and higher correlated methods. The results serve not only to show how comparison hybridization and resonance descriptors can be obtained from diverse wavefunctional forms, but also to exhibit their remarkable overall consistency with Pauling’s original intuitions dating back nearly nine Molecules 2021, 26, 4110 3 of 15

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.

2. 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 [75–77]. Further numerical details of optimizations, IRC evaluation, and NRT keyword settings are described in Supplementary Materials.

3. 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)

1 = √ + p hi s λi pθi (1) 1 + λi

λi of sp character, where pθi is a valence p orbital aligned with direction θi and the hybridization parameter λi 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 s- and p-character requires that

1 = 1 (2) ∑ + i 1 λi

λ i = 3 (3) ∑ + i 1 λi

where 1/(1 + λi) and λi/(1 + λi), respectively, represent the fractional s- and 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 first- order reduced density matrix Γ for any N-electron wavefunction ψ(1,2, ... ,N). This matrix has elements Z ∗ ˆ 0 0 0 Γij = χi (1) Γ 1 1 χj 1 d1 d1 (4) Molecules 2021, 26, x FOR PEER REVIEW 4 of 6

∗ Γ 𝜒 1Γ1|1 𝜒1 d1 d1′ (4)

for atom‐centered basis functions {𝜒} and density (integral‐) operator Γ1|1 ,

Γ1|1 𝑁𝜓1, 2, … , 𝑁 𝜓∗1,2,…,𝑁 d2 … d𝑁 (5)

Molecules 2021, 26, 4110 4 of 15 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 trans‐ formed to an orthogonal “natural atomic orbital” (NAO) representation, the details of for atom-centered basis functions {χ } and density (integral-) operator Γˆ (1|10), which are described elsewhere k[78]. NBO analysis then seeks the set of localized one‐ and Z 0 ∗ 0 two‐center orbitals,Γˆ 1 1 the= N naturalψ(1, 2,bond . . . , orbitalsN) ψ 1 ,(NBOs), 2, . . . , N thatd2 . best . . dN represent the(5) electron den‐ sity. The NHOs are the atomic components of these NBOs. WeNBOs assume are hereobtained that the from density eigenvectors matrix is represented of one‐ and in an two orthogonal‐center blocks basis. Ifof the density matrix.the basis The functions NBO are search instead procedure non-orthogonal, initially as searches is usually one the case,‐center the blocks, density isselecting first all eigen‐ vectorstransformed having to an occupancies orthogonal “natural (eigenvalues) atomic orbital” that exceed (NAO) representation,threshold (initially the details 1.90e, the “occu‐ of which are described elsewhere [78]. NBO analysis then seeks the set of localized one- pancyand two-center threshold”). orbitals, These the natural vectors bond are orbitals identified (NBOs), as that atomic best representcore and the lone electron pair orbitals, and thedensity. density The NHOs associated are the with atomic these components functions of these is projected NBOs. from the density matrix. The pro‐ cedureNBOs next are searches obtained from two eigenvectors‐center blocks of one- of and the two-center projected blocks density of the matrix, density ma-selecting eigen‐ vectorstrix. The NBOthat searchagain procedurehave occupancies initially searches exceeding one-center threshold. blocks, selecting These all two eigenvec-‐center vectors are tors having occupancies (eigenvalues) that exceed threshold (initially 1.90e, the “occupancy generally non‐orthogonal, and those vectors that overlap considerably (squared‐overlap threshold”). These vectors are identified as atomic core and lone pair orbitals, and the exceedingdensity associated 0.70) withat common these functions centers is projected are eliminated. from the density The matrix.remaining The procedure vectors are orthogo‐ nalizednext searches using two-center an occupancy blocks‐ ofweighted the projected symmetry density matrix,orthogonalization selecting eigenvectors procedure [78]. This yieldsthat again the have set of occupancies orthogonal, exceeding two‐center threshold. orbitals These (the two-center bonds), vectors each are A–B generally bonding orbital non-orthogonal, and those vectors that overlap considerably (squared-overlap exceeding 0.70) at common centers are eliminated. TheΩ remaining 𝑐ℎ vectors 𝑐ℎ are orthogonalized using (6) an occupancy-weighted symmetry orthogonalization procedure [78]. This yields the set of representedorthogonal, two-center as a linear orbitals combination (the bonds), of each atomic A–B bonding bonding orbital hybrids, hA, hB, with polarization coefficients cA, cB. The set of NHOs includes all one‐center NBOs and all hybrids, hA, hB, of the two‐center NBOs, along withΩAB extra= cAh‐Avalence+ cBhB Rydberg functions that complete(6) the span of the basis set. The one‐ and two‐center NBOs together often account for over 99.9% of represented as a linear combination of atomic bonding hybrids, hA, hB, with polarization thecoefficients calculatedcA, cB electron. The set of density. NHOs includes all one-center NBOs and all hybrids, hA, hB, of the two-centerFigure 1 NBOs, shows along representative with extra-valence bonding Rydberg hybrids functions for that the complete central theatoms span of CH4, SiH4, andof the GeH basis4. set. The The orbitals one- and depicted two-center in NBOs this figure together are often “pre account‐orthogonal” for over 99.9% because of the although they arecalculated orthogonal electron to density. all other hybrids on the central atom, each can strongly overlap the 1 s Figure1 shows representative bonding hybrids for the central atoms of CH 4, SiH4, orbital of the adjacent H atom to which the hybrid is directed. and GeH4. 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.

Figure 1. Pre-orthogonal bonding hybrids for CH4 (left), SiH4 (middle), and GeH4 (right).

FigureNHO 1. Pre character‐orthogonal is found bonding to be largely hybrids independent for CH4 ( ofleft the), SiH ab initio4 (middle or density), and functional GeH4 (right). method employed, as illustrated for the 15 main group hydrides of Table1. The p-character of theNHO bonding character hybrids is reportedfound to for be a rangelargely of computationalindependent methods, of the ab for initio densities or density func‐ calculated at the single-determinantal uncorrelated (RHF), multi-determinantal correlated tional method employed, as illustrated for the 15 main group hydrides of Table 1. The p‐ (SCGVB, CAS), and single-reference correlated (MP2, CCSD) levels, and with density characterfunctional theoryof the (B3LYP), bonding all athybrids ﬁxed MP2/aVTZ is reported optimized for a geometries.range of computational For each hydride, methods, for densitiesthe p-character calculated varies weakly at the across single the‐determinantal series of densities. uncorrelated Even for HBr, which (RHF), exhibits multi the‐determinantal correlatedlargest λ variation (SCGVB, (from CAS), 6.82 at and the RHF single level‐reference to 7.81 for correlated SCGVB), the (MP2, percent CCSD)p-character levels, and with densitychanges byfunctional only 1.4% theory (from 87.2% (B3LYP), to 88.6%). all at Notefixed speciﬁcally MP2/aVTZ that optimized the SCGVB geometries. hybrid For each descriptors of Table1 are generally in line with the near-Pauling results that are found both hydride, the p‐character varies weakly across the series of densities. Even for HBr, which at higher and lower computational levels, contrary to the conclusions of [57,58]. Thus, the

Molecules 2021, 26, x FOR PEER REVIEW 5 of 7

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.

Molecules 2021, 26, 4110 5 of 15 Table 1. Hybrid p‐character (λ) of X–H bonding hybrids for the first‐, second‐, and third‐row hy‐ drides a.

NBO user canRHF be conﬁdent thatB3LYP the hybrid descriptionSCGVB offeredCAS at one level ofMP2 theory will beCCSD BHlargely3 consistent2.00 with that obtained2.00 using nearly2.00 any other level,2.00 particularly2.00 for densities 2.00 CHfrom4 correlated2.99 or density functional3.00 calculations.2.99 2.99 2.99 2.99 NH3 2.89 2.90 2.96 2.99 2.93 2.94 Table 1. Hybrid p-character (λ) of X–H bonding hybrids for the first-, second-, and third-row hydrides a. H2O 3.24 3.31 3.42 3.47 3.45 3.44 HF 3.69RHF 3.86 B3LYP SCGVB4.05 CAS4.09 MP24.10 CCSD 4.08 AlH3 1.97 1.98 1.98 1.98 1.98 1.98 BH3 2.00 2.00 2.00 2.00 2.00 2.00 SiH4 CH4 2.96 2.992.98 3.00 2.992.96 2.992.96 2.992.96 2.99 2.96 NH 2.89 2.90 2.96 2.99 2.93 2.94 PH3 3 5.38 5.72 5.49 5.60 5.57 5.67 H2O 3.24 3.31 3.42 3.47 3.45 3.44 H2S HF5.55 3.695.83 3.86 4.055.88 4.096.00 4.105.83 4.08 5.93 HCl AlH3 5.79 1.975.93 1.98 1.986.46 1.986.55 1.986.17 1.98 6.24 GaH3 SiH4 1.99 2.962.00 2.98 2.961.99 2.961.99 2.961.99 2.96 1.99 PH3 5.38 5.72 5.49 5.60 5.57 5.67 GeH4 2.99 3.00 2.99 2.99 2.99 2.99 H2S 5.55 5.83 5.88 6.00 5.83 5.93 AsH3 HCl6.14 5.796.91 5.93 6.466.37 6.556.47 6.176.54 6.24 6.64 H2Se GaH3 6.43 1.997.11 2.00 1.997.00 1.997.05 1.996.95 1.99 7.06 HBr GeH4 6.82 2.997.27 3.00 2.997.81 2.997.80 2.997.43 2.99 7.52 AsH3 6.14 6.91 6.37 6.47 6.54 6.64 a aVTZ valuesH2Se calculated 6.43 at MP2/aVTZ 7.11 optimized 7.00 geometries. 7.05 6.95 7.06 HBr 6.82 7.27 7.81 7.80 7.43 7.52 a aVTZ values calculated at MP2/aVTZ optimized geometries. Figure 2 shows the character of the X–H bonds, including bond polarization (cX2) and

hybridization (λ) of the main group atom. As the electronegativity of X increases,2 the bond Figure2 shows the character of the X–H bonds, including bond polarization ( cX ) and increasinglyhybridization polarizes (λ) of the and main the group bonding atom. hybrid As the electronegativity gains p‐character, of X increases, as anticipated the bond by Bent’s rule [79,80].increasingly The polarizes Group and13 hydrides the bonding (XH hybrid3, X gains= B, Al,p-character, Ga) have as anticipatedtrigonal planar by Bent’s geometries 2 so thatrule the [79 central,80]. The atoms Group are 13 hydrides essentially (XH 3sp, X hybridized = B, Al, Ga) have (67% trigonal p), as confirmed planar geometries by the NHOs. 2 Similarly,so that the the centralGroup atoms 14 hydrides are essentially (XH4sp, Xhybridized = C, Si, Ge) (67% arep), tetrahedral as conﬁrmed with by the sp NHOs.3‐like hybrids 3 (75%Similarly, p), consistent the Group with 14 Pauling’s hydrides (XH original4, X = C, inferences Si, Ge) are tetrahedral from molecular with sp -likesymmetry. hybrids (75% p), consistent with Pauling’s original inferences from molecular symmetry.

2 Figure 2. Polarization (cX ) and hybridization (λ) of the X–H bonds of normal-valent hydrides as a function of the Pauling Figure 2. Polarization (cX2) and hybridization (λ) of the X–H bonds of normal‐valent hydrides as a function of the Pauling electronegativity of the main group atom (MP2/aVTZ). electronegativity of the main group atom (MP2/aVTZ). 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

Molecules 2021, 26, 4110 6 of 15

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 sp0.24 hybridized (80.4% s), so by conservation of hybrid character, only 19.6% s-character is available for the bonding hybrid (sp4.10). 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 (XH3, D3h), 14 (XH4, Td), and 17 (HX, C∞v) hydrides because symmetry requires alignment with the line-of-centers. Table2 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 spλ-hybridized orbitals, equivalent to the angle between the

pθi 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 Table2 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).

Table 2. Bond angles (∠HXH) and inter-hybrid angles (α and β) of the Group 15 and 16 hydrides a.

∠HXH α β

NH3 106.8 110.0 107.9 H2O 104.1 106.8 104.7 PH3 93.6 100.3 98.0 H2S 92.2 99.9 95.6 AsH3 92.5 98.8 98.8 H2Se 91.1 98.3 96.8 a MP2/aVTZ values in degrees.

All the foregoing results are qualitatively consistent with the intuitions that ani- mated Pauling’s original conception of hybridization, long before the availability of re- spectably 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].

4. Resonance Delocalization As mentioned above, Pauling’s original formulation of the theory of resonance in chemistry [13] was grounded in mesomerism concepts [36–43] 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 then- available 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 Molecules 2021, 26, x FOR PEER REVIEW 7 of 9

by encyclopedic familiarity with available chemical structural data, rather than then‐avail‐ able VB formulations (later shown to be significantly flawed [24]). As recounted by Eisen‐ berg [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 am‐ ide structure and reactivity in the framework of modern quantum‐chemical computations for a simple amide tautomerization reaction. In the NRT formulation [4], resonance weightings {wα} are obtained from convex‐ type (wα ≥ 0, α wα = 1) expansion of the electron density operator, with weightings chosen to optimally approximate the full quantum‐chemical density operator Γ for the chosen ab initio or density functional calculation. The corresponding resonance‐type Γ ex‐ pansion

Γ 𝑤 Γ (7)

is expressed as a weighted sum of localized density operators Γ, one operator for each idealized localized bonding pattern α contributing to the resonance hybrid. Resonance weights, wα, are variationally optimized subject to normalization and positivity con‐

Molecules 2021, 26, 4110 straints 7 of 15

𝑤 1; 𝑤 0 (8) 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 bycomputations minimizing for a simple amide the tautomerization Frobenius reaction. norm In the NRT formulation [4], resonance weightings {wα} are obtained from convex-type (wα ≥ 0, ∑α wα = 1) expansion of the electron density operator, with weightings chosen to ˆ optimally approximate the full quantum-chemical density operator ΓQCminfor the𝚪 chosen ab𝚪. (9) initio or density functional calculation. The corresponding resonance-type Γˆ NRT expansion

Γˆ = w Γˆ (7) An efficient andNRT parallelized∑α α α implementation of NRT is available in NBO 7.0 [69]. In

is expressed as a weighted sum of localized density operators Γˆ α, one operator for each additionidealized localized to bonding reporting pattern α contributing the details to the resonance of the hybrid. resonance Resonance hybrid (weights and structures), NRT calculatesweights, wα, are variationally “natural optimized bond subject toorders” normalization and positivity constraints ∑ wα = 1; wα ≥ 0 (8) α by minimizing the Frobenius norm 𝑏 𝑤 𝑏 (10) minkΓQC − ΓNRTk. (9) wα whereAn efﬁcient 𝑏 and parallelizedis the implementationinteger bond of NRT isorder available inofNBO the 7.0 [ 69A–B]. In atom pair of resonance structure α. addition to reporting the details of the resonance hybrid (weights and structures), NRT calculatesWe “natural illustrate bond orders” application of NRT by considering formamide (F)‐formimidic acid (FA)

tautomerization (Figure= 3).(α )Formamide is the simplest naturally occurring molecule that bAB ∑α wαbAB (10) features(α) the N‐C=O peptide bond. Its conversion to formimidic acid is catalyzed by sol‐ where bAB 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) venttautomerization molecules (Figure3). Formamide or by isanother the simplest naturally formamide occurring molecule molecule, that but we only examine here the uncata‐ lyzedfeatures the intramolecular N-C=O peptide bond. Its conversion reaction to formimidic that acid proceeds is catalyzed by solvent via a simple 1,3‐proton transfer mechanism. 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.

Figure 3. Schematic bonding patterns for tautomeric isomerization of formamide (F) to formimidic Figureacid (FA). 3. Schematic bonding patterns for tautomeric isomerization of formamide (F) to formimidic Figure4 shows the B3LYP/aVTZ energy proﬁle for tautomerization. Formimidic acid is 12.3(FA). kcal/mol less stable than formamide and is separated from formamide by a 47.7 kcal/mol barrier. The barrier is consistent with the 47.4 kcal/mol estimate calculated by Hazra and Chakraborty [83] at the MP2/6-311++G ** level. Figure 4 shows the B3LYP/aVTZ energy profile for tautomerization. Formimidic acid is 12.3 kcal/mol less stable than formamide and is separated from formamide by a 47.7 kcal/mol barrier. The barrier is consistent with the 47.4 kcal/mol estimate calculated by Hazra and Chakraborty [83] at the MP2/6‐311++G ** level.

Molecules 2021, 26, x FOR PEER REVIEW 8 of 10 Molecules 2021, 26, 4110 8 of 15

Figure 4. B3LYP/aVTZ energy profile for the tautomerization of formamide (F) to formimidic acid Figure(FA). IRC 4. = 0 B3LYP/aVTZ corresponds to the transition energy state. profile for the tautomerization of formamide (F) to formimidic acid (FA).NRT IRC analysis = 0 corresponds of formamide yieldsto the the transition resonance hybrid state. of Table3. 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 (NRTF1, the “naturalanalysis Lewis of structure”) formamide only contributes yields 43.8% the of resonance the resonance hybrid.hybrid of Table 3. Formamide is wellThe leading represented secondary structure, by just the four charge-transfer resonance form F2 structuresat 33.7%, stems that from collectively strong describe 99.3% of the π-type resonance as electron density from the N lone pair, nN, delocalizes into the πCO * resonanceantibond. An image expansion. of this donor–acceptor The molecule interaction inis Table sufficiently3 shows significant delocalized orbital that the dominant reso‐ nanceoverlap betweenform ( theF1 C, andthe N “natural atoms (on the Lewis right) that structure”) lends considerable only double-bond contributes 43.8% of the resonance character to the CN bond, while electron transfer into the πCO * antibond (on the left) hybrid.acts to reduce The CO leading double-bond secondary character. These structure, effects on bond the order charge are consistent‐transfer with form F2 at 33.7%, stems from the mixing of the F2 structure into the resonance hybrid. Perturbative analysis of the strong π‐type resonance as electron density from the N lone pair, nN, delocalizes into the Kohn-Sham matrix suggests that the nN → πCO * interaction alone stabilizes formamide πbyCO about * antibond. 62 kcal/mol. An Two additionalimage of interactions, this donor–acceptor both involving σ-type interaction delocalization in Table 3 shows significant orbitalof electrons overlap from an Obetween lone pair are the somewhat C and weaker N atoms (at 26.2 (on and 22.8 the kcal/mol) right) andthat lends considerable double‐ result in smaller, although still significant, contributions to the resonance expansion from bondstructures characterF3 and F4. to the CN bond, while electron transfer into the πCO * antibond (on the left) Table4 shows the corresponding analysis for formimidic acid. Like formamide, actsformimidic to reduce acid is fairly CO well double described‐bond by four character. resonance structures, These with effects weights totalingon bond order are consistent with the98.2%. mixing The Lewis of structure the F2 (FA1 structure) dominates the into resonance the expansionresonance at 64.2%, hybrid. and the Perturbative analysis of the leading charge-transfer form (FA2) at 20.2% arises from π-type delocalization of an O lone Kohn‐Sham matrix suggests that the nN →πCO * interaction alone stabilizes formamide by pair, nO, into the πCN * antibond. This charge-transfer interaction is stabilizing by about about40.4 kcal/mol. 62 kcal/mol. Two weaker Twoσ-type additional delocalizations, interactions, involving the N lone both pair andinvolving NH σ‐type delocalization of bond, contribute about 14% of the resonance hybrid. The resonance expansion clearly electronssuggests that from resonance an delocalization O lone pair effects are are somewhat somewhat weaker weaker in formimidic (at 26.2 acid and 22.8 kcal/mol) and result inthan smaller, in formamide, although which probably still accountssignificant, for the greater contributions stability (by ~12 to kcal/mol) the resonance of expansion from struc‐ the latter tautomer. turesNatural F3 and bond F4 orders. for formamide and formimidic acid are shown in Figure5, along with the optimized bond lengths. These bond orders are weighted-averages of the integer bond orders for the structures F1–F4 of Table3 and FA1–FA4a of Table4, respectively. TableIgnoring 3. the NRT proton resonance transfer, the hybrid principal of geometry formamide changes during. tautomerization are the lengthening of the CO bond (by 0.135 Å) and shortening of the CN bond (by 0.097 Å). # Structure Weight (%) Donor–Acceptor Interaction b

F1 43.8

nNπCO * F2 33.7 (61.9)

MoleculesMolecules 2021 2021, 26, 26, x, xFOR FOR PEER PEER REVIEW REVIEW 8 8of of 10 10

Molecules 2021, 26, 4110 9 of 15

FigureFigure 4. 4. B3LYP/aVTZ B3LYP/aVTZ energy energy profile profile for for the the tautomerization tautomerization of of formamide formamide (F) (F) to to formimidic formimidic acid acid These changes result from the loss of CO double-bond character (bond order decreasing (FA).(FA). IRC IRC = =0 0corresponds corresponds to to the the transition transition state. state. 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. NRTNRT analysis analysis of of formamide formamide yields yields the the resonance resonance hybrid hybrid of of Table Table 3. 3. Formamide Formamide is is We examined tautomerization more fully by performing NRT analysis at geometries wellwell represented represented by by just just four four resonance resonance structures structures that that collectively collectively describe describe 99.3% 99.3% of of the the across the reaction proﬁle of Figure4. To simplify the analysis, we limited the NRT expan- resonanceresonance expansion. expansion. The The molecule molecule is is sufficiently sufficiently delocalized delocalized that that the the dominant dominant reso reso‐ ‐ sion to only four resonance forms, including the two dominant structures of formamide (F1 nancenance form form ( F1(F1, the, the “natural “natural Lewis Lewis structure”) structure”) only only contributes contributes 43.8% 43.8% of of the the resonance resonance and F2) and the two dominant structures of formimidic acid (FA1 and FA2)[84]. These four hybrid.hybrid. The The leading leading secondary secondary structure, structure, the the charge charge‐transfer‐transfer form form F2 F2 at at 33.7%, 33.7%, stems stems from from structures alone constitute the minimal set required to simultaneously describe bond break- strongstrong π‐ π‐typetype resonance resonance as as electron electron density density from from the the N N lone lone pair, pair, n Nn,N delocalizes, delocalizes into into the the ing/formation during proton transfer and resonance effects in the π system (neglecting ππCOCO * *antibond. antibond. An An image image of of this this donor–acceptor donor–acceptor interaction interaction in in Table Table 3 3shows shows significant significant weaker σ-type resonance contributions). Figure6 shows the dependence of the resonance orbitalorbital overlap overlap between between the the C C and and N N atoms atoms (on (on the the right) right) that that lends lends considerable considerable double double‐ ‐ weights on reaction coordinate. bondbond character character to to the the CN CN bond, bond, while while electron electron transfer transfer into into the the π πCOCO * *antibond antibond (on (on the the left) left) bondF character→ FA conversion to the CN begins bond, with while formamide electron transfer electron into density the π described* antibond by (onF1 andthe left)F2 acts to reduce CO double‐bond character. These effects on bond order are consistent with actsinacts roughly to to reduce reduce 80%:20% CO CO double double proportion.‐bond‐bond character. character. As proton These These transfer effects effects begins on on bond bondπ resonanceorder order are are consistent strengthensconsistent with with as the mixing of the F2 structure into the resonance hybrid. Perturbative analysis of the thethethe mixingF2 mixingcontribution of of the the F2 F2 increases. structure structure Note into into thatthe the resonanceF2 resonancehas the hybrid. same hybrid. N=C-O Perturbative Perturbative bonding analysis analysis pattern of asof the the the Kohn‐Sham matrix suggests that the Nn N→ → πCOCO * interaction alone stabilizes formamide by KohnproductKohn‐Sham‐Sham Lewis matrix matrix structure suggests suggestsFA1 that, that although the the n n the→π latterπ * *interaction interaction only begins alone alone to contribute stabilizes stabilizes formamide importantly formamide by toby about 62 kcal/mol. Two additional interactions, both involving σ‐type delocalization of abouttheabout resonance 62 62 kcal/mol. kcal/mol. hybrid Two Two within additional additional close proximity interactions, interactions, to the both both transition involving involving state σ‐ σ‐ (IRCtypetype = delocalization 0). delocalization The transition of of electrons from an O lone pair are somewhat weaker (at 26.2 and 22.8 kcal/mol) and result electronsstateelectrons is strongly from from an an delocalized O O lone lone pair pair with are are somewhat nearly somewhat equal weaker weaker contributions (at (at 26.2 26.2 and (~28%) and 22.8 22.8 fromkcal/mol) kcal/mol)F1, F2 and, and and result resultFA1 . in smaller, although still significant, contributions to the resonance expansion from struc‐ inWhenin smaller, smaller, the reactionalthough although is still complete,still significant, significant, the formimidic contributions contributions acid to to is the roughlythe resonance resonance 90% expansionFA1 expansionand 10% from fromFA2 struc struc. ‐ ‐ turestures F3 F3 and and F4 F4. . Table 3. NRT resonance hybrid of formamide a. TableTable 3. 3. NRT NRT resonance resonance hybrid hybrid of of formamide formamide a. a. # Structure Weight (%) Donor–Acceptor Interaction b # # StructureStructure WeightWeight (%) (%) DonoDonor–Acceptorr–Acceptor Interaction Interaction b b

F1F1 F1 43.843.843.8

N CO n n→NnNππCOπCO ** * F2F2 F2 33.733.733.7 N CO MoleculesMolecules 2021 2021, 26,, ,26 x ,,FOR x FOR PEER PEER REVIEW REVIEW (61.9)(61.9)(61.9) 9 of9 of11 11

n n→OnσOσCNσ CN** * F3F3 F3 12.112.112.1 O CN (26.2)(26.2)(26.2)

n n→OnσOσCHσ CH** * F4F4 F4 9.79.79.7 O CH (22.8)(22.8)(22.8)

a b aB3LYP/aVTZB3LYP/aVTZa B3LYP/aVTZ values.values. values.b Donor–acceptor Donor–acceptor b Donor–acceptor interactions interactions interactionsinteractions of the of natural ofthe the natural Lewis natural structure Lewis Lewis structure (F1 structure) that are (F1 isomorphic(F1) that) that are are with isomorphictheisomorphicisomorphic secondary with resonance with the the secondary structures.secondary resonance Values resonance in parentheses structures. structures. are Values second-orderValues in ininparentheses parentheses estimates are of are the second second interaction‐order‐order strength esti esti‐ ‐ matesinmates kcal/mol. of ofthe the Imagesinteraction interactioninteraction depict strength the strength favorable in ininkcal/mol. overlapskcal/mol. Images of Images the donor–acceptor depict depict the the favorable NBO favorable pairs. overlaps overlaps of ofthe the donor– donor– acceptoracceptor NBO NBO pairs. pairs.

TableTable 4 4shows shows the the corresponding corresponding analysis analysis for for formimidic formimidic acid. acid. Like Like formamide, formamide, formimidicformimidic acid acid is isfairly fairly well well described described by by four four resonance resonance structures, structures, with with weights weights total total‐ ‐ inging 98.2%. 98.2%. The The Lewis Lewis structure structure (FA1 (FA1) dominates) dominates the the resonance resonance expansion expansion at at64.2%, 64.2%, and and thethe leading leading charge charge‐transfer‐transfer form form (FA2 (FA2) at) at20.2% 20.2% arises arises from from π‐ π‐ π‐typetype delocalization delocalization of ofan an O O lonelone pair, pair, n O n, Ointo,, into the the π π πCNCN * antibond.* antibond. This This charge charge‐transfer‐transfer interaction interaction is isstabilizing stabilizing by by aboutabout 40.4 40.4 kcal/mol. kcal/mol. Two Two weaker weaker σ‐ σ‐ σ‐typetype delocalizations, delocalizations, involving involving the the N Nlone lone pair pair and and NHNH bond, bond, contribute contribute about about 14% 14% of ofthe the resonance resonance hybrid. hybrid. The The resonance resonance expansion expansion clearly clearly suggestssuggests that that resonance resonance delocalization delocalization effects effects are are somewhat somewhat weaker weaker in informimidic formimidic acid acid thanthan in informamide, formamide, which which probably probably accounts accounts for for the the greater greater stability stability (by (by ~12 ~12 kcal/mol) kcal/mol) of of thethe latter latter tautomer. tautomer.

TableTable 4. 4.NRT NRT resonance resonance hybrid hybrid of offormimidic formimidic acid acid a. a..

# # StructureStructure WeightWeight (%) (%) DonoDonor–Acceptorr–Acceptor Interaction Interaction b b

FA1FA1 64.264.2

nOnOπCNπ CN* * FA2FA2 20.220.2 (40.4)(40.4)

nNnNσCHσ CH* * FA3FA3 10.110.1 (10.7)(10.7)

Molecules 2021,, 26,, x FOR PEER REVIEW 99 ofof 1511 MoleculesMolecules 20212021,, 2626,, xx FORFOR PEERPEER REVIEWREVIEW 99 ofof 1111 Molecules 2021, 26, x FOR PEER REVIEW 9 of 11

nO→σCN * F3 12.1 nnOOσσCNCN * * F3F3 12.112.1 nO(26.2)σCN * F3 12.1 (26.2)(26.2) (26.2)

nO→σCH * F4 9.7 nnOOσσCHCH * * F4F4 9.79.7 nO(22.8)σCH * F4 9.7 (22.8)(22.8) (22.8)

a B3LYP/aVTZ values. b Donor–acceptor interactions of the natural Lewis structure (F1) that are aa B3LYP/aVTZ B3LYP/aVTZ values.values. bb Donor–acceptorDonor–acceptor interactionsinteractions ofof thethe naturalnatural LewisLewis structurestructure ((F1F1)) thatthat areare a B3LYP/aVTZisomorphic with values. the secondaryb Donor–acceptor resonance interactions structures. of Values the natural in parentheses Lewis structure are second (F1) that‐order are esti ‐ isomorphicisomorphic withwith thethe secondarysecondary resonanceresonance structures.structures. ValuesValues inin parenthesesparentheses areare secondsecond-ordersecond‐‐orderorder estiesti-esti‐‐ isomorphicmates of the with interaction the secondary strength resonance in kcal/mol. structures. Images Values depict inthe parentheses favorable overlaps are second of the‐order donor– esti‐ matesmates ofof thethe interactioninteraction strengthstrength inin kcal/mol.kcal/mol. ImagesImages depictdepict thethe favorablefavorable overlapsoverlaps ofof thethe donor–donor– matesacceptor of the NBO interaction pairs. strength in kcal/mol. Images depict the favorable overlaps of the donor– acceptoracceptor NBONBO pairs.pairs. acceptor NBO pairs. Table 4 shows the corresponding analysis for formimidic acid. Like formamide, TableTableTable 444 showsshowsshows thethethe correspondingcorrespondingcorresponding analysisanalysisanalysis forforfor formimidicformimidicformimidic acid.acid.acid. LikeLikeLike formamide,formamide,formamide, formimidic acid is fairly well described by four resonance structures, with weights total-total‐ formimidicformimidicformimidic acid acidacid is isis fairly fairlyfairly well wellwell described describeddescribed by byby four fourfour resonance resonanceresonance structures, structures,structures, with withwith weights weightsweights total totaltotal‐‐‐ ing 98.2%. The Lewis structure (FA1) dominates the resonance expansion at 64.2%, and inginging 98.2%. 98.2%.98.2%. The TheThe Lewis LewisLewis structure structurestructure ( (FA1(FA1FA1)) ) dominates dominatesdominates the thethe resonance resonanceresonance expansion expansionexpansion at atat 64.2%, 64.2%,64.2%, and andand thethe leadingleading chargecharge-transfercharge‐‐transfertransfer formform ((FA2FA2)) atat 20.2%20.2% arisesarises fromfrom π‐ π‐π-typetypetype delocalizationdelocalization ofof anan OO thelone leading pair, nchargeO, into‐transfer the πCN form * antibond. (FA2) at This 20.2% charge arises‐transfer from π‐ interactiontype delocalization is stabilizing of an Oby lonelone pair,pair, nnOO,, intointo thethe π ππCNCN ** antibond.antibond. ThisThis chargecharge-transfercharge‐‐transfertransfer interactioninteraction isis stabilizingstabilizing byby lone pair, nO, into the πCN * antibond. This charge‐transfer interaction is stabilizing by about 40.4 kcal/mol. Two weaker σ‐σ-typetype delocalizations, involving the N lone pair and aboutaboutabout 40.4 40.440.4 kcal/mol. kcal/mol.kcal/mol. Two TwoTwo weaker weakerweaker σ‐ σ‐ σ‐typetypetype delocalizations, delocalizations,delocalizations, involving involvinginvolving the thethe N NN lone lonelone pair pairpair and andand NH bond, contribute about 14% of the resonaresonancence hybrid. The resonance expansion clearly NHNHNH bond, bond,bond, contribute contributecontribute about aboutabout 14% 14%14% of ofof the thethe resonance resonanceresonance hybrid. hybrid.hybrid. The TheThe resonance resonanceresonance expansion expansionexpansion clearly clearlyclearly Molecules 2021, 26, 4110 suggests that resonance delocalization effects are somewhat weaker in formimidic 10acid of 15 suggestssuggestssuggests that thatthat resonance resonanceresonance delocalization delocalizationdelocalization effects effectseffects are areare somewhat somewhatsomewhat weaker weakerweaker in inin formimidic formimidicformimidic acid acidacid Molecules 2021, 26, x FOR PEER REVIEWthan in formamide, which probably accounts for the greater stability (by ~12 kcal/mol) of 10 of 12 thanthanthan in inin formamide, formamide,formamide, which whichwhich probably probablyprobably accounts accountsaccounts for forfor the thethe greater greatergreater stability stabilitystability (by (by(by ~12 ~12~12 kcal/mol) kcal/mol)kcal/mol) of ofof the latter tautomer. thethethe latter latterlatter tautomer. tautomer.tautomer. Table 4. NRT resonance hybrid of formimidic acid a. Table 4. NRT resonance hybrid of formimidic acid a. TableTable 4.4. NRTNRT resonanceresonance hybridhybrid ofof formimidicformimidic acidacid aa.. Table 4. NRT resonance hybrid of formimidic acid a. # Structure Weight (%) Donor–Acceptor Interaction b # Structure Weight (%) Donor–Acceptor Interaction b ## StructureStructure WeightWeight (%)(%) DonoDonorr–Acceptor–Acceptor InteractionInteraction bb # Structure Weight (%) Donor–Acceptor Interaction b

FA1 64.2 FA1FA1FA1 64.2 64.264.2 FA1FA1 64.264.2

σNHσCO * FA4 3.7 (10.6)

nOπCN * FA2 20.2 nO→nnπOOCN→ππ*CNCN * * FA2FA2FA2 20.2 20.220.2 nnOO(40.4)ππCNCN * * FA2FA2 20.220.2 (40.4)(40.4)(40.4) (40.4)(40.4)

a B3LYP/aVTZ values. b Donor–acceptor interactions of the natural Lewis structure (FA1) that are isomorphic with the secondary resonance structures. Values in parentheses are second‐order esti‐ nNσCH * FA3mates of the interaction strength10.1 in kcal/mol.nnNN→σσCHCH * * Images depict the favorable overlaps of the donor– FA3FA3 10.110.1 nN→nnσNNCH(10.7)σσCH*CH * * FA3FA3 FA3 10.110.110.1 (10.7)(10.7) (10.7)(10.7)(10.7) Molecules 2021,, 26,, x FOR PEER REVIEWacceptor NBO pairs. 10 of 12

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. σ →σNHσ σ*CO * FA4IgnoringFA4 the proton transfer,3.7 3.7 theNH principalCO geometry changes during tautomerization are (10.6)(10.6)(10.6) 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

aa from 1.877 tobb 1.160) and gain of CN double‐bond character (increasing from 1.223 to 1.948) B3LYP/aVTZa B3LYP/aVTZ values. values. Donor–acceptorb Donor–acceptor interactionsinteractions of of the thethe natural natural Lewis Lewis structure structure (FA1 )((FA1 that are)) thatthat isomorphic are isomorphicisomorphicwithas the the secondary with resonance thethe resonancesecondary description structures. resonance Values structures. in morphs parentheses Values in arein from parentheses second-order mostly are estimates second F1 of‐‐ order theto interactionpredominantly esti‐‐ FA1. matesstrength of thethe in kcal/mol.interactioninteraction Images strength depict inin the kcal/mol. favorable ImagesImages overlaps depict of the thethe donor–acceptor favorablefavorable overlaps NBO pairs. of thethe donor– acceptor NBO pairs.

Natural bond orders for formamide and formimidic acid are shown inin Figure 5, along with the optimized bond lengths.lengths. These bond orders are weighted‐averages of the integerinteger 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 lengtheninglengthening of the CO bond (by 0.135 Å) and shortening of the CN bond (by 0.097 Å). These changes result from the lossloss 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..

Figure 5. Natural 5. Natural bond orders bond and optimizedorders bondand lengths optimized (in parentheses, bond in Å)lengths for formamide (in parentheses, and in Å) for formamide formimidic acid. and formimidic acid.

Figure 5. NaturalWe examined bond orders and tautomerization optimized bond lengthslengths more (in(in parentheses, fully byinin Å) performing forfor formamideformamide NRT analysis at geometries andacross formimidicformimidic the acid. reaction profile of Figure 4. To simplify the analysis, we limited the NRT ex‐

pansionWe examined to onlytautomerization four resonance more fully by forms,performing including NRT analysis the at geometries two dominant structures of forma‐ acrossmide the (reactionF1 and profile F2) ofand Figure the 4. twoTo simplify dominant the analysis, structures we limitedlimited of the formimidic NRT ex‐ acid (FA1 and FA2) [84]. pansion to only four resonance forms, includingincluding the two dominant structures of forma‐ mideThese (F1 and four F2) and structures the two dominant alone structures constitute of formimidic the minimal acid (FA1 andset FA2required) [84]. to simultaneously describe Thesebond four structuresbreaking/formation alone constitute the during minimal set proton required totransfer simultaneously and describe resonance effects in the π system bond breaking/formation during proton transfer and resonance effects inin the π π system (neglecting(neglecting weaker σ‐ σ‐weakertype resonance σ‐type contributions). resonance Figure contributions). 6 shows the dependence Figure of 6 shows the dependence of thethe resonance resonance weights onweights reaction coordinate. on reaction coordinate.

Molecules 2021, 26, x FOR PEER REVIEW 11 of 13

Molecules 2021, 26, 4110 11 of 15

Figure 6. Variation in resonance weights along the tautomerization IRC.

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 prod‐ uct 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 correlationsFigure 6. Variation around in resonance each weights integer along (single the tautomerization dominant IRC. NLS) or half‐integer

(two‐state bond‐shiftFigure [85])Figure bond6. 7Variation shows order the correlation within resonance connecting of natural weights bond curvatures order withalong bond to the length accommodate tautomerization for geometries the IRC. slightly different slopesacross of thedifferent IRC. The correlationsbond types, reveal but slight their S-shaped essential curvatures, linearity or more is specifically, suggested by the robust |χ|2 coefficients.near-perfect linear Such correlations correlations around eachstrongly integer (singlesupport dominant the NLS)useful or half-integer predictive (two-stateF → bond-shift FA conversion [85]) bond order begins with connecting with curvaturesformamide to accommodate electron the density described by F1 and F2 associations of NRT bondslightly orders different slopeswith ofexperimentally different bond types, measurable but their essential quantities, linearity is suggestedconsistent in roughly 80%:20%2 proportion. As proton transfer begins π resonance strengthens as the with well‐known empiricalby the robust relationships |χ| coefficients. connecting Such correlations a variety strongly of supportbond properties, the useful predictive includ‐ F2associations contribution of NRT bond increases. orders with experimentallyNote that measurableF2 has the quantities, same consistent N=C‐O bonding pattern as the prod‐ ing bond lengths [86–88],with well-known bond energies empirical relationships [89–92], connectingIR vibration a variety frequencies of bond properties, [93,94], including and NMR spin‐spin couplinguctbond Lewis constants lengths [structure86–88 [95].], bond energies FA1, [ 89although–92], IR vibration the frequencies latter only [93,94 ],begins and NMR to contribute importantly to the resonancespin-spin coupling hybrid constants within [95]. 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, includ‐ ing bond lengths [86–88], bond energies [89–92], IR vibration frequencies [93,94], and Figure 7. Bond order–bond length correlations for CO (circles), CN (squares), OH (filled circles), and NH (filled squares) Figurebonds, 7. showing Bond theorder–bond least-squaresNMR regression length spin line‐correlationsspin and corresponding coupling for CO Pearson constants (circles), |χ|2 correlation CN [95]. (squares), coefficient for OH each bond(filled type. circles), and NH (filled squares) bonds, showing the least‐squares regression line and corresponding Pear‐ son |χ|2 correlation coefficient for each bond type.

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 elec‐ tron‐pushing, curly‐arrow representation that the bench chemist would use to depict the

Figure 7. Bond order–bond length correlations for CO (circles), CN (squares), OH (filled circles), and NH (filled squares) bonds, showing the least‐squares regression line and corresponding Pear‐ son |χ|2 correlation coefficient for each bond type.

Molecules 2021, 26, x FOR PEER REVIEW 12 of 14

Molecules 2021, 26, 4110 12 of 15 reaction mechanism (Figure 8). Red arrows correspond to the bond/lone pair rearrange‐ Variations in the NRT weights for the four resonance structures across the IRC, as well asment concomitant associated changes in natural with bond orders, proton are entirely migration, consistent with the electron-and blue arrows represent the change in π elec‐ pushing, curly-arrow representation that the bench chemist would use to depict the reaction mechanismtron distribution (Figure8). Red arrows of correspond the peptide to the bond/lone bond. pair rearrangement associated with proton migration, and blue arrows represent the change in π electron distribution of the peptide bond.

Figure 8. Curly-arrow depiction of resonance-type electronic delocalizations in formamide tautomerization. Figure 8. Curly‐arrow depiction of resonance‐type electronic delocalizations in formamide tau‐ With this simple example, we have shown that NRT analysis provides a tool for easily obtainingtomerization. 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 representativeWith this of those simple obtained from example, numerically complex we quantumhave chemical shown that NRT analysis provides a tool for eas‐ wavefunctions at any reasonably current computational level.

5.ily Summary obtaining and Conclusions compact and chemical intuitive descriptors of molecular structure and reac‐ tivityContrary that to skepticism are fully that is sometimes consistent expressed [45 with,54,58], we the believe prescient that the mesomerism/resonance insights of Pau‐ present results conﬁrm the essential correctness and usefulness of Pauling’s hybridization andling, resonance Robinson, concepts, as consistently Ingold, found in NBO/NRTand other analysis ofbonding wavefunctions from pioneers, dating back to the pre‐quantum me‐ the best currently available quantum chemical methods. If anything, improved quantitative accuracychanical of the wavefunction era. tendsSimilar to enhance admirationto the of Pauling’s hybridization powerful intuitions, results presented above, the present developed long before numerically reliable solutions of Schrödinger’s equation became routinelyB3LYP/aVTZ available. results are fully representative of those obtained from numerically complex In closing this tribute, it may be appropriate to relate that E. Bright Wilson considered Johnquantum von Neumann chemical and Linus Pauling wavefunctions to be the only two authentic geniuses at any he ever reasonably met. current computational level. 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) all5. IRC Summary geometries, and (iii) and a sample Conclusions Gaussian input file, with NBO/NRT input for one of the IRC geometries. Author Contributions:ContraryF.W. conceived to skepticism this study; E.D.G. performed that the calculations;is sometimes E.D.G. and expressed [45,54,58], we believe that the F.W. analyzed the results; F.W. and E.D.G. wrote the paper. All authors have read and agreed to the publishedpresent version results of the manuscript. confirm the essential correctness and usefulness of Pauling’s hybridization Funding: Support for computational facilities at ISU was provided by the College of Arts and Sciencesand and resonance Office of Information Technology.concepts, Support foras computational consistently facilities at UW-Madison found in NBO/NRT analysis of wavefunctions wasfrom provided the in part bybest National Sciencecurrently Foundation Grant available CHE-0840494. quantum chemical methods. If anything, improved Data Availability Statement: The data presented in this study are available in Supplementary Materials. Conflictsquantitative of Interest: The authors accuracy declare no conflict of of the interest. wavefunction tends to enhance admiration of Pauling’s pow‐ Sampleerful Availability: intuitions,Not applicable. 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 in‐ put 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 Sci‐ ences 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 Ma‐ terials. Conflicts of Interest: The authors declare no conflict of interest. Sample Availability: Not applicable.

References

1. Seiler, P.; Weisman, G.R.; Glendening, E.D.; Weinhold, F.; Johnson, V.B.; Dunitz, J.D. Observation of an Eclipsed C(sp3)‐CH3 Bond in a Tricyclic Orthoamide: Experimental and Theoretical Evidence for C‐H...O Hydrogen Bonds. Angew. Chem. Int. Ed. 1987, 26, 1175–1177. 2. Glendening, E.D.; Faust, R.; Streitwieser, A.; Vollhardt, K.P.C.; Weinhold, F. On the Role of Delocalization in Benzene. J. Am. Chem. Soc. 1993, 115, 10952–10957.

Molecules 2021, 26, 4110 13 of 15

References 3 1. Seiler, P.; Weisman, G.R.; Glendening, E.D.; Weinhold, F.; Johnson, V.B.; Dunitz, J.D. Observation of an Eclipsed C(sp )-CH3 Bond in a Tricyclic Orthoamide: Experimental and Theoretical Evidence for C-H··· O Hydrogen Bonds. Angew. Chem. Int. Ed. 1987, 26, 1175–1177. [CrossRef] 2. Glendening, E.D.; Faust, R.; Streitwieser, A.; Vollhardt, K.P.C.; Weinhold, F. On the Role of Delocalization in Benzene. J. Am. Chem. Soc. 1993, 115, 10952–10957. [CrossRef] 3. Suidan, L.; Badenhoop, J.K.; Glendening, E.D.; Weinhold, F. Common Textbook and Teaching Misrepresentations of Lewis Structures. J. Chem. Educ. 1995, 72, 583–586. [CrossRef] 4. Glendening, E.D.; Weinhold, F. Natural Resonance Theory. I. General Formulation. J. Comput. Chem. 1998, 19, 593–609. [CrossRef] 5. Glendening, E.D.; Weinhold, F. Natural Resonance Theory. II. Natural Bond Order and Valency. J. Comput. Chem. 1998, 19, 610–627. [CrossRef] 6. Glendening, E.D.; Badenhoop, J.K.; Weinhold, F. Natural Resonance Theory. III. Chemical Applications. J. Comput. Chem. 1998, 19, 628–646. [CrossRef] 7. Pauling, L.; Wilson, E.B., Jr. Introduction to Quantum Mechanics; McGraw-Hill: New York, NY, USA, 1935. 8. Pauling, L. Nature of the Chemical Bond, 3rd ed.; Cornell University Press: Ithaca, NY, USA, 1960. 9. Heitler, W.; London, F. Wechselwirkung neutraler Atome und homöpolare Bindung nach der Quantenmechanik. Z. Phys. 1927, 44, 455–472. [CrossRef] 10. Pauling, L. The Application of the Quantum Mechanics to the Structure of the Hydrogen Molecule and Hydrogen Molecule-Ion and to Related Problems. Chem. Rev. 1928, 5, 173–213. [CrossRef] 11. Pauling, L. The nature of the chemical bond. Application of results obtained from the quantum mechanics and from a theory of paramagnetic susceptibility to the structure of molecules. J. Am. Chem. Soc. 1931, 53, 1367–1400. [CrossRef] 12. Slater, J.C. Directed valence in polyatomic molecules. Phys. Rev. 1931, 37, 481–497. [CrossRef] 13. Pauling, L. The theory of resonance in chemistry. Proc. R. Soc. Lond. Ser. A 1977, 356, 433–441. 14. Todd, A.R.; Cornforth, J.W. Robert Robinson, 13 September 1886–8 February 1975. Biogr. Mem. Fellows R. Soc. Lond. 1976, 22, 415–527. 15. Hund, F. Zur Deutung einiger Erscheinungen in den Molekelspektre. Z. Physik 1926, 36, 657–674. [CrossRef] 16. Mulliken, R.S. The Assignment of Quantum Numbers for Electrons in Molecules. I. Phys. Rev. 1928, 32, 186–222. [CrossRef] 17. Lennard-Jones, J.E. The Electronic Structure of Some Diatomic Molecules Trans. Faraday Soc. 1929, 25, 668–686. [CrossRef] 18. Hartree, D.R. The Wave Mechanics of an Atom with a Non-Coulomb Central Field. Part I. Theory and Method. Proc. Camb. Philos. Soc. 1928, 24, 89–110. [CrossRef] 19. Fock, V. Näherungsmethode zur Lösung des quantenmechanischen Mehrkörperproblems. Z. Phys. 1930, 61, 126–148. [CrossRef] 20. Slater, J.C. Note on Hartree’s Method. Phys. Rev. 1930, 35, 210–211. [CrossRef] 21. Coulson, C.A. Self-consistent ﬁeld for molecular hydrogen. Proc. Camb. Philos. Soc. 1938, 34, 204–212. [CrossRef] 22. Roothaan, C.C.J. New Developments in Molecular Orbital Theory. Rev. Mod. Phys. 1951, 23, 69–89. [CrossRef] 23. Mulliken, R.S. Spectroscopy, molecular orbitals, and chemical bonding (Nobel Prize, 1966). Science 1967, 157, 13–24. [CrossRef] 24. Norbeck, J.M.; Gallup, G.A. Valence-bond calculation of the electronic structure of benzene. J. Am. Chem. Soc. 1974, 96, 3386–3393. [CrossRef] 25. Goddard, W.A., III. Improved Quantum Theory of Many-Electron Systems. I. Construction of Eigenfunctions of S2 Which Satisfy Pauli’s Principle. Phys. Rev. 1967, 157, 73–78. [CrossRef] 26. Goddard, W.A., III; Dunning, T.H., Jr.; Hunt, W.J.; Hay, P.J. Generalized valence bond description of bonding in low-lying states of molecules. Acc. Chem. Res. 1973, 6, 368–376. [CrossRef] 27. Dunning, T.H.; Hay, P.J. Beyond Molecular Orbital Theory: The Impact of Generalized Valence Bond Theory in Molecular Science. In Computational Materials, Chemistry, and Biochemistry: From Bold Initiatives to the Last Mile; Shankar, S., Muller, R., Dunning, T., Chen, G.H., Eds.; Springer: Cham, Switzerland, 2021; Volume 284, pp. 55–87. 28. Foresman, J.B.; Frisch, A.E. Exploring Chemistry with Electronic Structure Methods, 3rd ed.; Gaussian, Inc.: Wallingford, CT, USA, 2015. 29. van’t Hoff, J.H. Sur les formules de structure dans l’espace. Arch. Neerl. Sci. Exactes Nat. 1874, 9, 445–454. 30. LeBel, J.-A. Sur Les Relations Qui Existent Entre Les Formules Atomiques Des Corps Organiques et Le Pouvoir Rotatoire de Leurs Dissolutions. Bull. Soc. Chim. Paris 1874, 22, 337–347. 31. Ramberg, P.J. Chemical Structure, Spatial Arrangement: The Early History of Stereochemistry, 1874–1914; Routledge: Abingdon, UK, 2017. 32. Bragg, W.H.; Bragg, W.L. The structure of the diamond. Nature 1913, 91, 557. [CrossRef] 33. Wierl, R. Elektronenbeugung und Molekülbau. Ann. Phys. 1931, 8, 521–564. [CrossRef] 34. Jensen, W.P.; Palenik, G.J.; Suh, I.-H. The history of molecular structure determination viewed through the Nobel Prizes. J. Chem. Educ. 2003, 80, 753–761. 35. Coulson, C.A. Valence, 2nd ed.; Oxford University Press: Oxford, UK, 1965. 36. Kermack, W.O.; Robinson, R.R. An explanation of the property of induced polarity of atoms and an interpretion of the theory of partial valencies on an electronic basis. J. Chem. Soc. Trans. 1922, 121, 427–440. [CrossRef] 37. Lowry, T.M. Studies of electrovalency. Part I. The polarity of double bonds. J. Chem. Soc. Trans. 1923, 123, 822–831. [CrossRef] Molecules 2021, 26, 4110 14 of 15

38. Allan, J.; Oxford, A.F.; Robinson, R.; Smith, J.C. The relative directive powers of groups of the forms RO and RR’N in aromatic substitution. Part III. The nitration of some p-alkoxyanisoles. J. Chem. Soc. 1926, 129, 401–411. [CrossRef] 39. Ingold, C.K.; Ingold, E.H. The nature of the alternating effect in carbon chains. Part V. A discussion of aromatic substitution with special reference to the respective roles of polar and non-polar dissociation; and a further study of the relative directive efficiencies of oxygen and nitrogen. J. Chem. Soc. 1926, 129, 1310–1328. 40. Ingold, C.K. Mesomerism and tautomerism. Nature 1934, 133, 946–947. [CrossRef] 41. Ingold, C.K. Principles of an electronic theory of organic reactions. Chem. Rev. 1934, 15, 225–274. [CrossRef] 42. Saltzman, M.D. CK Ingold’s development of the concept of mesomerism. Bull. Hist. Chem. 1996, 19, 25–32. 43. Kerber, R.C. If it’s resonance, what is resonating? J. Chem. Educ. 2006, 83, 223–227. [CrossRef] 44. Coulson, C.A.; Fischer, I. Notes on the Molecular Orbital Treatment of the Hydrogen Molecule. Phil. Mag. 1949, 40, 386–393. [CrossRef] 45. Grushow, A. Is it time to retire the hybrid orbital? J. Chem. Educ. 2011, 88, 860–862. [CrossRef] 46. Tro, N.J. Retire the Hybrid Atomic Orbital? Not So Fast. J. Chem. Educ. 2012, 89, 567–568. [CrossRef] 47. DeKock, R.L.; Strikwerda, J.R. Retire Hybrid Atomic Orbitals? J. Chem. Educ. 2012, 89, 569. [CrossRef] 48. Landis, C.R.; Weinhold, F. Comments on ‘Is It Time To Retire the Hybrid Atomic Orbital?’. J. Chem. Educ. 2012, 89, 570–572. [CrossRef] 49. Truhlar, D.G. Are Molecular Orbitals Delocalized? J. Chem. Educ. 2012, 89, 573–574. [CrossRef] 50. Hiberty, P.C.; Volatron, F.; Shaik, S. In Defense of the Hybrid Atomic Orbitals. J. Chem. Educ. 2012, 89, 575–577. [CrossRef] 51. Truhlar, D.G.; Hiberty, P.C.; Shaik, S.; Gordon, M.S.; Danovich, D. Orbitals and the Interpretation of Photoelectron Spectroscopy and (e,2e) Ionization Experiments. Angew. Chem. Int. Ed. 2019, 58, 12332–12338. [CrossRef][PubMed] 52. Graham, L.R. A Soviet Marxist view of structural chemistry: The theory of resonance controversy. Isis 1964, 55, 20–31. [CrossRef] 53. The Soviet Resonance Controversy. Available online: https://paulingblog.wordpress.com/2009/03/26/paulings-theory-of- resonance-a-soviet-controversy/ (accessed on 28 May 2021). 54. Frenking, G.; Krapp, A. Unicorns in the world of chemical bonding models. J. Comput. Chem. 2007, 28, 15–24. [CrossRef] 55. Landis, C.R.; Hughes, R.P.; Weinhold, F. Bonding Analysis of TM(cAAC)2 (TM = Cu, Ag, Au) and the Importance of Reference State. Organometallics 2015, 34, 3442–3449. [CrossRef] 56. Landis, C.R.; Hughes, R.P.; Weinhold, F. Comment on ‘Observation of alkaline earth complexes M(CO)8 (M = Ca, Sr. or Ba) that mimic transition metals’. Science 2019, 365, 553. [CrossRef] 57. Xu, L.T.; Thompson, J.V.K.; Dunning, T.H., Jr. Spin-coupled generalized valence bond description of group 14 species: The carbon, silicon and germanium hydrides, XHn (n = 1–4). J. Phys. Chem. A 2019, 123, 2401–2419. [CrossRef] 58. Xu, L.T.; Dunning, T.H., Jr. Orbital hybridization in modern valence bond wave functions: Methane, ethylene, and acetylene. J. Phys. Chem. A 2020, 124, 204–214. [CrossRef] 59. Foster, J.M.; Boys, S.F. Canonical configurational interaction procedure. Rev. Mod. Phys. 1960, 32, 300–302. [CrossRef] 60. Edmiston, C.; Ruedenberg, K. Localized atomic and molecular orbitals. Rev. Mod. Phys. 1963, 35, 457–464. [CrossRef] 61. Pipek, J.; Mezey, P.G. A fast intrinsic localization procedure applicable for ab initio and semiempirical linear combination of atomic orbital wave functions. J. Chem. Phys. 1989, 90, 4916–4926. [CrossRef] 62. Khaliullin, R.Z.; Head-Gordon, M.; Bell, A.T. An efficient self-consistent field method for large systems of weakly interacting components. J. Chm. Phys. 2006, 124, 204105. [CrossRef] 63. Jansík, B.; Høst, S.; Kristensen, K.; Jørgensen, P. Local orbitals by minimizing powers of the orbital variance. J. Chem. Phys. 2011, 134, 194104. [CrossRef][PubMed] 64. Glendening, E.D.; Landis, C.R.; Weinhold, F. Natural Bond Orbital Methods. WIREs Comput. Mol. Sci. 2012, 2, 1–42. [CrossRef] 65. Weinhold, F.; Landis, C.R.; Glendening, E.D. What is NBO Analysis and How is it Useful? Int. Rev. Phys. Chem. 2016, 35, 399–440. [CrossRef] 66. Glendening, E.D.; Landis, C.R.; Weinhold, F. Natural Bond Orbital Theory: Discovering Chemistry with NBO7. In Complementary Bonding Analysis; Grabowsky, S., Ed.; De Gruyer: Amsterdam, The Netherlands, 2021. 67. Weinhold, F.; Landis, C.R. Discovering Chemistry with Natural Bond Orbitals; John Wiley: Hoboken, NJ, USA, 2012. 68. Glendening, E.D.; Landis, C.R.; Weinhold, F. Resonance Theory Reboot. J. Am. Chem. Soc. 2019, 141, 4156–4166. [CrossRef] 69. Glendening, E.D.; Badenhoop, J.K.; Reed, A.E.; Carpenter, J.E.; Bohmann, J.A.; Morales, C.M.; Karafiloglou, P.; Landis, C.R.; Weinhold, F. NBO 7.0; Theoretical Chemistry Institute, University of Wisconsin: Madison, WI, USA, 2018. 70. Glendening, E.D.; Landis, C.R.; Weinhold, F. NBO 7.0: New Vistas in Localized and Delocalized Chemical Bonding Theory. J. Comput. Chem. 2019, 40, 2234–2241. [CrossRef][PubMed] 71. NBO. Affiliated Electronic Structure Systems and Protected Interfaces. Available online: https://nbo7.chem.wisc.edu/affil_css. htm (accessed on 2 July 2021). 72. Frisch, M.J.; Trucks, G.W.; Schlegel, H.B.; Scuseria, G.E.; Robb, M.A.; Cheeseman, J.R.; Scalmani, G.; Barone, V.; Petersson, G.A.; Nakatsuji, H.; et al. Gaussian-16, Revision C.01; Gaussian, Inc.: Wallingford, CT, USA, 2019. 73. Cooper, D.L.; Thorsteinsson, T.; Gerratt, J. Fully variational optimization of modern VB wave functions using the CASVB strategy. Int. J. Quant. Chem. 1997, 65, 439–451. [CrossRef] 74. Cooper, D.L.; Thorsteinsson, T.; Gerratt, J. Modern VB representations of CASSCF wave functions and the fully-variational optimization of modern VB wave functions using the CASVB strategy. Adv. Quant. Chem. 1998, 32, 51–67. Molecules 2021, 26, 4110 15 of 15

75. Werner, H.-J.; Knowles, P.J.; Knizia, G.; Manby, F.R.; Schütz, M. Molpro: A general-purpose quantum chemistry program package. WIREs Comput. Mol. Sci. 2012, 2, 242–253. [CrossRef] 76. Werner, H.-J.; Knowles, P.J.; Manby, F.R.; Black, J.A.; Doll, K.; Heßelmann, A.; Kats, D.; Köhn, A.; Korona, T.; Kreplin, D.A.; et al. The Molpro quantum chemistry package. J. Chem. Phys. 2020, 152, 144107. [CrossRef][PubMed] 77. Werner, H.-J.; Knowles, P.J.; Knizia, G.; Manby, F.R.; Schütz, M.; Celani, P.; Györffy, W.; Kats, D.; Korona, T.; Lindh, R.; et al. MOLPRO, 2021.1, A Package of Ab Initio Programs; Technologie-Transfer-Initiative GmbH an der Universität Stuttgart: Stuttgart, Germany, 2021. 78. Reed, A.E.; Weinstock, R.B.; Weinhold, F. Natural Population Analysis. J. Chem. Phys. 1985, 83, 735–746. [CrossRef] 79. Bent, H.A. Distribution of atomic s character in molecules and its chemical implications. J. Chem. Educ. 1960, 37, 616–624. [CrossRef] 80. Bent, H.A. An appraisal of valence-bond structures and hybridization in compounds of the ﬁrst-row elements. Chem. Rev. 1961, 61, 275–311. [CrossRef] 81. Alabugin, I.V.; Bresch, S.; Manoharan, M. Hybridization trends for main group elements and the Bent’s rule beyond carbon: More than electronegativity. J. Phys. Chem. A 2014, 118, 3663–3677. [CrossRef] 82. Eisenberg, D. The discovery of the α-helix and β-sheet, the principal structural features of proteins. Proc. Natl. Acad. Sci. USA 2003, 30, 11207–11210. [CrossRef] 83. Hazra, M.K.; Chakraborty, T. Formamide tautomerization: Catalytic role of formic acid. J. Phys. Chem. A 2005, 109, 7621–7625. [CrossRef] 84. Weinhold, F.; Glendening, E.D. NBO Program Manual; Theoretical Chemistry Institute, University of Wisconsin: Madison, WI, USA, 1996; p. B-82ff. Available online: https://nbo7.chem.wisc.edu/nboman.pdf (accessed on 2 July 2021). 85. Jiao, Y.; Weinhold, F. NBO/NRT two-state theory of bond-shift spectral excitation. Molecules 2020, 25, 4052. [CrossRef] 86. Coulson, C.A. The electronic structure of some polyenes and aromatic molecules. VII. Bonds of fractional order by the molecular orbital method. Proc. R. Soc. Lond. A 1939, 169, 413–428. 87. Coulson, C.A. Bond lengths in conjugated molecules: The present position. Proc. R. Soc. Lond. A 1951, 207, 91–100. 88. Mishra, P.C.; Rai, D.K. Bond order-bond length relationship in all-valence-electron molecular orbital theory. Mol. Phys. 1972, 23, 631–634. [CrossRef] 89. Johnston, H.S.; Parr, C. Activation energies from bond energies. I. Hydrogen transfer reactions. J. Am. Chem. Soc. 1963, 85, 2544–2551. [CrossRef] 90. Bent, H.A. Structural chemistry of donor-acceptor interactions. Chem. Rev. 1968, 68, 587–648. [CrossRef] 91. Bürgi, H.; Dunitz, J.D. Fractional bonds: Relations among their lengths, strengths, and stretching force constants. J. Am. Chem. Soc. 1987, 109, 2924–2926. [CrossRef] 92. Johnstone, R.A.W.; Loureiro, R.M.S.; Lurdes, M.; Cristiano, S.; Labat, G. Bond energy/bond order relationships for N-O linkages and a quantitative measure of ionicitiy: The role of nitro groups in hydrogen-bonding. ARKIVOC 2010, 2010, 142–169. [CrossRef] 93. Badger, R.M. A relation between internuclear distances and bond force constants. J. Chem. Phys. 1934, 2, 128–131. [CrossRef] 94. Boyer, M.A.; Marsalek, O.; Heindel, J.P.; Markland, T.E.; McCoy, A.B.; Xantheas, S.S. Beyond Badger’s rule: The origins and generality of the structure-spectra relationship of aqueous hydrogen bonds. J. Phys. Chem. Lett. 2019, 10, 918–924. [CrossRef] 95. Gründemann, S.; Limbach, H.-H.; Buntkowsky, G.; Sabo-Etienne, S.; Chaudret, B. Distance and scalar HH-coupling correlations in transition metal dihydrides and dihydrogen complexes. J. Phys. Chem. A 1999, 103, 4752–4754. [CrossRef]