Quantum dynamics of the O(3P)+CH4→OH+CH3 reaction: An application of the rotating bond umbrella model and spectral transform subspace iteration Hua-Gen Yu and Gunnar Nyman Citation: J. Chem. Phys. 112, 238 (2000); doi: 10.1063/1.480576 View online: http://dx.doi.org/10.1063/1.480576 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v112/i1 Published by the American Institute of Physics. Additional information on J. Chem. Phys. Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors Downloaded 20 Oct 2012 to 128.143.23.241. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions
Transcript
Quantum dynamics of the O(3P)+CH4→OH+CH3 reaction: An applicationof the rotating bond umbrella model and spectral transform subspaceiterationHua-Gen Yu and Gunnar Nyman Citation: J. Chem. Phys. 112, 238 (2000); doi: 10.1063/1.480576 View online: http://dx.doi.org/10.1063/1.480576 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v112/i1 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors
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JOURNAL OF CHEMICAL PHYSICS VOLUME 112, NUMBER 1 1 JANUARY 2000
Quantum dynamics of the O „
3P…1CH4˜OH1CH3 reaction:An application of the rotating bond umbrella model and spectraltransform subspace iteration
Hua-Gen Yu and Gunnar Nymana)
Department of Chemistry, Physical Chemistry, Go¨teborg University, S-412 96 Go¨teborg, Sweden
~Received 20 August 1999; accepted 7 October 1999!
We have applied the rotating bond umbrella~RBU! model to perform time-independent quantumscattering calculations of the O(3P)1CH4→OH1CH3 reaction based on a realistic analyticpotential energy surface. The calculations are carried out in hypercylindrical coordinates with alog-derivative method incorporating a guided spectral transform~GST! subspace iterationtechnique. A single sector hyperspherical projection method is used for applying the boundaryconditions. The results show that ground-state CH4 gives CH3 that is rotationally cold. For CH4initially vibrationally excited in the C–H stretch or the H–CH3 bending mode, a bimodal CH3
In recent years, dramatic progress has been madquantum reactive scattering theory. Reactions involving fatoms1–7 have been studied using quantum dynamics thries. Among them, two prototype reactions H21OH↔H1H2O6,8–13 and H21CN↔H1HCN14,15 have been ex-tensively investigated. Some of the full dimensional calcutions enable the study of state-to-state reaction properThe results from such calculations can be used to judgevalidity of various approximate quantum methods. Howeva unified theory for a general tetra-atomic reaction is stillfrom the completeness reached for triatomic systems.16–18
For example, for triatomics, all accessible arrangement chnels are computationally treated on the same basis forand nonzero total angular momentum. On the other handaccurately calculate state-to-state differential cross sectfor a four-atom reaction is a most challenging task.
For polyatomic reactions, reduced dimensionalmodels1,2,19–21are usually used. This avenue of research wstimulated by papers by Sun and Bowman22–24 and Brooksand Clary25 10 years ago. In particular, the rotating boapproximation ~RBA! of Clary26–28 has been used frequently. The RBA includes nonlinear geometries andbeen applied to many four-atomic and polyatomic reactiosee, e.g., Refs. 2 and 7 and references therein. RecentlyRBA has been extended to treat the umbrella CH3 vibrationin the gas-phaseSN2 reactions29–32
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-s.
her,r
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s
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where the umbrella motion was described as an angularordinate. In addition, the three-dimensional collinear tymodel of Sun and Bowman has been modified to studyH1CH4→H21CH3
34 and Cl1CH4HCl1CH335 reactions.
Here, the umbrella mode was treated as the vibrational cdinate of a pseudodiatomic molecule. Recently, we hhowever improved the description of the umbrevibration.36
In other recent work, we have developed a foudimensional rotating bond umbrella~RBU! model37 imple-mented with a spectral transform subspace iteratapproach.38,39 It has been used successfully to study tCl1CH4→HCl1CH3 and H1CH4→H21CH3 reactions.37,40
A noticeable effect on the rate constants was obtainedincluding the fourth dimension. In this work, we further dvelop the RBU and apply it to the O(3P)1CH4→OH1CH3 reaction. The role of an enhanced initiseed vector for the performance of the spectral transfoLanczos method is investigated, which will be presentedSec. III.
The O(3P)1CH4 reaction is a primary step in methancombustion. The kinetics of the reaction has been extsively studied experimentally41–45 and theoretically.33,46,47
The recommended expression45 for the rate constant over thtemperature range 300–2500 K is
k~T!51.15310215T1.56exp~24270/T! cm3 mol21 s21.
Ab initio calculations47–49 show that this reaction has a necollinear O•••H•••CH3 transition state due to the conicaintersection of the two lowest electronic states. An acceable classical barrier height seems to be around 13–14 kmol. There also exist a number of studies investigatingdynamics by quasiclassical trajectory calculations47,50 andexperimentally51,52 in addition to the quantum dynamicstudy of Clary.33
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239J. Chem. Phys., Vol. 112, No. 1, 1 January 2000 Quantum dynamics of O(3P)1CH4
In Sec. II we present the RBU Hamiltonianthe guided spectral subspace iteration methand formulas for reaction probabilities and thermal rate cstants. The potential energy surface used and numeaspects are given in Sec. III, while results adiscussion are presented in Sec. IV. Conclusions are founSec. V.
II. THEORY
A. RBU Hamiltonian
A four-dimensional RBU model37 has been used in thquantum scattering calculations. In this theory, for a genatom transfer reaction,A1BCD3AB1CD3 , the RBUmodel includes four internal physical motions: theB–C(R2) and A–B (R1) reactive bond stretches, the umbrevibrational mode, and a bending mode of theBCD3 frag-ment, which becomes a rotational mode of theCD3 frag-ment.zP@2R3 ,R3# is the umbrella coordinate which is thdistance of atomC to the center of mass ofD3 , as shown inFig. 1. The dynamics study is performed in the hypercyldrical coordinates (r,w,z,u), whereu5p2ac is illustratedin Fig. 1. The RBU Hamiltonian is expressed as36,37
H~r,w,z,u!
52\2
2m F 1
r3
]
]rr3
]
]r1
1
r2
]2
]w2G2\2
2
]
]zG~z!
]
]z
1\2 j 2
2I ~z!1
\2
2mr2 sin2 w~ j 22V2!
1\2
2mr2@J~J11!2V2#1V~r,w,z,u!, ~1!
with the volume element for the internal variables
dt5r3dr dw dzsinu du,
where
I ~z!5mDR32~12cosh!1
mDmC
mC13mDR3
2~112 cosh!,
cosh53z22R3
2
2R32
,
m5FmAmB~mC13mD!
mTG1/2
,
FIG. 1. Coordinates for the rotating bond umbrella model.
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,-al
in
al
-
mz53mCmD
mC13mD,
mT5mA1mB1mC13mD ,
G~z!5R3
22z2
mz~R3213mDz2/mC!
,
and the rotational angular momentum operator is
j 2521
sinu
]
]usinu
]
]u. ~2!
Here,mA , mB , mC , andmD are the masses of theA, B, C,andD atoms, respectively.r, w, andz are the hyper-radiusthe hyperangle, and the umbrella coordinate, respectiveluis an angle describing the rotation ofCD3 in a plane definedby A ~or B!, C, andX, whereX is the point of the center omass ofD3 . J andV are the total angular momentum andprojection on theZ-axis of the body-fixed frame~along theA–B bond!. Both are treated as good quantum numbersthis model. The other symbols have their usual meaninFor more details, the reader is referred to Refs. 36 and 3
In Eq. ~1!, V(r,w,z,u) is a four-dimensional potentiaenergy surface, whose value is obtained by optimizinghelicopter rotational motion ofCD3 for a given set of(r,w,z,u,R3). The effect on the rate constants of this suddapproximation which neglects the zero-point energies frthe degrees of freedom not treated in the RBU, is appromately accounted for by an energy-shifting approximationdescribed in Sec. II C. Further, in the scattering calculatiothe pseudoinversion symmetryV(r,w,z,u)5V(r,w,2z,p2u) was not utilized but was considered when evaluatreaction probabilities and rate constants.
B. Quantum scattering and spectral transformiteration
Scattering calculations are carried out using the quadiabatic log-derivative algorithm of Manolopoulos,53 imple-mented with a guided spectral transform~GST! Krylov sub-space iteration technique.37–39 The hyper-radiusr is dividedinto L sectorsi. For all values ofJ and V, the functionsucm
JV& with initial quantum statem are expanded in thecoupled channel form
ucmJV~r,w,z,u!&5r23/2(
i 51
L
(n51
N
hnm~r; i ,J,V!
3uHn~w,z,u; i ,V!&. ~3!
Then, the coupled channel equations are obtained
d2
dr2hnm~r; i ,J,V!1 (
n850
N
~DH!nn8hn8m~r; i ,J,V!50,
~4!
where the coupling matrix has elements
~DH!nn8522m
\2^HnuHS1
\2
2mr2 FJ~J11!2V213
4G2EuHn8&. ~5!
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240 J. Chem. Phys., Vol. 112, No. 1, 1 January 2000 H.-G. Yu and G. Nyman
Here, uHn(w,z,u; i ,V)& are the eigenstates of the surfaHamiltonianHS , i.e.,
HSuHn~w,z,u; i ,V!&5E nHuHn~w,z,u; i ,V!&, ~6!
with
HS52\2
2mr i2
]2
]w22
\2
2
]
]zG~z!
]
]z1
\2
2I ~z!j 2
1\2
2mr i2 sin2 w
~ j 22V2!1VH , ~7!
and
VH~w,z,u; i !5V~r5r i ,w,z,u!. ~8!
The eigenstates defined in Eq.~6! are calculated in adirect product discrete variable representation~DVR! basisin the variablesw, z, and u. It is denoted $uabc&;a51, . . . ,Nw ,b51, . . . ,Nz , andc51, . . . ,Nu%, where$ua&%and$ub&% are potential optimized DVRs~PO-DVRs!54 for wandz, respectively.$uc&% is the primitive DVR for the angu-lar coordinate, whose abscissas are given by the eigenvaof the cosu operator matrix represented in a set of normized Legendre polynomials$Pj (cosu);j5uVu,uVu11, . . . ,Nu
1uVu21%. The eigenvectors define the unitary collocatimatrix Q which transforms the wave functions between finbasis representation~FBR! and DVR in u. In the $uabc&%basis set, the matrix representation ofHS is given by37
@HS~r i !#a8b8c8,abc
5@Tw~r i !#a8adb8bdc8c1@Tz#b8bda8adc8c1$@Tu~zb!#c8c
1@TV~r i ,wa!#c8c%da8adb8b1@V~r i !#a8b8c8,abc , ~9!
with
@Tw~r i !#a8a5^a8u2\2
2mr i2
]2
]w2ua&, ~10!
@Tz#b8b5^b8u2\2
2
]
]zG~z!
]
]z1Vref~z!ub&, ~11!
@Tu~zb!#c8c5 (j 5uVu
Nu1uVu21j ~ j 11!\2
2I ~zb!Qjc8
* Qjc , ~12!
@TV~r i ,wa!#c8c5 (j 5uVu
Nu1uVu21@ j ~ j 11!2V2#\2
2mr i2 sin2 wa
Qjc8* Qjc ,
~13!
@V~r i !#a8b8c8,abc5@VH~wa ,zb ,uc ; i !
2Vref~zb!#da8adb8bdc8c , ~14!
whereVref(z) is a reference potential in z.One important issue which we wish to emphasize is th
in these calculations, a Fourier basis set has been emplas the primitive basis set inz in order to obtain a hermitianmatrix Tz in the DVR. The hermiticity of the matrix may blost in some DVR basis sets, such as the sine-DVR. Thidue to the approximation of DVR theory. Although it is posible to explicitly construct a hermitian form,55 it was found
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es-
t,ed
is
that such a treatment results in the problem of ‘‘ghoseigenstates. Fortunately, these ghosts do not affect the clation of true eigenstates. However, since they are stableder variation of the size of the subspace in a given basiswhich indicates that their behavior is different from the sprious ghost eigenstates appearing in the conventional Lazos algorithm, it is difficult to sort them out.
In calculating the eigenstates of the surface Hamiltoniwe have used the GST Lanczos method developed byauthors.37,38 Since this method was presented in detelsewhere,37,38 only a brief review is given here. The GSalgorithm utilizes the partial reorthogonalization56 Lanczosrecurrence57 with a hermitian spectral transform functionoperatorF(HS)
Diagonalizing the tridiagonal matrix formed bya j and b j ,we can obtain the approximate eigenvalues«n and the ex-pansion coefficientsCn of the eigenvectoruHn& in the Krylovsubspace$uqk&,k51, . . . ,j % such that
uHn&5 (k51
j
Cknuqk&. ~17!
If the subspace is big enough to converge a set ofN states ofF(HS), one can then obtain the correspondingN eigenstatesof the surface HamiltonianHS , which has the same eigenvectors but different eigenvaluesE n
H . The eigenvaluesE nH of
HS are evaluated from the equation«n5F(E nH). The roots
can easily be obtained by the Newton–Raphson method58
The spectral transform operatorF(HS) in Eq. ~15! hasbeen expressed as a series of Chebyshev polynomialTl
as37,38
F~HS!5(l 50
LC
Al~a!Tl~Hnorm!, ~18!
where Hnorm5@HS2H#/DH with DH50.5@Hmax2Hmin#
and H50.5@Hmax1Hmin# is the normalized surface Hamiltonian. Hmax and Hmin are, respectively, estimates of thmaximum and minimum eigenvalues ofHS . The expansioncoefficientsAl(a) are determined by the following integra
Al~a!522d l0
p E21
11 f ~E!Tl~E!
A12E2dE
522d l0
p E0
p
exp~2@aDH~1
1cosu!#g!cos~ lu!du, ~19!
wheref (HS) is a parent filter operator.f (HS) was suggestedas37,38
f ~HS!5exp~2@a~HS2Hmin!#g!, ~20!
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241J. Chem. Phys., Vol. 112, No. 1, 1 January 2000 Quantum dynamics of O(3P)1CH4
wherea.0 andg51;2 are two parameters. It has beeshown that the value ofLC in Eq. ~18! is quite small.37 Typi-cally, it ranges from 6 to 12, depending on the spectral ra(Hmax2Hmin).
In terms of the Chebyshev recurrence,59 theF(HS)-vector multiplication in Eq.~15! can be carried ouby using the sequential action ofHS on a vector uq&5(abccabcuabc&. Each such action can be efficiently calclated as
HSuq&5 (a8b8c8
H @V~r i !#a8b8c8,a8b8c8ca8b8c8
1(a
@Tw#a8acab8c81(b
@Tz#b8bca8bc8
1(c
@Tu~zb8!#c8cca8b8c
1(c
@TV~r i ,wa8!#c8cca8b8cJ ua8b8c8&, ~21!
due to the sparseness of the matrix defined in Eq.~9!. Wenote that this matrix-vector, multiplication is highly paralleizable.
Finally, the scattering matrixS has been extracted fromthe log-derivative matrixY(r i) at a large value ofr i usingthe hyperspherical projection method.37,60 The S-matrix isgiven by
SJV~E!52k1/2W21W* k21/2, ~22!
with
W5@Y~r i !X(1)2X(3)#1 i @Y~r i !X
(2)2X(4)#, ~23!
and k is a diagonal matrix with elements (k)n8n
5A2m(E2E nH)dn8n . The matricesX( i ) have been given in
Ref. 37.
C. Reaction probabilities and rate constants
The state-to-state reaction probabilities for the reactO1CH4(n3b ,nb ,n4)→OH~n!1CH3(n2 , j ) are given by
Pn3bnbn4→nn2 jJV 5uSn3bnbn4→nn2 j
JV u2, ~24!
where the vibrational quantum numbersn,n3b ,nb ,n2 , andn4 refer to the O–H stretch, the reactive H–C stretch,H–CH3 bending, and the umbrella modes in CH3 and CH4,respectively; andj is a CH3 rotational quantum number. ThRBU cumulative reaction probability~CRP!, N(E,J,V), iscalculated by summing the state-to-state reaction probaties over all final and initial states, i.e.,
N~E,J,V!5 (n3bnbn4
(nn2 j
Pn3bnbn4→nn2 jJV . ~25!
The thermal rate constant is calculated from40,61
k~T!5Q~T!Qrot
‡ ~T!exp$2DVaG/kT%
2p\Qr~T!E
2`
1`
dE
3exp$2E/kT%N~E,J5V50!, ~26!
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e
n
e
li-
whereQ(T) is the partition function at the transition state fall vibrational modes not explicitly treated in the RBU scatering calculations,Qrot
‡ (T) is the rigid rotor symmetric topapproximation for the rotational partition function of thOCH4 complex at the transition state, andQr(T) is the reac-tant partition function per unit volume. Here, theJ- andK-shifting approximations1 have been invoked. The termexp(2DVa
G/kT) approximately corrects the rate constant fthe difference,DVa
G , between the vibrationally adiabatiground-state barrier height of the full dimensional potenfunction and the four-dimensional one used in the dynamstudies.
The ground-state tunneling coefficient (hG) and trans-mission coefficient (kG) are calculated as usual.62 The tun-neling coefficient gives the ratio of the thermal rate constout of the ground state to the thermal rate constant out ofground state obtained if reaction probabilities below thebrationally adiabatic threshold are set to zero. The transmsion coefficient gives the ratio of the quantum mechaniground-state rate constant to the one where no tunnelingcurs and there is no reflection at energies above the adiaground-state barrier.
III. POTENTIAL ENERGY SURFACE AND NUMERICALASPECTS
A realistic analytic potential energy surface developby Espinosa-Garcı´a63 was used. This potential is essentialthat of Corchadoet al.,46 but symmetric in all the potentiaenergy terms pertaining to the four methane hydrogen atoBoth surfaces are fitted to theab initio results for the elec-tronic state3A9 of OCH4.46 The new potential gives a classical barrier height of 13.26 kcal/mol with a late saddle poand an energy difference of 6.16 kcal/mol between produand reactants. The harmonic normal mode frequencies ostationary points on the surface are listed in Table I. Usthese frequencies to take the zero-point energy into acco
TABLE I. Harmonic vibrational frequencies (cm21) of reactant, product,and transition state.
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242 J. Chem. Phys., Vol. 112, No. 1, 1 January 2000 H.-G. Yu and G. Nyman
the vibrationally adiabatic ground-state (VaG) barrier height is
found to be 10.06 kcal/mol; similarly, the reaction enthal~0 K! is found to be 2.42 kcal/mol.
For the O(3P)1CH4 reaction, there is a Jahn–Telleconical intersection along ‘‘collinear’’O–H–CH3 geometry.As a result, the saddle point for the electronic state3A8 hasnearly identical structure and energy to the one of the3A9state.46,47 It is also believed that both surfaces contributemost equally to the O(3P)1CH4 reaction.46 Since we haveperformed the quantum scattering calculations only on3A9 surface, an extra factor of 2 is introduced into the eltronic partition function of the transition state in ordercompare the calculated rate constants with experimentasults.
The other partition functions defined in Eq.~26! are cal-culated as described in Refs. 37 and 64. The electronicstructure of O(3P) has been included. The spin-orbit splting energies used65 are 158.29 and 226.99 cm21 for the3P1–3P2 and 3P0–3P2 states, respectively. For the evalution of rotational partition functions, the symmetric top trasition state moments of inertia were set to 11.469 a172.207 amua0
2, while that of the spherical top reactant CH4
was set to 11.487 amua02. These values were obtained fro
the analytic surface. In addition, the zero-point energy crection has the valueDVa
G510.0629.47650.587 kcal/mol,where Va
G59.476 kcal/mol is determined from the RBmodel.
The propagation of the coupled channel equationsperformed in an adiabatic basis of 250 functions using 1sectors fromr56.2a0 to r520.0a0 , where the asymptoticanalysis was made to extract theS matrix elements. It took35 CPU hours on a single SGI/195 MHz MIPS R10000 noto compute the adiabats and carry out the first energy pcalculation. This is the most time-consuming step in tquantum scattering study. All eigenpair calculations wdone in a DVR basis contracted from a direct product baset Nw3Nz3Nu550330330545 000. The critical poten-tial energy used to truncate the DVRs is 3.25 eV. The resing compact basis varies from 7500 to 21 000 DVRs,shown in Fig. 2.
As mentioned in our previous work,38 the efficiency ofthe spectral transform Lanczos algorithm can be further
FIG. 2. Number of contracted DVRs vs hyper-radiusr.
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s2
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-
proved by using an enhanced initial seed vector. For geality and simplicity, here we have taken a slightly perturberenormalized, summation of the lowest 250 eigenstates~i.e.,the propagated ones! from the preceding sector as the initivector in the next sector. Such a vector can be easily ppared with negligible costs. As can be expected, theprovement depends on the surface Hamiltonian differen
HS(r i 11)2HS(r i). The smaller the difference, the larger thincrease in efficiency.
Figure 3 shows the comparison of two Lanczos iteratiowithout and with an enhanced seed vector for calculating250 lowest eigenstates in each sector. All other parameare kept the same between the two runs. It can be seenthe reduction in the number of Lanczos iterations is notiable, especially at large hyper-radius, where the adiachange slowly withr as shown in Fig. 4. The enhancemendefined as the ratio of the reduced number of Lanczos ittions with the preconditioned initial vector to the numberiterations with a random initial seed vector, is about 5% aeraged over all sectors. Further, it can be seen that the sizthe Krylov subspace is in this application less than;3.5N,whereN is the number of eigenstates to be calculated. Tis, the subspace is rather small, which results from the usthe spectral transform technique as addressed before.38
A better seed vector might be obtained by taki
@HS(r i 11)2HS(r i)# as a perturbation toHS(r i) or usingother preconditioning methods.66 However, from the point ofview of computational efficiency, that is not recommendhere.
FIG. 3. The size of the Krylov subspace of the guided spectral transf~GST! Lanczos iteration method as a function of hyper-radiusr for calcu-lating the lowest 250 adiabats.~A! with random initial seed vector;~B! withenhanced initial vector.
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243J. Chem. Phys., Vol. 112, No. 1, 1 January 2000 Quantum dynamics of O(3P)1CH4
IV. RESULTS AND DISCUSSION
A. Vibrational enhancement and thermal rateconstants
Figure 5 shows cumulative reaction probabilities for tO1CH4 reaction as a function of total energy. Since thisan activated reaction with a high barrier, a pronounced tneling effect has been obtained as expected.67 Further, thereaction probabilities increase slowly at low energies. Hoever, once the energy is larger than 0.47 eV, the probabgrows rapidly. This results from vibrationally excited statof CH4 which begin to make substantial contributions. Thevibrationally excited states significantly enhance the reacity, as demonstrated in Fig. 6. Vibrationally exciting anmode of CH4 treated in the RBU model reduces the reactthreshold energy. The reaction probabilities increase quicas the translational energy is raised. The RBU model d
FIG. 4. A few of the lowest adiabats for total angular momentumJ50. Inthe asymptotic limit, some reactant and product states are indicateR(n3b ,nb ,n4) andP(n,n2 , j ), respectively.
FIG. 5. Cumulative reaction probabilities for the O(3P)1CH4
→OH1CH3 reaction vs total energy, measured with respect to the grovibrational state.J50. The arrow indicates the energy ofVa
G .
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e-
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not exploit the inversion symmetry. The initial state-selecreaction probabilities are therefore doubly counted andexceed unity.
The effect on the reactivity of vibrationally exciting CH4
was recently studied in detail by Clary33 using an extendedversion of the rotating bond approximation~RBA!. He alsofound that exciting either the umbrella mode or the C4stretching mode enhances the reaction.
Calculated thermal rate constants and ground-stateneling (hG) and transmission (kG) coefficients are given inTable II. The tunneling effects are significant. Even at a htemperature,hG is much larger than unity. It can also bseen that the recrossing (hG/kG) is large for this reactionover a wide temperature range.
Figure 7 shows a comparison of the RBU thermal rconstants with experimental measurements. The agreemegood over the temperature range 300–1250 K in spite ofuse of a reduced dimensionality model and theJ-shiftapproximation.1 This also implies that the potential is reasoably accurate. The difficulty of transition state theory~TST!to describe the rate constant at low temperature is causethe light atom transfer and the large barrier height, wherequantum tunneling plays an important role. Thus, the Trate constants are very sensitive to the method used fortunneling corrections.46
by
d
FIG. 6. Reaction probabilities as a function of translational energy forO(3P)1CH4(n3b ,nb ,n4)→OH1CH3 reaction summed over all producstates forJ50. The curves are labeled as (n3b ,nb ,n4).
TABLE II. Calculated thermal rate coefficients@k(T)/cm3 mol21 s21# andground-state tunneling (hG) and transmission (kG) coefficients for theO(3P)1CH4→OH1CH3 reaction.
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244 J. Chem. Phys., Vol. 112, No. 1, 1 January 2000 H.-G. Yu and G. Nyman
The present rate constants agree better with experimthan those of Clary.33 One reason is that the potential we uhas a slightly lower barrier. Another reason may be thatthis reaction the RBA model gives an incorrect exothermity, while the RBU model gives a quite accurate exotherm
FIG. 7. Comparison of calculated thermal rate constants~solid line! withexperimental results~solid diamonds!, taken from Refs. 41–45.
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ity. This can be seen from the adiabats in Fig. 4, whichlarge values of the hyper-radius reflect the exothermicat 0 K.
B. Vibrational and rotational distributions of products
It is interesting to study the state-to-state dynamics, frwhich insight into the reaction mechanism can be gainFigure 8 displays vibrational distributions of the producsummed over the rotational states of CH3, for four initialstate-selected processes. For the O1CH4(0,0,0) ground-statereaction, the OH product is hardly excited while CH3 is mod-erately excited. This result agrees with the experimenmeasurements for a hot oxygen atom reacting wmethane51 as well as with recent quasiclassical trajecto~QCT! calculations.47 Moreover, exciting the umbrella modof CH4 leads to higher umbrella mode excitation of the CH3
product. Again, the OH product is dominantly in its grounstate.
From the above we deduce that the umbrella mode isstrongly coupled with the O–H and H–C stretching motionsince the vibrational energy of the umbrella motion of CH4 ismainly released in the umbrella mode of the CH3 product. Asimilar tendency has also been obtained for the H1CH4
→H21CH3 reaction.40 The weak coupling between the um
FIG. 8. Vibrational distributions of the products vs translational energy for the initial state-selected reaction O(3P)1CH4(n3b ,nb ,n4)→OH(n)1CH3(n2)summed over the rotational states of CH3 for J50.
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245J. Chem. Phys., Vol. 112, No. 1, 1 January 2000 Quantum dynamics of O(3P)1CH4
FIG. 9. Rotational distributions of the CH3 product vs translational energy for the state-to-state reaction O(3P)1CH4(n3b ,nb ,n4)→OH~n50!1CH3(n2
50) for J50.
n-lla
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brella mode and the H–C stretch is evident from the pafor O1CH4(1,0,0), where the H–C bond is initially vibrationally excited. In this case, the distributions of the umbremode of CH3 are similar to those for O1CH4(0,0,0).
There is coupling between both the H–C stretch moand the H–CH3 bending of CH4 and the O–H vibration.Placing one quantum of energy in the H–C stretch orH–CH3 bending of CH4 can yield a noticeable population iOH(n51) as shown in the panels for O1CH4(0,1,0) andO1CH4(1,0,0).
As the reaction is ‘‘collinearly’’O–H–CH3 dominated,the rotational distribution of the CH3 product is rather coldfor O1CH4(0,0,0), as shown in Fig. 9. This is in gooagreement with recent experimental observations.52 Since thetorque is small for near-collinearO–H–Cgeometries, it wasargued52 that the CH3 product should have a cold rotationdistribution. Although the results of Suzuki and Hirota51 in-dicate that CH3 is significantly rotationally excited, theianalysis was not based on complete rovibrational statetributions of the CH3 fragment. To resolve this issue, furthcalculations and experiments would be valuable.
It can be seen from Fig. 9 that excitation of the umbremode of CH4 has limited effect on the rotational distributionof CH3. On the other hand, a remarkable effect has bobtained for the O1CH4(0,1,0) and O1CH4(1,0,0) reac-tions. They result in a bimodal CH3 rotational distribution.
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The most probable states move up toj 55 or 6. As a result,the CH3 product is relatively excited rotationally. This cabe understood by considering the vibrational momentumthe H–CH3 bending mode of CH4. Such a bending motionproduces a torque, leading to a rotating CH3 fragment. Evenso, compared to the largest accessible quantum numj max524, the rotational quantum number of the highest snificantly populated state is still small.
The large effect on the rotation of exciting a vibrationstretch motion is surprising. The bimodal structures ofrotational distributions in Fig. 9, which were also observfor the H1CH4 reaction,40 may be a result of the dynamicatreatment used here, in particular where they result fromciting the stretch motion. In the notation of Sec. II A, thRBU model forcesA, B, and the center of mass ofCX to beon a straight line. The rotational motion of this line (J.0), may induce a rotation ofCX. Further, this motioncould couple the H–CH3 bending and stretching motionsThis may be the reason for some similarities seen in Figs8, and 9 in the effects resulting from H–CH3 stretching orbending excitation. These effects depend on the potentialergy surface, but it is interesting to note that the samenamical treatment of a bend and a stretch is also used inRBA model.
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246 J. Chem. Phys., Vol. 112, No. 1, 1 January 2000 H.-G. Yu and G. Nyman
V. CONCLUSIONS
The rotating bond umbrella~RBU! model has been employed to study the reaction dynamics of the ground-soxygen atom with methane, based on a realistic analytictential energy surface of Espinosa-Garcı´a. Time-independenquantum scattering calculations were performed in hypspherical coordinates using a log-derivative method anguided spectral transform~GST! Krylov subspace iterationmethod developed by us. The acceleration of the congence obtained by using a preconditioned seed vector foLanczos iteration has been explored.
The results show that vibrational excitations of CH4 cansubstantially enhance the reaction. The CH3 product is rota-tionally cold and moderately vibrationally excited, while thproduct OH is dominantly in the vibrational ground staThis is in good agreement with experimental results. Althe calculated thermal rate constants are in good agreewith measurements. The obtained results demonstrateusefulness of the RBU model, which treats four importadegrees of freedom for a general atom transfer reactioA1BCD3AB1CD3 . The umbrella motion of theCD3
moiety is treated with fixed bond length, but is otherwiexactly described, i.e., it is not treated as a pseudodiato
All calculations are carried out using only the lowepotential energy surface. Thus, the Stuckelberg interferebetween the ground and first electronically excited stadue to the Jahn–Teller conical intersection, is excluded,though the effect on the reaction rate has been approximaconsidered. The Stuckelberg effect may play a role indynamics but we have not explored this.
It is also confirmed that the GST Krylov subspace itetion algorithm is quite useful in high-dimensional timindependent quantum scattering calculations. Comparethe shifted and inverted spectral transform Lancziteration,68,69 this algorithm can compute all eigenpairsinterest for the scattering calculations within a single sspace iteration and without any matrix factorization. Furthit does not require explicit storage of the matrix elementsthe surface Hamiltonian. Also, only matrix-vector multipcations are needed. Such operations can be efficientlyformed using the sparseness of the Hamiltonian matrixtained in the DVR70 or in a mixed grid/basis set.71
Consequently, this method allows us to use a large basisMoreover, the GST algorithm is independent of the Hamtonian structure and is thus a universal method.
The spectral transform technique not only acceleratesconvergence of the eigenstates, but also keeps the Krsubspace to a moderate size as shown above. The latteris similar to that of the implicitly restarted Lanczos meth~IRLM ! first suggested by Sorensen.72 The crucial point ofthe IRLM is a restarting technique which enhances the svector and minimizes its content of components of eigenvtors for the next restarted Lanczos iteration by using a pjection method. For more details, the reader is referredRef. 72. Recently, Pendergastet al.73 have applied the IRLMto quantum scattering calculations. Thus, it would be intesting to compare our GST Lanczos method with the IRLin the future.
Finally, it is interesting to note that the RBU model w
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reduce to the RBA model if one considers the fragmentCD3
as the unreactive diatomic in the RBA. Thus, the GST thedeveloped before and here37,38as well as the accurate singler hyperspherical projection method37 can be adapted to thRBA with only slight modifications. This will be discussein a forthcoming paper.74
ACKNOWLEDGMENTS
We would like to thank Joaquı´n Espinosa-Garcı´a forsupplying the potential energy surface and the normal mfrequencies in Table I. The calculations were performed oSilicon Graphics Power Challenge supercomputer at Chaers University of Technology and Go¨teborg University. Thisresearch has been supported by the Swedish Natural ScResearch Council~NFR!, the Swedish Council for Planningand Coordination of Research~FRN! and the WallenbergFoundation.
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