Refined Theoretical Estimates of the Atomization Energies and

J. Phys. Chem. A 2010, 114, 613–623

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Refined Theoretical Estimates of the Atomization Energies and Molecular Structures of Selected Small Oxygen Fluorides David Feller* and Kirk A. Peterson Department of Chemistry, Washington State UniVersity, Pullman, Washington 99164-4630

David A. Dixon Chemistry Department, The UniVersity of Alabama, Shelby Hall, Tuscaloosa, Alabama 35487-0336 ReceiVed: August 23, 2009; ReVised Manuscript ReceiVed: NoVember 10, 2009

The heats of formation at 298.15 K and structures of five small oxygen fluoride molecules have been determined with coupled cluster theory incorporating at least through quadruple excitations. Because the wave functions of several of these systems exhibit strong multiconfiguration character, correlation recovery beyond CCSD(T) was found to be essential for achieving accurate results. Comparison is made with multireference configuration interaction properties where appropriate. The final ∆Hf(298 K) values obtained in this study are FO (2Π3/2) ) 26.5 ( 0.2, FOO (2A′′) ) 6.4 ( 0.7, OFO (2B2) ) 125.0 ( 0.3, FOF (1A1) ) 5.9 ( 0.3, and FOOF (1A) ) 6.4 ( 0.7 kcal/mol. For FO, FOO, and FOF, the theoretical ∆Hf values with their accompanying error bars easily fall within the experimental error bars. In the case of OFO the heat of formation has not been determined experimentally. The best current ∆Hf for FOOF lies outside the NIST-JANAF experimental error bars (4.59 ( 0.5 kcal/mol) despite the use of very high levels of theory and the adoption of what is believed to be a conservative estimate of the theoretical uncertainty. Good agreement with experiment was found for the structures. I. Introduction A recent, extensive survey examined the performance of a multicomponent, high-level coupled cluster theory approach to predicting the heats of formation and structures of more than 100 small molecules.1 Excellent overall agreement was found with the best available experimental data. For example, when comparisons were restricted to the subset of molecules whose experimental heats of formation at 298.15 K (∆Hf(298K)) possessed an uncertainty of (1 kcal/mol or less, the theoretical mean absolute deviation (MAD) was found to be only 0.15 kcal/ mol (91 comparisons). For the smaller set of molecules with even tighter experimental uncertainties ((0.15 kcal/mol or less), MAD decreased to 0.10 kcal/mol (49 comparisons). Frozen core (FC), single-reference-based coupled cluster theory up through a quasiperturbative treatment of triple excitations, commonly known as CCSD(T)(FC) and without further corrections, fell somewhat short of achieving so-called chemical accuracy ((1 kcal/mol) by displaying an MAD of ∼2 kcal/mol. Corrections for a number of well-known deficiencies in CCSD(T)(FC) theory proved necessary in order to achieve the higher level of accuracy. In particular, for chemical systems whose wave functions displayed significant multiconfiguration character the “higher order” (HO) correlation correction becomes important. Although there are several ways to introduce HO effects, our survey accounted for it by explicitly treating quadruple (or higher) excitations in the coupled cluster expansion. In a limited number of cases it proved possible to perform explicit full configuration interaction (FCI) calculations. Fortunately, for well-behaved systems whose wave functions are dominated by the Hartree-Fock (HF) configuration (cHF g 0.93) the size of * To whom correspondence should be addressed. E-mail: dfeller@ owt.com.

the HO correction is small relative to other corrections. Thus, for these systems the computationally very demanding HO correction can be avoided in all but the most demanding studies. Among the problematic multiconfigurational systems were two oxygen fluoride molecules, FOO (2A′′) and FOF (1A1). The purpose of the present study is to apply a uniformly high level of theory, which exceeds even that of the 2008 study, to these two systems and to FO (2Π), OFO (2B2) and FOOF (1A). Reliable experimental observations are not available for OFO. In a 2003 coupled cluster/configuration interaction study we found that it was difficult to predict the properties of FOOF as reliably as one would like.2 Despite the passage of years, the many nominally “inactive” lone pairs present in the oxygen fluorides continue to cause computational difficulties which impede the accurate prediction of molecular properties in molecules composed of strongly electronegative atoms. Numerous previous studies have examined one or more of these five molecules. We will not attempt a comprehensive review of that literature. Besides the 2003 and 2008 studies already mentioned, the FOO radical was examined with secondand fourth-order Møller-Plesset perturbation theory, complete active space (CAS) self-consistent field, and quadratic configuration interaction (QCI) methods by Francisco et al.3 The authors reported a final ∆Hf(0 K) for FOO of 8.9 ( 3 kcal/mol obtained by combining information from isodesmic and isogyric reactions with QCI/6-311G(d,p) results. Ventura and Keininger studied FOO and FOOF with B3LYP/6-311++G(3df,3pd) density functional theory (DFT).4 Their recommended heats of formation (∆Hf(FOO, 0 K) ) 7.2 kcal/mol and ∆Hf(FOOF, 0 K) ) 8.2 kcal/mol) were based on an average of two procedures. In a pair of computational studies appearing in 2004, Denis and Ventura5 and Denis6 reported the heats of formation and structures of FO, FOO, and OFO.

10.1021/jp908128g  2010 American Chemical Society Published on Web 12/07/2009

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FOOF was also the subject of an early application of the CCSD(T)(FC) analytic gradient technique by Scuseria, who used a [5s,3p,3d,1f] basis set.7 The goal of the study was to test the method on “difficult” quantum chemistry problems. Lee et al.8 used CCSD(T)(FC) with a triple-ζ atomic natural orbital basis set and obtained a ∆Hf(0 K) for FOOF of 9.7 ( 2.0 kcal/mol. No correction for core/valence correlation effects was included. An early success of DFT within the local density approximation (LDA) was the prediction of the geometry of FOOF.9 The authors reported a best F-O bond length that was 0.012 Å shorter than experiment and a O-O bond length that was 0.003 Å too long. Jursic used a variety of DFT methods in an investigation of the structures of FOO, FOOF, and two other small oxygen fluorides.10 In 1998, Keininger et al. studied the same collection of oxygen fluoride molecules as examined in the present study by applying the B3PW91 DFT method in combination with the correlation-consistent cc-pVQZ basis set and in some cases with the aug-cc-pVQZ basis set.11 That study also reported Gaussian-2 thermodynamic properties. The ∆Hf(298 K) results for FOOF (DFT ) 7.3, G2 ) 4.1 kcal/mol) were sufficiently at odds with the NIST-JANAF12 value of ∆Hf(298 K) ) 4.6 ( 0.5 kcal/mol, which the authors felt justified in calling for a re-examination of the experimental thermochemical data. More recently, Karton et al. reported results for FO, FOO, FOF, and FOOF at the W4.n level of theory.13 They also performed CCSDT(Q) calculations with a polarized double-ζ basis set to get an estimate of the higher order correlation corrections for the geometry of FOOF. Their best ∆Hf(298 K) values were 26.43 ( 0.11 kcal/mol for FO (W4.4), 5.94 ( 0.14 kcal/mol for FOF (W4.2), 5.87 ( 0.16 kcal/mol for FOO (W4), and 7.84 ( 0.18 to 8.21 ( 0.18 kcal/ mol for FOOF (W4) using the CCSD(T)/cc-pVQZ and experimental geometries. II. Computational Approach The composite approach employed here closely follows what is described in our recent survey.1 It represents the current evolution of a strategy developed over the past decade and a half in collaboration with multiple co-workers and will be described very briefly.2,14-29 The one-particle expansion makes use of the diffuse function augmented valence correlationconsistent Gaussian basis sets, aug-cc-pVnZ (n ) D, T, ..., 7),30-38 and the corresponding weighted core/valence (CV) basis sets, cc-pwCVnZ (n ) D, T, ..., 5).34 Only the pure angular components, e.g., 5-term d’s, were retained. Scalar relativistic (SR) calculations utilized the cc-pVQZ_DK recontracted basis set proposed de Jong et al.39 SR corrections to the atomization energies and molecular structures were obtained from second-order Douglas-Kroll-Hess (DKH) CCSD(T)(FC) calculations.40,41 Experience has shown that the SR correction is generally insensitive to the correlation treatment and basis set. For example, increasing the basis set from cc-pVTZ_DK to cc-pVQZ_DK changes the magnitude of the atomization energy corrections by no more than 0.02 kcal/mol in any of the five molecules of this study, an amount that is considerably below the uncertainty associated with either the frozen core CBS or higher order correction error bars. The most extensive valence basis set, aug-cc-pV7Z, contains multiple shells of 15-term k functions (l ) 7). The current version of MOLPRO does not support k functions, and calculations with the few other programs that do support very high angular momentum functions proved to be too time consuming. Consequently, the small energy contribution arising from k functions in the basis set was estimated by exploiting the uniformity in the correlation energy convergence pattern as a

Feller et al. function of l, as described elsewhere.17,42,43 The contribution of k functions to the total SCF energy is on the order of 3µEh, and the contribution to SCF energy differences is even smaller. For the largest molecule in the current study, FOOF, k functions were estimated to contribute just 0.2 kcal/mol to the CCSD(T) atomization energy. Tests on smaller molecules with explicit inclusion of k functions suggest that our estimates should be accurate to approximately (0.02 kcal/mol for these systems. Spectroscopic properties (re, De, and ωe) for FO were obtained from a sixth-degree Dunham fit of the potential energy curve.44 All CCSD(T) calculations were performed with MOLPRO 2008.2.45 In general, geometries were optimized for every method/basis set combination. The only exceptions were very large CCSDT and CCSDTQ calculations that required multiple days per energy evaluation and CCSDTQ5 and CCSDTQ(5) calculations on polyatomic species, where CCSDTQ geometries were used. CCSDT and CCSDTQ geometry optimizations for the polyatomic species were accomplished with a sequential quadratic interpolation procedure in the internal coordinates. Open-shell calculations made use of the R/UCCSD(T) method, which begins with restricted open-shell Hartree-Fock (ROHF) orbitals but allows a small amount of spin contamination in the solution of the CCSD equations.46-48 It is requested in MOLPRO with the keyword “UCCSD(T)”. Full atomic symmetry on the orbitals was imposed when evaluating the atomic CCSD(T) energies. Atomic symmetry was not imposed in the CCSDT, CCSDTQ, and CCSDTQ5 calculations.49,50 All CCSDT, CCSDTQ, and CCSDTQ5 calculations were performed with the MRCC program of Ka´llay and co-workers51,52 interfaced to MOLPRO. The largest CCSDT(Q) calculation involved approximately 8 × 109 determinants, while the largest CCSDTQ calculation involved 3.6 × 109 determinants. In addition to CCSDT, CCSDTQ, and CCSDTQ5, the HO correction includes an estimate for the remaining error with respect to the exact, full configuration interaction (FCI) limit based on the continued fraction (cf) approximant, originally formulated by Goodson in terms of HF, CCSD, and CCSD(T) energies.53 In the present context, we used the CCSD, CCSDT, and CCSDTQ sequence of energies, or in some cases the CCSDT, CCSDTQ, and CCSDTQ5 energies. Despite the use of very large valence basis sets, the residual finite basis set error is large enough that it must be considered in some fashion. Estimates of the remaining correlation energy were obtained from an average of five complete basis set (CBS) extrapolation formulas. These include an exponential,54,55 a mixed Gaussian/exponential,56 two inverse powers of lmax, where lmax is the highest angular momentum present in the basis set,57-60 and an expression suggested by Schwenke.61 The spread among the five values was taken as an estimate of the uncertainty in the final result. An illustration of the performance of this approach can be seen in Figure 1, where the convergence of the electronic atomization energy, ΣDe, for FOOF is shown as a function of the underlying basis sets. All of the CBS extrapolations provide a significant improvement over the raw CCSD(T) values, and the average of the extrapolations exhibits relatively stable behavior with respect to improvements in the basis sets. As was the case with the atomization energies, bond lengths obtained with the aug-cc-pV7Z basis set contain a small residual basis set error, typically on the order of a few parts in 10-4 Å. An exponential formula was used to estimate the bond lengths in the CBS limit. For FOOF the largest bond length correction resulting from the extrapolation was -0.0005 Å. Bond angles

Structures of Selected Small Oxygen Fluorides

J. Phys. Chem. A, Vol. 114, No. 1, 2010 615 III. Results and Discussion Wave functions for the present five oxygen fluoride systems, expressed in terms of natural orbitals derived from CAS internally contracted multireference configuration interactions (iCAS-CI)65,66 expansions, are as follows

ψ(FO 2∏) ≈ 0.94(1σ22σ23σ24σ25σ21πx21πy22πy22πx1) 0.16(...5σ2 f 6σ2...2πx1) + ... (1a) ψ(FOO 2A″) ≈ 0.87(1a′22a′23a′24a′25a′26a′27a′28a′21a′′29a′22a′′210a′23a′′1) 0.30(...10a′2 f 11a′2...3a′′1) + ... (2) ψ(OFO 2B2) ≈ 0.87(1a122a121b223a124a122b221b125a123b222b121a226a124b22) 0.14(...6a12 f 4a225b21) - 0.11(...3b22 f 5b22) + ...

(3a)

ψ(OFO B1) ≈ 2

0.83(1a122a121b223a124a122b221b125a123b221a226a124b222b11) Figure 1. Convergence of the frozen core CCSD(T) electronic atomization energy of FOOF as a function of the valence basis set.

0.25(...1a224b22 f 7a11a212b124b21) + 0.17(...1a226a12 f 6a112b125b211a21) + ...

(3b)

ψ(FOF A1) ≈ 1

did not require extrapolation because they were already converged to approximately 0.01°. Accurate anharmonic zero-point vibrational energies (ZPEs) are needed for computing heats of formation. The ZPE for FO was obtained from the Dunham fit previously mentioned. The ZPE for FOO was obtained from a quartic fit of the R/UCCSD(T)(FC)/ aug-cc-pVTZ near-equilibrium potential energy surface. For OFO and FOF the ZPE was based on the expression

ZPE ) 0.5(ZPEH+ZPEF) + χ0- 0.25(Σχii)

(1)

where ZPEH ) 0.5(Σωi) and ZPEF ) 0.5(Σνi). The harmonic component was obtained from CCSD(T)(FC)/aug-cc-pVQZ calculations, and the anharmonic corrections were based on second-order Møller-Plesset perturbation theory, MP2(FC)/augcc-pVTZ, calculations with Gaussian 03,63 which includes a second-order perturbative treatment of such effects using finite difference evaluations of third and semidiagonal fourth derivatives. Finally, the anharmonic ZPE for FOOF was based on eq 1 and CCSD(T)/aug-cc-pVTZ harmonic frequencies. A first-order correction to the electronic energy due to the motion of the nuclei, known as the diagonal Born-Oppenheimer correction (DBOC), was obtained from Hartree-Fock calculations using the aug-cc-pVTZ basis set. These calculations were performed with a development version of PSI3.64 Hartree-Fock theory typically overestimates the magnitude of this correction compared to correlated levels of theory, but as will be seen, for the molecules in this study the DBOC corrections are very small. While the W4.n model chemistries used by Karton et al. in their recent study of small oxygen fluoride and oxygen chloride molecules made use of many of the same components as were used here, the W4.n approaches differ in their details.13 For example, in the determination of the CCSD(T)(FC) CBS limit, Karton et al. perform separate CCSD singlet and triplet pair energy extrapolations with basis sets up through aug-cc-pV6Z. The (T) piece is extrapolated from calculations through augcc-pV5Z. We have chosen to extrapolate the total CCSD(T)(FC) energies through aug-cc-pV7Z. We also make use of optimized geometries at every level of theory, with the exception of very large CCSDT and CCSDTQ calculations that require multiple days per energy evaluation. 62

0.91(1a121b222a123a122b224a121b125a123b221a224b226a122b12) + ...

(4) ψ(FOOF 1A) ≈ 0.83(1b21a22a22b23a23b24a24b25a25b26a27a26b27b28a28b29a2) 0.22(...8b29a2 f 8b19b19a110a1) - 0.14(...8b2 f 9b2) + ... (5)

In FO (2Π3/2) the unpaired spin density resides primarily on the oxygen, while in FOO (2A′′) it is on the terminal oxygen. The 2 B1 state of OFO corresponds to a low-lying, first-order transition state. To the extent the magnitude of the leading coefficients are a reliable indicator of difficulty for the single-reference CCSD(T) method, FOO, OFO, and FOOF all appear likely to pose problems as discussed in more detail below. CCSD(T)(FC) total energies, electronic atomization energies, harmonic ZPEs, and structural parameters are given in Table 1 for the six valence basis sets used in this study. Frequencies. With the small aug-cc-pVDZ basis set, the 2B2 state of OFO was predicted to be a first-order transition state with an imaginary frequency of 273.8i cm-1corresponding to a b2 asymmetric stretching mode. However, normal-mode analyses with larger basis sets found the 2B2 state to be a minimum with a lowest frequency b2 mode near 370 cm-1. The next higher lying state (2B1) of OFO was found to be a first-order transition state even at the R/UCCSD(T)(FC)/aug-cc-pVQZ level of theory with an imaginary frequency of 370i cm-1. Methods based on unrestricted Hartree-Fock (UHF) wave functions are plagued by extensive spin contamination in the case of the FOO radical. With an S2 of 1.8 (compared to 0.75 for a pure doublet), the UMP2(FC) method predicts an FO bond length in FOO that is roughly 1 Å too long. Increasing the amount of correlation recovery via UCCSD(T) partially compensates for the spin contamination, but the FO bond length then becomes ∼0.08 Å shorter than values predicted by R/UCCSD(T) or RCCSD(T). Since these in turn are shorter than the bond length predicted from RCCSDTQ, UCCSD(T) would appear to still suffer the effects of spin contamination. Denis and Ventura discussed the problems arising from spin contamination for FOO at the UCCSD(T) level of theory5 and concluded that CCSDT improves agreement with experiment. In contrast,

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TABLE 1: Frozen Core CCSD(T) Property Convergence with Respect to the Valence Basis Seta FO (2Π) basis

ETOT

De

harmonic EZPE

rFO

aug-cc-pVDZ aug-cc-pVTZ aug-cc-pVQZ aug-cc-pV5Z aug-cc-pV6Z aug-cc-pV7Zb

-174.542625 -174.685031 -174.729348 -174.744493 -174.749682 -174.752201

42.06 49.22 51.06 51.64 51.91 52.07

1.42 1.51 1.52 1.53 1.53 1.53

1.3784 1.3601 1.3536 1.3520 1.3513 1.3510

FOO (2A′′) basis

ETOT

ΣDe

harmonic EZPE

rOO

rFO

∠OOF


-249.584398 -249.785687 -249.849587 -249.870895 -249.878272 -249.881880

114.98 125.67 129.57 130.57 131.08 131.37

3.54 3.60

1.2010 1.1959 1.1908 1.1900 1.1896 1.1895

1.6917 1.6414 1.6357 1.6326 1.6314 1.6309

110.71 110.85 110.89 110.85 110.84 110.83

OFO (2B2) basis

ETOT

ΣDe

harmonic EZPE

rFO

∠OFO


-249.397851 -249.599026 -249.661025 -249.682102 -249.689333 -249.692875

-2.08 8.54 11.24 12.10 12.52 12.58

T.S.c 2.04 2.10

1.5159 1.4880 1.4830 1.4810 1.4802 1.4799

87.50 87.60 87.38 87.38 87.37 87.36

OFO (2B1) basis

ETOT

ΣDe

aug-cc-pVDZ aug-cc-pVTZ aug-cc-pVQZ aug-cc-pV5Z

-248.811524 -249.595479 -249.657317 -249.678351

-4.86 6.31 8.91 9.75

rFO

∠OFO

1.4788 1.4478 1.4417 1.4398

120.07 119.66 119.65 119.60

harmonic EZPE d

T.S. T.S.d T.S.d FOF (1A1)

basis

ETOT

ΣDe

harmonic EZPE

rFO

∠FOF


-274.146715 -274.375116 -274.446150 -274.470542 -274.478939 -274.483005

75.98 88.32 91.19 92.08 92.52 92.80

3.03 3.24 3.27

1.4325 1.4119 1.4057 1.4041 1.4034 1.4031

102.68 102.98 103.03 103.01 103.01 103.01

FOOF (1A) basis

ETOT

ΣDe

harmonic EZPE

rOO

rFO

∠OOF

∠FOOF

aug-cc-pVDZe aug-cc-pVTZ aug-cc-pVQZ aug-cc-pV5Z aug-cc-pV6Z aug-cc-pV7Zb

-349.156896 -349.442127 -349.532077 -349.562473 -349.572986 -349.578117

129.08 143.66 148.16 149.39 150.01 150.40

4.96 5.08

1.2095 1.2340 1.2289 1.2299 1.2303 1.2304

1.6279 1.5448 1.5390 1.5345 1.5325 1.5318

109.21 108.54 108.58 108.51 108.48 108.48

88.68 87.74 87.69 87.65 87.62 87.62

a Total energies (ETOT) at the optimized geometries are in Hartrees. Electronic atomization energies (ΣDe) and harmonic zero-point vibrational energies (EZPE) are in kcal/mol. The atomization energies are with respect to symmetry equivalenced R/UCCSD(T) atoms. Open-shell species were treated with the R/UCCSD(T) method. Structural parameters are in Angstroms and degrees. b The aug-cc-pV7Z basis set results include an estimate for the small effects of missing k functions, as described in the text. c With the aug-cc-pVDZ basis set, OFO (2B2) has a single b2 symmetry mode with an imaginary frequency of 273.8i cm-1 corresponding to dissociation to FO + O. d With the aug-cc-pVDZ basis set, OFO (2B1) has a single b2 symmetry mode with an imaginary frequency of 462.7i cm-1 corresponding to dissociation to FO + O. At the aug-cc-pVTZ basis set level this mode has a frequency of 357.1i cm-1, and with the aug-cc-pVQZ basis set it is 365.2i cm-1. e For comparison purposes, the iCAS-CI+Q/aug-cc-pVDZ structure is rOO ) 1.2133 Å, rFO ) 1.6409 Å, ∠OOF ) 109.30°, and ∠FOOF ) 88.62°.

we find the effects of iterative triples at the RCCSDT(FC)/ccpVQZ level of theory to be very small (-0.02 kcal/mol) and to be completely swamped by the much larger (2.90 kcal/mol) contribution from quadruple excitations. The spin contamination issue is distinct from the degree of multiconfigurational character in the wave function. While OFO (2B2) displays a significant

degree of multiconfigurational character, the magnitude of spin contamination in spin-unrestricted calculations on this molecule is very small. The theoretical fundamental frequencies for FO and FOO predicted in this study are in good agreement with the available experimental data. We predict the following values for FO ν )

Structures of Selected Small Oxygen Fluorides 1045.3 cm-1 (theory) vs 1048 (expt.),12 and for FOO νi ) 380.4, 597.8, and 1480.6 cm-1 (theory) vs 376.0, 579.3, and 1486.9 (expt.).67 The FOO values are based on an R/UCCSD(T)/augcc-pVTZ anharmonic fit of the potential surface. Atomization Energies. Table 2 contains the best composite theoretical estimates of the electronic atomization energies and structures together with the corresponding uncertainties for the frozen core, CV, and HO corrections. Of these three, the last is the most difficult to estimate with any degree of accuracy due to the very high computational cost of post CCSD(T) methods. In our 2008 survey,1 we discussed specific examples of the difficulties associated with balancing the triples and quadruples components, which are often of opposite signs and individually much larger than the sum of the two. In that work, we adopted the practice of computing the triples contribution with a basis set one step larger than was used for the quadruples in recognition of the slower convergence in the former. For example, we could use a CCSDT(FC)/cc-pVTZ + CCSDTQ(FC)/cc-pVDZ combination. Table 3 shows how this pairing of T/V(n+1)Z + Q/VnZ methods converges as the underlying basis sets are improved for H2O2 and three of the oxygen fluoride systems. We have chosen one-half the difference in the T/VTZ + Q/VDZ f T/VQZ + Q/VTZ corrections as an approximate uncertainty in the HO correction. This ignores any potential additional uncertainty arising from the much smaller continued fraction FCI component. Table 2 contains corrections for atomic and molecular spin-orbit (SO) effects. The contribution of atomic SO coupling must be considered for high-accuracy atomization energies. Our nonrelativistic atomic calculations describe an average multiplet state, requiring a shift in the energy of the atomic asymptotes which has the effect of decreasing the atomization energy. Although the atomic multiplet splittings could be obtained from theory, we have chosen to use the tabulated values of C. Moore, i.e., 0.216 kcal/mol for O and 0.385 kcal/mol for F.68 The molecular SO correction for FO, -0.26 kcal/mol, was obtained from a CI calculation with the cc-pVTZ basis set. The decision to base the error estimates for our final atomization energy results using a molecule-by-molecule criterion that encompasses the uncertainties in the major components is consistent with our 2008 study.1 In a worst-case scenario our uncertainties are additive. Therefore, reliance on fortuitous cancellation of error is not assumed. “Error” in the present context is measured with respect to the exact solution of the molecular Schro¨dinger equation. As already discussed, with the present state of development of our composite approach the principal contributions to that error arise from the spread in the FC and CV CBS extrapolations and the uncertainty associated with the major pieces of the higher order correction. This admittedly crude error estimate strategy is necessitated by a lack of formal error bars in conventional electronic structure calculations, which unlike stochastic experimental measurements are completely deterministic in nature. Unlike most electronic structure techniques, quantum Monte Carlo (QMC) methods69 do exhibit stochastic characteristics. Our reading of the QMC literature suggests that achieving an uncertainty of less than 1 kcal/mol in the atomization energy of a system like FOOF would be computationally very challenging and geometry optimization or normal-mode analyses would likely be beyond current capabilities. Further comments on QMC are beyond the scope of the present discussion. It is our belief that the current error estimates are conservative. As our approach continues to evolve, we expect to improve both the accuracy of our property predictions as well as the estimates of our uncertainty.

J. Phys. Chem. A, Vol. 114, No. 1, 2010 617 Other research groups have chosen distinctly different approaches for estimating the uncertainty in their theoretical predictions. Karton et al.13 based their uncertainties on the performance of Wn theory when applied to a set of 25 small molecules70 for which presumed accurate data was available from the Active Thermochemical Tables (ATcT) using the Core (Argonne) Thermochemical Network.71-73 For example, they assign error bars of (0.18 kcal/mol to their best ∆Hf(298 K) value for FOOF, which is approximately twice the root-meansquare (rms) deviation for W4 with respect to the ATcT values. The implicit assumption underlying this approach is that the performance of Wn for the small test set is sufficiently representative of molecules in general that the uncertainties can be used interchangeably. The uncertainties reported by Karton et al. are significantly smaller than the ones we report. In the case of FOOF, Karton et al. report two theoretical values. The first, 7.84 ( 0.18 kcal/mol, corresponds to the usual CCSD(T)/ cc-pVQZ geometry employed by W4 theory. The second, 8.21 kcal/mol, was obtained at the experimental geometry of Jackson.74 Since these values differ by 0.37 kcal/mol (because the default W4 geometry differs significantly from experiment), it raises the question of whether error bars as small as (0.18 kcal/ mol are realistic. Finally, the uncertainties given in the 2009 paper of Karton et al. were derived “For a set of 25 first- and second-row small molecules”,13 with MAD ) 0.066 kcal/mol and rms ) 0.085 kcal/mol (95% confidence interval ) 0.16 kcal/ mol), with reference made to an earlier 2006 paper by some of the same authors.70 Tables V and VI in the earlier paper include data on 30 molecules, with 26 accompanying ATcT values. The 2006 paper discusses the rms value of 0.085 kcal/mol as being associated with just the first row compounds and an increase in rms to 0.15 kcal/mol when second-row compounds were considered. For the five systems examined here, the CV correction is small, never exceeding 0.5 kcal/mol in absolute magnitude for ΣDe or -0.003 Å for bond lengths. For other systems of similar size, the CV correction can approach 4 kcal/mol.1 Oxygen fluorides are uncharacteristic of most first- and second-row molecules in that the CV correction slightly reduces ΣDe, whereas normally it increases the atomization energy. The DKH scalar relativistic correction is similarly small. The results in Table 2 demonstrate the importance of higher order (i.e., beyond CCSD(T)) correlation recovery. By a significant margin, it is the largest of the six electronic energy corrections for the systems considered here. Footnotes to Table 2 provide the details of the combination of methods and basis sets used for determining ∆HO for each system. In terms of the electronic atomization energies, ∆HO increases ΣDe by approximately 2% across all systems. Clearly, for small oxygen fluoride compounds, the higher order correction is essential for achieving chemical accuracy without undue reliance on fortuitous cancellation of error. For FO, FOO, and OFO, the CCSDT and CCSDTQ corrections are either of the same sign or the CCSDT correction is close to zero. For many small molecules the contributions to the atomization energy of triples and quadruples are of opposite sign and are individually much larger than the total ∆HO correction.1 In the case of FOF and FOOF, the triples and quadruples corrections are of opposite signs, with the latter dominating. Due to their magnitudes, converging the ∆HO corrections for FOO and FOOF presents significant computational difficulties. For FOOF, in particular, the situation is exacerbated by the relatively large number of electrons being correlated. Among the many systems we have examined to date, it is unusual to

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Feller et al.

TABLE 2: Best Composite, Vibrationless Atomization Energies and Equilibrium Structuresa FO (2Π3/2) component

De

rFO

CCSD(T)(FC)/CBS ∆CCSD(T)(CV)/CBS ∆CCSD(T)-DKH/VQZ ∆HO(CV)/pwCVDZ ∆HO(FC)b atomic + (mol S.O.)c ∆DBOC RHF/aVTZ

52.27 ( 0.10 -0.05 ( 0.01 -0.08 0.02 0.76 ( 0.05 -0.34 0.004 52.58 ( 0.2

1.3508 -0.0018 0.0004 NA 0.0046 NA NA 1.3540 1.3541

experimentald FOO (2A′′) component

ΣDe

rOO

rFO

∠OOF

CCSD(T)(FC)/CBS ∆CCSD(T)(CV)/CBS ∆CCSD(T)-DKH/VQZ ∆HO(FC)e atomic S.O. ∆DBOC RHF/aVTZ

131.72 ( 0.15 -0.01 ( 0.01 -0.16 2.40 ( 0.5 -0.82 0.003 133.13 ( 0.7

1.1894 -0.0020 0.0002 0.0050 NA NA 1.1926 1.200

1.6305 -0.0016 -0.0002 0.0291 NA NA 1.6578 1.649

110.83 0.00 0.01 0.08 NA NA 110.9 111.2

experimentalf

OFO (2B2) component

ΣDe

rFO

∠OFO

CCSD(T)(FC)/CBS ∆CCSD(T)(CV)/CBS ∆CCSD(T)-DKH/VQZ ∆HO(FC)g atomic S.O.

13.06 ( 0.16 -0.44 ( 0.02 0.03 2.76 ( 0.10 -0.82 14.59 ( 0.3

1.4799 -0.0029 0.0001 0.0167 NA 1.4938

87.36 0.03 -0.01 1.35 NA 87.73

FOF (1A1) component

ΣDe

rFO

∠OOF

CCSD(T)(FC)/CBS ∆CCSD(T)(CV)/CBS ∆CCSD(T)-DKH/VQZ ∆HO(FC)h atomic S.O. ∆DBOC RHF/aVTZ

93.16 ( 0.22 -0.17 ( 0.01 -0.11 0.87 ( 0.12 -0.99 0.01 92.77 ( 0.3

1.4025 -0.0017 0.0004 0.0049 NA NA 1.4061 1.4053

103.01 0.01 0.00 0.11 NA NA 103.1 103.1

experimentali

FOOF (1A) component

ΣDe

rOO

rFO

∠OOF

∠FOOF

CCSD(T)(FC)/CBS ∆CCSD(T)(CV)/CBS ∆CCSD(T)-DKH/VQZ ∆HO(FC)j atomic S.O. ∆DBOC RHF/aVTZ

150.89 ( 0.22 -0.21 ( 0.01 -0.16 2.91 ( 0.5 -1.20 0.02 152.25 ( 0.7

1.2305 -0.0018 0.0002 -0.0063 NA NA 1.2226 1.217 1.216

1.5313 -0.0020 0.0002 0.0561 NA NA 1.5857 1.575 1.586

108.48 0.00 0.00 0.92 NA NA 109.4 109.5 109.2

87.62 -0.01 0.01 0.20 NA NA 87.8 87.5 88.1

experimentalk experimentall

a Atomization energies are in kcal/mol. Structural parameters are in Angstroms and degrees. Open-shell states were treated with the R/UCCSD(T) method. NA ) not available. b FO (2Π) higher order correction based on CCSDT(FC)/cc-pVQZ + CCSDTQ(FC)/cc-pVTZ + CCSDTQ5(FC)/ cc-pVDZ + cf est. FCI/cc-pVDZ. c Molecular spin-orbit calculation based on a iCAS-CI/cc-pVTZ calculation. d Experimental FO bond length taken from Hammer et al.75 e FOO (2A′′) higher order correction based on CCSDT(FC)/cc-pVQZ + scaled CCSDT(Q)(FC)/cc-pVTZ(no f) + CCSDTQ5(FC)/cc-pVDZ(no d) + cf est. FCI/cc-pVDZ. f Experimental high-resolution IR FOO re structure taken from Yamada and Hirota.76 The reported uncertainties are rOO and rFO ( 0.013 Å, ∠OOF ( 0.36°. g OFO (2B2) higher order correction based on CCSDT(FC)/cc-pVQZ + CCSDTQ(FC)/cc-pVTZ + CCSDTQ5(FC)/cc-pVDZ(no d) + cf est. FCI/cc-pVDZ(no d). h FOF (1A1) higher order correction based on CCSDT(FC)/ cc-pVQZ + scaled CCSDT(Q)(FC)/cc-pVTZ + cf est. FCI/cc-pVTZ. i Experimental FOF structure taken from Morino and Saito.77 The reported uncertainties are rFO ( 0.0004 Å, ∠FOF ( 0.05°. j FOOF (1A) higher order correction for ΣDe based on estimated CCSDT(FC)/cc-pVQZ + CCSDTQ(FC)/cc-pVTZ + CCSDTQ(5)/cc-pVDZ(no d) + cf est. FCI/cc-pVDZ. The first piece was obtained from adding the CCSDT/cc-pVTZ f cc-pVQZ difference (-0.23) in FOO to the CCSDT(FC)/cc-pVTZ value in FOOF. The second piece was obtained by adding the CCSDT(Q) cc-pVDZ f cc-pVTZ difference (-0.70) in FOOF reported by Karton et al. to the scaled CCSDT(Q)(FC)/cc-pVDZ value in FOOF. The HO correction for the structure was based on CCSDT/cc-pVTZ + CCSDT(Q)/cc-pVDZ calculations. k Experimental FOOF microwave rs/r0 structure taken from Jackson.74 The reported uncertainties are rOO and rFO ( 0.003 Å and ∠OOF and ∠FOOF ( 0.5°. l Experimental FOOF electron diffraction rg values from Hedberg et al.79 The reported uncertainties are rOO and rFO ( 0.002 Å and ∠OOF ( 0.2° and ∠FOOF ( 0.4°.

Structures of Selected Small Oxygen Fluorides TABLE 3: Basis Set Sensitivity of the Triples and Quadruples Pieces of the Higher Order Correlation Correction to Atomization Energies (kcal/mol)a system

T/VTZ + Q/VDZ

T/VQZ + Q/VTZ

H 2 O2 FO FOO FOF

-0.39 + 0.74 ) 0.35 0.29 + 0.54 ) 0.83 0.21 + 2.90 ) 3.11 -0.36 + 1.39 ) 1.03

-0.48 + 0.65 ) 0.17 0.24 + 0.48 ) 0.72 -0.02 + 2.08 ) 2.06 -0.46 + 1.21 ) 0.75

a On the basis of CCSDT(FC)/cc-pV(n+1)Z + CCSDTQ(FC)/ cc-pVnZ combinations, with the exception of FOO where a CCSDTQ(FC)/cc-pVTZ(no f) calculation was substituted. Inclusion of the f functions for FOO would have resulted in 13 × 109 determinants, which was beyond the capabilities of our hardware.

encounter differences exceeding 0.3-0.4 kcal/mol in absolute magnitude as the ∆HO basis set expansion was improved from T/VTZ + Q/VDZ to T/VQZ + Q/VTZ, even in systems with pronounced multiconfigurational character. For example, with C2 the CCSDT/VTZ + CCSDTQ/VDZ combination produces a correction of -0.25 kcal/mol, whereas the improved CCSDT/ VQZ + CCSDTQ/VTZ combination yields -0.02 kcal/mol. Our estimate of ∆HO(C2) at the CBS limit is ∼0.09 kcal/mol. Of course, C2 possesses only 8 valence electrons as compared to the 26 valence electrons in FOOF. Karton et al.13 report a -0.70 kcal/mol decrease in the CCSDT(Q) correction when the basis set was increased from cc-pVDZ to cc-pVTZ. By incorporating their VDZ f VTZ correction, we will attempt to estimate the CCSDT + CCSDTQ piece of the HO correction at the CCSDT/ VQZ + CCSDTQ/VTZ level. Neither piece can currently be computed directly due to software and hardware limitations. With the smaller CCSDT/VTZ + CCSDTQ/VDZ basis set combination we obtained a correction of 3.21 kcal/mol. The CCSDT/VQZ correction for the atomization energy was estimated by adding the VTZ f VQZ change in the triples correction taken from FOO (-0.23 kcal/mol) to the FOOF CCSDT/VTZ value (-0.40 kcal/mol). Judging by the CCSDT convergence patterns for FO, FOF, and FOO, this may slightly underestimate the true value. Karton et al.13 report an estimated CBS triples correction of -0.74 kcal/mol using VDZ and VTZ basis sets. The CCSDTQ/VTZ component was estimated by combining the VDZ f VTZ change in the CCSDT(Q) correction for FOOF (-0.70) to our VDZ value (3.61 kcal/mol). Notice that the CCSDT(Q) VDZ f VTZ term was evaluated at the CCSD(T)/cc-pVQZ geometry of Karton et al.,13 while the CCSDTQ/cc-pVDZ was evaluated at the optimal CCSDT(Q)/cc-pVDZ geometry. It would have been preferable to evaluate the VDZ f VTZ term at something close to the CCSDT(Q)/cc-pVTZ optimal geometry, but at present such a calculation is beyond our capabilities. This situation certainly contributes to the uncertainty in our higher order correction but was unavoidable. Thus, our final CCSDT/VQZ + CCSDTQ/ VTZ estimate is 2.28 kcal/mol, a decrease of nearly 1 kcal/mol relative to the value obtained from the smaller VTZ and VDZ basis sets. The corresponding value (CCSDT + CCSDT(Q) + CCSDTQ correction) reported by Karton et al.13 in their Table 4 is 1.27 kcal/mol. However, the T-(T) + T4 component in W4 theory is 1.55 kcal/mol because T4 is based on a sum of four terms, all of which are empirically scaled by a factor of 1.10. Part of the difference with the value obtained in the present work is due to the use of different geometries when evaluating the higher order correction. For most molecules, ∆HO shows little sensitivity to the choice of geometry. Even for systems with pronounced multiconfigurational character, such as C2 and O3, the impact

J. Phys. Chem. A, Vol. 114, No. 1, 2010 619 of using a CCSD(T) geometry instead of a geometry optimized at a higher level of theory is small. For example, in the case of O3 use of the optimal CCSDTQ geometry leads to an elongation of the O-O bonds by 0.006 Å but the atomization energy increases by only 0.01 kcal/mol. However, for FOO and FOOF, where levels of theory beyond CCSD(T) produce changes in bond lengths exceeding 0.05 Å, the choice of what structure to use for computing ∆HO can have a significant effect on the magnitude of the correction. This is particularly true for FOOF. The approach we adopted in our 2008 study,1 as well as in the present work, relies on components evaluated at their respective optimized geometries in order to yield corrections for molecular structures and harmonic frequencies in addition to atomization energies. Conceptually, we seek to approximate the FCI result in the large basis set limit at the optimal FCI geometry. Of course, the size of the system places practical limitations on our ability to achieve this goal. In order to illustrate the sensitivity of the triples + quadruples portion of the higher order correction to the geometry, calculations were performed with the cc-pVDZ basis set at the W4 (CCSD(T)/cc-pVQZ) geometry and at the experimental geometry. Use of the W4 geometry yields a T+Q correction of 2.2 kcal/mol, while the experimental geometry produces a value of 3.0 kcal/mol. For comparison purposes, when separate CCSDTand CCSDT(Q)-optimized geometries are used for computing each component, the correction is 4.1 kcal/mol. Differences of this size emphasize the difficulty of accurately determining the T+Q part of the higher order correction and estimating the associated uncertainty in the atomization energy. The final piece of the higher order correlation correction is associated with excitations beyond quadruples and is meant to approximate the remaining difference with respect to FCI. As previously discussed, in the case of FO, OFO, and FOO we determined this component from CCSDTQ5 calculations using the cc-pVDZ (or cc-pVDZ(no d)) basis sets and a continued fraction approximant. Unfortunately, even with the small ccpVDZ(no d) basis set the number of determinants in a CCSDTQ5 calculation on FOOF exceeded 4.5 × 109 and was beyond the capabilities of our hardware. With the cc-pVDZ basis set, the cf(CCSD/CCSDT/CCSDTQ)-estimated FCI approximation yields a correction to CCSDTQ of 0.62 kcal/mol at the optimal CCSDT(Q) geometry, considerably larger than the 0.27 kcal/mol obtained by Karton et al.13 at the CCSDTQ(5)/ccpVDZ(no d) level of theory. A rough measure of the accuracy of the cf approximant can be obtained for systems (FO, OFO, FOF, and FOOF without d) where cf(CCSDT/CCSDTQ/ CCSDTQ5) numbers are available. The average error for these four systems is 0.11 kcal/mol. In three out of four instances the cf(CCSD/CCSDT/CCSDTQ) corrections overshoot the cf(CCSDT/CCSDTQ/CCSDTQ5) values. Using the optimal CCSDT(Q)/cc-pVDZ geometry, we obtained a scaled CCSDTQ(5)/ cc-pVDZ(no d) + cf estimated FCI correction of 0.63 kcal/ mol, essentially the same as what was predicted by the cf(CCSD/ CCSDT/CCSDTQ) approximate with the cc-pVDZ basis set. In light of the approximate nature of the current ∆HO estimate, we attach relatively large error bars ((0.5 kcal/mol). Table 4 contains a breakdown of the higher order correction determined in this study as well as the corresponding breakdown for the correction reported by Karton et al.13 Geometries. Agreement between theory and experiment for the bond lengths and bond angles in Table 2 is very close in light of the reported uncertainties in the experimental values and the lack of vibrational averaging/thermal effects in the theoretical results for polyatomic species. We predict re(FO) )

620


1.3541 Å compared to the observed value of 1.3540 Å from Hammer et al.75 For FOO, we predict an O-O bond distance 0.007 Å shorter than experiment and an F-O distance that is 0.009 Å longer than experiment. In light of the (0.013 Å experimental bond length uncertainties, it is impossible to draw any further conclusions.76 In the case of FOF, theory and experiment77 are in near perfect agreement. Because of the multireference character of OFO (2B2) noted above, it is interesting to compare the structures predicted by singlereference-based CCSD(T)/aug-cc-pVDZ + ∆HO and multireference-based iCAS-CI(FC)+Q/aug-cc-pVDZ theory, where the +Q notation implies the use of the multireference Davidson correction to account for the contribution of unlinked clusters.78 Despite the semiquantitative nature of the Davidson correction, the two structures are in remarkably close agreement with rFO bond lengths differing by only 0.006 Å and bond angles agreeing to better than 0.1°. While the partially optimized FOOF structure reported by Scuseria at the CCSD(T)(FC)/TZ2Pf level of theory agrees well with experiment ([rOO] ) 0.001 Å, [rFO] ) 0.014 Å), this agreement is likely due to a fortuitous cancellation of errors given the limitations inherent to frozen core CCSD(T) with a medium size basis set.7 For comparison purposes, the experimental microwave rs/r0 values of Jackson74 and the electron diffraction rg values of Hedberg et al.79 are provided in Table 2. Our current CCSD(T)(FC)/CBS estimates for rOO and rFO differ from the TZ2Pf values of Scuseria by 0.013 and -0.058 Å, respectively. A more recent theoretical investigation by Kraka et al.80 reported a CCSD(T)/CBS estimate for the structure corresponding to rOO ) 1.221 Å, rFO ) 1.547 Å, ∠OOF ) 108.8°, and ∠FOOF ) 88.0°. These values differ considerably from the CBS values determined in this study (see Table 2). Although the Kraka et al. CBS extrapolated bond lengths were based on values obtained with the aug-cc-pVTZ and aug-cc-pVQZ basis sets, it appears that all electrons were correlated in their calculations. The cc-pVnZ and aug-cc-pVnZ sequences of basis sets were designed for valence correlation recovery and will frequently produce misleading results when used in all electron calculations. An accurate all electron treatment requires basis sets specifically designed for core/valence correlation recovery, e.g., cc-pwCVnZ. Kraka et al. concluded that the CCSD(T) level of theory was incapable of correctly predicting the structure of FOOF. We agree with this conclusion if high accuracy is the goal, but we note that our CCSD(T)(FC)/CBS or (FC) + (CV) + DKH structure differs considerably from the structure they reported. Karton et al. demonstrated that higher order correlation effects are important in obtaining the geometry of FOOF.13 They estimated the impact of higher order correlation from the difference in the geometries at the CCSDT(Q)/cc-pVDZ and CCSD(T)/cc-pVDZ levels of theory and found corrections of +0.061 Å to the F-O bond length and -0.005 Å to the O-O bond. The best estimate re structure (rOO ) 1.221 Å, rFO ) 1.597 Å, ∠OOF ) 108.6°, ∠FOOF ) 88.1°) reported by Karton et al. combined a CCSD(T)/aug-cc-pV5Z set of bond lengths and angles from the partial optimization of Feller and Dixon2 with a CV correction based on the difference between CCSD(T)(CV)/ aug-cc-pwCVTZ and CCSD(T)(FC)/aug-cc-pVTZ and their HO correction. The sign of the CV correction for the FOOF dihedral angle in their Table 3 appears to be wrong. Their predicted rFO value is 0.011 Å longer than our best value, primarily due to the 0.008 Å difference in the frozen core components. Note that the FOOF structure of Karton et al. was not obtained from any of the 11 Wn variations. Like the more than four dozen

Feller et al. TABLE 4: Breakdown of the Higher Order Correction for FOOF (kcal/mol) component

this work

est. CCSDT/cc-pVQZ est. CCSDTQ/cc-pVTZb CCSDTQ(5)/cc-pVDZ(nod)c cf est. FCI

-0.63 2.91 0.54 0.09

total a

2.91

component CCSDT/CBS(DT) CCSDT(Q)/cc-pVTZd CCSDTQ/cc-pVDZe CCSDTQ(5)/ccpVDZ(no d)

Karton et al.a -0.74 3.03 -0.74 0.27 1.82

13

Karton et al. obtained this at the W4 (CCSD(T)/cc-pVQZ) geometry. b On the basis of scaled CCSDT(Q)/cc-pVDZ atomization energy correction (3.61 kcal/mol) plus the CCSDT(Q) cc-pVDZ f cc-pVTZ difference (-0.70 kcal/mol) from Karton et al. The CCSDT(Q)/cc-pVDZ component was computed at the optimal CCSDT(Q)/cc-pVDZ geometry; rOO ) 1.21026 Å, rFO ) 1.70196 Å, ∠OOF ) 110.35°, and ∠FOOF ) 88.44°. The higher order corrections used by Karton et al. were evaluated at the optimal CCSD(T)/cc-pVQZ geometry; rOO ) 1.2342 Å, rFO ) 1.5323 Å, ∠OOF ) 110.5°, and ∠FOOF ) 88.5°. c CCSDTQ(5) correlation energy was scaled by the ratio [E(CCSDTQ5 corr.)/E(CCSDTQ(5) corr.)] taken from FOF. The scale factor was 0.999979. d Reported as 1.10 × 2.75 kcal/mol. e Reported as 1.10 × [CCSDTQ/ cc-pVDZ - UCCSDTQ/cc-pVDZ].

fixed recipe “model” chemistries in the literature, the Wn methods are designed for treating thermochemical properties. In order to compare coupled cluster and CI structures, iCASCI(FC) and iCAS-CI+Q(FC) optimizations were performed on FOOF with the small aug-cc-pVDZ basis set. The multireference CI method scales very poorly when the basis set is enlarged. Even with such a small basis there are 18.2 × 106 internally contracted configuration state functions (CSFs), corresponding to 3.1 × 109 uncontracted CSFs due to the large number of active electrons. The rapid increase in the number of double excitations out of the reference space makes this method impractical for use with still larger basis sets for FOOF. The two multireference CI techniques, CI and CI+Q, predicted O-O bond lengths that were within 0.0013 Å of each other but were ∼0.009 Å longer than CCSD(T)(FC)/aug-cc-pVDZ + ∆HO, which we consider our best estimate of the FCI(FC)/aug-ccpVDZ value. The error with respect to the best value in the raw CCSD(T) O-O bond length is smaller than the error in the two CI values. Conversely, both CI F-O bond lengths, CI and CI+Q, were shorter than the CCSD(T) + ∆HO value by as much as 0.042 Å. This is not surprising as a simplistic view of FOOF might describe it as an O2 molecule with two F atoms rather weakly bonded to the electrons in the π* orbital in the diatomic (note the long F-O bond length and the short O-O bond length). Heats of Formation. Theoretical and experimental heats of formation at 298.15 K based on the present set of calculations are shown in Table 5 together with a selection of other theoretical values. Temperature conversion factors which are required for converting the 0 K heats of formation to 298.15 K were based on standard thermodynamic and statistical mechanics expressions within the rigid-rotor/harmonic oscillator approximation.81 Temperature corrections for the atoms were taken from Curtiss et al.82 The atomic heats of formation at 0 K were taken from NIST-JANAF: ∆Hf(O 0 K) ) 58.98 ( 0.02 kcal/mol and ∆Hf(F 0 K) ) 18.47 ( 0.07 kcal/mol.12 Ruscic et al. recommended revised values: ∆Hf(O 0 K) ) 58.997 ( 0.000 kcal/ mol and ∆Hf(F 0 K) ) 18.45 ( 0.06 kcal/mol, based on the Active Thermochemical Tables.71 Differences on the order of 0.02 kcal/mol are insignificant relative to other sources of error in our approach.

Structures of Selected Small Oxygen Fluorides

J. Phys. Chem. A, Vol. 114, No. 1, 2010 621

TABLE 5: Theoretical and Experimental Heats of Formationa molecule

∆Hf(0 K)

∆Hf(298 K)

b

B3PW91c

CCSD(T)d

2003 theorye

2004 theory

2009 theoryh

∆Hf(298 K)

∆Hf(0 K)

∆Hf(298 K)

∆Hf(298 K)

∆Hf(298 K)

FO ( Π) FOO (2A′′)

26.3 ( 0.2 6.9 ( 0.7

26.5 ( 0.2 6.4 ( 0.7

25.2 6.0

OFO (2B2) FOF (1A1) FOOF (1A)

125.2 ( 0.3 6.4 ( 0.3 7.4 ( 0.7

125.0 ( 0.3 5.9 ( 0.3 6.4 ( 0.7

134.4 5.0 7.3

2

27.9 ( 0.4 9.6 ( 0.6

9.7 ( 2.0

6.6 ( 0.5 9.6 ( 0.9

f

27.30 7.47 f 6.5 ( 1g 129.1f

26.43 ( 0.11 5.87 ( 0.16

5.94 ( 0.14 7.84 ( 0.18i 8.21j

expt. ∆Hf(298 K) 26.58 ( 0.11 6.1 ( 0.5 5.49 ( 0.40 6.24 ( 0.50 (90.5 ( 4.8) 5.86 ( 0.48 4.59 ( 0.5

a Values are in kcal/mol. The theoretical anharmonic zero-point energies are FO ) 1.48 kcal/mol, FOO ) 3.59 kcal/mol, 3.34 kcal/mol, and FOOF ) 4.72 kcal/mol. The experimental heats of formation were taken from Harding et al.89 for FO, NIST-JANAF12 Lyman and Holland85 and Pagsberg et al.90 (in descending order) for FOO and FOOF, and Burcat and Ruscic91 for FOF. The value for OFO listed in NIST-JANAF12 is based on an older theoretical calculation by Gosavi et al.83 b For comparison purposes, the 2008 study by Feller et al.1 reported the following ∆Hf(298 K) values: FO ) 26.6 ( 0.2 kcal/mol, FOO ) 5.8 ( 0.3 kcal/mol, and FOF ) 5.9 ( 0.3 kcal/mol. c Kieninger et al.11 density functional results. Basis sets are as follows: FO, FOF, FOO ) t-aug-cc-pVQZ; OFO ) cc-pVQZ; and FOOF ) aug-cc-pVQZ. d Lee et al.8 based on CCSD(T)(FC)/ANO4 calculations using a TZ2P basis set and isodesmic reactions. e Composite coupled cluster theoretical values from large basis sets reported by Feller and Dixon.2 f CCSD(T)(FC)/CBS + CCSDT(FC)/cc-pVTZ + CCSD(T)(CV)/CBS + CCSD(T)(FC)-DK/ cc-pVQZ_DK + atomic S.O. from Denis.6 g CCSD(T)(FC)/CBS + CCSDT(FC)/cc-pVTZ + CCSD(T)(CV)/cc-pwCVQZ “best” value reported by Denis and Ventura.5 h Composite coupled cluster theoretical values reported by Karton et al.13 The levels of theory were FO ) W4.4, FOO ) W4, FOF ) W4.2, and FOOF ) W4. i Evaluated at the CCSD(T)(FC)/cc-pVQZ geometry used by the W4 method. j Evaluated at the experimental geometry of Jackson.74

The predicted heats of formation of FO and FOF are in excellent agreement with experiment. These most recent predictions are also within 0.1 kcal/mol of the values in our 2008 study1 and the 2009 W4.n values of Karton et al.13 for FO (W4.4) and FOF (W4.2). The JANAF12 heat of formation for OFO is clearly in error, being not positive enough. The JANAF value was obtained from a calculation83 of the energy difference between FOO and OFO at the CI(SD+Q)/6-31G level and using the experimental value for FOO. The calculated value for FOO lies within the experimental error limits for the NIST-JANAF value. The analysis by Lyman84 of the kinetics85 of the reaction of F + O2 gave a value of 5.49 ( 0.40 kcal/mol, slightly smaller than our calculated value of 6.4 ( 0.7 kcal/mol at 298 K. Thus, we prefer the NISTJANAF value. The very close agreement between the ∆Hf(298 K) ) 6.4 ( 0.7 kcal/mol for FOO obtained in this work and the 6.5 ( 1 kcal/mol composite value reported in 2004 by Denis and Ventura5 is due in part to cancellation of error in the latter, which neglected scalar relativistic effects (reduces ΣDe by -0.16 kcal/mol) and atomic spin-orbit effects (reduces ΣDe by another -0.82 kcal/mol). The results from our study shows that the failure to include quadruple excitations in their study should have produced an even larger difference with respect to the current findings. We were unable to reproduce the 6.5 kcal/ mol value of Denis and Ventura using the data in their Table 2. Combining the CBS (AVTZ,AVQZ) value for the “contamination-free” wave function (8.0 kcal/mol) with the cc-pwCVQZ correction (-0.2 kcal/mol) and the CCSDT/cc-pVTZ correction (-1.0 kcal/mol) led to a final value of 6.8 kcal/mol. A subsequent report on FOO of ∆Hf(298 K) ) 7.47 kcal/mol by Denis, which did include scalar relativistic and atomic S.O. corrections, is in worse agreement with our value.6 The value of 5.87 kcal/mol obtained by Martin and co-workers at the W4 level is 0.3 kcal/mol higher than our value. The reliability of the NIST-JANAF heat of formation for FOOF was discussed in our 2003 study.2 It ultimately relies on a 1959 bomb calorimeter measurement of the decomposition of FOOF into F2 + O2 performed at 190 K by Kirshenbaum et al.86 The original study assumed the constant volume heat capacity of the reactants (FOOF) and products (O2 + F2) were equal over the temperature range from 190 to 298 K and estimated the heat of vaporization. Thirty years later, Lyman

TABLE 6: Adiabatic Bond Dissociation Energies at 0 K in kcal/mol molecule

products

BDE(F-O)

O2 FO FOO OFO FOF FOOF

O+O F+O F + O2 O + FO F + OF F + FOO

51.2 11.6 -39.9 38.4 18.0

BDE(O-O) 117.91 78.4 45.2

reanalyzed the data using the actual heat capacity difference and arrived at the 298 K value adopted by NIST-JANAF.84 However, NIST-JANAF increased the error bars from (0.2 to (0.5 kcal/mol. It is difficult to gauge the accuracy of the NISTJANAF value and the associated error bars. Suggestions have been made over the years for an experimental reinvestigation of this quantity.2 Our current best estimate for the heat of formation of FOOF at 298 K, 6.4 ( 0.7 kcal/mol, lies outside the upper limit of the experimental error bars by about 1 kcal/mol considering the lower limit of the calculated value. The 1996 theoretical value of 9.7 ( 2.0 kcal/mol reported by Lee et al.8 and the 2003 value of 9.6 ( 0.9 kcal/mol from Feller and Dixon2 are clearly too large because of their neglect of the higher order correction. In hindsight, the error estimate in the latter was overly optimistic. The value of 7.84 kcal/mol (or 8.21 at the experimental geometry) obtained by Martin and co-workers at the W4 level is larger than our best value by more than 2 kcal/mol. The major reason for this difference is that their estimate for the higher order correction is 2 kcal/mol lower than our value. The current values can be used to calculate adiabatic bond dissociation energies (BDEs), which are shown in Table 6. The molecule OFO is shown to be metastable with a negative F-O BDE of almost -40 kcal/mol. The largest F-O BDE is in FO followed by FOF, FOOF, and FOO. The F-O BDE in FOO is only 11.6 kcal/mol. The O-O BDE in FOO is 39.5 kcal/mol smaller than that in O2 and 33.2 kcal/mol larger than the O-O BDE in FOOF. The O-O BDE in FOOF of 45.2 kcal/mol is comparable to that in HOOH (48.4 kcal/mol).1 This is of interest as the theoretical O-O bond distance in FOOF is 1.223 Å and in HOOH is 1.451 Å. Thus, although the O-O bonding is quite different in FOOF and HOOH, the adiabatic O-O BDE is essentially the same.

622


IV. Conclusion Coupled cluster theory up through quadruple excitations (and in some cases quintuple excitations) with higher order correlation estimates were combined with large one-particle basis sets in an effort to predict heats of formation at 298.15 K and structures of five small oxygen fluoride molecules. Explicit consideration of quadruple excitations is essential for achieving accurate results and close agreement with experiment. The latter was achieved for FO, FOO, and FOF. However, in the case of FOOF the best theoretical heat of formation fell outside the experimental error bars. A related theoretical study by Karton et al.13 reported similar conclusions. A qualitative understanding of the issues with FOOF can be obtained from the following considerations. The O-O bond distance in FOOF (1.216 Å) is only slightly longer than that in diatomic O2 (3Σg-), where re ) 1.20752 Å.87 At the same time, the F-O distance in FOOF (1.586 Å) is considerably longer than the distance in FOF (1.4053 Å). If we adopt the F-O bond distance in FOF as a “normal” FO bond, then the F-O bond in FOOF is ∼0.18 Å longer than normal. Thus, FOOF can be viewed as two F (2P) atoms singlet coupled to the triplet ground state of O2. Each F-O bond is substantially elongated (partially dissociated), which gives rise to a significant amount of multireference character in each bond, partly accounting for the difficulty in predicting the energetics of this molecule. The BDEs are also consistent with this model as the first BDE in FOOF is 18.0 kcal/mol as compared to the first BDE in FOF of 38.4 kcal/mol. The average of the two F-O BDEs in FOOF is 15 kcal/mol, much smaller than the average BDEs in FOF of 45 kcal/mol, consistent with the above arguments in terms of the geometries. With the inclusion of the present results, we find MAD ) 0.20 kcal/mol with respect to experiment (100 comparisons) using all of the pertinent data in the Computational Results Database (CRDB).88 The latest version of the CRDB holds ∼102 000 theoretical and experimental entries covering 329 molecular species and 42 atoms. Bond lengths are similarly well described, with MAD ) 0.0004 Å for A-B nonhydrogen bonds (79 comparisons). These findings reinforce the conclusion that single-reference-based coupled cluster theory, when carried out to sufficiently high excitation levels, is a very powerful tool for the high-accuracy study of small molecules. Even systems with pronounced multiconfigurational character are amenable to the approach. Nonetheless, while recognizing the great strides that have been made in our ability to accurately model the molecular properties of small systems, FOOF remains a cautionary tale of the potential difficulties that persist. The principal disadvantage of the method is the steep scaling with system size and the slow convergence with respect to improvements in the one-particle basis set. The latter issue will be addressed in a forthcoming paper. Acknowledgment. This work was supported in part by the Chemical Sciences, Geosciences, and Biosciences Division, Office of Basic Energy Sciences, U.S. Department of Energy (DOE), under grant no. DE-FG02-03ER15481 (catalysis center program). D.A.D. also thanks the Robert Ramsay Chair Fund of The University of Alabama for support. K.A.P. would also like to gratefully acknowledge the support of the National Science Foundation (CHE-0723997). References and Notes (1) Feller, D.; Peterson, K. A.; Dixon, D. A. J. Chem. Phys. 2008, 129, 204105.

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