On the vertical excitation energy of cyclopentadiene - Semantic Scholar

JOURNAL OF CHEMICAL PHYSICS

VOLUME 121, NUMBER 11

15 SEPTEMBER 2004

On the vertical excitation energy of cyclopentadiene Yannick J. Bomble, Kurt W. Sattelmeyer, and John F. Stanton Departments of Chemistry and Biochemistry, Institute for Theoretical Chemistry, The University of Texas at Austin, Austin, Texas 78712

Ju¨rgen Gauss Institut fu¨r Physikalische Chemie, Universita¨t Mainz, D-55099 Mainz, Germany

共Received 11 May 2004; accepted 16 June 2004兲 The vertical excitation energy for the lowest valence ␲→␲* transition of cyclopentadiene is investigated. Using a combination of high-level theoretical methods and spectroscopic simulations, the vertical separation at the ground state geometry is estimated to be 5.43⫾0.05 eV. This value is intermediate between those calculated with coupled-cluster and multireference perturbation theory methods and is about 0.13 eV higher than the observed maximum in the absorption profile. © 2004 American Institute of Physics. 关DOI: 10.1063/1.1780159兴

ate 共MCQD兲 approaches in conjunction with a slightly larger basis set give 5.19 and 5.26 eV, respectively.8 The first and third of these values compare quite favorably with the ‘‘experimental’’ values and have therefore been assumed to attest to the accuracy of the corresponding theoretical methods. Calculations with a basis set similar to that of Ref. 7 using EOM-CC were reported, and gave a vertical separation of 5.65 eV at the coupled-cluster singles and doubles 共CCSD兲 level.9 However, addition of a noniterative correc˜ 兲共Ref. 10兲 tion for triple excitations known as EOM-CCSD共T lowered this value significantly; the resulting separation of 5.30 eV is indeed close to both ‘‘experiment’’ and the CASPT2 result. Subsequently, the closely-related symmetry adapted cluster-configuration interaction 共SAC-CI兲 method11 of Nakatsuji was applied to this problem using a relatively large basis set 共180 functions兲 and gives a value of 5.54 eV,12 0.21 eV below a result obtained years earlier with the same method in a smaller basis set.13 Hence, it seems that basis set augmentation acts to lower the excitation energy of cyclopentadiene and that methods based on coupled-cluster response theory provide predictions which are roughly 0.4 eV higher than CASPT2. In our view, the studies summarized above are not sufficient to determine the relative accuracies of multireference perturbation and coupled-cluster theories for this problem. Rather small basis sets have been used in all of the calculations discussed above and the only thing one can say is that CASPT2 in a small basis set gives an excitation energy that is closer to the experimental peak maximum than do the coupled-cluster approaches. This is a merit in the pragmatic sense, but does little in the way of providing unambiguous assessment of the intrinsic accuracy of the methods. The purpose of this paper is to present calculations that are the most comprehensive to date on the excited state of CP, and use them to make some comments on the accuracy achieved for this problem with various approaches.

I. INTRODUCTION

It is a common practice in quantum chemistry to calibrate the accuracy of excited state methods by comparing calculated vertical transition energies with the position of corresponding maxima in absorption spectra. As detailed elsewhere, this approach makes the implicit assumption that there is a fairly significant displacement in the excited state, so that the vibrational wave function has maximum amplitude in the region of the classical turning points.1 Of course, this assumption does not always hold, and the absorption maximum will then be displaced from the vertical excitation energy. One example where this is well recognized is the transition to the 1 B 2u state of ethylene, where the difference between the vertical excitation energy and the maximum of the absorption profile appears to be about 0.3–0.5 eV, with the former at higher energy.2 Given that accuracies of better than 0.5 eV are now attributed to quantum chemical methods such as equation-of-motion 共EOM兲共linear response兲 coupled-cluster 共CC兲 theory3 and various realizations of perturbation theory built upon a multireference zeroth-order wave function such as complete active space second order perturbation theory 共CASPT2兲,4 rigorous calibration of the accuracy of a given method should probably be based on something other than ‘‘vertical’’ energy differences associated with absorption maxima. A prototype for the electronic spectra of conjugated ␲ electron systems is cyclopentadiene 共CP兲, where the lowest valence ( 1 B 2 ← 1 A 1 ) ␲→␲* excitation has been investigated by one-photon absorption in the gas phase.5,6 The maximum found in the most recent gas-phase experiment is at 5.30 ⫾0.02 eV, the uncertainty owing to the fact that there are two closely spaced and partially resolved peaks that have comparable intensities at ca. 5.28 and 5.32 eV. Quantum chemical calculations by the Lund group using the CASPT2 method and a relatively modest basis set augmented with moleculecentered Rydberg functions found a vertical separation of 5.27 eV.7 In another study, multi-reference second order perturbation theory 共MRPT2兲 and the ostensibly more accurate 共and similar to CASPT2兲 multi-configuration quasidegener0021-9606/2004/121(11)/5236/5/$22.00

II. COMPUTATIONAL DETAILS

Vertical excitation energies of CP have been calculated at the ground state geometry optimized at the CCSD共T兲 level 5236

© 2004 American Institute of Physics

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J. Chem. Phys., Vol. 121, No. 11, 15 September 2004

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TABLE I. Geometry of cyclopentadiene optimized at the CCSD共T兲 level of theory with a cc-pVQZ basis set. The molecule is in the principal axis system, and the atomic Cartesian coordinates are given in bohr.

TABLE II. Geometry of cyclopentadiene optimized at the CCSD level of theory with a TZ2P basis set. The molecule is in the principal axis system, and the atomic Cartesian coordinates are given in bohr.

Atom

Atom

C C C C C H H H H H H

x

y

z

0.000 000 00 0.000 000 00 0.000 000 00 0.000 000 00 0.000 000 00 ⫺1.655 857 28 1.655 857 28 0.000 000 00 0.000 000 00 0.000 000 00 0.000 000 00

0.000 000 00 2.219 041 13 ⫺2.219 041 13 ⫺1.385 114 51 1.385 114 51 0.000 000 00 0.000 000 00 4.160 969 61 ⫺4.160 969 61 ⫺2.546 132 33 2.546 132 33

2.329 601 11 0.567 646 38 0.567 646 38 ⫺1.835 994 57 ⫺1.835 994 57 3.563 793 97 3.563 793 97 1.179 268 27 1.179 268 27 ⫺3.510 138 32 ⫺3.510 138 32

of theory14 with a cc-pVQZ basis set15 共see Table I兲 using various basis sets based on Dunning’s correlation consistent hierarchy.15,16 These include the cc-pVXZ (X⫽D,T,Q) and aug-cc-pVXZ series (X⫽D,T,Q); the latter includes diffuse functions required to describe Rydberg character in excited states. In the calculations with diffuse functions, the aug-cc-pVXZ set was used on the carbon atoms only. The hydrogen atoms were described with the corresponding cc-pVXZ basis; the composite basis is denoted as aug-cc-pVXZ⬘ in this work. The calculations using 100, 234, and 455 (cc-pVXZ) and 169, 368, and 580 (aug-cc-pVXZ⬘ ) basis functions include some of the largest yet reported at the EOM-CCSD level of theory. Triple excitations were calculated with two of the smaller basis sets 共cc-pVDZ and ccpVTZ兲 at the CCSDT-3 level of theory;17 the full CCSDT method18 was used in conjunction with the cc-pVDZ basis. It should be noted that calculations using the full EOMCCSDT method have not previously been reported by our group; this paper is the first in which our general-purpose implementation is used. The calculation for CP, which includes 100 basis functions, is the largest yet done at this level of theory. Apart from the cc-pVDZ/CCSDT and aug-cc-pVXZ⬘ /CCSD calculations, in which the core electrons were frozen, all electrons were correlated in the calculations. Effects of dropping core electrons were assessed at the cc-pVDZ/CCSD and cc-pVTZ/CCSD levels and found to be negligible 共⬍0.01 eV兲. All quantum chemical calculations used a local version of the ACESII program package.19 Spectroscopic simulations were done with the package developed by our group.20 It uses the vibronic coupling ap-

C C C C C H H H H H H

x

y

z

0.000 000 00 0.000 000 00 0.000 000 00 0.000 000 00 0.000 000 00 ⫺1.657 860 63 1.657 860 63 0.000 000 00 0.000 000 00 0.000 000 00 0.000 000 00

0.000 000 00 2.226 477 96 ⫺2.226 477 96 ⫺1.393 379 60 1.393 379 60 0.000 000 00 0.000 000 00 4.169 823 99 ⫺4.169 823 99 ⫺2.553 028 50 2.553 028 50

2.341 836 15 0.562 110 98 0.562 110 98 ⫺1.835 970 18 ⫺1.835 970 18 3.569 334 88 3.569 334 88 1.166 067 14 1.166 067 14 ⫺3.509 699 71 ⫺3.509 699 71

proach developed by Cederbaum and co-workers21 and is capable of handling linear or quadratic coupling between an arbitrary number of diabatic electronic states. With quadratic intrastate coupling and no interstate coupling, the approach is equivalent to a Franck-Condon simulation in the harmonic approximation with a complete treatment of Duschinsky mixing. The Hamiltonian was parametrized by calculating the first and second derivatives of the ground and excited state surfaces and transforming these to the dimensionless normal coordinate representation associated with the ground state. All derivatives required for the parametrization were calculated analytically,22 with the exception of second derivatives of the excited state, which were determined by finite difference of analytic first derivatives.23 The quantum chemical calculations were done at the CCSD and EOMCCSD level with the TZ2P 共Ref. 24兲 basis set at a geometry optimized at the CCSD 共Ref. 25兲 level with the same basis set 共Table II兲. As a part of this study, it was found that six vibrational modes of CP exhibit significant Franck-Condon activity and careful attention was given to how many harmonic oscillator functions in each mode are needed to achieve convergence in the appearance of the calculated spectrum. The simulated spectra in Figs. 2 and 3 were obtained with 63 000 and 50⫻106 basis functions, respectively, using 2500 Lanczos recursions.26 The raw peak positions and intensities were convoluted with a Lorentzian lineshape function with a width 共FWHM兲 of 0.05 eV. A complete set of parameters for the simulation is available upon request from the authors.27

TABLE III. Vertical transition energies 共in eV兲 for the 1 B 2 ← 1 A 1 transition of cyclopentadiene, calculated using linear-response CC methods and various basis sets. The calculations were obtained at the geometry given explicitly in Table I.

Basis set cc-pVDZ cc-pVTZ cc-pVQZ aug-cc-pVDZ⬘ aug-cc-pVTZ⬘ aug-cc-pVQZ⬘

Number of functions

CCSD

CCSDT-3

CCSDT

100 234 455 169 368 580

6.01 5.81 5.74 5.73 5.68 5.68

5.90 5.70

5.90


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Bomble et al.

III. RESULTS AND DISCUSSION

Results of the vertical excitation energy calculations are collected in Table III, where it can be seen that the CCSD level of theory converges to an excitation energy in the vicinity of 5.68 eV. However, triple excitation effects are fairly significant and very consistent in magnitude. A lowering of 0.11 eV is predicted in all three calculations; it is notable that the full CCSDT treatment agrees with the considerably cheaper CCSDT-3 model to within 0.01 eV when the ccpVDZ basis set is used. Hence, results obtained with CCSDT-3 should give results that would differ negligibly from the full EOM-CCSDT treatment for larger basis sets where the latter method’s cost becomes prohibitive. The effects of residual basis set incompleteness have been assessed by extrapolating the excitation energies using the formula advocated by Helgaker et al. for correlation energies, viz., E X ⫽E ⬁ ⫺

a X3

,

where X is the cc-pVXZ (D⫽2, T⫽3, etc.兲 energy and E ⬁ is the corresponding estimate in the complete basis.28 Using the cc-pVXZ series (X⫽T and Q兲, one obtains a result of 5.69 eV for the valence-only basis set limit,29 which then needs to be augmented with an estimate for diffuse functions. A plausible estimate is ⫺0.03 eV, which gives a total EOM-CCSD basis set limit estimate of 5.66 eV. Similarly, the aug-cc-pVTZ⬘ and aug-cc-pVQZ⬘ results can be extrapolated, which gives a value of 5.67 eV. Hence, it seems that the EOM-CCSD basis set limit is indeed very close to 5.66 eV. Given the consistency of data seen in the table, the triple excitation contribution at the basis set limit is probably around ⫺0.11 eV, so that a ‘‘best estimate’’ value of the vertical energy gap between the ground and 1 B 2 states of CP at the geometry given in Table I is 5.55⫾0.05 eV 共see Fig. 1 for a graphical comparison to other theoretical estimates兲 at the EOM-CCSDT level with an exhaustive basis set. This is still 0.20–0.25 eV above the absorption maximum, which is a very disappointing result if one assumes that the transition is one for which the vertical excitation energy can be equated with the absorption maximum. To address this question, we have carried out simulations of the electronic spectrum of CP using the theoretical framework developed by Cederbaum and co-workers.21 Because there is an appreciable vibronic coupling of the 1 B 2 state with the higher-lying valence 1 A 2 state that actually causes the 1 B 2 state to be nonplanar,31 simulations of the electronic spectrum with and without this coupling mechanism differ not only in the density of states but also slightly in the position of the absorption maximum 共see Figs. 2 and 3兲. One can notice that the simulation including the linear vibronic coupling reproduces the experimental spectrum more accurately, especially in the low energy part of the spectrum. In either case, a Franck-Condon simulation or one in which linear vibronic coupling between the 1 B 2 and higher valence 1 A 2 states is included, the absorption maximum is found to lie ca. 0.13 eV below the vertical excitation energy used in the model Hamiltonian.32 Hence, it seems likely that the true

FIG. 1. Comparison of coupled-cluster theory to other theoretical estimates. The black bar represents the vertical energy evaluated in this work with the corresponding error bars.

FIG. 2. Absorption spectrum simulation of cyclopentadiene without including linear vibronic coupling in eV 共top兲. Enlargement of the maximum of absorption region 共bottom兲.



Excitation energy of cyclopentadiene

5239

in the previous literature. CASPT2 is too low, and EOMCCSD is too high. The correct answer is to be found somewhere in the middle, which can in principle be approached systematically in both CASPT2 and EOM-CC frameworks 共although the machinery for doing so is far better established in the latter兲. That CASPT2 is too low is not surprising. The excitation energy predicted at the complete active space self consistent field 共CASSCF兲 level is more than 2 eV above the CASPT2 result, and second-order perturbation theory is notorious for overestimating correlation effects when they are significant in magnitude. ACKNOWLEDGMENTS

This work was supported by the Robert A. Welch and National Science Foundations 共Y.J.B., K.W.S., and J.F.S.兲, and the Fonds der Chemischen Industrie 共J.G.兲. E. R. Davidson and A. A. Jarzecki, Chem. Phys. Lett. 285, 155 共1998兲. C. Petrongolo, R. J. Buenker, and S. D. Peyerimhoff, J. Chem. Phys. 76, 3655 共1982兲. 3 J. F. Stanton and R. J. Bartlett, J. Chem. Phys. 98, 7029 共1993兲. 4 K. Andersson, P.-A. Malmqvist, B. O. Roos, A. J. Sadlej, and K. J. Wolinsky, J. Phys. Chem. 94, 5483 共1990兲; K. Andersson, P.-A. Malmqvist, and B. O. Roos, J. Chem. Phys. 96, 1218 共1992兲. 5 H. Stobbe and F. Dunnhaupt, Chemische Berichte 52, 1436 共1919兲; L. W. Pickett, F. Paddock, and E. Sackter, J. Am. Chem. Soc. 63, 1073 共1941兲; A. Sabljic and R. McDiarmid, J. Chem. Phys. 93, 3850 共1990兲. 6 Q.-Y. Shang and B. S. Hudson, Chem. Phys. Lett. 183, 63 共1991兲. 7 L. Serrano-Andres, M. Merchan, I. Nebot-Gil, B. O. Roos, and M. Fulscher, J. Am. Chem. Soc. 115, 6184 共1993兲. 8 H. Nakano, T. Tsuneda, T. Hashimoto, and K. Hirao, J. Chem. Phys. 104, 2312 共1996兲. 9 J. D. Watts, S. R. Gwaltney, and R. J. Bartlett, J. Chem. Phys. 105, 6979 共1996兲. 10 J. D. Watts and R. J. Bartlett, Chem. Phys. Lett. 258, 581 共1995兲. 11 H. Nakatsuji and K. Hirao, J. Chem. Phys. 68, 2053 共1978兲; H. Nakatsuji, K. Ohta, and K. Hirao, ibid. 75, 2952 共1981兲. 12 J. Wan, M. Ehara, M. Hada, and H. Nakatsuji, J. Chem. Phys. 113, 5245 共2000兲. 13 H. Nakatsuji, O. Kitao, and T. Yonezawa, J. Chem. Phys. 83, 723 共1985兲. 14 K. Raghavachari, G. W. Trucks, J. A. Pople, and M. Head-Gordon, Chem. Phys. Lett. 157, 479 共1989兲; R. J. Bartlett, J. D. Watts, S. A. Kucharski, and J. Noga, ibid. 165, 513 共1990兲. 15 T. H. Dunning, J. Chem. Phys. 90, 1007 共1989兲. 16 R. A. Kendall, T. H. Dunning, and R. J. Harrison, J. Chem. Phys. 96, 6796 共1992兲. 17 J. D. Watts and R. J. Bartlett, Chem. Phys. Lett. 258, 581 共1996兲; J. Noga, R. J. Bartlett, and M. Urban, ibid. 134, 126 共1987兲. 18 M. Ka´llay and P. R. Surja´n, J. Chem. Phys. 113, 1359 共2000兲; S. Hirata, M. Nooijen, and R. J. Bartlett, Chem. Phys. Lett. 326, 255 共2000兲; S. A. Kucharski, M. Wloch, M. Musial, and R. J. Bartlett, J. Chem. Phys. 115, 8263 共2001兲; H. Larsen, K. Hald, J. Olsen, and P. Jørgensen, ibid. 115, 3015 共2001兲; K. Kowalski and P. Piecuch, ibid. 115, 643 共2001兲; J. Noga and R. J. Bartlett, ibid. 86, 7041 共1987兲. 19 J. F. Stanton, J. Gauss, J. D. Watts, W. J. Lauderdale, and R. J. Bartlett, Int. J. Quantum Chem., Quantum Chem. Symp. 26, 879 共1992兲. 20 J. F. Stanton, K. W. Sattelmeyer, and M. Nooijen 共unpublished兲. 21 H. Ko¨ppel, W. Domcke, and L. S. Cederbaum, Adv. Chem. Phys. 57, 59 共1984兲. 22 A. C. Scheiner, G. E. Scuseria, J. E. Rice, T. J. Lee, and H. F. Schaefer, J. Chem. Phys. 87, 5361 共1987兲; J. Gauss and J. F. Stanton, Chem. Phys. Lett. 276, 70 共1997兲. 23 J. F. Stanton, J. Chem. Phys. 99, 8840 共1993兲; J. F. Stanton and J. Gauss, ibid. 100, 4695 共1994兲. 24 The TZ2P basis sets for carbon and hydrogen are fully documented elsewhere 关 J. Gauss, J. F. Stanton, and R. J. Bartlett, J. Chem. Phys. 97, 7825 共1992兲兴. 25 G. D. Purvis and R. J. Bartlett, J. Chem. Phys. 76, 1910 共1982兲. 1 2

FIG. 3. Absorption spectrum simulation of cyclopentadiene including linear vibronic coupling in eV 共top兲. Enlargement of the maximum of absorption region 共bottom兲.

vertical excitation energy of CP is above the absorption maximum at 5.30⫾0.02 eV by such an amount. This energy separation is therefore estimated to be 5.43⫾0.05 eV, where the assigned error bars are conservative. With the data at hand, it is appropriate to evaluate the accuracy of multireference perturbation theory and coupledcluster approaches for studying this particular electronic transition. It seems that the former is too low, and the latter probably too high. Assessing the magnitude of the error, however, is not entirely trivial. We believe that inclusion of connected quadruple excitations is probably needed to bring the CC results into quantitative agreement with experiment. On the other hand, CASPT2 is about 0.15 eV, too low. If one assumes that basis set effects in the CC and CASPT2 calculations are comparable 共which is a plausible approximation兲, then the difference between the extrapolated EOM-CCSD energy of 5.66 eV and that calculated with the same level of theory using the basis set from Ref. 7 共5.79 eV兲共Ref. 33兲 suggests that CASPT2 would give a value somewhere near 5.15 eV with a large basis set. It should be noted that this is about 0.3 eV below the vertical excitation energy estimated here. Hence, it seems that the magnitude of error obtained in CASPT2 and EOM-CCSD calculations is approximately the same despite the illusion created by comparisons to the absorption maximum for these methods using small basis sets


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Each Lanczos iteration was completed in less than 80 s when 50⫻106 basis functions were used. 27 To request the parameters used in the simulation, contact [email protected] or [email protected] 28 T. Helgaker, W. Klopper, H. Koch, and J. Noga, J. Chem. Phys. 106, 9639 共1997兲. 29 Effects associated with inadequate treatment of core correlation by the cc-pVXZ basis sets were assessed by comparing EOM-CCSD excitation energies calculated with the cc-pVTZ and cc-pCVTZ 共Ref. 30兲 basis sets and were entirely negligible 共0.005 eV兲. 30 D. E. Woon and T. H. Dunning, J. Chem. Phys. 103, 4572 共1995兲.

M. Z. Gierski and F. Zerbetto, J. Chem. Phys. 99, 3721 共1993兲. This estimate is based on the following. The simulations are faithful to the experiment in the sense that they give two closely spaced features with comparable intensities in the vicinity of the maximum in the absorption profile. Experimentally, the center of the two peaks is located at 5.30 eV, and the distance from the vertical excitation energy used in the simulation and the corresponding feature is 0.13 eV, with the latter at lower energy. 33 This calculation was carried out with a local version of the ACES II program package using EOM-CCSD level of theory with the basis set and the experimental geometry from Ref. 7. 31 32
