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CCSDT study of the fluoroperoxyl radical, FOO
Chemical Physics Letters 385 (2004) 292–297 www.elsevier.com/locate/cplett
CCSDT study of the fluoroperoxyl radical, FOO Pablo A. Denis *, Oscar N. Ventura CCPG, DEQUIFIM, Facultad de Quımica, UDELAR, CC 1157, Montevideo 11800, Uruguay Received 16 October 2003; in final form 12 December 2003 Published online: 22 January 2004
Abstract The FOO radical was studied at the coupled-cluster (CC) theoretical level, employing both the perturbative calculation of the contribution of triple excitations, CCSD(T), and the exact evaluation, CCSDT. Two solutions were found, one of them with a large spin contamination. It is shown that spin contamination is a problem at the CCSD(T) level, while it is largely inmaterial at the in reasonable agreement to experiment. The complete CCSDT level. The full CCSDT level affords an FO distance of 1.632 0.005 A 0 enthalpy of formation at the uncontaminated CCSDT level, Df H298 ðFOOÞ ¼ 6:5 1 kcal/mol, is also in very good agreement with the experimental value of 6.24 0.5 kcal/mol. Contrary to previous studies, CCSDT performs better than CCSD(T) in the calculation of the properties of this radical. Ó 2003 Elsevier B.V. All rights reserved.
1. Introduction Despite its smallness, and the presence of only firstrow atoms, the fluoroperoxyl radical (FOO) is one of the most difficult molecules to study at the computational level. The first pioneering research was performed at the SCF level using large sp basis sets by Hinchliffe [1]. Five years later, Gosavi et al. [2] studied again the problem, this time using CI calculations on geometries optimized at the SCF level. The attention on this molecule increased markedly in the last decade, when several studies were published. Due to the strong multiconfigurational character of FOO, Anderson and Ross selected this molecule to test their multiconfigurational second order perturbation theory, CASPT2, with great success [3]. Tozer et al. [4] used again the fluoroperoxyl radical as a benchmark for the spin restricted open shell Moller Plesset theory. Later, Francisco et al. [5] estimated the 0 Df H298 (FOO) as 10.7 3 kcal/mol, 5 kcal/mol larger than the two experimental values available, those of * Corresponding author. Address: Faculty of Chemistry, Department of Physics, Chemistry and Mathematics, University of Uruguay, Gral Flores 2124, Montevideo 1457, Uruguay. Fax: +598-20-29241906. E-mail addresses: [email protected] (P.A. Denis), oscar@fq. edu.uy (O.N. Ventura).
0009-2614/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2003.12.081
0 ðFOOÞ ¼ 5:49 0:40 Lyman and Holland [6], Df H298 0 kcal/mol, and of Pagsberg et al. [7], Df H298 ðFOOÞ ¼ 6:24 0:50 kcal/mol. Ventura and Kieninger [8] showed afterwards that the FOO radical could be studied successfully with hybrid density functional theory, obtaining a value of 5.98 kcal/mol for the enthalpy of formation at the B3PW91/ta-cc-pVQZ level. Finally, Alcami et al. [9] performed the most recent theoretical investigation on the fluoroperoxyl radical. They used several different exchange-correlation potentials at the DFT level, and QCISD(T) and CCSD(T) at the ab initio level, in all cases using PopleÔs 6-311+G(3df) basis set. The authors concluded, in agreement with Ventura and Kieninger [8], that DFT performed better than ab initio methods for some of the computed properties. For example, the F–O distance predicted by QCISD(T)/ shorter than the experimental 6-311+G(2d) was 0.084 A while a value determined by Mckellar et al. [10], 1.649 A simple B3LYP/6-311+G(3df) calculation exhibits half the error of the ab initio method. It is interesting to notice however, that Fernandez et al. [11], in disagreement with Alcami et al. [9], had calculated a distance of for the FO bond length. We will return later to 1.631 A this discrepancy between the calculations. Previous experience shows then that large basis sets and highly correlated ab initio methods are necessary for a good description of the FOO radical at the
P.A. Denis, O.N. Ventura / Chemical Physics Letters 385 (2004) 292–297
molecular orbital level. Therefore, we decided to apply both CCSD(T) and CCSDT methods to the study of this molecule, employing DunningÕs correlated consistent basis sets [12], a more complete treatment than any other single-configuration calculation available in the literature on this species. The use of CCSDT in addition to CCSD(T) requires some comment. In recent papers it was showed that CCSD(T) performs generally better than CCSDT for the calculation of enthalpies of formation of a large set of molecules like CO, H2 O, F2 , HF, N2 , studied by Bak et al. [13], first row diatomic hydrides studied by Feller and Sordo [14], some molecules in the G2/97 test set studied by Feller and Dixon [15], and Denis who studied 24 sulfur-containing molecules [16]. The general conclusion has been that, the better results obtained with CCSD(T) are due to error cancellation between the missing triple- and quadruple-excitations, as proved recently by Ruden et al. [17]. In this Letter we show however, that such is not the case of FOO, where an unexpectedly large contribution of the full treatment of triple excitations improves markedly the results obtained by QCISD(T). We show that this result is related to the spin contamination of the CCSD(T) calculations, a fact already described in the literature, especially by Gauss [18–20].
2. Theoretical methods Coupled cluster theory with single-, double- and perturbative treatment of triple excitations, CCSD[T] [21], and the full CCDST method including the complete treatment of triple excitations [22] were employed to study the properties of the FOO radical. The basis sets used for the ab initio calculations were Dunning ccpVXZ, (X ¼ T,Q,5), and aug-cc-pVXZ correlation consistent basis sets, X ¼ D,T,Q [12,23]. Core correlation was evaluated with the aid of Dunning cc-pwCVTZ basis set [24]. Extrapolation to the complete basis set (CBS) limit was done with the two-parameter equation E ¼ B þ C=L3 [25]. Complete geometry optimizations were performed with the CCSD[T] method for all basis sets, except quadruple zeta. The frozen core (fc) approximation was used for both coupled cluster methodologies unless otherwise noted (full treatment, i.e., inclusion of all electrons, is denoted as fu). Zero point energies were computed at the CCSD(T,fu)/cc-pwCVTZ level of theory. The normal UCCSD(T) calculation led to a spin.contaminated wavefunction. A different, spincontamination free wavefunction was obtained using the ROHF density as starting point for the UCCSD(T) calculations. The different characteristics of these two wavefunctions will be described later. The B3LYP density functional method [26,27] in conjunction with PopleÔs basis set 6-311+G(3df) [28] and DunningÕs aug-cc-pCV5Z [12] was employed for com-
293
parative purposes. All coupled cluster calculations were performed using the ACESII program of Bartlett and coworkers in Ref. [29] while the HF, MP2 and B3LYP calculations were performed using GA U S S I A N 98 [30].
3. Results and discussion 3.1. Structure of the FOO radical The geometrical parameters obtained for the fluoroperoxyl radical at different computational levels are reported in Table 1. The FOO species is predicted to be unstable at the UHF level, as shown in the curve of UHF/cc-pV5Z energy vs FO distance in Fig. 1 (the OO distance and FOO angle were optimized at each point). No minimum was found at this level, but there is an If a minimum inflexion point between 1.35 and 1.40 A. exists at all at this level of calculation, the potential energy surface must be very flat in its proximity. On the other hand, a stable FOO structure was found at the ROHF level (as reported previously [1,2]), with an F–O almost 0.3 A shorter than the distance of 1.369 A, experimental result (1.649 A). The ROHF curve is also shown in Fig. 1. Inclusion of electron correlation at the UMP2 or UMP4 levels does not improve the agreement with experiment, with the F–O distance still below This is in agreement with the recent study of 1.40 A. Kraka et al. [31] on FOOF, where the same failure of perturbation theory was pointed out. A substantial progress is obtained using CCSD(T), with a CCSDT/ still 0.1 A shorter than CBS FO distance of 1.534 A, experiment. This is an unusual situation. For instance, Bak et al. [32] studied 19 molecules at the CCSD(T)/ccpwCVQZ level and found that the mean and maximum respectively. absolute errors were 0.009 and 0.059 A, The explanation of this apparent exception was found in the spin-contamination of the wave function. Both in the calculations of Alcami et al. [9] and our own at the CCSD(T) level, the FO distance is much too short. In our work at least, the mean value of the S 2 operator at the CCSD(T) level is 1.4 instead of the theoretical 0.75. This shows that the ground state wavefunction is contaminated by a low lying quartet which we investigated separately (see later). The total atomic spin densities at the F, intermediate O and end O atoms are, respectively, )0.73, 0.79 and 0.83 electrons, showing a baa spin distribution arising from the collision of a doublet F and triplet O2 with opposite spins. Performing a ROHF calculation and using the electronic density as starting point for the CCSD(T) calculation and geometry optimizations, a second wavefunction (also 2 A00 ) is obtained. The optimum geometry, shown in Table 1, exhibits now the correct FO distance, the total energy at the minimum is 3.48 kcal/mol more stable than that of the contaminated function, and the spin distribution now,
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Table 1 Geometrical parameters, vibrational frequencies (in cm1 ) and zero point energies (in kcal/mol) obtained for the FOO radical at different levels of theory Method RHF-SCF UMP2 UMP4STDQ B3LYP ROHF
CCSD[T] CCSD[T]
CCSD[T,fc]a CCSD[T,fu]a CCSD[T,fc]b CCSDT Experimental
6-31G 6-31G* 6-311G(d) 6-311+G(2d) aug-cc-pCV5Z cc-pVTZ cc-pVQZ cc-pV5Z cc-pVTZ aug-cc-pVDZ aug-cc-pVTZ 1 cc-pwCVTZ cc-pwCVTZ cc-pVTZ cc-pVTZ
F–O
O–O
\FOO
1.440 1.381 1.383 1.381 1.619 1.371 1.369 1.369 1.553 1.633 1.563 1.534 1.550 1.547 1.634 1.632 1.649
1.333 1.255 1.250 1.238 1.185 1.242 1.240 1.239 1.211 1.208 1.208 1.208 1.211 1.209 1.198 1.200 1.200
106.7 107.7 109.6 109.4 111.2 108.1 108.2 108.2 110.3 110.0 110.2 110.3 110.3 110.3 111.0 110.3 111.2
FO stretch
FOO bend
OO stretch
Zpe
359 659 661 662 454
591 1073 1068 1068 675
1552 1252 1259 1260 1415
4.27 3.64
458 461 388 367 221
680 684 614
1412 1414 1515
3.64 3.66 3.60
417
1636
Reference [2] [2] [5] [5] This This This This This This This This This This This This [10]
work work work work work work work work work work work work
a
With spin contamination, hS 2 i ¼ 1.38. b Without spin contamination, hS 2 i ¼ 0:76.
ergy curve (the reduced potential energy curve at fixed d(OO) and h (FOO) parameters), from 1.50 to 1.80 A, were calculated at the CCSDT/cc-pVTZ level of theory. This is a very demanding calculation, since each point took more than 50 h of cpu time in a single node of an IBM SP3 machine. The results are presented in Fig. 2. only 0.017 A The F–O distance obtained is 1.632 A, shorter than the experimental value. A double checking was now performed, fixing the FO distance at 1.632 A 2.00
CCSDT/cc-pVTZ 1.50
Fig. 1. UHF/cc-pVTZ potential energy curve for the variation of the FO distance, with simultaneous optimization of the OO distance and FOO angle at each point. The relative energy (in kcal/mol) has been computed arbitrarily with respect to the lowest point displayed.
0.04, 0.13 and 0.94 electrons, is that of a doublet with the non-paired electron localized at the end oxygen atom. CCSDT automatic geometry optimizations are presently outside our computational capabilities. Thus, an alternative approach was chosen to investigate the improvement obtained by full inclusion of triple excitations. Fixing the OO distance and the FOO angle at their and CCSD(T)/cc-pVTZ optimum values of 1.208 A 110.3°, the energy of 14 points on the FO potential en-
E (Kcal/ mol)
Quadratic Fit
1.00
0.50 Spline Smoothing
0.00
-0.50 1.50
1.60
1.632
1.70
1.80
r(FO) (Å) Fig. 2. Potential energy curve for the variation of the FO distance, keeping the OO distance and FOO angle at the CCSD(T)/cc-pVTZ optimum values. The relative energy (in kcal/mol) has been computed with respect to the minimum of the curve. A quadratic fit was done around the minimum, to calculate the harmonic vibrational frequency.
P.A. Denis, O.N. Ventura / Chemical Physics Letters 385 (2004) 292–297
and the FOO bond angle again at 110.3°, and scanning then the O–O distance. The optimum O–O distance was found to be almost identical to that obtained at the The effect of core CCSD(T)/cc-pVTZ level, 1.208 A. correlation is small, shortening the F–O distance by and the O–O distance by 0.0013 A at the 0.0031 A CCSD(T)/cc-pwCVTZ level. The CCSDT calculation has a much smaller spin contamination than the CCSD(T) calculation, to the point that the results obtained starting from a contaminated UHF or an uncontaminated ROHF calculation are nearly the same. The result obtained shows that the CCSDT method predicts correctly the structure of FO2 . The CCSD(T) calculations, on the contrary, due to the use of perturbation theory to calculate the triple excitation contribution, gave an spin-contaminated wavefunction which results in an incorrect geometrical structure. This drawback can be corrected if an ROHF electronic density is used as input, something understandable on the basis of the curves shown in Fig. 1. The FO distance obtained in this way is then the one correctly obtained shorter previously by Fern andez et al. [11], only 0.017 A than the experimental one. The other two methodologies that describe reasonably well the F–O bond are B3LYP too short) and multiconfigurational (1.619, 0.030 A too long) [16]. It is interesting CASPT2 (1.669, 0.020 A to realize that the accuracy of the B3LYP calculation is largely related to the absence of any spin-contamination in this method. Since the determinant formed from the Kohn–Sham orbitals is not a wavefunction of the electronic Hamiltonian, the very same concept of spin contamination does not apply to DFT calculations. 3.2. Vibrational frequencies The vibrational frequencies of the fluoroperoxyl radical are also presented in Table 1. Error cancellation sometimes allow to obtain good frequencies even with bad geometries, but this is not the situation for the FOO radical. At the ROH level, only the O–O stretch is described reasonably well, being only 20% lower than the experimental result. Even CCSD(T) has still problems to describe the three vibrational modes, the O–O stretch again being best reproduced, with an underestimation of 14%. Due to the bad description of the bond, the F–O stretch is overestimated by 100%, and the FOO bend is 60% larger than the experimental value. Inclusion of core correlation does not affect the results much, changing frequencies by only 4–5 cm1 . These results are largely produced because of the incorrect underlying wavefunction. If the non-contaminated function is used instead, the errors are reduced by about 100 cm1 . The B3LYP DFT method provides the closest result to the experimental value, and is therefore the most costeffective method to use with these species (see the conclusions by Kraka et al. [31]). It would be desirable to
295
perform a complete potential energy surface scan at the CCSDT level, to obtain a force field and determine the spectroscopic constants of the FOO radical, but this has not been done in this Letter. At any rate, the F–O stretch frequency can be obtained numerically from the potential energy scan we performed at fixed OO distance and FOO angle. This value is reported in Table 1. We used a quadratic fit to the CCSDT/cc-pVTZ energy curve (dotted curve in Fig. 2) and obtained 367 cm1 as the value for the harmonic vibration. This value is 90 cm1 lower than that obtained at the CCSD(T)/ccpVTZ level with the spin-contaminated function, below the value obtained at the CCSD(T) level with the noncontaminated function, and very near to the B3LYP result. As can be seen in Fig. 2, the curve is not really harmonic and this may explain part of the remaining differences with respect to experiment. 3.3. Thermochemistry Another interesting property of FOO, which has been the subject of some discussion in the past, is its enthalpy of formation. To test the performance of CCSD(T) and CCSDT with respect to this property, we performed calculations on the atomization reaction. The results obtained are presented in Table 2. Notice that both calculations have been performed, employing the spincontaminated and the contamination free wavefunctions. A strong basis set dependence is observed, unusual for a species composed of only first-row atoms. The 0 computed Df H298 ðFOOÞ changes by 2.3 and 2.8 kcal/mol at the CCSD(T) and CCSDT levels, when the basis set is increased from aug-cc-pVTZ (already a respectably large basis set) to aug-cc-pVQZ. Both CCSD and CCSD(T) give a poor estimation of the enthalpy of formation at the CBS limit, 28.0 and 11.3 kcal/mol, respectively, with errors of about 22 and 5 kcal/mol with respect to the experimental results. This is not an unexpected result, since the FOO radical has a marked multiconfigurational character, as demonstrated by a large T1 value of 0.45 [33]. In some situations, like biradical ozone [34], CCSD(T) can perform correctly, but this is not the case with FOO. In fact, the problem arises again because of the spin contamination. The correct CCSD(T) wavefunction is able to give a further decrease of 3.48 kcal/ 0 mol with the result that the calculated Df H298 ðFOOÞ is very good. The CCSDT calculation gives this same result, showing that the main source of error is not the way in which the triple contributions is included (perturbative or iterative), but spin-contamination is. Considering the CCSDT value as the basic contribution, extrapolating the CCSD and CCSD(T) results to the complete basis set limit, and taking into account the core correlation contribution, 0.20 kcal/mol with the cc-pwCVQZ basis set, we estimate the enthalpy of formation of FOO as 0 Df H298 ðFOOÞ ¼ 6:5 1 kcal/mol. This estimation is
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Table 2 Calculated enthalpy of formation for the FOO radical, in kcal/mol Basis set
S2
HF
CCSD
CCSD(T)
CCSDT
HF–CCSD
CCSD–CCSD(T)
CCSD(T)–CCSDT
cc-pVTZ
1.38 0.77 1.41 0.77 1.40 0.77 1.52 1.40 0.77 1.41 0.77 0.77 1.41 0.77
144.4 169.8 142.8 169.0 142.6 168.6 139.8 143.0 169.1 142.5 168.6 168.5 142.0 168.4
37.0 37.4 32.3 32.6 28.9 29.1 40.7 33.4 33.6 30.3 30.7
22.2 18.7 16.4 12.9 12.2 8.7 28.6 17.9 14.1 14.1 10.6
18.7 17.7
107.4 132.4 110.2 136.4 112.2 139.3 99.1 109.6 135.5 112.2 138.3
14.8 18.7 15.9 19.7 16.7 20.4 12.1 15.5 19.5 16.2 20.1
3.5 1.0
28.0 28.6 36.9 37.0 32.2 32.3
11.3 8.0 21.7 21.5 16.2 16.0 11.0 7.5
114.1 140.3
16.7 20.5
cc-pVQZ 1 (TZ,QZ) aug-cc-pVDZ aug-cc-pVTZ aug-cc-pVQZ aug-cc-pV5Z 1 (ATZ,AQZ) cc-pwCVTZ-fc cc-pwCVTZ-full cc-pwCVQZ-fc cc-pwCVQZ-full Best result a b
Cont.a Non.b
7.5 6.5
Using the spin-contaminated wavefunction. Using the contamination-free wavefunction.
actually an upper limit to the computed enthalpy of formation. In fact, the CCSDT/cc-pVQZ and CCSD(T)/ cc-pV5Z calculations, necessary for the extrapolation to the CBS limit, could not be made with the available computational resources. If both calculations were carried out, one should expect a result closer to experiment, since an increase in basis set lowers the calculated enthalpy of formation. One should notice also that scalar relativistic effects and spin orbit splitting were not considered, but their influence ought to be very small because of the nature of he (first-row) atoms involved. The results obtained with the CCSD(T) method show the large effect of spin contamination (a decrease of 3.5 kcal/ mol) while the effect is much smaller in CCSDT (only 1 kcal/mol). The effect of the full inclusion of triples, in case one uses the correct wavefunction, is also only 1 kcal/mol. It is not inappropriate to remind here that the B3PW91/ta-pVQZ calculation of the enthalpy of formation of FOO, performed in [35], gives a value of 6.0 kcal/mol, almost as good as the CCSDT/CBS result and certainly less computationally expensive. As with respect 0 to the F–O distance, the DFT Df H298 ðFOOÞ is closer to experiment than the CCSD(T) one and only slightly worse than the CCSDT one.
3.4. Low lying quartet state Because of the multiconfigurational character of the ground state of FOO and the spin contamination observed in the reference wavefunction, we decided to investigate the lowest-lying excited quartet state, optimizing its geometrical structure at the CCSD(T)/ccpVTZ and B3LYP/6-311+G(3df) levels. Contrary to the doublet ground state, the quadruplet does not exhibit excessive spin contamination (3.80 instead of the theoretical 4). The same electronic structure is obtained starting from an UHF or an ROHF reference calculation. The optimized structure looks like a van der Waals an O–O complex with an F–O bond distance of 2.991 A, and an FOO bond angle of bond distance of 1.213 A 78.19°. This structure was confirmed to be a minimum on the quartet potential energy surface as indicated by the positive force constants. The doublet-quartet energy gap was calculated at the CCSD(T)/cc-pVTZ (7.4 kcal/ mol), CCSD(T)/cc-pVQZ (9.9 kcal/mol) and CCSDT/ cc-pVTZ (8.0 kcal/mol). Following the usual extrapolation procedure described before, the extrapolated CCSDT/CBS value would be 12.4 kcal/mol. This is comparable to the value obtained at the DFT B3LYP level, 11.7 kcal/mol, although this method predicts a
Table 3 Structure and vibrational frequencies of the quartet FOO Method CCSD(T) B3LYP
cc-pVTZ 6-311+G(3df)
F–O
O–O
\FOO
w1
w2
w3
Zpe
2.991 2.611
1.213 1.207
78.2 76.7
84 73
95 105
1580 1624
2.52 2.57
P.A. Denis, O.N. Ventura / Chemical Physics Letters 385 (2004) 292–297
an O–O structure with a shorter F–O bond, 2.610 A, bond of 1.207A, and an FOO angle of 76.7°, the later two very close to the coupled cluster result. We summarize the properties of the FOO in the quartet state in Table 3.
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Acknowledgements The authors acknowledge PEDECIBA, (UNESCO PNUD) and Dr. Raul E. Cachau for economic and scientific support. The authors acknowledge also useful suggestions by the first referee of this Letter.
4. Conclusions The full CCSDT method has been employed to study the fluoroperoxyl radical. The inclusion of full calculation of triple excitations makes a tremendous improvement in the computed properties of FOO with respect to the ones previously calculated at the CCSD(T) level by Alcami et al. [9]. The F–O distance is predicted to only 0.017 A shorter than the exbe 1.632 0.005 A, better than the periment 1.649 A, and almost 0.1 A, CCSD(T) estimation. The enthalpy of formation of 0 FOO was determined to be Df H298 ðFOOÞ ¼ 6:5 1 kcal/mol. Contrary to what was observed in previous investigations, CCSDT performs better than CCSD(T) in the calculation of this enthalpy of formation, the CCSD(T)–CCSDT difference being 3.48 kcal/mol. A similar effect was reported by Bak et al. [13] for methylene, another molecule with multiconfigurational character, but not as strong as in the case of FOO. However, the observed effect is fictitious, and due only to the fact that CCSD(T) calculations, unless started from the ROHF density with the correct spin distribution, lead to a spin contaminated wave function, with an erroneous geometry. If the calculations are started from the correct ROHF wavefunction, the geometry obtained is now much closer to the experimental one, the wavefunction is the same as obtained at the CCSDT level, and no important CCSD(T)–CCSDT energy difference appears. This situation should be kept in mind, because it is probably arising also in the cases of ClOO, BrOO and OFO, as our preliminary calculations indicate. Despite being a single reference method, CCSDT describes the chemical structure and thermochemistry of FOO as well as multiconfigurational based methods such us CASPT2 [3] or, to be fair, DFT methods such as B3LYP or B3PW91 [8,35]. The lowest-lying quartet state was also investigated. A looser structure than that of the doublet ground state was found and the doublet-quartet energy gap was estimated to be 12.4 kcal/mol at the CCSDT/CBS level and 11.7 kcal/mol at the B3LYP level. Astonishingly as it may appear at first glance, B3LYP/6-311+G(3df) calculations are again demonstrated to be as good as CCSDT calculations with extended basis sets in the particular case of FOO.
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CCSDT study of the fluoroperoxyl radical, FOO Pablo A. Denis *, Oscar N. Ventura CCPG, DEQUIFIM, Facultad de Quımica, UDELAR, CC 1157, Montevideo 11800, Uruguay Received 16 October 2003; in final form 12 December 2003 Published online: 22 January 2004
Abstract The FOO radical was studied at the coupled-cluster (CC) theoretical level, employing both the perturbative calculation of the contribution of triple excitations, CCSD(T), and the exact evaluation, CCSDT. Two solutions were found, one of them with a large spin contamination. It is shown that spin contamination is a problem at the CCSD(T) level, while it is largely inmaterial at the in reasonable agreement to experiment. The complete CCSDT level. The full CCSDT level affords an FO distance of 1.632 0.005 A 0 enthalpy of formation at the uncontaminated CCSDT level, Df H298 ðFOOÞ ¼ 6:5 1 kcal/mol, is also in very good agreement with the experimental value of 6.24 0.5 kcal/mol. Contrary to previous studies, CCSDT performs better than CCSD(T) in the calculation of the properties of this radical. Ó 2003 Elsevier B.V. All rights reserved.
1. Introduction Despite its smallness, and the presence of only firstrow atoms, the fluoroperoxyl radical (FOO) is one of the most difficult molecules to study at the computational level. The first pioneering research was performed at the SCF level using large sp basis sets by Hinchliffe [1]. Five years later, Gosavi et al. [2] studied again the problem, this time using CI calculations on geometries optimized at the SCF level. The attention on this molecule increased markedly in the last decade, when several studies were published. Due to the strong multiconfigurational character of FOO, Anderson and Ross selected this molecule to test their multiconfigurational second order perturbation theory, CASPT2, with great success [3]. Tozer et al. [4] used again the fluoroperoxyl radical as a benchmark for the spin restricted open shell Moller Plesset theory. Later, Francisco et al. [5] estimated the 0 Df H298 (FOO) as 10.7 3 kcal/mol, 5 kcal/mol larger than the two experimental values available, those of * Corresponding author. Address: Faculty of Chemistry, Department of Physics, Chemistry and Mathematics, University of Uruguay, Gral Flores 2124, Montevideo 1457, Uruguay. Fax: +598-20-29241906. E-mail addresses: [email protected] (P.A. Denis), oscar@fq. edu.uy (O.N. Ventura).
0009-2614/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2003.12.081
0 ðFOOÞ ¼ 5:49 0:40 Lyman and Holland [6], Df H298 0 kcal/mol, and of Pagsberg et al. [7], Df H298 ðFOOÞ ¼ 6:24 0:50 kcal/mol. Ventura and Kieninger [8] showed afterwards that the FOO radical could be studied successfully with hybrid density functional theory, obtaining a value of 5.98 kcal/mol for the enthalpy of formation at the B3PW91/ta-cc-pVQZ level. Finally, Alcami et al. [9] performed the most recent theoretical investigation on the fluoroperoxyl radical. They used several different exchange-correlation potentials at the DFT level, and QCISD(T) and CCSD(T) at the ab initio level, in all cases using PopleÔs 6-311+G(3df) basis set. The authors concluded, in agreement with Ventura and Kieninger [8], that DFT performed better than ab initio methods for some of the computed properties. For example, the F–O distance predicted by QCISD(T)/ shorter than the experimental 6-311+G(2d) was 0.084 A while a value determined by Mckellar et al. [10], 1.649 A simple B3LYP/6-311+G(3df) calculation exhibits half the error of the ab initio method. It is interesting to notice however, that Fernandez et al. [11], in disagreement with Alcami et al. [9], had calculated a distance of for the FO bond length. We will return later to 1.631 A this discrepancy between the calculations. Previous experience shows then that large basis sets and highly correlated ab initio methods are necessary for a good description of the FOO radical at the
P.A. Denis, O.N. Ventura / Chemical Physics Letters 385 (2004) 292–297
molecular orbital level. Therefore, we decided to apply both CCSD(T) and CCSDT methods to the study of this molecule, employing DunningÕs correlated consistent basis sets [12], a more complete treatment than any other single-configuration calculation available in the literature on this species. The use of CCSDT in addition to CCSD(T) requires some comment. In recent papers it was showed that CCSD(T) performs generally better than CCSDT for the calculation of enthalpies of formation of a large set of molecules like CO, H2 O, F2 , HF, N2 , studied by Bak et al. [13], first row diatomic hydrides studied by Feller and Sordo [14], some molecules in the G2/97 test set studied by Feller and Dixon [15], and Denis who studied 24 sulfur-containing molecules [16]. The general conclusion has been that, the better results obtained with CCSD(T) are due to error cancellation between the missing triple- and quadruple-excitations, as proved recently by Ruden et al. [17]. In this Letter we show however, that such is not the case of FOO, where an unexpectedly large contribution of the full treatment of triple excitations improves markedly the results obtained by QCISD(T). We show that this result is related to the spin contamination of the CCSD(T) calculations, a fact already described in the literature, especially by Gauss [18–20].
2. Theoretical methods Coupled cluster theory with single-, double- and perturbative treatment of triple excitations, CCSD[T] [21], and the full CCDST method including the complete treatment of triple excitations [22] were employed to study the properties of the FOO radical. The basis sets used for the ab initio calculations were Dunning ccpVXZ, (X ¼ T,Q,5), and aug-cc-pVXZ correlation consistent basis sets, X ¼ D,T,Q [12,23]. Core correlation was evaluated with the aid of Dunning cc-pwCVTZ basis set [24]. Extrapolation to the complete basis set (CBS) limit was done with the two-parameter equation E ¼ B þ C=L3 [25]. Complete geometry optimizations were performed with the CCSD[T] method for all basis sets, except quadruple zeta. The frozen core (fc) approximation was used for both coupled cluster methodologies unless otherwise noted (full treatment, i.e., inclusion of all electrons, is denoted as fu). Zero point energies were computed at the CCSD(T,fu)/cc-pwCVTZ level of theory. The normal UCCSD(T) calculation led to a spin.contaminated wavefunction. A different, spincontamination free wavefunction was obtained using the ROHF density as starting point for the UCCSD(T) calculations. The different characteristics of these two wavefunctions will be described later. The B3LYP density functional method [26,27] in conjunction with PopleÔs basis set 6-311+G(3df) [28] and DunningÕs aug-cc-pCV5Z [12] was employed for com-
293
parative purposes. All coupled cluster calculations were performed using the ACESII program of Bartlett and coworkers in Ref. [29] while the HF, MP2 and B3LYP calculations were performed using GA U S S I A N 98 [30].
3. Results and discussion 3.1. Structure of the FOO radical The geometrical parameters obtained for the fluoroperoxyl radical at different computational levels are reported in Table 1. The FOO species is predicted to be unstable at the UHF level, as shown in the curve of UHF/cc-pV5Z energy vs FO distance in Fig. 1 (the OO distance and FOO angle were optimized at each point). No minimum was found at this level, but there is an If a minimum inflexion point between 1.35 and 1.40 A. exists at all at this level of calculation, the potential energy surface must be very flat in its proximity. On the other hand, a stable FOO structure was found at the ROHF level (as reported previously [1,2]), with an F–O almost 0.3 A shorter than the distance of 1.369 A, experimental result (1.649 A). The ROHF curve is also shown in Fig. 1. Inclusion of electron correlation at the UMP2 or UMP4 levels does not improve the agreement with experiment, with the F–O distance still below This is in agreement with the recent study of 1.40 A. Kraka et al. [31] on FOOF, where the same failure of perturbation theory was pointed out. A substantial progress is obtained using CCSD(T), with a CCSDT/ still 0.1 A shorter than CBS FO distance of 1.534 A, experiment. This is an unusual situation. For instance, Bak et al. [32] studied 19 molecules at the CCSD(T)/ccpwCVQZ level and found that the mean and maximum respectively. absolute errors were 0.009 and 0.059 A, The explanation of this apparent exception was found in the spin-contamination of the wave function. Both in the calculations of Alcami et al. [9] and our own at the CCSD(T) level, the FO distance is much too short. In our work at least, the mean value of the S 2 operator at the CCSD(T) level is 1.4 instead of the theoretical 0.75. This shows that the ground state wavefunction is contaminated by a low lying quartet which we investigated separately (see later). The total atomic spin densities at the F, intermediate O and end O atoms are, respectively, )0.73, 0.79 and 0.83 electrons, showing a baa spin distribution arising from the collision of a doublet F and triplet O2 with opposite spins. Performing a ROHF calculation and using the electronic density as starting point for the CCSD(T) calculation and geometry optimizations, a second wavefunction (also 2 A00 ) is obtained. The optimum geometry, shown in Table 1, exhibits now the correct FO distance, the total energy at the minimum is 3.48 kcal/mol more stable than that of the contaminated function, and the spin distribution now,
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Table 1 Geometrical parameters, vibrational frequencies (in cm1 ) and zero point energies (in kcal/mol) obtained for the FOO radical at different levels of theory Method RHF-SCF UMP2 UMP4STDQ B3LYP ROHF
CCSD[T] CCSD[T]
CCSD[T,fc]a CCSD[T,fu]a CCSD[T,fc]b CCSDT Experimental
6-31G 6-31G* 6-311G(d) 6-311+G(2d) aug-cc-pCV5Z cc-pVTZ cc-pVQZ cc-pV5Z cc-pVTZ aug-cc-pVDZ aug-cc-pVTZ 1 cc-pwCVTZ cc-pwCVTZ cc-pVTZ cc-pVTZ
F–O
O–O
\FOO
1.440 1.381 1.383 1.381 1.619 1.371 1.369 1.369 1.553 1.633 1.563 1.534 1.550 1.547 1.634 1.632 1.649
1.333 1.255 1.250 1.238 1.185 1.242 1.240 1.239 1.211 1.208 1.208 1.208 1.211 1.209 1.198 1.200 1.200
106.7 107.7 109.6 109.4 111.2 108.1 108.2 108.2 110.3 110.0 110.2 110.3 110.3 110.3 111.0 110.3 111.2
FO stretch
FOO bend
OO stretch
Zpe
359 659 661 662 454
591 1073 1068 1068 675
1552 1252 1259 1260 1415
4.27 3.64
458 461 388 367 221
680 684 614
1412 1414 1515
3.64 3.66 3.60
417
1636
Reference [2] [2] [5] [5] This This This This This This This This This This This This [10]
work work work work work work work work work work work work
a
With spin contamination, hS 2 i ¼ 1.38. b Without spin contamination, hS 2 i ¼ 0:76.
ergy curve (the reduced potential energy curve at fixed d(OO) and h (FOO) parameters), from 1.50 to 1.80 A, were calculated at the CCSDT/cc-pVTZ level of theory. This is a very demanding calculation, since each point took more than 50 h of cpu time in a single node of an IBM SP3 machine. The results are presented in Fig. 2. only 0.017 A The F–O distance obtained is 1.632 A, shorter than the experimental value. A double checking was now performed, fixing the FO distance at 1.632 A 2.00
CCSDT/cc-pVTZ 1.50
Fig. 1. UHF/cc-pVTZ potential energy curve for the variation of the FO distance, with simultaneous optimization of the OO distance and FOO angle at each point. The relative energy (in kcal/mol) has been computed arbitrarily with respect to the lowest point displayed.
0.04, 0.13 and 0.94 electrons, is that of a doublet with the non-paired electron localized at the end oxygen atom. CCSDT automatic geometry optimizations are presently outside our computational capabilities. Thus, an alternative approach was chosen to investigate the improvement obtained by full inclusion of triple excitations. Fixing the OO distance and the FOO angle at their and CCSD(T)/cc-pVTZ optimum values of 1.208 A 110.3°, the energy of 14 points on the FO potential en-
E (Kcal/ mol)
Quadratic Fit
1.00
0.50 Spline Smoothing
0.00
-0.50 1.50
1.60
1.632
1.70
1.80
r(FO) (Å) Fig. 2. Potential energy curve for the variation of the FO distance, keeping the OO distance and FOO angle at the CCSD(T)/cc-pVTZ optimum values. The relative energy (in kcal/mol) has been computed with respect to the minimum of the curve. A quadratic fit was done around the minimum, to calculate the harmonic vibrational frequency.
P.A. Denis, O.N. Ventura / Chemical Physics Letters 385 (2004) 292–297
and the FOO bond angle again at 110.3°, and scanning then the O–O distance. The optimum O–O distance was found to be almost identical to that obtained at the The effect of core CCSD(T)/cc-pVTZ level, 1.208 A. correlation is small, shortening the F–O distance by and the O–O distance by 0.0013 A at the 0.0031 A CCSD(T)/cc-pwCVTZ level. The CCSDT calculation has a much smaller spin contamination than the CCSD(T) calculation, to the point that the results obtained starting from a contaminated UHF or an uncontaminated ROHF calculation are nearly the same. The result obtained shows that the CCSDT method predicts correctly the structure of FO2 . The CCSD(T) calculations, on the contrary, due to the use of perturbation theory to calculate the triple excitation contribution, gave an spin-contaminated wavefunction which results in an incorrect geometrical structure. This drawback can be corrected if an ROHF electronic density is used as input, something understandable on the basis of the curves shown in Fig. 1. The FO distance obtained in this way is then the one correctly obtained shorter previously by Fern andez et al. [11], only 0.017 A than the experimental one. The other two methodologies that describe reasonably well the F–O bond are B3LYP too short) and multiconfigurational (1.619, 0.030 A too long) [16]. It is interesting CASPT2 (1.669, 0.020 A to realize that the accuracy of the B3LYP calculation is largely related to the absence of any spin-contamination in this method. Since the determinant formed from the Kohn–Sham orbitals is not a wavefunction of the electronic Hamiltonian, the very same concept of spin contamination does not apply to DFT calculations. 3.2. Vibrational frequencies The vibrational frequencies of the fluoroperoxyl radical are also presented in Table 1. Error cancellation sometimes allow to obtain good frequencies even with bad geometries, but this is not the situation for the FOO radical. At the ROH level, only the O–O stretch is described reasonably well, being only 20% lower than the experimental result. Even CCSD(T) has still problems to describe the three vibrational modes, the O–O stretch again being best reproduced, with an underestimation of 14%. Due to the bad description of the bond, the F–O stretch is overestimated by 100%, and the FOO bend is 60% larger than the experimental value. Inclusion of core correlation does not affect the results much, changing frequencies by only 4–5 cm1 . These results are largely produced because of the incorrect underlying wavefunction. If the non-contaminated function is used instead, the errors are reduced by about 100 cm1 . The B3LYP DFT method provides the closest result to the experimental value, and is therefore the most costeffective method to use with these species (see the conclusions by Kraka et al. [31]). It would be desirable to
295
perform a complete potential energy surface scan at the CCSDT level, to obtain a force field and determine the spectroscopic constants of the FOO radical, but this has not been done in this Letter. At any rate, the F–O stretch frequency can be obtained numerically from the potential energy scan we performed at fixed OO distance and FOO angle. This value is reported in Table 1. We used a quadratic fit to the CCSDT/cc-pVTZ energy curve (dotted curve in Fig. 2) and obtained 367 cm1 as the value for the harmonic vibration. This value is 90 cm1 lower than that obtained at the CCSD(T)/ccpVTZ level with the spin-contaminated function, below the value obtained at the CCSD(T) level with the noncontaminated function, and very near to the B3LYP result. As can be seen in Fig. 2, the curve is not really harmonic and this may explain part of the remaining differences with respect to experiment. 3.3. Thermochemistry Another interesting property of FOO, which has been the subject of some discussion in the past, is its enthalpy of formation. To test the performance of CCSD(T) and CCSDT with respect to this property, we performed calculations on the atomization reaction. The results obtained are presented in Table 2. Notice that both calculations have been performed, employing the spincontaminated and the contamination free wavefunctions. A strong basis set dependence is observed, unusual for a species composed of only first-row atoms. The 0 computed Df H298 ðFOOÞ changes by 2.3 and 2.8 kcal/mol at the CCSD(T) and CCSDT levels, when the basis set is increased from aug-cc-pVTZ (already a respectably large basis set) to aug-cc-pVQZ. Both CCSD and CCSD(T) give a poor estimation of the enthalpy of formation at the CBS limit, 28.0 and 11.3 kcal/mol, respectively, with errors of about 22 and 5 kcal/mol with respect to the experimental results. This is not an unexpected result, since the FOO radical has a marked multiconfigurational character, as demonstrated by a large T1 value of 0.45 [33]. In some situations, like biradical ozone [34], CCSD(T) can perform correctly, but this is not the case with FOO. In fact, the problem arises again because of the spin contamination. The correct CCSD(T) wavefunction is able to give a further decrease of 3.48 kcal/ 0 mol with the result that the calculated Df H298 ðFOOÞ is very good. The CCSDT calculation gives this same result, showing that the main source of error is not the way in which the triple contributions is included (perturbative or iterative), but spin-contamination is. Considering the CCSDT value as the basic contribution, extrapolating the CCSD and CCSD(T) results to the complete basis set limit, and taking into account the core correlation contribution, 0.20 kcal/mol with the cc-pwCVQZ basis set, we estimate the enthalpy of formation of FOO as 0 Df H298 ðFOOÞ ¼ 6:5 1 kcal/mol. This estimation is
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Table 2 Calculated enthalpy of formation for the FOO radical, in kcal/mol Basis set
S2
HF
CCSD
CCSD(T)
CCSDT
HF–CCSD
CCSD–CCSD(T)
CCSD(T)–CCSDT
cc-pVTZ
1.38 0.77 1.41 0.77 1.40 0.77 1.52 1.40 0.77 1.41 0.77 0.77 1.41 0.77
144.4 169.8 142.8 169.0 142.6 168.6 139.8 143.0 169.1 142.5 168.6 168.5 142.0 168.4
37.0 37.4 32.3 32.6 28.9 29.1 40.7 33.4 33.6 30.3 30.7
22.2 18.7 16.4 12.9 12.2 8.7 28.6 17.9 14.1 14.1 10.6
18.7 17.7
107.4 132.4 110.2 136.4 112.2 139.3 99.1 109.6 135.5 112.2 138.3
14.8 18.7 15.9 19.7 16.7 20.4 12.1 15.5 19.5 16.2 20.1
3.5 1.0
28.0 28.6 36.9 37.0 32.2 32.3
11.3 8.0 21.7 21.5 16.2 16.0 11.0 7.5
114.1 140.3
16.7 20.5
cc-pVQZ 1 (TZ,QZ) aug-cc-pVDZ aug-cc-pVTZ aug-cc-pVQZ aug-cc-pV5Z 1 (ATZ,AQZ) cc-pwCVTZ-fc cc-pwCVTZ-full cc-pwCVQZ-fc cc-pwCVQZ-full Best result a b
Cont.a Non.b
7.5 6.5
Using the spin-contaminated wavefunction. Using the contamination-free wavefunction.
actually an upper limit to the computed enthalpy of formation. In fact, the CCSDT/cc-pVQZ and CCSD(T)/ cc-pV5Z calculations, necessary for the extrapolation to the CBS limit, could not be made with the available computational resources. If both calculations were carried out, one should expect a result closer to experiment, since an increase in basis set lowers the calculated enthalpy of formation. One should notice also that scalar relativistic effects and spin orbit splitting were not considered, but their influence ought to be very small because of the nature of he (first-row) atoms involved. The results obtained with the CCSD(T) method show the large effect of spin contamination (a decrease of 3.5 kcal/ mol) while the effect is much smaller in CCSDT (only 1 kcal/mol). The effect of the full inclusion of triples, in case one uses the correct wavefunction, is also only 1 kcal/mol. It is not inappropriate to remind here that the B3PW91/ta-pVQZ calculation of the enthalpy of formation of FOO, performed in [35], gives a value of 6.0 kcal/mol, almost as good as the CCSDT/CBS result and certainly less computationally expensive. As with respect 0 to the F–O distance, the DFT Df H298 ðFOOÞ is closer to experiment than the CCSD(T) one and only slightly worse than the CCSDT one.
3.4. Low lying quartet state Because of the multiconfigurational character of the ground state of FOO and the spin contamination observed in the reference wavefunction, we decided to investigate the lowest-lying excited quartet state, optimizing its geometrical structure at the CCSD(T)/ccpVTZ and B3LYP/6-311+G(3df) levels. Contrary to the doublet ground state, the quadruplet does not exhibit excessive spin contamination (3.80 instead of the theoretical 4). The same electronic structure is obtained starting from an UHF or an ROHF reference calculation. The optimized structure looks like a van der Waals an O–O complex with an F–O bond distance of 2.991 A, and an FOO bond angle of bond distance of 1.213 A 78.19°. This structure was confirmed to be a minimum on the quartet potential energy surface as indicated by the positive force constants. The doublet-quartet energy gap was calculated at the CCSD(T)/cc-pVTZ (7.4 kcal/ mol), CCSD(T)/cc-pVQZ (9.9 kcal/mol) and CCSDT/ cc-pVTZ (8.0 kcal/mol). Following the usual extrapolation procedure described before, the extrapolated CCSDT/CBS value would be 12.4 kcal/mol. This is comparable to the value obtained at the DFT B3LYP level, 11.7 kcal/mol, although this method predicts a
Table 3 Structure and vibrational frequencies of the quartet FOO Method CCSD(T) B3LYP
cc-pVTZ 6-311+G(3df)
F–O
O–O
\FOO
w1
w2
w3
Zpe
2.991 2.611
1.213 1.207
78.2 76.7
84 73
95 105
1580 1624
2.52 2.57
P.A. Denis, O.N. Ventura / Chemical Physics Letters 385 (2004) 292–297
an O–O structure with a shorter F–O bond, 2.610 A, bond of 1.207A, and an FOO angle of 76.7°, the later two very close to the coupled cluster result. We summarize the properties of the FOO in the quartet state in Table 3.
297
Acknowledgements The authors acknowledge PEDECIBA, (UNESCO PNUD) and Dr. Raul E. Cachau for economic and scientific support. The authors acknowledge also useful suggestions by the first referee of this Letter.
4. Conclusions The full CCSDT method has been employed to study the fluoroperoxyl radical. The inclusion of full calculation of triple excitations makes a tremendous improvement in the computed properties of FOO with respect to the ones previously calculated at the CCSD(T) level by Alcami et al. [9]. The F–O distance is predicted to only 0.017 A shorter than the exbe 1.632 0.005 A, better than the periment 1.649 A, and almost 0.1 A, CCSD(T) estimation. The enthalpy of formation of 0 FOO was determined to be Df H298 ðFOOÞ ¼ 6:5 1 kcal/mol. Contrary to what was observed in previous investigations, CCSDT performs better than CCSD(T) in the calculation of this enthalpy of formation, the CCSD(T)–CCSDT difference being 3.48 kcal/mol. A similar effect was reported by Bak et al. [13] for methylene, another molecule with multiconfigurational character, but not as strong as in the case of FOO. However, the observed effect is fictitious, and due only to the fact that CCSD(T) calculations, unless started from the ROHF density with the correct spin distribution, lead to a spin contaminated wave function, with an erroneous geometry. If the calculations are started from the correct ROHF wavefunction, the geometry obtained is now much closer to the experimental one, the wavefunction is the same as obtained at the CCSDT level, and no important CCSD(T)–CCSDT energy difference appears. This situation should be kept in mind, because it is probably arising also in the cases of ClOO, BrOO and OFO, as our preliminary calculations indicate. Despite being a single reference method, CCSDT describes the chemical structure and thermochemistry of FOO as well as multiconfigurational based methods such us CASPT2 [3] or, to be fair, DFT methods such as B3LYP or B3PW91 [8,35]. The lowest-lying quartet state was also investigated. A looser structure than that of the doublet ground state was found and the doublet-quartet energy gap was estimated to be 12.4 kcal/mol at the CCSDT/CBS level and 11.7 kcal/mol at the B3LYP level. Astonishingly as it may appear at first glance, B3LYP/6-311+G(3df) calculations are again demonstrated to be as good as CCSDT calculations with extended basis sets in the particular case of FOO.
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