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Assessment of density functional theory for the prediction of the nature of the oxirene stationary point
Journal of Molecular Structure (Theochem) 629 (2003) 263–270 www.elsevier.com/locate/theochem
Assessment of density functional theory for the prediction of the nature of the oxirene stationary point Robert C. Mawhinneya,*, John D. Goddardb a
Department of Chemistry and Biochemistry, Concordia University, Montreal QC, Canada H3G 1M8 b Department of Chemistry and Biochemistry, University of Guelph, Guelph ON, Canada N1G 2W1 Received 27 January 2003; accepted 27 February 2003
Abstract The nature of the stationary point of oxirene, cyclic C2H2O, has been examined using selected GGA and hybrid functionals in combination with a number of basis sets. All GGA functionals (BLYP, BP91, PBE, HCTH, HCTH147, and HCTH407) predict the stationary point to be a transition state and show very little basis set dependence. The hybrid functionals (B3LYP, B3P91, PBE0, B97, B97-1, and B3P86), on the other hand, show more basis set dependence and predict oxirene to be a minimum in a number of cases. However, these parameterized hybrid functionals are not consistent in their characterization of oxirene. The theoretically derived hybrid functional PBE0 predicts oxirene to be a minimum with 11 of the 12 basis sets used and results with the largest basis set are in good agreement with earlier predictions with the high level correlated method, CCSD(T). q 2003 Elsevier B.V. All rights reserved. Keywords: Oxirene; Density functional theory; Stationary point characterization
1. Introduction The C2H2O system has been examined intensively from a theoretical viewpoint, with computational studies dating back nearly three decades [1,2,3]. This anti-aromatic three-member ring system, oxirene, has been postulated by experimentalists to be an intermediate in the Wolff rearrangement. In that reaction, the photolysis of a diazoketone leads to the formation of a ketocarbene which ultimately rearranges to the product ketene [4,5]. Experiments in which the reactant carbonyl carbon was labelled showed an * Corresponding author. E-mail addresses: [email protected] (R.C. Mawhinney), [email protected] (J.D. Goddard).
equal distribution of 13C labelled ketene products [6]. To explain this result an epoxide species of C2v symmetry, oxirene, was proposed as an intermediate. The theoretical characterization of stationary points on certain potential energy surfaces can be demanding. While DFT methods have proven very useful for the prediction of many molecular properties, there seems to be a deficiency when applied to the characterization of certain difficult stationary points. The C2v stationary point of oxirene appears to be one of these problematic systems. It has been observed that the ring opening mode of oxirene, which is the lowest-energy vibration, is extremely sensitive to the theoretical model used. Therefore, depending on the choice of model chemistry, oxirene is predicted to be
0166-1280/03/$ - see front matter q 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0166-1280(03)00198-2
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either a minimum energy species or a transition structure [7 – 12]. Vacek et al. [7] very thoroughly examined the character of the C2v oxirene stationary point through variations in the level of theory and basis set. Their main conclusions regarding oxirene were that Hartree –Fock theory always predicts it to be a minimum, DFT yields a transition state structure, and for postHartree –Fock methods, such as Moller-Plesset and coupled cluster, the nature of the stationary point depends on the basis set. With their most elaborate CCSD(T) method, oxirene was a minimum with a ring-opening vibration mode of ca. 150 cm21 (an average of 139 cm21 from CCSD(T)-full/cc-pVTZ and 163 cm21 from CCSD(T)-frozen core/TZ2P(f, d)). Only recently has any DFT method predicted oxirene to be a minimum [12]. In that article, Wilson and Tozer showed that only the recently formulated hybrid functional, B97-2, predicts oxirene to be a minimum. All other published results from DFT [7,8,9] confirmed the conclusions of Vacek et al. [7]. However, in the examination of oxirene, these DFT studies have been somewhat limited in the choice of exchange-correlation functionals and the basis set dependence in DFT was not thoroughly examined. In this letter, we assess the character of the oxirene stationary point using a larger selection of functionals, both GGA and hybrid, and a greater range of basis sets.
2. Methods All calculations were performed using the CADPAC suite of programs [13]. A total of 13 functionals and 12 basis sets were used for geometry optimization and vibrational frequency analysis of the oxirene stationary point. C2v symmetry was enforced for oxirene. The most stable C2H2O isomer, ketene, also was computed to assess the relative stability of oxirene with various methods. We have included results with the BLYP [14,15], B3LYP [15,16], B97 [17] and B97-1 [18,19] functionals for completeness. These functionals were previously shown to predict oxirene to be a transition structure [7 –9,12]. A selection of basis sets [27 – 33] will be tested since the results on oxirene with conventional correlated
methods showed some basis set dependence. DFT is, of course, a correlated approach although its basis set dependence is usually considered to be slight. The version of CADPAC used did not have the B97-2 functional implemented. The remaining eight functionals used in this work consist of the GGA functionals BP91 [14,20], PBE [21,22], and the HCTH set (HCTH, HCTH147, and HCTH407) [19, 23,24] and the hybrid functionals B3P86 [16,25], B3P91 [16,20], and PBE0 [26,27]. The basis sets were taken from those internal to the CADPAC suite of programs [34]. These include the 631G split valence and 6-311G triply split valence sets with variations in the numbers of polarization functions added, [28 – 30] and the Huzinaga-Dunning double and triple zeta sets with and without polarization (DZ, DZP, PVDZ, TZ, TZ2P, and PVTZ) [31 – 33].
3. Results and discussion The geometries of oxirene, the ring opening vibrational frequency, and the relative energies with respect to the ketene minimum are given in Tables 1 –4. The relative energies are calculated from the total energies without consideration of the changes in zero point vibrational energies between the two isomers. The results for the GGA functionals are presented in Tables 1 and 2. Table 1 contains results for those GGA functionals which do not involve fits to experimental data (BLYP, BP91, and PBE). The GGA results in Table 2 are from functionals (HCTH, HCTH147, and HCTH407) which were parameterized with the use of experimental results. The results with hybrid functionals are given in Tables 3 and 4. Table 3 involves functionals with a three parameter fit in the exchange functional (B3LYP, B3P91, and B3P86). The results with functionals that have one parameter for exact exchange (B97, B97-1, and PBE0) are collected in Table 4. B3LYP (VWN-III) and B3LYP(VWN-V) [17] produced practically identical results and only the predictions from B3LYP(VWN-III) are reported in Table 3. The effects of basis sets can be addressed. For the doubly and triply split valence basis sets, 6-31G and 6-311G, when the number of polarization functions is increased, e.g. from 6-31G* to 6-31G** to 6-31G
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Table 1 Molecular structures, vibrational frequencies of the ring opening mode and the energy differences between ketene and oxirene as predicted by GGA functionals Level of theory
Frequency (cm21)
˚) rCO (A
˚) rCC (A
˚) rCH (A
aCOC (deg)
aCCH (deg)
DE (kcal mol21)
BLYP/6-31G* BLYP/6-31G** BLYP/6-31G(2d,p) BLYP/6-31GE BLYP/6-311G* BLYP/6-311G** BLYP/DZ BLYP/DZP BLYP/PVDZ BLYP/TZ BLYP/TZ2P BLYP/PVTZ BP91/6-31G* BP91/6-31G** BP91/6-31G(2d,p) BP91/6-31GE BP91/6-311G* BP91/6-311G** BP91/DZ BP91/DZP BP91/PVDZ BP91/TZ BP91/TZ2P BP91/PVTZ PBE/6-31G* PBE/6-31G** PBE/6-31G(2d,p) PBE/6-31GE PBE/6-311G* PBE/6-311G** PBE/DZ PBE/DZP PBE/PVDZ PBE/TZ PBE/TZ2P PBE/PVTZ
399i 397i 347i 330i 395i 387i 335i 395i 415i 273i 347i 347i 380i 377i 317i 294i 372i 363i 313i 366i 396i 248i 315i 311i 379i 376i 312i 290i 373i 363i 319i 368i 398i 248i 312i 310i
1.531 1.531 1.530 1.528 1.532 1.530 1.605 1.534 1.536 1.600 1.528 1.525 1.515 1.515 1.515 1.512 1.515 1.514 1.588 1.517 1.520 1.581 1.512 1.509 1.513 1.512 1.512 1.510 1.513 1.511 1.586 1.514 1.517 1.578 1.510 1.507
1.278 1.279 1.278 1.279 1.273 1.273 1.301 1.290 1.283 1.280 1.270 1.270 1.278 1.279 1.277 1.279 1.273 1.273 1.299 1.288 1.282 1.281 1.271 1.271 1.279 1.279 1.278 1.280 1.274 1.274 1.300 1.289 1.283 1.282 1.272 1.272
1.081 1.080 1.078 1.081 1.079 1.077 1.081 1.084 1.086 1.076 1.075 1.075 1.081 1.080 1.079 1.082 1.079 1.077 1.080 1.083 1.085 1.077 1.076 1.076 1.083 1.081 1.080 1.083 1.081 1.079 1.082 1.085 1.087 1.078 1.078 1.078
49.4 49.4 49.3 49.5 49.1 49.2 47.8 49.7 49.4 47.2 49.1 49.2 49.9 49.9 49.9 50.0 49.7 49.7 48.3 50.2 49.9 47.8 49.7 49.8 50.0 50.0 50.0 50.2 49.8 49.9 48.4 50.4 50.0 47.9 49.8 49.9
161.6 161.5 162.0 161.8 161.9 161.6 161.0 161.5 161.6 160.9 161.4 161.5 161.8 161.7 162.1 161.9 162.1 161.8 161.0 161.7 161.7 160.9 161.5 161.6 161.8 161.7 162.1 161.9 162.1 161.8 161.1 161.7 161.7 160.9 161.5 161.6
82.16 81.83 81.13 80.33 84.90 84.08 79.81 82.07 83.53 77.80 81.31 81.84 81.53 81.16 80.46 79.33 83.70 82.88 79.70 80.97 82.63 77.80 80.29 80.55 81.31 80.94 80.12 79.02 83.57 82.76 79.92 80.89 82.56 77.87 80.03 80.32
(2d,p) the oxirene stationary point becomes slightly more stable with respect to ketene. Polarization is more important in describing the strained three member ring as compared to the ketene. However, the effect is small, e.g. the oxirene becomes 1.03 kcal/mol more stable on changing from BLYP/6-31G* to BLYP/6-31G(2d,p). With the hybrid functionals the stabilizations in kcal/mol with the same change in basis set are only 0.37 B3P86, 0.30 B3LYP, 0.59 B97 and 0.31 PBE0. Oxirene also becomes vibrationally more stable with the addition
of more polarization functions i.e. the ring opening frequency if imaginary becomes smaller or if real (the PBE0 results in Table 4) becomes larger. A similar result was observed by Vacek et al. [7]. For the basis sets originating from the double and triple zeta sets, the addition of polarization functions to the unpolarized DZ or TZ set destabilizes the oxirene stationary point. When the CO distances are examined with the unpolarized DZ and TZ basis sets, these bond lengths are too ˚ . The destabilization long and greater than 1.55 A
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Table 2 Molecular structures, vibrational frequencies of the ring opening mode, and the energy differences between ketene and oxirene as predicted by the 15 parameter GGA functionals Level of theory
Frequency (cm21)
˚) rCO (A
˚) rCC (A
˚) rCH (A
aCOC (deg)
aCCH (deg)
DE (kcal mol21)
HCTH/6-31G* HCTH/6-31G** HCTH/6-31G(2d,p) HCTH/6-31GE HCTH/6-311G* HCTH/6-311G** HCTH/DZ HCTH/DZP HCTH/PVDZ HCTH/TZ HCTH/TZ2P HCTH/PVTZ HCTH147/6-31G* HCTH147/6-31G** HCTH147/6-31G(2d,p) HCTH147/6-31GE HCTH147/6-311G* HCTH147/6-311G** HCTH147/DZ HCTH147/DZP HCTH147/PVDZ HCTH147/TZ HCTH147/TZ2P HCTH147/PVTZ HCTH407/6-31G* HCTH407/6-31G** HCTH407/6-31G(2d,p) HCTH407/6-31GE HCTH407/6-311G* HCTH407/6-311G** HCTH407/DZ HCTH407/DZP HCTH407/PVDZ HCTH407/TZ HCTH407/TZ2P HCTH407/PVTZ
372i 368i 309i 284i 365i 355i 315i 357i 392i 250i 304i 298i 371i 368i 309i 284i 365i 355i 321i 359i 393i 254i 305i 300i 342i 338i 278i 243i 337i 326i 302i 327i 372i 222i 270i 260i
1.499 1.499 1.500 1.496 1.499 1.498 1.575 1.501 1.503 1.564 1.496 1.493 1.500 1.500 1.501 1.497 1.500 1.499 1.576 1.502 1.505 1.566 1.497 1.494 1.491 1.491 1.492 1.488 1.491 1.490 1.569 1.493 1.496 1.557 1.488 1.486
1.273 1.274 1.274 1.275 1.270 1.270 1.295 1.283 1.278 1.278 1.268 1.268 1.273 1.274 1.273 1.275 1.269 1.270 1.295 1.283 1.278 1.278 1.267 1.267 1.272 1.272 1.272 1.273 1.268 1.269 1.294 1.282 1.276 1.277 1.267 1.267
1.078 1.076 1.075 1.078 1.076 1.074 1.076 1.079 1.082 1.073 1.073 1.073 1.078 1.076 1.075 1.078 1.076 1.074 1.076 1.080 1.082 1.073 1.073 1.073 1.077 1.075 1.075 1.077 1.075 1.073 1.075 1.079 1.081 1.072 1.072 1.073
50.3 50.3 50.3 50.4 50.1 50.2 48.6 50.6 50.3 48.2 50.1 50.2 50.2 50.3 50.2 50.4 50.1 50.1 48.5 50.6 50.3 48.2 50.1 50.2 50.5 50.5 50.5 50.7 50.3 50.4 48.7 50.9 50.5 48.4 50.4 50.5
162.0 161.9 162.1 162.0 162.3 162.0 161.1 161.9 161.8 161.0 161.7 161.8 162.0 161.9 162.2 162.0 162.3 162.0 161.2 161.9 161.9 161.0 161.7 161.8 162.1 162.0 162.2 162.1 162.4 162.1 161.2 162.0 162.0 161.0 161.8 161.9
83.00 82.60 82.06 80.67 84.89 84.05 81.78 82.35 84.16 79.78 81.63 81.68 82.78 82.39 81.88 80.52 84.84 84.00 81.62 82.23 84.07 79.60 81.52 81.65 82.57 82.17 81.75 80.25 84.59 83.74 82.25 81.96 83.99 80.05 81.25 81.33
due to ring strain in the epoxide is artificially decreased at such long CO distances but increases when polarization functions lead to shorter bond lengths. The 6-31GE basis set, which is described in the CADPAC documentation [34] as starting from 6-31G and adding diffuse s and p functions and using two sets of slightly diffuse polarization functions, produces two interesting results. With the three parameter hybrid functionals, B3P86 and B3P91, the ring opening frequency is real but very small, 38 and 68 cm21, respectively.
Basis set effects play some role in the differing results for the structures, ring opening vibrational frequencies and relative energies with the various exchange-correlation functionals. However, basis set effects do not appear to be the primary cause of the different curvatures of the potential energy surface along the ring opening mode for oxirene. The different exchange correlation functionals are now considered. Note that all GGA functionals (Tables 1 and 2), without regard to basis set, predict oxirene to be a transition state. The imaginary
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267
Table 3 Molecular structures,vibrational frequencies of the ring opening mode, and the energy differences between ketene and oxirene as predicted by the B3—three parameter hybrid functionals Level of theory
Frequency (cm21)
˚) rCO (A
˚) rCC (A
˚) rCH (A
aCOC (deg)
aCCH (deg)
DE (kcal mol21)
B3P86/6-31G* B3P86/6-31G** B3P86/6-31G(2d,p) B3P86/6-31GE B3P86/6-311G* B3P86/6-311G** B3P86/DZ B3P86/DZP B3P86/PVDZ B3P86/TZ B3P86/TZ2P B3P86/PVTZ B3P91/6-31G* B3P91/6-31G** B3P91/6-31G(2d,p) B3P91/6-31GE B3P91/6-311G* B3P91/6-311G** B3P91/DZ B3P91/DZP B3P91/PVDZ B3P91/TZ B3P91/TZ2P B3P91/PVTZ B3LYP/6-31G* B3LYP/6-31G** B3LYP/6-31G(2d,p) B3LYP/6-31GE B3LYP/6-311G* B3LYP/6-311G** B3LYP/DZ B3LYP/DZP B3LYP/PVDZ B3LYP/TZ B3LYP/TZ2P B3LYP/PVTZ
181i 175i 119i 38 181i 159i 59 153i 221i 188 88i 53i 172i 165i 105i 68 171i 147i 83 141i 212i 196 67i 31 219i 214i 181i 137i 221i 205i 101i 208i 251i 144 166i 154i
1.495 1.495 1.495 1.492 1.495 1.494 1.568 1.496 1.500 1.558 1.491 1.489 1.494 1.494 1.494 1.491 1.494 1.492 1.567 1.495 1.499 1.557 1.490 1.488 1.505 1.504 1.504 1.501 1.505 1.503 1.579 1.506 1.510 1.570 1.501 1.498
1.268 1.268 1.267 1.268 1.263 1.263 1.288 1.277 1.273 1.271 1.261 1.261 1.267 1.268 1.267 1.268 1.263 1.263 1.288 1.276 1.272 1.270 1.260 1.260 1.267 1.268 1.266 1.268 1.262 1.262 1.289 1.277 1.272 1.269 1.260 1.259
1.076 1.074 1.074 1.076 1.074 1.072 1.074 1.078 1.080 1.071 1.071 1.071 1.075 1.073 1.073 1.075 1.073 1.071 1.073 1.077 1.079 1.070 1.070 1.070 1.074 1.073 1.072 1.075 1.072 1.070 1.073 1.077 1.079 1.069 1.069 1.069
50.2 50.2 50.2 50.3 50.0 50.0 48.5 50.5 50.2 48.1 50.0 50.1 50.2 50.2 50.2 50.3 50.0 50.1 48.5 50.5 50.2 48.2 50.0 50.1 49.8 49.8 49.8 50.0 49.6 49.6 48.2 50.2 49.8 47.7 49.6 49.7
162.0 161.9 162.2 162.0 162.3 162.0 161.3 161.9 161.9 161.1 161.7 161.8 162.0 161.9 162.2 162.0 162.3 162.0 161.3 161.9 161.9 161.1 161.7 161.8 161.9 161.8 162.2 162.0 162.2 161.9 161.3 161.8 161.8 161.1 161.6 161.7
82.28 81.95 81.91 80.65 84.67 83.87 80.64 81.72 83.82 78.65 81.47 81.67 82.31 81.97 81.96 80.67 84.60 83.81 80.70 81.73 83.78 78.72 81.49 81.65 83.08 82.76 82.78 81.74 85.82 85.02 81.02 82.86 84.73 79.01 82.55 82.93
frequencies with the GGA functionals range from 415i cm21 (BLYP/PVDZ) to 222i cm21 (HCTH407/ TZ). This range is due to basis set effects and not the different GGA functionals. The maximum range of lowest frequency with any one basis set (TZ2P) is 77 cm21 from HCTH407/TZ2P 270i cm21 to BLYP/ TZ2P 347i cm21. Changing the GGA functional, as in Table 1, or re-parameterizing it using larger ‘training’ sets, as in Table 2, does not have a large effect on the character, minimum or transition structure, of
the oxirene stationary point as judged by the lowest vibrational frequency. Most of the hybrid functionals (Tables 3 and 4) predict an imaginary frequency for the oxirene stationary point. This observation is consistent with previous findings [7,8,12]. However, some hybrid functionals predict oxirene to be a minimum, as reported recently for the B97-2 functional [12]. The unpolarized and generally less suitable DZ and TZ sets yield a small but real ring opening vibration with
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Table 4 Molecular structures, vibrational frequencies of the ring opening mode, and the energy differences between ketene and oxirene as predicted by one exact exchange parameter hybrid functionals Level of theory
Frequency (cm21)
˚) rCO (A
˚) rCC (A
˚) rCH (A
aCOC (deg)
aCCH (deg)
DE (kcal mol21)
B97/6-31G* B97/6-31G** B97/6-31G(2d,p) B97/6-31GE B97/6-311G* B97/6-311G** B97/DZ B97/DZP B97/PVDZ B97/TZ B97/TZ2P B97/PVTZ B97-1/6-31G* B97-1/6-31G** B97-1/6-31G(2d,p) B97-1/6-31GE B97-1/6-311G* B97-1/6-311G** B97-1/DZ B97-1/DZP B97-1/PVDZ B97-1/TZ B97-1/TZ2P B97-1/PVTZ PBE0/6-31G* PBE0/6-31G** PBE0/6-31G(2d,p) PBE0/6-31GE PBE0/6-311G* PBE0/6-311G** PBE0/DZ PBE0/DZP PBE0/PVDZ PBE0/TZ PBE0/TZ2P PBE0/PVTZ
269i 266i 226i 207i 266i 254i 115i 268i 283i 131 215i 221i 265i 261i 219i 199i 262i 249i 111i 263i 281i 136 208i 215i 39 63 127 180 34 94 179 104 124i 258 154 173
1.506 1.506 1.505 1.502 1.505 1.504 1.577 1.507 1.510 1.567 1.501 1.499 1.507 1.507 1.506 1.503 1.505 1.504 1.578 1.507 1.510 1.568 1.502 1.500 1.488 1.487 1.487 1.485 1.487 1.486 1.560 1.488 1.492 1.549 1.484 1.481
1.268 1.269 1.268 1.269 1.263 1.264 1.290 1.277 1.273 1.271 1.261 1.260 1.270 1.270 1.269 1.271 1.265 1.265 1.291 1.279 1.275 1.273 1.263 1.262 1.266 1.266 1.266 1.267 1.262 1.262 1.286 1.275 1.271 1.270 1.260 1.260
1.075 1.074 1.073 1.076 1.073 1.071 1.074 1.077 1.080 1.070 1.070 1.070 1.075 1.074 1.073 1.076 1.074 1.072 1.074 1.078 1.080 1.071 1.071 1.070 1.074 1.073 1.073 1.076 1.073 1.072 1.072 1.077 1.079 1.070 1.071 1.071
49.8 49.8 49.8 50.0 49.6 49.7 48.3 50.2 49.9 47.9 49.7 49.7 49.8 49.9 49.8 50.0 49.7 49.7 48.3 50.2 49.9 47.9 49.7 49.8 50.4 50.4 50.4 50.5 50.2 50.3 48.7 50.7 50.4 48.4 50.2 50.3
161.8 161.7 162.1 161.8 162.1 161.8 161.3 161.6 161.7 161.1 161.6 161.6 161.8 161.7 162.1 161.9 162.2 161.8 161.3 161.7 161.8 161.1 161.6 161.7 162.1 162.0 162.3 162.1 162.4 162.1 161.4 162.0 162.0 161.1 161.8 161.9
83.79 83.47 83.20 82.28 86.16 85.40 81.22 83.73 85.20 79.06 82.84 83.34 83.38 83.04 82.69 81.73 85.64 84.88 80.88 83.23 84.76 78.68 82.31 82.78 82.05 81.71 81.76 80.40 84.30 83.52 80.84 81.50 83.65 78.68 81.19 81.32
the B3P86 and B3P91 hybrid functionals. B97/TZ and B97-1/TZ also predict oxirene to be a minimum. Overall, the predicted lowest vibrational frequencies for the hybrid functionals range from 283i cm21 B97/PVDZ to 258 cm21 PBE0/TZ. The energy of the stationary point for oxirene, with respect to the ketene, is increased slightly by the inclusion of exact exchange, e.g. BLYP/6-311G** 84.08 kcal/mol vs. B3LYP/6-311G** 85.02 kcal/mol. This trend in the relative energy upon inclusion of exact exchange is different to that observed for basis set effects where
a more positive vibrational frequency correlated with a stabilization of the oxirene. The inclusion of exact exchange starts to correct or corrects the curvature of the potential energy surface along the ring-opening mode. Unlike the GGA functionals, the hybrid functionals show a large variation in the frequency of the ring opening mode within a given basis set. For example, the different hybrid functionals predict frequencies which range from 215i cm21 (B97) to 154 cm21 (PBE0) with the largest TZ2P basis set. For the three
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parameter functionals, B3, different predictions of potential energy surface curvature along the ring opening mode are observed. The B3P86 functional predicts that oxirene is a minimum with one polarized basis set (6-31GE) while with two of the polarized sets (6-31GE and PVTZ) the B3P91 functional predicts a real ring opening frequency. The parameterized B3, B97, and B97-1 hybrid functionals do not show large changes in the ring opening vibrational mode provided adequate polarized basis sets are used. The PBE0 hybrid functional, while earlier classified as a one exact exchange parameter functional, is not actually ‘parameterized’. Both the GGA exchange and correlation corrections and the amount of exact exchange were ‘theoretically’ derived, the latter using perturbation theory [26,27]. This functional predicts oxirene to be a minimum on the potential energy surface with all basis sets except one (PVDZ). Therefore, the PBE0 functional frequently yields positive curvature of the potential energy surface along the ring opening mode near the oxirene stationary point. The results with the largest TZ2P basis set are used to compare the PBE0 and B97-2 [12] functionals. PBE0 predicts a ring opening frequency of 154 cm21 for oxirene. This frequency is approximately the average value from the CCSD(T) predictions [7]. The B97-2/TZ2P result was slightly smaller at 107 cm21 [12]. Both PBE0 and B97-2 predict very similar geometries (1.484, 1.260, 1.071, 50.1, 161.8 and 1.486, 1.260, 1.069, 50.2, 161.9 for rCO , rCC , rCH , aCOC , aHCC, respectively). These bond lengths and bond angles are all slightly smaller than the corresponding CCSD (T) values. The relative energies of oxirene with respect to ketene, are within 0.32 kcal/ mol of each other with PBE0 and B97-2 and are about 1 kcal/mol larger than the average result from CCSD (T) of 80.02 kcal/mol [7]. The PBE0 functional suggests itself as a useful computational tool for the study of a difficult stationary point such as oxirene.
4. Conclusions GGA functionals consistently, and at variance with the highest level CCSD(T) methods, predict oxirene to be a transition state on the potential energy surface. With GGA functionals, variations in the energy
269
difference relative to ketene and the frequency of the ring opening mode are largely due to basis set effects. Different GGA functionals with the same basis set show very little variation in the size of the imaginary frequency. The inclusion of exact exchange reduces the magnitude of the imaginary frequency, and in a number of cases predicts oxirene to be a minimum. The inclusion of exact exchange, however, destabilizes the oxirene stationary point with respect to the ketene minimum. The most interesting finding involves the PBE0 hybrid functional, which predicts oxirene to be a minimum on the potential energy surface with 11 of the 12 basis sets employed. Furthermore, the predicted ring opening frequency, of 154 cm21, using PBE0 and the largest TZ2P basis set is nearly equal that obtained from high level CCSD(T) calculations. Similarly, the energy difference between the oxirene stationary point and the ketene global minimum is within ca. 1 kcal/mol of that predicted by CCSD(T).
Acknowledgements Financial support for this research by the Natural Sciences and Engineering Research Council of Canada (NSERC) is gratefully acknowledged. Computing facilities were provided by Lakehead University through the coordination of the C3.ca organization.
References [1] I.G. Csizmadia, H.E. Gunning, R.K. Gosavi, O.P. Strausz, J. Am. Chem. Soc. 95 (1973) 133. [2] M.J.S. Dewar, C.A. Ramsden, J. Chem. Soc. Chem. Commun. (1973) 688. [3] A.C. Hopkinson, J. Chem. Soc. Perkin.Trans 2 (1973) 794. [4] L. Wolff, Justus Liebigs Ann. Chem. 325 (1902) 129. [5] L. Wolff, Justus Liebigs Ann. Chem. 394 (1912) 23. [6] I.G. Csizmadia, J. Font, O.P. Strausz, J. Am. Chem. Soc. 90 (1968) 7360. [7] G. Vacek, J.M. Galbraith, Y. Yamaguchi, H.F. Schaefer III, R.H. Nobes, A.P. Scott, L. Radom, J. Phys. Chem. 98 (1994) 8660. [8] A.P. Scott, R.H. Nobes, H.F. Schaefer III, L. Radom, J. Am. Chem. Soc. 116 (1994) 10159. [9] V. Termath, D.J. Tozer, N.C. Handy, Chem. Phys. Lett. 228 (1994) 239.
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[10] N.L. Ma, M.W. Wong, Eur. J. Org. Chem. (2000) 1411. [11] C. Delamere, C. Jakins, E. Lewars, Can. J. Chem. 80 (2002) 94. [12] P.J. Wilson, D.J. Tozer, Chem. Phys. Lett. 352 (2002) 540. [13] R.D. Amos, I.L. Alberts, J.S. Andrews, S.M. Colwell, N.C. Handy, D. Jayatilaka, P.J. Knowles, R. Kobayashi, K.E. Laidig, G. Laming, A.M. Lee, P.E. Maslen, C.W. Murray, J.E. Rice, E.D. Simandiras, A.J. Stone, M.-D. Su, D.J. Tozer, CADPAC 6.5, The Cambridge Analytic Derivatives Package, 1998 [14] A.D. Becke, Phys. Rev. A 38 (1988) 3098. [15] C. Lee, W. Yang, R.G. Parr, Phys. Rev. B 37 (1988) 785. [16] A.D. Becke, J. Chem. Phys. 98 (1993) 5648. [17] S.H. Vosko, L. Wilk, M. Nusair, Can. J. Phys. 58 (1980) 1200. [18] A.D. Becke, J. Chem. Phys. 107 (1997) 8554. [19] F.A. Hamprecht, A.J. Cohen, D.J. Tozer, N.C. Handy, J. Chem. Phys. 109 (1998) 6264. [20] J.P. Perdew, J.A. Chevary, S.H. Vosko, K.A. Jackson, M.R. Pederson, D.J. Singh, C. Fiolhais, Phys. Rev. B 46 (1992) 6671. [21] J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865.
[22] J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 78 (1997) 1396. [23] A.D. Boese, N.L. Doltsinis, N.C. Handy, M. Sprik, J. Chem. Phys. 112 (2000) 1670. [24] A.D. Boese, N.C. Handy, J. Chem. Phys. 114 (2001) 5497. [25] J. Perdew, Phys. Rev. B 33 (1986) 8822. [26] J.P. Perdew, M. Ernzerhof, K. Burke, J. Chem. Phys. 105 (1996) 9982. [27] M. Ernzerhof, J.P. Perdew, K. Burke, Int. J. Quantum Chem. 64 (1997) 285. [28] P.C. Hariharan, J.A. Pople, Theor. Chim. Acta 28 (1973) 213. [29] R. Krishnan, J.S. Binkley, R. Seeger, J.A. Pople, J. Chem. Phys. 72 (1980) 650. [30] M.J. Frisch, J.A. Pople, J.S. Binkley, J. Chem. Phys. 80 (1984) 3265. [31] S. Huzinaga, J. Chem. Phys. 42 (1965) 1293. [32] T.H. Dunning, J. Chem. Phys. 53 (1970) 2823. [33] T.H. Dunning, J. Chem. Phys. 55 (1971) 716. [34] More complete discussions of the basis sets including the values of polarization exponents and the construction of the 631GE set are available online in the CADPAC documentation at: http://www-theor.ch.cam.ac.uk/software/cadpac.html.
Assessment of density functional theory for the prediction of the nature of the oxirene stationary point Robert C. Mawhinneya,*, John D. Goddardb a
Department of Chemistry and Biochemistry, Concordia University, Montreal QC, Canada H3G 1M8 b Department of Chemistry and Biochemistry, University of Guelph, Guelph ON, Canada N1G 2W1 Received 27 January 2003; accepted 27 February 2003
Abstract The nature of the stationary point of oxirene, cyclic C2H2O, has been examined using selected GGA and hybrid functionals in combination with a number of basis sets. All GGA functionals (BLYP, BP91, PBE, HCTH, HCTH147, and HCTH407) predict the stationary point to be a transition state and show very little basis set dependence. The hybrid functionals (B3LYP, B3P91, PBE0, B97, B97-1, and B3P86), on the other hand, show more basis set dependence and predict oxirene to be a minimum in a number of cases. However, these parameterized hybrid functionals are not consistent in their characterization of oxirene. The theoretically derived hybrid functional PBE0 predicts oxirene to be a minimum with 11 of the 12 basis sets used and results with the largest basis set are in good agreement with earlier predictions with the high level correlated method, CCSD(T). q 2003 Elsevier B.V. All rights reserved. Keywords: Oxirene; Density functional theory; Stationary point characterization
1. Introduction The C2H2O system has been examined intensively from a theoretical viewpoint, with computational studies dating back nearly three decades [1,2,3]. This anti-aromatic three-member ring system, oxirene, has been postulated by experimentalists to be an intermediate in the Wolff rearrangement. In that reaction, the photolysis of a diazoketone leads to the formation of a ketocarbene which ultimately rearranges to the product ketene [4,5]. Experiments in which the reactant carbonyl carbon was labelled showed an * Corresponding author. E-mail addresses: [email protected] (R.C. Mawhinney), [email protected] (J.D. Goddard).
equal distribution of 13C labelled ketene products [6]. To explain this result an epoxide species of C2v symmetry, oxirene, was proposed as an intermediate. The theoretical characterization of stationary points on certain potential energy surfaces can be demanding. While DFT methods have proven very useful for the prediction of many molecular properties, there seems to be a deficiency when applied to the characterization of certain difficult stationary points. The C2v stationary point of oxirene appears to be one of these problematic systems. It has been observed that the ring opening mode of oxirene, which is the lowest-energy vibration, is extremely sensitive to the theoretical model used. Therefore, depending on the choice of model chemistry, oxirene is predicted to be
0166-1280/03/$ - see front matter q 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0166-1280(03)00198-2
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either a minimum energy species or a transition structure [7 – 12]. Vacek et al. [7] very thoroughly examined the character of the C2v oxirene stationary point through variations in the level of theory and basis set. Their main conclusions regarding oxirene were that Hartree –Fock theory always predicts it to be a minimum, DFT yields a transition state structure, and for postHartree –Fock methods, such as Moller-Plesset and coupled cluster, the nature of the stationary point depends on the basis set. With their most elaborate CCSD(T) method, oxirene was a minimum with a ring-opening vibration mode of ca. 150 cm21 (an average of 139 cm21 from CCSD(T)-full/cc-pVTZ and 163 cm21 from CCSD(T)-frozen core/TZ2P(f, d)). Only recently has any DFT method predicted oxirene to be a minimum [12]. In that article, Wilson and Tozer showed that only the recently formulated hybrid functional, B97-2, predicts oxirene to be a minimum. All other published results from DFT [7,8,9] confirmed the conclusions of Vacek et al. [7]. However, in the examination of oxirene, these DFT studies have been somewhat limited in the choice of exchange-correlation functionals and the basis set dependence in DFT was not thoroughly examined. In this letter, we assess the character of the oxirene stationary point using a larger selection of functionals, both GGA and hybrid, and a greater range of basis sets.
2. Methods All calculations were performed using the CADPAC suite of programs [13]. A total of 13 functionals and 12 basis sets were used for geometry optimization and vibrational frequency analysis of the oxirene stationary point. C2v symmetry was enforced for oxirene. The most stable C2H2O isomer, ketene, also was computed to assess the relative stability of oxirene with various methods. We have included results with the BLYP [14,15], B3LYP [15,16], B97 [17] and B97-1 [18,19] functionals for completeness. These functionals were previously shown to predict oxirene to be a transition structure [7 –9,12]. A selection of basis sets [27 – 33] will be tested since the results on oxirene with conventional correlated
methods showed some basis set dependence. DFT is, of course, a correlated approach although its basis set dependence is usually considered to be slight. The version of CADPAC used did not have the B97-2 functional implemented. The remaining eight functionals used in this work consist of the GGA functionals BP91 [14,20], PBE [21,22], and the HCTH set (HCTH, HCTH147, and HCTH407) [19, 23,24] and the hybrid functionals B3P86 [16,25], B3P91 [16,20], and PBE0 [26,27]. The basis sets were taken from those internal to the CADPAC suite of programs [34]. These include the 631G split valence and 6-311G triply split valence sets with variations in the numbers of polarization functions added, [28 – 30] and the Huzinaga-Dunning double and triple zeta sets with and without polarization (DZ, DZP, PVDZ, TZ, TZ2P, and PVTZ) [31 – 33].
3. Results and discussion The geometries of oxirene, the ring opening vibrational frequency, and the relative energies with respect to the ketene minimum are given in Tables 1 –4. The relative energies are calculated from the total energies without consideration of the changes in zero point vibrational energies between the two isomers. The results for the GGA functionals are presented in Tables 1 and 2. Table 1 contains results for those GGA functionals which do not involve fits to experimental data (BLYP, BP91, and PBE). The GGA results in Table 2 are from functionals (HCTH, HCTH147, and HCTH407) which were parameterized with the use of experimental results. The results with hybrid functionals are given in Tables 3 and 4. Table 3 involves functionals with a three parameter fit in the exchange functional (B3LYP, B3P91, and B3P86). The results with functionals that have one parameter for exact exchange (B97, B97-1, and PBE0) are collected in Table 4. B3LYP (VWN-III) and B3LYP(VWN-V) [17] produced practically identical results and only the predictions from B3LYP(VWN-III) are reported in Table 3. The effects of basis sets can be addressed. For the doubly and triply split valence basis sets, 6-31G and 6-311G, when the number of polarization functions is increased, e.g. from 6-31G* to 6-31G** to 6-31G
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265
Table 1 Molecular structures, vibrational frequencies of the ring opening mode and the energy differences between ketene and oxirene as predicted by GGA functionals Level of theory
Frequency (cm21)
˚) rCO (A
˚) rCC (A
˚) rCH (A
aCOC (deg)
aCCH (deg)
DE (kcal mol21)
BLYP/6-31G* BLYP/6-31G** BLYP/6-31G(2d,p) BLYP/6-31GE BLYP/6-311G* BLYP/6-311G** BLYP/DZ BLYP/DZP BLYP/PVDZ BLYP/TZ BLYP/TZ2P BLYP/PVTZ BP91/6-31G* BP91/6-31G** BP91/6-31G(2d,p) BP91/6-31GE BP91/6-311G* BP91/6-311G** BP91/DZ BP91/DZP BP91/PVDZ BP91/TZ BP91/TZ2P BP91/PVTZ PBE/6-31G* PBE/6-31G** PBE/6-31G(2d,p) PBE/6-31GE PBE/6-311G* PBE/6-311G** PBE/DZ PBE/DZP PBE/PVDZ PBE/TZ PBE/TZ2P PBE/PVTZ
399i 397i 347i 330i 395i 387i 335i 395i 415i 273i 347i 347i 380i 377i 317i 294i 372i 363i 313i 366i 396i 248i 315i 311i 379i 376i 312i 290i 373i 363i 319i 368i 398i 248i 312i 310i
1.531 1.531 1.530 1.528 1.532 1.530 1.605 1.534 1.536 1.600 1.528 1.525 1.515 1.515 1.515 1.512 1.515 1.514 1.588 1.517 1.520 1.581 1.512 1.509 1.513 1.512 1.512 1.510 1.513 1.511 1.586 1.514 1.517 1.578 1.510 1.507
1.278 1.279 1.278 1.279 1.273 1.273 1.301 1.290 1.283 1.280 1.270 1.270 1.278 1.279 1.277 1.279 1.273 1.273 1.299 1.288 1.282 1.281 1.271 1.271 1.279 1.279 1.278 1.280 1.274 1.274 1.300 1.289 1.283 1.282 1.272 1.272
1.081 1.080 1.078 1.081 1.079 1.077 1.081 1.084 1.086 1.076 1.075 1.075 1.081 1.080 1.079 1.082 1.079 1.077 1.080 1.083 1.085 1.077 1.076 1.076 1.083 1.081 1.080 1.083 1.081 1.079 1.082 1.085 1.087 1.078 1.078 1.078
49.4 49.4 49.3 49.5 49.1 49.2 47.8 49.7 49.4 47.2 49.1 49.2 49.9 49.9 49.9 50.0 49.7 49.7 48.3 50.2 49.9 47.8 49.7 49.8 50.0 50.0 50.0 50.2 49.8 49.9 48.4 50.4 50.0 47.9 49.8 49.9
161.6 161.5 162.0 161.8 161.9 161.6 161.0 161.5 161.6 160.9 161.4 161.5 161.8 161.7 162.1 161.9 162.1 161.8 161.0 161.7 161.7 160.9 161.5 161.6 161.8 161.7 162.1 161.9 162.1 161.8 161.1 161.7 161.7 160.9 161.5 161.6
82.16 81.83 81.13 80.33 84.90 84.08 79.81 82.07 83.53 77.80 81.31 81.84 81.53 81.16 80.46 79.33 83.70 82.88 79.70 80.97 82.63 77.80 80.29 80.55 81.31 80.94 80.12 79.02 83.57 82.76 79.92 80.89 82.56 77.87 80.03 80.32
(2d,p) the oxirene stationary point becomes slightly more stable with respect to ketene. Polarization is more important in describing the strained three member ring as compared to the ketene. However, the effect is small, e.g. the oxirene becomes 1.03 kcal/mol more stable on changing from BLYP/6-31G* to BLYP/6-31G(2d,p). With the hybrid functionals the stabilizations in kcal/mol with the same change in basis set are only 0.37 B3P86, 0.30 B3LYP, 0.59 B97 and 0.31 PBE0. Oxirene also becomes vibrationally more stable with the addition
of more polarization functions i.e. the ring opening frequency if imaginary becomes smaller or if real (the PBE0 results in Table 4) becomes larger. A similar result was observed by Vacek et al. [7]. For the basis sets originating from the double and triple zeta sets, the addition of polarization functions to the unpolarized DZ or TZ set destabilizes the oxirene stationary point. When the CO distances are examined with the unpolarized DZ and TZ basis sets, these bond lengths are too ˚ . The destabilization long and greater than 1.55 A
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Table 2 Molecular structures, vibrational frequencies of the ring opening mode, and the energy differences between ketene and oxirene as predicted by the 15 parameter GGA functionals Level of theory
Frequency (cm21)
˚) rCO (A
˚) rCC (A
˚) rCH (A
aCOC (deg)
aCCH (deg)
DE (kcal mol21)
HCTH/6-31G* HCTH/6-31G** HCTH/6-31G(2d,p) HCTH/6-31GE HCTH/6-311G* HCTH/6-311G** HCTH/DZ HCTH/DZP HCTH/PVDZ HCTH/TZ HCTH/TZ2P HCTH/PVTZ HCTH147/6-31G* HCTH147/6-31G** HCTH147/6-31G(2d,p) HCTH147/6-31GE HCTH147/6-311G* HCTH147/6-311G** HCTH147/DZ HCTH147/DZP HCTH147/PVDZ HCTH147/TZ HCTH147/TZ2P HCTH147/PVTZ HCTH407/6-31G* HCTH407/6-31G** HCTH407/6-31G(2d,p) HCTH407/6-31GE HCTH407/6-311G* HCTH407/6-311G** HCTH407/DZ HCTH407/DZP HCTH407/PVDZ HCTH407/TZ HCTH407/TZ2P HCTH407/PVTZ
372i 368i 309i 284i 365i 355i 315i 357i 392i 250i 304i 298i 371i 368i 309i 284i 365i 355i 321i 359i 393i 254i 305i 300i 342i 338i 278i 243i 337i 326i 302i 327i 372i 222i 270i 260i
1.499 1.499 1.500 1.496 1.499 1.498 1.575 1.501 1.503 1.564 1.496 1.493 1.500 1.500 1.501 1.497 1.500 1.499 1.576 1.502 1.505 1.566 1.497 1.494 1.491 1.491 1.492 1.488 1.491 1.490 1.569 1.493 1.496 1.557 1.488 1.486
1.273 1.274 1.274 1.275 1.270 1.270 1.295 1.283 1.278 1.278 1.268 1.268 1.273 1.274 1.273 1.275 1.269 1.270 1.295 1.283 1.278 1.278 1.267 1.267 1.272 1.272 1.272 1.273 1.268 1.269 1.294 1.282 1.276 1.277 1.267 1.267
1.078 1.076 1.075 1.078 1.076 1.074 1.076 1.079 1.082 1.073 1.073 1.073 1.078 1.076 1.075 1.078 1.076 1.074 1.076 1.080 1.082 1.073 1.073 1.073 1.077 1.075 1.075 1.077 1.075 1.073 1.075 1.079 1.081 1.072 1.072 1.073
50.3 50.3 50.3 50.4 50.1 50.2 48.6 50.6 50.3 48.2 50.1 50.2 50.2 50.3 50.2 50.4 50.1 50.1 48.5 50.6 50.3 48.2 50.1 50.2 50.5 50.5 50.5 50.7 50.3 50.4 48.7 50.9 50.5 48.4 50.4 50.5
162.0 161.9 162.1 162.0 162.3 162.0 161.1 161.9 161.8 161.0 161.7 161.8 162.0 161.9 162.2 162.0 162.3 162.0 161.2 161.9 161.9 161.0 161.7 161.8 162.1 162.0 162.2 162.1 162.4 162.1 161.2 162.0 162.0 161.0 161.8 161.9
83.00 82.60 82.06 80.67 84.89 84.05 81.78 82.35 84.16 79.78 81.63 81.68 82.78 82.39 81.88 80.52 84.84 84.00 81.62 82.23 84.07 79.60 81.52 81.65 82.57 82.17 81.75 80.25 84.59 83.74 82.25 81.96 83.99 80.05 81.25 81.33
due to ring strain in the epoxide is artificially decreased at such long CO distances but increases when polarization functions lead to shorter bond lengths. The 6-31GE basis set, which is described in the CADPAC documentation [34] as starting from 6-31G and adding diffuse s and p functions and using two sets of slightly diffuse polarization functions, produces two interesting results. With the three parameter hybrid functionals, B3P86 and B3P91, the ring opening frequency is real but very small, 38 and 68 cm21, respectively.
Basis set effects play some role in the differing results for the structures, ring opening vibrational frequencies and relative energies with the various exchange-correlation functionals. However, basis set effects do not appear to be the primary cause of the different curvatures of the potential energy surface along the ring opening mode for oxirene. The different exchange correlation functionals are now considered. Note that all GGA functionals (Tables 1 and 2), without regard to basis set, predict oxirene to be a transition state. The imaginary
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Table 3 Molecular structures,vibrational frequencies of the ring opening mode, and the energy differences between ketene and oxirene as predicted by the B3—three parameter hybrid functionals Level of theory
Frequency (cm21)
˚) rCO (A
˚) rCC (A
˚) rCH (A
aCOC (deg)
aCCH (deg)
DE (kcal mol21)
B3P86/6-31G* B3P86/6-31G** B3P86/6-31G(2d,p) B3P86/6-31GE B3P86/6-311G* B3P86/6-311G** B3P86/DZ B3P86/DZP B3P86/PVDZ B3P86/TZ B3P86/TZ2P B3P86/PVTZ B3P91/6-31G* B3P91/6-31G** B3P91/6-31G(2d,p) B3P91/6-31GE B3P91/6-311G* B3P91/6-311G** B3P91/DZ B3P91/DZP B3P91/PVDZ B3P91/TZ B3P91/TZ2P B3P91/PVTZ B3LYP/6-31G* B3LYP/6-31G** B3LYP/6-31G(2d,p) B3LYP/6-31GE B3LYP/6-311G* B3LYP/6-311G** B3LYP/DZ B3LYP/DZP B3LYP/PVDZ B3LYP/TZ B3LYP/TZ2P B3LYP/PVTZ
181i 175i 119i 38 181i 159i 59 153i 221i 188 88i 53i 172i 165i 105i 68 171i 147i 83 141i 212i 196 67i 31 219i 214i 181i 137i 221i 205i 101i 208i 251i 144 166i 154i
1.495 1.495 1.495 1.492 1.495 1.494 1.568 1.496 1.500 1.558 1.491 1.489 1.494 1.494 1.494 1.491 1.494 1.492 1.567 1.495 1.499 1.557 1.490 1.488 1.505 1.504 1.504 1.501 1.505 1.503 1.579 1.506 1.510 1.570 1.501 1.498
1.268 1.268 1.267 1.268 1.263 1.263 1.288 1.277 1.273 1.271 1.261 1.261 1.267 1.268 1.267 1.268 1.263 1.263 1.288 1.276 1.272 1.270 1.260 1.260 1.267 1.268 1.266 1.268 1.262 1.262 1.289 1.277 1.272 1.269 1.260 1.259
1.076 1.074 1.074 1.076 1.074 1.072 1.074 1.078 1.080 1.071 1.071 1.071 1.075 1.073 1.073 1.075 1.073 1.071 1.073 1.077 1.079 1.070 1.070 1.070 1.074 1.073 1.072 1.075 1.072 1.070 1.073 1.077 1.079 1.069 1.069 1.069
50.2 50.2 50.2 50.3 50.0 50.0 48.5 50.5 50.2 48.1 50.0 50.1 50.2 50.2 50.2 50.3 50.0 50.1 48.5 50.5 50.2 48.2 50.0 50.1 49.8 49.8 49.8 50.0 49.6 49.6 48.2 50.2 49.8 47.7 49.6 49.7
162.0 161.9 162.2 162.0 162.3 162.0 161.3 161.9 161.9 161.1 161.7 161.8 162.0 161.9 162.2 162.0 162.3 162.0 161.3 161.9 161.9 161.1 161.7 161.8 161.9 161.8 162.2 162.0 162.2 161.9 161.3 161.8 161.8 161.1 161.6 161.7
82.28 81.95 81.91 80.65 84.67 83.87 80.64 81.72 83.82 78.65 81.47 81.67 82.31 81.97 81.96 80.67 84.60 83.81 80.70 81.73 83.78 78.72 81.49 81.65 83.08 82.76 82.78 81.74 85.82 85.02 81.02 82.86 84.73 79.01 82.55 82.93
frequencies with the GGA functionals range from 415i cm21 (BLYP/PVDZ) to 222i cm21 (HCTH407/ TZ). This range is due to basis set effects and not the different GGA functionals. The maximum range of lowest frequency with any one basis set (TZ2P) is 77 cm21 from HCTH407/TZ2P 270i cm21 to BLYP/ TZ2P 347i cm21. Changing the GGA functional, as in Table 1, or re-parameterizing it using larger ‘training’ sets, as in Table 2, does not have a large effect on the character, minimum or transition structure, of
the oxirene stationary point as judged by the lowest vibrational frequency. Most of the hybrid functionals (Tables 3 and 4) predict an imaginary frequency for the oxirene stationary point. This observation is consistent with previous findings [7,8,12]. However, some hybrid functionals predict oxirene to be a minimum, as reported recently for the B97-2 functional [12]. The unpolarized and generally less suitable DZ and TZ sets yield a small but real ring opening vibration with
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Table 4 Molecular structures, vibrational frequencies of the ring opening mode, and the energy differences between ketene and oxirene as predicted by one exact exchange parameter hybrid functionals Level of theory
Frequency (cm21)
˚) rCO (A
˚) rCC (A
˚) rCH (A
aCOC (deg)
aCCH (deg)
DE (kcal mol21)
B97/6-31G* B97/6-31G** B97/6-31G(2d,p) B97/6-31GE B97/6-311G* B97/6-311G** B97/DZ B97/DZP B97/PVDZ B97/TZ B97/TZ2P B97/PVTZ B97-1/6-31G* B97-1/6-31G** B97-1/6-31G(2d,p) B97-1/6-31GE B97-1/6-311G* B97-1/6-311G** B97-1/DZ B97-1/DZP B97-1/PVDZ B97-1/TZ B97-1/TZ2P B97-1/PVTZ PBE0/6-31G* PBE0/6-31G** PBE0/6-31G(2d,p) PBE0/6-31GE PBE0/6-311G* PBE0/6-311G** PBE0/DZ PBE0/DZP PBE0/PVDZ PBE0/TZ PBE0/TZ2P PBE0/PVTZ
269i 266i 226i 207i 266i 254i 115i 268i 283i 131 215i 221i 265i 261i 219i 199i 262i 249i 111i 263i 281i 136 208i 215i 39 63 127 180 34 94 179 104 124i 258 154 173
1.506 1.506 1.505 1.502 1.505 1.504 1.577 1.507 1.510 1.567 1.501 1.499 1.507 1.507 1.506 1.503 1.505 1.504 1.578 1.507 1.510 1.568 1.502 1.500 1.488 1.487 1.487 1.485 1.487 1.486 1.560 1.488 1.492 1.549 1.484 1.481
1.268 1.269 1.268 1.269 1.263 1.264 1.290 1.277 1.273 1.271 1.261 1.260 1.270 1.270 1.269 1.271 1.265 1.265 1.291 1.279 1.275 1.273 1.263 1.262 1.266 1.266 1.266 1.267 1.262 1.262 1.286 1.275 1.271 1.270 1.260 1.260
1.075 1.074 1.073 1.076 1.073 1.071 1.074 1.077 1.080 1.070 1.070 1.070 1.075 1.074 1.073 1.076 1.074 1.072 1.074 1.078 1.080 1.071 1.071 1.070 1.074 1.073 1.073 1.076 1.073 1.072 1.072 1.077 1.079 1.070 1.071 1.071
49.8 49.8 49.8 50.0 49.6 49.7 48.3 50.2 49.9 47.9 49.7 49.7 49.8 49.9 49.8 50.0 49.7 49.7 48.3 50.2 49.9 47.9 49.7 49.8 50.4 50.4 50.4 50.5 50.2 50.3 48.7 50.7 50.4 48.4 50.2 50.3
161.8 161.7 162.1 161.8 162.1 161.8 161.3 161.6 161.7 161.1 161.6 161.6 161.8 161.7 162.1 161.9 162.2 161.8 161.3 161.7 161.8 161.1 161.6 161.7 162.1 162.0 162.3 162.1 162.4 162.1 161.4 162.0 162.0 161.1 161.8 161.9
83.79 83.47 83.20 82.28 86.16 85.40 81.22 83.73 85.20 79.06 82.84 83.34 83.38 83.04 82.69 81.73 85.64 84.88 80.88 83.23 84.76 78.68 82.31 82.78 82.05 81.71 81.76 80.40 84.30 83.52 80.84 81.50 83.65 78.68 81.19 81.32
the B3P86 and B3P91 hybrid functionals. B97/TZ and B97-1/TZ also predict oxirene to be a minimum. Overall, the predicted lowest vibrational frequencies for the hybrid functionals range from 283i cm21 B97/PVDZ to 258 cm21 PBE0/TZ. The energy of the stationary point for oxirene, with respect to the ketene, is increased slightly by the inclusion of exact exchange, e.g. BLYP/6-311G** 84.08 kcal/mol vs. B3LYP/6-311G** 85.02 kcal/mol. This trend in the relative energy upon inclusion of exact exchange is different to that observed for basis set effects where
a more positive vibrational frequency correlated with a stabilization of the oxirene. The inclusion of exact exchange starts to correct or corrects the curvature of the potential energy surface along the ring-opening mode. Unlike the GGA functionals, the hybrid functionals show a large variation in the frequency of the ring opening mode within a given basis set. For example, the different hybrid functionals predict frequencies which range from 215i cm21 (B97) to 154 cm21 (PBE0) with the largest TZ2P basis set. For the three
R.C. Mawhinney, J.D. Goddard / Journal of Molecular Structure (Theochem) 629 (2003) 263–270
parameter functionals, B3, different predictions of potential energy surface curvature along the ring opening mode are observed. The B3P86 functional predicts that oxirene is a minimum with one polarized basis set (6-31GE) while with two of the polarized sets (6-31GE and PVTZ) the B3P91 functional predicts a real ring opening frequency. The parameterized B3, B97, and B97-1 hybrid functionals do not show large changes in the ring opening vibrational mode provided adequate polarized basis sets are used. The PBE0 hybrid functional, while earlier classified as a one exact exchange parameter functional, is not actually ‘parameterized’. Both the GGA exchange and correlation corrections and the amount of exact exchange were ‘theoretically’ derived, the latter using perturbation theory [26,27]. This functional predicts oxirene to be a minimum on the potential energy surface with all basis sets except one (PVDZ). Therefore, the PBE0 functional frequently yields positive curvature of the potential energy surface along the ring opening mode near the oxirene stationary point. The results with the largest TZ2P basis set are used to compare the PBE0 and B97-2 [12] functionals. PBE0 predicts a ring opening frequency of 154 cm21 for oxirene. This frequency is approximately the average value from the CCSD(T) predictions [7]. The B97-2/TZ2P result was slightly smaller at 107 cm21 [12]. Both PBE0 and B97-2 predict very similar geometries (1.484, 1.260, 1.071, 50.1, 161.8 and 1.486, 1.260, 1.069, 50.2, 161.9 for rCO , rCC , rCH , aCOC , aHCC, respectively). These bond lengths and bond angles are all slightly smaller than the corresponding CCSD (T) values. The relative energies of oxirene with respect to ketene, are within 0.32 kcal/ mol of each other with PBE0 and B97-2 and are about 1 kcal/mol larger than the average result from CCSD (T) of 80.02 kcal/mol [7]. The PBE0 functional suggests itself as a useful computational tool for the study of a difficult stationary point such as oxirene.
4. Conclusions GGA functionals consistently, and at variance with the highest level CCSD(T) methods, predict oxirene to be a transition state on the potential energy surface. With GGA functionals, variations in the energy
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difference relative to ketene and the frequency of the ring opening mode are largely due to basis set effects. Different GGA functionals with the same basis set show very little variation in the size of the imaginary frequency. The inclusion of exact exchange reduces the magnitude of the imaginary frequency, and in a number of cases predicts oxirene to be a minimum. The inclusion of exact exchange, however, destabilizes the oxirene stationary point with respect to the ketene minimum. The most interesting finding involves the PBE0 hybrid functional, which predicts oxirene to be a minimum on the potential energy surface with 11 of the 12 basis sets employed. Furthermore, the predicted ring opening frequency, of 154 cm21, using PBE0 and the largest TZ2P basis set is nearly equal that obtained from high level CCSD(T) calculations. Similarly, the energy difference between the oxirene stationary point and the ketene global minimum is within ca. 1 kcal/mol of that predicted by CCSD(T).
Acknowledgements Financial support for this research by the Natural Sciences and Engineering Research Council of Canada (NSERC) is gratefully acknowledged. Computing facilities were provided by Lakehead University through the coordination of the C3.ca organization.
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