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J. Phys. Chem. A 2011, 115, 94–98
Ab Initio Coupled Cluster Determination of the Equilibrium Structures of cis- and trans-1,2-Difluoroethylene and 1,1-Difluoroethylene David Feller,*,† Norman C. Craig,‡ Peter Groner,§ and Donald C. McKean| Department of Chemistry, Washington State UniVersity, Pullman, Washington 99164-4630, United States, Department of Chemistry and Biochemistry, Oberlin College, Oberlin, Ohio 44074, United States, Department of Chemistry, UniVersity of MissourisKansas City, Missouri 64110-2499, United States, and School of Chemistry, UniVersity of Edinburgh, Edinburgh EH9 3JJ, United Kingdom ReceiVed: October 6, 2010; ReVised Manuscript ReceiVed: NoVember 22, 2010
The equilibrium structures of cis- and trans-1,2-difluoroethylene and 1,1-difluoroethylene, C2H2F2, have been determined with high-level coupled cluster techniques combined with large basis sets, explicit consideration of core/valence, and scalar relativistic and higher order correlation effects. Excellent agreement was found with new semiexperimental structures, increasing the level of confidence in both approaches. Differences in bond lengths among ethylene and the fluoroethylenes are discussed. Introduction A recent report of the normal coordinate analysis of the gasphase infrared (IR) spectra of CF2dCHD and CF2dCD2 included new information on the microwave spectra of five isotopic species and a semiexperimental equilibrium structure for CF2dCH2.1 Comparison was made with a preliminary, highlevel theoretical structure. Direct comparison between (r0, rg, rR, and rz, etc.) structures for polyatomic molecules obtained from spectroscopic or diffraction-based experimental measurements2-5 and theoretical “bottom of the well” structures is typically limited by the inherent differences in the two sets of values. However, by incorporating vibration-rotation interaction constants (spectroscopic R’s) generated by density functional theory or low-level perturbation theory with experimental data, it is possible to produce semiexperimental structures that exhibit excellent agreement with the best electronic structure geometries.6-10 As with other research areas where theory and experiment overlap and complement each other, the different approaches provide a means of calibrating and challenging the other to achieve higher accuracy. The purpose of this work is to present very high level theoretical equilibrium structures for cis- and trans-1,2-difluoroethylene and 1,1-difluoroethylene and to compare the structures with the corresponding semiexperimental results. Sources of uncertainty in the calculations will be discussed insofar as they limit the accuracy that can be achieved. Procedure In qualitative terms, our approach to determining the gasphase equilibrium structures of the three isomers considered in this study (see Figure 1) begins with geometry optimizations at the highest affordable level of theory. These initial results are followed by a series of calculations intended to further reduce the largest residual sources of error. A final composite estimate is created in which all physically important factors have been * To whom correspondence should be addressed. † Washington State University. ‡ Oberlin College. § University of MissourisKansas City. | University of Edinburgh.
Figure 1. 1,1-Difluoroethylene (a), cis-1,2-difluoroethylene (b), and trans-1,2-difluoroethylene (c).
addressed and the error in each contribution has been analyzed. The rationale behind this somewhat complex strategy is to circumvent the need to perform a single calculation that would likely prove intractable. Nonetheless, skill and attention to detail are required to successfully decompose the original problem into manageable pieces without suffering an unacceptable loss in accuracy. An oftentimes precarious balance must be struck between accuracy and the speed of the calculations because, at this stage in the development of quantum mechanical methods, the available algorithms rapidly increase in computational cost as greater accuracy is sought. Effectively exact solutions of the relativistic molecular Schro¨dinger equation remain beyond our capabilities except in trivially small chemical systems. The procedure adopted for this study, to be described in subsequent sections, has been shown to yield highly accurate results for other small molecules.9-11 Unfortunately, this method involves several computationally expensive steps that are currently only possible for systems up to several dozen atoms. A less expensive alternative entails identifying a cheaper theoretical method that reproduces an experimentally wellestablished property similar to the (potentially unknown) property of interest. By making the assumption that good agreement in one instance delivers comparable accuracy in the unknown case, this approach avoids the expensive calculations but relies heavily on a fortuitous cancellation of error. For example, in the present context one might choose a method that reproduces the CdC bond length in ethylene and hope that this method performs equally well in reproducing the structures of difluoroethylenes. To give one specific example, frozen core (FC), second-order perturbation theory with a quadruple-ζ quality basis set yields a CdC bond length in ethylene of 1.3302
10.1021/jp109584k 2011 American Chemical Society Published on Web 12/09/2010
Equilibrium Structures of Various C2H2F2
Figure 2. Convergence of the optimized CdC and C-F bond lengths in 1,1-difluoroethylene as a function of the basis set size in frozen core CCSD(T) calculations. aVDZ ) aug-cc-pVDZ, etc.
Å. This compares to a semiexperimental value of rCC ) 1.3305 Å.12 While this level of agreement is excellent, perturbation theory calculations with other (either smaller or larger) basis sets destroy the delicate balance of error and lead to worse agreement with experiment. In the case of the isomers of difluoroethylene it will be shown that a low-cost combination such as second-order perturbation theory and a quadruple-ζ quality basis set is unable to reproduce the agreement observed in ethylene. Thus, while the simpler method may be adequate for predicting structures of closely related compounds with reasonable accuracy, it lacks the robustness of the more expensive and more generally applicable approach followed in this study. The dominant sources of error in most electronic structure calculations arise from the practical necessity of truncating the so-called 1-particle and n-particle expansions. The former is normally associated with the use of finite Gaussian basis sets, while the latter is associated with the use of wave functions that provide limited correlation recovery. An exact 1-particle expansion is reached in the complete basis set (CBS) limit. An exact n-particle expansion is reached in the full configuration interaction (FCI) limit. With current computer hardware and software, the most sophisticated n-particle method that can routinely be applied to chemical systems containing a few dozen atoms or less is coupled cluster theory with single and double excitation combined with a quasiperturbative treatment of triple excitations.13-16 This method, commonly known as CCSD(T), was demonstrated to be capable of reproducing bond lengths to (0.001 Å (between non-hydrogen atoms) and (0.003 Å (between a hydrogen and non-hydrogen) for a collection of roughly 50 small molecules when used in conjunction with several small corrections.10 CCSD(T) recovers a large enough fraction of the correlation energy that the error in the n-particle expansion is much smaller than the error in the 1-particle expansion. All CCSD(T) calculations were performed with MOLPRO 2009.1.17 Convergence of the CdC and C-F bond lengths in 1,1difluorethylene as a function of the basis set size is shown in Figure 2 at the CCSD(T)(FC) level of theory. Basis sets for frozen core calculations were taken from the diffuse function-
J. Phys. Chem. A, Vol. 115, No. 1, 2011 95 augmented correlation consistent basis sets of Dunning and co-workers.18-26 These are labeled aug-cc-pVnZ, (n ) D, T, Q, 5, 6), and the largest consists of an [8s,7p,6d,5f,4g,3h,2i] contraction on carbon and fluorine. As the basis set approaches completeness (from left to right in the figure) and the bond strength increases, the bond lengths contract by ∼0.020 Å. Although the aug-cc-pV6Z basis set is quite large by most standards, for the highest possible accuracy the remaining error with respect to the CBS limit must be estimated. For this purpose we use a simple exponential function in n, the basis set index. For the three isomers of difluoroethylene, this extrapolation results in a small additional contraction of the bond lengths by another ∼0.0001 Å. In other small molecules the extrapolation can result in a much larger contraction; e.g., ∆re ) 0.0006 Å in F2. Geometry optimizations on diatomic and triatomic molecules with even larger basis sets (aug-cc-pV7Z and augcc-pV8Z) confirm the effectiveness of the simple extrapolation procedure. No extrapolation of the bond angles was attempted because they are typically converged to (0.001° at the augcc-pV6Z basis set level. The extrapolated CCSD(T)(FC) structure parameters for the three isomers are listed in Table 1. Ignoring zero point vibrational effects, CCSD(T)/aug-cc-pV6Z calculations find the cis isomer to be 9.4 kcal/mol above the 1,1-isomer and the trans form is another 1.1 kcal/mol above the cis. The experimental value for the cis-/trans-difluoroethylene electronic energy difference is 1.086(22) kcal/mol, in excellent agreement with the calculations.27 If zero point vibrational energies are included, the trans form remains higher in energy than the cis, but the difference is reduced to 0.8 kcal/mol. Next in terms of importance for the structure of difluoroethylene is the need to address the effects of core/valence (CV) correlation, i.e., the contributions arising from the inner core (1s) electrons on carbon and fluorine. For this purpose we performed all-electron geometry optimizations with the ccpwCVnZ, (n ) D, T, Q) basis sets.22 The bond lengths in the all-electron sequence and the frozen core sequence were individually extrapolated to the CBS limit, and the difference in the bond lengths was taken as the ∆CCSD(T)(CV)/CBS correction listed in Table 1. The introduction of CV correlation further increases the bond strength and produces an additional contraction, on the order of -0.002 to -0.003 Å, of the bond lengths. The bond angle corrections were taken from the difference of the cc-pwCVQZ optimized values with all electrons correlated and within the frozen core approximation. Scalar relativistic (SR) corrections, which are not expected to be very large for elements such as light and C and F, were obtained from Douglas-Kroll-Hess (DKH) CCSD(T)(FC) calculations29,30 using the cc-pVTZ-DK basis set.31 The correction is relatively insensitive to the size of the basis set or level of theory. We have included the relativistic correction for the isomers of difluoroethylene for the sake of consistency in our general approach, which is designed to cover chemical systems where the correction is larger, e.g., HOCl ∆SR ) -0.001 Å and K2 ) ∆SR ) -0.014 Å. With heavier elements the size of the relativistic correction can grow even larger. In the present case the corrections do not exceed -0.0002 Å. The final set of corrections is intended to pick up the higher order correlation contribution to the structures. These arise from the difference between CCSD(T) and FCI. Explicit FCI calculations with even a modest VDZ size basis set are intractable for difluoroethylene. Instead, we performed CCSDT(FC)/cc-pVTZ and CCSDT(Q)(FC)/cc-pVDZ optimizations and used the sum of the CCSD(T) f CCSDT/VTZ and CCSDT f CCSDT(Q)/
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TABLE 1: Theoretical and Experimental Structures for Difluoroethylenesa r(CH)
R(CCF)
R(CCH)
1,1-Difluoroethylene 1.31717 -0.00201 0.00001 0.00010 0.00038 1.3157 1.3157(2) 1.315(2) 1.325(2)
1.07645 -0.00141 -0.00008 -0.00003 0.00008 1.0750 1.0754(1) 1.091(9) 1.074(6)
125.15 -0.01 0.00 0.00 -0.01 125.14 125.16(2)
119.36 0.02 0.00 0.01 0.00 119.39 119.40(1) 119.0(4)
1.32677 -0.00283 -0.00020 0.00001 0.00083 1.3246 1.323(1) 1.324
cis-1,2-Difluoroethylene 1.33508 -0.00204 0.00006 0.00018 0.00030 1.3336 1.334(1) 1.339
1.07862 -0.00137 -0.00008 -0.00008 0.00009 1.0772 1.075(1) 1.089
122.18 0.00 0.00 0.01 0.00 122.19 122.3(1) 122.0
122.74 -0.01 0.00 0.02 -0.01 122.76 122.5(1) 123.9
1.32606 -0.00298 -0.00015 0.00007 0.00085 1.3239 1.324(1) 1.318
trans-1,2-Difluoroethylene 1.34118 -0.00209 0.00005 0.00013 0.00032 1.3396 1.339(1) 1.351
1.07903 -0.00137 -0.00004 -0.00001 0.00009 1.0777 1.078 1.079
119.73 0.00 -0.01 0.00 0.00 119.72 119.8 119.2
125.38 -0.07 0.01 0.02 0.00 125.34 125.1(1) 126.3
method
r(CC)
CCSD(T)(FC)/CBS(Q56) ∆CCSD(T)(CV)/CBS ∆CCSD(T)(FC)-DKH/VTZ_DK ∆CCSDT(FC)/VTZ ∆CCSDT(Q)(FC)/VDZb best composite theory semiexperimental (re)c experimental (rg,rz)d experimental (r0)e
1.32051 -0.00300 -0.00016 0.00000 0.00074 1.3181 1.3175(4) 1.340(3) 1.313(3)
CCSD(T)(FC)/CBS(Q56) ∆CCSD(T)(CV)/CBS ∆CCSD(T)(FC)-DKH/VTZ_DK ∆CCSDT(FC)/VTZ ∆CCSDT(Q)(FC)/VDZb best composite theory semiexperimental (re) experimental (rs)f CCSD(T)(FC)/CBS(Q56) ∆CCSD(T)(CV)/CBS ∆CCSD(T)(FC)-DKH/VTZ_DK ∆CCSDT(FC)/VTZ ∆CCSDT(Q)(FC)/VDZb best composite theory semiexperimental (re) experimental (rs/r0)f
r(CF)
a
Bond lengths are in angstroms, and bond angles are in degrees. Experimental uncertainties in the final figure are given in parentheses. Includes (Q) f Q correction of -0.000 23 Å to r(CF) taken from CF2. c Semiexperimental values based on harmonic contributions to R’s computed after scaling of the force constants to the ωobs values, as reported by McKean et al.1 d Based on electron diffraction and microwave data reported by Mijlhoff et al.34 e Based on microwave data reported by Laurie and Pence.36 f Craig et al.35 b
VDZ corrections as our estimate of the total higher order correction. The (T) fT and TfQ contributions are sometimes of opposite signs and individually can be more than an order of magnitude larger than their sum. CCSDT replaces the quasiperturbative triple excitations in CCSD(T) with a fully iterative treatment and is known to converge more slowly than the quadruples correction. CCSDT(Q) includes a quasi-perturbative treatment of the slightly more than 3 × 109 quadruple excitation determinants. This number made it prohibitively expensive to perform optimizations at the full CCSDTQ level of theory. To account approximately for the difference between CCSDT(Q) and CCSDTQ, we included a (Q) f Q adjustment of -0.000 23 Å to the C-F bond length, which was adopted from CF2. The corresponding correction for the CdC bond length in ethylene was
