Natural Bond Orbital Theory of Pseudo-JahnâTeller Effects

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A: Molecular Structure, Quantum Chemistry, and General Theory

Natural Bond Orbital Theory of Pseudo Jahn-Teller Effects Frank Weinhold, and Davood Nori-Shargh J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.7b12810 • Publication Date (Web): 19 Apr 2018 Downloaded from http://pubs.acs.org on April 19, 2018

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The Journal of Physical Chemistry

Natural Bond Orbital Theory of Pseudo Jahn-Teller Effects Davood Nori-Shargha and Frank Weinholdb* a

Department of Chemistry, Arak Branch, Islamic Azad University, Arak, Iran ([email protected]); bTheoretical Chemistry Institute and Department of Chemistry, University of Wisconsin-Madison, Madison, Wisconsin 53706, USA ([email protected]) Abstract We describe a unified picture of symmetry-breaking electronic interactions that are usually described as “pseudo Jahn-Teller (PJT) effects” and attributed to vibronic coupling, but can also be associated with hyperconjugative donor-acceptor interactions in the framework of natural bond orbital (NBO) and natural resonance theory (NRT) analysis. We show how NBO/NRT descriptors offer a simplified alternative to the vibronic coupling picture of PJT effects that yields both improved cause-effect specificity and chemically enriched understanding of symmetry-breaking phenomena, but with no necessary input from ground-state vibrational or excited-state electronic properties. Comparative NBO/NRT vs. vibronic coupling analyses of PJT effects are illustrated for two well-known cases: trans-bending in Si2H4 and higher Group-14 homologs of ethylene, and chain-kinking in cyclopentadienylideneketene (C5H4CCO) and related cumulene ketones. The conceptual and practical advantages of the NBO-based hyperconjugative approach may be expected to extend to numerous PJT-type symmetry-breaking phenomena throughout the chemical sciences.

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Introduction The origins of symmetry and symmetry breaking provoke curiosity throughout the natural sciences.1 In the molecular world, Jahn and Teller2 (JT) first presented a conceptual framework for understanding the inherent structural instability of high-symmetry radical species in electronically degenerate states, where some allowed vibrational distortion must always serve to break the electronic degeneracy and lead to lower ground-state energy for a broken-symmetry geometry. However, similar distortions from an envisioned highsymmetry precursor can be recognized in a much broader variety of non-degenerate closedshell species that lie well outside conventional JT rationale, identified instead as “pseudo Jahn-Teller” (PJT) effects.3 As detailed below, such structural symmetry breaking is commonly attributed to vibronic mixing of ground and excited electronic states induced by displacements along a specific vibrational mode of the envisioned high-symmetry precursor species.4-12 In this manner, the PJT rationale similarly evokes the symmetry dependence of the original JT analysis, but in a broadened range of applicability. A representative example of PJT effects is provided by the heavier Group-14 homolog of ethylene (e.g., Si2H4) (Scheme 1). Unlike D2h-symmetric ethylene, the heavier homologs distort to trans-bent C2h-symmetric geometry, viz.

Scheme 1: Schematic representation of the ground states structures of Si2H4 and C2H4.

However, the electronic origin of the altered preference for D2h vs. C2h geometry has also been explained13-15 in the framework of Natural Bond Orbital (NBO) theory16 and


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NBO-based natural resonance theory (NRT),17-19 whose assumptions and methodology differ appreciably from those of JT or PJT theory. Briefly, NBOs are local eigen-orbitals of the first-order reduced density matrix (1-RDM), which condenses all possible 1e “orbital-type” information from an input wavefunction ψ (ab initio or DFT, variational or perturbative, uncorrelated or correlated,...), and NRT weightings are obtained by variationally matching the idealized NBO-based resonance structures and their weighted sum to the full 1-RDM. A related type of symmetry breaking is exhibited in the unexpected kinking deformations of cumulene ketones,20,21 which were similarly analyzed with NBO/NRT methods22 to provide a simple chemical rationale of the interesting deformation patterns (Scheme 2).

Scheme 2. Schematic PJT symmetry breaking in C5H4CCO

In still another example, open-shell NBO theory was shown to provide improved understanding of ordinary JT effects in radical species,23 based on simple spinaveraging of open-shell NBO structures that allows far-reaching extension and quantification of JT-type symmetry-breaking with no direct use of molecular geometry or symmetry information. Such complementary theoretical views of a structural effect ― despite numerous conceptual and numerical differences in the theoretical methodologies ― motivate the present search for deeper relationships between conventional vibronic-coupling and NBO/NRT approaches that clarify their similarities and differences.


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At the outset, it should be stressed that finding an alternative way to view PJT-type effects does not imply that one perspective is more “rigorous,” “correct,” or “fundamental” than the other. Proper treatment of the “vibronic” couplings of electron and nuclear (e/N) degrees of freedom (i.e., full electronic optimization for each nuclear displacement) is the essence of all current quantum chemical methods for describing the adiabatic ground-state potential energy surface (PES), including PJTtype distortions. The present work emphasizes the complementarity between adiabatic (quantum chemical) and diabatic 2-state coupling (spectroscopic) models by employing specific idealized “diabatic” NRT resonance structures (easily extracted from the quantum chemical calculations for the ground-state PES) as the explicit conceptual bridge to Marcus-like 2-state depiction of e/N interactions. Once such adiabatic-diabatic connections are recognized, it becomes purely a matter of taste whether to seek details of e/N couplings in the adiabatic ground state PES (where the envisioned 2-state couplings leave “small corrections”) or in some excited state PES (where the same couplings leave “high parentage”). In the following section, we first outline the distinct theoretical conceptions and methodology underlying traditional vibronic coupling vs. NBO/NRT analyses of the origin of PJT symmetry breaking and summarize the computational methods. In the ensuing section, we first compare the effectiveness of vibronic coupling and NBO/NRT methods as applied to specific examples of the symmetry breaking (Schemes 1, 2), then discuss the subtle relationships that seem to best explain both the apparent similarities and profound differences between these methods. The more general PJT perspective inferred from these specific examples is summarized in a final Summary and Conclusions section.


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Alternative Methods to Analyze Pseudo Jahn-Teller Effects Vibronic Coupling Analysis In the conventional approach to PJT effects,3 the expected (simple or unperturbed) system is in a high-symmetry nuclear configuration Rs, with corresponding electronic eigenstates {ψi(Rs)}, whereas the equilibrium nuclear configuration Ru is of unexpected low symmetry, with corresponding eigenstates {ψi(Ru)}. As the notation suggests, we can envision an adiabatic connection between electronic eigenstates at Rs and Ru (= Rs + Q) that enables each parametrized ψi(R) to be shifted continuously along a deformation path Q from Rs to Ru. In analogy to JT theory, the origin of PJT structural deformation is usually sought in a perturbative two-state model of the vibronic coupling between electronic ground ψ0 and a particular excited state ψn associated with a shift in nuclear configuration Q along a specific mode of vibration. If the product ψ0(∂H/∂Q)0 of the undistorted ground-state ψ0 has high overlap with a particular excited state ψn under geometry distortion Q, the electronic states ψ0, ψn are considered to be mixed by a vibronic coupling constant F0n (1) F0n = 〈ψ0|(∂H/∂Q)0| ψn〉 that alters the initial vibrational force constant K0 (intrinsically positive) to a decreased value K that is perturbatively estimated as (2) K = K0 ‒ (F0n2/∆0n) Here ∆0n is the energy difference of the two states and the initial force constant K0 is evaluated as (3) K0 = 〈ψ0|(∂2H/∂Q2)0|ψ0〉 in the ground-state ψ0 = ψ0(Rs) at undistorted geometry Rs. According to Eq. (2), a vibronic instability leading to deformation from the initial high-symmetry geometry occurs whenever K ˂ 0, that is, when F0n2/∆0n exceeds K0.


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The magnitude of the vibronic coupling constant F0n is in turn strongly dependent on proper matching of the symmetry of vibrational displacement Q with the symmetry of electronic states ψ0, ψn to give a non-vanishing transition integral in Eq. (1). Since the symmetry of ψ0 and the symmetry-breaking coordinate Q is generally fixed, the excited state ψn for 2-level description must be selected to give a totally symmetric integrand in Eq. (1). In multilevel problems, the effects of many such excited states may contribute to the distortion and their effects on K are summed up.3 Although the starting point of vibronic analysis is a chosen multi-configurational (e.g., CI, CAS, CIS, TD-DFT,...)24 description that includes both ground and excited states of the system in question, the final states of interest are usually expressed in terms of a common set of ground-state molecular orbitals (MOs). In this case, the symmetry restrictions on ψ0, ψn are expressible in terms of corresponding restrictions on MOs of ground and excited determinants, often involving the highest-occupied (HOMO) and lowest-unoccupied (LUMO) MOs of initial high-symmetry geometry. Orbital-based aspects of PJT vibronic coupling analysis are illustrated in a subsequent section of comparative chemical applications. NBO-Based Analysis In the NBO-based approach to PJT effects, the expected high-symmetry starting point is generally that associated with an idealized Lewis-structural picture of the bonding pattern, whereas the actual low-symmetry geometry includes the many effects of resonance-type donor-acceptor interactions. The starting point for NBObased perturbation theory analysis25 is the determinant of doubly-occupied Lewis-type (donor) NBOs {χi(L)}, the “natural Lewis structure” (NLS) wavefunction ψ0(L), (4) ψ0(L) ≡ det|(χ1(L))2(χ2(L))2...(χN/2(L))2| In Hartree-Fock (HF) or density functional theory (DFT), the energy EL ≡ 〈ψ0(L)|H|ψ0(L)〉 of the NLS determinant provides the zero-order approximation to the full HF or DFT Efull ≡ 〈ψfull|H|ψfull〉 of system Hamiltonian H. Higher-order


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perturbative corrections incorporate the effects of excitations to non-Lewis NBOs {χj*(NL)}. Specifically, the energy lowering (in magnitude) associated with donoracceptor excitation (χi(L))2 → (χj*(NL))2 from occupied χi(L) “donor” to unoccupied χj*(NL) “acceptor” NBO is estimated by the well-known 2nd-order perturbation theory formula (5) ∆E(2)i→j* = 2〈χi(L)|F|χj*(NL)〉2/(εj* − εi) where εj* = 〈χj*(NL)|F|χj*(NL)〉, εi = 〈χi(L)|F|χi(L)〉 are NBO energies and F is the 1electron effective Hamiltonian (Fock/Kohn-Sham operator) of HF/DFT description. Note that each (χi(L))2 → (χj*(NL))2 (i→j*) hyperconjugative26 excitation corresponds to a contributing resonance structure27 whose NRT weighting can be independently calculated (by a non-perturbative algorithm) for comparison with the orderings suggested by Eq. (5). Alternatively, the energetic effect of a particular i→j* donor-acceptor interaction, or of all possible interactions involving non-Lewis NBO χj*(NL), or of any combination of such individual i→j* and collective j* contributions, can be evaluated by deleting such interaction(s) from the evaluation of Efull with the help of $DEL-keylist options.28,29 For each such deletion, the calculated E$DELi→j* is raised by variational loss of i→j* stabilization, with resulting energy difference ∆E$DELi→j* (6) ∆E$DELi→j* ≡ E$DELi→j* − Efull that provides an alternative estimate of i→j* stabilization for comparison with perturbative estimate (5). Importantly, each E$DELi→j* evaluation can be combined with standard geometry optimization (OPT) and vibrational frequency (FREQ) calculations, allowing one to determine the altered potential energy surface, and thus the full structural and spectroscopic consequences, of a particular resonance-type i→j* interaction. Systematic $DEL evaluations thereby allow determination of specific “smoking gun” cause-effect relationships between a particular structural feature of interest and a chosen set of i→j* interactions. The general mapping of each


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NBO i→j* interaction onto an associated NRT resonance structure in turn provides enriched chemical understanding of the target structural effect in familiar resonancetheoretic terms. Other NBO program options30 allow one to obtain the explicit NBO→MO transformation matrices that map onto delocalized molecular orbital (MO) constructs. Note that in contrast to the vibronic coupling approach, NBO/NRT algorithms depend only on the 1-RDM of the specified ground electronic state,16 with no direct input of molecular structure or symmetry, vibrational modes, or electronic excited state properties. The numerical applications below illustrate how NBO/NRT analysis can yield increased specificity and understanding of PJT effects with reduced input demands and simpler numerical implementation. Computational Methods For Si2H4, we employ standard DFT-type calculations24 at the B3LYP/aug-ccpVTZ level for the ground-state and corresponding TD-B3LYP/aug-cc-pVTZ calculations for excited-state properties required by vibronic coupling analysis. For C5H4CCO, where only ground-state $DEL calculations are required, we employ the more economical B3LYP/6-311++G** level in light of the observed insensitivity of symmetry-breaking properties to theory level in related species.22 All calculations were performed with the G09 Rev. E01 program system31 and interactive NBO/NRT analyses and $DEL optimizations were performed with the NBO 6.0 program.32 Orbital visualizations were prepared with the NBOView module of the NBOPro6@Jmol program.33

Comparative Applications to Chemical Species Trans-Bending in Si2H4 The vibronic coupling approach to Si2H4 starts from symmetry classification of the nuclear distortion coordinate Q and electronic states ψ0, ψn of Eqs. (1-3). According to general group-theoretic considerations,34 symmetry reduction from D2h to C2h requires ACS Paragon Plus Environment

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a nuclear displacement Q belonging to the b2g irreducible representation of the initial high-symmetry geometry, while the electronic ground state ψ0 belongs to the totally symmetric Ag representation (with upper vs. lower case distinguishing electronic vs. vibrational labels). A non-vanishing vibronic coupling constant F0n [Eq. (1)] therefore requires that a candidate excited state ψn must be of B2g symmetry. More specifically, Q(b2g) = Qim must correspond to the unique normal mode of vibration with imaginary frequency (K < 0), the unstable Hessian eigenvector of harmonic frequency analysis [cf. Eq. (3)]. The search for a vibronically coupled excited state ψn therefore involves examining electronic excited-state energies {Ek} (obtained, e.g., from TD-DFT description) as a function of Qim extension, looking for a suitable ψn of B2g symmetry that appears to be the upper member “twin state” of a 2state avoided crossing, i.e., with ψn trending upward to mirror the energy-lowering downward trend of ψ0 under Qim distortion. The energy denominator ∆0n of Eq. (1) suggests the preference for the lowest-lying ψn that exhibits proper symmetry and repulsive twin-type coupling with respect to ψ0. Figure 1 displays the calculated Ek vs. Qim curves for the lowest four TD excited states of Si2H4, showing the ground E0(Ag) state (solid line) and lowest E3(B2g) excited state (dashed line) as well as nearby states of other symmetry (dotted lines). As shown in the figure, the dashed E3(B2g) is indeed trending upward, as expected for the twin state in a 2-state perturbative model. Accordingly, conventional PJT analysis would attribute trans-bending in Si2H4 to vibronic coupling F03 of ψ0 with excited state ψ3.


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Figure 1. TD-B3LYP/aug-cc-pVTZ spectrum of ground and excited states of Si2H4 along the nuclear displacement Q(b2g), showing energy differences ∆E (kcal mol-1) with respect to the highsymmetric planar configuration. The ground state (circle, heavy black solid line) and three lowlying lowest excited states (brown box, yellow dashed line) are compared with the apparent twin B2g state ψ3 (red circle, heavy red solid line).

Deeper chemical understanding of the vibronic coupling results may then be sought in MO compositions or other details of ψ0 and ψ3. Figure 2 displays the energy curves for E0(Ag) and E3(B2g) states on a scale that better shows their respective shapes despite the wide energy separation (ca. 108 kcal mol−1), with side panels that illustrate the distinguishing MOs of each state. As shown in Fig. 2, the strongest orbital contributions to vibronic coupling between ψ0 and ψ3 seem to involve the B3uACS Paragon Plus Environment

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symmetric HOMO and B1u-symmetric SUMO (LUMO + 1) orbitals. To complete the analysis, comparable spectral features of Si2H4 might then be compared with those of C2H4 to finally seek rationalization for why disilene buckles to trans-bent geometry but ethylene does not.

Figure 2. TD-B3LYP/aug-cc-pVTZ calculated energy curves for E0(Ag) and E3(B2g) states along the nuclear displacement Q(b2g), showing energy differences ∆E (kcal mol-1) with respect to the high-symmetric planar configuration. The side panels illustrate the distinguishing B3u-symmetric HOMO and B1u-symmetric SUMO (LUMO + 1) orbitals that contribute to vibronic coupling between ground ψ0 and excited ψ3.

The NBO analysis of PJT symmetry breaking in Si2H4 differs in many respects. No vibrational frequency or TD-type excited-state calculations are required. Indeed, only two single-point ground-state calculations are required: (i) that for final equilibrium trans-bent C2h geometry, and (ii) that for initial idealized D2h geometry (i.e., optimized in constrained planar geometry), each followed by NBO analysis.


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The graphical forms of valence bonding (σSiSi, πSiSi, σSiH) and antibonding (σ*SiSi, π*SiSi, σ*SiH) NBOs of Si2H4 are displayed in Figure 3. A glance at the figure suggests the many donor-acceptor interactions [i→j*; cf. Eq. (5)] that are vanishing by symmetry in idealized D2h geometry but non-zero (and hence, candidate symmetry breakers) in trans-bent geometry: σSiSi→π*SiSi, πSiSi→σ*SiSi, πSiSi→σ*SiH (4) and σSiH→π*SiSi (4). The full Qim dependence (or only the direct D2h vs. C2h difference) of ∆E$DELi→j* evaluations [Eq. (6)] can be used to assess the net stabilization energy associated with each such i→j* interaction. Table 1 summarizes numerical values of E(2)i→j, ∆E$DELi→j* for the four types of symmetry-breaking NBO interactions in transbent C2h geometry.


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σSiSi

πSiSi

σSiH

σ*SiSi

π*SiSi

σ*SiH

Figure 3. Valence-level bond and antibond NBOs of Si2H4 (D2h), in contour (middle) and rendered surface (right) views.


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Table 1. NBO 2nd-order perturbation-theory and $DEL-deletion energy estimates (E(2), ∆E$DEL, kcal/mol) for leading donor-acceptor interactions in the trans-bent equilibrium geometry of Si2H4. (All such interactions vanish in planar D2h geometry.)

donor acceptor E(2) ∆E$DEL πSiSi σ*SiSi 3.57 3.38 σSiSi

π*SiSi

3.21

3.71

πSiSi

σ*SiH

2.14

2.52

σSiH

π*SiSi

2.13

2.30

It is immediately recognizable that the πSiSi and σ*SiSi NBOs of the symmetrybreaking πSiSi→σ*SiSi NBO interaction (Fig. 3) closely resemble the HOMO(B3u) and SUMO(B1u) orbitals that were previously identified (Fig. 2) as distinguishing MOs of the ψ0(Ag)-ψ3(B2g) vibronic coupling. This recognition suggests an underlying connection between NBO donor-acceptor and vibronic coupling conceptions of the symmetry-breaking phenomenon. Specifically, each NBO donor-acceptor interaction in Table 1 corresponds to an alternative electronic bonding pattern (resonance structure) that contributes only weakly to the adiabatic ground-state wavefunction ψ0, but can also serve as “diabatic precursor” of an adiabatic excited state ψn that is apparently the repulsive twin-state of ψ0 near a 2-state avoided crossing . If the apparent association of an excited state ψn with each diabatic resonancestructural contribution to ψ0 (Table 1) is valid, we can anticipate that the vibronic picture must include coupling to an additional excited state ψn′ that serves as adiabatic counterpart of the important σSiSi→π*SiSi NBO interaction in Table 1. This σSiSi→π*SiSi-like excited state must again have B2g symmetry and lie higher in energy than the πSiSi→σ*SiSi-like state ψ3 previously identified (Fig. 2) as important in symmetry breaking. Accordingly, we are led to search a still larger range of TD excited states for contributions to a more complete vibronic coupling picture, as shown in Figure 4. The figure shows the expanded Ek(Qim) scan for the lowest nine TD states, now


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including two additional states of B2g symmetry, k = 7 and 9. The E7(Qim) profile has incorrect (downward) curvature, but the E9(Qim) profile (in red) is evidently the “missing” σSiSi→π*SiSi-like excited state that was omitted in the initial two-state vibronic-coupling representation of symmetry breaking.

Figure 4. TD-B3LYP/aug-cc-pVTZ spectrum of ground and excited states of Si2H4 along the nuclear displacement Q(b2g), showing energy differences ∆E (kcal mol-1) with respect to the highsymmetric planar configuration. The ground state (circle, heavy black solid line) and seven lowlying lowest excited states (brown box, yellow dashed line) are compared with two apparent twin B2g states ψ3 , ψ9 (red circle, heavy red solid line).


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Figure 5 compares the Q-displacement curves for ψ0, ψ9 on an expanded vertical scale, with side panels illustrating the leading MO difference between these states (cf. Fig. 2). As shown in the side-panels, the apparent MO excitation is from the SOMO (HOMO−1) of ψ0 to the LUMO of ψ9, corresponding most closely to the σSiSi and π*SiSi NBOs of Fig. 3, respectively. This result is consistent with the expected importance of hyperconjugative σSiSi→π*SiSi NBO interaction in the symmetry-breaking phenomenon, and offers further support for a general conceptual association of each ground-state i→j* donor-acceptor interaction with a specific orbital promotion and “vibronically coupled” excited state.

Figure 5. TD-B3LYP/aug-cc-pVTZ calculated energy curves for E0(Ag) and E9(B2g) states along the nuclear displacement Q(b2g), showing energy differences ∆E (kcal mol-1) with respect to the high-symmetric planar configuration. The side panels illustrate the distinguishing Ag-symmetric HOMO-1 and B2g-symmetric HOMO orbitals that primarily contribute to vibronic coupling between ground ψ0 and excited ψ9 states.


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The weaker πSiSi→σ*SiH and σSiH→π*SiSi hyperconjugative contributions of the NBO/NRT picture (Table 1) might also be sought in still higher TD excited states to add to a more complete multi-state vibronic coupling picture. However, increased configurational mixing makes such multi-state searching increasingly problematic, unlikely to discern the associations with localized πSiSi→σ*SiH and σSHi→π*SiSi hyperconjugations that could add further chemical insight to the origin of symmetry breaking. These difficulties are in clear contrast to the simplicity with which symmetry-breaking hyperconjugative interactions are routinely recognized, quantified, and chemically interpreted in the framework of ground-state NBO analysis. How can we be certain that the principal πSiSi→σ*SiSi, σSiSi→π*SiSi, ... hyperconjugations identified by NBO/NRT analysis (Table 1) are the true physical “cause” of the PJT symmetry breaking “effect”? Such cause-effect relationships can be unambiguously established with the help of ∆E$DELi→j* deletion techniques28,29 [Eq. (6)], combined with full reoptimization of molecular geometry to determine the precise structural consequences of a chosen deletion. For example, deleting either of the πSiSi→σ*SiSi and σSiSi→π*SiSi hyperconjugations [with respective resultant energies E$DEL(πSiSi→σ*SiSi) and E$DEL(σSiSi→π*SiSi)] and reoptimizing Si2H4 geometry (from initial equilibrium C2h starting point) leads to the fully D2h-symmetric geometry as shown schematically in Figure 6. The figure displays the shapes and interactions of the various σSiSi, σ*SiSi, πSiSi and π*SiSi NBOs in both high- (left) and low-symmetry (right) geometries, with corresponding 2nd-order E(2) estimates of donor-acceptor interaction strength for each symmetry-breaking type, πSiSi→σ*SiSi (upper) or σSiSi→π*SiSi (lower). The $DEL reoptimizations rigorously demonstrate the direct cause-effect relationship between hyperconjugative interactions and the symmetrybreaking structural feature of interest, and the NBO contour diagrams of Fig. 6 show the chemical origin of the favorable hyperconjugative stabilizations (E(2) ≈ 3.2-3.6 kcal/mol; cf. Table 1) that reward symmetry-breaking in each case.


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Figure 6. NBO contour and surface diagrams for πSiSi→σ*SiSi (upper panels) and σSiSi→π*SiSi (lower panels) interactions, showing perturbative estimates E(2) of donor-acceptor stabilization (kcal/mol) in planar D2h (left) and trans-bent C2h (right) geometry. $DEL deletion of either interaction restores the high-symmetry geometry, where all such symmetry-breaking interactions are vanishing.

Similar $DEL deletions and reoptimizations for the weaker πSiSi→σ*SiH (4) or σSiH→π*SiSi (4) interactions of Table 1 lead similarly to planarized D2h-symmetric geometry, showing that removal of each such class of hyperconjugations is sufficient in this case to restore the high-symmetry geometry and thus remove the perceived PJT structural “effect”. However, even if the structural PJT effect appears satisfactorily “explained” by any one (or any combination) of symmetry-breaking hyperconjugations in Table 1, each of these distinct hyperconjugative classes may give rise to differential spectroscopic PJT effects, particularly on the IR frequency of the key Q(b2g) vibrational mode that is featured in the vibronic coupling picture. Such


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IR spectroscopic effects can be rigorously evaluated by carrying out correponding frequency (FREQ) calculations at the E$DEL-reoptimized geometry for each hyperconjugative deletion. Figure 7 displays the calculated harmonic vibrational frequencies for various classes of $DEL deletions (including “None” for the full calculation with no deletions), showing how the low-frequency Q(b2g) vibrational mode (enlarged red dots) is progressively shifted relative to other frequencies (blue dots) by specific symmetry-breaking $DEL-deletions (a)-(e) (see figure caption) or by “Lewis” deletion of all hyperconjugative (non-Lewis) interactions. As noted above, each such hyperconjugative deletion is sufficient to raise νQ to positive values characteristic of D2h-symmetric stability, but the overall frequency shift (by ca. 800 cm−1) gains different increments from different hyperconjugative classes, confirming the assessments (Table 1) of differential donor-acceptor stabilization. Although the inferences drawn from NBO/NRT descriptors of hyperconjugative stabilization require no input from vibrational calculations, they serve to illuminate deeper details of IR shifts that are beyond the vibronic coupling picture of conventional structural PJT effects.


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Figure 7. B3LYP/aug-cc-pVTZ calculated vibrational frequencies ν (cm-1, all calculated in D2hsymmetric stationary geometry of Si2H4, with imaginary-valued frequencies plotted as negative) for various $DEL hyperconjugative deletions: “None”: full calculation with no deletions (at D2h-symmetric transition state) (a) Delete the four πSiSi→σ*SiH interactions (b) Delete the four σSiH→π*SiSi interactions (c) Delete the σSiSi→π*SiSi interaction (d) Delete the πSiSi→σ*SiSi interaction (e) Delete both σSiSi→π*SiSi and πSiSi→σ*SiSi interactions “Lewis”: delete all non-Lewis interactions Enlarged red dots denote the b2g vibrational frequency (Q-mode with imaginary frequency in the D2h-symmetric Si2H4 transition state geometry) and blue dots denote other vibrations that are little affected by deletions.

Still deeper understanding of the surprising differences between C2H4 and heavier M2H4 homologs (M = Si, Ge,…) can be found in the details of πMM→σ*MM, σMM→π*MM and other resonance-type contributions throughout the Group 14 series, with energetic spacings and interaction strengths varying in expected ways with size, electronegativity, and related valence-shell properties of heavier homologs.13 All such hyperconjugative rationalizations and quantifications can be obtained from NBO/NRT descriptors of the ground-state electronic density matrix, with no recourse to symmetry or other excited-state or vibrational mode descriptors that are characteristic of the vibronic coupling model of PJT effects.


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Cumulene Kinking in C5H4CCO As a second example of NBO/NRT hyperconjugative analysis of PJT effects, we examine related aspects of the interesting kinking (or non-kinking) of …C=C=C… cumulene chains in C5H4CCO and related mono- and diketones (Scheme 2).22 For the C5H4CCO case that has been explored most thoroughly by theoretical20 and experimental21 means, the calculated low-symmetry equilibrium geometry and C symmetric transition state geometry (with imaginary νQ = −82i cm−1) are displayed in Figure 8. Neglecting the cyclopentadienyl moiety, the leading NBO donor-acceptor interactions of the O=C′=C″ tail (as judged by E(2) or ∆E$DEL estimates) are found to to be of πC′C″→σ*C′C″ (d1), σC′C″→π*C′C″ (d2), or nO(π)→σ*C′C″ (d3) type, as summarized in Table 2. Here, nO(π) denotes the off-axis p-rich oxygen lone pair oriented perpendicular to the O=C′ axis, whereas nO(σ) is the inert s-rich on-axis lone pair (no rabbit ears!).35

Figure 8. Comparison of structural bond lengths (Å) and angles (°) of the kinked (Cs) and highsymmetry (C2v) configurations of cyclopentadienylideneketene.


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Table 2. NBO 2nd-order perturbation-theory and $DEL-deletion energiy estimates (E(2), ∆E$DEL, kcal/mol) for leading NBO donor-acceptor interactions (d1)-(d3) in the kinked O=C′=C″ tail of equilibrium C5H4CCO.

donor acceptor E(2) ∆E$DEL d1. πC′C″ σ*C′C″ 1.35 1.52 d2.

σC′C″

π*C′C″

1.26

0.95

d3.

nO(π)

σ*C′C″

0.95

0.83

We can carry out similar variational E$DEL optimizations to exhibit the decisive cause-effect relationships between kink-type distortions and selected symmetrybreaking hyperconjugations of the ketene tail atoms. Starting from the kinked equiibrium geometry (Fig. 8, left), $DEL-type reoptimization with deletion of any one of the interactions of Table 2 is found to yield the C2v-symmetric geometry (Fig. 8, right). Thus, each interaction of Table 2 may be considered an “origin” of PJT-type symmetry breaking, sufficient (by its removal) to induce linearization of the O=C′=C″ tail and thus quench all remaining symmetry-breaking interactions. As in the Si2H4 case, structural symmetry breaking in C5H4CCO involves a manifold of hyperconjugative interactions whose resonance-structural depictions would require a corresponding multiplicity of excited-state contributions for vibronic coupling rationalization. Once again we can probe deeper aspects of PJT-type structural stability and their cause-effect relationship to NBO descriptors of hyperconjugative interactions by extending $DEL-type deletion studies to IR spectroscopic properties. Figure 9 displays the relationship between E$DEL-based frequency of the kinking mode (νQ) vs. corresponding ∆E$DEL assessment of interaction strength, both for individual donoracceptor interactions (d1, d2, d3) as well as composite deletion (d1-d3) of all three interactions in Table 2. As shown in Fig. 9, each individual deletion lifts νQ from imaginary to positive real values, insuring equilibrium stability of the C2v-symmetric structure in each case. However, the vibrational stability of this structure is ACS Paragon Plus Environment

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progressively enhanced with increasing hyperconjugative interaction strength(s), viz., up to νQ = +51 cm−1 for d1 deletion, or up to +82 cm−1 for composite d1-d3 deletions. Thus, simple NBO-based E(2) or ∆E$DEL estimates can be used to predict which donoracceptor interaction (or equivalently, which resonance-structural correction) contributes most strongly to the ∆νQ shift associated with vibrational stability or instability.

Figure 9. IR frequency (νQ, cm−1, with negative values denoting imaginary frequencies) of kinkdistortion mode Q vs. NBO deletion energy (E$DEL, kcal/mol) for various deletions (individual d1, d2, d3; sum total d1-d3) of symmetry-breaking donor-acceptor interactions (Table 2) in C5H4CCO.

As described more completely elsewhere,22 the chemical logic of hyperconjugative symmetry-breaking is straightforward. Although Lewis-type hybridized bonding interactions prefer perfect linear alignment for pure sigma-type (end-on) or pi-type (edge-on) bonding, such interactions are only slightly weakened by minor “bondbending” departures from linear symmetry. However, in the crowded double-bonding geometry of cumulene ketone chains, the strong onset of symmetry-breaking π-σ* or σ-π* interactions (viz, the nO(π)-σ*C′C″ d3 interaction of Table 2) is only prevented by exact maintenance of linear symmetry. Slight departures from linear symmetry are


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therefore more strongly rewarded by onset of π-σ* or σ-π* hyperconjugation than penalized by loss of direct σ- or π-type bonding overlap, resulting in spontaneous symmetry breaking in the plane of π-type influence (necessarily that of nO(π) and the cyclopendienyl moiety in C5H4CCO). Specific structural consequences of a particular NBO hyperconjugative interaction are easily inferred from the corresponding resonance structure depiction, and the relative weighting of each resonance correction can be predicted from crude E(2), ∆E$DEL estimates or accurate NRT evaluation. NBO analysis also provides a variety of other options for quantifying hyperconjugative effects on specific chemical properties of interest.16 Summary and Conclusions The present work shows the connection between the NBO/NRT description and conventional vibronic coupling description as two theoretical approaches to analyzing the origin of symmetry-breaking distortions that are commonly attributed to “pseudo Jahn-Teller (PJT) effects.” The results illustrate how the principal hyperconjugative interactions identified by NBO/NRT analysis can be demonstrated to be the authentic physical “cause” of the PJT symmetry-breaking “effect” with the help of ∆E$DELi→j* deletion techniques, combined with full reoptimization of molecular geometry and frequency evaluation. The distortions from high-symmetric configurations of H2Si=SiH2 and C5H4C=CO can both be attributed to symmetry-breaking hyperconjugative interactions of π-σ* and σ-π* type that are obtained from NBO/NRT analysis of the 1st-order density matrix for ground-state electronic wavefunctions of arbitrary single- or multi configurational form, independent of any required input from excited-state calculations or other vibronic-type data. An apparent connection can be drawn between the NRT-based resonance-structural depiction of a specific ground-state hyperconjugative correction and the envisioned vibronic excited state in which this resonance structure serves as “parent” rather than “small correction.” Expressed in


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other terms, the vibronic couplings of conventional PJT analysis could be envisioned as referring to a lower-level (pre-resonance “diabatic”) theoretical description in which resonance-mixing effects (quantifiable by NBO/NRT descriptors) are not yet incorporated in the ground-state wavefunction. However, the present results indicate that PJT-type symmetry-breaking typically involves a manifold of hyperconjugative interactions that render any simple associations with 2-state vibronic coupling models misleading. Even if certain general diabatic vs. adiabatic conceptual associations can be recognized between the two approaches, we believe that NBO-based hyperconjugative analysis26 offers significant conceptual and practical advantages over conventional vibronic coupling analysis for a vast majority of the “PJT effects” encountered in organic, inorganic, and bioorganic species. We have also found a striking relationship between the hyperconjugative interactions and spectroscopic details of interest by means of subsequent frequency calculation at the E$DEL-reoptimized geometry, which exhibits the precise hyperconjugative cause-effect relationship to vibrational frequency shifts. In this manner, NBO/NRT-based analysis of symmetry-breaking allows extension to the associated “spectroscopic PJT effects” of IR frequency shifts that dictate the stability or instability of symmetry-breaking structural deformations. NBO-based analysis can therefore serve both to supplant and extend conventional vibronic-coupling approaches to PJT-type structural and spectroscopic phenomena, despite its reduced computational demands. It should be noted that the present work serves to generalize the even simpler use of Lewis-structural concepts to analyze symmetry breaking in free radicals,23 such as NO2 and related species.36 In such open-shell cases, the structural and spectroscopic properties may be pictured in terms of “competing preferences” of the majority (alpha) and minority (beta) spin sets for different 1c/2c bond patterns (reflecting distinct Coulomb and exchange potentials), with resulting intermediate spin-averaged properties of competing spin sets. Such a “different Lewis structures for different


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spins” picture37 thereby connects to analogous structural differences between corresponding closed-shell species, such as NO2+ (linear) vs. NO2‒ (116º bent), or isoelectronic CO2 (linear) vs. O3 (117º bent), each of which is commonly explained in terms of Bent’s rule38 or related Walsh diagrams39 of elementary chemical bonding theory. The intermediate bending angle of the NO2 radical (134º) is in qualitative accord with the estimate provided by such elementary closed-shell comparisons, as is the IR bending frequency (765 cm‒1, vs. 612 for NO2+ and 800 for NO2‒, or 664 for CO2 and 897 for O3). Supporting Information The Supporting Information is available free of charge at XXX. Gaussian input files containing optimized coordinates, vibrational frequencies, ZPE corrections, and other details for all stationary-state species in this work, including NBO keyword input and results for $DEL-optimization jobs.

Acknowledgment: Computational facilities for this research were supported in part by National Science Foundation Grant CHE-0840494.

References (1) Heilbronner, E.; Dunitz, J. D. Reflections on Symmetry in Chemistry ...and Elsewhere; Wiley-VCH: New York, 1992. (2) Jahn, H A.; Teller, E. Stability of polyatomic molecules in degenerate electronic states. I. Orbital degeneracy. Proc. R. Soc. A 1937, 161, 220-235. (3) Bersuker, I. B. Pseudo-Jahn–Teller effect—A two-state paradigm in formation, deformation, and transformation of molecular systems and solids. Chem. Revs. 2013, 113, 1351−1390.


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(4) Garcia-Fernandez, P.; Liu, Y.; Bersuker, I. B.; Boggs, J. E. Pseudo Jahn–Teller origin of cis–trans and other conformational changes. The role of double bonds. Phys. Chem. Chem. Phys. 2011, 13, 3502-3513. (5) Zou, W.; Filatov, M.; Cremer, D. Bondpseudorotation, Jahn-Teller, and pseudo Jahn-Teller effects in the cyclopentadienyl cation and its pentahalogeno derivatives. Int. J. Quantum Chem. 2012, 112, 3277-3288. (6) Jose, D.; Datta, A. Understating the buckling distortions in silicene. J. Phys. Chem. C 2012, 116, 24639−24648. (7) Kayi, H.; Garcia-Fernandez, P.; Bersuker, I. B.; Boggs, J. E. Deviations from Born–Oppenheimer theory in structural chemistry: Jahn–Teller, pseudo Jahn–Teller, and hidden pseudo Jahn–Teller effects in C3H3 and C3H3−. J. Phys. Chem. A 2013, 117, 8671-8679. (8) Garcia-Fernandez, P.; Aramburu, J.A.; Moreno, M.; Zlatar, M.; Gruden-Pavlovic, M. A practical computational approach to study molecular instability using the pseudo-Jahn-Teller effect. J. Chem. Theory Comput. 2014, 10, 1824-1833. (9) Pratik, S. M.; Datta, A. 1,4-dithiine − puckered in the gas phase but planar in crystals: Role of cooperativity. J. Phys. Chem. C 2015, 119, 15770−15776. (10) Pratik, S. M.; Chowdhury, C.; Bhattacharjee, R.; Jahiruddin, S.; Datta, A. Pseudo Jahn-Teller distortion for a tricyclic carbon sulfide (C6S8) and its suppression in Soxygenated dithiine (C4H4(SO2)2). Chem. Phys. 2015, 460, 101−105 and other articles in this JT-related Chem. Phys. special issue. (11) Jalali, E.; Nori-Shargh, D. Symmetry breaking in the axial symmetrical configurations of enolic propanedial, propanedithial, and propanediselenal: pseudo Jahn-Teller effect versus the resonance-assisted hydrogen bond theory. Can. J. Chem. 2015, 93, 673-684. (12) Liu, Y.; Wang, Y.; Bersuker, I. B. Geometry, electronic structure and pseudo Jahn-Teller effect in tetrasilacyclobutadiene analogues. Sci. Reports 2016, 6, 23315.


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(13) Landis, C. R.; Weinhold, F. Origin of trans-bent geometries in maximally bonded transition metal and main group molecules, J. Am. Chem. Soc. 2006, 128, 7335-7345. (14) Nori-Shargh, D.; Mousavi, S. N.; Boggs, J. E. Pseudo Jahn-Teller effect and natural bond orbital analysis of structural properties of tetrahydridometallenes M2H4 (M = Si, Ge, and Sn). J. Phys Chem. A 2013, 117, 1621-1631. (15) Kouchzkzadeh, G.; Nori-Shargh, D. Symmetry breaking in the planar configurations of disilicon tetrahalides: Pseudo Jahn-Teller effect parameters, hardness and electronegativity. Phys. Chem. Chem. Phys. 2015, 17, 29251-29261. (16) Weinhold, F.; Landis, C. R. Discovering Chemistry with Natural Bond Orbitals; Wiley-VCH: Hoboken NJ, 2012. (17) Glendening, E. D.; Weinhold, F. Natural resonance theory: I. General formalism. J. Comp. Chem. 1998, 19, 593-609. (18) Glendening, E. D.; Weinhold, F. Natural resonance theory: II. Natural bond order and valency. J. Comp. Chem. 1998, 19, 610-627. (19) Glendening, E. D.; Badenhoop, J. K.; Weinhold, F. Natural resonance theory: III. Chemical applications. J. Comp. Chem. 1998, 19, 628-646. (20) Scheiner, A. C.; Schaefer, H. F. Cyclopentadienylideneketene: Theoretical consideration of an infrared spectrum frequently mistaken for that of benzene. J. Am. Chem. Soc. 1992, 114, 4758-4762. (21) Radziszewski, J. G.; Kaszynski, P.; Friderichsen, A.; Abildgaard, J. Bent cyclopenta-2,4-dienylideneketene: Spectroscopic and ab initio study of reactive intermediate. Collect. Czech. Chem. Commun. 1998, 63, 1094-1106. (22) Weinhold, F. Why do cumulene ketones kink? J. Org. Chem. 2017, 82, 1223812245. (23) 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.


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(24) Foresman, J. B.; Frisch, Æ Exploring Chemistry with Electronic Structure Methods: A Guide to Using Gaussian, 3rd ed.; Gaussian Inc.: Pittsburgh, PA, 2015. (25) Weinhold, F.; Landis, C. R. Discovering Chemistry with Natural Bond Orbitals; Wiley-VCH: Hoboken NJ, 2012, Sec. 5.2. (26) Alabugin, I.V.; Gilmore, K.M.; Peterson, P.W. Hyperconjugation. WIRES Comput. Mol. Sci. 2011, 1, 109-141. (27) Weinhold, F.; Landis, C. R. Discovering Chemistry with Natural Bond Orbitals; Wiley-VCH: Hoboken NJ, 2012, Sec. 5.6. (28) Weinhold, F.; Glendening, E. D. NBO 6.0 Program Manual; Theoretical Chemistry Institute: UW-Madison, 20113, Sec. B.6.10. Note that such NBO orbital deletions (with subsequent re-optimization to determine altered structural and vibrational properties) could in principle be carried out with any higher-level ab iniitio wavefunction, but only at the expense of full 4-index basis transformations. The much simpler $DEL-deletion method, as implemented for single-determinant HF or DFT levels, is more practical and representative of current practice in molecular calculations. (29) Weinhold, F.; Landis, C. R. Discovering Chemistry with Natural Bond Orbitals; Wiley-VCH: Hoboken NJ, 2012, Sec. 5.3; http:/nbo6.chem.wisc.edu/tut_del.htm (accessed 1 April 2018). (30) Additional keywords (CMO, NBOMO) for directly extracting the unitary transformation matrices that relate localized NBOs and delocalized MOs are described in Weinhold, F.; Glendening, E. D. NBO 6.0 Program Manual; T3, p. B-144ff and B194ff. (31) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Scalmani, G.; Barone, V., Mennucci, B.; Petersson, G. A. et al. Gaussian 09, Revision E.01, Gaussian, Inc.: Wallingford CT, 2009. (32) Glendening, E. D.; Badenhoop, J. K.; Reed, A. E.; Carpenter, J. E.; Bohmann, J. A.; Morales, C. M.; Landis, C. R.; Weinhold, F. NBO 6.0, Theoretical Chemistry


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Institute, University of Wisconsin-Madison: Madison, WI, 2013, http://nbo6.chem.wisc.edu/ (accessed 1 April 2018). (33) Weinhold, F.; Phillips, D.; Foo, Z.Y.; Hanson, R. M. NBOPro6@Jmol, Theoretical Chemistry Institute, University of Wisconsin-Madison: Madison, WI, 2018. (34) Cotton, F. A. Chemical Applications of Group Theory. Wiley; New York, 1963. (35) Clauss, A. D.; Nelsen, S. F.; Ayoub, M.; Landis, C. R.; Weinhold, F. Rabbit ears hybrids, VSEPR sterics, and other orbital anachronisms. Chem. Educ. Res. Pract. 2014, 15, 417-434. (36) Jackels, C. F.; Davidson, E. R. The two lowest 2A' states of NO2, J. Chem. Phys. 1976, 64, 2908-2917. (37) Weinhold, F.; Landis, C. R. Discovering Chemistry with Natural Bond Orbitals; Wiley-VCH: Hoboken NJ, 2012, Sec. 4.5. (38) Bent, H.A., Distribution of Atomic s Character in Molecules and Its Chemical Implications, J. Chem. Ed. 1960, 37, 616–634. (39) Walsh, A.D. The electronic orbitals, shapes, and spectra of polyatomic molecules. I. AH2 molecules, J. Chem. Soc. 1953, 1953, 2260-226.


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Natural Bond Orbital Theory of Pseudo-JahnâTeller Effects

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Natural Bond Orbital Theory of Pseudo-JahnâTeller Effects