5098
J. Phys. Chem. 1983, 87,5098-5106
We will use the dipole moment and polarizability changes measured here for ATR to estimate this interaction in bacteriorhodopsin. Earlier electric field studiesg show that the dipole moment change upon excitation for the all-trans protonated Schiff base (lap1 = 1 2 f 2 D) is equal within experimental error to that for ATR. It is also reasonable to assume that the polarizabilities of the aldehyde and the protonated Schiff base are similar. For n2 = 2.2, Ah = 13 D, and Aa, = 600 A3, the calculated change in energy is 5900 cm-l. This corresponds to a shift in the absorption maximum from 440 to 595 nm, very close to that observed in bacteriorhodopsin (-570 nm). The large excited-state polarizability contributes in two ways. First, the excitedstate dipole moment more than doubles in magnitude upon going from gas phase to solution phase, while the ground-state dipole moment, which depends only on the much smaller ground-state polarizability for its change upon solvation, undergoes a much smaller increase. The resultant large dipole moment increase upon excitation produces a shift of 2900 cm-l. Second, there is a large red shift due to the change in polarizability upon excitation (3000 cm-l). Although higher order terms should be included in eq 2 1 a t the high field strengths considered here,6c these results provide a useful guide for understanding the interaction of retinal’s excited state with its environment. A number of model calculations have been performed by using a charged amino acid residue to perturb the electronic structure of the retinal chromophore.6c,6b-d
Because of the large increase in excited-state dipole moment and polarizability upon solvation, these theoretical models should be parameterized to reproduce solutionphase rather than gas-phase values of the dipole moments and polarizabilities unless the protein dielectric environment is explicitly considered.sa The electric field experiments described here are particularly important in this regard, because they provide a well-defined perturbation and environment with which to test these condensed-phase calculations. For example, in PMM a shift of 220 cm-’ in retinal’s transition energy will result from an internal field of 1 X lo6 V/cm due to Ah, while the shift due to Aa, is about 1order of magnitude smaller (17 cm-l). These shifts provide an excellent experimental parameter with which to judge the accuracy of calculations of electrostatic perturbations. Acknowledgment. We thank Dr. Lubert Stryer for loaning us the apparatus used to initiate these experiments and for providing the demethylretinoic acid derivatives (synthesized by Hoffman-La Roche). We also thank A. Myers and R. R. Birge for providing a more detailed summary of their previously published refractive index results for all-trans-retinal. This research was supported by grants from the NSF (CHE-7911306 and CHE-8116042) and the NIH (EY 02051). R.M. is a Research Career Development Awardee (EY 00219). Registry No. DPB, 538-81-8; DPH, 17329-15-6; DPO, 22828-29-1; DPD, 20576-64-1; all-trans-retinal, 116-31-4.
Photodissociation Angular Distribution of Diatomics in Intense Fields Andr6 D. Bandrauk” and Gilies Turcottet Dhpartment de chimie, Facult6 des sciences, Universit6 de Sherbrooke, Sherbrooke, Quebec, Canada J 1K 2R 1 (Received: May 12, 1983)
A formal quantum treatment of bound-to-continuum transitions is developed which takes into account the lifetime of the initial state of a diatomic molecule in interaction with a strong laser. We derive general expressions for the photodissociation angular distribution valid for weak and intense fields. Numerical calculations are presented for Arzf at field intensities >lo9 W/cm2. It is shown that, at such high intensities, anisotropies will occur in the angular distribution due to the different lifetimes of individual M sublevels of a particular initial J rotational level. At saturation, the angular distributions approach statistical behavior. Highly nonstatistical distributions are obtained at particular field strengths where photodissociation delay occurs due to the formation of laser-induced bound states.
I. Introduction Since photodissociation and photopredissociation are the simplest molecular dissociative processes, they are amenable to exact calculations. The predicted angular distributions of the resulting photofragments offers us a qualitative way of interpreting the experimental r e ~ u l t s . l - ~ Firsborder time-dependent perturbation theory is the basis for this interpretation, and its applicability is restricted to weak fields and short-time limit^.^ T o interpret molecular dissociation experiments induced by an intense laser or by continuous laser radiation, one must go to nonperturbative methods. From energy conservation considerations, one realizes that it is more useful to treat
* 1982-4
Killam (Canada Council) Research Fellow.
NSERC and ME$ Graduate Fellow. 0022-385418312087-5098$0 1.5010
the total system, molecule and photons, together, since the total energy for such a system is constant. Multiphoton processes are but a redistribution of energies between the molecule and the laser field. This viewpoint gives rise to the dressed molecule or electron field picture of multiphoton processes, transposed from atomic5B to molecular physic^.^^^ Direct photodissociation in this representation (1) R. N. Zare, Mol. Photochem., 4, l(1972). (2) S. Mukamel and J. Jortner, J . Chem. Phys., 61, 5348 (1974).
(3) C. Pernot, J. Durup, J. B. Ozenne, J. A. Beswick, P. C. Cosby, and J. T. Moseley, J. Chem. Phys., 71, 2387 (1979). (4) J. M. Yuan and T. F. George, J . Chem. Phys., 68, 3040 (1978). ( 5 ) C. Cohen-Tannoudji in “Frontiers in Laser Spectroscopy”, R. Balian and S. Haroche, Ed., North-Holland, Amsterdam, 1975. (6) C. R. Stroud, Phys. Reu. A, 3, 1044 (1971). (7) T. F. George, I. H. Zimmerman, J. M. Yuang, J. R. Laing, and P. L. Devries, Acc. Chem. Res., 10, 449 (1977), and references therein.
0 1983 American Chemical Society
Angular Distribution of Diatomics in Intense Fields
The Journal of Physical Chemistry, Vol. 87, No. 25, 1983 5099
can now be considered as a direct laser-induced predissociation and thus both weak- and strong-field limits can be treated4rs-10simultaneously. Of note is that in strong-field photodissociation, the initial state is considerably broadened at field strengths greater than los W / C ~ ~In. addition, ~ J ~ the M sublevels of a rotational level with J quantum number have different Stark shifts and also acquire different laser-induced widths.l0J’ As shown by us recently,12 this causes considerable depolarization in light scattering (resonance Raman effect) by the photodissociated continuum. Furthermore, as pointed out originally by Voronin and Samokhin,13at high field strengths, laser-induced resonances should lead to delay of the multiphoton dissociation process and should thus introduce a substantial change in the angular distribution of the photofragments. We have previously correlated this delay in dissociation as a result of the formation of new adiabatic states by the strong laser field.lOJ1 In the next section we present a formal quantum treatment of bound-to-continuum transitions. A timedependent approach, similar to Mower’s treatment of stimulated transitions between unstable states,14 is used, thus taking properly into account the broadening of the initial molecular state by the incident laser. The expression for the dissociation probability is general and can be used to calculate the angular distribution irrespective of the laser intensity. In section 111, we show how one can recover the weak-field limit in agreement with the traditional results,l provided certain experimental conditions are satisfied. In section IV, we present results of coupled equation calculations in the dressed molecule representation at high field intensities (I > lo9 W/cm2). We present the numerical procedure and the calculated angular distributions for the particular case of Ar2+,a system which we previously used to show the intensity dependence of resonance Raman scattering.11J2 For Ar gas, the laser breakdown threshold occurs above 10l2 W/cm2 at 1 atm pressure and 1-ns pulse d ~ r a t i 0 n . l ~Thus in a crossed molecular-laser beam experiment, one should be able to measure dissociation products for intensities up to about 1O’O W/cm2 and be assured that the dissociation is essentially completed in the laser beam.4 In fact, with Ar2+, one has the advantage that all other excited potentials emanating from atomic excited states are at least three photons away from the photodissociated continuum state (see Figure 1). Thus one can safely neglect multiphoton ionizations at I I 1O1O W/cm2 which cause breakdown at I > IO1*W/cm2.15 11. Theory
‘P +‘P __ 6
~
2h w
Is+P‘
4
2h w
\
Figure 1. Schematic representation of the electronic field potentials for direct photodissociation of Ar,’ submitted to a XeF laser beam (w 28 328 cm-’). Resonant and nonresonant molecular states and the asymptotic atomic states are indicated. E, is the initial state energy.
Both states being coupled by the laser radiation, one has a situation analogous to predissociation.16 The initial state becomes a resonance which reflects the energies and widths of the “dressed states”. When the Fermi golden rule is used, the line widths are given by r = 2*1v~b1~, where v a b is the radiative coupling. One therefore expects intuitively that these will depend upon the relative orientation of the laser and the molecule. Making use of the dipole approximation, we can express this dependence in a simple mathematical form: if we assume that the molecule is initially in a state with a well-defined angular momentum J whose projection upon the Z laboratory axis is M and upon the internuclear axis is 0, the radiative coupling is given by1’J7 (J’M’Q’lVJJMQ)=
y = 1.17 X 10-7[1( W / C ~ ~ ) ](cm-l/au) ’/~
We consider a physical situation where a molecule initially in some bound state absorbs a photon from a single mode laser field and ends up in a repulsive excited state. In the dressed molecule p i ~ t u r e , ~ ~the ’ - ~zeroth-order ~ dressed states are obtained by adding the field energy to the molecular energy and, as shown in Figure 1,the electronic field potential of the initial molecular state with n photons crosses the repulsive state with ( n - 1)photons. ~
~~
(8) T. F. George, J. Phys. Chem., 86, 10 (1982). (9) A. D. Bandrauk and M. L. Sink, Chem. Phys. Lett., 57,569 (1978). (10)A. D.Bandrauk and M. L. Sink, J. Chem. Phys., 74,1110 (1981). (11) A. D. Bandrauk and G. Turcotte, J. Chem. Phys., 77,3867 (1982). (12) G. Turcotte and A. D. Bandrauk, Chem. Phys. Lett., 94, 175 (1983). (13)A. I. Voronin and A. A. Samokhin, Sou. Phys. J E W , 43,4 (1976). (14)L. Mower, Phys. Rev., 142, 799 (1966). (15) G. M.Weyl, D. I. Rosen, J. Wilson, and W. Seka, Phys. Reo. A , 26,1164 (1982).
where u is the polarization axis in the laboratory frame, y is a constant depending upon the field intensity, (abcdlef) are Clebsch Gordan coefficients, and d, is the xth component of the transition moment along the molecular axis. The values of u and M describe the orientation of the laser polarization and of the molecule with respect to laboratory axes. The angular dependence of the coupling is provided by the first Clebsch Gordan coefficient, along with the selection rule J’ = J, J f 1. Even in this simple situation, the total system includes three continua, one for each allowed value of J ! Moreover, each continuum is coupled back to three bound states with J” = J’, J’ f 1, which are themselves coupled to three con(16) A. D. Bandrauk and M. s. Child, Mol. Phys., 19, 95 (1970). (17) 0. Atabek, R. Lefebvre, and M. Jacon, J . Chem. Phys., 72, 2670 (1980).
5100
The Journal of Physical Chemistry, Vol. 87, No. 25, 1983
tinua, etc. We have shown in ref 11 that this "ladder effect" becomes important for strong fields ( I > IOs W/ cm2). For the photodissociation process, we write the total Hamiltonian as H = Horn+ P HR V R where the field-free molecular Hamiltonian is expressed as the sum of a zeroth-order Hamiltonian Hornand an intramolecular coupling P; HRis the pure radiation Hamiltonian and VR is the radiative coupling. The field is described in the Fock representation by eigenstates In) with eigenenergies n h w , where n and w are respectively the number and frequency of the photons present in the state. The molecular eigenstates IEa,a)satisfy the equation
+
+
HOrnlEa,a)= EaJa) (2) By writing the Hamiltonian as H = Ho + V with Ho = Horn + HR,V = V" + V Rone can describe the total system by linear combinations of the eigenstates of Ho HoIEa,a,na) = (Ea + nahU)lEa,a,na) (3) with IEa,a,na) = IEa,a)lna) (4) If we assume that the system is described at time t = 0 by I$(O)), then, in the Schroedinger picture, it will evolve according to
~ $ ( t )=) e-rHt/hl$(0))
Bandrauk and Turcotte
It can be shown18J9that (PGQ)may be expressed as PGQ = g p v QGQ (13) where g p = P(E+- PHP)-'P (14) (15) QGQ = &[E+- Ho - QtQI-lQ t = v + vgpv (16) The energy dependence of PGQ is then provided by the evaluation of the restricted operators gp, t , and QGQ. The operator g p only mixes the open channels and, for the sake of simplicity, we assume that there is no direct coupling between the continua so that we obtain
The properties o f t are best seen through the evaluation of an element tba. Making use of (17), we have tba
=
+
c,n,
1
vbc vca
E+ - E c - nc)iw
(18)
which can be written as (PP = principal part)
(5)
iTs
which can be written as
Vbc Vca 6(E - E, - nchw) (19)
G(E+)is the Green's operator, G(E+)= (E+- H)-l and E+ = lim,,+ (E + k). Making use of the basis states defined in (4), we have
I$(t)
) =
IEb,b,nb)[
E,,a,n,
Iba(t)(Ea,a,nal$(O)) 1
(7)
where the coefficients (Ea,a,nal$(0))describe the initial state while Iba(t) = ( Eb,b,nblI(t)lEa,a,na) is the transition amplitude between the basis states IEa,a,na)and IEb,b,nb), i.e.
In order to solve this last integral, we need the energy dependence of the Green's function. For this purpose, we have recourse to the well-known partition technique.18J9 We introduce a projection operator P for the basis states whose molecular energy belongs to a continuum (open channels) and for Q for the states whose molecular energy is discrete (closed channels). P and Q are written as
Q
=
C IEa,a,na)(Ea,a,nal
(10)
tba is thus a complex, andytic function of the r e d variable E. The last step is to obtain QGQ in a manageable form. We take linear combinations of the bound states to form a new basis (IL)]which diagonalizes the effective Hamiltonian Q(Ho+ QtQ)Q,18J9 i.e. (LIQWo + QtQ)QIL)= (EL- ~ ~ L / ~ P (20) LL' and IL)(LI QGQ = C L E+ - EL irL/2
+
The operator Ho + QtQ can be viewed as an effective, energy-dependent (via t ) Hamiltonian in the subspace of the closed channels. Strictly speaking, its eigenstates form a biorthogonal b a s i ~ . ' We ~ ~ retain ~ ~ here the simpler notation o f (20) but recall that IL), EL, and rLare energy dependent. The dressed states we introduced a t the beginning of this section are here formally defined by (20). Making use of the fact that Q = cLIL)(LI,while Q2 = Q, we can substitute (13), (17), and (21) in (12) to write the transition amplitude as Iba(t) = 1 (Eb,b,nblVIL)(LIEa,a,na) e-lEt/h 21ri L (E+- Eb - nbhw)(E - EL + i r ~ / 2 )
-cJ a
E,,a+,
and satisfy the usual relations1' P+Q=l PQ=QP=O P2=P (11) PHoQ = Q H 8 = 0 Q2 = Q For a photodissociation process, the initial state is entirely contained in Q l $ ( O ) ) while the final state is given by PI$(t)). The transition amplitude is then (18)B. W. Shore, Reu. Mod. Phys., 39, 439 (1967). (19) J. P. Laplante, A. D. Bandrauk, and C. Carlone, Can. J . Phys., 55, 1 (1977).
(22)
By writing (E+- Eb - n,hw)-' as iei(E+-Eb-nbhw)t/h l-mt/h du e-i(E+-Eb-nbhw)U, we have21 lim Iba(t)= e-i(Ebfnbhw)t/hX -t
(20) M. Sternheim and J. Walker, Phys. Rev. C, 6, 114 (1972).
Angular Distribution of Diatomics in Intense Fields
The Journal of Physical Chemistry, Vol. 87, No. 25, 7983 5101
It should be pointed out that (22)and (23)are the exact expressions for the bound-to-continuum transition amplitude. The physics of the problem is clearly seen through each parameter in (23). The initial state IE,,a,n,) redistributes itself among the dressed states (via the amplitude (LIE,,a,n,)) and the dissociation is controlled by the coupling between the continua and the dressed state (((Eb,b,nblqL)), while resonances in the processes are given by the energies of the dressed states. For small values of rL(narrow resonances), we can approximate the values of IL),EL, and rLas constant over the range where I(E - E L + irL/2)-l1 is significant. Under these conditions, the integral in (22) can be solved by contour integration as one gets contributions from two poles: E = Eb nbhw and E = EL - i r L / 2
We take the 2 axis along the polarization axis so that in (l), we have v = 0 and M’ = M . For a given initial state IE,,a,n) = IEaPaJ,M,O$ann), all channels have M = Ma and the sum over M may be dropped in (30). Making use of the fact that for radiative transitions AJ fixes we may also drop the index p . The transition amplitude from IEa,a,n,) to a particular electronic field state IKPQn) can then be written for a particular M component for all J
Iga(t)= C [ 2 a ( 2 J + 1)]”20;l;n(rC.,,0,,o)l~(t)
(31)
J
I x ( t ) = e-’*JC(JMn[VILM) (ML1J,Mna) x
+
We now use the explicit coordinate representation of the wave functions and look a t the angular distribution of photofragments. The molecular wave functions are written in the Born-Oppenheimer approximation as
where we have explicitly written the dependence of the dressed states upon M . To obtain the total probability P(OK,t)of observing the fragments in the direction of K, we average the square of the amplitude with the ( 2 J , + 1)values of M , and then integrate the result over the final energies EK of the fragments. After contour integration P(BK,t)is then given by 2a P(O,,t) = D$*(+,,OK,o)o~,(+~,OK,o) x (2Ja + 1) MJJ’ e-i(arsi)[( 2 J 1)(25’ + 1 ) ] ” 2 c (JnlVlLM)(LMIJaMn,)(J’nlVIL’M)* x LL’ [1 - e - i ( h d L ’ M ) t / t ~ ~ - ( r ~ ~ + r ~ ’ ~ ) t / z t ~ ] ( L ’MIJaMna) * (33) ELM- ELJM+ i ( r L M+ rLlM)/2
c
+
with
a
where are the electronic coordinates and r,O,+ are the internuclear axis coordinates in the laboratory frame; rC.aa(Q:r) is the electronic wave function giving rise to the potential Ubn(r); Ff&*(r) is the nuclear radial wave function and D$,(+,O,O) is a Wigner rotation matrix. With p = fl, the molecular wave functions defined in (25) have a definite parity. The radial functions have the usual asymptotic behavior, namely, F&&) = 0 for a bound state and F&,(r) (1/k1I2) sin (k,+ 65,) for a continuum, where
111. Weak-Field Limit In the weak-field limit, we can assume that no significant mixing of the bound states occurs to that ILM) N IJ,Mn,). Furthermore, in experiments where the fragments have a high kinetic energy, the phase shift is almost independent of J so that e-L(aJ-sJ) = 1. From these facts, the probability function becomes
-
and 6, is the phase shift. For the present study, it is convenient to take linear combinations of the continuum wave functions with the same asymptotic kinetic energy so that the resultant wave contains as its-outgoing part an outgoing plane wave with wave vector k where k=
kK
(29)
Assuming that the perturbation V is purely radiative, i.e., V” = 0, then the J dependence of (JnlVIJ,Mn,) is essentially contained in the angular part (cf. eq I), we write
and
K being a unit vector pointing along the final direction of
the fragments. The projection of the electronic angular momentum upon K is i2.l It is shown in Appendix I that such functions are given by
(30)
where +K and OK are the angles defining the direction of K in the laboratory frame. In order to avoid cumbersome notations in the formulae describing the angular distribution, we can immediately make some simplifications. (21) C.Cohen-Tannoudji, B. Diu, and F. Lid&, ‘MBcanique Quantique I”, Herman, Paris, 1973.
~
~~
(22) J. M. Brown et al., J . Mol. Spectrosc., 55, 500 (1975).
The Journal of Physical Chemistry, Vol. 87, No.
5102
25, 1983
Bandrauk and Turcotte
TABLE I: Coefficient of the Legendre Polynominal Calculated from Eq 42 with x = 0, R = 1/2 1 J
0
0.5 1.5 2.5 3.5
1.5000 1.7857 1.9585 2.0852
2
4
6
1
-0,7143
-1.1463
0.2172 0.4556
-1.4629
""'
J-1.5
\
-0.0617
In the limit r M t / h 0, A J = -1 photodissociation is enhanced. Since y is essentially an energy shift, it is usually positive for levels above the crossing point (Figure 116*28). Registry No. Ar2+, 17596-58-6. (28) A. D. Bandrauk and J. P. Laplante, (1976).
J. Chem. Phys.,
65, 2592
Energy Partitioning to Product Translation in the Infrared Multiphoton Dissociation of Diethyl Ether L. J. Butler, R. J. BUSS,+R. J. Brudzynskl, and Y. T. Lee" Materials and Molecular Research Dlvision, Lawrence Berkeley Laboratory and Department of Chemistry, University of California, Berkeley, California 94720 (Received: June IO, 1983)
The infrared multiphoton decomposition of diethyl ether (DEE) has been investigated by the cross laser-molecular beam technique. The center-of-mass product translational energy distributions (P(E9)were measured for the two dissociation channels: (1)DEE C2H60+ C2H5and (2) DEE C2H50H+ C2H4.The shape of the P(E9 measured for radical channel 1 is in agreement with the predictions of statistical unimolecular rate theory. The translational energy released in concerted reaction 2 peaks at 24 kcal/mol; this exceedingly high translational energy release with a relatively narrow distribution results from the recoil of the products from each other down the exit barrier. Applying statistical unimolecular rate theory, we estimate the average energy levels from which DEE dissociates to products using the measured P(E9 for radical channel 1.
-
-
I. Introduction Previous molecular beam investigations of unimolecular reaction dynamics have shown that potential energy barriers in the exit channel beyond the endoergicity have a large effect on the asymptotic product translational energy
distributions. For almost all of the simple fission reactions studied, in which a single bond is broken without an exit barrier and no new bonds are formed, the products have statistical translational energy distributi0ns.l However, for all the complex fission reactions studied, in which
t Present address: Sandia National Laboratory, Division 1811, Albuquerque, NM 87185.
(1) Aa. S. Sudbo, P. A. Schulz, E. R. Grant, Y. R. Shen, and Y. T. Lee, J . Chem. Phys., 70, 912 (1979).
0022-3654/83/2087-5 106$01 S O / O
0 1983 American Chemical Society
