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Quantum mechanical calculations on the decomposition of H2O2
Volume 149, number 3
CHEMICAL PHYSICS LETTERS
19 August 1988
QUANTUM MECHANICAL CALCULATIONS ON THE DECOMPOSITION OF H202 K. NISHIKAWA and S.H. LIN Department of Chemutry, Arizona Stale University, Tempe, AZ 85287-1604,
USA
Received 7 January 1988; in final form 28 June 1988
We have calculated the transition probability for dissociation in overtone-excited HzOz. The probability of direct dissociation from the vibrationally excited O-H mode to the continuum of the O-O mode is very small; but the relatively highly excited O-O mode has a large dissociation probability. IVR processes following local-mode excitation play a very important role in inducing the dissociation; consequently we investigated the roles of the other vibrational mode and possible reaction paths which deposit energy to the reaction coordinate, Applying the density matrix method, we studied the time evolution of the overtone excited state, and obtained a time-resolved spectrum.
1. Introducton The unimolecular reactions of small systems have attracted considerable experimental and theoretical attention. Recently, to study the unimolecular reaction dynamics of an isolated molecule, experiments on highly vibrationally excited hydrogen peroxide [ l-31 have been carried out to investigate the mechanism of intramolecular energy flow and the coupling which controls the energy redistribution. Crim and co-workers [3] measured the vibrational overtone spectra of both predissociative and bound states of hydrogen peroxide with LIF product analysis in a supersonic beam experiment, and found that the lifetime of the O-H overtone excited state (6~,,_H) is of the order of picoseconds by analyzing the spectral width. Zewail and co-workers [ 21 obtained the time-resolved spectrum of OH fragment buildup by Hz02 dissociation initiated by Y&H=5 excitation, and analyzed the transient with a best-fit biexponential rise. They concluded that the buildup rate of the fast component is related to the homogeneous linewidth (about 0.09 cm-‘), and the energy redistribution time from the O-H mode to the O-O mode is less than 60 ps in their experiment. Motivated by these experiments, a number of theoretical studies [ 4-6 ] have been done to investigate the intramolecular vibrational energy flow and overtone-induced dissociation in H202. Uzer et al. [ 41 examined in detail the non-linear effect and various couplings which affect intramolecular energy flow, and found the dissociation lifetimes to be of the order of picoseconds, while Sumpter and Thompson [ 5 ] treated a wide class of four-atom molecules, and discussed the relation between chaotic motion and dissociation and/or rate of energy flow. Both groups obtained considerable information about the intramolecular dynamics of hydrogen peroxide. However, their treatments were based on a classical trajectory method. In this work, we shall study the unimolecular dissociation problem quantum mechanically. For this purpose, we also choose the HZ02 molecule as our test system. We will focus on the direct dissociation from the fifth overtone (6 vOH) of the O-H stretching mode to the continuum of the O-O stretching mode and apply to socalled local-mode approximation to the O-H and O-O stretching modes, expressed as Morse oscillators. First, we derive a general expression for the dissociation transition probability [ 71. We then apply it to H202; to our knowledge calculations of this type have never been reported for H202. We find that the direct dissociation probability from the fifth overtone state to the O-O continuum is too small lo induce the dissociation into two OH radicals. However, analyzing the general expression numerically, it was found that the transition amplitude corresponding to dissociation becomes significant if the O-O stretching mode is relatively highly excited. This 0 009-2614/88/$ 03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division )
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led us to conclude that the dissociation of Hz02 takes place following IVR with lifetimes of the order of picoseconds. In order to study IVR in detail, we have explicitly calculated the required element of the perturbative G and F matrices. We then investigated the reaction path from the O-H mode to the O-O reaction mode through the other vibrational mode, taking into account energy conservation. Finally, applying the density matrix method [ 8,9] to the above IVR process, we have analyzed the time evolution of the initially prepared OH mode, and have obtained the time-resolved behavior of the dissociation.
2. Model Hamiltonian and transition probability for direct dissociation First, we will construct the model Hamiltonian of Hz02 according to the treatment by Uzer et al. [ 41. In the dissociation process from highly excited vibrational states, the electronic state does not change. We treat the vibrational motion in the local-mode picture, and in this preliminary work we assume the rotational degrees of freedom as a thermal bath. Fig. 1 gives a schematic diagram for H202 indicating the numbering of each local mode. The stretching modes are described by Morse oscillators, the bending modes by harmonic oscillators and the torsion mode by a hindered rotation following Sumpter and Thompson [ 51. Furthermore, we approximate the G and F matrices as the static ones at the equilibrium configuration. The total Hamiltonian is then given by
(1) where v,(R,)=oi{1-exp[-Ui(R;-RP)]}2,
VH(Ri)=ik;(Ri-RP)2,
VT(&)=
1 A,COS(&) n
,
(2)
and all parameters are determined according to spectroscopic data. For the stretching modes required in later calculations, we choose the dissociation energy Di and potential width parameter ai to reproduce the BirgeSponer relation for transition frequencies, and these quantities are given in table 1. Next, due to the fact that the couplings between local modes are weak, we calculate the dissociation rate by using Fermi’s golden rule. We are interested in a direct transition from the overtone excited O-H mode to the O-O continuum; so we will for simplicity consider only the two local modes (O-H and O-O stretching modes)
I
H
/
Oh
6 Fig. 1. Numbering of the Hz02 vibrational degrees of freedom used in this work. R,, R3 and Rs represent stretching modes, R2 and R, bending modes and R6 the torsional mode in the localmode picture.
244
Table 1 Parameters for the stretching vibrational modes O-H and 0-O of H,Oz in the local-mode picture. p represents the reduced mass of two modes, w and Aware the vibrational constant for the BirgeSponer relation, and these values are chosen to give good agreement between the fundamental frequency and the experimental transition frequency. Using these constants, we calculated the dissociation energy R, potential width parameter a, and number parameter d, which determines the number of discrete states of the Morse oscillator
p(g) 0 (cm-‘) Aw (cm-‘) D (eV) a (au-‘) d
O-H
o-o
1.574x 1o-24 3791.5 90.5 4.92 1.194 20.95
1.338x 1O-23 1000.0 14.0 2.21 1.369 35.71
CHEMICAL
Volume 149, number 3
19August1988
PHYSICSLETTERS
expressing the basis set as the product states 1n, m) = 1r~)~~) m)o_o. First, we take the off-diagonal G matrix (g= Gy, ) as the perturbation: Hp=gP,P,=
cos’yoP,P3, m.
where cxOis the OOH equilibrium angle and m, the oxygen mass. The physical meaning of this term is that the vibrational motion of the central oxygen atom transfers energy from one mode to the other. The transition probability from 1mt) to ) ns) may be evaluated following Rosen’s method [ 71. Here Is) indicates the O-O continuum normalized with respect to energy, and the parameter s is calculated to satisfy energy conservation E
fi2aSs
_
,~ -
2
2m
2
EEO,-~ -Ej;‘-H + EF-O _
(4)
Then we get 2a x ~g2(fia,)2~J(n,
r,(ns:ml)=~I(nsl~lrnt)l*
m)J(s, t)=0.190X1012Z(n,
m)J(s,
t) ,
(5)
where Z(n, m) = ab,b,,,
m!r(2d, -m) n!T( 2d, - n)
(6A)
and J(s
t)_lb
’ - 2 t
IRf+d*+~)IZ
t!r(2d* -t)
sinh(2ns) 8 sinh*(ns) tcos2[Ic(dz+&)]
1
(6B)
where r(z) is the gamma function, ,u2 is the reduced mass of the O-O bond, the parameter b,=2( d-n1_)and suffixes 1 and 2 represent the O-H and O-O mode, respectively. In the above expression, the factor I( n, m) is related to the discrete-discrete transition probability of the O-H mode, while J(s, t) is the transition probability from discrete I t} to continuum 1s> for the O-O mode. The second factor J(s, t) controls the dissociation. Secondly, we will consider the off-diagonal F matrix (f=F$ ) as the perturbation Hk=fR,R,=-0.083x10SR,R~.
(7)
Then we get the transition probability for dissociation as (fia,aid,
d2)
0~ m) J(s, 2) Rf(n, m) Rk
t)
x0.387x lO”Z(n, m)J(s, t) R:(n, m) R:(s, t) ,
(8)
where r(n-j+l) k=,
I=O,=O
r(2dl -m-j+/r(2d, -m)
1) r(2d, -n) r(2d, -n-j)
’
(9) In expression (8), extra factors RI (n, m) and R2(3, t) appear comparing with expression (5). This is due to 245
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CHEMICAL PHYSICS LETTERS
19 August 1988
the Taylor series expansion of the coordinate r with respect to the parameter z ( = 2dexp [ -a ( r- ro) ] ) around an equilibrium distance z=2d (i.e I=Y~). From the order of magnitude estimation of the ratio I-,/T,, we found that the two transition probabilities r, and r, are of the same order of magnitude for the same transition. Therefore, we shall use the r, expression for unimolecular dissociation to analyze the system numerically. First, we note that the experimental lifetime of the O-H overtone excited state (m = 6) is of the order of a picosecond [ 31 and its lifetime is generally given by the inverse of the dissociation rate r (i.e 7= l/r). Using the parameters for H202 in table 1, we have calculated the dissociation rate using eq. ( 5 ) and a part of the numerical results is shown in rig. 2a, where the O-O energy parameter s and the logarithm of r, I and J are plotted against the quantum number t of the O-O mode. There, we fixed the initial O-H excited state as m=6, and selected the final state with n= 5 and 3. That is, we considered the direct dissociation from I6t) to 1ns) . The O-O initial state (i.e. t) fulfils s> 0 according to energy conservation, which fixes the minimum quantum number t = 22 and 11 corresponding to’the 6-, 5 and 3 O-H transitions, respectively. From this calculation, we find that the ratio of the factors related to the O-H mode transition is I( 5,6) /I( 3,6) x 10’; the
dl’20.95 d2 - 3571
. --- 160 -+I39 Il~*l6t>-,l5s)
. . ’
. *
.s3. * * . . .
L
‘ .w-3 . .... ................ ’
~~~1~~1.57
c
.
8
‘
.
I
I
,
.
I
.,,.,.*.............I
10913:-0.22
I
.
.
.
a
.
.
.
, I . I La. , , , , , w5
‘
*
-4.
* *NJ3 . . . * . . . . . . . . . ..w...*.
I
654321 I
15
I
I
25
20
I
30
I
m
35
t Fig. 2. (a) Plot of r( ns, 6th I( n, 6), J(s, t) and s versus t in the dissociative transition ( I6t) --*1ns), n= 5 and 3). The initial O-O state given by the quantum number t commences at t=22 and 11 corresponding to the O-H transition 6 + 5 and 3, respectively, because of the energy conservation of the 160 - 1ns> transition. This figure shows that the factor I( n, 6) is larger than J(s, t), and this latter factor is a major factor determining the magnitude of the transition probability r. (b) Plot of I(n, m). IN indicates the factor I(m-N, m), I, is always larger than the other factors I,., (N# I ), and plays an important role in the dissociation.
246
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similar ratio for the O-O transition .Zs(t, s)/J,(t, s)x 105, and the ratio of n=5 1(5,6)/J(t, s) x 103-105. The ratio of the dissociation rates is r( 5s: 6t) /r( 3s: 6t) PSlo’, where the main difference comes from the factor J. The transition probability Z-‘3s: ( 6t) is too small to dissociate into two fragments. This is consistent with Uzer’s classical results indicating that there is no direct transfer from the O-H to O-O mode. However, the dissociation rates r( 5s: 6t) with t x 22-25 (corresponding to the hot molecule) become comparable in magnitude with the observed ones. This means that we need some other mechanism to generate the highly excited O-O state before dissociation. This may be caused by collisional excitation, virtual transition via the strong laser field or the concurrence of IVR and dissociation. The large difference between .ZSand J3 with the same t is due to the value of S, which contributes to J as an exponential form. Two factors I and J contribute independently to the transition probability using eq. ( 5 ). First, we examine the behavior of I shown in fig. 2b. The factor I, corresponding to the transition m+m- 1 is varying slowly, and is always larger than the other Z,. The larger the difference between initial and final states, the smaller is the transition rate. For example, the ratio of N= 1 and 6 is I( 5,6)/Z(O, 6) % 106. Therefore, the m-m- 1 transition plays an important role in the dissociation and IVR processes. Next, we examine the J factor in more detail, which plays a more important role in the dissociation process. Uzer et al. [ 41 also stressed the presence of a resonance which increases the energy transfer dramatically, and calculated the trajectories by alternating the force constant and coupling constant artificially. In order to find the situation where the rate becomes large, we have evaluated the rate-determining factor J changing the dissociation energy D and potential parameter a, which are shown in figs. 3a and 3b, respectively. In fig. 3a, we plotted J(t, s) (t=O and 5) and the number parameter d against a with D=2.0 eV and s=2.5, which should be appropriate for Hz02. There is a strong correlation between log J( t, s) and d in this range of a. The ratio of J(0, s) and J(5, s) is J(S,s)/J(O,s) E lo*-105. Thus we find that a large a value (i.e. small d)and large t are preferable for dissociation. On the other hand, fig. 3b shows that logJ( t, s) is almost proportional to the parameter d and the .Zratio is .J( 5, S) /J( 0, s) = 10 5-108 in this range of D. Thus, dissociation may easily occur d D = 2.0 eV s =2.5
25
d
0=1.2aud
sz2.5
20
20
40. I ,
15
. x
In
.
t=o :,*‘
2 _!$ - la
.
30.
,’
x
.
.
. .
;
5
: J
.
*
20.
ei .
I
.
t=o
, . I
3
. ,
30
.
2 - 10‘.-
‘t-5 10.
**
0.5
”
’
L
*
15
1 40.
.
I ”
.
’
I
”
*
.
.
.
50.
”
I
1.0
1
I
1.5
2.0
0 eV.
*.
5
. .
- 10.
’. . .
1
I
I
I
I
1.0
1.5
20
2.5
30
Q”-’
Fig. 3. (a) Plot ofb(s, 1) and dversus a with 0=2.0 eV and s=2.5. From the potential parameter dependence of the factor J(s, 1) (k0 and 5) and the number parameter d, we found a strong correlation between log J(s, t) and din this range of a, and a large value of a (i.e., small d) is preferable for the dissociation. (b) Plot of J(s, t) and d versus D with a= 1.2 au-’ and s=2.5. This figure shows that log J(s, t) is almost proportional to the number parameter d, and dissociation can occur easily in systems with small dissociation energy (i.e. small d). 247
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CHEMICAL PHYSICS LETTERS
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in systems with small dissociation energy. This situation may be realized in a weak van der Waals complex [ lo] between a diatomic molecule and an atom, for example.
3. IVR and time evolution of the system Now that the possibility for direct dissociation is shown to be very small, we have to consider the role of the other vibrational mode and find the reaction path to dissociation. In IVR processes the electronic state does not change and the system remains in the ground state. The originally unexcited O-H mode does not accept energy, and does not participate in this IVR process, according to Uzer et al. [4]. We assume the rotational degrees of freedom as a heat bath. Thus, we can remove these degrees of freedom from the basis set, and describe it as [No”, NooH,, NooH*, Noo, N,,,). Next, we find the reaction path through which the deposited energy can flow. In order to couple effectively and transfer its energy efficiently, the intermediate state should be energetically close to the initial state. Moreover, taking into account the binary interaction (off-diagonal G matrix elements), we selected the state shown in fig. 4 (the explicit notation is given in table 2 ). There are two continuum states IO, 1, 0,S, 0) and IO, 0,0,S, 1) which couple to two different discrete states IO, 2,0,26,0) and 11, 0,0,22,1>, respectively. Both continuum states have some discrete vibrational energy which transfers very rapidly into rotational energy of the resultant OH radicals. Therefore, we can regard the generation of both continua as the formation of OH radicals. Both transition probabilities to the continuum are r6= 3.06 x 10” s- ’ and r9= 3.84 x 10’ s- ‘, respectively. This difference in the time scale between two different channels causes the peculiar time behavior in the OH buildup spectrum [ 21. We next investigate the time evolution of the system after preparing the overtone excited state 16, 0, 0,0,0). In order to study the time behavior we apply the density matrix method to this system. Eliminating the heat bath (rotational degrees of freedom) and the “irrelevant space” (continuum, etc.), the master equation of the reduced density matrix for the “relevant space ” in the Markoff approximation can be expressed as [ 8,9], (A= 1)
kl
Table 2 Relevant states and energies. States 1a) and (b) are continuum states and belong to state 110) in fig. 4. The parameter s is the continuum energy parameter defined in eq. (4) E,
In>
II> 12) 13) 14) 15) Fig. 4. Two energy transfer channels in IRV. The states shown here belong to the relevant space; explicit expressions are given in table 2. Coupling strengths between two states are also shown here in lo-* cm-‘. The states 16) and 19) are coupled to continuum 1IO); their dissociation lifetimes are 0.33 ps and 0.26 ns, respectively.
248
16) 17) IS>
19) la> lb)
16,0, 0, 0, 0) 14, 2, 0, 3,O> l3,2, I,% 0) 12,3,0,9,0 > 11,3,0. 14,O) IO, 2,0,X, 0) 14, 1,0,4,1> 12, l,O, 13, 1) I 1,0,0,22, 1) IO, 1,0,&O) IO,O,O,s, 1)
(cm-‘)
0.0 15.0 -38.5 -5.5
-115.0 - 1.0 -98.1 317.9 -35.1 0.0 s=3.78 0.0 s=9.26
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CHEMICAL PHYSICS LETTERS
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where r is the damping operator due to interaction with the heat bath and R the relaxation matrix due to the “irrelevant space”. Substituting the explicit matrix elements of r and R, we obtain
where W,,, represents the decay constant from 1k) to 1n) through the heat bath, yn= Ck W,,, means the total decay rate from 1n) to the other state and yrnnis the dephasing constant. P represent the effective interaction between the relevant state through irrelevant states, and r,, the decay rate from (n) through irrelevant space. In order to obtain the time evolution of population pnn(t ), we assume that the decay rate for long and short channels is 0.5 x 10m2and 0.2 X lo-‘, and the population exchange rate 0.5 X 10T2 and 0.1 x 10m2,respectively, in r, units. Other data are given in table 2. The results are shown in figs. 5a-5d, where the time unit is r6-l = 0.33 ps. The initially prepared state 16, 0, 0, 0, O} decays exponentially with a lifetime of c 30 ps, shown in fig. 5a. The population of the dissociating states 16) and 19) is shown in lips. 5b and 5c. The effective dissociation rate due to IVR is given by r6p66(t ) + r9bpg9(t) so that in short time, its rate is mainly determined by p66, and p99 determines the long-time behavior. The integrated yield of the continuum is given by plo, 1o(t), which indicates that OH radicals begin to build up at x 50 ps and become constant at x 300 ps. After this time, the curve increases slowly due to the decay of state I9), i.e. r,p,,(t).
200
400
600
800 TIME
1000
1200
1400
200
400
600
BOO
1000
1200
1400
TIME
Fig. 5. (a) Time evolution of population 11 ), i.e., p,,(t). Time is measured in t-6’=0.33 ps. (b) Plot ofp,,(t). The dissociation rate from state 16) is given by r6p&r)_ (c) Plot ofp,,(r). The dissociation rate from state 19) is given by r9p&t). (d) Plot ofp,,, ,,,(t). This curve shows the integrated yield of continuum 1IO), and is determined by the populations pc6( t ) and pg9( t ). The buildup of OH radicals starts at = 50 ps and continues to rise until = 300 ps.
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From the above considerations we give a brief summary of our results: ( 1) The possibility of direct dissociation from I6t) to 1ns) corresponding to the direct energy flow from the O-H to O-O mode is negligible except when the molecule is in the hot state. (2) One possibility for the required dissociation is by the indirect mechanism shown in fig. 4. The first process is a fast IVR which relaxes the deposited O-H energy into the O-O mode through the OOH mode. Then, the rate-determining dissociation process occurs in the resultant highly excited O-O mode. (3) Because of the presence of two channels with different decay rates, fast and slow components appear in the continuum buildup rate. Zewail and co-workers [ 21 observed this phenomenon in their time-resolved spectra for the OH buildup. In our simple model, we can easily understand these time behaviors.
Acknowledgement
This work was supported by the NSF. We thank Dr. A. Boeglin for a number of useful discussions and for help with the computational calculations.
References [ I] T.M. Ticich, T.R. Rizzo, H.-R. Dtlbal and F.F. Grim, J. Chem. Phys. 84 ( 1986) 1508: H.-R. Diibal and F.F. Crim, J. Chem. Phys. 83 (1985) 3863; T.R. Rizzo, C.C. Hayden and F.F. Grim, J. Chem. Phys. 81 (1984) 4501. [2] N.F. Scherer and A.H. Zewail, J. Chem. Phys. 87 (1987) 97; N.F. Scherer, F.E. Doany, A.H. Zewail and J.W. Perry, J. Chem. Phys. 84 (1986) 1932. [3] L.J. Butler, T.M. Ticich, M.D. Likar andF.F. Crim, J. Chem. Phys. 85 (1986) 2331. [4] T. Uzer, J.T. Hynes and W.P. Reinhardt, J. Chem. Phys. 85 (1986) 5791; Chem. Phys. Letters 117 (1985) 600. [5] B.G. Sumpter and D.L. Thompson, J. Chem. Phys. 82 (1985) 4557; 86 (1987) 2805. [6] L. Brouwer, C.J. Cobos, J. Troe, H.-R. Dilbal and F.F. Crim, J. Chem. Phys. 86 (1987) 6171. [7] N. Rosen, J. Chem. Phys. 1 (1933) 319; C.L. Pekeris, Phys. Rev. 45 (1934) 98; P.M. Morse, Phys. Rev. 34 ( 1929) 57. [8] R.W. Zwanzig, J. Chem. Phys. 33 (1960) 1338. [9] S.H. Lin and H. Eyring, Proc. Natl. Acad. Sci. US 74 (1977) 3623; B. Fain and S.H. Lin, Surface Sci. 147 (1984) 497. [IO] T. Kokubo and Y. Fugimura, J. Chem. Phys. 85 (1986) 7106; J.A. Beswick and J. Jortner, J. Chem. Phys. 68 (1978) 2277.
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QUANTUM MECHANICAL CALCULATIONS ON THE DECOMPOSITION OF H202 K. NISHIKAWA and S.H. LIN Department of Chemutry, Arizona Stale University, Tempe, AZ 85287-1604,
USA
Received 7 January 1988; in final form 28 June 1988
We have calculated the transition probability for dissociation in overtone-excited HzOz. The probability of direct dissociation from the vibrationally excited O-H mode to the continuum of the O-O mode is very small; but the relatively highly excited O-O mode has a large dissociation probability. IVR processes following local-mode excitation play a very important role in inducing the dissociation; consequently we investigated the roles of the other vibrational mode and possible reaction paths which deposit energy to the reaction coordinate, Applying the density matrix method, we studied the time evolution of the overtone excited state, and obtained a time-resolved spectrum.
1. Introducton The unimolecular reactions of small systems have attracted considerable experimental and theoretical attention. Recently, to study the unimolecular reaction dynamics of an isolated molecule, experiments on highly vibrationally excited hydrogen peroxide [ l-31 have been carried out to investigate the mechanism of intramolecular energy flow and the coupling which controls the energy redistribution. Crim and co-workers [3] measured the vibrational overtone spectra of both predissociative and bound states of hydrogen peroxide with LIF product analysis in a supersonic beam experiment, and found that the lifetime of the O-H overtone excited state (6~,,_H) is of the order of picoseconds by analyzing the spectral width. Zewail and co-workers [ 21 obtained the time-resolved spectrum of OH fragment buildup by Hz02 dissociation initiated by Y&H=5 excitation, and analyzed the transient with a best-fit biexponential rise. They concluded that the buildup rate of the fast component is related to the homogeneous linewidth (about 0.09 cm-‘), and the energy redistribution time from the O-H mode to the O-O mode is less than 60 ps in their experiment. Motivated by these experiments, a number of theoretical studies [ 4-6 ] have been done to investigate the intramolecular vibrational energy flow and overtone-induced dissociation in H202. Uzer et al. [ 41 examined in detail the non-linear effect and various couplings which affect intramolecular energy flow, and found the dissociation lifetimes to be of the order of picoseconds, while Sumpter and Thompson [ 5 ] treated a wide class of four-atom molecules, and discussed the relation between chaotic motion and dissociation and/or rate of energy flow. Both groups obtained considerable information about the intramolecular dynamics of hydrogen peroxide. However, their treatments were based on a classical trajectory method. In this work, we shall study the unimolecular dissociation problem quantum mechanically. For this purpose, we also choose the HZ02 molecule as our test system. We will focus on the direct dissociation from the fifth overtone (6 vOH) of the O-H stretching mode to the continuum of the O-O stretching mode and apply to socalled local-mode approximation to the O-H and O-O stretching modes, expressed as Morse oscillators. First, we derive a general expression for the dissociation transition probability [ 71. We then apply it to H202; to our knowledge calculations of this type have never been reported for H202. We find that the direct dissociation probability from the fifth overtone state to the O-O continuum is too small lo induce the dissociation into two OH radicals. However, analyzing the general expression numerically, it was found that the transition amplitude corresponding to dissociation becomes significant if the O-O stretching mode is relatively highly excited. This 0 009-2614/88/$ 03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division )
243
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CHEMICAL PHYSICS LETTERS
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led us to conclude that the dissociation of Hz02 takes place following IVR with lifetimes of the order of picoseconds. In order to study IVR in detail, we have explicitly calculated the required element of the perturbative G and F matrices. We then investigated the reaction path from the O-H mode to the O-O reaction mode through the other vibrational mode, taking into account energy conservation. Finally, applying the density matrix method [ 8,9] to the above IVR process, we have analyzed the time evolution of the initially prepared OH mode, and have obtained the time-resolved behavior of the dissociation.
2. Model Hamiltonian and transition probability for direct dissociation First, we will construct the model Hamiltonian of Hz02 according to the treatment by Uzer et al. [ 41. In the dissociation process from highly excited vibrational states, the electronic state does not change. We treat the vibrational motion in the local-mode picture, and in this preliminary work we assume the rotational degrees of freedom as a thermal bath. Fig. 1 gives a schematic diagram for H202 indicating the numbering of each local mode. The stretching modes are described by Morse oscillators, the bending modes by harmonic oscillators and the torsion mode by a hindered rotation following Sumpter and Thompson [ 51. Furthermore, we approximate the G and F matrices as the static ones at the equilibrium configuration. The total Hamiltonian is then given by
(1) where v,(R,)=oi{1-exp[-Ui(R;-RP)]}2,
VH(Ri)=ik;(Ri-RP)2,
VT(&)=
1 A,COS(&) n
,
(2)
and all parameters are determined according to spectroscopic data. For the stretching modes required in later calculations, we choose the dissociation energy Di and potential width parameter ai to reproduce the BirgeSponer relation for transition frequencies, and these quantities are given in table 1. Next, due to the fact that the couplings between local modes are weak, we calculate the dissociation rate by using Fermi’s golden rule. We are interested in a direct transition from the overtone excited O-H mode to the O-O continuum; so we will for simplicity consider only the two local modes (O-H and O-O stretching modes)
I
H
/
Oh
6 Fig. 1. Numbering of the Hz02 vibrational degrees of freedom used in this work. R,, R3 and Rs represent stretching modes, R2 and R, bending modes and R6 the torsional mode in the localmode picture.
244
Table 1 Parameters for the stretching vibrational modes O-H and 0-O of H,Oz in the local-mode picture. p represents the reduced mass of two modes, w and Aware the vibrational constant for the BirgeSponer relation, and these values are chosen to give good agreement between the fundamental frequency and the experimental transition frequency. Using these constants, we calculated the dissociation energy R, potential width parameter a, and number parameter d, which determines the number of discrete states of the Morse oscillator
p(g) 0 (cm-‘) Aw (cm-‘) D (eV) a (au-‘) d
O-H
o-o
1.574x 1o-24 3791.5 90.5 4.92 1.194 20.95
1.338x 1O-23 1000.0 14.0 2.21 1.369 35.71
CHEMICAL
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PHYSICSLETTERS
expressing the basis set as the product states 1n, m) = 1r~)~~) m)o_o. First, we take the off-diagonal G matrix (g= Gy, ) as the perturbation: Hp=gP,P,=
cos’yoP,P3, m.
where cxOis the OOH equilibrium angle and m, the oxygen mass. The physical meaning of this term is that the vibrational motion of the central oxygen atom transfers energy from one mode to the other. The transition probability from 1mt) to ) ns) may be evaluated following Rosen’s method [ 71. Here Is) indicates the O-O continuum normalized with respect to energy, and the parameter s is calculated to satisfy energy conservation E
fi2aSs
_
,~ -
2
2m
2
EEO,-~ -Ej;‘-H + EF-O _
(4)
Then we get 2a x ~g2(fia,)2~J(n,
r,(ns:ml)=~I(nsl~lrnt)l*
m)J(s, t)=0.190X1012Z(n,
m)J(s,
t) ,
(5)
where Z(n, m) = ab,b,,,
m!r(2d, -m) n!T( 2d, - n)
(6A)
and J(s
t)_lb
’ - 2 t
IRf+d*+~)IZ
t!r(2d* -t)
sinh(2ns) 8 sinh*(ns) tcos2[Ic(dz+&)]
1
(6B)
where r(z) is the gamma function, ,u2 is the reduced mass of the O-O bond, the parameter b,=2( d-n1_)and suffixes 1 and 2 represent the O-H and O-O mode, respectively. In the above expression, the factor I( n, m) is related to the discrete-discrete transition probability of the O-H mode, while J(s, t) is the transition probability from discrete I t} to continuum 1s> for the O-O mode. The second factor J(s, t) controls the dissociation. Secondly, we will consider the off-diagonal F matrix (f=F$ ) as the perturbation Hk=fR,R,=-0.083x10SR,R~.
(7)
Then we get the transition probability for dissociation as (fia,aid,
d2)
0~ m) J(s, 2) Rf(n, m) Rk
t)
x0.387x lO”Z(n, m)J(s, t) R:(n, m) R:(s, t) ,
(8)
where r(n-j+l) k=,
I=O,=O
r(2dl -m-j+/r(2d, -m)
1) r(2d, -n) r(2d, -n-j)
’
(9) In expression (8), extra factors RI (n, m) and R2(3, t) appear comparing with expression (5). This is due to 245
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the Taylor series expansion of the coordinate r with respect to the parameter z ( = 2dexp [ -a ( r- ro) ] ) around an equilibrium distance z=2d (i.e I=Y~). From the order of magnitude estimation of the ratio I-,/T,, we found that the two transition probabilities r, and r, are of the same order of magnitude for the same transition. Therefore, we shall use the r, expression for unimolecular dissociation to analyze the system numerically. First, we note that the experimental lifetime of the O-H overtone excited state (m = 6) is of the order of a picosecond [ 31 and its lifetime is generally given by the inverse of the dissociation rate r (i.e 7= l/r). Using the parameters for H202 in table 1, we have calculated the dissociation rate using eq. ( 5 ) and a part of the numerical results is shown in rig. 2a, where the O-O energy parameter s and the logarithm of r, I and J are plotted against the quantum number t of the O-O mode. There, we fixed the initial O-H excited state as m=6, and selected the final state with n= 5 and 3. That is, we considered the direct dissociation from I6t) to 1ns) . The O-O initial state (i.e. t) fulfils s> 0 according to energy conservation, which fixes the minimum quantum number t = 22 and 11 corresponding to’the 6-, 5 and 3 O-H transitions, respectively. From this calculation, we find that the ratio of the factors related to the O-H mode transition is I( 5,6) /I( 3,6) x 10’; the
dl’20.95 d2 - 3571
. --- 160 -+I39 Il~*l6t>-,l5s)
. . ’
. *
.s3. * * . . .
L
‘ .w-3 . .... ................ ’
~~~1~~1.57
c
.
8
‘
.
I
I
,
.
I
.,,.,.*.............I
10913:-0.22
I
.
.
.
a
.
.
.
, I . I La. , , , , , w5
‘
*
-4.
* *NJ3 . . . * . . . . . . . . . ..w...*.
I
654321 I
15
I
I
25
20
I
30
I
m
35
t Fig. 2. (a) Plot of r( ns, 6th I( n, 6), J(s, t) and s versus t in the dissociative transition ( I6t) --*1ns), n= 5 and 3). The initial O-O state given by the quantum number t commences at t=22 and 11 corresponding to the O-H transition 6 + 5 and 3, respectively, because of the energy conservation of the 160 - 1ns> transition. This figure shows that the factor I( n, 6) is larger than J(s, t), and this latter factor is a major factor determining the magnitude of the transition probability r. (b) Plot of I(n, m). IN indicates the factor I(m-N, m), I, is always larger than the other factors I,., (N# I ), and plays an important role in the dissociation.
246
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similar ratio for the O-O transition .Zs(t, s)/J,(t, s)x 105, and the ratio of n=5 1(5,6)/J(t, s) x 103-105. The ratio of the dissociation rates is r( 5s: 6t) /r( 3s: 6t) PSlo’, where the main difference comes from the factor J. The transition probability Z-‘3s: ( 6t) is too small to dissociate into two fragments. This is consistent with Uzer’s classical results indicating that there is no direct transfer from the O-H to O-O mode. However, the dissociation rates r( 5s: 6t) with t x 22-25 (corresponding to the hot molecule) become comparable in magnitude with the observed ones. This means that we need some other mechanism to generate the highly excited O-O state before dissociation. This may be caused by collisional excitation, virtual transition via the strong laser field or the concurrence of IVR and dissociation. The large difference between .ZSand J3 with the same t is due to the value of S, which contributes to J as an exponential form. Two factors I and J contribute independently to the transition probability using eq. ( 5 ). First, we examine the behavior of I shown in fig. 2b. The factor I, corresponding to the transition m+m- 1 is varying slowly, and is always larger than the other Z,. The larger the difference between initial and final states, the smaller is the transition rate. For example, the ratio of N= 1 and 6 is I( 5,6)/Z(O, 6) % 106. Therefore, the m-m- 1 transition plays an important role in the dissociation and IVR processes. Next, we examine the J factor in more detail, which plays a more important role in the dissociation process. Uzer et al. [ 41 also stressed the presence of a resonance which increases the energy transfer dramatically, and calculated the trajectories by alternating the force constant and coupling constant artificially. In order to find the situation where the rate becomes large, we have evaluated the rate-determining factor J changing the dissociation energy D and potential parameter a, which are shown in figs. 3a and 3b, respectively. In fig. 3a, we plotted J(t, s) (t=O and 5) and the number parameter d against a with D=2.0 eV and s=2.5, which should be appropriate for Hz02. There is a strong correlation between log J( t, s) and d in this range of a. The ratio of J(0, s) and J(5, s) is J(S,s)/J(O,s) E lo*-105. Thus we find that a large a value (i.e. small d)and large t are preferable for dissociation. On the other hand, fig. 3b shows that logJ( t, s) is almost proportional to the parameter d and the .Zratio is .J( 5, S) /J( 0, s) = 10 5-108 in this range of D. Thus, dissociation may easily occur d D = 2.0 eV s =2.5
25
d
0=1.2aud
sz2.5
20
20
40. I ,
15
. x
In
.
t=o :,*‘
2 _!$ - la
.
30.
,’
x
.
.
. .
;
5
: J
.
*
20.
ei .
I
.
t=o
, . I
3
. ,
30
.
2 - 10‘.-
‘t-5 10.
**
0.5
”
’
L
*
15
1 40.
.
I ”
.
’
I
”
*
.
.
.
50.
”
I
1.0
1
I
1.5
2.0
0 eV.
*.
5
. .
- 10.
’. . .
1
I
I
I
I
1.0
1.5
20
2.5
30
Q”-’
Fig. 3. (a) Plot ofb(s, 1) and dversus a with 0=2.0 eV and s=2.5. From the potential parameter dependence of the factor J(s, 1) (k0 and 5) and the number parameter d, we found a strong correlation between log J(s, t) and din this range of a, and a large value of a (i.e., small d) is preferable for the dissociation. (b) Plot of J(s, t) and d versus D with a= 1.2 au-’ and s=2.5. This figure shows that log J(s, t) is almost proportional to the number parameter d, and dissociation can occur easily in systems with small dissociation energy (i.e. small d). 247
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in systems with small dissociation energy. This situation may be realized in a weak van der Waals complex [ lo] between a diatomic molecule and an atom, for example.
3. IVR and time evolution of the system Now that the possibility for direct dissociation is shown to be very small, we have to consider the role of the other vibrational mode and find the reaction path to dissociation. In IVR processes the electronic state does not change and the system remains in the ground state. The originally unexcited O-H mode does not accept energy, and does not participate in this IVR process, according to Uzer et al. [4]. We assume the rotational degrees of freedom as a heat bath. Thus, we can remove these degrees of freedom from the basis set, and describe it as [No”, NooH,, NooH*, Noo, N,,,). Next, we find the reaction path through which the deposited energy can flow. In order to couple effectively and transfer its energy efficiently, the intermediate state should be energetically close to the initial state. Moreover, taking into account the binary interaction (off-diagonal G matrix elements), we selected the state shown in fig. 4 (the explicit notation is given in table 2 ). There are two continuum states IO, 1, 0,S, 0) and IO, 0,0,S, 1) which couple to two different discrete states IO, 2,0,26,0) and 11, 0,0,22,1>, respectively. Both continuum states have some discrete vibrational energy which transfers very rapidly into rotational energy of the resultant OH radicals. Therefore, we can regard the generation of both continua as the formation of OH radicals. Both transition probabilities to the continuum are r6= 3.06 x 10” s- ’ and r9= 3.84 x 10’ s- ‘, respectively. This difference in the time scale between two different channels causes the peculiar time behavior in the OH buildup spectrum [ 21. We next investigate the time evolution of the system after preparing the overtone excited state 16, 0, 0,0,0). In order to study the time behavior we apply the density matrix method to this system. Eliminating the heat bath (rotational degrees of freedom) and the “irrelevant space” (continuum, etc.), the master equation of the reduced density matrix for the “relevant space ” in the Markoff approximation can be expressed as [ 8,9], (A= 1)
kl
Table 2 Relevant states and energies. States 1a) and (b) are continuum states and belong to state 110) in fig. 4. The parameter s is the continuum energy parameter defined in eq. (4) E,
In>
II> 12) 13) 14) 15) Fig. 4. Two energy transfer channels in IRV. The states shown here belong to the relevant space; explicit expressions are given in table 2. Coupling strengths between two states are also shown here in lo-* cm-‘. The states 16) and 19) are coupled to continuum 1IO); their dissociation lifetimes are 0.33 ps and 0.26 ns, respectively.
248
16) 17) IS>
19) la> lb)
16,0, 0, 0, 0) 14, 2, 0, 3,O> l3,2, I,% 0) 12,3,0,9,0 > 11,3,0. 14,O) IO, 2,0,X, 0) 14, 1,0,4,1> 12, l,O, 13, 1) I 1,0,0,22, 1) IO, 1,0,&O) IO,O,O,s, 1)
(cm-‘)
0.0 15.0 -38.5 -5.5
-115.0 - 1.0 -98.1 317.9 -35.1 0.0 s=3.78 0.0 s=9.26
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where r is the damping operator due to interaction with the heat bath and R the relaxation matrix due to the “irrelevant space”. Substituting the explicit matrix elements of r and R, we obtain
where W,,, represents the decay constant from 1k) to 1n) through the heat bath, yn= Ck W,,, means the total decay rate from 1n) to the other state and yrnnis the dephasing constant. P represent the effective interaction between the relevant state through irrelevant states, and r,, the decay rate from (n) through irrelevant space. In order to obtain the time evolution of population pnn(t ), we assume that the decay rate for long and short channels is 0.5 x 10m2and 0.2 X lo-‘, and the population exchange rate 0.5 X 10T2 and 0.1 x 10m2,respectively, in r, units. Other data are given in table 2. The results are shown in figs. 5a-5d, where the time unit is r6-l = 0.33 ps. The initially prepared state 16, 0, 0, 0, O} decays exponentially with a lifetime of c 30 ps, shown in fig. 5a. The population of the dissociating states 16) and 19) is shown in lips. 5b and 5c. The effective dissociation rate due to IVR is given by r6p66(t ) + r9bpg9(t) so that in short time, its rate is mainly determined by p66, and p99 determines the long-time behavior. The integrated yield of the continuum is given by plo, 1o(t), which indicates that OH radicals begin to build up at x 50 ps and become constant at x 300 ps. After this time, the curve increases slowly due to the decay of state I9), i.e. r,p,,(t).
200
400
600
800 TIME
1000
1200
1400
200
400
600
BOO
1000
1200
1400
TIME
Fig. 5. (a) Time evolution of population 11 ), i.e., p,,(t). Time is measured in t-6’=0.33 ps. (b) Plot ofp,,(t). The dissociation rate from state 16) is given by r6p&r)_ (c) Plot ofp,,(r). The dissociation rate from state 19) is given by r9p&t). (d) Plot ofp,,, ,,,(t). This curve shows the integrated yield of continuum 1IO), and is determined by the populations pc6( t ) and pg9( t ). The buildup of OH radicals starts at = 50 ps and continues to rise until = 300 ps.
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From the above considerations we give a brief summary of our results: ( 1) The possibility of direct dissociation from I6t) to 1ns) corresponding to the direct energy flow from the O-H to O-O mode is negligible except when the molecule is in the hot state. (2) One possibility for the required dissociation is by the indirect mechanism shown in fig. 4. The first process is a fast IVR which relaxes the deposited O-H energy into the O-O mode through the OOH mode. Then, the rate-determining dissociation process occurs in the resultant highly excited O-O mode. (3) Because of the presence of two channels with different decay rates, fast and slow components appear in the continuum buildup rate. Zewail and co-workers [ 21 observed this phenomenon in their time-resolved spectra for the OH buildup. In our simple model, we can easily understand these time behaviors.
Acknowledgement
This work was supported by the NSF. We thank Dr. A. Boeglin for a number of useful discussions and for help with the computational calculations.
References [ I] T.M. Ticich, T.R. Rizzo, H.-R. Dtlbal and F.F. Grim, J. Chem. Phys. 84 ( 1986) 1508: H.-R. Diibal and F.F. Crim, J. Chem. Phys. 83 (1985) 3863; T.R. Rizzo, C.C. Hayden and F.F. Grim, J. Chem. Phys. 81 (1984) 4501. [2] N.F. Scherer and A.H. Zewail, J. Chem. Phys. 87 (1987) 97; N.F. Scherer, F.E. Doany, A.H. Zewail and J.W. Perry, J. Chem. Phys. 84 (1986) 1932. [3] L.J. Butler, T.M. Ticich, M.D. Likar andF.F. Crim, J. Chem. Phys. 85 (1986) 2331. [4] T. Uzer, J.T. Hynes and W.P. Reinhardt, J. Chem. Phys. 85 (1986) 5791; Chem. Phys. Letters 117 (1985) 600. [5] B.G. Sumpter and D.L. Thompson, J. Chem. Phys. 82 (1985) 4557; 86 (1987) 2805. [6] L. Brouwer, C.J. Cobos, J. Troe, H.-R. Dilbal and F.F. Crim, J. Chem. Phys. 86 (1987) 6171. [7] N. Rosen, J. Chem. Phys. 1 (1933) 319; C.L. Pekeris, Phys. Rev. 45 (1934) 98; P.M. Morse, Phys. Rev. 34 ( 1929) 57. [8] R.W. Zwanzig, J. Chem. Phys. 33 (1960) 1338. [9] S.H. Lin and H. Eyring, Proc. Natl. Acad. Sci. US 74 (1977) 3623; B. Fain and S.H. Lin, Surface Sci. 147 (1984) 497. [IO] T. Kokubo and Y. Fugimura, J. Chem. Phys. 85 (1986) 7106; J.A. Beswick and J. Jortner, J. Chem. Phys. 68 (1978) 2277.
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