Ab Initio Probing of the Ground State of Tetraradicals: Inattention to the

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Inattention to the Strictest Form of Hund’s Rule is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S.


Sudip Kumar Chattopadhyay

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Revision-III M/S ID:jp-2018-105142-R2

Ab initio Probing of the Ground State of Tetraradicals: Breakdown of Hund’s multiplicity rule Sudip Chattopadhyay1, ∗ 1

Department of Chemistry, Indian Institute of Engineering Science and Technology, Shibpur, Howrah 711103, India (Dated: February 18, 2019)

Abstract The electronic structure of organic σ-type polyradical including 2,4,6-tridehydropyridine radical cation (246-TDHP) and three isomers of tetradehydrobenzene(TDHB) have been studied using a computationally robust and cost-effective second-order multireference perturbative model which provides a balanced treatment of nondynamic and dynamic contributions to the electron correlation problem in the ground or excited electronic states which are imperative for predicting structural properties (e.g., ground state multiplicity, energy gaps between high-spin and low-spin states, etc.) of polyradicals. Energy gaps are useful to capture insight into the degree of interaction between the radical sites. An important finding of this study is that the tetraradicals considered here possess singlet ground states, contrary to Hund’s rule. Present findings are in close agreement with the available high-level ab initio estimates at attainable cost implying that a perturbative description of the systems is adequate. The impact of N+ on the nature of ground state for the 246-TDHP have also analyzed. The singlet-triplet energy gaps for 1245- and 1234-TDHB are smaller than for ortho-benzyne mainly due to the ring strain. 1235-TDHB is 14.42 and 11.05 kcal/mol lower in energy than 1245- and 1234-isomers, respectively. IVO-SSMRPT predicts 1 A1 −3 B2 and 1 A1 −5 B2 gaps of 25.84 and 105.15 kcal/mol, respectively for 246-TDHP cation.

1



I.

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INTRODUCTION

The understanding of different facets of electronic properties of aromatic carbon-centered σ, σ, σ, σ-tetraradicals (such as tetradehydrobenzenes and 2,4,6-tridehydropyridine radical cation) and reactive intermediates with four formally unpaired electrons, is an intriguing challenge for both experimentalists and theoreticians. Owing to the complex electronic structures arising from the counter-intuitive inter-play of (formally) unpaired electrons (located on four different atomic centers) distributed over nearly-degenerate orbitals and atypical bonding patterns between radical centers, tetraradicals are difficult to investigate both experimentally and theoretically1–5 . The development of robust theoretical methods for the inspection of radical remains an active realm of research in quantum chemistry. Recent review of Kenttämaa and coworkers1 provides a serviceable adviser for navigation through the multitude of problems created by openshell systems. Akin to the di- and tri-radicals, launching positive charge may produce a notable effect on the structural properties (e.g., ground state multiplicity, energy gaps between high-spin and low-spin states, etc.) of the tetraradicals; however, the consequences of introducing heteroatoms have not yet been intimately explicated. The intricate counter intuitive interactions between the unpaired electrons (ranging from strongly antibonding to almost bonding) provide notable structural and chemical signatures of spectroscopic interest. Indeed, the bonding interactions between the radical sites lead to rigid structures along with the reduction of reactivity. Note that the extent of interaction between unpaired electrons can also be analyzed by the measurement of energy gaps between different electronic states. Note that electronic degeneracies and high density of states are even more prominent in tetraradicals compared to mono-, di- and tri-radicals. Due to strong electronic degeneracy, the wave functions of tetraradicaloids carry different leading configurations which provide multireference(MR)/multiconfigurational(MC) wave functions (often termed as strongly correlated systems). Therefore, the hierarchy of approaches designed on the single-determinantal zero-order wave function is inadequate by construction. Rather than employing a brute-force protocol, an accurate account of such systems can be reached when all the configurations are treated on an equal footing and dynamical correlation is included. The reliability and consistency of the estimates acquired via any MR-based method is strongly dependent on how well the approach encapsulates a balanced treatment of both static and dynamic correlations6–12 .

2


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Multireference perturbation theory (MRPT) has long become a standard computationally affordable protocol to provide a reasonable approximation to the solution of the time independent Schrödinger equation when a single reference wavefunction is not a good zerothorder function. Over these last few years, we have witnessed a resurgence of MRPT methods under various variants for the description of electronic systems that simultaneously show chemically relevant static and dynamical electron correlation13–58 . All of these MRPTs attribute different trade-offs between accuracy and computational viability that permit them to handle differently sized molecules with varying precision. However, a proper implementation of these MRPT approaches is often found to be painful due to the structural/formal features of the working equations (vide infra). In MRPT methods, account of both types of correlation effects, essentially, opens up three options such as (i)‘perturb-then-diagonalize’, (ii), ‘diagonalize-then-perturb’ and (iii) ‘diagonalize-then-perturb-then-diagonalize’. Here we primarily focus on the single-root/state-specific MRPT methods (that are relevant for this work) where one targets one specific state of interest by starting out with a MR model space. The toolbox of state-specific MRPT contains a great variety of approaches that can also be classified broadly into two categories: (i) size-inextensive and (ii) size-extensive. At present, the most popular size-extensive MRPTs are second-order NEVPT2 (n-electron valence perturbation theory) of Angeli-Malrieu-coworkers18–20 and state specific multireference perturbation theory (SSMRPT) of Mukherejee et al.22 . The unitary group approach (UGA) based spin-adaptation of Mukhereje’s SSMRPT (Mk-MRPT)22 has been developed and applied recently by Mao et al.25,26 . They have given a proof of the size-extensivity for the UGA-SSMRPT method where the configuration state functions (CSFs) are obtained from the action of unitary generators. The protocol used for the spin-adaptation is a crucial issue to come up with an explicitly size-extensive formulation. The other category, referred to as size-inextensive MRPTs are MRMP2 (second-order MR Møller-Plesset perturbation theory) of Hirao13,14 , CASPT2 (complete-active-space second-order perturbation theory) of Roos et al.15–17 . Despite the many predictive successes, both MRMP213,14 and CASPT215–17 (belongs to the so-called ‘diagonalize-then-perturb’ approach) sometimes suffer from numerical instability problems due to the well-known intruder states. The problem of intruder states, which is an“Achilles’s heel ” for the MRPT, can be cured in a pragmatic manner through the introduction of parameters using various techniques59–61 which are theoretically not justifiable and may invite arbitrariness in the final results/predictions62–64 . The size3



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consistency issue of MRPT methods has been discussed in more detail by van Lenthe and co-workers48,65 who argued that the choice of projection operators to define the zeroth-order Hamiltonian is crucial to maintain the size-consistency of MRPT. They pointed that the use of projectors in zeroth-order Hamiltonian need not necessarily violate size-consistency. The problem of size-consistency/size-extensivity of MRPT has also been discussed by Rintelman et al.66 using fully uncontracted MRMP2 method. It should also be mentioned that NEVPT2 methods of Malrieu and coworkers18–20 , which are also being widely used, are geared to guarantee size-extensivity. In NEVPT2 approach18–20 , the occurrence of intruder is smartly resolved by invoking a partially bielectronic definition of the zeroth-order Hamiltonian52 that encompasses two-electron interactions within the active orbitals. Due to the “diagonalize-then-perturb” philosophy, CASPT2 and MRMP2 methods do not revise the model-space component of the wave function under the effect of dynamical correlation effect. Therefore, the CAS functions are fixed in CASPT2 and MRMP2 calculations and not affected by the perturbation operator (unrelax description). However, in the MR calculations, the relative contributions of the configurations in reference- and correlated- wavefunctions vary notably. Multistate (MS) target space formulations of CASPT2 and MRMP2 although alleviate the problem of revision of the model space coefficients as a result of coupling with the virtual functions53–57 , the problems of size-inextensivity and intruder state persist. Whatever be its degree of contraction in the perturber space, NEVPT2 methods (assume the “diagonalize-then-perturb” strategy) do not account for reference relaxation effects when dynamical correlation is included. A quasidegenerate MS extension of NEVPT2 (quasi-degenerate NEVPT2) has also been suggested recently by Angeli et al.58 which allows the treatment of states in which electronic structures of different types (mixed electronic states) contribute strongly to the wave function. It is worth mentioning that all of MS versions of CASPT2, MRMP2 and NEVPT2 are of the “diagonalize-then-perturb-then-diagonalize” type. If the computational method does not permit modification of the reference space coefficients in the correlation calculation a correct description of systems with near avoided crossings or conical intersections cannot be anticipated. In an attempt to overcome at least some of the problems of widely used MRPTs that prevent their routine use in electronic structure computations, Mukherjee and coworkers proposed the second-order SSMRPT22 which emanates as a perturbative approximant to 4


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the SSMRCC (state-specific multireference coupled cluster)67 . It is worth stressing that most of the MRPTs to date have originated from physical considerations. In standard SSMRPT calculations, the reference functions have usually been generated via complete active space self-consistent field (CASSCF) calculations and its applicability has been explored in various situations and directions22–28 . Prompted by the success of the perturbative method of Mukherjee and coworkers (Mk-MRPT), very recently, Chattopadhyay et al.68 have proposed an SSMRPT approach using an improved virtual orbital complete active space configuration interaction (IVO-CASCI)69 (where the unoccupied valence orbitals are obtained by using an IVO generation technique) rather than the CASSCF functions. It should be mentioned here that the IVO-modification of SSMRPT theory of Mukherjee and coworkers25,26 , identified with the abbreviation IVO-SSMRPT, inherits all the vital redeeming features of the parent SSMRPT approach23,25,26 that makes it a serious candidate for efficient handling of electron correlation in the presence of quasidegeneracies of varying extent. The reference (zeroth order) wave function, IVO-CASCI, may comprise a large number of configurations and rectify the shortcomings of the standard CASSCF method. The former approach is computationally simpler than the latter one, because of the fact that the IVO-CASCI computations do not involve iterations beyond those in the initial SCF step, nor do they possess traits that create convergence difficulties (along with multiple solutions corresponding to minima) with increasing size of the CAS, as is often encountered in the orthodox CASSCF calculations. Thus, the absence of nonlinear nature of the orbital optimization step in IVO-CASCI leads not only to a reduction in computational expense but also to an improved qualitative description of the energy surface. The IVO-CASCI based MR method is attractive in terms of its applicability to bigger systems compared to the CASSCF-based MR ones. While the IVO-CASCI captures the static part of correlation energy, it cannot effectively take care of the dynamical correlation70 . Efficiently recovering the missing correlation energy is a an important component of any perturbation theory, like SSMRPT. Various applications of SSMRPT and IVO-SSMRPT appear quite encouraging because these methods are endowed with several redeeming formal features. Size-extensivity is explicitly satisfied in IVO-SSMRPT as that of the parent SSMRPT methods22,25,26 . It can be size-consistent if a correct set of orbitals is employed (localized on the separate fragments). However, the SSMRPT method shows very small size-consistency errors even for delocalized orbitals. IVO-SSMRPT method can be applied to ground and excited states 5



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with various spin multiplicities (due to the spin-free nature of the method). Irrespective of whether the state of interest is ground or excited, the IVO-SSMRPT method avoids the obstacle of intruder-states in a natural manner (without invoking ad hoc parameter or increasing the size of the active space). This security of numerical stability is a major driver for the various implementations of the SSMRPT protocols.Therefore, radicaloid states can be treated with uniform accuracy which is one of the aim of the present work. It has been shown that the IVO-SSMRPT method gives energies, optimized geometrical parameters and relative energies, which are at least as accurate as corresponding estimates obtained with the CASSCF-SSMRPT calculations and more reliable than the ‘gold standard’ CCSD(T) approach for molecules where the wave function has some MR nature68,71–73 . It has also been explicated that the IVO-SSMRPT, akin to the parent one, yields competitive results, compared to different alternative single-root MRPT approaches68,71–73 . In addition, SSMRPT method is generalizable to incomplete model while all the applications still considered are based on complete model space. Unlike the “diagonalize-thenperturb” philosophy based MRMPT methods (such as CASPT2, MRMP2, and NEVPT2), IVO-SSMRPT protocols allow the revision of zeroth-order wave function. In IVO-SSMRPT modus operandi, dynamical correlation effect is folded into the second-order effective Hamiltonian. The energy of target state after convergence of the first-order cluster finding equations is obtained by diagonalizing this effective Hamiltonian within the reference space. The diagonalization process effectively accounts for mixing (and relaxing) of the reference space functions in the presence of “effective interactions” that include dynamic correlation effects. The relaxed coefficients in IVO-SSMRPT formulation appear as the parameters of the perturbed wave function. As a result of this, the IVO-SSMRPT recipe is capable of describing the surface crossings involving states of same and different symmetries74 . For IVO-SSMRPT method, the unrelaxed coefficients can also be employed to provide the target energy as an expectation value of the second order effective operator. It should be noted that unrelaxed IVO-SSMRPT method belongs to the ‘diagonalize-then-perturb’ philosophy, while the relaxed IVO-SSMRPT version can be described as a ‘diagonalize-then-perturbthen-diagonalize’ scheme comprising of three steps: (i) get the zero-order solutions, (ii) construct the state specific effective Hamiltonian, and (iii) diagonalize the effective Hamiltonian and bring about the desired mixing among the zero-order states. The IVO-SSMRPT is a second-order Rayleigh-Schrödinger perturbation theory with a reference dependent multi6


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partitioning Møller-Plesset scheme75 . Therefore, the IVO-SSMRPT model comprises almost all the preferable features that a genuine MR methodology should have. From the very mode of structural nature of the cluster amplitude finding equation for IVO-SSMRPT (and also SSMRPT) method, it is found that there is no coupling among the different excitation components in cluster operators, and the coupling exists with only those cluster operators which yield the same excitation as that obtained by the product of excitation operators for the specific cluster amplitude under consideration. No cluster amplitudes require to be stored in the IVO-SSMRPT calculations. This feature of the equation furnishes a very appealing computational protocol. As that in other MR-based methods76–78 , the choice of active orbitals to construct CAS is an important issue (that needs a notable trial and error) of IVO-SSMRPT approach. As that in other single-root approaches, IVO-SSMRPT is mainly tailored for the ground state; excited states can only be estimated as lowest states of an irreducible representation of the molecular point group or spin by exploiting root homing strategy. One should keep in mind that although the root honing scheme used here to converge to the desired root might seem handy, in practice, it is not so in general, for the situations where the coefficients rapidly change sign viz. mixed electronic states (sensitive to root flipping effects). The IVO-SSMRPT scheme has been demonstrated to describe excited states (where many other standard electronic structure methods tend to be unusable) with similar accuracy to other methods that employ a much larger set of configurations to construct the CAS71–73 . Note that for the excited state investigations, the SSMRPT method with CASSCF and IVOCASCI orbitals often face convergence difficulty. It generally needs more iterations than does the ground state and, in some situations, only proper convergence, employing a weaker threshold, can be obtained. At this point it is worth discussing these issues are also applicable in the case of CASPT2 and NEVPT2 calculations. It is noteworthy that most of the MRPT protocols (particularly CASPT2 and NEVPT2) mentioned above have also proven to be productive avenues to deal with electronically excited states together with the ground state. One should also note that separate computations targeting different roots of the effective Hamiltonian are needed in single-root MR calculations (each determinant needs to be picked and individually optimized for each state), but in multiroot MR-procedures, all roots (as the dimension of the reference space) are generated in a single computation53–57 . As mentioned before, the target IVO-SSMRPT state is spin-pure in nature and thus it 7



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is competent to deal with electronic states circumventing the nuisance of spin contamination. In the SSMRPT method, the perturbed target state is characterized by state-specific parametrization of Jeziorski and Mokhorst (JM) expansion79 (which uses individual Slater determinants as perturbers) where the wave operator acts on the CAS function. The spinadaptation of JM Ansatz-based methods is a challenging task. As the basic formalism for the spin-free IVO-SSMRPT approach is identical to that for the UGA-SSMRPT method of Mao et al.25,26 , IVO-SSMRPT protocol enjoys exactly the same formal (basic algebraic) structure as the parent. Note that no transition density matrix elements are needed in the computations of the IVO-SSMRPT amplitude equations is due to the use of sufficiency conditions, as in the theory of Mao et al.. However, the transition density matrices is to be needed to construct the matrix elements of the effective Hamiltonian (in fact upto threebody transition density matrices). It is important to stress here that Li and Paldus80 have developed the first general UGA formulation of an MRCC theory using the JM Ansatz. Other intruder-free MRPT formulations the include multiconfiguration perturbation theory (MCPT)29 which allows reference relaxation effects. In the case of MCPT, sizeextensivity is manifested in one of the variants. Very recently, Giner et al.51 put forward a novel second-order MRPT based on the JM Ansatz (dubbed as JMMRPT2) which has the noteworthy property of avoiding the intruder state problem and is explicitly size-extensive. JMMRPT2 has close kinship with SSMRPT model. One should note that the effective Hamiltonian version of JM-MRPT2 permits the relaxation of the coefficients of the reference function. It is also pertinent to note that the GVVPT2 (generalized Van Vleck perturbation theory) method proposed by Khait-Hoffmann21 circumvented intruder problem by the collection of potential intruder states into the secondary reference space and using nonlinear denominator shifts that vary continuously. GVVPT2 is subspace specific in nature, yielding energies of all the states belonging to the primary reference space. The size-extensivity property is not apparent in GVVPT2 owing to the explicit use of Hilbert space projectors. The final diagonalization step in GVVPT2 accounts for the relaxation of coefficients in the unperturbed function. Spin-flip protocols, suggested by Krylov81–83 form another important theoretical framework for accurate quantum chemical description of quasidegenerate systems. Note that methods tailored to handle radicaloid systems represent different parametrization of the wave function with certain benefits and drawbacks for specific systems. Various attempts have been published to generalize density functional 8


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theory (DFT) to handle systems beset by quasidegenerate situations84 . The multireference approach retaining the excitation degree perturbation theory (where GVVPT21 protocol is used to set up the required perturbation equations) of Fink46,47 has several unique features of a good ab initio method by virtue of the fact that it is size consistent, unitary invariant with respect to orbital transformations within the inactive, active, and virtual orbital spaces. Moreover it supplies a systematic prescription to converge to the exact solution of the Schrödinger equation. However, it suffers from the perennial problem of intruder states. Here, we also mention that the block-correlated perturbation theory30 is fully size-extensive. Other developments connected to perturbative scheme include fully size-extensive product state linearized CC (MPS-LCC) method85 where the wave function is constructed using first-order perturbation theory and ingenious zero-order Hamiltonian as originally proposed by Fink46 . In Fink’s formulation, the zeroth order Hamiltonian is not allowed to interact with the electronic states with different number of electrons in the active, core and virtual spaces. The main merit of Fink’s scheme is that it not only converts to the LCC equations at the first order perturbation theory but higher order corrections can also be formulated in a systematic way. Although elegant, SSMRPTs face a number of limitations. Similar to NEVPT2 methods, the non-invariance of SSMRPT with respect to the transformation among the active orbitals may be problematic for local correlation. Hence, it needs the use of localized orbitals to ensure and demonstrate manifest size-consistency property for these methods. At this point, it needs to be noted that the JM Ansatz applied to a CAS reference function does not show such orbital invariance issue. Such aspect has been addressed by Mukherjee and coworkers49,50 using their orbitally noninvariant UGA-SSMRPT to elucidate the fragmentation process of di- and triatomic molecular systems. Moreover, a very small value of reference coefficient may yield rather large cluster amplitudes [due to the appearance of a term proportional to the ratio of two reference coefficients,

cν cµ

in the cluster amplitudes finding equations] which

in turn invites instability in the computed energy. Several sophisticated solutions have been proposed to address this defect12,26 . To avoid or at least to attenuate numerical ill-effects, removal of cluster amplitudes or coefficients below threshold in magnitude can be used. It is worth emphasizing that the amplitude determining equations of IVO-SSMRPT method do not face any such numerical instability for the different systems treated till now. The generality of IVO-SSMRPT is required for solving many chemical problems with 9



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pronounced MR-character. The IVO-SSMRPT method68,71–73 has shown promising capabilities for studying transition states, excitation energies, barrier heights, bond dissociation, and optimized geometrical parameters at affordable computational cost. The IVO-SSMRPT formulation has the advantage that it can describe bond dissociation process (of both single and multiple bonds) in a formally and physically faithful and correct manner. In our previous works68,72 we found that the IVO-SSMRPT method gives uniform descriptions along the entire potential energy surfaces (PESs) for systems with various spin multiplicities and extracts molecular properties of spectroscopic interest that are close to reference or experimental findings (if available). IVO-CASCI reference function incorporates all neardegenerate configurations, which accounts for nondynamical electron correlation, for example, in a bond breaking/making process. This wave function is then used to include dynamic correlation effects using SSMRPT, as we have already mentioned. As a result, the IVO-SSMRPT wave function will be quantitatively accurate for various chemical processes, which can be an energy surface for a chemical reaction, a photochemical process, etc. In Ref.68, we have studied ground state PES of challenging, multiconfigurational F2 , Be2 , and N2 systems which provide a serious challenge to any MR method owing to the presence of an intricate interplay of nondynamical and dynamical electron correlations effects for the entire PES. The present work demonstrates that the results obtained at IVO-SSMRPT level of calculations accord with those at the CASSCF-SSMRPT one as is evident from the data assembled in figures and tables in Ref.68. This issue migh be elaborated by considering Be2 and F2 systems. Be2 has received an extensive attention, mostly because of the peculiar topology of the PES due to the donor-acceptor interaction between occupied 2s and vacant 2p orbitals. For the ground state Be2 system, the IVO-SSMRPT(2,2)/ccpV6Z and CASSCF-SSMRPT(2,2)/cc-pV6Z calculations provide dissociation energies (De ) of 1.75 and 1.77 kcal/mol. The equilibrium bond length (Re ) and vibrational frequency (ω) obtained by IVO-SSMRPT(2,2)/cc-pV6Z are 2.646 ˚ Aand 198 cm−1 . The corresponding CASSCF-SSMRPT(2,2)/cc-pV6Z values are 2.646 ˚ Aand 196 cm−1 , respectively. For another challenging system, F2 , CASSCF-SSMRPT(2,2)/cc-pVQZ calculations provide Re =1.4109 ˚ A, De = 41.50 kcal/mol, and ω= 931 cm−1 . The corresponding values provided by IVOSSMRPT(2,2)/cc-pVQZ scheme are 1.4110 ˚ A, 39.52 kcal/mol, and 921 cm−1 . It has also been observed that nonparallelity errors provided by IVO-SSMRPT method are usually satisfactory relative to full configuration interaction surfaces. Therefore, one may argue that 10


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the IVO modification of the SSMRPT scheme cherishes all the advantages of the parent SSMRPT method without sacrificing its accuracy. It is pertinent to emphasize that the number of iterations needed for IVO-SSMPRT to converge is always lower than the corresponding CASSCF-SSMRPT one for the variety of systems that we have studied till now. Once again, it should be stressed that the IVO-protocol does not require tedious and costly CASSCF iterations beyond those in an initial SCF step. The IVO-SSMRPT method is not only suitable to calculate an isolated energy surface but also competent to optimize geometries. In the work carried out so far, we have employed IVO-SSMRPT to study mono-, di- and tri-radicals. More examinations on the suitability for radicaloid systems still need to be done. Molecules like tetraradicals can serve as vehicles for evaluation of the usefulness of a method designed to treat systems plagued by neardegeneracies. As the studies on tetraradicals are very limited, in the present work, we intend to study 2,4,6-tridehydropyridine radical cation and three isomers of tetradehydrobenzenes or benzdiynes through the estimations of the optimized geometrical parameters, vibrational frequencies, relative orders, and energy gaps between different states using IVO-SSMRPT method.

II.

RESULTS AND DISCUSSION

As a consequence of its usefulness, as well as its low computational cost compared to that of coupled-cluster (CC), it is desirable to be able to analyze the results of IVO-SSMRPT calculations of structural properties of spectroscopic interest for tetraradicals. Inspite of different investigations, the chemical properties of aromatic carbon-centered σ-tetraradicals are not known very well. Note that tetradicals with feebler spinspin interaction owing to the larger distance between the radical centers and/or inopportune placement of the nonbonding MOs, may exhibit different nature, however their chemical properties are not well studied. Kenttämaa and co-workers1,86–88 argued that reactivity predictions for σ, σ, σ, σtype radicals cannot be compiled based solely on the chemical nature of analogous mono/di-radicals, even when the interaction between the radical centers is weak. The present work reports the exploration on tetraradical such as 2,4,6-tridehydropyridine radical cation and three-isomers of tetradehydrobenzene. They are unusually troublesome to investigate experimentally. The experimental investigations on these systems are complemented by the11



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oretical studies. The reactivity of radical ions has been of interest for decades. The large heats of formation of tetraradicals are probably one of the reasons why they are strenuous to isolate experimentally2,3 . However, the computational studies on such tetraradicals can be complemented by experimental investigations and corresponding analysis. The counter intuitive modes of interaction between different radical centers have direct effects on the structures of the benzdiynes and its analogue that deviate appreciably from the regular hexagon structure of benzene. Although mono-, and di-radicals are well studied, the related σ, σ, σ-triradicals and σ, σ, σ, σ-tetraradicals remain elusive1,86–88 . IVO-SSMRPT computations have been accomplished with the GAMESS electronic structure package89 . At this point we want to mention that the leading computational labor of the cluster ampli2 tude equation in spin-adapted IVO-SSMRPT method is primarily controlled by N2o Nv2 NCAS

[where o=occupied orbitals and v=virtual orbitals]. The disk space involved for computing the amplitude finding equation is very small compared to space necessary for storing two-electron integrals. Thus, IVO-SSMRPT provides manageable cost/accuracy ratio. In IVO-SSMRPT calculations, no cluster amplitudes need to be stored as there is no coupling between the various excitation components in cluster operators. It may be useful to note that IVO-SSMRPT method is computationally more demanding than its competitors CASPT2, NEVPT2, MRMP2, MCPT, and GVVPT2 due to the cost of multiple cluster amplitude sets and an iterative process leading to the PT corrections in SSMRPT protocol. Moreover, the relaxed description results in computationally demanding codes and various widely used MRPT approaches take benefit from the simplifications brought by internal contraction. Finally, although SSMRPT protocols attain a number of essential formal criteria, one should not forget the chief merit of MRMP2, CASPT2 and NEVPT2 (belongs to the family of internally contracted MRPTs) methods is the availability of far developed, efficient codes that can handle large model spaces. The basis sets used here have been obtained from the EMSL database90 . Here, the notation CAS(a,b) describes a CAS with ‘a’ electrons distributed in ‘b’ orbitals. Although the NEVPT2 approach has been implemented in various quantum chemistry software packages, NEVPT2 (strongly contracted) calculations [using default geometry convergence threshold (tolerances)] reported here have been done using the ORCA quantum chemistry package developed by Neese91 . NEVPT2 analytic gradient scheme is not available, but numerical gradient protocol is implemented in ORCA that deliver promising results. Here, IVO-SSMRPT geometry optimizations have been performed with numeri12


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cal energy derivatives technique. In the IVO-SSMRPT gradient approach, the convergence threshold for geometry optimizations employed is 10−5 . On the other hand, the systems are treated using analytical gradient scheme for the CASPT2 geometry optimization which is based on the analytical expressions of the response functions to compute the required derivatives. Although, both protocols have found various use, ideally, energy gradient should be computed analytically. Note that evaluating derivatives numerically is conceptually simple and can be correct in many situations. However, it is generally accepted that analytic derivative techniques are advantageous, and in various cases needed, when one wishes to get accurate gradients and hessians. The numerical scheme, however, needs a costly number of (single-point) computations, and its correctness must be inspected with respect to the finite difference step size. The analytical derivative technique needs much fewer energy calculation steps for the estimation of a single property compared to the numerical one. It should be emphasized that the use of numerical gradient protocol can invite artificial symmetry breaking, especially for MR wave functions. It must be highlighted that the relative computational labor of analytic gradient and energy computations is constant, whether one treats a diatomic molecule or a larger one. On the other hand, the computational cost of numerical differentiation of the energy scales with the number of atoms92 . Consequently, solution of the numerical energy gradients can be time consuming, and need considerable computer resources. Therefore, analytic derivative methods although more mathematically-involved, are favored due to lower computational labor and higher accuracy. As a result of this, it is worthwhile to develop analytical energy gradients for IVO-SSMRPT method to characterize the stationary points on Born-Oppenheimer potential energy surfaces. It should be mentioned that much less work has been reported on analytical gradient techniques for state specific MR-based methods. This is because either the derivative equations are complicated to implement effectively or the computational costs are high, or both. CASPT2 calculations have been carried out with the MOLPRO program93 . The modest aim of the present work is not to furnish any elaborate dissection of the performance of MRPT method but rather to appraise qualitative differences (if any) between the 2,4,6-tridehydropyridine radical cation and the tetradehydrobenzene. The interaction between four formal radical centers in tetraradicals can be explicated by close teamwork between experimental and theoretical researches. In a nut-shell, we have modestly tried to address the following relevant questions by 13



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applying IVO-SSMRPT method which provides estimates of interest by focusing on one state within the model space at a time: • What is the nature of the ground state of these tetraradicals? • What is the extent of the energy gaps between the multiplets? • When replacing carbon by nitrogen, does identity of the ground state remain unchanged? • What type of alterations in the order of electronic states could be anticipated under the effect of the positive charge?

A.

Benzdiynes: Tetradehydrobenzene (TDHB)

Compared to benzyne (with two radical centres), benzdiynes (formally obtained by dehydrogenation of the benzynes) have presented a challenge to both experimental and theoretical chemists4 . Compared to didehydrobenzenes(C6 H4 ) and tridehydrobenzenes(C6 H3 ), tetradehydrobenzdiynes (benzdiynes, C6 H2 ) have only received limited attention theoretically94–99 . The addition of a fourth radical center in tridehydrobenzenes to form tetradehydrobenzenes (TDHB) (see Figure 1) yields in more-intricate electronic structures with various close-lying low- and high-spin states which always create much challenge to theory used to study such systems2,3,94,96–99 . The relative ordering of the lowest singlet and triplet states for TDHB isomers vary unusually across the different levels of methods. The different locations of the radical centers in TDHB-isomers have a major impact on their chemical nature. According to the findings of Yabe and co-workers2,3 , electron-withdrawing substituents destabilize the benzdiyne more than they destabilize the corresponding benzenes, whereas electron-donating moieties have the opposite effect. Both 1245-TDHB and the ortho form, 1234-TDHB, possess two triple bonds in an aromatic ring which produces a large ring strain and an extremely high electron density in the ring. Spectroscopic investigations on the benzdiynes, in conjunction with ab initio calculations support the MR character of benzdiynes4 . The contributions of HF configurations in CASSCF solutions shows that the MR-nature of 1245- and 1234TDHBs is larger compared to the 1235-TDHB one. The MR character of 1245-TDHB is larger than that of the 1234-one which causes problems in calculations using a single determinant. The schematic representations of the atomic connectivity for all three isomers 14


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of TDHB considered here are depicted in Figure 1. The frontier molecular orbitals (MOs) (Figure S1) and electrostatic potential charge distribution (Figure S2) of three isomers of TDHB are given in Supporting Information. Here, we have used Dunning’s correlation-consistent cc-pVDZ basis set and CAS(10,10) which included the six benzene π orbitals and the four σ orbitals, and the ten electrons in those orbitals. Structures together with the geometrical parameters obtained at IVOSSMRPT(10,10)/cc-pVDZ level of calculations are assembled in Figure 2. The energy gaps (∆EST , kcal/mol) between the three isomers of singlet and triplet tetradehydrobenzene at IVO-SSMRPT level are also estimated which provide a quantitative picture of the electronic state ordering. The energy gaps calculated between the singlet and triplet (ST) for three isomers at various levels of theory are listed in Table 1. In this work, we have used optimized orbitals for each state independently to compute geometrical parameters and energy gaps, ∆EST . The extent of the interaction between the electrons of radical centers is reflected in the value of energy separation between the states of different spin-multiplicity. For example, the smallest overlap between the orbitals carrying unpaired electrons arises in p-benzyne, which has the smallest ∆EST , 3.22 kcal/mol. The interaction between radical centers is larger in m-benzyne, in which the ∆EST is 20.06 kcal/mol. A partial bond is created between the two adjacent radical centers in o-benzyne, leading to ∆EST of 38.28 kcal/mol71–73 . Table 1 shown that IVO-SSMRPT and CASPT2 calculations favor the singlet ground state, thus violating Hund’s rule of maximum multiplicity. Present IVO-SSMRPT calculations indicate that ∆EST in 1234-, 1245-, and 1235-TDHBs are 19.95,15.90, and 36.55 kcal/mol, respectively. The ST splitting in 1234-, 1245-, and 1235-TDHBs obtained with the CASPT2(10,10)/cc-pVDZ calculations are 17.52, 16.82, and 28.80 kcal/mol which are in close proximity with values obtained by IVO-SSMRPT(10,10)/cc-pVDZ level of calculations (except for 1235-TDHB). Schaefer and coworkers98 investigated the three isomers of TDHBs through the estimations of ST-energy gaps and vibrational frequencies using both ab inito [CCSD(T)/TZ2P//CCSD/DZP] and DFT (density functional theory) methods. Their studies suggested that all of the cyclic isomers have singlet ground states. At CCSD(T)TZ2P//CCSD/DZP level of calculations98 , the ST splittings were computed to be 26.3, 18.3, and 37.5 kcal/mol, respectively. At this point it is worth noting that CCSD(T) method is not suitable for capturing static correlations in general. Although CCSD(T) happens to be sufficient for the present case; however, it would be injudicious to trust such 15



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findings without substantiating the estimates against a more suitable methodology. Although some types of open-shell structures can be handled by standard CCSD/CCSDD(T) wave function; however, complications due to spin-contamination may arise which is unwelcome, specially if one is interested in the energy gaps of states of different spin multiplicity. In many situations, the problem can be mitigated by using ROHF-like references. Unfortunately, this strategy does not resolve the difficulty, and even brings new one, such as relatively poor convergence. Moreover, it has long been discerned that the cluster operators defined with respect to spin-orbitals for spin-adaptation usually invites spin-broken solutions for the nonsinglet electronic states. The spin-broken solutions and spin-contamination phenomena usually affect the correctness and the quality of results. It should be noted that the spin-contamination requires to be carefully monitored as large deviations from spin-pure values indicating potential MR-character100 . In passing, we note that for high-spin states that are not strongly influenced by orbital degeneracy and spin contamination and can thus be described by single-configuration HartreeFock wave functions. Note that the isomer 1235TDHB shows ∆EST as large as ortho-benzyne [∆EST = 38.14 kcal/mol]. On the other hand ∆EST values for 1245- and 1234-TDHBs are close to that of meta-benzyne [∆EST = 21.2 kcal/mol]. The order of different singlet and triplet TDHB isomers provided by the IVOSSMRPT method vis-a-vis other established levels of calculations is very encouraging. The 1245 (ortho) and 1234 (para) singlet TDHBs have C≡C bonds analogous to ortho-benzyne (1.262 ˚ A), but the ST splittings are notably lower than in ortho-benzyne, generally owing to increased ring strain. The ground-state multiplicity along with the energy gaps between multiplets is important for the design of novel magnetic materials as these parameters can be used as a measure of electronic coupling between the formal radical centers in the radicals. Therefore, these quantities also delineate a meter of the stabilizing or destabilizing interactions between the unpaired electrons on the radical centers. The extent and nature of interaction between the unpaired electrons is primarily depend on the property of the MOs containing these unpaired electrons. In the present investigations of TDHB cation and three isomers of TDHB, the singlet state appears as the electronic ground state, in contrast to the prediction of Hund’s rule (suggests that the lowest energy state is the one with the highest spin-multiplicity). This clearly indicates the signature of bonding interaction between the radical centers. Note that generally contravention of Hund’s rule emerges when the singly occupied NBMOs (nominally 16


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nonbonding MOs) are generally confined on different locations of the molecule (disjoint in nature) and thus the extent of the interactions among the radical centers are not significant. As a result of this, the states of different spin-multiplicity with the same spatial configuration are quaisdegenerate in nature. Here, it is worth mentioning that for a long time, the utility of high-spin polyradical species has been well recognized in development of high-spin molecular magnets (and spintronics, paramagnetic relaxation reagents), whereas low-spin polyradicals have gained much less attention from both experimental and theoretical chemists for the same. Present results and findings [see Figure 2] for TDHB isomers are generally consistent with those of the previously published results2,3,98,99 . The length of the formal triple bonds in 1245-TDHB and the ortho form, 1234-TDHB as calculated at the IVO-SSMRPT level are considerably shorter than the CC bond length in benzene. The IVO-SSMRPT estimates of the geometrical parameters are in a reasonable agreement with the present CASPT2 values as well as QICISD, and CCSD(T) of Schaefer and coworkers98 . From the work of Bettinger et al.98 we found that increasing the sophistication of correlation treatment (from CCSD/DZP to CCSD(T)/TZ2P) used in the geometry optimization of 1245-TDHB does not alter the values of the geometrical parameters notably. The optimized geometry for the 1234-TDHB in the triplet state obtained by IVO-SSMRPT numerical method advocates the existence of a σ-allyl like radical group with the vicinal radical center. Similar to the CASPT2 calculations, the C≡C bonds in 1245-TDHB are smaller than in 1234-isomer. In the benzene ring plane of 1234-TDHB, the four electrons are delocalized over the four σ orbitals, similar to the delocalization of π system in butadiene. This fact leads to the lengthening of terminal C-C bonds along with the abridging of intervening C-C bond length in 1234-TDHB compared to the corresponding C-C bond distances in 1245-isomer where the existence of the two CH moieties prevents σ conjugation. Due to the existence of the σ (in-plane) conjugation, 1234-TDHB is more stable than 1245-isomer by 3.4 kcal/mol. Similar to that of previously published estimations at the level of CC98 , present estimations also indicate that the C≡C bonds of 1234-TDHB and congeners 1245-TDHB are larger than the 1.24 ˚ Aortho-benzyne dehydro-CC bond lengths predicted experimentally101 . IVO-SSMRPT level of calculations, thus, provide a reliable C≡C bond length for both 1234- and 1245-TDHBs. IVO-SSMRPT calculations support that one of the dehydro-CC bonds in the singlet state is notably shorter than in the triplet one for 1245-TDHB indicating ring strain is lower for triplet 1245-TDHB. 17



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Considering the findings of all the methods treated here, the structures for 1,2,3,5tetradehydrobenzene (1235-TDHB) is established to be bicyclic in nature (generated via through-space orbital overlap). Bicyclic structure of 1235-TDHB with partial σ bonds between C1 and C3 (1.81 and 1.77 ˚ Afor singlet and triplet state, respectively) adopts in IVO-SSMRPT(10,10) level of calculations, in which the ]CCC are only 85 and 89◦ , respectively. Structure (bicyclic form) and geometrical parameters of 1235-TDHB yielded at the CASPT2 level are generally in good agreement with the present method. In the CASPT2(10,10) geometry, this angle is small, namely 83◦ . C1-C3 distances for singlet and triplet states provided by using CASPT2(10,10) level of calculations are 1.81 nand 1.74 ˚ A, respectively. Studies of Schaefer and coworkers98 using CC methods also suggest bicyclic structure. Similar finding was also reproduced in the studies of Arulmozhiraja-Sato-Yabe2,3 and Sattelmeyer-Stanton99 . Therefore, present work provides further evidence for the existence of a bicyclic form. Note that the choice of density functional and basis set have a substantial effect on the shape (monocyclic or bicyclic structure) of the 1235-TDHB. It can be stated that the conventional DFT with the approximate exchange-correlation functionals generally disappoints to describe the MR nature102,103 . It was found that DFT (BLYP and B3LYP), CASSCF, CCSD, and CCSD(T) calculations predict C1-C3 distance for 1235TDHB varying from 1.76 to 1.84 ˚ A2,3,98,99 . In the geometry of 1235-TDHB, at B3LYP/ccpVTZ level of calculation provides the C1-C3 distance of 1.772 ˚ A. Additionally, a smaller C1-C3 separation in 1235-TDHB than in meta-benzyne designates a bicyclic structure of 1235-TDHB. It is especially noteworthy that the structure of meta-benzyne has been the subject of great debate, since various ab initio protocols suggest bicyclic anti-Bredt structure rather than the correct monocyclic one. Interestingly, it has been reported that state-of-art calculations disagree with the existence of a bicyclic form of meta-benzyne102–104 . Previous IVO-SSMRPT calculation also argued in favor of monocyclic (biradical) structure with two radical centres for meta-benzyne71 rather than a bicyclic structure. Due to a substantial degree of ‘back-lobe’ orbital interaction, the formal radical centers have a meaningful extent of stabilization, which leads to relatively short dehydrocarbon atom separation and a large ST energy gap. The bicyclic shape of 1235-TDHB emerges due to the two factors such as (i) bonding like interaction between the meta dehydro centers, and (ii) σ-type delocalization over C-atoms as that of the delocalization effect in π allyl moiety. An intense attractive interaction of C5 radical center with the σ-allyl moiety is indicated by a large radical sta18


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bilization energy of 30.5 kcal/mol98 . The radical character of 1235-TDHB is significantly reduced relative to its other isomers because of through-bond coupling between the radical centers. It should be noted that the 1235-TDHB with C2v symmetry is more stable the Cs conformer98 . The self-consistent field (SCF) level of calculations favor distorts from C2v symmetry and adopts an effectively bicyclic geometry for the ortho structure 1234-TDHB which is more than 10 kcal/mol lower than the C2v form. However, incorporation of correlation effects at reasonably high-level remove this feature2,3,98,99 . It should be noted that IVOCASCI calculations also predict bicyclic geometry with C1-C3 distance of 1.95˚ Aindicating that the IVO-CASCI(10,10) method exaggerates the radical character and fails to correctly account for a partial bond formation between the two centers. Here, it should be noted that bicyclic 1235-TDHB can be formally viewed as a diradicaloid meta-isomer and the radical character of 1235-TDHB is rather small because a large HOMO-LUMO gap is observed for the 1235-TDHB system. It is pertinent to stress that if the degeneracy between orbitals enhances, the MR-nature of the lowest singlet will swell. In 1245- and 1234-TDHBs, the relatively large dehydrocarbon atom distance indicates that the interaction between radical centers is almost zero which is consistent with their monocyclic form. Relative energies estimated by IVO-SSMRPT and CASPT2 for the three isomers of TDHB mirror those obtained by previous high-level methods2,3,98,99 . The energy differences between the ground state of the three isomers obtained by IVOSSMRPT(10,10)/cc-pVDZ scheme along with other methods are assembled in Table 2. As the two triple bonds in 1234-TDHB and 1245-TDHB (analogous to ortho-benzyne) introduce a significant amount of ring strain, the 1235-TDHB might be energetically more stable in the tetradehydrobenzene series. IVO-SSMRPT findings compare well to both multistate CCSD(T)/TZ2P//CCSD/DZP and B3LYP/cc-pVTZ data of Yabe and coworkers2,3 . Note that 1235-TDHB lies 14.42 kcal/mol below 1245-TDHB and 11.05 kcal/mol below 1234-TDHB as per present IVO-SSMRPT(10,10) calculations.

The corresponding val-

ues are 14 and 7.8 kcal/mol at the CCSD(T)/TZ2P//CCSD/DZP level of calculations2,3 . The most stable 1235-TDHB is 10.32 and 15.74 kcal/mol more stable than 1234- and 1245-TDHBs at the CASPT2(10,10)/cc-pVDZ level, respectively. Energy differences of 1235-TDHB relative to 1234- and 1245-TDHBs of 8.47 and 14.32 kcal/mol have been obtained at CCSD(T)/cc-pVTZ//B3LYP/cc-pVTZ2,3 . The 1235-TDHB isomer is more stable by 13.07 and 14.32 kcal/mol than 1245-TDHB one at the B3LYP/cc-pVTZ and 19



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CCSD(T)/ccpVTZ//B3LYP/cc-pVTZ2,3 , respectively. In addition to the above calculations, CCSD(T)/6-31G∗∗ //CASSCF(4,4)/6-31G∗∗ ab initio calculations suggest that 1234TDHB benzdiyne is more stable than 1245-TDHB by 4.8 kcal/mol105 . Isomer 1235-TDHB with C2v point group is more stable than 1234-TDHB by 8.9 kcal/mol. Previous published ab initio studies on TDHBs propose that the para isomer 1245 is higher in energy than the ortho compound 1234 by 1.2 and 11.5 kcal/mol due to CASSCF(10,10)/3-21G96 and MP2/631G∗97,106 calculations, respectively. The CCSD(T)/TZ2P level of calculations suggest that 1235-TDHB is 11 kcal/mol more stable than 1245-TDHB. The corresponding values provided by B3LYP/TZ2P and BLYP/TZ2P schemes are 16.9 and 19.4 kcal/mol, respectively. This fact indicates that the DFT energy gap is markedly different from the CCSD(T) one. In contrast with this finding98 , Arulmozhiraja-Sato-Yabe2,3 work indicated that the energy gap between 1245-, and 1235-TDHBs obtained with the B3LYP functional agrees well with that of the QCISD and CCSD(T) calculations. Relative energies (in kcal/mol) of the 1235-TDHB in the ground state with respect to 1234- and 1245- calculated at the CCSD(T)/cc-pVTZ level are 8.5 and 14.4, respectively2–4 . In the work of Bettinger et al.98 , the CASSCF(10,10) calculations suggest singlet 1245-TDHB to be the lowest in energy in contrast to CCSD(T) and IVO-SSMRPT calculations. It is now well-documented for the benzyne (didehydrobenzenes) that ortho-benzyne is markedly lower in energy than its meta- and para-isomers. The disagreement of CASSCF (MP2) approach with MRPT and CCSD(T) might be due to CASSCF (MP2) inability of incorporation of dynamical (nondynamical) electron correlation in an appropriate manner98 . This fact argues in favor of a balanced treatment of both correlation to get to the correct answer. In the case of triplet states, MR-based mean-field like calculations yield correct order. Note that in-plane σ-conjugation in 1234-TDHB can be considered as the energy difference of 1234-TDHB with respect to its 1245-isomer98 . The present studies discussed above disclose that 1235-TDHB with C2v symmetry is more stable than 1234-TDHB and 1245-TDHB, despite the fact that 1235-TDHB is the one least well described by a traditional Lewis structure. Note that 1245-TDHB is the least stable among the three isomers. The results obtained here, together with those reported earlier indicate that the stability order: 1235 > 1234 > 1245-TDHB. To calibrate present IVO-SSMRPT findings, we also consider the results obtained by NEVPT2 method (with CASSCF orbitals) which do a good job of describing both static and dynamic correlations similar to the IVO-SSMRPT one. Selected optimized geometrical 20


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parameters of the lowest singlet and triplet 1235-TDHB obtained by NEVPT2(10,10)/ccpVDZ numerical gradient scheme are also summarized in Figure 2. It is worth addressing here that NEVPT2 gradient calculation converges fairly well for the triplet state compared to singlet one. NEVPT2(10,10) gradient calculations also favor the bicyclic form for both singlet and triplet 1235-TDHB with partial σ bonds between C1 and C3 (1.861 and 1.693 ˚ Afor singlet and triplet state, respectively). Further, it was also found that for the 1235TDHB in singlet and triplet states, the deviations of NEVPT2 geometrical parameters from the other methods reported here are not significantly large. The energy obtained from NEVPT2(10,10)/cc-pVDZ gradient calculations predicted that the lowest excited triplet state of 1235-TDHB is 38.05 kcal/mol above the singlet ground state which is in very good agreement with the IVO-SSMRPT and CASPT2 gradient estimates. Therefore, the structural predictions of NEVPT2 for the 1235-TDHB are conclusive. For the other two isomers, NEVPT2 gradient calculations suffer convergence problem, the reason for which is not clear to us at present. At this stage it should be mentioned that to deal with the excited state, one might often require a greater number of iterations to “home” to the root of interest while solving the NEVPT2 equations, similar to other single-root MRPT methods. For TDHBs, the utility of the IVO-SSMRPT approach can also be illustrated by comparing IVO-SSMRPT energy gaps with the NEVPT2 values (through single-point energy calculation) that are collected in Table 1. Note that NEVPT2 single-point state energy calculations are carried out with the optimized IVO-SSMRPT(10,10)/cc-pVDZ geometries. The NEVPT2 findings are quite parallel to IVO-SSMRPT and CASPT2 methods. As shown in the table, NEVPT2 ∆EST for TDHBs computed using the IVO-SSMRPT(10,10)/cc-pVDZ geometries also strongly argue in favor of a singlet ground state. The NEVPT2(10,10)/ccpVDZ level of calculations provide the same energy ordering as that of IVO-SSMRPT and CASPT2 methods and yield the ∆EST values of 18.62, 15.50, and 40.28 kcal/mol for 1234TDHB, 1245-TDHB and 1235-TDHB, respectively showing acceptable agreement with the corresponding estimates provided by IVO-SSMRPT and CASPT2 methods. Note that the NEVPT2 estimate of the ST energy gap obtained using the numerical gradient (optimization) scheme of the individual states followed by the calculation of the energy difference is in agreement with the gap obtained by single-point NEVPT2 calculations for the singlet and triplet states using the IVO-SSMRPT(10,10)/cc-pVDZ optimized geometries for the two states(vide supra). The NEVPT2 energy calculations disclose that 1235-TDHB is more 21



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stable than 1245- and 1234-isomers and that 1245-TDHB is the least stable among the three isomers. It is instructive to emphasize the fact that the most stable 1235-isomer is 13.55 and 19.50 kcal/mol more stable than 1234- and 1245-TDHBs at the NEVPT2(10,10)/ccpVDZ//IVO-SSMRPT(10,10)/cc-pVDZ level of calculations similar to other MRPT and CC methods considered here. The NEVPT2 method behaves well even when compared with CC calculations. Therefore, all the NEVPT2 findings obtained using optimized IVOSSMRPT geometries agree with the present and previously published high level theoretical results. In our experience, this is an indication of a good (quantitatively) description of TDHB systems provided by IVO-SSMRPT numerical gradient scheme. In order to gain insights into substituent effects on the electronic structural properties of tetraradicals, we have carried out a computational investigation on 2,4,6-tridehydropyridine radical cation.

B.

Charged Aryl Tetraradical: 2,4,6-tridehydropyridine radical cation

Next we study 2,4,6-tridehydropyridine (246-TDHP) radical cation, an analogue of 1235TDHB yet extremely difficult to study experimentally5 . A substantial apprehension of how structural changes affect the electronic and magnetic interactions in organic high-spin systems is one of the essential issue for the development of improved metal-free magnetic organic materials and for fine-tuning of their natures. Recent attentiveness in the understanding on the factors controlling various reactivity of radical cations has led Kenttämaas group to different distinctive radical systems1,86–88 . The first reactivity study of the 246-TDHP radical cation in the gas phase (by using Fourier-transform ion cyclotron resonance mass spectrometry) was reported by Gallardo et al.5 . Their study suggests that this tetraradical cation is highly electrophilic and quickly reacts with various nucleophiles by reducing the C-N ortho-benzyne unit, yielding a relatively unreactive meta-benzyne analogue. They also stated that the tetaradicals display substantially lower reactivity as compared to the monoand triradicals. The tetraradical, 246-TDHP cation exhibited no radical reactions with allyl iodide but instead only charge-driven reaction(s) due to an even-electron resonance structure [see Figure 3]5 . The presence of multiple resonance structures that hinder radical reactivity which makes the 246-TDHP cation highly electrophilic. Moreover, it quickly interacts with various nucleophiles by quenching the NC orthobenzyne part, thereby yielding a relatively 22


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unreactive meta-benzyne analogue. The findings of Kenttämaa-Cramer and co-workers86–88 on the 2,4,6-tridehydropyridinium cation indicate that the chemical nature of this triradical has close kinship with the related monoradicals rather than related biradicals. Note that the frontier MOs as well as potential charge distributions for the 246-TDHP radical cationic system have been compiled respectively in Figures S3 and S4 of the Supporting Information. The 246-TDHP radical cation has received little attention from computational chemists which inspired us to take a second look at the IVO-SSMRPT level of calculations to study it electronic structural properties. Here, we have used CAS(10,10) and cc-pVDZ basis set for our MR-based calculations for the ground 1 A and excited 3 B2 and 5 B2 states. Among the three states having different multiplicities, 1 A1 and 5 B2 have similarities in the charge distribution pattern. Note that the N atom bears higher negative charge in the 5 B2 246TDHP radical cation. For 246-TDHP radical cation in 3 B2 state, the C atoms attached to H atoms have negative charge with opposite to 1 A1 and 5 B2 . For the 246-TDHP radical cation, the atom numbering scheme is shown in Figure 4. To investigate the sensitivity of the results on the basis set used, we also supply the results with cc-pVTZ bases using the geometries obtained in the IVO-SSMRPT(10,10)/cc-pVDZ optimizations. Table 3 summarizes predictions of energy gaps, ∆Eij computed for different states of 246-TDHP cation at various levels of protocol. The notation ∆Eij describes energy gap between the ith and the j th states:∆Eij = Ej − Ei where i=1 A1 (S) and j=3 B2 (T) and 5 B2 (Q). There are not enough data in the literature to judge IVO-SSMRPT estimates. Generally the strong bonding interaction stabilizes low-spin states, on the other hand, weak interaction often prefers highspin ones. The closed agreement of ∆Eij values obtained by the IVO-SSMRPT(10,10) and CASPT2(10,10) calculations (zero-point energy correction is not included) clearly demonstrate the efficacy of IVO-SSMRPT method. At IVO-SSMRPT(10,10) levels of theory, the singlet state of 246-TDHP cation is calculated to be the ground state similar to TDHB systems. The works of Gallardo et al.5 also indicated that the ground state of 246-TDHP radical cation is evidently 1 A. The perturbation due to the presence of the N atom with positive charge does not change the nature of the ground state. This is due to the resonance structures shown in Figure 3 which allow greater charge delocalization away from the nitrogen atom. This delocalization stabilizes the singlet state over the triplet one. This fact explains why 246-TDHP cation in singlet state undergoes radical reactions slower than the triplet one1,5 . Note that the existence of the N+ in the aromatic moiety destabilizes 3 B2 23



Page 30 of 48

state by restricting the interaction between the radical centers at C2 and C6 [see Figure 5] and this does not canceled to any considerable extent by the charge delocalization process offered by resonance effect. However, the presence of the N+ has reasonable impact on the computed ST energy gaps compared to TDBH isomers. Present theoretical calculations suggest that 246-TDHP cation has a singlet ground state lying 25.84 kcal/mol below the triplet 3

B2 state in agreement with the earlier studies5 . It is worth noting that the first ab initio

study on 246-TDHP cation using RHF-UCCSD(T)/cc-pVTZ//UBPW91/cc-pVDZ due to Kenttämaa and coworkers5 also support the singlet electronic ground state. A slight state preference is found for the ground state, apparently violating Hund’s rule as the ground state of the molecule should be the triplet state. This effect is associated with the splitting of the orbital energies and the delocalization of the positive charge in cation strongly alters the electronic distribution and some structural features. In UCCSD(T)/cc-pVTZ calculations, the singlet state is well split from triplet, 3 A0 and lies below the 3 A0 . Theoretical estimates for energy gaps discussed here do not vary widely depending on the nature of the methods used. Note that the presence of nitrogen and charge have generally dramatic effects on the relative energies of the triplet and singlet states107 . Dougherty and co-workers108 argued that although changing benzene by pyridinium ring in a meta-xylylene does not influence the energy gap between the ground triplet and excited singlet states, protonation of the pyridinium moiety flips the ordering of the ground and the excited states thereby altering the chemical identity of the molecule. At the IVO-SSMRPT/cc-pVDZ level, the computed ∆E1 A−3 B2 and ∆E1 A−5 B2 are 25.84 and 105.15 kcal/mol, respectively. The corresponding value provided by RHF-UCCSD(T)/cc-pVTZ//UBPW91/cc-pVDZ level of calculations are 31.2 and 109.2 kcal/mol indicating the close accordance with the IVO-SSMRPT estimates. The ST and SQ energy gaps are 22.93 and 101.22 kcal/mol computed by CASSCF-CASPT2(10,10) method, respectively. Note that IVO-SSMRPT(10,10) and CASPT2(10,10) methods provide very similar values for ∆Eij gaps. Generally, the dependence on the basis set of the IVO-SSMRPT is not very significant. For the 1 A →3 B2 transition, enhancing the basis set size changes the IVO-SSMRPT excitation energy by 0.17 kcal/mol, while the corresponding change for 1 A →5 B2 transition is around 1.77 kcal/mol. In addition to the MRPT studies, CC calculations at the CCSD(T)/cc-pVDZ level using the geometry of IVO-SSMRPT/ccpVDZ have also been done to evaluate the relative stabilities of 3 B2 and 5 B2 against the singlet ground state. The CCSD(T) theoretical estimates are 34.05 and 96.09 kcal/mol 24


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for ∆E1 A−3 B2 and ∆E1 A−5 B2 , respectively suggesting the incorporation of the nondynamic correlation energy plays a significant role in this system. In the NEVPT2(10,10)/cc-pVTZ calculations for energy gaps, the geometries of the ground- and the excited-states are taken as those optimized at the IVO-SSMRPT(10,10)/ccpVDZ level. Present NEVPT2 results summarized in Table 3 concur with the IVO-SSMRPT and CASPT2 investigations exhibiting the 2,4,6-TDHP radical cation to be characterized by a singlet ground state. Regarding the values of the ∆E1 A−3 B2 and ∆E1 A−5 B2 separations, while IVO-SSMRPT computations provided gaps of 25.84 and 105.15 kcal/mol, the NEVPT2 method results in smaller gaps of 24.84 and 97.88 kcal/mol. The relative energies computed with NEVPT2(10,10) agree well with the CASPT2(10,10) results obtained with the same basis set. The NEVPT2 estimated splittings for the 246-TDHP agree reasonably well at the levels of CC methods employed here. The closeness of NEVPT2(10,10)/ccpVTZ gaps with other methods described here is an indication of a good quality of the optimized geometries offered by IVO-SSMRPT(10,10)/cc-pVTZ gradient calculations. The CCSD(T)/cc-pVDZ//SSMRPT(10,10)/cc-pVDZ results also clearly suggest that the use of optimized IVO-SSMRPT geometry is effective for correct descriptions of the ST gaps in 246-TDHP radical cations. Structures together with the geometrical parameters acquired at various levels of strategies are collected in Figure 5 for the singlet ground state along with 3 B2 and 5 B2 states of 246-TDHP cation. The 1 A1 ground state of 246-TDHP radical cation is characterized by a shorter C2-C6 bond length compared to the triplet and quintet states. On the other hand, the relatively large C2-C4 dehydrocarbon atom distance recommends that the interaction between these two radical centers is reasonably faint. At this point we want to mention that the bonding interactions lead to shorter bond lengths between the radical centers, more rigid structures, and decreased reactivity for the singlet state. Therefore, the relative stability of singlet 246-TDHP cation, results from the formation of partial bonds between the radical centers which lead to (nearly) bicyclic structure that would introduce a significant amount of bond angle strain. While the charge delocalization through the resonance structures probably counterbalance this ring strain to some extent, its impact does not emerge to be appreciable enough to compensate for the relatively weak interaction between the radical centers at C2 and C6. The presence of hetero-atom reduces bonding interactions between the C2 and C6 radical centers compared to the corresponding TDHB systems. In the excited 25



Page 32 of 48

state, triplet and quintet, the fairly large C2-C6 dehydrocarbon atom separation advocates that there is no interaction between these two radical centers. Due to the bicyclic structure, N-C2 (and N-C6) bond distance in the singlet state is larger than the monocyclic structures of 3 B2 and 5 B2 states at all level of theoretical calculations considered here. The comparison between CASPT2 and IVO-SSMRPT results lead to similar conclusions. From the previous findings of other workers, it is found that there is no obvious difference in the geometries and structural parameters provided by different DFT methods with various functionals [see supplementary material in Ref.5]. It is believed that the present computed geometrical parameters for the 246-TDHP cation are reliable too as IVO-SSMRPT provides noticeable agreements with CASPT2(10,10) approaches. Of course, a detailed comparison with the IVO-SSMRPT values will have to wait until experimental values are available. In passing, we want to mention that the findings provided by DFT methods are in good accord with high-level wavefunction based ab initio calculations. However, the bicyclic geometries for meta-benzyne, with a CC partial bridging bond (1.6 to 1.8 ˚ A) due to DFT calculations can be considered as artifacts because for meta-benzyne considerably longer distances between the radical centers are obtained with high-level ab initio methods1,68 . Present along with previous works suggested that the states of different multiplicity of tetraradicals are handled nicely at the IVO-SSMRPT level and thus the method has been found to provide quantitative descriptions for the gaps between multiplets in various tetraradicals. The present work also indicates that the use of IVO-SSMRPT optimized geometries may be sufficient for arriving at accurate predictions of the energy gaps in muliplets. Correctly predicting the energy gaps of MR systems is a frontier area in the realm of electronic structure theories. The multiplicity of the ground state and the energy splitting between the ground and excited states are the decisive properties to design the molecular magnets. The notable difference between the optimized (equilibrium) geometries of both singlet and triplet states might be the crucial trait to identify the multiplicity of the ground state of complexes where these tetraradicals are incorporated. As discussed above, both tetraradicals considered here elect low-spin, singlet ground states over lowest triplet ones. At this point, one can argue that the limited size of the employed basis sets does not allow one to claim this with certainty. However, a good agreement of the present as well as the previous IVO-SSMRPT findings with current generation calculations bolster our belief about the faithfulness of the present IVO-SSMRPT estimates. The incorporation of dynam26


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ical correlation effect is pivotal for quantitative estimations of the energy gaps between the ground and excited states of tetraradicals. The geometrical constants and the relative state energies estimated at the CASPT2 and IVO-SSMRPT levels of theory agree reasonably well for both radical systems, indicating the relaxation (modification) of reference space function and size-extensive property which are the main merits of IVO-SSMRPT against CASPT2 does not have a great influence on the calculated properties. Improving the accuracy of the present findings can be made possible using MRCC methods but comes at a huge computational labor. However, more sophisticated treatment of correlation effects does not affect the general findings presented above for the individual tetraradicals. More work is needed to further refine the estimates obtained by the present work. Moreover, it will be enthralling to study to what extent the trends observed here can be extrapolated to other variety of tetraradical systems. A well accepted general objection with CAS-based MRPT methods is that the computational cost grows rapidly with the number of reference functions and becomes soon a bottleneck in the CAS-based calculations. To overcome this problem, a number of interesting and useful strategies have been developed39–45,109,110 .

III.

CONCLUSION

Polyradicals, (say tetrardicals) have become the focus of substantial research due of their usefulness in a multitude of applications, including photocatalysis, biological systems, organic electronics, and so on. However, predicting the electronic structural properties of radical systems specially tetraradicals that display intricate inter-play of static and dynamic correlation effects (MR nature) continues to persist a challenge to the practitioners of electronic structure theory. Moreover, organic polyradicals possess a combination of unique and intriguing features that are not only of central attentiveness but also could have appreciable technological utilities and understanding ground and excited states properties of these systems becomes increasingly important. Notably few theoretical and experimental investigations have been devoted to these prototypical tetraradicals till recently. The results of investigation illustrates a greater complexity for the tetraradical species compared to the related mono- and diradical ones. The spin-free IVO-SSMRPT (an IVO strategy of the virtual orbitals within the framework of the spin-free UGA-SSMRPT of Mukherjee et al.) proves to be robust and is found 27



Page 34 of 48

to be widely applicable to diverse chemical situations/processes. As the IVO scheme reduces computational cost and IVO-CASCI retains the attractive features of the CASSCF wave function, expanding the limit of application of the UGA-SSMRPT method. The IVOSSMRPT method is not restricted to ground state calculations only. Can we extend this scheme to tackle polyradicals, which are of interest in the context of molecular magnets? There is every cause to trust that it can. To demonstrate it, we use IVO-SSMRPT to investigate the σ,σ,σ,σ-tetraradicals including three isomers of tetradehydrobenzene and 2,4,6-tridehydropyridine radical cation using very challenging chemical configurations. Owing to their intrinsically MR-nature, these radicals can only properly be tackled by employing correlated approaches of MR-discipline. The lack of availability of accurate geometrical parameters, energetics and other electronic structural properties makes them good test beds for benchmarking investigations of newly suggested approaches designed to handle neardegenerate situations of arbitrary complexity. The IVO-SSMRPT results obtained here have been compared with those of other widely used MRPT computational methods such CASPT2 and NEVPT2. IVO-SSMRPT is similar in scope to the popular MRPT methods, but in our opinion, features a number of advantages such as strict guarantee of size-extensivity, explicit intruder free nature, and the scope of applying it within the relaxed (decontracted) and unrelaxed (contracted) descriptions of the model space functions. Here, we demonstrate that one can estimate the optimized geometries and energy ordering of the different states unambiguously, starting with a second-order MRPT method, IVO-SSMRPT. Given its reasonably robust effective Hamiltonian based MR-based treatment of electron correlation, we take the IVO-SSMRPT approach as a method of selection for estimations of the relative state energies. On the basis of present studies on the tetraradicals considered here, singlet state is favored as a ground state over the triplet one, in contrast to the suggestion of the Hund’s rule. The presence of N+ group in the aromatic ring of 2,4,6-tridehydropyridine radical cation does not affect the energy ordering of the low-lying singlet and triplet states due to the resonance structure which allows charge delocalization away from the N atom. The through-bond coupling of the in-plane electrons stabilizes the singlet ground states. Violations of Hund’s rule are frequently observed when the exchange interaction between the electrons is weak. In such situations, instantaneous (dynamic) correlation effects often inspire low-spin ground states. Present estimates agree reasonably well at all the levels of theory addressed here. Observing the estimates, one can note that the MRPT methods 28


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considered here deliver very close performances. Current work along with other levels of theory suggest that 2,4,6-tridehydropyridine radical cation and 1,2,3,5-tetradehydrobenzene adopt the bicyclic geometry (results from the formation of σ bond between the C1 and C3 radical centers) in their ground state. Here, IVO-SSMRPT estimates for the energy splittings fully endorse the other established theoretical prediction. The ground state 2,4,6tridehydropyridine radical cation and 1,2,3,5-tetradehydrobenzene can be described as diradicaloid whereas, the other isomers of tetradehydrobenzene in their ground state act as true tetraradicals with four reactive radical sites. The accuracy and consistency of the IVOSSMRPT estimates for tetraradicals is crucial for investigations on large radical systems with strong quasidegenerate nature. Nevertheless, with the current set of calculations, the level of quantitative accuracy cannot be expected keeping in mind the second-order nature of the method and small size of the basis set used. The present investigation indicates that the state-specific parametrization of JeziorskiMonkhorst Ansatz with improved virtual orbitals is sufficiently flexible and a useful tool for the study of singlet and non-singlet states of polyradical systems with appreciable neardegenerate and non-degenerate character. The paper ends wit the hope that the IVOSSMRPT method could be used as a promising candidate to probe the stability for atypical intermediates

Acknowledgments

The Department of Science and Technology of India [grant no. EMR/2015/000124] is acknowledged for the generous financial support. Finally, the author expresses his gratitude to the anonymous reviewers for their valuable suggestions and comments which have helped the author to enrich the presentation of the paper.

Conflict of Interest

The authors declare no competing financial interest.

29



Page 36 of 48

Supporting Information

The frontier molecular orbitals and electrostatic potential charge distributions for the isomers of TDHB and 246-TDHP radical cation. This material is available free of charge via the Internet at http://pubs.acs.org.

30


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Table 1: Energy gaps, ∆EST (kcal/mol) between singlet and triplet states for benzdiynes at different levels of methods. A positive value indicates that the singlet state lies lower. Reference

Methods 1234-TDHB 1245-TDHB 1235-TDHB

Present work

IVO-SSMRPT(10,10)/cc-pVDZ

19.95

15.90

36.55

CASPT2(10,10)/cc-pVDZ

17.52

16.82

28.80

NEVPT2(10,10)/cc-pVDZ//IVO-SSMRPT(10,10)/cc-pVDZ

18.62

15.50

40.28

CCSD(T)TZ2P//CCSD/DZP

26.3

18.3

37.5

Ref.98

Table 2: Energy separations (kcal/mol) of 1235-tetradehydrobenzene (TDHB) relative to its other two isomers in the ground state at various levels of calculations. Reference Present work

Ref.2,3

Methods 1234-TDHB 1245-TDHB IVO-SSMRPT(10,10)/cc-pVDZ

11.05

14.42


10.32

15.74

B3LYP/cc-pVTZ

8.06

13.07

CCSD(T)/TZ2P//CCSD/DZP

7.8

14.0

CCSD(T)/cc-pVTZ//QCISD/cc-pVDZ

8.29

14.37

CCSD(T)/cc-pVTZ//B3LYP/cc-pVTZ

8.47

14.32

Table 3: Comparison of computed relative energies (kcal/mol) of the excited 3 A0 and 3 B2 states of 2,4,6-tridehydropyridine radical cation relative to the ground state 1 A, ∆Eortho/para obtained by various methodologies. Reference Present work

Ref.5

Methods ∆E1 A−3 B2 ∆E1 A−5 B2 IVO-SSMRPT(10,10)/cc-pVDZ

25.84

105.15

IVO-SSMRPT(10,10)/cc-pVTZ//SSMRPT(10,10)/cc-pVDZ

26.01

106.92


22.93

101.22

NEVPT2(10,10)/cc-pVDZ//SSMRPT(10,10)/cc-pVDZ

24.84

97.88

CCSD(T)/cc-pVDZ//SSMRPT(10,10)/cc-pVDZ

34.05

96.09

RHF-UCCSD(T)/cc-pVTZ//UBPW91/cc-pVDZ

31.2

109.2

31



Page 38 of 48

Figure Caption: Figure 1: Schematic representation of three isomers of tetradehydrobenzenes, C6 H2 considered in the present work. Figure 2 : Structure and selected optimized geometrical parameters of the lowest singlet and triplet tetradehydrobenzene isomers computed at various methods. Bond lengths are in angstroms (˚ A) and the angles are in degrees (◦ ). Figure 3: Resonance representation 2,4,6-tridehydropyridine radical cation electronic structure that hinder radical reaction. Figure 4: The atom numbering scheme of 2,4,6-tridehydropyridine radical cation. Figure 5:

Structure and selected optimized geometrical parameters of 2,4,6-

tridehydropyridine radical cation for the ground (1 A1 ) as well as excited (3 B2 and 5 B2 ) states. Bond lengths are in angstroms (˚ A) and the angles are in degrees (◦ ).

32


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∗

Electronic address: [email protected]

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