Resonance Natural Bond Orbitals: Efficient Semilocalized Orbitals for

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Resonance Natural Bond Orbitals: Efficient Semilocalized Orbitals for Computing and Visualizing Reactive Chemical Processes E. D. Glendening† and F. Weinhold*,‡ †

Department of Chemistry and Physics, Indiana State University, Terre Haute, Indiana 47809, United States Theoretical Chemistry Institute and Department of Chemistry, University of WisconsinMadison, Madison, Wisconsin 53706, United States

J. Chem. Theory Comput. 2019.15:916-921. Downloaded from pubs.acs.org by UNIV AUTONOMA DE COAHUILA on 04/06/19. For personal use only.

‡

S Supporting Information *

ABSTRACT: We describe a practical algorithm for calculating NBObased “resonance natural bond orbitals” (RNBOs) that can accurately describe the localized bond shif ts 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 multichannel 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 multicenter 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.

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INTRODUCTION Previous versions of the Natural Bond Orbital (NBO) program1 have provided a sequence of orthonormal localized and semilocalized 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. 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 “Lewiscentric” 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 multicenter character of transition-state species or other far-from-equilibrium species along a chosen reactive pathway. As a result, such orbitals are precluded from serving © 2019 American Chemical Society

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-def ined 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 multichannel 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 resonancetype concepts of chemical reactivity that trace back to Robinson’s prequantal “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 Claisentype rearrangement reactions.12 RNBO/PRNBO construction is fully implemented in the current NBO 7.0 version of the general NBO analysis program.13 Received: September 18, 2018 Published: January 6, 2019 916

DOI: 10.1021/acs.jctc.8b00948 J. Chem. Theory Comput. 2019, 15, 916−921

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METHOD RNBOs are a further extension of NLMOs5 for incorporating effects of resonance-type donor−acceptor delocalizations from the initial localized NBOs.14 As shown in Figure 1, each such

each contributing bonding pattern R, R′. For continuity between distinct bonding patterns, the ordinal index i and phase of each Θ̃i(R) in eq 1 are chosen to properly match those of Θ̃i(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 preorthogonal (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 NAObased decomposition of each occupied RNBO (similar to that for Lewis-type NLMOs) which allows ready quantitation of multicenter 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 Klein17 (related to Pipek-Mezey localization18 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 wave functions of any form or accuracy. Silvi and co-workers20,21 have described alternative real-space density “bonding evolution” in terms of the electron 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 wave functions to exhibit parallels to curly arrow depictions. Among these varied efforts to capture localized or semilocalized 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 (Figure 1), including NRT-based 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, 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 first-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

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.

donor−acceptor (DA) interaction can be uniquely associated with a resonance structure (R) and associated curly arrow representation. The corresponding stabilization energy of each such delocalization (EDA(2)) can be estimated by second-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 the closest possible resemblance to the parent NBO. 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 {Θ̃i(R)} of each contributing resonance structure to obtain resonance-averaged orbitals {ωi} of high-occupancy and nearorthogonal character, viz. ωi =

∑ wR Θ̃i R

(R)

(1)

These are subsequently transformed to orthonormal RNBOs {Ωi} by Löwdin symmetric orthogonalization.16 Note that contributing Θ̃i(R), Θ̃i(R′) orbitals in eq 1 are generally nonorthogonal, because they arise from separate optimizations for 917

DOI: 10.1021/acs.jctc.8b00948 J. Chem. Theory Comput. 2019, 15, 916−921

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Journal of Chemical Theory and Computation 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 {Θ̃i(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. || ΓNLS − ΓQC ||2 = min

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 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 reference-level 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 ensure that any resonance structure contributing appreciably in the transition-state region

(2)

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)} ΓNRT =

∑ wR Γ(R)

(3a)

R

that are constrained by the convexity conditions all wR ≥ 0, with ∑ wR = 1 R

(3b)

and provide optimal weightings to satisfy the NRT variational minimization criterion, viz. || ΓNRT − ΓQC ||2 = min

(3c)

10b

As described elsewhere, the density matrix-based NRT formulation ensures that empirical resonance-type assumptions are satisfied for all one-electron properties, in contrast to Pauling-type (wave function-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 doubleoccupancy for closed-shell single-determinant 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 (occupancydetermined) parent NBOs of contributing bonding patterns that insures their own uniquely determined forms, even in the single-determinant 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 valence-level 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 wave functions, 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

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 balland-stick geometry (upper) and Lewis-structural bonding pattern (lower). 918

DOI: 10.1021/acs.jctc.8b00948 J. Chem. Theory Comput. 2019, 15, 916−921

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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 (Figure 2).

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 bondorder shifts accompanying this remarkable rearrangement are described elsewhere.12 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 bondshifting RNBOs, which can be identified (from left to right; see column headings) as πC(1)C(2), σO(3)C(4), and πC(5)C(6) in reactant R labeling or σC(1)C(6), πC(2)O(3), and πC(4)C(5) in product P labeling. Each bond-shifting RNBO exhibits conspicuous multicenter character near the energetic transition state, whereas the

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 the Supporting Information (SI).

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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 919

DOI: 10.1021/acs.jctc.8b00948 J. Chem. Theory Comput. 2019, 15, 916−921

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Journal of Chemical Theory and Computation remaining “spectator” bonds retain localized 2-center NBOlike character throughout the entire reaction pathway. As a particular example, consider the middle column of Figure 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 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) and 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 Figure 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 semilocalized 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 Webbased animation of the three bond shifts can be viewed at http://nbo7.chem.wisc.edu/claisen.ppsx.

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-bypoint 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 nonlinear 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 wave function as well as efficient basis sets of occupied and virtual semilocalized orbitals for constructing improved wave functions of general applicability.

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ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jctc.8b00948. Gaussian 16 and NBOPro@Jmol input and output files for Claisen rearrangement reaction (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

F. Weinhold: 0000-0002-9580-054X Funding

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Support for computational facilities was provided in part by National Science Foundation Grant CHE-0840494.

CONCLUDING DISCUSSION RNBOs (and their PRNBO visualization counterparts) provide an important addition to the family of localized and semilocalized 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 NBOrelated 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 multicenter and metallic bonding, whether as equilibrium species or reactive intermediates. Many such species have been successfully described by the AdNDP (adaptive natural 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 multicenter 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

Notes

The authors declare no competing financial interest.

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REFERENCES

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