J. A m , Chem. SOC.1987, 109, 4193-4198
4193
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Calculated Barrier Heights for OH C2H2and OH C2H, Using Unrestricted Merller-Plesset Perturbation Theory with Spin Annihilation Carlos Sosat and H. Bernhard Schlegel*$ Contribution f r o m the Department of Chemistry, Wayne State University, Detroit, Michigan 48202. Received October 6, 1986
Abstract: The reactions of O H radical with C2H2and CzH4 have been studied with ab initio molecular orbital techniques. Reactants, loose clusters, transition structures, and products were optimized at UHF/3-21G and UHF/6-3 1G*. The barrier heights have been computed by using unrestricted Hartree-Fock and Mdler-Plesset perturbation theory up to fourth order, including single, double, and quadruple excitations. Spin contamination in the U H F wave function has been corrected by annihilating the largest spin contaminant. The vibrational frequencies were computed by using analytical derivative methods at the UHF/3-21G level. The barrier heights for both reactions are overestimated by 7-15 kcal/mol at the UMPZ, UMP3, and UMP4 levels. Annihilation of the largest spin contaminant lowers the barrier heights by 7-15 kcal/mol. Calculations at the PMP4/6-3 l G * level are in good agreement with the estimated experiment barrier heights.
open-shell molecules, estimated corrections for contamination from The reactions of OH with acetylene and ethylene are known higher spin states are also included. to be important in hydrocarbon combustion as well as atmospheric and OH In previous work we have shown that the barrier heights for chemistry.'-5 Experimental studies on OH C2H21-12 hydrogen addition to ethylene and formaldehyde are overestimated + c2H4'-43'1-25have shown that near room temperature the by 6-10 kcal/mol a t the U M P 4 level due to spin contamination predominant mechanism involves the electrophilic addition of the in the U H F wave function. The difficulties in computing barrier OH radical to the 7r bond, forming an activated complex which heights with unrestricted Mdler-Plesset perturbation theory can can be collisionally stabilized. be overcome by annihilation of the largesr spin contaminant. This OH + C2H2, C2H2,0H* ( n = 1, 2) (1) leads to a much improved description of bond dissociation curves28 C2H2,0H* + M --* C2H2,OH + M (2) In early experimental work on OH C2H2, no pressure dependence of the rate constant was found a t low temperature^.^^^ ( I ) Combustion Chemistry; Gardiner, W. C., Jr., Ed.; Springer-Verlag: New York, 1984. However, recent studies covering a wider range of temperature (2) Hucknall, D. J. Chemistry of Hydrocarbon Combustion;Chapman and and pressure, and using flash photolysis/resonance fluorescence Hall: New York, 1985. and laser pyrolysis/laser fluorescence techniques,8-10indicate that (3) Glassman, I. Combustion; Academic Press: New York, 1977. the rate constant does depend on the pressure, consistent with eq (4) Kerr, J. A,; Parsonage, M. J. Evaluated Kineric Data on Gas Phase Addition Reactions; Butterworths: England, 1972. 1 and 2. Analysis of the temperature and pressure dependence (5) Williams, A,; Smith, D. B. Chem. Rev. 1970, 70, 267. . ' ~ on yields an activation energy of 1.3 =! 0.1 k c a l / m ~ l , ~based (6) Breen, J. E.; Glass, G. P. Int. J . Chem. Kinet. 1970, 3, 145. an estimated heat of reaction of -36 f 6 kcal/mol for OH + C2H2 (7) Pastrana, A,; Carr, R. W., Jr. Int. J . Chem. Kinet. 1974, 6 , 587. C2H20H. At temperatures higher than 1000 K, the addition (8) Perry, R. A,; Atkinson, R. A,; Pitts, J. N., Jr. J . Chem. Phys. 1977, 67, 5577. channel becomes less important because of competition from (9) Michael, J. V.; Nava, D. F.; Borkowski, R. P.; Payne, W. A,; Stief, I. hydrogen abstraction (estimated activation energy 6-8 kcal/ J. J . Chem. Phys. 1980, 73, 6108. m01).~J~ (IO) Smith, G. P.; Fairchild, P. W.; Crosley, D. R. J . Chem. Phys. 1984, The OH + C2H4 system has also received considerable atten81, 2667. ( I 1 ) Smith, I. W. M.; Zellner, R. J . Chem. Soc., Faraday Trans. 2 1973, t i ~ n . " - ~In~ accord with eq 1 and 2, the rate for OH C2H4 is 69, 1617. found to be dependent on the total pressure. However, unlike (12) Davis, D. D.; Fischer, S.; Schiff, R.; Watson, R. T.; Bollinger, W. J . addition to acetylene, a small negative activation energy has been Chem. Phys. 1975, 63, 1707. 0.2,'3*14-0.7 f 0.3,15 -0.6 f 0.316 kcal/mol). found (-0.9 (13) Greiner, N. R. J . Chem. Phys. 1970, 53, 1284. Several explanations have been given for this, including the for(14) Tully, F. P. Chem. Phys. Lett. 1983, 96, 148. (15) Atkinson, R.; Perry, R. A.; Pitts, J. N., Jr. J . Chem. Phys. 1977, 66, mation of a weakly bound complex.26 Estimates of the heat of 1197. reaction for the addition process range from -29 to -32 kcal/ (16) Zellner, R.; Lorenz, K. J . Phys. Chem. 1984, 88, 984. m0I.~3'~-'*Mass spectral methods have been used to observe (17) Howard, C. J. J . Chem. Phys. 1976, 65, 4771. directly the primary adduct, C 2 H 4 0 H , and to study its decom(18) Bartels, M.; Hoyerman, K.; Sievert, R. Symp. (Inr.) Combus. [Proc.] 1982, 19, 6 1. position into CH, + CH20 and H CH3CH0.18,'9At higher (19) Morris, E. D., Jr.; Stedman, D. H.; Niki, H. J. Am. Chem. Soc. 1971, temperatures, hydrogen abstraction becomes the dominant 93. 3570. pathway (estimated activation energy ca. 3 k c a l / m ~ l ) . ' , ~ ~ ~ ~ ~ (20) Morris, E. D., Jr.; Niki, H. J . Phys. Chem. 1971, 75, 3640. In an earlier theoretical study, Melius, Binkley, and Koszy(21) Lloyd, A. C.; Darnall, K. R.; Winer, A. M.; Pitts, J. N., Jr. J . Phys. Chem. 1976, 80, 789. kowskiZ7examined the reactions of OH with C2H2,C2H4, and (22) Atkinson, R.; Darnall, K. R.; Lloyd, A. C.; Winer, A. M.; Pitts, J. H C N . Geometries were optimized a t the HF/6-31G* level and N., Jr. Adv. Photochem. 1979, 1 1 , 375. energies were calculated by using fourth order unrestricted (23) Klein, Th.; Barnes, I.; Becker, K. H.; Fink, E. H.; Zabel, F. J. Phys. Mdler-Plesset perturbation theory (UMP4/6-3 lG**). The heats Chem. 1984, 88, 5020. (24) Bradley, J. N.; Capey, W. D.; Fair, R. W.; Pritchard, D. K. Int. J . of reaction and barrier heights were estimated by applying bond Chem. Kinet. 1976, 8, 549. additivity corrections to the U M P 4 calculations (BAC-MP4). (25) Westbrook, C. K.; Dryer, F. L.; Schug, K. P. Symp. (Inr.) Combust. These corrections were determined by a least-squares fit to ca. [Proc.] 1982, 19, 153. 50 molecules with well-established heats of formation. For (26) Singleton, D. L.; Cvetonovic, R. J. J . Am. Chem. SOC.1976, 98, 6812.
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'Present address: Quantum Theory Project, University of Florida, Gainesville, Florida 32605. $Camille and Henry Dreyfus Teacher-Scholar.
0002-7863 /87/ 1509-4193$01.50/0 , I
,
Mozurkewich, M.; Benson, S. W. J . Phys. Chem. 1984, 88, 6429, 6435. (27) Melius, C. F.; Binkely, J. S.; Koszykowski, M. L. 8th International Symposium on Gas Kinetics; Nottingham: England, 1984, and unpublished results.
0 1987 American Chemical Society
4194 J . Am. Chem. SOC.,Vol. 109, No. 14, 1987
Sosa and Schlegel
and much better agreement between experimental and theoretical barrier heights.29 The present study uses these same techniques to investigate the structures and energetics of the reactants, loose clusters, transition states, and radical intermediates for the reactions O H + C,H2 C 2 H 2 0 Hand OH + C2H4 C 2 H 4 0 H .
-
-
Method
I
Ab initio molecular orbital calculations were performed with the GAUSSIAN 82 system of programs.30 The restricted H a r tree-Fock method ( R H F ) was used for closed shell systems and the unrestricted Hartree-Fock method ( U H F ) for open shell system^.^' All equilibrium geometries and transition structures were fully optimized at the Hartree-Fock level by using analytical gradient with split-valence ( 3-21G)33and split-valence plus polarization (6-31G*)34basis sets. In addition, several points were calculated along the reaction paths for the OH addition by fixing the hydroxy-reactant distance, R(C-0), and minimizing the energy with respect to all other parameters. Vibrational frequencies and zero-point energies were obtained from analytical second derivative^^^ calculated at the HF/3-21G level. Electron correlation energy was calculated by using fourth order Mdler-Plesset perturbation theory in the space of single, double, and quadruple excitations (MP4SDQ, frozen core). The effects of spin contamination were examined by annihilation of the next highest spin component in the U H F wave function. In terms of the Lowdin spin projection operator36
P,
=
s2
I1 k#sS(S
- k ( k +1) 1) - k ( k + 1)
=
(P,*olHIP,*o)
s2
Since commutes with H and P, is indempotent, the projected Hartree-Fock energy can also be written as E p r o j UHF
=
I
121.8 121.8'
1.076.
This simplification was used in our first two and on the basis of eq 5 it yields the following formulae for the projected Hartree-Fock energy and wave function.
c ('EolHl!U (+ll'k+ll*o)
IZO
( ' ~ o ~ f i ~+o )
(~ol~5+ll~o)
('Eolfil+L) (11.,I~21'Eo)
C
+
= (PoIHpo)
iZo
(*OlHl+i)
=
(+iIfjsI'Eo)
('EolPs'Eo)
+
- (s
+ l)(s + 2)
('Eopl'E0)- (s +
l)(s
+ 2)
(6)
(28) Schlegel, H. B. J . Chem. Phys. 1986, 84, 4530. (29) Sosa, C.; Schlegel, H. B. Int. J . Quantum Chem. 1986, 29, 1001. (30) Binkely, J. S.; Frisch, M. J.; DeFrees, D. J.; Raghavachari, K.; Whiteside, R. A.; Schlegel, H. B.; Fluder, E. M.; Pople, J . A. GAUSSIAN 82; Carnegie-Mellon University: Pittsburgh, 1983. (31) Pople, J. A,; Nesbet, R. K. J . Chem. Phys. 1954, 27, 571. (32) Schlegel, H. B. J . Comput. Chem. 1982, 3, 214. (33) Binkely, J. S.; Pople, J. A.; Hehre, W. J. J . Am. Chem. Soc. 1980, 102, 939. (34) Hehre, W. J.; Ditchfield, R.; Pople, J. A. J . Chem. Phys. 1972, 56, 2257. Hariharan, P. C.; Pople, J. A. Chem. Phys. Lett. 1972, 66, 217. (35) Pople, J. A.; Krishnan, R.; Schlegel, H. B.; Binkley, J. S. Int. Quantum Chem., Quantum Chem. Symp. 1979, 13, 225. (36) Lowdin, P.-0. Phys. Reu. 1955, 97, 1509.
(*olHl'Eo) +
l#O
(*,p1'E0) - (s = EUHF + AEPUHF
(5)
Because the U H F wave function satisfies Brillouin's theorem and H contains only 1 and 2 operators, the summation over +ican be restricted to all double excitations. Often, the largest contribution to the spin contamination comes from only the next highest spin, Le., s 1. Under such circumstances, the full projection operator can be approximated by the first term in eq 3, k = s + 1. The result is no Longer a projector (not idempotent) but an annihilation operator, A,+I, that removes the s + 1 spin contaminant (thepenominator is chosen to ensure intermediate normalization of A,+,'Eo). 5 2
2.633.
I
EPUHF=
('EolHP,l'Eo)
=
2.535
I
(*OI~s*O)
&+I
'I
Figure 1. Geometries of the 2B2luube clusters: HF/3-21G optimized (no superscript), HF/6-31G* optimized (asterisk), in A and deg.
(4)
( ~s'EolPs'Eo)
2.592 2.602'
I
(3)
the projected Hartree-Fock energy can be written as E p r o j HHF
I'
c +,(+,ls21'Eo)
As+l'EO= 'E,
+,
+ l)(s + 2)
+ ('EOlS2l'EO)- (s + l)(s + 2) = ' E o + IZO
(7)
5., (8)
For GI, runs over all single excitations and all ab-type double excitations. Perturbation corrections for electron correlation, 'El, 'E2, etc., consist of single, double, and higher excitations. As a first aperoximation to spin-projected UMPn energies, the spin correction 'E, must be reduced by the amount already contained in 'El, 'E2, etc. This leads to the following approximate formulae for the projected MPn energies.
Calculated Barrier Heights for OH
-+ C2H2and OH + C2H4
J . Am. Chem. SOC.,Vol. 109, No. 14, 1987 4195 20
h
-
15-
'\
1.937 \ 1.955'
-
117.0
162.9 158.1.
2E
1.055 1.060'
5-
Y
,r
0-
z
1.056 1.061.
P 6
147.8 1244* 147.3.
A -5-
X / / / /
-
/
f 100.5 \\ 100.7.
\
-2oJ
I -
1.8
1.9
2
2.1
2.2
2.3
2.4
2.6
2.5
C-0 Distance in Angstroms
1.896
Figure 4. Energy profiles along the reaction path for addition of O H C2H2computed with the 3-21G basis set.
+
20 7
"
1.071 1.073'
Figure 2. Geometries of the O H + C2H2and O H + C2H, transition states: HF/3-21G optimized (no superscript), HF/6-31G* optimized (asterisk), in A and deg.
10 2
-0
5B
5-
c z
g
0-
Y C
1.385 1.353.
109.9 m
r
-10-
,
, ,
B
MP3/6-31Go
X p M P ~ 6 1 3 1 ~ 4 MP4/S-3lG0 P_MP@-JlG'
1
