Resonance Natural Bond Orbitals: Efficient Semilocalized Orbitals for

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Resonance Natural Bond Orbitals (RNBOs): Efficient Semi-Localized Orbitals for Computing and Visualizing Reactive Chemical Processes Eric D Glendening, and Frank Weinhold J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.8b00948 • Publication Date (Web): 06 Jan 2019 Downloaded from http://pubs.acs.org on January 7, 2019

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Journal of Chemical Theory and Computation

Resonance Natural Bond Orbitals (RNBOs): Efficient Semi-Localized Orbitals for Computing and Visualizing Reactive Chemical Processes E. D. Glendeninga and F. Weinholdb aDepartment

of Chemistry and Physics, Indiana State University, Terre Haute, Indiana 47809 USA; bTheoretical Chemistry Institute and Department of Chemistry, University of Wisconsin-Madison, Madison, Wisconsin 53706 USA Abstract We describe a practical algorithm for calculating NBO-based “resonance natural bond orbitals” (RNBOs) that can accurately describe the localized bond shifts of a reactive chemical process. Unlike conventional NBOs, the RNBOs bear no fixed relationship to a particular Lewis-structural bonding pattern, but derive instead from the natural resonance theory (NRT)-based manifold of all bonding patterns that contribute significantly to resonance mixing (and associated multi-channel reactivity) at a chosen point of the potential energy surface. The RNBOs typically retain familiar localized Lewis-structural character for stable near-equilibrium species, yet they freely adopt multi-center character as required to satisfy Pople’s prerequisite that no allowed computational basis set should be inherently biased toward a particular nuclear arrangement or bonding pattern. A simple numerical application to intramolecular Claisen rearrangement demonstrates the computational and conceptual advantages of describing reactive bond-shifts with RNBOs rather than other conventional NBO- or MO-based expansion sets.

Introduction Previous versions of the Natural Bond Orbital (NBO) program1 have provided a sequence of orthonormal localized and semi-localized orbital basis sets for chemical analysis purposes: Natural Atomic Orbitals (NAOs),2 Natural Hybrid Orbitals (NHOs),3 Natural Bond Orbitals (NBOs),4 Natural Localized Molecular Orbitals (NLMOs).5 Each orthonormal analysis set is complemented by the corresponding “pre-orthogonal” set (PNAO, PNHO, PNBO, PNLMO) for visualization purposes.6 Such chemically oriented orbitals serve as a useful conceptual bridge7 between the chosen atomic orbital basis functions (AOs) and delocalized canonical molecular orbitals (CMOs) of a standard quantum chemistry calculation.

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Despite widespread usage for analysis and visualization purposes, the NBO-based orbital sets are intrinsically tied to the localized Lewis-structural picture of 2-center (2c)/ 2-electron (2e) bonding. Although well adapted to describing the stable molecular species of principal chemical interest, such “Lewis-centric” 2c/2e orbitals, depicting the Natural Lewis Structure (NLS) bonding pattern, inherently lack the generality required for a broader range of chemical applications. Specifically, the NBO-based sets cannot smoothly (i.e., without numerical discontinuities) describe arbitrary chemical transformations from one 2c/2e bonding pattern to another, nor can they accommodate the multi-center character of transition-state species or other farfrom-equilibrium species along a chosen reactive pathway. As a result, such orbitals are precluded from serving as the basis for any computational method that can satisfy Pople’s first prerequisite for an acceptable model chemistry, namely: “the method should be well defined and applicable in a continuous manner to any arrangement of nuclei and any number of electrons.”8 Such basis independence from any particular Lewis-structural bonding motif is essential for describing multi-channel potential energy surfaces of complex chemical reactions. In the present work we describe construction and application of a novel “resonance” (or “reactive”) NBO-type basis set of “RNBOs” that satisfy Pople’s prerequisite and provide a useful tool for computing and visualizing reactive chemical phenomena. RNBOs are closely tied to resonance-type concepts of chemical reactivity that trace back to Robinson’s pre-quantal “curly arrow” mnemonic.9 Algorithmic construction of RNBOs is based on recent enhancement10 of the Natural Resonance Theory (NRT)11 method for determining optimal numerical resonance weightings for a chosen level of quantum chemical description. The computational algorithms for constructing RNBOs and their companion PRNBO visualization orbitals are outlined below and illustrated for simple numerical application to intramolecular Claisen-type rearrangement reactions.12 RNBO/PRNBO construction is fully implemented in the current NBO 7.0 version of the general NBO analysis program.13 Method RNBOs are a further extension of NLMOs5 for incorporating effects of resonancetype donor-acceptor delocalizations from the initial localized NBOs.14 As shown in Figure 1, each such donor-acceptor (DA) interaction can be uniquely associated with a resonance structure (R) and associated curly-arrow representation. The corresponding 2 ACS Paragon Plus Environment

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stabilization energy of each such delocalization (EDA(2)) can be estimated by 2nd-order perturbation theory. The DA effects are manifested in the forms of NLMOs as weak “delocalization tails” that bring the NLMO to full double-occupancy while retaining closest possible resemblance to the parent NBO.

Figure 1. General mapping of NBO donor-acceptor types (left) onto associated curly-arrow representation (center) and resultant “charge transfer” (CT) resonance structure depiction (right) for a variety of NLS bonding motifs.

In the NRT formulation, however, the “parent” NLS and each “daughter” DA structure are considered as independent contributors to the NRT manifold (each with its own optimal NBOs and NLMOs). The optimal NRT resonance weightings are therefore obtained without perturbative approximations or other presumptions of dominant NLS parentage. The final NRT weightings {wR} can be combined with the NLMOs {Θi(R)} of each contributing resonance structure to obtain resonanceaveraged orbitals {ωi} of high-occupancy and near-orthogonal character, viz., 3 ACS Paragon Plus Environment


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(1) ωi = ΣR wR Θi(R) These are subsequently transformed to orthonormal RNBOs {Ωi} by Löwdin symmetric orthogonalization.16 Note that contributing Θi(R), Θi(Rʹ) orbitals in Eq. (1) are generally nonorthogonal, because they arise from separate optimizations for each contributing bonding pattern R, Rʹ. For continuity between distinct bonding patterns, the ordinal index i and phase of each Θi(R) in Eq. (1) are chosen to properly match those of Θi(R′) for each alternative structure R′. Proper matching is enforced by a simple overlap criterion, starting (rather arbitrarily) from NLMOs of the leading structure of the NRT expansion (often the NLS). Unlike the NLMOs, which are intrinsically tied to a single bonding pattern, the RNBOs are free to evolve smoothly and “democratically” toward a variety of alternative bonding patterns, dependent on chosen reaction path. RNBOs therefore extend resonance-type orbital conceptions far beyond the weak-perturbation limit associated with NLMOs. The final RNBOs can be expressed as the unitary matrix U from orthonormal NAOs to orthonormal RNBOs {Ωi}. With this same transformation matrix, one easily obtains the corresponding pre-orthogonal (PRNBO) visualization orbitals {pΩi} by merely replacing NAOs by PNAOs in the transformation. The NBO 7.0 program provides full keyword support for evaluating transformation matrices from input basis AOs to final RNBOs or PRNBOs. Program output includes NAO-based decomposition of each occupied RNBO (similar to that for Lewis-type NLMOs) which allows ready quantitation of multi-center character. RNBOs may be compared with several recent proposals17-19 for visually depicting curly-arrow orbital or density fluxes in chemical reactions. The intrinsic bond orbital (IBO) method of Knizia and Klein,17 (related to Pipek-Mezey localization,18 but employing a different power of the orbital separation distance) apparently reduces or removes LMO discontinuities that often appear along paths between distinct bonding patterns. The method of Vidossich and Lledos19 focuses on the LMO orbital centroid (rather than details of orbital shape) along a reaction path. Each such MO-based method depends on the top-down (localization of initial CMOs) rather than bottom-up (delocalization of initial NBOs) strategy of NLMO/RNBO construction. The latter is distinguished in making no use of CMOs and applying more generally to correlated or uncorrelated wavefunctions of any form or accuracy. Silvi and coworkers20,21 have described alternative real-space density “bonding evolution” in terms of the electron 4 ACS Paragon Plus Environment

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localization function (ELF), while Proud et al.22 have described “extracule density” variations in a related manner. Most recently, Liu et al.23 employed a novel real-space representation of “nodal pockets” of MO or post-MO wavefunctions to exhibit parallels to curly-arrow depictions. Among these varied efforts to capture localized or semi-localized aspects of reactive orbital or density shifts, we believe the RNBOs are clearly distinguished by their freedom from restrictive CMO-type associations and their unique link to general resonance-theoretic concepts (Fig. 1), including NRTbased bond orders that are expected (and demonstrated7) to exhibit useful correlations with a broad variety of structural and spectroscopic bond properties (bond length, frequency, energy, NMR shieldings and 1J spin couplings, etc.). The crux of the RNBO advance is the new convex-solver NRT,10a which permits a far more thorough and far-reaching automated search for candidate structures than the earlier implementation. Elementary mathematical aspects of the NRT algorithm may be briefly sketched as follows. All NBO-related orbital sets derive from the 1st-order reduced density matrix,15 ΓQC, of a given quantum chemistry (QC) description of the system of interest. After initial import of this matrix from the host electronic structure system, ΓQC is first transformed to the orthonormal NAO basis in preparation for the block diagonalizations of NBO construction. The optimal forms of NBOs are chosen to satisfy a maximum-density criterion, which corresponds to an idealized NLS density matrix ΓNLS [composed from doubly-occupied NBOs {Θi(NLS)} for the specified 2c/2e NLS bonding pattern] that exhibits maximum resemblance to the true ΓQC (expressed as the Frobenius norm of deviations between the two matrices), viz., (2) ||ΓNLS ‒ ΓQC||2 = min In similar terms, the “best” resonance-type approximation to ΓQC can be formulated in terms of the convex combination of resonance weightings {wR} and idealized resonance-structural density matrices {Γ(R)} (3a) ΓNRT = ΣR wR Γ(R) that are constrained by the convexity conditions (3b) all wR ≥ 0, with ΣR wR = 1 and provide optimal weightings to satisfy the NRT variational minimization criterion, viz., 5 ACS Paragon Plus Environment


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(3c) ||ΓNRT ‒ ΓQC||2 = min As described elsewhere,10b the density matrix-based NRT formulation ensures that empirical resonance-type assumptions are satisfied for all one-electron properties, in contrast to Pauling-type (wavefunction-based) resonance formulations. Recent recognition10a that (3c) can be efficiently solved by convex quadratic programming techniques has opened the door to a broad new range of NRT-based applications. In formal terms, the RNBOs can be clearly distinguished from NLMOs, CMOs, or natural orbitals (NOs), but certain relationships between RNBOs and other orbital constructs may be noted for limiting theoretical levels or specific chemical species. Because Lewis-type NLMOs have exact double-occupancy for closed-shell singledeterminant methods (and nearly so in correlated methods), the RNBOs naturally inherit high (NO-like) occupancy at any theoretical level. However, RNBOs retain the close relationship to the unique (occupancy-determined) parent NBOs of contributing bonding patterns that insures their own uniquely determined forms, even in the singledeterminant limit. Although the shapes of RNBOs and NLMOs differ significantly along a bond-shifting reaction pathway, their resemblance will generally increase in “spectator” bonds (not directly involved in the reactive bond shift) or limiting reactant or product species where resonance is weak. Note also that the allowed NRT contributions (as in those of Pauling-type resonance theory) refer only to valencelevel bonding patterns (and associated determinants), so the NRT expansion cannot serve as a “complete set” to reduce the variational minimum in (3c) to zero. Note finally that RNBOs, even if obtained from single-determinant wavefunctions, are almost always visually distinct from CMOs. Both RNBOs and CMOs evolve smoothly along a reaction pathway, but the RNBOs tend to depict orderly 3-center shifts (“morphing” of an initial 2-center bond to an adjacent center) rather than the chaotic “complete delocalization” suggested by CMOs. In principle, it might appear that resonance weightings {wR} from pre-NBO7 implementations of NRT could be used to compute RNBOs, but in practice the results could not be comparable to those of the current NRT algorithm. The impracticality stems from the fact that pre-NBO7 NRT often required step-by-step user guidance ($NRTSTR keylist input) to traverse a reaction path, with numerical uncertainties compounded by ineptness of available nonlinear solvers and possible effects of user subjectivity. Still more serious differences arise from the former NRT distinction between structures of “reference” vs. “secondary” importance, resulting in numerical 6 ACS Paragon Plus Environment

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discontinuities when a given structure passes from one level of importance (and approximation) to the other. Such difficulties are avoided in the current NBO7 implementation, because all structures are consistently treated with full referencelevel detail and improved $CHOOSE-type optimization, leading to desired numerical continuity (see below) and provably optimal resonance weightings. We may also comment briefly on some practical computational aspects of RNBO determination. The default settings of numerical RNBO implementation in NBO7 allow many applications to be successfully completed with no special user input or guidance, other than specifying “RNBO” (or “NRT PLOT”) in $NBO keylist input. Of course, the final details of numerical RNBOs depend on those of the underlying NRT search, which the new convex-solver method allows to be much more thorough and accurate than pre-NBO7 NRT even with default keyword settings. However, for reaction path animation (as described below) where visual continuity is most critical, both the smoothness of the calculated reaction path (e.g., chosen step-size, use of analytic derivatives, etc.) and thoroughness of the NRT search are important. To this end, we recommend that reaction-path calculations be performed with high precision and consistent coordinate-frame orientation at each point, and that NRT searches (i) be allowed to run to completion (rather than the default 3-cycle limit), and (ii) include the transition-state $NRTSTR keylist to insure that any resonance structure contributing appreciably in the transition-state region be retained as candidate structure throughout the reaction path search (or, if necessary, include a still larger composite $NRTSTR keylist that includes all structures found to contribute significantly at any point of the reaction path) . Additional numerical details of NBO7 program input/output for NRT and RNBO evaluation are provided in Supporting Information (SI). Numerical Application: Intramolecular Claisen Rearrangement As a representative numerical and visual application, we consider a model Claisen rearrangement reaction24 at the B3LYP/6-311++G** theory level,25 as previously described in terms of NRT bond orders.12 Figure 2 displays the energetic profile for the chosen (intrinsic, IRC) reaction coordinate path, with insets depicting the optimized geometry (upper) and nominal bonding pattern (lower) for the reactant ether (R; left) and product aldehyde (P; right). Further NRT details of the bond-order shifts accompanying this remarkable rearrangement are described elsewhere.12 7 ACS Paragon Plus Environment


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Figure 2. Energy profile (B3LYP/6-311++G** level) for intrinsic reaction coordinate of model Claisen rearrangement reaction,12 showing reactant ether (R) and product aldehyde (P) species in optimized ball-and-stick geometry (upper) and Lewis-structural bonding pattern (lower).

To examine the Claisen bond-shifts in deeper orbital detail, we calculate RNBOs at a succession of points (frames) along the IRC for visual animation. Figure 3 displays five such frames (at IRC ≈ ‒15, ‒6, 0, +6, +15) for the three primary bond-shifting RNBOs, which can be identified (from left to right; see column headings) as πC(1)C(2), σO(3)C(4), πC(5)C(6) in reactant R labelling, or σC(1)C(6), πC(2)O(3), πC(4)C(5) in product P labelling. Each bond-shifting RNBO exhibits conspicuous multi-center character near the energetic transition state, whereas the remaining “spectator” bonds retain localized 2-center NBO-like character throughout the entire reaction pathway.


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IRC

πC(1)C(2)→σC(1)C(6)

σO(3)C(4)→πC(2)O(3) πC(5)C(6)→πC(4)C(5)

m50 (≈‒15)

m20 (≈‒6)

ts (0.00)

p20 (≈+6)

p50 (≈+15)

Figure 3. Visualizations of the three principal bond-shifting RNBOs (columns) for five sequential points (at IRC ≈ ‒15, ‒6, 0, +6, +15; rows) along the IRC for intramolecular Claisen rearrangement (Fig. 2).

As a particular example, consider the middle column of Fig. 3 for the “σO(3)C(4) → πC(2)O(3)” RNBO. In the “m50” geometry (IRC ≈ ‒15) of the reactant species R, this RNBO is well localized on the σO(3)C(4) ether bond, as 9 ACS Paragon Plus Environment


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shown in the upper panel. As the reaction progresses to “m20” geometry (IRC ≈ ‒6), the central lobe of the sigma-type RNBO distends noticeably toward the adjacent C(2) atom. The pronounced 3c C(2)-O(3)-C(4) character continues to grow toward a maximum near the transition state (IRC = 0.0), then diminishes toward increasing 2c C(2)-O(3) π-type character as the reaction progresses to “p20” (IRC ≈ +6) geometry, where only a small “delocalization tail” distortion toward C(4) is visible in the panel (4th row). Finally, in the “p50” geometry (IRC ≈ +15) of the equilibrium product species P, this RNBO is easily recognized as the aldehyde πC(2)O(3) bond. Each of the other two columns of Fig. 3 similarly displays the continuous RNBO metamorphosis that shifts a localized 2c/2e bond from reactant to product position [i.e., π(R)C(1)C(2) → σ(P)C(1)C(6) or π(R)C(5)C(6) → π(P)C(4)C(5)] through a semi-localized 3c/2e intermediate. In this manner, the energetic transition state is decomposed into three distinct 2e RNBO bond shifts that comprise the overall “elementary” reaction mechanism. Each such 2e bond shift may be considered to pass through its own “natural” transition state (i.e., where the NRT bond-order shift is half-complete) at different IRC values, allowing deeper mechanistic insight into rate-determining effects of chemical substitution on each bond shift.12 A web-based animation of the three bond shifts can be viewed at c:\nbo7.chem.wisc.edu\claisen.ppsx. Concluding Discussion RNBOs (and their PRNBO visualization counterparts) provide an important addition to the family of localized and semi-localized basis sets produced by the NBO program. Unlike previous members of this family (NAO, NHO, NBO, NLMO), the RNBOs are not produced by default, because they intrinsically depend on NRT weightings (and keyword specification) that involve more intensive calculations than other NBO-related sets. Nevertheless, we believe that RNBO calculations are eminently practical for a broad variety of chemical reactions, and that the conceptual utility of animated RNBO reaction sequences will often justify their somewhat higher computational cost. RNBOs also fill an important need for describing more exotic types of multi-center and metallic bonding, whether as equilibrium species or reactive intermediates. Many such species have been successfully described by the AdNDP (adaptive natural 10 ACS Paragon Plus Environment

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density partitioning) method of Zubarev and Boldyrev26 and its solid-state (SSAdNDP) generalization.27 Aside from the well-known 3c/2e hypovalent bonding in borane-type species, strong computational and experimental evidence has been found for higher-order multi-center bonding in a broad variety of metallic clusters and other species of unusual charge, spin multiplicity, and coordination pattern.28 The RNBO approach adheres somewhat more closely to the localized few-center strategy of general NBO/NRT description (rather than fully delocalized MO-like description for partitions that fail to yield well-localized NBOs), but can be expected to closely resemble or complement AdNDP description when both methods are applicable. We may also mention the potential utility of RNBOs as a uniquely compact and efficient basis set (nbasis ≈ Nelec) for high-accuracy correlated calculations of general chemical reaction dynamics. As stressed above, the RNBOs intrinsically satisfy Pople’s prerequisite for uniform applicability to reactive potential energy surfaces of arbitrary complexity. Point-by-point determination of RNBOs along a reactive pathway (e.g., from starting low-level DFT description) offers a powerful method for dynamical basis optimization, including the important non-linear expansions and contractions that accompany charge shifts. Due to their intrinsic few-center character, RNBOs can exploit the well-known advantages of localized orbitals29 in achieving linear (or low-order) scaling with respect to system size. RNBOs may therefore offer both improved options for analyzing and visualizing a given wavefunction as well as efficient basis sets of occupied and virtual semi-localized orbitals for constructing improved wavefunctions of general applicability. Acknowledgment Support for computational facilities was provided in part by National Science Foundation Grant CHE-0840494. References (1)] Glendening, E. D.; Landis, C. R.; Weinhold, F. NBO 6.0: Natural bond orbital analysis program, J. Comput. Chem. 2013, 34, 1429-1437 [Erratum: J. Comput. Chem. 2013, 34, 2134]. (2) Reed, A.E.; Weinstock, R.B.; Weinhold, F., Natural population analysis, J. Chem. Phys. 1985, 83, 735-746.



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(3) Foster, J.P.; Weinhold, F., Natural hybrid orbitals, J. Am. Chem. Soc. 1980, 102, 7211-7218. (4) Reed, A.E.; Weinhold, F. Natural bond orbital analysis of near-Hartree-Fock water dimer, J. Chem. Phys. 1983, 78, 4066-4073. (5) Reed, A.E.; Weinhold, F., Natural localized molecular orbitals, J. Chem. Phys. 1985, 83, 1736-1740. (6) Weinhold, F.; Landis, C. R. Discovering Chemistry with Natural Bond Orbitals; John Wiley, Hoboken NJ, 2012, p.13ff. (7) 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. (8) Pople, J.A., Int. J. Quantum Chem. 1990, 38, 349-371; Hehre, W.J.; Radom, L.; Schleyer, P.v.R.; Pople, J.A., Ab Initio Molecular Orbital Theory; Wiley-Interscience, New York, 1986, p. 31. (9) Robinson, R., J. Soc. Chem. Ind., 1924, 43, 1297; O’Hagen, D.; Lloyd, D., Chem. World (31 March, 2010); http://www.ch.imperial.ac.uk/rzepa/blog/?p=7234 (H. Rzepa blog). (10) (a) Glendening, E.D.; Wright, S.J.; Weinhold, F., Efficient optimization of natural resonance theory weightings with convex programming, J. Chem. Theor. Comp. (in preparation); (b) Glendening, E.D.; Landis, C.R.; Weinhold, F., Resonance theory reboot, J. Am. Chem. Soc. (in preparation). (11) Glendening, E.D.; Weinhold, F. Natural resonance theory. I. General formulation, J. Comput. Chem. 1998, 19, 593-609; Glendening, E.D.; Weinhold, F. Natural resonance theory. II. Natural bond order and valency, J. Comput. Chem. 1998, 19, 610-627; Glendening, E.D.; Badenhoop, J.K.; Weinhold, F., Natural resonance theory. III. Chemical applications, J. Comput. Chem. 1998, 19, 628-646. (12) Glendening, E.D.;Weinhold, F., Natural resonance theory of chemical reactivity, with application to intramolecular Claisen rearrangement, Tetrahedron 2018, 74, 4799-4804. (13) 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: Natural Bond Orbital Analysis Programs, Theoretical Chemistry Institute, U. Wisconsin, Madison WI, 2018; http://nbo7.chem.wisc.edu/. 12 ACS Paragon Plus Environment

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(14) Weinhold, F.; Landis, C.R., Valency and Bonding: A Natural Bond Orbital Donor-Acceptor Perspective; Cambridge University Press, Cambridge UK, 2005. (15) Davidson, E.R., Reduced Density Matrices in Quantum Chemistry; Academic Press, New York, 1976. (16) Löwdin, P.-O., On the non-orthogonality problem connected with the use of atomic wave functions in the theory of molecules and crystals, J. Chem. Phys. 1950, 18, 365-375. (17) Knizia, G.; Klein, J.E.M.N., Electron flow in reaction mechanisms - revealed from first principles, Angew. Chem., Int. Ed. 2015, 54, 5518-5522. (18) Among CMO-localization methods of this type, see Foster, J.M.; Boys, S. F., Canonical configuration interaction procedure, Rev. Mod. Phys. 1960, 32, 300-302; C. Edmiston, C.; Ruedenberg, K., Localized atomic and molecular orbitals, Rev. Mod. Phys. 1963, 35, 457-465; 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. (19) Vidossich, P.; Lledos, A., Computing the arrows of chemical reactions, Chem. Texts 2017, 3, 17. (20) Andres, J.; Berski, S.; Silvi, B., Curly arrows meet electron density transfers in chemical reaction mechanisms: from electron localization function (ELF) analysis to valence-shell electron-pair repulsion (VSEPR) inspired intepretation, Chem. Commun. 2016, 52, 8183-8195. (21) Andres, J.; Gonzalez-Navarrete, P.; Safont, V.S.; Silvi, B., Curly arrows, electron flow, and reaction mechanisms from the perspective of the bonding evolution theory, Phys. Chem. Chem. Phys. 2017, 19, 29031-29046. (22) Proud, A.J.; Mackenzie, D.E.C.K.; Pearson, J.K., Exploring electron pair behaviour in chemical bonds using the extracule density, Phys. Chem. Chem. Phys. 2015, 17, 20194-20204. (23) Liu, Y.; Kilby, P.; Frankcombe, T.J.; Schmidt, T.W., Calculating curly arrows from ab initio wavefunctions, Nature Comm. 2018, 9, 1436. 13 ACS Paragon Plus Environment


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carried out by local pair natural orbital based coupled-cluster methods, Chem. Soc. Rev. 2014, 43, 5032-5041.



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Resonance Natural Bond Orbitals: Efficient Semilocalized Orbitals for

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