J. Phys. Chem. 1996, 100, 8731-8736
8731
The Torsional Barrier of ClOOCl P. C. Go´ mez Departamento Quı´mica Fı´sica I, Facultad de CC. Quı´micas, UniVersidad Complutense de Madrid, E-28040 Madrid, Spain
L. F. Pacios Departamento Quı´mica y Bioquı´mica, ETSI Montes, UniVersidad Polite´ cnica de Madrid, E-28040 Madrid, Spain ReceiVed: May 31, 1995; In Final Form: March 4, 1996X
Ab initio internal rotation barrier heights Vcis and Vtrans for the peroxide form of Cl2O2 have been calculated using shape-consistent effective core potentials (ECP) and triple-ζ valence-only basis sets specially optimized for the ECP recently developed by us (Pacios, L. F.; Go´mez, P. C. Int. J. Quantum Chem. 1994, 49, 817). These basis sets are augmented with standard polarization functions, and correlation is accounted for at the MPn (n ) 2, 4), CCSD, and CCSD(T) levels of theory. All calculations consistently produce a cis barrier higher than the trans one, being our highest level results (CCSD(T)): Vcis ) 3538 cm-1 and Vtrans ) 1890 cm-1. This disagrees with previous experimental estimates, and the discrepancy is discussed here. Internal rotation potential and torsional constants gββ(β) are calculated for a grid of points at the MP2 level allowing for full relaxation of the geometry, fitted to a Fourier series, and used to calculate some of the lowest torsional energy levels. A single-point calculation potential at the CCSD(T) level has also been calculated and used for this purpose. The torsional levels are presented, and the torsional fundamental transition is found to be in good agreement with the experimental values. Optimized geometry, rotational constants, harmonic frequencies, and dipole moments are presented as a test of the reliability of our calculations for future studies on other halogen peroxides.
Introduction Chlorine peroxide (ClOOCl) and its isomeric forms chlorine chlorite (ClOClO) and chloryl chloride (ClClOO) have received1-13 a good deal of attention over the last 6 years in an attempt to determine the role that the ClO-dimer plays in the proposed1 catalytic cycle leading to ozone depletion in the Antarctic stratosphere and, as reported recently,14-17 in the Arctic as well. Yet this is not the only threat to the ozone layer: three other cycles involving different compounds as key species such as ClO plus BrO,14,21 ClONO2,18,19,20,23 and HOCl20,22 have also been proposed and are supposed to act as cooperating mechanisms in the destruction of O3. ClO + BrO has been predicted to proceed through intermediates such as ClOOBr and OClOBr,24 and the photoisomerization of OClOBr has been studied25 recently in a matrix isolation experiment. Also, a new feature observed at 312 nm in the UV spectrum of the self-reaction of BrO has been attributed to BrOOBr26 that is believed to play a role similar to ClOOCl in ClO self-reaction. Early ab initio calculations at the MP2 level of the theory2 pointed to the peroxide form of Cl2O2 as the lowest lying of the three isomers, lower in energy than chloryl chloride by 1 kcal/mol, whereas other possible isomeric forms considered were found to be clearly higher in energy. More recent high-level ab initio calculations have shown12 some deficiencies of MP2 in determining the relative energetics of isomers with different types of bonding, especially when hypervalent forms are involved, but led to estimates that also suggested ClOOCl being the most stable isomer. On the other hand, from the experimental point of view, sub-millimeter measurements4 identified ClOOCl as the dominant species in the self-reaction of ClO and permitted the determination of its geometry for both the X
Abstract published in AdVance ACS Abstracts, May 1, 1996.
S0022-3654(95)01493-6 CCC: $12.00
ground and first torsional states. Recent calculations found13 excitation energies of ClOOCl for the third pair of excited electronic states 1A, 1B to be close to 5.20 eV, with oscillator strengths summed for both transitions at about 0.1, which seems in agreement with the experimental finding5 of an absorption band in the UV spectrum of Cl2O2 having its maximum at 246 nm. It was also suggested that the most likely products in the photodissociation of ClOOCl are ClOO plus Cl, as required for ClOOCl to be the intermediate species in the catalytic cycle mentioned above. In addition, IR bands3,4 have been discussed in view of ab initio calculations and assigned to ClOOCl. All the evidence above points to the peroxide form of Cl2O2 as the one involved in the mentioned catalytic cycles and makes this molecule a case of great interest. However, very little work has been done on the torsional potential of ClOOCl, probably as a consequence of the experimental difficulties in directly measuring torsional transitions because of their low intensity. In particular, there was some uncertainty of the values of cis and trans barriers for which the only available experimental results were those by Birk et al.4 We present here the first, as far as we know, calculation which includes electronic correlation of the torsional potential function and cis and trans barrier heights Vc and Vt. Torsional constants and a few transitions between the lowest lying levels for 35ClOO35Cl and its isotopomers containing 37Cl are also presented. The relative heights Vc > Vt and the qualitative behavior of the geometrical parameters as functions of the torsional angle obtained here are in agreement with previous ab initio calculations carried out using a double-ζ basis set (6-31G*) at the HF level by Samdal et al.27 We also plan to extend this study to the other peroxides mentioned above, in particular ClOOBr and BrOOBr. In this regard, the methodology used in the present paper has been © 1996 American Chemical Society
8732 J. Phys. Chem., Vol. 100, No. 21, 1996
Go´mez and Pacios
TABLE 1: Geometrical Parameters (β Is the Torsional Angle), Rotational Constants A, B, C, and Dipole Moments µ MP2 rClO (Å) rOO (Å) RClOO (deg) β (deg) A (MHz) B (MHz) C (MHz) µ (D) a
EXT fb
ECP-TZDP(f)a
TZDP(f)b
ECP-TZDPa
TZDPb
6-31G* c
ECP-TZPa
CCSD(T) TZDPb
exptd
1.705 1.412 109.3 82.5 13104 2455 2180 0.80
1.712 1.407 108.9 82.8 13048 2444 2170 0.79
1.711 1.407 108.9 83.0 13096 2441 2170 0.84
1.738 1.412 108.9 84.1 12878 2361 2110 0.73
1.731 1.421 108.9 83.7 12881 2375 2115 0.86
1.741 1.420 109.0 85.0 12967 2333 2085 0.92
1.751 1.387 110.3 84.4 13088 2308 2061 0.93
1.753 1.411 109.5 84.7 12871 2309 2062 0.76
1.704 1.426 110.1 81.0 13109 2410 2140
Present work. b Ab initio calculation in ref 12. c Ab initio calculation in ref 2. d Experimental data in ref 4.
tested and found promising to deal with these computationally demanding systems in further applications currently in preparation. Methods In a previous work we have presented valence-only (5s5p) Gaussian basis sets contracted to [3s3p], optimized for shapeconsistent averaged relativistic effective core potentials (ECP)28 for main-group elements B-Ne, Al-Ar, and Ga-Kr. These triple-ζ (TZ) basis sets are intended for valence-only ab initio calculations with inclusion of correlation when ECP accounting for relativistic corrections are employed in standard (nonrelativistic) treatments. The performance of these basis sets was initially tested in atomic calculations by comparison with HFlimit results and in molecular calculations for a few selected diatomics including radicals such as ClO, GaO, and BrO. Correlation at MP2 and CISD levels of theory was considered, and properties such as bond distances, harmonic frequencies, dipole moments, and atomization and ionization energies were determined. The test was satisfactory and allowed the demonstration of the reliability of the TZ basis sets in ECP valenceonly correlated molecular calculations. The main advantage in using this methodology for the purpose of the present work is that energies calculated allowing full relaxation of the geometry (except, of course, the torsional angle) can be obtained for a good number of points at a moderate computational effort. As far as the core electrons are properly treated by means of the ECP, the computational attention is focused on the valence electrons only, and thus similar systems containing atoms heavier than chlorine, such as Br2O2 and ClOOBr, can be studied at the same cost, while implicitly accounting for the main relativistic corrections included in the effective potential. For the calculations on ClOOCl presented here, three different sets of polarization functions have been chosen in addition to the TZ valence segment given in ref 28. The first one consists of one set of d polarization functions on both atoms with exponents Rd ) 0.850 for oxygen and Rd ) 0.619 for chlorine: this choice results in the basis set denoted TZP. The next larger basis set is designated TZDP and uses two sets of d polarization functions with exponents Rd ) 2.314, 0.645 for oxygen and Rd ) 1.072, 0.357 for chlorine. This basis is finally augmented with one set of f polarization functions with exponents Rf ) 1.428 for oxygen and Rf ) 0.743 for chlorine; this larger basis set is hereafter denoted TZDP(f). All the exponents in the three sets were also employed by Lee et al.12 and taken from several standard references on basis set tabulations. Since our TZ basis set correctly mimics the corresponding all-electron valence segments,28 no reoptimization of polarization exponents has been attempted. Electron correlation has been accounted for at the secondorder many-body perturbation theory (MP2) in order to deter-
mine the equilibrium geometry and to evaluate the torsional potential as a function of the torsional angle. Despite the relative simplicity of this method, there is a considerable amount of information demonstrating the reliability of MP2 in optimizing geometries; in fact some of the best ab initio geometries available for Cl2O2 have been obtained by using this method.12 Since the type of bonding does not change during internal rotation, it is reasonable to expect that the problems found in determining the energy differences among ClOOCl, ClClOO, and ClOClO, and mentioned in the previous section, will not occur in this case. Yet the computational effort required to calculate the torsional barrier remains moderate using this method. Values of cis and trans barriers (Vc and Vt) have been calculated also at CCSD and CCSD(T) levels of theory using the TZDP(f) basis set and a MP2/ECP-TZDP(f) optimized geometry. All the calculations in this work were performed using the Gaussian 92 package.29 To assess the reliability of the methodology used in this work, some molecular properties such as equilibrium geometries, harmonic frequencies, and dipole moments have been calculated. Geometrical parameters have been optimized for the minimum of the potential energy using both basis sets. Results are summarized in Table 1, which also contains some of the most accurate ab initio geometries up to now along with the experimental values. It can be noticed that MP2/ECP-TZDP and MP2/ECP-TZDP(f) perform comparably to their all-electron counterparts, the differences being smaller than one-hundredth of an angstrom for rOO and rClO, smaller than 0.5° for the torsional angle, and negligibly small for the ClOO angle. On the other hand, these differences, except for rOO, are clearly smaller than the differences with the experimental values. The 0.009 Å difference between the MP2/ECP-TZDP rOO and its corresponding all-electron value (in columns 5 and 6) may not be significant since the agreement of the latter value with the experimental looks probably too good for the general performance of the method and basis set used, particularly if it is considered that this excellent result is spoiled when a larger basis set including f functions is used. In any case, differences are always smaller than 0.5%, and since MP2/ECP provides good agreement with the experiment, comparable to that in CCSD(T) results, the performance of the model in determining the optimized geometries can be considered as rather satisfactory. Calculations of harmonic frequencies at the MP2/ECP level using both TZDP and TZDP(f) basis sets are presented in Table 2, along with the experimental values. For comparison purposes other MP2 and CCSD(T) calculations included in Table 1 are also shown here. Our IR intensities for TZDP and TZP are given in columns 4 and 6. Frequencies calculated in this work are found to be very close to their all-electron counterparts, the differences being smaller than 3% except for the difficult case of ω2, where they are slightly larger. MP2/ECP-calculated
Torsional Barrier of ClOOCl
J. Phys. Chem., Vol. 100, No. 21, 1996 8733
TABLE 2: Harmonic Frequencies (cm-1), Anharmonic Corrected Frequencies (in Parentheses) (See Text), and Infrared Intensities (in brackets) (km mol-1) expt mode ω1 OO str (A1) (h) ω2 ClO str (A1) ω3 ClOO bnd (A1) ω4 ClOOCl tor (A1) ω5 ClO str (B2) ω6 ClOO bnd (B2)
ECP-TZDP(f)a TZDP(f)b 777 (761) 637 (631) 338 120 695 (688) 459
775 650 338 117 702 456
MP2 ECP-TZDPa 763 (747) [6.4] 602 (596) [11.2] 328 [0.0] 121 [0.3] 657 (650) [20.0] 443 [2.2]
TZDPb ECP-TZPa 751 624 330 118 676 444
797 [12.1] 649 [28.3] 327 [0.5] 127 [0.4] 591 [18.8] 439 [7.1]
CCSD(T) TZDP(f)b
6-31G* c 765 [5.92] 633 [12.7] 328 [0.0] 119 [0.35] 679 [19.0] 446 [1.14]
MI d
e
IRf
sub-mmg
[9.3]i
791 (764) 752 754 752 596 (585) [13.3] 648 543 560 329 [0.1] 116 [0.3] 127 ( 20 656 (644) [24.1] 650 648 653 438 [4.5] 418
a Present work. b Ab initio calculations in ref 12. c Ab initio calculations in ref 2. d Matrix isolation experiment, ref 3. e Matrix isolation experiment, ref 31. f Results in ref 5. g Results in ref 4. h Symmetry species correspond to molecular symmetry group G4. i Infrared intensities were calculated at the CCSD(T)/TZDP level; see text in ref 12.
frequencies for ω2 and ω5, which are two of the firmly assigned bands, are larger than the experimental values, as was to be expected for harmonic frequencies, and to be compared with the actual measured values, anharmonic corrctions must be done. Although the calculation of the anharmonic contributions escapes the objective of this work, it is reasonable, for the purpose of having an estimate of the amount of this effect, to consider, following Lee et al.,12 the anharmonic correction for OO to be related to the same magnitude calculated for the OO stretch in hydrogen peroxide and the correction for the ClO stretch to be close to the corresponding value in the monomer. We have chosen, however, to make anharmonic corrections to the OO stretch in a percent amount similar to the one proposed30 for the MP2/TZDP ω3 in H2O2, rather than use the same absolute value. In doing so, a 2% reduction of ω1 will yield a result of 761 cm-1 for the TZDP(f) basis set and 747 cm-1 for the same basis set without f functions, this last result being in excellent agreement with the 752 cm-1 experimental value. In a similar manner, a 1% anharmonic correction of the harmonic symmetric and antisymmetric ClO stretches will provide for the same basis sets 631 and 596 cm-1 for ω2 and 688 and 650 cm-1 for ω5. Again the ω5 MP2/ECP-TZDP value is in very good agreement with the experimental data. There has been some controversy on the symmetric ClO stretch, for which two experimental values were proposed: a matrix isolation experiment performed by Cheng et al.3 lead to 648 cm-1, whereas gas phase measurements by Burkholder et al.5 provided a value of 560 cm-1. The subject was also examined by Lee et al.12 from the ab initio point of view. Very recent matrix isolation spectroscopical work by Jacobs et al.31 has come up with a value of 543 cm-1, which is closer to the result by Burkholder et al. Since ω2 is near to 560 cm-1, all our harmonic frequencies show larger values than the experimental ones, allowing the anharmonic corrections to lead to a better agreement with the experimental values, in particular in the case of ECP-TZDP results. Finally ECP-TZDP and ECP-TZP ω6 values, 443 and 439 cm-1, respectively, lie reasonably close to the new 418 cm-1 experimental result by Jacobs et al. so as to expect a good agreement when anharmonic correction is taken. The torsional fundamental obtained using our ECP-TZDP basis set, 121 cm-1, is clearly below the 127 cm-1 found experimentally. According to our calculations using the MP2/ ECP-TZDP internal rotation potential to solve the Schro¨dinger equation, which will be presented in the last section, the torsional fundamental lies at 119 cm-1, which allows us to evaluate the anharmonic correction for the torsion, which turns out to be 2 cm-1, actually the smallest anharmonicity in the four measured bands. We thus conclude that the torsional fundamental should be found most likely within the interval 121 ( 7 cm-1 if a 6% error, comparable to the one in ω2 and several times larger than that in the other bands, is assumed for our torsional results.
TABLE 3: Coefficients of the ab Initio Torsional Potential and gββ Term Fittings to a Fourier Series in the Form A ) ∑Ai cos iβ (A ) V, gββ) Obtained by Means of MP2/ ECP-TZP, MP2/ECP-TZDP, and CCSD(T)/ECP-TZDP// MP2/ECP-TZDP Methods (See Text) (All Values in cm-1) MP2/ECP-TZP A0 A1 A2 A3 A4 A5 A6
MP2/ECP-TZDP
CCSD(T)/ECP-TZDP// MP2/ECP-TZDP
V
gββ
V
gββ
V
1591 679 1424 413 -135 -52 -1
0.371 0.220 0.189 0.095 0.052 0.026 0.014
1405 590 1240 358 -135 -61 -11
0.354 0.193 0.170 0.081 0.042 0.019 0.009
1619 545 1460 363 -141
As for the dipole moment (see Table 1), most of the ab initio calculations carried out so far provide values ranging from 0.76 to 0.93 D, our largest basis sets results being 0.73 D (MP2/ ECP-TZDP) and 0.79 D (MP2/ECP-TZDP(f)). Since the CCSD(T) is the highest level result in Table 1, one may consider the 0.76 D value by Lee et al. to be probably the most reliable one; however, a different of 0.03 D with our MP2/ECP-TZDP(f) value is probably too small to be conclusive in this respect. The Torsional Barrier The torsional barrier was computed at the MP2/ECP level by performing calculations at different values of the torsional angle β from 0 to 2π, using a 20 deg interval and full optimization of the geometrical parameters at every point with TZP basis sets. The same calculations were repeated with a TZDP basis set using a 30 deg interval. Finally, a series of CCSD(T)/ECP-TZDP//MP2/ECP-TZDP single-point calculations were carried out. Nine points (0° ) 360°, 60° ) 300°, 84.1° ) 275.9°, 120° ) 240°, and 180°, where 84.1° is the MP2/ECP-TZDP minimum) were included in this set. The obtained energy values were fitted, in each case, to a Fourier series in the form
V(β) ) ∑Vicos iβ
(1)
i
Table 3 contains the Vi coefficients of the fitting for TZP and TZDP basis sets and shows the series to be rapidly convergent. Barrier heights Vc and Vt have been computed as the energy differences between V(β)π) - V(β)βeq) and V(β)0) V(β)βeq) obtained using fully optimized MP2 geometries for all three conformers. Calculations were carried out at the MPn (n ) 2,4), CCSD, and CCSD(T) levels of theory using TZP, TZDP, and TZDP(f) basis sets, as mentioned earlier. Ab initio calculations of torsional barriers implicitly assume either that the exact value of the energy for each point in the
8734 J. Phys. Chem., Vol. 100, No. 21, 1996
Go´mez and Pacios
TABLE 4: Cis and Trans Barriers Calculated in This Work Compared to the Experimental Result in Ref 4 (All Values in cm-1)
Vt Vc
HF ECP-TZP
ECP-TZP
MP2 ECP-TZDP
ECP-TZDP(f)
MP4(SDQ) ECP-TZDP(f)
CCSD ECP-TZDP(f)
CCSD(T) ECP-TZDP(f)
expt
1423 4413
1836 3917
1612 3388
1526 3049
1844 4144
1891 4194
1890 3538
5660 3020
barrier is known or at least that errors due to noncomputed contributions to the energy remain unchanged when going from one conformation to another. If geometries are fully optimized for each value of the torsional angle, as is the case here, the main sources of error in calculating torsional barriers, and in particular barrier heights, are deficiencies in the treatment of the electron correlation and insufficiencies related to ther finite size of the basis set used. It is thus assumed that the relativistic contribution to the energy is the same in all the conformers. This is not exactly true, but to a good approximation, relativistic contributions can be considered as the sum of atomic contributions and, in addition, it amounts to a small percentage of the atomic energy itself in the case of light atoms such as oxygen and a somewhat bigger percentage for Cl. It is nevertheless unlikely to modify its value just because of conformational changes. In the present work this contribution has been implicitly included for Cl by means of the ECP. Results are summarized in Table 4. It can be seen that both calculated Vc and Vt consistently decrease when enlarging the basis set used (Table 4, columns 3-5), reaching a value for Vc very close to the experimental one when using TZDP(f) (agreement within 1%). It is also clear that these basis sets are not equally suitable for the three conformers because Vc and Vt vary when changing from TZP to TZDP(f). Unfortunately there is not a “variational” principle that guarantees that the accuracy of energy differences will improve as the basis set is increased, but the trend shown in Table 4 is that Vc as well as Vt differences with the previous value tend to be smaller as the basis set is enlarged. The improvement of the correlation scheme (Table 4, columns 6-8) has a remarkable effect on Vc and Vt, which increase with respect to MP2 values. That is just the opposite trend to the one observed when enlarging the basis set and gives rise to the interesting result that cis and trans barriers obtained using the shortest basis set and the poorest correlation scheme in this paper are in very close agreement with the CCSD(T)/ECP-TZDP(f) results. In any case, Table 4 clearly shows that all methods and basis sets used in this work systematically yield Vc larger than Vt (almost twice as large), which is in disagreement with the experimental available data. Even though there could still be a moderate basis set effect, as suggested by comparison of columns 3 and 5 of Table 4, that might lead to a somewhat lower Vc and Vt for CCSD(T) calculations if a larger basis set (including g functions on the chlorine atoms, for instance) were used, it is very unlikely that this would reverse the qualitative fact of Vc being much larger than Vt. We conclude, therefore, in view of the general trend in Table 4 and considering in particular our best calculations, CCSD(T)/ECP-TZDP(f) (column 8 of Table 4), that the experimental Vt is by far too large, while Vc seems to be a very likely value. The reason for the discrepancy for Vt can be as follows: determining the internal rotation potential exclusively from the torsional fundamental forces one to use only three parameters in the Fourier series. Therefore, since the second derivative of the potential with respect to the torsional angle must be positive at the equilibrium geometry, and equals 4 sin2 βeq times V2, this coefficient V2 turns out to be positive as well, and as a
Figure 1. Variation of bond distances Cl-O and O-O (upper) and ClOO angle (lower) with the torsional angle for ClOOCl. Results from ab initio calculations at the MP2/ECP-TZDP level of theory.
consequence, V1 ) -4V2 cos βeq has its negative sign predetermined if βeq < π/2, as is the case in ClOOCl. This analysis always leads to Vt > Vc no matter the nature of the molecular system, provided that βeq is smaller than 90°. In the case of FOOF, for example, a similar estimate, using βeq ) 87.5 obtained by Jackson32 and νt ) 202 cm-1 from Kim et al.,33 produces a Vt ) 6738 cm-1 and a smaller Vc ) 5658 cm-1. This disagrees with recent ab initio density functional theory calculations34 that provide values larger than those estimates by a factor greater than 2 and Vc > Vt (Vt ) 14 060 cm-1 and Vc ) 14 515 cm-1), these results being in fairly good agreement with the old experimentally based estimate by Jackson. In the present case of ClOOCl, what happens is that coefficient V1 obtained by fitting the ab initio points is positive and has a smaller absolute value than that found by Birk et al. Another factor which can help in understanding the internal rotation barrier is the dependence of the geometrical parameters on the torsional angle along the minimum energy path. For this purpose, the dependence of the bond angles and bond distances on β calculated by using the TZDP basis set is shown in Figure 1. It can be noticed that rClO shows a very small variation with β compared to rOO and more clearly when compared to RClOO. In fact, assuming a similar accuracy in rClO for βeq and for any other conformation (i.e. 0.03 Å), the largest variation in this bond distance is only twice the estimated uncertainty for this magnitude. However, rOO presents a larger
Torsional Barrier of ClOOCl
J. Phys. Chem., Vol. 100, No. 21, 1996 8735
TABLE 5: Five Lowest Torsional Energy Levels for the Three Isotopomers of ClOOCl Corresponding to 35Cl and 37Cla MP2/ECP-TZP
MP2/ECP-TZDP
CCSD(T)/ECP-TZDP//MP2/ECP-TZDP
t
35-35
35-37
37-37
35-35
35-37
37-37
35-35
35-37
37-37
0 1 2 3 4
0 124 248 370 489
0 124 246 366 484
0 123 244 363 480
0 119 235 350 461
0 118 233 346 457
0 117 231 343 453
0 122 242 361 478
0 121 240 357 474
0 119 238 354 469
a Values were obtained using MP2/ECP-TZDP and MP2/ECP-TZP potentials and kinetic g ββ in the Hamiltonian given by eq 6 (see text). Levels in each case are referred to the ground torsional state. All values in cm-1. The experimental value for t ) 1 is 127 ( 20 cm-1 (see ref 4).
variation with β and reaches the shortest value for the conformation of minimum potential energy and the largest value for the cis conformation, with a somewhat shorter value for the trans one. Finally RClOO exhibits a large β-dependence, decreasing from cis to trans and showing a small shoulder for the equilibrium conformation. As far as short distances correspond to increased bonding, it is possible to use the rOO and rClO β-dependence to follow the OO and ClO bonding character as β varies from 0 to π, while larger values of RClOO may be viewed as a consequence of higher repulsion between chlorines. Thus, the cis barrier can be considered as the result of a weakened OO bonding plus an increased ClCl repulsive character, whereas there exists a balance for the trans barrier between the lower ClCl repulsion and a small OO bonding. Under these circumstances a trans barrier lower than the cis one should be expected, which is consistent with our numerical results for Vc and Vt. In turn, the shortest OO distance and the shoulder in the RClOO curve for the minimum energy conformation coincide. All this suggests the OO bonding to be the dominant factor in the torsional barrier, while the role played by ClO bonding seems to be less important that the OO one and qualitatively shows the opposite trend.
Both gij and g are periodic functions of β. Expression 2 consists of three terms, Hr-t ) Hr + Ht + Hcoup, where the torsional Hamiltonian is
∂ ∂ ln g ∂ ln g 2 p2 ∂ 1 p2 1 ∂ Ht ) - gββ + gββ g 2 ∂β ∂β 2 4 ∂β ββ ∂β 16 ∂β V′(β) (3)
[(
)]
(
and the rotation-torsional coupling term is
Hcoup ) 1/2∑(Jβgiβ) Ji + 1/2∑giβ(JβJi + JiJβ) i
(4)
i
In this particular case (4) reduces to
Hcoup ) 1/2(Jβgyβ)Jy + 1/2gyβ(JyJβ + JβJy)
(5)
as pointed out by Flaud et al.,35 when the torsional axis system is chosen, namely, the z-axis is parallel to the OO bond and the y-axis is the instantanteous C2 symmetry axis. This coupling term can be removed from the kinetic operator by means of a contact transformation,
Teff ) S*TS
Torsional Energy Levels Torsional energy levels have been calculated using the internal rotation potentials and kinetic gββ terms obtained in the present work and are shown in Table 3. In the case of CCSD(T) levels, where no optimization of the geometry at the very level of the theory was carried out, the corresponding MP2 gββ has been used. No terms coupling the torsion with other vibrational modes have been included in the Hamiltonian. This approximation is based on the fact that according to Table 2, the closest fundamental to the torsional one is ω3, and, although no directly measured experimental value for this frequency is known, it appears from our TZDP calculations that it is some 207 cm-1 above the torsional. This value is actually farther apart than are ω1 and ω2 from each other where no evidence of Fermi resonance has been reported. Thus, the possible influence of vibrations on the torsional levels is probably second-order effects, showing an influence on them smaller than that produced by modifications on the torsional barrier height or on the kinetic gββ terms. Therefore, as far as the interaction between the internal rotation and the remaining internal motions can be neglected, we can focus on the rotation-torsion Hamiltonian:
Hr-t ) 1/2∑i,j(g /4Jig- /4)gij(g- /4Jjg /4) + Vt(β) 1
)
1
i,j ) x,y,z,β
1
1
(2)
∂ Jβ ) -ip ∂β where gij stands for the element i,j in the inverse of the generalized matrix of inertia and g symbolizes its determinant.
where S ) exp(is(β)Jy). That transformation moves the effect of the coupling into the rotational part, whereas gββ remains unchanged. In the present case the correct expression for s(β) is
s(β) ) -
gyβ gββ
2 and gyy changes to geff yy ) gyy + 2s(β)gyβ + s(β) gββ. Then Hr-t can be factored out in a rotational term, and a pure torsional one can be written in a convenient way as
1 1 ∂ ∂ ln g ∂ ∂ 1 ∂ ln g 2 - F + V(β) Ht ) - F F hc ∂β ∂β 4 ∂β ∂β 16 ∂β (6)
[(
)
(
)]
where F ) (h/8π2c)gββ and V(β) is in units of cm-1. The term in brackets is a multiplicative factor that does not contain any derivative term of the wave function and can be considered, in fact, as a kinetic contribution to the torsional potential. It has been evaluated in the present case and found to be negligibly small compared to V(β). The torsional Schro¨dinger equation has been solved variationally using a free rotor basis set. The size of the basis set was chosen long enough to ensure that no variation on the eigenvalues was found upon further improvement. In Table 5 the five lowest torsional states for 35ClOO35Cl, referred to the torsional ground state, obtained using the three potentials calculated here, are given. All three t ) 0 f t ) 1 transitions computed are rather coincident, particularly those found using the TZDP basis set, and are within the experimental interval.
8736 J. Phys. Chem., Vol. 100, No. 21, 1996 These results are in support of the previous conclusion of the torsional fundamental being probably within the mentioned 121 ( 7 cm-1 interval. Torsional levels for the less abundant 35ClOO37Cl and 37ClOO37Cl are also presented in Table 5. It can be noticed that practically no splittings are appreciable, nor staggering in the torsional levels, as is the case also with F2O2, indicating a comparatively high barrier with respect to H2O2. This is in agreement with the comment and finding by Birk et al. Levels for 35ClOO37Cl seem to lie somehow in between the corresponding levels of 35Cl and 37Cl peroxides, which is a result of using the same geometry and internal rotation potential for all three isotopomers, as is customary in ab initio calculations.
Go´mez and Pacios provides, among other results, values at the QCISD(T)/6-31+G(2df) level of the theory for Vc ) 10.6 and Vt ) 6.2 kcal mol-1. This is in good agreement with out CCSD(T) results (Table 4) of Vc ) 10.1 and Vt ) 5.4 kcal mol-1. References and Notes
Acknowledgment. We gratefully acknowledge financial support from the Spanish Direccio´n General de Investigacio´n Cientı´fica y Te´cnica (DGICYT) under Project Nos. PB90-167 (P.C.G.) and PB92-0331 (L.F.P.). We are grateful to an anonymous referee for bringing to our knowledge refs 27 and 31.
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Note Added in Proof: Upon completion of the revision of this paper, a work on this subject by Francisco appeared36 that
JP951493A
Concluding Remarks In view of the above results, the general performance of MP2/ ECP-TZDP(f) calculations can be considered rather satisfactory in determining geometries and frequencies. Even though MP2 results may benefit to some extent from certain cancellation of errors, as suggested in ref 12, the fact is that for the nonhypervalent chlorine peroxide the method looks reliable and provides a uniform level of accuracy for these two properties. The use of ECP in conjunction with the optimized TZ basis sets employed has shown a behavior very similar to the corresponding all-electron MP2 calculations. It must be emphasized that besides the lower computational cost in valenceonly calculations the use of ECP operators permits a consistent treatment of the inner (core) electrons. As far as the valence space is the same for all the halogens, the ECP treatment allows us to look forward to dealing with other molecules containing atoms heavier than chlorine without any increase in the computational effort while keeping the calculations at the MP2/ TZDP(f) level. The internal rotation potential has been calculated by using MP2/ECP-TZDP at a specified grid of points, where the geometrical parameters were optimized, and also by using CCSD(T)/ECP-TZDP at a subset of the previous geometries. Cis and trans barriers have been calculated, using fully MP2 optimized geometries at minimum, cis, and trans conformations, at both MP2 and CCSD(T) levels of theory, and using three basis sets of increasing size. All the methods and basis sets used (including semiempirical AM1 and PM3 calculations not shown here) agree in providing a cis barrier height larger than the trans one to different extents, our highest level result being Vc ) 3538 cm-1 and Vt ) 1890 cm-1. Although more precise values for the barrier heights would require a thorough study of additional basis set effects at the CCSD(T) level, we feel that the main conclusion obtained here of Vc being larger than Vt is almost surely correct, and no qualitative changes should be expected.
