J. Phys. Chem. 1988, 92, 4277-4285 the emission a t long times comes from a uniform distribution of directions 0 for fiF, so that r(m)
= ( 2 / 5 ) ~ d % ( c o e) s P2(c0s O ) / l ”0d ( c o s 0)
(2)
The main point is that the decay of the anisotropy at times very short compared with the overall motions of the polymer corresponds to nearly unrestricted motion of the transition dipole (for unrestricted motion 0, = 90°) away from the initial distribution of orientations. We now consider what motional properties of the transition dipole fiF(t)might account for the observed decay on the nanosecond time scale. In the case of poly(phenylmethylsi1ane) we showed previously that there existed a relatively fast (tens of picoseconds) decay of the anisotropy to a plateau value. A similar behavior is seen for poly(di-n-hexylsilane), but in this case, because of the much longer fluorescence lifetime, the slow decay of the so-called plateau region is easily seen. Overall, the decay functions are nonexponential, but beyond about 100 ps the form of r(t) has a characteristic decay time of about 2 ns. Whatever the important structural aspects of the disordered solution and ordered crystalline phases might be, they apparently do not affect the anisotropy decay at long times: the decay of the fluorescence anisotropy is essentially the same in suspension and solution. Nor is the anisotropy decay eeatly changed at liquid nitrogen temperature: there seems to be no significant barrier to the processes responsible for polarization decay. We reject cross-chain relaxation as the cause of anisotropy decay. The suspension settles on standing and must have a density typical of a solid 10 M in Si(C6H&. The density of solution polymer we may estimate from hydrodynamic radii ( 100 nm) determined by light-scattering measurementsL2to be at least an order of magnitude smaller. Forster-Dexter energy transfer, varying as the sixth power of the chain separation, would then be a t least 2 orders of magnitude slower in the solution phase. We may reject, too, torsional motions of the polymer. It seems
-
-
4277
difficult to believe that the same motions could be occurring in the glass a t 78 K as in the solution, and restricted motions such as these would lead in any case to an incomplete loss of anisotropy. A plausible cause of the decay, considered in ref 1, is that it represents the intersegment motions of the excitations. All that is required is that the segments occupy a random distribution of orientations in the laboratory frame. The segments themselves could be terminated by regions where there are severe reorientations of the polymer backbone which may involve a range of trans, gauche, or intermediate structures. Based on the excitation bandwidth” and the expected energy shiftsLsfor different local polymer configurations, we expect these regions separating the larger radiative segments to consist of a significant number of silicon atoms. If only one or two atoms were involved, the superexchange of excitation between different segments would be expected to be considerably faster than nanoseconds.
Summary We have shown that the fluorescence lifetime of the polysilane, poly(di-n-hexylsilane), is significantly increased in crystalline forms. This is attributed to a decrease in the rate of nonradiative processes accompanied by an increase in the radiative lifetime. We do not see this lifetime lengthening in a glass at 77 K and regard this as evidence of the existence of an isolated ordered all-trans rodlike form in this medium. The fluorescence anisotropy, however, is comparatively independent of phase and decays with a characteristic time of 2 ns attributed to motion of the excitation among the polymer segments. Acknowledgment. This research was supported by NSFDMR-85 19059 and by the Sandia National Laboratories, supported by the U S . Department of Energy under Contract No. DEAC04-76-DP00789. (17) Takeda, K.; Teramae, H.; Matsumoto, N. J . Am. Chem. SOC.1986, 208, 8186.
FEATURE ARTICLE The Hydrated Electron: Quantum Simulation of Structure, Spectroscopy, and Dynamics Peter J. Rossky* and Jurgen Schnitker Department of Chemistry, University of Texas at Austin, Austin, Texas 78712 (Received: March 30, 1988)
The rapidly advancing ability to study quantum mechanical behavior in condensed phase systems via molecular-level simulation is discussed and illustrated in the context of the hydrated electron system. The recently developed models and techniques are outlined, and applications to equilibrium structure, steady-state optical spectroscopy, and aspects of electronic relaxation dynamics are described. The a priori simulation approach reveals not only an average structure consistent with earlier inferences from experiment but also significant fluctuations which are demonstrated to play a critical role in determining the energetic distribution of electronic states and the characteristic, featureless absorption spectrum. Studies of the transient electronic relaxation of initially created excess electrons in water via electronically adiabatic dynamics are presented which permit direct contact with ultrafast time-resolved, optical spectra. The results indicate that the dynamics of electron solvation per se does not dominate the experimentally observed rate of appearance of the equilibrium hydrated species.
I. Introduction The hydrated electron, e,, is a ubiquitous transient species in irradiated aqueous systems and plays a central role in solution photochemistry and photoelectrochemistry. It has been the subject of continuous experimental and theoretical study since its iden-
tification more than 25 years ago.’ The pertinent literature is extensive, as is that for the analogous species in ammonia, and we Will make no attempt to wm”m-Ie it here.’ (1) Hart, E. J.; Boag, J. W. J . Am. Chem. SOC.1962, 84, 4090.
0022-365418812092-4277$01.50/0 0 1988 American Chemical Society
4278 The Journal of Physical Chemistry, Vol. 92, No. 15, 1988
The physical nature of eq- is generally accepted to be. analogous to that of solvated atomic anions.' That is, the electron occupies a cavity or void in the solvent and is surrounded, and solvated, by the water molecules. The immediate issues are to, first, describe and understand the structure of e,- at a molecular level and then to proceed to unravel the physics underlying both the properties of the species and the physical probes available to study it. The equilibrium steady-state optical electronic spectrum is a basic example. The experimentally measured spectrum is both exceptionally broad (-0.85 eV) and featureless and is readily reproduced by a wide range of mutually exclusive ad hoc models for the electronic eigen~pectrum.~In such a situation, a first principles approach is necessary to determine the veracity of the alternative views. To carry out this plan is not straightforward. For simple atomic ions, many aspects of these phenomena have been addressed via classical computer ~ i m u l a t i o n . These ~ classical techniques have become practically routine in recent years as a method for investigating, at a molecular level, the behavior of liquids and In such studies, the statistical behavior of a sample of molecules interacting through a model potential function is examined in a specified thermodynamic state. Both structural and dynamical information regarding the nature of solvent and solute spatial distribution are accessible by such a route. The issues of solvation structure, reaction mechanisms, and reaction dynamics in solution are thus now accessible to direct molecular-level investigation. However, for the system of interest here, and for a wide variety of more complex cases, the essential role of quantum mechanics in the description of the solute distribution is obvious. The standard methods of computer simulation are based on classical (Newtonian) mechanics and are therefore not immediately applicable to these important problems. In the past few years, there has been a qualitative advance in the ability to carry out correspondingly detailed studies for quantum mechanical systems, and the solvated electron can be viewed as the prototype of a quantum solute.* The essential feature which can now be included in such a treatment is the intimate coupling between the quantum system (here, the electron) and the surrounding medium, which in many cases can be treated classically. Even within the restricted class of systems characterized by quantum solutes in an essentially classical solvent bath, there are an enormous number of interesting chemical issues to be addressed, including such exciting current topics as solution and biological electron transfer. In this context, one should view the solvated electron as the "hydrogen atom" of the area. As such, the full study of this problem and the development of satisfactory models (2) For reviews see (a) Hart, E. J.; Anbar, M. The Hydrated Electron; Wiley: New York, 1970. (b) Metal-Ammonia Solutions; Lagowski, J. J., Sienko, M. J., Eds.; Butterworths: London, 1970. (c) Electrons in Fluids; Jortner, J., Kestner, N. R., E%.; Springer: Berlin, 1973. (d) Electron-Soluent and AnionSolvent Interactions; Kevan, L., Webster, B. C., Eds.; Elsevier: Amsterdam, 1976. (e) Thompson, J. C. Electrons in Liquid Ammonia; Clarendon: Oxford, 1976. (f) Schindewolf, U. Angew. Chem., In?.Ed. Engl. 1978, 17, 887. (g) Webster, B. C. Annu. Rep. Prog. Chem. See. C 1979, 76, 287. (h) Feng, D.-F.; Kevan, L. Chem. Reu. 1980, 80, 1. (3) This view is still contested by some workers: (a) Golden, S.; Tuttle, Jr., T. R. J . Phys. Chem. 1978,82,944. (b) Hameka, H. F.; Robinson, G. W.; Marsden, C. J. J . Phys. Chem. 1987, 91, 3150. (4) Kajiwara, T.; Funabashi, K.; Naleway, C. Phys. Rev. A 1972, 6, 808. Mazzacurati, V.; Signorelli, G. Lett. Nuouo Cim. 1975, 12, 347. Webster, B. J . Phys. Chem. 1980, 84, 1070. Bogdanchikov, G. A,; Burshtein, A. J.; Zharikov, A. A. Chem. Phys. 1986,107,75. Bartczak, W. M.; Hilczer, M.; Kroh, J. J . Phys. Chem. 1987, 91, 3834. See also Kestner, N. R., in ref 2d, P 1. (5) See, for example, (a) Geiger, A. Eer. Bunsen-Ges. Phys. Chem. 1981, 85.52. (b) Impey, R. W.; Madden, P. A,; McDonald, I. R. J . Phys. Chem. 1983, 87, 5071. (c) Heinzinger, K. Pure Appl. Chem. 1985, 57, 5031. (6) (a) Articles by Valleau, J. P.; Whittington, S. G.; by Valleau, J. P.; Torrie, G. M.; by Erpenbeck, J. J.; Wood, W. W.; and by Kushick, J.; Berne, B. J. In Statistical Mechanics, Parts A and B; Beme, B. J., Ed.;Plenum: New York, 1977. (b) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Clarendon: Oxford, 1987. (7) Rossky, P. J. Annu. Reo. Phys. Chem. 1985, 36, 321. (8) Chandler, D. J . Phys. Chem. 1984, 88, 3400.
Rossky and Schnitker and of methods for evaluating static and dynamic properties serves as an essential goal in itself. The goals of the present article are to, first, outline some important techniques in the growing area of quantum simulation, including methods for simulating equilibrium structure and relatively newer techniques for simulating equilibrium and nonequilibrium dynamics. Second, we will describe selected results from three different calculations that have been carried out in our laboratory for the specific system of eaq-at room temperature and that are illustrative of the type of information that is now accessible. These calculations are the simulation of the equilibrium structure of e,-, the computation and resolution of the equilibrium optical absorption spectrum, and a study of the relaxation dynamics of an excess electron injected into pure liquid water. The emphasis in describing the results will be on developing a physical picture of the solvent and solute system, including the excess electronic eigenstates, and understanding the degree to which an analogy to simple ion solvation is profitable. It should be kept in mind here that the origin of localized excess electronic states in polar fluids is not trivial. As for the case of simple ionic solutes, an attractive potential well is developed when the solvent orientation is polarized by the predominantly electrostatic influence of the electron. Competing in this localization, however, is the large quantum kinetic energy of the confined very light particle, that alone would favor an extended state. The typical structure then represents a balance between these effects. Necessarily, one must expect some degree of fluctuation in electronic structure that is correlated with fluctuations in solvation structure. In fact, it will be clear from the calculated results that a proper account of fluctuations is of utmost importance for a complete understanding of these phenomena. We should mention that considerable parallel work is being carried out in other groups on negatively charged water clusters, and such systems may provide particularly important theoretical test case^.^^'^ However, it should be noted that the current evidence indicates that for small and intermediate size clusters ( n I32 molecules) the bound electron exists in a surface or external (orbiting) state, qualitatively different from the s o h tion-phase species.I0 Hence, such smaller systems do not directly address the same issues as those of interest here. In section 11, we will discuss the methodology of the area. Section I11 addresses the three specific problems of eaq-behavior described above. The conclusions are given in section IV. 11. Methods
There are basically three elements to the molecular-level simulation of any system, namely, the prescription of a set of intermolecular or interparticle potential functions, the assembly of an appropriate molecular system, and the implementation of an appropriate sampling algorithm, either dynamical or essentially statistical in nature.6 For the most part, these elements are common to both classical and quantum mechanical simulations. In particular, modeling of intermolecular interactions among the solvent molecules is a relatively well developed area although there is substantial opportunity for quantitative improvement.' We have used the so-called SPC, or simple point charge, model for water, which consists of a single Lennard-Jones sphere with three embedded p i n t charges located at the nuclear sites." The molecular unit is treated as rigid. The interaction between a pair of molecules is then a sum of nine Coulombic terms and a single Lennard-Jones 6-12 term. This form is typical of the available models for water, and it is particularly computationally efficient. The development of an electron-solvent interaction potential is a more subtle issue and one that is still in its infancy in a relative (9) Thirumalai, D.; Wallqvist, A,; Beme, B. J. J. Stat. Phys. 1986, 43, 973. Wallqvist, A,; Thirumalai, D.; Berne, B. J. J . Chem. Phys. 1986, 85, 1583. (10) Landman, U.; Barnett, R. N.; Cleveland, C. L.; Scharf, D.; Jortner, J . J . Phys. Chem. 1987, 91, 4890. Barnett, R. N.; Landman, U.; Cleveland, C. L.; Jortner, J. Phys. Rev. Lett. 1987, 59, 811; J . Chem. Phys. 1988, 88, 4429. (1 1) Berendsen, H . J. C.; Postma, J. P. M.; Van Gunsteren, W. F.; Hermans, J. In Intermolecular Forces; Pullman, B., Ed.; Reidel: Dordrecht, 1981; p 331.
The Journal of Physical Chemistry, Vol. 92, No. 15, 1988 4219
Feature Article sense. In a practical calculation, it is necessary to treat the electrons associated with solvent molecules in the simulation implicitly and develop a so-called pseudopotential description for the electron-molecule interaction.’* Such a potential function, well-known for electron-atom interactions in solid-state theory, treats the contributions from a fixed “core” of electrons in terms of an effective potential acting on the “valence” set.13 In the present context, the pseudopotential represents the solvent electrons (and associated nuclei), and the “valence” electron is solely the single excess electron present in the solvent. In our pseudopotential, the solvent contribution is taken to be that obtained from each water molecule individually in its electronic ground state, and we do not explicitly consider any electronic relaxation contributions. The details of the potential have been given in the literature, and we do not repeat them here.12 The potential includes three contributions. The first is a purely electrostatic term which is taken to be that produced by the charge distribution of the SPC water model.’’ This part of the potential is in reasonable accord with that calculated directly from an a b initio molecular wave function, except quite close to the nuclei. The electrostatic potential used is somewhat stronger due to the polarization of the solvent by other solvent molecules which is implicit in the charge distribution of the model water.I4 The second term is a spherically symmetric polarization term, referred to the oxygen nucleus, taken from electron-molecule scattering technology.15 It is of the form (.0/2rW
- exP[(-r/ro)611
where ‘yo is the isotropic part of the molecular polarizability and r, is a cutoff parameter. For r,, we use the sum of the OH bond length and the Bohr radius. The third and most subtle term is an effective, repulsive, potential included to account for the requirements of orthogonality between the one-electron wave function describing the excess electron and those comprising the water molecular wave function. Our potential is analogous to that used some time ago to describe the electron-helium interaction.I6 Under reasonable assumptions of smoothness of the excess electron wave function, one can approximate this core term in a local form, in close analogy to the Slater-type local exchange approximation^.'^ The final form of this part of the potential is evaluated with the s-type basis functions of a doubler multicenter ab initio molecular wave function.ls The repulsive core potential used consists of nine exponential terms, two centered at each hydrogen nucleus and five centered at the oxygen nucleus. Exchange terms were estimated in a local density a p p r o ~ i m a t i o nand ’ ~ found to be quite small compared to other uncertainties in the potential except in regions of high solvent electron density; such effects are therefore omitted. Several other pseudopotentials for electron-water interactions (and for the closely related ammonia system) have been considered.20-23 They are all based on similar concepts, although some are derived on relatively ad hoc grounds20-21while others follow the relatively first principles procedures outlined here.’2,22-23One of the most recent considers the electronic polarization of solvent (12) Schnitker, J.; Rossky, P. J. J. Chem. Phys. 1987, 86, 3462. (13) Szasz, L. Pseudopotential Theory of Atoms and Molecules; Wiley: New York, 1985. (14) Berendsen, H. J. C.; Grigera, J. R.; Straatsma, T. P. J . Phys. Chem. 1987, 91, 6269. (15) Gianturco, F. A.; Thompson, D. G. J . Phys. B 1980, 13, 613. (16) Kestner, N. R.; Jortner, J.; Cohen, M. H.; Rice, S. A. Phys. Reo.
-.
1965. -.--,-140. A%. ----
by the electron and other solvent molecules in an explicit selfconsistent manner, which may well be particularly important in the quantitative treatment of this and other ionic solution problems.22 The rather remarkable feature that has been observed so far is that the structural descriptions of the hydrated electron do not appear very sensitive to the choice of potential, and it appears that more probing properties such as the optical spectrum are necessary to refine these interactions. It should be emphasized in this context that in the development of such a pseudopotential one is forced to make fairly strong, but well-defined, approximations in order to obtain a relatively simple, and local, potential.12 It will be clear from the data presented in the next section that improvements are desirable in order to obtain more accurate quantitative agreement with experiment. It is likely that such improvements will follow a largely empirical route, starting from a firmly based form as that outlined here. It is such a process that has characterized the entire area of liquid state and biophysical modeling, and it has been generally succeSsf~1.7,11,1424 Considering that the relevant potential functions are early in their development, it is appropriate at this stage to emphasize qualitative physical descriptions rather than quantitative ones. It is that emphasis that characterizes the work described in this paper. The assembly of an appropriate system for simulation is essentially the same as that for a classical system. In the present context, one employs a basic system comprised of several hundred solvent molecules and a single (electronic) solute. The calculations described below use from 200 to 500 water molecules. An infinite system is mimicked by employing so-called periodic boundary conditions so that no free liquid surfaces are present.6 All pair interactions are truncated at a distance of 8 A. Such a truncation scheme may produce some substantial errors in absolute energies, but it appears that the structural ramifications of long-range polarization are fairly limited.25 In any case, a viable route to avoiding such truncation in a manageable sample size is not at hand, and further quantitative investigation of this point for systems of the type considered here is still required. The recent evolution of simulation techniques for quantum systems is at the heart of the ability to examine such systems as the hydrated electron. The quantum simulation methods differ most significantly from classical approaches and from one another in the means of representing the electronic distribution. We will assume that readers are familiar with the basic qualitative features of classical computer simulation of l i q ~ i d s ,and ~ , ~ here we will address the quantum simulation techniques at a similar, qualitative level. Two basic approaches have evolved for representation of the electronic distribution. The first is the so-called Feynman path integral representation of the thermal electronic density26and is by far the more highly developed. We will not go into the origins of the technique or most applications in any detail; a recent review of this information is available.27 The critical advantage of the path integral approach lies in the fact that the simulation involved is equivalent to that for a completely classical (but different) system, so that the techniques of classical simulation can be immediately applied. It is the focus of attention on this quantum-classical “isomorphism”28that leads to the facile representation of quantum particles in computer simulation^.^^ For the case at hand, the equilibrium distribution of the electronsolvent system is obtained if the quantum particle is replaced, in the simulation, by a cyclic chain polymer consisting of P (pseudo)particles, each connected to its two nearest neighbors by a harmonic potential whose force constant is related to the
(17) Hara, S. J . Phys. SOC.Jpn. 1967, 22, 710. (18) Arrighini, G. P.; Guidotti, C.; Salvetti, 0.J . Chem. Phys. 1970, 52, in17
(19) (20) 209. (21) (22) 6404.
Salvini, S.; Thompson, D. G. J . Phys. B 1981, 14, 3797. Jonah, C. D.; Romero, C.; Rahman, A. Chem. Phys. Lett. 1986, 223, Sprik, M.; Impey, R. W.; Klein, M. L. J . Stat. Phys. 1986,43,967. Wallqvist, A,; Thirumalai, D.; Berne, B. J. J . Chem. Phys. 1987, 86,
(23) Barnett, R. N.; Landman, U.; Cleveland, C. L.; Jortner, J. J . Chem. Phys. 1988, 88, 4421.
(24) Jorgcnsen, W. L.; Chandrasekhar, J.; Madura, J. D.; Impey, R. W.; Klein, M. L. J . Chem. Phys. 1983, 79, 926. (25) Andrea, T. A.; Swope, W. C.; Andersen, H. C. J. Chem. Phys. 1983, 79,4576. Linse, P.; Andersen, H. C. J . Chem. Phys. 1986,85, 3027. Brooks, 111, C. L. J . Chem. Phys. 1987,86, 5156. (26) Feynman, R. P. Statistical Mechanics; Benjamin: Reading, 1972; Chapter 3. (27) Berne, B. J.; Thirumalai, D. Annu. Reo. Phys. Chem. 1986, 37, 401. (28) Chandler, D.; Wolynes, P. G. J . Chem. Phys. 1981, 7 4 , 4078.
4280 The Journal of Physical Chemistry, Vol. 92, No. 15, 1988
thermal de Broglie wavelength of the electron. Each pseudoparticle interacts with the solvent particles via the specified electron-solvent potential but reduced by a factor of P’.A simulation of the classical system, with sufficiently large P, provides the desired spatial distributions, and related quantities, if one interprets the polymer pseudoparticle distribution as the thermal quantum density, Le., the thermally averaged probability density to find the excess electron at a particular point in space. The path integral description given above can also be generalized to more complex cases such as quantized molecular motion and vibration. A rapidly expanding list of applications including electrons in fluids and clusters, as well as low-temperature molecular and atomic liquids and clusters, have now been considered.27329s30 An obvious alternative to the path integral method is a direct description of the electron in terms of wave functions. Efficient methods have indeed been developed for evaluating the BornOppenheimer, or adiabatic, electronic states associated with a given solvent configuration. The solution for the oneelectron eigenstates following from the rather complex potential surface generated by the collection of water molecules is not trivial but can be readily accomplished by techniques employing an expansion in either plane w a ~ e s ~orl -distributed ~~ G a u ~ s i a n s . ~Such ~ approaches provide sufficiently flexible representations that the electronic states can be described accurately and without the inappropriate bias that would result from the use of atomiclike basis functions. In the first of these methods, the plane wave expansion can be readily combined with a spatial grid representation of the electronic state, under conversion between the two representations by Fourier transformation. The propagation of the wave function in or equivalently the evaluation of the eigenstate spectrum for a given solvent c ~ n f i g u r a t i o n ,can ~ ~ then , ~ ~ be carried out by using only simple multiplicative operations and repeated fast Fourier transforms (FFTs). No explicit matrix diagonalizations have to be performed, so that the calculations are particularly efficient. It is this approach which we have used for all eigenstate calculations presented below. The principal difficulty encountered with the explicit wave function representation is that the appropriate thermal average (density matrix) is not conveniently represented, from a computational viewpoint, in such terms. If, at equilibrium, a distribution of electronic states are accessible, the path integral method thus has clear-cut advantages. For the present system, the ground state is very well separated from the first excited state ( > 4 0 k ~ T )so , for many purposes the thermally excited electronic states can be neglected in describing the equilibrium system. There are, however, situations where the wave function representation appears essential. A first example is provided by optical spectroscopy which is an important characteristic probe of the system. With an explicit eigenstate description, the calculation of optical spectra is straightforward and computationally convenient, in contrast to the situation encountered with a path integral d e ~ c r i p t i o n . ~ ~Given , ~ ’ an ensemble of solvent configurations and the respective electronic eigenstate manifolds, the directly accessible spectroscopic quantities are the vertical electronic excitation energies from the ground state and the corresponding dipole transition matrix elements. The envelope of the set of such electronic absorption intensities as a function of excitation energy provides the first approximation to the optical (29) 949. (30) (31) 412. (32) (33)
Rossky, P. J.; Schnitker, J.; Kuharski, R. A. J . Stat. Phys. 1986, 43,
Rossky and Schnitker absorption spectrum, although contributions due to differences in the ground- and excited-state Born-Oppenheimer potential surface for the solvent are neglected. Since the dominant contribution to the spectral behavior in the present case is expected to be related to the existence of a number of excited states and the inhomogeneous broadening corresponding to the fluctuations in the solvent environment, this is a very valuable approach, as will be clear below. An explicit eigenstate description is also desirable for the treatment of timedependent phenomena, in both equilibrium and nonequilibrium situations. The time propagation of wave functions is a convenient, and familiar, approach. A corresponding method for the direct treatment of the time-dependent density matrix has not been presented. While significant progress is being made toward the direct evaluation of equilibrium average time correlation functions using path integral methods, these have only been exploited for model systems, and their usefulness for systems as complex as those of interest here remains to be i n ~ e s t i g a t e d . ~ ~ ~ The dynamics to be considered in this paper will be limited to the adiabatic time evolution of the electronic ground state in the solution. That is, we assume that the Born-Oppenheimer approximation is valid so that the electronic state is only a parametric function of the solvent coordinate^.^^*^^,^^ The time dependence of the electronic state then arises solely from the time dependence of the nuclear coordinates. Correspondingly, the set of forces on the nuclei F required to propagate the solvent nuclear positions Q in time is evaluated (via the Hellmann-Feynman theorem) as
where $o denotes the electronic ground-state wave function for specified solvent coordinates Q,H i s the full system Hamiltonian, the gradient is taken with respect to the solvent coordinates, and the integration is over electronic coordinate^.^ With such an algorithm, the wave function need not be explicitly propagated in time. The new state can be evaluated by solving the time-independent one-electron problem for each new solvent configuration in the time-dependent sequence. The corresponding electronic absorption spectrum in the sense described above can be evaluated simultaneously if desired. 111. Results
We will discuss three aspects of the hydrated electron behavior, namely, the equilibrium structure, the equilibrium electronic absorption spectra, and finally, nonequilibrium transient spectroscopy, explored by using the adiabatic dynamics method. A . Equilibrium Structure. We consider both the electronic structure and the hydration structure in this section, as obtained from a path integral simulation analogous to that published elsewhere.43 The results for structure, and for steady-state spectroscopy, described here are obtained by using a system consisting of 500 water molecules and one electron (P = 1500 pseudoparticles) at T = 300 K, by using cubic periodic boundary conditions to mimic an infinite system. The simulation of the equivalent classical system is done by using molecular dynamics simulation (classical dynamics) to sample the configurations. The 60-ps run using a time step of 0.002 ps requires 7 h on a Cray X-MP. We will note elsewhere in the paper other timings to give the reader a feeling for the scale of t h e computations. We consider first the electronic structure per se.
Doll, J. D.Adu. Chem. Phys., in press. Feit, M. D.;Fleck, Jr., J. A.; Steiger, A. J . Comput. Phys. 1982.47,
Kosloff, R.; Tal-Ezer, H. Chem. Phys. Lett. 1986, 127, 223. (a) Selloni, A.; Carnevali, P.; Car, R.; Parrinello, M. Phys. Rev. Left. 1987, 59, 823. (b) Selloni, A.; Car, R.; Parrinello, M.; Carnevali, P. J. Phys. Chem. 1987, 91,4947. (34) Schnitker, J.; Motakabbir, K.; Rossky, P. J.; Friesner, R. A. Phys. Rev, Lett. 1988, 60, 456. (35) Sprik, M.; Klein, M. L. J. Chem. Phys. 1987, 87, 5987. (36) Nichols, 111, A. L.; Chandler, D. J . Chem. Phys. 1987, 87, 6671. (37) (a) Thirumalai, D.;Berne, B. J. J . Chem. Phys. 1983, 79, 5029. (b) Thirumalai, D.; Berne, B. J. Chem. Phys. Lett. 1985, 116, 471.
(38) Behrman, E. C.; Jongeward, G. A.; Wolynes, P. G. J . Chem. Phys. 1983, 79, 6777.
(39) (a) Doll, J. D.; Coalson, R. D.; Freeman, D. L. J . Chem. Phys. 1987, 87, 1641. (b) Chang, J.; Miller, W. H. J. Chem. Phys. 1987, 87, 1648. (40) (a) Makri, N.; Miller, W. H. Chem. Phys. Lett. 1987, 139, 10. (b) Wolynes, P. G . J . Chem. Phys. 1987,87, 6559. (c) Doll, J. D.; Freeman, D. L.;Gillan, M. J. Chem. Phys. Lett. 1988, 143, 277. (41) Pechukas, P. Phys. Rev. 1969, 181, 174. (42) Thirumalai, D.; Bruskin, E. J.; Berne, B. J. J . Chem. Phys. 1985,83, 230. (43) Schnitker, J.; Rossky, P. J. J . Chem. Phys. 1987, 86, 3471.
Feature Article
The Journal of Physical Chemistry, Vol. 92, No. 15, 1988 4281
12.33 A
T 5A
I
Figure 1. Potential energy surface for a solvated excess electron in liquid
water. The potential is repulsive in the dotted regions (cores of the solvent molecules) and attractive elsewhere. The bold contour line in the center denotes an energy of -4.5 eV. For this particular configuration, the absolute minimum occurs at -6.75 eV. A feeling for the degree of complexity in the potential surface experienced by the electron, and correspondingly of the typical degree of distortion in the electronic structure, can be obtained from a plot of a typical instantaneous potential surface. Figure 1 shows a planar section through the electronic center of mass for one such case. The isopotential contour lines, spaced by 0.5 eV, indicate regions of favorable (negative) potential, with the absolute minimum occurring at -6.75 eV. The dotted regions are at unfavorable (positive) potentials, in the core region of water molecules. The contour marked with a bold line occurs at -4.5 eV, which is 2.25 eV above the absolute minimum, corresponding to a typical electronic kinetic energy. This contour therefore corresponds approximately to the classical turning point. It is clear that the potential surface is only crudely spherical. The fluctuations in electronic density can be visualized by examining representative pseudoparticle distributions during the simulation. (More quantitative analysis is given below.) Such a set of distributions is given in Figure 2, where seven examples equally spaced throughout the simulation are shown. The 1500 pseudoparticle positions are connected sequentially by straight lines in the figure. The 'optical density" viewed in these examples corresponds to the electron density in the respective positions. It is evident from the figure that the fluctuations in size and shape are not large, but, a t the same time, they are clearly discernible. Nevertheless, the electron is clearly characterizable by a localized, roughly spherical, density distribution. The average radius, characterized by the radius of gyration, is found here to be 2.05 A with a mean-square fluctuation of about 0.1 A, in reasonable accord with some earlier estimates.2 We now turn to the solvation structure. The orientational structure can be analyzed in much the same way as for ordinary atomic ~ o l u t e s , by * ~evaluating ~~ the distribution of solvent dipole directions and OH bond directions with respect to the electron position. The result shows that the solvent is bond-0riented.4~ This structure is perhaps predictable based on the potential12 but contrasts with the dipolar orientation frequently assumed in simplified theoretical treatments.2 The distribution is, in fact, essentially the same as that observed in earlier negative ion solvation studies.*c The radial distribution of solvent is shown in Figure 3. The radial correlations are clearly different in first appearance from
0
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10000
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steps
Figure 2. Sample distributions of an excess electron in liquid water as
obtained from a path integral simulation. The electronic distribution is on the average spherical, but there are fluctuations evident in both radius and shape.
-0
2
4
6
8
10
r/A Figure 3. Radial pair correlation functions between electroniccenter of
mass and either oxygen or hydrogen nuclei of the solvent. ionlike behavior. For a hydrated ion, the radial correlations manifest sharply defined solvation layer structure.* For example, for a typical model of C1- in aqueous solution,*b one finds that the chloride-oxygen correlation function exhibits a distinct first layer peak at about 3.3 A with a peak height of about 2.8. The first ion-hydrogen peak occurs about 1 A closer to the ion at about 2.3 A, with a peak height of close to 2.5. The number of nearest-neighbor water molecules, or coordination number, obtained from radial integration of the distribution function is about 7 . For electron hydration, the bond orientation of the solvent is still apparent (hydrogen approaching 1 A closer to the electronic center than does oxygen), but both hydrogen and oxygen peaks are strongly broadened. Nevertheless, the electron is solventcoordinated in an ioniclike manner; the coordination number obtained from integration of the oxygen radial correlation function shown in Figure 3 is about 6, depending on the choice of radial position used to determine the radius of the first solvation shell. Considering the well-defined orientational correlations found and the reasonable, and relatively small, coordination number observed, it is reasonable to attribute the apparent diffuse nature of the radial correlations to the fluctuations in the shape and radius of the excess electron, rather than to a lack of structure. It is in this respect that the solution containing an excm electron differs most from that with a simple ion. The electron exhibits the solvation structure that would be expected of an ion which had
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4282 The Journal of Physical Chemistry, Vol. 92, No. I S . 1988
the additional freedom to fluctuate in size and shape, while remaining compact and roughly spherical. Finally, it is of interest to compare, as far as possible, the calculated structure to experimental results. While inferences have been drawn +?om a variety of measurements,2 one direct structural analysis has been carried Using electron spin echo measurements to determine a set of solvent proton populations and distances in an aqueous glass, Kevan has proposed a specific idealized structure for the hydrated electron. Keeping in mind that the glass is formed at high salt concentration and low temperature, we can compare the liquid-state results described above to the experimental assessment. Kevan finds that his data are best fit by a (glassy) solvent with six nearest-neighbor protons at a distance of 2.1 8, and a second set of six protons at a distance of 3.5 8, from the electronic center implying an electron-oxygen distance of 3.1 8, and bond-oriented solvent.44 The present results are in good accord with these results. We find a roughly six-coordinate, bond-oriented solvent, with the nearest protons centered at 2.3 8,and oxygen atoms centered at 3.3 A (see Figure 3). Considering the difference in system composition, the limited information which can be extracted from the experiment, and the uncertainties in our present potential, the agreement in structural results is very satisfactory. Not surprisingly, the present results clearly show a substantial degree of dispersity in the detailed solvation structure which cannot be a priori extracted from the experimental data. We note that the structural results reported here differ somewhat from those reported earlier by us for this system$3 with the present results manifesting somewhat better defined solvation structure. The two simulations differ somewhat in technical details. Of potentially greatest significance, the present simulation sample is larger (500 vs 300 water molecules) but the interactions of electron and water are truncated at a smaller distance (8 A from pseudoparticle to water oxygen, compared to 12 8, from electron center of mass to water oxygen in the earlier Clearly the role of such differences in determining quantitative results for this type of system (where the solute has no inherent radius) needs to be explored more fully. B . Steady-State Optical Spectrum. As indicated in the Introduction, the physical description underlying the observed broad and featureless optical absorption spectrum of the hydrated electron is a longstanding i s s ~ that e ~ can ~ ~ be directly addressed by the present theoretical studies. The calculation^^^ consist of the determination of the ground electronic state and first nine excited states in the Born-Oppenheimer approximation for each of 600 configurations of the solvent sampled from the path integral simulation just described. The spectrum is then the envelope of the 5400 lines with intensities proportional to the corresponding electronic transition dipole matrix elements. That this calculation is not numerically trivial is evidenced by the complexity of the potential surface shown in Figure 1. Nevertheless, this spectral calculation required only 3 h on a Cray X-MP using the combined plane wave/grid method3*and the most recent Fourier transform codes available.45 In Figure 4, we show the calculated34 and e ~ p e r i m e n t a l ~ ~ spectral behavior. The amplitude of the calculated spectrum is normalized to unity at its maximum, as is the experimental result.& In the upper half of the figure, experimental and simulated spectra are labeled, with the dashed curve showing the experimental data shifted to higher energy by 0.7 eV. These results show that the calculated spectrum, in fact, reflects the exceptional breadth and also the asymmetry evident in the experimental result, indicating f origins of these observations via the thethat an analysis t ~the oretical results is vel1 founded. It is, however, also clear that the \ theoretical spectrum exhibits significant quantitative shortcomings; namely, the excitation energies are clearly too large compared to experiment and the high-energy tail is not fully developed in
-
(44) Kevan, L. J . Phys. Chem. 1981, 85, 1628. (45) Nobile, A,; Roberto, V. Compui. Phys. Commun. 1986,40, 189; Ibid. 42, 233. (46) Jou, F.-Y.; Freeman, G. R. J . Phys. Chem. 1979, 83, 2383
Rossky and Schnitker 1
A Amax
0
1
A
A,,,
0 1
2
3
4
A E / eV
Figure 4. Upper panel: optical absorption spectra of the hydrated electron from experiment and simulation. The dashed line is obtained after shifting the experimental spectrum to higher energies by 0.7 eV. Lower panel: individual s-p subbands that contribute to the simulated absorption spectrum.
the calculation. It is reasonable to attribute these aspects primarily to the approximate nature of the electronsolvent pseudopotential. The theoretical spectrum per se is itself of little direct interest. Rather it is the analysis of the underlying states and the physics leading to their energetic distribution that is informative. We consider this next. The nature of the electronic states can be directly determined without calculation of the spectrum. It is found that, almost without exception, a roughly spherical s-like localized ground state is followed at higher energy by a triple of p-like states that are also bound and localized.34 Above the p-like states lies a band of apparently unbound delocalized states of indefinite symmetry. The spectrum can then be understood in terms of strongly allowed excitations from the ground state to the three p states and to the higher energy band. If the three p states are ordered by energy, the spectrum can then be further decomposed into the contributions from each excitation. If the potential surface were spherically symmetric, then these three contributions would be equivalent due to the corresponding p-state degeneracy, although each excitation would still be broadened by any fluctuations in the radial potential. The result observed is shown in the lower part of Figure 4. The three excitations to p states dominate the spectrum. The excitation energies are clearly substantially nondegenerate, and each excitation is substantially inhomogeneously broadened by the variability of the solvent surroundings. In fact, the deviation among the most probable excitation energies for the three s-p transitions contributes comparably to the breadth of each transition in determining the overall spectrum. Thus it is immediately evident that fluctuations in the solvent configuration are of critical importance to the spectral behavior. One can then ask for a more detailed physical picture of the underlying solvent fluctuations reflected in this result. In particular, one can directly examine the correlation between relatively simple measures of the shape of the potential surface and the observed spectroscopic energies. The magnitude of the transition dipole moments for the dominant s to p transitions is found to be insensitive to these considerations. In order to describe the potential surface in a way unbiased by arbitrary definitions, we use the ground-state electronic density distribution as an a priori manifestation of this shape. Hence, the “radial” size of a solvent cavity h’measured via the radius of gyration of the ground-state electron density distribution and the deviation from spherical symmetry is measured via the corresponding moment of inertia
The Journal of Physical Chemistry, Vol. 92, No. 15, 1988 4283
Feature Article 3.2
1
1.8
2.0
2.2
0.0 I 0.0
24
r/A
Figure 5. Correlation between electronic ground-state radius and average s-p excitation energy, (3).
tensor. Specifically, we consider, the defined asymmetry parameter 7 given by 7 E
[Imax - Iminl/Imax
(2)
where the values are the maximum and minimum elements in the (diagonal) principal axis frame. Since the three p states would be degenerate for a spherically symmetric, purely radial, potential we consider, first, the average of the three s-p transition energies AE’ = (AEs-p,+
+ AE,)/3
(3)
and correlate this value against the radius discussed above. The result is shown in Figure 5 , where the 600 examples available are each plotted. The observed correlation is quite remarkable. That is, the average s-p excitation energy is rather accurately predictable from the average radial extent of the ground-state wave function. This result also suggests that the deviation from experiment found in the peak position in the spectrum (see Figure 4) can be rather easily remedied by a softening of the repulsive short-ranged part of the electron-water pseudopotential. We note that the width of the distribution of radii is only about 0.2 A, or about 10% of the average value. The corresponding width in the distribution of AE’is about 0.6 eV, substantially less than the observed spectral width in the simulated spectrum (Figure 4). The role of asymmetry in making up the remaining width is explored in Figure 6, where the energetic splitting among the p-states, E3 - E , , is correlated against the simple asymmetry parameter 7 defined in (2). Although the correlation is not quite as good as for the radius (Figure 5), particularly (and not surprisingly) for the largest asymmetry, the correlation is impressive. That is, the splitting follows the asymmetry closely, as might be expected, for example, by analogy to a particle in a noncubic box. However, the slope observed here is roughly twice that that would be obtained for the box problem with parameters chosen to correspond to the average observed values. Two quantitative aspects are noteworthy. First, the typical deviation from spherical symmetry corresponds to only about 8% variation in the axis lengths characterizing the electronic distribution ( ( 7 ) 0.16). Thus, the deviations are relatively small but are of critical importance. Second, due to the fact that asymmetric configurations of the solvent so vastly outnumber symmetric arrangements, the probability of observing the symmetric case ( q = 0) is vanishingly small. Correspondingly, the most probable configurations have sizeable splittings; the most probable result is 0.8 eV. It is therefore not surprising that less detailed models which have focused on spherically symmetric potentials have failed to generate sufficient spectral widths.2h The basic method outlined above fot numerically evaluating the electronic absorption spectra has now also been applied to another model Hamiltonian for the hydrated electron by Berne and c o - ~ o r k e r s . ~The ~ two models differ in a number of ways,
-
OJ
02
03
I
7 Figure 6. Correlation between electronic ground-state asymmetry, (2), and p-state energy splitting.
but those of most likely significance include a quantitatively rather different short-range repulsive electron-water pseudopotential, the introduction of solvent polarization self-consistently, and the use of a different model for the solvent-solvent interaction. It is impossible to predict the separate influence of these various differences on the resulting spectrum. This alternative model yields a spectrum47with a peak located at 1.7 eV in accord with experiment, but the spectrum is markedly less asymmetric than the calculated result shown in Figure 4. Considering that the present result fails to reproduce the peak position well, it is not now possible to place a figure of merit on the relative performance of the two interaction potentials, and no objective procedure for shifting or scaling the calculated spectra is available to assist in such an evaluation. However, an important conclusion that can be drawn from this comparison is that optical spectroscopy provides an essential and sound means of discriminating among models that is not directly available via structural analysis alone. C. Electronic Solvation Dynamics. The methods for simulation of electronically adiabatic dynamics (Born-Oppenheimer dynamics of the solvent) outlined in section I1 can be applied to a variety of problems. The approach is only strictly appropriate if one expects negligible participation in the physical process by other electronic states. Thus, one expects validity for electronic diffusion in polar liquids in e q u i l i b r i ~ m , ~although ~ ~ * ~ * ?not ~ ~in many solids or in liquids which only weakly localize the electron. Also, aspects of transient spectroscopy of the equilibrated electron are directly acce~sible.~~ Here, we consider a somewhat different application, namely, the test of a scenario for relaxation of an initially high-energy electron to its equilibrium state in water. The process of electron localization has been considered on numerous occasions.51 The experiment consists of creation, by one of a number of methods, of a high-energy, presumably delocalized, electron in a liquid, presumably initially in its bulk equilibrium state. The electron is trapped and equilibrates to its final equilibrium state via solvent configurational rearrangement combined with nonradiative relaxation of the electronic energy.51 The issue is the detailed physical description of the relaxation process. The prevailing view has described the relaxation as proceeding via an initial very fast “thermalization” of the excess electronic energy via transfer to solvent modes to form a localized, but not equilibrated, state. The localization site may be determined by preexisting structure in the fluidS2and/or by induced polar(47) Wallqvist, A.; Martyna, G.; Berne, B. J. J . Phys. Chem. 1988, 92, 1721. (48) Schnitker, J.; Motakabbir, K. A.; Rossky, P. J., manuscript in prep-
aration.
(49) Sprik, M.; Klein, M. L. J . Chem. Phys., in press. (50) Motakabbir, K. A.; Schnitker, J.; Rossky, P. J., manuscript in preparation. (51) (a) Mozumder, A. In ref 2d, p 139. (b) Walker, D. C. J . Phys. Chem. 1980,84,1140. (c) Kenney-Wallace, G. A. Adu. Chem. Phys. 1981,47,535. (d) Kenney-Wallace, G. A.; Jonah, C. D. J . Phys. Chem. 1982, 86, 2572. (52) Schnitker, J.; Rossky, P. J.; Kenney-Wallace, G. A. J . Chem. Phys. 1986, 85, 2986.
4284
.KOSSKY - . * ana. acnnitKer ”. ..
The Journal of Physical Chemistry, Vol. 92, No. 15, 1988
ization by the electron (self-trapping). This state would be only shallowly trapped, as can be inferred from time-resolved spectra in alcohols51dand most recently in water.53 The further equilibration would proceed under the dominant control of solvent configurational rearrangement. The best available experiments in terms of spectral range and time resolution are those by Migus et al. for water.53 These experiments provide a detailed description of the time-dependent evolution of the optical absorption spectra. The critical feature of the experiment is the observation of an apparently stepwise solvation process. The first step is characterized by the rise of absorption in the infrared (peaking at X > 1250 nm) with a s ) . ~This ~ rise is characteristic time scale of 110 fs (1.1 X compatible with a continuous blue shift of the absorption maximum, with the peak absorption remaining above 1250 nm, the instrumental cutoff. The rise has been tentatively assigned to the thermalization and initial localization of the electron. The second step is manifested by a coincident disappearance of the initially formed state and the appearance of the fully formed equilibrated state (absorption maximum at 720 nm),with a characteristic time scale of 240 fs.53 The experimental evidence for electrons in alcohols,51dalthough somewhat less conclusive, suggests a similar kind of description. Thus the spectra do not exhibit a continuous wavelength shift during the second stage. As noted by Migus et al.53 available theories which describe the equilibration via solvent relaxation would, in fact, predict a continuous spectral shift.54 Migus et al. therefore tentatively suggest that the second observed step may correspond to an electronic transition or dephasing between the initially formed localized electronic state and a state more characteristic of the final solvated electron, with a rate that is nevertheless reflective of the solvent dynamics.53 The calculation described here is designed to provide insight into the likelihood of various interpretations of the experiments by directly testing one specific relaxation process scenario. Specifically, we evaluate the time-dependent optical absorption spectra that follows from the dynamical evolution of a system where, initially, the solvent is characteristic of pure water at equilibrium and the electron is placed in its electronic ground state with no solvent relaxation. We take 100 examples of initial solvent configurations, selected a t 1-ps intervals from a dynamical simulation of water. The electronic ground state is evaluated for each configuration and then the system is evolved in time adiabatically; Le., the electronic state is forced to be the Born-oppenheimer ground state following from the solvent configuration as outlined in section 11. This presumes that thermalization occurs much more quickly than solvent configurational relaxation and that no electronic excited states play a dynamical role; Le., the electronic relaxation is forced to correspond to the physical picture characterizing the preuailing view of the slower events in the relaxation p r o ~ e s s . ~ ’ One has no a priori basis for presuming that the dynamical prescription used is correct for the physical process, but by evaluating the consequences of the prescription, in direct comparison to experiment, one can,in any case, form clear conclusions about the nature of the process that is being experimentally observed. For a 200 water molecule system, the propagation of 100 initial states for 90 fs each (time step of 2 fs) requires about 4 h on a Cray X-MP. We also calculate the first 20 excited states at 7 distinct time points for each of the 100 solvent configurations. This takes about 9 Cray X-MP hours. We first state some general observations and then discuss the optical spectra. It is observed that, without exception, the electron localizes to nearly its equilibrium radius on a time scale of -30 fs in a preexisting cavity in the liquid and that coincidentally the solvation (53) Migus, A.; Gauduel, Y.; Martin, .I. L.; Antonetti, A. Phys. Rev.Lett. 1987, 58, 1559. (54) Calef, D. F.; Wolynes, P.G . J . Chem. Phys. 1983,78,4145. Zusman, L. D.; Helman, A. B. Chem. Phys. Lett. 1985, 114, 301.
%oh
0
2000
1000
3000
i/nm
Figure 7. Simulated time-resolved spectra for adiabatic (ground-state) trapping scenario of an excess electron in liquid water. The subsequent spectra are vertically shifted as indicated, and the normalization is always with respect to the equilibrium absorption spectrum (Figure 4). In the dashed parts of the early spectra, the 20 transitions included in the calculations may not account for all of the intensity in the respective wavelength region, but the shown trends should be qualitatively correct due to the Thomas-Reiche-Kuhn sum rule. The solid line regions of the spectra correspond to wavelengths longer than the largest wavelength (lowest energy) encountered in any example examined; see also ref 34.
energy (including the quantum kinetic energy of the electron) drops by about 2.5 eV. This necessarily implies that only very small solvent displacements are required to produce a reasonably deep trap and that the preexistence of cavities, rather than deep traps, is characteristic of the process. Examination of selected trajectories for longer times shows a second time scale of about 200 fs that is representative of both heat dissipation out of the immediate solvation shell of the electron and of the (modest) translational reordering of solvent required to basically form the hydrated electron solvation structure shown in Figure 3. The corresponding optical spectra calculated at selected times during this relaxation are shown in Figure 7. We note at the outset that the results shown in Figure 7 are changed very little even if one selects from the initial states only those for which the electron is well localized from the start. This further implies that the search for a relatively deep localization site is unlikely to be a significant contributor to the dynamics observed in the experimental spectra. In particular, the sequence of simulated spectra shows the following features: 1. At the earliest time, the simulated spectrum is consistent with that observed e ~ p e r i m e n t a l l y .Although ~~ the wavelength region of most accuracy here does not overlap the experiment, the oscillator strength sum rule assures that the shorter wavelength region looks similar to the dashed data shown in our figure. This consistency is insensitive to selection of initial states according to degree of electron locali~ation.~~ This certainly does not imply that the experiment is sensing ground-state electrons, but rather that even localized ground-state electrons produce this diffuse infrared absorption, in the absence of solvent relaxation. 2. The calculated spectra then show a continuous blue shift in the optical absorption maximum with a gradual narrowing of the spectrum. Such a behavior is consistent with the early time evolution of the experimental absorption spectrum. 3. There is a further evolution of the simulated spectrum which has not been followed in detail. The spectrum after 80 fs is still somewhat broader than that a t equilibrium and has about 80% of the full intensity at its maximum. As noted earlier, a second slower time scale in the range of 200 fs is evident in the calcu~
~~
~~~
( 5 5 ) Motakabbir, K.A,; Rossky, P. J., submitted for publication in Chem.
Phys.
Feature Article lations. The key point here is that the spectral shape and its peak position shift to be comparable to that at equilibrium on a much shorter time scale, of the order of 140 fs, so that the spectral manifestation of the solvation process is only large in its earliest stages, and the remainder produces only modest (although observable) effects. The speed with which the electron is “solvated”, as manifested spectroscopically, is remarkable. Although the model solvent dynamics may be somewhat faster than real water, based on calculated pure water dynamics,” the calculated rate of spectral evolution suggests that the time scale governing the initial step in the experiment may well be electron solvation in largely complete form, as far as spectral changes are concerned. Considering the observations made above, a consistent interpretation of the experimentally observed process follows in general ~ we have grounds terms that expressed by Migus et ~ 1 . :although to be somewhat more concrete. It is consistent to postulate that, in the first step (1 10-fs time scale), the electron is solvated in an effective competition with nonradiative energy loss so that the electron is not in its ground state. Otherwise the spectrum would, as we observe, transit quickly to nearly the equilibrium spectrum. The shifted infrared spectrum observed experimentally a t early times (- 100 fs) is then characteristic of a solvated excited state. The second step process would then be characterized by the time scale for nonadiabatic transition between this solvated excited state and the ground state, with a time scale characteristic of the liquid. However, the ground state thus formed would nor have the full equilibrium absorption spectrum, since the solvent would not be relaxed around this new state. However, as we have seen, the transit time for the spectrum to continuously blue shift from this new state to the equilibrium state should be short compared to that associated with the rate-limiting step. The observation of this step would be only a somewhat enhanced intensity at wavelengths intermediate between those dominated by the initial excited electronic state and those dominated by the equilibrated ground state. The experimental data, in fact, show such enhancement, and it was noted that this was beyond a simple two-state spectral analysis.53 Of course, the description given here of the solvation process remains speculative. However, the calculations provide tangible support for the general view tentatively expressed based on the experimental data alone53and provide additional concrete elements in the development of a complete physical description. Most importantly, the calculations imply that the straightforward interpretation of the equilibration process in terms of the dominance
The Journal of Physical Chemistry, Vol. 92, No. 15, 1988 4285 of solvent configurational relaxation that is prevalent in the earlier literature5’ is not a tenable one.
IV. Conclusions We have described elements of the rapidly evolving methods of quantum simulation as applied to electronic structure in liquids, with particular application to the hydrated electron. It is clear that the techniques are advancing quickly and that wholly new avenues of investigation into solution chemistry are becoming accessible. Progress in the near future will likely include fully molecular treatments of electron-transfer dynamics in bulk solution, at interfaces, and in biological systems,s6 ground- and excited-state electron transport, and excited-state relaxation processes in solutions, all a t a level of realism not previously approached. To realize these expectations, there will necessarily need to be significant progress in the development of appropriate pseudopotentials, particularly for electronic states of molecular solutes. The development of efficient algorithms capable of handling electronic state transitions (nonadiabatic behavior) on an equal footing with other dynamical processes is clearly of utmost importance as well, and such development is under way in our laboratory, as well as in others. Such an algorithm is, for example, necessary in order to finalize the description of the electronic relaxation discussed here, but also for the general description of charge-transfer reactions, and any process involving competitive nonradiative relaxation. That such techniques, and the fruits that would follow from them, will be forthcoming in the near future seems very likely considering the high interest among theoretical groups and the rate at which development has occurred in the recent past. Acknowledgment. We are grateful to K. A. Motakabbir for his assistance in the evaluation of optical spectra. The work discussed here has been carried out with the generous support of the Center for High Performance Computing of the University of Texas system. Other research support has been provided by the Robert A. Welch Foundation and the National Institute of General Medical Sciences. P.J.R. is the recipient of an N S F Presidential Young Investigator Award, a Dreyfus Foundation Teacher-Scholar Award, and an N I H Research Career Development Award from the National Cancer Institute, DHHS. ( 5 6 ) Kuki, A.; Wolynes, 1647.
P. G . Science (Washington, D.C.) 1987, 236,
