Spectroscopy and photodissociation dynamics of a two-chromophore

J. Phys. Chem. 1991,95,6129-6141

6129

Spectroscopy and Photodissociation Dynamics of a Two-Chromophore System: Raman Scattering as a Probe of Nonadiabatic Electronic Coupling J. Z. Zhang,*.f E. J. Hellel,* D. Huber, and D. G. Imre Department of Chemistry, University of Washington, Seattle. Washington 981 95 (Received: May 6. 1991)

This paper presents an investigation of a model system with two identical, dissociative chromophores. Each chromophore, if left electronically uncoupled to the other, directly dissociates. When we allow electronic coupling, there is a competition between two time scales: the vibrational time scale of the chromophore dissociation dynamics and the electronic time scale for energy exchange between chromophores. The vibrational dynamics and the d a t e d spectroscopiesare extremely sensitive to the winner of this competition. Both electronic absorption and resonance Raman spectra carry distinct signatures of the dissociation dynamics and the coupling between the two chromophores. We give a general formalism for constructing a model potential energy surface for a two chromophore system and for calculating its absorption and Raman spectra. The model is tested by simulating the photodissociation dynamics and electronic absorption as well as resonance Raman spectra and good agreement is found between our calculation and experimental results. The calculated for the water molecule, H20, results also agree well with simulations performed by using ab initio potential energy surfaces.

I. Introduction We consider the molecular dynamics and spectroscopy of molecules containing more than one Such systems can be treated as having multiple interacting diabatic electronic states, one for the locally excited state of each chromophore. In this paper we treat the special case of two identical chromophores. In our picture this gives rise to two equivalent, coupled, diabatic electronic states, which are degenerate whenever the chromophore bond lengths are equal. We focus on the fastest events (several femtoseconds) and observe the consequence of the exchange of electronic excitation between the two chromophores on the early time dynamics, or equivalently on the spectroscopy that probes these dynamics. The subject of this paper is the photodissociation dynamics and spectral signatures for a two-identical-chromophoresystem. We will show that both absorption and emission (Raman) spectra carry information about the nuclear as well as electronic dynamics in such systems. The study is performed by using the timedependent wave packet formalism. Some very interesting work on similar subjects has been reported by Metiu,'v2 Tannor,' Coalson,' and their co-workers recently. Coalson and Kinsey' first presented an extension of the wave packet theory for a single excited surface system developed by Heller and ~o-workers~~J*~' to systems involving two excited electronic states coupled nonadiabatically. They used perturbation theory to deal with the nonadiabatic coupling of electronic and nuclear motion and gave the formulation for computing electronic absorption and Raman spectra. They illustrated the effect of nonadiabatic coupling on the spectrum and the wave packet dynamics on two bound excited states through a one-dimensional test system. Metiu et aL13have presented an extensive study on the electronic absorption as well as Raman spectra for a model H3+ system using the time dependent formalism. The system was treated as having two dissociative, diabatic states coupled with a nonadiabatic interaction in two dimensions. They demonstrated the effect of nonadiabatic coupling on both the absorption and the Raman spectra. The effect of the relative orientation of the transition dipoles on the Raman spectrum was also discussed. They propagated the wave packet numerically by using a Fourier transform method developed by Fleck et al.zz and extended to curve-crossing problems by Alvarellos and MetiumZ3Very recently, Das and Tannor3 extended the time-dependent formulation to systems with chemical branching. They treated the molecule C2F41Bras a two-nonidentical-chromophore system with nonadiabatic coupling and calculated successfully the absorption spectrum as well as the branching ratio of final products due to photodissociation. h n t addreas: Department of Chemistry, University of California, Berkeley, California 94720.

The emphasis of our present paper differs from these previous studies in several aspects. First of all, we are interested in understanding real molecular systems such as H 2 0 and CHJ2 for which experimental data are available for testing the theory. Second, we emphasize the dynamical aspect of the issue as well as the spectral signatures. We present a systematic study of the electronic as well as the nuclear (dissociation) dynamics of a two-identical-chromophore system with a variety of coupling strengths. Third, we give a general formalism for constructing potential energy surfaces (PESs) for two-chromophore systems based on the information available from the corresponding single chromophore. This is useful for molecules for which PESs are not available from first principles such as ab initio calculations. Finally, because of the local-mode nature of the PES we choose for the ground state of the system, the Raman spectra become easy to analyze and the dynamics associated with the Raman process simple to understand. It is worthwhile analyzing the problem first on a qualitative level. From earlier work on absorption and Raman spectroscopyl 7-21 we know that low-resolution absorption spectroscopy or preresonant Raman spectroscopy effectively cuts off the influence of the dynamics on the spectra in a time determined by the un(1) Jiang, X. P.; Heather, R.; Metiu, H.J. Chcm. Phys. 1989,90,2555. (2)Heather, R.;Mctiu, H.J. Chcm. Phys. 1989,90,6903. (3) Das, S.;Tannor, D. J. Chem. Phys. 1989,91,2324. (4)Coalson, R. D.; Kinscy, J. L. J. Chem. Phys. 1986,85,4322. (5) Dwuter-Leomte, M.; Dehanng, D.; Leyh-Nihant, E.; Praet, M.Th.; Lorquct, A. J.; Lorquct, J. C. J . Phys. Chcm. 1985,89,214. (6)Person, M. D.; Lao, K. Q.; Eckbolm, E. J.; Butler, L. J. J. Chem. Phys. 1989,91,812. (7) Domcke, W.;Kbppel, H.; Cedcrbaum, L. S. Mol. Phys. 1981,43,851. ( 8 ) Schneider, R.;Domcke, W.; Kbppcl, H. J. Chem. Phys. 1990,92,1W5. (9)Seo, K.; Kono, H.; Fujimura, Y. Chem. Phys. Lerr. 1980,74, 549. (IO) Gregory, A. R.; Henneker, W. H.; Siebrand, W.; Zgicnki, M. 2.J. Chem. Phys. 1915,63,5415. (1 1) Hcnneker, W. H.; Sicbrand, W.; Zgicnki, M. 2.Chem. Phys. Lerr. 1976,43, 1I. (12) Goutcrman, M.J . Chem. Phys. 1965,42,351. (13) Fulton, R. L.;Gouterman, M. J. Chcm. Phys. 1964,41,2280. (14)Wjtkowski, A.; Mofftt, W. J. Chcm. Phys. 1960,33,872. (15) Hint, D. M.Porenrfal Ewrm - Surfaces: - Taylor & Francis: London and Philadelphia, 1985. (16) Fisher, G. Vibronic Coupling, Academic Prcss: London, 1984. (17)Heller, E.J.; Sundbcrg, R. L.; Tannor, D. J. Phys. Chcm. 1982,86, *e** IOLL.

(18) Imre, D. G.; Kinsey, J. L.; Sinha, A,; Knnos, J. J. Phys. Chcm. 1984, 88,3956. (19)Heller, E. J. Acc. Chcm. Res. 1981, 14,368. (20)Zhang, J. Z.;Heller, E. J.; Huber, D.; Imrc, D. G.; Tannor, D. J. Chem. Phys. 1988,89,3602. (21) Lee, S.Y.; Heller, E. J. J . Chem. Phys. 1979,71. 4777. (22) Fleck, J. A., Jr.; Morris,J. R.; Feit, M. D. Appl. Phys. 1976,IO, 129. (23)Alvarellos, J.; Metiu, H. J . Chem. Phys. 1988, 88,4957.

0022-3654/91/2095-6 129$02.50/0 Q 1991 American Chemical Society

6130 The Journal of Physical Chemistry, Vol. 95, No. 16, 19‘91

certainty principle. Even if the molecule remains bound in the excited state, the low-resolution absorption spectrum and the highor low-resolution preresonant Raman spectrum contain only early time dynamical information. Here we take a model in which the chromophores dissociate in the excited state. Dissociation gives another time cutoff that allows us to probe the molecule by resonance Raman spectroscopy without fear of resonant enhancements that can complicate the analysis. Let us consider an A-B-A molecule with two equivalent AB-bond chromophores. In the usual first-order treatment of the interaction of the molecule with the radiation field, only one of the two chromophores can be initially excited by a photon. (However there is an amplitude for each to be the excited one, and the role of the interferenceof these two amplitudes is a major effect that we consider in this paper.) For the sake of argument we introduce two artificial constraints, both of which are relaxed later. We can either (1) clamp the nuclei in place or (2) restrict the electronic excitation to one chromophore. The latter restriction defines a diabatic electronic state, and it amounts to setting the electronic coupling VI, between diabatic states 1 and 2 to zero. There is one diabatic excited state for each chromophore. If we introduce the electronic coupling VI, between the diabatic states, the eigenstates, which result from the 2 x 2 eigenvalue problem of coupled diabatic states, are two Born-Oppenheimer adiabatic states. If we initially excite one chromophore, for clamped nuclei and equal bond lengths, the electronic energy hops A-B-*A) sinusoidally with a freback and forth (A*-B-A quency given by the adiabatic state splitting, i.e., twice the diabatic coupling VI,. This frequency is one of the two competing time scales that play a central role in the problem we will address. The second time scale is revealed by unclamping the nucleii and applying the constraint localizing the electronic excitation to one chromophore. The excited-bond chromophore experiences new forces, in the direction of longer bond length. (The other A-B bond keeps the same local potential as in the ground state.) The force causes the bond to lengthen, and to break the electronic degeneracy of the diabatic states. Thus, even if the diabatic restriction is removed after one bond is sufficiently lengthened, it will be too late for the electronic energy to transfer; the splitting between the diabatic surfaces will be greater than the coupling. This happens at a distance x’ from the crossing point, where (dVl/dx)x’ = VI2,which takes a time t d = (2pV12)1/2/(dV,/dx) to reach, with p the reduced mass for A-B. Following photoabsorption, the competition ensues between hopping (characteristic time 1, EC: h/4VIz), and motion of the bond chromophore out of the regime where it can hop (characteristic time t d ) . The story of localization of electronic energy or indeed an electron itself is a familiar one in many contexts in spectroscopy and electron transfer. Our model is ancient in this respect, but we hope to introduce some dynamical insights that have direct footprints in the absorption and Raman spectroscopies. The paper is organized as follows. In section 11, a two-chromophore model system is constructed and presented in the diabatic as well as in the adiabatic representation. The corresponding Hamiltonian for this system is discussed. The theoretical method and the numerical technique for the calculation are given in section 111. In section IV, we present our results: we first look at the one-photon probe of the excited state dynamics and compare the absorption spectra as well as the dynamics for a variety of coupling strengths; then in the second part of this section the two-photon (Raman) spectra are investigated and applications to real molecules are demonstrated.

-

11. Model

In the interest of keeping the parameters in our model realistic, we choose the H20 molecule as an example. This will also provide us with a test of the model, since the photodissociation dynamics of H20has been very well studied and experimental data are available for both the absorptionUand Ramanz spectra. We have (24) Wang, H. T.;Felp, W. S.;McGlynn. S.P. J . Chem. Phys. 1977,67, 2614.

Zhang et al.

->3

1

9 r 1 .o

3.0

5.0

7.0

x (a4 Figure 1. Contour plot of the two diabatic potential energy surfaces VI (dashed lines) and V, (solid lines) in the internal coordinates. The shaded area indicates the FC region. shown that it is possible to account for the observed spectra by treating the molecule in two degrees of freedom.26 We treat the molecule in the two stretch coordinates with the bond angle, labeled as 8, fixed in the calculation. To perform the numerical study, we first need the ground-state potential energy surface; it determines the initial wave packet, and later on the Raman spectrum. We adopt the ground-state potential energy surface of the H 2 0 molecule as developed by Watts et al.,27*28 which was shown to reproduce experimental observables very well. Next we need to construct the potential energy surfaces for the excited states. We have developed a simple scheme for constructing potential energy surfaces for two-chromophore systems; it has been used before in the study of CH2IPm We first assume that there is no electronic coupling between the two chromophores AI-B and B-A,; Le., we treat them as two separate molecules. Using X and Y to represent the AI-B and B-A2 bonds, respectively, if one of these two chromophores, e.g. X,is exciied by a photon while Y remains in the ground state, we have Al-B-A2. The potential energy surface for such a system may be written as VI = ae-b(X-1333)+ D [1 - e-P(p(Y-1.81)]2 (1) where the exponential part ae-Mx-1.833) is the repulsive interaction for the excited state for A*-B. The parameters a and b can usually be chosen by fitting to the electronic absorption spectrum of B-A as we have demonstrated for CH212. Here we choose a and b such that the a_bsorpGonspectrum will have a width similar to that of the H 2 0 A X absorption spectrum. We use a = 0.08 au and b = 0.8u,,-I. The second term corresponds to the potential energy curve of the ground state B-A, which is assumed to be a Morse oscillator. We choose De = 0.162 au and p = 1.2255ao-’, which corresponds to a vibrational frequency of around 3700 cm-I and a reduced mass of 1 (atomic mass unit). The Morse parameters are chosen to fit the OH molecule. By symmetry, if the chromophore B-A2 (Y) is excited and X is unexcited, the potential energy surface for this electronic state is given by ae-b(r-1.833)+ D [1 - e-p(p(x-l.81)]2 I/2 (2)

-

A contour plot of these two diabatic surfaces is shown in Figure 1.

There is a clear resemblance of our procedure for constructing the potentials to the ‘diatomics-in-molecules” approach.s31 The (25) Sension, R. J.; B ~ d ~ p ~R.k J.; i ,Hudson, B.S. Phys. Reu. LLtt. 19E8,

61, 694.

(26) (a) Zhang, J. 2.;Imre, D.G.J. Chem. Phys. 1989, 90, 1666. (b) Imre, D. 0.;Zhang. J. 2.Chem. Phys. 1989,139, 89. (27) Coker, D. F.; Watts, R. 0.J . Phys. Chem. 1987, 91, 2513. (28) Coker, D. F.; Miller, R. E.; Watts, R. 0. J . Chem. Phys. 1985, 82, 3554. (29) Tully, J. C. In Potential Energy Surfaces; Lawley, K. P., Ed.;Wiley: New York, 1980. (30) Olson, J. A.; Garrison, B. J. J . Chem. Phys. 1985,83, 1392. (31) Tully, J. C. In Semiempirical Mefhods of Electronic Structure Calculations; Segal, G.A., Ed.;Plenum: New York, 1977; Part A.

Spectroscopy and Dynamics of a Two-Chromophore System

The Journal of Physical Chemistry, Vol. 95, No. 16, 1991 6131

2

1

)( ....'

I

2

2

1

P92

9 c 1.o

M p r e 2. (Top) Schematic cut of the excited-state potential energy surfaces along the reaction coordinate. VI*is the coupling between the two diabatic surfaces 1 and 2. Surfaces a and b are adiabatic potential energy surfaces. (Bottom) schematic diagram of the transition moments in the diabatic (pgl and p@) and the adiabatic pictures (pmand pgb)for 0 = oo.

method may be viewed as semiempirical, an attempt to put in the "physics" of the chromophore coupling in a simple parametricway. By construction we see that each one of these two diabatic surfaces corresponds to a situation where one of the B-A bonds is in the ground electronicstate, a Morse potential, and the other bond in the electronically excited state, which is repulsive. In the pure diabatic limit, the molecule cannot cross from one surface to the other, and therefore the dissociation is purely along one local mode. If there is coupling between the two chromophores, the excitation may transfer from one bond to the other. The rate of surface hopping depends on the coupling strength. When the coupling VI, between the two potential energy surfaces is nonzero, there is another representation of the problem. We can obtain two adiabatic surfaces from the two diabatic surfaces by diagonalizing a 2x2 matrix. We label these two adiabatic surfaces V, and V,:

Vb = 1/2(VI + V 2 ) - 1/2

qCV1- V2)2+ 4% v 2

(1 + ev(X-s.0))+ ( 1 + eu(Y-s.0))

7.0

x (a4

Figure 3. Contour plot of the two adiabatic potential energy surfaces V,

(dashed lines) and V, (solid lines) for coupling f12 = 0.02 au.

B-A bond lengths and P& , are the conjugate momenta of X and Y. In keeping in with the H 2 0 system, we use mA = 1 and mB = 16 in atomic mass units. The cross term in eq 6 introduces the kinematic coupling between these two chromophores, which is zero for @ = 90° and small for a light-heavy-light (LHL) system such as water. This term turns out to be nonneglegible in certain cases and is kept throughout the calculation. The excited-state Hamiltonian for the vibrational motion is then i= I

where lei) with i = 1,2 are the two excited diabatic states. We assume that they are orthonormal and lel ) (e21commutes with the kinetic energy operator T and we also use VI,= Since we are interested in using one- and two-photon spectroscopy between the ground and excited electronic states as our probe, the last ingredient we need for the study is the transition moment that couples the ground- and excited-state surfaces. We construct two transition moments p,, and pg2 in the diabatic representation

(4)

The coupling term is chosen to be a function of the two internal coordinates: '12

5.0

3.0

(5)

where y = 1.Oa$ and C2is a constant representing the coupling strength. f12is varied as we study the system from weak to strong couplings. This function ensures that when the two chromophores are far apart the coupling is zero, since the electronic excitation can hardly hop between AI and A2 when the two are far apart. On the other hand, VI,is nearly constant with respect to X and Yin the Franck-Condon (FC) region where the reaction begins. For a moderate coupling strength, the upper adiabatic surface, V,, appears to be bound, the lower adiabatic surface, Vb, dissociative. The minimum separation between them is 2VI2. This is illustrated in Figure 2 with a schematic one-dimensional cut along the reaction coordinate for the two potential energy surfaces in both the diabatic and adiabatic representations. For f12 = 0.02 au, for example, a contour plot of the two adiabatic surfaces is shown in Figure 3. Up to this point, we have discussed the potential part of the Hamiltonian. We now need the kinetic energy operator. If we freeze the A-B-A bending motion, the vibrational kinetic energy in the internal coordinate system can be written as

where p l = mgmA/(mB + mA),p2 = mBcos B; X,Yare the two

where 7 = 1.5a{', pel and Cglare the unit vector and the magnitude of the transition moment pal. p I couples the ground state with the surface yielding AI + BAI. A i s functional form for pBl ensures an only slightly varying transition moment over the ground-state equilibrium geometry, yet it goes to zero in the asymptotic region. This corresponds to dissociation into ground-state products. p I transfers the molecule to surface VI. Since the B-A2 bond (3) is unexcited on this surface, pSl is constant with respect to this coordinate. By symmetry we have for Pg2

(9) is constant with respect to the AI-B bond length ( X ) and it goes to zero as the B-A2 bond length increases to infinity.

pa

111. Theoretical Method and Numerical Procedure

A time-dependent formalism has been developed to treat the electronic absorption and Raman spectra as well as the photodissociation dynamics for processes involving one excited electronic ~urface.~~J"' Here we extend the method to include two coupled electronically excited states, as has been done by a few others.'+ For completeness of presentation,a derivation of the formulation is outlined in Appendixes A and B. Here we give only the final results and some physical interpretations of the equations. The time-dependent formalism of absorption and emission presents a very simple picture for the interaction of light with molecules. The photon of polarization 21 produces a wave packet given by

6132 The Journal of Physical Chemistry, Vol. 95, No. 16. 1991 2

14) = Eler)(Per.~l)C,lxo) = 141) + 142) i= I

(10)

where we define 14,) = le,)(ji&)C,lxo). Previously, 14) was treated as being on one of the excited states. Here we find that when more than one state is involved, the photon prepares a superposition of wave packets, each residing on one of the excited electronic states. The picture for excitation is then very intuitive. There are two wave packets each moving on its electronic potential energy surface according to one of the two Hamiltonians. This simple picture is correct as long as no amplitude can be transferred between the two, i.e. VI, = 0. As soon as the amplitude can move from one surface to the other, interference can take place. It is clear then that the relative signs of the wave packets on surfaces VIand V2and the sign of the coupling VI, determine whether the interference is constructive or destructive. This in turn changes the dynamics and consequently results in changes in the spectrum. The wave packet prepared is then propagated by the excited-state Hamiltonian He

Zhang et al. On the ground state these two Raman wave functions can interfere with each other. The total Raman spectrum is given by

where

= e-'HI'I@R(wl) ) (17) with H,the ground-state Hamiltonian. The shape of the Raman wave function is determined by the dynamics of I4(t)) on the excited states. As we discussed earlier, the rate of electronic excitation exchange between AI-B and B-A2 governs much of the dynamics. In the diabatic limit, i.e. low rate of excitation will move mainly along the local A-B coordinate. exchange, I@(?)) The Raman wave function will develop amplitude along this coordinate. The Raman spectrum is expected to show a progression in the pure local mode. In the adiabatic limit the excitation hops from AI-B to B-A2 very quickly, and the molecule ends up moving along the symmetric stretch coordinate. The I W ) = e-'H+$) (11) Raman spectrum will then be expected to show a progression in Here He propagates 14) on the two surfaces simultaneously and the symmetric stretch. This is true regardless of the nature of transfers amplitude from one surface to the other through VI*, the ground-state Hamiltonian, be it a local-mode molecule as H 2 0 the coupling term. or a normal-mode molecule. The Raman spectrum is determined To calculate the absorption spectrum, we first need to calculate by the dynamics on the excited state as well as the ground state. the autocorrelation function (t$I#(t)). The absorption spectrum Excited-state dynamics determine which of the vibrational eiis then given by17*19*21 genstates carries spectral intensities, while the ground electronic surface determines band positions and widths. e(w) = CwS+mel(Y+")f(6(0)14(t)) dt (12) To calculate both the absorption and the Raman spectra, we solve the time-dependent Schradinger equation to generate the where wo is the ground-state energy and C is a constant given by time evolution of either I4(t)) or I @ R ( q , t ) ) . The numerical C = 2/hceo with w the photon frequency, eo the vacuum susmethod used is a FFT grid technique developed initially by Kosloff ceptibility, and c the speed of light. The calculation of the auand K o s l ~ f f . ~ ~ -In ~ ' practice, our calculation is performed as tocorrelation function is simple for the case with only one excited follows. First, we calculate the ground vibrational eigenstate state; it becomes more involved when two excited states are present (typically u" = 0) for the ground electronic state. Then this as is the case here. The autocorrelation function (414(t))is given generated eigenstate, multiplied by pgl,and p p is transferred to by the excited state to produce 14). I4(t)) is obtained by propagating 2 2 14) with He, the coupled excited-state Hamiltonian, according to (4l#(t)) = C C ( c t ~ ? ) ( P ~ ~ , ) ( x o l C g , o e - ' * ~ f(1C3)~ x o ) the timedependent Schradinger equation by use of the numerical i=1j-1 method mentioned above. We observe the dynamics by recording the wave packet moving on diabatic surface "snapshots" of 141(t)), In general, the molecules are randomly oriented with respect VI and I&(t)), the wave packet moving on surface V2. We then to the laser field. Therefore, to calculate the absorption and calculate the autocorrelation function (t#$(t)) and Fourier Raman spectra we need to take an average over all orientations. transform this function to obtain the absorption spectrum. For The calculation of the angle-averaged transition moments is given the Raman spectra we project out the Raman wave function at in Appendix A. which then yields the desired laser frequency to generate I\kR(~l)), As we have done with the absorption spectrum, we follow the I@R(OI,O)). We propagate Io~(w1,O))on the ground state to obtain single-surface time-dependent formalism for the Raman specthe emission (Raman) spectrum. Readers interested in the details trum17J9*21 and extend it to include two coupled surfaces. In this of the calculation are referred to ref 35. picture the incident photon prepares a timeindependent state that is termed the Raman waue function. For two coupled surfaces IV. Results and Discussion it is given by There are two variables to consider. The first is the coupling 2 strength VI2,which controls the rate of electronic energy transfer. ( * R ( ~ I ) ) = ~ + m e x p ( i w l dt t ) Cexp(-iH,t)(~,.?)(e,) C,lxo) I= I The second is less obvious and is related to the relative phase and amplitude of the two wave packets IC#J~(O)) and 1&(0)), which (14) depends on the relative orientation of the two transition dipole where wl is the incident laser frequency and the rest of the moments, i.e. the angle between them (Appendixes A and B). quantities are the same as before. This total Raman wave function Because of the close connection between the absorption spectrum is a sum of two Raman wave functions, one on each of the two and the excited state dynamics, we discuss fvst the dynamics and diabatic surfaces. As long as these two wave functions are on the absorption spectra for two extreme cases where the two dipoles excited states, they are orthogonal to each other since (elle2)= are parallel (B =)'0 or antiparallel (@ = 180°), respectively. As 0. The Raman (emission) spectrum is a result of an electronic shown in Appendix A, we can then obtain the absorption spectrum transition from the excited surfaces back to the ground state. pal for any configuration (8) from the absorption spectra calculated transfers the Raman wave function on surface VIto the ground for these two cases. We choose the case of @ = 90° as an illusstate, while pa2takes the Raman wave function from surface V2. tration. We then examine the effect of nonadiabatic coupling on What reaches the ground state is then given by the Raman spectrum for the @ = 90"configuration (section 1V.B). I@R(aIrt) )

-OD

(32) Kosloff, R.; Kosloff, D.J . Chem. Phys. 1983, 79, 1823. (33) Gerber, R. B.; Kosloff, R.; Berman, M.Compur. Phys. Rep. 1986, 5, 59. (34) Kosloff, D.; Kosloff, R. J . Compur. Phys. 1983, 52, 35. (35) Zhang, J. 2.Ph.D. Dissertation, University of Washington, Seattle, WA, 1989.


rl

The Journal of Physical Chemistry, Vol. 95, No. 16, 1991 6133 t=2.9

B-A (au)

B-A (au) ,1114.5

Figure 4. Moving wave packet I4,(t))on one of the two diabatic surfaces VIfor zero coupling; is., = 0. Time is in femtoseconds and the gray lines in the first frame indicate the V, potential energy surface.

The relation between the Raman spectrum and the excited-state dynamics as well as the ground-state dynamics will be addressed. A. L ~ Y M ~ Cand S Absorption Spectrum. 1. Parallel Transition Dipoles. We start with the parallel dipole case (B = OO). This can happen in a number of ways, for example when the laser field vector Z, bifurcates the angle between the two bond-chromophore transition moments. This can be realized by fixing a molecule in a crystal and choosing the laser polarization to match this condition. Another case is a molecule with two parallel bond transition moments with the molecule randomly oriented in a laser field, as in the gas phase. Both cases will satisfy the conditions required for fl = Oo as defined in Appendix A. It should be pointed out that even though our calculation is performed from the diabatic basis we can look at the problem in either the diabatic or the adiabatic picture. Furthermore, we can take advantage of the symmetry of the molecule and present results only for one of the diabatic surfaces, Exactly the same process is happening on the other diabatic surface, by symmetry. As a reference point, we start with an examination of the zero-coupling case. Figure 4 shows the wave packet dynamics on one of the diabatic surfaces VI for zero coupling between the two chromophores. As expected, the dynamics is very simple, a wave packet is prepared at t = 0 in the FC region; as time goes on it starts to move toward larger displacement along one of the local B-A bonds while slightly spreading at the same time. The wave packet never comes back to the FC region, and it is well into the asymptotic region after about 23 fs. Since there is no coupling between the two diabatic surfaces, VIand V,, the evolving wave packet on V , never "feels" the existence of V,. Thus, this is equivalent to the situation of direct dissociation for a single chromophore. Since the wave packet never revisits the FC region, the autocorrelation function (dashed lines in Figure 9) shows a simple monotonic decay. Therefore, no structure is observed in the absorpion spectrum (dashed line in Figure lo), since the spectrum is simply given by the Fourier transform of the autocorrelation function. It is clear that in this case we can predict the spectra and dynamics from our experience with the studies of single-chromophore Next, we turn on the coupling between the two diabaticsurfaces. We first look at the weak-coupling case (0.0025 au or 550 cm-I). (36) Williams, S.0.;Imre, D.G. J . Phys. Chem. 198s, 92, 3363.

-

k11.6

k14.5

-

Figure 5. Moving wave packet Idl(t)) on one of the two diabatic surfaces = 0.0025 au and /3 = Oo. VI for coupling

Typically a spectroscopist would consider 550 cm-I for two degenerate electronicstates as strong coupling. In our example the dissociation is over in 23 fs, and the relevant time is actually even shorter, less than 5 fs, as discussed in the Introduction. For the coupling term to have substantial effect, the exchange rate must be faster than 5 fs. A coupling strength of 550 cm-I implies a frequency with a period of about 30 fs. This means that very little amplitude crosses from one surface to the other before the molecule dissociates. The dynamics for this case is shown in Figure 5. We find a slight difference between this case and the zero-coupling case. There is clearly a small piece of the wave function that develops motion along the symmetric stretch. This piece will take slightly longer to dissociate. This is also evident in the autocorrelation function (Figure 9). A slightly slower decay as compared to the zero-coupling case is observed due to the piece that lingers in the FC region. The center of the spectrum (Figure 10) shows a shift of 0.0025 au to the blue side, while the overall band shape is still very similar to that in the zero-coupling case. As we will show later, Figure 5 represents a wave packet motion on V,, the bound adiabatic surface. Obviously the adiabatic picture is not a good representation in this case. When we use a larger coupling, e.g., 0.01 au, the dynamics (Figure 6) as well as the absorption spectrum (Figure 10) changes drastically. First, the dynamics is much more complex. Part of the wave packet still follows the dissociation path as in the zero-coupling case, while part of the wave packet is trapped near the FC region for a long time, moving back and forth along the symmetric stretch mode. While it is vibrating in the symmetric stretch, pieces of this wave packet keep leaking out along the local stretch coordinate. The dynamics appear as if there is a very "leaky" quasi-bound state that traps part of the wave packet for some time. The dynamics of the trapped pieces of the wave packet appear to be occurring on a surface like V, in Figure 3, which is the upper of the two adiabatic surfaces, while the pieces that leak out move as if they were on a dissociative surface like Vk Obviously in this case the diabatic picture is no longer a good representation of the dynamics, but neither is the adiabatic one as is evident from the rapid dissociation on the V, surface. We are in a regime of intermediate coupling, where the wave packet hops between the two diabatic surfaces numerous times before dissociation occurs. It appears that the wave packet or at least part of it is born on the upper of the two adiabatic surfaces, which is "bound". If this picture were a good representation, we would

6134 The Journal of Physical Chemistry, Vol. 95, No. 16, 1991

Zhang et al.

El 1-2.9

c

1.0

5.0

9.0 13.0

B-A (au)

1.0

111 t4.7

5.0 X (au)

7.0

3.0

Figure 7. Initial wave packets (solid lines) prepared on the excited-state potential energy surfaces (dashed lines) in the diabatic (VIand Vz)or the adiabatic (V,and Vb)pictures. The wave function on Vbis enlarged 20 times. This is for 4 = Oo and flz= 0.02 au.

-

Figure 6. Moving wave packet on one of the two diabatic surfads V,for coupling f12 = 0.01 au and 4 Oo.

end up with the piece that was born on V, remaining on that surface for a long time. Instead this surface predisociates through the lower surface V,. The correlation function (Figure 9) shows that indeed the molecule lingers in the FC region for a long time. We even observe one period of vibration in the symmetric stretch, which takes on the order of 19 fs. The absorption spectrum (Figure 10) shows a structured band blue-shifted from the reference zero-coupling spectrum. The structure in the absorption spectrum is a symmetric stretch progression in the adiabatic V, state. The absorption spectra in Figure 10 for various couplings indicate that as the coupling strength is increased only one of the two adiabatic states carries appreciableoscillator strength. From Figure 2 we may expect for large coupling, where the adiabatic picture is appropriate, to observe two electronic transitions, one to V, and the other to V, Our rcaults clearly show that for parallel transition moments the transition to the lower dissociative surface Vb becomes electronically forbidden. When the two transition moments are antiparallel, as we will show, the selection rules reverse and electronic transition to Vb is allowed, while V, is electronically forbidden. As expected, when the two chromophorcs are perpendicular to each other we find that both V, and Vbare electronically allowed. Even for the parallel/antiparalleI cases, the "forbidden" transition is actually just very weak, giving the common spectroscopic observation of an electronically allowed transition next to a vibronically allowed one. To gain some insight into this phenomenon, we first look at the wave functions prepared by the transition moment on the excited states in both the diabatic and adiabatic pictures. In the diabatic representation what reaches the state le,) ( i = 1, 2) is 14,) = &Jx0). Since pgl and pg2 are nearly constants over Ixo), I&) and )412) are almost identical with Ixo), and the probability of being on surface VI is equivalent to that for being on surface V2. The same wave function can be represented in the adiabatic picture where we have

14.) = rg,lxo) PgbkO)

(18) (19)

where p and L(#b are the transition moments in the adiabatic picture &Mom of Figure 2). They determine which transition is electronically allowed and which is forbidden. 14,) and (+b) for the parallel case with q2= 0.01 au are shown in Figure 7.

1-2.9

8-A (au)

II tr8.7

6

t=l 1.6

L 1-14.5

Figure 8. Moving wave packet on one of the two diabatic surfaces V,for coupling f12 = 0.02 au and 4 = Oo.

We find that I&,) has a node along the antisymmetric stretch. The amplitude of I&) is two orders of magnitude smaller than that of 14,); this is the uibronically allowed transition. We find that the intensity of the vibronically allowed transition is very sensitive to the shape of the two diabatic surfaces and the strength of the coupling. The general rules are that if one follows the adiabatic curves (Figure 2) and finds a rapid change in electronic wave function with nuclear geometry from one diabatic state to the other, the vibronically allowed transition will be strong. This is expected to occur when the two surfaces meet at a very steep angle or/and when the coupling is small. If the electronic character does not vary much over Ixo), as would be the case for strong coupling where V, is a mixture of equivalent parts of VI and V2over Ixo),the vibronically allowed component will be very weak. Thus, the relative strength of the vibronically allowed transition contains important information about the shape of the



ijU k2.9

y'$=.0025 au

.4

1.0

_ -5.0 -- 9.0 13.0

r--

B-A (au)

0.0

10

20

30

time (fs)

Figure 9. Autocorrelation functions for b = Oo with different coupling strengths as indicated. The dashed lines are for zero coupling.

Figure 11. Moving wave packet on one of the two diabatic surfaces V, for coupling f12 = 0.01 au and @ = 1 8 0 O . Time is in f e m t m n d s and

the lines in the first frame indicate the V, potential energy surface.

.oo

.05

.IO

.20

Frequency (au)

Flgm 10. Absorption spectra for B = Oo and different coupling strengths VI2as indicated. The dashed lines are for zero coupling.

potential energy surfaces and the coupling. If we make the coupling even larger (e& 0.02 au), we observe more pronounced changes in both the dynamics (Figure 8) and the absorption spectrum (Figure 10). For the dynamics, the dissociation is even slower and a larger fraction of the original wave packet remains trapped in the FC region vibrating along the symmetric stretch. We are starting to approach the adiabatic limit. The dynamics appears to be better described as being on V, with slight prediwciation by Vb This can be seen also in the autocorrelation function (Figure 9) which shows that a larger portion of the wave packet revisits the FC region after about 19 fs. The vibrational structures in the absorption spectrum is therefore more pronounced and the entire spectrum is shifted further toward the blue as one may predict. When q2= 0.03 au, less than 1% dissociates within each vibrational period. We consider this the adiabatic limit. To summarize, we find that when the electronic hopping rate is smaller than the dissociation rate, the dynamics is mostly along the local mode. For coupling corresponding to a hopping period on the order of 4 fs the dynamics is dominated by an adiabatic, symmetric stretch motion. The parallel dipoles result in a simple

selection rule. If the coupling VI2is positive, the upper of the two adiabatic states is electronically allowed, while the lower one is vibronicaily allowed. 2. Antiparallel Transition Dipoles. Next we address the situation where two initial wave packets prepared on the two diabatic surfaces have opposite phases (8 = 180O). This can be considered as a molecule with two antiparallel transition moments randomly oriented in a laser field. We will show that both the dynamics and the absorption spectrum are very different from those of the 8 = Oo case. We start by examining the weak-coupling case (0.0025 au). As before, we find very small changes in the dynamics compared to the zero-coupling case. The absorption spectrum shown in Figure 14 is shifted to the red this time, which is the opposite trend of that seen in the parallel dipole s stem. We next examine the case with = 0.01 au. The dynamics on one of the diabatic surfaces is shown in Figure 11. It should be noted again that we are examining just one of the diabatic surfaces VI,and exactly the same process is happening on the other surface V2by symmetry. The dissociation is slightly slower but still follows a simple path similar to that of the zero-coupling situation. Closer inspection shows that a small piece of the wave packet stays longer in the FC region, while most of the wave packet is heading out along the dissociation channel. This suggests that the dynamics happen mostly on a surface that is dissociative but not "purely" repulsive. This is consistent with the dissociative adiabatic surface Vbcoupled to a bound state V,. The absorption spectrum (Figure 14) indicates that transition to the lower of the two adiabatic surfaces is allowed, while the upper surface V, is electronically forbidden. In a way very similar to that of the parallel dipole case, we find that in this case transition to V, becomes vibronically allowed. The autocorrelation function (Figure 13) also indicates a slightly slower dissociation. The absorption spectrum (Figure 14) is shifted to the red and more importantly the spectrum is no longer structureless. It appears that a very weak and diffuse vibrational structure is starting to appear on top of the absorption band. This becomes obvious when we go to an even larger coupling (0.02 au). In this case the dynamics (Figure 12) shows clearly that the wave packet does not dissociate following a simple local-mode pattern. The structure in the absorption spectrum becomes more apparent (Figure 14). This suggests that a piece of the wave packet revisits the FC region

d2

pI-7


Zhang et al.

/ - -

,;---

-

I__-

9 v

1.0

5.0

9.0 13.0

6-A (au)

.oo

.05

.10

.15

.20

Frequency (au)

Figure 12. Moving wave packet on one of the two diabatic surfaces V, for coupling flz= 0.02 au and Y, = 180’.

0.0 0.0

10

20

Figure 14. Absorption spectra for 6 = 180’ and different coupling strengths flzas indicated. The dashed line is for zero coupling.

30

time (fs)

FIpre 13. Autocorrelation functions for 6 = 180“ with different coupling strengths as indicated. The dashed lines are for zero coupling.

before the molecule dissociates. This is obvious when we look a t the autocorrelation function shown in Figure 13. There we observe a very weak recurrence after about 19 fs, which is a symmetric stretch vibrational period. The structure in the absorption spectrum is due to the vibrational motion along the symmetric stretch on the vb surface this time. The same phenomenon was found in the HzOA k absorption spectrum.26 = 0.03 au) are also shown in Results for strong coupling Figures 15 and 16. 3. Perpendicular Transition Dipoles. If the angle between the two transition dipoles 6 is neither zero or 180°, we find populations on both surfaces in either the diabatic or adiabatic representation, which makes it difficult to view the dynamics. Appendix A gives a treatment for any arbitrary angle. It is shown that the absorption spectrum for any angle can be calculated as a simple sum of that for the parallel and antiparallel cases. Figure 15 shows the absorption spectra for perpendicular transition dipoles, i.e. B = 90°. Clearly transitions to both states become allowed. In general we have found that, as the rate of excitation hopping between the two chromophores increases, the featureless absorption band splits and two electronic transitions become apparent. The spacing

-

(e2

Frequency (au)

6 = 90° and different coupling as indicated. The dashed lines are for zero coupling. Each strengths flz of the spectra is a sum of the corresponding spectra in Figures 10 and Figure IS. Absorption spectra for 14.

between the center of the two bands is twice the coupling. Both bands develop vibrational structures. The higher energy band shows very pronounced progression; even for f12 = 0.01 au the vibrational line widths are very narrow. As the vibrational structures in the higher energy band sharpen, some weak structures appear in the lower band as well. The progressions in both are of very similar frequencies. The sharp structures in the upper band for large indicate motion in a bound potential V,. The diffuse vibrational structures for the lower band are results of motion on a repulsive potential vb. The vibrational frequencies in the two bands are those of the symmetric stretches. 4. AppUention to Water Absorptioh We now apply the model to study the photodissociation process and the absorption spectrum



I

I '

Figure 16. Raman wave function for fl = 90° with incident frequency o1at the center of the lower absorption band. The coupling strengths and the laser frequenciesare indicated.

-

of HzO.The experimental absorption spectrum for the A transition in H20shows some weak diffuse structure on top of the absorption band.,' This structure is attributed to the weak oscillation along the symmetric stretch coordinate on the A-state surface from the wave packet study using a b initio If we use the above-developed model PES to perform our calculation for H,O,the only undetermined parameter is the coupling strength. Thus we vary this parameter to reproduce the experimental ab= 0.02 au or (about 4400 sorption spectrum and find that cm-l) gives the best result as compared with the experimental data as well as the calculated result using ab initio PES. This coupling strength implies an electronic energy exchange with a period on the order of 4 fs. For comparison, a vibrational period in the symmetric stretch on this surface is about 19 fs, since the vibrational frequency along the symmetric stretch is found to be around 1800 cm-I. More importantly, when looking a t the dynamics, we find that it takes the wave packet about 10 fs to leave the interaction region. This means that the two chromophores get a chance to exchange energy more than once before dissociation occurs. This puts H,O near the adiabatic limit. The Raman spectra, as we will show next, can serve as an independent test of this coupling parameter. B. Two-Photon Robe: Resonance Raman Spectroscopy. Information about the individual chromophores determines much of the surface, but the coupling strength is the crucial parameter. Since the spacing between the two adiabatic absorption bands at the symmetric configuration is twice the coupling, it seems then trivial to measure VI,from the spectrum. But when we are faced with a real molecule of reasonable size, most often the upper state is embedded in a large number of other electronic states. In many cases it cannot be identified. Moreover its spectral position may be altered due to interaction with the higher lying states. In H20 we were able to estimate the coupling just by inspection of the vibrational structure in the lower absorption band. It is possible to do this for the special case of two light oscillators with high vibrational frequencies. A change in frequency by only a factor of two would result in a disappearance of the structure altogether. What we would like is another spectroscopic probe that bypasses the above-mentioned difficulties. It has been shown l ~ e f o r e I ~ - ~ l * ~ that resonance Raman spectroscopy can be used as a sensitive tool to probe photodissociation dynamics on ultrafast time scales. This suggests that the same technique might be suitable to probe the interaction between the two chromophores. The advantage of Raman spectroscopy is that the ground electronic surface is used to probe the motion on the excited state. Since the ground state is bound, sharp structures are usually observed in the Raman spectrum, which is in contrast to the typically structureless absorption spectrum for a dissociative molecule.

Gz

.oo

.03

a i = 0.08au

j I

.06

01

= 0.02au = 0.07 au

.09

.12

Frequency shift (au) Figure 17. Raman spectra for fl = 90° with coupling strengths f12 and laser frequenciesindicated. For the Raman spectrum, it is important to consider the effect of the incident laser frequency. Since this effect has been discussed thoroughly in ref 36, we will address this issue only briefly. We have chosen the incident frequency always at the center of the low-energy adiabatic electronic absorption band. Tuning to the center of the band ensures relatively long-time dynamics and many overtones in the Raman spectrum. We did probe the high-energy electronic state as well, but only for one case. By the time the two electronic absorption bands are resolvable, the upper-state lifetime is so long that the spectrum is akin to a normal fluorescence spectrum. Moreover, in most molecules it is easy to identify the lower of the two states, while the higher energy state is often mixed with other electronic states. Thus we focus our investigationon the lower repulsive state. Thus, as the coupling is changing, we vary the incident frequency so as to remain on resonance with the repulsive excited state, to remove detuning effects. We perform the study only for the fl = 90° case, since it acceSSeS both adiabatic surfaces. By tuning exactly on resonance with the lower state, we maximize its influence on the Raman spectrum. To calculate the Raman spectrum, we return to eqs 15-17. Equation 15 gives the recipe for creating a Raman wave function at incident frequency wI. The prescription is: follow the moving wave packet I+(?)) on each of the two electronic surfaces separately, multiply each by a phase factor determined by wI,and add the wave packet at each new time step to the previous one. We obtain a wave function, I@(R,q)),with amplitude obviously only in regions where ) + ( r ) ) has visited. Figure 16 shows four Raman wave functions for couplings from qz= 0 . 0 3au. The zero-coupling limit shows amplitude only in the two local modes. As flzincreases, it becomes clear that I+(?))is headed into the symmetric stretch before it spreads into the two dissociation channels. The effect of this initial motion where both bonds increase simultaneously is retained throughout the dissociation process. It will eventually be manifested in the vibrational distribution of the diatomic product. Since the Raman spectrum is determined by both the excited state and the ground state, for a given Raman wave function (determined mainly by the excited-state dynamics), the Raman spectral features will depend only on the nature (e.g., local-mode or normal-mode character) of the ground-state surface. For instance, if the eigenstates for the ground surface are of the = 0 single progression spaced local-mode type, we will see for by the local-mode frequency. Since H20is known to be a

e2


Zhang et al.

0

VI 2=0.03au a i = 0.125 au

/?

iiA i +

H S 171.4 nm

,A, ,

Uovonumbor

Figure 18. Experimental resonance Raman spectrum of H20with excitation at 171.5 nmJ6 Note the good agreement with Figure 17 for f12 = 0.02 au and wI= 0.07 au. The first peak in this spectrum corresponds to the second high-resolutionpeak in the calculated spectrum.

local-mode molecule and we are using the H20ground-state surface, the zero-coupling spectrum (shown in Figure 17), as we predicted, shows a single progression. If the ground state on the other hand were of normal-mode character, we would observe a very different spectrum with multiple lines due to various overtones of the symmetric stretch, the antisymmetric stretch, and their combination modes. For larger coupling, the Raman spectrum becomes more complex (Figure 17). The lower panel in Figure 17 shows the Raman spectrum for f12 = 0.03 au, the large coupling limit. When we compare to the top panel for f12 = 0, we find many more lines in the spectrum. This is because the ground state is of local-mode character, while the Raman wave function is almost purely normal mode (symmetric stretch). It takes a superposition of a few local-mode states to make a normal-mode excitation, resulting in many more local-mode states carrying oscillator strength. Since the spectrum is a function of ground and excited surfaces, it appears that to interpret the spectrum for the excited-state dynamics we must know much about the ground state. This requirement is undesirable since often we do not know the nature of the higher vibrational levels of the ground electronic state. Moreover, in most molecules of reasonable size it would be impossible to resolve all the individual eigenstates. Thus in practice we would like a measure of the excited-state dynamics that is as independent of the details of the ground-state surface as possible. The key to seeing such a simplified measure is to consider the low-resolution Raman spectrum (Figure 17), which is determined by shorter time dynamics compared to the high-resolution spectrum.” For the strong-coupling case, since the Raman wave function has more normal-mode character in it, the low-resolution spectrum shows a normal-mode symmetric stretch progression on the local-mode ground state. This is exactly the case considered by Heller and Gelbart,” who showed that a low-resolution normal-mode spectrum could be built up from a pure local-mode high-resolution spectrum, if a normal-mode “pluck” of the molecule is provided. In some sense this low-resolution spectrum is a probe of the ground state along the symmetric stretch. The spacings between the low-resolution peaks give a frequency and anharmonicity that can be used to obtain an effective one-dimensional potential. Since we know that what we plucked is a pure symmetric stretch, this one-dimensional potential is a cut through the ground surface along the symmetric stretch. The frequency and anharmonicity of these low-resolution peaks correlate with a potential whose dissociation limit is A, B + A2. Returning to the weak-coupling limit, where we “pluck” the pure local mode, we see a pure local-mode progression. The line spacings correlate with a potential whose dissociation limit is A + B-A, i.e. close to half the symmetric stretch dissociation limit. To guide the eye, we have added a vertical dashed line in Figure 17 to illustrate that although the eigenstates appear a t the same

+

10

.06

.03

.12

.09

Frequency shift (au)

Figure 19. Resonance emission for excitation into the upper electronic state with wI = 0.125 au and f12 = 0.03 au. Note that the spectrum appears like a typical fluorescence spectrum.

1 .o

3.0

5.0

7.0

X (au)

Figure 20. Raman wave function for incident frequency as in Figure 19. It shows that the laser prepares an eigenstateof the V, potential surface.

energy, as they should, for all the spectra the low-resolution peaks move to higher energy with coupling, corresponding to a decrease in anharmonicity or equivalently an increase in the dissociation limit. Therefore, we have been able to map the motion on the excited state onto a simple spectroscopic observable. We look a t the low-resolution spectrum, then treat it as one-dimensional and obtain a dissociation limit for the potential. Depending on the motion on the excited state, the apparent dissociation limit will vary between the dissociation into A + B-A all the way to an energy twice this dissociation limit in the large-coupling limit. Thus far we have only studied the Raman spectra for excitation into the center of the low-energy electronic state. When the coupling is large and two distinct electronic absorption bands can be identified, it becomes interesting to look at the effect of tuning the incident frequency to the higher energy state. When the two electronic absorption bands are resolvable, the adiabatic picture becomes more appropriate. We recall that the upper adiabatic surface V, is “bound” and predissociative. We present here one example for tuning to V, at large coupling ( f12 = 0.03 au). In general we find that the Raman (fluorescence) spectra become extremely sensitive to small changes in frequency as the coupling increases. This is consistent with the longer lifetime and the sharper spectral features inducing resonance effects. In Figure 19 we show a spectrum obtained when we tune the laser to the second sharp peak in the corresponding absorption spectrum (0, = 0.125 au). We find a spectrum that looks like a typical fluorescence spectrum. The Raman wave function for this case is shown in Figure 20. It appears like an eigenstate with one quantum of excitation in the symmetric stretch. This confirms our earlier assignment of this spectrum as being the symmetric stretch on the V, surface. We now apply this general model to two molecules. We have already calculated the absorption spectrum for the H20A transition and concluded that f12 for this molecule is on the order +

(37) Heller, E. J.; Gelbart, W.G. J . Chcm. Phys. 1980, 73,626.

Spectroscopy and Dynamics of a Two-Chromophore System of 0.02 au. We are now in position to compare the Raman spectra as well. Figure 18 shows the experimental Raman spectrum as obtained by Hudson and c o - w ~ r k e r s . ~We ~ find a remarkable fit between our calculation and the experiment (compare the q2 = 0.02 au case in Figure 17 to Figure 18) for coupling on the order of 0.02 au, consistent with the absorption study. We can also reproduce this Raman spectrum with just the lower adiabatic surface, V,, suggesting that the upper state, V,,is not important for the Raman spectrum at this frequency. This calculation is also consistent with the results calculated by using an ab initio PES.26 In the purely diabatic case it takes about 10 fs to leave the interaction region (see Figure 3). A coupling matrix element of 0.02 au implies surface crossing on a 2-fs time scale, Le., the wave packet gets a chance to hop between the two diabatic surfaces manyjimes before it dissociates. This explains why treating the 2 transition in the adiabatic limit and ignoring the H20 A upper V, surface altogether was successful. The second example is the photodissociation of CH2I2.I4We have shown that it is possible to create a model for this system based on the two single C-I chromophores as determined by CHJ, leaving only the nonadiabatic coupling as a free parameter. On the basis of the absorption and Raman data, we determine a coupling between the two C-I chromophores at 0.024 au, which is again in the strong-coupling limit. We find it interesting that for the two molecules, CH212and H20, with quite different chromophores the electronic exchange rate is very similar. In both molecules the two chromophores are on the same atom. It will be interesting to compare the coupling as the chromophores are separated further as in 1,2-diiodobenzene, or 1,3-diiodobenzene, etc. We expect that as the two chromophores are moved apart we eventually recover the single-chromophore limit, i.e. f12 close to zero. Another interesting approach for tuning the exchange rate is to break the degeneracy of the two chromo hores. This can be done by a small perturbation such as in HODL b or a larger one as in I-CF2-CF2-Br. The last example was recently treated as a curve-crossing problem by Das and Tannor.) Other molecules should be ripe for .study by this methodology. For example Butler et ala6have performed very interesting emission spectroscopy experiments on H#, which they have interpreted in terms of wave packet dynamics on the excited potential energy surface.

-

V. Conclusions We have presented a systematic study of a model two-chromophore system where each of the excited chromophores photodissociates rapidly when left uncoupled to each other. When we allow electronic coupling, there is exchange of electronic excitation between them. The exchange rate is a crucial parameter, which is determined by the coupling strengths. Both the spectroscopy and the dynamics are found to be sensitive to this parameter. We examined a range of coupling strengths from weak coupling, where the diabatic picture is appropriate, through an intermediate to a strong-coupling adiabatic limit. We have examined the dynamics on two coupled diabatic surfaces and found that for the weak-coupling case the dynamics is dominated by local-mode motion. As the coupling is increased, the dynamics turns to symmetric stretch motion and the dissociation slows down. For nonzero coupling there are two distinct dissociation mechanisms/lifetimes, one 'for the lower energy regime and one for the higher. This becomes mast obvious at large coupling where the adiabatic description is good and higher energy corresponds to long lifetime in the upper bound electronic state. On the other hand the wave packet in the lower state reaches the asymptotic region only slightly later than that in the zero-coupling diabatic case. These dynamics are manifested in the absorption spectra through the appearance of two electronic absorption bands, the upper one highly structured and the lower one showing a very diffuse vibrational progression. In both spectra the vibrational structures are associated with motion in the symmetric stretch. (38) h e i o n , R. J.; Brudzynski, R. J.; Hudson, B. S.; Zhang, J. Z.; Imrc, D.G. Chcm. Phys. 1990, I l l , 393.

The Journal of Physical Chemistry, Vol. 95, NO. 16, 1991 6139 The general trend is that as the coupling is increased the two absorption bands move apart with a spacing twice the coupling. The other trend is that as the coupling is increased the vibrational structures sharpen in both absorption bands. Turning to the Raman spectrum, we were able to find a very simple spectral indicator of the dynamics in the excited state. If the coupling is weak (Le., there is slow exchange), the low-resolution emission spectrum shows a progression whose frequency and anharmonicityare of local-mode character. The dissociation energy extracted from the progression corresponds to the dissociation limit of a single A-B bond. As we increase the coupling, this apparent dissociation limit increases. In the strong-coupling limit the spectral progression is purely along the symmetric stretch whose dissociation limit is twice that of the single A-B bond. The excellent agreement between this simple model and the experimental data for H20was a pleasant surprise. A second application to CH212has already been published, where we treated the system in the adiabatic limit. We have also used O2potential curves as the two chromophores to construct the 0,IB2 surface. We find at least qualitatively very good agreement with ab initio surfaces. The formalism presented in this paper has been general enough that it can be easily extended to study other problems. For example, it can be used to study systems involving more than two coupled excited states, as occur frequently in molecules. Interaction between ultrafast laser pulses and molecules is an subject of strong current interest and is well suited for study with the formalism presented.

Acknowledgment. We gratefully acknowledge the financial support for this project by the National Science Foundation (Grant Nos. CHE-8707168and CHE-9014555) and the donors of the Petroleum Research Fund, administered by the American Chemical Society. We thank David Tannor and Horia Metiu for helpful comments on our paper. Appendix A Autocorrelation Function and Absorption Spectrum In this appendix we outline a derivation of the autocorrelation function and absorption spectrum for a system involving two excited electronic states, which is an extension of the formalism for one excited state system~.~'J~ Similar results have been derived by Coalson and Kinsey" and Metiu et al.;'J we have included this here for completeness and for consistency of notation in our paper. The wave packet prepared at t = 0 on the excited states after the electronic transition for such a system can be written as

or

where Ixo) is the initial vibrational eigenstate in the ground electronic state Ig), le,) with i = 1,2 are the two excited electronic states, Zl is the polarization vector of light, and C, and J, are the magnitude and the direction of the transition moments from the ground electronic state lg) to the excited electronic state le,). C de nds on the nuclear coordinate, while ji@is a unit vector. 'Thiznitial wave packet is propagated according to the timedependent Schriidinger equation by the Hamiltonian of the excited-state surfaces He (in atomic unit h = 1):

1W)) = e-'"++)

(A31

where 2

He = Z l e , ) ( T + Vi)(e,l I= 1

+ le2)V21(e1l+ lq)VI2(e2l

(A4)

We assume le,) with i = 1, 2 are orthonormal and le,)(e,l commutes with the nuclear kinetic energy operator T. From eqs A3 and A4 we can obtain

6140 The Journal of Physical Chemistry, Vol. 95, No. 16, 1991 2

I4(t)) = i=C1exp(-iHet) ler)( i i 8 r Z ~c8,(xo) )

(A5 )

From eqs A2 and A5 we can calculate the autocorrelation function

..

Zhang et al. If we choose the polarization vector of the light to be along the

Z axis in the spacefmed coordinate system, then it is easy to derive expressions for

iii1.9= sin(j3/2) sin 8 sin J. + cos 8 cos (j3/2) ii524= sin (j3/2) sin 8 sin J. + cos 0 cos (j3/2)

(ji;l-9)(3;2.9) = -sin2 (j3/2) The absorption spectrum is given by

(A14 (A151

sin2 8 sin2 J. + cos2 e cos2 (j3/2)

Then to find the angle averages d - m

((iigl*6)2)

Therefore, we obtain t(w)

=

=

1(iigl'z1)2

dQ

= (1 /8a2) x ' x 2 * x 2 ' ( s i n (j3/2) sin 8 sin J. +

cos~8cos (8/2)) sin 8 de d 4 dJ. = !I3 (A17) (A81 where wo is the ground-state energy and C is a constant given by C = 2/hceo with w the photon frequency, to the vacuum susceptibility, and c the speed of light. In eq A6 there are two cross terms of the form ( xolCg,*e-'HJC~lxo)

2 i= 1

= (1/8a2)x'L2'L2'(sin (j3/2) sin 8 sin $ +

649)

where the propagatQracts on the vibrational state of the electronic state le ) and produces the vibrational state on the electronic state le,). d e propagation includes the coupling VI, between the two diabatic electronic states. In the case VI2 = 0, eq A6 reduces to

(4l4W)

Similarly, we find

cos 8 cos (8/2)) sin 8 de d 4 dJ, =

(All)

=

(j3/2) sin2 8 sin2 J.

+

cos2 8 cosz (812)) sin 8 d8 d 4 d$

COS^ (j3/2) - sin2 (/3/2)]

= !I3cos j3

(A19)

Now let us come back to the discussion of the absorption spectrum. First we shall look at two special cases: parallel ' 0 and antiparallel transition moments transition moments (j3 = ) (/3 = 180'). From eq A8 for parallel transition moments the averaged absorption spectrum is

For antiparallel transition moments (j3 = 180') the averaged spectrum is

(A211

Then for perpendicular transition moments (Le., j3 = 90') the average over the cross term ((ii8,49)(ii2-Z1)) is zero, and the averaged absorption spectrum is ZgO(W) = ~ 3 C w-m~ + m e x p [ + i(w Eo)rl dt

sin 43/21

(A18)

and

= (1 /8a2) x ' x 2 ' x 2 ' ( - s i n (ii8rz,)2(X O J C ~ ~ * ~ ~ ' ~ (A ~ C10)~ , ~ X ~ )

Then, the absorption spectrum is just the sum of the absorption spectrum to state lel) and state le2). This is true also for the cases where the average of (Z&)(iig(Z!) over all orientations is zero. This happens when the two transition moments are perpendicular to each other, as will be shown next. The equation for calculating the absorption spectrum given above is for a molecule with fixed orientation. Since in most applications the molecules are randomly oriented in space with respect to the polarization vector ZIof light, we have to take average over all orientations in space. There are two averages we must consider: ((ii~.Z,)~)and ( (ii&I)(ii& ). Here we are going to digress a little bit to show how to calculate these two averages. If we take the molecule fixed (body-fixed) coordinate system such that the z axis bifurcates the angle j3 between ii and iiBl and the two transition moments are in the x-z plane, tken in the molecule-fixed coordinate system we have for the two transition moments

X

2

C

i- I (xolC~*exp(-iH,W8,Jxo)

('412)

(AW

The transformation matrix from molecules to space-fixed coordinates is39

From eqs A20-A22 it is clear that the averaged absorption spectrum for the 8 = 90' case is the sum of the two spectra for the parallel and antiparallel dipole cases, i.e.

A-' =

590(w)

cosycos$ - cosOsin$sinyl -sinylcos$ - cos8sin$cosyl s i n e s i n $ UM yl sin $ + UM 0 cos $sin yl -sin yl sin $ + cos e cos $ cos yl -sin e cos sin 9 sin yl sin 9 cos yl me (A 13)

(39)Goldatein, H.Classical Mechonfcs;Addison-Wesley: Reading, MA,

1980.

= !!2[?O(@)

+ ?180(~)1

(A23

It is also easy to show that for any arbitrary angle fl the spectrum can be calculated from Z~(W)

= Fw(w)

+ [ZO(W)

- S~O(W)] COS j3

(A24)

This means that if we calculate the absorption spectra for fl = ' 0 and for j3 = 180°, then we can calculate the spectrum for any angle j3.


Appendix B Raman Wave Function and R a m Spectrum In this appendix, we give derivations for the calculation of the Raman spectrum for two coupled surfaces. We concentrate on the Raman spectrum although this derivation is suitable for any two-photon process. We will look at the state the first photon prepares, which we call the Raman wave function, and the Raman spectrum. To calculate the Raman spectrum we first need the Raman wave function which is transferred by the transition moment to the ground electronic state and is then propagated on this surface. First, we want to obtain in general the Raman wave function on the excited surface. By definition, we have

where o1is the incident frequency and Heis the excited state Hamiltonian as given in eq A4 in Appendix A. By substituting I&(t)) from eq A5 into the above equation we can get

Then this Raman wave function I*R(~I)) is transferred back to the ground state, which corresponds to the emission of a photon, and a new wave function is generated, which is given as

The Journal of Physical Chemistry, Vol. 95, No. 16, 1991 6141 for

= 180°

I@R,l80(~1)) =

and then for /3 = 90'

Y ~+-Ae x ~ ( dt~ Cexp(-iHet))g)C~Cdlxo) ) 2

I@R.~(oJ II

i- 1

(B6) It is easy to show that the following relation holds. I@R,90(WI)) = f/2[1@R,0(WI)) + I@R,180(@I) )I For the general cases we get I@Rs(wI)) = I @ R , w ( ~ ) + [ I @ R , o ( ~ )- I @ R , ~ ~ I ))I

(B7) COS

B

(B8) Therefore, if we calculate the Raman wave functions for = ' 0 and for B = 180°, we can obtain the Raman wave functions for B = 90' or any other configuration /3 from them. Finally, to calculate the Raman spectrum we need to propagate ( @ R ) on the ground state according to the time-dependent Schrcidinger equation I*R(W) ) = exp(-iHgt))@~(wd)

(ii&) CpiCpllxo ) (B9) where H is the Hamiltonian for the ground-state surface. Then we calcutate the autocorrelation function ( * ~ l @ ~ ( t ) ) (@R(~~@R(w)) =

This formalism is general up to this point. In order to simplify our discussion for the moment we assume Za = Zl; Le., the scattered light is polarized in the same direction as the incident light. Just as in the case for the absorption spectrum, we need to take an average over all different orientations in space. Furthermore, we will consider first some special situations such as parallel or antiparallel transition moments. For B = Oo, we obtain (after taking the average over different orientations)

From this we can calculate the total Raman spectrum according to$q 19 I(w&'l)

a

w3~r-"'(@,k)l@R(wI~f))dt

(B11)

note eqs B9-B11 are for general cases. They can be simplified for specific cases such as = 90° or f = ZI,and that is the example we used in our discussions.

Spectroscopy and photodissociation dynamics of a two-chromophore

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