Download Quantum mechanical calculation of the rate constant for the reaction

Quantum mechanical
H+02+OH+0
calculation
of the rate constant
for the reaction
Claude Leforestie?) and William H. Miller
Department of Chemistry, University of California, and Chemical SciencesDivision, Lawrence Berkeley
Laboratory, Berkeley, California 94720
(Received 13 July 1993; accepted26 October 1993)
Rigorous three-dimensional quantum mechanical calculations have been carried out for the
cumulative reaction probability of the titled reaction for total angular momentum, J=O. Higher
values of J are included approximately, and the resulting thermal rate constant agreeswell with
experimental values for 8OO(T<3O@JK.
The recent review by Miller, Kee, and Westbrook’has
emphasizedthe central importance of the title reaction for
modeling combustion kinetics (“...the single most important reaction in combustion...“). A very thorough summary of recent theoretical and experimental work is also
given by Varandas, Brandao, and Pastrana2in their classical trajectory study of this reaction and its isotopic variants. The reaction is complicated becausethe potential energy surface (PES) has a deep well ( -2 eV)
corresponding to the stable radical HO2 [cf. Fig. 1(a)]
which leads to formation of a collision complex, and the
small number of atoms and/or disparity of their masses
seemsto invalidate the use of simple statistical models for
a quantitative description of the reaction rate.’
The purpose of the present paper is to report the results of quantum mechanical calculations for the thermal
rate constant of the reaction over the range T = 800-3000
K, using the DMBE IV potential energy surface of Pastrana et al3 Rigorous calculations-which should be essentially exact for the assumedpotential energy surfacehave been carried out for total angular momentum J=O of
the three-body system, with the effectsof higher J included
approximately (vide infra).
The theoretical approach is that of Seideman and
Miller,4 with specific further developmentsintroduced by
Manthe and Miller.’ Thus the cumulative reaction probability (CRP) N(E) is calculated directly (i.e., without
having to solve the complete state-to-statereactive scattering problem), and the thermal rate constant given in terms
of it by a Boltzmann averageover total energy E,
k(T)=[2tiQr(T)]-‘J:m
dEe-“kriV(E),
(1)
where Q, is the reactant partition function (per unit volume). Manthe et aL5 expressionfor the CRP is
ME) =TrE&E) I= ~PAE),
(24
where the reaction probability operator @ is
ii(E) =4Z;‘2&E)+ip&(E)6;‘2.
(2b)
Here 6(E) = (E+ir!-I?)
-’ is the Green’s function, where
l? is the Hamiltonian operator and c=&+Zp an absorbing
potential which enforces outgoing wave boundary conditions, t?,(gpP,being the part of the absorbingpotential in the
J. Chem. Phys. 100 (l), 1 January 1994
0021-9606/94/l
reactant (product) region. Manthe et al. showed that a
convenient way to evaluatethe trace of&E) in Eq. 2(a) is
to determine its eigenvalues f&(E)},
the eigenreaction
probabilities, whose values all lie between 0 and 1.
Any convenient basis set can be used to represent the
operators in Eq. (2); it is necessaryonly that the ( L2)
basis span the interaction region between the absorbing
potentials (and somewhat into them) and that the absorbing potentials be located outside the region where the reaction dynamics is actually determined.4The present calculation is challenging because of the deep well in the
interaction region [cf. Fig. 1(a)] which necessitatesthe use
of a large basis set. Jacobi coordinates of the reactants [cf.
Fig. 1(b)] are used to specify a set of grid points for a
discrete variable representation ( DVR)6 of the following
size:
r: 60 sine-DVR7 points in the interval (0,6);
R: 213 sine-DVR points in the interval (2,lO) contracted to 87 points by meansof the HEG method,*
using the eigenfunctions of { ?R + V,,( R ) }, where
V,dR) =min,e VW,@;
r: 32 symmetry-adapted DVR points built from the
restricted Legendre basis set {P1(cos 0);
I= 1,3,5,...,63;only odd functions are used because
the ground state of the O-O-H system is odd upon
the interchange of the two oxygen atoms
This primitive basis is then contracted via the sequential
adiabatic reduction scheme of Bacic and Lightg: at each
grid point Rp, the (r,e) Hamiltonian is diagonalized and
only the eigenfunctions q5,(r,t3;Rp) retained for which the
corresponding eigenvalueE,,(R,) ~3.6 eV (relative to the
bottom of the O-O-H well). The final basis ] (b,(R,) ) 1RP)
consists of 4338 functions. The absorbing potentials were
chosen to be quartic functions, e,(Q) =A (Q- Q0)4/
(Q,,,- Q0)4 for a=r and p. For the reactant absorbing
potential the coordinate Q is r [cf. Fig. 1(b)], with rc=4.75
ao, rmaX=6.0ao, and A=3.2 eV, and for the product absorbing potential the coordinate Q is R [cf. Fig. 1(b)], with
Ro=8.5 ao, R,,,= 10 ao, and A=2.2 eV. Extensive convergencetests were carried out, varying both the strength
and location of the absorbing potentials, to ensure that the
values obtained for N(E) are independentof them. It was
00(1)/733/3/$6.00
@ 1994 American institute of Physics
733
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Letters to the Editor
734
OH+0
0.58
H+02
0.8
0.6
Ei
2
0.4
(a)
0.0
0.6
0.8
E W)
0.9
1
FIG. 2. Eigenreaction probabilities--i.e., the eigenvalues of the matrix
P(E) of Eq. (2)-as
a function of total energy E.
N(E) = 2 (W+ l)N(E,J),
(3)
J
FIG. 1. (a) Energetics (in eV) of the H-O-O potential energy surface.
(b) Jacobi coordinates used to describe the H+O,-OH+0
reaction.
necessary,e.g., for R,, (the location of the absorbingpotential in the product O+OH valley) be as large as it is in
order to get the reaction threshold correct.
The*eigenreactionprobabilities bk} i.e., the eigenvalues of P were computed by the Lanczos procedure described in Ref. 5. Thus if there are n nonzero eigenvalues
bk} which contribute to the sum in Eq. 2(a), then -n
Lanczos iterations (plus maybe one or two) are required,
and this in turn meansthat -2n operationsof the Green’s
function onto a vector are required. The operation of the
Green’s function is carried out by LU decomposition.”
For each energy E, the LU decomposition needs to be
carried out only once (requiring - 248 s on&theCray C90),
and then every subsequentoperation of G onto a vector
requires only back substitution (requiring - 0.2 s) . Therefore, the computation of N(E) requires - (248+0.4n) s
for each energy, and n varies from two to ten for energies
E from threshold up to - 1.1 eV.
Figure 2 shows the individual eigenreactionprobabilities Cpk(E)} as a function of energy, and Fig. 3 their sum
N(E) . Resonancestructure resulting from a collision complex causedby the deep potential well appearsin both the
individual probabilities and in the CRP. The fact that the
reaction probabilities do not all rise promptly to unity with
increasing energy is also a manifestation of dynamics that
violates the basic tenet of transition state theory; i.e., all
trajectories-using the languageof classical mechanicswhich enter the interaction region do not proceedto products, but some turn around and becomenonreactive.
One should calculate the CRP separately for each
value of total angular momentum J,iV( E,J), and then sum
over J,
but for the present only calculations for J=O have been
carried out. Higher J values are taken into account by the
“J-shifting” approximation” (essentially a molecular version of the modified wave number approximation12 of
atomic physics),
N(E,J) zz 2
N(E+,O),
(44
K= -J
where I$,, is the rotational energy of the O-O-H complex.
It is then easyto show that the integral over E in Eq. ( 1),
with sum over J in Eq. (3)) simplifies to
5.0
EW
FIG. 3. The cumulative reaction probability, given by Eq. (2), as a
function of energy E for total angular momentum J=O.
J. Chem. Phys., Vol. 100, No. 1, 1 January 1994
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Letters to the Editor
10”
7
2.x.X.
10”
:Li Icy
B
%.
.
; 10”
IO’O
0.3
0.5
0.7
1000/T (OK)
0.9
1.1
FIG. 4. Arrhenius plot of the thermal rate constant, as given by Eq. (6))
over the range T= 750-3ooO K. The solid curve connecting the circles is
the experimental result (Ref. 13), the dashed curve connecting the
squares the present results with the G-G-H intermediate geometry used
for o’, and the long-short dashed curve connecting the diamonds the
present result with the transition state geometry used for &, .
dE ewElkT $ (W+ l)N(E,J)
=&ot,,(
T) I--COdE e-E’kTN(E,J=O),
(4b)
where Q& is the rotational partition function for the
O-O-H complex,
i
e-E::/kT.
(&I
K= --J
Also, the reactant partition function in Eq. ( 1) is given by
ef,,( T) = E
(W+ 1)
QA T) = QelQvibQrotQtrans 9
(5)
where Qviband Q,,, are the standard vibrational and rotational partition functions for O,, Q,r=3 is the electronic
degeneracy,and Q,, = (pkT/2n-# ) 3’2is the translational
partition function (per unit volume) for relative motion of
H and O2 (CLbeing the relevant reduced mass).
Since this reaction is not dominated by a simple transition state, it is not obvious what geometry one should use
in Eq. 4(c) to determine the rotation energy levels and
thus gm, for the J-shifting approximation. Therefore Fig. 4
shows the rate constant given by Eq. ( 1) with Eq. (4)) i.e.,
k(T)~:[2~Q,(T)l-‘ejbt(T)
m
dE e-E’kTN(E,J=O),
X
I -02
(6)
for the temperaturerange T r 800-3000 K, for two choices
of the (O-O-H)t geometry: the geometry of the O-O-H
stable intermediate [i.e., the bottom of the potential well in
Fig. 1(a)], and the geometry of the transition state (which
735
lies on the product side of the intermediate). The two results differ by an approximately constant factor of -2.5
and are seen to bracket the latest experimental results of
Du and Hessler’3 (and also the classical trajectory calculations of Ref. 2).
Further analysis of these rigorous quantum calculations will shed more light on the qualitative featuresof the
dynamics of this important reaction, and calculations for
J>O will be useful to eliminate the uncertainty due to the
J-shifting approximation. The very fact that the present
calculations could be carried out, however, is evidenceof
the progressbeing made in theoretical reaction dynamics.
Note added in proo$ We call attention to the very recent paper14which has also treated this reaction, using the
same potential energy surface. Specifically, they have carried out coupled channel scattering calculations for J=O,
producing state-to-statereaction probabilities and thus also
the cumulative reaction probability. Their results for the
latter quantity are in reasonable agreement with our
present values (Fig. 3).
W.H.M. is pleasedto acknowledgesome helpful comments of Dr. Russell T Pack. This work has been supported by the Director, Office of Energy Research,Office
of Basic Energy Sciences,Chemical SciencesDivision of
the U.S. Department of Energy under Contract No. DEAC03-76SFOOO98.
The calculations were carried out on
the Cray C-90 at the National Energy ResearchSupercomputer Center (NERSC) at the Lawrence Livermore National Laboratory. CL has been supported in part by National ScienceFoundation Grant No. CHE89-20690.
“)Permanent address: Laboratoire de Chimie ThCorique, Universid de
Paris-Sud, Bltiment 490, 91405 Orsay Cedex, France.
‘J. A. Miller, R. J. Kee, and C. K. Westbrook, Annu. Rev. Phys. Chem.
41, 345 (1990).
*A. C. Varandas, J. Brandao, and M. R. Pastrana, J. Chem. Phys. 96,
5137 (1992).
‘M. R. Pastrana, L. A. M. Quintales, J. BrandXo, and A. J. C. Varandas,
J. Phys. Chem. 94, 8073 ( 1990).
4 (a) T. Seideman and W. H. Miller, J. Chem. Phys. 96,4412 (1992); 97,
2499 (1992); (b) W. H. Miller, Act. Chem. Res. 26, 174 (1993).
‘U. Manthe and W. H. Miller, J. Chem. Phys. 99, 3411 ( 1993).
6J. V. Lill, G. A. Parker, and J. C. Light, J. Chem. Phys. 85,900 (1986).
‘C. Leforestier, J. Chem. Phys. 94, 6388 (1991).
‘D. 0. Harris, G. G. Engerholm, and W. D. Gwinn, J. Chem. Phys. 43,
1515 (1965).
9Z. Bacic and J. C. Light, J. Chem. Phys. 85, 4594 (1986); 87, 4008
(1987).
“W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling,
Numerical Recipes (Cambridge University, Cambridge, 1986).
“J. M. Bowman, J. Phys. Chem. 95, 4960 ( 1991).
I2 (a) K. Takayanagi, Prog. Theor. Phys. 8, 497 ( 1952); (b) see also, E.
E. Nikitin, Elementary Theory of Atomic and Molecular Processes in
Gases, translated by M. J. Kearsley (Oxford University, Oxford, 1974),
pp. 58-60.
13H. Du and J. P. Hessler, J. Chem. Phys. 96, 1077 (1992).
14R T Pack, E. A. Butcher, and G. A. Parker, J. Chem. Phys. 99, 9310
(l.993).
J. Chem. Phys., Vol. 100, No. 1, 1 January 1994
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