J . Phys. Chem. 1988, 92,4348-4351
4348
by Marinelli and Johnstom6 However, as mentioned above, the probability that such a metastable CIONOz state exists is considered to be small. In the stratosphere the products NO, and ClONO will both be rapidly photolyzed: NO,
+ hv
-
NO
- + + - + NOz
ClONO
hv
C10
+ O2
O(3P)
XZOa1620 nm X20b
C1 + NO2
NO
XZlb
(20a)
I 590 nm XZla 11670 nm (21a)
I1093 nm
(21b)
Uncertainties in the NO yield in the NO3 photolysis exist,13but in the atmosphere channel 20b, which generates 0 atoms, dominates. Provided that reaction 21a dominates over reaction 21b, the photolyses of CIONOz via pathways 2 and 4 followed by that of NO, and CION0 are equivalent in the stratosphere. If reaction 21b occurs, then the products C10 and N O are generated instead of C1 and NO,. As all these radicals react to remove odd oxygen (0 and 0,) in the stratosphere, to a first approximation the photolysis of C 1 0 N 0 2 via channel 2 or channel 4 has the same effect. Due to the low concentrations of C10N02, any interaction between the proposed metastable state of C10N02 and CIONOZ cannot be of stratospheric importance. However, N, or Ozmight also react with such a species. It appears from this work that any metastable state of CIONOz formed by absorption of 254-nm photons at 298 K does not react significantly with 20 Torr of N 2 or Oz. In the stratosphere CIONOz absorbs radiation at wavelengths which are longer than 254 nm. Production of a metastable CIONO, is presumably wavelength and/or temperature dependent. At wavelengths closer to the threshold of production, the lifetime of such a molecule might be longer. It is interesting in this respect to note that Margitan7 found to be 0.8 f 0.05 at 238 K in the 355-nm photolysis of C1ONOz using argon as buffer gas. CIONOz is generated in the stratosphere by the pressure-dependent reaction between C10 and NOz. Its maximum concentration is observed at about 30 km.zO In the MPI-Mainz twodimensional model the two principal loss mechanisms considered for CION02 in the stratosphere are photolysis and reaction of 0 atoms with C10N02.21 The reaction of O H with ClONO,, (20) Zander, R.; Rinsland, C. P.; Farmer, C. B.; Brown, L. R.; Norton, R. H.Geophys. Res. Lett. 1986, 13, 757. (21) Schmailzl, U.; Crutzen, P. J., private communication.
although its products are uncertain, can be considered as the next most significant loss process: OH C 1 0 N 0 2 products (22)
+
-
The majority of the C1ONOz in the stratsophere lies between 12 and 30 km. At 5 5 O N the model values for the photolysis rate and the first-order removal of C1ONOZby of CIONOz, JCIONO2, 6.2 X reactions 16 and 22, k I 6 [ 0 ]and kzz[OH],are 6.8 X lo”, and 1.5 X lod s-I at 30 km and 3.5 X 2.6 X and s-l at 12 km, respectively. Production of metastable 2.6 X ClONO,, which re-forms ClONO,, in the photolysis of CIONOz will reduce JCION02, which in turn will lead to a corresponding increase in the stratospheric concentration of C10N02. Higher concentrations of C1ONO2 in the stratosphere than currently predicted imply that less C1 is present as CIO,. Lower concentrations of C10, in the stratosphere should reduce the predicted O3 depletion by the tropospheric release of chlorofluorocarbon compounds. However, as NO, would also be somewhat reduced by an increase in C1ONO2, detailed computer simulations need to be performed to take into account the coupled nature of stratospheric chemistry, before predicting such effects. Although the uncertainty in the recent CIONOz concentration profile measurement is in the range 30-45%, it is interesting to note that this profile is approximately 25% higher than model predictions.z0 In conclusion, for a better understanding of the atmospheric behavior of C1ONOZ,it is necessary to investigate the photolysis of C1ONO2 molecules generated by absorption between 280 and 360 nm over temperature and pressure ranges of relevance to the upper atmosphere. Acknowledgment. The authors thank Dr. J. J. Margitan for detailed discussions of his experiments. This work was funded by the Max Planck Gesellschaft and by the Deutsche Forschungs Gemeinschaft through its Middle Atmosphere Program (MAP). Registry No. CION02, 14545-72-3; NO,, 12033-49-7; NO2, 1010244-0; ClONO, 65283-98-9; O,,10028-15-6; CI, 22537-15-1; 0, 1777880-2; N205, 10102-03-1. (22) Molina, L. T.; Molina, M. J. Geophys. Res. Lett. 1977, 4 , 83. (23) Graham, R. A.; Johnston, H. S. J. Phys. Chem. 1978,82, 254. (24) Burrows, J. P.; Tyndall, G. S.; Nowak, U.; Moortgat, G. K. In Proceedings of the Physico-Chemical Behauiour of Atmospheric Pollutants Workshop, 30-31 10 1984, Orleans, France; Commission of the European Communities; drussels, Belgium, 1985; Report XII/ENV/2/85. (25) Ravishankara, A. R.; Wine, P. H.;Smith, C. A,; Barbone, P. E.; Torabi, A. J. Geophys. Res. 1986, 91, 5361. (26) Burrows, J. P.; Tyndall, G. S.;Moortgat, G. K. Chem. Phys. Lett. 1985, 119, 193.
Control of the Yield of Combetlng Unlmolecular Reactions through Double-Resonance Coherent Trapping Alessandro Lami* and Giovanni Villani Istituto di Chimica Quantistica ed Energetica Molecolare del CNR, via Risorgimento, 35, 56100 Pisa, Italy (Received: May 28. 1987; In Final Form: February 17, 1988) It is shown that simultaneous irradiation by two laser frequencies can be used to control the relative yields of products for a certain class of unimolecular reactions. These must involve a “doorway state”, which can decay either directly or indirectly (Le., after transferring energy to other intermediate states). The infrared photopredissociation of the complex H-F-Li to give H-F + Li or H + Li-F is studied as an example, on the basis of a simple model. Introduction A certain number of processes in physics and chemistry can be schematized as in Figure 1. Starting from the ground state Ig), the quantum system under study can be excited to state 11) by absorption of a photon q.Two channels are then open: (i) 0022-3654/88/2092-4348%01.50/0
the direct decay into the continuum ( I t l ) ) (process AI); (ii) the indirect (Le., via the state 12)) decay into the continuum ( 1 ~ ~ ) ) (process A2). In this paper we show how it is possible to achieve some control on the relative yields of processes A I and A2 by using another laser 0 1988 American Chemical Society
The Journal of Physical Chemistry, Vol. 92, No. 15, 1988 4349
Yield of Competing Unimolecular Reactions A2
A1
for coherent trapping; in fact if w2 = E2 - E3 it is easy to verify that one of the three dressed states (obtained by diagonalizing the excited manifold photons w2) is a linear combination of 11) and 13) and thus cannot decay into the A2 channeL2 By tuning 0' to match the energy of this trapped state, we are then able to increase the yield of A I versus that of A2.
+
Theory and Calculations To be more precise let us consider the relevant molecules 8 photon states:
= Is) 11;O) = 1 1 )
h l )
12;O) = 12)
13;O) E 13) Figure 1. The level scheme discussed in this paper. The two lasers wI and w2 can be utilized to control the ratio of the amount of flow into the
continuum {ltl)l (process A,) to that into continuum {lez)) (process A2).
(1)
In first order for the field wl and taking h = 1 , the transition amplitude from 1s) to p) (j= 1, 2, 3) is
B7XA-B-c
where q10 is the transition amplitude from 11) to p) without the field wl (but with the field w2) and V I Eis the radiative interaction due to the field wl Vlg = ( 1IP.Eo(')lg) The amplitude qIo can be obtained by solving the time-dependent Schroedinger equation for the excited manifold (see Appendix) u.O ( t - t 3 = xCjk&('-'? (3)
V
11
Figure 2. A schematic reaction path with two minima corresponding to different isomers and two continua, corresponding to different dissociative channels. Also indicated are the four levels involved in the coherent trapping.
source a t a frequency w2 to couple the state 12) to a third bound state 13) (see Figure 1 ) . In the following we will call this "coherent trapping control" (CTC) for reasons that will become clear below. As a possible chemical application let us consider the schematic reaction path of Figure 2, where a' is a predissociation and A2 is a predissociation after an isomerization (for example, H C N H C N or HCN C N H C N H. Analogous situations in which an almost direct breaking is in competition with a rearrangement followed by a different reaction are quite common in photochemistry.' The way we suggest for controlling is based on the well-known coherent trapping for a three-level system under double-resonance irradiatione2 Before going into more detail let us explain qualitatively what happens, supposing that the wlfield is weak with respect to the internal perturbation coupling 11) and 12) and to the field w2 coupling 12) and 13). This weak wl-field assumption is not strictly necessary but has the merit of simplifying the problem in such a way that the physical meaning of each step is not obscured by algebraic and numerical manipulations. Furthermore it can be well justified for many situations, including that illustrated in Figure 2. Under such an assumption the ground state Ig) plays simply the role of a source of population for the excited manifold. We suppose here that the continua are not directly accessible by absorption so that they contribute only by adding an imaginary part to the energies of state 1 1 ) (rl)and 12) (y2) (for more details on the effective Hamiltonian formalism applied to problems of trapping in three-level systems continuum, see ref 3). By looking at Figure 1 it appears clearly that l l ) , 12), and 13) have a A-configuration and the only difference with the case discussed in ref 2 is that V12is nonradiative. This is irrelevant
-
+
- -
+
+
~
~~
See,e.g.: Burn, T. L.; Baer, T. J . Chem. Phys. 1986,85, 6361. Lami, A. Chem. Phys. 1987, 1 I S , 399. (2) Alzetta, G.; Gozzini, A.; Moi, L.; Orriols, G. Nuouo Cimenro 1976, B36, 5. Arimondo, E.; Orriols, G. Nuouo Cimento Lert. 1976, 17, 333. Whitley, R. M.; Stroud, C. R. Jr. Phys. Rev. 1976, AI4, 1498. ( 3 ) Lami, A.; Rahman, N. K. Phys. Rev. 1986, A33.782; 1986, A34.3908. (1)
k
where the complex energies X k = E k - i r k are the eigenvalues of the effective Hamiltonian ( V I ,and V2,are supposed to be real) Hcff I
El - irl V12 0 VI, E 2 - ir2 V2, (0 v 2 3 E3 + w 2
)
(4)
and the Cjkare given in the Appendix. Substituting eq 3 into eq 2 we have e-r(E,+wl)r
W t )
- e-iEkt-rki
VIET". ( E , + w l - Ek)+ irk
(5)
The yield of A I , which we call Y l ,can be defined as
Y2(t,w1) = 1 - Y'(t,U')
(7) Pc,(t,wl)is the total population of the continuum 1 at time t , which for fixed w2 is a function o f t and wl:
P c , ( ~ , ~=I2) ~ ~ ~ ' 1 ~ ~dt' , ( ~ 31 I=' 1 , 2
(8)
As one can see the transition amplitude from the ground state to the excited state V) (j = 1, 2, 3), eq 5, is the sum of a purely oscillating part and a damped part (contributing only to the transient). Suppose now that the two fields are switched on for a time t >> T k , ( T k = l / r k , k = 1, 2 , 3). The transients in eq 5 then disappear and the long-time amplitudes become
4350 The Journal of Physical Chemistry, Vol. 92, No. 15, 1988
extending our study to include bending. The preliminary results indicate that it plays a minor role since (1) the direct transformation of one quantum of the H-F stretching into many quanta of the bending ( u r 10) has a weak probability; (2) the matrix elements coupling states in which all the three modes vary their number of quanta are typically quite small. Hence in this preliminary report we considered only the two coupled stretchings. The potential energy surface given in ref 5 has been projected in two dimensions by fixing 0 = 114’. The resulting bidimensional surface function of x = r(H-F) - r,(H-F) and y = r(Li-F) - rq(Li-F) can be written as
-4.4
-4.8
s 5 x
Lami and Villani
-5.2
-5.6
:
-6.0
V k Y ) = V(X,O)
-6.4
1.0
2.8
3.0 WLI-F I A )
4.0
5.0
6.0
Figure 3. The Morse potentials utilized to represent the sections of the potential energy surface along the H-F (a) and LCF (b) distances for LHFLi = 1 1 4 O . Also indicated by circles are some points of the potential energy surface in ref 5 . As one can see, the Morse potentials used are adequate only in the zone near the minimum.
and C,, is given in Appendix. Equations 10 and 11 are the basic results of this paper; in fact the yield Yl of process A, can be computed from eq 6 and the yield of the competing process A2 can be obtained as 1 - Y,. Equation 10 shows that the amount of dissociation into a given channel grows linearly with time and is proportional to the intensity of the field wI(through VI:) and to the width of the state decaying in that channel. The factor Q,eq 11, becomes 1 when the second field is switched off, as one can easily verify. It gives the modification of the dissociation produced by the field w2.
Photodissociation of the H-F-Li Complex In Figure 2 we illustrated the general features of the proposed CTC method by using a reaction-path picture. This is certainly the most useful way of looking a t chemical reactions in polyatomics. For illustrating the present approach with a specific example we have chosen a simple triatomic system, the H-F-Li complex, in which the reaction coordinate may be approximately identified with the local vibrational modes. The ab initio calculations4 for the H-F + Li reactive collision have indicated that the H-F-Li complex is stable (-4.6 kcal/mol) with the following equilibrium structure: r,(H-F) = 0.942 %.; rq(F-Li) = 1.947 A; O,(H-F-Li) = 114O. Taking into account the zero-point energy, even the “ground state” state of the complex has an energy above the threshold for dissociation into H-F(u = 0) + Li. Thus, strictly speaking, our system has no bound states. However, the lifetime of the lowest states is quite long, since they must decay by tunneling of the Li atom through an extended potential energy barrier (see Figure 3). In the following we refer to a beam experiment where it is supposed that a significant number of H-F-Li complex is present for a sufficient time to probe our theoretical results. Suppose that we tune our first laser source to excite the u = 2 overtone for the H-F local mode in the complex. By looking at the potential energy surface obtained by fitting the ab initio data’ one sees that the excited overtone can decay (i) by tunneling to give H + Li-F and (ii) by transferring energy to the Li-F mode which dissociate to give H-F + Li. To utilize the CTC for controlling the above competition we have simply modeled the reaction dynamics: (i) the molecule has been supposed to be nonrotating; (ii) the role of the bending motion has been neglected. The assumption (i) is not so drastic if one is not interested in fine details as the angular distribution of the fragments. Assumption (ii) is more questionable since the bending should in principle play a role in the energy redistribution. We are in fact (4) Chen, M. M. L.; Schaefer 111, H. F. J . Chem. Phys. 1980, 72,4376. (5) Carter, S.; Murrel, J. N. Mol. Phys. 1980, 41, 567.
+ V(0,Y)+ V’(X,Y)
(12)
The one-dimensional sections V(x,O)and V(0,y)can be fitted quite well (Figure 3) below the dissociation threshold bv Morse potentials with the following parameters: DH-F = 0.058803 au; DLi-F = 0.02600 au;
CYH-F
= 2.339 au-I
CYL~-F=
0.83 au-’
The zero-order Hamiltonian for our two-mode molecule an be written as (w E reduced masses)
(13) and its eigenstates (products of Morse eigenfunctions) are taken as unperturbed states and indicated by lul,u2). The full Hamiltonian is
H = Ho + V’(X,Y)+ Txy
(14)
where
( Txyis the stretching-stretching kinetic coupling.)
In the following, since in the relevant states the stretching H-F is not very excited, we use a simplified version of V’(x,y),obtained by expanding it around x = 0 V’(x,y) z -V(O,O)+ k(x,y)x
(16)
where
k(x,y) has been calculated numerically from the surface in ref 5. Let us excite the second overtone of the H-F stretching (2,0), which is near-degenerate with the state (1,s) (1 quantum of stretching H-F is transformed into 8 quanta of stretching Li-F). The latter is obviously degenerate with the continuum states 10,~). The following quantities enter into the three-level effective Hamiltonian: EI E E2,o = (2,01f42,0) E2 3 E1,8
E,
E
= (1,8lHl1,8)
Eo,8 = (0,8lHl0,8)
v,, = (2,01~11,8); rl = r2,0 r2
r1,8 =
~1(1,8)HIO,t)~
(18)
The above matrix elements, which factorize into a part involving the H-F stretching and a part involving the Li-F stretching, have been calculated from the Morse eigenfunctions: E l = E,
+ 8600 cm-I; E2 = Eg + 8608 cm-’ E , = E , + 5083 cm-’ VI, = 62 cm-I;
r2= 7 cm-’
The Journal of Physical Chemistry, Vol. 92, No. 15, 1988 4351
Yield of Competing Unimolecular Reactions
1.8
8.75
c
8.58
0.25
I
-50
1
50
4
Figure 4. The stationary yield Yl (defined as the fraction of complexes excited by the first laser to u(H-F) = 2 dissociatinginto H + Li-F) as a function of the detuning AI = Eg + wI-El for w2 = E2 - E3 (see Figure 1). The four curves refer to different values of V23: (a) V2, = 0; (b) V23 = 1 cm-I; (c) V,, = 10 cm-I; (d) V2, = 25 cm-l. Notice that the maximum yield is obtained at AI = 0 and that the region where the enhancement is more effective is quite small for weak w2-field(curves b and
c).
(Eg is the ground state energy.) The last relevant quantity is the tunneling width of state 12,0), Le., rl. This has been estimated by utilizing semiclassical results on the basis of the parameters characterizing the energy barrier in the fitted surface. We obtained
rl = 3 cm-' The state 13) in Figure 1 can be identified, for the present case, with the 10,8)overtone, by choosing suitably w,: w,
1.
188
= E , - E3 = 3525 cm-'
1
I
I
I
2.
3.
4.
5.
4 Figure 5. An enlarged view of the central part of the Figure 4.
theoretical investigations on the decay of C-H overtones6indicate that this will hopefully realized in a nonnegligible number of cases. If the above basic requirement is satisfied we think that the present three-level results will be perturbed by the other degrees of freedom but not substantially modified. This will be the argument of a future, more complete, investigation.
Appendix UjlO(t)can be derived from the resolvent through the relation
where
G ( E ) = (EI- Hcff)-'
(A21
From eq 4 and A2
The results of computations are summarized in Figures 4 and 5.
Notice that we have_ studied the tunnzling yield for various values of the coupling (d is the dipole and E, the @,-field strength) V,, = (0,81&*211,8)= (OlZy(il1) Since (Oldll) (which we do not know exactly) is of the order of atomic units, the maximum field strength used for calculations is around 10" au, corresponding to a megawatt laser. Figures 4 and 5 show clearly that the tunneling yield can be made close to unity by suitably tuning the w1 laser for a frequency range depending on the strength of the field w,.
Conclusions Our numerical example shows that the proposed CTC may be very effective and the aim of this paper is mainly to signal this interesting possibility to laser chemists (a further advantage is that a moderate light intensity is required). The present CTC method cannot be applied, however, to all unimolecular reactions. A certain number of requirements must be satisfied. The basic one is that the decay of the initial state into channel 2 (see Figure l), Le., the indirect dissociation, could be analyzed in terms of a predominant sequence of Fermi resonances. This may appear to be a strong requirement, but recent
and
gl(E) = ( E - E2
+ ir,)(E - E3 - a,)
g,(E) = - Y d E - E3
- V23'
- w2)
From eq A l , utilizing the residue theorem one has (the A's are the roots of the secular equation for Hen, eq 4)
and comparing with eq 3 cjk
=
gj(xk)
n
I#k
(xi
- 1,)
Registry No. HF, 7664-39-3; Li, 7439-93-2; H, 12385-13-6; LiF,
7789-24-4. ( 6 ) Silbert 111, E. L.; Reinhardt, W. P.; Hynes, J. T. J. Chem. P h p . 1982, 81, 1115. Hutchinson, T. S.;Hynes, J. T.; Reinhardt, W. P. J . Phys. Chem. 1986, 90, 3528.
