Computer Modeling of Matter

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Computer Modeling of Matter

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.


Computer Modeling of Matter Peter Lykos, EDITOR Illinois Institute of Technology

Based on a symposium sponsored by the ACS Division of Computers in Chemistry at the 175th Meeting of the American Chemical Society, Anaheim, California, March 14,

1978.

ACS SYMPOSIUM SERIES 86

AMERICAN CHEMICAL SOCIETY WASHINGTON, D. C.

1978


Library of Congress CIP Data Computer modeling of matter. (ACS symposium series: 86 ISSN 0087-6156) Includes bibliographies and index. 1. Molecular theory—Data processing—Congresses. 2. Matter—Properties—Data processing—Congresses. I. Lykos, Peter George. II. American Chemical Society. Division of Computers in Chemistry. III. Series: American Chemical Society. ACS symposium series; 86. QD461.C632 542'.8 78-25828 ISBN 0-8412-0463-2 ASCMC 8 86 1-272 1978

Copyright © 1978 American Chemical Society All Rights Reserved. The appearance of the code at the bottom of thefirstpage of each article in this volume indicates the copyright owner's consent that reprographic copies of the article may be made for personal or internal use or for the personal or internal use of specific clients. This consent is given on the condition, however, that the copier pay the stated per copy fee through the Copyright Clearance Center, Inc. for copying beyond that permitted by Sections 107 or 108 of the U.S. Copyright Law. This consent does not extend to copying or transmission by any means—graphic or electronic—for any other purpose, such as for general distribution, for advertising or promotional purposes, for creating new collective works, for resale, or for information storage and retrieval systems. The citation of trade names and/or names of manufacturers in this publication is not to be construed as an endorsement or as approval by ACS of the commercial products or services referenced herein; nor should the mere reference herein to any drawing, specification, chemical process, or other data be regarded as a license or as a conveyance of any right or permission, to the holder, reader, or any other person or corporation, to manufacture, reproduce, use, or sell any patented invention or copyrighted work that may in any way be related thereto. PRINTED IN ΤΗE UNITED STATES OF AMERICA


ACS Symposium Series Robert F. Gould,

Editor

Advisory Board Kenneth B. Bischoff

Nina I. McClelland

Donald G . Crosby

John B. Pfeiffer

Jeremiah P. Freeman

Joseph V . Rodricks

E. Desmond Goddard

F. Sherwood Rowland

Jack Halpern

Alan C. Sartorelli

Robert A . Hofstader

Raymond B. Seymour

James P. Lodge

Roy L. Whistler

John L. Margrave

Aaron Wold


FOREWORD The ACS SYMPOSIUM SERIES was founded in 1974 to provide

a medium for publishin format of the Series parallels that of the continuing ADVANCES IN CHEMISTRY SERIES except that in order to save time the papers are not typeset but are reproduced as they are submitted by the authors in camera-ready form. Papers are reviewed under the supervision of the Editors with the assistance of the Series Advisory Board and are selected to maintain the integrity of the symposia; however, verbatim reproductions of previously published papers are not accepted. Both reviews and reports of research are acceptable since symposia may embrace both types of presentation.


PREFACE he ever increasing power of computers and the continuing decrease in their cost enable the chemist to construct increasingly sophisticated mathematical models of bulk matter—both at equilibrium and changing in time—from a molecular perspective. Thanks to the pioneering work of Bernie Alder and others who developed the method of molecular dynamics and to the Monte Carlo method of Metropolis used for equilibrium data, as the size and speed of computers increased i Frank Stillinger (/. Chem. of molecular dynamics to the most important and complex bulk matter of all—liquid water. That seminal paper sparked a great interest in modeling on the part of chemists. The important discovery that a mathematical model whereby one averages the individual properties of a few hundred interacting molecules suffices to assess the bulk properties of many important systems suggests that chemists now can build more effective bridges between atomic and molecular physics on the one hand and surface chemistry and chemical thermodynamics and kinetics on the other. In the recent "Science Update: Physical Chemistry" in Chemical and Engineering News ((1978) June 5, p 20-21), the impact of the "computer primarily as an aid to modeling was highlighted as the most pervasive and important factor influencing physical chemistry today. In that article Mitch Waldrop wrote, "So important have the big computers become that number crunching and theoretical chemistry often seem synonymous. The applications can be divided into three broad areas: quantum chemistry, chemical dynamics, and statistical mechanics. If computers were big enough, those three would form a logical sequence for the complete ab initio calculation of the properties of bulk matter." The number and range of computer-based models of bulk matter has increased rapidly. An important international conference (with proceedings), "Computational Physics of Liquids and Solids" held April, 1975, at Queen's College in Oxford, involved 34 papers that displayed a broad range of chemical phenomena being studied in this manner. Just a year and a half later the Faraday Division of the Chemical Society, London, held a two-day symposium on "Newer Aspects of Molecular Relaxation Processes" at the Royal Institution, London. The so-called 'experimental technique of examining motions in a computer-generated ix In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

liquid* was considered together with experimental methods, with theoretical models of relaxation processes, and with the hydrodynamics of rotation in fluids. Two books have appeared which, while not presenting a unified view of the field nor a critical assessment of the literature, do provide the interested scientist with an entree to the use of computer simulation of liquids ("Theory of Simple Liquids," J. P. Hansen, I. R. McDonald, Academic, 1976; "Atomic Dynamics in Liquids," N . H . March, M . P. Tosi, Halsted (Wiley), 1977). The impact on statistical mechanics as a discipline is reflected in the two-volume work, "Statistical Mechanics," B. J. Berne, Ed., Part A, Equilibrium Techniques, and Part B, TimeDependent Processes, Plenum, 1977. Indeed, in his Nobel prize address, I. Prigogine made several references to the use of computer simulations as an aid to developmen macroscopic and microscopic aspects of the second law of thermodynamics (Science (1978) 201,777-785). Enough experience has been gained through the design and testing of such models that chemists interested in gaining insight into particular chemical systems are beginning to apply the models in a rather sophisticated manner. For example, William Jorgensen, a theoretical organic chemist interested in solvation effects in organic chemistry, has begun by looking at liquid hydrogen fluoride in a paper to be published in December, 1978, /. Am. Chem. Soc. Indeed theorists are organizing and presenting their theories with careful attention to how they might be applied using a computer (for example, see "Simulation of polymer dynamics 1. General theory and 2. Relaxation rates and dynamic viscosity," M. Fixman, /. Chem. Phys. (1978) 69, p 1527, 1538). The call for contributions to this symposium has resulted in an interesting collection of eighteen chapters which provide the reader with a variety of touchstones ranging from the first chapter, a straight-forward application by K. Heinzinger of the Rahman-Stillinger method to an aqueous solution of sodium chloride, to the last chapter, an overview from microphysics to macrochemistry via discrete simulations including chemical kinetics. The other chapters include examination of an algorithm assessing the importance of three-body interactions and of algorithms reducing machine requirements with regard to size of store and speed. Most of the work done to date has been based on classical mechanics. The chapter by M . H . Kalos brings home the fact that one needs to examine the validity of such models from within the framework of quantum mechanics. The interesting phenomenon of collective modes of motion is exemplified by M . L . Klein's chapter on collective modes in x In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

solids. The liquid-vapor interface is addressed in two chapters, one by Rao and Berne and the other by Thompson and Gubbins. Sundheim's chapter on high field conductivity emphasizes that the modeler is free to impose whatever simulated physical conditions he pleases and, in fact, can achieve in his computer experiment conditions which would be extremely difficult, if not impossible, to achieve in the laboratory. Finally, the Chester, Gann, Gallagher, and Grimison chapter demonstrates that the computer mystique is being displaced by a hard-headed appraisal by chemists of what computer enhancements are possible with existing technology. Through the addition of a peripheral device (designed to do floating-point arithmetic very rapidly) to the campus largescale data-processing machine, it was possible to improve the cost effectiveness of the total syste simulation of matter. The growing sophistication of matter modeling as a result of advances in computer size, speed, and cost effectiveness may well have as its greatest impact, however, the enhancement of communication among those interested in any particular physical system. Illinois Institute of Technology Chicago, Illinois October 13, 1978

PETER LYKOS

xi In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

1 Molecular Dynamics Simulations of Liquids with Ionic Interactions K. HEINZINGER, W. O. RIEDE, L. SCHAEFER, and GY. I. SZÁSZ Max-Planck-Institut fuer Chemie (Otto Hahn-Institut), Mainz, Germany

Molecular dynamics (MD) simulations of liquids with ionic interaction have so far been performed for molten salts and aqueous electrolyte solutions. The characteristic problem for this kind of simulation are the far ranging Coulombic forces. The f i r s t preliminary MD calculations for molten salts have been reported by Woodcock in 1971 (1). The large amount of work published in the meantime has been reviewed by Sangster and Dixon (2). In the case of aqueous electrolyte solutions only preliminary results for various alkali halide solutions have been published so far (3). The more advanced state of the art for the molten salts leads to a concentration of the effort on the improvement of the algorithm for the integration of the equation of motion necessary to calculate dynamical properties with still higher accuracy. One example where very high accuracy is needed is the calculation of the isotope effect on the diffusion coefficients. In the section on molten salts below one way to improve the algorithm is d i s cussed and the progress is checked on the mass dependence of the diffusion coefficients in a KCl melt. Results with a certain degree of r e l i a b i l i t y from MD simulations of aqueous solutions reported up to now are restricted to structural properties of such solutions. In the section on aqueous solutions below very preliminary velocity autocorrelation functions are calculated from an improved simulation of a 0.55 molal NaCl solution. The problem connected with the stability of the system and the different cut-off parameters for ion-ion, ionwater and water-water interactions are discussed. Necessary steps in order to achieve quantitative results for various dynamical properties of aqueous electrolyte solutions are considered. Molten Salts The Hamilton differential equation system can be solved 0-8412-0463-2/78/47-086-001$07.00/0 © 1978 American Chemical Society In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

2

COMPUTER MODELING OF MATTER

numerically for a N particle system with a simple "leap-frog" algorithm as used by Verlet (4) or by a predictor-corrector a l gorithm. The most important criterion for the choice of the a l gorithm is the numerical s t a b i l i t y . The final decision follows from necessary accuracy with which for example the energy is preserved, which in turn is determined by the desired accuracy of the transport properties, say self diffusion coefficient. In the simulated system of molten KC1 i t has been investigated how the total energy, the total momentum, and the velocity autocorrelation functions varies with different algorithms. The MD calculation for the KCl-system was carried out at the melting point (1043K). The basic cube contains N=2-108 particles and the cube length S is than calculated to be 20.6A. For the pair potential the Born-Mayer-Huggins potential ip(r) = + er"

1

+ bexp(-Br) + Cr" + Dr' 6

(1)

8

is used with the parameter in Table I. Table I Parameters for the Born-Mayer-Huggins Potential b

+

K

+

-

K

+

- CI"

CI"'-

cr

D

C

10~ erg

A-1

1991.67768

2.967

1224.32459

2.967

-

4107.95500

2.967

- 145.5

12

K

B

10" erg A 12

-

6

10" erg A8 12

24.3

-

24.0

48.0

-

73.0

- 250.0

Three different algorithms were investigated. In the f i r s t version (I) the Coulombic energies and forces were evaluated with the use of the erfc part of the Ewald method (6) only. In the other two versions (II and III) the f u l l Ewald method was employed. In a l l versions the separation parameter T), the number of nearest neighbours n~ , and the maximum value of the reciprocal lattice vectors h] (not occuring in version I) were chosen to be: i\ = 0.175,

n^ = 6,

h

1

= 8.

The algorithm which was used in version I is a so called "leapfrog" algorithm: Ii(n+1) = r.(n-1) + 2At v.(n-1) + (At /m.) JF.(n) 2


(2)

1.

HEINZINGER ET AL.

MD Simulations of Liquids with Ionic Interactions

3

v.(n) = v^n-1) +[v.(n-1) - v.(n-2)] + (A^/m^F.fn) Here r.-j( )> v.i( )> n

n

Lj( )

a n c l

n

a

r

e

t

h

e

- ^(n-1)]

(3)

position, velocity and force

of particle i at time t = nAt respectively and m-j are the masses -14 of the ions. The time step length was chosen to be 0.5-10 sec. The behavior of the total energy for version I is shown in F i gure 1 where (AE/E) =6 - 10-3. A change from the "leap-frog" algorithm to a predictorcorrector algorithm has been made in version II. For the position vectors one can write: r?(n+1) = r?(n-1) + 2 At v.(n-1) + r?(n+1) = r?(n-1) + 2 At v.(n-1) + (At /4m.)[ F.(n+1) + 4F.(n) + 3F.(n-1)] 2

(5)

Here the indices p and c indicate predicted and corrected values respectively. The formula for the velocities has been changed slightly compared to the formula (3): ^ ( n ) = v.(n-1) + (At/2m.)[F (n) + F.(n-1)] i

(6)

With this version II, two simulations with different time steps were carried out starting from the same configuration (0.5 10"1^s and 0.25 10~14s) over a time interval of 1.2 ps. The total energies for both runs are shown in Figure 1 . The value (AE/E)Q ^ has decreased compared to the one of version I by one order of magnitude. Moreover AE/E decreased further by a factor 0.5 when the time step was shortened to 0.25.10"14 . In version III the positions and velocities have been treated with the same predictor-corrector algorithm. This means that F-j ( t ) , the derivative of the forces with respect to time, appears in the formula for the velocities. s

rP(n+1) = r?(n-1) + 2 At v?(n-1) + (2At /3 ) [2F (n) + F.(n-1)] 2

mi

i


(7)


J

1 01

Etotal

1 •

1 05

.

.

3

.

.

, 1 psec

.

.

•

3

Figure 1. Total energy of version I ('upper curve) in units of 10 J/mol and of version II flower curve) with a energy scale enlarged by a factor of 4 A. The length marked by (jimm) in the upper scale corresponds to the interval of the lower scale. The time scale starts after an equilibration run over 7.2 psec from a roughly equilibrated starting configuration.

627 6

633-

1.

HEINZINGER ET AL.


r?(n+1) = r + 1 (-w)(-1) (-2)

V

d

+

2

w w

=(

+

>

2

0=1

H )

J ^

e

I

a

+

B

/ d

a

a (-a)

d

+

V (r) = + 2 / r ++

e


B

16


The switching function, S ^ r ) has been introduced to reduce unr e a l i s t i c Coulomb forces between very close water molecules ( 1 2 ) . e is the elementary charge and q = 0.23 e is the charge in the ST2 water model, d and r denote distances between point charges and LJ centres, respectively. The choice of a and 6, odd for positive and even for negative charge yields the correct sign. The LJ parameters for water-water interaction are taken from the ST2 model, and for cation-cation from the isoelectronic noble gases (Y3). The anion-anion parameters are derived from the cation ones on the basis of the Pauling radii as shown in detail in a previous paper (14). The LJ parameters between different part i c l e s are calculated by application of Kong's combination rules (J5J. A l l e and o are given in Table III. Table

III

Lennard - Jones parameter in a NaCl solution is H0

Na

3.10

2.92

4.02

54.84

30.83

2.73

3.87

5937

28.32

-

27.87

2

H0 2

Na cr

+

52.61

-

+

CI"

4.86

The classical equations of motion are integrated in time steps of At = 1.09-10-16 (22).In order to take account of the different strengths of the various interactions and to keep the computer time in reasonable limits, different cut off parameters are chosen for ion-ion, ion-water and water-water interactions. The Ewald summation (6), which gives the correct Coulombic energy for an unlimited number of image ions, is used for ion-ion interactions. As the ions residing in the f i r s t and second neighbour image boxes are included in the direct calculation of the so called error function part, a suitable choice of the separation parameter allows to neglect the Fourier part of the Ewald sum. The ion-water interaction is cut off at 9.1 A, half the sidelength of the basic periodic box. The water-water interaction is cut off at 7.1 A which means that only f i r s t and second nearest neighbour molecules are included in the calculation. The structural properties of the NaCl solution calculated from this simulation are not significantly different from preceding ones. The characteristic distances an heights of the ionwater and water-water radial pair correlation function remain a l most unchanged whether a temperature control mechanism is built s


1.

HEINZINGER ET AL.

MD Simulations of Liquids with Ionic Interactions 17

in or not. The radial pair correlation functions from the simulation of a 2.2 molal NaCl solution (200 water molecules and 8 ions of each kind) are compared in detail with X-ray investigations in a previous paper (14). The problem connected with the determination of hydration numbers have also been discussed previously (16). In addition i t has been concluded from the MD s i mulations of alkali halide solutions that in the f i r s t hydration shells a lone pair orbital of the water molecules is directed towards the cation and a linear hydrogen bond is formed with the anion (1?). This result is also in agreement with conclusions from experimental and other theoretical investigations. The real aim of MD simulations i s , of course, the calculation of the dynamical properties of the liquid. The velocity autocorrelation function have been calculated from this 10.000 time step run. In a l l cases the functions have been averaged over 360 starting vectors. The s t a t i s t i c a l quality is much better, of course, in the case of average extends over 20 of the ions. In Figure 8 the angular (solid line) and the translational (dashed line) velocity autocorrelation functions of the water molecules are shown. The correlation time for the angular velocity is as expected much shorter than for the translational velocity. In Figure 9 the velocity autocorrelation functions for the cation (solid line) and the anion (dashed line) are drawn. If these autocorrelation functions are integrated, in spite of a l l s t a t i s t i c a l uncertainties i t is found that the corresponding diffusion coefficients are of the same order of magnitude as is known or estimated from experimental investigations. Therefore i t is j u s t i f i e d to continue the investigation along this line. It is obvious that much longer simulation times are required to make quantitative statements about the dynamical properties. But not only an increase in simulation time is necessary, also the simulation i t s e l f needs further improvement. The system simulated corresponds to a microcanonical ensemble and consequently i t should have a constant total energy after an equilibration time of suitable length. In Figure 10 the total energy (solid line) and the potential energy (dashed line) for the system are shown over the 10.000 time steps run. I can be seen that the increase of the total energy is equally distributed between kinetic and potential energy. The unavoidable cumulative errors due to the algorithm, which were discussed above in detail in the case of the molten s a l t s , are not the main source of energy increase in the simulation of aqueous solutions. Responsible for the problems here are mainly the cut-off r a d i i . Figure 11 shows the variations of the total energy of the basic box E during a simulation run of 270 & t, which is essentially caused by the cut-off radii R and Ri in the evaluation of the potential energy E t ( t ) . Let us consider the expression: w

p o


18


Figure 9. Normalized velocity autocorrelation functions qp(t) for cations (solid) and anions (dashed in a 0.55m NaCl solution)


1. HEINZINGER ET AL.

M D Simulations of Liquids with Ionic Interactions 19


20


-232 AE

-233

50

100

150

200

250 At

Figure 11. Variation (cut-off jumps) of the total energy of the basic box in units of 10 erg for 200 water molecules and eight Ntf CZ. The time step is 2.18 • 16 sec and AE/E = 3 • 10 . 12

16

4

s


1.

E

MD Simulations of Liquids with Ionic Interactions 21

HEINZINGER ET AL.

pot

( t +

A t

>- pot E

•

tV (r (t^t)-V (r (t))}

( t ) =

i

1 j

1 j

i j

^(r^tt^t))-

1 j

^(r^t))

Z

(28)

where the summation extends over i and j such that I

r^.JtJ^R and ^ .(t+At)

p

o w r

§

I

4^

1. HEINZINCER ET AL.

1


1 0 25

r 0.5

1 075

psec

Figure 14. Kinetic energy of the total system ( )> the cations ( anions ( )in Kelvin as a function of simulation time


1

1

—

j, and

26


depends both on the accuracy of the programmed force routines and, as F(t) is calculated from predicted values, on the accuracy of the predicted com coordinates and Euler angles. Simulations of aqueous solutions are characterized by the fact that the changes in the coordinates in one time step depend strongly on the type of motion. The choise of At for such a s i mulation is essentially determined by the very fast changes in the Euler angle coordinates, and At is thus in principle too small for the integration of the translational motion. A At of 1.09-10-16 sec leads to a contribution from the corrector of lArl i s the f o r c e between the atoms induced by t h e solvent molecules. L J

s o l v

In t a b l e 1, we r e p o r t how the solvent induced force < F > v a r i e s with r . The e r r o r s are estimated from the magnitude o f the f o r c e components p e r p e n d i c u l a r t o the l i n e j o i n i n g a and b . S t r i c t l y speaking, these should be zero. C l e a r l y much longer runs are necessary t o improve the accuracy o f the work reported here. We found i t d i f f i c u l cause at the l a r g e s t s e p a r a t i o s t a n t i a l , making t h e t a s k o f i n t e g r a t i n g impossible. S O l v

The i n t e r e s t i n g f e a t u r e o f our r e s u l t s i s t h e o s c i l l a t i o n i n ; t h i s i n d i c a t e s t h a t when t h e two Xe atoms are a d i s t a n c e g r e a t e r than 5.5A a p a r t , a water molecule i s l i k e l y t o take a p o s i t i o n between them. T h i s i s c o n s i s t e n t with the two Xe atoms r e s i d i n g i n two cages with a monolayer o f water between them. A f t e r completion o f t h i s work we l e a r n e d that A. Geiger, et a l (h ) s t u d i e d the h y d r a t i o n o f two Lennard-Jones s o l u t e p a r t i c l e s . T h e i r r e s u l t s show a q u a l i t a t i v e s i m i l a r i t y t o ours, but they do not give a q u a n t i t a t i v e r e s u l t f o r t h e p o t e n t i a l o f mean f o r c e . Our work supports the theory o f P r a t t and Chandler (2) on the hydrophobic e f f e c t . Previous Monte Carlo work (3) along s i m i l a r l i n e s d i s p l a y e d no o s c i l l a t i o n i n , although i n t h a t study the authors explored values o f r smaller than 7. Table

F

L J

o

2e

O

w

and e

-7.55 -U.30 -1.19 -0.58 -0.29

w

a W

2e w

5..35 6,.07 7..U6 8..30 9..18

I

w

19.89 ± 5 . 2 -20,20 ±k.9 3.23 ±6.2 35.88 ±10.2 33.7^ ± 5 . 8

w

2e w

12.3 * -5.2 -2^.50 ±k.9 2.05 ±6.2 36.h6 ±10.2 3^.03 ± 5 . 8 1

r e f e r t o the parameters f o r the ST2 p o t e n t i a l .


34


Literature 1.

F. H.

Stillinger,

Cited

and A. Rahman, J. Chem. Phys. 6 0 , 1545

(1974) 2.

L. P r a t t and D. Chandler, J. Chem. Phys. 67, 3683

3.

V.G. Dashevsky and G.N. Sarkisov, Mol. Phys. 27, 1271

4.

A. Geiger, A. Rahman, and F.H.

RECEIVED August 15,

Stillinger,

(1977)

(Preprint)

1978.


(1974)

4 Applying the Polarization Model to the Hydrated Lithium Cation CARL W. DAVID Department of Chemistry, University of Connecticut, Storrs, CT 06268

Electrostatic model chemistry. The Polarization Model has a genealogy which extends back to the work of Born and Heisenberg (1) and Rittner (2) i n which simple electrostatics was applied to ionic compounds. Modern examples of such models include the recent work of E t t e r s , Danilowicz, and Dugan (3), and, i n Inorganic Chemistry, the work of Guido and Gigli (4). Since H e a v i s i d e ' s theorem precludes purely electrostatic stability of ionic compounds, a repulsive potential of some sort must be included to stabilize an isolated ionic molecule in the gas phase. The freedom to introduce such an ad hoc repulsive potential suggested to various people that even wider lattitude might be had i n order to extend electrostatic models to include covalently bonded compounds. Our concern here i s with water and its interactions, and we specialize to that substance now. Early models of water, attempting to account for polymeric properties of the species using quasi-electrostatic schemes, placed imaginary negative charges i n the vicinity of the oxygen atom. The l a t e s t such venture is, already, almost a classic. The Ben-Naim S t i l l i n g e r ( B N S ) p o t e n t i a l (5) was used i n d i s c u s s i n g liquid water up to and i n c l u d i n g d i r e c t numerical i n t e g r a t i o n of Newton's equations of motion for water molecules (6). The p o t e n t i a l was an outstanding success. Several other p o t e n t i a l s for the water-water i n t e r a c t i o n have r e c e n t l y appeared, as has a r e v i s i o n of the BNS p o t e n t i a l known as ST2 (7, 8, 9, 10). Tn addition, several model p o t e n t i a l s have appeared which incorporate p o l a r i z a t i o n effects (11, 12). Each of these p o t e n t i a l s seems quite capable of being 0-8412-0463-2/78/47-086-035$06.75/0 © 1978 American Chemical Society In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

36


used t o compute p r o p e r t i e s o f m a c r o s c o p i c samples o f water. These e l e c t r o s t a t i c p o t e n t i a l s f o r water a l l s u f f e r a f a t a l d e f e c t , as they a p p l y t o f r o z e n w a t e r not i c e - b u t water m o l e c u l e s w i t h fixed (i.e., f r o z e n ) geometry. I n an a t t e m p t t o e n l a r g e t h e c l a s s of water p o t e n t i a l s , Lemberg and S t i l l i n g e r f!3) introduced a central force model w h i c h a l l o w e d " u n f r e e z i n g " t h e water m o l e c u l e . Here ?id hoc atom p a i r i n t e r a c t i o n s are introduced which, i n t o t o , bind the water m o l e c u l e t o g e t h e r at i t s eouilihrium geometry while simultaneously allowing water molecules t o i n t e r a c t p r o p e r l y among t h e m s e l v e s . T h i s model was t e s t e d i n a r e c e n t m o l e c u l a r dynamics s i m u l a t i o n (14) q u i t e s u c c e s s f u l l y . The c e n t r a l f o r c e model does n o t s u f f e r from t h e f r o z e n geometry defec models. The m o l e c u l dissociates. However, i t i s i n t e r n a l l y u n a f f e c t e d by e x t e r n a l e l e c t r o s t a t i c f i e l d s , and t h e r e f o r e does n o t f u l l y mimic r e a l w a t e r . An e l e c t r o s t a t i c model o f t h e water monomer, which included polarizability i n some r e a s o n a b l e manner m i g h t a l l o w f o r complete s i m u l a t i o n o f l i o u i d water and o f i o n i c solutions. The s t a t i c d i e l e c t r i c properties of this ubiquitous s o l v e n t would, in a good model, be a u t o m a t i c a l l y c o r r e c t , both on t h e microscopic and on t h e m a c r o s c o p i c l e v e l . The thermodynamic p r o p e r t i e s p r e d i c t e d by such a model w o u l d , e x c e p t f o r m i n o r c a l i b r a t i o n e r r o r s , be c l o s e to q u a n t i t a t i v e . Having no s e r i o u s doubts about m i m i c k i n g water a t l i q u i d d e n s i t i e s , one can p r o c e e d to u s i n g more s o p h i s t i c a t e d models o f t h e w a t e r molecule. The r a i s o n d ' e t r e need no l o n g e r bo j u s t i f i c a t i o n o f t h e methodology, and one can s e t t l e down t o p r e d i c t i o n s and c o r r e l a t i o n s confident that the e r r o r s i n t h e p r e d i c t i o n s a r e i n a c c u r a c i e s i n t h e model. The P o l a r i z a t i o n

Model

The P o l a r i z a t i o n Model r e g a r d s a water m o l e c u l e as a c l u s t e r o f 2 p r o t o n s ( w h i c h r e p e l each o t h e r ) and an o x i d e i o n ( w h i c h i s p o l a r i z a b l e ) . The b i n d i n g o f the o x i d e i o n t o the p r o t o n s i s e f f e c t e d through an empirical potential function. The d i p o l e moment o f the m o l e c u l e i s adjusted t h r o u g h two special e l e c t r o s t a t i c m o d i f i e r f u n c t i o n s , K ( r ) and L ( r ) . The p r o t o n s g e n e r a t e an e l e c t r i c f i e l d at the o x i d e a n i o n w h i c h , i n r e a l e l e c t r o s t a t i c s , would have


4.

Polarization Model and the Hydrated Lithium Cation

DAVID

the

37

form 3

F - e r /

(1

r

where e i s t h e charge on t h e p r o t o n . Fince the electrons o f r e a l water prevent t h e proton from actually s e t t i n g up such a f i e l d , and s i n c e t h e p r o t o n s would s e t up no f i e l d i f r=0, i t is r e a s o n a b l e t o i n v e n t a s h i e l d i n g f u n c t i o n which w i ] l account f o r these two f a c t s , i . e . , modify electrostatics so t h a t i t i s applicable i n this a r t i f i c i a l model. The f u n c t i o n c h o s e n , K f r ) has l i m i t s such t h a t , a t l a r g e r , K ( r ) goes t o z e r o , thereby returning t h e e l e c t r o s t a t i c s which i s e x p e c t e d . However, a t s m a l l r , t h e f u n c t i o n K f r ) a c t s to c u t o f f t h e e f f e c f i e l d so t h a t a t r the f i e l d c o m p l e t e l y . The f o r m a l d e f i n i t i o n o the function i s : 3

( 1 - K(r) )

G - e r/ r

(2

G i v e n t h e e f f e c t i v e e l e c t r i c f i e l d (G) a t the o x i d e i o n due t o t h e p r o t o n s , one has, i f the oxide i o n i s p o l a r i z a b l e , t h e i n d u c t i o n o f a d i p o l e moment in the anion. The magnitude o f t h i s induced moment is 3

p

= a e r/ r

( 1 - K(r))

(3

Associated with this i n d u c e d d i p o l e moment on t h e o x i d e i o n ( w h i c h opposes t h e n u c l e a r d i p o l e moment) i s a p o l a r i z a t i o n energy = (1/2)

U

u . E

(4

pol Fere, we choose to again modify normal electrostatics. Instead o f using the true e l e c t r i c field i n E q u a t i o n 4, and i n s t e a d o f usir> r

o where ri2 i s the v e c t o r from the center of molecule 1 to that o f

0-8412-0463-2/78/47-086-062$05.00/0 © 1978 American Chemical Society In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

5.

MURAD

AND

63

Molecular Dynamics Simulation of Methane

GUBBINS

molecule 2, = 4>i0^^ i s the o r i e n t a t i o n o f molecule i , r g i s the distance between s i t e a of molecule 1 and s i t e 8 o f molecule 2, r = 10.025 A i s the c u t o f f d i s t a n c e , and a

Q

-

W

(2)

4 e

where e g and a g are the Lennard-Jones parameters (see Figure 1). Using t h i s p o t e n t i a l model, f i v e sets o f parameters (e 3>o* $) have been used to evaluate thermodynamic p r o p e r t i e s ( c o n f i g u r a t i o n a l energy, pressure and s p e c i f i c h e a t ) . The best o f these parameter sets i s used i n making a more d e t a i l e d study o f methane. This best set gives r e s u l t s s u p e r i o r to those o f a recent study (10) u t i l i z i n g the Williams (11) s i t e - s i t e exp-6 model, and i s also s u p e r i o r to the r e s u l t s obtained by Hanley and Watts (12) using an i s o t r o p i c m-6-8 a

a

a

a

Method The quaternion method has been described i n d e t a i l i n recent p u b l i c a t i o n s CO so that we only give a b r i e f summary o f the method here. The quaternion parameters x> n, £ C can be defined i n terms of G o l d s t e i n E u l e r angles (13) by ?

X

= cos

(6/2)

n

= s i n (0/2)

c o s ( * + )/2, ^ c o s ( * - <j>)/2,

(3)

> 5

= s i n (0/2)

s i n ( * - )/2,

C

= cos

sin(i|i + (J>)/2.

(0/2)

j

From (3) i t can be seen that X

2

+ n

2

+

C + C =1

(4)

2

The r o t a t i o n matrix A (13) defined by V = A • V -principal - -lab

(5)

i s given by

-S A =

2

I -2«n L

+

n

2

- C + x,

+ cx),

2(nc - 5x).

2Ux - 5n),

2

2

S - n 2

2

- c

-2(5c + nx).

2

+ x, 2

-S

2

2(nc + £x)

(6)

2(nx - SO

n

2

+ c

2

+ x

2


64

COMPUTER

MODELING

OF

MATTER

The p r i n c i p a l angular v e l o c i t y u>p i s r e l a t e d to the quaternions by the matrix equation W

' Px\ (7)

0 / The i n v e r s e of (7) i s simple to f i n d s i n c e the 4 x 4 matrix i s orthogonal. The equations of motion f o r these parameters are thus f r e e of s i n g u l a r i t i e s . A system o f 108 model methane molecules was s t u d i e d using standard p e r i o d i c boundary c o n d i t i o n s ( 1 ) . The i n t e r m o l e c u l a r forces and torques wer system. The torques wer Tp, using the r o t a t i o n matrix (equation ( 6 ) ) . P r i n c i p a l torques and f o r c e s F were used to evaluate the time d e r i v a t i v e s of the p r i n c i p a l angular v e l o c i t i e s and c a r t e s i a n p o s i t i o n s r of the molecules using the equations du) -77^ = T d t

/I Pa P

(a = x,y,z)

(8)

a

2

d r dP"

(9)

F/m

A t h i r d order p r e d i c t o r - c o r r e c t o r method (14) was used to i n t e grate the center of mass motion (equation ( 9 ) ) , while a second order method was used f o r o r i e n t a t i o n a l equations of motion of the molecules (equations (8) and ( 7 ) ) . A l l r e s u l t s were based on 2,000 time steps of 1.47 x 1 0 " seconds, a f t e r e q u i l i b r a t i o n . The energy and pressure were c o r r e c t e d f o r the long range p o t e n t i a l c o n t r i b u t i o n i n the usual way ( 1 ) , by p u t t i n g g g = 1 f o r r g > r . The s h i f t i n the p o t e n t i a l a l s o a f f e c t s the energy, but not the pressure or s p e c i f i c heat. The energy was c o r r e c t e d f o r t h i s p o t e n t i a l s h i f t by w r i t i n g U = + U h i f t > where Ujfl) i s the value corresponding to equations (1) and (2) and obtained i n the s i m u l a t i o n , and U h i f t i s the c o r r e c t i o n , given by 15

a

a

Q

s

s

where g ( r ) i s the s i t e - s i t e c o r r e l a t i o n f u n c t i o n . n

n


5.

MURAD

65


A N D GUBBINS

Energy conservation was w i t h i n about 0.75% per 10,000 time steps. One time step took approximately 25 seconds f o r a 108 molecule system on the PDP 11/70 minicomputer. Results The p o t e n t i a l o f equations (1) and (2) i n v o l v e s the para meter set ( e , a , ecc/ HH> Cc/°HH> ^CH* ^CH)» where ncH CcH g i the departure from the usual combining r u l e s f o r and CCH, e

c c

a

a

n

d

c c

v e

Q

CH

=

n

(a

* CH CC ke

CH

^CH CC

+

a

HH

(10)

}

(11)

HH

In t h i s work we have s t u d i e e

e

a

n

d a

a

T

i

CC^ HH CC^ HH* * and are put equal to those given by Williams (11) i n h i s para meter set VII. Table I shows the Lennard-Jones parameter sets used i n t h i s work. In the f i r s t f i v e sets ZQQ and OQQ are kept f i x e d , and the r a t i o s ^QQ/O^H and ecc/ HH are v a r i e d from one set to the next. The parameters f o r set 1 were obtained by p u t t i n g the Lennard-Jones o* g and e g values equal to those used by Williams (11) i n h i s parameter set VII f o r the exp-6 model. Parameter sets 2-5 show increases i n £HH o f e i t h e r 10 o r 20%, and decreases i n of e i t h e r 0, 1, o r 2%. In a l l f i v e sets ncH n d were kept f i x e d at 0.972 and 1.132, r e s p e c t i v e l y , as given by Williams parameter set VII. For each o f the 5 parameter sets given i n Table I three s t a t e points were s t u d i e d and the mean squared d e v i a t i o n s between molecular dynamics and experimental r e s u l t s (15) were c a l c u l a t e d for the c o n f i g u r a t i o n a l energy, pressure and s p e c i f i c heat. The three s t a t e points used were: (a) T = 200 K, p = 10.00 mol fc" , (b) T = 225 K, p = 18.50 mol Z" , (c) T = 125 K, p = 25.27 mol ST . The f i r s t o f these c o n d i t i o n s i s i n the moderately dense super c r i t i c a l r e g i o n , the second i s i n the dense s u p e r c r i t i c a l r e g i o n , and the t h i r d i s i n the dense l i q u i d r e g i o n . Table I I gives the root mean squared d e v i a t i o n s found f o r each parameter s e t . Set 5 gave the best r e s u l t s . For t h i s parameter set s i x a d d i t i o n a l s t a t e c o n d i t i o n s were s t u d i e d . The r e s u l t i n g molecular dynamics r e s u l t s are compared with experiment i n Table I I I . The molecular dynamics r e s u l t s shown i n Table I I I may be used to r e s c a l e the values of SQQ and OQQ> keeping f i x e d CC/ HH> CC/ HH> ^CH ^CH* ^ t h o d o f c a r r y i n g out t h i s r e s c a l i n g i s i l l u s t r a t e d ( f o r two s t a t e c o n d i t i o n s ) i n Figure 2. For a s i n g l e s t a t e c o n d i t i o n (e.g. 1 i n Figure 2) i t i s p o s s i b l e to f i n d a set o f (OQQ CQQ) values that give exact agreement between molecular dynamics and experiment f o r the c o n f i g u r a t i o n a l e

a

a

a

1

1

e

G

a

a

a

n

d

T

h

1

e m

9


66

COMPUTER

MODELING

OF MATTER

Figure 1. Site-site and center-center distances for two CH, molecules at orientations o> and w t

t

2

Table I P o t e n t i a l Parameter Sets Studied

1 2 3 4 5 Rescaled 5

cc

CH (K)

HH (K)

"CH (X)

HH (A)

350 350 350 350 350

023 023 008 008 2.995

2.870 2.870 2.841 2.841 2.813

48.760 48.760 48.760 48.760 48.760

20.690 21.700 21.700 22.665 22.665

850 535 535 220 220

3.350

2.995

2.813

51.198

23.798

8.631

CC (A)

Parameter Set

e

/k

(K)

£

/ K

E

/ K

The b o n d l e n g t h i n parameter sets 1-5 i s kept f i x e d at 1.10 A. The parameter s e t " r e s c a l e d 5" a l s o has t h i s same value o f the bond l e n g t h , s i n c e the r e s c a l e d a's are unchanged. o


5.

MURAD

AND


GUBBINS

Table I I Root Mean Squared Deviations For Models Studied

Root Mean Squared Deviation"*"

Parameter Set

C

U /NkT

P/pkT

Cy/Nk

1 2 3 4 5

0.45 0.41 0.26 0.18 0.22

0.46 0.41 0.34 0.21 0.14

0.43 0.35 0.44 0.37 0.37

0.45 0.40 0.30 0.21 0.21

Rescaled 5

0.08

0.07

0.22

0.10

V

= C V

- C

V

i d g

a

s

, where C

V

and C

i d g

T o t a l ft

a

s

are a t the

v

V

same T. We expect the e r r o r s i n our MD values to be c

no greater than 0.03 f o r U /NkT and 0.08 f o r P/pkT.

ft

For the t o t a l d e v i a t i o n , energy, pressure and s p e c i f i c heat were given the weights 4:2:1.

Table I I I Results f o r Parameter Set 5 Before R e s c a l i n g

M „ Temperature Density (K) (mol I ) n

C

U /NkT MD

Ex£

r

P/pkT MD

Ex£

C /Nk v MD Ex£

198 228 250

10.00 10.00 10.00

-1.49 -1.70 0.48 0.35 0.37 0.70 -1.28 -1.45 0.70 0.55 0.15 0.55 -1.13 -1.25 0.77 0.66 0.12 0.39

223 248 276

18.50 18.50 18.50

-2.45 -2.59 0.92 0.80 0.32 0.39 -2.15 -2.28 1.22 1.08 0.37 0.37 -1.92 -1.99 1.45 1.25 0.52 0.40

123 151 182

25.27 25.27 25.27

-6.68 -6.84 -0.10 0.0 0.55 0.85 -5.28 -5.41 1.23 1.15 0.73 0.98 -4.27 -4.33 2.19 1.91 0.72 0.79


67

68

COMPUTER

MODELING O F MATTER

Figure 2. Rescaling the potential parameters acc and e r, keeping the ratios VCC'-VCH'-VHH and tcc'>*cH-*nHfixed.Curves 1 and 2 correspond to different state conditions. C

Table IV Results f o r Parameter Set 5 A f t e r R e s c a l i n g

c

U /NkT

r

P/pkT

C /Nk v MD Exp

Temperature (K)

Density (mol J T )

MD

Exp

MD

Exp

208 239 263

10.00 10.00 10.00

-1.49 -1.28 -1.13

-1.59 -1.34 -1.12

0.48 0.70 0.77

0.41 0.60 0.71

0.37 0.15 0.12

0.70 0.44 0.35

234 260 290

18.50 18.50 18.50

-2.45 -2.15 -1.92

-2.45 -2.15 -1.92

0.92 1.22 1.45

0.90 1.13 1.34

0.32 0.37 0.52

0.37 0.37 0.43

129 158 191

25.27 25.27 25.27

-6.68 -5.28 -4.27

-6.52 -5.15 -4.18

-0.10 1.23 2.19

0.30 1.35 2.20

0.55 0.73 0.72

0.89 0.96 0.80

1


5.

MURAD


A N D GUBBINS

O-O 100

180

260

340

T(K)

Figure 3. Configurational internal en&J from molecular dynamics (points) and experiment (lines)

eT

1

1 00

i

180

1 T(K)

i

260

1

1—

3 40

Figure 4. Pressure from molecular dynamics (points) and experiment (lines)


69

70

COMPUTER

MODELING

OF MATTER

energy. This y i e l d s a l i n e on a p l o t of a^ vs. £QC* A second, but d i f f e r e n t , s e t o f (OQQ, cc) values i s obtained by equating molecular dynamics and experimental values f o r the pressure at the same s t a t e c o n d i i t o n . T h i s y i e l d s a second l i n e on the CC * CC P l « The i n t e r s e c t i o n o f these two l i n e s gives a unique s e t o f ( QQ, values f o r the s t a t e c o n d i t i o n s t u d i e d . By t r e a t i n g the molecular dynamics data f o r the other s t a t e c o n d i t i o n s i n a s i m i l a r way i t i s p o s s i b l e to estimate a best set o f (GQQ, cc) l from the r e s u l t i n g " i n t e r v a l o f c o n f u s i o n " on a OQQ V S . €QQ p l o t . T h i s best s e t of r e s c a l e d parameters i s given i n Table I , and the r e s u l t i n g mean squared d e v i a t i o n s are given i n Table I I . Table IV and Figures 3 and 4 compare the molecular dynamics values with experiment f o r t h i s rescaled p o t e n t i a l . The molecular dynamics r e s u l t s found from parameter s e t 1 are almost the same as those found f o r the Williams p o t e n t i a l using h i s s e t V I I , and f o r the r e s c a l e d Lennard-Jone s t a n t i a l l y b e t t e r , as seen from Table I I . We p l a n to study f u r t h e r refinements of t h i s p o t e n t i a l model, i n c l u d i n g the a d d i t i o n o f an octupole moment, and we are a l s o i n the process of i n v e s t i g a t i n g other p r o p e r t i e s , i n c l u d i n g the o r i e n t a t i o n a l c o r r e l a t i o n f u n c t i o n s , s e l f - d i f f u s i o n c o e f f i c i e n t s , and time correlation functions. c

e

a

v s

e

o t

0

e

v

a

u

e

s

Acknowledgments We thank the American Gas A s s o c i a t i o n , the N a t i o n a l Science Foundation, and the Petroleum Research Fund, as administered by the American Chemical S o c i e t y , f o r support o f t h i s work. L i t e r a t u r e Cited 1.

S t r e e t t , W. B. and Gubbins, K. E., Ann. Rev. Phys. Chem., (1977), 28, 373.

2.

S t i l l i n g e r , F. H., Adv. Chem. Phys., (1975), 12, 107.

3.

McDonald, I . R. and 64, 4790.

4.

Kushick, J . and Berne, B. J . , J . Chem. Phys., (1976), 64, 1362.

5.

Evans, D. J . and Watts, R. O., Mol. Phys., (1976), 32, 93.

6.

Ryckaert, J . P. and Bellemans, 30, 123.

Klein,

M. E., J . Chem. Phys.,

(1976),

A., Chem. Phys. L e t t . ,


(1975),

5.

MURAD

A N D GUBBINS


71

7.

Evans, D. J., Mol. Phys., (1977), 34, 317; Evans, D.J.and Murad, S., ibid., (1977), 34, 327.

8.

S t r e e t t , W. B., T i l d e s l e y , D. J . and (1978), 35, 639.

Saville,

G., Mol. Phys.,

9.

S t r e e t t , W. B., T i l d e s l e y , D. J. and t h i s volume.

Saville,

G., (1978),

10.

Murad, S., Evans, D. J., Gubbins, K. E., S t r e e t t , W. B. and T i l d e s l e y , D. J., Mol. Phys., (1978), submitted.

11.

W i l l i a m s , D. E., J. Chem. Phys.,

12.

Hanley, H. J. M. and Watts, R. O., Mol. Phys., 1907; Aust. J . Phys., (1975), 28, 315.

13.

G o l d s t e i n , H., " C l a s s i c a Wesley, (1971).

14.

Gear, C. W., "Numerical Initial Value Problems i n Ordinary Differential Equations", P r e n t i c e - H a l l , Englewood Cliffs (1971).

15.

Goodwin, R. D., "The Thermodphysical P r o p e r t i e s o f Methane from 90 to 500 K at Pressures to 700 Bars", N.B.S. T e c h n i c a l Note No. 653 (National Bureau o f Standards, Washington, DC, 1974).

RECEIVED

(1967), 47, 4680. (1975), 29.,

August 15, 1978.


6 Structure of a Liquid-Vapor Interface M. RAO and B. J. BERNE Columbia University, New York, NY 10027

A l i q u i d c o e x i s t s wit critical temperature Tc. separated by a small t r a n s i t i o n region c a l l e d the i n t e r f a c e . Recently there has been much t h e o r e t i c a l e f f o r t devoted t o an understanding of p r o p e r t i e s o f the interface(1,2,3,4). I t i s a l s o p o s s i b l e now to shed some l i g h t on the s t r u c t u r e and dynamics o f the i n t e r f a c e s using computer simulation(5,6,7,8). With t h i s information one can begin t o evaluate v a r i o u s t h e o r e t i c a l approximations and provide a q u a n t i t a t i v e framework f o r the microscopic phenomenology o f the l i q u i d - v a p o r i n t e r f a c e . In this paper we present some r e s u l t s on the l i q u i d - v a p o r i n t e r f a c e obtained from a Monte Carlo s i m u l a t i o n . Our model system c o n s i s t s o f 2 0 4 8 p a r t i c l e s , i n t e r a c t i n g via a classical mechanical Lennard-Jones (6,12) p o t e n t i a l (Mr)

= V(r) - V ( r )

0

< r < r

=0

r

Q

Q

where V(r) = 4 e [ ( ^ )

1 2

6

- (^) ] and r

Q

Q

< r

= 2.5a.

The p a r t i c l e s are

placed i n a f u l l y p e r i o d i c box o f s i z e 1 4 . 7 a * 1 4 . 7 a * 2 5 . 1 a . The d e t a i l s o f c r e a t i n g an inhomogeneous system without any ex t e r n a l f i e l d i n t h i s p e r i o d i c box are presented elsewhere (5^) . The s t a r t i n g c o n f i g u r a t i o n thus generated i s a two sided f i l m with l i q u i d density n^ i n the middle o f the box and vapor d e n s i t y n^ on e i t h e r s i d e .

Using the standard Metropolis

scheme the f i l m

i s e q u i l i b r a t e d a t a temperature o f 1 1 0 K. A step s i z e o f 0 . 2 a i s used f o r the random walk which gives an acceptance r a t i o o f 0.5.

At e q u i l i b r i u m the vapor d e n s i t y has an average value o f 0 . 0 5 and the l i q u i d d e n s i t y has an average value o f 0 . 6 5 ( r e duced u n i t s are used throughout). The symmetrized d e n s i t y pro f i l e i s shown i n f i g u r e 1 with dots. The o r i g i n i s chosen t o be



1

X

o

-6.0 -5.0

1

X

o

1 -4.0

X

O

x

°

-3.0

1

X

1

X

0

-2.0

o

-1.0

1

O X

z

C)

0.2 -

O 0.4 -

0.6 -

X

0.8 •

1.0 -

1.2 -

o

X

1.0

1

o

X

0

1 2.0

X X X

o o o 1 1 3.0 4.0

X

X

o 1 5.0

Figure 1. Density and structure factor profiles of a planar sheet of Lennard-Jonesium fluid in equilibrium with its own vapor at 110 K. The origin is chosen to be the Gibbs equimolecular dividing surface. The curves are obtained from Monte Carlo simulation using one million configurations. Circles denote the density profile and the crosses denote the structure factor, r.

-7.0

1

X

°

O 0 n (z) XX s(k,z

14 -

74

COMPUTER

the Gibbs equimolecular L z

d i v i d i n g surface given

MODELING OF

MATTER

by

(n - n )

9_

= 2

g

(n

-

£

n) g

where i s the length of the box i n Z d i r e c t i o n and n i s the average d e n s i t y (=N/V). The two components of the pressure ten sor ( Z ) (the l o n g i t u d i n a l component) and P (Z) (the transverse P

N

T

component) are determined using 1 m i l l i o n c o n f i g u r a t i o n s gen erated during the random walk. The surface t e n s i o n y i s obtained from the pressure tensor L

Z/2

Y = J

[P (Z) - P ( Z ) ] d Z N

T

" Z/2 L

The value of y obtained from the s i m u l a t i o n i s 0 . 4 2 reduced u n i t s . The s u r f a c e t e n s i o n y i n curved i n t e r f a c e s i s r e l a t e d to the y i n a plane i n t e r f a c e £y Tolman(j)) Y

= Y/d

+

26/r)

c where r i s the r a d i u s of the curved i n t e r f a c e and 6 i s the cur vature dependence d i s t a n c e . The d e t a i l s o f e s t i m a t i n g 6 from computer s i m u l a t i o n are presented e l s e w h e r e ( 1 0 ) . The value obtained f o r 6 i s 1 . 0 a . In f i g u r e 1 we a l s o present the transverse s t r u c t u r e f a c t o r S (k,Z) p r e v i o u s l y s t u d i e d ( 6 ) i n a s i m i l a r system a t lower temp e r a t u r e . S (k,Z) i s defined as T

T

N(AZ)

S (k,Z) = < T

I e x p [ - i k - ( r . - r .) ] >/ i,j=l 1

where k =

0)

and

(0,

3

p a r a l l e l to the s u r f a c e , i and

j

running over the N ( A z ) atoms i n a s l i c e of volume L x L * A z centered at Z. The enhancement of S (k,Z) i n the i n t e r f a c e r e g i o n i s a t t r i b u t e d to the c a p i l l a r y waves that are thermally a c t i v a t e d . The genesis of the s i n g u l a r low-k transverse be h a v i o r has been d i s c u s s e d r e c e n t l y by Wertheim(_2) , Weeks (1) and Kalos, Percus and R a o ( 6 ) . The emerging p i c t u r e of the i n t e r f a c e from these analyses i s t h a t the t r a n s i t i o n from l i q u i d to vapor i s q u i t e abrupt i n the i n t e r f a c e — of the order of one d i a meter — but t h i s i n t e r f a c e f l u c t u a t e s markedly i n space and time. I t i s these f l u c t u a t i o n s that broaden the i n t r i n s i c den s i t y p r o f i l e which i s q u i t e sharp. T h i s broadening a l s o depends on the s i z e of the system. Such a broadening has r e c e n t l y been observed i n computer s i m u l a t i o n ( 8 ) . T


6.

RAO A N D BERNE

Structure of a Liquid-Vapor

Interface

75

I t i s now p o s s i b l e t o simulate not only plane i n t e r f a c e s but a l s o s p h e r i c a l d r o p l e t s i n e q u i l i b r i u m with vapor(11). The s t r u c t u r e and dynamics o f these i n t e r f a c e s should throw some l i g h t on the phenomenon o f n u c l e a t i o n .

Abstract

The structure of the interface of an argon like fluid in equilibrium with its own vapor at 110 K is studied using the Monte Carlo method. The geometry of the interface is chosen to be planar and both longitudinal and transverse correlations are investigated. The longitudinal density profile shows no significant structure. From the measurement of the pressure tensor, the surface tension γ and its curvature dependence distance δ are determined. The transverse correlations exhibit very long range order in the interfac behavior. Literature Cited 1.

Weeks, J. D., J. Chem. Phys. (1977) 67, 3106.

2.

Wertheim, M. S., J . Chem. Phys. (1976) 65, 2337.

3.

L o v e t t , R. A., DeHaven, P. W., Viecelli, P., J. Chem. Phys. (1973) 58, 1880.

4.

Singh, Y. and Abraham, F. F., J. Chem. Phys. (1977) 67, 537.

5.

Rao, M. and Levesque, D., J. Chem. Phys. (1976) (65, 3233.

6.

Kalos, M. H., Percus, J . K. and Rao, M., Jour. S t a t . Phys. (1977) 17, 111.

7.

Miyazaki, J . , Barker, J . A. and Pound, G. M., J. Chem. Phys. (1976) 64, 3364.

8.

Chapela, G. A., S a v i l l e , G., Thompson, G. M. and Rowlinson, J . J . , J. Chem. S o c i e t y , Faraday T r a n s a c t i o n s I I (1977) 73, 1133.

9.

Tolman, R. C., J. Chem. Phys. (1949) 17, 333.

J.

J.

and Buff, F.

10. Rao, M. and Berne, B. J. (to be p u b l i s h e d ) . 11. Rao, M., Berne, B. J. and Kalos, M. H., J. Chem. Phys. (1978) 68, 1325. RECEIVED

August 15, 1978.


7 Computer Simulation of the Liquid-Vapor Surface of Molecular Fluids S. M. THOMPSON and K. E. GUBBINS School of Chemical Engineering, Cornell University, Ithaca, NY 14853

The r e s u l t s describe program designed to stud vapor surfaces of systems composed o f polyatomic s p e c i e s . Both computer s i m u l a t i o n and p e r t u r b a t i o n theory methods are the t o o l s used to o b t a i n these r e s u l t s . Here we report molecular dynamics (MD) computer simulations o f two homonuclear diatomic l i q u i d s , each at a temperature in the r e g i o n of the triple p o i n t . The l i q u i d - v a p o r s u r f a c e of systems of monatomic molecules has been the subject o f considerable i n v e s t i g a t i o n (1-7) i n the past few years, but as f a r as we are aware, t h i s is the first work of t h i s type on molecular s p e c i e s . This work is initially d i r e c t e d towards o b t a i n i n g both e q u i l i b r i u m and non-equilibrium p r o p e r t i e s in the inhomogeneous s u r f a c e zone f o r one component systems. E q u i l i b r i u m p r o p e r t i e s i n c l u d e the d e n s i t y - o r i e n t a t i o n profile, which provides i n f o r mation on p r e f e r r e d o r i e n t a t i o n s ( i f any) i n the surface zone, and s u r f a c e tensions and energies. Non-equilibrium p r o p e r t i e s i n c l u d e t r a n s l a t i o n a l and r e - o r i e n t a t i o n a l v e l o c i t y a u t o c o r r e l a t i o n functions and t h e i r a s s o c i a t e d memory f u n c t i o n s , l e a d i n g to information on the d i f f u s i o n o f molecules both perpendicular to and p a r a l l e l to the plane o f the i n t e r f a c e . Here only e q u i l i b rium p r o p e r t i e s are presented. Future extensions i n c l u d e the study o f binary mixtures of molecules of v a r y i n g complexities and the behaviour of r e l a t i v e l y massy s u r f a c t a n t molecules i n the surface region. I.

The Simulation

Each system c o n s i s t s o f 216 molecules confined to square prisms o f dimensions x , V L ( = X ) and Z ( > X L ) with the usual p e r i o d i c boundary c o n d i t i o n s i n the (xy) plane of the s u r f a c e . The l i q u i d was confined to the centre o f the c e l l with vapor phases a t e i t h e r end. The two surfaces are s t a b l e without the i n c l u s i o n o f e x t e r n a l forces (2,_3>4_»j3) . I n i t i a l l y the center of mass i s f i x e d i n the center ( z = z / 2 ) o f the c e l l . The L

l

L

C 0 M

L


7.

THOMPSON AND

Liquid-V apor Surface of Molecular Fluids

GUBBINS

77

r

t r a n s l a t i o n a l and angular v e l o c i t i e s o f the molecules a r e r a n domly chosen subject to the c o n s t r a i n t s that the centre o f mass has no r e s u l t a n t t r a n s l a t i o n a l or angular v e l o c i t y and that the d e s i r e d k i n e t i c energy i s obtained, t h i s being d i v i d e d between t r a n s l a t i o n and r o t a t i o n a l according to the e q u i p a r t i t i o n law. Random o r i e n t a t i o n s were assigned. The molecular centers o f mass are assigned coordinates on an f . c . c . l a t t i c e s t r u c t u r e o f the appropriate l i q u i d d e n s i t y , while the 'vapor phases a r e 'empty . As the s i m u l a t i o n proceeds, the l a t t i c e s t r u c t u r e melts and a vapor phase develops. When the r e s u l t a n t density p r o f i l e has become s t a b l e , t h i s i n i t i a l e q u i l i b r a t i o n period ( t y p i c a l l y 10* i n t e g r a t i o n steps) i s r e j e c t e d and the run proper begins. A vapor molecule which attempts to e x i t the c e l l by means o f one of the end w a l l s i s returned to the c e l l by simply bouncing i t o f f the w a l l . The low vapor d e n s i t y ensures that t h i s procedure r e s u l t s i n t o t a l l y n e g l i g i b l e d i s t o r t i o n s to the surface s t r u c ture. The c e l l s were eac ( Z L = 19a), and the widt twice the p o t e n t i a l c u t - o f f d i s t a n c e . This l a t t e r quantity was 2.5 molecular diameters plus the molecular elongation (see below). Thus the c e l l widths were 5.6584 and 6.2160 molecular diameters f o r N and C l r e s p e c t i v e l y . Each f l u i d was modelled by means o f a s i t e - s i t e or atomatom p o t e n t i a l (8). Each molecule i s assumed to c o n s i s t o f two i n t e r a c t i o n centers (commonly assumed p o s i t i o n a l l y c o i n c i d e n t with the atomic n u c l e i ) . The intermolecular p o t e n t i a l i s then the sum o f four s h i f t e d Lennard-Jones (12,6) i n t e r a c t i o n s (see f i g u r e 1) 1

1

2

2

4 u(r u)ia>2) = 12

I ^JS^K^ K l

^

=

where \jS

- \j

M

M

~ LJ u

( r

c>

+

U

i j

(

r

c

)

(

r

c

"

r

)

2

and ^ ( r )

- 4e[(a/r)

x a

6

- (a/r) ]

(3)

Where r i s the p o t e n t i a l c u t - o f f d i s t a n c e , the prime denoting d i f f e r e n t i a t i o n , z i s the w e l l depth and a the molecular c o l l i s i o n diameter o f the Lennard-Jones i n t e r a c t i o n , co^ = { S ^ ^ } are the p o l a r angles s p e c i f y i n g the o r i e n t a t i o n o f molecule i , and = i - 2- The f i n a l term i n (2) ensures that the p o t e n t i a l and i t s d e r i v a t i v e a r e continuous a t the c u t - o f f d i s t a n c e , l e a d i n g to b e t t e r dynamics. The four atom-atom distances are given by: c

12


78

COMPUTER

r! - r f 2

r

2

= r

i

3

= ri

2

X

2

12

2

2r

2

2

= r

MATTER

2

+ £| + 2 r ( £ i C O S 6 i + £ cos 6 ) + 2*ifc f(wiw )

2

2

OF

2*f + 2£iri (cos 6 i - cos 0 ) - 2£if(u)i )

2

2

r

2

MODELING

2

i 2

2

2

2

2

(£ cos 9 i + £iCos 0 ) + 2£i£ f (o)ia) ) 2

2

2

2

2£| - 2 r i £ ( c o s 0 i - cos 0 ) - 2fc|f (u>i(i>) 2

2

2

2

(4) with f(u)iU) ) = cos 0 i cos 0 2

+ s i n 0i s i n 0

2

2

cos at 172.0 K po


84

COMPUTER MODELING

Table I

Averages Obtained

During

OF

MATTER

Simulations

N2_

Cl

2

40.00 x 10

3

Number of time steps

42.75

Temperature, K

66.52 + 2.01

172.0

0.608 + 0.024

0.526 ± 0.023

Surface thickness , d/a

1.79

1.14

L o c a t i o n o f Gibbs d i v i d i n g surface, z / a

4.20

4.00

Surface t e n s i o n ,

11.98

L i q u i d d e n s i t y , p*^

s

i

m

X

10

3

± 5.0

Q

JnT

x

2

10

+ 0.44

37.35 ± 0.83

+ 3

Experimental c o e x i s t i n g l i q u i d density, P* (12) L,exp —

0.664 a t 66.5 K

0.55 at 172.0 K

Experimental s u r f a c e t e n s i o n , Jm" x 1 0 (13)

11.41 at 66.5 K

39.06 at 172.0 K

2

3

z ^ 4a and z ^ 5a i s genuine s t r u c t u r e , being free o f these features. The corresponding curve f o r N i s not p l o t t e d because of poor s t a t i s t i c s f o r t h i s n e a r - s p h e r i c a l molecule. A pro nounced tendency to adopt p r e f e r r e d o r i e n t a t i o n s i s i n d i c a t e d , t h i s tendency being height dependent. In the l i q u i d phase at z - z = 0.85a (corresponding to the maximum i n p ( z ) ) the mole cules have a tendency to o r i e n t with t h e i r axes v e r t i c a l ( 9 = 0 ) , while at the Gibbs s u r f a c e ( c l o s e to the minimum i n p ( z ) ) a r e v e r s a l takes place and the molecules bend to o r i e n t with t h e i r axes p a r a l l e l to the i n t e r f a c i a l plane. This r e s u l t i s q u a l i t a t i v e l y i n agreement with the p r e d i c t i o n s o f f i r s t order p e r t u r b a t i o n theory f o r a n i s o t r o p i c overlap forces (14). A p e r t u r b a t i o n treatment using the f u l l atom-atom p o t e n t i a l i s under way. The p a i r p o t e n t i a l i s c u r r e n t l y being modified by the i n c l u s i o n of a quadrupole-quadrupole p o t e n t i a l , and the e f f e c t of s u r f a c e area on s u r f a c e thickness found i n (7) f o r monatomic molecules i s being i n v e s t i g a t e d . 2

Q

2

2

Acknowledgment It i s a pleasure to thank the N a t i o n a l Science Foundation and the Petroleum Research Fund, administered by the American Chemical S o c i e t y , f o r grants i n support of t h i s work, and S o h a i l Murad f o r a copy of h i s s i m u l a t i o n program f o r hydrogen c h l o r i d e .


7. THOMPSON AND GUBBiNs

Liquid-Vapor Surface of Molecular85Fluids

Abstract An application of the molecular dynamics method to simulate the liquid-vapor surface of molecular fluids is described. A predictor-corrector algorithm is used to solve the equations of translational and rotational motion, where the orientations of molecules are expressed in quaternions. The method is illustrat ed with simulations of 216 homonuclear (N and Cl ) diatomic molecules. Properties calculated include surface tensions and density-orientation profiles. 2

2

Literature Cited 1. Chapela, G. Α., Saville, G. and Rowlinson, J. S., Faraday Disc. Chem. Soc. (1975) 59, 22. 2. Lee, J. Κ., Barker (1974) 60, 1976. 3. Abraham, F. F., Schreiber, D. E. and Barker, J. Α., J. Chem. Phys. (1975) 62, 1958. 4. Opitz, A. C. L., Phys. Letters A (1974) 47, 439. 5. Liu, K. S., J. Chem. Phys. (1974) 60, 4226. 6. Rao, M. and Levesque, D., J. Chem. Phys. (1976) 65, 3233. 7. Chapela, G. Α., Saville, G., Thompson, S. M. and Rowlinson, J. S., Trans. Far. Soc. II (1977) 73, 1133. 8. Sweet, J. R. and Steele, W. Α., J. Chem. Phys. (1967) 47, 3029. 9. Cheung, P. S. Y. and Powles, J. G., Mol. Phys. (1975) 30, 921. 10. Singer, Κ., Taylor, A. and Singer, J. V. L., Mol. Phys. (1977) 33, 1757. 11. Evans, D. J. and Murad, S., Mol. Phys. (1977) 34, 327. 12. Vargaftik, Ν. Β., Tables on the Thermodphysical Properties of Liquids and Gases, 2nd ed., Halsted Press Div., Wiley, New York (1975). 13. Jasper, J. J., J. Phys. Chem. Ref. Data (1972) 1, 841. 14. Haile, J. M., Gubbins, Κ. E. and Gray, C. G., J. Chem. Phys. (1976) 64, 1852. RECEIVED

August 15, 1978.


8 High Field Conductivity BENSON R. SUNDHEIM Department of Chemistry, New York University, 4 Washington Place, New York, NY 10003

In the computatio transpor propertie molecular dynamics data, both direct and indirect (fluctuation-regression) means have been used. The indirect method deals with systems at equilibrium whereas the direct methods do not. Here we study the transport properties associated with electrical fields by the examination of the steady-state properties of a simulated fused salt exposed to a uniform f i e l d . The latter is large by laboratory standards in order to produce stati s t i c a l l y useful displacements. This means that the upper portion of the linear response is explored in a way that is not readily accomplished in the laboratory. Since large fields imply that there must be substantial heat dissipation, i t is necessary to "thermostat" the system so that, as potential energy is withdrawn from the electric f i e l d , kinetic energy is withdrawn from the system. By monitoring this withdrawal, an alternative determination of the conductance can be made. Several interesting by-products of this computer "experiment" are discussed below, including the rate at which kinetic energy in one degree of freedom is "thermalized" into others, the properties of the fluctuating dipole moment per unit volume and the relation between experiments in the laboratory and the computer results. A technical modification in the means of computations has led to the use of a relatively large (40 A) cell so that the relatively long wave length propagating modes can be explored. The molecular dynamics calculation of electrical conductivity in ionic fluids has been approached in several different ways. The autocorrelation function of the current may be related to the conductivity by 0-8412-0463-2/78/47-086-086$05.00/0 © 1978 American Chemical Society In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

8.

SUNDHEIM

High Field

Conductivity

Kubo type relations (1_}. By expansion i t may be separated into terms containing only the autocorrelation function of a given particle and terms contain crosscorrelation functions. The latter can be identified as being the source of deviations from the NernstEinstein equation. (_2) A second, quite novel method, is to treat the electric f i e l d as producing a small perturbation to the particle trajectories (3). The molecular dynamics calculation is carried out twice, once without the f i e l d and once with the f i e l d applied as a step function or as an impulse function. By comparison of the two sets of calculations, i t is possible to obtain an estimate of the electrical conductivity. The accuracy and, indeed, the fundamental justification fcr the method has not yet b e e n well established. Finally, there is Remembering that the motion produced b y an external electrical f i e l d is a small perturbation on the Brownian motion, it can be seen that very high fields can be applied without sensibly altering the gross properties. In computer experiments, millions of volts per centimeter can b e applied without concern for electrode processes and thermostatting can b e supplied to prevent significant temperature changes. Consequently, it is possible to examine the range of applicability of Ohm's Law and the details of the transfer of energy from the f i e l d to one component of the kinetic energy and hence to a l l three degrees of freedom and ultimately into the thermostatic bath. The conductivity can be determined both by determining the rate at which heat is extracted from the system and by determining the particle mobility in the applied f i e l d . It is this high f i e l d method which is the f i r s t subject of this communication. Computational Details For convenience, the potential energy of interaction, aside from the coulombic term, was chosen to be the same for a l l particles, being a rough approximation to that appropriate to molten KC1. Its form was that recommended by Woodcock (!5). The symmetry of the potential plus the absence of ionic polarization terms means that the results are not specific to any real salt but rather pertain only to this model. On the other hand, the main results are meant to refer to the properties associated with dense ionic fluids in general and not to unique properties associated with various anomalies in the


88

COMPUTER

MODELING

OF MATTER

potential energy of interaction. The coulombic contribution was evaluated by the Ewald method (5,J5). The direct force contribution was evaluated Tor a pair of particles by using a precalculated force/distance table based on the potential energy function. Pairs with distances (either direct or to the nearest image) greater than 6.6 A were omitted by use of a "link" system (7) as adapted to fused salt systems by Woodcock (8). The applied electrostatic f i e l d was simulated by adding a constant force term (of opposite sign for the two ionic species) to the direct force on each particle. This results in a flow of energy into the system. Compens a t o r thermostatting was applied at each time step (10" sec) by determining the ratio of the net kinetic energy to that required by the nominal temperature and dividing eac of this ratio. The Verle for the integration. Autocorrelation functions were computed by the Fourier transform method.(10). The rescaling of the velocities in order to maintain the desired rms velocity in each cartesian coordinate is an approximation to thermostating.A . better procedure would be to scale in such a way as to maintain a Boltzman distribution. The linear scale factor utilized here represents the leading term in a power series expansion in such a Boltzman rescaling. Noting that the average correction coefficient is less than 0.998 even for very high f i e l d s , i t may be readily seen that the relative deviation between the distribution achieved and a Boltzman distribution is quite small and unimportant in these studies. The rescaling plays no role in the momentum balance so that the system may be described as a constant temperature, constant potential gradient, constant momentum, constant density ensemble. (The density is constant over a spatial "graining" of the orjjer of the cell volume, here approximately 40x40x40 A, and the temperature is constant over the temporal graining of 10 sec.) Experimental measurements of electrical conductivity in real systems ignore heating effects wherever possible, or remove them by extrapolation to zero applied f i e l d . In our computations a similar extrapolation to zero applied f i e l d was made. Results We turn now to an examination of the results the computations which illustrate the method.

of


SUNDHEIM

High Field

Conductivity

The average current flow over 500 steps in systems which had previously come to a steady state was used to calculate the electrical conductivity. The results are summarized in F i g . l . A slight apparent positive slope to the least squares line i s , in fact, well within the 95% confidence level of zero slope (dashed line) and we conclude that up to 15 megavolts per cm the isothermal conductivity is constant for this model. (Further decreases in the uncertainty of the slope, obtainable by longer runs and/or more points, did not seem to be especially valuable.) The underlying reason for the constancy of the conductivity i s , of course, the fact that the external force is only a tiny fraction of the rms fluctuating force experienced by a particle. There are two other means of obtaining the elect r i c a l conductivity fro amount of heat extracted from the system at each step was recorded and divided by the square of the current. The resulting quantity is also the conductivity. Values obtained in this way are also shown in F i g . l (+'s) . Finally the power spectrum of the electrical current in a f i e l d free system was computed and the autocorrelation function derived therefrom used to obtain the electrical conductivity via the well-known relationship (1,2). The value so obtained is entered on F i g . l as an open c i r c l e . AH of these methods are, in principle, equivalent. The highest accuracy is attached to the direct measurement of the mean current. The diffusion coefficient can be obtained by the limiting slope of the graph of the mean square displacement per particle versus time. Alternatively, the velocity autocorrelation function may be utilized for the calculation (1,2).The s t a t i s t i c a l reliability of the velocity autocorrelation function is quite high for this large system since i t represents the mean of the autocorrelation function for 1728 particles over several sets of 500 steps. Comments There are two points peculiar to the computation of electrical conductivity that should be discussed here. One has to do with the mode of the electrostatic current waves and the other with dielectric shielding. Both transverse and longitudinal current oscillations occur in the molecular dynamics computations (6). The periodic boundary conditions mean that the


90

COMPUTER

MODELING

10

I10

6

O F MATTER

20

V/cm )

Figure 1. Computed conductivity as a function of applied field. The solid line represents the least squares fit to the high-field method points and the dashed line the mean value of these points. The open circle is the zero field point obtained by the current correlation function method. The crosses are the points obtained by the heat dissipation method.


SUNDHEIM

High Field

Conductivity

current will be divergenceless for wavelengths equal to or greater than the dimensions of the elementary c e l l . When the DC current is obtained by averaging over the c e l l , longitudinal currents are eliminated. (The zero k vector wave does not occur since the periodic boundary conditions imply a "toroidal" space which is also divergenceless at + infinity and hence has no planes of charge accumulation.) These calculations therefore refer to a zero k vector transverse wave, which is the quantity measured on a real salt with reversible electrodes and direct current. The dielectric properties are c l a s s i c a l l y pictured in terms of "free" and "bound" charges associated with an electrical f l u i d . In a fused salt system, charge is carried by both types of particles and, for non-polar particles which can be bound. Nevertheless, i t is possible to polarize the system of ions by inducing a charge separation. In the simplest terms, an external field biases the local distribution functions, giving rise to a reaction f i e l d which modifies the mean f l u c t uating force on each particle. We may cast this in the conventional form by the following argument. The linear expression for the current-voltage relation is (11) J =E/at) In an isotropic f l u i d *~ andX are scalars. polarization per elementary cell of edge S is

The

P= 2 z . r.-n.S I I I I where n is the (integral) number of displacements through the unit cell wall experienced by a particle which was originally in the cell centered at (0,0,0). Then fc

PP/H) = £ z . v . - Sg>n./>t) c Since ^z^v- = J, 1

the net current observed by averaging over the cell and the "mobile charge current", j may be written as j = S£"0n./»t)


92

COMPUTER

MODELING

O F MATTER

Then J = j + PP/At) Thus, pP/^t) plays the role of the "bound charge current". Of course, in the steady state J = j . By comparison, since j = «"E p -yi by setting E = D -

4 P7r

D =£E so that ,

where the angular bracket denotes a s t a t i s t i c a l average which w i l l be evaluated by the c l a s s i c a l molecular dynamics method (MD). The f i r s t c a l c u l a t i o n s o f t h i s type were c a r r i e d out f o r l i q u i d r a r e gases near the t r i p l e p o i n t (h) . Since then a v a r i e t y o f f l u i d s have been s t u d i e d i n c l u d i n g l i q u i d metals (5.), l i q u i d n i t r o g e n (6), the c l a s s i c a l one-component plasma (7.)» molten s a l t s (8) and water (9.). Because l i q u i d s are i s o t r o p i c S(g,a)) depends only on |g| whereas f o r s o l i d s t h i s i s not t r u e . A f u r t h e r d i s t i n c t i o n a r i s e s because f o r many s o l i d s under a wide v a r i e t y o f s t a t e c o n d i t i o n s the c o n s t i t u e n t p a r t i c l e s execute small amplitude v i b r a t i o n s about w e l l defined e q u i l i b r i u m p o s i t i o n s (noteworthy exceptions are p l a s t i c c r y s t a l s ) . If this s i t u a t i o n p e r t a i n s i t i s p o s s i b l e t o express r ^ ( t ) = R$+u£(t) the instantaneous p o s i t i o n o f p a r t i c l e £ i n terms o f i t s mean p o s i tion and i t s time dependent displacement uj^(t) . I f U£ i s i n some sense small with respect t o R^ one can develop F(§,t) as a power s e r i e s i n §*u. This i s the s o - c a l l e d phonon expansion (10)


96

COMPUTER

F(Q,t) = Fo + F + F . x

n t

MODELING

OF MATTER

+F + 2

where the successive terms d e s c r i b e the e l a s t i c (zero-phonon) s c a t t e r i n g one-phonon i n e l a s t i c s c a t t e r i n g , the i n t e r f e r e n c e between one and two phonon s c a t t e r i n g , and the two-phonon s c a t t e r i n g , e t c . For a c l a s s i c a l s o l i d these i n d i v i d u a l c o r r e l a t i o n f u n c t i o n s can be evaluated u s i n g the MD method as can the f u l l F(Q,t). From a t e c h n i c a l p o i n t o f view one can proceed as f o l l o w s . One can evaluate F(Q,t) f o r a given Q and then take the F o u r i e r transform. A l t e r n a t i v e l y , one can proceed v i a the F o u r i e r Laplace transform o f the d e n s i t y operator PQ(CO) u s i n g the formula T

T

N S(Q,o)) = l i m j e T

i u ) t

p (t)dt j ?

e~

i ( A ) t

' p _ ( t ' )dt ?

J^

3

~* °

= lim|p

(w)\ / 2

X

Both methods have been used i n the l i t e r a t u r e (11). Other methods i n v o l v i n g p e r t u r b a t i o n s o f t r a j e c t o r i e s have a l s o been used t o study c o l l e c t i v e modes i n s o l i d s (12). The one-phonon approximation t o the dynamical s t r u c t u r e f a c t o r S i can be w r i t t e n (13) N S!(>V5. o• o o ,a cs oo •H

< •H

>>

i» -P -P d O •H •H aJ CO - p •p

^cu cu CJ o o cs C * -p •H

OJ O ON ON ON ON 0 0 ON O CO oo — t C*— > l— VO — t — t CO o ooo o o oo o

o w Eh

LA ON L A C"-— r l CO r l O H O H 4 V O V D

ON L A 0 O L A OO ON O OJ

o ooo o

o oo o

H -=r ON oo H

H

0 0 0 0 0

O

OJ OJ

O

1-3

O

rH

LA

o a* CO

pq

L A t— ON

o

EH

o 52; w r—I H H (—i

Figure 3.

ln(r ) VS. c /(l - r , and R i 2 - c ' H s i t e - s i t e i n t e r a c t i o n s c o n t r i b u t e to secondary f o r c e s . These conventions are i l l u s t r a t e d i n F i g u r e 3. Computer Storage. Increases i n computer storage t o accom odate MTS methods are given i n u n i t s o f the number o f molecules, N, and the number o f s i t e s , N , i n the system (e.g., i n a system o f N=108 methane molecules, N =540). In every case the storage r e q u i r e d f o r the l i s t o f primary neighbors i s ^6N. For MTS methods based on the T a y l o r s e r i e s expansion, equation ( 3 ) , there i s an added requirement o f ^5N words f o r each term i n equation (3). Thus f o r a 2nd order T a y l o r s e r i e s (three terms) the storage i n c r e a s e s by ^15N . The l i n e a r e x t r a p o l a t i o n method (equation 4) r e q u i r e s only about 25N a d d i t i o n a l words o f s t o r a g e . Examples o f storage requirements f o r s e v e r a l MTS programs are given i n the t a b l e below. Computing Time. In molecular dynamics s i m u l a t i o n s most o f r

m

n

a

a

s

s

S

S


154


the computing time i s spent i n c a l c u l a t i n g i n t e r a c t i o n s between p a i r s o f molecules that l i e w i t h i n a s p e c i f i e d c u t o f f d i s t a n c e , hence computing speed v a r i e s i n v e r s e l y with the average number o f p a i r i n t e r a c t i o n s c a l c u l a t e d per time step. Let us c o n s i d e r as a base f o r comparison a conventional molecular dynamics pro gram i n which t h i s average number i s N . Using MTS methods, the s i m u l a t i o n i s c a r r i e d forward i n blocks o f n steps where, during the f i r s t one or two steps the usual N i n t e r a c t i o n s are c a l c u l a t e d , together with the d e r i v a t i v e s i n equation (3) or (4), while during the remaining steps the i n t e r a c t i o n s between a l e s s e r number o f primary neighbors, Np, are c a l c u l a t e d . N i s determined by the c u t o f f d i s t a n c e r , and Np by the d i s t a n c e r used to d i s t i n g u i s h between primary and secondary neighbors. For the t e t r a h e d r a l model d e s c r i b e d here we have used a r.. was greater than r = 2.5a: jk c Q

r

o

r

1 J

&

u

AT

( . . ,r.. ,r.. ) = 0 IJ ik' jk

i f r . . , r , or r , > r i j ' lk'

r

c

A s i m i l a r t r u n c a t i o n of t h i s p o t e n t i a l was (12). 3.

(6)

used by Barker, et a l .

Molecular Dynamics with Three-Body I n t e r a c t i o n s

3.1 Conventional Molecula i n t e r a c t i o n s are e x p l i c i t l y i n c l u d e d i n molecular dynamics simu l a t i o n s , the equations of motion to be solved take the form: 2

d r. at F, = F . — i —

2 B

2B F. =2 B

F. _

1

3B

+ F. l — i r I i*j

9

v v = - H j . *J ^ i 3 u

( r

(9)

r

i1» ik , ^± 3

l

k

, r

lk "

)

lk

(10)

3

where _F., n^, r_^ are the f o r c e , mass and p o s i t i o n v e c t o r of mass center r o r p a r t i c l e i , r e s p e c t i v e l y , and s u p e r s c r i p t s 2B and 3B denote two and three body c o n t r i b u t i o n s , r e s p e c t i v e l y . In the work reported here these equations of motion were solved using a p r e d i c t o r - c o r r e c t o r scheme due to Gear (27). D e t a i l s of a p p l i c a t i o n of t h i s method to molecular dynamics s i m u l a t i o n s are given elsewhere (28, 29). The geometry of the system was taken to be a cube of s i d e L and p e r i o d i c boundary c o n d i t i o n s were used with the u s u a l minimum image c r i t e r i o n (30). In molecular dynamics with only p a i r i n t e r a c t i o n s , the speed of program execution i s l i m i t e d by e v a l u a t i o n of the f o r c e on each p a r t i c l e , eq. (9). Since simulations u s u a l l y u t i l i z e p a i r p o t e n t i a l s truncated at some p a i r s e p a r a t i o n r r > r i j * ik' jk L c # 1

T

(11)

and the t r u n c a t i o n o f eq. (6) a p p l i e s . T h i s a l g o r i t h m i n which a neighbor l i s t i s used f o r both two and three body f o r c e e v a l u a t i o n and i n which three body f o r c e s are c a l c u l a t e d e x p l i c i t l y at each time step s h a l l be r e f e r r e d to as the c o n v e n t i o n a l mole c u l a r dynamics (CMD) method. U n f o r t u n a t e l y , even with the neighbor l i s t a p p l i e d to the three body f o r c e s , the CMD program f o r 108 Lennard-Jones plus A x i l r o d - T e l l e r p a r t i c l e s executes about 15 times slower than the same program f o r 108 Lennard-Jones p a r t i c l e s . In the work r e ported here, source programs were w r i t t e n i n FORTRAN IV u s i n g s i n g l e p r e c i s i o n a r i t h m e t i c , compiled on an IBM FORTRAN IV-G compiler, and executed on the IBM 370/165 a t Clemson U n i v e r s i t y . 3.2 M u l t i p l e Time Step (MTS) Method A p p l i e d to Three Body Forces. In order to improve the execution speed of simula tion programs with three body i n t e r a c t i o n s i n c l u d e d , the mul t i p l e time step method of S t r e e t t , e t a l . (5) has been a p p l i e d to the e v a l u a t i o n of the three body f o r c e s . The m u l t i p l e time step method attempts to take advantage of the f a c t that molecular motions i n a f l u i d may be reduced to components which operate on very d i f f e r e n t time s c a l e s . More p r e c i s e l y , one can o f t e n i d e n t i f y components o f the f o r c e on a molecule which have r e l a t i v e l y l a r g e d i f f e r e n c e s i n t h e i r r a t e s of change with time. I t i s the q u i c k l y v a r y i n g component of the f o r c e which l i m i t s the s i z e of the time step At which must be used to o b t a i n s t a b l e s o l u t i o n s


180

COMPUTER

MODELING OF MATTER

to the d i f f e r e n t i a l equations of motion. In the MTS method these q u i c k l y v a r y i n g components of the f o r c e are e x p l i c i t l y evaluated at each time step i n the u s u a l way; however, the slowly v a r y i n g components are only e x p l i c i t l y evaluated at longer i n t e r v a l s , say, every n time steps. At time steps between e x p l i c i t determination, the slowly v a r y i n g components of the f o r c e are estimated by e x t r a p o l a t i o n from the previous e x p l i c i t e v a l u ation. The e x t r a p o l a t i o n may be l i n e a r (LMTS method) or, i f one i s more ambitious, based on a T a y l o r s e r i e s expansion. The degree to which the MTS method improves execution speed of the program depends on the amount of computation avoided i n e x t r a p o l a t i n g the slowly v a r y i n g f o r c e components r a t h e r than e x p l i c i t l y c a l c u l a t i n g them. I f we consider a p p l y i n g t h i s MTS method to s i m u l a t i o n s with three body i n t e r a c t i o n s , eq. (8) already represents the f o r c e on molecule i d i v i d e d i n t o two p a r t s Barker and Henderson have noted that the three bod v a r y i n g (21). To t e s s i m u l a t i o n f o r 108 p a r t i c l e s i n t e r a c t i n g with Lennard-Jones plus A x i l r o d - T e l l e r p o t e n t i a l s u s i n g the conventional molecular dyna mics algorithm. Figure 3 shows a t y p i c a l comparison of a compo nent of the two and three body f o r c e s on one p a r t i c l e during a segment of the s i m u l a t i o n . The f i g u r e i n d i c a t e s that the three body component of the f o r c e does indeed have a slower r a t e of change than the two body f o r c e component. Encouraged by F i g u r e 3, the MTS method was then a p p l i e d to the s i m u l a t i o n i n the f o l l o w i n g manner. The e n t i r e three b o d y ^ f o r c e i s considered to be slowly v a r y i n g , t h e r e f o r e a l l of _F. i s e x p l i c i t l y evaluated only at p e r i o d i c i n t e r v a l s i n ^ t h e simu lation. We choose to use l i n e a r e x t r a p o l a t i o n of F. for, although S t r e e t t , et a l . f i n d that a t h i r d order T a y l o r s e r i e s e x t r a p o l a t i o n i s more accurate, the a n a l y t i c e v a l u a t i o n s of the time d e r i v a t i v e s o^ the A x i l r o d - T e l l e r f o r c e are h o p e l e s s l y tedious. Thus, F\ i s e x p l i c i t l y evaluated at two s u c c e s s i v e times, t and (t + A t ) , and then l i n e a r l y e x t r a p o l a t e d over the next n time steps. T h i s gives the three body f o r c e on p a r t i c l e i at any intermediate time step ( t + k At) as: fi

Q

Q

F. — i

3 B

(t + k At) = F . o —1

3 B

3 B

( t ) + k [ F . ( t + At) - F . o — i o —x

3 B

( t )] o

(12)

In a s i m u l a t i o n program using a p r e d i c t o r - c o r r e c t o r a l g o rithm, eq. (12) would appear i n the f o r c e e v a l u a t i o n step. The only a d d i t i o n a l storage r e q u i r e d f o r the MTS method over that f o r the CMD method i s ^ s p a c e f o r 6N numbers - 3N l o c a t i o n s f o r the components of _F. ( t ^ ) * 3N l o c a t i o n s f o r the components of the d i f f e r e n c e term i n brackets i n eq. (12). We experimented with v a r i o u s values of the time step At and number of e x t r a p o l a t i o n steps n. A compromise among s t a b l e s o l u t i o n s of the d i f f e r e n t i a l equations, conservation of system energy, and execution speed was obtained using n = 8 and a n c


15.

HAILE

MD Simulations of Fluids with Three-Body Interactions

181

At = 2.15 (10 ) s e c . T h i s value of the time step i s about onef i f t h that used by V e r l e t (31) and Rahman (34) i n s i m u l a t i o n s of the p a i r w i s e a d d i t i v e Lennard-Jones f l u i d . Note t h a t , i n the MTS method, any property which i s being c a l c u l a t e d as a running ensemble average a t each time step during the s i m u l a t i o n must a l s o be d i v i d e d i n t o slowly (three body) and q u i c k l y (two body) v a r y i n g c o n t r i b u t i o n s . The MTS method i s a p p l i e d to the slowly v a r y i n g components of the p r o p e r t i e s v i a equations analogous to eq. (12). T h i s comment a p p l i e s , f o r example, to the i n t e r n a l energy and pressure i n the work reported here. Two p a r a l l e l s i m u l a t i o n s f o r 108 p a r t i c l e s i n t e r a c t i n g with Lennard-Jones p l u s A x i l r o d - T e l l e r p o t e n t i a l s have been performed. The f i r s t c a l c u l a t i o n u t i l i z e d the CMD method i n which the f o r c e s were e x p l i c i t l y evaluated a t each time step. In the second run the two body f o r c e s were determined i n the standard way and the LMTS metho body f o r c e s . Both run t i c l e p o s i t i o n s and v e l o c i t i e s and both were continued f o r 1650 time steps. A comparison of the p r o p e r t i e s obtained from the two c a l c u l a t i o n s i s given i n Table I . In a d d i t i o n to the pro p e r t i e s l i s t e d i n Table I , r a d i a l d i s t r i b u t i o n f u n c t i o n s , v e l o c i t y , speed, and f o r c e a u t o c o r r e l a t i o n f u n c t i o n s , and atomic mean squared displacements (from which d i f f u s i o n c o e f f i c i e n t s were obtained) were c a l c u l a t e d . For a l l of these p r o p e r t i e s , the LMTS values were w i t h i n 0.1% of the values obtained by the CMD method. Figure 4 shows the per cent d e v i a t i o n i n the i n s t a n taneous t o t a l energy o f the two c a l c u l a t i o n s . Since, i n the LMTS method d e s c r i b e d above, the three body f o r c e s a r e e x p l i c i t l y evaluated only two of every ten time s t e p s , we might expect the LMTS method to be about 5 times as f a s t as the CMD method. The r e s u l t s i n Table I confirm t h i s ; i . e . , u s i n g the LMTS method, a s i m u l a t i o n of 108 Lennard-Jones p l u s A x i l r o d - T e l l e r p a r t i c l e s r e q u i r e s about 20 CPU minutes f o r 2000 time steps on an IBM 370/165. T h i s execution speed remains about 2.5 times slower than a CMD method, p a i r w i s e a d d i t i v e LennardJones s i m u l a t i o n . The r e s u l t s from the LMTS method and the CMD method a r e i n good agreement f o r those p r o p e r t i e s which a r e averaged over the s i m u l a t i o n . However, a f t e r about 1000 time steps, the i n s t a n taneous components of the three body f o r c e s on i n d i v i d u a l par t i c l e s i n the LMTS c a l c u l a t i o n begin to d e v i a t e from those i n the CMD method, as i n d i c a t e d i n Table I . These d e v i a t i o n s a r e due to gradual accumulation o f i n a c c u r a c i e s from the l i n e a r e x t r a p o l a t i o n of the three body f o r c e . I t should be emphasized that while these d e v i a t i o n s occur, the t o t a l energy and momentum of the LMTS system remain c o n s e r v a t i v e and agree c l o s e l y with those of the CMD method (see F i g u r e 4 ) . This disagreement i n the three body f o r c e s can be i n t e r p r e t e d as a d r i f t i n g o f the



182

Figure 3. Comparison of z-component of two-(2B) and three-body (3B) forces on particle #27 in conventional molecular dynamics simulation of Lennard-Jones plus Axilrod-Teller fluid. At = time step = 2.15 (10' ) sec, a = 0.65, kTV = 1.033. Note that the scale for three-body force is an order of magnitude smaller than that for the two-body force. 15

P

1

3

%

-0.1 L

1 1400

1

1 1600

I

I 1800

I

1 2000

At

Figure 4. Percent deviation in instantaneous total system energy calculated by MTS method from that determined by CMD method for 108 particles interacting with Lennard-Jones plus Axilrod-Teller forces. System parameters same as those in Figure 3. Percent deviation calculated as: (E MD — EUTS)100/E UD' C

C


15. HAILE


183

Table I Test of the L i n e a r M u l t i p l e Time Step Method A p p l i e d to the A x i l r o d - T e l l e r P o t e n t i a l i n Molecular Dynamics Simulations of Lennard-Jones p l u s A x i l r o d - T e l l e r I n t e r a c t i o n s . CMD Method

Number P a r t i c l e s , N

108

LMTS Method

108

3 0.65

0.65

2.15

2.15

1.0325

1.0326

Number d e n s i t y , pa Time step, At (10"^)se CPU time, mins. Temperature,

kTe ^ -4.356

-4.356

0.104

0.103

-2.795

-2.797

3B 3 -1 a e

-0.363

-0.373

3B 3 -1 a e

0.125

0.120

3B 3 -1 ae

0.0465

0.0519

xa

1.457

1.462

v e c t o r f o r p a r t i c l e 27

ya

4.981

4.987

at time step 1650

za

1.358

1.364

Average Conf. I n t . Energy, U(Ne) Average Pressure, P(pkT) ^ Instantaneous T o t a l Energy E(Ne) at time step 1650 Instantaneous Components of 3-body Force

p

on p a r t i c l e 27 a t time

r

F

x

F y

step 1650

Components of p o s i t i o n

r

F

z



P(pkT)

-1

-1

-1

U(Nc)

kTe

Number Time Steps

Time Step ( 1 0 ) sec

1 5

Number P a r t i c l e s

3

= 0.65

-0.11

-4.52

1.036

1200

9.6

864

-0.05

-4.54

1.052

2000

2.15

108

Verlet This ( r e f . (31)) Work

Lennard-Jones

pa

+0.10

-4.36

1.033

1650

2.15

108

MTS Method

L J + AT

pa

-0.20

-5.90

+0.38

-5.64

0.746

1730

2.15

108

MTS Method

LJ + AT

= 0.817

0.746

2000

2.15

108

This Work

Lennard-Jones

P r e l i m i n a r y R e s u l t s Showing E f f e c t of Three Body A x i l r o d - T e l l e r Force on I n t e r n a l Energy and Pressure

Table I I

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185

LMTS system onto a d i f f e r e n t t r a j e c t o r y i n phase space from that followed by the system evolved by the CMD method. Hence, there i s no s t r i c t l y Newtonian connection between widely separated phase p o i n t s ; but then, there i s no true Newtonian connection between widely separated phase p o i n t s i n the system generated by the CMD method due to round o f f e r r o r s and the truncated poten tial. The c o n c l u s i o n i s that the phase space t r a j e c t o r i e s become d i f f e r e n t i n the two c a l c u l a t i o n s , but one i s not n e c e s s a r i l y more wrong (or r i g h t ) than the other. Hoover and Ashurst have discussed t h i s p o i n t i n another context with s i m i l a r c o n c l u s i o n s (35). The r e s t r i c t i o n which t h i s imposes i s that the LMTS method cannot be used to study very l o n g - l i v e d phenomena; such as the long t a i l of the v e l o c i t y a u t o c o r r e l a t i o n f u n c t i o n . 4.

Preliminary

Results

for Equilibrium

Properties

The l i n e a r m u l t i p l has been used to simulat Jones {jlus A x i l r o d - T e l l e r p o t e n t i a l s at two s t a t e c o n d i t i o n s : (a) p a =0.65, kTe~ = 1.036, which i s one of the c o n d i t i o n s of the Lennard-Jones f l u i d s t u d i e d by V e r l e t (31) and by Singh, et a l . (14, 15), (b) p a = 0.817, kTe = 0.746, which i s c l o s e to the c o n d i t i o n of the Lennard-Jones f l u i d s t u d i e d by Rahman (34). From these two c a l c u l a t i o n s the c o n f i g u r a t i o n a l i n t e r n a l energy, pressure, and r a d i a l d i s t r i b u t i o n f u n c t i o n have been determined. For comparison, s i m u l a t i o n runs f o r a p u r e l y Lennard-Jones f l u i d have a l s o been performed at s t a t e c o n d i t i o n s c l o s e to the above. E q u i l i b r i u m property r e s u l t s at both s t a t e c o n d i t i o n s f o r the Lennard-Jones and Lennard-Jones plus A x i l r o d - T e l l e r f l u i d s are compared i n Table I I . Long range c o r r e c t i o n s f o r the truncated two body p o t e n t i a l have been added to the i n t e r n a l energy and pressure; however, long range c o r r e c t i o n s f o r the truncated three body p o t e n t i a l have been neglected i n the v a l u e s given i n Table I I . These r e s u l t s confirm the e a r l i e r work by Barker, et a l . i n that the three body e f f e c t i s only a few percent on the i n t e r n a l energy, but i s s i g n i f i c a n t l y l a r g e r on the pressure. Note t h a t , at the s t a t e c o n d i t i o n s s t u d i e d , the net e f f e c t of the A x i l r o d - T e l l e r i n t e r a c t i o n i s r e p u l s i v e , as evidenced by p o s i t i v e c o n t r i b u t i o n s to the energy and pressure. In Figure 5 the r a d i a l d i s t r i b u t i o n f u n c t i o n s f o r the LennardJones and Lennard-Jones plus A x i l r o d - T e l l e r f l u i d s are compared at the high d e n s i t y s t a t e c o n d i t i o n . I t appears t h a t the three body i n t e r a c t i o n s have no e f f e c t except i n the f i r s t peak r e g i o n where the t r i p l e t i n t e r a c t i o n s lower g(r) by about 3%. However, we estimate the s t a t i s t i c a l e r r o r i n g(r) f o r these s i m u l a t i o n s i s about +3%, so i t i s unclear to what extent the observed effect i s real. We note that Schommers has reported s m a l l three body e f f e c t s on the d i s t r i b u t i o n f u n c t i o n f o r the two dimensional f l u i d , as w e l l (16). However, the p e r t u r b a t i o n theory c a l c u l a 3



Figure 5. Effect of Axilrod-Teller potential on radial distribution function at pa = 0.817. For Lennard-Jones fluid, kTY = 0.746; for Lennard-Jones plus Axilrod-Teller fluid, kTY = 0.740. 3

1

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Figure 6. Comparison of trajectories of particle #27 in Lennard-Jones and Lennard-Jones plus Axilrod-Teller fluids at pa = 0.65. Both runs were started from the same point in phase space and trajectories shown are from time steps 5002000 in each simulation. For this calculation the Axilrod-Teller strength constant v was assigned a value of three times that for argon given in Section 2. Note that the circles represent positions of the center of mass of the atom, not the atomic diameter. 3


187

188

COMPUTER

MODELING OF MATTER

1 1

t i o n s of Singh and Ram show " s i g n i f i c a n t e f f e c t of the t r i p l e t i n t e r a c t i o n s on g(r) at low temperatures and high d e n s i t i e s (14). The g(r) f o r the low d e n s i t y c a l c u l a t i o n shows e s s e n t i a l l y no e f f e c t due to the three body f o r c e s . Singh and Ram a l s o f i n d l i t t l e e f f e c t at low d e n s i t i e s . Although some dynamic p r o p e r t i e s have been c a l c u l a t e d from these s i m u l a t i o n s , we f i n d , i n g e n e r a l , that 2000 time steps f o r 108 p a r t i c l e s do not provide s u f f i c i e n t data to o b t a i n s t a t i s t i c a l l y meaningful dynamic p r o p e r t i e s . Work i s now i n pro gress aimed at i n c r e a s i n g the number of p a r t i c l e s i n the system and the l e n g t h of the run to improve the s t a t i s t i c a l p r e c i s i o n of dynamic property c a l c u l a t i o n s . In an e f f o r t to g a i n f u r t h e r enlightenment as to the e f f e c t s of the A x i l r o d - T e l l e r p o t e n t i a l , we have compared i n d i v i d u a l p a r t i c l e t r a j e c t o r i e s f o r the same p a r t i c l e over the same time segment from two d i f f e r e n t s i m u l a t i o n s . In one s i m u l a t i o n the intermolecular p o t e n t i a the p o t e n t i a l was Lennard-Jone c u l a t i o n s were done at the same d e n s i t y and were s t a r t e d from the same c o n f i g u r a t i o n and p a r t i c l e v e l o c i t i e s . F i g u r e 6 com pares the t r a j e c t o r i e s obtained from such a study. In the Lennard-Jones f l u i d the p a r t i c l e shown i n F i g u r e 6 i s e x h i b i t i n g v i b r a t o r y motion due to c o l l i s i o n s with neighboring p a r t i c l e s . In the Lennard-Jones plus A x i l r o d - T e l l e r f l u i d , the same p a r t i c l e during the same time segment i s undergoing an extended t r a j e c t o r y of l a r g e l y d i f f u s i o n a l motion. I t may be that study of three body f o r c e s on dynamic p r o p e r t i e s w i l l r e v e a l f a i r l y strong e f f e c t s on i n d i v i d u a l p a r t i c l e motion but that these e f f e c t s tend to cancel when averaged to o b t a i n bulk dynamic p r o p e r t i e s . Hence, Schommers two dimensional s i m u l a t i o n s show strong three body e f f e c t s on the v e l o c i t y a u t o c o r r e l a t i o n f u n c t i o n but i n s i g n i f i c a n t e f f e c t on the s e l f d i f f u s i o n c o e f f i c i e n t (16). 1

5.

Conclusions

A m u l t i p l e time step method has been s u c c e s s f u l l y developed f o r performing molecular dynamics simulations i n which three body i n t e r a c t i o n s are e x p l i c i t l y i n c l u d e d . The method has been t e s t e d f o r 108 p a r t i c l e s i n t e r a c t i n g with a Lennard-Jones plus A x i l r o d - T e l l e r p o t e n t i a l . When compared with a conventional molecular dynamics program u s i n g the same p o t e n t i a l model, the MTS program executes - 4 . 5 times f a s t e r while g i v i n g e s s e n t i a l l y the same values f o r e q u i l i b r i u m and dynamic p r o p e r t i e s , f o r runs of up to 2000 time steps. At t h i s p o i n t , the MTS method i s of d o u b t f u l r e l i a b i l i t y f o r studying l o n g - l i v e d phenomena. Based on t h i s study, i t seems that the m u l t i p l e time step methods could provide a mechanism f o r extending molecular dynamics simu l a t i o n s to a v a r i e t y of phenomena which i n v o l v e components that i n h e r e n t l y operate on d i f f e r e n t time s c a l e s ; f o r example, study of phase t r a n s i t i o n s or long-chain molecules.


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In p r e l i m i n a r y s t u d i e s of three body e f f e c t s w i t h the MTS program, we have confirmed the work of Barker, et_ a l . (11, 12) that the A x i l r o d - T e l l e r c o n t r i b u t i o n i s only a few per cent f o r the i n t e r n a l energy but s i g n i f i c a n t l y l a r g e r f o r the p r e s s u r e . The e f f e c t of the A x i l r o d - T e l l e r i n t e r a c t i o n on the r a d i a l d i s t i r b u t i o n f u n c t i o n i s found to be s m a l l . This r e s u l t i s i n agreement w i t h Schommers' two dimensional s i m u l a t i o n s (16); but disagrees w i t h the t h e o r e t i c a l c a l c u l a t i o n s of Singh, et_ a l . at high d e n s i t y (14, 15). F i n a l l y , we note that the work reported here begins a study of long range, three body i n t e r a c t i o n s but does not i n c l u d e p o s s i b l e short range, three body e f f e c t s . P. A. E g e l s t a f f i s studying such e f f e c t s u s i n g neutron s c a t t e r i n g and Monte C a r l o s i m u l a t i o n (36). Acknowledgments The author i s g r a t e f u f o r v a l u a b l e d i s c u s s i o n s on the m u l t i p l e time step method; H. W. Graben f o r d i s c u s s i o n s on three body f o r c e s ; P. A. E g e l s t a f f and J . A. Barker f o r i n s t r u c t i v e correspondence. T h i s work was supported, i n p a r t , by a grant from the F a c u l t y Research Committee, Clemson U n i v e r s i t y . The Clemson U n i v e r s i t y Computer Center generously provided the computer time used i n t h i s work. Literature Cited 1. Hansen, J . P. and McDonald, I . R., "Theory of Simple L i q u i d s , " Academic P r e s s , New York, 1976. 2. Watts, R. 0. and McGee, I. J., " L i q u i d State Chemical P h y s i c s , " J . Wiley and Sons, New York, 1976. 3. M e t r o p o l i s , M., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. N., and T e l l e r , E., J. Chem. Phys. (1953) 21, 1087. 4. A l d e r , B. J . and Wainwright, T. E., J. Chem. Phys. (1957) 27, 1208. 5. S t r e e t t , W. B., T i l d e s l e y , D. J., and Saville, G., Molec. Phys. (1978) in p r e s s . 6. Copeland, D. A. and Kestner,N.R.,J.Chem. Phys. (1968) 49, 5214. 7. Casanova, G., D u l l a , R. J., Jonah, D. A., Rowlinson, J . S., and S a v i l l e , G., Molec. Phys. (1970) 18, 589. 8. D u l l a , R. J., Rowlinson, J. S., and Smith, W. R., Molec. Phys. (1971) 21, 299. 9. Sherwood, A. E. and P r a u s n i t z , J .M.,J.Chem. Phys. (1964) 41, 413. 10. Fowler, R. and Graben, H. W., J. Chem.Phys. (1972) 56, 1917. 11. Barker, J . A., F i s h e r , R. A., and Watts, R. O., Molec.Phys. (1971) 21, 657.


190


12.

Barker, J . A., Henderson, D., and Smith, W. R., Molec. Phys. (1969) 17, 579. 13. Y a r n e l l , J . L., Katz, M. J . , Wenzel, R. G., and Koenig, S. H., Phys. Rev. A (1973) 7, 2130. 14. Ram, J . and Singh, Y., J. Chem. Phys. (1977) 66, 924. 15. Sinha, S. K., Ram, J . , and Singh, Y., J . Chem. Phys. (1977) 66, 5013. 16. Schommers, W., Phys. Rev. A (1977) 16, 327. 17. B e l l , R. J., J. Phys. B (1970) 3, 751. 18. A x i l r o d , B. M. and Teller, E., J. Chem. Phys.(1943) 11, 299. 19. Muto, Y., Proc. Phys. Math. Soc. Japan (1943) 17, 629. 20. Jansen, L. and Lombardi, E., Faraday D i s c . Chem. Soc. (London) (1965) 40, 78. 21. Barker, J . A. and Henderson, D., Rev. Mod. Phys. (1976) 48, 587. 22. V e r l e t , L., Phys. Rev. (1968) 165, 201. 23. F i s h e r , R. A. and 25, 529. 24. Singh, Y., Molec. Phys. (1975) 29, 155. 25. Shukla, K. P., Ram, J . , and Singh, Y., Molec. Phys. (1976) 31, 873. 26. Stogryn, D. E., J . Chem. Phys. (1970) 52, 3671. 27. Gear, C. W., "Numerical Initial Value Problems in Ordinary Differential Equations," Prentice-Hall, Englewood Cliffs, New Jersey, 1971. 28. Cheung, P. S. Y. and Powles, J . G., Molec. Phys. (1974) 30, 921. 29. H a i l e , J . M., "Surface Tension and Computer Simulation of Polyatomic F l u i d s , " Ph.D. D i s s e r t a t i o n , U n i v e r s i t y of F l o r i d a , G a i n e s v i l l e , 1976. 30. Wood, W. W., i n "Physics o f Simple L i q u i d s , " H. N. V. Temperley, J . S. Rowlinson, and G. S. Rushbrooke, eds, North-Holland, Amsterdam, 1968. 31. V e r l e t , L., Phys. Rev. (1967) 159, 98. 32. S c h o f i e l d , P., Comput. Phys. Comm. (1973) 5, 17. 33. Quentrec, B. and Brot, C., J. Comput. Phys. (1973) 13, 430. 34. Rahman, A., Phys. Rev. (1964) 136, 405. 35. Hoover, W. G. and Ashurst, W. T. in " T h e o r e t i c a l Chemistry," vol. 1, H. E y r i n g and D. Henderson, eds., Academic Press, New York, 1975. 36. E g e l s t a f f , P. A., p r i v a t e communication (1978). RECEIVED September 7, 1978.


16 Monte Carlo Studies of the Structure of Liquid Water and Dilute Aqueous Solutions DAVID L. BEVERIDGE, MIHALY MEZEI, S. SWAMINATHAN, and S. W. HARRISON Chemistry Department, Hunter College of the City University of New York, 695 Park Avenue, New York, NY 10021 The advent o f t h i r gether with recent advance molecular quantum mechanics and statistical mechanics combine to make the s t r u c t u r e of molecular l i q u i d s a c c e s s i b l e to t h e o r e t i c a l study at new l e v e l s of r i g o r v i a computer s i m u l a t i o n . There is p r e s e n t l y broad based research a c t i v i t y in this area from q u i t e d i v e r s e points of view in physics and chemistry. A s e r i e s of Monte Carlo computer s i m u l a t i o n s t u d i e s of the s t r u c t u r e and p r o p e r t i e s of molecular l i q u i d s and s o l u t i o n s have r e c e n t l y been c a r r i e d out in t h i s Laboratory. The c a l c u l a t i o n s employ the canonical ensemble Monte C a r l o - M e t r o p o l i s method based on a n a l y t i c a l pairwise p o t e n t i a l functions r e p r e s e n t a t i v e of ab initio quantum mechanical c a l c u l a t i o n s o f the i n t e r m o l e c u l a r i n t e r a c t i o n s . A number of thermodynamic p r o p e r t i e s i n c l u d i n g in ternal energies and radial d i s t r i b u t i o n functions were determined and are reported h e r e i n . The r e s u l t s are analyzed f o r the s t r u c ture o f the statistical s t a t e o f the systems by means o f q u a s i component d i s t r i b u t i o n f u n c t i o n s f o r c o o r d i n a t i o n number and binding energy. S i g n i f i c a n t molecular s t r u c t u r e s c o n t r i b u t i n g to the statistical s t a t e o f each system are identified and d i s p l a y e d i n stereographic form. This a r t i c l e reviews the main r e s u l t s of our most recent work and deals s p e c i f i c a l l y with l i q u i d water, the d i l u t e aqueous s o l u t i o n of methane, and d i l u t e aqueous s o l u t i o n s o f monatomic cations and anions. The background f o r these s t u d i e s i s surveyed i n Section I , followed by general considerations on the method ology and computational parameters. Sections III-V c o l l e c t the i n d i v i d u a l r e s u l t s system by system, followed i n Section VI by a general d i s c u s s i o n and conclusions. 1-4

I.

Background The motional degrees o f freedom a v a i l a b l e to molecules i n l i q u i d s and solutions mandate t h e o r e t i c a l s t u d i e s o f these systems to be problems i n s t a t i s t i c a l mechanics and dynamics. The most fundamental approach to problems i n t h i s area i s t o t r e a t each


192


system as simply an assembly of molecules i n t e r a c t i n g under a c o n f i g u r a t i o n a l p o t e n t i a l , and t o attempt to solve the c o r r e sponding many-body problem i n c l a s s i c a l s t a t i s t i c a l mechanics or k i n e t i c theory. When the intermolecular i n t e r a c t i o n s are r e l a t i v e l y strong, as i n most molecular l i q u i d s , general s o l u t i o n s cannot be r e a d i l y developed i n a n a l y t i c a l form. However, recent s t u d i e s show i t i s p o s s i b l e t o approach these problems by numeri c a l methods using large s c a l e d i g i t a l computers. In s t a t i s t i c a l mechanics, the system can be t r e a t e d by c o n f i g u r a t i o n a l averaging u s i n g the Monte C a r l o - M e t r o p o l i s method, whereas i n the k i n e t i c approach, molecular dynamics, the i n d i v i d u a l molecular t r a j e c t o r i e s are c a l c u l a t e d by simultaneous s o l u t i o n of the Newton-Euler equations. The Monte C a r l o and molecular dynamic methods are c o l l e c t i v e l y r e f e r r e d t o as "computer s i m u l a t i o n s " . In p r i n c i p l e , the computer s i m u l a t i o n methods are able t o accommodate a l l the thermodynamic and r e l a t e d p r o p e r t i e s of an e q u i l i b r i u m s y s tem, and molecular dynamic systems as w e l l . The s i m u l a t i o b i l i t y of c a r r y i n g out computer experiments on the system to e l u c i d a t e the e f f e c t s of v a r i o u s d e f i n a b l e c h a r a c t e r i s t i c s on the results. There remain s i g n i f i c a n t assumptions i n computer simu l a t i o n , mainly regarding the proper form of the conf i g u r a t i o n a l p o t e n t i a l , and i n d i v i d u a l c a l c u l a t i o n s themselves are r e l a t i v e l y lengthy undertakings i n terms of computer time. However, the i n i t i a l r e s u l t s on molecular l i q u i d s and s o l u t i o n s are encourag ing and are p r o v i d i n g new i n s i g h t s i n t o the s t r u c t u r a l chemistry of these systems. A l a r g e body of the computer simulation work has been r e ported on model systems such as hard d i s c s , spheres or LennardJones p a r t i c l e s . Here the i n t e r p a r t i c l e p o t e n t i a l i s known and can be used t o r a p i d l y c a l c u l a t e the c o n f i g u r a t i o n a l energy of the system as required f o r Monte C a r l o s t u d i e s or the c o n f i g u r a t i o n a l f o r c e on a p a r t i c l e as required f o r molecular dynamics. A great d e a l of systematic i n f o r m a t i o n has been developed from model systems which can be q u a l i t a t i v e l y a p p l i c a b l e t o r e a l sys tems. A s e r i e s of d e f i n i t i v e reviews of the Monte C a r l o method and r e s u l t s on model systems have been prepared by Wood-*. The dynamics approach was i n i t i a l l y c h a r a c t e r i z e d i n the s e r i e s of papers by A l d e r , Wainwright and coworkers^. A comprehensive r e view of l i q u i d s t a t e theory was r e c e n t l y published by Barker and Henderson . The a p p l i c a t i o n of computer s i m u l a t i o n methods to molecular systems i s i n a r e l a t i v e l y e a r l y stage. The Monte C a r l o s t u d i e s on water by Barker and Watts (1969) and Sarkisov e t . a l . (1974) and the molecular dynamics study of water by Rahman and S t i l l i n ger (1971) ^ were the forerunners of computer s i m u l a t i o n work on chemical problems. Progress has g e n e r a l l y been slow i n t h i s area due t o the magnitude of computer f a c i l i t i e s required and the l i m i t e d a v a i l a b i l i t y of p o t e n t i a l f u n c t i o n s f o r d i v e r s e chemical ap plications. Quite r e c e n t l y s e v e r a l computer s i m u l a t i o n s t u d i e s 7

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on l i q u i d water have appeared, due to L a d d , Owicki and Scheraga,12 d Swaminathan and Beveridge.^ Clementi and coworkers have reported extensive r e s u l t s on w a t e r ^ and ion-water sys t e m s ^ and have pioneered the use of intermolecular p o t e n t i a l functions derived from ab i n i t i o quantum mechanical c a l c u l a t i o n s . A systematic approach to the determination of a n a l y t i c a l poten t i a l f u n c t i o n s was contributed from t h i s Laboratory.15 The dy namics of ion-water i n t e r a c t i o n s f o r small c l u s t e r s has r e c e n t l y been described, and computer simulations have j u s t been reported on l i q u i d benzene, ^ l i q u i d nitrogen*? and l i q u i d ammonia. The d i l u t e aqueous s o l u t i o n of methane has been treated by Owicki and S c h e r a g a ^ and Swaminathan, Harrison and Beveridge.^ A d d i t i o n a l current a p p l i c a t i o n s to molecular l i q u i d s are i n the newly pub l i s h e d monograph by WattsÔ d f course i n the companion papers i n t h i s volume. a

n

a

n

Q

A p p l i c a t i o n s of computer s i m u l a t i o n methods to biochemical and b i o l o g i c a l problem a l groups are i n t e r e s t e conformation based on computer s i m u l a t i o n . Clementi and co workers are working on water-amino a c i d p o t e n t i a l functions and 9 1 Zi

r e l a t e d problems d e a l i n g with molecular a s s e m b l i e s . Scheraea and co-workers are i n v o l v e d i n complementary studies of water*^ and methane-water s o l u t i o n , ^ and are e x p l o r i n g the use of Monte Carlo c a l c u l a t i o n s i n other problems of b i o p h y s i c a l i n t e r e s t . Karplus i s c u r r e n t l y using molecular dynamics to study p r o t e i n folding i n solution.^2 The o p p o r t u n i t i e s f o r t h e o r e t i c a l s t u d i e s of chemical and b i o l o g i c a l processes using computer s i m u l a t i o n are extremely broad and d i v e r s e , and we f u l l y expect that t h i s approach w i l l u l t i m a t e l y have a broad impact on t h e o r e t i c a l chem i s t r y , biochemistry and b i o l o g y . Each of the systems i n d i v i d u a l l y under c o n s i d e r a t i o n i n t h i s review has an extensive s c i e n t i f i c h i s t o r y . The background ap p r o p r i a t e to each system i s developed i n considerable d e t a i l i n our i n d i v i d u a l papers and, except f o r p a r t i c u l a r points and r e ferences to the most recent relevant work, i s not repeated here. The focus h e r e i n i s thus on p r e s e n t a t i o n of c o l l e c t e d current Monte Carlo r e s u l t s from t h i s Laboratory obtained on a s e r i e s o f important systems, with the computational procedures and analyses c a r r i e d out on a u n i f i e d and coherent b a s i s . II.

Calculations

The methodology of Monte Carlo c a l c u l a t i o n s e s p e c i a l l y f o r p o l a r and i o n i c systems i s c u r r e n t l y an a c t i v e area of study i n computer s i m u l a t i o n theory. Aspects such as p o t e n t i a l f u n c t i o n s , boundary c o n d i t i o n s , sampling a l g o r i t h m s , convergence c r i t e r i a and t r u n c a t i o n e r r o r s are a l l r e c e i v i n g considerable research a t t e n tion. The s e n s i t i v i t y of r e s u l t s to assumptions i n the c a l c u l a t i o n s i s discussed p a r t i c u l a r l y i n a recent paper by Levesque, Patey and W e i s s . ^


194


A l l c a l c u l a t i o n s described herein are based on s t a t i s t i c a l thermodynamic computer s i m u l a t i o n under canonical ensemble condi t i o n s on the system, with temperature T, number of p a r t i c l e s N, and volume V assumed and constant. Configurational integrations f o r each system are c a r r i e d out by Monte-Carlo methods i n v o l v i n g a s t o c h a s t i c walk through c o n f i g u r a t i o n space with c o n f i g u r a t i o n s s e l e c t e d on the b a s i s of t h e i r p r o b a b i l i t y i n the ensemble by the method suggested by Metropolis e t . al.^4 The N - p a r t i c l e system i s given a l i q u i d phase environment by the appropriate choice of number density and the use of image c e l l s , i . e . p e r i o d i c boundary conditions. The formalism employed i s developed f u l l y i n Refer ences 1 and 3. A l l c a l c u l a t i o n s unless otherwise i n d i c a t e d are based on 125 p a r t i c l e s i n a cubic c e l l at a temperature at 25°C. The volume of the c e l l i s determined by density and s p e c i f i e d i n d i v i d u a l l y f o r each system. The c a l c u l a t i o n s i n v o l v e a s p h e r i c a l c u t o f f i n the p o t e n t i a l f u n c t i o n at h a l f the c e l l edge used i n conjunction with th c r i t e r i a and e r r o r bound i n the s i m u l a t i o n are developed i n terms of c o n t r o l functions as defined by Wood.5 The c o n f i g u r a t i o n a l energy of the system i n each of the Monte Carlo c a l c u l a t i o n s discussed below i s developed under the assump t i o n of the pairwise a d d i t i v i t y of i n t e r m o l e c u l a r i n t e r a c t i o n s by means of a n a l y t i c a l f u n c t i o n s r e p r e s e n t a t i v e of ab i n i t i o quantum mechanical c a l c u l a t i o n s of the i n t e r m o l e c u l a r i n t e r a c t i o n energy. There are of course a number of l i m i t i n g assumptions i n v o l v e d i n the c o n s t r u c t i o n of such f u n c t i o n s , such as the s i z e and s p e c i f i c a t i o n of b a s i s sets i n the quantum mechanical c a l c u l a t i o n s , but they are formally w e l l defined and t h e i r c a p a b i l i t i e s and l i m i t a t i o n s with respect to the pairwise i n t e r a c t i o n can be developed in d e t a i l . The f u n c t i o n s can be r e a d i l y defined on the r e q u i s i t e regions of c o n f i g u r a t i o n space, and a n i s o t r o p i c s i n the i n t e r molecular i n t e r a c t i o n s are of course a u t o m a t i c a l l y i n c l u d e d . The i n d i v i d u a l terms i n the a n a l y t i c a l p o t e n t i a l f u n c t i o n should not be a s c r i b e d any p h y s i c a l s i g n i f i c a n c e ; they are simply a means f o r an i n t e r p o l a t i o n based on a l i m i t e d number of d i s c r e t e quantum-mechanically c a l c u l a t e d i n t e r m o l e c u l a r i n t e r a c t i o n energies. The major assumptions inherent i n the c o n f i g u r a t i o n a l energy c a l c u l a t i o n s i n t h i s study are thus the neglect of three-body and higher order c o n t r i b u t i o n s , t r u n c a t i o n e r r o r s i n the quantum me c h a n i c a l c a l c u l a t i o n s of pairwise i n t e r a c t i o n energies, and s t a t i s t i c a l e r r o r s i n the multidimensional curve f i t t i n g i n the ana l y t i c a l potential. A l l c a l c u l a t e d q u a n t i t i e s reported, with the exception of the f r e e energy f o r l i q u i d water, are based on simple ensemble averages and are produced i n a s t r a i g h t f o r w a r d manner i n the M e t r o p o l i s procedure. The a n a l y s i s of r e s u l t s i s based p a r t i c u l a r l y on guasicomponent d i s t r i b u t i o n f u n c t i o n s as introduced by Ben-Nairn.25 Quasicomponent d i s t r i b u t i o n functions are defined on the s t a t i s t i c a l s t a t e of the system and give the d i s t r i b u t i o n of


16.

BEVERIDGE ET AL.


195

p a r t i c l e s ( i n terms o f mole f r a c t i o n ) with any p a r t i c u l a r w e l l defined c h a r a c t e r i s t i c . The c h a r a c t e r i s t i c s r e l e v a n t to t h i s work are a) c o o r d i n a t i o n number, the number o f p a r t i c l e s w i t h i n a given radius of the center of mass of a given p a r t i c l e , and b) b i n d i n g energy, the i n t e r a c t i o n energy of a given p a r t i c l e with a l l other p a r t i c l e s i n the system. The quasicomponent d i s t r i b u t i o n s are of course obtained as averages over a l l i d e n t i c a l p a r t i c l e s and a l l c o n f i g u r a t i o n s o f the system. III.

L i q u i d Water

The extensive use of water as a solvent i n chemical systems, the d i r e c t p a r t i c i p a t i o n o f water i n many chemical and biochemi c a l processes and the unique f u n c t i o n o f water as a b i o l o g i c a l l i f e support system combine to make the s t r u c t u r e of l i q u i d water a matter of c e n t r a l importance i n understanding many chemical, biochemical and b i o l o g i c a uid water has been studie dynamics based on ad hoc models f o r the system, and there has been a long-standing controversy i n the s c i e n t i f i c l i t e r a t u r e as to whether the s t r u c t u r e of water i s best represented by a mix ture model, whereby the system i s viewed as a composite o f ener g e t i c a l l y d i s t i n c t c l u s t e r s of p o s s i b l y d i v e r s e s i z e and s t r u c ture, or by an e n e r g e t i c continuum model, wherein l o c a l molecular environments with a continuous d i s t r i b u t i o n of p r o g r e s s i v e l y bent hydrogen bonds are featured. Arguments pro and con f o r each mod e l based on both t h e o r e t i c a l a n a l y s i s and experimental evidence are summarized i n the recent reviews by Davis and Jarzynski26 and K e l l ^ ? and general p e r s p e c t i v e on the problem i s developed i n the s e r i e s e d i t e d by F r a n k s ^ and a recent review by Gorbunov and Naberukhin.29 L i q u i d water was the f i r s t molecular l i q u i d to be e x t e n s i v e l y s t u d i e d using computer s i m u l a t i o n methods. Experimental data on d i v e r s e thermodynamic p r o p e r t i e s are a v a i l a b l e and Narten, Levy and co-workers have obtained r a d i a l d i s t r i b u t i o n functions for the system by d i f f r a c t i o n methods.30 This system thus serves as a s e n s i t i v e t e s t on the q u a l i t y of the c o n f i g u r a t i o n a l poten t i a l used i n a computer s i m u l a t i o n . The c a l c u l a t e d oxygen-oxygen r a d i a l d i s t r i b u t i o n f u n c t i o n of l i q u i d water has been reported for q u i t e a v a r i e t y of pairwise a d d i t i v e e m p i r i c a l and a n a l y t i c a l p o t e n t i a l f u n c t i o n s . Among the e m p i r i c a l f u n c t i o n s , the ST2 pot e n t i a l ^ l i s c u r r e n t l y widely adopted. Clementi and c o - w o r k e r s ^ c a r r i e d out extensive s t u d i e s o f p a i r w i s e p o t e n t i a l f u n c t i o n s r e p r e s e n t a t i v e of ab i n i t i o quantum mechanical c a l c u l a t i o n s on the water dimer, and best agreement with the observed r a d i a l d i s t r i b u t i o n f u n c t i o n was obtained using a f u n c t i o n r e p r e s e n t a t i v e of a s e l f c o n s i s t e n t f i e l d (SCF) c a l c u l a t i o n plus moderately l a r g e i n t e r m o l e c u l a r c o n f i g u r a t i o n i n t e r a c t i o n (CI) on the sys tem. The f u n c t i o n was reported by Matsuoka, Clementi and Y o s h i mine (MCY)32 and i s henceforth r e f e r r e d to h e r e i n as the MCY-CI


196


potential. Recent s t u d i e s on l i q u i d water i n t h i s Laboratory * have been d i r e c t e d toward an a n a l y s i s of the s t r u c t u r e of the system from Monte Carlo computer based on the MCY-CI p o t e n t i a l f u n c t i o n . The c a l c u l a t i o n s were c a r r i e d out f o r 125 p a r t i c l e s at a temperature of 25°C and a volume commensurate with a density of 1 gm/cm . The ensemble averages f o r the system were based on 500K c o n f i g u r a t i o n s s e l e c t e d on the b a s i s of the M e t r o p o l i s method. Conver gence was checked by comparing the r e s u l t s of two independent runs, one beginning at a c o n f i g u r a t i o n with r e l a t i v e l y high ener gy and another beginning at a c o n f i g u r a t i o n very near the minimum energy f o r the system. The c a l c u l a t i o n s converged to the same i n t e r n a l energy w i t h i n 1% and apparently no p r a c t i c a l nonergodici t i e s were encountered.33 3

The c a l c u l a t e d thermodynamic i n t e r n a l energy f o r the system obtained from Monte Carlo computer s i m u l a t i o n based on the MCYCI p o t e n t i a l was -8.58 value of -9.9 kcal/mol mainly to the assumption of pairwise a d d i t i v i t y i n the c o n f i g u r a tional potential. The c a l c u l a t e d heat c a p a c i t y , corrected f o r i n t e r n a l modes, i s 17.9 cal/mol • deg as compared with the e x p e r i mental value at 25° of 18 cal/mol • deg. The c a l c u l a t e d oxygen-oxygen r a d i a l d i s t r i b u t i o n f u n c t i o n for l i q u i d water obtained from Monte Carlo computer s i m u l a t i o n based on the MCY-CI p o t e n t i a l i s shown i n F i g . 1. There i s q u i t e good accord between the c a l c u l a t e d and observed values f o r the p o s i t i o n of a l l three main peaks. In the region of the f i r s t hy d r a t i o n s h e l l , the shape of the c a l c u l a t e d peak agrees w e l l with experiment, although the c a l c u l a t e d maximum i s s l i g h t l y too high. For the second s h e l l , the p o s i t i o n of the maximum i s c o r r e c t but the shape i s biased towards short distance s i d e . A shoulder at ca. 3.5 8 i s c l e a r l y i n d i c a t e d . The t h i r d s h e l l appears g e n e r a l ly well described. An a n a l y s i s of the s t r u c t u r e of l i q u i d water was c a r r i e d out i n terms of quasicomponent d i s t r i b u t i o n f u n c t i o n s f o r coordina t i o n number and b i n d i n g energy and a l s o by examining stereographi c views of s i g n i f i c a n t molecular s t r u c t u r e s . Coordination num ber i s c a l c u l a t e d based on R^ = 3.3, the f i r s t minimum i n the r a d i a l d i s t r i b u t i o n f u n c t i o n , and t h i s d i s t r i b u t i o n f u n c t i o n i s e s s e n t i a l l y an a n a l y s i s of the f i r s t hydration s h e l l . The c a l c u l a t e d r e s u l t , d i s p l a y e d as mole f r a c t i o n of p a r t i c l e s x ( K ) vs. c o o r d i n a t i o n number K i s shown i n Figure 2. The d i s t r i b u t i o n ranges from K=2 to K=6, biased towards higher c o o r d i n a t i o n num bers, with K=4 predominant at 47%. The average c o o r d i n a t i o n num ber was found to be 4.1. The quasicomponent d i s t r i b u t i o n f u n c t i o n f o r b i n d i n g energy, d i s p l a y e d as mole f r a c t i o n of p a r t i c l e s x^(is) vs. b i n d i n g energy V , i s given i n Figure 3. The d i s t r i b u t i o n i s unimodal with a maximum at -17.7 kcal/mole. As discussed by Ben-Nairn,^5 mix ture model f o r the system should give r i s e to a bimodal or p o l y c

a


BEVERiDGE


ET AL.

2.8 r

I

= ft 08 04 0 I—

m

16 2.4 3.2 4 0 4 8 5 6 6 4 7 2 8 0

R (A) —

Figure 1. Calculated points on the oxygen-oxygen radial distribution function g(R) vs. center of mass separation R for liquid water at 25 °C. Experimental data (solid line) from Narten, Danford, and Levy (SO).

0.8r

|

0.6 -

2

0.4 -

o * 0.2 0

Figure 2. Calculated quasicomponent distribution function x (K) vs. coordination number K for liquid water c


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modal d i s t r i b u t i o n of binding energies while a unimodal d i s t r i b u t i o n i s c o n s i s t e n t with the idea of an e n e r g e t i c continuum model. Thus the modality of the c a l c u l a t e d d i s t r i b u t i o n f u n c t i o n f o r b i n d i n g energy supports the continuum model f o r l i q u i d water. Examining these r e s u l t s from the point of view of hydrogen bond energy, an i c e - l i k e environment with four l i n e a r hydrogen bonds would be expected to have a binding energy o f ^ 2 2 kcal/mol. This energy i s on the lower edge of the c a l c u l a t e d d i s t r i b u t i o n and not h e a v i l y favored. The region of higher p r o b a b i l i t y must therefore correspond to bent hydrogen bonds. To examine t h i s point f u r t h e r , we extracted from the c a l c u l a t i o n s a number of low energy, high frequency s t r u c t u r e s . Stereographic views of two of them, chosen such that the c e n t r a l molecule has K=4, are shown i n Figures 4 and 5. There i s a notable prevalence of bent hydro gen bonds throughout the s t r u c t u r e . C o l l e c t i v e l y , these r e s u l t s support the e n e r g e t i c continuum model f o r l i q u i d water p r e v i o u s l y mentioned molecula other t h e o r e t i c a l work on t h i s problem, Kauzmann ^ has demon s t r a t e d that a 2-state mixture model f o r the system cannot formal l y account f o r a l l the observed data on the system. Vibrational s p e c t r a l data supporting a mixture model based on an i s o s b e s t i c point i n the Raman spectra as a f u n c t i o n of temperature have been reported by W a l r a f e n . ^ This problem has been reexamined recent l y by Scherer, Go and K i n t ^ who showed that the apparent isosbes t i c point i n Raman data that has not been decomposed i n t o iso t r o p i c and a n i s o t r o p i c parts i s f o r t u i t o u s . Rice and coworkers f i n d t h e i r d e t a i l e d a n a l y s i s of the 0-H s t r e t c h i n g region of the v i b r a t i o n a l s p e c t r a of amorphous s o l i d water c o n s i s t e n t with a s l i g h t l y bent hydrogen bond model. The r e l a t i o n s h i p between the observed f a r i n f a r e d spectrum of water and the molecular dynamics computer simulation r e s u l t s has been developed by Curnette and Williams. The recent r e c o n s i d e r a t i o n of s p e c t r o s c o p i c aspects of the water s t r u c t u r e problem due to Gorbunov and N a b e r u k h i n ^ f i r m l y supports the continuum model. The Monte Carlo-Metropolis method f o r computer simulation i s i d e a l l y s u i t e d to produce ensemble averages of the p r o p e r t i e s of the system. However i n the case of f r e e energy, the ensemble av erage expression i s not convenient f o r computational purposes due to the i l l - c o n d i t i o n e d nature of the integrand and the concomitant convergence p r o b l e m s . ^ We have r e c e n t l y c a r r i e d out a Monte Carlo c a l c u l a t i o n of the f r e e energy of l i q u i d water based on a procedure which follows from e a r l y work by Kirkwood ^ i n v o l v i n g a numerical quadrature over the i n t e r n a l energy o f the system de\eloped i n terms of an a u x i l i a r y parameter.^ The c a l c u l a t i o n was found to be computationally t r a c t a b l e and r e s u l t e d i n a Helmholz c o n f i g u r a t i o n a l free energy of -4.31- 0.07 kcal/mol compared with an observed value of -5.74 kcal/mol;^ the corresponding entropy was found to be -14.44 - 0.09 cal/deg mol against an observed val ue of -13.96 cal/deg mol at 25°C. The e r r o r i n free energy being 3

3

3

3 7

3 8

3


16.

BEVERiDGE


ET AL.

0.08

199

r

Figure 3. Calculated quasicomponent distribution function x (v) vs. binding energy for liquid water It

v

Figure 4. Stereographic view of a fragment of a significant molecular structure contributing to the statistical state of liquid water


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MATTER

of the same order of magnitude as the e r r o r i n the i n t e r n a l ener gy, we a s c r i b e t h i s discrepancy a l s o to the neglect higher order c o n t r i b u t i o n s to the c o n f i g u r a t i o n a l p o t e n t i a l . IV.

The D i l u t e Aqueous S o l u t i o n of Methane

The d i l u t e aqueous s o l u t i o n of methane i s a system of promi nent i n t e r e s t i n molecular l i q u i d s as the prototype of a non-po l a r molecular solute d i s s o l v e d i n l i q u i d water, and i s one of the simplest molecular systems where the hydrophobic e f f e c t ^ i s manifest. Moreover, a d e t a i l e d knowledge of the s t r u c t u r e of the methane-water s o l u t i o n at the molecular l e v e l can provide leading information on the i n t e r a c t i o n of water with d i s s o l v e d hydrocar bon chains i n general, and thereby c o n t r i b u t e to the t h e o r e t i c a l b a s i s f o r understanding the r o l e of water i n maintaining the s t r u c t u r a l i n t e g r i t y of b i o l o g i c a l macromolecules i n s o l u t i o n General backgroun has been r e c e n t l y reviewe important work on these systems i s due to E l e y ^ ^ d Frank and E v a n s . ^ Methane has been i d e n t i f i e d as a "structure-maker" i n aqueous s o l u t i o n i n the language of Frank and Wen.^5 The nature of s t r u c t u r a l changes i n solvent water by d i s s o l v e d hydrocarbons has f o r sometime been discussed as by T a n f o r d ^ i terms of water c l a t h r a t e formation, based on work p a r t i c u l a r l y by G l e w ^ and analogies drawn from a number of hydrate c r y s t a l s t r u c t u r e s of non-polar species, known to i n v o l v e water c l a t h r a t e s t r u c t u r e s of order 20 and 24. Key papers on t h i s t o p i c include the review by Kauzmann^? and work by Scheraga and coworkers^**. E a r l y computer simulation work on the methane-water system was reported by Dashevsky and Sarkisov.^^ Recent t h e o r e t i c a l s t u d i e s of the methane-water system are the ab i n i t i o molecular o r b i t a l c a l c u l a t i o n s of the methane-water pairwise i n t e r a c t i o n energy by Ungemach and SchaeferÔ and the Monte Carlo computer s i m u l a t i o n on the d i l u t e aqueous s o l u t i o n i n the isothermali s o b a r i c ensemble by Owicki and S c h e r a g a . ^ A recent Monte Carlo study of s t r u c t u r e of the d i l u t e aque ous s o l u t i o n of methane from t h i s Laboratory^ i n v o l v e s one meth ane molecule and 124 water molecules at 25°C at l i q u i d water d e n s i t y . The c o n f i g u r a t i o n a l energy of the system i s developed under the assumption of pairwise a d d i t i v i t y using p o t e n t i a l func t i o n s r e p r e s e n t a t i v e of ab i n i t i o quantum mechanical c a l c u l a t i o n s f o r both the water-water and methane-water i n t e r a c t i o n s . For the water-water i n t e r a c t i o n we have c a r r i e d over the MCY-CI p o t e n t i a l f u n c t i o n used i n our previous study of the s t r u c t u r e of l i q u i d water reviewed i n the preceeding s e c t i o n . For the methane water i n t e r a c t i o n energy, we have r e c e n t l y reported-** an a n a l y t i c a l p o t e n t i a l f u n c t i o n r e p r e s e n t a t i v e of quantum mechanical c a l c u l a t i o n s based on SCF c a l c u l a t i o n s and a 6-31G b a s i s s e t , with cor r e l a t i o n e f f e c t s included v i a second order M o l l e r - P l e s s e t (MP) corrections,52 T h i s f u n c t i o n was used f o r the methane-water cona n

n


16.

BEVERIDGE ET AL.


201

t r l b u t i o n s to the c o n f i g u r a t i o n a l p o t e n t i a l presented h e r e i n . Ensemble averages were c a l c u l a t e d on the b a s i s of a 650K stochas t i c walk. The c a l c u l a t e d p a r t i a l molar i n t e r n a l energy f o r t r a n s f e r of methane from gas phase to water was -23.3- 6.6 kcal/mol, compared with an experimental value of -2.6 k c a l / m o l ^ The c a l c u l a t e d r e s u l t i s seen to be negative as expected f o r the hydrophobic e f f e c t , but an order of magnitude too low. This discrepancy i s discussed f u r t h e r below. The c a l c u l a t e d r a d i a l d i s t r i b u t i o n f u n c t i o n f o r the center of mass of water molecules with respect to the methane carbon atom i s shown i n Figure 6. We f i n d a broad, unstructured f i r s t peak with a minimum i n the region o f 5.3&. I n t e g r a t i n g g(R) up to t h i s point gives an average coor d i n a t i o n number of 19.35. An a n a l y s i s of the s t r u c t u r e o f the d i l u t e aqueous s o l u t i o n of methane was a l s o developed i n terms of quasicomponent d i s t r i bution functions and stereographi structures. The c o o r d i n a t i o was c a l c u l a t e d on the b a s i s of R = 5.38, f i x e d at the f i r s t minimum i n the methane-water r a d i a l d i s t r i b u t i o n f u n c t i o n . A p l o t of the mole f r a c t i o n of methane molecules x ( K ) vs. t h e i r corresponding water c o o r d i n a t i o n number i s given i n Figure 7. The x^(K) obtained i s a broad unimodal d i s t r i b u t i o n ranging from K=16 to K=22 with a maximum i n the region K=19 and 20, biased s l i g h t l y i n shape towards higher c o o r d i n a t i o n numbers. The c a l c u l a t e d quasicomponent d i s t r i b u t i o n f u n c t i o n f o r b i n d i n g energy, the mole f r a c t i o n of methane molecules x ( v ) as a f u n c t i o n of methane b i n d i n g e n e r g y v i s shown i n Figure 8. There i s some i n c i p i e n t s t r u c t u r e i n the curve but the e r r o r bounds on the f u n c t i o n are too l a r g e to a s c r i b e t h i s any p h y s i c a l significance. Comparing the average value of t h i s q u a n t i t y , -23.31" 6.6, with the c a l c u l a t e d p a r t i a l molar i n t e r n a l energy of t r a n s f e r f o r methane confirms that the s i g n and magnitude o f t h i s l a t t e r term are due to water s t a b i l i z a t i o n e f f e c t s . I f we now assume the ~20 kcal/mole discrepancy between the c a l c u l a t e d and observed values of the i n t e r n a l energy comes from the ~20 water molecules found i n the f i r s t h y d r a t i o n s h e l l of methane, t h i s i s an e r r o r of ~1 kcal/mole per water molecule or a f r a c t i o n of a kcal/mol per pairwise i n t e r a c t i o n or hydrogen bond. Con s i d e r i n g the approximations inherent i n the c a l c u l a t i o n of p a i r wise i n t e r a c t i o n energies as enumerated i n S e c t i o n I I , t h i s d i s crepancy appears to be commensurate with the c a p a b i l i t i e s and l i m i t a t i o n s of the c o n f i g u r a t i o n a l energy c a l c u l a t i o n . The magnitudes and d i s t r i b u t i o n of c o o r d i n a t i o n numbers found f o r methane i n the s t a t i s t i c a l s t a t e of the d i l u t e aqueous s o l u t i o n are g e n e r a l l y c o n s i s t e n t with the presence of water c l a t h r a t e cages as shown i n Figure 9. In a number of other s t r u c tures q u a s i c l a t h r a t e regions could be i d e n t i f i e d , but d e f e c t s and d i s t o r t i o n s were more prevalent than i n Figure 9. One such s t r u c t u r e , r e p r e s e n t a t i v e of other, i s given i n Figure 10. The 3

M

c

f i


202

COMPUTER


Figure 5. Stereographic view of a fragment of another significant molecular structure contributing to the statistical state of liquid water

2.8 2.4

20 |

I .6

CP

- MP 0.8 0.4 0.0

—

I .8

t

24

—

32

i

—

40

i

—

i

48 R (&)

—

i

—

i

—

i

56 6.4 7 2

—

i

—

8.0

i

8.8

—

Figure 6. Calculated methane-water radial distribution gfRj vs. center of mass separation R from Monte Carlo computer simulation for the dilute aqueous solution of methane atT = 25°C


BEVERIDGE E T A L .

1

1

14


1 15

1 16

T 1 17

1 1 18

i1 19

i1 20

i1 21

1r — '—nI

22

23

Figure 7. Calculated quasicomponent distribution function XcfK) vs. methane coordination number K for the dilute aqueous solution of methane

v ( kcal/mol)

—

Figure 8. Calculated quasicomponent distribution function x (v) vs. methane binding energy v for the dilute aqueous solution of methane n


COMPUTER


b Figure 9. Stereographic view of methane and its first hydration shell taken from a significant molecular structure contributing to the statistical state of the system. (Top) disposition of centers of mass of water molecules about methane (shaded) with the quasiclathrate cage delineated; (bottom) disposition of water molecules about methane in the same structure.


16. BEVERIDGE ET AL.


b Figure 10. Stereographic view of methane and its first hydration shell taken from another significant molecular structure contributing to the statistical state of the system. (Top) disposition of centers of mass of water molecules about methane (shaded) with the quasiclathrate cage delineated; (bottom) disposition of water molecules about methane in the same structure.


205

206

COMPUTER MODELING OF

MATTER

emergent d e s c r i p t i o n of the aqueous s o l u t i o n environment of meth ane i s that of a d i s t o r t e d and d e f e c t i v e continuum c l a t h r a t e structure. E f f e c t s of s o l u t e methane on solvent water s t r u c t u r e were developed i n terms of the d i f f e r e n c e i n Xg(K) and x^(i/) c a l c u l a t e d f o r the s o l u t i o n and f o r the pure l i q u i d . The e f f e c t s of bulk water are removed by the d i f f e r e n c i n g , allowing the d i r e c t d i s p l a y of s t r u c t u r a l changes i n solvent water. The d i f f e r e n c e p l o t s r e v e a l under high s t a t i s t i c a l e r r o r an increase i n 4-coordinate species and a s l i g h t but c l e a r l y discernable s h i f t t o ward lower binding energy f o r the solvent molecules, p r o v i s i o n a l l y consistent with general ideas of " s t r u c t u r e making". F u r t h e r work i s c u r r e n t l y i n progress on t h i s point. V.

D i l u t e Aqueous Solutions

of Monatomic Cations and

Anions.

The p a r t i c u l a r s i g n i f i c a n c chemistry makes the s t r u c t u r monatomic cations and anions a l s o a t o p i c of fundamental i n t e r e s t . Moreover, the sodium and potassium ions i n p a r t i c u l a r f i g ure prominently i n biochemical membrane p o t e n t i a l phenomena, and t h e i r hydration s t a t e i n aqueous s o l u t i o n i s an important f a c t o r i n ion s e l e c t i v i t y and membrane permeability i n b i o l o g i c a l sys tems. Modern t h e o r e t i c a l studies of these systems date from the c l a s s i c paper of Bernal and Fowler i n 1933. ^ The current s t a t e of both experimental and t h e o r e t i c a l research on i o n i c s o l u t i o n s i s the subject of s e v e r a l recent comprehensive reviews p a r t i c u l a r l y by Friedman and c o - w o r k e r s . The prevalent d e s c r i p t i v e ideas about the l o c a l aqueous s o l u t i o n environment of ions stems from the work of Frank and Wen,^5 h o p a r t i t i o n e d the solvent i n to three regions. In the immediate v i c i n i t y of the i o n , region A, water molecules are t i g h t l y bound and h i g h l y o r i e n t e d . Region C at l a r g e distance from the ion was considered as e s s e n t i a l l y bulk water, and the i n t e r v e n i n g region B i s a region of s t r u c t u r a l ambiguity i n t e r f a c i n g regions A and B. Recently Kistenmacher, Popkie and dementi *' reported analy t i c a l intermolecular p o t e n t i a l functions f o r ion-water i n t e r a c t i o n s r e p r e s e n t a t i v e of near Hartree-Fock molecular o r b i t a l c a l culations. These functions along with water-water p o t e n t i a l s of commensurate q u a l i t y were ised to study the s t r u c t u r e and s o l v a t i o n numbers of ion-water c l u s t e r s based on energy o p t i m i z a t i o n . Monte Carlo simulation work based on these functions have been reported by Mruzik, Abraham, Scheiber and P o u n d focussing on free energy c a l c u l a t i o n s , and by WattsÔ d coworkers on ion p a i r s i n large water c l u s t e r s . McDonald and Rasaiah have c a r r i e d out very large order studies of ion p a i r s i n s o l u t i o n based on model p o t e n t i a l f u n c t i o n s . Molecular dynamics on i o n water systems have been reported by Briant and B u r t o n ^ and Heinzinger and Vogel.^0 5

55

w

5

57

a n

5


BEVERIDGE E T A L .


207

12 0

10.0

|

8.0

_

6.0

&

4.0

R ( £]

Figure 11. Calculated cations-water radial distribution function vs. center of mass separation R for the dilute aqueous solution of lithium atT = 25°C

Figure 12. Calculated cation—water radial distribution function vs. center of mass separation R for the dilute aqueous solution of sodium atT = 25° C


208

COMPUTER MODELING O F M A T T E R

Figure 13. Calculated cation-water radial distribution function vs. center of mass separation R for the dilute aqueous solution of potassium at T = 25°C

4.8

r

4.0

!" _ 2.4 cr ^

I .6

— 0.8 0.0 08

. v I

I

1.6 2 4

^ ^ ^ / N ^ -

1

F

' 1

1

3.2 4 0

48

1

1

1

5 6 6.4 7.2

1

8.0

R (&)

Figure 14. Calculated anion-water radial distribution function vs. center of mass separation R for the dilute aqueous solution of fluoride at T = 25°C


BEVERIDGE ET AL.

Structure of Liquid Water and Dilute Solutions 209

2.8 2.4 2.0

I

1 6

5

I 2

cn

* 0.8

•

Xs/KT

V

0.4

o.o | 0.8

I | i 1 1 1 1 1 1 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0

R [&)—+>

Figure 15. Calculated anion-water radial distribution function vs. center of mass separation R for the dilute aqueous solution of chloride at T = 25°C


210

COMPUTER

08 0.6

R

MODELING OF

MATTER

= 3 30 &

M

(X

(K))

c

=6 77 ± 0 . 1 8

0 4 0.2 0.0 0.8

t

1

Xo

0.6

R =3 M

00&

(X (K))

-- 5 9 8 ± 0

C

22

0.4 0.2 "^"]

00

1 0 0.8

-i

R = M

2.65

1

—

No —

4

r

I

( X ( K ) ) =5 0 5 + 0 0 7 C

0.6 0.4 0.2 Li* 0.0

1

T"

4 K

5 —

Figure 16. Calculated quasicomponent distribution functions x fK) vs. ion coordination number K for dilute aqueous solutions of alkali metal cations r


16.

BEVERIDGE ET AL.


211

1.0 0.8

R

=4.10 &

M

(x

c

(K)) =7.69 ± 0 . 2 9

0.6 0.4 0.2 cr

| 0.0

1

1

1

1—

h—

R =3.30 £ M

| .0

<X (K)) = 5 . 4 6 ± 0 . 2 4 C

x 0.8 0.6 0.4 0.2 0.0

F~ —I——1~

3

4

i

i

5 K

6

1 9

r~

1

10 11

Figure 17. Calculated quasicomponent distribution functions x (K) vs. ion coordination number K for dilute aqueous solutions of halide anions c


212

COMPUTER

0.050


A

0025 0.0

^ cn x

1

1

1

1

I

|

1

1

1

1

I

1

0050 0025 0.0

1

1

0.050 0 025 0 0 -300

-250

-275

-225

175

-200

-150

v ( kcal /mol)

Figure 18. Calculated quasicomponent distribution functions x (v) vs. ion binding energy for dilute aqueous solutions for alkali metal cations B

v

0050 0025 0.0

V , c r 1

1

1

-250

-225

i

i

i

-175

-150

0.050 0.025

V

0.0 -275

-200

v ( kcal /mol) —

Figure 19. Calculated quasicomponent distribution functions Xn(v) vs. ion binding energy for dilute aqueous solutions for halide anions v


16.

BEVERIDGE E T A L .

Structure of Liquid Water and Dilute Solutions 213

Figure 20. Stereographic view of a significant molecular structure contributing to the statistical state of the dilute aqueous solution of KS

Figure 21. Stereographic view of a significant molecular structure contributing to the statistical state of the dilute aqueous solution of F~


214

COMPUTER MODELING O F M A T T E R

We have r e c e n t l y c a r r i e d out Monte Carlo computer simulation of d i l u t e aqueous s o l u t i o n s of the monatomic cations L i , Na and K and the monatomic anions F~" and C l ~ using the KPC-HF functions f o r the ion-water i n t e r a c t i o n and the MCY-CI p o t e n t i a l f o r the water-water i n t e r a c t i o n . The temperature of the systems was taken to be 25° and the density chosen to be commensurate with the p a r t i a l molar volumes as reported by M i l l e r o . ^ l The c a l c u l a t ed average q u a n t i t i e s are based on from 600- 900K c o n f i g u r a t i o n s a f t e r e q u i l i b r a t i o n of the systems. The c a l c u l a t e d ion-water r a d i a l d i s t r i b u t i o n functions are given f o r the d i l u t e aqueous s o l u t i o n s o f L i , K , Na+, F~ and C I " i n Figures 11-15, respec tively. An a n a l y s i s of the s t r u c t u r e of the d i l u t e aqueous s o l u t i o n s of monatomic cations and anions was developed i n a manner analo gous to that presented above f o r pure water and the d i l u t e aque ous s o l u t i o n of methane. The d i s t r i b u t i o n of coordination num bers x^,(K) vs. K i s presente f o r the anions i n Figur and centered i n the region K=5, 6 and 7. The average water co o r d i n a t i o n number of each ion i s found by determining the i n t e g r a l of the f i r s t peak i n the ion-water r a d i a l d i s t r i b u t i o n func t i o n , and these values are also recorded on Figures 16 and 17. The binding energy analyses are given i n Figures 18 and 19 f o r cations and anions r e s p e c t i v e l y ; each one i s continuous and u n i modal. Representative stereographic s t r u c t u r e s are given f o r the c a t i o n K i n Figure 20 and f o r the anion F~ i n Figure 21; the other cations and anions introduce no e s s e n t i a l l y new q u a l i t a t i v e features of the s t r u c t u r e . These r e s u l t s should be considered i n the context of observ ed discrepancy of 10-40% between the c a l c u l a t e d and observed i n t e r n a l energy o f t r a n s f e r due to the assumption o f pairwise a d d i t i v i t y i n the c o n f i g u r a t i o n a l p o t e n t i a l and the t r u n c a t i o n o f the p o t e n t i a l i n c o n f i g u r a t i o n a l energy c a l c u l a t i o n s . The e f f e c t of higher order terms i n the c o n f i g u r a t i o n a l p o t e n t i a l on the a n a l y s i s o f s t r u c t u r e as presented h e r e i n has yet to be determin ed. +

+

+

+

VI.

Summary and Conclusions

The s e r i e s of studies of molecular l i q u i d s presented h e r e i n c o l l e c t r e s u l t s on a diverse s e t of chemically relevant systems from a uniform t h e o r e t i c a l point of view: ab i n i t i o c l a s s i c a l s t a t i s t i c a l mechanics on the (T,V,N) ensemble with p o t e n t i a l functions representative of ab i n i t i o quantum mechanical c a l c u l a t i o n s of pairwise i n t e r a c t i o n s and s t r u c t u r a l a n a l y s i s c a r r i e d out i n terms of quasicomponent d i s t r i b u t i o n f u n c t i o n s . The l e v e l of agreement between c a l c u l a t e d and observed q u a n t i t i e s i s quoted to i n d i c a t e the c a p a b i l i t i e s and l i m i t a t i o n s to be expect ed o f these c a l c u l a t i o n s and i n that perspective we f i n d a num ber of s t r u c t u r a l features of the systems p r e v i o u s l y discussed on


16.

BEVERIDGE ET AL.


215

an e m p i r i c a l b a s i s to be q u a n t i f i e d . The present r e s u l t s point p a r t i c u l a r l y to a) a continuum model f o r l i q u i d water, b) a d i s t o r t e d , d e f e c t i v e c l a t h r a t e environment f o r methane i n aqueous s o l u t i o n , and c) d i s t i n c t l y d i f f e r e n t d i s t r i b u t i o n s o f e q u i l i b rium c o o r d i n a t i o n numbers f o r N a and K i n water which may be of s i g n i f i c a n c e i n biomembrane phenomena. U l t i m a t e l y i t i s hoped that a d e s c r i p t i v e chemistry of molecular l i q u i d s u s e f u l i n d i v e r s e a p p l i c a t i o n s can be developed on a r i g o r o u s b a s i s u s i n g the r e s u l t s of computer s i m u l a t i o n s and we c o n t r i b u t e these s t u d i e s as a step i n that d i r e c t i o n . +

+

VII. Acknowledgement Support f o r t h i s research comes from NIH Grant //1-R01NS12149-03 from the N a t i o n a l I n s t i t u t e s o f N e u r o l o g i c a l and Com municative Diseases and Stroke and a CUNY F a c u l t y Research Award. D.L.B. acknowledges U.S.P.H.S 6TK04-GM21281 from th Studies.

VIII. References 1. 2. 3. 4. 5.

6.

7. 8. 9. 10. 11. 12. 13.

S. Swaminathan, and D.L. Beveridge, J . Am. Chem. Soc., 99, 8392 (1977). M. Mezei, S. Swaminathan, and D.L. Beveridge, J . Am. Chem. Soc., 100, 3255 (1978). S. Swaminathan, S.W. H a r r i s o n , and D.L. Beveridge, J . Am. Chem. Soc., in press. M. Mezei and D.L. Beveridge, MS in preparation. W.W. Wood and J . J . Erpenbeck, Am. Rev. Phys. Chem., 27, 319 (1976). J . J . Erpenbeck and W.W. Wood, in "Statistical Mechanics, Pt B", B.J. Berne, ed., Plenum Press, New York, N.Y. 1977 p 1. B.J. A l d e r and T.E. Wainwright, J . Chem. Phys., 31, 459 (1959); 33, 1439 (1960). B.J. A l d e r , D.M. Gass, and T.E. Wainwright, J . Chem. Phys., 53, 3013 (1970). J.A. Barker and D. Henderson, Rev. Mod. Phys., 48, 587 (1976). J.A. Barker and R.O. Watts, Chem. Phys. L e t t . , 3, 144 (1969). G.N. Sarkisov, V.G. Dashevsky, and G.G. Malenkov, Mol. Phys. 27, 1249 (1974). A. Rahman and F.H. Stillinger, J. Chem. Phys., 55, 336 (1971). A.J.C. Ladd, Mol. Phys., 33, 1039 (1977). J.C. Owicki and H.A. Scheraga, J . Am. Chem. S o c . , 99, 7403 (1977). E. Clementi, " L i q u i d Water S t r u c t u r e " , Springer V e r l a g , New York (1976).


216

COMPUTER

MODELING

OF MATTER

14. J . Fromm, E. Clementi, anf R.O. Watts, J . Chem. Phys., 62, 1388 (1975). E. Clementi, R. B a r s o t t i , J . Fromm and R.O. Watts, Theoret. Chem. Acta (Berl.) 43, 101 (1976). 15. S. Swaminathan, R.J. Whitehead, E. Guth and D.L. Beveridge, J . Am. Chem. Soc., 99, 7817 (1977). 16. D.J. Evans and R.O. Watts, Mol. Phys., 32, 93 (1976). 17. P.S.Y. Cheung and J.G. Powles, Mol. Phys., 32, 1383 (1976). 18. I.R. McDonald and M.L. K l e i n , J . Chem. Phys., 64, 1790 (1976); P.K. Mehrotra and D.L. Beveridge, MS. i n p r e p a r a t i o n . 19. J.C. Owicki and H.A Scheraga, J . Am. Chem. Soc., 99, 7413 (1977). 20. R.O. Watts, " L i q u i d State Chemical Physics", John Wiley & Sons, New York (1976). 21. E. Clementi, F. L a v a l l o n e , and R. Scordamoglia J . Am. Chem. Soc., 99, 5531 (1977); R. Scordamoglia, F. Cavallone, and E. Clementi, J . Am Chem Soc. 99 5545 (1977); G B o l i s and E. Clementi, J . Am 22. P. Rossky, M. Karplus tion. 23. D. Levesque, G.N. Patey and J . J . Weis, Mol. Phys. 34, 1077 1091 (1977). 24. N.A. M e t r o p o l i s , A.W. Rosenbluth, M.N. Rosenbluth, A.H. Tell er and E. Teller, J. Chem. Phys., 21, 1087 (1953). 25. A. Ben-Nairn, "Water and Aqueous S o l u t i o n s " , Plenum Press, New York, N.Y. 1974. 26. C.M. Davis and J . J a r z y n s k i i n "Water and Aqueous S o l u t i o n s " R.A. Home, ed., John Wiley and Sons, New York, N.Y. 1972. The mixture model emerged from the earliest thoughts about the s t r u c t u r e of liquid water as a composite of ice-like (bulky) c o n t r i b u t i o n s i n e q u i l i b r i u m with a dense component. Highly developed forms of t h i s theory are the interstitial model, (O.Ya. Samoilov, Zh. Fiz. Khim. 20, 1411 (1946). V.A. Mikhailov, Zh. S t r u c t . Khim. 8, 189 (1967), 9, 397 (1968); A.H. Narten and H.A. Levy, Science 165, 3892 (1969). and the f l i c k e r i n g c l u s t e r model, (A.T. Hagler, H.A. Scherga and G. Nemethy, Ann. N.Y. Acad. S c i . 204, 51 (1973). 27. G.S. K e l l i n "Water and Aqueous S o l u t i o n s " R.A. Home, ed., John Wiley and Sons, New York, N.Y. 1972. The e n e r g e t i c continuum model f o r liquid water was developed during d i s s e r t a t i o n research by J.A. Pople, Proc. Roy. Soc. (London), Ser A, 205, 163 (1951). 28. F. Franks, ed., "Water-A Comprehensive T r e a t i s e " V o l s . I - I I I , Plenum Press, New York, N.Y. 1972-73. 29. B.Z. Gorbunov and Yu.I. Naberukhin, J . S t r u c t . Chem. 16, 703 (1975). 30. A.H. Narten and H.A. Levy, J . Chem. Phys., 55, 2263 (1971). A.H. Narten, J . Chem. Phys., 55, 5681 (1972). 31. F.H. Stillinger and A. Rahman, J . Chem. Phys., 60, 1545 (1974).


16.

BEVERIDGE ET AL.


217

32. O. Matsuoka, E. Clementi, and M. Yoshimine, J . Chem. Phys., 64, 1351 (1976). 33. B. Larsen and S.A. Rodge, J . Chem. Phys., 68, 1309 (1978). C. P a n g a l i , M. Rao, and B.J. Berne, Chem. Phys. L e t t . , in press. 34. W. Kauzmann, Colloqes I n t e r n a t i o n a l du C.N.R.S. #2461, 63 (1976). 35. G. Walrafen i n "Water - A Comprehensive T r e a t i s e " ,Vol.I, F. Franks, ed., Plenum Press, New York. 1972, p.151. 36. J.R. Scherer, M.K. Go and S. K i n t , J . Phys. Chem. 78, 1304 (1974). 37. S.A. Rice, W.G. Madden, R. McGraw, M.G. Sceats, and M.S. Berger, J . Chem. Phys. i n p r e s s . 38. B. Curnutte and D. W i l l i a m s , i n S t r u c t u r e of Water and Aqueous Soln, W.A.B. Luck, ed. ISBN 3-527-25588-5 (1974). 39. J.G. Kirkwood, i n "Theory of L i q u i d s " B.J A l d e r Ed. Gor don and Breach, Ne 40. C. Tanford, "The Hydrophobi New York, N.Y. 1973. 41. F. Franks i n "Water-A Comprehensive T r e a t i s e " , I I , F. Franks ed. Plenum Press, New York,N.Y.1972,p.l. 42. A Ben-Naim i n "Water and Aqueous S o l u t i o n , " R.A. Horne, ed., W i l e y - I n t e r s c i e n c e , New York, N.Y. 1972, p. 425. 43. D.D. Eley, Trans. Faraday Soc., 35., 1281 (1939). 44. H.S. Frank and M.W. Evans, J . Chem. Phys., 13, 507 (1945). 45. H.S. Frank and W.Y. Wen, Disc. Faraday Soc. 24, 133 (1957). 46. D.N. Glew, J . Am. Chem. Soc., 66, 605 (1962). 47. W. Kauzmann, Adv. P r o t e i n Chem., 14, 1 (1959). 48. G. Nemethy and H.A. Scheraga, J . Chem. Phys. 36, 3382, 3401 (1962). 49. V.G. Dashevsky and G.N. S a r k i s o v . , Mol. Phys., 27, 1271 (1974). 50. S.R. Ungemach and H.F. Schaefer I I I , J . Am. Chem. Soc., 96, 7898 (1974). 51. S.W. H a r r i s o n , S. Swaminathan, D.L. Beveridge and R. D i t c h f i e l d Int. J . Quantum Chem., i n press. 52. J.A. Pople, J.S. B i n k l e y , and R. Seeger, I n t . J . Quantum Chem., Symp. No. 10, 1 (1976). 53. M. Yaacobi and A. Ben-Nairn, J . Phys. Chem., 78, 175 (1924); H. Edelnoch and J.C. Osborne, J r . , Advan. P r o t e i n Chem., 30, 188 (1976). 54. J.D. Bernal and J . Fowler, J . Chem. Phys., 1, 515 (1933). 55. H.L. Friedman and W.D.T. Dale i n "Modern T h e o r e t i c a l Chemis try", Vol IV, Statistical Mechanics, pt A, B.J. Berne, ed., Plenum Press, New York, N.Y. 1977. 56. H. Kistenmacher, H. Popkie and E. Clementi, J . Chem. Phys. 61, 799 (1974). 57. M.R. Mruzik, F.F. Abraham, D.E. S c h r e i b e r , and G.M. Pound, J . Chem. Phys., 64, 481 (1976).


218


58. 59.

I.R. McDonald, J.C. Rasaiah Chem. Phys. L e t t . 34, 382. (1975). C.L. B r i a n t and J . J . Burton, J . Chem. Phys., 60, 2849 (1974). K. Heinzinger and P.C. Vogel, Z. N a t u r f o r s c h . , 29a, 1164 (1974); P.C. Vogel and K. Heinzinger, Z. N a t u r f o r s c h . , 30a, 789 (1975). F.J. M i l l e r o i n "Water and Aqueous S o l u t i o n " , R.A. Horne, ed., W i l e y - I n t e r s c i e n c e , New York, N.Y. 1972, p. 519.

60.

61.

RECEIVED September 7, 1978.


17 Computer Modeling of Quantum Liquids and Crystals M. H. KALOS, P. A. WHITLOCK, and D. M. CEPERLEY Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY 10012

There are a number of many-body systems which exhibit quantum effects on a macroscopic scale. These include liquid and crystal states of both He-3 and He4, the electron gas, and neutron matter which probably constitutes the interior of pulsors. In addition, "nuclear matter" - a hypothetical extensive system of nucleons has been studied for the insight one may gain into the nature of finite nuclei. The theoretical studies of these systems have by now a long history, but are by no means concluded. In the last few years, significant advances have been made. This has come in part from the maturity of and gradual unification of many-body theory, in part from the development and application of powerful new expansion procedures, especially varieties of hypernetted-chain equations (1) and finally to the growing power of computer simulation methods for quantum systems. This article is intended as a review of some recent development in computational methods for extensive quantum systems, and of the relation between results so obtained to the evolution of other theoretical work. C o m p u t a t i o n a l m o d e l l i n g o f quantum many-body s y s tems i s n o t e s p e c i a l l y n o v e l . The e a r l y h i s t o r y o f M o n t e C a r l o m e t h o d s i n c l u d e d many p r o p o s a l s f o r t h e solution of Schrodinger s equation with intended a p p l i c a t i o n t o t h e many-body p r o b l e m . Unfortunately, the c o m p u t a t i o n a l p o w e r a v a i l a b l e was n o t a d e q u a t e t o d o more t h a n s i m p l e e x e r c i s e s . T h e f i r s t work i n w h i c h a s i g n i f i c a n t c o n t r i b u t i o n t o t h e o r y was made was t h a t o f W. L . M c M i l l a n (2_) who n o t e d t h a t t h e g e n e r a l s a m p l i n g a l g o r i t h m o f M e t r o p o l i s e t a l . (3)developed to t r e a t e q u i l i b r i u m c h e m i c a l systems c o u l d be used e q u a l l y w e l l t o o b t a i n v a r i a t i o n a l estimates o f the energy o f a m a n y - b o d y s y s t e m when t h e t r i a l f u n c t i o n h a s a product form. S i n c e t h e n , a l a r g e number o f s i m i l a r 1


220

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c a l c u l a t i o n s have been c a r r i e d o u t t r e a t i n g extensive s y s t e m s o f atoms f r o m h y d r o g e n t o n e o n . For a thorough r e v i e w o f t h e s e c a l c u l a t i o n s see r e f e r e n c e ( £ ) . I t i s l i k e l y t h a t c a l c u l a t i o n s o f t h i s k i n d w i l l be e v e n more u s e d i n t h e f u t u r e s i n c e t h e y a r e w e l l s u i t e d f o r modern m i n i c o m p u t e r s . We w o u l d l i k e t o e m p h a s i z e h e r e some a d d i t i o n a l m e t h o d o l o g i c a l developments and t h e i r r e s u l t s . The f i r s t i s the v a r i a t i o n a l treatment of f u l l y antisymmetrized t r i a l functions (5). The second i s the G r e e n ' s f u n c t i o n Monte C a r l o a l g o r i t h m ( £ , 6) w h i c h h a s , i n e f f e c t , made p o s s i b l e t h e n u m e r i c a l i n t e g r a t i o n of the S c h r o d i n g e r e q u a t i o n . F e r m i o n Monte C a r l o Consider

a

Hamiltonia

H = - ^ X Z v

i

+

1

Z I v ( r ±<J

)

.

(1)

L e t R s t a n d f o r the c o o r d i n a t e s o f a l l p a r t i c l e s and ^ E

,

|^ (R)| dR 2

T

T

p o i n t s {R } w e r e d r a w n density function m

2

T

average

2

H^ (R)dR/

=|* (R)| /|

T

(R)| dR

(2)

the ground s t a t e o f the system. E q . (2) w e r e r e w r i t t e n as

|* (R)| * (R) and dom

|^

**(R)H* (R)dR/

T

the

at

ran-

(4)

|* (R)| dR 2

T

(3)

population of

the

take

have

quantity

m=l i s Eip. F u r t h e r m o r e duct form: * (R,A) T

if

we

="TTf(r..,A) i<j 1

J

ip (R) T

= exp{z

the

pro-

u ( r . . , A ) } (6)

\ YZ i<j

then sampling p i s exactly analagous Boltzmann d i s t r i b u t i o n s i n c e u(r^j,A) T

to

J

to is

sampling the a repulsive


17.

KALOS ET AL.

Quantum Liquids and Crystals

221

f u n c t i o n w h i c h s e r v e s t o keep t h e p a r t i c l e s a p a r t s i m i l a r t o the r o l e o f t h e p o t e n t i a l , v ( r ^ j ) , f o r c l a s s i c a l s y s t e m s . The s a m p l i n g may be a c c o m p l i s h e d by t h e Monte C a r l o method o f M e t r o p o l i s e t a l . (3^)- T h i s method i s a u s e f u l s i m u l a t i o n method f o r Bose l i q u i d s l i k e He-4. C r y s t a l s can a l s o be s t u d i e d w i t h a t r i a l f u n c t i o n o f t h e form *

(R) = exp{-

\

u ( r . .)} TT * (

r

J

(7)

where (r) i s a s i n g l e p a r t i c l e o r b i t a l w h i c h l o c a l i z e s p a r t i c l e I c l o s e t o l a t t i c e s i t e s^. U s u a l l y i s s e t as 2

| i n a way t h a t was c o m p u t a t i o n a l l y econom i c a l i n s p i t e o f the n e c e s s i t y o f e v a l u a t i n g s e v e r a l d e t e r m i n a n t s a t each s t e p of t h e random w a l k . They found t h a t the c o n v e r g e n c e o f t h e p e r m u t a t i o n expan s i o n appears s a t i s f a c t o r y a t o n l y one d e n s i t y - t h a t for which the pressure i s zero. At other d e n s i t i e s ,


222

COMPUTER


or for the k i n e t i c or p o t e n t i a l energies separately, the convergence i s poor. F o r e x a m p l e a t pcP = 0 . 4 1 4 , t h e f e r m i o n Monte C a r l o g i v e s an e n e r g y o f 2 . 8 4 ° K p e r atom w h i l e t h e f i r s t o r d e r p e r m u t a t i o n e x p a n s i o n g i v e s 1.77°K. In fact the f i r s t order gives only h a l f the f u l l antisymmetrization correction. I t s h o u l d be s t r e s s e d t h a t t h e s e c a l c u l a t i o n s s t i l l do n o t g i v e c o r r e c t l y the observed energy of l i q u i d He-3 ( - 1 . 3 ° K instead of - 2 . 5 ° K ; of course there i s considerable c a n c e l l a t i o n ; the p o t e n t i a l energy i s about - 1 1 ° K per atom). T h i s c a n be a s c r i b e d t o an i n c o r r e c t p o t e n t i a l b u t t h e r e i s e v e n more s e r i o u s d o u b t a b o u t t h e a c c u r a cy o f the p r o d u c t t r i a l f u n c t i o n . We w i l l d i s c u s s i n t h e n e x t s e c t i o n some c o n s e q u e n c e s o f t h e n e g l e c t o f t h r e e body c o r r e l a t i o n s i n the t r i a l f u n c t i o n . R e f e r e n c e (_5) a l s o t r e a t e d f e r m i o n s y s t e m s w h i c h model n e u t r o n and n u c l e a plified pair potential c o m b i n a t i o n o f Yukawa f u n c t i o n s : v(r)

(ID l

The W u - F e e n b e r g e x p a n s i o n a l w a y s u n d e r e s t i m a t e d t h e e n e r g y c a l c u l a t e d by the f e r m i o n Monte C a r l o method. In t r e a t i n g c r y s t a l phases of f e r m i o n systems i t is known t h a t t h e e f f e c t o f a n t i s y m m e t r y ( t h e e x c h a n g e energy) i s very s m a l l . E q s . (7) a n d (8) may t h e n b e used for the t r i a l f u n c t i o n s . The e q u a t i o n o f s t a t e of fermions i n t e r a c t i n g w i t h a s i n g l e r e p u l s i v e Yukawa was d e t e r m i n e d f o r b o t h l i q u i d a n d c r y s t a l p h a s e s and a c r i t i c a l s t r e n g t h d e t e r m i n e d f o r t h e existence of a phase t r a n s i t i o n . It i s also worth mentioning that r e c e n t l y d e v e l o p e d i n t e g r a l e q u a t i o n s (9_, !10) t h a t e x t e n d t h e HNC m e t h o d t o i n c l u d e a n t i s y m m e t r i z a t i o n ( t h e FHNC e q u a t i o n s ) g i v e a good a c c o u n t o f the n e u t r o n f l u i d energ i e s a t low d e n s i t i e s . At high d e n s i t i e s , different e x p r e s s i o n s f o r t h e k i n e t i c e n e r g y t h a t w o u l d be e q u a l in a correct variational calculation give rather d i f f e r e n t answers i n d i c a t i n g t h a t the expansions are not w e l l behaved. I n t e r e s t i n g l y , one o f t h e e x p r e s s i o n s f o r t h e k i n e t i c e n e r g y , t h a t due t o P a n d h a r i pande and Bethe (1), g i v e s good agreement w i t h the f e r m i o n Monte C a r l o r e s u l t s i n a l l c a s e s . This fact i s not yet e x p l a i n e d . C e p e r l e y (11) has a p p l i e d the methods d i s c u s s e d h e r e t o t h e t r e a t m e n t o f a n e l e c t r o n g a s i n b o t h two and t h r e e d i m e n s i o n s w i t h a u n i f o r m n e u t r a l i z i n g b a c k ground. He c o n s i d e r e d t h r e e p o s s i b l e s t a t e s : fluid with h a l f the spins up; f l u i d with a l l spins a l i g n e d ;


17.


KALOS E T A L .

223

and c r y s t a l p h a s e s . He found t h a t a t low d e n s i t y t h e c r y s t a l phase i s f a v o r e d . A t i n t e r m e d i a t e d e n s i t i e s the t o t a l l y p o l a r i z e d f l u i d i s most s t a b l e and a t h i g h d e n s i t i e s theequilibrium state i s that of the unpolarized f l u i d . I n t h r e e d i m e n s i o n s t h e t r a n s i t i o n den s i t i e s o c c u r a t (5.4±0.4) x 1 0 / c m and a t (9.2±1.8) x lol^/cm . fi d good agreement f o r t h e u n p o l a r i z e d f l u i d w i t h o t h e r t h e o r e t i c a l work (!L2, 1J3) a t low d e n s i t i e s b u t n o t a t h i g h d e n s i t y where t h e Monte C a r l o i s a t i t s most a c c u r a t e . The e q u a t i o n o f s t a t e f o r the c r y s t a l i s i n agreement w i t h an anharmonic expan s i o n method (.14) . We b e l i e v e t h a t f e r m i o n Monte C a r l o w i l l f i n d s i g n i f i c a n t f u t u r e a p p l i c a t i o n s , p o s s i b l y i n Quantum Chemistry. 1 3

3

3

H

e

n

s

Green's F u n c t i o n Mont T h i s i s a c l a s s o f a l g o r i t h m s w h i c h makes f e a s i b l e on c o n t e m p o r a r y computers an e x a c t Monte C a r l o s o l u t i o n o f the Schrodinger equation. I t i s exact i n the sense t h a t a s t h e number o f s t e p s o f t h e random w a l k becomes l a r g e t h e computed energy t e n d s toward t h e ground s t a t e energy o f a f i n i t e system o f bosons. I t s h a r e s w i t h a l l Monte C a r l o c a l c u l a t i o n s the p r o b l e m of s t a t i s t i c a l e r r o r s and (sometimes) b i a s . In the s i m u l a t i o n s o f e x t e n s i v e systems, i n a d d i t i o n , there i s t h e a p p r o x i m a t i o n o f a u n i f o r m f l u i d by a f i n i t e p o r t i o n w i t h (say) p e r i o d i c boundary c o n d i t i o n s . The l a t t e r a p p r o x i m a t i o n appears t o be l e s s s e r i o u s i n quantum c a l c u l a t i o n s t h a n i n c o r r e s p o n d i n g c l a s s i c a l ones. We c a n g i v e h e r e o n l y a s k e t c h o f the b a s i s o f the method; f o r more d e t a i l s c o n s u l t r e f e r e n c e (£) and (£> • 3N I n d i m e n s i o n l e s s form and i n t h e space R f o r an N body s y s t e m , S c h r o d i n g e r ' s e q u a t i o n may be w r i t t e n as 2

-[V +V(R) ]*(R) = E<MR)

(12)

where V(R) i s t h e t o t a l p o t e n t i a l energy. Suppose t h e p o t e n t i a l energy i s bounded from below (13)

V(R) > - V . Q

Then Eq.

(12) may be r e w r i t t e n a s (-V +V(R)+V )iMR) = (E+V )iMR) 2

Q

0


(14)

224

COMPUTER

Consider Green's which s a t i s f i e s

function

for

the

(-V +V(R)+V )G(R,R ) 2

0

Q

=


operator

on the

6(R-R )

left,

( 1 5 )

Q

a n d a l l a p p r o p r i a t e b o u n d a r y c o n d i t i o n s f o r \p ( R ) . For example, i n s i m u l a t i n g an e x t e n s i v e system u s i n g a f i n i t e number o f p a r t i c l e s and p e r i o d i c b o u n d a r y c o n d i t i o n s , G must be p e r i o d i c w i t h r e s p e c t t o e v e r y c o ordinate. Equation 1 3 implies that G ( R , R Q ) i s always n o n n e g a t i v e and c a n be i n t e r p r e t e d as a d e n s i t y function. With the help o f such a Green's f u n c t i o n , E q . ( 1 4 ) may b e f o r m a l l y i n t e g r a t e d t o g i v e *(R)

=

(E+V )

G(R,R')iHR')dR'

Q

•

(16)

T h i s e q u a t i o n may b f u n c t i o n ip(n) t the n iterate. Asymptotically i p ( ) i s p r o p o r t i o n a l t o I ^ Q , t h e g r o u n d s t a t e wave function. The c o e f f i c i e n t is asymptotically constant i f E i n E q . (16) i s set equal to the ground s t a t e energy E Q . I t i s n o t d i f f i c u l t t o see t h a t i f a t any s t a g e n o n e h a s a p o p u l a t i o n o f p o i n t s {R][ ^ } d r a w n a t r a n d o m f r o m \p( ' , a n d i f o n e s a m p l e s {Rj[ +1) } a t t

a

n

n

n

n

n

random u s i n g t h e d e n s i t y f u n c t i o n ( E + V Q ) G ( R ^ ^ ) , R ^ ^ ) , t h e n t h e e x p e c t e d d e n s i t y o f t h e new p o p u l a t i o n n e a r R is 1

(E+V )ij; Q

( n )

(R')dR'

E i^

( n + 1 )

(R)

n

(17)

T h i s d e f i n e s one s t e p o f a random w a l k whose asympt o t i c d e n s i t y i s ijjg (R) . G ( R , R o ) i s n o t known e x p l i c i t l y (or by q u a d r a t u r e ) f o r any b u t t h e most s i m p l e (and u n i n t e r e s t i n g p r o b lems) . But i t i s c l e a r l y r e l a t e d to the s o l u t i o n of a d i f f u s i o n problem for a p a r t i c l e s t a r t i n g at R Q i n a 3 N d i m e n s i o n s p a c e and s u b j e c t t o a b s o r p t i o n p r o b a b i l i t y V(R) + V Q per u n i t time. We t h e r e f o r e expect t o be a b l e t o sample p o i n t s R f r o m G ( R , R Q ) conditional on R o . I t t u r n s o u t t o b e p o s s i b l e b y means o f a r e c u r s i v e random walk i n w h i c h e a c h s t e p i s drawn from a known G r e e n ' s f u n c t i o n f o r a s i m p l e s u b d o m a i n o f t h e f u l l space f o r the wavefunction. References (£) and (6) c o n t a i n a t h o r o u g h d i s c u s s i o n o f t h i s essential t e c h n i c a l p o i n t , and a l s o o f t h e methods w h i c h p e r m i t t h e a c c u r a t e c o m p u t a t i o n o f t h e e n e r g y and o t h e r q u a n tum e x p e c t a t i o n s s u c h a s t h e s t r u c t u r e f u n c t i o n , m o mentum d e n s i t y , B o s e - E i n s t e i n c o n d e n s a t e f r a c t i o n , a n d


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crystal structure. One t e c h n i c a l p o i n t i s w e l l w o r t h r e i t e r a t i n g h e r e : i f o n e m o d i f i e s E g . (16) by m u l t i p l y i n g t h r o u g h by ^ T ( R ) , a t r i a l f u n c t i o n o f the k i n d used i n v a r i a t i o n a l methods, t h e n the Monte C a r l o v a r i a n c e o f a l l q u a n t i t i e s may b e c o n s i d e r a b l y r e d u c e d . When t h i s t r a n s f o r m a t i o n o f E q . (16) is carried out, i t i s p o s s i b l e t o e s t a b l i s h an e s t i m a t o r f o r the e n e r g y whose v a r i a n c e v a n i s h e s i n t h e l i m i t ~* ^0* I n p r a c t i c e we f i n d t h a t o p t i m u m v a r i a t i o n a l w a v e f u n c t i o n s always r e d u c e the v a r i a n c e o f the energy by large ratios. L i m i t e d e x p l o r a t i o n h a s a l s o shown that s i g n i f i c a n t departures from such t r i a l functions may c h a n g e t h e v a r i a n c e ( u s u a l l y , but not always, for the worse) but t h a t w i t h i n s t a t i s t i c s the answers agree. Results

Obtained wit

A number o f s y s t e m s h a v e b e e n s t u d i e d w i t h a l gorithms o f the c l a s s d e s c r i b e d here. A l l t h o s e t o be d i s c u s s e d here used the a c c e l e r a t i o n technique o u t l i n e d a t t h e end o f the p r e v i o u s s e c t i o n by e m p l o y i n g a ip w h i c h h a d t h e f o r m o f E q . (6) f o r f l u i d s a n d Eqs. ( 7 , 8) f o r crystals. C e p e r l e y , C h e s t e r , a n d K a l o s c a r r i e d o u t two studies (15_, 16_) o f b o s o n s y s t e m s i n w h i c h p a i r Y u k a wa forces (one t e r m w i t h e > 0 i n E q . ( 1 1 ) ) were used. F o r t h e most p a r t , t h e e n e r g y v a l u e s a g r e e d rather well with variational results, usually lying o n l y 1% l o w e r . I n c e r t a i n c a s e s d i s a g r e e m e n t s o f up t o 4% w e r e n o t e d . On t h e o t h e r h a n d , t h e r a d i a l d i s t r i b u t i o n f u n c t i o n was f o u n d t o b e s i g n i f i c a n t l y sharpe r i n t h e GFMC c a l c u l a t i o n s : t h e f i r s t p e a k o f the r a d i a l d i s t r i b u t i o n was u s u a l l y 10% m o r e p e a k e d t h a n i n the c o r r e s p o n d i n g v a r i a t i o n a l calculation. The g e n e r a l l y g o o d a g r e e m e n t w i t h t h e variational e n e r g i e s i n d i c a t e s t h a t the e q u a t i o n s o f s t a t e o f f l u i d s a n d c r y s t a l s w i t h Yukawa a n d s i m i l a r forces a r e g i v e n a d e q u a t e l y f o r most p u r p o s e s by the o r d i n a r y v a r i a t i o n a l method. F o r the s t r o n g l y c o u p l e d e l e c t r o n fermion f l u i d ( i . e . a t low d e n s i t i e s ) one c a n e s t i m a t e t h a t the change i n energy from the c o r r e s p o n d i n g Bose fluid is small. T h i s suggests t h a t the e r r o r i n u s i n g a p r o d u c t t r i a l f u n c t i o n a s i n E q . (9) i s c o r r e s p o n d ingly small. B u t t h e r e i s one p o i n t a b o u t t h e c o m p a r i s o n o f GFMC t o v a r i a t i o n a l r e s u l t s t h a t seems g e n e r a l l y a p p l i c a b l e and i m p o r t a n t : v a r i a t i o n a l r e s u l t s for the c r y s t a l e n e r g i e s are c l o s e r t o the e x a c t n u m e r i c a l r e s u l t s than are c o r r e s p o n d i n g r e s u l t s f o r the fluid. T h a t means t h a t t h e e s t i m a t i o n o f m e l t i n g a n d f r e e z i n g T


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d e n s i t i e s from v a r i a t i o n a l r e s u l t s w i l l have a s y s t e matic bias favoring c r y s t a l states. K a l o s , Levesque, and V e r l e t (6) r e p o r t e d a s t u d y on t h e hard-sphere f l u i d and c r y s t a l . They found t h e energy about 3-5% deeper t h a n had been o b t a i n e d v a r i a t i o n a l l y (17) and a s t r u c t u r e f u n c t i o n some 10-20% s h a r p e r . The a u t h o r s a l s o d e v e l o p e d a p e r t u r b a t i o n t h e o r y c o n n e c t i n g h a r d - s p h e r e and o t h e r s t r o n g l y r e p u l s i v e p o t e n t i a l s . Using t h i s r e l a t i o n they estima t e d a minimum energy f o r f l u i d He-4 w i t h L e n n a r d Jones p o t e n t i a l s (Eq. (10)) as -6.8±0.2°K/atom o c c u r i n g a t 1.0±.l o f t h e e x p e r i m e n t a l l y o b s e r v e d d e n s i t y , P0C u r r e n t l y , W h i t l o c k e t a l . (18) have been t r e a t i n g t h e Lennard-Jones f l u i d and FCC c r y s t a l by means o f t h e GFMC method. The a n a l y s i s and c a l c u l a t i o n o f certain correction l i m i n a r y r e s u l t s suppor sults. The e q u i l i b r i u m f l u i d i s found a t p/po = 1.03 w i t h an energy o f -6.85°K/atom. T h i s i s i n s t r i k i n g c o n t r a s t w i t h t h e v a r i a t i o n a l t r e a t m e n t based on a p r o d u c t w a v e f u n c t i o n , Eq. (6) f o r w h i c h no c a l c u l a t i o n has g i v e n a r e s u l t deeper t h a n -6°K. The e x p e r i m e n t a l r e s u l t i s -7.14°K. Thus t h e L e n n a r d - J o n e s p a r a m e t e r s o f Eq. (10) g i v e s u b s t a n t i a l l y b e t t e r e q u a t i o n o f s t a t e t h a n had been supposed on the b a s i s o f v a r i a tional calculations. Variational calculations with t h e p r o d u c t t r i a l f u n c t i o n g i v e r a t h e r c r u d e upper bounds t o the e q u a t i o n o f s t a t e and hence r a t h e r l i m i t e d i n f o r m a t i o n about t h e He-He p o t e n t i a l . Two t h e o r e t i c a l s t u d i e s have shed l i g h t on t h e d i s c r e p a n c y between the v a r i a t i o n a l and GFMC e n e r g i e s f o r l i q u i d He-4. Chang and C a m p b e l l (.19) e s t i m a t e d by a p e r t u r b a t i o n a l t h e o r y t h a t about h a l f t h e d i s c r e p a n cy between the two r e s u l t s c o u l d be a s c r i b e d t o t h e n e g l e c t o f three-body c o r r e l a t i o n s i n t h e t r i a l f u n c t i o n . More r e c e n t l y , P a n d h a r i p a n d e (20) used a t r i a l f u n c t i o n w i t h a three-body c o r r e l a t i o n corresponding t o a l i n e a r i z e d " b a c k - f l o w " (21). W i t h i n t h e frame work o f i n t e g r a l e q u a t i o n s o f t h e HNC t y p e , he c a l c u l a t e d a minimum energy o f (-6.72±0.2)°K a t a P / P Q o f 1.05. These t h r e e - b o d y c o r r e l a t i o n e f f e c t s are un d o u b t e d l y i m p o r t a n t i n He-3 as w e l l . A s i g n i f i c a n t p a r t o f t h e d i s c r e p a n c y between t h e s t r u c t u r e f u n c t i o n deduced from v a r i a t i o n a l c a l c u l a t i o n s and from e x p e r i m e n t can a l s o be a s c r i b e d t o t h e n e g l e c t o f three-body c o r r e l a t i o n s i n the t r i a l f u n c tions. F i g u r e 1 shows a c o m p a r i s o n o f e x p e r i m e n t a l , v a r i a t i o n a l , and GFMC e s t i m a t e s o f S ( k ) . I t i s clear t h a t t h e l a t t e r agrees b e t t e r w i t h e x p e r i m e n t t h a n t h e



Figure 1. Comparison of structure functions for He-4 at equilibrium density. The solid line shows the smoothed experimental data of Achter and Meyers (Phys. Rev. (1969) 188, 291) with bars indicating one standard deviation. S(K) computed variationally is shown by triangles; the circles show the results computed using GFMC.

-a

to to

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variational result. I t i s a l s o c l e a r , i n s p i t e o f the improvement o f e q u a t i o n o f s t a t e and o f S(k) t h a t r e s u l t s from the use o f GFMC, t h a t the L e n n a r d - J o n e s p o t e n t i a l c a n n o t be the c o r r e c t one. Three body p o t e n t i a l e f f e c t s a r e s m a l l i n He-4; we e s t i m a t e t h a t a t p = P Q the t r i p l e d i p o l e f o r c e c o n t r i b u t e s about 0.16°K/atom, f a i r l y c l o s e t o the r e s u l t g i v e n by Murphy and B a r k e r (22) . There i s good r e a s o n t o b e l i e v e i n g e n e r a l t h a t t h e e f f e c t o f t h r e e body f o r c e s on t h e e q u a t i o n of s t a t e w i l l be s m a l l as s u g g e s t e d by t h i s p a r t i c u l a r r e s u l t . T h i s i s a consequence o f the h i g h energy o f t h e f i r s t e x c i t e d s t a t e of helium. I t i s not u n r e a s o n a b l e t o hope t h a t two body f o r c e s d e t e r m i n e d from s c a t t e r i n g , v i r i a l and t r a n s p o r t d a t a s h o u l d a l s o be c o n s i s t e n t w i t h the p r o p e r t i e s o f h e l i u m l i q u i d s and c r y s t a l s . Conclusions I n t h i s b r i e f r e v i e w we have chosen t o concen t r a t e upon the c h a r a c t e r o f some new methods f o r t h e Monte C a r l o m o d e l l i n g o f quantum systems. I n so d o i n g we have emphasized c e r t a i n d e f i c i e n c i e s o f the o l d e r method w h i c h r e s t s upon t h e p r o d u c t t r i a l f u n c t i o n i n a v a r i a t i o n a l expression. I t i s n e c e s s a r y t o remark t h a t t h i s l a t t e r t e c h n i q u e remains u s e f u l : i t i s a r e a s o n a b l e g u i d e t o t h e phenomena i n quantum systems and f o r s o f t - c o r e systems g i v e s r e s u l t s f o r the equa t i o n o f s t a t e o f l i q u i d s and c r y s t a l s w h i c h are ade quate f o r most p u r p o s e s . The e x t e n s i o n o f the Monte C a r l o v a r i a t i o n a l method t o i n c l u d e t h r e e - b o d y c o r r e l a t i o n s i s s t r a i g h t f o r w a r d but c o m p u t a t i o n a l l y s l o w ; i t s h o u l d be done t o p r o v i d e r e l i a b l e c h e c k s on t h e t h e o r e t i c a l work on s u c h e f f e c t s i n He-4. The s t u d y o f inhomogeneous systems and m i x t u r e s remains l a r g e l y unexplored. The f e r m i o n and Green's F u n c t i o n Monte C a r l o a r e i m p o r t a n t i n t h e m s e l v e s and i n t e r e s t i n g as h i n t s t o t h e r i c h n e s s o f methodology t h a t can be b r o u g h t t o b e a r on the c o m p u t a t i o n o f quantum systems. I n t h e s h o r t t e r m we e x p e c t t o broaden the s p e c i f i c t r i a l f u n c t i o n s used i n f e r m i o n Monte C a r l o t o p e r m i t the more a c c u r a t e s t u d y o f He-3 and t h e t r e a t m e n t o f more r e a l i s t i c models o f n u c l e a r and n e u t r o n m a t t e r . We e x p e c t a l s o t o t r y a v a r i a n t i n Quantum C h e m i s t r y problems. The most immediate r e s e a r c h we p l a n w i t h the Green's F u n c t i o n Monte C a r l o i s t h e e x p l o r a t i o n o f the e q u a t i o n of s t a t e of He-4 l i q u i d s and c r y s t a l s w i t h two body p o t e n t i a l s w h i c h f i t more d a t a t h a n the


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Lennard-Jones-de B o e r - M i c h e l s form we have used. The e x p l o r a t i o n o f inhomogeneous systems i s a l o g i c a l n e x t s t e p . A t t h e moment o n l y GFMC o f f e r s an unambiguous approach t o t h e c a l c u l a t i o n o f t h e d e n s i t y p r o f i l e a t an i n t e r f a c e , f o r example. F o r t h e f u t u r e we a n t i c i p a t e t h e development o f p r a c t i c a l methods f o r t h e t r e a t m e n t o f systems a t f i n i t e t e m p e r a t u r e s (some o f t h e t e c h n i c a l problems o f e x t e n d i n g GFMC t o t e m p e r a t u r e s g r e a t e r t h a n z e r o have r e c e n t l y been s o l v e d f o r t h e two-body h a r d sphere p r o b l e m (23)) . An e x t e n s i o n t o p e r m i t t h e e x a c t o r v e r y r e l i a b l e t r e a t m e n t o f f e r m i o n systems seems p o s s i b l e and u s e f u l . Acknowledgment T h i s work was s u p p o r t e o f Energy, C o n t r a c p a r t by t h e N a t i o n a l S c i e n c e F o u n d a t i o n under G r a n t DMR-74-23494 t h r o u g h t h e M a t e r i a l S c i e n c e C e n t e r , Cornell University. Literature Cited 1 2 3 Teller, 4 Statistical 5 6 7 8 9 10 11 12 13 14 15

P a n d h a r i p a n d e , V.R. and B e t h e , H.A., Phys. Rev. C, (1973), 7, 1312. M c M i l l a n , W.L., Phys. Rev., (1965), 138 A, 442. M e t r o p o l i s , N., R o s e n b l u t h , A.W., R o s e n b l u t h , M.N., A.H., and Teller, E., J. Chem. Phys., (1953) 2 1 , 1087. C e p e r l e y , D.M. and Kalos, M.H., "Quantum ManyBody P r o b l e m s " , Chap. IV o f Monte Carlo Methods in Physics, K. B i n d e r , e d . ; in press, Springer-Verlag. C e p e r l e y , D.M., Chester, G.V., and Kalos, M.H., Phys. Rev. B, (1977), 16, 3081. K a l o s , M.H., L e v e s q u e , D. and Verlet, L., Phys. Rev. A., (1974), 9, 2178. Wu, F.Y. and Feenberg, E., Phys. Rev., (1962), 128, 943. S c h i f f , D. and Verlet, L., Phys. Rev., (1967), 160, 208. Z a b o l i t z k y , J., Phys. Rev. A., ( 1 9 7 7 ) , 16, 1258. Day, B., Rev. Mod. Phys., in press. C e p e r l e y , D.M., to be published. S t e v e n s , F.A. and P o k r a n t , M.A., Phys. Rev. A, (1973), 8, 990. Freeman, D.L., Phys. Rev. B, (1977), 15, 5513. C a r r , W.J., Phys. Rev., (1961), 122, 1437. C e p e r l e y , D.M., C h e s t e r , G.V. and Kalos, M.H.,



230

16 17 18 19 20 print, 21 22 23

Phys. Rev. D, in press. C e p e r l e y , D.M., C h e s t e r , G.V. and Kalos, M.H., Phys. Rev. D, (1976), 13, 3208. Hansen, J.-P., L e v e s q u e , D. and Schiff, D., Phys. Rev. A, (1971), 3, 776. W h i t l o c k , P.A., K a l o s , M.H., C h e s t e r , G.V. and C e p e r l e y , D.M., to be published. Chang, C.C. and C a m p b e l l , C.E., Phys. Rev. B, (1977), 15, 4238. Pandharipande,V.R., University of Illinois pre December, (1977), I L L - ( T H ) - 7 7 - 4 1 . Feynman, R.P. and Cohen, M., Phys. Rev., (1956), 102, 1189. Murphy, R.D. and B a r k e r , J.A., Phys. Rev. A, (1971), 3, 1037. Whitlock, P.A. and K a l o s , M.H., s u b m i t t e d t o J. Comp. P h y s . , (1978)

RECEIVED September 7, 1978.


18 From Microphysics to Macrochemistry via Discrete Simulations JACK S. TURNER Center for Statistical Mechanics and Thermodynamics, University of Texas, Austin, TX 78712 Historically there are two distinct classes of problems in chemistry to which discret applied widely and with considerable success: At one extreme the bulk physical properties of atomic and molecular fluids are studied as the "exact" dynamical evolution of a collection of representative particles is followed in "computer experiments" using the well-established method of molecular dynamics (1-10). At the other extreme similar techniques (11-16) are used to explore the chemical transformations which may occur in an isolated collision between potentially reactive species (e.g., in a very dilute gas). Between these mutually exclusive limits lies an increasingly important area of chemistry in which macroscopic properties are a direct consequence of cooperative interplay between molecular motion and chemical dynamics. Of immediate interest in this area are chemical systems which may undergo nonequilibrium i n s t a b i l i t i e s and transitions between different regimes of macroscopic physicochemical behavior (See Ref. 17 for a recent survey of the f i e l d ) . Fully analogous to equilibrium phase transitions and c r i t i c a l phenomena, these "nonequilibrium phase transitions" present even more formidable d i f f i c u l t i e s in both microscopic theory and laboratory experiment. Hence an obvious need exists for detailed computer simulations to provide "experimental" data for theorists and "theoretical" guidance for experimentalists. Since the development of appropriate simulation methods in this area is in its infancy, the purpose of this paper will be to provide motivation for such investigations in the more familiar context of molecular dynamics in classical f l u i d s , to review feasible methods at two levels of description which have emerged in the last few years (18, 19, 20, 21, 22), and to i l l u s trate their application using two simple models which exhibit chemical i n s t a b i l i t i e s and transitions. In a l l three classes of problems the systems of interest are characterized by the interaction of large numbers of individual degrees of freedom. It is this feature which leads to great theoretical d i f f i c u l t i e s in both classical and quantum systems, 0-8412-0463-2/78/47-086-231$08.25/0 © 1978 American Chemical Society In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

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and hence provides strong motivation for numerical "experiments" using high-speed digital computers. Appropriate simulations i n volve molecular dynamics [MD] (1-1J3) or Monte Carlo [MC] [7_ 23, 24, 25) methods which require the details of microscopic interactions and processes. During the course of a simulation this essential information is obtained from a potential energy function (or surface), and i t is our limited a b i l i t y to provide this i n formation which serves to delineate the two extreme problem classes for practical application. In the f i r s t problem class mentioned above (hereinafter called class A), a collection of particles (atoms and/or molecules) is taken to represent a small region of a macroscopic system. In the MD approach, the computer simulation of a laboratory experiment is performed in which the "exact" dynamics of the system is followed as the particles interact according to the laws of classical mechanics. Used extensively to study the bulk physical properties of classical f l u i d s , such MD simulations can yield information about equilibrium (See Ref. ^ for a review) in addition to the equation of state and other properties of the system at thermodynamic equilibrium (7,, 8_ for example). Current a c t i v i t i e s in this class of microscopic simulations is well documented in the program of this Symposium. Indeed, the state-of-the-art in theoretical model-building, algorithm development, and computer hardware is reflected in applications to relatively complex systems of atomic, molecular, and even macromolecular constituents. From the practical point of view, simulations of this type are limited to small numbers of particles (hundreds or thousands) with not-toocomplicated inter-particle force laws (spherical symmetry and pairwise additivity are typically invoked) for short times (of order 10"' to 10"^ second in liquids and dense gases). In direct contrast to simulations of physical properties dominated by inter-particle forces, the second problem class (class B) concerns interactions among intramolecular and intermolecular degrees of freedom in an event which produces a chemical change in the participants. Severely limited by the need for an accurate quantum mechanical potential energy surface, simulations in this area typically involve classical trajectory ( i . e . , MD) calculations for the simplest chemical reactions 9

2

(11-16).

It is not surprising that the two main classes of microscopic simulations have evolved quite independently. Aside from the obvious problem of calculating potential energy functions (surfaces), the greatest computational d i f f i c u l t y arises in treating systems with multiple time scales. Dynamical simulations within class A are feasible because the bulk properties of interest can be determined on a time scale corresponding to a computationally f i n i t e number of molecular c o l l i s i o n s . When the most important events are rare on this time scale, one rapidly reaches the limits of f e a s i b i l i t y for detailed molecular dynamics


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simulations. With rare exceptions, collisions in which the p a r t i cipants are chemically transformed are indeed rare dynamical events. Hence the combination of potential energy surface and time-scale d i f f i c u l t i e s means in particular that complex chemical reactions in a many body system are beyond the reach of truly microscopic simulations. That i s , the systems of immediate interest in the intermediate class (class C) cannot be treated by straightforward extension (and combination) of class A and class B methods. This conclusion is certainly nothing new to workers in either of the two main areas of simulation. Indeed much effort in recent years has been directed at precisely the problem which is addressed here in the specific context of chemical i n s t a b i l i ties and nonequilibrium phase transitions: How can we simulate the evolution and steady state behavior of systems in which the events of greatest interest are rare on the time scale of i n dividual c o l l i s i o n s , bu collective many-body dynamics Multiple Time Scales, Bottlenecks and Many-Body Dynamics Systems in which problems of multiple time scales and of "bottlenecks in phase space" occur are hardly exceptional, and promising methods of theory and discrete-event simulation have been developed recently in several important areas. In a marriage of molecular dynamics (and Monte Carlo methods) to transition state theory (2&), for example, Bennett (2Z) has developed a general simulation method for treating a r b i t r a r i l y infrequent dynamical events ( e . g . , an enzyme-catalyzed reaction process). An entirely different class of problems has been the recent focus of Adelman, Doll, and coworkers (28, 29, 30, 31_, 32J, who have investigated the microscopic dynamics of scattering, sorption, and reaction at the gas-surface interface. By taking the manybody l a t t i c e dynamics into account in a novel theoretical model based on nonequilibrium s t a t i s t i c a l mechanics, these authors have developed a new method of molecular dynamics involving generalized Langevin equations of motion. The new simulation techniques of Bennett (27.) and Adelman and Doll (28_, 29) i l l u s t r a t e well several important aspects of the present problem. The former considers relatively fast but infrequent events, includes a l l relevant degrees of freedom expressible in a potential energy surface, and concentrates on the dynamical bottlenecks separating regions of the system's state (phase) space (e.g., products from reactants). In contrast, the latter method treats only a subset of degrees of freedom in detail (e.g., the colliding atom and those surface atoms and neighbors most directly involved), while including in an average way the effect of coupling to other degrees of freedom of the bulk l a t t i c e . It is important to realize that both methods treat individual events or sequences of events in great d e t a i l , yielding s t a t i s t i -


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cal information about a single overall process. Like the class B chemical simulations mentioned above, these methods serve to characterize the "elementary processes" (that i s , the reactive events) with which the problems of class C are concerned. In other words, the "fundamental units" of interest in the intermediate class of systems are not simply the atoms or molecules themselves, but rather the reactive collisions and transformations. As we shall see, i t is the space-time dynamics and interactions of these new "units" (mediated, of course, by other degrees of freedom via elastic and inelastic c o l l i s i o n s , intramolecular dynamics . . . ) which determine ultimately the macroscopic behavior of such systems. In this respect we see more clearly the connection to the problems of class A. At e q u i l i brium phase transitions in classical f l u i d s , for example, the important "units" are the clusters or nuclei of the new phase embedded in the i n i t i a l phase. These units are strongly coupled to their surroundings (e.g., via c o l l i s i o n s ) , and in addition are characterized by their ow The time-scale characteristic of size and/or composition changes in individual nuclei is obviously much longer than that of the detailed microscopic dynamics. In particular, the timedelay associated with spontaneous development of a "critical nucleus" has been a source of great computational d i f f i c u l t y in simulations of both 1iquid+solid (33) and vapor+1 iquid (34-42) phase transitions in simple classical fluids. Before turning to an analogous situation at the "nonequilibrium phase transitions" which are of immediate interest, therefore, let us f i r s t illustrate the key problems and innovations in simulation which are already evident in MD studies of low-order clustering and homogeneous nucleation in the vapor phase (34-42). Molecular Dynamics in Dilute Gases In MD simulations of dense f l u i d s , the classical many-body equations of motion can be integrated e f f i c i e n t l y using a singletime-step method (e.g., Refs. 3^ and 4J. This is so because at any instant of time a l l particles are interacting strongly with a number of others. In a dilute gas, however, only a small fraction of the particles are strongly interacting at any time. If a Lennard-Jones 6-12 (LJ) potential is assumed for a model of f l u i d Argon, for example, most atoms are at most very weakly interacting via the long-range attractive t a i l of the potential. With a timestep selected for the few colliding pairs of atoms, the efficiency of the usual MD methods is lost. Recognizing that each atom spent most of its time in essentially free flight between binary c o l l i sions, Harrison and Schieve (34_, 36J introduced a cutoff ( R „ ) in the range of the LJ potential and combined the Alder ( 1 ) freeflight method for "isolated" atoms with a Rahman (3_) or Verlet (4) continuous-potential method for the remaining atoms. (For details of the dilute gas algorithm see Ref. 36_.) The result was


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a hybrid MD algorithm with which simulations of a dilute c l a s s i cal gas could be performed e f f i c i e n t l y in two and three dimensions (34, 35, 36, 37, 38). Low-Order Clustering in a Computer Model of Argon Vapor. Using the algorithm which decoupled isolated atomic motion from collisions between atom pairs, Harrison and Schieve (34, 35j f i r s t studied the approach to equilibrium in a 2D gas of (LJj Argon. At lower temperatures and higher pressures, low-order clustering was observed, beginning with the formation of bound states (dimers) by the three-body channel (35, 36^, 37, 38) A + A J A* 2

A* + A + A 2

2

+A

(1)

In the f i r s t step an unstabl the dimer is formed whic energy is then removed by a third atom in the second step to form the stable dimer A . A typical sequence is shown in Figure 1, in which a dimer 12 is created in a c o l l i s i o n with 3 and destroyed after a few vibrations in a c o l l i s i o n with 6. Detailed MD studies of the dimer formation "reaction" have been carried out in two and three dimensions (_35, 36, ^37, 38). The result of greatest interest for the present study is the time-scale problem i n troduced by the occurrence of the "chemical reaction" (Eq. 1). Since the dimer-formation process is a dynamically rare event, lengthy computer runs (of order 40,000 collisions in 2.4 x 10"' sec. model time) were required to obtain s t a t i s t i c a l l y reliable values for the dimer formation rates, lifetimes, and other properties. The problem to be faced in this situation is a classic one in computer simulation: What level of microscopic detail must be included in order to retain the essential features of the system at a not unreasonable cost? The solution of this problem in the dimer study illustrates well the general approach of this paper: Beginning with the most detailed simulation possible (which then serves as a benchmark), develop a hierarchy of simulations in which the computer model at each level incorporates, in an average way, at least, those features of lower levels relevant to the questions of interest. In this way the qualitative behavior and properties of a system can be mapped with great efficiency, and areas of particular interest so identified could be subjected to more careful, even quantitative, study in lowerlevel simulations. For the dimer simulation two simplifications were made: First was the obvious step of restricting the study to two dimensions, at least i n i t i a l l y . Second, the continuous LJ potential was replaced by a square-well hard-core (SWHC) potential with parameters determined empirically from equation-of-state data for Argon (cf, Ref. _43, p. 158). With these two refinements i t 2


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was possible not only to improve the s t a t i s t i c s of the dimer c a l culations, but also to consider s t i l l lower temperatures and higher pressures at which trimers, tetramers and other low-order clusters could be formed (39, 40., 41). Molecular Dynamics Evidence for Vapor-Liquid Nucleation. Specializing the MD model now to a 2D system of 100 SWHC "Argon" atoms, let us review briefly relevant aspects of these simulations. The SWHC potential is defined with well-depth e = 2.305 x 10-14 erg, hard core radius o\ = 2.98 x 10" cm, and range of attractive interaction a = 1.96 o\. The atomic mass is m = 6.628 x 10" g. With periodic boundary conditions, the system is adiabatic, at constant volume, total energy, and number of atoms. I n i t i a l l y the atoms are displaced slightly from the sites of a square lattice 3.334 x 10" cm (112-08 a ) on a side, and are given a fixed speed with random velocity orientation. This fixes the total energy of the ( i n i t i a l l y non-interacting) system to be the i n i t i a are formed (with negative potential energy), therefore, the kinetic energy ( i . e . , temperature) increases accordingly until an equilibrium distribution of monomers, dimers, trimers, . . . is eventually reached. A quite remarkable feature of the MD studies of Zurek and Schieve (40, 41, 42) using the above computer model is the direct observation oT~nucTeation in the vapor-liquid phase transition region. The phase transition i t s e l f is clearly indicated in Figure 2; the heat capacity C = 3E(T*)/8T* is seen to increase dramatically near Tgq * 0.5. Here T* = kT/e and the total energy E* = E/e (or Pfn) is plotted against the equilibrium kinetic temperature Tj£q. (Initial conditions with negative total energies are obtained by decreasing the temperature in the final state of higher-energy runs.) In Figure 3 a "supercritical" cluster is clearly evident in a "snapshot" of the system at Tf =0.2 (41J. This qualitative indication that nucleation has occurred is made quantitative in terms of the physical cluster distributions plotted in Figure 4. Here a dramatic change in the distribution is seen as the phase transition region is entered in 4c and4d. The gap which appears separates clusters into sub- and superc r i t i c a l classes, and the width of the gap provides bounds on the number of atoms in the "critical" cluster. It is important to realize that nucleation has introduced yet another time scale into the MD simulation--the time required for spontaneous generation of the "critical" nucleus which, in a non-adiabatic system, initiates an irreversible transition from the metastable i n i t i a l vapor state to the stable liquid phase. Although the implications of this time scale have not been treated in the Argon studies, they will be examined in the simulations of nonequilibrium phase transitions which follow. (Note: These nucleation results have now been verified in a recent MD/MC study by Rao, Berne, and Kalos, Ref. 54.) 8

2

23

6

x

v

n


TURNER


237

Journal of Chemical Physics

Figure 1. Plot of particle trajectories involved in the formation and destruction of a dimer (12) during a molecular dynamics simulation of 2-D (LJ) Argon (35)

7

~r~*

Journal of Chemical Physics

Figure 2. The spatial distribution of the particles in the system at a given instant shows a well-developed droplet containing 19 particles. The size of the points in the picture corresponds to the size of the outer wall of the potential (a ). T*ISIT 2

ss -0.2, T*j5 s£ 0.5 (41). Q


COMPUTER

238


1.2

0.8

0.4

L0

05

Figure 3. The E(T*) [T* dependence, showing the characteristic shape for T* ^ 0.5. T* has already evolved in the higher (T* i 0) temperature for some time. The slope is the specific heat c . After Ref. 42. IXIT

IX T

u

II I I I II I

I I ] I I I I

i

01

a.) E «0.5 ,

T.0.66

b.) E*-0.3,

r-0.56

c.)

r-0.53

#

I

1

1

1

' I

0.01 0.001

01 A

0.01 0001

01

E*-0 ,

001 0001

01

d.) E*--0.2,f-0.51

001 0.001 1

*

1

1

1

1

* • •

2 3 4 5 6 7 8 910

1

1

*

1

1

15

* •

i i i7 i 1

20

25

30 1

Figure 4. Mole fractions, m for clusters formed in the adiabatic system at different energies, E*. After Ref. 42. h


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Microphysics to Macro chemistry via Discrete Simulations

239

Chemical Instabilities, Nonequilibrium Phase Transitions, and Dissipative Structures Let us now digress briefly from matters of simulation to i n troduce the kinds of chemical systems in which nonequilibrium phase transitions can occur. This is most conveniently done at the macroscopic level of deterministic chemical kinetics. Consider a chemical system under open or closed conditions so that i t is possible to maintain a time-independent macroscopic state. At or near thermodynamic equilibrium, the macroscopic description of such a system yields a unique stationary solution that is always stable with respect to arbitrary perturbations. S u f f i ciently far from equilibrium, however, the governing kinetic equations for nonlinear chemical systems may admit solutions other than the continuous extension of the equilibrium solution into the nonequilibrium domain. Beyond a c r i t i c a l distance from e q u i l i brium, moreover, this thermodynamic branch may become unstable in response to fluctuation point of i n s t a b i l i t y , the response of the system to an i n f i n i t e s i mal disturbance leads ultimately to a new operating regime characterized by organization in space, time, or function. Whatever the outcome of such a macroscopic i n s t a b i l i t y , therefore, the i n stability i t s e l f originated in the response of the system to a fluctuation. Consequently a purely deterministic description of the system in terms of mean values alone is no longer adequate, and i t is essential to supplement the "average" description with a theory of fluctuations extended to nonlinear systems under farfrom-equilibrium conditions. At thermodynamic equilibrium fluctuations constitute a negligible correction to the macroscopic description of matter except near instability points such as phase transitions or c r i t i c a l points. Similarly, in nonequilibrium situations as well one expects that fluctuations become important only near points of nonequilibrium instability (see Ref. ]_7. for a detailed discussion). For nonlinear systems in which such i n s t a b i l i t i e s and transitions can occur, the role of fluctuations is even more dramatic than at equilibrium, due to the existence in general of many distinct "phases" compatible with the external conditions imposed on the system. This fact has been known for several years now, since the f i r s t (numerical) demonstration on simple chemical models of multiple solutions to the governing macroscopic differential equations (44_, 45_, _46L, 47). The variety of possible solutions includes tirne-independent homogeneous and inhomogeneous states as well as states which are organized in time (homogeneous chemical oscillations) or in time and space (travelling chemical waves). Which of these "dissipative structures" (V7) will arise under given conditions depends crucially on the specific type of perturbation to which the original state is i n i t i a l l y unstable. In terms of spontaneous fluctuations, which are present in general over a range of wavelengths, both the final state and the evolu-


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MODELING

OF MATTER

tion toward that state will be determined by the space-time characteristics of the fastest-growing unstable mode. As one would expect, developments in the theory of such phenomena have employed chemical models chosen rorc for analytical simplicity than for any connection to actual chemical reactions. Due to the mechanistic complexity of even the simplest laboratory systems of interest in this study, moreover, application of even approximate methods to more r e a l i s t i c situations is a formidable task. At the same time a detailed microscopic approach to any of the simple chemical models, in terms of nonequilibrium s t a t i s t i cal mechanics, for example, is also not feasible. As is well known, the method of molecular dynamics discussed in detail a l ready had its origin in a similar situation in the study of c l a s s i cal fluids. Quite recently, the basic MD computer model has been modified to include inelastic or reactive scattering as well as the elastic processes of interest at equilibrium phase transitions (18), and several applications of this "reactive" molecular dynamics~lRMD) method t cal i n s t a b i l i t i e s have been reported (1_8> 1_9, 22_). A variation of the RMD method will be discussed here in an application to a first-order chemical phase transition with many features analogous to those of the vapor-liquid transition treated earlier. Reactive molecular dynamics may be viewed as a microscopic computer experiment inasmuch as a l l relevant particle processes can be included to some degree. In particular the method treats elastic as well as inelastic or reactive collisions between p a r t i cles and, indeed, requires that the latter be relatively rare events in order to simulation actual chemical kinetics in a r e a l i s t i c way. When large enough systems are considered (e.g., thousands of particles) i t should be possible to "measure" temporal and spatial correlation functions, for example, and to make quantitative the notion of c r i t i c a l size and amplitude of fluctuations necessary to nucleate a macroscopic transition. A f i r s t step in the latter direction will be reported here. Applications to date of reactive molecular dynamics methods demonstrate f e a s i b i l i t y for the study of hundreds or thousands of particles involved in chemical reactions for which reactive proba b i l i t i e s do not vary too widely. The latter condition is essent i a l i f s t a t i s t i c a l l y significant numbers of a l l possible events are to occur within a practical computation period. Even i f the requirement for a large excess of elastic collisions is relaxed, however, reaction rates typical of experimental chemical systems demand simulation run times which approach the feasible limit except for quite small numbers of particles. Turning therefore to a higher level method for this type of system, one may treat numbers of particles in a small cell or volume element rather than individual particles, and invoke a Monte Carlo procedure for determining which reactive event will occur, how much time will elapse between events, and in which "cell" of the system the


18. TURNER


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process will occur. Transport between adjacent cells is handled in a similar fashion. In such a method, only the events of greatest interest (reaction or transport) are treated, so that widely different rate constants need not be a problem. In the change from reactive molecular dynamics to stochastic simulation, much larger systems may be considered, and computational efficiency is gained at the expense of molecular detail. In addition, problems related to numerical "stiffness" associated with the deterministic d i f f e r ential equations (48, 49J are completely avoided in a stochastic simulation. Finally, a number of implementations of such techniques have been presented which give qualitatively reasonable results. Of particular significance is a recent formulation due to Gillespie (20J, in which the transition probabilities and Monte Carlo implementation are rigorous consequences of the very assumptions from which chemical master equations of the birth-anddeath type are derived. This fact is of great importance for the development and testin periments based on the latter formulation simulate directly a stochastic chemical evolution satisfying a master equation of the type which is central to theoretical developments in this area (17). A report of i n i t i a l investigations of chemical i n s t a b i l i ties using stochastic simulation methods will also be presented. In order to fix the main ideas most e f f i c i e n t l y , the microscopic chemical simulations will be presented in applications to simple chemical models which together exhibit most of the known types of i n s t a b i l i t i e s and transitions. As will be seen, both RMD and stochastic simulations treat the elementary c o l l i s i o n and transport processes which occur in a chemical mixture instead of the concentration (or populations) of the constituents themselves. In the language of a Markovian stochastic description of chemical kinetics (cf, Ref. 17), both of these methods may be expressed in terms of the probabTTity P ( T , y ) 6 x that the next event will occur in the infinitesimal time increment 6x following an interval T and will be an event of type y. Such a probability leads naturally to a procedure for simulating a system's time-evolution in a "computer experiment." Each method is distinguished therefore by an algorithm for evaluating P ( x , y ) and for implementing an i t e r ative procedure to simulate an evolution of a chemical system. Microscopic Chemical Simulations:

Reactive Molecular Dynamics

In designing a simulation or "computer experiment" in a particular context, one must f i r s t determine the level of detail needed to describe the phenomena of interest. For chemical reactions, in which the fundamental interactions are between atoms and molecules, one would expect to begin at the level of microscopic dynamics. Simulations of this kind follow the details of classical many-body dynamics derived from assumed intermolecular interactions, and constitute the by now familiar method of


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molecular dynamics. As we have seen, such microscopic simulations of chemical reactions are impractical at present, largely due to the enormous number of "uninteresting" dynamical events which must be followed in detail between the reactive collisions of greatest concern. One way of avoiding part of the d i f f i c u l t y is to adopt a molecular "pseudodynamics" approach in which molecules travel in free f l i g h t between instantaneous hard-sphere collisions at which reactive processes may occur. Such methods have been employed recently by Portnow (18), by Ortoleva and Yip (19), and by the present author (22) and are applied here to some questions raised in the deterministic and stochastic formulations of chemical i n s t a b i l i t i e s . First Order Chemical Phase Transition in a Cooperative Isomerization Reaction. A convenient model of a f i r s t order transition is provided by a reversible isomerization reaction in a macroscopically homogeneous system (2)

B

having activation energies e for the forward reaction and n/N per molecule of B present in the system for the reverse reaction, where N is the total number of molecules. The activation energy for the reverse reaction depends on the number Ng of B molecules present, and therefore introduces a cooperativity into the mechanism. In physical terms, this cooperativity expresses the stabilizing influence on a B isomer of intermolecular interactions with other isomers of the same type. Aside from being a convenient way to introduce nonlinearity into the simplest chemical model, this procedure yields a model which is isomorphic to a Bragg-Williams (mean field) model for monomolecular adsorption on a uniform surface. The latter model of the equilibrium twodimensional l a t t i c e gas is known to exhibit a first-order phase transition (cf, Ref. _50) and provides a convenient vehicle for an especially intuitive discussion of the stochastic theory of metastability and nucleation at chemical first-order transitions (22, 51). (The use here of this equilibrium example is motivated by practical considerations of theory and simulation, and does not limit the generality of the results and conclusions.) If the cooperativity is introduced into the rate constant for the reverse reaction via a "mean-field" dependence on Ng, then the equilibrium constant for the isomerization reaction takes the form K = exp (-e + nN /N) B

(3)

where the activation energies are expressed in units of kgT, with kg the Boltzmann constant and the T the absolute temperature. If N is fixed (closed system), then the kinetics of the reaction is given by a single macroscopic rate equation


18. TURNER

243


W- - 0< k

N

- B> " N

e

" 0 B "

£

k

N

e

n N B / N

4

Defining a new time variable T = k e" t and the mole fraction B - B / N gives 0

X

=

X

e

N

*1 = i -

x

- xe " e

(5)

nx

A linear stability analysis of the steady state XQ of this system implies stability whenever the real normal mode frequency u is negative, and instability otherwise, where xu> = -1 + nx - nx . Points of marginal s t a b i l i t y correspond to a> = 0 and, according to the quadratic expression for a), there may be 0, 1, or 2 points. It is easy to see that there is no instability for n < n 4. For n > nc» however, there is a region of e in which three steady states are possible. Typica Figure 5. Here the curve passing through the c r i t i c a l point (nc»£c>xc) (4,2,1/2) separates regions of 1 and 3 steady states. In the three state region, the upper and lower branches are stable, the middle branch unstable, according to linear stability analysis. If i t is supposed for convenience that e is an externally variable parameter of the model, then for n > n the transition between stable branches of the deterministic solution should occur discontinuously at the points of marginal stability (arrows, Figure 5). The similarity between this picture and the pressurevolume diagram for the equilibrium liquid-vapor transition is apparent. Indeed, according to the macroscopic van der Waals model of a nonideal gas (50) the liquid-vapor phase change in a homogeneous f l u i d should occur at a lower pressure than the reverse transition. That a single equilibrium transition pressure (e.g., dashed l i n e , Figure 5, an "equal-areas" construction) is observed regardless of the direction of change is a direct consequence of the response to spatially localized, finite-amplitude fluctuations in the i n i t i a l f l u i d state. Moreover, when such fluctuations are of insufficient size or amplitude to provide a stable nucleus for the transition, then states beyond the e q u i l i brium transition point may be realized. Such states are termed metastable, since they are stable only until a large enough fluctuation appears to form a "supercritical" nucleus. At which point along the metastable branch will destabilizing fluctuations appear, and how long will the metastable state remain "stable" prior to occurrence of c r i t i c a l fluctuations? Clearly such questions are beyond the scope of the deterministic description, and will be addressed here from the viewpoint of RMD simulation. 2

c

=

=

c

Reactive Molecular Dynamics. A microscopic computer model of a chemically reacting mixture is constructed with the following


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essential features: (1) Hard-sphere dynamics. The key assumption is that the most important chemical effects occur upon energy and momentum transfer at molecular c o l l i s i o n s , the net effect of attractive intermolecular forces being to alter the collision crosssection. Hpnce the microscopic molecular dynamics is simplified enormously by replacing a r e a l i s t i c intermolecular potential by a hard core repulsion. The molecules are then approximated by rigid spheres, and in the present model are assumed in addition to have no internal degrees of freedom. As in theMD model of a dilute gas discussed earlier (34_, 36), the molecular dynamics reduces then to free-flight motion between collisions* and results in a computational efficiency which makes i t possible to include chemistry at a l l . (2) Reactive transitions. Chemistry is introduced into the model by assigning reaction probabilities to members of a colliding pair at the instant of c o l l i s i o n . This process is handled conveniently, and in a theoretically consistent fashion, by assuming an activatio nel, and selecting in a followed. If a single reaction is possible, for example, the condition for reaction is simply that the relative kinetic energy of the colliding pair exceed the activation energy for the process. Once a particular reaction channel is selected, molecular identities and other properties of the reacting molecules are altered according to the laws governing the particular chemical transformation. All particles then move in free flight until the next c o l l i s i o n occurs and the channel-selection process is repeated. It is easy to see that these two components of the RMD simulation provide the information contained in the reaction probab i l i t y function P ( x , y ) . The molecular dynamics determines the type of c o l l i s i o n C|< and the elapsed time T , while the probability of reactive process R occurring upon c o l l i s i o n C^ is given independently in the second step. y

A Discrete Model of Cooperative Isomerization. The algorithm which implements the method of reactive molecular dynamics (RMD) is understood best in the context of a specific application. Therefore let us specialize now to the simple isomerization reaction (Eq. 2) to address the questions of "chemical nucleation" raised in that context. Only the main ideas are stressed here. A more detailed account of these studies appears in Ref. Z?. Consider a binary mixture of constituents A and B, and assume that the reversible isomerization is a collision-induced reaction. There are then four elementary reactive processes which may occur upon binary c o l l i s i o n in the system: (a)

A + A

+

A +B


(6a)

18.

TURNER

Microphysics to Macrochemistry via Discrete Simulations (b)

k

i

245

B+ B

(6b)

A + A

(6c)

B+ A

(6d)

A + B Z (c) (d)

B+ B

->

Since the system is closed, the total number N E N/\ + N 3 is constant, and i t is easy to verify that in this case the determinist i c rate equation for reduces to the form of Eq. 4, where the constant N is absorbed into the pre-exponential factor kg of the latter equation. The sequence of c o l l i s i o n s , and hence a partial determination of reaction channel, is generated by the hard-sphere molecular dynamics algorithm. Fo is clear, and i t remain action, given the appropriate c o l l i s i o n , is satisfied. For type A-B collisions there are two possible channels from which to choose. Thus transition probabilities W (C^) are assigned which give the probability for reaction R to occur given a c o l l i s i o n of type C|

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