The Hydrated Electron Quantum Simulation of Structure, Spectroscopy, and Dynamics

Prévia do material em texto

J. Phys. Chem. 1988, 92, 4277-4285 4277 
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 s e) P2(c0s O ) / l ” d ( c o s 0 0) (2) 
The main point is that the decay of the anisotropy at times very 
short compared with the overall motions of the polymer corre- 
sponds 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 f iF(t) might account for the observed decay on the nano- 
second 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 es- 
sentially 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 po- 
larization 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 measurementsL2 to 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 
FEATURE ARTICLE 
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 reorien- 
tations of the polymer backbone which may involve a range of 
trans, gauche, or intermediate structures. Based on the excitation 
bandwidth” and the expected energy shiftsLs for 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 su- 
perexchange 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 a t 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 NSF- 
DMR-85 19059 and by the Sandia National Laboratories, sup- 
ported by the US . Department of Energy under Contract No. 
DEAC04-76-DP00789. 
(17) Takeda, K.; Teramae, H.; Matsumoto, N. J . Am. Chem. SOC. 1986, 
208, 8186. 
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 7871 2 (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 
we Will make no attempt to wm”m-Ie it here.’ 
extensive, as is that for the analogous species in ammonia, and 
(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,- a t 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 e igen~pectrum.~ In 
such a situation, a first principles approach is necessary to de- 
termine 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 ~ imula t ion .~ These classical techniques have 
become practically routine in recent years as a method for in- 
vestigating, a t 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 in- 
vestigation. 
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 computersimulation are based on classical 
(Newtonian) mechanics and are therefore not immediately ap- 
plicable 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 
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 im- 
portant techniques in the growing area of quantum simulation, 
including methods for simulating equilibrium structure and rel- 
atively newer techniques for simulating equilibrium and none- 
quilibrium 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 dy- 
namics 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 I 32 molecules) the bound electron exists in a surface or 
external (orbiting) state, qualitatively different from the soh- 
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 sim- 
ulation of any system, namely, the prescription of a set of in- 
termolecular 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 em- 
bedded 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.; Her- 
mans, J . In Intermolecular Forces; Pullman, B., Ed.; Reidel: Dordrecht, 1981; 
p 331 . 
(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. 
Feature Article 
sense. In a practical calculation, it is necessary to treat the 
electrons associated with solvent molecules in the simulation im- 
plicitly 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 pseudopo- 
tential, 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 con- 
tributions. 
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 po- 
tential 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 / 2 r W - 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, po- 
tential 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 ap- 
proximate 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 
appro~imat ion’~ and 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 consid- 
ered.20-23 They are all based on similar concepts, although some 
are derived on relatively ad hoc grounds20-21 while others follow 
the relatively first principles procedures outlined here.’2,22-23 One 
of the most recent considers the electronic polarization of solvent 
The Journal of Physical Chemistry, Vol. 92, No. 15, 1988 4219 
(12) Schnitker, J.; Rossky, P. J. J. Chem. Phys. 1987, 86, 3462. 
(13) Szasz, L. Pseudopotential Theory of Atoms and Molecules; Wiley: 
(14) Berendsen, H. J. C.; Grigera, J. R.; Straatsma, T. P. J . Phys. Chem. 
(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. 
New York, 1985. 
1987, 91, 6269. 
1965. 140. A%. --, - - . - - - - -. 
(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) Salvini, S.; Thompson, D. G. J . Phys. B 1981, 14, 3797. 
(20) Jonah, C. D.; Romero, C.; Rahman, A. Chem. Phys. Lett. 1986, 223, 
(21) Sprik, M.; Impey, R. W.; Klein, M. L. J . Stat. Phys. 1986,43,967. 
(22) 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. 
209. 
6404. 
Phys. 1988, 88, 4421. 
by the electron and other solvent molecules in an explicit self- 
consistent manner, which may well be particularly important in 
the quantitative treatment of this and other ionic solution prob- 
lems.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 suc- 
Considering that the relevant potential functions are early in 
their development, it is appropriate a t 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 es- 
sentially 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 a t 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 density26 and 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 com- 
pletely 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”28 that 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 
ceSsf~1.7,11,1424 
(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, 74, 4078. 
4280 The Journal of Physical Chemistry, Vol. 92, No. 15, 1988 Rossky and Schnitker 
thermal de Broglie wavelength of the electron. Each pseudo- 
particle 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 gener- 
alized 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 mo- 
lecular and atomic liquids and clusters, have now been consid- 
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 Born- 
Oppenheimer, 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 ~ l - ~ ~ or 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 calcu- 
lations 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 compu- 
tational 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 rep- 
resentation 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 cal- 
culation 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 configu- 
rations and the respective electronic eigenstate manifolds, the 
directly accessible spectroscopic quantities are the vertical elec- 
tronic excitation energies from the ground state and the corre- 
sponding dipole transition matrix elements. The envelope of the 
set of such electronic absorption intensities as a function of ex- 
citation energy provides the first approximation to the optical 
ered.27329s30 
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 con- 
tribution 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 corre- 
lation 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 ap- 
proximation 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 spec- 
troscopy, 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) a t 
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 the computations. 
(29) Rossky, P. J.; Schnitker, J.; Kuharski, R. A. J . Stat. Phys. 1986, 43, 
(30) Doll, J . D. Adu. Chem. Phys., in press. 
(31) Feit, M. D.; Fleck, Jr., J . A.;Steiger, A. J . Comput. Phys. 1982.47, 
(32) Kosloff, R.; Tal-Ezer, H. Chem. Phys. Lett. 1986, 127, 223. 
(33) (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. 
We consider first the electronic structure per se. 949. 
412. (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, 
(43) Schnitker, J.; Rossky, P. J. J . Chem. Phys. 1987, 86, 3471. 
230. 
Feature Article - 12.33 A 
The Journal of Physical Chemistry, Vol. 92, No. 15, 1988 4281 
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 
a t unfavorable (positive) potentials, in the core region of water 
molecules. The contour marked with a bold line occurs a t -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 sim- 
plified theoretical treatments.2 The distribution is, in fact, es- 
sentially 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 
T 
5A 
I 
0 
steps 
5000 
steps 
10000 15000 
steps steps 
20000 
steps 
25000 
steps 
30000 
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 electronic center 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 near- 
est-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 solvent- 
coordinated 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 
4282 
the additional freedom to fluctuate in size and shape, while re- 
maining 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 mea- 
surements 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 tem- 
perature, 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 com- 
position, the limited information which can be extracted from the 
experiment, and the uncertainties in our present potential, the 
agreement in structural results is verysatisfactory. Not sur- 
prisingly, 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 In- 
troduction, the physical description underlying the observed broad 
and featureless optical absorption spectrum of the hydrated 
electron is a longstanding i s s ~ e ~ ~ ~ that 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 
that an analysis t ~ f the origins of these observations via the 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 
The Journal of Physical Chemistry, Vol. 92, No. I S . 1988 
\ 
Rossky and Schnitker 
(44) Kevan, L. J . Phys. Chem. 1981, 85, 1628. 
(45) Nobile, A,; Roberto, V. Compui. Phys. Commun. 1986,40, 189; Ibid. 
(46) Jou, F.-Y.; Freeman, G. R. J . Phys. Chem. 1979, 83, 2383 
42, 233. 
1 
A 
Amax 
0 
1 
A 
A,,, 
2 3 4 
0 
1 
AE / 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 exci- 
tation is substantially inhomogeneously broadened by the varia- 
bility 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 de- 
termining the overall spectrum. Thus it is immediately evident 
that fluctuations in the solvent configuration are of critical im- 
portance to the spectral behavior. 
One can then ask for a more detailed physical picture of the 
underlying solvent fluctuations reflected in this result. In par- 
ticular, 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 
Feature Article The Journal of Physical Chemistry, Vol. 92, No. 15, 1988 4283 
3.2 1 
1.8 2.0 2.2 2 4 
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 
( 2 ) 7 E [Imax - Iminl/Imax 
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 asfor the radius (Figure 5), particularly (and not sur- 
prisingly) 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 distri- 
bution ( ( 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 sym- 
metric 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, 
0.0 I I 
0.0 OJ 02 0 3 
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 spectrum47 with a peak located at 1.7 eV in accord with ex- 
periment, 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 
a c c e ~ s i b l e . ~ ~ 
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 re- 
laxation 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 fluidS2 and/or by induced polar- 
(47) Wallqvist, A.; Martyna, G.; Berne, B. J. J . Phys. Chem. 1988, 92, 
(48) Schnitker, J.; Motakabbir, K. A.; Rossky, P. J., manuscript in prep- 
(49) Sprik, M.; Klein, M. L. J . Chem. Phys., in press. 
(50) Motakabbir, K. A.; Schnitker, J.; Rossky, P. J., manuscript in prep- 
aration. 
(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. 
1721. 
aration. 
. - . * . ” . . . 4284 The Journal of Physical Chemistry, Vol. 92, No. 15, 1988 
%oh 
ization by the electron (self-trapping). This state would be only 
shallowly trapped, as can be inferred from time-resolved spectra 
in alcohols51d and most recently in water.53 The further equil- 
ibration 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 
characteristic time scale of 1 10 fs (1.1 X s ) . ~ ~ This rise is 
compatible with a continuous blue shift of the absorption max- 
imum, 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,51d although 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 a t 
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 thermalizationoccurs 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 pro~ess .~’ 
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 com- 
parison to experiment, one can, in any case, form clear conclusions 
about the nature of the process that is being experimentally ob- 
served. 
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 a t 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. 
(54) Calef, D. F.; Wolynes, P. G . J . Chem. Phys. 1983,78,4145. Zusman, 
1987, 58, 1559. 
L. D.; Helman, A. B. Chem. Phys. Lett. 1985, 114, 301. 
KOSSKY ana acnnitKer 
- 
0 1000 2000 3000 
i / n m 
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 a t 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 exper- 
imental 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~per imenta l ly .~~ 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 local i~at ion.~~ 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 a t 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 The Journal of Physical Chemistry, Vol. 92, No. 15, 1988 4285 
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 ob- 
servable) 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 com- 
plete form, as far as spectral changes are concerned. 
Considering the observations made above, a consistent inter- 
pretation of the experimentally observed process follows in general 
terms that expressed by Migus et ~ 1 . : ~ although we have grounds 
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 a t 
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 en- 
hancement, 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 alone53 and provide additional concrete elements 
in the development of a complete physical description. Most 
importantly, the calculations imply that the straightforward in- 
terpretation of the equilibration process in terms of the dominance 
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. Itis 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 so- 
lution, a t 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 pseudo- 
potentials, 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 im- 
portance 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 a t 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 Devel- 
opment Award from the National Cancer Institute, DHHS. 
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1647.