Geometry Optimization of Radicaloid Systems Using Improved Virtual

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Geometry Optimization of Radicaloid Systems Using Improved Virtual Orbital-Complete Active Space Configuration Interaction (IVO-CASCI) Analytical Gradient Method Sudip Chattopadhyay* Department of Chemistry, Bengal Engineering and Science University, Shibpur, Howrah 711103, India

Rajat K. Chaudhuri† Indian Institute of Astrophysics, Bangalore 560034, India

Karl F. Freed‡ The James Franck Institute and Department of Chemistry, University of Chicago, Chicago, Illinois 60637, United States ABSTRACT: The improved virtual orbital (IVO) complete active space (CAS) configuration interaction (IVO-CASCI) method is a simplified CAS self-consistent field (SCF), CASSCF, method. Unlike the CASSCF approach, the IVO-CASCI method does not require iterations beyond an initial SCF calculation, rendering the IVO-CASCI scheme computationally more tractable than the CASSCF method and devoid of the convergence problems that sometimes plague CASSCF calculations as the CAS size increases, while retaining all the essential positive benefits of the CASSCF method. Earlier applications demonstrate that the IVO-CASCI energies are at least as accurate as those from the CASSCF and provide the impetus for our recent development of the analytical derivative procedures that are necessary for a wide applicability of the IVO-CASCI approach. Here we test the ability of the analytic energy gradient IVO-CASCI approach (which can treat both closed- and open-shell molecules of arbitrary spin multiplicity) to compute the equilibrium geometries of four organic radicaloid species, namely, (i) the diradicals trimethylenemethane (TMM), 2,6-pyridyne, and the 2,6-pyridynium cation and (ii) a triradical 1,2,3-tridehydrobenzene (TDB), using various basis sets and different choices for the active space. Although these systems and related molecules have fascinated theoretical chemists for many years, their strong multireference character makes their description quite difficult with most standard many-body approaches. Thus, they provide ideal tests to assess the performance of the IVO-CASCI method. The present work demonstrates consistent agreement with far more expensive benchmark state-of-the-art ab initio calculations and thereby indicates that this new gradient method is able to describe the geometries of various radicaloids very accurately, even when small, but qualitatively correct, reference spaces are used. For example, the IVO-CASCI method leads to a monocyclic structure for the 2,6-isomers of the didehydropyridinium (pyridynium) cation and of didehydropyridine (pyridyne), while SCF and single-reference CCSD computations predict an incorrect bicyclic structure. The IVO-CASCI structures and relative stability for the ground 2A1 and excited 2B2 states of TDB also accord with the experimentally observed IR spectra and with other highly sophisticated theoretical calculations. The blend of accuracy and reduction in computational cost offered by the present IVOCASCI analytical gradient method clearly demonstrates that the method provides a practical avenue for studying the geometries of various radicaloid species of different levels of complexity.

I. INTRODUCTION The presence of degeneracy or quasi-degeneracy is ubiquitous to a host of phenomena in chemistry [such as in the computation of potential energy surfaces (PES) and certain types of excited states, the location of transition state(s) of diradical/triradical character, bond breaking, and the mapping of complete reaction paths, etc.]. Considerable effort has been devoted to adapting single-reference (SR) methods to treat these situations, but as of now, these SR methods remain challenged by the presence of strongly multiconfigurational contributions in zeroth order.1-4 r 2010 American Chemical Society

Multireference (MR) approaches employ more than one reference determinant and, consequently, are currently the most effective means for describing these chemical situations.5-9 The most commonly used MR method is the (mean-field) complete active Special Issue: Graham R. Fleming Festschrift Received: April 19, 2010 Revised: June 14, 2010 Published: June 29, 2010 3665

dx.doi.org/10.1021/jp103536w | J. Phys. Chem. A 2011, 115, 3665–3678

The Journal of Physical Chemistry A space self-consistent field (CASSCF) method10,11 that includes contributions from nondynamical electron correlation and that affords the strategic advantages (i) of being applicable to both ground and excited states within a single framework, (ii) of yielding a size consistent energy, and (iii) of including nondynamical correlation in an effective way for closed- and open-shell electronic wave functions. CASSCF calculations combine an initial self-consistent field (SCF) calculation with a full configuration interaction computation within a subset of the molecular orbitals that defines the active space. The CASSCF procedure is iterative, and each cycle of the CASSCF orbital optimization requires a partial integral transformation from the atomic orbital basis to the molecular orbital basis. When large active spaces are used, a substantial CI eigenvalue problem must also be solved in each cycle. These factors make the CASSCF procedure relatively demanding computationally. Another shortcoming of the CASSCF method is the intruder state problem (especially for large CASs),12 which significantly limits its wide applicability by, in certain circumstances, introducing irregular behavior that often renders the method unusable for a specific study. Thus, it is worthwhile to search for conceptually simple, well-defined, computationally effective alternatives that avoid, or at least minimize the difficulties associated with the CASSCF method, and various schemes have been suggested as a remedy.13-25 It is worth mentioning that the modern CASSCF algorithms are applied in an attempt to either alleviate or at least attenuate the difficulties that are intricately associated with the conventional CASSCF method.26 The improved virtual orbital-complete active space CI (IVOCASCI) scheme of Freed and co-workers15-17 is one outcome of the search for replacements to the CASSCF procedure. The IVO-CASCI method defines the active virtual orbitals as those obtained by diagonalizing a Fock operator constructed from the field of (N - 1) electrons, and a noniterative CASCI follows to provide the first approximation in a new generation of MR meanfield approaches. The IVO method bears similarity to approaches proposed long ago by Silverstone and Yin,13 Huzinaga and Arnau,14 Hunt and Goddard,27 and later discussed in detail by Morokuma and Iwata.28 Some work has considered using average virtual orbitals in the CI method,29 and the GVB-RP spincoupled valence bond method30 of Goddard and co-workers31 is a related quite promising alternative to generate reliable MCSCF wave functions and accurate PES for quasi-degenerate situations that are not properly represented by the GVB-PP model. Advantages of using the IVO-CASCI method15-17 rather than the CASSCF approach include the absence of iterations after the initial SCF calculation (thus reducing the computational expense drastically), the lack of convergence difficulties due to the intruder states, the need for only a single CASCI, and the fact that the IVOs are variationally optimized using a simple unitary transformation rather than being computed using an iterative procedure. Thus, the IVO-CASCI method provides a manageable cost/accuracy ratio for accurately dealing with large multiconfigurational systems with varying degrees of quasi-degeneracy and real or avoided curve crossings.15-17 Indeed, previous applications15-17,32 demonstrate that use of IVO-CASCI calculations as a leading approximation followed by the use of IVO orbitals in a MRPT treatment32 yields good correlation energies with small reference spaces as well as a balanced description for the electronic structures of molecules and potential energy surfaces of chemical reactions.

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The most serious challenge associated with CAS type methods lies in choosing a meaningful active space for describing a given chemical problem. Both CASSCF and IVO-CASCI approaches are, in principle, open-ended, in the sense that more accurate calculations can always be obtained by increasing the number of correlated orbitals, although the number of configurations grows dramatically such that only a fraction of dynamical correlation can be included. However, a more efficacious procedure involves improving the predictions from the IVO-CASCI method with MRPT33 or MCQDPT (multiconfiguration quasi-degenerate perturbation theory)34 perturbation methods to recover additional contributions from dynamic electron correlation, as evidenced by several recent applications.32 Moreover, Paldus and co-workers35 have shown that the CASCI or CASSCF corrected CCSD method can effectively account for higher than pair clusters while requiring only a small additional computational effort over that of the standard CCSD approach, and this provides an interesting case for future use of IVOs. The development of methods for the evaluation of energy gradients is necessary to facilitate the search for minima and saddle points in the potential energy surface as well as to enable the computation of geometries and vibrational frequencies, the study of chemical reactions, energy relaxation processes, and the dynamics of molecules and molecular systems. However, analytic gradient methods are unavailable for many MR methods because of the much greater complexity of the formalisms compared to the SR case.36-38 Thus, few studies exist with even numerical gradients of MR wave functions.39-51 Chaudhuri et al.39 have successfully treated equilibrium geometries and vibrational frequencies of highly correlated MR systems using the numerical gradient IVO-CASCI approach whose results are comparable to or better than those provided by CASSCF treatments with the same size of CAS. Although the numerical gradient scheme is straightforward, the implementation is computationally demanding and is only practicable for small systems. However, the satisfactory quality of the results from the numerical approaches has provided a strong impetus to extend the analytical gradient formalism for geometry optimizations of electronic states with pronounced MR character. Recently, Chaudhuri and his collaborators52 have developed and applied an analytical gradient scheme for the IVO-CASCI approach and have obtained very encouraging results for systems with varying degrees of strong quasi-degeneracy. The present paper is devoted to testing the applicability of the IVO-CASCI gradient method to accurately describe the geometry of radicaloid species, a test motivated in part because the lack of iterations in the IVO-CASCI method renders the analytical gradient calculations significantly faster than the corresponding CASSCF gradient scheme.52 In summary, the IVO-CASCI gradient approach is particularly attractive because it is (i) formally simple, (ii) applicable to both closed- and open-shell states of any spin multiplicity, (iii) welldefined throughout the PES of a chemical reaction for an appropriately described active space, (iv) computationally inexpensive, (v) rapidly convergent, (vi) multistate in nature (can compute several states at once, in contrast to state-by-state formulations), and (vii) it is built on the variational scheme, which suggests that the derivatives can be easily obtained via the Hellmann-Feynman theorem.52 In common with the CASSCF method, the IVO-CASCI wave function may be constructed for virtually any type of electronic structure, closed or open shell, ground or excited state, neutral or ion, etc. and is invariant under 3666

dx.doi.org/10.1021/jp103536w |J. Phys. Chem. A 2011, 115, 3665–3678

The Journal of Physical Chemistry A orbital transformations within each subspace, thereby simplifying optimization and enabling use of natural or localized orbitals. The present scheme thus opens a new avenue toward the accurate geometry optimization of small to large molecular systems at low computational expense. The IVO-CASCI method has been implemented within the GAMESS(US) program system,53 and the IVO-CASCI analytical gradient codes have been supplied for incorporation into the distributed version of GAMESS(US).53 This paper continues our preceding studies52 to test the ability of the IVO-CASCI analytical gradient method to provide accurate geometries of four complex molecular radicaloid species of various sizes and character, namely, (i) the diradicals trimethylenemethane (TMM) and 2,6-pyridyne and the 2,6-pyridynium cations and (ii) a triradical 1,2,3-tridehydrobenzene (TDB) with various basis sets. These benchmark studies enable illustrating the issues discussed in the beginning of this section. The four systems are also of considerable intrinsic interest as potential antitumor agents. Additional interest in molecules with one or more unpaired electrons is fueled by possible use as building blocks in plastic magnetic materials (their spin-spin interactions can promote electrical conductivity or ferromagnetism), but their special characteristics make their experimental and theoretical investigation challenging for, e.g., determining the geometries of systems that exhibit pronounced radical character.51,54-58 The results provided by the IVO-CASCI gradient method are compared with results from the CASSCF and other computationally facile alternatives as well as several levels of theory, including multireference second-order perturbation theory, density functional theory (DFT), and coupled-cluster theory. Comparisons with single-reference methods are included to illustrate multireference effects in the computation of geometries. The IVO-CASCI method is devoid of dynamical correlation and is thus not on an equal footing with dynamically correlated SR or MR CCSD methods, but the comparison is nevertheless useful to calibrate our computed values. While it is difficult to achieve the accuracy of CC approaches with mean-field methods, the close agreement of IVO-CASCI analytical gradient geometries for both the ground and the excited state of the four radicaloid systems is quite encouraging. Calculations are presented for the energy gap between the ground and excited states that provides very useful information for designing molecular magnets. Comparisons are also provided with the spin-flip CC and DFT theories of Krylov and co-workers25 to treat states of multireference character with MS < S as an spin changing excitation from the high spin MS = S component, which is well described using a single reference configuration. Detailed discussions of the IVO-CASCI method and the algorithm for the evaluation of analytic energy gradients are provided in our previous paper.52 The four test cases of diradicals and triradicals are presented in the next section followed in the last section by some general conclusions and summarizing remarks.

II. RESULTS AND DISCUSSION This section presents illustrative examples that demonstrate the applicability of the IVO-CASCI approach for determining the equilibrium geometries for a wide range of difficult quasidegenerate systems. Our aim is to investigate whether the computationally efficient IVO-CASCI approach yields geometries of comparable accuracy to those from the CASSCF method but at far reduced computational cost. All four molecules contain (nearly) degenerate orbitals, leading to zeroth order wave

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functions that require a MR description. Although the standard SR-based methods are highly successful for systems that are not quasi-degenerate, they often fail to describe radicaloid states properly because, in general, the importance of higher than pair operators (and, hence, greater cost) in SR approaches increases with the degree of quasi-degeneracy or, equivalently, with the MR character of the state. The extent of the MR character incorporated by the various theories for the radicaloid species can be conveniently and quantitatively assessed by the magnitude of — C2-N1-C6 and — C1-C2-C3 bond angles and C2-C6 and C1-C3 bond lengths, where C1/N1 designates the radical center. The C2-C6 and C1-C3 bond lengths in benzene are approximately 2.4 Å. Thus, for a hexagonal monocyclic system, the C2-C6 and C1-C3 bond lengths are expected to lie between 2.0-2.4 Å. We also report the adiabatic energy gap between the low lying singlet and triplet states, which provides an essential link between the electronic structures and the reactivity of the systems, such as their DNA damage potential. Because of the scarcity of reliable experimental data for these four transient/short-lived species, the quality of the IVO-CASCI geometries must be assessed by comparison with the best available state-of-the-art many-body methods. The far less costly analytical gradient IVO-CASCI geometries are as good as or somewhat better than the analytical gradient CASSCF geometries. Because one or two molecular examples with very low dimensional CASs are insufficient to provide an adequate test, we consider four chemically interesting and theoretically challenging systems using different basis sets and CASs with several dimensions since the most obvious conceptual challenge for CAS-based methods is to choose a meaningful active space for describing a given chemical problem. All the IVO-CASCI gradient calculations have been performed by interfacing our code with the GAMESS(US)53 ab initio package. The calculations employ the correlation consistent polarized basis sets cc-pVXZ (X = D, T) of Woon and Dunning59 and the 6-311G*60 basis (obtained from the EMSL database61) for optimizing the geometry and for illustrating the dependence of the computed geometries on the basis sets. Core electrons are not correlated since inclusion of their contributions has only a marginal effect. A. Diradicals. The first application of the IVO-CASCI approach considered here is a prototypical diradical [which Salem defines as a molecule with two unpaired electrons occupying two quasi-degenerate orbitals62] whose predicted electronic structure differs qualitatively between various theoretical studies because of the complexity emerging from the presence of closely lying electronic states with different bonding patterns, thus providing a good testing ground for MR methods. The theoretical investigation of radicaloid species are also motivated by their potential role as intermediates in DNA cleavage reactions. 1. Trimethylenemethane (TMM). The trimethylenemethane (TMM) diradical is a nonalternate conjugated hydrocarbon, even though it is a non-Kekule system because the molecule has two or more formal radical centers. Synthesis and observation of this reactive molecule generally proceeds using matrix isolation techniques. The ground state of TMM is a triplet (3A02, D3h symmetry) as specified by Hund’s rule since four π electrons are delocalized over the four π -type molecular orbitals, two (e0 ) of which are exactly degenerate in D3h symmetry (Figure 1). This description is confirmed by both EPR and IR experiments.63,64 Figure 1 also depicts the geometry and labeling of atoms of TMM system. Previous theoretical studies65-69 also corroborate the triplet ground state character of TMM. Although the theoretical 3667


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Figure 1. Molecular structure of trimethylenemethane (TMM) and labeling of atoms. The π-electronic configuration of TMM (has three adjacent sp2-hybridized radical centers) of the two lowest states. The 3B2 state carries the label 3A02 in D3h symmetry.

description of the MS = 0 component of the triplet state is difficult due to inherent MR character, we consider this MS = 0 component for the 3A02 state to demonstrate the capabilities of the methods. The next lowest lying singlet state is the planar 1A1 state with C2v symmetry. All four π-orbitals are close in energy, and hence a high level of correlation is required for describing this system.65,66 Prior calculations establish that an accurate geometry can be obtained with suitable MR methods. Hence, by comparing our results with the previous state-of-the-art calculations, we can gauge the applicability of the IVO-CASCI method to biradicals. The IVO-CASCI geometry optimizations are carried out with 6-311G*, cc-pVDZ, and cc-pVTZ basis sets to assess the sensitivity of the computed geometrical parameters to the basis set. The active space used in the calculations comprises four π and two σ bonding electrons. Table 1 compares the geometries for the 3A02 and 1A1 states from the IVO-CASCI approach with other rigorous ab initio many-body methods, such as MCSCF and DFT,65 SF-DFT [spin-flip-DFT],66 SF-based,66 and RMRCCSD(T) [reduced multireference-CCSD(T)]68 methods. Table 1 exhibits a very weak dependence of the geometry on the basis set for the IVO-CASCI method, and for purposes of the comparison, a similar weak dependence is assumed for the other methods. Table 1 suggests that the structure of 1A1 singlet excited state can be characterized by one short and two long C-C bonds. The closed-shell singlet planar C2v geometry is stabilized by a shortening of the C-C (r1) bond in agreement with previous theoretical studies.65,68 The ground state geometry is

closest to that of the RMR-CCSD(T) (reduced multireference coupled-cluster method with singles, doubles and triples)68 study, with deviations in C-C bond lengths ∼0.01-0.02 Å and in C-H bond lengths by perhaps ∼0.01 Å. The MCSCF(10,10) treatment also agrees with the CC geometry except for a larger ∼0.03 Å deviation for the CdC bond length, while the SF calculation has the C-C bond length off by ∼0.04 Å. The IVOCASCI geometry for the 3A02 state closely mimics that of the SFDFT/6-31G* one.66 Our computed CdC bond length for the lowest triplet state is very close to the corresponding value for ethylene in its ground state. The IVO-CASCI computed C-C bond length of the 3A02 state with different basis sets is slightly shorter by ∼0.01 Å than the RMR-CCSD(T) length and betters those from the MCSCF(10,10), BP86 (DFT calculations with Becke’s gradient corrected expression for the exchange energy and Perdew’s gradient corrected expression for the correlation energy),65 and SF approaches whose deviations are ∼0.02, 0.02, and 0.03 Å, respectively. The IVO-CASCI C-H bond lengths are perhaps systematically short by ∼0.01 Å with the largest cc-pVTZ basis, but the orbital optimization by the MCSCF study obtains the same C-H bond lengths as the IVO-CASCI with the same basis (cc-pVDZ) and size of reference space. We have also optimized the geometry of TMM in both the states by CASSCF(6,6) gradient approach. The CASSCF results with CAS(6,6) are reported in Table 1 along with those with the IVO-CASCI(6,6)/ cc-pVDZ results (provided within the parentheses in the same column). We note that the IVO-CASCI(6,6)/cc-pVDZ gradient 3668



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Table 1. Selected Structural Parameters of the 3A02 and 1A1 States of Trimethelynemethane (TMM) Calculated Using the IVOCASCI Method with 6-311G*, cc-pVDZ, and cc-pVTZ Basis Setsa

state 3 0 A2

parameter

A1

IVO-

IVO-

RMR-

SF-

CASCI-

CASCI-

MCSCF(10,10)c

BP86c

CCSD(T)d

DFT

6-311G*

cc-pVDZb

cc-pVTZ

cc-pVDZ

cc-pVDZ

6-31G*

6-31G*e

R(C-C) (r1)

1.405

1.404 (1.405)

1.400

1.438

1.424

1.419 (1.415)

1.4021

R(C-C) (r2)

1.410

1.409 (1.408)

1.404

1.438

1.424

1.419

1.4021

R(C-H) (r3)

1.070

1.081 (1.088)

1.071

1.081

1.100

1.081

1.0755

R(C-H) (r4)

1.070

1.081 (1.080)

1.071

1.081

1.100

1.081

1.0755

1.100

1.081

R(C-H) (r5)

1

IVOCASCI-

1.070

1.081 (1.080)

1.071

1.081

1.0755

— C-C-C(a1)

120.1

120.1 (120.1)

120.3

120.0

120.0

120.0

120.00

— C-C-H(a2)

121.2

121.1 (120.1)

121.1

120.9

121.0

120.9

121.16

R(C-C) (r1) R(C-C) (r2)

1.334 1.477

1.336 (1.335) 1.476 (1.476)

1.329 1.470

1.370 1.496

1.381 1.450

1.349 (1.345) 1.486 (1.492)

1.3384 1.4526

R(C-H) (r3)

1.071

1.082 (1.080)

1.073

1.082

1.104

1.082

1.0773

R(C-H) (r4)

1.070

1.080 (1.080)

1.071

1.080

1.097

1.080

1.0749

1.098

1.080

R(C-H) (r5)

1.070

1.080 (1.080)

1.070

1.080

1.0744

— C-C-C(a1)

117.7

118.4 (119.0)

117.8

121.1

114.1

121.1

120.90

— C-C-H(a2)

121.7

121.4 (121.4)

121.4

121.2

121.4

121.2

121.54

a

Bond lengths and bond angles are given in angstroms (Å) and degrees, respectively. IVO-CASCI geometry optimization is carried out with a CAS(6,6) constructed from 8a1, 1b1, 2b1, 5b2, 6b2, and 1a2 orbitals, where the 1a2, 1b1, and 5b2 orbitals are occupied at the SCF level. b CASSCF(6,6) results obtained with the cc-pVDZ basis set are given within parentheses in the column containing the IVO-CASCI/cc-PVDZ results. c MCSCF(10,10) and BP86.65 d RMR-CCSD(T) (results obtained with the cc-pVTZ basis set are given in parentheses).68 e SF-DFT.66

scheme preserves an accuracy comparable with that of the CASSCF(6,6)/cc-pVDZ method at the expense of a low computational cost, thus bringing forth the benefit of the IVO-CASCI over the costly CASSCF method. To further illustrate the trustworthiness of the present method, we also compare the S-T energy separation (ΔES-T) between the 3A02 and 1A1 states with the large number of previous studies pursued because these energy splittings are very useful for investigating the chemical properties of the diradicals. Finding a balanced description of both states is not straightforward because earlier calculations indicate that the singlet state requires a MR description whereas the high spin component of the triplet state is essentially single-reference in nature. Despite the tremendous methodological developments, the need for an MR description implies that models capable of providing reliable multiplet splitting in non-Kekule systems, such as TMM, still remain a challenge despite numerous attempts. The relative importance of the selection of the active space and basis set in MR methods can be assessed by comparing the various theoretical estimated S-T gap against experiment. The MS = 0 solutions for the singlet and triplet states can be treated within the same MR framework and thus should provide a more balanced and accurate description of the S-T splitting. The S-T gaps estimated from the IVO-CASCI method using different basis sets are compared in Table 2 with those of established MR methods, selected SR methods, and experiment. The ΔES-T determined from IVO-CASCI computations deviates by 6 kcal/mol from experiment (when corrected for the zero point energy) despite the omission of dynamical correlation, while the closest agreement is achieved with the far more costly 4R-BWCCSD approach that contains extensive dynamical correlation. Table 2 exhibits a strong dependence of the estimated ΔES-T on the method employed. Li and Paldus68 observe that the CCSD method, as expected, yields a very unsatisfactory ΔES-T for TMM (indeed, the worst result). The completely renormalized coupled cluster method, CR-CC(2,3), likewise

fares poorly despite using the complete form of the triply excited CCSD(T) moments and renormalizing the correction using the left ground eigenstate of the similarity-transformed CCSD Hamiltonian,70 and the performance of the RMR-CCSD71 is likewise not encouraging even though both CR-CC(2,3) and RMRCCSD methods represent efficient alternatives to overcome the failure of the CCSD/CCSD(T) approach in quasi-degenerate situations. These findings thereby stress the clear benefit of using MR approaches in view of the fact that the singlet state possess a reasonable degree of MR character. The IVO-CASCI computed ΔES-T is comparable to the calculations from the BW-MRCCSD (Brillouin Wigner multireference coupled cluster), MCQDPT (multiconfiguration quasi-degenerate perturbation theory), CASPT2 (complete active space with second-order perturbation theory), and highly correlated single reference EOM-SF-CC(2,3) (equation of motion spin flip coupled cluster method) as well as the SS-EOM-CEPA (state selective equation of motion coupled electron pair approximation) and SS-EOMCCSD (state selective equation of motion coupled cluster singles and doubles) methods. The IVO-CASCI method underestimates the S-T gap, whereas the correlated standard many-body methods provide an overestimate. In general, with the same basis set, the IVO-CASCI calculations are an order of magnitude faster than the MCQDPT, CASPT2, BWCCSD, SS-EOM-CEPA, SS-EOMCCSD, and EOM-SF-CC(2,3) calculations. We now turn attention to the CASSCF/cc-pVDZ and IVO-CASCI(6,6)/cc-pVDZ ΔES-T results with CAS(6,6), given in Table 2 to illustrate the efficacy of the IVO-CASCI method. ΔES-T values obtained by these two methods (zero-point energy correction not being included for both) are 0.49 and 0.50 kcal/mol, respectively, which clearly demonstrates a good agreement among themselves. The IVOCASCI and CASSCF ΔES-T results show values that corroborate well with the standard database values. Although, the IVO-CASCI and CASSCF value for the S-T gap is pretty good, dynamical correlation should be appended for a final quantitative comparison to experiment (0.787 kcal/mol). 3669



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Table 2. Singlet-Triplet Adiabatic Energy Separation [ΔE E(1A1) - E(3A02)] of TMM As Obtained with Various Basis Sets and Methods reference

method

present work IVO-CASCI(6,6)/cc-pVDZ

ref 66

ref 65

ref 67

ref 68

ref 69

ΔE (eV) 0.50

IVO-CASCI(6,6)/cc-pVTZ

0.52

CASSCF(6,6)/cc-pVDZ

0.49

SF-DFT(50,50)/6-31G*

0.866

SF-CIS/DZP SF-CIS(D)/DZP

0.882 0.885

SF-OD/DZP

0.936

MCSCF(4,4)/DZP

0.843

MCQDPT2(4,4)/DZP

0.863

MCSCF(10,10)/DZP

0.834

MCQDPT2(10,10)/DZP

0.824

MCSCF(4,4)/cc-pVTZ

0.841

MCQDPT2(4,4)/cc-pVTZ MCSCF(10,10)/cc-pVTZ

0.862 0.832

MCQDPT2(10,10)/cc-pVTZ

0.828

MCSCF(2,2)/cc-pVDZ

0.45

MCSCF(4,4)/cc-pVDZ

0.87

MCSCF(10,10)/cc-pVDZ

0.83

MCSCF(10,10)/cc-pVTZ

0.82

CASPT2N(2,2)/cc-pVDZ

0.99

CASPT2N(4,4)/cc-pVDZ CASPT2N(10,10)/cc-pVDZ

0.87 0.83

CASPT2N(10,10)/cc-pVTZ

0.83

4R-BWCCSD/cc-pVDZ

0.800

4R-BWCCSD it/cc-pVDZ

0.791

4R-BWCCSD/cc-pVTZ

0.779

4R-BWCCSD it/cc-pVTZ

0.771

CR-CC(2,3)/cc-pVTZ

1.38

RMR-CCSD(T) SU CCSD

1.03 0.86

EOM-SF-CCSD/cc-pVTZ

0.93

EOM-SF-CC(2,3)/cc-pVTZ

0.79

SS-EOM-CCSD[þ2]/cc-pvDZ (MCSCF)

0.859

SS-EOM-CCSD[þ2]/cc-pvDZ (MCSCF)

0.857

SS-EOM-CCSD[þ2]/cc-pvDZ (Brueckner) 0.853 SS-EOM-CCSD[þ2]/cc-pvTZ (Brueckner) 0.850 SS-EOM-CEPA[þ2]/cc-pvDZ (MCSCF) SS-EOM-CEPA[þ2]/cc-pvTZ (MCSCF)

0.844 0.834

SS-EOM-CEPA[þ2]/cc-pvDZ (MCSCF)

0.858

SS-EOM-CEPA[þ2]/cc-pvDZ (Brueckner) 0.832 SS-EOM-CEPA[þ2]/cc-pvTZ (Brueckner) 0.826 ref 64

experiment

0.6996 ( 0.006

experiment - ΔZPE

0.787

Table 2 exhibits a quite weak basis set dependence of the IVOCASCI ΔES-T for TMM in accord with the observation of Li and Paldus68 and with Cramer and Smith65 who find a negligible change of ΔES-T in CASPT2 and DFT studies upon increasing the basis set from cc-pVDZ to cc-pVTZ. 2. 2,6-Pyridyne (C5NH3). The determination of the equilibrium geometry for the ground state of the 2,6-pyridyne biradical (see Figure 2) is a challenging task for ab initio methods because this system is well accepted to possess a singlet ground state

Figure 2. Geometry of 2,6-pyridyne (the H in NH is absent) and the 2,6-pyridynium cation to present the atom-numbering scheme.

rather than a triplet as predicted by Hund’s rule. Additionally, the presence of two quasi-degenerate frontier orbitals makes this molecule another interesting test case for understanding of the applicability of MR methods. Although naturally occurring enediynes are toxic, nitrogen-containing didehydropyridines (pyridynes or azabenzynes) are biologically inactive in the neutral medium of normal cells. Formally, it can be synthesized by replacing one CH group in benzyne with an NH group. Introducing a nitrogen atom into the ring perturbs the structures and energies of the system and complicates its theoretical treatment. Recent theoretical studies on this system provide a wealth of new physical data.51,54-56,58 A comprehensive analysis by Prochnow et al.51 using a state specific multireference coupled cluster method including single and double excitations (termed as SSMRCCSD or Mk-MRCCSD)72 provide bond lengths and angles ( — ) for pyridyne that can be used as reference for comparison, so the same basis is employed as in the calculations of Prochnow et al.51 The IVO-CASCI calculations consider the reference spaces CAS(2,2), CAS(8,8), and CAS(12,12) to delineate the effect of the reference space size on the equilibrium geometrical parameters. CAS(8,8) is constructed from (1-4) b1, (1-2) a2, 11a1, and 7b2 orbitals where 11a1, 1b1, 2b1, and 1a2 are occupied orbitals at the SCF level. CAS(12,12) is constructed by adding 10a1 and 6b2 occupied (SCF) orbitals and (12-13) a1 unoccupied (in SCF) orbitals to the CAS(8,8). Tables 3 and 4 compare the IVO-CASCI calculations using different basis sets with various other methods. These tables exhibit several interesting features: The biradical can exist in either an open-shell monocyclic structure or closed-shell bicyclic one as best corroborated by the values of the — C1-N-C6 and the C2-C6 bond length. All the HF-SCF, DFT(BPW91), QCISD, and CCSD [see refs 51 and 58] methods predict an incorrect bicyclic structure (with a three-membered ring formed by the two radical centers) for the singlet ground state of 2,6pyridyne, results clearly implying that single determinantal methods or correlated methods based on a HF single determinantal wave function cannot describe this molecule appropriately. Consequently, it is not surprising that the standard CCSD method yields the wrong geometry. The recent Mk-MRCCSD/cc-pVTZ calculations51 yield mono- and bicyclic forms with the open-shell monocyclic structure [with C2-C6 = 2.014 Å and — C1-N-C6 = 98.01] lower in energy than the closedshell bicyclic one. A previous CCSD(T)/cc-pVTZ51 study finds 3670



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Table 3. Selected Structural Parameters for the Ground State of 2,6-Pyridyne (C5NH3)a parameter

CCSDb

CCSD(T)b

Mk-MRCCSDb

BPW 91c

CASSC(8,8)b

R-MRCCST(T)d

B3LYPd

IVO-CASCI

N-C2

1.3473

1.3596

1.3489

1.356

1.332

1.360

1.337

1.3215 (1.3275)

C2-C3

1.3892

1.3963

1.3896

1.390

1.388

1.395

1.380

1.3777 (1.3726)

C3-C4

1.4056

1.4098

1.4046

1.397

1.411

C2-C6

1.9140

1.818

2.014

C3-H

1.0916

1.0940

1.0920

C4-H

1.0979

1.0995

1.0967

— C1-N-C6

90.52

98.15

98.70

— N-C2-C3 — C2-C3-C4

143.20 115.85

137.09 117.23

136.91 116.97

— C3-C4-C5

111.40

113.22

113.55

— C2-C3-H

119.91

119.48

119.72

1.561

2.131

1.3892 (1.3896) 2.1225 (2.0562) 1.0790 (1.0798) 1.0829 (1.0810)

100

69

106.84 (101.51) 131.67 (135.21) 116.49 (116.64)

118

111

116.84 (114.80) 120.50 (119.98)

a

All calculations use the cc-pVDZ (99 CGTOs) basis set. Bond lengths and bond angles are given in angstroms (Å) and degrees, respectively. IVOCASCI and Mk-MRCCSD geometry optimizations employ a CAS(2,2) constructed from the a1 (HOMO) and b1 (LUMO) orbitals. The entries in parentheses correspond to the IVO-MRMP calculations. b Reference 51. c Reference 54. d Reference 58.

Table 4. Selected Structural Parameters for the Ground State of 2,6-Pyridyne (C5NH3)a parameter

CCSDb

CCSD(T)b

Mk- MRCCSDb

IVO-CASCI

N-C2 C2-C3

1.3313 1.3734

1.3479 1.3801

1.3361 1.3735

C3-C4

1.3936

1.3964

1.3908

C2-C6

1.818

2.014

2.017

C3-H

1.0771

1.0800

1.0777

1.0702

C4-H

1.0838

1.0855

1.0825

1.3156 1.3712

B3LYP

MP2

1.3314 1.3716

1.3466 1.3784

1.3836

1.3919

1.3960

2.1139

1.7454

2.0346

1.0800

1.0809

1.0747

1.0874

1.0864

— C1-N-C6

86.15

96.80

98.01

106.90

81.91

98.13

— N-C2-C3

146.94

138.12

137.45

131.66

150.75

136.82

— C2-C3-C4 — C3-C4-C5

114.52 110.93

117.01 112.93

116.80 113.49

116.43 116.91

112.95 110.70

118.03 112.17

— C2-C3-H

120.74

119.51

119.78

120.52

122.11

119.07

a

All calculations use the cc-pVTZ (222 CGTOs) basis set. Bond lengths and bond angles are given in angstroms (Å) and degrees, respectively. IVOCASCI geometry optimization employs a CAS(2,2) constructed from the a1 (HOMO) and b1 (LUMO) orbitals. b Reference 51.

that the open-shell diradiacl (monocyclic) structure is lower in energy in the singlet state with C2-C6 = 2.014 Å and — C1-NC6 = 96.80. The RMR-CCSD(T)58 approach also yields a monocyclic singlet state with — C1-N-C6 = 100. The IVOCASCI/cc-pVDZ(cc-pVTZ) calculation produces the C2-C6 bond length and — C1-N-C6 as 2.1225 (2.1139) Å and 106.84(106.90), respectively [see Table 3]. Table 4 displays the IVO-CASCI and the large scale Mk-MRCCSD calculations as differing by 0.03, 0.01, 0.01, 0.09, and 0.01 Å for the N-C2, C2-C3, C3-C4, C2-C6, and CH bond lengths, respectively. The three C-C bond lengths determined from the CASSCF(8,8) theory agree well with the present simpler CASSCF(2,2) [or TCSCF] results,51 while the RMR-CCSD(T) results58 incur slightly larger deviations from the Mk-MRCCSD bond lengths. The geometries from the IVO-CASCI(8,8) and IVO-CASCI(12,12) calculations in Table 5 exhibit a very minor dependence on the size of the reference space since both larger CASs agree closely with the (2,2) treatment. The IVO-CASCI(2,2) geometry is very close to that of the presumably most accurate MkMRCCSD and is even closer to the CAS(12,12) reference case. This behavior is consistent with prior findings that the IVOCASCI method often provides well converged geometries with rather small CASs, thus reemphasizing the usefulness of the method.

To illustrate the additional contributions from dynamical correlation in affecting the geometry, the entries in parentheses in Table 3 summarize the selected geometrical parameters computed using the IVO-MRPT method with a numerical gradient scheme. The use of the IVO-MRMP32 [essentially an IVOCASCI variant of Hirao’s MRPT approach33] method improves the C2-C6 bond length as compared to the IVO-CASCI approach. Essentially we observe that this bond length is only ∼0.04 Å away from the corresponding result of the Mk-MRCCSD method. The IVO-CASCI approach yields a C2-C6 bond length that deviates by ∼0.11 Å. This genuinely supports the use of the IVO-MRMP approach and clearly reveals that the inclusion of dynamical correlation in IVO-CASCI approach via the MRMP method33,32 is very important for a correct quantitative description of the electronic structure of diradicals. 3. 2,6-Pyridynium Cation (C5NH4þ). The challenging diradical, the 2,6-pyridynium cation (see Figure 2), can normally be generated by replacing one CH group in the corresponding benzynes by an NHþ group. Charged diradicals are even more difficult to treat than the neutrals because they possess singlet ground states that require MR treatments instead of the diradical triplet states that are more amenable to SR methods. The earlier computational studies of the ground state equilibrium geometry of C5NH4þ have established quite well the ubiquity of implementing 3671



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Table 5. Ground State Structural Parameters of 2,6-Pyridyne (C5NH3) and the 2,6-Pyridynium Cation (C5NH4þ) Using CAS(8,8) and CAS(12,12) Active Space Based IVO-CASCI Calculations with Various Correlation Consistent Basis Setsa cc-pVDZ parameter

CAS(8,8)

CAS(12,12)

cc-pVTZ CAS(8,8)

CAS(12,12)

C5NH3 System N-C2

1.3362

1.3467

1.3287

1.3413

C2-C3

1.3877

1.3883

1.3775

1.3809

C3-C4

1.3974

1.3965

1.3817

1.3903

C2-C6

2.1173

2.0962

2.1030

2.0903

C3-H

1.0800

1.0800

1.0710

1.0709

C4-H

1.0830

1.0833

1.0749

1.0745

— C1-N-C6 — N-C2-C3

104.80 132.63

102.21 134.12

104.63 132.62

102.38 134.03

— C2-C3-C4

117.34

117.51

117.20

117.43

— C3-C4-C5

115.26

114.52

115.73

114.71

— C2-C3-H

119.69

119.23

119.78

119.34

C5NH4þ System N1-C2

1.3324

1.3327

1.3269

1.3273

C2-C3

1.3745

1.3712

1.3675

1.3650

C3-C4

1.4024

1.4132

1.3967

1.4056

C2-C6 N1-H

2.2324 1.0075

2.2373 1.0068

2.2198 0.9998

2.2236 0.9995

C3-H

1.0799

1.0801

1.0708

1.0712

C4-H

1.0814

1.0829

1.0726

1.0735

— C2-N1-C6

113.80

114.14

113.54

113.78

— N1-C2-C3

126.64

126.86

126.85

127.11

— C2-C3-C4

117.55

116.95

117.47

116.81

— C3-C4-C5

117.81

118.22

117.83

118.37

— C2-C3-H

119.63

120.46

119.69

120.67

a

Bond lengths and bond angles are given in angstroms (Å) and degrees, respectively.

MR approaches to such situations with significant degeneracy of the frontier orbitals. Thus, this particular system serves as another good example to assess the performance of the IVO-CASCI approach. The IVO-CASCI geometry optimization employs the simplest CAS(2,2), and calculations with CAS(8,8) and CAS(12,12) investigate the performance enhancement with more reference configurations. The benchmark for the IVO-CASCI predictions is again chosen as the geometry from the expensive Mk-MRCCSD calculations by Prochnow et al.51 Tables 6 and 7 compare the IVO-CASCI optimized equilibrium bond lengths and angles for the singlet state of the pyridynium cation with results of previous SR and MR methods and portray the utility of the IVOCASCI approach. Tables 6 and 7 indicate that although the IVO-CASCI(2,2) and CASSCF(8,8) procedures yield rather similar results (within ∼0.01 Å), the former alternative is less costly and most appealing. The tables illustrate two particularly large differences between the SCF and CAS treatments. The SCF/cc-pVDZ calculation incorrectly predicts the singlet to have a bicyclic structure with the C2-C6 bond length and — C1-N-C6 of 1.4959 Å and 70.29, respectively. On the other hand, Tables 6 and 7 clearly demonstrate that the erroneous bicyclic structure opens to the

correct monocyclic one upon incorporation of nondynamical correlation. The IVO-CASCI procedure correctly maintains a monocyclic ring structure with C2-C6 = 2.2281 Å and — C1N-C6 = 113.95 for cc-pVDZ and C2-C6 = 2.2176 Å and — C1-N-C6 = 113.79 for cc-pVTZ bases, again correcting a fundamental failure of standard HF-SCF approaches. The CCSD and DFT B3LYP methods predict a monocyclic structure as the global minimum in contrast to their errors for the 2,6-pyridyne system, although the CCSD method is generally not reliable even in the presence of a small extent of quasi-degeneracy, as evident from the significant changes in geometry in moving from the CCSD f CCSD(T) approximations, where the latter results are very close to those of the Mk-MRCCSD calculation, indicating the importance of higher excitations. Recently, Prochnow et al.51 emphasize that from a purely methodological viewpoint, the CCSD and CCSD(T) does not represent the most accurate approach while high level correlated theories yield a monocyclic structure for the 2,6-pyridynium cation, the far cheaper IVOCASCI(2,2) calculation suffices. Tables 6 and 7 display the IVO-CASCI bond lengths (apart from C2-C6) for C5NH4þ as generally within ∼0.01 Å of those from the CCSD(T) and Mk-MRCCSD treatments, but generated at a fraction of the computational cost. Table 5 summarizes the CAS(8,8) and CAS(12,12) geometries that again closely parallel the IVO-CASCI(2,2) calculations and slightly improve the agreement with the Mk-MRCCSD theory. Thus, the IVOCASCI method performs reasonably well in this case despite the lack of dynamical correlation, incurring acceptably modest departures from other high level ab initio methods. To assess the contribution of the dynamical correlation to the geometrical parameters calculated here, we have applied the IVO-MRMP method [in which dynamical electron correlation is incorporated by the MRMP calculation33 due to Hirao and coworkers] implemented in GAMESS(US). Contributions from dynamical correlation are included for illustration in Table 6 using the numerical oriented IVO-MRMP(2,2)/cc-pVDZ gradient scheme. The main outcome of the IVO-MRMP approach is to improve the description of the C2-C6 bond length and leave the rest of the geometry unchanged. Even though the IVO-CASCI results show good agreement with the database values for the 2,6-pyridyne and -pyridynium cation, a consideration of dynamical correlation via the MRMP approach33,32 enhances this agreement significantly. Finally, our group is currently exploring the applicability of the IVO-MRMP gradient method to various challenging types of molecular systems, and preliminary results indicate that their equilibrium geometries compare favorably to the those from the state-of-the-art Mk-MRCCSD approach.51 B. Triradical: 1,2,3-Tridehydrobenzene (TDB). Finally, the IVO-CASCI gradient method is employed to investigate the two lowest doublet states (2A1 and 2B2) of the 1,2,3-tridehydrobenzene (1,2,3-C6 H3, termed as a TDB; see Figure 3) triradical species and to study their relative ordering. Triradicals73 pose a significant challenge for any ab initio method due to orbital degeneracies and the resultant complex electronic structure that contains several close lying low spin (doublet) and high spin (quartet) states and that consequently requires an MR treatment. Investigations of triradicals are less numerous than for biradicals despite their importance in organic and bio-organic chemistry. However, recently the electronic structure of TDB has been investigated by a variety of ab initio methods.74-77 Observations by Sander and co-workers77 establish that the two lowest doublets 3672



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Table 6. Selected Structural Parameters for the Ground State of 2,6-Pyridynium Cation (C5NH4þ)a parameter

SCFb

TCSCFb

CCSDb

CCSD(T)b

Mk-MRCCSDb

B3LYP

BPW91c

CASCF(8,8)c

IVO-CASCI

N1-C2

1.2993

1.3297

1.3370

1.3502

1.3418

1.3291

1.347

1.332

C2-C3

1.3592

1.3587

1.3738

1.3814

1.3734

1.3602

1.371

1.375

C3-C4

1.4290

1.3979

1.4127

1.4174

1.4127

1.4129

C2-C6

1.4959

2.2299

2.1633

2.2387

2.1983

2.1015

N1-H

1.0170

1.0060

1.0215

1.0236

1.0214

1.0242

1.0063 (1.0039)

C3-H

1.0763

1.0796

1.0929

1.0949

1.0929

1.0933

1.0794 (1.0797)

C4-H

1.0844

1.0820

1.0958

1.0973

1.0949

1.0955

1.3291 (1.3258) 1.3596 (1.3569) 1.3987 (1.3989)

2.242

2.235

2.2291 (2.1949)

1.0826 (1.0779)

— C2-N1-C6 — N1-C2-C3

70.29 164.11

114.09 126.75

107.99 130.81

112.12 127.65

111.26 128.43

104.47 133.59

113.98 (114.74) 126.83 (128.30)

— C2-C3-C4

103.90

116.67

117.30

117.72

117.43

116.88

116.67 (117.06)

— C3-C4-C5

113.69

119.07

115.79

117.14

117.02

114.59

119.01 (117.53)

— C2-C3-H

128.65

120.47

119.78

119.84

120.01

120.54

120.46 (120.01)

a

All calculations use the cc-pVDZ (104 CGTOs) basis set. Bond lengths and bond angles are given in angstroms (Å) and degrees, respectively. IVOCASCI and Mk-MRCCSD geometry optimizations employ a CAS(2,2) constructed from the a1 (HOMO) and b1 (LUMO) orbitals. Entries within parentheses are from IVO-MRMP calculations. b Reference 51. c Reference 54.

Table 7. Selected Structural Parameters for the Ground State of 2,6-Pyridynium Cation (C5NH4þ)a parameter

CCSDb

CCSD(T)b

Mk-MRCCSDb

IVO-CASCI

B3LYP

MP2

N-C2

1.3235

1.3375

1.3292

1.3239

1.3226

1.3322

C2-C3

1.3570

1.3647

1.3568

1.3524

1.3510

1.3679

C3-C4 C2-C6

1.3994 2.122

1.4041 2.204

1.3992 2.183

1.3933 2.2178

1.4068 2.0850

1.4005 2.2065

N-H

1.0105

1.0129

1.0102

0.9988

1.0165

1.0166

C3-H

1.0788

1.0811

1.0788

1.0703

1.0828

1.0829

C4-H

1.0818

1.0834

1.0807

1.0738

1.0852

1.0835

— C1-N-C6

106.54

110.93

111.43

113.78

104.04

111.83

— N-C2-C3

131.94

128.51

129.05

127.01

133.98

127.51

— C2-C3-C4

117.02

117.59

117.27

116.58

116.72

118.79

— C3-C4-C5 — C2-C3-H

115.54 119.95

116.87 119.90

116.94 120.12

119.05 120.53

114.57 122.72

115.58 119.45

a

All calculations use the cc-pVTZ (236 CGTOs) basis set. Bond lengths and bond angles are given in angstroms (Å) and degrees, respectively. The IVOCASCI geometry optimization employs a CAS(2,2) constructed from the a1 (HOMO) and b1 (LUMO) orbitals. b Reference 51.

of TDB are adiabatically quasi-degenerate and well separated vertically. The significant difference between the vertical and adiabatic energy gaps implies that the bonding patterns of TDB in these two doublet states are quite different. Previous studies by various computational models75,76 have demonstrated that all three isomers of TDB prefer low spin doublet ground states rather than quartet, implying that the aufbau principle prevails over Hund’s rule in determining the ground state electronic configuration. Formally, TDB can be derived from benzene by removing three hydrogen atoms. The interaction among the three unpaired electrons enhances bonding and leads to the fact that the triradical is tighter than the parent neutral benzene molecule (the stabilizing interactions range from 0.5 to 32 kcal/mol).75,76 The three unpaired electrons in TDB are distributed over three nearly degenerate orbitals [10a1 (bonding), 7b2 (nonbonding), and 11a1 (antibonding) under C2v symmetry], which leads to unusual bonding patterns and features due to even more extensive electronic degeneracies than diradicals [see ref 75]. Because of a bonding interaction between the unpaired electrons, the 2A1 state has the leading electronic configuration (10a1)2(11a1)1,

while (10a1)2(7b2)1 is the leading electronic configuration for the 2B2 state. Therefore, the two lowest doublet electronic states of TDB are closed-shell in nature. Following earlier investigations,74-77 it is found that the 2A1 state exhibits a C1-C3 bonding interaction at a small distance (around 1.69 Å), whereas the 2B2 state has a larger separation (∼2.37 Å) [see Figure 4]. As shown in Figure 4, the transition state 2A0 represents an intermediate between these two doublet situations. Krylov’s comprehensive calculations show that (a) the 2A1 state is energetically less stable than the 2B2 state (the global minimum) and (b) the 2 A1 state is monocyclic whereas the 2B2 state is bicyclic. The same observations emerge from our present IVO-CASCI gradient studies using various basis sets. TDB has been isolated and characterized by Venkataramani et al.77 using vibrational spectroscopy in cryogenic neon matrices at 3 K. By comparing various calculated results and three experimentally measured vibrational transitions, they conclude that a stable bicyclic 2A1 state of TDB is formed under the conditions of matrix isolation. The selected structural parameters of 1,2,3-tridehydrobenzene from the IVO-CASCI method using a CAS(8,7) with the ccpVDZ and cc-pVTZ basis sets are summarized in Table 8 for the 3673


The Journal of Physical Chemistry A two lowest doublet states. Our calculations are compared with those by Krylov and co-workers75,76 using a wide range of expensive to naive computational approaches (both wave function and density function varieties), including the R-CCSD(T), ROHFCCSD(T), SF-CCSD, SF-DFT, BYLP, and B3LYP methods with a cc-pVTZ basis. The comparisons again clearly indicate the usefulness of the IVO-CASCI approach. Of particular interest is the distance between the meta radical centers in the 2A1 and 2B2 states, which Krylov and co-workers76 compute with standard ROHF-CCSD(T)/cc-pVTZ and R-CCSD(T)/cc-pVTZ methods, and the — C1-C2-C3 for the 2A1 and 2B2 states that they obtain from ROHF-CCSD(T)/cc-pVTZ and R-CCSD(T)/cc-pVTZ calculations vary within the ranges 76.9-77.3 and 130.9-131.1, respectively. Table 8 clearly exhibits quite different equilibrium structures as emerging from the IVO-CASCI calculations for the 2A1 and 2 B2 doublet states due to the different bonding patterns and

Figure 3. Molecular structure of 1,2,3-tridehydrobenzene (TDB) triradical and labeling of atoms.

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differences in the separation between the C1 and C3 atoms. The IVO-CASCI calculations concur with the experimentally supported view77 of a bicyclic 2A1 state with a relatively short C1-C3 bond of 1.707-1.686 Å, whereas the 2B2 state has a monocyclic structure with a C1-C3 distance of 2.337-2.330 Å characteristic of a benzene derivative. The IVO-CASCI values of — C1-C-C3 for both the 2A1 (78.7-78.2) and 2B2 (131.7-130.8) states also support the bicyclic and monocyclic structures, respectively. As already mentioned, when the separation is small, the a1 orbital is bonding between these two carbon centers as found in the 2A1 state. At larger separations between C1 and C3, the b2 orbital, with antibonding character between C1 and C3, becomes lower in energy than the a1 orbital and determines the structure of the 2B2 state [see Figure 4]. The adiabatic near-degeneracy between the 2 A1 and 2B2 states emerges from the competition between — C1-C2-C3 and C1-C3 interactions. The geometry of TDB provided by IVO-CASCI calculations also indicates the extent of partial bond formation between the radical centers. The IVO-CASCI/cc-pVDZ and IVO-CASCI/cc-pVTZ geometries for TDB in Table 8 are almost identical, in accord with the insensitivity to basis found for the biradicals. Considering the R-CCSD(T)/cc-pVTZ calculations as the benchmark, the far less expensive IVO-CASCI/cc-pVTZ bond lengths mostly agree (generally an underestimate) to ∼0.01 Å, with ∼0.02 Å deviations for the 2A1 state C4-C6 and C1-C2 and ∼0.04 Å for the 2 B2 state C1-C3 bonds. Thus, the IVO-CASCI predicted geometries display the same trends appearing in the studies of Krylov and co-workers,76 namely, the C5-C6 bond length in the 2 A1 state is shorter than that of the 2B2 state, and the C4-C6 bond length in the 2A1 state is slightly longer than the C4-C6 bond length in the 2B2 state. Table 8 also displays the close agreement of the IVO-CASCI geometry for both the states with the those obtained from BLYP/cc-pVTZ and B3LYP/cc-pVTZ methods. The adiabatic energy gap (ΔEadb) between the 2A1 and 2B2 states has been used as a stringent test for the validity of newly developed methods because ΔEadb is very small due to geometrical relaxation, the near degeneracy of the two states, and the strongly differing radical character of the two doublets. The precise value of the adiabatic gap is also an important parameter for analyzing studies of the reactivity of this TDB. Table 9 compares the calculated ΔEadb with recent CCSD(T), MR-CI (multireference configuration interaction),78 CASSCFRS2/RS3 [RS2 and RS3 stands for second and third order

Figure 4. Molecular orbitals of the two lowest electronic states of TDB as a function of separation between the C1 and C3 radical centers. Energy as a function of C1-C3 length. 3674



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Table 8. Comparison of Selected Geometrical Parameters for the 2A1 and 2B2 States of 1,2,3-Trihydrobenzene As Determined from the IVO-CASCI Method with cc-pVDZ, and cc-pVTZ Basis Setsa state 2

A1

2

B2

IVO-CASCI-

IVO-CASCI-

R-CCSD(T)b-

B3LYPb-

SF-CCSDb-

SF-DFTb-

cc-pVDZ

cc-pVTZ

cc-pVTZ

cc-pVTZ

6-31G(d)

6-311G(d)

R(C5-C6) R(C1-C6)

1.408 1.380

1.402 1.373

1.410 1.380

1.404 1.371

1.404 1.376

1.395 1.368

R(C1-C2)

1.346

1.336

1.354

1.342

1.352

1.332

R(C1-C3)

1.707

1.686

1.692

1.692

1.642

1.629

R(C5-H8)

1.082

1.073

1.085

1.084

1.078

1.089

R(C4-H9)

1.078

1.069

1.079

1.078

1.083

parameter

1.072

— C4-C5-C6

113.1

113.0

113.1

113.1

113.0

112.7

— C3-C4-C5

110.0

109.7

109.6

109.8

108.7

109.3

— C2-C3-C4 — C1-C2-C3

154.1 78.7

154.6 78.2

155.2 77.3

154.6 78.2

157.0 74.8

157.0 75.4

R(C5-C6)

1.414

1.409

1.412

1.406

1.408

1.398

R(C1-C6)

1.404

1.397

1.402

1.394

1.395

1.385

R(C1-C2)

1.281

1.271

1.300

1.287

1.295

1.281

R(C1-C3)

2.337

2.310

2.367

2.342

2.360

2.326

R(C5-H8)

1.084

1.075

1.084

1.083

1.089

1.077

R(C4-H9)

1.080

1.071

1.082

1.081

1.075

1.086

— C4-C5-C6 — C3-C4-C5

123.2 115.3

123.1 115.0

123.1 116.0

122.9 115.9

123.0 116.1

122.7 115.9

— C2-C3-C4

117.2

118.0

116.8

117.2

117.8

117.4

— C1-C2-C3

131.7

130.8

131.1

130.9

129.1

130.3

a

Bond lengths and bond angles are given in angstroms (Å) and degrees, respectively. IVO-CASCI geometry optimization uses a CAS(7,8) constructed from the 1b1, 2b1, 1a2, 2a2, 3b2, 4b2, 11a1, and 7b2 active orbitals, where the 1b1, 2b1, 1a2, and 11a1 orbitals are occupied at the SCF level. b Reference 76.

Rayleigh-Schr€odinger perturbation theory respectively],79 AQCC (multireference average-quadratic coupled cluster)80 and ACPF (multireference averaged coupled pair functional)81 studies by Krylov and co-workers.76 We begin analysis of Table 9 by assessing the suitability of MR approaches for correctly predicting the relative order of the 2B2 and 2A1 states and their adiabatic energy gap. Although the MRCISDþQ(Davidson correction) and AQCC methods lead to a nearly vanishing ΔEadb, the ACPF calculation correctly places the 2 A1 state above the 2B2 (a moderate preference of 0.03 eV for the 2 A1 state). Krylov and co-workers76 observe that both CASSCF(9,9) and MR-CISD treatments predict the 2B2 state to be more stable than the 2A1 state. The EOM-SF(2,2) approach also favors the 2B2 over the 2A1 state.76 However, Krylov and co-workers76 have demonstrated that the inclusion of triples in the EOMSF(2,2) scheme reverses the state ordering. While the CASSCF approach places the 2B2 state lower than the 2A1 state, this error is corrected with the CAS-RS2 and CAS-RS3 methods.76 Thus, the proper inclusion of dynamical correlation significantly affects the predicted relative order of these two doublet states. The adiabatic energy gap estimated by R-CCSD, R-CCSD(T), and ROHF-CC calculations with various basis sets [see ref 76] favor the 2A1 state. Two doublet isomers are separated by a barrier, and ΔEadb (including zero point energy) is about 0.03-0.09 eV. Table 9 indicates that the IVO-CASCI method yields this adiabatic gap as the large overestimate of 0.50 eV. Nevertheless, the IVO-CASCI calculation correctly identifies the bicyclic 2A1 isomer as the ground state, while the CASSCF computation erroneously predicts the monocyclic 2B2 state as the global minimum on the ground state PES. In summary, the IVO-CASCI method provides the correct energy ordering of the 2B2 and 2A1 doublet states of TDB, but an accurate treatment of the gap requires the

Table 9. Adiabatic Energy Separation [ΔEadb (eV) E(2A1) - E(2B2)] of 1,2,3-Tridehydrobenzene As Obtained with Various Basis Sets and Methods reference

method

cc-pVDZ

cc-pVTZ

cc-pVQZ

present work

IVO-CASCI

-0.53

-0.50

ref 76a

CASSCFb

0.66

0.68

0.68

CASSCFb

0.69

0.71

0.71

MR-CISDb

0.18

0.20

0.20

MR-CISDþQb

0.00

0.01

0.01

AQCCb

0.00

0.00

0.00

R-CCSDb

-0.22

-0.23

-0.22

R-CCSDc R-CCSD(T)b

-0.22 -0.10

-0.23 -0.10

-0.23 -0.09

R-CCSD(T)c

-0.09

-0.10

-0.09

CAS-RS 2b

-0.02

-0.03

-0.04

CAS-RS 2c

-0.01

-0.03

-0.04

CAS-RS 3b

-0.03

-0.04

-0.05

CAS-RS 3c

-0.02

-0.04

-0.05

ACPF UB3PW91

-0.03 0.24

-0.03 0.19

-0.03

ref 77

UBPW91

-0.24

-0.18

UB3LYP

-0.18

-0.08

UBLYP

-0.12

-0.04

a

All multireference calculations employ a CASSCF(9,9) reference space. b B3LYP/cc-pVTZ equilibrium geometry. c R-CCSD(T)/cc-pVTZ equilibrium geometry.

inclusion of dynamical correlation. The geometries of both doublet states are obtained with an accuracy close to that from far more expensive high level SR and MR methods. 3675


The Journal of Physical Chemistry A The CCSD(T) method should only be applied with caution to systems with reasonable degrees of MR character, such as biradicals or triradicals where large amplitude single excitations can enter (typically these systems have low symmetry). Evidently, the direct inclusion of nondynamical correlation is essential for these cases rather than attempting its treatment as a byproduct of a description designed for dynamical correlation. The IVO-CASCI analytical gradient method with a simple CAS and without any explicit account of dynamical electron correlation provides geometries for radicaloid species comparable to those obtained via expensive, highly correlated state-of-the-art calculations [such as Mk-MRCCSD, CCSD(T), BWP91 and so on]. Of course, as with any MR method, some numerical experimentation is necessary to establish an optimal reference space. Our calculations again display a weak dependence on the size of the reference space and of the basis sets. Obtaining a uniformly accurate description of reaction intermediates (such as radicaloid species), transition states, and electronic excited states requires a method with a well balanced description of dynamic and nondynamic correlation effects. We are investigating IVO-CASCI based correlated gradient methods [such as IVO-MRMP] as a possible source of these efficient and reliable approximations. The results in ref 32 suggest that our strategy of avoiding the CASSCF step and using IVO-CASCI orbitals can be useful for many situations.

III. CONCLUSION The shortcomings of the conventional CASSCF method can be essentially ameliorated or at least attenuated with the IVOCASCI method, which accounts very effectively for the nondynamic correlation. Since its introduction, the IVO-CASCI approach has been advertised as a very useful alternative to the CASSCF approach for treating systems with states containing significant multireference character. The IVO-CASCI method has been demonstrated to provide accuracies at least as good as the CASSCF scheme but with reduced computational cost because the former does not require the expensive iterative procedure involved with CASSCF calculations. The extension of the IVOCASCI scheme to enable geometry optimization via analytical gradients is demonstrated to provide geometries of complex MR systems with accuracy equal to or exceeding that from CASSCF calculations and generally with the use of smaller reference spaces. This paper focuses on assessing the efficacy of the recently developed analytical gradient IVO-CASCI formalism to describe the geometry of various radicaloid species [specifically, TTM, 2,6-pyridyne, the 2,6-pyridynium cation, and TDB] which often play important roles as intermediates in chemical reactions. The IVO-CASCI gradient calculations are performed using our in house code which we have interfaced with the GAMESS(US) package. The systems considered here possess considerable MR character and therefore are challenging to describe accurately. Nevertheless, the IVO-CASCI gradient approach using a simply chosen (often quite small) active space provides a good depiction of the geometry of systems with quasi-degeneracy while enhancing the ease of use and retaining advantages of the CASSCF gradient scheme at even lower computational costs. Comparisons are provided both with CASSCF calculations and with far more expensive and sophisticated theoretical results. The results illustrate that the overall performance of IVO-CASCI approach for biradical and triradical systems is identical to or exceeds that

ARTICLE

of the CASSCF and is often comparable to far more costly approaches that include nondynamical and dynamical correlation effects in a balanced and efficient manner. The present IVO-CASCI calculations exclusively favor the open-shell monocyclic form of the 2,6-pyridyne and 2,6-pyridynium cation in accord with the state-of-the-art Mk-MRCC and R-RMR-CCSD results. Likewise, the IVO-CASCI description of the geometry of the biradical TMM agrees well with the highly accurate BWCCSD, SS-EOM-CEPA, SS-EOM-CCSD, and EOM-SF-CC(2,3) benchmarks at significantly lower computational cost, and the IVO-CASCI geometry even for the very challenging doublet TDB system is comparable to the expensive state-of-the-art calculations with SR and MR methods. Predicting the relative ordering of the 2A1and 2B2 states of TDB has been a particularly thorny problem that has been addressed by calculations at all levels of theory.76 In contrast to the erroneous ordering by the CASSCF(9,9), MR-CISD, and AQCC predictions, the estimated adiabatic energy gap with the IVO-CASCI analytical gradient method correctly favors the 2A1 state over the 2B2 state. Most of the highly correlated theories successfully predict the 2A1 ground state with a splitting of 0.03-0.09 eV (or 0.69-2.07 kcal/mol). As with the CASSCF result, the IVO-CASCI gap is generally quite large and can be attributed to the lack of sufficient dynamical correlation within the IVO-CASCI wave function. In summary, the geometries predicted by the IVO-CASCI method are fairly insensitive to the size of the basis set and of the reference space. Thus, the IVO-CASCI analytical gradient scheme provides a robust and economic protocol for determining realistic geometries for states with significant multireference character. Although the IVO-CASCI analytical gradient calculations only include static correlation effects, dynamical correlation can be appended using the IVO-MRMP approach, but this method is currently available only with numerical gradients. Incorporating the dynamical electron correlation, in the present work, we find that the MRMP2 correction to the IVO-CASCI reference (using IVOMRMPT gradient approach) shows good agreement with the literature values. We are now engaged in the development of the IVO-MRMP based analytical gradient approach.

’ AUTHOR INFORMATION Corresponding Author

*Electronic address: sudip_chattopadhyay@rediffmail.com. Notes †

Electronic address: [email protected].

’ ACKNOWLEDGMENT It is a great pleasure to present this work in an issue dedicated to Professor Graham R. Fleming, an indefatigable forerunner in the field of modern chemical physics. Continued support by the Department of Science and Technology, India (grant SR/S1/ PC-32/2005), is gratefully acknowledged. This research is also supported, in part, by an NSF grant (CHE-0749788). ’ REFERENCES (1) Paldus, J.; Li, X. Adv. Chem. Phys. 1999, 110, 1. (2) Crawford, T. D.; Schaefer, H. F., III In Reviews in Computational Chemistry; Lipkowitz, K. B., Boyd, D. B., Eds.; Wiley: New York, 2000. (3) Bartlett, R. J. In Theory and Applications of Computational Chemistry: The First Forty Years; Dykstra, C. E., Frenking, G., Kim, K. S., 3676


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Notes ‡

Electronic address: [email protected].

3678


Geometry Optimization of Radicaloid Systems Using Improved Virtual

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