Variations on the Bergman Cyclization Theme: Electrocyclizations of Ionic Penta-, Hepta-, and Octadiynes

The Bergman cyclization of (Z)-hexa-3-ene-1,5-diyne to form the aromatic diradical p-benzyne has garnered attention as a potential antitumor agent due to its relatively low cyclization barrier and the stability of the resulting diradical. Here, we present a theoretical investigation of several ionic extensions of the fundamental Bergman cyclization: electrocyclizations of the penta-1,4-diyne anion, hepta-1,6-diyne cation, and octa-1,7-diyne dication, leveraging the spin-flip formulation of the equation-of-motion coupled cluster theory with single and double substitutions (EOM-SF-CCSD). Though the penta-1,4-diyne anion exhibits a large cyclization barrier of +66 kcal mol–1, cyclization of both the hepta-1,6-diyne cation and octa-1,7-diyne dication along a previously unreported triplet pathway requires relatively low energy. We also identified the presence of significant aromaticity in the triplet diradical products of these two cationic cyclizations.


A. Electronic Structure Approaches for Diradical Molecules
We are interested in characterizing Bergman-like electrocyclization reactions, in which the reactants are closed-shell, ground state singlets and the products are (presumably) aromatic diradicals.Diradicals are theoretically challenging molecules 1,2 in which two electrons occupy two quasi-degenerate orbitals, the electronic states of which may be approximately described within the context of a two-level system, visualized schematically in Fig. S-1.For sufficiently small orbital energy splitting, the resulting six electronic configurations (Fig. ), each of these electronic states are multiconfigurational, requiring more than one Slater determinant to adequately describe the total wavefunction; the contribution to the total electronic energy of these additional determinants in the total wavefunction is typically referred to as static electron correlation.
While a relatively compact, two-configurational self-consistent field (TCSCF) wavefunction would be sufficient to capture the static correlation in this isolated two-level system, real diradicals are not so simple: in addition to the possibility of other quasidegenerate orbitals, whereby a more general, multiconfigurational self-consistent field (MCSCF) wavefunction is required, the contributions of instantaneous, pairwise electron-electron repulsion must also be captured.This dynamical electron correlation necessitates the use of a post-MCSCF, multireference approach, e.g., MR-PT, MR-CI, MR-CC, etc. in order to adequately describe the properties and chemical behavior of these molecules.Determining the reference active space (Fig. S-1.a.i) for molecules such as these is complicated, and MR approaches (Fig. S-1.a.ii-iii; e.g., MR-CI or MR-CC), are computationally intensive for even small systems with only a few heavy (i.e., non-hydrogen) atoms; for reviews of these theoretical approaches, see Refs 3 and 4, respectively.Furthermore, if the transition states in these Bergman-like cyclizations are product-like, they too may possess significant diradical character; due to the additional complications inherent to transition-state searches, the application of MR approaches to quantitatively examine diradical-producing reactions has also been limited.

FIG. S-1. Schematic representations of two different approaches to study diradical molecules,
within the context of a simple two-level system comprised of two electrons (labeled 1 and 2) in two quasidegenerate orbitals φ 1 and φ 2 .Permutations of the orbital occupations in this two-level system produces six distinct electronic configurations, represented above as a Slater determinant | φ 1 (i)φ 2 (j) ≡ | ij with electron i occupying the spatial orbital φ 1 and electron j occupying φ 2 and where an overbar denotes an electron has β spin.
Alternatively, spin-flip (SF) formulations of excited state methods based on singlereference theories like equation-of-motion coupled-cluster (SF-EOM-CC; abbreviated SF-CC) [5][6][7][8][9] or time-dependent density functional theory (SF-TDDFT) [10][11][12][13] have been developed that approach these systems in a different way, visualized in iii).In this manner, the problem of applying a post-MCSCF multireference approach to capture both the static and dynamical electron correlation in these systems is reduced within the framework of a relatively more straightforward, computationally tractable single-reference theory where dynamical electron correlation is included "for free" (i.e., not requiring a subsequent correlated computation on top of the spin-flip treatment generating the multiconfigurational states of interest).Recently, the computational investigations of the canonical Bergman cyclization by Luxon et al. leveraging SF-CCSD have exhibited good agreement with experimental reaction barriers and thermodynamic quantities, 14 the success of which inspires us to apply this approach for the ionic extensions of this chemistry examined here.

B. Towards a General Approach for Placing NICS Probes
As discussed in the Methods section of the main text, the standard location for the placement of isotropic single-point NICS probes for planar, symmetric aromatic molecules is in the center of the ring plane [for NICS(0)] and 1 Å above and below the ring plane [for NICS(±1)], all three of which must be equidistant from all ring atoms.For the highly nonsymmetric and nonplanar cyclic molecules examined here, however, there are two problems with this convention, namely (i) there does not exist a point (or points) which are mutually equidistant from all ring atoms and (ii) the notions of "above" and "below" are ill defined in the absence of a ring plane.While the first concern is easily circumvented by taking inspiration from the literature 3 where the NICS(0) probe location was originally defined to coincide with the non-mass-weighted geometric centroid of the ring, the second is still an open question with several conventions existing in the literature for symmetric, non-planar molecules.4Rather than performing a much more involved and costly analysis based on mapping the isotropic NICS values along a grid surrounding our molecules, we have instead determined a system which is capable of unambiguously placing these probes in a manner analogous to NICS(±1) for our nonsymmetric molecules that will recover previously reported conventions when applied also to symmetric molecules.To derive where the NICS(±1) probes should be placed in the more general case of a nonsymmetric, nonplanar molecule, it is intuitively sufficient to begin by examining where these probes are placed for the conceptually simplest aromatic molecule: benzene.In benzene, the NICS(0) probe is placed in the center of the ring plane, with NICS(±1) probes placed 1 Å perfectly above and below the molecular plane (see Fig. S-2.a for visualization).By connecting these points, it is clear that they lie on a line which coincides with the principal axis of rotation (i.e., the C 6 axis of the D 6h point group).While not all molecules possess a principal rotation axis defined by their symmetry, all molecules do possess a rotational reference frame defined by the rotational moments of inertia as originally outlined in the classic text by Wilson, Decius and Cross.5 For symmetric molecules, the principal axis of rotation (by symmetry) does coincide with one of the vectors defining this rotational frame, namely the principal moment of inertia.These moments of inertia are defined to be the S-6 eigenvectors of the moment of inertia tensor, I, a 3 × 3 matrix with elements given by where α n , β n , and γ n are mass-weighted Cartesian coordinates for atom n, within which the origin of the coordinate frame is defined to be the molecular center-of-mass.The three moments of inertia which define the rotational reference frame, {| i n : n = a, b, c}, are the eigenvectors which diagonalize the moment of inertia tensor, I: It is worth noting that the eigenvalues {I n : n = a, b, c} are related to the conventional (i.e., spectroscopic) rotational constants N = A, B, C according to To place the NICS(±1) probes for a general, non-symmetric and non-planar molecule, we must take several additional considerations into account, namely 1. the atoms being considered in the construction of the moment of inertia tensor are only the ring atoms (i.e., the eight carbon atoms of 8cTL), 2. the origin of the coordinate system should be the non-mass-weighted centroid of the ring atoms, rather than the center-of-mass, and 3. we are only interested in the principal moment of inertia, which coincides with the principal symmetry axis for symmetric molecules.
Therefore, we may simply diagonalize the non-mass-weighted moment of inertia tensor Ĩ, constructed within the non-mass-weighted coordinate frame defined by coordinates α, β, γ and whose origin is placed at the ring centroid according to the eigenequation at which point our NICS(±1) probes may be placed 1 Å in either direction along the principal moment of inertia in this reference frame, | ĩa , as visualized in Fig. S-2.b for species 8cTL.For the convenience of the reader and as a service to the community, we have provided a script capable of automating this process written in the highly readable Python programming language.See https://github.com/Parish-Lab/Bergman-Variations-578 for instructions to download, install, and use this script.

S-II. Supplementary Results & Discussion
A. Cyclization of the Penta-1,4-diyne Anion The reaction energy profile for the cyclization of the penta-1,4-diyne anion is shown in with energy relative to reactant 5a of only +41.9 kcal mol −1 makes the triplet state of this conformer a potentially attractive synthetic target for use in anti-tumor applications.Taken together, these results show that the TCS pathway most closely resembles the canonical Bergman cyclization of (Z)-hexa-3-ene-1,5-diyne, which is not surprising as that cyclization proceeds via a TCS pathway.
a States named according to their generating excitation operator in the given point group; see text.
d Energies computed at the C 2v -symmetry geometry optimized at the UCCSD/cc-pVDZ level of theory on the triplet surface for 7c.
e Energies of the reference determinant computed at the UCCSD/cc-pVDZ level of theory.
f Energies of all singlet excited states computed at the EOM-SF-UCCSD/cc-pVDZ level of theory.
As discussed above in Sec.S-I A, the spin-flip procedure works by applying a spin-flip excitation operator onto a high-spin triplet reference wavefunction.By closely examining the excitation amplitudes output from the spin-flip procedure, it is possible to construct the wavefunction for the lowest singlet state of 7c at its singlet-optimized C 1 geometry: where we have included the two leading configurations in the multiconfigurational state (the next configuration only contributes to the total wavefunction with a weight of 0.027 = 2.7%), or again more compactly as Based on its orbital occupations and the signs on its excitation amplitudes, this state clearly for the TCS of 7c, relative to the ground state of p-benzyne, indicates that through-bond coupling is less significant in 7c, in keeping with Hoffmann's original observation that trans 1,4 arrangement of radical lobes favors such coupling, of which 7c only has one such pathway but two are present for p-benzyne.

S-19
E. Nucleus-Independent Chemical Shift Data

S-21
TABLE S-6.Nucleus-independent chemical shifts (NICS) along an xy scan between the NICS(+1) probes above the two ring moieties of 7d and 8d, remaining +1 Å above the molecular plane along the scan, computed using Gaussian09 at the B3LYP/6-311++G** level of theory with a closed-shell, restricted reference determinant.a Initial NICS probe centered above the cyclopentyl and cyclohexyl ring moieties of 7d and 8d, respectively.
b Final NICS probe centered above the cyclobutyl ring moieties of 7d and 8d.
-1.a.i) combine to produce four symmetry-adapted electronic states: two closed-shell, twoconfigurational singlets (Ψ TCS + and Ψ TCS − ), one open-shell singlet (Ψ OSS ), and three degenerate components of the triplet (Ψ T Ms=−1 , Ψ T Ms=0 , and Ψ T Ms=+1 ), provided in Fig. S-1.a.ii-iii.Aside from the high-spin components of the triplet (Ψ T Ms=±1 Fig. S-1.b.In the SF approach, a high-spin (M s = ±1) triplet reference is constructed with an open-shell self-consistent field approach using either a restricted or unrestricted reference (UHF/ROHF or UK-S/ROKS; Fig. S-1.b.i), before spin-flip excitations being performed to generate all low-spin S-4 (M S = 0) excited determinants (Fig. S-1.b.ii).These determinants may then be combined and symmetry-adapted using the squares of the transition amplitudes generating each spinflipped determinant as expansion coefficients to generate multiconfigurational states for which both static and dynamical electron correlation is captured (Fig. S-1.b.

Fig. S- 3 .
Fig. S-3.Energetics for this cyclization were computed at the SF-EOM-CCSD/cc-pVDZ level of theory using SF-EOM-CCSD/cc-pVDZ optimized low-spin geometries (inset in Fig. S-3).For the reactant species 5a, the C 1 -C 2 and C 1 -C 3 bonds are observed to be shorter than would be expected for a typical C-C single bond.Likewise, the acetylenic C 2 -C 4 and C 3 -C 5 bonds are longer than would be expected for a typical C-C triple bond.These bond compressions and elongations are likely due to π-electron donation from the formal π lone pair at C 1 into the π * orbitals of the acetylene moieties.To put it another way, the perpendicular π system of this molecule is that of a pentadienyl anion, with attendant delocalization.Thanks to the symmetry specification required by the implementation of SF-CC in Q-Chem, both open-shell singlet (OSS) and two-component, closed-shell singlet (TCS) pathways could be found for this cyclization.Along the OSS pathway, the C 4 • • • C 5 distance changes from 4.72 Å (reactant, 5a) to 2.02 Å (transition state, 5a→5c), before finally becoming 1.45 Å in the OSS product (5c).Both transition state 5a→5c and product 5c assume a nonplanar, C s -symmetry structure reminiscent of the âĂĲenvelopeâĂİ conformer of cyclopentane, with central carbon C 1 puckered out of the ring plane by approximately 11 • in 5a→5c and 5 • in 5c.While the ∠C 3 C 1 C 2 bond angle narrows from 123.2 • in the reactant (5a) to 111.0 • in the transition state (5a→5c), the OSS product 5c actually exhibits a widening of this bond angle to 115 • .This ∠C 3 C 1 C 2 widening in 5c relative to 5a→5c accompanies a lengthening of the distance between C 2 • • • C 3 from 2.35 Å as the radical centers are forming in 5a→5c to 2.41 Å between the radical centers in 5c, a geometric change that reduces the Coulombic repulsion between the singly occupied radical orbitals of the OSS electronic state.With ac-
represents the constructive combination of the two-component closed-shell singlet | Ψ TCS + visualized in Fig. S-1.Unlike p-benzyne, 7c seems to have a high-spin triplet ground state -lower in energy than this two-component closed-shell singlet by 2.53 kcal mol −1 .This raises the question of whether the high-spin triplet is the ground state only at the singlet-optimized C 1 structure, or if it is the ground state also at the triplet-optimized C 2v structure.To address this possibility, where the A and S radical orbitals correspond to σ * and σ in Fig. S-5, respectively.The reduced weight of the (A) 2 configuration and increased weight of the (S) 2 configuration

a 1 A 1
Singlet-triplet gap taken to be the energy of transition from singlet to triplet surface.b Energies computed at the CCSD/cc-pVDZ level of theory with a restricted, closed-shell reference determinant c Energies computed at the EOM-SF-CCSD/cc-pVDZ level of theory with an unrestricted, high-spin open-shell reference determinant d S 2 values computed for Hartree-Fock reference determinant e S 2 values computed for low-spin excited states of EOM-SF-CCSD procedure f Energies computed at the UCCSD/cc-pVDZ level of theory with an unrestricted, high-spin open-shell reference determinant g Relative energies computed with respect to X state of 7a S-17

c 1 )
FIG. S-6.Relative electronic energies (kcal mol −1 ) computed along the frozen-string pathway connecting the left-twisted conformation of the singlet structure of 8a to the bicyclic species 8d, computed at the UCCSD/cc-pVDZ level of theory with an unrestricted reference determinant.

aFIG. S- 7 .
FIG. S-7.NICS probe locations for the xy-scans between the NICS(+1) probes above the two ring moieties in each of 7d and 8d; all probes along the scan coordinate remain +1 Å above the molecular plane.

TABLE S -
1. Total energies (kcal mol −1 ) for indicated electronic states of 7c at the given molecular geometry, relative to the energy of the heptaen-1,6-diyne cation (7a).

TABLE S -
2. Absolute (E h ) and relative (kcal mol −1 ) energetics for all species in the cyclization of the penta-1,4-diyne anion (Scheme 3, top panel in the main text).Energies were computed at the indicated levels of theory at geometries optimized and frequency confirmed at this same level for the given electronic state within each species' point-group symmetry.Singlet-triplet gap taken to be the energy of transition from singlet to triplet surface.Energy computed at the CCSD/cc-pVDZ level of theory with a restricted, closed-shell reference determinant c S 2 values computed for Hartree-Fock reference determinant d Energies computed at the EOM-SF-CCSD/cc-pVDZ level of theory with an unrestricted, high-spin open-shell reference determinant e S 2 values computed for low-spin excited states of EOM-SF-CCSD procedure a b

TABLE S -
3.Absolute (E h ) and relative (kcal mol −1 ) energetics for all species in the singlet and adiabatic triplet cyclization pathways of the heptaen-1,6-diyne cation (Scheme 3, middle panel in the main text).Energies were computed at the indicated levels of theory at geometries optimized and frequency confirmed at this same level for the lowest-energy singlet (top) and adiabatic triplet (bottom) surfaces within each given point-group symmetry.

TABLE S -
4.Absolute (E h ) and relative (kcal mol −1 ) energetics for all species in the cyclization of the octadien-1,7-diyne dication (Scheme 3, bottom panel in the main text).Energies were computed at the indicated levels of theory at geometries optimized and frequency confirmed at this same level for the lowest-energy singlet (top) and adiabatic triplet (bottom) surfaces within each given point-group symmetry.
a S 2 values computed for Hartree-Fock reference determinant b Singlet-triplet gap taken to be the energy of transition from singlet to triplet surface.

TABLE S -
5. Nucleus-independent chemical shifts (NICS) for all cyclic species considered here, computed using Gaussian09 at the indicated level of theory.