3201
J . Phys. Chem. 1992,96, 3201-3213
FEATURE ARTICLE Photodissociation of Water in the First Absorption Band: A Prototype for Dissociation on a Repulsive Potential Energy Surface V. Engel, Fakultat fur Physik, Albert- Ludwigs- Uniuersitat, D- 7800 Freiburg, Germany
V. Staemmler, Lehrstuhl fur Theoretische Chemie, Ruhr- Unioersitat, 0-4630 Bochum, Germany
R. L. Vander Wal, F. F. Crim, Department of Chemistry, University of Wisconsin, Madison, Wisconsin 53706
R. J. Sension,+B. Hudson, Department of Chemistry and Chemical Physics Institute, University of Oregon, Eugene, Oregon 97403
P. Andresen, Fakultat fur Physik, Universitat Bielefeld, 0-4800 Bielefeld, Germany
S. Hennig, K. Weide, and R. Schinke* Max- Planck- Institut fur Stromungsforschung, 0-3400 Gottingen, Germany (Received: October 25, 1991;
In Final Form: December 26, 1991)
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The photodissociationof water in the first absorption band, H20(%)+ hw H20(A1B1) H(2S) + OH(*II), is a prototype of fast and direct bond rupture in an excited electronic state. It has been investigated from several perspectives-absorption spectrum, final state distributions of the products, dissociation of vibrationally excited states, isotope effects, and emission spectroscopy. The availability of a calculated potential energy surface for the A state, including all three internal degrees of freedom, allows comparison of all experimental data with the results of rigorous quantum mechanical calculations without any fitting parameters or simplifying model assumptions. As the result of the confluence of ab initio electronic structure theory, dynamical theory, and experiment, water is probably the best studied and best understood polyatomic photodissociation system. In this article we review the joint experimental and theoretical advances which make water a unique system for studying molecular dynamics in excited electronic states. We focus our attention especially on the interrelation between the various perspectives and the correlation with the characteristic features of the upper-state potential energy surface.
I. Introduction Advances in theoretical and experimental techniques for studying photodissociation now permit extremely detailed inquiry into the decomposition of molecules in electronically excited states1** Photodissociation is practically important in areas as diverse as atmospheric chemistry and the study of interstellar masers, but it has a fundamental allure as well. Fully quantum mechanical calculations of the dissociation dynamics of a molecule on an ab initio potential energy surface (PES) offer the opportunity to understand the dissociation process in complete detail, and fully resolved experiments provide the means of testing that insight.j Such a stringent comparison requires several ingredients. The theorist must calculate the PES as well as solve the equations of motion to predict the course of the decomposition at the level of individual quantum states, and the experimentalist must perform the equivalent measurement with full quantum state resolution in excitation and detection. Theory and experiment together provide a m a n s of unravelling the dissociation dynamics of small polyatomic molecules at the most fundamental level. 'Present address: Department of Chemistry, University of Pennsylvania, Philadelphia, PA 19104.
Dissociation of water following excitation to its first electronically excited state
H20(k1A!) + hw
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-
H20(A1BI)H(2S) + OH(Xzll,nj)
(1) provides a prototypical example of the complete theoretical analysis and experimental observation of the fragmentation of an electronically excited triatomic molecule. Water is a particularly attractive molecule for detailed investigations because of the relative simplicity of its first excited-state PES and the attendant dissociation dynamics. The electronic absorption spectrum for water shown in Figure 1 suggests two simplifying aspects of its photodissociation. T_he transition to the lowest electronically excited singlet state (A) appearing near 165 nm (Ephoton = 7.5 eV) is well sepalated from the transitions to the next two singlet states (B and C) and has ( 1 ) Dynamics of Molecular Photofragmentation. Furuday Discuss. Chem. SOC.1986, 82. (2) Molecular Photodissociation Dynamics; Ashfold, M. N. R., Baggott, J. E., Eds.; Royal Society of Chemistry: London, 1987. (3) Schinke, R. Photodissociation Dynamics; Cambridge University Press: Cambridge, 1992.
0022-3654/92/2096-3201.$03.00/00 1992 American Chemical Society
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The Journal of Physical Chemistry, Vol. 96, No. 8, 1992
Engel et al. (a)
\ABSORPTION DETECTION
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u 122nm
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Figure 1. The a-bs?rption_spectrumof water as function of the photon energy Ephoton. A, B, and C indicate the three lowest absorption bands. Redrawn from Giirtler, et al. Chem. Phys. Lett. 1977, 51, 386.
little structure. The simplest interpretation of this isolated transition is that it reaches a single state and, hence, that the dissociation dynamics is not complicated by the interaction of multiple electronic states. The corresponding spectrum shows only broad and shallow oscillations (see Figure 5 for a-spectrup with better resolution) whereas the spectra for the B- and C-state transitions are both structured. The absence of structure suggests that the dissociation in the A state is direct, proceeding rapid!y to form products after initial excitation; Le., the lifetime of H,O(A) is smaller than an internal vibrational period. Measurements and ab initio calculations have provided a wealth of information on the photodissociation of water (and its isotopes) through the A state. In increasing level of detail, the points of comparison are as follows: 1. The absorption spectrum with its slight structures; 2. The distribution of the products among their quantum states; 3. The effect of different initial rotational and vibrational excitations on the course of the dissociation; 4. The OH/OD branching ratio in the dissociation of HOD; 5. The Raman spectrum that probes the molecule as it evolves toward fragments. Each point provides distinct information about the fragmentation dynamics and the combination of all different types of observables leads to a most complete picture. Because spectroscopic detection of the OH fragment provides the relative populations of its vibrational, rotational, spin-orbit, and A-doublet states, it is possible to compare theoretical calculations with fully state resolved photodissociation measurements. One can perform calculations and measurements at comparable level of detail for process (1) by specifying the initial state prior to electronic excitation and measuring the relative populations of the product quantum states. Thus, there are measurements of the product state distributions for the photodissociation of single rotational, stretching, and bending states of water. The combination of experimental and theoretical tractability has motivated a considerable amount of fruitful collaboration during the past decade involving several research groups. In this review, we summarize the theoretical and experimental work that has made the photodissociation of water a prototype of direct decomposition in a polyatomic molecule. The earlier review of Andresen and Schinke4 focused primarily on the rotational state distributions of the OH product and the time-independent quantum mechanical calculations. The experimental advances since 1987 (dissociation of highly excited states of water, the dissociation of HOD, and the emission spectrum of the complex while breaking apart) as well as the application of the time-dependent quantum mechanical picture5 make a (final) review very timely. The example of water illustrates how the various facets of a half-collision can be investigated with modern experimental methods and how the confluence of theory and experiment advances the general understanding of molecular dynamics. 11. Experimental Approaches Figure 2 illustrates several experimental approaches schematically on an one-dimensional cut along the 0-H coordinate (4) Andresen, P.; Schinke, R. In ref 2. ( 5 ) Imre. D. G.;Zhang, J. Chem. Phys. 1989, 139, 89
(c I
\
INITIAL STATE SELECTION-
\\
Id1
L.,.RAMAN SCATTERING
Figure 2. Illustration of the four different experimental approaches employed to investigate the photodissociation of water in the first absorption band. Schematically shown are one-dimensional cuts through the ground and the first excited-state potential energy surfaces. The distribution on the left-hand side of each figure represents the thermal distribution of the ro-vibrational states in the ground electronic state.
on the ground and first electronically excited potential energy surfaces. The simplest experimental view of the dissociating water molecule is through its ultraviolet absorption spectrum, as illustrated in Figure 2a, The breadth of the feature corresponding to excitation of the A state (seeFigure 1) indicates that the excited molecule rapidly dissociates. The distribution of initial states of water from which the absorption occurs also potentially influences the spectrum since it is an average of the contributions from all of the populated states. Experimentally investigating the dissociation in greater detail requires more highly resolved measurements, which ideally determine the quantum states of the products while specifying the initial state from which the photoexcitation occurs. The next level of detail comes from measuring the population of the quantum states of the OH product by laser-induced fluorescence (LIF) as indicated in Figure 2b. For example, Andresen and mworkers have monitored the O H fragments from the 157-nm photolysis of water at room temperature6*' and cooled in a supersonic expansion' in order to learn about vibrational, rotational, and A-doublet state populations of the products. The latter, in particular, have provided insight into the geometry and forces in the dissociation. However, even if these experiments are performed in a beam the inevitable averaging over at least a few overall rotational states of water in the ground electronic state leads to substantial suppression of finer details which, in principle, are necessary to get to the bottom of the bond-breaking process. Photodissociation from a selected initial state, illustrated in Figure_2c, avoids the averaging over several quantum states of H,O(X)and therefore sharpens the comparison with theory even further. Andresen and c o - w ~ r k e r s have * ~ ~ prepared a single rotational state of the antisymmetric stretching vibration by infrared excitation and photolyzed molecules in that state with 193-nm light to determine the pathways of photodissociation from a single ro-vibrational state. This experiment involving three lasers-one to preexcite the water molecule, one to promote it into the A-state continuum, and a third laser to probe the O H product-permits the clearest comparison with the theoretical predictions. An extension of this approach using vibrational overtone excitation'OJ' ( 6 ) Andresen, P.; Rothe, E. Chem. Phys. Lerr. 1982, 86, 270. (7) Andresen, P.; Ondrey. G. S.; Titze, B.; Rothe, E. W. J . Chem. Phys.
1984, 80, 2548. (8) Andresen, P.; Beushausen, V.; Hausler, D.: Liilf, H. W.: Rothe, E. W. J . Chem. Phys. 1985, 83, 1429. (9) Hausler, D.; Andresen, P.; Schinke, R. J . Chem. Phys. 1987,87, 3949. (IO) Vander Wal. R. L.; Crim, F. F. J . Phys. Chem. 1989, 93. 5331.
Feature Article allows comparison among vibrational states, some having very similar energies but rather different nuclear motions, to demonstrate the control that the eigenstate from which the excitation occurs exerts on the dissociation. It is even possible to exploit this control to break a chemical bond selectively, as demonstrated in ultraviolet dissociation experiments that use vibrational overtone excitation1*J3or stimulated Raman scatteringI4 to excite the 0-H bond in HOD in order to cause its preferential cleavage in the photolysis. Raman scattering experiments15J6(Figure 2d) in which the tiny amount of light emitted from the dissociating molecule is dispersed go beyond probing the asymptotic states and offer a view of the state evolution in the course of the photodissociation. Different excitation wavelengths probe the dynamics at different times during the state evolution. For example, excitation far from the maximum in the spectrum shows only a single feature reflecting early times, but excitation more nearly in resonance with the excited state produces structured emission showing the evolution of stretching excitation on the way to dissociation. The absorption spectrum, resonance emission studies, and asymptotic state measurements probe a hierarchy of time scales in the dissociation: The short-time dynamics control the absorption spectrum, intermediate-time behavior dominates the Raman experiments, and the long-time dynamics influence the final state distributions. As the results described in the following sections show, the directly dissociative character of the excited-state surface is the singular feature of water that allows us to understand the behavior at all of the stages of the dissociation.
The Journal of Physical Chemistry, Vol. 96, No, 8, 1992 3203 2.5
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111. Potential Energy Surfaces and Transition Dipole Moment
Function The quantum mechanical treatment of the nuclear dynamics of water requires the potential energy surfaces for the XIAI and the AIBI electronic states as well as the associated coordinatedependent transition dipole moment function jiXA. The ground-state PES is needed to, calculate the initial bound-state wave functions of the H20(X) parent molecule. In most calculations the semiempirical potential of Sorbie and Murrell" was employed. Other ground-state potentials'* have been also used, but unlike infrared spectra the influence of the detailed form of the ground-state PES on the observables measured in dissociation experiments with ultraviolet light have been found to be insignificant. Essential for a realistic description of fragmentation processes is a global PES for the excited state including all three internal degrees of freedom. It must cover the entire range in coordinate space sampled by the molecule as it breaks apart, from the Franck-Condon (FC) region out into the asymptotic channel where the fragments are irreversibly broken up. As p_ointedout in the Introduction and manifested by Figure 1, the AIB, state is well separated from other electronic states. This simplifies the quantum chemical calculation of the potential considerably since curve crossings are absent and nonadiabatic coupling is negligibly small. Furthermore, the total number of electrons of H20is small enough to permit electronic structure calculations at a high level of accuracy. ( I I ) Vander Wal, R. L.; Scott, J. L.; Crim, F. F. J. Chem. Phys. 1991, 94, 1859. (12) Vander Wal, R. L.;Scott, J. L.; Crim, F. F. J. Chem. Phys. 1990, 92, 803. (13) Vander Wal, R. L.; Scott, J. L.; Crim, F. F.; Weide, K.; Schinke, R. J. Chem. Phys. 1991, 94, 3548. (14) Bar. 1.; Cohen, Y.; David, D.; Rosenwaks, S.;Valentini, J. J. J. Chem. Phys. 1990, 93, 2146. Bar, I.; Cohen, Y.; David, D.; Arusi-Parpar, T.; Rosenwaks, S.; Valentini, J. J. J. Chem. Phys. 1991, 95, 3341. (15) Sension, R. J.; Brudzynski, R. J.; Hudson, B. S. Phys. Reo. Left. 1988, 61, 694. (16) Sension, R. J.; Brudzynski, R. J.; Hudson, B. S.; Zhang, J.; Imre, D. G. Chem. Phys. 1990, 1 4 1 , 393. (17) Sorbie, K. S.;Murrell, J. N. Mol. Phys. 1975, 29, 1387; 1976, 31, 905. (18) Reimers, J. R.; Watts, R. 0. Mol. Phys. 1984, 52, 357. Coker, D. F.; Watts, R. 0. J. Phys. Chem. 1987, 91, 2513. Coker, D. F.; Miller, R. E.; Watts, R. 0. J. Chem. Phys. 1985.82. 3554.
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Figure 4. Angular dependence of the h a t e potential energy surface for several H-OHseparations R. The Jacobi coordinates R, r, and y are defined in the text. Shown is the interaction potential VI = V - V(R=m) for fixed OH separation r. Also displayed is the wave function &(y) for the lowest bending state of H,O(X). Adapted from ref 2 5 .
The A'B,-state PES was determined by Staemmler and Palma19 using quantum chemical ab initio methods with electron correlation effects being included at the CEPA level. Energies for about 250 nuclear configurations were calculated with both OH separations, ROH,varied between 1.6 and 4.0 a. and HOH bending angles ranging from 0' to 180'. The points were fitted to an analytical expression which was utilized ic the subsequent dynamics calculations. Figure 3 depicts the A-state PES as a function of the two OH separations for a fixed bending angle of 104.5', which is the equilibrium bond angle in the X'Al ground state. The potential is sym-metric with respect to the C, symmetry line. In contrast to the X state which has a deep minimum when all three atoms are close together, the A-state potential possesses a barrier located on the symmetry line at an OH distance of 1.09 A. The barrier height relative to the H + OH(r,) asymptote is 1.98 eV. Since the potential is purely repulsive and rather steep in the FC (19) Staemmler, V.; Palma, A. Chem. Phys. 1985, 93, 63.
Engel et al.
3204 The Journal of Physical Chemistry, Vol. 96, No. 8, 1992
region, the dissociation is direct and fast giving rise to the broad absorption spectrum. The arrow in Figure 3 illustrates a typical classical trajectory which starts in the FC region. Figure 4 depicts the angular dependence of the potential. Shown is the interaction potential V,(R,y,r,)= V(R,y,r,)- V(R=-,y,r,) for several values of the dissociation coordinate R. Here, R is the distance between the recoiling H atom and the center of mass of OH, re is the (fixed) equjlibrium distance of OH, and y is the angle between the vectors R and P. R , r, and y are the scattering of Jacobi coordinates appropriate to describe the fragmentation into H and OH. The variation of the potential with y (the anisotropy) determines the degree of rotational excitation of the O H fragment during the break up. Although the A-state PES is generally quite anisotropic, it is remarkable that the equilibrium bond angle in the FC region R = r = 1 A is almost the same as in the electronic ground state. This has the consequence that the molecule_does not noticeably change its bond angle when excited to the A state. Since the anisotropy near the ground-state equilibrium remains small throughout the fragmentation for all values of R, it follows that OH is dominantly produced in low rotational states. The transition dipole moment function ,iiMwas also cal$ulat$ quantum chemically, but only a t the S C F level.20 The X A transition is a perpendicular transition and therefore only the component of the transition dipole moment vector perpendicular to the molecular plane is nonzero. Near the FC region, where the electronic transition takes place, jiXAdepends only weakly on the nuclear coordinates such that the Condon approximation of a constant transition dipole function is trustworthy. (In what follows we will view the transition dipole function as Wing a scalar function.) This holds true, howevy, only for the excitation of the lowest vibrational state of H,O(X) whose extension in the coordinate space is rather limited. The dissociation of excited vibrational states or the calculations of the emission spectrum of the dissociating water molecule requires the full coordinate dependence of PXA. In conclusion, the &state PES has a barrier at short 0-H separations and is purely repulsive along the dissociation path. It is rather flat near the equilibrium angle in the electronic ground state such that the torque dV/ay is small throughout the collision. Because of the low degree of excitation in the bending (or rotational) coordinate one can safely separate the OH stretching from the bending motion. This allows one to freeze the angular degree of freedom for rotationally summed cross sections and to perform dynamical calculations which include only the two O H stretches. This reduction from 3D to 2D simplifies the numerical calculations significantly.
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IV. Dynamical Calculations In what follows we will use atomic units in which h = 1. A. Absorption Spectrum and Finnl State LXshibutions. Within first-order perturbation theory for the light-matter interaction the partial dissociation cross section for absorbing a photon with frequency w and producing the fragments in a particular internal state is given by Fermi's Golden Rule3*21~22 oh(Elnje) a wl(q,(EInie)Ir,Iqi)12 (2) Here, 13,)is the initial bound-state wave function with energy Ei in the electronic ground state with the index ( i ) collecting all quantum numbers which specify the particular vibrational and rotational state. p, denotes the component of the transition dipole moment function PxA along the polarization vector of the electric field. The continuum wave function le,(Elnje)) is an eigenstate of the Hamiltonian HA in the excited electronic state with total energy E = Ei + w . It describes the dissociation into a particular chemical channel designated by the index 8-H + OD or D + OH in the dissociation of HOD, for examplewith the diatomic (20) Engel, V.;Schinke, R.; Staemmler, V. J . Chem. Phys. 1988,88, 129. (21) Weissbluth, M.Atoms and Molecules; Academic Press: New York, 1978. (22) Loudon, R. The Quantum Theory of Light; Oxford University Press: Oxford, U.K.,1983.
fragment being produced in an internal vibrational (n), rotational (j), and electronic ( e ) quantum state. Both wave functions are solutions of the time-independent SchrMinger equation including all nuclear degrees of freedom. While (ei) is a bound wave function Iq8(Elnje))is a dissociative wave function with welldefined boundary conditions in the limit that the H-OH separation goes to infinity.23*24 The total dissociation cross section (Le., the spectrum) for any initial state ( i ) is the sum of all partial cross sections,
a'(,?) = Cd@lnje) 8njc
(3)
It depends only on the energy E and the initial state of the parent molecule, but not on the final state of the fragments. Therefore, it contains less information about the bond rupture than the partial cross sections. The probability for producing the O H fragment in a particular final state is (4)
Finally, one may calculate the thermally averaged total cross section of water for a given temperature T defined by CAE) = Cgi(T)d(E) i
(5)
where the coefficients gi( T) measure the thermal population of the eigenstate The definition for the partial cross sections is equivalent. The computation of the continuum wave functions Iq8(Elnje)) is by far the most demanding step in the calculation of dissociation cross sections. Practical methods employing hyperspherical coordinates (not including the electronic degree of freedom of OH(Q)) have been described in ref 20. As a consequence of the low degree of rotational excitation of OH one can utilize the so-called infinite-order sudden approximation in which the angular degree of freedom is decoupled from the two stretching degrees of freedom.25 As an alternative to the time-independent approach one can calculate the various photodissociation cross section also in the time-dependent picture pioneered by HelleraZ6 In the time-dependent method one solves the time-dependent SchrMinger equation for the wavepacket @,(t)in the excited electronic state with the initial condition @i(0)= p,B,. The wavepacket must be propagated in time until it travels freely in the fragment channels. The total absorption cross section is then proportional to the Fourier transform of the autocorrelation function c ( t ) , Le.,
with c ( t ) being
(7) The autocorrelation function is the scalar product of the initial wavepacket at time t = 0 and the evolving wavepacket at time t. Equation 6 establishes the interrelation between the time and the frequency domain. The autocorrelation function is the link between the time-dependent molecular dynamics on the one hand and the spectrum on the other hand. The various partial dissociation cross sections can be also obtained from the single wavepacket by projecting @,(t)in the limit t onto the stationary eigenfunctions of the asymptotic Hamiltonian. The latter are products of outgoing plane waves in the dissociation coordinate R and the vibrational-rotational eigenfunctions of the OH fragment.j The time-independent and time-dependent approaches are strictly equivalent and yield the same cross ~ections.~ They merely provide different views and means of calculation and interpretation.
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(23) Shapiro, M.;Bersohn, R. Annu. Reu. Phys. Chem. 1982, 33, 409. (24) Schinke, R. In Collision Theoryjor A t o m and Molecules; Gianturco, F. A., Ed.; Plenum Press: New York, 1989. (25) Schinke, R.; Engel, V.;Staemmler, V. J . Chem. Phys. 1985,83,4522. (26) Heller, E. J. J . Chem. Phys. 1978,68, 2066, 3891; Acc. Chem. Res. 1981, 14, 368; in Potential Energy Surfaces and Dynamics Calculations; Truhlar, D. G . , Ed.; Plenum Press: New York, 1981.
The Journal of Physical Chemistry, Vol. 96, No. 8, 1992 3205
Feature Article Both methods have been applied to the dissociation of water in Being an initial value problem rather the first than a double-ended boundary value problem, the time-dependent method is generally simpler to interpret. This holds Rarticularly for systems with short survival times such as H20(A) because under such circumstances the evolving wavepacket follows essentially a classical trajectory. Finally, we should add that for a directly dissociating molecule the purely classical approach in which swarms of trajectories are run on the excited-state PES gives also quite reliable absorption spectra and final vibrational-rotational state distributions for ~ a t e r . ~Moreover, * ~ ~ running classical trajectories always provides the simplest picture of a fast dissociation process. B. Raman Spectra. Since Raman processes involve two photons, their description requires second-order perturbation theory for the light-matter interaction. In the time-independent picture, Raman cross sections are calculated according to the KramersHeisenberg-Dirac expression2'
The transition occurs between the two bound states I\ki) and l\kl) within the electronic ground state. It is mediated by excitation of the continuum states I\k&Elnje)) in the excited electronic state. w and w' are the frequencies of the absorbed and emitted radiation. The Raman amplitude contains the same kind of matrix elements as the absorption cross section and therefore the same computer programs can be used. In analogy to the absorption cross section one can rewrite the time-independent expression for the Raman amplitude in a time-dependent form,33-35
eiw'a,/(t)
where aif(t)is the cross-correlation function defined as
aiht) =
- theory
--- experiment
u)
r C
3
P
e
1500
1600
1700
1800
1900
A [AI Figure 5. Comparison of the measured spectrum of W a n g et and the calculated spectrum of Engel et Also shown are the vibrationally resolved partial dissociation cross sections. The total cross section is the sum of all partial cross sections. Reprinted with permission from ref 20. Copyright 1988 American Institute of Physics.
where the amplitude has the form
t 8 w ) = Jmdt
,
(p,\kje-""A'lpc,\kj) = ( @ A O ) l @ i ( r ) )
(1 1)
The Raman amplitude is thus proportional to the half Fourier transform of the cross correlation function. The latter is similar to the autocorrelation function except that the state at t = 0 corresponds to a state of the parent molecule that is different from the state from which the evolving wavepacket originates. V. Absorption Spectrum According to the hierarchy of measurements outlined in section I1 we begin the comparison between experiment and ab initio theory with the total absorption cross section. The spectrum of water in the first absorption band has been known for more than 40 years.36 Figure 5 shows the comparison between the measured spectrum of Wang et aL3' and the calculated spectrum of Engel et alez0It consists of a broad continuum with a maximum around (27) Henriksen, N . E.; Zhang, J.; Imre, D. G. J. Chem. Phys. 1988, 89, 5607. (28) Zhang, J.; Imre, D. G. J . Chem. Phys. 1989, 90. 1666. (29) Zhang, J.; Imre. D.G.; Frederick, J. H. J. Phys. Chem. 1989, 93, 1840. (30) Engel. V . ; Schinke, R. J. Chem. Phys. 1988, 88, 6831. (31) Engel, V.; Schinke. R.; Staemmler, V. Chem. Phys. L e r r . 1986, 130, 413. (32) Guo, H.; Murrell, J. N. Mol. Phys. 1988, 65, 821; J . Chem. Soc.. Faraday Trans. 2 1988, 84, 949. (33) Lee, S.-Y.; Heller, E. J. J . Chem. Phys. 1979, 7/, 4777. (34) Tannor, D. J.; Heller, E. J. J . Chem. Phys. 1982, 77, 202. (35) Heller, E. J.; Sundberg, R. L.; Tannor, D. J. J . Phys. Chem. 1982, X6 - -, 1822 .- - - . (36) Watanabe, K.; Zelikoff, M. J . Opr. SOC.Am. 1953, 43, 753. (37) Wang, H.-T.; Felps, W. S.; McGlynn, S . P. J . Chem. Phys. 1977,67, 2614.
165 nm with a superimposed progression of shallow but regular undulations. The comparison of the calculated with the measured spectrum, which contains no adjustable parameter, is almost perfect and thus underlines the high accuracy of the ab initio PES in both, the vertical energy separation from the ground state and the shape in the FC region. The position on the energy axis, the width, and even the weak undulations are satisfactorily reproduced by the calculation. The slight deviation a t long wavelengthsjs very likely due to absorption by excited rotational states of H20(X) in the experiment which was performed at room temperature while only the rotational and vibrational ground state was included in the theory. A Boltzmann average according to eq 5 makes the agreement even better.20 The same good agreement is found for D20.20 The broad spectrum points to a direct and fast dissociation mechanism and indeed the autocorrelation function calculated by Henriksen et al.27decays to zero within a few femtoseconds. Classical trajectories launched in the FC region (a typical example is displayed in Figure 3) slide down the potential slope and dissociate immediately. The slight undulations, however, indicate short-time trapping of a small part of the wavepacket or a tiny fraction of the classical trajectories which results in a small recurrence of the autocorrelation function after about 19 f ~ . ~ ~ According to the basic properties of the Fourier transformation the recurrence time is related to the energy spacing AE of the undulations in the frequency domain by T = 2rjA.E. What type of internal molecular motion is temporarily excited in the transient complex and thereby leads to these diffuse vibrational structures? This is one of the most intriguing questions if the absorption spectrum of a polyatomic molecule shows slight undulatory structures.38 Wang et al.37interpreted them as the result of selective excitation of the bending mode in the electronically excited molecule. The ab initio calculations definitely rule out this assignment since the structure do not vanish even if we artificially fix the bending angle! On the contrary, the diffuse structures become yet more pronounced if the bending angle is frozen because the averaging over angle within the sudden approximation, which naturally tends to damp any structures, is suppressed.20 The weak undulations are due to temporary excitation_of the symmetric stretching mode on top of the barrier of the A-state PES as predicted by in the time-independent picture, and by Heller," in the time-dependent picture, for general symmetric ABA molecules. The wavepacket calculations of Henriksen et al.*' clearly verify this picture for the dissociation of water in the first absorption band. A very small portion of the evolving wa(38) Schinke, R.; Weide, K.; Heumann, B.; Engel, V. Faraday Discuss. Chem. Soc., in press. (39) Pack, R. T. J. Chem. Phys. 1976, 65, 4765. (40) Heller. E. J. J. Chem. Phys. 1978, 68, 3891.
3206 The Journal of Physical Chemistry, Vol. 96, No. 8, 1992
"1
1
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I theory
5
n
Figure 6. Calculated vibrational state distributions for several photolysis wavelengths.20The original distribution of Andresen et al.' for 157 nm had been later corrected in ref 4. Experiment and theory have been normalized to the same value for n = 0.
vepacket performs one symmetric stretch oscillation along the barrier separating the two dissociation channels. This leads to a recurrence after about 19 fs which in turn gives rise to the undulations in the spectrum. The curvature along the antisymmetric stretch coordinate (or, more precisely, along the minimum energy path from one H + OH channel to the other) determines the number of recurrences and therefore the diffuseness of the vibrational structures. Very recently, Braunstein and Pack4' applied the model of Pack using parameters derived from the ab initio PES and obtained very good agreement with the calculated spectrum of Engel et aLzo
VI. Vibrational Distributions Summing the partial dissociation cross sections over all fragment quantum numbers except for the vibrational quantum number (n) leads to the vibrationally resolved dissociation cross sections .'(,!?in) = Ca$Elnje)
(12)
Bje
which are also shown in Figure 5. This figure lucidly illustrates how the total spectrum is composed of the individual partial cross sections each of which consists of a broad background and basically one single undulation. The shift of the partial cross sections on the energy (or wavelength) axis reflects the quantization of the symmetric stretch motion on top of the barrier of the upper-state PES and can be qualitatively explained in an adiabatic-type picture.39 Adding the various @In) leads to the quite regular progression of diffuse structures in the total spectrum. Within the time-dependent picture the undulation of each partial cross section can be interpreted as an interference between the dominant part of the wavepacket which dissociates immediately and the much smaller part of the wavepacket which is trapped in the barrier region for about one vibrational period. Figure 6 depicts calculated vibrational-state distributions of the O H product for several excitation wavelengths.20 They are simply cuts through the partial cross sections of Figure 5 at the respective wavelengths. The degree of vibrational excitation clearly increases with decreasing photolysis wavelength and the evident variation with total energy is in accord with the trajecJory picture: Excitation with energies below the barrier of the A-state PES launches the classical trajectories in either of the two H + O H channels. They at once rush down into the exit channel and because the minimum energy path is not much curved at distances slightly away from the barrier they lead to vibrationally unexcited OH radicals. Increasing the energy has the effect of starting the trajectories further up the inner slope of the transition-state region, more and more displaced from the saddle point toward ever smaller O H separations. Thus, in the very first moments after the trajectories are released both O H separations elongate simultaneously (symmetric stretch motion) and lead to acceleration in a direction perpendicular to the reaction path. One OH bond is ruptured and the other one is left vibrationally excited. It must be underlined at this point that purely classical Monte Carlo calculations give vibrational-state distributions in perfect agreement with the quantum result^.^^*^^ (41) Braunstein, M.; Pack, R. T. J . Chem. Phys. 1992, 96, 891
Engel et al. Until recently, measurements of vibrational-state distributions were available for comparison with theory for only two photolysis wavelengths, 193 and 157 nm.719942 Due to predissociation in the OH(2Z) state merely the n = 0:n = 1 branching ratio rather than complete distributions could be determined by the normal LIF method. In accord with the classical arguments of above, only OH(n=O) fragments have been detected at the long wavelength. At 157 nm, however, which is well beyond the maximum of the absorption spectrum, Andresen et al.' found a n = 0:n = 1 ratio of about 1:l in very good agreement with the calculation. Very recently, Mikulecky et al.43 reconsidered the 157-nm photolysis using nondiagonal pumping to probe also the higher vibrational states of OH(211) and thereby measured the complete vibrational distribution. They obtained a much "colder" distribution which decays to zero remarkably faster than the calculated distribution. Their n = 0:n = 1 ratio is approximately 1.8 in striking disagreement with the calculation" as well as the earlier measurement of Andresen et ale7 Furthermore, two-photon excitation with 355 nm (corresponding to 177 nm) yields within the experimental uncertainty no measurable OH(n= 1) products" while the calculation gives a substantial amount of vibrationally excited O H fragments (see Figure 6). We will come back to this puzzle in the summary section.
VII. Rotational Distributions The question of rotational excitation of the OH radical has been investigated from several perspectives. It is ideally suited to elucidate some fundamental aspects of direct dissociation which are likewise applicable for other systems. The final distribution of rotational states of the OH product is determined by three 1. The overall rotation of H 2 0 ( k ) before the photoabsorption. 2. The bending vibration in the ground electronic stcte. 3. The anisotropy, i.e., the angular dependence of the A-state PES, which generates a torque aV/&ywith y being the orientation angle of H with respect to OH. Figure 4 depicts the upper-state potential V(R,r,y) as a function of y for several H-OH distances R and fixed internuclear separation of the O H fragment, r. It is rather flat in the FC region around 104' for all H-OH distances which has the consequence that rotational excitation or deexcitation in the exit channel is very The so-called exit channel interaction (concerning rotational excitation) is very weak throughout the fragmentation. The final rotational state distribution of O H reflects therefore essentially overall rotation and internal bending vibration of the parent molecule in the ground electronic state. A. Tbe Franck-Condon Picture. In the absence of translational-rotational coupling in the exit channel the final rotational state distribution can be described within the FC limit.2s-a-51 Let us first discuss the case without overall rotation. The (unnormalized) rotational state distribution of O H is then approximately
where &(y) is the bending wave function of H 2 0 ( g ) in bending state k = 0, 1, ..., and Yp(y,O) is the free-rotor wave function of OH after the fragmentation. Assuming that the bending motion (42) Grunewald, A. U.; Gericke, K.-H.; Comes, F. J. Chem. Phys. Letf. 1987, 133, 501. (43) Mikulecky. K.; Gericke, K.-H.; Comes, F. J. Chem. Phys. Left.1991, .1x2. - -, -290 - -. (44) Mikulecky, K.; Gericke, K.-H.; Comes, F. J. Ber. Bunsenges. Phys. Chem. 1991, 95, 927. (45) Schinke, R. J. Phys. Chem. 1986, 90, 1742. (46) Schinke, R. Annu. Rea. Phys. Chem. 1988, 39, 39. (47) Schinke, R. Comments At. Mol. Phys. 1989, 23, 15. (48) Freed, K. F.; Band, Y. B. In ExciredSfafes;Lim, E. C., Ed.; Academic Press: New York, 1978; Vol. 3. (49) Beswick, J. A.; Gelbart, W. M. J . Phys. Chem. 1980, 84, 3148. (SO) Morse, M. D.; Freed, K. F. Chem. Phys. Lefr. 1980. 74, 49. ( 5 1 ) Schinke, R.; Vander Wal, R. L.; Scott, J. L.; Crim, F. F. J . Chem. Phys. 1991. 94, 283.
The Journal of Physical Chemistry, Vol. 96, No. 8,1992 3207
Feature Article
o,3
1
k =2
0 exp.
103- 2 >
0.2%
j
Figure 7. Measured final rotationabstate distributions of OH for disso-
ciation of different initial H,O(X) vibrational states as indicated (local-mode assignment) with k being the bending quantum number (open circles). In all cases the total angular momentum of water in the ground electronic state is zero and the antisymmetricA-doublet state of the 2113,2spin-orbit state of OH is probed. For comparison also the FC distributions obtained from eq 14 without the sinusoidal factor are shown (closed circles). Reprinted with permission from ref 51. Copyright 1991 American Institute of Physics. is harmonic (with frequency wHoand ‘mass” f i defined by 1 f i = (l/p2,) (l/mR,Z)the FC distribution is approximatelyks’
+
Re,re,and yeare the equilibrium Jacobi coordinates of the ground electronic state and p and m are the reduced masses of OH and H OH, respectively. Equation 14 states that the average rotational state distribution is proportional to the angular distribution of the corresponding bending wave function with the angle y being replaced by the argument j / w H o t . The sinusoidal term causes superimposed fast oscillations which we will ignore for the time being. The FC model can be considered as a zeroth-order approximation. Any deviation of measured distributions from the FC model manifests the influence of exit channel coupling. Figure 7 shows the FC distributions defined in eq 14 without the sin2 [...I term for the three lowest bending states. The distribution for k = 0 is simply a Gaussian centered at j = 0 whose width is inversely related to the breadth of the bending wave function. The FC distributions following the dissociation of excited bending states, on the other hand, show distinct structures which reflect the nodal pattern of the corresponding bending wave functions in the angular space. Dissociation of the k = 0 state probes only a narrowJegion around the equilibrium angle where the anisotropy of the A-state PES is negligibly small with the result that the rotational-state distribution of OH does not significantly change during the fragmentation. The final rotational distribution consequently resembles closely the initial Franck-Condon distribution.2s The measured datas1 shown in the upper part of Figure 7 for the dissociation of the lowest bending state clearly confirm this prediction. (The experiments of Schinke et aLsl were performed with thegoal of investigating the influence of bending excitation of H20(X) on the OH rotational-state distribution. The additional excitation by three or four quanta of OH stretching excitation in the experiment is unimportant because of the apparent sepa-
+
ration between OH stretching and r ~ t a t i o n . ~ ) In the dissociation of excited bending states the wavepacket or the trajectories sample a wider range of orientation angles and therefore the anisotropic portion of the upper-state PES becomes gradually more influential. Consequently, the measured distributions differ noticeably from the FC model. The deviations between the FC theory and the experimental distributions for the dissociation of the k = 1 and 2 bending states, depicted in the middle and the lower part of Figure 7, elucidate the effect of increasing the final state interaction. Interestingly, the torque imparted to the rotor decreases rather than increases the degree of rotational excitation relative to the FC prediction. Analysis of classical trajectories provides a simple explanation of this we note that quantum surprising o b s e r v a t i ~ n . ~Incidentally ~,~~ mechanical calculations incorporating the full anisotropy of the upper-state PES reproduce the experimental distributions satisfactorily and especially reproduce the trend with increasing k.sl According to eq 14 the rotational distribution following the dissociation of the lowest bending state is, on the average, approximately a Gaussian function in the rotational quantum number, Le., e-j2/d0Ho.Plotting this Gaussian distribution in a Boltzmann representation yields, except for the smallest rotational quantum numbers, j I 3, roughly a straight line corresponding to a “temperaturen of about 400 K. This is in good accord with the experimental data obtained by Andresen et ala7for the photodissociation in a molecular beam at 157 nm, 210 and 475 K, for the two A-doublet states of OH. With increasing temperature of H 2 0 higher and higher total angular momentum states participate in the fragmentation process which has the net result that the rotational excitation of OH and therefore its rotational ‘temperature” rises as well. Photodissociation in the bulk (300 K) yields an OH ‘temperature” of about 900 K.7 The effect of parent internal rotation on the O H rotational-state distribution has been investigated by Levene and Valentinis3within the impulsive model. It is important to stress, however, that the good description of the measured rotational-state distributions by a linear Boltzmann plot does not manifest energy randomization in the sense of statistical mechpics in the excited electronic state before it breaks apart. The A complex decays much too rapidly for energy randomization to occur. The linear Boltrmann representation in the case of water merely reflects the H20bending and overall rotation wave function in the ground electronic state. The applicability of the FC approximation, which is the consequence of the lack of an appreciable torque in the A state, implies that, unlike the vibrational distribution, the OH rotational-state distribution does not significantly depend on the photolysis wavelength.2s Dissociation at 193,42177,44and 157 nm7 clearly confirms this prediction. Lowering the photolysis wavelength increases the rotational ‘temperature” of OH only slightly. B. Population of Electronic Fine-StructureStates. So far we have considered only the nuclear rotation of the OH fragment. The photodissociation of water in the first continuum produces OH in the 211 electronic state which has one unpaired p~ electron in the outer shell with electronic angular momentum A = 1 and spin S = The coupling of A and S leads to two fine-structure and ’II3/2, which on the average are split by about states, 2111/2 139 cm-l. Within the experimental uncertainty, the two finestructure states are statistically populated, i.e., the ratio of the populations in the 2111/2. and the 2113/2manifolds roughly equals the ratio of their statistical weights N / ( N + 1). Here, h! = j for 2111/2and 2113/t,respectively, with j being and N = j the total angular momentum of OH including the nuclear and the electronic angular momentum as well as the spin of the electron. A statistical 2111/2/2113/2 branching ratio has been found for dissociation in the beam and in the bulk as well as for the dissociation of single rotational states of water which we discuss below. Moreover, the statistical population is independent of the photolysis wavelength.
+
(52) Kiihl, K.;Schinke, R. Chem. f h y s . Letr. 1989, 158, 81. (53) Levene, H . B.; Valentini, J. J . J . Chem. f h y s . 1987, 87, 2594.
The Journal of Physical Chemistry, Vol. 96, No. 8, 1992 OLI
I
,
I
I
",&
"
I
1
rotational state j 8. Rotational-state distributions of OH in the symmetric Adoublet state of the 211,,2 spin-orbit state for four distinct initial rota-
tional states of water all having the same total angular momentum quantum number J = 4 but different projections on the u- and c-axis. The normalization between experiment (open circles) and FC theory (closed circles) is made for each case separately and indicated by the arrows. Adapted from ref 9. If OH rotates, the orbital of the unpaired electron can be oriented either in the plane of rotation or perpendicular to it. This leads to an additional splitting into two A-doublet components, orbital is ,'I which are only about 0.05 cm-I apart. The symmetric with respect to the plane of rotation whereas the IIorbital is antisymmetric to it. This has far-reaching consequences for the population of the two A-doublet states. Photodissociation in the beam leads to a selective population of the antisymmetric A-doublet state with the ratio II-/II+ depending strongly on the OH angular momentum N.' In order to explain this selectivity we must recall the electronic structure of water in the ground state. The two most weakly bound electrons are in a p~ orbital perpendicular to the nuclear plane. The photon promotes one of them to the next higher orbital which is symmetric with respect to the molecular plane and leaves the other one in the original orbital. When the water molecule breaks apart, the OH fragment starts to rotate in the molecular plane with one unpaired electron remaining in the orbital perpendicular to the plane of rotation; i.e., the electronic reflection symmetry is conserved from the parent to the fragment molecule. This adiabatic-type transition rationalizes qualitatively the strong propensity for the antisymmetric Adoublet state. In order to make these arguments more quanfifafiue, one must take into account, however, the coupling between the nuclear and the electronic angular momenta according to Hund's coupling case (a).' Overall rotation of the parent molecule before the absorption destroys the planarity of the dissociation process with the result that the two A-doublet states of O H gradually mix. As a consequence, the propensity for the antisymmetric state diminishes with rising temperature of the H 2 0 ample.^ Photodissociation in the bulk leads to an equal population of both A-doublet states.' C. Photodissociation of Single Rotational States. Employing the two-color excitation scheme described in section 2 (see also Figure 2) it is possible to study photodissociation on a truly state-to-state level: A first laser excites resonantly a single vibrational-rotational eigenstate within the ground electronic state. Subsequently, a second laser promotes this selected state into the A continuum. In this way one can study the influence of initial overall rotation of the parent molecule on the rotational state distribution of the fragment^.^,^.^^ Figure 8 depicts final rotational state distributions of OH(2113/2) in the lower A-doublet state following th_e dissociation of four selected initial rotational states of H20(X).9 In each case, the total angular momentum of water is J = 4. Only the projections of J on the a-axis, K,,and on the c-axis, KO are different. Water is an asymmetric-top molecule and any total angular momentum state splits into 2J + 1 nondegenerate substates. Each substate
Engel et al. is described by a distinct rotational wave function which leaves, according to the FC limit, its fingerprint on the final rotational distribution. In order to compare with these highly resolved experimental distributions, the simple FC theory described above must be extended to correctly include the electronic structure of both products, H(2S) and OH(211), as well as the coupling between the molecular angular momentum and the orbital angular momentum of H with respect to OH.S49SSThe resulting formulas are in essence similar to eq 13, but obviously much more involved. They include the exact rotational wave function of water in a given rotational state and a lot of angular momentum coupling elements. The general FC theory of B a l i n t - K ~ r t ihas ~ ~ been applied to water9qssand the results are compared to the experimental distributions in Figure 8. It is applicable, however, only if the rotational-translational coupling in the exit channel is negligibly small. Therefore, one cannot automatically employ it for the dissociation of excited bending states or other systems. Figure 8 reveals several interesting aspects. First, the measured distributions agree, within the experimental uncertainties, very nicely with the theoretical FC predictions. Second, the truly state-resolved distributions exhibit slight oscillations as a function of N . Unlike the beam and the bulk results they cannot be described by a Boltzmann distribution! The oscillations are in principle remnants of the sinusoidal term in eq 14 which in turn reflects the displacement of the equilibrium angle in the ground state from the linear c o n f i g ~ r a t i o n . ~They ~ * ~ ' would be absent for a nearly linear molecule. However, inclusion of all nuclear and electronic angular momenta strongly damps these oscillations. Third, the distributions depend uniquely on the initial rotational state of water. They reflect, in the sense of the FC limit, the rotational wave function of the respective state which obviously changes with J , K,,and K,. Additionally, we note that the propensity for populating the antisymmetric A-doublet state, found in the nozzle beam, does not strictly hold if a single rotational state is dissociated. [See the review article by: Andresen, P. In Frontiers of Laser Spectroscopy of Gases; Alves, A. C. P., et ai., Eds.; Kluwer Academic: New York, 1988) for a more detailed discussion.] Furthermore, the FC theory predicts, in full accord with experiment, that the two spin states, 2111,2and 211,/2,are populated according to their statistical weights. The experimental data in Figure 8 are obtained for the dissociation of water involving one quantum of the OH vibrational stretching mode. Similar results by Crim and collaborators" obtained with four quanta in the precursor state agree well with the data of Andresen et aL8 and with the theoretical predictions. In the latter case the vibrational motion has not been taken into account anyway. The independence of the OH rotational-state distributions of the initial vibrational state of water as well as the final vibrational state of OH7 manifests that rotation and vibration essentially decouple. Since the extended FC theory satisfactorily reproduces all facets of the fully state resolved experiment, one can actually synthesize the beam and the bulk data by averaging over all initial rotational states according to a Boltzmann distribution with temperature THzO.This procedure bears out the following main observations: First, averaging over merely a few initial rotational states smears out all of the finer structures seen in Figure 8. For temperatures of water greater than 50 K or so one obtains a reasonably linear Boltzmann plot corresponding to a "temperature" TOH which steadily rises with increasing THZo.Second, the population ratio II-/II+gradually decreases with the temperature of water as has been found experimentally. In conclusion, the photodissociation of single rotational states of water reveals a wealth of surprising details. Virtually all experimental findings are reproduced by an extended FC theory including the full electronic structure of the reactant as well as the products. Only the torque generated by the dissociative PES (54) B a h t - K u r t i , G. G. J. Chem. f h y s . 1986, 84, 4443. ( 5 5 ) Schinke. R.; Engel, V.; Andresen, P.; Hausler, D.; Baht-Kurti, G . f h y s . Reu. Left. 1985, 5 5 , 1180.
G.
The Journal of Physical Chemistry, Vol. 96,No. 8, 1992 3209
Feature Article
1.0
2.0 ROH[aO1
1.0
2.0 ROH[aO1
3.0
200
220
240
260
ZEO
Wavelength 12 (nm)
-
Figure 9. Contour plots of wave functions for the 140-) and the 13 I-) states of water in the electronic ground state multiplied by the A transition dipole function. The heavy arrows indicate the route of the evolving wavepacket in the exit channel of the excited-state potential
energy surface. can be safely neglected. The rotational and electronic finestructure distributions reflect the wave function of the parent molecule as well as the wave functions of the free products. Averaging over only a few initial states rapidly smears out finer details and yields BoltmUM-type distributions which unjustifiably points to a 'temperature". MIL Dissociation of Excited Vibrational States In the last section we discussed the dissociation of selected excited bending and overall rotational states of water and elucidated how the corresponding wave function of water is reflected in the OH rotational-state distribution. We now turn to the fragmentation of vibrationally excited states and illustrate how the absorption spectrum and the final-vibrational-state distribution of OH depend on the initial state of water. Excitation of excited vibrational states of the parent molecule acceSSeS different portions of the upper-state PES. Ry selecting the initial state one can, at least in principle, "steer" the outcome of the fragmentation. The experiments of Crim and co-workers'*-" are performed in the same way as described in yction VI1.C with the*exception that the fmt laser prepares H20(X) in excited states corresponding to four rather than only one OH stretching quanta. In the local mode assignments6 the states excited are 131-) and 140-) where we ignore the bending quantum number which is zero throughout this section. In short, the )31-) vibrational state is an antisymmetric linear combination of the states 131) and 113) which themselves are products of two one-dimensional OH stretching wavefunctions with one and three quanta of excitation, respectively. Figure 9 depicts the wavefunctions for 1W)and 13I-) as calculated with the Sorbie and MurrellI7 ground-state PES. They are multiplied by the c ( transition ~ ~ dipole function. By construction these wave functions are antisymmetric with respect to the C , symmetry line and therefore have a node on this line. Because of the symmetry requirement that wave functions for water in highly excited vibrational states can have a bizzare appearance. Figure 10 depicts the low-energy portion of the measured" and the calculateds7 absorption spectrum for water being initially in the (407state. The agreement is astonishingly good. In the same spirit as the spectrum shown in Figure 5 for the loo+) vibrational ground state reflects the associated wave function-essentially a two-dimensional Gaussian-type function-the oscillations in Figure 10 mirror the nodal behaviour of the 140-) wave function along the dissociation path from the FC region out into the fragment channel.s7 These reflection structures have the same origin as in emission spectra of diatomic molecules from a bound upper to a repulsive lower state.s* Because the absorption cross section is proportional to the modulus square of the overlap of two oscillatory two-dimensional wave functions, one describing the vibrationally excited parent molecule and the other one r e p resenting the dissociating H-OH molecule, the interpretation and (56)Child, M.S.Acr. Chem. Res. 1985, 18, 45. (57)Weide, K.;Hennig, S.;Schinke, R.J . Chem. Phys. 1989,91, 7630. (58) Tellinghuisen, J. In Phorodissociation and Phoroionizarion;Lawlcy, K. P., Ed.; Wiley: New York, 1985.
Figure 10. Measured and calculated absorption spectrum for the (40-) state of H,O(X). Adapted from ref 11.
identification of the reflection structures is not always clear. Each excited vibrational state has its own distinct absorption spectrum. It should be underlined a t this point that contrary to the dissociation of the vibrational ground state the coordinate dependence of the transition dipole function has to be taken into account. It should come as no surprise that the final-vibrational-state distribution of the OH product also depends uniquely on the particular vibrational state of water in the ground electronic state. Vander Wal et al." considered the products of dissociation from the 140-)and 131-) vibrational state. Both states have almost the same energy content but the energy is distributed differently between the two stretching modes. The fragmentation has been studied at three wavelengths, 266, 239.5, and 218.5 nm, and the OH(n=O):OH(n= 1) branching ratio has been measured in each case. The correspon-ding total energies in the upper state are below the barrier of the A-state PES for the two largest wavelengths and just above the barrier for the shortest one. Thus, excitation with the 266- and 239.5-nm photons samples, according to the FC principle, essentially those portions of the ground-state wave functions which extend relatively far out into either of the two fragment channels. In the 266-nm photolysis Vander Wal et al. found a n=O:n=l branching ratio of !:O for 140-) but literally no dissociation products for 131-). The lack of absorption and subsequent dissociation in the latter case reflects the fact that the 131-) wave function does not extend sufficiently far out into the H OH channel to have significant overlap with the continuum wave function corresponding to 266 nm. Dissociation at the other two wavelengths gives dominantly either OH(n-0) products (0.99:O.Ol for 239.5 nm and 0.91:0.09 for 218.5 nm) for the 140-) state or OH(n=l) products (0.16:0.84 for 239.5 nm and 0.060.94 for 218.5 nm) for the 131-) state. The theoretical calculations almost quantitatively reproduce these experimental data.s7 The consistent production of OH in either the n = 0 or the n = 1 state reflects the essentially different behavior of the 140-) and the 131-) wave functions at larger H-OH separations (>2.0 a. or so) as can be seen in Figure 9. It is this part of the wave functions which overlaps the continuum wave functions at the relatively long photolysis wavelengths used in the experiment. In this region, the 140-) wave function behaves in the direction perpendicular to the dissociation path roughly like an OH(n=O) wave function and therefore yields primarily OH(n-0) products.s7 Similarly, the 131-) state has the overall appearance of an OH( n = l ) wave function and thus leads mainly to OH(n=l). In conclusion, the vibrational branching ratios reflect rather directly the general shape of the parent wavefunction a t relatively large H-OH distances. The interpretation of vibrational-state distributions at shorter wavelengths, however, is much more involved because the simple adiabatic-type arguments do not hold anymore and quantum mechanical interference effects are strong.s7
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IX. Isotope Effects: Tbe Dissociation of HOD In principle, a polyatomic molecule can dissociate into several distinct product channels. The chemical branching ratio is an important quantity of considerable practical interest and contains a wealth of dynamical information about the actual bond fission
3210 The Journal of Physical Chemistry, Vol. 96, No. 8, 1992 102,
II
1
I
I
I
O D 1 OH
.-0
I
O
L
10'
n
100 I
-3 0
Figure 11.
uH+OD/UD+OH
-2 5
I
1
-2 0
-1 5
-10
E Lev1 branching ratio for the dissociation of vibra-
tional states In=O;m) of H O D where n and m denote the number of quanta in the O D and in the OH bond, respectively (short dashes, m = 0; long dashes, m = 2; solid line, m = 4). The open circle is the measured result of Shafer et ala6'for the photolysis of the ground state of HOD ( n = m = 0) at 157 nm. The filled circle is the result for 104) and 218.5 nm and the upward arrow indicates the lower limit for 104) and 239.5 cm. The arrow on the energy axis marks the barrier energy of the A-state potential energy surface. Energy normalization is such that E = 0 corresponds to three ground-state atoms, H 0 D. Reprinted with permission from ref 13. Copyright 1991 American Institute of Physics.
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mechanism. Mode selectivity in unimolecular reactions is still a -hot* topic in molecular dynamics studies (see, for example, the thematic issue edited by Manz and Parmenter, Chem. Phys. 1989, 139 and references therein). Again, H 2 0and its isotopic variant HOD play a key role because detailed measurements and rigorous ab initio calculations are feasible and can be compared on a quantitative level. In the following we discuss the OD/OH branching ratio and its dependence on the wavelength and the initial vibrational state of the HOD parent molecule.s~29~30*59 Figure 11 depicts the OD/OH branching ratio for three distinct initial states Jn=O;m)of HOD, where n and m denote the number of quanta of excitation in the OD and the OH bond, respectively. Let us first focus the discussion on the dissociation of the ground state, n = m = 0. The OD/OH branching ratio is at all energies larger than unity, but varies considerably with E . The dominance of OD at long wavelengths (low energies) reflects the asymmetry of the corresponding ground-state wave function which as a consequence of the lighter mass of H extends further out into the H-OD channel. In a classical Monte Carlo treatment the modulus square of the quantum mechanical wave function is the weighting function for classical trajectories,a@' and thus the asymmetry has the effect that at lower energies more trajectories start in the H-OD rather than in the HO-D channel. Since the trajectories dissociate directly without any multiple bounces in the inner region of the PES one obtains more OD than OH products. With increasing energy the branching ratio decreases, reaches the kinematical limit30of mD!mH = 2 at energies slightly above the upper-state potential barrier, and then gradually rises again. The increase at higher energies can be also explained by simple classical arguments.I3 The experimental result of Shafer et aL61 obtained for the 157-nm dissociation with a single photon confirms the theoretical prediction. If one pumps, for example, four quanta of stretching excitation into the H O bond before the photolysis laser promotes HOD into (59) Zhang, J.; Imre, D. G . Chem. Phys. Lett. 1988, 149, 233. (60) Goursaud, S.; Sizun, M.; Fiquet-Fayard, F. J . Chem. Phys. 1916.65, 5453. (61) Shafer, N.; Satyapal, S.; Bersohn, R. J . Chem. Phys. 1989, 90, 6807.
Engel et al. the h a t e continuum (vibrationally mediated photodissociation as described in sections VI1 and VIII) one intuitively expects that the H O bond breaks even more preferentially and that the branching ratio increases compared to the dissociation of the ground state. This naive picture is only true at low energies where indeed the branching ratio increases by several orders of magnitude compared to the m = 0 case (seeFigure 1l).13 At energies about 0.5 eV above the bamer in the upper state, however, the branching ratio becomes approximately 1; Le., roughly the same amounts of O H and OD products are formed despite the fact that the H O bond was initially excited by four quanta while the OD bond was in the ground vibrational state. This result is more astonishing than the extremely large branching ratio at long wavelengths. The latter can be easily understood in terms of the bound and the scattering wave function and their confined overlap if the total energy in the upper state is small. The branching ratio of about unity for intermediate energies has no simple explanation, however, The full overlap of two highly oscillatory two-dimensional wave functions, one describing the excited HOD state and the other one representing the continuum wave function dissociating either into the D OH or the H + O D channel, cannot be easily surmised because positive and negative portions partly cancel each other. The two-color experiment of Vander Wal et al." fully verifies the astonishing theoretical prognosis.
+
X. Resonance Raman Studies Resonance Raman spectroscopy can be viewed as a measure of the dynamics of a molecule following excitation to an excited electronic state.62 For the case of a directly dissociative excited-state PES, such as that of water, the resonance Raman spectrum reflects the initial motion of atoms away from the equilibrium coordinates of the ground electronic state in a direction which leads to dissociati0n.6~ The spatial extent of the upper-state PES probed by emission spectroscopy depends on the coordinate variation of the electronic transition matrix element between the ground and the excited state. The transition moment surface for water, pXA,approaches zero for large displacements along the reaction coordinate. This limits the snapshot picture of dissociation dynamics to the initial phase of the process. The dynamical picture obtained by emission spectroscopy is complementary to that provided by other means, the absorption spectrum or final state distributions. The development of experimental methods in ultraviolet resonance Raman spectroscopya has resulted in the ability to apply this method to simple molecules formed from light atoms where vacuum ultraviolet radiation is req~ired.6~ One of the first applications of this experimental method has been to the case of water.I5J6 Resonance Raman spectra of HzO, DzO, and HOD have been obtained with excitation at 174.6, 171.4, 160.0, and 150.0 nm (resonant with the A state). These spectra confirm the basic features of the early time dynamics of the photodissociation (62) Imre, D. G.; Kinsey, J. L.; Field, R.; Katayama, D. H. J . Phys. Chem. 1982, 86, 2564. Imre, D. G.; Kinsey, J . L.; Sinha, A.; Krenos, J . J . Phys. Chem. 1988.88, 3956. Sundberg, R. L.; Imre, D. G.; Hale, M. 0.; Kinsey, J. L.; Coalson, R. D. J . Phys. Chem. 1986, 90, 5001. Hale, M. 0.; Galica, G . E.; Glogover, S. G.; Kinsey, J. L. J . Phys. Chem. 1986,90,4997. Johnson, B. R.; Kinsey, J. L. J . Chem. Phys. 1987, 87, 1525. (63) Heather, R.; Jiang, X.-P.; Metiu, H. Chem. Phys. Lett. 1987, 142, 303. Rama Krishna, M. V.; Coalson, R. D. Chem. Phys. 1988, 120, 327. (64) Ziegler, L. D.; Hudson, B. J . Chem. Phys. 1981, 74, 982. Ziegler, L. D.; Hudson, B. J . Chem. Phys. 1983, 79, 1197. Ziegler, L. D.; Hudson, B. J . Phys. Chem. 1984, 88, I 1 10. Ziegler, L. D.; Kelly, P. B.; Hudson, B. J . Chem. Phys. 1984,81,6399. Kelly, P. B.; Hudson, B. Chem. Phys. Lert. 1985, 114, 451. Gerrity, D. P.; Ziegler, L. D.; Kelly, P. B.; Desiderio, R. A,; Hudson, B. J . Chem. Phys. 1985,83, 3209. Hudson, B.; Kelly, P. B.; Ziegler, L. D.; Desiderio, R. A.; Gerrity, D. P.; Hess, W.; Bates, R. Far Ultraviolet Resonance Raman Studies of Electronic Excitations. In Aduances in Laser Spectroscopy, Garetz, B. A., Lombardi, J. R., Eds.;Wiley: New York, 1986; Vol. 3, pp 1-32. Hudson, B. Vacuum Ultraviolet Resonance Raman Spectroscopy. In Recent Trends in Raman Spectroscopy;Jha, S.s., Banerjee, S. B., Eds.; World Scientific Press: Singapore, 1989; pp 368-385. (65) Hudson, B.; Sension, R. J . Far Ultraviolet Resonance Raman Spectroscopy: Methodology and Applications. In Vibrational Spectra and Strucrure; Bist, H. D., Durig, J. R., Sullivan, J. F., Eds.; Elsevier: Amsterdam, 1989; Vol. 17A, pp 363-390. Sension, R. J.; Hudson, B. J . Chem. Phys. 1989, 90, 1377.
The Journal of Physical Chemistry, Vol. 96, No. 8, 1992 3211
Feature Article
"O
1I
iI
tI
20.0
E 0.64
HzO, 160 nm
110.0
I
I
I
2080
4000
I
6000
8000
I
I
I
1
I0000
12000
14000
16008
Roman shift
Icm
I
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-' 1
Figure 12. Resonance Raman spectrcof isotopic species of water obtained with radiation resonant in the A-electronic state. (a) H20,obtained with 160-nmexcitation; (b) D20,obtained with 171-nmexcitation; and (c) HOD, obtained with 175-nm excitation. Bands due to spurious excitation components have been removed. The labels are based on a local-mode description. The HOD spectrum naturally includes H20and D20lines. Only the HOD peaks are labeled. The ordering of the labels in the case of HOD is OD stretch, bend, and OH stretch.
process expected on the basis of the calculated h a t e PES. Some representative spectra are presented in Figure 12. The main points of this experimental study are the following. (1) The general shape of the resonance Raman spectrum does not appreciably change with excitation wavelength. This is the behavior expected from resonance with a single, directly dissociative excited electronic state as occurs for water in the first absorption band. J2) There is no activity in the bending mode for excitation in the A state. From this we may directly conclude that electronic excitation in the A state does not result in a significant force along this coordinate direction. -This is consistent with having a potential energy minimum in the A state at the same angle as that of the ground state. The low degree of rotational excitation of the OH fragment discussed in section VI1 and the lack of bending excitation in the Raman experiment have the same origin. (3) A long progression (up to six quanta) is seen in the symmetric stretching degree of freedom for all isotopomers. This is consistent with the early-time motion in the direction of the symmetric stretching mode on top of the barrier of the upper-state PES as discussed in section VII. The long progression in the symmetric stretching mode is thus compatible with the existence of the diffuse structures in the spectrum as well as the substantial degree of vibrational excitation of OH. (4) The only activity seen in the asymmetric stretching degree of freedom occurs for two quanta of this motion, primarily in combination with one or two quanta of the symmetric stretching mode. This may be viewed as due to a Darling-Dennison resonance between the stretching levels of the same symmetry or, equivalently, as due to the local-mode nature of the more highly excited stretching vibrations of water. In order to calculate a resonance Raman spectrum to be compared to experiment, it is necessary to know the excited-state PES in the vicinity of the vertical region, the PES of the ground electronic state for displacements along all of the coordinates that result in enhanced Raman transitions, and the transition moment surface for the radiative excitation between the two electronic states. All these ingredients are known for water. In addition, excitation in the first absorption band has the advantage that the bending angle can be frozen a t its equilibrium value within the calculation thus reducing the dynamical problem to only two dimensions. Raman spectra for water have been calculated in the time-independenP as well as the time-dependent16.28,29 approach. (66) Hennig,
1988, 149,455.
S.;Engel, V.; Schinke, R.; Staemmler, V. Chem. Phys. Lett.
Raman shift [cm-ll Figure 13. Comparison of calculated and observed resonance Raman spectrum for H20 for excitation at 171 nm. Theory and experiment are normalized to each other at the (200) line (normal-mode assignment; the corresponding local mode assignment is given in Figure 12). Cases for which the experimental intensitiesare smaller than the theoretical ones are indicated by hatched histograms. Reprinted with permission from ref 66. Copyright 1988 Elsevier Science Publishers.
1
2
3
i
5
6
N Figure 14. Comparison of calculated and measured relative Raman intensities for H20as a function of the total stretching number N. Left, unshaded bars: averaged experimental intensities. Middle, hatch marked: theoretical results from ref 16. Right, dark bars: theoretical results from ref 66. The arrows indicate theoretical results obtained within the Condon approximation, Le., the transition dipole function is set to a constant. Reprinted with permission from ref 16. Copyright 1990 Elsevier Science Publishers.
The results of the comparison between experiment and ab initio theory are shown in Figures 13 and 14. The actual information obtained from experiment that can be compared with theory is the relative intensity of overtone and combination transitions for different quantum states (for a more detailed discussion on the correction of the experimental results for a comparison with theory see ref 16). Theory and experiment have been normalized to each other a t the first overtone of the symmetric stretch. Figure 13 compares "high-resolution" theoretical66 and experimental15 spectra. The splitting of the overtone and combination doublets of the form (n,O,O)/(n-2,0,2), Le., the Darling-Dennison splitting, is due to the form of the ground-state potential. Here, we use the normal-mode assignment; i.e., the first and the third quantum numbers correspond to the symmetric and antisymmetric stretch modes, respectively, and the second quantum number specifies
3212 The Journal of Physical Chemistry, Vol. 96, No. 8, 1992
the number of bending quanta. Similarly, the relative intensity of the individual components of these polyads depends primarily on the nature of the ground-state potential. To a first approximation the relative intensity is a measure of the amount of symmetric stretch excitation in each band. The agreement is quite good also in this case. In order to make a comparison of the calculated and observed resonance Raman spectra that emphasizes the excited-state dynamics, it is useful to sum the intensity of each component within a polyad before making this comparison. This is what has been done in Figure 14. The figure contains the “low resolution” experimental spectrum which was obtained by integration of the total intensities in the regions corresponding to different total quanta of stretching excitation. The results of the time-independent calculation of Hennig et and the time-dependent calculations of ref 16 are included in the figure. The overall agreement between theory and experiment for both wavelengths displayed is very satisfactory. The differences in the two calculations result from the use of different ground-state electronic surfaces and especially the use of different transition dipole functions. To test the influence of the latter, calculations within the Condon approximation were performed as well (coordinateindependent transition dipole moment).66 The results are indicated by the arrows in the figure. Even these results compare rather well with experiment. However, since pxA diminishes with increasing H-OH distance, the spectrum calculated with the coordinate-dependent transition dipole function falls off more rapidly. In conclusion, both the overall behavior of the emission spectra and subtle details are well reproduced by the ab initio calculations. The information about the dissociation dynamics obtained from the Raman studies is consistent with the general picture derived from the other measurements.
XI. Final Discussion The photodissociation of water in the first absorption band is a relatively simple process coepared to other polyatomic systems. After the promotion into the A continuum the fragmentation into H and O H takes place immediately and involves only one electronic state. This characteristic feature makes it an ideal candidate to compare detailed experimental data with rigorous theoretical calculations including an ab initio PES. The photodissociation of water with excitation around 165 nm has been investigated in unprecedented detail including (1) the absorption spectrum for unexcited H20(X); (2) the absyption spectrum for a highly excited vibrational state of H,O(X); (3) the vibrational distribution of OH; (4) the rotational distribution of OH measured in the bulk and the supersonic beam including the four A-doublet states; (5) the rotational distribution of O H following the dissociation of single rotational states of water; (6) the rotational distribution of O H for excited bending states of H20(X); (7) the vibrational-state distributions of O H following the dissociation of two highly excited vibrational states of water; (8) the OH/OD branching ratio in the dissociation of HOD initially in the ground vibrational state; (9) the branching ratio and the final vibrational-state distributions of O H and OD following the dissociation of vibrationally excited HOD; (IO) the Raman spectra for H 2 0 , D 2 0 , and HOD excited with several photolysis wavelengths; (1 l)_the emission spectra for H 2 0 and D 2 0 from the C state to the A ~ t a t e . ~The ’ ~ ~last ~ point has not been discussed in this review. The combination of all these different perspectives gives a complete picture of this simple bond fission process. If there would not be the most recent experiments of Gericke and c o - ~ o r k e r swe ~ ~could . ~ without risk claim that the ab initio calculations-which do not involve a single adjustable parameter-reproduce essentially all experimental data on a quantitative level. This underlies impressively the present day abilities of modem physical chemistry if electronic structure theory, (67) Engel, V . ; Meijer, G.; Bath, A.; Andresen, P.;Schinke, R . J Chem Phys. 1987, 87, 4310. (68) Docker, M . P.;Hodgson, A.;Simons, J . P. Mol. Phys. 1986, 57. 129.
Engel et al. dynamical theory, and experiment work closely together. In the last sentence of ref 43 Mikulecky et al. tentatively speculate that -a shift in the upper potential surface by =0.5 eV could explain the deviation between experimental and theoretical results”. Such a large shift would have dramatic effects on the absorption cross sections for the ground and the IN-)vibrationally excited states of water p i n t s 1 and 2 of above) and the emission spectrum for the C A transition (point 11) as well as the energy dependence of the branching ratio in the dissociation of HOD (points 8 and 9). A shift of half an electronvolt would completely ruin the agreement observed in Figures 5 , 10, and 11. On the contrary, a decrease of the upper-state PES by 0.05-0.1 eV, which is just the expected upper limit of the uncertainty of the ab initio calculation, would slightly improve the agreement. The explanation of the disagreement in the vibrational state distributions has to be more subtle. A stronger curvature of the upper-state PES along the antisymmetric stretch coordinate in the FC and the barrier region, as for H2S for example,69would possibly lead to significantly less vibrational excitation. On the other hand, it would also affect the breadth of the spectrum as well as the strength of the diffuse structures superimposed to the broad background. A narrower potential barrier would inevitably make the unstable periodic orbit oscillating along the symmetric stretch coordinate on top of the barrier more fragile and therefore it would damp the diffuse structures even more. Variations of the form of the barrier would also have consequences for the Raman spectra. A large sharpening of the potential for the antisymmetric stretching motion would result in a progression in the even overtones of this motion. The disagreement with the experimental results of Mikulecky et al. concerning the degree of vibrational excitation is a puzzle which we cannot unravel at the present time. While waiting for a third independent experimental check we will investigate the sensitivity of the various cross sections on slight changes of the potential energy surface, especially in the FC and the barrier region. The rotational-state distributions will not be affected by such modifications of the PES. At the end of this review it is appealing to compare briefly the dissociation of water in the first band with other fragmentation processes. Dissociation following the excitation of H 2 0 into the B state (second absorption band) behaves completely differently. The corresponding PES has a deep well at the linear H-0-H configuration and a strong dependence on the bending angle in the FC region.70 The latter leads to extremely strong rotational excitation of the O H f r a g m e n t ~and ~ ~ to , ~the ~ population of excited bending states in the Raman p r o c e s ~ . ~The ~ *degree ~ ~ of vibrational excitation, on the other hand, is only modest.74 The superimposed diffuse vibrational structures (see Figure 1) are due to bending rather than symmetric stretch vibration as in the case of the first coltinuum. In striking contrast to process 1 the dissociation in the B band involves three electronic states. Excitations of water in the first and in the second absorption band are completely different processes. On the first glance one would be tempted to expect that the dissociation of H2S in the 193-nm band proceeds in a simily fashion as process 1. That is not the case, however. While H20(A) dissociates on a single PES the fragmentation of H2S involves the two lowest excited electronic states, IB,and IA2 in CZusymmetry.3s,69.77.78One state is dissociative while the other one is bound.
-
(69) Heumann. B.; Diiren, R.; Schinke, R. Chem. Phys. Leu. 1991, 180, 583.
(70) Theodorakopoulos, G . ; Petsalakis, I . D.; Buenker, R . J. Chem. Phys. 1985, 96, 217.
(71) Carrington. T. J. Chem. Phys. 1966, 41, 2012. (72) Segev, E.; Shapiro, M. J . Chem. Phys. 1982, 77, 5604. Weide, K.; Schinke, R . J . Chem. Phys. 1987, 87, 4627. (73) Sension, R . J.; Brudzynski, R . J.; Hudson, B. S. Unpublished results. (74) Heumann, B.; Kiihl, K.; Weide, K.; Diiren, R.; Hess, 8.; Meier, U.; Peyerimhoff, S. D.; Schinke. R . Chem. Phys. Left. 1990, 166, 385. (75) Weide. K.:Schinke. R . J . Chem. Phys. 1989, 90, 7150. (76) Weide, K.; Kiihl, K.; Schinke. R . J . Chem. Phys. 1989, 91, 3999. (77) Weide. K.;Staemmler, V.; Schinke, R . J . Chem. Phys. 1990,93,861. ( 7 8 ) Theodorakopoulos. G.; Petsalakis, I . D. Chem. Phys. Lett. 1991, 178, 47s.
J . Phys. Chem. 1992,96, 3213-3217 A conical intersection occurs between these states right in the FC region which makes a dynamical treatment extraordinarily elaborate.79 The initial excitation is to the bound state which couples, however, strongly to the dissociative state. The diffuse structures superimposed to the broad absorption background are, as in the case of water, due to symmetric stretch motion, but in the potential well of the bound state rather than on top of the barrier of the dissociative ~ t a t e . ~ The ~ * 'involvement ~ of two electronic states has also consequences for the Raman spectra. In contrast to process 1, the Raman spectrum of H2S shows activity in the bending coordinate and it exhibits a distinct dependence on the excitation wavelengthaS0 Despite some similarities, the dissoci~
~~~~
(79) Heumann, B.; Weide, K.; Daren, R.; Schinke, R. To be published.
(80) Brudzynski, R. J.; Sension, R. J.; Hudson, B. Chem. Phys. Lerr. 1990, 165$487.
3213
ations of H2Sand H 2 0 , both excited in the first absorption band, behave quite differently.
Editor's Note. This Feature Article is longer than normal as it contains additional reviews in a field that the editors felt to be important. Acknowledgment. R.L.V.M. and F.F.C. gratefully acknowledge the support of the Division of Chemical Sciences, Office of Basic Energy Sciences of the United States Department of Energy. R.J.S. and B.H. acknowledge the assistance of Dr. R. J. Brudzynski in performing the resonance Raman experiments and the support of the US National Science Foundation via grant CHE8816698. V.E., S.H., K.W., and R.S. are grateful to the Deutsche Forschungs Gemeinschaft for continuous financial support.
ARTICLES Calculation of Electron-Transfer Matrix Elements of Bridged Systems Using a Molecular Fragment Approach Prabha Siddarth and R. A. Marcus* A . A . Noyes Laboratory of Chemical Physics,+ California Institute of Technology, Pasadena, California 91 125 (Received: August 23, 1991; In Final Form: December 20, 1991)
A perturbation method for calculating the electronic coupling for electron-transfer reactions between a donor and an acceptor separated by large or small bridges is developed. In this approach the intervening bridge is subdivided into smaller molecular fragments, thereby enabling calculations on larger systems. This method of molecular fragments is tested for a series of polyproline bridged systems. The results obtained for the electron-transfer matrix element are compared with those obtained from direct diagonalization of the full bridge and with experimental results. Previously, the result for the direct diagonalization of the bridge had been shown to agree with that obtained from diagonalization of the entire donor-bridgeacceptor system. The vertical donor-bridge orbital energy difference is estimated with the aid of a donor-bridge charge-transfer spectrum.
Introduction The effective electron-transfer matrix element HDA from a donor D to an acceptor A has been expressed in terms of individual atom-atom matrix elements Hij, Coulomb integrals Hii, and overlap integrals S,, both in perturbative and nonperturbative formalisms. In an early such formulation, McConnell showed that a perturbation approximation to the relevant matrix element is given by1
where Tis the matrix element between the donor (acceptor) and the nearest bridge atom in a sequentially connected series of identical bridge atoms, t is the bridge atom-bridge atom matrix element, W is the energy difference between a bridge atom orbital energy and donor or acceptor atom energy, and n is the number of bridge atoms. Extension of this result to a sequence of nonidentical interacting atoms and energy differences yields2,'
'Contribution No. 8490.
where Ei is the Coulombic energy Hii of the ith atom, ED is that of the donor (which equals that of the acceptor EA a t a resonant electron transfer), and the Hi, are atom-atom matrix elements. It has been pointed out24 that this expression could be further extended by allowing the i to denote the ith collection of atoms, e.g., a molecular fragment, in the bridge. In the extended Hiickel treatment, the atom-atom matrix element in eq 2, Hij, of which HD1,Hi,i+l,and HnA are examples, is given by5
(3) where K is taken to be 1.75. One method of extending the i in eq 2 from atoms to molecular fragments, adopted by Ulstrup and co-workers6 for proteins, is ( I ) McConnell, H. M. J. Chem. Phys. 1961.35, 508. (2) Larsson, S.J . Am. Chem. SOC.1981, 103, 4034. (3) Newton, M. D. Chem. Reu. 1991, 91, 767. (4) Ratner, M. A. J . Phys. Chem. 1990, 91,4877. ( 5 ) Hoffmann, R. J . Chem. Phys. 1963, 39, 1397; 1964, 40, 2474,2745. (6) Christensen, H . E. M.; Conrad, L. S.; Mikkelsen, K.V.;Nielsen, M. K.;Ulstrup, J. Inorg. Chem. 1990. 29, 2808.
0022-3654/92/2096-3213$03.00/00 1992 American Chemical Society
