Resonance Theory Reboot Eric D. Glendening,a Clark R. Landis,b

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Resonance Theory Reboot Frank Weinhold, Clark R. Landis, and Eric D Glendening J. Am. Chem. Soc., Just Accepted Manuscript • DOI: 10.1021/jacs.8b12336 • Publication Date (Web): 11 Feb 2019 Downloaded from http://pubs.acs.org on February 11, 2019

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Journal of the American Chemical Society

Resonance Theory Reboot Eric D. Glendening,a Clark R. Landis,b and Frank Weinhold*b aDepartment

of Chemistry and Physics, Indiana State University, Terre Haute, Indiana 47809; bDepartment of Chemistry, University of Wisconsin-Madison, Madison, Wisconsin 53706. ABSTRACT

What is now called “resonance theory” has a long and conflicted history. We first sketch the early roots of resonance theory, its heritage of diverse physics and chemistry conceptions, and its subsequent rise to reigning chemical bonding paradigm of the mid-20th-century. We then outline the alternative “natural” pathway to localized Lewis- and resonance-structural conceptions that was initiated in the 1950s, given semi-empirical formulation in the 1970s, recast in ab initio form in the 1980s, and successfully generalized to multi-structural “natural resonance theory” (NRT) form in the 1990s. Although earlier numerical applications were often frustrated by ineptness of then-available numerical solvers, the NRT variational problem was recently shown to be amenable to highly efficient convex programming methods that yield provably optimal resonance resonance weightings at a small fraction of previous computational costs. Such convexity-based algorithms now allow a full “reboot” of NRT methodology for tackling a broad range of chemical applications, including the many familiar resonance phenomena of organic and biochemistry as well as the still broader range of resonance attraction effects in the inorganic domain. We illustrate these advances for prototype chemical applications, including (i) stable nearequilibrium species, where resonance mixing typically provides only small corrections to a dominant Lewis-structural picture, (ii) reactive transition-state species, where strong resonance mixing of reactant and product bonding patterns is inherent, (iii) coordinative and related supramolecular interactions of the inorganic domain, where sub-integer resonance bond orders are the essential origin of intermolecular attraction, and (iv) exotic long-bonding and metallic delocalization phenomena, where no single “parent” Lewis-structural pattern gains preeminent weighting in the overall resonance hybrid.

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INTRODUCTION: PIONEER RESONANCE-THEORETIC CONCEPTIONS What can now be recognized as “resonance” concepts in chemistry can be traced to the triumphs of Kekulé’s 19th-century structural theory1 and the consternation provoked by its occasional failures in aberrant species such as benzene. Although disubstitution of either of the two equivalent Kekulé structures of benzene was expected to yield six distinct isomers, the non-existence of all but three such isomers implies a type of equivalence or “averaging” between assigned single- vs. double- linkages of the alternative structural diagrams. As the century turned, Thomson’s discovery of the electron2 and Bohr’s electronic rationalization of atom structure and chemical periodicity3 brought important physical insights into the chemical domain. In particular, these insights inspired G. N. Lewis’s remarkable recognition4 of shared electron pairs as the essential linkages of Kekulétype structural diagrams. The resulting Lewis-structural dot diagrams successfully rationalized the atomic ratios and other properties of many closed-shell molecules, providing the foundation of chemical bonding pedagogy to the present day. However, still left open were questions about the striking exceptions presented by benzene and the growing list of species whose properties appeared similarly averaged between those expected from alternative Lewis-structural depictions. In the decade following Lewis’s ground-breaking discovery, the unknown quantum mechanical laws governing electronic behavior were still subjects of speculation and confusion in the physics community. However, this pre-quantal era saw development of X-ray diffraction5 and other structural and kinetic tools that enabled a new category of physical organic chemists to address electronic bonding and reactivity questions in increasing mechanistic detail, complementing the simultaneous advances of synthetic organic chemists. Outstanding practitioners of this era included Christopher K. Ingold, Thomas M. Lowry, and Robert Robinson, all of whom were deeply involved in development of the “mesomerism” concept6 (now identified as “resonance”) that could be distinguished from classical isomerism or tautomerism. In this conception, each alternative “electromer” (electronic isomer) contributes proportionately to the actual electronic behavior, yielding properties that appear as intermediate averages of those expected from individual Lewis-structural representations. Of course, none of these chemically inferred concepts (nor the Lewis-structural concept itself!) could be reconciled with then-known physics. However, the undeniable power of mesomeric conceptions served to enrich the theory and practice 2 ACS Paragon Plus Environment

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of organic chemistry, including its nomenclature and pedagogy. Indeed, Robinson’s “curly arrow” notation,6b depicting the shifting patterns of electron-pair bonds along a reactive pathway, carries over intact into today’s organic textbooks, with uncanny power to predict preferred directions of electronic flow in complex chemical transformations. The 1925-6 discovery of quantum mechanics7 fundamentally altered the structure of physics and future of theoretical chemistry. Indeed, the restructuring of physical concepts ran so deeply that pioneer theoreticians struggled for words to suggest even vague analogies to known classical phenomena. In particular, the famous two-slit thought experiment8 focused attention on the deeply counter-intuitive aspects of wave-like superposition (“interfering alternatives”9) that underlies wavefunction description of all quantum phenomena. As expressed by Feynman, such superposition is “the greatest mystery, the only mystery” of quantum mechanics; “Any other situation in quantum mechanics, it turns out, can be explained by saying, ‘You remember the case of the experiment with the two holes? It’s the same thing.’”10 The word “resonance” (Resonanz) was first introduced by Heisenberg11 to describe a perceived analogy between quantal superposition and the familiar behavior of weakly coupled oscillators such as pendula suspended from a string. The oscillators are observed to exchange vibrational motions in a periodic manner, thus exhibiting time-averaged collective features that reflect aspects of each oscillator in isolation. Heisenberg’s quantum mechanical description of He was soon extended by Heitler and London12 to give the first successful theory of chemical bond formation in H2. The impact of quantum theory on chemical bonding conceptions was profoundly re-shaped by Linus Pauling, who came to Munich in 1926 as a young postdoc to join Sommerfeld’s group in the exciting theoretical developments of European physics. Based on his extensive chemical background, Pauling recognized how Heisenberg’s Resonanz conception of wave-mechanical superposition might be mapped onto the mesomerism concepts then emerging in organic chemistry. His first paper on the quantum-mechanical “resonance theory” of chemical bonding appeared in 1928,13 followed shortly14 (in parallel with John Slater15) by related formulation of atomic orbital hybridization (mixing of s, p angular types) that allowed extension of simple Heitler-London “valence-bond” (VB) functions to general polyatomic molecules. The resulting Heitler-London-Slater-Pauling perfect-pairing valence bond (HLSP-PP-VB) method was to dominate theoretical perceptions of the chemistry community for the 3 ACS Paragon Plus Environment


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next quarter century. As stated by Pauling in later years,16 “The theory of resonance is now treated in essentially every textbook of chemistry and is used by essentially every chemist.” Nevertheless, the tight coupling of Pauling-type resonance concepts to the inflexible 2-center/2-electron (2c/2e) VB form led to increasing conflicts17 as more accurate ab initio computational evaluations became available. Further details of these conflicts and their relationship to more general NBO conceptions are described elsewhere. 18 Deeper questions concerning Pauling-type resonance concepts can be based on an intrinsic flaw in the mapping onto empirical mesomerism concepts when the total wavefunction Ψ is assumed to be expressible as a linear combination of resonance structures {ψR}, viz., (1) Ψ = ΣR cRψR For each chemical property associated with quantum mechanical operator 𝓕, the expectation value 𝓕 is evaluated as (2) 𝓕 = ∫Ψ*𝓕 Ψ dτ = ΣR |cR|2∫ψR*𝓕ψR dτ + ΣR ΣS cR* cS∫ψR*𝓕ψS dτ The first summation on the right-hand side conforms properly to a weighted average, with weighting wR = |cR|2 for resonance structure ψR, but the second (involving all possible cross terms between distinct structures ψR, ψS) does not. The errant crossterms can be reassigned to presumed parent squared terms in Mulliken-like fashion,19 but neither the original appearance of these terms nor their rather arbitrary reassignment (independent of chosen property 𝓕) suggests robust theoretical foundation for the supposed mesomeric associations. Furthermore, the special case of the Hamiltonian operator (𝓕 = 𝓗), where the cross-terms might be assumed to be near-vanishing (because ψR, ψS are diabatic “near-eigenstates” of 𝓗), is also the case where resonance-mixing should lead to lowering (not averaging) the energies of ψR, ψS. The Pauling-type assumption that each resonance structure contribution corresponds to a linear term ψR in Ψ is therefore severely restrictive and limits the properties 𝓕 for which mesomerism-type averaging could be expected to hold. We return to these issues below.


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The first decade after the quantum revolution also saw development of the delocalized molecular orbital (MO) method of Hund, Mulliken, and Lennard-Jones20 which exhibits no obvious relationship to localized Lewis- or resonance-structural concepts. In the MO formulation, the orbital building blocks φi are delocalized linear combinations of AOs (LCAO-MOs) that can spread over the entire molecule, without apparent regard for Lewis-structural patterns, (3a) φi = ciAχA + ciBχB + ... The total MO wavefunction for a closed-shell species is merely a single Slater determinant of doubly-occupied MOs, with no apparent provision for multi-resonance character, (3b) ΨMO = det|φ1↑(1)φ1↓(2)φ2↑(3)φ2↓(4)...φN/2↑(N‒1)φN/2↓(N)| Yet despite its decidedly simplistic and “un-chemical” form, the MO wavefunction proves numerically superior in computational efficiency and variational accuracy to multi-resonance HLSP-PP-VB wavefunctions for every known chemical species but H2. As explosive growth of computational quantum chemistry in the post-war period made these advantages increasingly evident, VB-type approaches were reduced to a niche role. Despite the ascendance of MO theory (and related density functional theory), several lines of research still aim to recover VB-like connections to the Lewis- and resonance-structural concepts of empirical chemical practice and the chemistry classroom. These include (i) improved ab initio implementation of classical VB theory itself21 or VB-inspired methods that incorporate single- or multi-configuration self-consistent-field (MCSCF) improvements of a VB-type initial form;22 (ii) methods for extracting Pauling-type resonance descriptors from MO wavefunctions;23 (iii) graph-theoretical analysis of aromaticity and other resonance phenomena;24 and (iv) unitary transformations of MOs to localized molecular orbital (LMO) form.25 ALTERNATIVE “NATURAL” PATH TO LOCALIZED LEWIS AND RESONANCE CONCEPTS: NATURAL BOND ORBITAL AND NATURAL RESONANCE THEORY Persistent problems with VB-based methodology eventually led to an alternative “natural” path to localized Lewis and resonance aspects of chemical bonding, to be described herein. Paradoxically, this alternative retains no associations with VB5 ACS Paragon Plus Environment


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based methodology and indeed is independent of any particular wavefunction form (such choice being left entirely to the user), yet it reliably recovers the strong “VBlike” associations with empirical Lewis and resonance/mesomerism concepts, even if applied to wavefunctions of MO form. This path to “resonance without VB theory” will first be sketched in heuristic mathematical terms, including some technical details of its current implementation in popular electronic structure programs. [The reader with little appetite for mathematical or technical detail may prefer to skip forward to the numerical applications presented below, which illustrate the breadth and usefulness of modern bond order descriptors for a wide variety of chemical phenomena.] The alternative approach to extracting chemical insights from molecular wavefunctions was initiated by studies of Husimi26 and Löwdin27 in the post-war period. Husimi first showed that any question involving p-electron properties of Ψ (i.e., those associated with operator ℱ(p) whose contributing terms involve only p electrons at a time) could be rigorously answered in terms of the corresponding pthorder reduced density matrix (p-RDM) Γ(p), (4) Γ(p) = (N!/p!q!)∫Ψ*(rp′,rq)Ψ(rp,rq)drq Here, N = p + q is the total number of electrons, and rp lists the space-spin coordinates of surviving p electrons that are not “averaged out” by integration of remaining electronic coordinates rq over all space. Husimi’s construction of Γ(p) follows von Neumann’s formulation of quantum statistical mechanics28 in terms of the (unreduced) integral operator Γ(N) = Ψ*Ψ. From the definition (4), it can be shown that the expectation value ℱ(p) = ∫Ψ*ℱ(p)Ψ dτ of any p-electron operator ℱ(p) in N-electron state Ψ can be rigorously evaluated as the trace (diagonal sum) of the matrix product of ℱ(p) and Γ(p), viz., (5) ℱ(p) = ∫Ψ*ℱ(p)Ψ dτ = Tr{ℱ(p)Γ(p)} Such equations show that the reduced-dimensional p-RDM condenses the necessary information from Ψ to exactly describe the quantum mechanics of p-electron subsystems of the full N-electron system. Because the non-relativistic electronic Hamiltonian 𝓗 involves only operators of 1-electron (kinetic energy; electron-nuclear attraction) and 2-electron (electronelectron repulsion) type, Löwdin recognized that a complete many-electron theory 6 ACS Paragon Plus Environment

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could be reformulated in terms of Γ(1), Γ(2) alone. Moreover, the 1-RDM Γ(1), whose “diagonal element” in matrix representation is the electron density ρ, is itself sufficient for exact representation of any aspect of MO theory (or any theory dependent on ρ alone). Löwdin focused on the eigenfunctions {θi} of Γ(1), the socalled “natural orbitals” (NOs) satisfying (6) Γ(1)θi = Qi θi with bounded eigenvalues (electronic occupancies) 0 ≤Qi ≤ 2, in accordance with the Pauli principle. He also established the maximum-occupancy and other mathematical properties of NOs as optimal expressions of orbital-type aspects of the wavefunction — literally the eigen-orbitals of Ψ itself. In MO theory, the NOs are unitarily equivalent to MOs, and suffer from the same indeterminacies (non-uniqueness) as MOs, due to their degenerate doubleoccupancies (Qi = 2, i = 1,2,...,N/2). Such degeneracy allows any linear combination of solutions of Eq. (6) to be considered equally “natural” and delocalized [cf. Eq. (3a)]. In an AO-based matrix representation, however, one can also consider Γ(1) to be expressible in terms of local atomic blocks whose electronic occupancies can (and do!) vary in the manner predicted by Lewis-structural dot diagrams, thus breaking the degeneracies to yield unique natural orbitals for each localized bonding domain. How such atomic blocks are optimally defined is itself answerable in terms of the corresponding eigenvalue problem for the local (1c) NOs of each atomic block {θi(A)}, viz., (7) ΓA(1)θi(A) = Qi(A) θi(A) Here, ΓA(1) is the sub-block of Γ(1) over the complete basis space of atom A, and the associated “natural atomic orbital” (NAO) occupancies Qi(A) are again Paulicompliant (0 ≤ Qi(A) ≤ 2). The NAO algorithm29 involves a final overall orthogonalization step that removes residual Pauli-violating overlap effects while preserving maximal “natural” character of the final NAOs. Once Γ(1) is transformed to the basis of NAOs, one can proceed to solve the analogous 2-center eigen-orbital problem for diatomic A-B blocks ΓAB(1), viz., (8) ΓAB(1)Ωi(AB) = Qi(AB) Ωi(AB)



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to obtain the natural bond orbitals (NBOs)30 {Ωi(AB)}, each associated with electronic occupancy Qi(AB). Finally, one can diagonalize the weak interactions between NBO blocks to yield the natural localized molecular orbitals (NLMOs),31 which complete the NAO→NBO→NLMO sequence of natural localized orbitals. In MO theory, NLMOs achieve full double-occupancy and are thus unitarily equivalent to canonical MOs, even though they retain the uniqueness of their NBO parentage. In contrast to other LMO methods, construction of NLMOs requires no MO input whatsoever, and is intrinsically bottom-up (i.e., least-delocalized NBOs) rather than top-down (i.e., most localized MOs). Note that all operations described above can be performed on the 1-RDM of any Ψ, up to and including the exact solution of the many-electron Schrödinger equation. Thus, NAO/NBO/NLMO algorithms are intrinsically applicable to HLSP-PP-VB, MO, or any other computational theory level. As might be expected, the actual historical sequence of NBO evolution was somewhat more circuitous than sketched above. The idea of re-formulating MO theory in terms of bond orbitals (LCBO-MO theory) was first implemented32 and applied33 in semi-empirical CNDO (complete neglect of differential overlap) framework.34 The semi-empirical “AO basis” is creatively ambiguous, but assumed to satisfy the orthonormality property that NAO construction29 is designed to recover from common non-orthogonal basis sets. Although the basis BOs were initially constructed with Pauling-type prescriptions for hybridization and polarization, it was soon recognized30 that fully optimized NBOs could be obtained merely by optimizing the hybridization and polarization parameters to maximize electronic occupancy, thereby exploiting the intrinsic maximum-occupancy properties of eigenvalue problems such as Eq. (8). This early NBO implementation35 was employed in a variety of semi-empirical applications36 before the NAO-based extension was developed for general NHO/NBO/NLMO analysis of ab initio wavefunctions. This NBO 3.0 version3] was initially interfaced to (and later absorbed by) the Gaussian 82 electronic structure program, where it attracted numerous applications.38 The min-max properties of eigenvalue problems allow one to cast the NBO algorithms into alternative forms. In analogy to Eq. (3b), we can envision construction of an idealized natural Lewis structure (NLS) wavefunction ΨNLS (9) ΨNLS = det|Ω1(L)↑(1)Ω1(L)↓(2)Ω2(L)↑(3)Ω2(L)↓(4)...ΩN/2(L)↑(N‒1)ΩN/2(L)↓(N)| 8 ACS Paragon Plus Environment

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from the leading Lewis-type NBOs {Ωi(L)}. The wavefunction ΨNLS can in turn be reduced as in Eq. (4) to give the corresponding 1-RDM ΓNLS(1) for this idealized bonding pattern. The best-possible NBOs can be equivalently described as those that minimize the root-mean-square difference (in ||...|| matrix norm notation) between idealized ΓNLS(1) and the true 1-RDM Γtrue(1) of wavefunction Ψ, viz., (10) best{Ωi(L)}  min|| Γtrue(1) ‒ ΓNLS(1)|| In a similar manner, if we have a collection of idealized resonance-structural 1-RDMs {ΓR(1)}, we can seek the resonance weights {wR} (satisfying wR ≧ 0, ΣR wR = 1) that lead to optimal resonance-type representation of Γtrue(1), viz., (11) best{wR}  min||Γtrue(1) ‒ ΣR wR ΓR(1)|| The variational functional (11) defines the natural resonance theory (NRT)39 algorithm for determining optimal resonance weights of an incoherent (RDM-based) resonancetype assumption for Γtrue(1). From the variational minimum (13) we achieve the closest approximation of Γtrue(1) by a resonance-weighted sum of density matrices for idealized resonance structures R, (12) Γtrue(1) = ΣR wR ΓR(1) (subject to wR ≧ 0, ΣR wR = 1) For any 1-electron operator 𝓕, we can then use Eqs. (5), (12) to write the expectation value 𝓕 as the resonance-weighted average of property values 𝓕R in the idealized resonance structures, viz., (13) 𝓕 = ΣR wR*Tr{𝓕ΓR(1)} = ΣR wR 𝓕R As shown in Eq. (13), the NRT-type resonance-structure formulation contains no cross-terms and applies to every 1-electron operator, in contrast to the corresponding Pauling-type (wavefunction-based) formulation of Eq. (1). Note that the specific double-occupancy configuration interaction (DOCI) wavefunction form40 lacks the cross-terms of Eq. (1) and thereby allows resonance descriptions derived from (1) and (12) to agree, but the Γ(1)-based definition of resonance should be generally preferred for obviating such cross-terms in any allowed form of Ψ. The NRT variational algorithm (11) generalizes and extends the NBO algorithm (10) for choosing the best possible single Lewis structure and NLS determinant, Eq. 9 ACS Paragon Plus Environment


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(9). The latter forms the starting point for 2nd-order perturbation theory analysis of “non-Lewis” donor-acceptor (DA) corrections to the zeroth-order NLS picture.41 In this picture, each 2nd-order EDA(2) = Ei→j*(2) is the energetic stabilization associated with excitations (determinantal substitutions) from an occupied Lewis-type NBO Ωi(L) of the formal Lewis structure to an unoccupied non-Lewis-type NBO Ωj*(NL), as depicted schematically in the perturbative diagram of Fig. 1.

Figure 1. 2e-stabilizing interaction between a filled donor orbital Ωi(L) and vacant acceptor orbital Ωj(NL), leading to energy lowering ΔEij(2).

As shown in the diagram, any such 2-electron Ωi(L)→Ωj*(NL) delocalization is intrinsically stabilizing (energy-lowering), the universal quantum-mechanical reward for choosing the lower of the interfering alternatives resulting from two-state Ωi(L)Ωj*(NL) superposition mixing.42 Only in quantum mechanics can one achieve a state of lower energy by “mixing” with a state of higher energy! By drawing the new resonance structure associated with formal 2-electron Ωi(L)→Ωj*(NL) promotion from the parent ΨNLS determinant, Eq. (9), one recognizes that each EDA(2) donor-acceptor interaction maps onto a corresponding NRT resonance structure in Eq. (12). Each such NBO Ωi(L)→Ωj*(NL) donor-acceptor interaction and NRT resonance structure can in turn be mapped onto the corresponding curly arrow mnemonic, as shown in the “Rosetta stone” of NBO donor-acceptor interactions in Fig. 2. NRT evaluations thereby complement and extend conventional NBO 2nd-order EDA(2) perturbative analysis, and can be used even when the latter descriptors are unavailable (i.e., higher-level correlated wavefunctions, where no available 1-electron 10 ACS Paragon Plus Environment

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effective Hamiltonian operator of Fock or Kohn-Sham type provides orbital energetics). NRT evaluations are also fully consistent with the general 3c/4e ionicresonance picture43 of main-group hypervalency44 and metallic-like long-bonding,45 as well as the corresponding “12-electron rule” extensions of localized hybridization, bonding, and resonance concepts in transition metal chemistry.46

Figure 2. “Rosetta stone” of translations from NBO donor-acceptor interaction (left) to associated curly-arrow depiction of electronic shifts (center) from parent natural Lewis structure (NLS) to associated charge-transfer (CT) resonancestructural contribution (right).

It is evident from comparison of (10), (11) that the NBO description is but a special case of the more general NRT framework. Although the limiting case of a single dominant Lewis-structural contribution (zeroth-order NLS picture) has special importance in the context of chemical history, it by no means exhausts the range of chemically important possibilities in the broader framework of resonance 11 ACS Paragon Plus Environment


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superposition phenomena. A number of surprising “sub-molecular” bonding motifs will be illustrated in NRT numerical applications to follow NUMERICAL NRT “REBOOT” WITH CONVEX PROGRAMMING Although the formal NRT framework suggests broad applicability, its practical utility as a discovery tool in chemistry has been severely limited by numerical issues. While relatively small systems with few resonance centers could be handled successfully, larger species with more highly coupled resonance networks tended to encounter convergence failures or excessive demands for time, memory allocation, or user intervention. The initial (now “legacy”) NRT implementation offered three alternative numerical solvers for non-linear optimizations,47 but none proved satisfactory in many cases of interest. However, it was recently recognized48 that the NRT optimization problem can be recast into Gram-functional form with intrinsic convexity (bowl-shaped) properties that allow highly efficient optimizations with interior-points convex programming methods.49 The recent NBO 7.0 release of the Natural Bond Orbital Analysis program50 features a customized QPNRT solver for efficient Gram-based active-set solution of the constrained quadratic NRT functional (11). QPNRT-based minimization (11) offers numerous advantages over legacy NRT methods: (i) Legacy NRT considered only a small number of allowed “reference” structures [default: 20] for which all elements of each ΓR(1) are considered in the variational minimization, whereas only diagonal elements were considered (by a simple perturbative approximation) for remaining “secondary” structures . This restriction often results in numerically significant discontinuities when a structure is promoted from secondary to reference status (e.g., in reaction-path calculations). In contrast, the QPNRT solver treats all resonance structures consistently in full reference-like detail, while also allowing a much larger number of bonding patterns to be considered within allocated memory and maximum number (default: 10,000) of resonance structures, both keyword-extensible. (ii) Legacy NRT required expensive tests for possible multiple minima. In contrast, the QPNRT algorithm moves monotonically downward to a provably unique solution.


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(iii) Legacy NRT typically required only a small fraction of the total computer time to prepare candidate ΓR(1) structures for inclusion in the demanding numerical optimization of resonance weightings. In contrast, the QPNRT optimization time is generally negligible compared to that invested in generating the many ΓR(1) candidate structures that may enter the numerical optimization. In all respects, the QPNRT solver exhibits improved efficiency, robustness, and numerical reliability compared to legacy NRT numerical methods. Ref. [48] further documents the numerical improvements for a large number of test cases. The following section illustrates cases that were well beyond reasonable treatment by legacy NRT methods, but are now treated without user intervention (i.e., without $NRTLST keylist input47 or other keyword control beyond optional storage expansion). ILLUSTRATIVE QPNRT-BASED NRT APPLICATIONS Numerical Methods The ensuing examples are each treated at B3LYP/6-311++G** computational level with the Gaussian 16 program51 interfaced to NBO 7.0. All species are fully optimized, then analyzed with the QPNRT-based solver for optimal NRT weightings and bond orders. Note that NBO/NRT algorithms only require input of the 1-RDM Γtrue(1) (and AO descriptors for the basis in which Γtrue(1) is represented), but no information about molecular geometry, symmetry, Kohn-Sham MOs, orbital energies, or other numerical descriptors that might be correlated with experimental properties. Thus, NRT bond orders provide suitable blind predictors for expected correlations with bond lengths,52 bond frequencies (Badger’s rule),53 bond energies,54 and the like. Molecular Equilibrium Species: Adrenaline Figure 3 shows calculated NRT bond orders for adrenaline, taken as an example of a simple normal-valent biomolecular species. The QPNRT solver finds 67 contributing resonance structures, led by the two near-equivalent Kekulé structures (with weightings 14.95%, 14.63%) and a variety of remaining small hyperconjugative contributions (ranging up to 5.49%). One can see at a glance that the “single bond” bCH bond orders are slightly weakened when oriented in vicinal antiperiplanar arrangement to the N lone pair, as in C(1)-H(16) and C(3)-H(19), in accordance with general stereochemical principles .55 In the slight bOH differences between O(9)H(23) vs. O(11)-H(24) one can also see the expected effect of weak O(11)···H(23)13 ACS Paragon Plus Environment


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O(9) H-bonding interaction in ortho diols, tending to lengthen O(9)-H(23) and redshift its IR frequency.56

Figure 3. NRT bond orders for adrenaline (B3LYP/6-311++G** level).

The principal variations of Fig. 3 involve the bCC bond orders, which are expected to exhibit correlations with RCC bond length, stretching frequency, NMR spin-spin couplings, and other properties. The bCC-RCC correlations for this species are exhibited in Figure 4, showing the reasonable correlation coefficient (|χ|2 = 0.92) and dashed regression line. All these results concur with time-honored expectations for the general usefulness of theoretical bAB bond orders in rationalizing empirical properties of A-B bonds.


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Figure 4. NRT bond order-bond length correlation for RCC bonds of adrenaline (cf. Fig. 1), showing correlation coefficient (|χ|2 = 0.92) and least-squares regression fit (dashed line).

Reactive Transition-State Species: Claisen rearrangement A recent paper57 describes the application of convex-solver NRT to the complex bond shifts along the intrinsic reaction coordinate (IRC) for a model intramolecular Claisen rearrangement. Figure 5 displays some numerical details of computed NRT weightings along the Claisen IRC, showing the expected rise and fall of resonance contributions that underlie the three distinct bond shifts between reactant and product species. Even the small wiggle near IRC = ‒5 is a numerically persistent feature of the computed NRT weightings, indicating the favorable numerical characteristics of the QPNRT solver.



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Figure 5. Calculated NRT weightings for various resonance contributions (as indicated in insets) along the IRC for intramolecular Claisen rearrangement [57].

At a still deeper electronic level, the reactive bond shifts can be described with “resonance NBOs” (RNBOs), composed of resonance-averaged mixtures of NBOs58 that vary continuously along the IRC. Figure 6 employs pre-orthogonal (P)RNBO visualization orbitals to display the specific σO(3)C(4)→πC(2)O(3) bond shift that converts the σO(3)C(4) NBO of the reactant ether to the πC(2)O(3) NBO of the product carbonyl species (one of the three concerted bond shifts found in the transition state region57). The frames of Fig. 6 show stepwise [Δ(IRC) = 0.29] snapshots of this bond shift as the rearrangement proceeds from IRC = ‒2.33 to +2.04 across the energetic transition state (IRC = 0.00). The RNBO-based depiction thereby provides a vivid and accurate semi-localized image of “curly arrow” resonance delocalization (Fig. 2) at the orbital level.


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Figure 6. (P)RNBO visualization of σO(3)C(4)→πC(2)O(3) bond shift, showing evolution [in Δ(IRC) = 0.29 steps] from σO(3)C(4)-like (upper left; IRC = ‒2.33) to πC(2)O(3)-like character (lower right; IRC = +2.04), with increasingly pronounced multi-center character near the transition state (IRC = 0.00).

Coordinative and Supramolecular Resonance Mixing: Fe2(CO)9 Among the vexing species that have long defied legacy-NRT description, the diiron carbonyls offer particularly strong evidence for improved QPNRT numerics. 17 ACS Paragon Plus Environment


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Here we briefly describe new NRT results for the D3h-symmetric Fe2(CO)9 species,59 focusing on the question of metal-metal (bFeFe) and metal-carbonyl (bFeC: terminal, bridging) bonding interactions. In this case, keyword-specified storage expansion was required to handle the large number of resonance structure considered in the automated NRT search algorithm (free of keylist guidance or other user input). Figure 7 displays one of the many equivalent “leading” structures (upper panel) as well as the composite NRT bond orders from the final 102-term resonance expansion (lower panel).


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Figure 7. NBO/NRT analysis of Fe2(CO)9, showing one of twelve equivalent “leading” Lewis-structural bonding patterns (ca. 4% each; upper panel) and the final symmetry-distinct bFeFe, bFeC, bCO NRT bond orders (lower panel). (Presuperscripts in the upper panel denote number of non-bonding lone pairs.)



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As suggested in both panels of Fig. 7, NRT analysis finds no evidence of direct metal-metal bonding (bFeFe = 0.00). However, each metal shows significant coordinative bonding (bFeC = 0.32) to each of the three formaldehyde-like (bCO = 2.51) bridging carbonyls, as well as stronger coordinative bonding (bFeC = 0.57) and evident π-backbonding (bCO = 2.94) to terminal carbonyls. In addition to leading interactions shown in Fig. 5, a variety of weaker carbonyl-carbonyl interactions contribute smaller bond orders to overall aggregation [including bC(3)C(19) = 0.018, bC(3)C(15) = 0.015, bC(3)O(6) = 0.009, bC(3)C(7) = 0.008, and symmetric-equivalent values]. In the present case, the twelve symmetry-equivalent leading resonance structures (4.35% each) account for ~58% of the NRT resonance weighting, and the next most important twelve (1.35% each) another ~16%. However, the significant remainder (~25%) derives from 78 surviving structures (of more than 16,000 considered!) in the final QPNRT cycle. Fe2(CO)9 therefore exhibits daunting aspects of the resonance delocalization limit where no single Lewis structure contributes appreciable percentage to the overall resonance hybrid, but only the bond orders (robustly converged from the myriad contributions of hundreds or thousands of resonance structures) provide fuzzy links to the idealized integer “bond sticks” of the chemist’s Lewis structure diagram. Long Bonding, Bridge Bonding, and the Metallic Limit: B(BeH)6+ As a final example, we illustrate the effectiveness of new NRT for a still more exotic example of resonance-dominated bonding, far from any parent Lewis-structural limit. The example chosen is the B(BeH)6+ “beryllium star” cation60 of D6h symmetry, as shown (in default Jmol ball-and-stick representation) in Figure 8a. The optimal NBO bonding pattern is depicted in Figure 8b, but it is immediately evident that many alternative symmetry-equivalent structures could be drawn that are equally justified (or unjustified!) to be considered the parent of the composite resonance hybrid.


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Figure 8. (a) Schematic coordination diagram (left) and (b) one of six equivalent leading resonance contributions (right) of D6h-symmetric B(BeH)6+ cation.

The NRT weightings confirm that any envisioned parent 2c/2e Lewis-type bonding pattern is superficial and misleading in the case of B(BeH)6+. Indeed, the combined weighting of the six symmetry-equivalent leading resonance structures is less than 25%, and the 48 smallest contributions of the 114-term expansion (each in the 0.20.3% range) have combined weighting that is more than three times greater than any single structure such as Fig. 8b. The broad spectrum of low-level resonance weightings naturally results in a scattering of fractional (sub-integer) bond orders, as displayed in Figure 9. Most strikingly in this case, the NRT bond orders display negligible like-atom bonding (bBeBe = bHH = 0.000) but quite appreciable values between unlike atoms (bBBe = 0.294, bBeH = 0.445, bBH = 0.109). The B^H “long bond” (shown as an arching linkage in Fig. 9) is particularly counter-intuitive to conventional chemical bonding concepts, but can be readily understood45 in terms of related 3c/4e long-bonding phenomena whose chemical importance has been demonstrated in exotic noble gas and metallic bonding motifs.61



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Figure 9. Graphical display of fractional NRT bond orders for symmetry-distinct atom pairs in B(BeH)6+.

From a broader perspective, one can recognize that atom clusters such as B(BeH)6+ are representative of an interesting sub-molecular domain of material aggregation that is now open to NRT exploration. As an operational definition, one may classify an array of atoms {A, B,...} as a “molecule” if, and only if, each atom A is found to be “chemically bonded” to another atom B of the array with NRT bond order exceeding a near-unit threshold value, say, bAB ≥ ½. By this criterion, one would conclude that B(BeH)6+ has no “chemical bond” or “molecule” character whatever! Nevertheless, the resonance-type (fractional) bonding interactions of B(BeH)6+ are direct analogs of the familiar H-bonds of water clusters and related biochemical aggregation phenomena that share common quantum-superposition origins with the covalent bonding of molecules,42 but lie in the sub-integer range (bAB < ½) of bonding attractions that are often attributed (mistakenly!) to “non-covalent” forces of classical type. New NRT now makes it straightforward to transition smoothly into this expanded inorganic domain of metallic-like resonance attraction. CONCLUDING REMARKS The foregoing discussion supports traditional localized VB-like and resonance structure conceptions62 in contrast to the “completely delocalized” picture that is sometimes advocated from MO-based perspective. However, we have described how the NBO-based natural resonance theory (NRT) provides a more rigorous and practical computational tool for revival (“reboot”) and extension of resonance concepts in a wide variety of chemical and materials applications. The computational applicability extends to sub-molecular metallic aggregation and related phenomena where a multiplicity of resonance-type contributions may dominate over those of any


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perceived parent Lewis-structural bonding pattern, contrary to the usual assumptions of the molecular chemist. The applications described above are representative of many cases that would be hopelessly impractical with legacy (pre-NBO7) NRT, but are successfully handled in routine fashion with the current convex-solver algorithm of the NBO 7.0 program.50 We believe that newstyle NRT analysis can be a valuable addition to the theoretical toolkit in numerous areas of chemical research, as well as in the chemistry classroom. AUTHOR INFORMATION Corresponding Author *[email protected] REFERENCES (1) C. A. Russell, The History of Valency, Leicester U. Press, Leicester, UK, 1971, (2) J. J. Thomson, Cathode rays, Phil. Mag. 1897, 44, 293. (3) N. Bohr, The spectra of helium and hydrogen, Nature 1913, 92, 231. (4) G. N. Lewis, The atom and the molecule, J. Am. Chem. Soc. 1916, 38, 762. (5) W. Friedrich, P. Knipping, M. Laue, Interferenz-Erscheinungen bei Röntgenstrahlen, Ann. Phys. 1913, 346, 971. (6) (a) See, e.g., W. O. Kermak, R. R. Robinson, 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. 1922, 121, 427; T. M. Lowry, Studies of electrovalency. Part I. The polarity of double bonds, J. Chem. Soc. 1923, 1923, 822; J. Allan, A. F. Oxford, R. Robinson, and J. C. Smith, The relative directive powers of groups of the forms RO and RR’N in aromatic substituton. Part III. The nitration of some p-alkoxyanisoles, J. Chem. Soc. 1926, 1926, 401; C. K. Ingold and E. H. Ingold, 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, 1926, 1310; C. K. Ingold, Mesomerism and tautomerism,Nature 1934, 133, 946; C. K. Ingold, Principles of an electronic theory of organic reactions, Chem. Rev. 1934, 15, 225; M. D. Saltzman, C. K. Ingold’s development of the concept of mesomerism, Bull. Hist. Chem. 1996, 19, 25; Lord 23 ACS Paragon Plus Environment


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TOC Graphic


Resonance Theory Reboot Eric D. Glendening,a Clark R. Landis,b

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