J. Phys. Chem. 1996, 100, 7691-7697
7691
Electron Localization in Liquid Methanol. Lifetime of the Pre-existing Electron Traps Maria Hilczer†,‡ and M. Tachiya*,‡ Institute of Applied Radiation Chemistry, Technical UniVersity, Wro´ blewskiego 15, 93-590 Ło´ dz´, Poland, and National Institute of Materials and Chemical Research, Tsukuba, Ibaraki 305, Japan ReceiVed: October 30, 1995; In Final Form: February 8, 1996X
The structure of liquid methanol, obtained by computer simulation, has been analyzed in order to identify the regions of attractive potential that can serve as pre-existing sites for primary localization of an excess electron. Only about 4% of the regions of attraction produce the binding level and are able to accomodate the electronsthese regions have been considered as pre-existing electron traps. The static and dynamical properties of the traps have been investigated and described in terms of the statistical distributions of their geometrical parameters and their lifetimes, respectively. The analysis performed for liquid methanol at room temperature shows that the average lifetime of the electron traps is on the order of a few tens of femtoseconds. However, the concentration of the pre-existing traps that can survive over a period of several hundred femtoseconds is sufficiently high to justify the commonly assumed “trap-seeking” mechanism of electron trapping in polar media.
I. Introduction The mechanism of the early stages of an excess electron localization in amorphous media is still actively discussed in literature.1 The basic question is whether the electron localization should be considered as a self-trapping process (“trapdigging” mechanism) or as being trapped by the so-called preformed or pre-existing traps (“trap-seeking” mechanism). The idea of the latter mechanism was formulated 27 years ago by Tewari and Freeman,2 and then it was supported experimentally for excess electrons in liquid alcohols by Kenney-Wallace and Jonah.3 The existence of preformed trapping sites (or cavities arising from structural fluctuations inside a molecular liquid) where the potential produced by matrix molecules attracts the excess electron can be predicted in light of liquid structure theory and the fluctuation theory of statistical mechanics. This was first shown by Tachiya and Mozumder4,5 in their pioneering work. The early path-integral computer simulations6-10 have proved that an excess electron is localized in a void space between molecules. The results of the classic molecular dynamics simulations11-15 employed to study the possible trapping sites in water and methanol show that the structural fluctuations in these liquids produce energetically favorable regions that are sufficiently large and abundant enough for primary electron localization. Moreover, the femtosecond experiments16-18 show that the process of electron capture in water and methanol is much faster than the dielectric relaxation of solvent molecules. All these facts seem to confirm the “trap-seeking” mechanism of the primary electron localization in polar liquids. To supplement the discussion on the validity of one or another mechanism, we should address the question of the relation between the primary localization time of excess electrons and the persistence times of the preformed traps in polar liquids. In other words, we should ask if the lifetimes of the structures identified as electron traps are at least comparable with the experimentally estimated time of transition of the delocalized electron in the conduction band to the localized state in the trap. * Corresponding address. † Technical University. ‡ National Institute of Materials and Chemical Research. X Abstract published in AdVance ACS Abstracts, April 1, 1996.
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The primary localization time in liquid water at room temperature is reported as 110 fs16 or 180 ( 40 fs.17 The initially localized electron is characterized by the IR absorption band and evolves to the visible-absorbing equilibrated electron, eaq-, on a subpicosecond time scale. The evolution is not observed as a continuous blue shift of the absorption maximum but as a decay of the IR band and a simultaneous rise in intensity of the visible band. The experimental results were reproduced by the two-state model17,19 according to which the quasi-free electron is initially localized in the excited p-state of the trap and then relaxes to the ground s-state via a radiationless transition. This model was also supported by the adiabatic10,20 and the nonadiabatic21,22 quantum simulations of the electron solvation dynamics in water. However, recent experimental and theoretical papers show that the excess electron localization cannot be accounted for using a simple two-state model.23-25 They postulate a modification of the two-state model to include other processes such as solvation of the ground or excited electronic states or solvent cooling after a nonadiabatic electron transition to its ground state. The role of these processes in the observed spectral dynamics of the aqueous electron is now the question open to discussion. Particularly, some experiments26 suggest the existence of the second relaxation channel of the excess electron. This channel does not contribute to the formation of eaq- but is involved in the formation of a hybrid electron-radical pair whose appearance time and mean lifetime are very close to those of the presolvated electron. The simple two-state model also seems to be insufficient to reproduce the localization dynamics of the quasi-free electrons in liquid alcohols.18,27-30 On the basis of the femtosecond kinetic measurements of an excess electron in liquid methanol, Pe´pin et al.18 have suggested that the electrons become distributed in two localized states, a weakly bound state and a strongly bound one, between which there is a stepwise transfer mechanism. Both species are assumed to relax, which is reflected in the blue shift of their transient absorption bands. The time of the electron capture into the primarily localized states has not been determined in ref 18. However, the authors have postulated that about half the electrons are found in deeply localized states within the first picosecond.18,28 The presence of the very localized electrons at early times has been also indicated by Shi et al.30 in their femtosecond studies of electron © 1996 American Chemical Society
7692 J. Phys. Chem., Vol. 100, No. 18, 1996 solvation dynamics in a series of neat linear alcohols. Unfortunately, these authors could not determine accurately the primary localization time of excess electrons because the measured absorption involved an interfering, ultrafast transient signal that they attributed to either an alcohol cation or an excited state of the neutral alcohol molecule. The dynamics and, consequently, lifetimes of the preformed electron traps in polar liquids are directly related to the solvent relaxation dynamics. In order to determine the persistence times of traps, it is necessary to consider the dynamics of a liquid on a microscopic level. In the present study the problem of the trap lifetime has been addressed explicitly. We investigate the structure of the computer-generated sample of liquid methanol at room temperature, and we record, over the time interval of several tens of picoseconds, the changes in the locally averaged electrostatic potential in the regions that we identify as electron traps. From these data we calculate the time dependence of the concentration of the pre-existing traps and the statistical distribution of their lifetimes. The results obtained from such an analysis are to some extent dependent on the assumed definition of the trap. In the present approach, similarly as in ref 12, we base our definition on the locally averaged electrostatic potential in cavities between solvent molecules. Taking into account the spatial extension of the electron charge, the averaged potential seems to be more appropriate, in comparison with a point value of the potential at any particular place inside cavity, for classifying a cavity as a trapping site. Moreover, in the trap identification algorithm we consider the so-called quantum strength of the trap, i.e., we assume that the electron trap has to be deep and large enough to accomodate the quasi-free electron. The trap elimination procedure that we applied (see section IIC) is similar to the elimination method proposed by Mozumder31 in his discussion of the results obtained by Schnitker et al.11 on the preexisting trapping sites in the computer-simulated liquid water. Schnitker et al.11 identified the trapping sites in a water matrix as local minima of the potential inside cavities between water molecules. Such an approach resulted in a very high trap concentration of 4.4 M (this value was reduced later by Mozumder31 to 0.74 M). Although their definition of the electron trap connects trap depths with values of the potential at its instantaneous local minima, it does not connect trap sites to any particular places of the matrix. Consequently, this definition is not precise enough for performing an extensive analysis of the trap dynamics. Schnitker, Rossky, and KenneyWallace were, however, the first who have estimated the trap persistence time in liquid water.11 In the presentation that follows we begin in section II with a description of the details of the performed computer simulation and next we present the employed procedures for selection of cavities and for identification of electron traps in the molecular matrix. In section IIIA we discuss the statistical distributions of shape, size, and energy of the pre-existing traps and compare them with the corresponding results for cavities with negative average energy and also with some results derived in ref 12 on the basis of the random field theory. It is worth noting that the trap distributions in the present paper are obtained from the pure statistical analysis of the trapping regions found in the computersimulated methanol matrix. Finally, in section IIIB we present the dynamical properties of traps (described in terms of the time correlation functions of fluctuations in the trap energy) and we address the issue of persistence times of the pre-existing electron traps in polar liquids.
Hilczer and Tachiya II. Details of Calculations A. Molecular Dynamics Simulation. The calculations were performed for a methanol matrix at room temperature. Configurations of the liquid sample were generated by a classical molecular dynamics simulation. The system of 256 methanol molecules, interacting via a 3-center H1-model intermolecular potential of Haughney et al.,32 was confined to a cubic box of side length 25.864 Å. The usual periodic boundary conditions and the Ewald summations of the electrostatic interactions were employed. The equations of motion were integrated with a time step of 1 fs using the leapfrog and leapfrog/quaternion algorithms for the translational and the angular motions of methanol molecules, respectively.33 Equilibration of the system was performed for a period of 9 ps, and during this period the translational and rotational kinetic energies of the system were rescaled to suit the assumed temperature of 298 K. Starting from the equilibrated methanol configuration, the simulation was extended over 34 ps. A quantity of 290 configurations separated by 0.1 ps were used for static analysis of geometry and energy of the pre-existing electron traps. The regions that were identified as electron traps in these configurations (see section IIC) were employed as input data for the investigation of the trap dynamics. The trap lifetimes and the time correlation functions for the energy of trapping regions were determined by tracing the energy of these regions through consecutive methanol configurations each separated by 2 fs from the next. B. Cavity Selection. We assumed that excess electrons are localized in cavities between molecules.4,5,11-13 Thus, our procedure for the pre-existing traps identification started from a search for cavities in the computer-simulated methanol sample. For each of 290 equilibrium molecular configurations we performed the Delaunay tessellation of space.34 In our analysis the positions of the oxygen atoms were considered to be vertices of the Delaunay tetrahedra. The four vertices of a Delaunay tetrahedron defined a circumscribing sphere that could be considered as a cavity or a particle-free space. Strictly speaking, there was no oxygen atom inside such a sphere. The mean number of spheres found per one methanol configuration was 1641. For each sphere we evaluated the locally averaged potential energy Eav of the excess electron, which was treated as a negative test charge (see section IID). We eliminated the spheres with zero or positive average potential energy, and from the remaining spheres in a given molecular configuration we constructed the compact regions of negative energy by merging the overlapping spheres. The merging algorithm assumed that the two nearest neighboring spheres could be treated as different cavities only if the distance between their centers was larger than half the radius of the larger sphere. Otherwise, the spheres were treated as two parts of the same larger cavity. Using this procedure for every pair of neighboring spheres with negative energies, we reduced the number of cavities per one methanol configuration from, on the average, 919 to about 141. Less than 30% of the finally obtained cavities were composed of only one sphere. Thus, the mean number of spheres per one multiple-sphere region of negative potential energy was equal to 6. C. Trap Identification Procedure. Among the compact regions with Eav < 0 we selected the electron traps, i.e., the cavities that could accomodate an excess electron. We employed the capture condition in the simplest approximation of a square potential well (in atomic units):
-Eav R2 > π2/8
(1)
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where R is the cavity radius. In our case, however, most of the regions of negative energy were not spherically symmetric and we had to estimate the radius of each multiple-sphere region by a numerically obtained value of its gyration radius Rg. D. Electron Average Potential Energy. We assumed a pure electrostatic form of the interaction potential between a solvent molecule and the excess electron. The charge distribution of the methanol molecules was described by the partial charges of the H1-model potential and the long-range electrostatic interactions were accounted for by using an anisotropic approximation to the Ewald summation as proposed by Adams and Dubey.35 The average electrostatic (Ewald) potential energy Eav of the negative test charge in a sphere with the radius Rj is given by the formula N
n
Eav [Rj] ) - ∑ ∑ qRi ∫|br |eR Ψ(b rj - b r Ri) db rj i)1 Ri
j
Figure 1. Example of a pre-existing trap in the computer-simulated sample of liquid methanol. The region of space enclosed within this surface is composed of 21 overlapping spheres of negative potential energy.
(2)
j
where the summation is performed for all the partial charges of all the methanol molecules in a simulation box, i.e., n ) 3 and N ) 256. b rj is the position of the test charge inside the jth sphere, and b rRi stands for the position of a partial charge qRi of the ith methanol molecule. The Ewald potential Ψ for a pair of point charges distant by b r is taken to be35
1 Ψ(b) r ) + a1r2 + a2r4 + a3r6 + a4r8 + a5T4 + a6T6 + r a7T8 + a8T4r2 + a9T6r2 + a10T4r4 (3) with Tn ) xn + yn + zn and r ) |r b| ) (x2 + y2 + z2)1/2. Distances in eqs 2 and 3 are expressed in units of side length of the simulation box, and the constants a1 to a10 have the following values 2.094 395, -4.506 792, 6.651 269, -10.866 13, 7.511 320, 17.071 59, 60.539 89, -23.279 44, -113.007 8, and 65.196 80, respectively. The value of Eav for each of the multiple regions was taken to be the mean value of negative average potential energies Eav[Rj] calculated for all the spheres that belonged to a given region. We assumed that the energy of each component sphere contributes to the value of Eav for the entire region with the weighting factor proportional to the volume of this sphere. III. Results and Discussion A. Static Properties of the Pre-existing Electron Traps. We identify as the electron traps only 4.45% of the regions with a negative average energy Eav. An example of the pre-existing trap found in the simulated methanol sample is presented in Figure 1. Only 3% of traps are composed of one sphere, and the multiple trapping regions with Eav < 0 contain, on the average, 23 spheres. The region shown in the figure comprises 21 spheres. Although the limited set of 1722 traps found in all analyzed methanol configurations does not allow us to construct smooth statistical distribution functions, the histograms shown in the next figures characterize fairly well some basic properties of the pre-existing electron traps. In order to measure the volume of the compact cavities, we employed the standard Monte Carlo method of calculation of multiple integrals. A quantity of 2.5 × 106 points were randomly generated for each considered region of negative energy. The random points were also employed for estimation of the spatial extent of the trapping regions. The shape of multiple cavities was described by the gyration tensor. We calculate the components of the tensor for a given cavity using values of the x, y, and z coordinates of the points randomly
Figure 2. Histogram of the statistical distribution of volume of the pre-existing electron traps compared to the volume distribution of the compact regions in a methanol sample where the average electrostatic potential energy Eav is negative (curve 1).
distributed inside the cavity. After the Jacobi diagonalization procedure we obtained the principal components of the tensor Rx, Ry, and Rz and, consequently, the cavity gyration radius Rg ) (Rx2 + Ry2 + Rz2)1/2. Figure 2 shows the histogram of the statistical distribution of the pre-existing trap volume V compared to the volume distribution that was calculated for all regions with Eav < 0. The latter function is quite smooth, since it was obtained based on 40 923 regions. The average volume of the negative energy region is 272 ( 2 Å3, and this value increases to 339 ( 3 Å3 if we consider only those cavities with Eav < 0, which are composed of more than one sphere. The mean size of the electron trap is 〈V〉 ) 960 ( 22 Å3, but the histogram has two maxima. The first maximum at ∼190 Å3 corresponds to the single-sphere traps, and the second one at about 750 Å3 is connected with the multiple-sphere cavities. It is worth noting that value of the average volume 〈V〉 of an electron trap obtained in the present calculations is very close to the value of 〈V〉 predicted analytically for traps in liquid methanol in ref 12. The statistical distributions of the x, y, and z components of the gyration tensor are nearly identical both for the regions with Eav < 0 and for the electron traps. For traps we have obtained the mean values 〈Rx〉 ) 2.92 ( 0.05 Å, 〈Ry〉 ) 2.85 ( 0.05 Å, and 〈Rz〉 ) 2.87 ( 0.05 Å, exhibiting isotropic distributions of the randomly generated points around the trap centers. These values could suggest the spherical symmetry of the electron traps. We have found, however, that most of the electron traps (about 95%)36 are not spherically symmetric and the values of Rx, Ry, and Rz for a particular trap can differ from each other
7694 J. Phys. Chem., Vol. 100, No. 18, 1996
Hilczer and Tachiya
(a)
(b)
Figure 4. Histogram of the distribution of the electrostatic potential energy Eav of the pre-existing electron traps compared to the energy distribution for the compact regions with Eav < 0 (curve 1).
(a)
Figure 3. (a) Histogram of the statistical distribution of the gyration radius Rg obtained for the pre-existing electron traps in liquid methanol and the gyration radius distribution for the compact regions with negative potential energy (curve 1). (b) Statistical distributions of radii R(V) of the spheres with volumes equal to the volumes of the regions described in part a.
even up to 30 or 40%. As one can expect, the components of the gyration tensor averaged over all the regions with Eav < 0 are smaller and they are 〈Rx〉 ) 1.843 ( 0.008 Å, 〈Ry〉 ) 1.836 ( 0.008 Å, and 〈Rz〉 ) 1.838 ( 0.008 Å. The statistical distributions of the gyration radius Rg for the electron traps and for the negative energy regions are given in Figure 3a. The mean value of Rg for the latter regions equals 3.210 ( 0.007 Å, whereas for the traps we obtain 〈Rg〉 ) 5.15 ( 0.05 Å. It is interesting to compare the virtual size of a multiple cavity (cf., Figure 2) with its gyration radius. Supposing that each cavity of volume V were a sphere of the same volume, we could calculate the statistical distribution of the radius R(V) of such a sphere. The R(V) distributions obtained for cavities with Eav < 0 and, separately, for electron traps are both presented in Figure 3b. The ensemble-averaged value of the sphere radius is 5.96 ( 0.05 Å for electron traps compared to 3.791 ( 0.009 Å for the regions with negative energy. For each multiple cavity the sphere radius R(V) was found to be greater than Rg. Thus, we consider the gyration radius to be a radius of the spherical “core” of the trap, and therefore, we employ R ) Rg in the trap identification condition given by eq 1. The statistical distributions of the electrostatic potential energy for cavities in liquid methanol are presented in Figure 4. The probability of finding the region with a given negative value of Eav increases with the increase of energy and reaches a maximum at Eav close to 0 eV. The ensemble-averaged value of the energy for cavities equals -0.238 ( 0.002 eV, whereas for the pre-existing traps 〈Eav〉 ) -0.543 ( 0.008 eV. The
(b)
Figure 5. (a) Distribution of the lifetime of the pre-existing electron traps in liquid methanol at room temperature. (b) Decay of the number of regions identified at t ) 0 as pre-existing traps.
maximum of the trap energy distribution is shifted toward more negative values of Eav, and we can say that in liquid methanol there is no pre-existing traps shallower than -0.2 eV. A similar result was obtained in ref 12 where the potential energy surface of the methanol matrix was analyzed in terms of the random field theory. B. Dynamical Properties of the Pre-existing Electron Traps. The results presented in Figure 5 concern the lifetime of the pre-existing electron traps in liquid methanol at room temperature. The persistence time for a given trap was determined by tracing its energy over consecutive steps of the computer simulation. We assume that the electron trap, or the region of a given size that was originally selected as a trap, decays when the average energy Eav of this region becomes
Pre-existing Electron Traps
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higher than -π2/(8 Rg2). Thus, the time step of the simulation at which condition 1 for a considered trap is first violated determines the lifetime of this trap. The lifetimes recorded for 1722 traps allowed us to construct the statistical distribution that is plotted in Figure 5a. Using this distribution function, we estimate the average lifetime for the pre-existing traps in liquid methanol to be 41 fs. Figure 5b shows the decay of the number of regions identified at t ) 0 as the pre-existing traps in methanol. The initial concentration of traps is determined to be 0.57 M. After 0.1 ps the trap concentration decreases to 0.06 M. About 3.4% of the traps survive the next 0.1 ps, but even after 0.3 ps we still have 7.9 × 10-3 M of the electron traps. We should note that, although the trapping time of the excess electrons in methanol has not yet been determined, its value for liquid water is smaller than 0.18 + 0.04 ) 0.22 ps.17 Assuming that the initial concentration of the quasi-free electrons in the methanol sample is on the order of 5 µM16,3 and that it takes 0.5 ps for the quasifree electron to be trapped, we can say that each excess electron has 530 pre-existing traps to choose from as the localization site. The quoted numbers suggest that pre-existing traps exist sufficiently long and are abundant enough to play an important role in electron capture. This result seems to support the “trapseeking” mechanism of electron localization in polar liquids. To complete our discussion on the properties of the preformed electron traps, we should address two questions. (1) How is the rate of the trap decay connected with solvent dynamics? (2) Does the procedure of energy averaging considerably influence the obtained distribution of lifetimes of the electron traps? To answer these questions, we employ a time correlation function CR(t) of fluctuations in the electrostatic potential energy ER
CR(t) )
〈δER(0) δER(t)〉 〈δER2〉
(4)
where δER(t) denotes the instantaneous fluctuation of ER(t) from its mean equilibrium value and 〈δER2〉 is the variance of the ER distribution. The time correlation function 4 with the electrostatic potential energy of a solute, ER ) Ev, is commonly used in the studies of dynamical aspects of solvation in polar liquids, which are based on the linear response formalism.37-39 It was shown that the function Cv(t), although obtained from results of equilibrium computer simulations, reflects the two-part character of the nonequilibrium solvation response function. The fast component of Cv(t) for liquids at room temperature is characterized by a time scale of tens of femtoseconds (e.g., ∼25 fs in water, 50-100 fs in acetonitrile, and 30-50 fs in methanol40) and results mostly from small amplitude inertial motions of solvent molecules in the close vicinity of a solute. This component is well described by a Gaussian of the form
1 Cv(t) ≈ exp - ωv2t2 2
(
)
(5)
with the solvation frequency ωv expressed as
ωv2 )
( ) ∂2Cv(t) ∂t2
(6) t)0
The slower component, which decays on a time scale of 1-5 ps, reflects larger amplitude motions involving the reoorganization and translation of solvent molecules that is diffusive in nature. The latter component is strongly dependent on the
Figure 6. Time correlation functions Cav(t) (curve 1) and Cp(t) (curve 3) of fluctuations in the cavity-averaged electrostatic potential energy, Eav, and in the electrostatic energy at the cavity center-of-mass, Ep, respectively. Curve 2 is the solvation time correlation function for an atomic solute, taken from ref 39. All curves concern liquid methanol at room temperature and were obtained from MD simulation data.
solvent’s degree of association. The hydrogen bonding affects the solvation dynamics at shorter times as wellsthe response functions for solvents such as water and methanol exhibit oscillations for times up to 0.2 ps because of intermolecular vibrational motions (O‚‚‚H-O librations). This oscillatory behavior of Cv(t) is, as we can expect, less pronounced in methanol (cf., Figure 6) than in water. It is worth noting that the validity of the linear response approximation and, consequently, the application of formula 4 in studies of the solvation dynamics in liquid methanol were recently criticized by Fonseca and Landanyi.41 However, in the case of atomic solutes, single rigid ions,39,40 and also some larger molecules (e.g., dimethylaniline42), the linear response theory seems to work correctly. The two-part character of solvent relaxation is reflected in the dynamics of the pre-existing traps. This can be deduced from a comparison of the time correlation functions Cav(t) (curve 1) and Cv(t) (curve 2), which are presented in Figure 6. The latter function concerns the solvation response of liquid methanol at room temperature and was calculated for an atomic solute.39 The former function describes the decay of fluctuations in the cavity-averaged electrostatic potential energy Eav. To obtain the function Cav(t), we have selected 289 cavities with negative initial values of Eav in the independent, equilibrium methanol configurations and then we have recorded the values of Eav for these regions over 9000 consecutive time steps of simulation (18 ps). As can be seen in Figure 6, the function Cav(t), which reflects the dynamics of the electron traps, is very similar, especially at short times, to the solvation response function Cv(t). The value of ωav, which is calculated for Cav(t) using eq 6, equals 33 ps-1 compared with ωv ) 30 ps-1 for the atomic solute.39 Thus, the characteristic times for the decay of the Gaussian components of both time correlation functions, Cav(t) and Cv(t), are equal to 30 and 33 fs, respectively. This implies that the molecular mechanism responsible for the initial decay of the pre-existing electron traps is the same as the mechanism of the initial solvent relaxation. Therefore, if we take into account the time dependence of the trap concentration (cf., Figure 5b), we can say that small amplitude inertial motions of solvent molecules in the first solvation shell of traps result in very fast decay of nearly 70% of the electron-trapping sites in liquid methanol. The equilibrium solvation response function Cv(t) for methanol was fitted in ref 39 by the sum of a Gaussian and an exponential term of the form 0.68 exp[-0.5(t/0.071)2] + 0.32
7696 J. Phys. Chem., Vol. 100, No. 18, 1996 exp(-t/2.4), where t is expressed in units of picoseconds. This formula predicts the time constant for the slower component of the solvent response to be equal to 2.4 ps. To quantify the longtime behavior of the function Cav(t), we have approximated it by a similar formula. The best approximation has the form Cav(t) ≈ 0.593 exp[-0.5(t/0.060)2] + 0.407 exp(-t/1.297) and was obtained using the nonlinear regression algorithm of Levenberg-Marquardt with the goodness of fit defined by the coefficient of determination equal to 0.9684. Thus, the slower component of the time correlation function for the electron traps decays with a time constant that is a factor of 1.85 smaller than the characteristic decay time of the exponential part of Cv(t). It seems that the trap dynamics is more sensitive, in comparison with the solvation response, to diffusive reorientations and translations of methanol molecules in the trap neighborhood. Curve 3 in Figure 6 corresponds to the time correlation function Cp(t) of fluctuations in the electrostatic potential energy Ep that was evaluated at the center-of-mass of cavities in the computer-generated methanol sample. The energy data that are necessary to obtain this function have been collected during the simulation runs in the same way (the same cavities and the same time steps) as the values of the averaged electrostatic energy Eav used in the calculation of Cav(t) (see above). The inertial component of Cp(t) is characterized by a time constant of 26 fs. The long-time decay constant of Cp(t), as estimated from the Gaussian-exponential fitting formula, is however considerably smaller than that of the time correlation function Cav(t) and equals 0.522 ps. Thus, if the electron trap were defined based on the point energy value Ep instead of the average energy Eav, the distribution of the lifetime of the electron traps would be more compressed. It means that a smaller fraction of traps could live longer than, for instance, 0.5 ps. However, even if the trap dynamics were described by Cp(t) instead of Cav(t), we could expect that the concentration of traps that survive 0.5 ps would still be about 300 times higher than the initial concentration of the excess electrons. The point values of energy were employed in studies of trapping site dynamics in liquid water by Schnitker et al.11 These authors obtained the persistence times for a majority of sites in the range 0.04-0.08 ps, which corresponds to the time scale characteristic of the librations of the water molecules. Moreover, analyzing the water configurations separated by 0.1 ps, they inferred that a small fraction of traps can survive up to a few picoseconds. The present work does not confirm the latter conclusionsthe longest lifetime that was found for the set of traps inspected in our study was 0.962 ps. However, our definition of a trapping site is quite different from the definition used in ref 11. We connect the pre-existing electron trap with a particular region of space, and we determine its lifetime by checking how long the average electrostatic energy of this region remains lower than the energy required by condition 1. In ref 11 the trapping site is assumed to move in a void space between molecules, and its survival time is determined by the number of simulation steps for which the distance between the positions of the site (i.e., the positions of two local minima identified as trapping sites) in every two successive configurations is smaller than a specified critical distance. Such an approach can produce longer lifetimes compared to our results, and if the temporal spacing between the molecular configurations is too large, it even introduces some arbitrary correlations. IV. Concluding Remarks The present calculations show that in polar liquids we can find compact regions of the average negative potential energy that are large enough to produce the binding level for an excess
Hilczer and Tachiya electron. These regions can be treated as the pre-existing electron traps that are directly related to the “trap-seeking” mechanism of the primary electron localization. We have found that the lifetimes or, more generally, the dynamics of the pre-existing electron traps is strictly connected with the dynamics of the solvent. The average persistence time that has been estimated for the traps in liquid methanol is equal to 41 fs. Nearly 70% of the traps disappear (with a characteristic time of 30 fs) as a result of small amplitude inertial motions of solvent molecules in the close vicinity of the trapping regions. The concentration of the “long-living” traps is, however, high enough for primary electron trapping. The statistical description of the electron traps in a given solvent can be performed either based on the extensive numerical analysis of the trapping regions found in the computer simulated matrix or by means of analytical tools from the random field theory.12 These two methods seem to be complementary, and the quantitative characteristics of the trap properties that can be calculated independently by both methods are very similar. Acknowledgment. M.H. thanks Dr. M. Sopek and Professor W. Bartczak for helpful discussions. References and Notes (1) Schiller, R. In Excess Electrons in Dielectric Media; Ferradini, Ch., Jay-Gerin, J.-P., Eds.; CRC: Boca Raton, FL, 1991. (2) Tewari, P. H.; Freeman, G. R. J. Chem. Phys. 1968, 49, 954. (3) Kenney-Wallace, G. A.; Jonah, C. D. J. Phys. Chem. 1982, 86, 2572. (4) Jonah, C. D.; Romero, C.; Rahman, A. Chem. Phys. Lett. 1986, 123, 209. (5) Tachiya, M.; Mozumder, A. J. Chem. Phys. 1974, 60, 3037. (6) Tachiya, M.; Mozumder, A. J. Chem. Phys. 1974, 61, 3890. (7) Schnitker, J.; Rossky, P. J. J. Chem. Phys. 1987, 86, 3462. (8) Wallqvist, A.; Thirumalai, D.; Berne, B. J. J. Chem. Phys. 1987, 86, 6404. (9) Wallqvist, A.; Martyna, G.; Berne, B. J. J. Phys. Chem. 1988, 92, 1721. (10) Rossky, P. J.; Schnitker, J. J. Phys. Chem. 1988, 92, 4277. (11) Schnitker, J.; Rossky, P. J.; Kenney-Wallace, G. A. J. Chem. Phys. 1986, 85, 2986. (12) Hilczer, M.; Bartczak, W. M. J. Phys. Chem. 1993, 97, 508. (13) Hilczer, M.; Bartczak, W. M. Radiat. Phys. Chem. 1992, 39, 85. (14) Hilczer, M.; Bartczak, W. M.; Sopek, M. Radiat. Phys. Chem. 1990, 36, 199. Bartczak, W. M.; Sopek, M.; Kroh, J. Radiat. Phys. Chem. 1989, 34, 93. (15) Motakabbir, K. A.; Rossky, P. J. Chem. Phys. 1989, 129, 253. (16) Migus, A.; Gauduel, Y.; Martin, J. L.; Antonetti, A. Phys. ReV. Lett. 1987, 58, 1559. (17) Long, F. H.; Lu, H.; Eisenthal, K. B. Phys. ReV. Lett. 1990, 64, 1469. (18) Pe´pin, C.; Goulet, T.; Houde, D.; Jay-Gerin, J.-P. J. Phys. Chem. 1994, 98, 7009. (19) Gauduel, Y.; Martin, J. L.; Migus, A.; Yamada, N.; Antonetti, A. In Ultrafast Phenomena V; Fleming, G. R., Siegman, A. E., Eds.; SpringerVerlag: New York, 1986; p 308. (20) Barnett, R. N.; Landman, U.; Nitzan, A. J. Chem. Phys. 1989, 90, 4413. (21) Webster, F. J.; Schnitker, J.; Friedrichs, M. S.; Friesner, R. A.; Rossky, P. J. Phys. ReV. Lett. 1991, 66, 3172. (22) Neria, Y.; Nitzan, A.; Barnett, R. N.; Landman, U. Phys. ReV. Lett. 1991, 67, 1011. (23) Messmer, M. C.; Simon, J. D. J. Phys. Chem. 1990, 94, 1220. (24) Schwartz, B. J.; Rossky, P. J. J. Phys. Chem. 1994, 98, 4489. (25) Kimura, Y.; Alfano, J. C.; Walhout, P. K.; Barbara, P. F. J. Phys. Chem. 1994, 98, 3450. Alfano, J. C.; Walhout, P. K.; Kimura, Y.; Barbara, P. F. J. Chem. Phys. 1993, 98, 5996. (26) Gauduel, Y.; Pommeret, S.; Migus, A.; Antonetti, A. J. Phys. Chem. 1991, 95, 533. Gauduel, Y.; Pommeret, S.; Antonetti, A. J. Phys. Chem. 1993, 97, 134. Gauduel, Y. In Ultrafast Dynamics of Chemical Systems; Simon, J. D., Ed.; Kluwer: Dordrecht, 1994; p 81. Gauduel, Y.; Gelabert, H.; Ashokkumar, M. J. Mol. Liq. 1995, 64, 57. (27) Walhout, P. K.; Alfano, J. C.; Kimura, Y.; Silva, C.; Reid, P. J.; Barbara, P. F. Chem. Phys. Lett. 1995, 232, 135. (28) Sander, M.; Brummund, U.; Luther, K.; Troe, J. Ber. Bunsenges. Phys. Chem. 1992, 96, 1486.
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