J. Phys. Chem. 1995, 99, 162 10- 16216
16210
Quantum Photochemistry, The Competition between Electronically Nonadiabatic Reaction and Electronic-to-Vibrational,Rotational, Translational Energy Transfer in Br* Collisions with H2 Steven L. Mielke and Donald G. Truhlar" Department of Chemistv, Chemical Physics Program, and Supercomputer Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431
David W. Schwenke NASA Ames Research Center, Mail Stop 230-3, Moffett Field, California 94035-1000 Received: June 23, 1995@
Electronic-to-vibrational energy transfer and nonadiabatic chemical reaction from an electronically excited reagent are studied by accurate quantum dynamics calculations for Br* H? HBr H or Br H2, where an asterisk denotes electronic excitation, and the lack of an asterisk denotes the electronic ground state. We present details of the formalism for calculating the scattering matrix in a diabatic representation in which the coupling between potential energy surfaces does not vanish asymptotically. The inelastic results are discussed in terms of the translational energy gap, which is often assumed to be a dominant factor in electronic-tovibrational-rotational energy transfer. The quenching of Br* by H? produces principally Hz with one quantum of vibrational excitation; this process satisfies a near-resonance condition for electronic-to-vibrational energy transfer. Reaction to produce HBr is the next most likely branching pathway, followed by rotationaltranslational energy transfer. Examination of the rotational distributions of each final vibrational level shows that the rotational states nearest to resonance are not the most heavily populated; Le., the resonance condition is more important for vibration than for rotation. The reactive results are discussed in terms of the relative efficiencies of various forms of reagent energy in overcoming the endoergicity and intrinsic barrier to reaction.
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1. Introduction Electronic-to-vibrational energy transfer'.' is a very important mode of energy relaxation that has only recently become amenable to simulation by accurate quantum dynamics calculat i o n ~ . ~Converged ,~ quantum dynamics calculations have also been reported recently for electronically nonadiabatic reactions,'-' colloquially designated "photochemical reactions". In the present paper we report quantum dynamical computations of the competition of energy relaxation and nonadiabatic reaction in collisions of Br* with H2 (an asterisk denotes electronic excitation). The computations are carried out by the generalized Newton variational principle'"-" (GNVP) and by another algebraic variational method, the outgoing wave variational p r i n ~ i p l e ~ . " (OWVP), ~ ~ - ' ~ that uses mixed basis sets consisting of both half-integrated Green's functions and (? functions. Br Hz is a very interesting system because the excited state of Br has an excitation energy of 0.457 eV, which is reasonably close to the Hz vibrational excitation energy, 0.5 14 eV, and a little less than the endoergicity, 0.722 eV, of the reaction Br H2(u=j=O) H + HBr(v'=j'=O), where v and j denote vibrational and rotational quantum numbers and primes denote final values. Thus, collisions of Br* with Hr, where Br* denotes the atom in the electronically excited 2Pl,?state and Br denotes the atom in the electronic ground state ?Pi,?, should require appreciable less translational energy to react than collisions of Br; furthermore, there should be an interesting competition between electronically enhanced reaction and near-resonant, nonreactive electronic-to-vibrational energy transfer. I9 Previous accurate quantum calculations on the reverse reaction, H t HBr Br H?, showed that reactive production of
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Abstract published in Ad~'uizceACS ilh.st~cicr.c.October 15. 1995.
0022-365419512099-162 10$09.0010
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Br* (i.e., reaction onto the upper surface) is small (about 0.5%) over a wide energy range.8 This result is in qualitative agreement with the fact that spin-orbit-excited halogen atoms typically have not been observed in infrared chemiluminescence experiments on reactions of hydrogen halides or, when they were observed in the F HCI HF C1*, F HBr HF Br*, and F DBr DF Br* reactions, had a relative probability in the range of only 1 - 10% of that for reaction onto the lower surface.20 In addition, it has been found that electronic excitation of Br inhibits its reaction with HI.?' The interested reader is referred for background to early work on the formulation of the quantum mechanical scattering theory for electronically nonadiabatic reactive to our own recent book chapter on this subject.18 and to several papers'.4. I k . 3 0 - 3 7 on the relationship between electronically adiabatic and diabatic representations of wave functions. Section 2 reviews the theoretical formulation and presents computational details not presented in our most recent previous papers4.' on details of the computational methods employed in our current computer program for linear algebraic variational calculations on atom-molecule collisions. Section 3 presents the calculations and results, section 4 has discussion, and section 5 contains a summary.
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2. Theory and Computational Details 2.1. Potential Energy Surfaces. The potential energy surfaces used in the present work are described elsewhere.8,3x In brief, we consider two diabatic electronic states which are taken to be the adiabatic electronic states in the absence of spinorbit coupling. The ground state is (within a constant) equal to a fit to the results of large basis set electronic structure
0 1995 American Chemical Society
Energy Transfer in Br* Collisions with Br
+ H,
J. Phys. Chem., Vol. 99, No. 44, 1995 16211
H2
section 2.1. Each vibronic diabatic basis function corresponds to single definite arrangement a (a= 1 for Br H2, a = 2 for HBr BI', and a = 3 for HBr' Br). A diabatic basis function for all degrees of freedom in arrangement a may be written4*'*
along MEP
+
200
+
i1 t
where B is a generic basis function label, u, j , Q, J , M ,and P are quantum numbers, x denotes the electronic coordinates, $ denotes a diabatic electronic basis function, $avj is a vibrational function, ra denotes the radial vibrational coordinate in channel a, Ra denotes the radial atom-diatom separation coordinate, and @, 0, and y denote the other four barycentric body-frame nuclear coordinates in arrangement a. (Barycentric coordinates are coordinates in which the center of mass of a system is fixed at the origin.) In this paper, when we say diabatic or adiabatic we are always referring to electronically diabatic or electronically adiabatic, respectively. A pure diabatic basis is one in which the matrix elements of nuclear momentum operators vanish. In general, such a basis does not exist except for the trivial and useless case of electronic functions fixed in space.36 In the present case our two-state basis is pure diabatic because we take the invariant space model,3O discussed elsewhere,36 as our defining model. The adiabatic basis is defined, at any geometry (specified by ra, Ra, and any third independent internal coordinate), by diagonalizing the 2 x 2 diabatic potential matrix at that geometry. In general, the adiabatic states are coupled by the nuclear momentum operators, and they contain contributions from diabatic basis functions of all arrangements. The asymptotically diagonal basis is defined by diagonalizing the asymptotic Hamiltonian in the asymptotic region (i.e., large Ra) for that arrangement not only in the space of x but also in the space of ra and the third internal coordinate. In the asymptotic regions, arrangement quantum number a, the diatomic rotational angular momentum j , and the relative translational angular momentum 1 are good quantum numbers, but electronic and vibrational degrees of freedom are fully coupled. In any asymptotic region the asymptotically diagonal basis, unlike the adiabatic one, is fully uncoupled, but all bases are coupled when three-body interactions are nonzero. As mentioned above, the potential matrix in the diabatic basis is denoted V., In previous work",'* (except the communication of ref 8), we considered systems defined such that Vlz vanishes in asymptotic regions. In such cases, the diabatic, adiabatic, and asymptotically diagonal bases are identical in asymptotic regions, although they differ where V12 f 0. When V12 f 0 asymptotically, then the adiabatic and asymptotically diagonal basis are generally different from each other (and from the diabatic basis) even in asymptotic regions. It is useful to contrast the adiabatic and asymptotically diagonal bases more completely for the present problem. Consider, in particular, the asymptotic regions of arrangements 1 and 2. (The discussion for arrangement 2 also applies to arrangement 3, by symmetry.) In the a = 1 asymptotic region, Vll-V22 and Vl2 are both constants. Thus, the diabatic-toadiabatic transformation is independent of r,, and the adiabatic and asymptotically diagonal representations are the same, just as when Vl2 vanishes asymptotically. But in the a = 2 asymptotic region, the diabatic-to-adiabatic transformation depends on r,, and a further diagonalization in the vibrational space is required to decouple the states. In the asymptotic region of arrangement 2, an adiabatic basis function has the form
x,
50
0
t
t
t
1
-4
-3
-2
-1
0
1
2
3
4
Distance along MEP (bohr)
Br
+ H,
along MEP
200 -
I 1
-4
-3
-2
-1
0
1
2
3
4
Distance along MEP (bohr)
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Figure 1. Potential energy surfaces along a path from reactants to products for Br H2 H HBr: (a) diabatic potentials; (b) adiabatic potentials. 1 kcdmoi = 0.0434 eV.
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calculations with scaled external correlation, the excited state is equal to a constant plus a pairwise sum of Morse and antiMorse functions, and the spin-orbit coupling potential is assumed constant and equal to 0.215 eV. The spin-orbit splitting and vibrational energies for this surface agree well with experiment, but the endoergicity is about 0.15 eV too high. Cuts though the potential surfaces along the minimum-energy reaction path are shown in Figure 1. The diabatic potential matrix is labeled V,,, where and are diabatic electronic quantum numbers equal to 1 or 2. 2.2. Vibronic Representations. We will consider three different basis sets for the vibronic degrees of freedom. These will be called the adiabatic (a), diabatic (d), and asymptotically diagonal (b) representations. The latter representation is labeled b because of the critical role it plays in the boundary conditions; it may also be called the boundary-condition basis. The problem is defined by the two-state electronic diabatic basis corresponding to the potential surfaces summarized in
x
x'
Mielke et al.
16212 J. Phys. Chem., Vol. 99, No. 44, 1995 whereas a basis function in the asymptotically diagonal representation has the form L %'
implicitly by working in a diabatic representation. Diagonalizing the Hamiltonian in electronic and vibrational degrees of freedom at large R, yields
J
where Q is a basis index. In the present work we do not use an adiabatic basis at all, except in the asymptotic region of arrangement a = 1 where it is identical to the asymptotically diagonal basis. 2.3. Scattering Theory. Accurate quantal dynamics calculations were carried out both by the GNVP10-i4and by the outgoing wave variational p r i n ~ i p l e . ~ ~ ~We ~ ' ~use - ' *our previous analysisI6 by which the GNVP is interpreted as a special case of an OWVP calculation in which the basis functions are half-integrated Green's functions for a distortion potential, and we discuss both types of calculations in the OWVP language. In this language, the trial wave function in these methods would be written in the asymptotically diagonal representation as the sum of two terms: (i) a nonreactive coupled-channel distorted wave with incoming wave in channel no and arrangement CQ and (ii) a trial outgoing wave with nonzero components 0; in all channels YZ of all arrangements a. In closed channels the components 0; are square integrable (decaying as real exponentials at large subsystem separations) whereas in open channels they are complex exponentials. Rather than carry out the calculation in the asymptotically diagonal representation though, each channel component 0; is expanded in diabatic basis functions C$ with linear coefficients, and the calculation is actually carried out in the diabatic representation. We neglect the coupling of electronic and nuclear angular momentum. In the diabatic representation, the problem then reduces to three nuclei moving on two potential energy surfaces coupled by a scalar potential, and the goal of the present work is to obtain converged quantum dynamics for that three-body problem and to transform the resulting diabatic scattering matrix to the asymptotically diagonal representation, which is the representation in which physical observables can be calculated. Complete details of the variational formalisms for multiple electronic states and a single arrangement are given in ref 4, and the formalisms are generalized to reactive collisions in ref 18. Taken together those papers contain or give refere n c e ~ ~ ~ for - ~all~details , ~ ~required . ~ ~ for . ~the~ present calculations except for one change that we have made for computational convenience. P r e v i o ~ s l y , ~we . ' ~transformed from the diabatic basis to an asymptotically diagonal basis prior to calculating the scattering matrix S. In the present work we calculate the scattering matrix in the diabatic representation and then transform it to the asymptotically diagonal representation. This involved modifications in the treatment of the boundary conditions and the distortion potentials, as explained next, and puts a constraint on the basis, namely, that all diabatic basis functions that are coupled asymptotically (Le., all those with a given a,j , and 1 or a given a, j , and projection Q of j on a body-frame axis) must have the same radial translational basis. In the present paper we satisfy this constraint by using the same radial translational basis in all channels. In particular, the radial translational basis in every channel n consists of mg diabatic half-integrated Green's functions { g ~ d , }and ~ ~ me l distributed Gaussians . The basis functions consist of products of diabatic electronic functions, vibrational functions, rotational-orbital functions, and relative translational functions. Geometric phases4' are included
(4) where YZ denotes a diabatic channel, p denotes a channel in the asymptotically diagonal (b) representation, and Le is the transformation matrix, whose elements are the coefficients in eq 3a. The transformation mixes electronic and vibrational quantum numbers, but it is block diagonal in rotational-orbital functions. Based on (4),the extended scattering matrix in the asymptotically diagonal representation is given by se
= LeTSed L'
(5)
in terms of the extended scattering matrix Sedin the diabatic representation. As discussed elsewhere,' the open channelopen channel submatrix of the extended scattering matrix Se is the usual scattering matrix S. Similarly, we define L as the rectangular matrix consisting of the open-channel columns of Le. It will similarly be useful to define an extended version" Ke of the reactance matrix K. Thus, S and K are Nopenx Nc,pe,psn, where Nopenis the number of open channels in the asymptotically diagonal representation, S' and Ke are N x N , where N is the total number of channels, L is N x Nopen,and Le is N x N. Furthermore, we will define diabatic versions of Seand K' called Sedand Ked,which will also be N x N . We will not, however, be able to define diabatic versions of S and K. One can speak of open and closed channels only in the asymptotically diagonal basis. Thus, any quantity that has a different definition for open and closed channels and that is required in the diabatic basis must be evaluated directly in the asymptotically diagonal basis and then transformed to the diabatic basis. This will effect the computational strategy since S and K as well as their distorted wave and Bom contributions and the matrices X and A defined in ref 13 all have different definitions for open and closed channels. The OWVP for the extended scattering matrix in the diabatic representation is sed
= SeOd
+
SsBd + B d T C d - ' B d
(6)
where a superscript d denotes a quantity in the diabatic representation, Seed is the contribution to the diabatic extended scattering matrix from the coupled-electronic-state distorted wave, SeBdis the distorted-wave Born contribution of the coupling potential to the extended scattering matrix, and the two remaining matrices are defined by
and
where E is the total energy, H is the Hamiltonian, and v'ldis a diabatic distorted w a ~ e . ~ .The ~ ' complex quantities SeBd,Bd, and Cd may be related to their analogs with real, reactance matrix boundary conditions analogously to the procedures employed previously13.'7.'0 as summarized below. The OWVP for the asymptotically diagonal scattering matrix is
Energy Transfer in Br* Collisions with H2
s = SO
+ SB + LTBdTCd-'BdL
J. Phys. Chem., Vol. 99, No. 44, 1995 16213
where L is the N x Nopensubmatrix of the N x N matrix Le,
so = L T s ~ O ~ L
xdm = L~xmL
(9)
(10)
and
~ m ~= , 1 , 2 , ..., mg
for a set of half-integrated Green's functions {gf:}&,=,. Repeat this mg times, where mg is the number of half-integrated Green's .functions in each channel. Arrange the Xdmmatrices side-by-side to form
Yd = (Xd1Xd2*..XdmB) The objective is to calculate eq 9. The steps in the scattering calculation are as follows: 1. Solve for real regular solutions P(R) and real halfintegrated Green's functions gd of the distorted wave blocks with reactance-matrix boundary conditions obtained by transformation of the asymptotically diagonal ones. Recall that any given basis function in the asymptotically diagonal representation is composed entirely of diabatic basis functions from a single arrangement. The transformation matrices are obtained by diagonalizing the problem in the asymptotic region of each arrangement. Calculate the ddmmatrix defined by
where A,,, is a Kronecker delta that is one if and only if channel n' is the same distortion block as channel n; and calculate the diabatic distorted-wave extended reactance matrix KeOdfrom the asymptotic form of P(R). Then calculate KeO
= LeT KeOd Le
db" = LeTdd" Le, m = 1 , 2 , ..., mg
(13) (14)
and
So = (1
+ iKo)(l - iKo)-'
(15)
where KO is the Nopenx Nopensubmatrix of Keo. 2. Calculate the diabatic distorted-wave reactance matrix KeBdand the real Bd and Cd mat rice^'^,'^ in the diabatic representation. These calculations are performed together since they share some intermediate matrices. Fold the My'square integrable functions of each channel by using eqs 52-54 of ref 40 to produce the real folded matrices KeBdf,Bdf, and Cdf. Note: in the present paper My'= me. 3. Calculate A from Keo by eq 77 of ref 14, calculate the complex dbmfrom the real dbmby eq 78 of ref 14, and calculate Xm from dbmT by eq 79 of ref 14. Here A is the matrix relating complex and real solutions to the distortion problem,I4 and Xm is a transformation matrix used to generate complex distortedwave Green's functions from real ones.I4 Form
and
SBf = WT KeBdf w
(17)
Equation 16b implies, since L is an orthogonal matrix, that the N x Nopenmatrix W consists of the open-channel columns of LeA, and eq 17 follows from
SeBdf = AdTKeBdfAd
4. Calculate the N x N matrix Xdmby
(18)
(19)
(20)
Thus Yd is N x (Nmg). 5. Transform to complex, diabatic Bdbfin which row labels are diabatic and column labels are in the asymptotically diagonal basis: Bdbf
= (gdf
+
yd"KeBdf)W
(21)
Calculate the complex diabatic C matrix: Cdf
= Cdf - g d f y d - y d T g d " + y d f @
where the real matrix
- ydTKeBdfyd
(22)
@ is related to ddmby
6. Calculate the desired scattering matrix by rewriting (9) as
In addition to the above changes that allow the treatment of systems for which the diabatic surface coupling does not vanish in asymptotic regions, we also changed the computation of both the reactive and nonreactive matrix elements in order to make these parts of the calculation more efficient. Briefly, the change involved the order in which the B and C matrix elements are formed from the auxiliary matrices. The change involves reversing the order of the two radial integrations. When we use the OWVP, this permits the elimination of certain of the auxiliary matrices (Le., fewer are required) and simplifies several other ones. (When we use the GNVP, the change of the order of the radial integrals has no direct effect.)
3. Calculations The energies of representative asymptotic states for the assumed set of potential surfaces are given in Table 1. All dynamics calculations presented here are for total angular momentum J = 0 and total energies E in the range 0.8-1.5 eV, where the zero of energy corresponds to Br infinitely far from H2 at its classical equilibrium geometry and all atoms at rest. To convert to energies relative to classical H -I- HBr, one must subtract 0.976 eV. The values of the masses used are 147 501.4 and 1837.151 au for 8'Br and H, respectively. The calculations are decoupled into two subproblems by exploiting the permutation symmetry of the two identical H atom^,'^,^* and all results presented here are for even permutational symmetry, Le., for para-H2. The diabatic distorted waves were coupled in all vibrational and electronic quantum numbers, u and respectively, for a given rotational quantum number j in a given arrangement a (a = 2 or 3 for HBr H, and a = 1 for Br HI). This is the maximum possible decoupling consistent with the correct boundary conditions; Le., because diabatic electronic and vibrational quantum numbers are not good quantum numbers even at infinite atom-diatom separations, all channels with a given rotational-orbital arrangement label must appear in the same distortion block.
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Mielke et al.
16214 J. Phys. Chem., Vol. 99, No. 44, 1995
TABLE 2: Reaction Probabilities for Br + H~(u, j) HBr and Br* + H z ( v , j ) H + HBP
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TABLE 1: Total Energy of Selected Asymptotic States arrangement LI j E (ev) Br + H? 0 0 0.269 0 0 0 0 0 0 1 1
2
Br*
+ HI
H
HBr
A
2
0 0 1 0 0 1 1
2 4 6 8 10 12 0 10 0 4 0 10 0 0 18 0 6
Br
E(eV)
0.313 0.415 0,570 0.774 1.020 1.303 0.783 1 ,500 1.265 1.399 0.725 1.477 1.239 1.139 1.487
0 2 4
L'=O
u= 1 1,1(-2) 1.9(- I ) 2.7i-1) 2.9(-4)
8
1.6(-4)" 1.7(-3) 22-2) 3.7 ( - 2) 1.3(-2)
10
2.2( -5)
0 2 4
1.0(-3) 1.0(-2)
1.2
6
1.3
6
8 10
0
1.4
2 4 6
1.453 1.495
8
7.8(-2) 6.3(-2) 1 .O(- 1 ) 6.7(-3) 3.1(-3) 3.1(-2) 1.3(-1) l,O(-l) 2.0(-1) 7.8(-2j 2.0(-7) 6.2(-3 j 6.8(-2) 1.6i-1) 1.5(- 1 J 2.3(-1) 3.0(-1) 1.5(-3)
-H +
Br" L J = ~
z,=O
L ' =1
4.2(-4) 1.2(-3) 6.6(-4) 1.3(-6)
23-2) 3.6(-1) 6.6(- 1 ) 9.7(-2) 31-11)
1.8(-8)
3.5(-2) 5.oi- I ) 6.7(- 1 )
2.0(-3)
6.4(-4)
1.4(-3)
1(-10) 3(-11)
1.3(-3) 2.2(-4) 4(-12)
1.2(-4) 3(-12)
4.8(- I j 1.3(-3)
6.8(-4) 13(-3) 1.8(-3j 9.7(-4) 3.71-6)
1.4(-6) 1.8(-7) 1(-12)
The basis used to expand the outgoing wave consists of 10 multiarrangement body-frame functions basis with channel 12 quantum numbers, a, u, j , and y , with J = 0, where y is a 4.2(-2) 1.3(-1) 6.8(-4) 2.8(-3) 1.5 0 2 3.1(-3) 5.8(-1) 5.7(-1) 1.3-3) label specifying a particular translational basis function. We 4 2.1(-5 j 6.7(- 1 ) 5.0(-3) 1.7(-3) need not specify Q since we will only consider J = 0 in this 6 9.6(-4) 53-1 j paper, in which case R = 0. Each channel has ma translational 2.6( - 1) 53-4) 8 basis functions, in particular m: distributed G a u s ~ i a n s ~ ~ , " ' - ~ ~ 3(-13) 10 12 along the radial coordinate and m: distorted-wave multichannel half-integrated Green's functions,I2-l4 16.18.40 where All results are for a given initial state and summed over final states. Read as 1.6 x lo-'. ma = m: im:
x,
('
The half-integrated Green's functions were computed by a finite difference scheme with a 13-point approximation to the radial kinetic energy operator at the center of the grid and an eightpoint approximation at the ends of the grids. All vibrationalrotational-orbital basis functions were used on both electronic surfaces, and the functions themselves were taken from the asymptotic eigenfunctions of the lower adiabatic surface. The original calculations were camed out with the GNVP, which is equivalent to expanding the outgoing wave entirely in nonreactive coupled-channels half-integrated Green's functions. Later calculations were carried out in the OWVP, and in those calculations the basis for every channel includes both distributed Gaussians and coupled-channels half-integrated Green's functions. The basis sets and numerical parameter^^,'^.'',^^ of the calculations were varied to demonstrate convergence. Two basis and parameter sets used for this purpose are illustrated in Table S-I (in supporting information). Table 1 gives the total energies of selected asymptotic states. Table 2 shows reaction probabilities for various initial states as functions of initial intemal energy for four total energies E. All total energies are with respect to the potential energy of Br H? at infinite separation. Figure 2 shows total reaction probabilities for 19 different initial states, 13 with Br unexcited and 6 with Br*. Figure 3 shows competing transition probabilities for Br* H?(u=Oj) to yield either quenched Br H? with final vibrational quantum number u' = 0 or 1 or to react. Figure 4 shows the final intemal energy distribution of the quenched Br H? for four total energies.
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4. Discussion The saddle point on the ground-state potential energy surface occurs in the exit channel at an energy 0.046 eV above that of products, which is consistent with the small (0.03 eV)47
i; l0-",2
' 0 4 'OI.6
0.8 '1.0 '1'.!"'1'4
E internal (eV)
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Figure 2. Reaction probabilities for Br + H ? ( v j ) H + HBr and Br* + H z ( v j ) H + HBr. All results are for a given initial state and are plotted as a function of initial energy. They are summed over final state.
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experimental activation energy for the reverse reaction. The variational transition state48at 0 K has a total energy of 1.220 eV, as compared to a total energy of 1.139 eV for the ground state of the products. (As mentioned in section 3, all total energies are measured with respect to the overall zero of energy, corresponding to Br H?.) Note that the spin-orbit coupling is primarily effective along the Br* H2 approach coordinate and has a negligible effect by the time that the system reaches the transition state. As a consequence of the rapid separation of the excited-state surface from the ground-state one as Br or Br* approaches H?, the probability of an electronically nonadiabatic reaction is small. In fact, Table 2 and Figure 2 show that for a wide variety of intemal and total energies this probability is about 2 orders of
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J. Phys. Chem., Vol. 99, No. 44, 1995 16215
Energy Transfer in Br* Collisions with H2
+ H2(v= 0, j = 0) + Br + H,(v')
Br'
Br'
+ H,(v
= 0, j = O)+ Br + H, (v', 1')
or
+ H2(v = 0, j = 0) -HBr i +H
Br' 10'
" "
"0.8
'
" "
0.9
Br'
'
1
"
' I '
"
'
I '
"
' I
" "
1 ' "
1.0 1.1 1.2 1.3 1.4 energy (eV)
'
-6
1.5 -6
+ H2(v= 0, j = 2) + Br + H,(v')
"
' ' I"
' '
I "
"
1
I
" ' 1
"
'
'
I '
"
'
I
-8 0.2
0.4
0.6
-
10
energy (eV)
--
0.8
1.0
,I 1.2
h i (ev) Figure 4. Electronic-to-vibrational-rotational energy transfer probBr Hz(v'j') as functions of the abilities for Br* + H*(u=Oj=O) final internal energy in Hz for four total energies. The vertical dotted lines indicate the initial internal energy and hence indicate the final internal energy corresponding to perfect resonance.
" "
10'
Figure 3. Transition probabilities for electronic-to-vibrationalenergy transfer Br* Hz(v=Oj) Br H~(v'),summed over j', and for reaction Br* Hz(v=Oj) H HBr, summed over v' and j', as functions of total energy E. (a) j = 0, (b) j = 2.
+ +
,
e
E = l . l eV
or Br" + H2(v = 0, j = 2) --f HBr + H 10.1
I,-\*:
+ +
magnitude smaller than the probability of an electronically adiabatic reaction. The small reactivity of Br* is consistent with the experimental results mentioned in the Introduction.20*2'In all cases the reaction probability for either Br or Br* is very small for internal energies greater than about 1.2 eV, roughly independent of total energy. Figure 3 shows that the branching probabilities for Br* H2( v=Oj) into the two possible vibrational levels of quenched H2 and into the reaction products are remarkably similar for j = 0 and j = 2. The most probable outcome is quenching into H2(~'=1), the near-resonant process. The reactive process shows a threshold at about 1.2 eV, as expected from the energy of the ground state of the variational transition state (see above). Above this threshold the reaction probability for Br* H2(v=Oj) is greater than the probability for off-resonant quenching into the v' = 0 state of H2 by a roughly constant factor of about 3'12 for both j = 0 and j = 2. The dominance of quenching into the v' = 1 state is consistent with a very universal propensity in collisional energy transfer for minimizing the transfer of energy into or out of translational
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m0tion.4~ It is interesting to see whether this propensity can also explain the rotational distributions of quenched products. If the quenching of H2 in collisions of Br* with ground-state HZ were purely resonant, Le., no net change in relative translational energy, the internal energy of quenched products would have to be 0.725 eV, the same as the internal energy before the collision. The closest available quantum states are v' = 0, j' = 8 and v' = 1, j' = 0 with energies of 0.774 and 0.783 eV, respectively. Figure 4 shows that these states do not dominate the quenched molecule state distributions. Furthermore, in most cases the j' distribution for a given E and v' does not peak at the closest-to-resonance j' state. We also carried out some calculations in which we "tuned" the value of the spin-orbit coupling parameter such that selected vj states in Br*(v=O) and Br(v=l) were exactly resonant. We observed that even in this case the resonant transitions were still not the dominant transition probabilities. We conclude that dynamical factors such as internal angular momentum are more important than conservation of intemal energy in establishing the final state distribution when we look at individual quantum states of the quenched molecules, Le., when both 'v andj' are resolved. Our new ability to calculate such detailed properties of electronically nonadiabatic reactions will allow us in the future to test approximate theories of such processes and develop a better understanding of the role of resonance and other qualitative aspects of photochemical processes. 5. Summary
As discussed in this paper and in a preceding letter,8 we have generalized our quantum scattering code to treat reactive electronically nonadiabatic atom-diatomic collision processes. Such reactions are prototypes of the important, yet largely unexplored, class of photochemical reactions. In this paper we present converged results for an electronically nonadiabatic chemical reaction initiating with an excited-state atom. The
Mielke et al.
16216 J. Phys. Chem., Vol. 99, No. 44, 1995 potential energy function is defined in a diabatic representation, and the algorithm we use employs a linear variational method (the outgoing wave variational principle) to obtain the scattering matrix. In particular, the formalism has been generalized to permit the treatment of systems for which the coupling between the two adiabatic electronic states remains nonzero at infinite atom-molecule separation. This involved modifications in the treatment of the boundary conditions and distortion potentials. We have used the generalized formalism with a two-state diabatic representation to treat spin-orbit coupling in the reaction of Br with H2: Br(*P,,,)
+ H2
-
HBr f H
Accurate quantal state-to-state transition probabilities have been calculated for this system for the case of total angular momentum J = 0 over a wide range of total energies (0.8 eV 5 E 5 1.5 eV). Over the range of energies studied, the quenching of spin-orbit excited Br (Le., Br('P112)) with ground state H? produces principally Hl with one quantum of vibrational excitation. This process satisfies a near-resonance condition for electronic-to-vibrational energy transfer. Reaction to produce HBr is the next most likely quenching pathway, followed by rotational-translational energy transfer. Examination of the rotational distributions of each final vibrational manifold in the quenching process shows that at the state-to-state level of detail the states nearest to resonance are not the most heavily populated.
Acknowledgment. The authors are grateful to Thomas C. Allison for providing Figure 1 and other assistance. This work was supported in part by the National Science Foundation under Grants CHE89-22048 and CHE94-23927. Supporting Information Available: A table of basis set and numerical parameters (4 pages). Ordering information is available on any current masthead page. References and Notes (1) Nikitin, E. E. Theory of Elementan Atomic and Molecular Processes in Gases; Oxford University Press: London, 1974. ( 2 ) Yardlev. J. T. Introduction to Molecirlar Enerrv Transfer. Academic: New york, 1980. ( 3 ) (a) Schwenke. D. W.: Mielke. S. L.: Tawa. G . J.: Friedman, R. S.: Halvick. P.: Truhlar, D. G. Chem. Phys. Lett. 1993, 203, 565. (h) Mielke. S. L.: Tawa. G. J.: Truhlar. D. G.: Schwenke. D. W. J . Am. Chem. SOL.. 1993. 115. 6436. (c) Mielke, S. L.: Tawa, G. J.: Truhlar. D. G.: Schwenke, D. W . In!. J . Qirantum Chem. S j m p . 1993. 27, 621. (4)Tawa. G. J.; Mielke, S. L.: Truhlar, D. G.: Schwenke, D. W. J . Chem. Phjs. 1994, 100. 5751. (5) Gilihen. M.: Baer, M. J. P/zy.r. Chem. 1994, 98. 12822. (6) Schwenke. D. W.: Mielke, S. L.; Truhlar, D. G. 208th ACS National Meeting. Washington. DC, Aug 21-25, 1994: Book of Abstracts, paper PHYS 381. (7) Schatz. G. C.: Komotos, M. 208th ACS National Meeting. Washington. DC, Aug 21-25. 1994: Book of Abstracts. paper PHYS 188. (8j Mielke, S. L.: T a w , G. J.: Truhlar, D. G.: Schwenke. D. W. Chem Phys. Lett. 1995, 234, 57. (9) Schatz, G . C. J . Phys. Chem. 1995. 99. 7522. I,.
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