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9 Stochastic Physical Modeling in Chemistry Peter Clifford and N. J. B. Green Oxford University 9.l Introcluction How can corrosion be controlled in the cooling system of a nuclear reactor? What is the most efficient design for a solar cell? How do you build an arti- ficial enzyme? These are just some of the important practical questions that lie behind the prolific research activity taking place in physical chemistry departments around the world. As a branch of science, physical chemistry is defined not so much by the circumscription of its subject matter as by its method of approach, appli- cable to a wide diversity of problems arising from physics and chemistry on the one hand to biology and materials science on the other. From a statis- tician's perspective, a familiar thread within the densely woven fabric of physical theory, mathematical development, and experimental technique is the constant concern with finding simple and expedient models, frequently of a stochastic nature (van Kampen, 1981; Wax, 1954~. Thus, although physical theory may in principle provide a complete microscopic description of the problem at hand, in practice the intractability of the mathematical development prevents useful predictions from being made. A classic illus- tration from physics is that of modeling the motion of a dust particle on the surface of a raindrop. The dust particle moves as a result of collisions with the water molecules. A typical raindrop will contain 102 molecules whose deterministic equations of motion can be formulated as a HamiTtonian sys- tem. The solution of the equations is clearly impracticable. The motion of in

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160 the dust particle is therefore unresolved. However, a stochastic approx~ma- tion can be derived, namely, Einstein's theory of Brownian motion, which provides good agreement with experimental observation. It should be noted that although the model fits the data on an observational scale, the trajec- tories of theoretical Brownian motion contradict physical laws, since infinite acceleration is required. When chemistry is introduced, things become more complicated. Con- sider, for example, the effect of a pulse of radiation on the water droplet. Radiation creates chemically reactive species distributed throughout the droplet. Chemical reactions occur when reactive species approach each other as a result of molecular motion. As in the case of Brownian motion it is natu- ral to Took for a stochastic approximation to the reaction process, but here we must track the motion of a large number of atomically small reactive species. One approach is to use the heuristics of statistical mechanics, pioneered by Gibbs, to provide joint distributions for molecular positions and velocities. ~. . . ~ , .. . ~ ne progress of cnem~ca~ events ~onow~ng radiation can then be treated as a stochastic process, but on an enormous state space. The stochastic behavior can be viewed as a manifestation of the chaotic character of the solutions of the nonlinear equations of motion. The skill of the physical chemist is to derive and validate-parsimonious approximations of the reaction process while attempting to fit experimental data. There are therefore close analo- gies between the activities of physical chemists and the role of statisticians in applied science, in that the physical chemist must construct models that on the one hand are reasonably faithful to the laws of physics (the client) and on the other are amenable to mathematical manipulation and eventual experimental verification. 9.2 Diffusion Controlled Reaction A classical theoretical problem in the analysis of the reaction rates in so- lutions is the modeling of diffusion controlled reactions. In these reactions, molecules of non-zero size diffuse and react instantaneously if they encounter one another. In a typical experiment, very reactive particles are produced randomly in space and essentially instantaneously by, for example, using a pulse of light. These particles diffuse and react with other species already in solution. The number or concentration of the reactive particles is ob- served in real time, by, for example, optical absorption methods. Computer simulations of liquids can provide insight into the reaction process, but the

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161 results are necessarily subject to statistical error. A great deal of theoretical work has been devoted to deriving and validating good analytical approx~- mations; see for example Balding and Green (1989) in the one-dimensional case. The original theory, owing to Smoluchowski (1917) fixed the coordi- nate system on a single particle and made the implicit approximation that all other particles diffuse independently in this frame of reference (Noyes, 1961~. We will refer to this as the independent pairs (IP) approximation. While this is probably a good approximation for a central, slowly moving molecule surrounded by faster moving molecules (e.g., colloid coagulation), it is certainly not true for the converse problem (fast central molecule in a sea of static traps). In three dimensions the Smoluchowski theory gives the same result for both cases, namely, the survival probability of the central particle is Q(t) = Q(O) exp t- JO kit') ~t'cs] where c', is the density of traps, ~ ~ ~ Veldt] a (9.1) a is the encounter radius, and D is the diffusion coefficient of the mobile molecuTe~s). For the latter case, where the Smoluchowski theory might be expected to break down because the intermolecular distances are highly correlated, the survival probability is related to the volume of the Wiener sausage, swept out by the diffusing molecule in the course of its trajectory. This is because the molecule will survive to time t if and only if there is no trap in the volume swept out, Vast), and if the traps are distributed according to a Poisson point process with intensity cs, the survival probability of a given trajectory is exp[-Vatt) Csi. The observed survival probability will be the expectation of this random variable Q = Etexp[-va~t) Csi] - Donsker and Varadhan (1975) have obtained precise asymptotic results for expectations of this kind. They show lam t(Dt)~/~+2) Cs2/~+2) in Q] =no, where ~ is the dimension and n~ is a positive constant. This is very dif- ferent from the simple exponential decay at long times predicted by the

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162 1 0 0 9 08 07 Q 06 0 5 04 03 02 01 ~ 00 _ 00 05 1~ 15 20 Log ( tips ) \'\.\.\ ~ . '\.W - it, 25 30 35 FIGURE 9.~: Survival probability Q for a particle diffusing in a sea of static traps. Comparison of Monte Cario simulation to o o), Donsker-Varac~han asymptotics ~- -), and the {P approximation ~). Smoluchowski theory. Since there is no obvious analytic way to assess the time scale on which the asymptotic behavior will be found, we have de- veloped a simulation technique for this purpose. Early simulation results indicate much better agreement with the Smoluchowski theory than with the Donsker-Varadhan result. The reasons for this observation are not clear at present. See Figure 9.1. 9.2.1 Racliation Spurs Diffusion controlled reactions are the fastest reactions that occur in solution. Experimental observations of the rate of reaction contain information about the initial spatial distribution of the reactive species. A substantial amount of research has been devoted to the analysis of radiation tracks. if radiation interacts weakly with the liquid (e.g., fast ,B-particles), the track consists of small isolated spurs, which are clusters of highly reactive particles; the spurs subsequently relax by diffusion and within each spur the particles react with each other on encounter. The problems of describing reactions in clusters can be illustrated by reference to a number of mode! systems with simplified chemistry.

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163 The Two-Species Spur The simplest system contains two types of particles, A and B. which react on encounter to form products AA, AB, or BB. The particles are identical in all but name and have identical spatial distributions. The classical method of dealing with such a system is to make a continuum approximation. The two spatially dependent concentrations (which are identical) obey macroscopic continuum equations of the form d Itch = d FACE = DA V CAkA A CAkA B CA CB DB V2 CBkBB CBkAB CA CB where the first terms on the right-hand sides represent diffusive spreading of the concentration profiles, and the remaining terms represent local depletion by reaction; the rate coefficients k are given by Smoluchowski's theory (cf. equation (9.1~. Although these equations are perfectly satisfactory when applied to macroscopic problems, they are not appropriate when dealing with the small number of particles in a spur of finite extent. There are two reasons for this: (1) the small number of particles in the spur ought to be treated as a discrete variable, and (2) the Smoluchowski rate constant is appropriate for a particle initially surrounded by a homogeneous Poisson _ fit ~ ~ ~ . . 1 ~ . 1 1 1 ~ 1 ~ 1 1 ~ ~ 1 ~ ~ held ot reactants as opposed to the highly clustered dlstrlblltlon in a spur. The necessity for a correct stochastic theory is easily demonstrated in this simple system. if there are initially NA particles of type A and NB particles of type B. then simple probabilistic arguments permit us to show that the TABLE 9.1: Typical Product Yield Ratios NA NB 1 2 2 3 3 4 4 3 10 10 9 oo oo 1 2 AA AB BB 0 1 0 1 4 1 1 3 8 20 1 3 9 1

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164 expected product yields, for Al times, are in the ratio NAA: NAB: NBB = 2NA(NA11: NANB: ~NB(NB11. .z Typical ratios are given in Table 9.1. The continuum approximation always predicts a ratio of 1:2:1 since it corresponds to the case of infinitely many particles. The independent pairs approximation can be used to provide a stochastic theory of spur kinetics. If the state of the spur at time ~ is labelled with M, N where M iS the number of A particles and N iS the number of B particles, then PMN(t), the probability of being in state M, N. satisfies the following forward equations: dt PMN (t) = 2 [(M + 2)(M + 1 )PM+2,NM(M1)PMN] AAA (t) 2 [(N + 2)(N + 1)PM,N+2N(N1)PMN]ABB(t) + [(M + 1)(N + 1)PM+1,N+1MNPMN])AB(t) ~ where the it's are the reaction rates for isolated pairs of particles whose initial spatial separation is equivalent to that in a cluster. These equations can be solved analytically in special cases, for example, when the particles are identical. In general, though, they must be solved numerically. Comparisons between this approximation, the continuum approximation, and the full Monte Carlo simulation of sample trajectories are given in Figure 9.2, taken from Clifford et al., (1987a). It is evident that the stochastic independent pair (TP) model is in very good agreement with the simulations. The ionic System The two species system can be generalized by including long-range forces between the particles, such as the Coulombic force between ions. Such forces act attractively between A and B particles, but like particles repel each other. The form of the it's is modified because the survival probability of a pair depends on the force between the particles. The it's must now be calculated approximately if the forces are weak (Clifford et al., 1987b) or numerically if they are strong (Green et al., 1989~. The introduction of forces would be expected to make the {P approximation worse, because of complicated interactions in the many-body system. Comparisons of the {P approximation and the results of full simulations of sample trajectories are shown in Figure 9.3. It is seen that even when the forces are so strong that

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165 1 6 ^ 12 . _ ~ 08 o A v 04 ~.` .. _~ ._._. _._._,_._,_, _,_ Am. AD By,_ _. _ ~ ! 20 30 40 0 10 Log (t/psJ FIGURE 9.2: Kinetics of two species spur. Initial configuration spherical Gaussian. Monte CarIo simulation (o o o); IP approximation ~ ); contin- uum approximation ~~ ). Reprinted, by permission, from Clifford et al. (1987a). Copyright (I) 1987 by the Royal Statistical Society. the AA and BB encounters are effectively impossible, the IP approximation is still remarkably accurate. The Scavenging System The simplest such system is A+A ~ AA A+S ~ AS, (9.2) where the species A iS clustered in a spur, whereas the species S exists in large numbers uniformly distributed over an extended volume. There is competition between intraspur recombination, the AA interaction, and scavenging, the AS interaction. The relative abundance of the ultimate yields of AA and AS provides information about the scavenging process. In the IP model the forward equation becomes d. PN = ~ ) ~(N+21(.N+ ~ )PN+2N(NT)PN]+kASCS L(\N+ ~ )PN+1NPN],

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166 016 0 14 A 0 1/ ~ 0'10 +~: 00E O 006 an v 004 2! (a) 1 6 12 ; ~ ~ \ z 08 04r 0-0 1 -I Do 10 20 30 40 Log (t/ps) ,-"' Fo v 10 - . . D9 ,,-~ 08 ,,, ' ~ 07 ,~ c ~6 "" ~ ''D05 zO3 02 D0- 05 1D 15 20 25 30 35 40 00 Log (t/ps) (b) of, Ad, WOK .. . . 1 05 10 1 5 2 0 2 5 3( 3 5 40 20 > 1 5 z 1 0 05 _ 00- ~ ;~ me, 00 10 20 30 40 50 60 70 Log (gyps) FIGURE 9.3: (a) Tonic reactions in high permittivity solvents. Average number of reactions in a spur containing two ion-pairs: A+,B-. Monte Cario simulation (o o a), simulation using the TP approximation ( ), and continuum theory (- - -). Reprinted, by permission, from Clifford et al. (1987b). Copyright (I) 1987 by the American Chemical Society. (b) Tonic reactions in Tow permittivity solvents. Average number of surviving pairs in a spur containing two ion-pairs: A+, B- . Left panel: homogeneous disper- sion; right panel: heterogeneous dispersion. Monte CarIo simulation (o 0 o) and simulation using the {P approximation (). Reprinted, by permission, from Green et al. (1989). Copyright ~ 1989 by the American Chemical Society.

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167 where kAS(t) is given by the Smoluchowski theory Of equation (9.1~) and N is the number of A particles remaining. The continuum approximation gives the equation 0ICA = DV CAkAA(~)CAkAS(t)CACS- Typical results are shown in Figure 9.4. Again, the continuum model fails to reproduce the results of a fuD Monte CarIo simulation, and the {P ap- ~ prox~mat~on Is superior. 9.3 Computer Simulation of Liquids Although a full description of a liquid system must be quantum mechanical, almost ah liquids (except those containing very small molecules such as he- lium) can be described adequately using a completely classical deterministic mode! (McQuarrie, 1976~. If we tag and follow one molecule in a computer simulation of such a deterministic system, its motion appears random. If several molecules are followed, their spatial configuration evolves as a spa- tial point process, marked by the individual velocities. What we would like to do is to find stochastic rules that indicate where the molecules will be, and how fast they will be traveling, as time goes on. For example, in ra- diation chemistry, where the effect of radiation is to create reactive species distributed throughout the liquid, we are interested in the time taken for such species to encounter and react with each other. 9.3.1 Some History In physics and chemistry, classical liquids are simulated by one of two tech- niques. The first relies on the laws of mechanics to provide the equations of motion of a finite system of interacting molecules (Goldstein, 1980~. This is known as the molecular dynamics approach. The second technique makes use of statistical arguments that were originally given by Gibbs. This is generally called the Monte CarIo approach. A rigorous and detailed account of the statistical treatment of mechanics can be found in RueDe (1969~. The book Computer Simulation of Liquids by Allen and Tildesley (1987) contains a comprehensive history of simulation methodology from the first computer experiments through to the latest ideas. Numerical simulation is the technique most widely used in recent years to study the properties

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168 3 (N. / ( 2 NAA) / (Nits) -,''''<,~/' /~' - _. Log ( tips) FIGURE 9.4: Average number of reactions in a spur containing four radicals A, A, A, A with scavenging by S molecules. Initial distribution spherical Gaussian. {P approximation ~ ); continuum theory (- - -I. Reprinted, by permission, from Clifford et al. (1987a). Copyright ~ 1987 by the Royal Statistical Society. Of liquids. It is now an extremely large research area in both physics and chemistry, with many hundreds of research groups involve(l. The classical mechanics of a system of Or structureless molecules is speci- fied by a HamiTtonian At, which is the sum of the kinetic and potential energy of the system: Air, p) = K(p) + V(r) . Hamilton's equations of motion are (Goldstein, 1980~: p = VrV ri = Pi/mi ~ (9.3) where p and r are the vectors of momenta and positions of the molecules. The kinetic energy is given by N (p) = ~p2/2mi, i=1 (9.4) where mi is the mass of molecule i and Pi is the magnitude of its momentum. The potential energy V(r) depends on the positions and orientations of the

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169 particles. It is usually sufficient to assume that V only depends on the interactions of particles in pairs, and to use a spherical average of the pair potential, although the pair potential may have to be modified to correct for higher order effects. In the absence of an external field, the potential energy then becomes V_~Z,2(rij), i j>i where rid is the distance between particles i and j. The computer simulation of liquids and gases was initiated by Metropolis et al. (1953), who used Monte CarIo methods to simulate the Gibbs equi- librium distribution of molecular configurations. Their aim was to derive values for stationary (i.e., equilibrium) physical properties such as expected energy and expected pressure. Early work was concerned with the case of a hard sphere potential, z,2(r) = oo for r < a and z,2(r) = 0 otherwise. In order to obtain dynamic properties, Alder and Wainwright (1959) developed a method by which the simultaneous equations of motion for many molecules are solved numerically. They illustrated their method by simulations using both hard sphere and square well potentials. Their paper is the first example of molecular dynamics simulation. Simulations using a realistic potential were made by Rahman (1964~. In particular, Rahman estimated the pair-correlation function girl = V ~(r' /`r) (9.5) where ntr, Ar) is the time averaged number of molecules at a distance be- tween r and r + Ar from a given molecule, and N/V is the average density of molecules. He showed that the system has spatial structure that decays slowly over time. 9.3.2 Sampling From Configuration Space Let us consider a system of N molecules in three-dimensional space, subject to a potential as previously described. We can think of the simultaneous positions and momenta, or equivalently positions and velocities, as coordi- Hates in 6N-dimensional space, X. We denote a point in this space by x. We call X the state space and refer to x as a state; in the physics literature, X is caped the phase space. Let ~ be a function of x. We refer to ~(x) as an instantaneous evaluation of the property . For example, K(p) in equation (9.4) is an instantaneous evaluation of kinetic energy.

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170 A large system can be thought of as the union of many smaller systems. At any instant of time, each small system will have a particular state. A macroscopic property of the large system is an average of the property eval- uated over the subsystems. There are two basic ways in which the state space can be explored. The first is to build a dynamic description of molec- ular motion that will move through the state space according to acceptable physical principles. This is the approach of molecular dynamics. As noted earlier (9.3.1), the equations of motion are those considered by Hamilton in classical mechanics. The required average is then taken over a succession of times for a single small system; arguing that if the time period is sufficiently large a representative sample of configurations win be obtained. The second method of sampling states relies on the validity of Gibbs's probabilistic analysis of large mechanical systems. The method involves Monte CarIo simulations. The state of each small system is treated as a random variable drawn from the Gibbs distribution, which is constructed! to have maximum entropy subject to certain constraints, giving an explicit form for the density of the distribution. The task of sampling the state space for this method is reduced to that of choosing a random sample, X1,X2,.. . ,~n, from a specified density pox): x ~ X. In typical applications the form of the density lends itself to sampling by the Metropolis method. The required estimate of the macroscopic property is then given by IF = (~(x~) + + {(xn)~/n. Since this can be interpreted as an average taken over a number of subsystems, it is usually referred to as a space-average. Both Monte CarIo and molecular dynamics simulations can be used to sample from the equilibrium distribution of a many-particle system. In prin- ciple it is therefore possible to test empirically whether Gibbs's theory agrees with the results of molecular dynamics simulation. In practice, simulations must be run for a large number of time steps until equilibrium is attained. The development of tests for spatial point patterns is an active research area in statistics (Diggle, 1983; Ripley, 1987; Besag and Clifford, 1989~. One of our aims is to link methods used by probabilists and statisticians in the study of spatial processes with methods used by physicists and chemists in their study of liquids. 9.3.3 A Typical Computer Experiment Lynden-Bell et al. (1986) investigated the behavior of carbon tetrafluoride CF4 near its triple point by carrying out a Molecular Dynamics simulation of 256 molecules in a cubic box with periodic boundary conditions, using a variant of the lLennard-Jones potential. They were interested in the struc-

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171 tore of the velocity autocorrelation function, i.e., the empirical correlation between the velocity of a molecule in a particular direction and its velocity in the same direction at some time in the future. For typical liquids, the velocity autocorrelation is strongly positive at short lags, since molecules tend to continue with the same velocity, negative at moderate lags, since molecules eventually bounce off their neighbors, and then slowly approach zero as the lag tends to infinity. In order to explain certain anomalies in the behavior of the velocity au- tocorrelation function, Lynden-Bell et al. conjectured the existence of "local cages" of molecular configurations. A molecule is said to be in a local cage if its motion is restricted by the proximity of neighboring molecules. They first estimated the density of cos Am), where (T) is the angle between the velocity of a molecule at time t, and the velocity of the same molecule at the later time t + A. They observe that at moderate values of ~ the estimated density is approximately uniform. Plotting the height of the estimated clen- sity as a function of time T. they notice that the shape of the curve closely follows that of the velocity autocorrelation function. Stratifying the molecu- lar trajectories by initial kinetic energy, I,ynden-Bell et al. then repeat their analysis, but for initially fast and slow molecules separately. The results are different for the two croups. The structure is the same, but the mag- I ~7 ~ nitude of the effect is much higher for fast molecules. They suggest that high-energy molecules rattle back and forth in cages, while slower molecules diffuse. Lynden-BeD et al. finish by looking at the velocity autocorrelation function for the two stratified groups of molecules. They show that the ve- locity autocorrelation function of slower molecules has, surprisingly, a more pronounced negative portion than that of the faster molecules. In the simplest statistical mechanical mode! of a liquid, molecules have independent velocities chosen from the Maxwell-Boltzmann distribution, i.e., multivariate normal. In Atkinson et al. (1990), it is shown that, with the possible exception of the last result, ah the results of Lynden-Bell et al. are consistent with a simple description in which each molecule moves along a random trajectory in such a way that the velocity components in three fixed orthogonal directions are independent Gaussian processes. It is not necessary to propose the existence of local cages. To see this, notice that costly is essentially a correlation coefficient for three pairs of velocity components. With the Gaussian assumptions above, the density of R = costly is then given by fR(r, pi) = 2 (1 _ p2~3/2 ~ r2 (8 + 3) (2pr'3 (9 6)

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72 (a) / o cos / ~ (b) 0 14ps O- 2 8 DS 038ps n bins 0 86ps 1 56ps 1 -1 o COS ~ FIGURE 9.5: Histogram of cow after various time lags: (a) simulated (Lynden-BeH et al., 1986) and (b) calculated from equation (9.6~. The curves are displaced, as in Lynclen-Bell et al.; in each case the horizontal dashed line represents a uniform density. Reprinted, by permission, frown Lynden-Bell et al. (1986~. Copyright ~ 1986 by Taylor and Francis, Ltd. where p = p(T) is the theoretical counterpart of the empirical velocity auto- correlation. In the discussion of their results, Lynden-Bell et al. observe that at moderate lags the distribution of R is nearly uniform. If R has a uniform distribution, then the molecule is equally likely to be moving in any direction at this time lag, regardless of its initial velocity. The authors also observe that, at longer time lags, the distribution of R becomes skewed, indicating that the particle's velocity is opposite to the original direction. Lynden-Bell et al. consider these results to be paradoxical, however, this is precisely what is predicted by the form of the theoretical density. More importantly, the theoretical predictions give good qualitative fits to the computer-simulated results. See Figure 9.5. To throw further light on the causes of the nearly uniform distribution of R at the time lag for which the velocity autocorrelation function is zero,

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,1\'---~ '--~;-~q !;j Tl , ~ ,1 i ' 1 - 1 ~ 0 0 05 10 15 t/p s - 173 1 ' 1 1 00 05 10 15 tips FIGURE 9.6: Time-dependence of the probability density of cost at ~ = 0, 90, and 180, conditioned on initial kinetic energy. (a) Molecu- lar dynamics simulation (Lynden-Bell et al., 1986) and (b) calculated from equation (9.7~. The continuous lines refer to the 7.2~o of particles with the highest kinetic energy, and the dashed lines to the 12~o with the lowest en- ergy. Reprinted, by permission, from I,ynden-Bell et al. (1986~. Copyright ~ 1986 by Taylor and Francis, Ltd. Lynden-Bell et al. stratified the trajectories of molecules by their kinetic energy at time zero. Their idea, as noted earlier, was that the observed effect was due to a balance between high-energy molecules rattling back and forth in local cages, while Tow-energy molecules were diffusing through the whole space. Taking v to be the velocity of a molecule at time t = 0, and writing ivy = or, the conditional density of R given or obtained under the Gaussian assumptions is ~ 2 2 2 ~ (5 + 3) ~ ( laps )s As is shown in Figure 9.6, the fit with the data of Lynden-BeB et al. is again excellent. The final observations of Lynden-BeH et al. are difficult to reproduce. We have tried unsuccessfully to confirm these results using a molecular dy- namics program. The program was run many times using different numbers

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174 of molecules (32, IDS, 256) and a variety of different computers (Vex main- frame, MassComp workstation, Sun workstation). Comparisons of double and single precision calculations were made, and the effect of optimizing compilers was also examined. Different computers, different precision, and the use of an optimizing compiler ad substantially changed the configura- tions and velocities of the molecules. At lags for which the velocity auto- correlation is positive, there was however, a consistent effect with the slow molecules having a slightly greater positive autocorrelation at a fixed time lag than the fast molecules. This is not consistent with the simple hypothesis of Gaussian velocity components. 9.4 Discussion ant} EMiure Directions The main areas of investigation in physical chemistry can be classified as follows: 1. physical properties of matter in equilibrium, 2. clynamical and transport properties of matter, 3. properties of atoms and molecules, 4. statistical mechanics linking the above, 5. energet~cs and dynamics of chemical reactions, and 6. complex systems. 9.4.1 Physical Properties of Matter in Equilibrium The properties of bulk matter are reasonably well understood on a quaTi- tative level, and if the substance is made up of simple molecules or atoms, such as the liquid inert gases, numerical simulations of small systems, based on the known intermolecular forces and involving of the order of a thou- san(1 molecules, are quite successful in reproducing the observed proper- ties. Molecular dynamics simulations become more complicated when the molecules are neither spherical nor rigid. A great deal of work is still in progress in this area. A typical simulation is a realization of a chaotic spa- tial temporal process, involving the position and velocity of several hundred molecules for perhaps 10,000 time steps. There are a number of outstanding statistical problems in the design and analysis of these computer experi- ments. In particular, it is of interest to determine when a simulation has

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175 reached equilibrium. This problem is complicated, in small systems, by the effects of phase transition. An additional complication, which is receiving attention, is the inclusion of quantum mechanical effects in liquids such as helium. The properties of polymers and biological compounds are also the subject of research activity. Attention has turned in recent years to the study of interfaces between different phases of matter. Monte Cario and molecular dynamics simulations have concentrated on bubbles and droplets, and a number of experimental techniques have been devised for studying the gas-solid, gas-liquid, and solid- liquid interfaces. This work has relevance to the understanding of catalytic and electrochemical processes. Recent advances in experimental metallurgy have enabled detailed anal- yses of the atomic structure of metallic alloys to be carried out. Data are becoming available that record the position, subject to quantifiable error, of up to 60Xo of the atoms in small three-dimensional regions of a given sample. The analysis of this enormous data base, in particular the task of reconstructing the atomic lattice from partial observations, is a challenging statistical problem, which can be approached by combining simulated an- nealing as an optimization technique and realistic annealing as a description of the aging process in the atomic lattice. 9.4.2 Dynamical and Transport Properties of Matter The transport properties of matter, such as viscosity, thermal conductivity, and diffusion, involve transfer of energy or momentum from one molecule to another during a collision. The theoretical relationship between the trans- port properties of gases and the intermolecular forces has been known for a Tong time. Recently, physical chemists have attempted to tackle the inverse problem of estimating the intermolecular forces from detailed experimental observations of the viscosity-temperature curve. There is increasing interest in this type of statistical exercise, in particular there are unresolved ques- lions of identifiability. In solids and liquids, it has become clear that there is a wealth of in- formation about the dynamics of the molecules from light-scattering and Lt the information is in a form that is dif- One promising line in this respect is the neutron-scattering experiments, be ficult to extract and interpret. use of computer simulations as idealized experiments, both to develop tools for the analysis of data and to construct Monte Cario estimates of dynam- ical quantities. Applications to more complex systems include the stu(ly of the motion of polymers in solution, with particular reference to enzyme activity.

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176 9.4.3 Properties of Atoms ant! Molecules A great deal of physical chemistry is involved with investigating the prop- erties of isolated atoms and small molecules. Spectroscopy of these species, using radiation ranging from radio frequency through infrared and the visi- ble spectrum to X-rays, provides basic information about the energies of the accessible quantum states and the symmetries of the corresponding wave- functions, molecular size and geometry, nuclear spin, dipole moments, mag- netic moments, polarizabilities, and so on. Spectroscopy can therefore be used to test the predictions of the great variety of quantum mechanical approximations employed to calculate molecular properties. 9.4.4 Statistical Mechanics The fundamental molecular properties and their interactions as measured by spectroscopists are the data required by statistical mechanics for the de- scription of bulk matter. Statistical mechanics is the central unifying theory of physical chemistry as it relates the properties of isolated molecules with the bulk. The reconciliation of the statistical mechanical approach with modern theories of chaos in dynamical systems is a problem of outstand- ing interest to mathematicians. Large deviation theory was used in early attempts to provide a probabilistic interpretation. Recent work on infinite particle systems, has given insight into the phenomenon of phase transition in the classical Gibbs distributions of statistical mechanics. Some of the important applications have been covered in previous sections. 9.4.5 Dynamics of Chemical Reactions Chemical reactions occur when molecules are transformed by the rearrange- ment of electrons and nuclei. Physical chemistry concerns itself not only with the energetics of chemical reaction, but also with their rates and with the distribution of energy in the products. Gas phase chemical reactions generally occur as a result of simple collisions between isolated molecules. The classical theory of these processes has recently been revolutionized by experiments in which molecules are produced in collimated mono-energetic beams, which allow many of the parameters of the colliding particles (e.g., speed, quantum state, and orientation) to be fixed, and the energy en cl an- gular distributions of the products to be analyzed, thus giving very detailed information about the collision dynamics and the flow of energy between and within molecules. ~7~

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177 Classical descriptions of these dynamics have been proposed, which show regions of regular behavior and regions of chaos. It is still not clear how such phenomena wiB transfer to quantum mechanical descriptions. Reac- tions in solution are not understood in such a detailed way, although there are quantum mechanical theories of electron transfer reactions and the like. An additional problem of reactions in liquids is that particles diffuse slowly through the liquid and can only react on encounter. There are therefore two limiting cases of reaction: diffusion control, where the rate depends only on the transport through the solution and the rate of encounter; and activation control, where reactive encounters are rare, so that many encounters wiD take place before reaction can occur. Theories of the former type of reac- tion have been developed for a long time, but are only now being tested, by numerical solution of the associated stochastic differential equations. For activation-controlled reactions more detailed modeling of the encounter com- plex is required. 9.4.6 Complex Systems As well as the fundamental research described above, physical chemistry is involved with description of more complex systems, particularly the evolu- tion of these systems. Frequently, the problems of interest have important spatial aspects that have to be taken into account. Atmospheric Chemistry Depletion of Ozone Layer A realistic mode! must incorporate chemistry and transport in the atmo- sphere. It also requires an understanding of interfaces such as those be- tween the air and cloud droplets and ice crystals, which act as sinks for active chemicals. Combustion Since the 1950s there has been a series of revolutionary changes in explo- sives technology, which has resulted in safer but sIower-reacting explosive products. In order to maintain product performance, much attention must now be given to understanding the detonation process. The initiation and establishment of the critical conditions for detonation have been subject to little detailed realistic investigation; although of course there are obvious analogies with the stochastic theories of spatial epidemics. The necessary cooperative interaction between small numbers of initiating sites suggests

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178 that a treatment based on macroscopic deterministic approximations may be inappropriate: the number of reacting species being just too small for the averaging implicit in standard treatments. Here again, there is the pos- sibility of nonlinear kinetics producing oscillations and chaotic behavior. Radiation Chemistry . When a liquid is exposed to ionizing radiation, reactive species are generated in an initially localized spatial distribution. For Tow linear energy transfer (LET) radiation, isolated clusters, caned spurs, are formed. A significant proportion of the chemical reaction following radiation occurs on a short time scale, when the localized distribution has not yet relaxed by diffusion. The chemical process can be treated successively using stochastic methods. Currently, there is interest in extending these results to the products of higher LET radiation, which are formed along linear tracks. Surface Kinetics and Electrochemistry The theory of surface kinetics seeks to explain effects such as etching, disso- Jution of crystals and the formation of corrosion pits. Stochastic models of growth and dissolution have been studied. An interesting class of problems concerns the description of flocculation processes, in which the growth of an aggregrate is limited by diffusion from the surrounding medium. Electrochemistry is concerned with the understanding of chemical effects pro(luce(1 at electrodes. Recent debates about the feasibility of cold fusion, hinged on the estimation of the probability of favorable molecular encounters at an electrode. Electrochemistry is clearly a branch of physical chemistry in which probabilistic calculations play an important role.

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