Theoretlcal Studies of the Hydrogen Peroxtde Potential Surface. 1. An

for product formation with time constants strongly dependent on the excitation ... overtone excitation populates excited bend and torsion states in ad...
1 downloads 0 Views 1MB Size
Download PDF
8004

J . Phys. Chem. 1989, 93, 8004-8013

Theoretlcal Studies of the Hydrogen Peroxtde Potential Surface. 1. An ab Initio Anharmonic Force Fleld Lawrence B. Harding Theoretical Chemistry Group, Chemistry Division, Argonne National Laboratory, Argonne. Illinois 60439 (Received: April I O , 1989; In Final Form: June 9, 1989) Ab initio GVB + 1 + 2 calculations, employing a (4s3p2dlf/3s2p) basis set, have been used to characterize the ground-state potential energy surface of hydrogen peroxide. An anharmonic force field that is quartic in the nontorsional degrees of freedom and higher order in the torsion is reported. A second-order perturbation theory calculation of the vibrational energy levels using the ab initio force field yields fundamental frequencies that are within 25 cm-’ (3%) of experiment. A reaction path analysis of the torsional mode is also presented. The predicted tunneling splitting between the lowest pair of torsional states is 11.214 cm-’. Also calculated are tunneling splittings for excited vibrational states for both H202 and D202.

I. Introduction Hydrogen peroxide has become an important testing ground for theoretical studies of unimolecular reactions. The combination of its small size and an increasingly large array of detailed experimental studies on the unimolecular dissociation of hydrogen peroxide makes it an ideal candidate for theoretical studies. To date, however, theoretical studies of hydrogen peroxide dynamics have been hampered by the lack of an accurate potential surface. This is the first in a series of papers reporting the results of accurate ab initio calculations on the ground-state potential surface of hydrogen peroxide, the ultimate goal of these studies being the development of an accurate ground-state global potential surface. As mentioned above, there have been many experimental studies of the unimolecular dissociation of hydrogen peroxide. Crim and co-workers’-’ have reported a series of studies in which the 00 bond is broken by pumping energy into high overtones of the OH stretch. Detailed information on the product state distribution has been obtained in these experiments using laser-induced fluorescence on the OH radicals. Zewail and S ~ h e r e r have ~.~ recently reported a refinement of the vibrational overtone induced dissociation experiment using picosecond laser pulses. When exciting the fourth O H overtone, they find a biexponential rate for product formation with time constants strongly dependent on the excitation wavelength. Simons and co-workersI0 have also reported a vibrational overtone induced dissociation experiment. By probing the A-doublet populations, recoil velocities, and rotational angular momentum of the product OH radicals, they obtain information about the interaction between the ground-state potential surface with excited-state surfaces in the long-range region (approaching the O H + OH asymptote). They also find that intramolecular vibrational redistribution (IVR) following OH overtone excitation populates excited bend and torsion states in addition to the 00 overtones. Luo et al.” have also reported a refinement of this experiment, in which they state-selectively excite ( I ) Rizzo, T. R.; Hayden, C. C.; Crim, F. F. Faraday Discuss. Chem. Soc. 1983, 75, 223.

(2) Rizzo. T. R.; Hayden, C. C.; Crim, F. F. J . Chem. Phys. 1984, 81, 4501. (3) Ticich, T. M.; Rizzo, T. R.; Diibal, H.-R.; Crim, F. F. J. Chem. Phys. 1986, 84, 1508. (4) Ticich, T. M.; Likar, M. D.; Diibal, H.-R.; Butler, L. J.; Crim, F. F. J . Chem. Phys. 1987,87, 5820. ( 5 ) Sinha, A.; Vander Wal, R. L.; Butler, L. J.; Crim, F. F. J . Phys. Chem. 1987, 91, 4645. (6) Likar, M. D.; Sinha, A.; Ticich, T. M.; Vander Wal, R. L.; Crim, F. F. Ber. Bunsen-Ges. Phys. Chem. 1988, 92, 289. (7) Likar, M. D.; Baggott, J. E.; Sinha, A.; Ticich, T. M.; Vander Wal, R. L.; Crim, F. F. J . Chem. SOC.,Faraday Tram. 2 1988, 84, 1483. (8) Scherer, N. F.: Doany, F. E.; Zewail, A. H.; Perry, W. J. J . Chem. Phys. 1986, 84, 1932. (9) Scherer, N. F.; Zewail, A. H. J . Chem. Phys. 1987, 87, 97. (10) Brouard, M.; Martinez, M. T.: OMahoney, J.; Simons, J. P. J. Chem. SOC.,Faraday Trans. 2, in press. ( 1 1 ) Luo, X.; Rieger, P. T.; Perry, D. S.; Rizzo, T. R. J . Chem. Phys. 1988, 89, 4448.

hydrogen peroxide by using a double-resonance technique, thus allowing them to study the unimolecular dissociation dynamics at a state-to-state level. In addition to the overtone-induced dissociation experiments noted above, a number of researchers have reported photodissociation studies of hydrogen peroxide using UV light.I2-l8 In these experiments the molecule is excited to a repulsive potential energy surface and the OH product state distribution is again analyzed by using laser-induced fluorescence. The key to interpreting the above experiments (and more detailed experiments now in progress) is the potential energy surface. The hydrogen peroxide potential surface has been studied with m i c r o w a ~ e , far-infrared,21-26 ~~.~~ ir~frared,~’-’~ and Raman35 spectroscopy. From these experiments, much is now known about the equilibrium geometry:638 torsional potentia1,39v40and low-lying

(12) Docker, M. P.; Hcdgson, A.; Simons, J. P. Chem. Phys. Lett. 1986, 128, 264. (13) Docker, M. P.; Hodgson, A,; Simons, J. P. Faraday Discuss. Chem. SOC.1986, 82, 25. (14) Jacobs, A.; Wahl, M.; Weller, R.; Wolfrum, J. Appl. Phys. B 1987, 42, 173. (15) Gericke, K.-H.; Klee, S.; Comes, F. J.; Dixon, R. N. J . Chem. Phys. 1986,85, 4463. (16) Klee, S.; Gericke, K.-H.; Comes, F. J. J . Chem. Phys. 1986, 85, 40. (17) Comes, F. J.; Gericke, K.-H.; Grunewald, A. U.; Klee, S. Ber. Bunsen-Ges. Phys. Chem. 1988, 92, 273. (18) Klee, S.; Gericke, K.-H.; Comes, F. J. Ber. Bunsen-Ges. Phys. Chem. 1988, 92,429. (19) Massey, J. T.; Bianco, D. R. J . Chem. Phys. 1954, 22,442. (20) Massey, J. T.; Hart, R. W. J . Chem. Phys. 1955, 23, 942. (21) Hunt, R. H.; Leacock, R. A.; Peters, C. W.; Hecht, K. T. J . Chem. Phys. 1965, 42, 1931. (22) Hunt, R. H.; Leacock, R. A. J. Chem. Phys. 1966, 45, 3141. (23) Oelfke, W. C.; Gordy, W. J . Chem. Phys. 1969, 51, 5336. (24) Helminger, P.; Bowman, W. C.; De Lucia, F. C. J . Mol. Spectrosc. 1981, 85, 120. (25) Bowman, W. C.; De Lucia, F. C.; Helminger, P. J . Mol. Specrrosc. 1981, 87, 571. (26) Hillman, J. J. J . Mol. Spectrosc. 1982, 95, 236. (27) Giguere, P. A. J . Chem. Phys. 1950, 18, 88. (28) Zumwalt, L. R.; Giguere, P. A. J. Chem. Phys. 1941, 9, 458. (29) Bain, 0.;Giguere, P. A. Can. J . Chem. 1955, 33, 527. (30) Giguere, P. A.; Bain, 0. J . Phys. Chem. 1952, 56, 340. (31) Chin, D.; Giguere, P. A. J . Chem. Phys. 1961, 34, 690. (32) Redington, R. L.; Olson, W. B.; Cross, P. C. J . Chem. Phys. 1962, 36, 1311.

(33) Olson, W. B.; Hunt, R. H.; Young, B. W.; Maki, A. G.; Brault, J. W. J . Mol. Spectrosc. 1988, 127, 12. (34) Hillman, J. J.; Jennings, D. F.; Olson, W. B.; Goldman, A. J . Mol. Spectrosc. 1986, 117, 46. (35) Giguere, P. A.; Srinivasan, T. K. K. J . Raman Spectrosc. 1974, 2, 125

0022-3654/89/2093-8004$01.50/00 1989 American Chemical Society

Theoretical Studies of the H 2 0 2Potential Surface vibrational states. A brief, incomplete summary of these studies follows. Zumwalt and Giguerez8first observed a splitting of an infrared band in the hydrogen peroxide spectrum and attributed this splitting to a double minimum torsional potential. The particular band they observed was the third overtone of the OH stretch. G i g ~ e r later e ~ ~observed a similar splitting in the second overtone. Redington et al.32measured the IR spectrum at higher resolution and obtained torsional splittings for several vibrational states, including the fundamentals of the OH stretches. They also reanalyzed the microwave data of Massey and Bianco19 to obtain the torsional splitting of the ground vibrational state (1 1.44 cm-’). Hunt et aL2’ reported the far-infrared spectrum of hydrogen peroxide, parts of which had been previously observed at low resolution by Giguere and c o - ~ o r k e r s .They ~ ~ were able to assign seven transitions between different torsional states. In this way they were able to derive the energies of the five lowest excited torsional states. Since then various parts of the spectrum have been greatly refined, most notably by Helminger, Bowman, and De working in the far-infrared region, by Hillman et in in the region of the V6 fundamental, and by Olson et the region of the u5 band. Dubal and Crim4I have recently reported a study of HOOH and HOOD using what was termed vibrational overtone predissociation spectroscopy. With this technique they were able to measure torsional splittings for very high overtones of the OH stretch (up to u = 6). They analyzed these results (along with previous data on the fundamentals and lower overtones) by using a model in which the vibrations and torsion are adiabatically separated. They were able to show that the trans torsional barrier height increases near-linearly with the number of quanta in the O H stretch. Butler et ale4*have extended this method to bound vibrational states (the fourth overtone of the OH stretch) by using a two-photon technique and a supersonic expansion. Krachkuruzov and Przhevalskii have attempted to derive from experimental data the equilibrium g e ~ m e t r y $force ~ * ~ constants$5 anharmonicity constants$648 and vibration-rotation coupling constants49for hydrogen peroxide and its isotopomers. This process was hampered by the small number of assigned vibrational transitions, and consequently it was necessary to make assumptions based on analogies with related molecules, O H and H20. The potential surface has also been probed with ab initio calculation^.^^ Most of the calculations reported to date have (36) Giguere, P. A,; Srinivasan, T. K. K. J . Mol. Spectrosc. 1977,66, 168. (37) Koput, J. J . Mol. Specrrosc. 1986, 115, 438. (38) Cremer, D.; Cristen, D. J . Mol. Spectrosc. 1979, 74, 480. (39) Hirota, E. J . Chem. Phys. 1958, 28, 839. (40) Ewig, C. S.; Harris, D. 0. J. Chem. Phys. 1970, 52,6268. (41) Diibal, H.-R.; Crim, F. F. J . Chem. Phys. 1985,83, 3863. (42) Butler, L. J.; Ticich, T. M.; Likar, M. D.; Crim, F. F. J . Chem. Phys. 1986,85, 2331. (43) Krachkuruzov, G. A.; Przhevalskii, I. N . Opt. Specrrosc. 1974, 36, 172. (44) Krachkuruzov, G. A,; Przhevalskii, I. N . Opt. Spectrosc. 1979, 46, 586. (45) Przhevalskii, I. N.; Krachkuruzov, G.A. Opt. Spectrosc. 1974, 36, 175. (46) Krachkuruzov, G. A.; Przhevalskii, I. N . Opt. Spectrosc. 1972, 33, 127. (47) Krachkuruzov, G. A.; Przhevalskii, I. N . Opt. Specrrosc. 1972, 33, 434. (48) Krachkuruzov, G. A,; Przhevalskii, I. N . Opt. Spectrosc. 1976, 41, 323. (49) Krachkuruzov, G.A.; Przhevalskii, I. N . Opt. Spectrosc. 1973, 35, 216. ( 5 0 ) Dunning, T. H., Jr.; Winter, N . W. Chem. Phys. Lerr. 1971,11, 194. (51) Dunning, T. H., Jr.; Winter, N . W. J . Chem. Phys. 1975,63, 1847. (52) Botschwina, P.; Meyer, W.; Semkov, A. M. Chem. Phys. 1976, 15, 25.

(53) Cremer, D. J . Chem. Phys. 1978, 69,4440. (54) Rogers, J. D.; Hillman, J. J. J . Chem. Phys. 1981, 75, 1085. (55) Rogers, J. D.; Hillman, J. J. J . Chem. Phys. 1982, 76, 4046. (56) Rogers, J. D. J . Phys. Chem. 1984,88, 526. (57) Block, R.; Jansen, L. J . Chem. Phys. 1985,82, 3322. (58) Carpenter, J. E.; Weinhold, F. J . Phys. Chem. 1986, 90, 6405. (59) Carpenter, J. E.; Weinhold, F. J . Phys. Chem. 1988, 92, 4295.

The Journal of Physical Chemistry, Vol. 93, No. 24, 1989 8005 focused on the torsional potential. One exception is the calculations of Botschwina, Meyer, and Semkow (BMS).52 They derived a quadratic force field for hydrogen peroxide by using small basis set Hartree-Fock calculations and then scaled the diagonal force constants to match estimated harmonic frequencies. In this way they obtained what it now considered to be one of the most accurate characterizations of the quadratic force field. Also of relevance to this study are calculations reported by Carpenter and W e i n h ~ l d . ~ ~These ” ~ authors focused on the interaction between the OH stretch and the torsion in order to model the vibrational overtone predissociation spectra of Dubal and Crim.41 The wealth of experimental studies on this molecule has stimulated a number of theoretical studies of the dynamics. These have required the development of empirical or semiempirical global surfaces. Among the first were studies by Uzer et al.61”3 who used the semiempirical force field of BMSS2as a starting point in the development of a global surface. Sumpter and Thompson@ independently developed a ground-state global surface, making use of spectroscopic data. Bersohn and S h a p i r ~have ~ ~ developed global surfaces for both the ground and excited electronic states to model the UV photodissociation experiments. Brouwer et a1.66 have modeled the thermal dissociation of hydrogen peroxide with statistical adiabatic channel calculations, using a very simple model for the potential. Nishikawa and L i d 7 have used quantum-mechanical calculations to study the overtone-induced dissociation of H202on a simple, model potential surface. Schinke and Staemmler68v69have reported classical trajectory studies of the photodissociation of H202using a potential derived from ab initio calculations. Ezra70 has refined the potential surface of Uzer et a1.,62 correcting the long-range (OH + OH) behavior of the surface. In this paper, we present the results of large basis set, correlated calculations on the anharmonic force field of hydrogen peroxide. These calculations include through quartic terms in the potential for the nontorsional degrees of freedom and higher order terms in the torsional mode needed to describe the low-lying cis and trans barriers to internal rotation. In section 11, the details of the electronic structure calculations and the fitting of the force field are discussed. In section 111, the results on both the characterization of the force field in the vicinity of the minimum and the torsional potential are presented. Finally, in section IV, the results are summarized. 11. Calculational Details A . Electronic Structure Calculations. The electronic structure calculations reported here employ a (1 2s6p2dlf/6s2p) set of primitive Gaussian functions contracted to (4s3p2dlf/3s2p). The oxygen s and p functions and the hydrogen s functions are general contraction^^^ optimized with ground-state R H F calculation^.^^ The polarization function exponents used are as follows: oxygen d, a1 = 2.3 14 and a2 = 0.645; oxygen f, CY = 1.428; hydrogen p, cy1 = 1.32 and a2 = 0.30. The oxygen polarization exponents were optimized for a RHF + 1 2 calculation on the 3Pstate of atomic oxygen.72 The hydrogen polarization functions were optimized for a CASSCF 1 + 2 calculation on the 211state of the OH radical.

+

+

(60) Carpenter, J. E.; Weinhold, F. J . Phys. Chem. 1988, 92, 4306. (61) Uzer, T.; Hynes, J. T.; Reinhardt, W. P. Chem. Phys. Lett. 1985,117, 600. (62) Uzer,T.; Hynes, J. T.; Reinhardt, W. P. J . Chem. Phys. 1986, 85, 5791. (63) Uzer, T.; MacDonald, B. D.; Guan, Y.; Thompson, D. L. Chem. Phys. Lett. 1988, 152, 405. (64) Sumpter, B. G.; Thompson, D. L. J . Chem. Phys. 1985, 82, 4557. (65) Bersohn, R.; Shapiro, M. J. Chem. Phys. 1986, 85, 1396. (66) Brouwer, L.; Cobos, C. J.; Troe, J.; Diibal, H.-R.; Crim, F. F. J . Chem. Phys. 1987,86, 6171. (67) Nishikawa, K.; Lin, S. H. Chem. Phys. Lett. 1988, 149, 243. (68) Schinke, R.; Staemmler, V. Chem. Phys. Lett. 1988, 145, 486. (69) Schinke, R. J. Phys. Chem. 1988, 92, 4015. (70) Ezra, G. Presented at the 196th ACS National Meeting, Los Angeles, CA, September 25-30, 1988. (71) Raffenetti, R. C. J . Chem. Phys. 1973, 58, 4452. (72) Dunning, T. H., Jr., private communication.

The Journal of Physical Chemistry, Vol. 93, No. 24, 1989

8006

Harding

With this basis set, perfect pairing, generalized valence bond TABLE I: Calculated Energies and Geometries for Hydrogen (GVB) calculations were carried out in which the 00 u bond is Peroxide explicitly correlated. This is the minimum wave function that GVB+ GVB+l+ correctly dissociates into two ground-state O H radicals. ConGVB 1+2 2 + QC figuration interaction calculations (CI) were then performed, using Minimum the orbitals obtained from the GVB calculation. The CI calcu-150.890968 -1 51.312607 -1 5 1.352673 lations include all single and double excitations relative to the GVB 1.484 1.456 1.462 configurations (a total of 432 040 configurations for geometries 0.964 0.944 0.956 with no symmetry). In order to correct for the effects of higher 99.5 99.9 99.6 order (quadruple) excitations, a multireference, D a v i d ~ o n ~ ~ , ~ ~ 121.4 115.0 113.4 correction of the form Erslkcal/mol

is added onto the GVB + 1 + 2,CI energies. In this formula AEsD is the difference between the GVB and GVB + 1 + 2 energies and C is the sum of the squares of the coefficients of the GVB configurations, in the CI wave function. All of the electronic structure calculations were carried out with the Argonne QUEST codes7s on an FPS/164 attached processor. The electronic structure calculations described above were carried out a t approximately 480 different geometries, in which the coordinates were varied over the following ranges

2.3 < Roo < 3.2 au, 1.6 < RoH < 2.0 au

RonI A HOO angle, deg E,,, kca I / mol Roo, A ROH,

A

HOO angle, deg

1.06 1.472 0.963 98.3

Cis Transition State 7.39 7.46 1.491 1.465 0.944 0.956 104.4 104.4

7.38 1.47 1 0.963 104.1

TABLE 11: Calculated Harmonic Normal Mode Frequencies (in cm-I) for Hydrogen Peroxide (HOOH) GVB+ GVB+l+ GVB 1+2 2+ Minimum w1 (sym OH str) 4134 3939 3816 w2 (sym HOO bend) 1486 1458 1429 887 w j (00 str) 779 887 w4 (torsion) 333 378 383 w 5 (antisym OH str) 4131 3939 3820 w6 (antisym HOO bend) 1359 1351 1328

oc

60' < H O O angles < 140' 0' < HOOH dihedral angle < 180' centered about the geometry of the minimum. These points were then fit to an analytic potential function as described in the next section. B. Force Field Fitting. The determination of the normal-coordinate force field for hydrogen peroxide was undertaken in three step^.^^,^^ First, the 480 calculated energies were fit to an internal-coordinate, potential function. The internal-coordinate expansion was then used in a normal-coordinate analysis, and finally the potential was reexpanded as a function of these normal coordinates. In this section we first discuss the internal-coordinate potential function, later used to determine the normal-coordinate force field. The internal-coordinate potential function consists of a Fourier series expansion in the dihedral angle coordinate and SimonsParr-Finlan78*79and Taylor series expansions in the bond length and bond angle coordinates, respectively. The form of the potential function is as follows: 8

v = n=O c [iJ,k,l,m c i j k / m , S ' ( R o , @ , ) g ( R O , H , ) S ~ ( ~ O ~ ~ ) "'H,o.@,"mHboboal

Roo, A

Trans Transition State 0.54 0.94 1.493 1.466 0.943 0.955 98.6 98.7

'Os

('+Hood

where

S i ( R ) = ( R - R,)'/R' is the Simmons-Parr-Finlan coordinate for the bond length, R, with equilibrium distance Re, the a's refer to the H-0-0 bond angles, and + H m H is the dihedral angle between the two HOO planes. The polynomial includes all terms satisfying the condition i+j+k+l+m+n54 In addition, the Fourier series expansion of the dihedral angle includes terms up to n = 8; the higher terms, n = 4-8, are not

Trans Transition State wI (sym OH str) 4147 3960 w2 (sym HOO bend) 1559 1545 w3 (00 str) 774 883 w4 (torsion) 2451' 2941' w 5 (antisym OH str) 4143 3965 w6 (antisym HOO bend) 1296 1279

3839 1520 884 302i 3852 1254

Cis Transition State w I (sym OH str) 4148 3951 w2 (sym HOO bend) 1424 1405 w 3 (00 str) 767 876 w4 (torsion) 5691' 591i w5 (antisym OH str) 4111 3899 ( 0 6 (antisym HOO bend) 1485 1473

3831 1381 877 590i 3772 1449

coupled to the other coordinates. Altogether 214 terms are included in this potential function, of which 124 are unique (not related to other terms by symmetry). Although over half of the calculated points lie within 10 kcal/mol of the minimum, some points range in energy up to almost 250 kcal/mol above the minimum. Because the total number of calculated points exceeds the number of coefficients by almost a factor of 4, the function will not exactly reproduce the energies of the calculated points. In order to assure that the region near the minimum is described accurately, a weighted, least-squares fit was used to determine the coefficients cUk/mn, the weights being chosen such that the points lowest in energy have the highest weights. In this procedure the weighting factor for ~ the ith point is taken t o be proportional to R Twhere T . = Ei - Emin ' I W I ~

(73) Langhoff, S. R.; Davidson, E. R. Int. J . Quantum Chem. 1974.8, 61. (74) Davidson, E. R.; Silver, D. W. Chem. Phys. Letr. 1978, 52, 403. (75) Shepard, R.; Bair, R. A.; Eades, R. A.; Wagner, A. F.; Davis, M. J.; Harding, L.B.; Dunning, T. H., Jr. Inr. J. Quantum Chem. Symp. 1983, No. 17, 613. (76) Harding, L. B.; Ermler, W. C. J . Comp. Chem. 1985, 6, 13. (77) Ermler, W. C.; Hsiuchin, C. H.; Harding, L. B. Comput. Phys. Commun. 1988, 51, 257. (78) Simons, G.; Parr, R. G.; Finlan, J. M . J. Chem. Phys. 1973,59, 3229. (79) Simons, G. J . Chem. Phys. 1974, 61, 369.

E, is the energy of the ith point, E& is the energy at the minimum, and W is chosen such that Ti is between 0 and 1 for all points. In this scheme the ratio between the largest and smallest weighting factors is R. R was selected by trial and error to be 106. For points lying within 10 kcal/mol of the minimum, the least-squares fit reproduces the calculated energies with a maximum error of 0.12 kcal/mol and a root-mean-square error of 0.03 kcaljmol. For higher lying points the errors are larger.

The Journal of Physical Chemistry, Vol. 93, No. 24, 1989 8007

Theoretical Studies of the H 2 0 2Potential Surface TABLE 111: Internal-Coordinate Force Constants for Hydrogen Peroxide BMS force GVB GVB 1 constant' GVB 1 2 2 QC ab initiob scaledb 8.133 8.834 8.009 f,,, mdyn/A 9.530 8.660 4.342 4.347 6.255 4.322 f R R , mdyn/A 3.402 0.957 1.078 0.894 fau, mdyn A 0.981 0.978 0.039 0.028 0.037 fM, mdyn A -0.014 -0.020 f,,, mdyn/A -0.001 -0.006 -0.092 -0.083 frR, mdyn/A -0.135 -0.100 mdyn -0.033 -0.025 -0.028 -0.040 La,, mdyn 0.008 0.005 0.003 -0.001 0.575 0.605 fRur mdyn 0.538 0.579 0.076 0.125 0.079 faa,,mdyn A 0.088 0.078 f R 4 , mdyn -0.016 -0.014 4.012 0.020 f Q ,mdyn 0.012 0.019 0.026 f a 4 , mdyn 0.021 0.026

+

+

+

+ +

'The internal coordinates are r,r' = RoH,R = R, angles, I#I = HOOH dihedral angle. Reference 52.

@,a' =

HOO

TABLE IV: Calculated Fundamental Frequencies and Anharmonicity Constants (cm-') for Hydrogen Peroxide (Experimentally Derived Results Are Shown in Parentheses)

HOOH DOOD Fundamental Frequencies 3616 (3599)' 1392 (1 387)b 859 (875)b 3613 (3611)' 1271 (1264.6)c

111. Discussion A . Equilibrium Geometry. A large number of attempts have been made to determine the equilibrium geometry of hydrogen peroxide using both experimental and theoretical techniques. The most recent experimental determination of the equilibrium structure is by K o p ~ t Koput . ~ ~ reports four structures each for a different, assumed r,(OH). In each case a set of observed transition energies were fit with a least-squares procedure, the parameters of which determine the three remaining geometric parameters in addition to several potential function parameters. The largest of the four O H distances considered in Koput's analysis, 0.965 A, is very close to that derived from the present GVB 1 2 QC calculations, 0.964 A. Using this assumed value, Koput derived an 00 distance of 1.464 A, an HOO angle of 99.4', and a dihedral angle of 111.8'. All compare very well with the GVB 1 2 QC results, 1.462 A, 99.6O, and 113.4', respectively. The most effort and controversy has focused on the equilibrium dihedral angle. A summary of previous experimental and theoretical determinations is given in Table VII, along with the results obtained from the present study. On the experimental side, neutron diffraction studies,80-81of crystalline hydrogen peroxide

+ + +

+ + +

(80) Busing, W. R.; Levy, H. A. J . Chem. Phys. 1965, 42, 3054. (81) Prince, E.; Trevino, S.F.; Choi, C. S.; Farr, M. K. J . Chem. Phys. 1975, 63. 2620.

3615 2674 1338 981 (981)' 860

Anharmonicity Constants XI I x22 x33

xss x66 XI2 XI3

XIS XI6

Since the dihedral angle is expanded in a Fourier series, the function used here accurately reproduces the energies of points in the regions of the cis and trans transition states for internal rotation, in addition to points near the minimum. A summary of the calculated energies and geometries of these three stationary points is given in Table I and the harmonic normal mode frequencies in Table 11. The quadratic, internal-coordinate force constants at the minimum are listed in Table 111. The internal-coordinate force field, determined by the procedure outlined above, is then reexpanded in terms of the normal coordinates by using the program SURVIB.76977 In this expansion, all terms that contribute to the second-order, perturbation theory, expressions for the molecular anharmonicity constants are included (there are 84 such terms). The calculated anharmonicity constants and fundamental frequencies for HOOH, DOOH, and DOOD are summarized in Table IV, the normal-coordinate force field parameters are given in Table V, and the vibration-rotation coupling constants are given in Table VI. Note that throughout the tables and text the modes are numbered in the conventional way, all of the A symmetry modes first, highest to lowest, followed by the modes of B symmetry. For HOOH and DOOD this means the symmetric and antisymmetric OH stretches are modes 1 and 5, respectively. For HOOD, all modes are of the same symmetry and so are just numbered highest to lowest. For HOOD then, the O H and OD stretches are modes 1 and 2, respectively.

2677 (2667)" 1025 (1028)" 861 (869)" 2672 (2661)e 949 (947)e

HOOD

x23 x2S x26 x3s x36 x56

-46.8 (-90.9 f 0.5)b -7.6 (-10 f 5) -6.2 (-10 f 5) -49.4 (-90.2 f 0.2) -6.7 (-3 f 3) -2.2 (-11 f 3) 0.5 (-11 f 2) -184.7 (-167.6 f 0.4) -25.5 (-3 f 2) -17.7 (-7 f 4) -3.9 (-11.5 f 1.5) -19.3 (-4 f 3) 0.9 (-11.1 f 1.1) -14.0 (-2 f 2) -27.4 (-3 f 2)

-24.4 (-48 f 5 ) b -4.0 (-5 f 4) -5.9 (-10 f 5) -26.1 (-47 f 5) -3.9 (-2 f 2) -2.2 (-6 f 4) 0.7 (-8 f 4) -97.2 (-88 f 5) -13.7 (-2 f 2) -14.3 (-5 f 3) -2.8 (-6 f 4) -9.2 (-2 f 2) 1.2 (-8 f 4) -11.4 (-1 f 1) -14.7 (-2 f 2)

-92.9 (-9O)g -48.8 (-47) -10.0 (-1 1) -4.9 (-6) -6.0 (-10) -3.8 (-121) -27.8 (-10) -2.2 (-3) 0.7 (-16) 1.8 (-10) -18.1 (-4) 0.9 (-12) -6.4 (-5) -16.0 (-7) -12.7 (-2)

"Reference 46. bReference48. CReference34. "References 35 and eReferences29, 30, and 48. 'Reference 29. EReference 45.

48.

find dihedral angles very close to 90'. However, since the torsional mode is very flat, one must be concerned about the effects of crystal-packing forces on the equilibrium angle. In fact, Jonsson et aL8* have shown, using a Monte Carlo simulation, that the dihedral angle decreases by approximately 35' on going from vapor phase to dilute aqueous solution. Studies of the rotational spectra of hydrogen peroxide are also somewhat ambiguous since four geometric parameters are needed to specify the geometry but only three rotational constants exist. Prior to Koput's analysis, the two most recent experimental determinations favored relatively large dihedral angles of approximately 120°, although several earlier studies favored smaller angles. Another difficulty in extracting an equilibrium dihedral angle from experimental data is made clear by the rotational analysis of Oelfke and G ~ r d y ?They ~ report vibrationally averaged dihedral angles for the ground vibrational state and its tunneling split, torsional pair. The two states differ in energy by only 11 cm-I but the dihedral angles differ by 4'. The difference is due to the fact that the vibrational amplitude of even the lowest state covers a substantial range of dihedral angles. The classical turning points of the lowest torsional state occur at angles of approximately 90' and 140'. Thus zero point motion covers a range of 50'. Consequently, in order to extract the equilibrium dihedral angle from experimental data, one must make assumptions concerning the form of the torsional potential. The most common assumption is to expand the potential in a Fourier series and truncate the series after the third term. It is not clear what effect this truncation will have on the resultant dihedral angle. As will be discussed below, this large-amplitude motion is also affected by the other, small-amplitude modes of the molecule. These effects must also be taken into account in order to derive a true equilibrium dihedral angle from observed spectra. On the theoretical side, Dunning and Winter,so,slusing R H F calculations, showed that a relatively large basis set, including d polarization functions, is needed to accurately characterize the torsional potential. Without d functions they obtained an optimum dihedral angle of 138'. Addition of one set of d functions lowered this to 113.7". Cremers3 has shown that electron correlation significantly affects the optimum dihedral angle, increasing it relative to an R H F calculation by approximately 5' (to 119'). (82) Jonsson, B.;Romano, S.; Karlstrom, G. Inr. J . Quanrum Chem. 1984, 25, 503.

8008

The Journal of Physical Chemistry, Vol. 93, No. 24, 1989

Harding

TABLE V Calculated Normal-Coordinate Force Fields for HOOH. DOOD. and HOOD (AU Ouantities in Atomic Units

normal mode displacement exponents 123456 200000 020000 002000 000200 000020 000002 300000 2 10000 20 1000 200 100 2000 10 20000 1 120000 11 1000 110100 110010 110001 102000 101 100 101010 101001 100200 1001IO 100101 100020 100011 100002 030000 02 1000 020100 0200 IO 02000 1 0 12000 01 1100 011010 01 1001 0 10200 0101 10 010101 0 10020 01001 1

X

lo6)

normal mode displacement

exponents

HOOH

DOOD

HOOD

123456

HOOH

DOOD

HOOD

151.15496 21.194 20 8.161 42 1.51994 151.441 87 18.297 85 3.222 10 -0.199 73 -0.014 09 -0.050 98

80.34100 11.379 86 8.12304 0.812 15 80.294 08 10.003 55 1.242 42 -0.10661 -0.002 27 -0.020 48

-0.034 68 -0.070 42 0.000 39

0.002 35

0.000 85

-0.008 83 0.647 93 0.01639

-0.002 21 0.354 79 0.006 46

-0.262 89 0.286 75 -0.001 89

-0.082 85 -0.101 68 0.054 15

-0.030 61 0.1 11 86 0.027 07

-0.105 07 0.170 74

-0.091 24 -0.057 03

-1.373 90

-0.532 68

9.59469 0.182 13 -0.977 19 -0.034 42 0.13876 -0.053 43

3.701 16 -0.083 72 -0.369 18 -0.01 5 79 -0.077 04 -0.026 06

-0.026 47 -0.020 50

-0.025 51 0.01 5 59

-0.042 26

-0.013 18

-0.174 33 1.804 86

-0.10255 -0.615 16

010002 003000 002 100 0020 10 002001 00 1200 001 110 001101 001020 001011 00 1002 0002 10 00020 1 000120 000111 000102 000030 00002 1 000012 000003 400000 220000 202000 200200 200020 200002 040000 022000 020200 020020 020002 004000 002200 002020 002002 000400 000220 000202 000040 000022 000004

-0.094 75 0.07203 0.002 20

-0.824 80 -0.588 39 0.029 30

151.29743 80.321 26 19.836 57 10.599 09 8.14093 1.166 72 4.53 1 52 -0.026 93 0.201 44 -0.025 80 0.013 70 0.058 32 0.009 29 -0.019 72 0.004 5 1 -0.004 83 0.01506 -1.245 75 -0.428 33 -0.59497 0.095 08 -0.034 67 -0.1 04 73 0.066 28 -0.069 9 1 0.131 96 -1.27682 -1.747 55 0.077 67 -0.095 7 1 -0.003 30 0.03 1 02 0.005 82 -0.041 63 0.006 26 0.040 49 0.438 20 -0.331 97 0.037 89 0.069 24 -0.034 68

0.043 96 -0.039 20 -0.004 47 -0.057 39 0.258 67 -0.055 9 1 0.000 73 0.000 90 0.004 18 -0.039 76 0.008 68 0.000 38 0.000 53 -0.004 24 0.001 10 0.003 06 -0.057 68 0.004 48 0.041 47 -0.056 85 0.001 49

0.01227 -0.009 27 -0.002 52 -0.016 10 0.072 16 -0.01550 0.000 17 0.000 50 0.000 99 -0.009 31 0.001 94 0.000 5 1 0.000 32 -0.002 41 0.000 73 0.000 87 -0.01621 0.001 29 0.01 108 -0.01 5 74 0.00041

0.259 66 0.045 34 0.005 30 -0.124 65 0.008 10 0.006 45 0.01282 -0.078 75 0.01964 0.01727 -0.018 14 -0.066 13 -0.008 93 -0.017 50 -0.008 89 -0.006 62 -0.071 08 -0.001 15 -0.001 00 -0.01 1 05 0.084 59 -0.002 01 -0.093 47 -0.002 92 -0.003 94 -0.075 02 0.023 88 0.000 66 -0.025 11 -0.002 66 -0.01 1 21 0.002 73 0.000 24 0.001 12 0.005 54 0.000 62 0.000 68 0.000 99 0.000 5 1 0.000 49 0.002 89

TABLE VI: Vibration-Rotation Coupling Constants (em-') about the A Axis for HOOH mode theory expmt" 1 0.187 0.234 2 -0.119 -0.104 3 0.017 -0.034 4 0.032 5 0.178 0.234 -0.135 -0.174 6

"Reference 49. Note the constants reported by these authors are opposite in sign to the standard convention used here.

Most recently, Carpenter and Weinhold:using correlated wave functions, concluded that more than one set of polarization functions are required to accurately predict the dihedral angle. Their most extensive calculations yielded an optimum angle of 1 1 1.1 '. The present calculations agree quite well with those of Carpenter et predicting a relatively small dihedral angle of 113.4'. B. 00 Bond Dissociation Energy. The calculated 00 equilibrium bond dissociation energies are 31.7, 49.0, and 52.0 kcal/mol for the GVB, GVB + 1 2, and GVB + 1 + 2 + QC calculations, respectively. The corresponding Do values are 26.0, 43.2, and 46.3 kcal/mol, respectively. Using heats of formation, AHf(Oo),from the JANAF83tables of -31.0 and 9.2 kcal/mol

+

(83) Chase, M. W., Jr.; Davies, C. A.; Downey, J. R., Jr.; Frurip, D. J.; McDonald, R. A.; Syverud, A. N. J . Phys. Chem. ReJ Data 1985, 14.

for hydrogen peroxide and OH, respectively, we obtain an experimental Doof 49.4 kcal/mol, or 3 kcal/mol above the most accurate (GVB 1 2 QC) calculations. The GVB dissociation energy is found to be approximately one-half the experimental value. CI reduces this error to 6 kcal/mol (1 5%). The quadruples correction leads to a further reduction in the error of 3 kcal/mol. The majority of the remaining error in the GVB 1 + 2 + QC dissociation energy can probably be attributed to limitations in the size of the basis set used. C . Quadratic Force Constants. The quadratic internal-coordinate force constants are summarized in Table 111. Also listed in Table I11 are the ab initio and scaled force constants of BMSS2 The results from the highest level of calculation in the present study (GVB + 1 + 2 + QC) are in good agreement with the scaled force constants of BMS, in which the diagonal a b initio force constants were scaled to match estimated harmonic frequencies. The most significant difference occurs for thefaa force constant, for which the GVB + 1 + 2 QC result is 7% larger than the BMSS2scaled result. Part of this difference can be attributed to the fact that BMS scaled the bend frequencies to match the observed fundamental frequencies rather than the harmonic frequencies. As will be seen below, the present calculations predict differences between the harmonic and fundamental frequencies for the symmetric and antisymmetric bends of 37 and 57 cm-I, respectively. BMS52also report a strong dependence of the fRa force constant on the dihedral angle, with values of 0.772,0.605, and 0.552 mdyn for dihedral angles of ,'O 1 19.1°, and 180°, respectively. The present calculations show both a smaller variation and a maximum

+ + +

+

+

The Journal of Physical Chemistry, Vol. 93, No. 24, 1989 8009

Theoretical Studies of the H 2 0 2Potential Surface TABLE VII: Summary of Experimental and Theoretical Determinations of the Equilibrium Dihedral Angle and the Cis and Trans Barriers to Internal Rotation bHmH, cis barrier, trans barrier, kcal/mol deg kcal/mol Experiment Redington et al. (1962)' 119.8 3.71 0.85 Hunt et al. (1965)* 111.5 7.03 1.10 Ewig et al. (1970)c 112.8 7.57 1.10 Krachkuruzov 119.1 Giguere et al. (1977)c 120 Koput (1986)J 111.8 7.1 1.1 ~~

Theory Dunning and Winter (1975)g RHF/(4s3pld/2sl p) Cremer (1978)* RHF/(4s3pl d/2s 1 p) RHF/(I ls6p2d/6s2p) RS-MP/(4s3p 1d/2s 1p) Carpenter and Weinhold (1 986)l MP2/6-31GS MP2/6-31 IG(2dp) MP2/6-31 IG(3d2p) present work GVB/(4~3p2dlf/3~2p) GVP+ 1 + 2 /(4s3p2dl f/3s2p) GVB+ 1 + 2 + QC/(4s3pZdl f/3sZp)

113.7

8.35

1.10

114.9 111.2 119.3

8.04 7.69 8.13

0.75 1.44 0.94

119.4 116.0 111.1

0.60 0.89 1.28

121.4 1 15.0

7.39 7.46

0.54 0.94

113.4

7.38

1.06

OReference 32. bReference 21. eReference 40. dReference 43. Reference 36. fReference 37. 8 References 50 and 51. *Reference 53. 'References 58-60.

+

value for a skewed geometry. The GVB 1 + 2 + QC results are 0.497, 0.575, and 0.563 mdyn for the cis transition state, skewed minimum, and trans transition-state geometries, respectively. The large difference for the cis geometry may be due in part to differences in geometry. In the calculations of BMSS2only the torsional angle was varied whereas in the present calculations all geometrical parameters are allowed to relax. The primary geometrical changes in going from the skewed to the cis geometry are increases in Roo and a (see Table I). Both of these changes are expected to reduce the fRa force constant. D. Normal Mode Frequencies. The calculated normal mode frequencies are listed in Table 11. Although many attempts have been made to derive harmonic frequencies for hydrogen peroxide from experimental data, all of these attempts involve unjustifiable assumptions necessitated by a lack of available data. Because of these ambiguities in the experimentally derived harmonic frequencies, comparisons between the experimental and theoretical harmonic frequencies cannot be assumed to be a reliable measure of the accuracy of the calculations. For this reason the discussion in this section will focus on the differences between the theoretical calculations, and a detailed comparison with experiment will be delayed until the next section where a comparison will be made between calculated and observed fundamental frequencies. Comparing first the GVB and the GVB 1 2 frequencies, one finds the GVB 00 stretching frequency to be approximately 100 cm-I below the CI result, while the OH stretching frequencies are approximately 200 cm-l above the CI results. This is because in the GVB wave function used here the 00 bond is correlated but the OH bonds are not. The GVB 00 stretch frequency is too small because the GVB 00 bond energy is much too small (see above). The GVB OH stretching frequencies are too large due to the lack of correlation in these bond pairs, and in fact these frequencies are expected to be essentially equal to the HartreeFock values. Smaller differences are found between the GVB and GVB + 1 + 2 bending and torsional frequencies. The quadruples correction is found to further lower the O H stretching frequencies, by approximately 100 cm-' relative to the GVB + 1 + 2 frequencies, and has a much smaller effect on all of the other frequencies.

+ +

Similar trends are found for the two torsional states. It is worth noting the variations in the imaginary frequencies. The most significant difference is in the GVB, trans, imaginary frequency, which is approximately 50 cm-' below the GVB 1 + 2 and GVB + 1 + 2 + QC results. This is presumably a reflection of the fact that the GVB trans barrier height is also significantly below the GVB 1 + 2 and GVB + 1 2 + QC predictions (see Table 1). E . Anharmonicity Constants. The calculated anharmonicity constants, as derived from second-order perturbation theory and the GVB 1 2 + QC force field, are listed in Table IV, along with the predicted fundamental frequencies. In the derivation of these constants, the torsional mode is not included. The torsional mode is a highly anharmonic, large-amplitude mode, for which an harmonic oscillator based perturbation theory treatment is not appropriate. The torsional mode will be discussed in Section G. Krachkuruzov and Przhevalskii (KP) have derived several sets of anharmonicity constants for the three isotopomers HOOH,46948 DOOD,"7948and HOOD$5 the most recent of which are listed in Table IV. It should be noted that the analysis of KP suffers from a lack of experimental data (the number of assigned overtone and combination bands used to derive the anharmonicity constants is less than the number of constants obtained). The agreement between the calculated anharmonicity constants and those derived from experiment is poor. For example, for xI1and xss (the diagonal anharmonicity constants for the symmetric and antisymmetric O H stretches), KP report values of -90.9 and -90.2 cm-I, respectively, whereas the present calculations predict values of -46.8 and -49.4 cm-'. The difference exhibits a fundamental problem with anharmonicity constant expansions for molecules like hydrogen peroxide. The values derived by KP are characteristic of local mode O H stretches and are approximately equal to the corresponding anharmonicity constant, w,x,, of the O H radical, 84.965 cm-'. The values derived from theory correspond to symmetric and antisymmetric combinations of the two local modes and for this reason are approximately half that expected for local modes. For comparison, the corresponding diagonal anharmonicity constants for the symmetric and antisymmetric stretches in H 2 0are -42.6 and -47.6 cm-', respecti~ely.~~ These are quite close to those predicted here for hydrogen peroxide. The experimental anharmonicity constants are derived by fitting the energies of observed overtone and combination bands. Since the coupling between the two OH stretches in hydrogen peroxide is very weak, even the lowest stretch overtones exhibit local mode behavior. Consequently the anharmonicity constants derived from these overtones are characteristic of local modes (as found by KP). For HOOD one might guess that this would not be a problem since the normal modes look like local modes. There is indeed good agreement between the calculated diagonal anharmonicity constants and those derived by W 5 for HOOD. There is however a large disagreement for the constant that couples the O H and OD stretches, xlz. KP derived the constants for HOOD by assuming a proportionality relationship with the constants of HOOH and DOOD (very few bands in the HOOD infrared spectrum have been assigned). Thus errors in the HOOH and DOOD constants are carried over directly to the HOOD constants. The best gauge of the accuracy of the calculated force field comes from comparing the calculated fundamental frequencies to those derived from experimental spectra. This comparison is given in Table IV. Note that the calculated frequencies here do not include any contributions from torsional motion. These contributions are small and will be discussed in Section G. The maximum error for HOOH is 17 cm-', for the OH symmetric stretch, with the calculations predicting this frequency to be too large. In general, the calculations predict all frequencies except the 00 stretch to be too large. The 00 stretch is predicted to be 16 cm-' too small. This trend is expected since only the 00 bond is explicitly correlated in the GVB (l/PP) reference wave function. The total root-mean-square error for the five nontor-

+

+

+

+ +

(84) Benedict, W. S.; Gailar, N.; Plyler, 1139.

E.K.J . Chem. Phys.

1956, 24,

8010 The Journal of Physical Chemistry, Vol. 93, No. 24, 1989

Harding

TABLE VIII: Fourier Expansion Coefficients for the Frequencies and Geometry on the Reaction Path for Internal Rotation normal mode frequencies, cm-I OH antisym sYm antisym str bend bend 3804.4 1423.9 1358.6 -43.3 -66.0 95.7 -8.3 7.4 26.7 3.4 -3.1 1.8 1.1 -0.2 0.7 0.1 3.4 0.0

OH sym str

constant cos (6) cos (24) cos (34) cos (4@) cos (54)

3820.0 -7.6 15.2 3.8 -0.2 0.1

00 str 882.6 -3.9 -2.5 0.7 0.8 -0.5

geometry ROH, au 1.8214 0.0015 -0.001 7 -0.0008

Roo, au 2.7699 -0.0004 0.01 I 9 -0.0002

HOO angle, deg 100.9 2.8 0.4

OH stretches

anti-sym

3750 1550

I

j

1500 "

n

7

1450

E & 1400

3

1350 1300

t

AR

950

,.-,

1

trans , , . , . . / .

0

25

cis

1

50

,

, . , .

75

, , . . . , , . . . / , . . .

100

125

150

I 175

Reaction Coordinate (au) Figure 1. Variation in Roo, aHoo,and &,mH along the torsional reaction path. The trans saddle point is located at s = 0 and the cis saddle point at s = 165.82, and the minimum is indicated by the vertical line at s = 63.16. The reaction coordinate has units of distance times the square root of mass, both in atomic units.

sional modes is 11 cm-l. Comparable accuracy is also found for the fundamentals of DOOD, with a maximum error in the calculated frequencies of 11 cm-'. For HOOD, there is exact agreement between the calculations and the one fundamental that has been observed. Calculated vibration-rotation coupling constants, about principal inertial axis A , are given in Table VI. The calculated coupling constants about the B and C axes are all small (10.01 cm-I). Experimentally derived49 vibration-rotation coupling constants are also shown in Table VI and are found to agree quite well with the theoretical results. The coupling constants for the two O H stretches are both predicted to be positive. This is because exciting either of these modes increases the averaged length of the OH bonds, which in turn increases the moment of inertia about the A axis (the A axis is approximately coincident with the 00 bond axis). Both bends are predicted to have negative vibration-rotation coupling constants. This is because exciting either of these modes increases the averaged aHw,which decreases the moment of inertia about the A axis. F. Torsional Potential. Hydrogen peroxide is the simplest molecule having an internal rotation, and as such, the potential for this rotation has received a great deal of attention, particularly the cis and trans barrier heights. A summary of previous experimental and theoretical determinations of the barrier heights is given in Table VII. There is now reasonably good agreement between theory and experiment on both barriers. The present

850

00 stretch

!,

0

trans I

I

,

I

I

25

I

I

50

..

75

I

cis I

I

100

I '

I " . '

125

I ' ' '

150

I

175

Reaction Coordinate (au) Figure 2. Variation in normal-coordinate frequencies along the torsional reaction path. The reaction coordinate is as defined for Figure 1. Frequencies of A symmetry are shown with solid lines, those of B symmetry with dashed lines.

GVB + 1 + 2 + QC calculations predict a trans barrier height of 1.06 kcal/mol, in excellent agreement with all of the more recent experimental determinations. The experiments are less sensitive to the cis barrier height. Most recent experiments have yielded cis barrier heights in the range 7.0-7.5 kcal/mol in good agreement with the present calculations, 7.38 kcal/mol. However, Olson et al.33have recently reported new high-resolution infrared spectra of hydrogen peroxide, which they interpret to imply a cis barrier height much larger than 7 kcal/mol. This conclusion appears to be inconsistent with the theoretical results. Since the potential function used to fit the a b initio points accurately describes this torsional mode, this potential can be used to derive a torsional reaction path.85 This is accomplished with two steepest descent calculations (in mass-weighted, atomic, Cartesian coordinates), one starting from the trans saddle point and one from the cis saddle point. The two paths meet at the hydrogen peroxide minimum. The reaction path calculations were carried out with the program P O L Y R A T E . ~ ~ Properties of this reaction path are depicted in Figures 1 and 2. The variation in the 00 bond length, the HOO bond angles, and the torsional angle as a function of progress along this reaction path is shown in Figure 1. The normal mode frequencies perpendicular to the reaction (85) Miller, W. H.; Handy, N. C.; Adams, J . E. J . Chem. Phys. 1980, 72, 99. (86) Isaacson, A. D.; Truhlar, D. G.; Rai, S . N.; Steckler, R.; Hancock, G. C.; Garrett, B. C.; Redmon, R. J. Compui. Phys. Commun. 1987, 47, 91.

Theoretical Studies of the H 2 0 2Potential Surface path are shown in Figure 2. Fourier expansion coefficients of the normal mode frequencies and geometrical parameters as a function of the dihedral angle on the reaction path are given in Table VIII. Not surprisingly, Figure 1 shows that the dominant change in geometry that occurs along the reaction path is the change in the dihedral angle, which is found to vary linearly with the reaction coordinate. The HOO bond angles are found to increase monotonically from the trans saddle point, through the minimum, to the cis saddle point. The 00 distance exhibits a shallow minimum when the two O H bonds are nearly perpendicular. Although not shown in Figure 1, the O H distances are also allowed to vary along the reaction path, but the changes are predicted to be very small (