Ab initio study of the vibrational spectra of the allyl and the cyclopropyl radicals

Journal of Molecular Structure (Theochem), 188 (1989) 79-94 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

79

AB INITIO STUDY OF THE VIBRATIONAL SPECTRA OF THE ALLYL AND THE CYCLOPROPYL RADICALS

C. COMETTA-MORINI,

T.-K. HA* and J.F.M. OTH

Laboratory of Organic Chemistry and Laboratory of Physical Chemistry, Swiss Federal Institute of Technology, ETH-Zentrum, CH-8092 Ziirich (Switzerland) (Received 4 October 1988)

ABSTRACT Ab initio geometry optimisations and computations of the vibrational spectra of ally1 and cyclopropyl radicals are presented and compared with experiment. The 6-31G*, 6-31G** and D95** basis sets were used at the Hartree-Fock (HF) level. Electron correlation was taken into account by using perturbation theory carried to the second order (MP2). Three different scaling methods, applied in order to improve the accordance between the computed and experimental vibrational frequencies, are discussed. In the case of the cyclopropyl radical, the results of our computation suggest, for two vibrational modes, a different assignment from the one obtained in previous theoretical work.

INTRODUCTION

With the development of matrix-isolation techniques [ 11, it has become possible to record the infrared (IR) spectra of short-lived species, such as small organic radicals. Of the two radicals studied, the matrix IR spectrum of the ally1 radical, the simplest conjugated hydrocarbon, has been reported in three different studies [Z-4], of which the last and most complete one [4] also includes the IR spectra of four isotopomers. Conversely, until recently the cyclopropyl radical had eluded all studies made in this direction. In 1982, Dupuis and Pacansky [ 51 tried and failed to trap cyclopropyl radical in an argon matrix. Holtzhauer and co-workers [ 4a,6] finally succeeded in recording for the first time the IR spectrum of the cyclopropyl radical (argon matrix, 18 K) following a completely new approach: the photochemically induced ring closure of the ally1 to cyclopropyl radical in an argon matrix at very low temperatures. In accordance with the limited number of experimental data available, only a few theoretical studies have been made on the vibrational spectrum of ally1 and cyclopropyl radicals. An ab initio vibrational analysis of cyclopropyl at the *Author to whom correspondence should be addressed.

0166-1280/89/$03.50

0 1989 Elsevier Science Publishers B.V.

80

Hartree-Fock (HF) SCF level with the 4-31G basis set was presented in 1982 by Dupuis and Pacansky [ 5 1. A complementary work by Takada and Dupuis [ 7 J on the structure and vibrational spectrum of ally1 at the MCHF level appeared shortly after. Scaled ab initio and MINDO/S open-shell calculations of the vibrational frequencies have been used to facilitate the assignment of the observed fundamental frequencies of ally1 [ 41. Preliminary ab initio SCF computations with the 6-31G* basis set were performed to aid the inte~retation of the cyclopropyl radical spectrum 161. The calculated harmonic frequencies were about lo-12% higher than the corresponding observed values, a result which has been shown to be typical of results obtained using the HF/6-31G* method in previous computations for many small molecules [8,9]. This is generally true for the HF approach and is due, on the one hand, to the deficiencies of the theoretical treatment, such as an insufficient basis set and neglect of electron correlation, and on the other hand to the neglect of the anharmonic contributions in the theoretical frequencies. Calculations taking into account electron correlation and using a large basis set have been shown to give results which are in better agreement with experiment, the main source of error remaining the neglect of anharmonicity. For example, MP2/6-3lG* frequencies are typically only 5-7% larger than the observed ones [lo]. However, because the computation of the vibrational spectrum requires a fair amount of computer time, it may be necessary to work at the SCF level and/or limit the size of the basis set used. To reduce the discrepancies between the calculated and the observed vibrational frequencies different scaling approaches have been suggested. The most straightforward one is the scaling of the raw theoretical results, that is of the computed vibrational frequencies. In this approach, the scale factors can be generated in two ways: (i) a universal value for a given level of theory is obtained as the average ratio between experiment and theory for a large number of vibrational frequencies [&lo] (scaling I, Sl ), or (ii) the scale factor is computed on a mode-by-mode basis [ll,l2] (scaling 2, S2). As stressed by Pulay [ 133, the comparison between theoretical and experimental data is best done at the level of potential constants and not at the level of vibrational frequencies. Thus it is preferable to scale the theoretical force field as done by Blom and Altona f14 1, who proposed an elaborate scheme of scaling factors for the 4-31G basis set, or by using the method of Fogarusi and Pulay [ 15 1, who used a set of standard diagonal scaling factors for their 4-21G basis set (Scaling 3,531. In view of the above, we decided to recompute the vibrational spectrum of ally1 and cyclopropyl radicals using a fairly large basis set (6-31G” or 631G**) at both the HF and MP2 levels of theory in order to study the effect of the introduction of electron correlation on the accuracy of the computed vibrational frequencies. Since Takada and Dupuis [ 7 J used a multiconfiguration Hartree-Fock wave function (MCHF) to describe the ally1 radical [5], we

thought it would be of interest to optimize the geometry and compute the vibrationat spectrum of ally1 radical using also an MGHF wave function. Moreover, we applied the above”mentioned scaling method witb the intention of comparing the accuracy obtainable with each method. In all cases, the adopted scaling factors were obtaiu~ without using the e~e~menta~ data on the two radicals, but with data on propene and ~y~lopropaue. This was done in order to obtain the computed spectral data for ally1 and cyclopropyi radicals which have a totally unbiased predictive value. COMPUTATIONAL DETAlI.23

The ~au~rn~~herni~al peculations were parformed using the GAUSSIAN 82 1161and GAMESS f 17f programs. The equilibrium geometries were determined at two different levels of theory. At the ~a~ree-Fo~k (HF) level the split-valence or double-zeta basis sets with polarization functions 6-31G* [ 183:, &3fG** 119) and IX%** [ZO) have been employed. The effect on the hornets and the total energy of including the el~t~n-correlation energy was investigated by applying M0ller-~lesset pe~~bation theory to the second order (MP2) [Zl] with the 6-31G* basis set. The MCHF wave unction used to describe the ally1 ra&cal included the conjurations obtained considering all the possible double and single excitations of the three 7celectrons of the ally1 radical from the bonding to the nonbonding and antibonding n orbitafs. Computation of the force constants was performed at the HF and MP2 optimized geometries, respectively. The force const~ts were obtained in both methods by numerical ~ffere~tiation, whereby the geometrical parameters were changed by 0.005 bohr or ~.~~~ rad (except for cy~loprop~l where the change was + 0.01bohr ) . The calculation of the vibrational frequencies was carried out using the ASYMZO program fZZ], Force constants and vibrational frequencies were also determined directly at the HF/6-31G* level in a single calculatiun by means of anal~i~al second-derivative techniques [ 231.

The geometries of the ally1 and cyclopropy~ radicals optimized under the C,, and C, symmetry constr~nts, ~espe~tively~ are shown in Tables 1 and 2. In eaeh case the OF-optimized geometries j6-31G*, 6-31G** and IX%**) do not depend very much on the basis set employed. The ~~~~F-optimi~d geometry for the ally1 radical as well as the energy are also very similar to the HP values. The MP2/6-31G* method provides slightly longer C-H bond lengths for both radicals, shorter C-C bond lengths for the ally1 and longer C-6 bond lengths

82 TABLE 1 Ab initio optimized geometry for the ally1 radical with C,, symmetrya Parameter Bond length (A) H&P C*C, C~HS C,H, Bond angle (“) C&x4 H&C, H&&z

6-31G*

6-31G**

1.078 1.390 1.074 1.076

D95**

1.079 1.390 1.074 1.076

124.6 121.4 121.2

MP2/6-31G*

1.079 1.396 1.076 1.077

124.5 121.4 121.2

1.088 1.377 1.082 1.084

124.5 121.1 121.1

124.4 121.8 121.0

MCHF

1.078 1.377 1.073 1.075 124.7 121.4 121.3

1I

“The atom numbering adopted is 5

6 2

4

3

ti 7

8

TABLE 2 Ab initio optimized geometry for cyclopropyl radical with C, symmetry+ Parameter Bond length (.$J CC* CC, C,H, CzH, CzH7 Bond angle (“) &CC3 C&H, C&&H, C1C*C& CCCH 123 7

6-31G*

1.470 1.517 1.072 1.078 1.078

6-31G**

1.469 1.517 1.072 1.078 1.078

D95**

1.477 1.523 1.074 1.078 1.078

MP2/6-31G*

1.469 1.526 1.081 1.087 1.087

62.2 41.1

62.2 40.8

62.1 41.1

62.6 41.1

118.9 118.5 106.5 - 107.4

118.9 118.5 106.5 - 107.7

118.7 118.3 106.4 - 107.8

119.1 118.7 105.3 - 108.2

“The atom numbering adopted is

83 TABLE 3 Total ab initio energy (E,,) of the ally1and cyclopropyl radicals Radical

State

Emt (a.u.) 6-31G*

Czv

2&

Cyclopropyl, C,

A’

Ally&

6-31G**

D95**

MP2/6-31G*

- 116.4681 - 116.47699 - 116.49362 - 116.82429 - 116.41554 - 116.42438 - 116.43993 - 116.79204

MCHF - 116.48359

for the cyclopropyl radical. The variation in bond angles in going from the HF to the MP2 approach is even smaller. Conversely, the inclusion of electron correlation has a large effect on the total energy (Table 3), which is lowered by 0.3562 a.u. for the ally1 and 0.3765 a.u. for the cyclopropyl radical. Vibrational analysis The vibrational spectra of the ally1 and cyclopropyl radicals were first computed at the UHF/6-31G* level for the UHF/6-31G* optimized geometry in a single calculation, in which the force constants were determined through analytical second differentiation. The results of these calculations, already presented previously [ 61, are given again in Tables 4 and 5, under the heading “Analytical”, for the sake of completeness. As can be seen from these tables, there is practically no difference between the frequencies obtained with analytically or numerically computed force constants. The vibrational spectra of the ally1 and cyclopropyl radicals were also computed at the 6-31G**/6-31G** and D95**/D95** levels. The theoretical frequencies so obtained do not differ substantially from the 6-31G*/6-31G* ones. More interesting are the results of the MP2 computations. The force field used in this case was obtained in the following way: the diagonal force constants were computed through numerical second differentiation at the MP2/ FC/6-31G* level using the MP2/6-31G* optimized geometry, while the offdiagonal force constants were those obtained at the 6-31G*/6-31G* level. This last approximation is justified by the observation that basis sets and electron correlation effects are, in general, more important for diagonal force constants than they are for most off-diagonal force constants [ 241. The frequencies obtained in this way are, overall, in better agreement with the experimentally determined ones (Tables 4 and 5) than are the HF computed ones. In the case of the ally1 radical (Table 4)) the mean average error in the computed frequencies is 9.3% for the HF values and 7.6% for the MP2 results. The ordering of the calculated frequencies corresponds to the ordering of the experimental ones, except for the asymmetrical C-C stretching mode ( v,C-C: observed 1284 cm-l; calculated 1251 cm-‘) and for the symmetrical CH,

)

3422 3418 3336 3325 3324 1647 1637 1537 1251 1341 1015 1035 1005 799 780 572 547 451

9.3

7.0

10.1 9.9 9.3 10.1 10.1 11.4 11.8 10.6 2.6 7.9 9.1 5.2 24.1 0.4

3421 3417 3316 3334 3325 1647 1638 1537 1254 1343 1077 1035 1006 799 780 572 547 452 9.3

7.0

10.0 9.9 8.7 10.4 10.1 11.4 11.9 10.6 2.3 8.1 8.9 5.2 24.2 0.4

Numerical 3045 3041 2951 2967 2959 1466 1458 1368 1116 1195 958 921 895 711 694 509 487 402

St

52

5.5

4.6

3221 3221 3126 3142 3133 1487 1490 1372 1171 1220 989 1052 885 775 733 546 3.9 511 404

0.4 0.2 1.3 0.2 0.4 1.2 0.7 0.5 8.7 2.0 14.9 7.7 12.5 15.0

53

4.6

0

3.7 3.7 2.5 4.1 2.9 0.6 1.8 1.2 8.8 1.8 16.4 6.9 8.7 5.9

% 3333 3329 3222 3246 3238 1595 1579 1444 1230 1301 1048 1123 9X8 823 801 595 557 437

Numerical

Sl

7.6

3200 3196 3093 3116 3108 1531 1516 1386 1181 1249 1006 1078 881 790 769 571 9.0 535 419

7.2 7.1 5.6 7.5 7.3 7.9 7.8 3.9 4.2 4.8 11.4 14.1 13.3 2.6

%

S2

4.6

3113 3109 3016 3028 3021 1480 1465 1385 1179 1255 1005 1088 886 794 773 624 4.7 584 452

2.9 2.8 1.4 3.2 2.9 3.6 3.5 0.2 8.0 0.6 14.9 9.5 8.7 1.5

%

off-diagonal force constants UHF/6-31G* computation.

3120 3116 3011 3014 3006 1461 1453 1382 1172 1217 1007 908 911 682 666 513 4.7 491 424

2.0 2.2 3.3 1.7 2.0 0.8 0.4 1.5 13.1 3.8 19.0 6.4 10.5 11.3

“IR inactive. bNot observed, “Diagonal force constants UMP2/FC/6-31G*;

Mean average error (%

511 b

3109 3109 3051 3019 3019 1478 1464 1389 1284 1242 1183 984 810 802 B a

Analytical

UMP2/FC”

UHF/6-31G*

[41

Obs.

Vibr. mode

Symm.

for the ally1 radical; the error in the computed frequencies is given in %

Vibrational frequencies (cm-‘)

TABLE 4

Analytical %

4.1

10.0

3434 10.5 3416 9.9 3353 9.9 3328 10.2 3317 9.9 1654 7.9 1628 11.2 1532 10.3 6.6 1199 9.0 1354 1089 7.9 1030 4.7 1004 23.9 716 10.7 670 585 8.0 14.3 552 447

0.1 0.0 1.1 0.3 0.1 0.1 0.1 0.3 8.2 1.1 15.0 10.6 8.6 1.0

%

MCHF

A’ A’ A” A’ A” A’ A” A’ A” A” A’ A” A’ A” A’ A” A’ A’

vCH vCH, vCH, vCH, vCH, 6CH, 6CH, v&CC r.CHz yCH

3118 3049/3042 3033 2960 2965 1440 1416 1237 (L 1085 1077 1037 997 B 8271824 777 743 a

Obs. [4]

12.0

12.5 9.0 13.1

9.7 14.2 16.6 17.3

3032 2998 2986 2930 2925 1464 1428 1191 1145 1064 1109 1076 1040 897 828 754 761 627

2.8 1.5 1.5 1.7 1.4 1.7 0.8 3.7

2.9

0.4 3.1 2.4

1.9 3.0 3.8 4.3

3121 3086 3073 3015 3010 1449 1428 1225 1143 1049b 1113 1058 1011 916 852 783 785 625’

0.1 1.3 1.3 1.2 1.5 0.6 0.8 1.0

1.8

3.3 0.8 5.6

3.3 3.3 2.0 1.4

3224 3179 3167 3100 3095 1495 1453 1254 1199 1072 1106 1117 1064 937 841 740 794 617 4.0

1.9 4.8 6.9

1.2 2.7 7.7 6.7

3.4 4.4 4.4 4.0 4.4 3.8 2.6 1.4

6.0 7.2 7.3 7.2 7.6 10.0 9.0 5.9

3305 3269 3253 3196 3190 1584 1544 1311 1195 1116 1147 1154 1072 976 869 764 767 626 6.0

5.3 1.7 3.2

2.8 6.5 11.3 7.5

%

Numerical 3173 3138 3123 3068 3062 1521 1482 1259 1147 1071 1101 1108 1029 937 834 733 736 601

Sl

3.1

1.9 5.7 0.9

1.3 2.2 6.8 3.2

1.8 3.0 2.9 2.9 3.3 5.6 4.7 1.8

% 3100 3066 3051 2998 2992 1446 1425 1224 1114 1081” 1096 1086 996 922 812 775 793 600’

S2

off-diagonal force constants UHF/6-31G* computation. Scaling factor from propene computation.

12.0

12.8 9.0 13.1

10.2 15.7 16.6 17.3

9.3 10.6 10.6 10.5 10.8 14.2 13.3 8.2

3407 3369 3355 3292 3286 1645 1604 1339 1287 1196 1246 1209 1169 1008 931 847 855 705

9.2 10.4 10.6 10.4 10.8 14.2 13.3 8.2

3406 3367 3354 3291 3285 1645 1604 1339 1287 1190 1230 1209 1169 1007 930 847 855 704

%

s3

Numerical

Analytical s2

UMP2/FCb

UHF/6-31G* S1

for the cyclopropyl radical; the error in the computed frequencies is given in %

“Not observed. “Diagonal force constants UMP2/FC/6-31G*;

Mean average error %

WCHZ ~J.CH, v.cc &CC pCHs+FH pCH,+GCH 6CH

7A-W.

Symm.

Vibr. mode

Vibrational frequencies (cm-‘)

TABLE 5

1.4

1.6 0.3 6.7

0.4 1.8 4.7 0.1

0.6 0.6 0.6 0.6 0.9 0.4 0.6 1.1

%

86

rocking mode (@Hz: observed 1242 cm-l; calculated 1341 cm-‘). The vibrational frequencies computed using the MCHF wave function are slightly worse than the UHF/6-31G* computed ones (mean average error 10.0% ). This’ average error of 10% is mainly due to a high discrepancy in the computed frequencies for the asymmetric C-C stretching and for the symmetrical out-ofplane vibration (wagging) of the CH:, group. For the cyclopropyl radical, the mean average error in the HF computed vibrational frequencies is slightly higher (12%) than for the ally1 radical. This is mainly due to a different ordering of the computed frequencies for the arCH wag ( yCH), the symmetrical CH, twisting ( 7$Hz) and the asymmetrical CH2 wagging (o,CH,) modes. The introduction of electron correlation reduces the deviation from observed values, but does not restore the ordering of the frequencies. Some comment is needed here about our assignment of two of the vibrational modes of cyclopropyl radical, namely the CH2 symmetrical rocking (p&Hz) and the CH2 symmetrical twisting ( 7$H2), The initial assignment of the observed frequencies [4a] was done with the help of the only computational results available at the time [5]. Dupuis and Pacansky [5] assigned the frequencies computed at 1241 cm-l and 870 cm-’ to the CH, symmetrical rocking and the CH, symmetrical twisting, respectively. They explained the resulting wide splitting of the two CH2 rocking modes (psCH2 1241 cm-’ and p,C!Hz 893 cm-‘) and of the two twisting modes ( 7,CH2 1301cm-l and 7$H, 870 cm-‘) as a result of some contamination from the motion of the aCH bond. As already briefly discussed [ 61, in examining the results of our computations we arrived at a different assignment of the p,CH, and z&Hz modes. We found the p&H2 mode at a lower (855 cm-‘) and the z&Hz mode at a higher (1230 cm-‘) frequency. To try to clarify this point we repeated the computation of Dupuis and Pacansky [ 51. We reoptimized the geometry of cyclopropane using the 4-31G basis set, starting from the 4-31G optimized geometry given by Dupuis and Pacansky [ 51, and then computed the vibrational spectrum. The results of these computations confirmed our assignment, which we have, therefore, adopted. Scaling procedure The application of the first scaling method (Sl ) requires the availability of a uniform scaling factor 0. This is defined as the average of the ratios dG (Oi) (experiment ) /vi (theory) for a large number of vibrational frequencies, dG ( Oi) being the observed O+ 1 energy difference for the ith vibrational mode and vi (theory) its harmonic theoretical vibrational frequency. For the two levels of theory employed here (HF/6-31G* and MP2/6-31G*), such scaling factors have been obtained by Hout et al. [lo] based on the average ratio for 165 frequencies in molecules constructed from first-row elements only. Their values are onF/6_310*= 0.89 and oMp2/&310*= 0.96. Multipli-

81 TABLE 6 Ab initio optimized geometry for propene with C, symmet@ Parameter Bond length (A) fx, C2C3 C,H, C,H, CzHs CzH7 &Ha Bond angle (“) CJX, H&C, H&C2 H&K& H&C2 H&H9 H&C&,

RHF/6-31G*

1.318 1.503 1.077 1.076 1.079 1.084 1.087

125.3 121.8 121.7 118.9 111.4 107.0 - 120.6

MP2/6-31G*

1.337 1.498 1.086 1.085 1.089 1.093 1.095

124.6 121.6 121.7 119.0 111.0 107.0 - 120.5

“The atom numbering adopted is

cation of the HF/6-31G* and MP2/6-31G* theoretical frequencies with the corresponding scaling factor gives the scaled frequencies presented in Tables 4 and 5 under the heading “Sl”. The results of this first type of scaling are surprisingly good when one takes into account the fact that the ratio dG ( Oi) (experiment) / Vi(theory) is not constant so that the universal average value is inappropriate for some vibrational modes. For example this is the case for the two stretching C-C modes ( v,CC and v,CC) in the ally1 radical (Table 4). For these two modes, the HF as well as the MP2 computed values are lower than the experimental ones. As a whole it was found that application of the scaling factor proposed for the HF/6-31G* level of theory leads to frequencies which, after the scaling, mostly lie below the observed ones. This is probably due to the fact that the applied average scaling factor has been derived through comparison of RHF/ 6-31G* computations with experiment and thus, inevitably, is not corrected for the effect of electron correlation in our UHF/6-31G* computations. The mode-by-mode scaling procedure (S2) avoids some of the above-men-

88 TABLE 7 Ab initio optimized geometw for cyclopropane with C,, symmetryS Parameter Bond length (A) GIG W-b Bond angle f”) H&J-b HX,C,

w,C&,

MP2/6-31G*

RHF/&31G*

1.502

1.497 1,076

1.084

114.0

114.2

118.1 - 108.0

118.1 - 107.9

“The atom numbering adopted is

6

8

TABLE 8 Ab initio energy (E,,) of propene (C,) and cyclopropane ( C3h) structure

Propene, C. Cyclopropane, C,,

State

‘A’ ‘A’

%, (a=) RHF/6-31G*

MP2/6-31G*

- 117.07147 - 117.05889

- 117.46966 - 117.46283

tioned disadvantages. Propene and cyclopropane were chosen as reference compounds. The IR spectra of these compounds are experimentally well known [ 25-271 and has been theoretically investigated at the ab initio level by Dupuis and co-workers [ 571 and by Blom and Altona [ 14a,b]. To our knowledge, there are no available ab initio computations at the 63lG* level of the vibrational spectrum of propene and cyclopropane. Therefore, it was necessary to reoptimize the geometry of the two molecules at the HF/6-31G* and MP2/6-31G* levels in order to compute their vibrational spectra. The calculated geometries are given in Tables 6 and 7 and the corresponding total energies in Table 8. The computed vibrational spectra are given in Tables 9 (propene) and 10 (cyclopropane} together with the experiments values taken from refs. 25-27.

89 TABLE 9 Vibrational frequencies (cm-‘) Vibr. mode”

Symm.

for propene C.; the error in the computed frequencies is given in % Obs.

[251

KS% uCH

~2% ~2% vaCH, U-X vc=c W-L W-L W-L WH, 6CH

&CH, P.CHB

yCH+rCH, 6&H, -pCH,

UC-C aI% &HZ &CCC CH,rot

A' A’ A’ A’ A” A’ A’ A’ A” A’ A’ A’ A’ A” A” A’ A’ A” A” A’ A”

Mean average error (% )

3091 3014 2991 2973 2953 2932 1653 1458 1442 1414 1378 1298 1174 1045 990 934 919 912 575 428 188

MP2/FCb numerical

RHF/6-31G* Analytical

Numerical

S3

3405 3333 3320 3273 3242 3195 1881 1639 1625 1594 1555 1443 1297 1187 1128 1029 985 1068 641 455 210

3391 3321 3307 3261 3230 3183 1876 1639 1626 1594 1555 1443 1296 1188 1129 1029 983 1068 641 456 211

3197 2129 3117 3075 3047 2998 1741 1514 1503 1442 1398 1294 1178 1072 975 930 913 777 566 405 419

10.2 10.6 11.0 10.1 9.8 8.9 13.8 12.4 12.7 12.7 12.8 11.2 10.5 13.6 13.9 10.2 7.2 17.1 11.5 6.3 11.7 11.3

9.7

10.2 10.6 9.7 9.4 8.6 13.5 12.4 12.7 12.7 12.8 11.2 10.5 13.6 13.9 10.2 6.9 17.1 11.5 6.3 11.7 11.2

3.4 3.8 4.2 3.4 3.2 2.3 5.3 5.7 4.2 2.0 1.5 0.3 0.3 2.6 1.5 0.4 0.6 14.8 1.6 5.4 3.3

3293 3219 3207 3186 3160 3115 1751 1618 1603 1523 1482 1353 1217 1105 1021 929 958 650 548 414 207’

6.5 6.8 7.2 7.0 7.0 6.2 5.9 11.0 11.2 7.7 7.5 4.2 3.7 5.7 3.1 0.5 4.2 28.7 4.7 3.3 10.1 7.2

“Assignment of the vibrational modes is according to refs. 7 and 14b. ‘Diagonal force constants MP2/FC/63lG*, off-diagonals force constants RHF/6-31G*. ‘See text.

It should be mentioned that the MP2/FC computations on propene led to a negative eigenvalue of the force-constant matrix, corresponding to the vibrational mode consisting in the free rotation of the CH3 group. In examining the force-constant matrix, we observed a near degeneracy of the three diagonal terms corresponding to the torsion of the methyl protons. This near degeneracy, due to the higher local symmetry of the CH, group, is more pronounced for the MPB/FC than for the HF force constants. Accordingly, it was decided to substitute the three diagonal force constants under question with the corresponding HF terms and a satisfactory value for the CH, rotation mode was obtained in this way. By comparison of the computed and the observed values, it was possible to obtain a set of scaling factors to be applied to the HF/6-31G* and MP2/631G* computed vibrational frequencies of ally1 and of cyclopropyl radicals (Tables 11 and 12). Note that when the computed frequency is lower than the observed one the corresponding scaling factor is larger than unity as is the case for the MP2 scaling factors for the CH, bending mode, the CH, twisting mode

90 TABLE 10 Vibrational frequencies (cm-’ ) for cyclopropane (C,, ); the error in the computed frequencies ia given in % Vibr. mode”

vCH vCH vCH XH2 ZH* cD-iz

vccc GJW d&Z &HZ &CCC 0% pCH2

Symm?

Obs. i261

A”(A;) E”(E”) A’@;) A’@;) E’(E)

E”(E”) A’(A;) A”(A;) A’tA;) E’(E’) E’(E) A” (A;) E” (E”)

Mean average error ( % )

3038 1479 1438 1188 1188 1126 1070 1029 868 854 739

MP2fFC” numerical

RHF/6-31G* Analytical

Numerical

3403 3380 3320 1679 1616 1330 1302 1268 1224 1191 961 923 806

3390 3369 3308 1678 1616 1330 1299 1268 1223 1191 958 923 805

9.7 9.7 9.3 13.5 12.4 11.9 9.6 12.6 14.4 15.7 10.7 8.1 9.1 11.1

53 9.3 9.3 8.9 13.5 12.4 11.9 9.3 12.6 14.3 15.6 10.0 8.1 8.9 10.9

3196 3197 3118 1531 146411463 1214 1213 1164 1127 1105 905 831 736

3.0 3.0 2.6 3.5 1.8 2.2 2.1 3.4 5.3 6.8 1.3 2.8 0.4 1.9

3310 3287 3230 1620 1558 1243 1272 1208 1137 1110 928 842 714

6.7 6.6 6.3 9.5 8.3 4.6 7.1 7.3 6.3 7.7 5.9 1.4 3.2 6.2

“Assignment of the vibrational modes according to refs. 26 and 27. “fhe symmetry of the vibrational modes corresponds to a structure with Cs, symmetry (as found in the calculations). The corresponding symmetry in the symmetry group I&, (as found in ref. 18) is given in parentheses. I)iagonal force constants MPZ/F~/631G*; off-diagonals force constants RHF/6-31G*.

and the CCC bending mode in propene (Table 11) and for the HF scaling factor of the two CH2 rocking modes in cyclopropane (Table 12). We employed scaling factors from Table 11 to scale the corresponding theoretical frequencies of the ally1 radical. The scaling factors applied to cyclopropyl are taken almost exclusively from Table 10, with the exception of those for the CH wagging and bending modes ($H and 6CH) which were taken from the computation on propene, because there are no corresponding modes of this type in cyclopropyl. For the C-H stretching modes the averages of the three values, of the four C-H scaling factors in Table 11 and of the four scaling factors in Table 12, were used. The S2 scaled frequencies of the ally1 and cyclopropyl radicals (given in Tables 4 and 5 under the heading “S2”) show a better agreement with the observed values than do the results of the Sl scaling. This is particularly true for the C-H stretching modes in both radicals and for the C-C stretching modes in cyclopropyl radical, confirming that the good transferability of computational results for the stretching modes, which are effectively localized in a particular bond. The S2-scaled values are slightly better than the Sl values for the C-C stretching modes in the ally1 radical, mostly because the employed scaling factor for the YC-C modes (HF 0.935; MP2 0.959) is larger than the average one.

91 TABLE 11 HF/6-31G* and MP2/FC/6-31G* Vibrational modeb

*u&H, * uCH *v.CH~ v&H, v,CH, v,CH, UC-C &CHs &CH, *6&H, &CHs *6CH *6&H, ACH, *yCHzCH, &CH, -pCH, *vc-c *6CH, *ICH, *&xc CH,rot.

scaling factors a; obtained from the computations on propene

at RHF/6-31G*

MP2/FC/6-31G*

0.912 0.908 0.904 0.912 0.914 0.921 0.881 0.889 0.887 0.887 0.886 0.899 0.906 0.880 0.877 0.908 0.935 0.854 0.89’7 0.939 0.891

0.934 0.936 0.933 0.935 0.934 0.941 0.944 0.901 0.899 0.928 0.930 0.959 0.965 0.946 0.969 1.005 0.959 1.403 1.049 1.034 0.908

a01= AC (Oi) / Vi (theory). bScaling factors for vibrational modes denoted by * are employed for theoretical frequencies of the ally1 radical.

The scaling of the force constants (S3) was done using the scaling factors proposed by Blom and Altona for the 4-31G basis set [14]. As the present computations were done using a larger basis set with d-polarization functions for carbon, it was thought necessary to check the transferability of the 4-31G scaling factors to our computations. Thus we applied the scaling factors obtained by Blom and Altona at the 4-31G level for propene and cyclopropane to the HF/6-31G* force constants obtained by us for the same molecules. The vibrational frequencies calculated from the scaled force constants are in very good agreement with the observed values (Tables 9 and 10) with the lower twisting mode of the CH3 group in propene being the only exception (Table 9). Encouraged by these results, we then applied the scaling factors of Blom et al. [ 281 to the UHF/6-31G* computed force constants of the ally1 and cyclopropyl radicals. The specific scaling factors used for the different vibrational modes are give in Table 13. The resulting scaled force field led to IR vibrational frequencies which are

92 TABLE 12 HF/6-31G* and MP2/FC/6-31G* cyclopropane Vibrational mode

vCH uCH vCH vCH M=H* EH, #Hz l&cc r&Hz @Hz w.CHz v&CC pCHZ &Hz

scaling factors at obtained

from the computations

on

ai

RHF/6-31G*

MP2/FC/6-31G*

0.915 0.915 0.918 0.918 0.881 0.890 0.893 0.915 0.888 0.875 0.865 0.909 0.925 0.918

0.937 0.934 0.940 0.941 0.913 0.923 0.956 0.934 0.932 0.941 0.929 0.945 1.014 1.034

‘ai = dG(Oi)/vi(theory). TABLE 13 The scale factors used in scaling the force constants Vibrational mode

Allyl”

Cyclopropylb

C-C stretching C-H stretching Bending8 C3H6 wagging Torsion Interaction

0.909 0.888 0.797

0.887 0.889 0.793 0.731 0.872 0.818

0.895 0.819

“From ref. 14b. bFrom ref. 14a. in good agreement with the observed ones (Tables 4 and 5 ), the mean average error being 5.1% for the ally1 and only 4.0% for the cyclopropyl radical. In the ally1 radical, the agreement is particularly good for the frequencies above 1400 cm-’ and still acceptable for the ones below this number, with the only exception being the computed C-C symmetrical stretch mode ( v,CC), the frequency of which lies 16.4% or 194 cm-l below the observed one. For the cyclopropyl radical, there is overall good agreement between the observed and the experimental spectrum, the largest discrepancy resulting from the computed asym-

93

metrical CH2 wagging mode which lies 7.7% or 80 cm-’ above the observed value. For the allyl, as well as for the cyclopropyl radical, the scaling of the force consents does not lead to such good agreementas that found for propane (Table 9 ) and cyclopropane (Table 10) 1Resides the different level of approximation, which has been shown to play a limited role, the major reason for the lower accuracy of the ally1and cy~lopropyl radicals compu~tion~ results is probably due to the fact that the scalingfactors have been obtained for closedshell molecules,whereashere they are applied to open-shell systems. It shouldbe noted that the scaling factors were es~blished by comp~ing ab initio computationswith vibrational frequenciesmeasuredon the model compounds (propene and ~y~lopropane)in the gas phase. The scalingmethod was then applied to IR data obtained with the ally1 radical and for cyclopropyl radicalstrapped in rare gas matrix. The matrix effects have not been noted in this discussionbecause the IR frequencyshift due to host-matrix interactions is smallerthan the scaling corrections introduced CONCLUSIONS

The theoretical calculations show that the use of a fairly largebasis set enables good a~eement between experimen~l and computed vibrational frequencies to be obtained. The introduction of electron correlation lowers the frequencies and further improves the agreementwith experiment. These results show that extension of the used basis set togetherwith the applicationof a method which takes into account electron correlationis the correct approach to obtaining accuratetheoreticalvibrational spectra. The three scaling methods applied also give satisfactory results. We have noticed, however,that the scaling of vibrational frequenciesusing a uniform scaling factor leads to an underestimationof the computed f~quen~ies, particularly at the HF level. A mode-by-mode scaling seems to be more adequate and gives very good results, especially in the case of cyclopropyl radical. The third scaling method, which we applied only at the HF level, also gives good reproductionof the experimentalspectra. The degree of agreement,which is very good for propene and cyclop~op~e, is limited by the fact that the scaling factors used were derivedfor closed-shellsystems and with a smallerbasis set. The only vibrational modes for which neither introduction of electron correlationnor scaling lead to an improvementin the HF resultsare the symmetric and asymmetriccombination of CC stretchesin the ally1radical. Even the use of a better wavefunction (MC comput&tions)does not improvethe results of the undoubtedlyless-adequa~ HF wave~n~tion. Finally, it should be stressed that the overall good agreementbetween the computed and experimentalspectra for the cyclopropyl radicalis only possible if the present assi~ment of the vibrations modes, and not that of Dupuis and Pacansky [ 5 1, is accepted.

94

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15 16

17

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