Basis Set and Correlation Effects on Transition State Geometries and

Publication Date (Web): October 17, 1996 ... We present an investigation of basis set and correlation effects on transition state geometries and prima...
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16892

J. Phys. Chem. 1996, 100, 16892-16898

Basis Set and Correlation Effects on Transition State Geometries and Kinetic Isotope Effects Sanne S. Glad and Frank Jensen* Research School of Chemistry, Australian National UniVersity, Canberra, ACT 0200, Australia, and Department of Chemistry, Odense UniVersity, CampusVej 55, DK-5230 Odense M, Denmark ReceiVed: May 29, 1996; In Final Form: August 16, 1996X

We present an investigation of basis set and correlation effects on transition state geometries and primary and secondary kinetic isotope effects within an E2 model system. Four different ab initio methods have been employed: Hartree-Fock (HF), second- and partial-fourth-order Møller-Plesset perturbation theory, and quadratic configuration interaction with single and double excitations. Calculations have also been performed using density functional theory methods, but they perform poorly on this system. Eleven different basis sets up to 6-311++G(2df,2p) have been employed at the HF and MP2 levels. Both geometries and kinetic isotope effects are influenced by the theoretical level, and the HF method gives results significantly different from the correlated methods. More important, the HF method produces wrong relative values of both the primary and one of the secondary kinetic isotope effects. A uniform scaling of the HF frequencies is shown to have varying effects, improving the performance for PKIEs significantly, but sometimes increasing the disagreement for SKIEs with the best theoretical method. To obtain results comparable to the largest calculations done, it is necessary to include electron correlation and employ a basis set of at least 6-31+G(d) quality.

Introduction

SCHEME 1

Kinetic isotope effects (KIEs) are ratios of rate constants for pairs of isotopically different molecules. They are widely used as tools for elucidating reaction mechanisms and have been investigated both experimentally and theoretically.1 A theoretical prediction of kinetic isotope effects requires optimized geometries of the reactant and transition state (TS) and corresponding frequencies. Obviously, the accuracy of kinetic isotope effects will depend on the theoretical method and the size of the basis set used. There have been several investigations on the performance of ab initio and density functional methods on ground state geometries and energies,2 as well as on frequencies,2a,g,h,3,4 but only one where isotope effects were considered by calculating fractionation factors (equilibrium isotope effects).5 Equilibrium isotope effects depend on the reactant and product properties, i.e., on minimum-energy structures. In contrast, kinetic isotope effects depend on the reactant and transition state. There can be large differences in the performance of a theoretical method on ground state and saddle point structures. It is therefore of interest to investigate what level of theory is required to produce reliable transition state geometries, frequencies, and thereby kinetic isotope effects. As a model system, we have chosen the E2 reaction of CH3CH2X + Nu-, using chloride and bromide as leaving groups, and fluoride and chloride as nucleophiles (Scheme 1). The reactions are named by the leaving group and the nucleophile as XNu, and the four TSs investigated are BrF, ClF, BrCl, and ClCl. A thorough investigation of the sensitivity to the theoretical level is performed for the ClF reaction, whereas the other reactions are included for analyzing relative values. Within the E2 system it is possible to investigate the primary kinetic isotope effect (PKIE) from hydrogen/deuterium exchange of the transferred hydrogen and two different R-secondary kinetic isotope effects, one for exchange of the hydrogens (H2, * To whom correspondence should be addressed at Odense University. X Abstract published in AdVance ACS Abstracts, October 1, 1996.

S0022-3654(96)01559-6 CCC: $12.00

H3) at C1 with deuterium (C1-SKIE) and one for the exchange of the nontransferred hydrogens (H4, H5) at C2 (C2-SKIE). Systems of the present type have been used in previous investigations concentrating on correlations between kinetic isotope effects and transition state geometries.6,7 Computational Details All calculations have been performed using the Gaussian-94 program package.8 Geometries have been fully optimized, and kinetic isotope effects have been calculated from standard statistical mechanics using harmonic vibrational frequencies.9

kH/kD ) exp((∆GD - ∆GH)/RT) ∆G ) Gq - Gr Four different levels of ab initio theory have been applied in this investigation: Hartree-Fock theory (HF), with electron correlation included by Møller-Plesset perturbation theory terminated at second (MP2) or fourth order (excluding the contributions from triple excitations, MP4(SDQ)), as well as quadratic configuration interaction with single and double excitations (QCISD). In addition, several density functional theory (DFT) methods have been tried (SVWN, BLYP, BPW91, B3LYP, and B3PW91). © 1996 American Chemical Society

TS Geometries and Kinetic Isotope Effects

J. Phys. Chem., Vol. 100, No. 42, 1996 16893

TABLE 1: Geometry Parameters (Scheme 1) for the ClF Transition Structurea HF/6-31G(d) 6-31+G(d) 6-31++G(d,p) 6-31++G(2d,2p) DH+(d) DH++(d,p) aug-cc-pVDZ 6-311+G(d) 6-311++G(d,p) 6-311++G(2d,2p) 6-311++G(2df,2p) MP2/6-31G(d) 6-31+G(d) 6-31++G(d,p) 6-31++G(2d,2p) DH+(d) DH++(d,p) aug-cc-pVDZ 6-311+G(d) 6-311++G(d,p) 6-311++G(2d,2p) 6-311++G(2df,2p) MP4(SDQ)/6-31G(d) 6-31+G(d) 6-31++G(d,p) 6-311+G(d) QCISD/6-31G(d) 6-31+G(d) 6-31++G(d,p) a

RCC

RCCl

RCH

RFH

θC1

θC2

1.462 1.426 1.435 1.434 1.438 1.442 1.430 1.412 1.436 1.433 1.437 1.464 1.422 1.427 1.437 1.438 1.440 1.441 1.424 1.444 1.440 1.438 1.465 1.425 1.430 1.428 1.466 1.428 1.433

1.997 2.185 2.134 2.128 2.138 2.110 2.176 2.261 2.112 2.115 2.085 1.930 2.114 2.076 2.053 2.078 2.054 2.070 2.092 2.001 2.015 2.000 1.956 2.152 2.111 2.117 1.960 2.150 2.110

1.338 1.359 1.359 1.364 1.378 1.371 1.348 1.347 1.368 1.361 1.378 1.327 1.432 1.425 1.391 1.425 1.413 1.391 1.453 1.433 1.375 1.393 1.349 1.427 1.426 1.457 1.338 1.416 1.414

1.203 1.198 1.189 1.186 1.174 1.173 1.207 1.214 1.174 1.183 1.167 1.232 1.155 1.140 1.173 1.160 1.143 1.182 1.137 1.125 1.183 1.163 1.207 1.155 1.135 1.129 1.217 1.161 1.142

65.09 70.68 69.36 69.29 69.41 68.70 70.80 72.70 68.75 69.00 68.16 62.22 67.80 66.87 66.60 66.98 66.38 67.06 67.19 64.84 65.50 65.37 62.92 68.77 67.75 67.84 63.05 68.74 67.75

61.76 65.48 64.56 64.59 64.50 64.00 65.33 67.08 64.33 64.38 64.04 62.22 66.78 66.09 65.36 65.66 65.11 65.76 67.04 64.58 64.53 64.77 61.92 66.35 65.64 66.27 61.90 65.97 65.32

Bond lengths are in angstroms and angles in degrees.

Geometries and kinetic isotope effects have been calculated using basis sets derived from the standard 6-31G10 and 6-311G11 basis sets, augmented with diffuse and polarization functions.12,13 For bromine the Binning and Curtis contraction14 of the (14s,11p,5d) primitive basis15 has been used. In addition, the Dunning/Huzinaga (DH) double-zeta basis set16 has been employed, as well as Dunning’s correlation consistent polarization valence double-zeta basis set augmented with diffuse functions, aug-cc-pVDZ.17,18 The basis sets are listed in Table 1, where the standard + and (d,p) notations have been used for diffuse and polarization functions, respectively.2a All kinetic isotope effects have been calculated at a temperature of 298.15 K. Results and Discussion The kinetic isotope effects presented below are calculated with respect to individual reactants, but without inclusion of solvent and tunneling effects. As such, they cannot be directly compared to experimental (solution or gas phase) data. Solvation influences the reaction barrier significantly, and the reaction profile is fundamentally different in the two phases. Activation energies can be found as Supporting Information; for ClF and BrF they are in general close to zero or negative, which means that some of the reactions will be collision controlled in the gas phase. In such cases, the kinetic isotope effect becomes negligible. For BrCl and ClCl the activation energies are 20-30 kcal/mol. Quantum mechanical tunneling is known to affect kinetic isotope effects,19-21 but due to difficulties in calculating tunneling contributions, either Wigner’s or Bell’s formula for one-dimensional tunneling has often been applied.22-24 It is not clear, however, that these corrections necessarily lead to better results. A further cause of possible disagreement between theory and experiment is the competition between the E2 and the SN2 reaction pathways found experi-

mentally. This issue has been addressed recently by Hu and Truhlar where the authors conclude that a quantitative prediction of the kinetic isotope effect is a theoretical difficult problem.25 Owing to these problems, and the fact that there are no experimental results for the present employed reactions, neither in solution nor in gas phase, we have chosen to concentrate only on differences between theoretically obtained results. For experimental results in related reactions we refer to previous papers.6 Geometries. The geometry of the ethyl chloride reactant is insensitive to the theoretical level and the size of basis set. The bond lengths vary by less than 0.02 Å (with the MP2/aug-ccpVDZ level consistently giving the largest values) and the dihedral angles of the ethyl chloride hydrogens by less than 1°. As the performance of theoretical methods for minimum-energy structures is well-known, these results will not be discussed in any detail. The geometries can be found as Supporting Information. The geometry of the ClF transition state is more dependent on the theoretical approach. The obtained values for the bond lengths RCC, RCCl, RCH, and RFH and the dihedral angles of the non-transferred hydrogens θC1 and θC2 can be found in Table 1. In general, the two most sophisticated methods, MP4(SDQ) and QCISD, give very similar results. The forming double bond, RCC, is essentially constant except for calculations with the 6-31G(d) basis set, which give significantly larger values at all levels. The three breaking/forming single bonds are influenced by the different methods to varying degree. The bond to the leaving group, RCC1, varies between 1.93 and 2.26 Å, where the HF method consistently overestimates the bond length. The 6-31G(d) basis set again provides results significantly different from all other calculations. Addition of diffuse or polarized functions on hydrogen atoms to the basis set generally decreases the bond length at the MP2 level, whereas inclusion of more electron correlation has the opposite effect. The two bonds around the transferred hydrogen, RCH and RFH, vary between 1.33-1.46 and 1.12-1.23 Å, respectively, where the MP2/6-31G(d) values deviates most. Inclusion of diffuse functions on non-hydrogen atoms in the basis set is seen to increase the RCH bond length and decrease the RFH bond length. Further enlargements of the basis set have varying but smaller effects. The two dihedral angles of relevance for the secondary kinetic isotope effects vary by 10° for θC1 and 5° for θC2, depending on the theoretical level. Inclusion of diffuse functions or triple-splitting the sp basis gives the most significant changes, irrespective of the theoretical method. In addition, the values of the two dihedral angles differ much more at the HF level than by the correlated methods. From the geometries it is clear that inclusion of diffuse functions on non-hydrogen atoms is important for these anionic systems. Comparing the MP2/6-31++G(d,p) ClF geometry with those obtained at the MP2/6-311++G(2df,2p) or QCISD/ 6-31++G(d,p) levels shows that basis set enlargement and electron correlation give effects which are in opposite directions. This suggests that the MP2 method with a medium-sized basis set, such as the 6-31+G(d) or the 6-31++G(d,p), should represent a cost-efficient approach for the E2 systems. Primary Kinetic Isotope Effects. The calculated primary kinetic isotope effects are given in Table 2. Inclusion of electron correlation by the MP2 method results in a significant decrease of the isotope effects. The MP4(SDQ) and QCISD methods give very similar results, which are slightly larger than the MP2 values. The overestimation of the KIEs by the HF method is primarily due to overestimation of the frequencies, as discussed below.

16894 J. Phys. Chem., Vol. 100, No. 42, 1996

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Figure 1. Primary kinetic isotope effects for the ClF reaction as function of basis set at the MP2 level.

Figure 2. Secondary kinetic isotope effects for the ClF reaction as function of basis set at the MP2 level.

TABLE 2: Primary Kinetic Isotope Effects for the ClF Reaction

TABLE 4: C2-Secondary Kinetic Isotope Effects for the ClF Reaction

basis set

HF

MP2

MP4(SDQ)

QCISD

basis set

HF

MP2

MP4(SDQ)

QCISD

6-31G(d) 6-31+G(d) 6-31++G(d,p) 6-31++G(2d,2p) DH+(d) DH++(d,p) aug-cc-pVDZ 6-311+G(d) 6-311++G(d,p) 6-311++G(2d,2p) 6-311++G(2df,2p)

5.358 5.612 5.394 5.304 5.558 5.378 5.398 5.803 5.277 5.330 5.259

4.604 5.040 4.813 4.844 5.087 4.808 4.967 5.734 4.530 5.092 4.896

4.783 5.185 5.010

4.740 5.180 5.055

6-31G(d) 6-31+G(d) 6-31++G(d,p) 6-31++G(2d,2p) DH+(d) DH++(d,p) aug-cc-pVDZ 6-311+G(d) 6-311++G(d,p) 6-311++G(2d,2p) 6-311++G(2df,2p)

1.359 1.310 1.320 1.330 1.366 1.352 1.305 1.286 1.344 1.336 1.360

1.318 1.321 1.341 1.333 1.342 1.350 1.328 1.313 1.386 1.345 1.356

1.330 1.304 1.336

1.319 1.301 1.329

5.630

TABLE 3: C1-Secondary Kinetic Isotope Effects for the ClF Reaction basis set

HF

MP2

MP4(SDQ)

QCISD

6-31G(d) 6-31+G(d) 6-31++G(d,p) 6-31++G(2d,2p) DH+(d) DH++(d,p) aug-cc-pVDZ 6-311+G(d) 6-311++G(d,p) 6-311++G(2d,2p) 6-311++G(2df,2p)

1.055 1.089 1.083 1.081 1.101 1.092 1.100 1.131 1.094 1.088 1.076

1.066 1.130 1.111 1.109 1.114 1.109 1.108 1.106 1.114 1.107 1.096

1.073 1.132 1.115

1.072 1.128 1.112

1.073

Figure 1 shows the basis set dependence of the PKIEs using the MP2 method. Inclusion of diffuse functions on both nonhydrogen and hydrogen atoms is seen to be important and give opposite effects. Addition of diffuse or a single set of polarization functions to the 6-311G basis set gives a much worse performance than with the 6-31G basis set. This suggests that basis set balance is important for these systems; i.e., the 6-311++G(d,p) basis set is underpolarized. The split-valence basis set with polarization and diffuse functions added to all atoms, 6-31++G(d,p), seems to be a good choice for these calculations. As the inclusion of electron correlation beyond MP2 has the opposite effect as basis set enlargement, the smaller 6-31+G(d) basis set actually gives results in closer agreement with the values obtained at the QCISD/6-31++G(d,p) level. Secondary Kinetic Isotope Effects. The secondary kinetic isotope effects of exchange of both hydrogens at C1 with deuteriums are given in Table 3. Except for the 6-311+G(d) basis set, inclusion of electron correlation increases the C1-SKIE, and the MP4(SDQ) and QCISD methods again give very similar results. The differences between the HF and correlated methods are much smaller than for the PKIEs. Figure 2 shows the C1-

1.349

SKIE for the MP2 method using different basis sets. It can be seen that the basis set dependency is very small, and only inclusion of diffuse functions on non-hydrogen atoms is crucial. Exchange of the two non-transferred hydrogens at C2 with deuterium produce the C2-SKIEs given in Table 4 and depicted in Figure 2. It is seen that there are larger fluctuations as a function of theoretical level than for C1-SKIE, especially at the HF level. Inclusion of diffuse functions on hydrogen increases the SKIE, while splitting the polarization functions into two sets decreases it. Basis set balance is again seen to be important for the C2-SKIE. For both SKIEs the MP2/6-31++G(d,p) method appears to be a good compromise between accuracy and computer cost. The fluctuations in absolute values, however, are much smaller than for the PKIEs, and only the smallest basis set, 6-31G(d), and the two imbalanced 6-311+G(d) and 6-311++G(d,p) basis sets seem inappropriate. RelatiWe Values. The absolute values of calculated isotope effects are usually hard to compare to experimental values as effects from solvent and tunneling can be difficult to estimate, and relative values are therefore often more important. To investigate the effect of different theoretical approximations, we compare the results obtained for four different combinations of nucleophiles and leaving groups. The four reactions fall into two groups with BrF and ClF being relatively early, whereas BrCl and ClCl represent more productlike transition structures. All calculations here are done using the 6-31+G(d) basis set, as this was shown for the ClF reaction to be a reasonable compromise between cost efficiency and accuracy. Geometries. Tables 5-7 show geometries for the three additional transition states BrF, BrCl, and ClCl using different levels of electron correlation employing the 6-31+G(d) basis set. The same trends are seen as for the ClF transition structure described above. The HF method overestimates the bond length

TS Geometries and Kinetic Isotope Effects

J. Phys. Chem., Vol. 100, No. 42, 1996 16895

TABLE 5: Geometry Parameters for the BrF Transition Structure Using the 6-31+G(d) Basis Set (Bond Lengths in angstroms and Angles in degrees) HF MP2 MP4(SDQ) QCISD

RCC

RCBr

RCH

RFH

θC1

θC2

1.425 1.428 1.428 1.431

2.343 2.250 2.303 2.303

1.327 1.360 1.361 1.352

1.231 1.218 1.211 1.217

71.32 67.90 69.24 69.27

65.92 66.12 66.10 65.78

TABLE 6: Geometry Parameters for the BrCl Transition Structure Using the 6-31+G(d) Basis Set (Bond Lengths in angstroms and Angles in degrees) HF MP2 MP4(SDQ) QCISD

RCC

RCBr

RCH

RClH

θC1

θC2

1.364 1.380 1.381 1.383

2.903 2.620 2.693 2.715

1.405 1.448 1.428 1.424

1.626 1.546 1.571 1.572

83.61 77.60 79.21 79.65

76.54 73.54 73.48 73.24

TABLE 7: Geometry Parameters for the ClCl Transition Structure Using the 6-31+G(d) Basis Set (Bond Lengths in angstroms and Angles in degrees) HF MP2 MP4(SDQ) QCISD

RCC

RCCl

RCH

RClH

θC1

θC2

1.364 1.378 1.380 1.382

2.757 2.488 2.540 2.561

1.421 1.494 1.468 1.463

1.609 1.514 1.540 1.542

83.74 77.76 79.00 79.43

76.40 73.96 73.66 73.43

to the leaving group and the bond between the nucleophile and the transferred hydrogen, while it underestimates the carbonhydrogen bond. Also, the dihedral angles are seen to be different from the correlated ones. Overall, inclusion of electron correlation results in geometries that are slightly more E1cblike, i.e., a shorter bond to the leaving group and a longer bond bond between C2 and the transferred hydrogen. The HF method accordingly produces geometries significantly different from the correlated methods for all four transition structures. The orderings of the HF bond lengths are the same as by the correlated methods, but the ClF/BrF and the ClCl/ BrCl pairs of transition structures are differentiated much less, especially around the transferred hydrogen. This is of significance when establishing connections between geometries and primary kinetic isotope effects, since these usually are taken to depend on the RCH and RNuH bond lengths. Secondary kinetic isotope effects are expected to depend on the dihedral angles of the non-transferred hydrogens. The angle θC1 is more sp2 hybridized than θC2 for all transition structures at all levels. In addition, the ClF and BrF transition structures are predicted to have practically identical θC1 and θC2, which also is the case for the ClCl and BrCl transition structures. Kinetic Isotope Effects. Table 8 shows calculated kinetic isotope effects for the four reactions, where the values calculated from scaled frequencies are given in parentheses. The scaling is done in accordance with recent recommendations by Scott and Radom,3b with factors of 0.9200 for HF, 0.9676 for MP2, and 0.9805 for QCISD. These scale factors were derived by fitting zero-point energies to experimental data. As kinetic isotope effects are primarily determined by zero-point energies, we have used these values instead of the common scale factors2a derived from frequencies. Scott and Radom also reported scaling factors for calculating enthalpies and entropies. Calculating isotope effects by scaling the individual contributions (ZPE, ∆Hvib, and Svib) differently produces results which differ by less than 1%. Figure 3 shows the primary kinetic isotope effects as a function of the theoretical method. Without frequency scaling, the HF method significantly overestimates the primary kinetic isotope effects, and the order of PKIEs for ClF and BrF is

reversed as compared to the results of the correlated methods. Primary kinetic isotope effects are expected to depend mostly on the bond lengths around the transferred hydrogen. As the order of these bond lengths is the same for all methods, the HF method predicts a correlation between PKIEs and transition state geometries which is opposite to the correlated methods. The difference is small, however, the ratio of PKIE values from scaled frequencies for ClF and BrF being 1.006, 0.957, and 0.989 at the HF, MP2 and QCISD levels, respectively. The MP2 method performs well, producing the right order of the four TSs, although differentiating slightly more between the PKIEs of BrF/ClF than the higher correlated methods do. Scaling the frequencies reduce the absolute PKIE values and improves the performance of the HF method significantly. In terms of absolute values the scaled HF method outperforms the scaled MP2 results with respect to the scaled QCISD calculations, but the incorrect ordering of BrF and ClF is of course unchanged. Figures 4 and 5 show the C1- and C2-secondary kinetic isotope effects, respectively. While the HF method consistently overestimates all PKIEs, the results are much less clear for the SKIEs. The HF values may be either smaller or larger than with the three correlated methods, and the MP2 values are in general closer to the QCISD values. In several cases, a scaling of the HF frequencies increases the discrepancy with the QCISD results. In contrast, a scaling of the MP2 frequencies consistently provides results closer to the scaled QCISD numbers. The HF method predicts the reverse ordering of the C2-SKIE for BrCl and ClCl, although the difference is small. A wrong ordering of secondary kinetic isotope effects by the HF method has been found also for an SN2 system.21 In summary, the HF method shows problems for both primary and secondary kinetic isotope effects for predicting trends connected with the transition state geometries. The MP2 method seems to be in good agreement with results at higher levels, provided that a basis set of at least 6-31+G(d) quality is employed. For minimum-energy geometries the HF method usually gives results of reasonable accuracy. Scott and Radom3 recommended the HF/6-31G(d) and the B3-based/6-31G(d) density functional procedures as the most cost-efficient methods for calculating fundamental frequencies and thermodynamic properties. On the basis of the present results, inclusion of electron correlation seems necessary for calculations of kinetic isotope effects, and DFT methods provide geometries and kinetic isotope effects significantly different from the ab initio values, as shown below. In a theoretical study of harmonic vibrational frequencies and deuterium isotope fractionation factors for small molecules,5 the final recommendation was to calculate isotope effects from scaled MP2 frequencies with either the 6-311G(d,p) or the 6-311+G(d,p) basis set. This is a somewhat surprising result, since these basis sets are not well-balanced. For kinetic isotope effects in E2 reactions well-balanced basis sets seem to be important, and the present investigation does not support those recommendations. Density Functional Theory Methods. We have tested several DFT methods, but their performance is poor for the present systems. The local density SVWN method fails to locate any transition states, and the gradient corrected BPW91 method locates only some of the transition structures. Using the 6-31+G(d) basis set, BLYP and the methods which include exact exchange (B3LYP and B3PW91) were capable of locating all TSs, although one of these has C1 symmetry (BLYP for BrCl). The geometry parameters are given in Table 9. Except for the C-C bond length, most of the parameters are signifi-

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TABLE 8: Kinetic Isotope Effects of Four E2 Reactions Calculated with the 6-31+G(d) Basis Set (Values Using Scaled Frequencies in Parentheses) PKIE

C1-SKIE

C2-SKIE

reaction

HF

MP2

MP4(SDQ)

QCISD

ClF BrF BrCl ClCl ClF BrF BrCl ClCl ClF BrF BrCl ClCl

5.612 (4.944) 5.575 (4.913) 8.603 (7.273) 8.664 (7.321) 1.089 (1.078) 1.085 (1.074) 1.285 (1.244) 1.332 (1.285) 1.310 (1.281) 1.285 (1.259) 1.133 (1.122) 1.128 (1.117)

5.040 (4.798) 5.277 (5.016) 7.666 (7.181) 7.738 (7.247) 1.130 (1.126) 1.097 (1.092) 1.261 (1.246) 1.320 (1.305) 1.321 (1.308) 1.279 (1.269) 1.181 (1.174) 1.191 (1.184)

5.185 5.285 7.584 7.743 1.132 1.109 1.269 1.317 1.304 1.265 1.153 1.162

5.180 (5.028) 5.237 (5.083) 7.526 (7.243) 7.688 (7.395) 1.128 (1.124) 1.108 (1.104) 1.276 (1.266) 1.326 (1.315) 1.301 (1.294) 1.262 (1.256) 1.151 (1.148) 1.162 (1.159)

Figure 3. Primary kinetic isotope effects for four E2 reactions, calculated with the 6-31+G(d) basis set. Solid lines and symbols refer to PKIEs from unscaled frequencies, and dashed lines and open symbols refer to PKIEs from scaled frequencies; see text.

Figure 5. Secondary kinetic isotope effects for exchange of two hydrogens at C2 with deuterium for four E2 reactions, calculated with the 6-31+G(d) basis set. Solid lines and symbols refer to SKIEs from unscaled frequencis, and dashed lines and open symbols refer to SKIEs from scaled frequencies; see text.

TABLE 9: Geometry Parameters for the Four Transition Structures Using Various DFT Methods and the 6-31+G(d) Basis Set (Bond Lengths in angstroms and Angles in degrees) reaction

method

RCC

RCX

RCH

RNuH

θC1

θC2

ClF

BLYP B3LYP B3PW91 BLYP B3LYP B3PW91 BLYPa B3LYP B3PW91 BLYP B3LYP B3PW91

1.446 1.431 1.429 1.455 1.435 1.434 1.374 1.372 1.365 1.370 1.370 1.363

2.194 2.171 2.130 2.285 2.287 2.240 2.985 2.821 2.835 2.913 2.708 2.734

1.283 1.330 1.355 1.252 1.308 1.324 1.693 1.564 1.618 1.766 1.589 1.647

1.335 1.248 1.213 1.384 1.272 1.247 1.445 1.486 1.448 1.420 1.474 1.435

68.89 69.06 68.17 67.79 68.60 67.62 84.77 82.19 83.13 86.27 82.90 84.02

64.95 65.78 65.82 64.17 65.46 65.31 78.65 76.38 78.06 80.10 76.73 78.53

BrF BrCl ClCl Figure 4. Secondary kinetic isotope effects for exchange of two hydrogens at C1 with deuterium for four E2 reactions, calculated with the 6-31+G(d) basis set. Solid lines and symbols refer to SKIEs from unscaled frequencies, and dashed lines and open symbols refer to SKIEs from scaled frequencies; see text.

cantly different from the ones obtained by the ab initio methods. The early transition structures (ClF and BrF) are predicted earlier and the late transition structures (BrCl and ClCl) much later by the DFT methods. In addition, is should be noted that the imaginary frequencies are very low as compared to the ab initio method values. The kinetic isotope effects obtained using BLYP, B3LYP, and B3PW91 and the 6-31+G(d) basis set are given in Table 10. A comparison with Table 8 shows significant differences between DFT and ab initio methods, where BLYP gives the most deviating results. In general, all three DFT methods result in small primary kinetic isotope effects. This can to some

a The transition structure has C symmetry; θ 1 C1 and θC2 are average values.

degree be explained by the more extreme transition structures and is also indicated by the very low imaginary frequencies for the reaction coordinate, which involves much more movement of atoms other than the transferred hydrogen. Intuitively, these results do not seem correct. In addition, the predicted ordering of BrCl and ClCl is reversed as compared to the ab initio results. The absolute values of the secondary kinetic isotope effects for the early transition states ClF and BrF are close to the ab initio ones, whereas the results for the two late transition states BrCl and ClCl are significantly larger. These figures can be explained by the almost complete sp2 hybridization predicted, but the ordering does not correlate with the geometries or with the ab initio results. It has been found also by others that there are

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J. Phys. Chem., Vol. 100, No. 42, 1996 16897

TABLE 10: Kinetic Isotope Effects of Four E2 Reactions Calculated Using Various DFT Methods and the 6-31+G(d) Basis Set PKIE

C1-SKIE

C2-SKIE

reaction

BLYP

B3LYP

B3PW91

ClF BrF BrCl ClCl ClF BrF BrCl ClCl ClF BrF BrCl ClCl

3.114 2.548 4.219 3.790 1.154 1.087 1.422 1.468 1.244 1.259 1.243 1.246

4.563 4.290 6.321 5.879 1.139 1.096 1.531 1.358 1.262 1.267 1.229 1.200

4.463 4.268 4.765 4.434 1.131 1.089 1.880 1.370 1.282 1.283 1.325 1.214

large discrepancies between DFT and ab initio methods for E2 reactions,7i,j,26 while for other reactions DFT methods have proved very efficient.27 Conclusion We have made a series of ab initio and DFT calculations on geometries and kinetic isotope effects for an E2 model system. The best DFT results are obtained with the methods which include exact exchange (B3LYP and B3PW91). However, these give transition structures characterized by reaction coordinates significantly different from the ab initio ones, resulting in small primary kinetic isotope effects. The secondary kinetic isotope effects of the early transition structures are close to the ab initio values, whereas the SKIEs of the late TSs are too large. For geometries, primary, and secondary kinetic isotope effects the MP2 method with a split-valence basis set augmented with diffuse and polarized functions on non-hydrogen atoms (631+G(d)) or on all atoms (6-31++G(d,p)) seems to be a costefficient choice. At least for the systems studied here, the smaller of these basis sets reproduces the trends of isotope effects and transition state geometries as accurately as the larger. A uniform scaling of the HF frequencies is shown to have varying effects, improving the performance for PKIEs significantly, but sometimes increasing the disagreement for SKIEs with the best theoretical method. The absolute value of the primary kinetic isotope effect is almost exclusively determined by one stretching vibration in the reactant.6a As stretching frequencies are consistently overestimated, scaling significantly improves the absolute value of the PKIE. The absolute values of the secondary kinetic isotope effects, however, are mainly determined by bending frequencies,6a where the errors are not nearly as uniform as for stretching frequencies. Uniform scaling of the frequencies therefore does not necessarily improve the absolute values of the secondary kinetic isotope effects. As kinetic isotope effects are determined by ratios of frequencies, this observation is perhaps not surprising, but it stands in sharp contrast to the common practice of scaling. Finally, it is important to realize that calculating absolute values for comparison with experimental kinetic isotope effects requires inclusion of other effects, and improvement of the theoretical approximation will most likely not be sufficient in this respect. Acknowledgment. This work was supported by grants from the Danish Natural Science Research Council. F.J. thanks RSC, ANU, for a visiting fellowship. Supporting Information Available: Tables including reactant geometries and activation energies (3 pages). Ordering information is given on any current masthead page.

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