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LO^WC-RANG F FORCES COMPUTER SIMULATION CONDENSED MEDIA
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Hi r* PROCEEDINGS
OF THE WORKSHOP
THE PROBLEM OF LONG-RANGE FORCES IN THE
COMPUTER SIMULATION OF CONDENSED MEDIA
Sponsored by the National Resource for Computation in Chemistry
Lawrence Berkeley Laboratory Berkeley, California 94720
Held at
Vallambrosa Center Menlo Park, California
January 8-11, 1980
NRCC Proceedings No. 9
Edited by: David Ceperely -m-
CONTENTS
Workshop Participants vii
Foreword xi Introduction xii
SESSION I. CONTROLLED STUDIES OF LONG-RANGE FORCE PROBLEM IN SIMULATION OF IONIC SYSTEMS Review Talk--The Problem of Coulombic Forces in Computer Simulation J. P. Valleau 3 Perturbation Theory for Hard Spheres B. Larsen and S. A. Rogde 9 Periodic Boundary Conditions in Simulation of Ionic Solids E. R. Smith 10 An Alternative to the Ewald Summation? D. J. Adams 11
Periodic, Truncated-Octahedral Boundary Conditions D. J. Adams 13
Summary of Session I B. Larsen 14
SESSION II. CONTROLLED STUDIES OF LONG-RANGE FORCE PROBLEM IN SIMULATION OF DIPOLAR SYSTEMS Review Talk—Dipolar Fluids G. N. Patey 19 Calculation of (k, ) by Computer Simulations of Permanent Dipolar Systems E. L. Pollock 23 Periodic Boundary Conditions for Dipolar Systems E. R. Smith 24 Computer Simulations of Hydrogen-bonded Liquids W. L. Jorgensen 25 -Tv-
Dielectric Theory for Polar Molecules with Fluctuating Polarizability G. Stell 27 Summary of Session II J. J. Weis 28
SESSION III. EFFECT OF LONG-RANGE FORCES ON THE SIMULATION OF SOLVATION AND ON BIOPOLYMER HYDRATION: COULOMB AND HYDRODYNAMIC FORCES Review Talk P. G. Wolynes 31 Long-Range Forces and Ion Transport Across Membranes K. R. Wilson 33 Ion Triples in 2-2 Electrolytes P. G. Rossky and J. D. Dudowicz 34 The Structure of Some Models for Aqueous Nickel Chloride Solutions in the High Concentration Range H. L. Friedman and J. B. Dudowicz 37 Summary of Session III P. J. Rossky 41
SESSION IV. SPECIAL TECHNIQUES FOR LONG-RANGE FORCE SIMULATION Reaction Field Method for Polar Fluids J. A. Barker 45 A Monte Carlo Study of Electric Polarization in Water R. 0. Watts 47 Lattice Sums for Periodic Boundary Conditions E. R. Smith 49 Summary of Session IV R. L. Fulton 50 -V-
SESSION V. PLASMAS: ESPECIALLY THOSE ACCURATE RESULTS THAT MAY BE USED FOR TESTING VARIOUS SIMULATION METHODS Review Talk—Simulation of Plasmas I. R. McDonald 55 Summary of Session V H. C. Andersen 59
SESSION VI. LONG-RANGE FORCE PROBLEMS IN SIMULATION OF SURFACE PHENOMENA
Surface Simulations and Long-Range Forces J. D. Doll 63 Some Exact Results and Some Hypernetted Chain Approximation Results for an Electric Double Layer D. Henderson and L. Blum 64 Treatment of Long-Range Forces in Monte Carlo Calculations on Electrical Double Layers J. P. Valleau and G. M. Torrie 65 Summary of Session VI D. Henderson . . . 67 WORKSHOP PARTICIPANTS
Dr. David J. ADAMS Prof. David CHANDLER Dept. of Chemistry Dept. of Chemistry University of Southampton University of Illinois Southampton S09 5NH Urbana, IL 61801 United Kingdom (217)333-2554 703-559122 Prof. Carl DAVID Prof. S. A. ADELMAN Dept. of Chemistry Dept. of Chemistry University of Connecticut Purdue University Storrs, CT 06268 W. Lafayette, IN 47907 (203) 86-3226 (317) 749-2039 Dr. J. D. DOLL Prof. Hans C. ANDERSEN Los Alamos Scientific Laboratory Dept. of Chemistry CNC 2/MS-738 Stanford University Los Alamos. NM 87545 Stanford, CA 94305 (505) 667-4686 (415) 497-2500 Dr. Donald L. ERMAK Dr. Tariq A. ANDREA L-40 Dept. of Chemistry Lawrence Livermore Laboratory Stanford University P.O. Box 808 Stanford, CA 94305 Livermore, CA 94550 (415) 497-2502 (415) 422-7410 Dr. John A. BARKER Prof. H. L. FRIEDMAN IBM Research Laboratory Dept. of Chemistry 5600 Cottle Road State University of New York San Jose, CA 95030 at Stony Brook (408) 256-7658 Stony Brook, NY 11794 (516) 246-5067 Prof. Bruce J. BERNE Dept. of Chemistry Prof. R. FULTON Columbia University Dept. of Chemistry New York, NY 10027 The Florida State University (212) 280-2186 Tallahassee, FL 32306 (904) 644-6449 Prof. L. BLUM Physics Dept. Prof. C. GRAY University of Puerto Rico Rio Piedras, PR 00931 Physics Dept. (809) 764-000, Ext. 2367 University of Guelph Guelph, Ontario Canada Dr. David CEPERLEY (519) 824-4120 NRCC Lawrence Berkeley Laboratory Dr. Arnold T. HAGLER Berkeley, CA 94720 University of California (415) 486-^22 San Diego La Jolla, CA 92093 (714) 452-3023 Dr. James HAILE Dr. James LEWIS Chemical Engineering Dept. Chemistry Board NSII Clemson University University of California Clemson, SC 29631 Santa Cruz, CA 95064 (803) 656-3055 (408) 429-4007
Prof. Douglas HENDERSON Prof. Ian R. MCDONALD IBM Research Laboratory Dept. of Physical Chemistry San Jose, CA 95193 University of Cambridge (408) 256-7662 Lensfield Road Cambridge, CB2 1EP Dr. Joseph B. HUBBARD England National Bureau of Standards (0223) 66499 Washington, D.C. 20760 Dr. George NEMETHY Mr. Larry JOHNSON Dept. of Chemistry NRCC Cornell University Lawrence Berkeley Laboratory Ithaca, NY 14853 Berkeley, CA 94720 (507) 256-4672 (415) 486-6722 Dr. Gren PATEY Prof. William J. J0RGENSEN Division of Chemistry Dept. of Chemistry National Research Council Purdue University Ottawa K1A 0R6 W. Lafayette, IN 47907 Canada (317) 494-8824 (613) 992-2627 Dr. Mike KLEIN Dr. E. L. POLLOCK Chemistry Division Lawrence Livermore Laboratory National Research Council L-71 Ottawa K1A 0R6 P.O. Box 808 Canada Livermore, CA 94550 (613) 9922627 (415) 422-4088 LT. Peter A. K0LLMAN Prof. L. R. PRATT University of California Dept. of Chemistry School of Pharmacy University of California San Francisco, CA 94143 Berkeley, CA 94720 (415) 666-4637 (415) 642-1040
Dr. A. J. C. LADD Dr. Annessur RAHMAN Dept. of Applied Science Argonne National Laboratory Walker Hall Argonne, IL 60439 University of California (312) 972-5528 Davis, CA 95616 Dr. Peter J. R0SSKY Dr. Bjorn LARSEN Dept. of Chemistry Univ. of Trondheim University of Texas at Austin Tranev. 16 Austin, TX 78712 N-7082 Katten (512) 471-3555 Norway (075) 93992 -IX-
Prof. K. SINGER Dr. Paul WEINER Dept. of Chemistry Dept. of Pharmaceutical Chemistry Royal Holloway College School of Pharmacy Egham 5351, University of California Surrey TW20 OEX San Francisco, CA 94143 England (415) 666-4637 (44) 7843-2397 Dr. Jean Jacques WEIS Dr. Edgar SMITH Laboratoire de Physique Theorique University of Melbourne et Hautes Energies Parkville Victoria 3052 Universite Paris Sud Australia 91405 Orsay France Prof. George STELL 941-7743 Dept. of Mechanical Engineering State University of New York Prof. Kent WILSON at Stony Brook Chemistry Dept. Stony Brook, NY 11794 University of California (516) 246-7057 at San Diego Mail Code B014 Dr. Glen TORRIE La Jolla, CA 92903 Dept. of Mathematics (714) 452-3283 Royal Military College of Canada Kingston, Ontario Prof. Peter WOLYNES Canada KL7 2W3 Dept. of Chemistry (613) 545-7477 Harvard University Cambridge, MA 02138 Dr. Sam TRICKEY (617) 830-1716 Quantum Theory Group University of Florida Dr. Sidney YIP Gainesville, FL 37611 Massachusetts Institute of Technology Prof. John P. VALLEAU 23-211 M.I.T. Dept. of Chemistry Cambridge, MA 02139 University of Toronto (617) 253-3809 Toronto, Ontario Canada M5S 1A1 (416) 978-3595 Prof. W. F. van GUNSTEREN Dept. of Chemistry Harvard University 12 Oxford Street Cambridge, MA 02138 (617) 495-1768 Prof. R. 0. WATTS Dept. of Chemistry The Australian National University P.O. Box 4 Canberra 2600 Australia (062) 493259 -XI-
FOREWORD
The National Resource for Computation in Chemistry (NRCC) was established to make information on existing and developing computational methodologies available to all segments of the chemistry community, to make state-of-the- art computational facilities (hardware and software) accessible to the chemistry community, and to foster research and development of new computa tional methods for application to chemical problems. Workshops form an integral part of the NRCC's program. Consultation with key workers in the field led us to the conclusion that a timely workshop for 1980 would be one on "The Problem of Long-Range Forces in the Computer Simulation of Condensed Media." The NRCC is indebted to Profs. George Stell and Harold Friedman of the State University of New York at Stony Brook for organizing the scientific program. They were assisted in their efforts by Dr. David Ceperley of the NRCC. Their combined efforts and those of the session summarizers guided the preparation of these Proceedings.
The NRCC is a division of Lawrence Berkeley Laboratory and is funded jointly by the Basic Energy Sciences Division of the U.S. Department of Energy (Contract W-7405-ENG-48) and the National Science Foundation (Grant CHE-7721305).
William A. Lester, Jr. Director, NRCC -Xll-
IHTRODUCTUItl raises difficult and en a 1 iengi mi prohlei'S. purpose of trie wnr'r snon wa' fo bring fogefhe A workshop entitled 'The Problem Of Lour, P.iinqe group of scientists, all of whom share a \fr Force; in Computer Simulation of Condensed Media'' diroct interest in clear!/ formulating and r via'-, held at the Vallomhrosa Center in Menlo Part . ing these problems. Thei>' wee1 a6 parrK'0' California frnr. January '- through Januar/ 11. 1SSSO '•'est of whom have Seer, arrive!, r-r.g.j' e-1 v It was sponsored hy the National Resource for tlons of Hami Honian rrudeis of cors'lr r^' rs! ''d Co-putat on in Chemistry (NRCC). A few participants were scientists .-ino a'- •
Simulation (both Monte Carlo and molecular who are deeply involved in the thr ,f uynan ical) has become a powerful and nearly ••^r i ne tool in the study of classical s,s'e • '•'an.' snad''S of i-pieee-. .i"d ;. of 'ia rticles interacting with short-range pair emer led at rhe worl. shop, pat it sr pot en tials. i.e.. potentials that decay faster tnae "iaf rherr. was genon> 1 a.., , ,<• -r>t • s r - • , wnere d is dimension and r the particle distance. (Prototypal n'ji'.r-lrs are •'.••"i.ir'iJ-.'oines and hard-sphere potentials.) ilbn.'l! ed'ails In'' tirpl; more l'0t p-l! d'. 1 :•' • tility of much simulation hinges on the tise (or' hardware. soifware. or fun firm', . surpr ising fact that many of the interesting hull detailed undecst andies "f fee -"','.',''- r' i ties of infinite systems can he infc red with erope of t host system's ar t ,a 1 i y oetng i'sir'-.. rn'.lr kable accuracy from s imula t i oris involving a sma 11 , per-iodir al ! y r .. interacting "leri'iodyriamir limp; that an. I rail i t •: ona wi rh a pair potential that is truncated beyond pi' imary inr ores ' to' ht>o, ••' i > iair . three particle diameters or less. To be sure, relation procedures "'list often be used fo .1 p. c , a f e 1; infer the i. inite-system untrunoated- ' ial properties, but conceptually they are of wo.-' strop he'.piHI atively -"'id imern ar / nature.
fetal i s . we jnv i te for- systems involving long-range forces (e.g.. ' oul ombi i . dipolar, hydrodynami c ) if is a different st "• Relating infinite-system properties to the re*.!*, ot lompu'er simulation involving relativel,. '.' a 1t numbers of partvlos, periodica'ly replicaterl. SESSION I
CONTROLLED STUDIES OF LCNG-RANGE FORCE PROBLEM IN SIMULATION OF IONIC SYSTEMS -3-
REVIEW TALK
THE PROBLEM OF COULOMBIC FORCES IN COMPUTER SIMULATION
J. P. Valleau
Lash Miller Chemical Laboratories University of Toronto Toronto, Ontario, Canada
A REVIEW cause. The experimental studies that pertain to these problems are discussed in this context. Two difficulties arise from the fact that we A serious problem is that no exact results are can carry out simulations only for small systems, available by which to evaluate unambiguously the of a few hundred particles. One is that in a techniques. Tirally we report some results (done closed box, all of the particles would be near in collaboration with Bjfirn Larsen) for one- the surface so that no bulk properties could be dimensional systems. found. At least for short-ranged forces, periodic boundary conditions are remarkably successful in Spherical Cutoff (SC). Ho interactions fur solving that problem. A second difficulty is that separatiofTr^reater than a cutofi distance, usually we can't find the particle distributions for L/2. This truncation (Figure 2) means that for separations bigger than the box allows, and so r :• L/? the potential energy of counterions is can't calculate correctly the long-ranged part too high: they will be forced within the cutoff; of the potential energy. This problem is crucial the coions will be drawn outside the cutoff. This for coulombic or polar forces, and the question is will give a gross distortTorTof-the structure and whether any of the current techniques is able to an energy much too low. There are results (Curd overcome it in order to produce results appropriate and Valleau.3) confirming this; even for low-!' to systems in the "thermodynamic limit." cases, where the other methods agree, cutoff gives
Coulombic forces are the longest-ranged physical forces in bulk systems, but even for them th^c are cases for which the problem disappears: (1) At fairly low density Debye shielding gives a good description of the system and if its range K"l is small compared to the box length L the long-ranged contributions all cancel. (2) At very high density, near close-packing, the distribution will be close to that of a regular lattice. In that limit the Ewald summation becomes correct, and the "minimum image" approximation is equivalent to the Evjen crystal summation, which is again Cc v-irt zmorvS essentially exact for typical sample sizes. (3) At high temperature the long-range ordering becomes unimportant. Brush, Sahlin and Teller* were the first to study these matters. For the one-component plasma they concluded that the MI and Ewald procedures gave similar, and presumably adequate, results if the plasma parameter r was 10. Hoskins and Smith seem to confirm this foe the symmetrical two-component case, r = [2(Ze)v Hifurc 1 EkT] (~p/'o)l/3, where !Ze| is the magnitude of the ionic charge, c the dielectric constant, and p the number density. C^rcrf Except in these limiting cases the various energy summation techniques lead to discrepant results, however. This is not surprising, since the distance from a particle to the face of the box surrounding it will be quite small. A typical calculat: n might involve N = 216 hard-core par ticles in a cubical box of length L at a density .:* I'JD-^ = ,669 (where D is the hard-core diameter), [n this case half the box length, L/2, is only 3.4D! Figure 1 shows the coulombic pair potential '•: it remains very substantial at_the box face, and even at the box corners (r = • 3 L/2). We review the various energy approximations so far attempted, in each case considering how the effective pair potential differs from this physical one and trying to understand the effects this difference will -4- entirely wrong structure and energy. This is This seems associated with a tendency to form because the individual ion doesn't "see" a compen- clusters; it is not a comment on the accuracy of satini charqe - the ions can escape effective charge the method, of course, since it may be that "real" neutrality." systems would also cluster at these densities.
i5 3ari.r'- and others have suggested adding a Adam's "Corrected" Cutoff. To ensure exact rearnor. field due to a dielectric surrounding the charge compensation, Adams" proposed a model in Cit-fJ sphere, as an approximate way of including which all charge outside a spherical cutoff was '•".ng-range energies. This seems like a good idea, regarded as residing on the cutoff sphere. This =.T: '-'•' been done for polar systems; as usually is exactly equivalent to a pair potential jf the r.'-cjectcd, ncw&ver, a step-function in $will form shown in Figure 4. It is clear that this '.-.'"'• occur and the structural problems will not will lead to structural errors just opposite to :,-: a.-^ded. those of a simple cutoff: now there wil1 be too many counterions in the corners, too few coions. Minimum Image (MI). Each particle sees only These structural errors are bound to be large particle images within a box centered on itself. (and will lead to anisotropy of the ion distribu "nis is sometimes referred to as a "cubical trunca tions extending inside the cutoff). The effect tion", cut is very different from cutoff since on the energy is less clear a priori, since the one can't actually "lose" any charge: each ion approximation draws counterions outside the low- r.u't see exact charge compensation. There is energy region but then ascribes to them an arti e.'.dence, however, that, under HI, the counterions ficially lowered energy. The N- dependence are somewhat repelled from the box sides, colons reported by Adams5 indicates however that the attracted, resulting in an energy which is too low. net effect is too high an energy (although it is Since the pair potentials are correct throughout slightly lower than Ewald results). the box these errors cannot be understood in terms of single-particle statistics. Effectively the Periodic Boundary Conditions (PBC). Includes pair potential has a kink at the boundary, as shown all periodic images of every particle, and has (for three directions) in Figure 3. This means been the most popular technique. This can be that the counterion populations beyond the box will accomplished using Ewald sums (Brush, Sahlin be higher than for an infinite system, the coion and Teller1 and many others; see recent study by populations smaller. The system copes with this De Leeuw, Perram and Smith?) or using multipole excess charge by creating an oscillation in the expansions (LaddS). distribution, with a counterion minimum at the box edge and a coion maximum (see results reported It has been argued that PBC includes long- below). range effects. In an electrically neutral system, however, the dominant term in the long-range The effect of this is to enhance charge effects involve dipolar polarization (cf. Figure 5) oscillations in the system and of course to lower of a kind well-known (5(a)) and entirely different the energy. The N- dependence observed for MI from that occurring under PBC (5(b)). In fact results (Valleau and Card,3 Hoskins and Smith,2 (in three dimensions), the dipole effects in PBC Valleau and Cohen) support the latter conclusion. just cancel exactly by symmetry, so that the The effect will be greatest along the principal dominant long-range effect is entirely missing axes, and probably explains the tendencies to from PBC results. simple cubic (in the one-component plasma, Brush Sahlin and Teller1) and CsCl (Hoskins and Smith?) We can examine the likely structural effects structures. of PBC by looking at the effective pair potentials between the particles in the central cell (each A further problem of HI is that for interme one of course dragging along an infinite lattice diate densities quasi-ergodic problems often prevent of images). It is evident from symmetry (Valleau convergence (Hoskins and Smith,2 Torrie,^ Adams5). and Whittington^) that a particle will exert no force on another situated (at /J L/2) on a corner of its surrounding box. The pair potential will -5- be flat at those points, as shown in Figure 6 (where some irrelevant infinities have been discarded). This flattening of $ will tend to pull counterions towards the edge of the box, push coions toward the center. The structure will therefore be distorted in the same directions as for Adams's approximation, and inJeed there is a rough similarity in the structures obtained •B (Adams5). This suggests that the energy is probably too high as well.
The discrepancy in 4> now extends however to (a) (b) very short ranges, and no doubt distorts the short- range structure. Another interesting feature is Ojnti.^-i between (at rt:.il .- pi>ljii/si)im ol ihe that the effective pair forces are now non-central. mnrru-til tn !hc central bin ni: 1.. Ilu- I * .tppmiimatmn Figures 7 and 8 show some calculations of the Figure 5 relative magnitudes and directions of the physical pair forces and those of the PBC. (The calculations are actually for the 1/r2 potential, but the effects will be similar for 1/r.) It is clear that, in addition to the radial structural effects mentioned before, there will be a strong tendency to sweep counterions towards the principal axes, and coions towards the diagonals. This may lead to a markedly anisotropic distribution, and probably explains the tendency to NaCl structures found under PBC Jimti (e.g., Hoskins and Smith2).
At moderately high densities the only extensive results use PBC (Hansen, Woodcock and Singer, Larsen, etc.). The distributions and energies agree fairly well with HNC calculations, but we don't know how accurate they may be. Larsen^ obtained Monte Carlo results for a Yukawa potential of sufficiently short range to avoid these energy summation problems. He then sought to obtain structures for the coulombic system using perturba tion theory with the Yukawa reference system; he obtains fair agreement with PBC Monte Carlo results. For various reasons this very clever approach seems Figure 6 to me to leave the questions still unresolved, however.
A radically different approach was proposed by Friedman; in this periodic boundary conditions are abandoned in favour of a fixed spherical cavity surrounded by a dielectric. The reaction field Y * o T.\.—./«; fu^ is approximated by "image forces". This avoids most of the problems cited above, but introduces instead the surface effects associated with the wall. This has been examined for polar systems ,-' -TEC (Minns and Valleau10), but the problems seem ///\ severe.
In sum, I am not optimistic that we can con vince ourselves that the methods so far proposed will lead to reliable results for thermodynamic systems.
ONE-DIMENSIONAL SYSTEMS
We can hope to study one-dimensional fluids by simulations of systems large enough to obtain exact results. These can be used to test the effects of approximations such as MI or PBC in smaller systems. We have been looking (with Bjtfrn Larsen) at one-dimensional systems of hard-core particles X/L~> with alternating "charges" 2. The one-dimensional Figure 7 -6-
x - Y ?..,...! fl.
•*7- / i, .A
/
/
/
y/L - V/L - !'i i:uro ana'ogue of the coulombic potential has •'jja x^j, even for large boxes. This distortion at the box the particle separation: for this, the forces are edge is MI is also exaggerated in the sense that independent of the configuration, so that features for three-dimensional systems it will be largely such as screening and polarization do not occur. smoothed-out by angular averaging. We have been studying instead a Logarithmic Model
•-H ,S - 11l j• loqyl (ij x '. ./0'i and an !nverse-X Model
'iV 'HS SI „, IJ
The former varies rapidly even for large distances and exaggerates the problems that occur with an inverse-distance potential. The MI pair potentials (as given below) and the PBC effective pair potentials (again dropping irrelevant infinities) are compared in Figure 9 for the logarithmic model; (for 1/x the difference is much less dramatic). Simulations have been carried out for both models at a variety of temperatures and dens'ties and for various sizes N. We report so">e 'es.'ts for the logarithmic model with •» = ;,. * J = 0.625. Figure 10 shows the energies -.'•' r "jre 11 the mean square energies. It 's '.'-.*.' :rat the two methods disagree substarv a'"/, -• '. "he energies are certainly incrre'/. •"'-' '• '.' ":63 = 4096). At N = 64 the agree~ier". '. »*.' r the standard devia tion of the '•es,'.'.. ~r~ -r»rriJ errors are in the directions pred".*.'-" i;:*-..
The structures z'-,•!"•.'. \~' - 14) show the expected bo*-edge effect ' v MI - highly magnified V. for the logarithmic ootential since it gives a ! very sharp kink in : rf, Figures 3 and 9) at I/? Kmure 9 -7-
It is very discouraging to find substantial structural differences between f-l and PBC even 1-D L««. ,;i for N = 64. (It is worth recalling that (64)3 = v 262,144.) It is clear that the smaller systems f»; give quite incorrect structures using either MI or PBC. h The inverse-x potential (which may be more V n' • O.t-'J' like the 3-D coulombic) is less discouraging. Here the MI and PBC energies often agree within r statistical uncertainty for N = 8. Finally, ii Figure 15 compares the structure in such systems. (For which the MI and PBC energies agree: Su/N -- -5.13 -0.04 and 5.07 m.04, respectively.) Minor structural discrepancies remain.
•^ _j3i< /v I. y..t i- /•'-* . j> - a C- j"
/,-- lif
lo
r 111 u 11' 11.'
1 1 •I Fiqure 10 £<£>Mi"' \ N -8- 1-D I-TV-.TC - * J'- if** 1 /K\. ^^V--/--" \\ —•- PBC f ! \ "'~/Ix V I \,VV < I Figure 14 REFERENCES 6. D. J. Adams, Chem. Phys. Lett. 62, 329 (1979). 1. S. G. Brush, H. L. Sahlin and E. Teller, 7. S. de Leeuw, J. W. Perram and E. R. Smith, J. Chem. Phys. 45, 2102 (1966). in press. 2. C. S. Hoskins and E. R. Smith, Research 8. A. J. C. Ladd, Hoi. Phys. 36, 463 (1978). Report No. 19, Mathematics Department, University of Newscastle, Australia, (1978). 9. J. P. Valleau and S. G. Whittington, in Modern Theoretical Chemistry, Vol. 5A, ed. B. Berne, 3. 0. N. Card and J. P. Valleau, J. Chem. Phys. Plenum, New York, 1977. 52, 6232 (1970). 10. C. S. Minns and J. P. Valleau, unpublished. 4. J. A. Barker and R. 0. Watts, Chem. Phys. Lett. 2, 14« (1969). 11. B. Li.rsen and S. A. Rogde, J. Chem. Phys. (in press). 5. G. H. Torrie, reported in ref. 9. PERTURBATION THEORY FOR CHARGED HARD SPHERES B. Larsen S. A. Rogde Department of Inorganic Chemistry and Department of Chemistry University of Trondheim University of Bergen N-7034 Trondheim-NTH, Norway N-5014 Bergen University, Norway The radial distribution function for a given We have used eq. (3) to calculate the corre system is related to that for a reference system by lation functi-iis, for symmetric charged hard spheres (the restricted primitive model, RPM) in & way which does not involve any of the usual methodological g(r) = g°(r) exp [- iBu(r) + Av ir) + AB(r) ] (1) problems in computer simulation of systrms with long-range forces.'' The idea is to calculate properties for a short-range system in the thermo where 6u(r) is the pair potential divided by kT, dynamic limit first and then make the forces long- >(r) is the sum of series diagrams (= h(r) - c(r)), ranged rather than the other way around, which and B(r) is the sum of bridge diagrams. Superscript is the common procedure. Our reference system "o" denotes a reference-system quantity and "A" differs from the RPM only in that the charges are denotes a difference between a perturbed-system and screened by a function e"^r. The screening is a reference-system quantity. A simplified and chosen just strong enough so that the system is practicable approximation to (1) is the so-called short ranged compared to L/2, and by a judicious reference hypernetted chain (RHNC) approximation^; choice of thermodynamic parameters we can optimize the perturbation theory. The results for gij(r) for the RPM are in good agreement with MC results g(r) = g°(r) exp [ - ARu(r) + A-y(r)] , (2) obtained with the Ewald method. A more elaborate analysis based on eq. (2) is in progress. obtained by neglecting AB(r). An even simpler REFERENCES approximation is obtained by considering only the leading terms in the cluster expansion of >(r)2; 1. F. Lado, Phys. Rev. 135, A1013 (1961). 2. H. C. Andersen and D. Chandler, J. Chem. Phys. i+j 9ij(r) = g°..(r) exp [(-l) A6(r) ] , (3) 57, 1918 (1972). 3. E. Waisman and J. L. Lebowitz, J. Chem. Phys. where S(r) is the chain sum known from the solution 56, 3086 (1972); E. /Jaisman, J. Chem. Phys. of the mean spherical approximation.3 In (3), 59, 495 (1973). we have also used the subscripts i and j to distinguish between the various rdfs occurring 4. B. Larsen and S. A. Rogde, J. Chem. Phys. for ionic salts. (in print). -10- PFRIODIC BOUNDARY CONDITIONS IN SIMULATION OF IONIC SOLIDS E. F. .Smith Department of Mathematics University of Melbourne Parkvillo, Victoria, Australia 7 -. lstti:o i^ums required in evaluating the pends on • ' , decreasing to zero as .'-*•*. - • - : \r. for a simulation of an ionic system in ii: :<.iundary conditions are conditionally For simulation of ionic solids with N~Cl type r '••:•'.z. Thus the result obtained can defend structure, this may be expected to make little dif .r"'ler wt summation. Summing by cubic or ference, since the UaCl unit cell has zero dipole •_ -:sl shel Is qives i resul t di f fering from the moment. However, for solids with CsCl structure, . o:.e by an amount proportional to the square where the unit cell does have a dipole moment, the '.*_- dipole momeiit of the configuration. If we Madolung constant with :' = 1 is about £ that of .>J!' the rut;ion outside the large sphere Iin its value with '-*"'. This suggests that without ;:is-. of summation by spherical she! Is) of properly accounting for surface effects, the struc /in. copies to be a continuum with dielectric ture into which a given ionic system solidi fies ;jnt . ', then the size OL this correction de may not be accurately predicted. -11- A!( ALTERNATIVE TO THE EWALD SUMMATION? D. J. Adams Department of Chemistry, The University, Southampton, UK A comparative study of summation methods and at the cut-off radius and at twice the cut-off boundary conditions for Monte Carlo calculations radius such that the total second moment obeys with the restricted primitive model (RPM) at high the second-moment condition while electro charge density has established the following1: neutrality is maintained. 1) The nearest-image summation is faulty The improved correction does little to because the configurations generated are biased remove the number dependence of the truncation away from radial symmetry by the shape of the method. However, the rdfs obtained with 512 periodic cell. ions in the periodic cube are even closer to those obtained with the Ewald summation and the 2) Physically reasonable RPM liquid differences are now largely within the s listical structures can be obtained by summing over near- error of the Monte Carlo calculations. The rdfs neighbour ions with a spherical cut-off, provided obtained using the improved spherical truncation that a simple correction is made for the method, which I call ST2, with 861 ions in electroneutrality condition. periodic truncated-octahedral boundary conditions are also very close to the Ewald results. 3) The Ewald summation did not show obvious effects of the periodicity2, but gave radial Thia now establishes even more stronglv than distribution functions (rdfs) similar to those the original calclations that the fwald found with a large spherical cut-off. The Ewald summation is perfectly satisfactory for the study rdfs showed very little number dependence. of systems of free ions. It does not introduce the spurious effects due to the periodicity of An extension to that study has been made by the simulation which had been thought tn occur-. using an improved correction to the spherical The ST2 method is also established as a truncation method. The electroneutrality or zero possible alternative o the Ewald summation, its moment correction was made by summing the total advantage is greater ease of pronramming and net charge within each truncation sphere and faster execution. In the present case with 51? assuming an extra opposing charge situated at the ions it is 1. •» times as fast as the Ewald method cut-off radius. A more sophisticated correction using 125 vectors in the Fourier sun. Its has been made that satisfies not only the electro disadvantage is that it shows a dependence on the neutrality condition but also the Stillinger- 3 truncation radius which is in turn limited by the Lovett second moment condition . To do this the system size. It is therefore superior to the extra term r. r? . Z. is calculated for Ewald summation only when a very large system is considered necessary. A large system using each ion i. The second moment correction is made 5T2 will almost certainly be slower than a smaller by assuming additional extra and opposite charges system using the Ewald summation. Monte Carlo -esults for RPM, 0.3503 and g = 35.48. Method (cl a !,-) Ewald summation 64 26. 4 12.6 P. P9 512 26. 5 12.9 0. 10 spherical truncation 64 26,, 4 8.8 p. 3« +0th order correction 512 27., 5 11.5 0. 06 spherical truncation 64 25,, 6 11.2 0. 27 +2nd order correction 512 27,, n 12.2 n. 14 -12- REFERENCES 1. D. J. Adams, Diem. Phys. Lett. 52, 329 (1979). 2. J. P. Valleau and S. Hhittinqton, Statistical Mechanics. Part A. Equilibrium Methods , en, b. d. Berne (Plenum. New York, iy//), Chap. 4. 3. F. H. Stillinper and R. Lovett, J. Chen. Phys. 4B, 3858 M96R). r/cr -13- PERIODIC, TRUNCATED-OCTAHEDRAL BOUNDARY CONDITIONS D. J. Adams Department of Chemistry, The University, Southampton, UK The cube has been used almost exclusively for DX = X(I) - X(J> the periodic cell in computer simulation. Wang DX = DX - AINT(2.*DX) ana Krumhansl used the rhombic dodecahedron,1 the Wigner-Seitz cell of the FCC lattice, because its DY = Yfl) - Y(J) more spherical shape gave it an advantage over the cube for computing the triplet distribution DY = DY - AINT(2.*DY) function. However, this shape gave problems DZ = Z(I) - Z(J) in calculating the nearest periodic images and their Monte Carlo program ran at only half the DZ = DZ - AINT(2.*DZ) speed of the cubic version. IF(ABS(DX) +ABS(DY) + ABS(DZ).LT.O. 75)G0T0 1 Such difficulties with the periodic boundary DX = DX - SIGN(0.5, DX) conditions seem to have deterred others from DY = DY - SIGN(0.5, DY) using any other shape but the cube, even though r tne need to explore possible shape effects has DZ = DZ - SIGN(0. -, DZ) been recognized.?>3 Sangster and Dixon have 1 CONTINUE pointed out that one of the problems with using the Ewald summation for simulating molten salts is that with a cubic periodic cell the effective Figure 1. Calculation of the vector (DX, DY, DZ) potential between two ions has cubic symmetry and between the nearest-neighbor images of particles that its anisotropic component is large and I and J. The containing cube is of unit length. difficult to approximate by polynomicals.3 In the cube comers the errors become large and they suggested that an alternative shape of periodic cell might reduce these difficulties by reducing the anisotropy. DX = X(I) - X(J) An alternative to the rhombic dodecahedron is the truncated octahedron (TO), the Wigner- DX = ABS(DX - AINT(2.*DX)) Sietz cell of the body-centred-cubic lattice. DY = Y(I) - Y(J) The inscribed sphere of the TO is smaller than that of the rhombic dodecahedron but in other DY = ABS(DY - AINT(2.*DY)) respects the TO is the more spherical shape. DZ = Z(I) - Z(J) Both shapes have the following advantages over the cube: DZ = ABS(DZ - AINT(2.*DZ)) R2 = DX**2 + DY**2 + DZ**2 1) They are more nearly spherical and a spherical shape is the ideal for modelling + AMIN1(0.0, 0.75 -(DX + DY + DZ)) liquids which are spherically symmetric. 2) For a given volume they give larger distances between a particle and its nearest Figure 2. Calculation of the square of the distance, periodic images, and thus also a larger usable R2, between the nearest-neighbor images of particles radius when calculating distribution functions, I and J. etc. 3) They minimize, though they will not necessarily eliminate, the inherent bias of the minimum image method.4 REFERENCES 4) They may permit a more efficient or accurate evaluation of the Ewald summation. 1. S. S. Wang, and J. A. Krumhansl, J. Chen?. Phys. 56, 4287 (1972). The great advantage of TO periodic boundary conditions is that they are 2. J. P. Valleau, and S. G. Whittington, comparatively simple to program, only slightly Statistical Mechanics. Part A. Equilibrium more involved than the cube. The extra effort Methods, ed. B. J. Berne (Plenum, New York. in programming TO boundary conditions is Ty//)', Chap. 4. sufficiently small, and the possiole gain so larae when the interactions are long-ranged, 3. (1. J. L. Sangster, and M. Dixon, Adv. Phys. that it seems worthwhile using them even though 25_, 247 (1976). in many cases the actual improvement will prove to be minor. 4. D. J. Adams, Chem. Phys. Lett., ^2, 329 (1979). -14- SUHHARY OF SESSION I B. Larsen Department of Inorganic Chemistry Univ. of Trondheim - Norwegian Inst, of Technology N-7034 Trondheim-NTH, Norway "he purpose of computer simulations of ionic namic limit for icnic systems has been proved by •'. js is to provide "experimental" data for model Lieb and Lebowitz,! and it is quite possible that :• > items that either are of theoretical interest or important progress, which may lead to the solution -er'-esent a similar real system of experimental of our problem, will be made along the same lines. '"•terest. In bo^h cases, one is normally aiming Theoretical studies of periodic systems should ;: simulation results in the thermodynamic limit therefore be encouraged and supported by computer N • • , V -' ", N/V = p) by an extrapolation of simulation, aiming at a deeper understanding of ""esults for an actual model system, which for the behavior of periodic systems and their N and practical reasons has to be small (N = 102 - 10*). shape dependence in particular. "Ms is done by imposing periodic boundary condi tions in a space-filling way. The methodological Some systematic studies of ionic systems have problems associated with this extrapolation for already been carried out, notably by Adams,2 Hansen systems with long-range forces are clearly non- et al.^ and by Valleau and Larsen.* trivial on the grounds of principle that have been outlined in Valleau's review. It is encouraging In a very interesting study, Adams did Monte that these problems are now being recognized and Carlo calculations on charged hard spheres of sym approached in a scientific way in the community metric charge and equal diameters (the restricted of computer simulation people, but it is also primitive model) using different cell shapes, dif disappointing that we do not seem to be anywhere ferent cell sizes, and different energy summation closer to a solution of these problems now than methods at a density and temperature corresponding we were a decade ago. Actually, the practical to a molten salt. In addition to the usual cubic situation is in many cases better than it would elementary cell, Adams considered the truncated seem at first sight. The limiting cases indicated octahedron (TO) and the rhombic dodecahedron, which by Prof. Valleau in his review are some examples. are the Wigner-Ceitz cells of the body-centered Also, there is a growing interest in the properties cubic and the face-centered cubic structures, of small systems for which one does not need the respectively. The advantage of these cells is thermodynamic-limit results. An example is biolog that they are more "spherical" than the cube, as ical cellular systems. In this summary, I shall shown in Table 1, and hence reduce the effect of tocus on some ways that have been discussed here non-shperical symmetry of the effective pair poten in which the problems can be approached. They tial used for cubic elementary cells. It is rela are not specific to ionic systems although they tively easy to combine the TO (shown in Figure 1) seem to have received particular attention for with a minimum-image energy summation method and such systems. the results for the internal energy and radial distribution functions for this combination fall between the Ewald and minimum image results for The problem of extrapolating to the thermo 2 5 dynamic limit has two aspects; one is the practical the cubic elementary cell. ' problem of how to evaluate the effect of the un wanted periodicity of the structure introduced by The spherical cut-off method for the energy the boundary conditions, and the other is the more calculation is of course no good for ionic systems subtle question whether the thermodynamic limit is since one is not guaranteed that the electroneu- unique. (Is the limit N, V-*™ , N/V =p.for a peri trality condition (the Stillinger and Lovett zero- odic system with N particles in the elementary cell moment condition)^ is satisfied in the sphere. identical to the thermodynamic limit for a non- Adams has recently devised a correction to the periodic system?) The existence of the thermody spherical cut-off method such that both Stillinger- Table 1. Comparison of cube, truncated octahedron (TO), and rhombic dodecahedron (RD). (Table by D. Adams.) :j~sphere radius Inscribed sphere volume Circumsphere volume jhape •;ed sphere radius Volume volume Cube • ': - 1.73 n/6 = 0.52 VTIT/Z = 2.72 TO T7T '- 1.29 Vlft/8 * 0.68 5V'5~TT/24 * 1.46 RD \'T = 1.41 %1>*/6 * 0.74 2TT/3 * 2.09 -15- r/o- Figure 2. Radial distribution functions for the restricted primitive model as obtained with a spheri cal cut-off method corrected so that Stillinger and Lovett's zeroth and second moment conditions are satisfied. The rdf's obtained with the usual Ewald Figure 1. A trunca'ed octahedron and its contain method are also shown. The thermodynamic state is ing cube. (Figure by D. Adams.) the same as in Table 2, and corresponds to a molten salt. (Figure by D. Adams). Lovett conditions are satisfied.7 This method has been applied to the cubic elementary cell. The results, shown in Table 2, show a substantial N- retical analysis of the N-dependence. The main dependence, but the 512-particle results are in conclusion to be drawn from the computer results good agreement with those obtained by the Ewald is that one must use very large systems in order method. The corresponding radial distribution to eliminate the different effects of the two sum functions are shown in Figure 2. mation methods. A special way of suppressing the surface Unfortunately, these pieces of work cannot be effects is to confine the d-dimensional model to regarded as conclusive with respect co our problem the closed surface of a d+1-dimensional body. This because of the fact that a limited variation in idea was used by Hansen et al.,^ who did molecular shape and size cannot yield the wanted extrapolated dynamics calculations of the two-dimensional elec quantity. For example, the apparently small In tron gas on the surface of a (three-dimensional) dependence in the results for charged hard spheres sphere. Their results are in good agreement with obtained with the Ewald method for N = 8, 64, 216, those obtained with the conventional Ewald method. 512° at intermediate densities does not necessarily mean there is a smooth extrapolation to N="> (Inci The effect of varying N in the usual mininum- dentally I would like to see results for N = 218 or image and Ewald methods for a one-dimensional ionic some other number that does not correspond to a system has been described by Valleau in his review crystal structure like those commonly used!) How talk. An interesting aspect of this model is that ever, I hope these and other studies will create it does not seem prohibitively difficult to comple interest in the problems involved and lead to com ment the computer simulation results with a theo- plement1' y theoretical studies. Table 2. Monte Carlo results for internal energy and contact valuc-. of the radial distribution functions for the restricted primitive model, r = No3ir/6V = 0.350J ind q = (ze)2/kTc = 35.48. (Table by D. Adams.) Method ikT> g - yq I -1 -'u/f + a) + + Ewald summation 64 26 4 12 6 o 09 512 26 5 12 9 0 10 Spherical truncation 64 26 4 8 8 34 + zeroth order correction 512 2/ 5 11 5 o 06 Spherical truncation 64 25 6 11 ? c 2? + second order correction 51? 2? 0 12 ? 0 14 -16 Another approach is to use a given size and as circumstantial evidence in favor of the Ewald shape of the elementary cell, but rather than having method. 1 do not think the methodological problems exact replicas of it, one could use replicas that have been solved yet for ionic systems, and hope have been slightly distorted. It is not clear how fully further theoretical progress will complement this can be worked out in practice, but presumably tne empirical approaches taken so far. the distortions must be made such that known condi tions on the correlation functions are satisfied. REFERENCES Even with a given size and shape of the cell 1. E. H. Lieb and J. L. Lebowitz, "Lectures on and the usual periodic boundary conditions, the the Thermodynamic Limit for Coulomb Systems," computation of bulk properties of ionic systems Battelle Recontres in Mathematics and Physics, is not unambiguous. This stems from the fact that summer 1971. the sum over Coulombic pair interactions is condi tionally convergent, and it consequently depends 2. D. i. Adams, Chem. Phys. Lett. 62, 329 (1979). on the way the summation is carried out. The ob served differences between the various energy sum mation methods reviewed by Prof. Valleau, even when 3. J. P. Hansen, D. Levesque, and J. J. Weis, they are carried out for relatively large systems, "Self Diffusion in the Two-Dimensional Classical are related to this feature of ionic systems. The Electron Gas," preprint. usual Ewald method corresponds to a particular way of arranging the sum; others have been discussed 4. J. P. Valleau, "The Problem of Coulombic Forces by de Leeuw et al." The effect of this dependency in Computer Simulation," this proceedings seems to be especially profound for certain solidslO (J. P. Valleau and B. Larsen), unpublished in which there is no reason to hope that it will results. cancel out in the configurational average. 5. D. J. Adams, "Periodic, Truncated-Octahedral An entirely different approach based on pertur Boundary Conditions," this proceedings. bation-theory ideas has been taken by Ceperley and Chester for the one-component plasma.H The prin 6. F. Stillinger and R. Lovett, J. Chem. Phys. ciple is to simulate a short-range system, and then 48, 3858 (1968). make the system long range by perturbation theory. This approach has also been applied recently to 7. D. J. Adams, "An Alternative to the Ewald charged hard spheres.12 Again, the results ob Summation?," this proceedings. tained are not conclusive because of the perturba tion treatment, which is necessarily approximate, 8. B. Larsen, Chem. Phys. Lett. 27, 47 (1374). but it gives another handle at the problem and may thus elucidate it. 9. S. de Leeuw, J. W. Perram, and E. R. Smith, "Simulation of Electrostatic Systems in Peri I think one can conclude from the discussion odic Boundary Conditions. I. Lattice Sums presented in this session that we have at present and Dieletric Constants," preprint. no way of obtaining exact results for non-periodic ionic systems in the thermodynamic limit. What we 10. E. R. Smith, "Periodic Boundary Conditions in do have are a number of approximations to them. Simulation of Ionic Solids," this proceedings. The validity of these approximations is unknown, but they certainly depend on the thermodynamic 11. D. H. Ceperley and G. V. Chester, Phys. Rev. state of the system. The various approximations A15, 755 (1977). used are very different in nature, and yet they all seem to confirm the results obtained with the 12. B. Larsen and S. A. Rogde, "Perturbation Theory usual Ewald method for the energy summation. There for Charged Hard Spheres," this proceedings, is a tendency, therefore, to regard these results and _iii 0. Chem. Phys. (in print). -19- RLvIEW TALK DIPOLAR FLUIDS G. H. Patey Division of Chemistry National Research Council of Canada Ottawa, Canada KlA 0R6 GENERAL REMARKS SIMPLE DIPOLAR FLUIDS Monte Carlo (MC) and molecular dynamics (MD) Simple dipolar fluids are defined by the calculations on polar systems have been done with potential a variety of different boundary conditions. Spherical cutoff (SC),1"3 minimum image (MI),1'3''* 1 3 5 5 6 9 0 reaction field (RF), ' » * Ewald " and other* UZ (3u) 3 1Z, = °°°( ) * (12 methods have been applied. It is now clear .8,9 U( u r -K that none of these methods yields the pair correla tion function for a non-periodic infinite system where and comparison with approximate theories is difficult and often ambiguous. Empirically it 3 11Z is found that the thermodynamic properties such * (12) = 3(1^-^ (u2-r) - (u1-u2) (3b) as the internal energy, etc. do not depend strongly upon the boundary conditions. However, it is not clear why this should be so and in some casts3 different boundary conditions do give significantly u\, U£ and r = (r, - r_)/|r. r,| are unit vectors different results. and uu00(r) is a spherically symmetric short-range The dielectric constant,c, poses more difficult potential. Dipolar hard spheres and Stockmayer problems. There are two basic questions. First particles which correspond to choosing u°°0(r) to of all the relationship between e and the mean be the hard sphere and Lennard-Jones interactions, square moment will depend upon the boundary condi respectively, have been the most extensively tions and must of course be known if r is to be studied. For such systems it is useful to expand obtained. For SC boundary condition1; the relevant the pair correlation function, h(12), in the form formula appears3 to be 000 110 110 h(12) = h (r) + h (r) * (12) E-l E+2 = yg (1) + h112(r) *112(12) (4a) where y = 4trpu /9kT and g = For an infinite system the dielectric constant 2 6 B&4 is given by 106 09 •*/ oe A --l)(2e + 1) (110) (5a) 9E yg 07 J oe where 05 04 03 .(HO) 1 + r? h110(r) dr (5b) 02 01 F cprrelati I 2 3 4 3 6 7 B 9 10 II 12 13 14 13 which contribute to c we calculate gt ' using the LHNC results for dipolar hard spheres. The ratio gp'^^'/goo is plotted in Figure 1 for the Figure 1. gn /9» for dipolar and hard spheres at p* = 0.8. The vertical lines are drawn at L/2 reduced density c* - po^ = 0.8 and several values for N = 108, 256, 500, and 864. From top to bottom of the reduced dipole moment defined by u*2 = the curves are for y*2 = 1, 1.5, 2, 2.25, 2.5, and u2/icTc3 where i is the hard sphere diameter. We 2.75. see that the significant correlation range becomes very large as u 2 is increased. At large values of u*^ the dielectric constant is very sensitive to the "long-range" part of hH°(r). This is illustrated by the large discrepancy between the LHNC and QHNC results13 at the higher values of u*? which can be traced to rather small differences in the "tail" of hH°(r). Similarly, if E is sensitive to a correlation range which is much greater than L/2 as the theories suggest, care should be taken in interrupting computer results for "small" periodic systems. It is not at all obvious that the dielectric constant so obtained will lie close to that of a non-periodic infinite system. For an infinite system z can also be obtained from the limiting expression J12 (r, - (E-1) 1 lr) 3iey~ ^ (6) Figure 2. g^ /g„ for dipolar hard spheres at p* = 0.8. The vertical lines are drawn at L/2 s for N = 108, 256, 500, and 864. From top to bottom 9R /9«° ^ plotted in Figure 2. Again as 2 described above, it is obvious that long-range the curves are for y* = 1, 2.25 and 2.75. correlations play a crucial role in determining e- Another method" of examining the significant k behavior correctly. Indeed, Figures 3-5 bear a correlation range is to calculate striking qualitative resemblance to the HD (Ewald) results of Pollock and Alder.8 These authors find that for small dipole moments (u*2 < 1) the HD h110(r) sirHcr dr and QHNC calculations are in excellent agreement 9R(*)=1+^ for all k. For larger values of u*2 good agreement (7) is obtained for k ? 2ir/L but serious discrepancies are found at k = Q which is exactly where the long- range part of hllO(r) becomes important. where gjk) is the Fourier Transform and g^tO) is just the gJH°' appearing in Eq. (5). We would e>.ppct the small k behavior of g„(k) to be MORE REALISTIC SYSTEMS sensitive to the long-range part"of hll"{r). To illustrate this we calculate %i^) for values The LHNC and QHNC approximations have also been solvedl^ for fluids of hard spheres with of B corresponding to L/2 for N= 108, 256, 500, dipoles and linear quadrupoles. It is foundl^ and 864 particles. The LHNC results for dipolar that the quadrupole moment greatly decreases the hard spheres at p* = 0.8 are shown in Figures 3-5. dielectric constant and that for large quadrupoles It can be seen that for u*z > 1 the tail of h110(r) a is effectively reduced to ~1 such that Onsager's mckes a significant contribution at small k and theory becomes quite accurate. Physically this again it is not obvious that computer calculations is easily understood since T-like configurations using small periodic systems will obtain the small = are v for which yi-y2 ° fa <"*ed by quadrupolar -21- ... 1 ._ ; , 20 p- =0.8 ( • 108 o 256 \ • \ 1.0 ker Figure 3..„gp(k) for dipolar hard spheres at '* - 0.8 and u*2 8 1. The solid line is gjk). R = L/2 and the solid and open dots are for N = 108 and 256, respectively. From left to right the vertical lines indicate ?T*O/L for N = 108, 256, 500, and 864. J M" -Z 4- »[ p' = 0.8 °\ • 108 •S o 256 3^ A " a 500 • m 864 2 X s." * 1 ^ Figure 4. gp{k) for dipolar hare spheres at p* = 0.8 and i*? - 2. The open and soiid squares are for N = 500 and 864, respectively. The remaining symbols are as in Figure 3. -22- /x'E= 2.75 p' = 0.8 • 108 o 256 a 500 • 864 -8—e—. 3 kcr Figure 5. cjn(k) for dipolar hard spheres at c" 0.8 and L*? = 2.75. The symbols are as in Figur 3 and 4. interactions. Also , the effect of different 5. J. A. Barker and R. 0. Watts, Chem. Phys. boundary conditions upon h( 12) is much less severe Lett. 3, 144 (1969); Molec. Phys. 26, 789 than that observed' i" for purely dipolar systems. (1973). E-.en relatively sma 11 quadrupole moments tend to dominate n If the d ipolar correlations and trun- 6. D. Adams, Molec. Phys. (in press), 1979-SO). eating the potentia 1 has little effect upon h(12). We note that these remarks apply only to axially 7. V. M. Jansoone, Chem. Phys. 3, 79 (1974). symmetric molecules such as HC1 or NH3. The situation may be qu ite different for molecules 8. E. L. Pollock and B. J. Alder (UCRL preprint, .:* cpfferent symmet ry such as HjO. 1980). 9. S. De Leeuw, J. W. Perram and E. B. Smith Proc. Roy. Soc. (in rress). 10. A. J. C. Ladd, Holec. Phys. 33, 1039 (1977); 36, 463 (1973). .'. - . ;»""ea- jnd 5. 3. Whitunqton, Modern ~ne:-eti:a" .'nen-stry. Vol. V, Statistical 11. B. V. Felderhof, Physica (in press). "^eVa" cs, D"-i J, edited by B. Berne (Plenum 12. G. N. Patev, Molec. Phys. 34, 427 (19771; 35, 1413 (1978). «e*s, wo'ec. Phys. 23, 13. G. N. Patev, D. Levesoue and J.-J. Weis, Molec. Phys. 38, 219 (1979). •. ". _evesnje. 3. V =itey and J.-J. Weis, Mo'rc. =n..-s. 3£, 10" 19"?'i. 14. The LHNC is equivalent to the 5SCA of Wertheim. [M. S. Wertheim, Molec. Phys. 25, 11 (1973'j. 4. u. N. 'atev and J. ?. Valleau, J. Chem. Phys. 61, 53* 1974'.; Chen. Phys. Lett. 21, 297 15. G. N. Datey and J. P. Valleau, J. Chem. Phys. TT9 73). 64, 170 (1976). -23- CALCULATION OF c(k,u) BY COMPUTER SIMULATIONS OF PERMANENT DIPOLAR SYSTEMS E. L. Pollock Lawrence Livermore Laboratory University of California Livermore, CA 9455Q The relations between the dielectric constant "parallel plate" summation whe>-e A = 4", B = C = 0 and the fluctuations of the polarization density so that in a dielectric sample depend strongly, because of the long-range nature of the dipole interaction, on the boundary conditions.^ 'T computer simula 1 (f-l)(?>+l) tions using a truncated electrostatic dipole •gr interaction these fluctuations are suppressed and large values of c cannot be accurately calculated. Hence it is necessary to use the untruncated and the use of the Ewald potential alone where interaction. A = B = C = 0 so that For infinite periodic systems the conditional convergence of the dipole-dipole interaction sum n - l €-1 makes it necessary to specify the order in which 9-y-T the terms are summed. For most cases of interest this summation order can be given by considering a finite system of some shape which can grow Finite wavelength fluctuations effectively decouple arbitrarily large while preserving this shape. from the boundary effects and all the periodic For an ellipsoidal shape the dipole electrostatic systems can be shown to satisfy (k = ?-•/> f 0) energy, per periodic cell of volume i> containing N dipoles f, is (M. von Laue, 1940) (Mk) - 1) (2c(k) * 1) y g(k) 5FTD Ewald i ? ? p U = U + I^n (AM-. + BM^ + CMp where where the first term is the Ewald potential, A, ik-r 8, and C are the depolarization factors with g(k) N7 ij) A + B + C = 4TT, and M = £ The polarization and k = 2TT/L (nx + my + SLZ) with (n, m, f.) integer. i = l This is also true at short wavelengths even for the spherically truncated potential and becomes a good approximation once the wavelength is less fluctuations are then related to r by (B. U. than the truncation radius. The relations for Felderhof, 1979). E(k,u) are obtained by replacing g or g(k) in the above formulas with: (k,t) ' Hx > ' "kT TO -iu,t •1 dt y K TfKB REFERENCE AND FOOTNOTE 1. These fluctuation relations are applied to .•HZ> -kT T+RT calculating >(k) for the Stockmayer fluid in Pollock and Alder, "Static Dielectric Properties of Stockmayer Fluids" (preprint where K - (t-r,/4_. Some special cases of this 1980, Physica, in press). The formulae result are: spherical summation where A = B = C attributed to B. U. Felderhof are derived in 4^/3 so that "Fluctuation Theorems for Dielectric with Periodic Boundary Conditions" (preprint 1979, Physica, in press). E. R. Smith's work on P2RI0DIC BOUNDARY CONDITIONS FOR DIPOLAR SYSTEMS E. R. Smith Department of Mathematics University of Melbourne Parkville, Victoria Australia '. revert or. some work carried out with J. W. We have carried out simulations of 256 hard " :: ••-- .•: ± •. Ze Leeuw of the Institute of Mathe- sphere particles with embedded point dipoles, "a::;: ; ier.se University, Denmark. averaging over 1 - 2 * 10 moves, at po =0.8. A; with ionic systems, the use of periodic - . _-.iiry conditions means that the Hamiltonian „, ;:~;:osed of conditionally convergent lattice 2.234 1 100 6 14.9 s-T.~. If we sum by spherical shells we find an 2 2 2.234 100 6 15.5 sx::a pair interaction [4-;..'U-/3L ] [l-«. (•.'-1) / 2 2.234 15 300 25 18 2.75 3.071 300 25 40 : '-1)1 where L is the side of the simulation sar.tle and the region outside the sphere of copies :f the sample is assumed to be a continuum of di- where x = U /a kT, n is the number of lattice •^Lectric constant ' . This extra term is not vectors summed over in the reciprocal space resent in the Ewald-Kornfeld sum unless = *-**». r lattice sum in the Hamiltonian for PBC and A is The dielectric constant of the simulation sample the acceptance rate of configurations during the is qiven by run. For the two runs at X = 2, n = 100, the P - ^ {2E. + 1 + Ge'y g(-')}/{2c'+l - 3yg(C')} Kirkwood g-factors agreed very well with the for 2 where y = 4 ~o;;**/9kT, r being the particle density mula and i. the magnitude of the dipole moments of the g(D = gH/(i + ygM). particlos. A perturbation expansion using the We note that the number of lattice vectors used above shift in the interparticle potential gives, in the reciprocal space sum in the Hamiltonian exactly, in the thermodynamic limit can affect the value of z quite strongly. This g(c"> = g(£*)/{l- \{€",-z')q(z')/3} is not surprising since the shift in interaction ] where X(E»,E-) = *Y<~[ ~ S^L ' potential caused by ignoring some lattice vectors Thus to calculate the dielectric constant it is of the form of a Kac potential and while weak is necessary to decide on a value of C then use is long ranged, so that it can have a considerable the correct Hamiltonian in a simulation to measure effect. For both the n = 300 runs (x=2, K=2.75) r i + g(.') and finally use the correct formula linking we found that 910n ~ ) and g (G+) were larger with g(G'). Long-range tails in the function than or equal to g {0+). This suggests that the r.,(r) appear naturally if z * 7s z. The tail is multipole expansion does not converge very rap r.^gative for -l' = 1 and positive for E' -*-00. The idly. Integral equations which do not take accu ir.tc-rr.^1 er.ergy ar.d specific heat are unaffected rate account of higher order multipole terms may ~y ' tr. the theme-dynamic limit (L-*-00). not give reliable results. -25- COHPUTER SIMULATIOHS OF HYDROGEN BONDED LIQUIDS William L. Jorgensen Department of Chemistry Purdue University West Lafayette, Indiana 47907 Monte Catlo statistical mechanics simulationonss ddefinitio n of a hydrogen bond is available from the have been carried out for liquid water,1 ammonia,a,^ ddistribution s of dimerization energies in the hydrogen fluoride,3 and methanol.'' Efficient proro- - 1liquids . This permitted a thorough analysis of cedures were developed to obtain the necessary tth e hydrogen bonding in terms of distributions of intermolecular potential functions from ab initiio hhydroge n bond numbers and angles. Agreement with quantum mechanical calculations on dimers. Stud3y ianalyse s of infrared data is good. Ammonia, hydro- of the basis set dependence of the results revealealed cge n fluoride, and methanol are all found to contain that reasonable potential functions could be genn-- vwindin g hydrogen bonded chains with an average of erated from minimal basis set calculations with m ttw o hydrogen bonds per monomer. Significant frac or minor modification. ttion s of monomers in one (chain ends) and three (Y junctions) hydrogen bonds are present (see Detailed structural and thermodynamic resultIts ifigures) . Liquid water has a more complex network were obtained for the liquids. Some key data arree vwit h many interconnecting rings and an average of shown in Table 1 and Figures 1 and 2. An energetiletic c 3.;5 hydrogen bonds per monomer at 25°C. Table 1. Heats Vaporization and Capacities flH° (kcal/mol) Cv (cal/mol-deg) Liquid T(C°) Calc. Expt. Calc. Expt. HF 0 6.2 7.2 17.1 <17.0 NH3 -33 5.0 5.7 13.3 12.7 H20 25 9.6 10.7 15.1 17.9 CHo0H 25 6.7 9.1 16.7 15.9 HYDROGEN BONO DISTRIBUTIONS 012345012345 XX RflDIHL DISTRIBUTION FUNCTION Figure 1 Figure 2 -26- REFERENCES 2. W. L. Jorgensen and H. Ibrahim, J. Am. Chem. Soc. 102, in press (1980). 3. W. L. Jorgensen, J. Chem. Phys. 70, 5888 (1979). 1. W. L. Jorgensen, J. Am. Chem. Soc. 101, 2011, 2016 (1979); Chem. Phys. Letts., in press 4. W. L. Jorgensen, J. Chem. Phys. 71, 5034 (1979); (1980). J. Am. Chem. Soc. 102_, in press TT980). -27- DIELECTRIC THEORY FOR POLAR MOLECULES WITH FLUCTUATING POLARIZABILITY G. Stell Departments of Mechanical Engineering and Chemistry State University of New York Stony Brook, New York 11794 The dielectric properties of a classical a corresponding approximation defined for dipolar system of interacting particles, each bearing a particles with nonfluctuating polarizability. In permanent dipole moment and thermally fluctuating the limit of a closed-packed system of nonpolar polarizability, is considered. It is shown how particles with cubic symmetry, it is found that a useful class of approximations initially defined the resulting approximations for r. all reduce to for a system of nonpolarizable particles can be the Claussius-Mossotti result, which is known to generalized to include fluctuating polarizability. be exact for systems of such symmetry in the case For the case in which the mean polarization of an of either fluctuating or constant polarizability. isolated particle is linear in applied field (i.e., The inclusion of nonlinear effects i: discussed harmonic fluctuations) it is further shown that briefly; in particular it is noted that a permanent the dielectric constant can be explicitly computed dipole moment is equivalent to a certain limiting in these approximations, which include the mean case of anharmonic fluctuations. spherical approximaion and the single super-chain approximation (equivalent to the "reference" ver The above work, done jointly by J. S- Hjlye sion of the linearized hypernetted chain approxi and G. Stell, is available from the latter author mation). The resulting e is identical to that in as CEAS Report #333. -28- SUMMARY OF SESSION I! J. J. WeiS Laboratoire de Physique Theorique et Hautes Energies Universite Paris Sud 91405 Orsay, France 941-7743 The main contributions to this session are The correlation functions one obtains by concerned with the use of periodic boundary the Ewald-Kornfeld method (E1 = •=) are different conditions (PBC) (Ewald method) to derive the from those of an infinite nonperiodic system so dielectric properties of dipolar systems. The that comparison with theory is meaningful only dielectric constant c is commonly obtained through if the latter can be solved for periodic boundary its relation to the mean square moment REVIEW TALK EFFECT OF LONG-RANGE FORCES ON THE SIMULATION OF SOLVATION AND ON BiOPOLYiCR HYDRATION SIMULATIONS, COULOMB AND HYDRODYNAMIC FORCES P. G. Wolynes School of Chemical Sciences University of Illinois Urbana, IL While long-range forces might be defined from however, if the simulation does not have transla- a practical viewpoint as those whose range is tional invariance (e.g., a true drop) there can longer than the range you wish to use in a computer be additional slow variables which couple to the simulation, special problems are connected with motion of an ion—there will be a restoring force forces which die away spatially no faster than pushing the ion toward or away from the center r-d, where d is the dimensionality of space. of the box, thereby coupling its velocity and Examples of such forces include: position. By explicitly including this force, one may be able to circumvent this problem. Dynamical 1. Electrical forces in charged or polar perturbation theories are not as sophisticated systems. in their development as static theories. A Mori (or other) continued fraction expansion of the 2. Hydrodynaraic forces which exist in all Fourier transform of a time-correlation function may systems. allow one to find tail corrections to dynamical quantities since the expansion coefficients can 3. Forces coming from broken symmetries, be related to equilibrium quantities. e.g., stresses in solids or capillary forces at interfaces. There are as yet only a few examples of simulations of single ion-solvent systems. Pollack The first two types of forces play a special role and Alder? have simulated an ion in a polarizable in understanding solvation and biopolymer hydra monatomic solvent. Ciccotti and Jacucci-^ have tion. performed a nonequilibrium simulation on a similar system in which the computed drag coefficient The issues are: compares favorably with experiments. Gosling ard Singer4 have simulated an ion in a solvent of 1. How do we describe the thermodynamic polar diatomic molecules. limit for these systems? In many-ion systems, electrical forces are 2. How rapidly, as we increase the size of screened at equilibrium at finite concentrations. the system, do we approach the thermo Nevertheless, care must be taken for dynamic dynamic limit? properties. As Fulton has most recently pointed out,5 if charges are moving, the ionic atmospheres 3. How can we speed up this approach? cannot adjust infinitely rapidly so that there are effective velocity-dependent ion-ion interac The answers to these questions depend on whether tions of dipolar range. This makes the evaluation we are focusing on static or dynamic problems. of the conductivity from correlation functions just as subtle as the dielectric phenomena which For single ion thermodynamic properties the occupy so much of the rest of the conference. approach to the thermodynamic properties is quite slow. A calculation based on the Born charging At low concentration one quantity of great model for a drop indicates its free energy differs interest is the solvent-averaged ion-ion potential from the free energy in the thermodynamic limit of mean force. Simulations have so far been some by an amount inversely proportional to the radius, what unsuccassful at obtaining these.6 Rc. While the correction is large, it is easily calculable (at least to lowest order). Thermo dynamic perturbation theory should allow one to Hydrodynamic forces are long-range and are correct simulation results reliably for static dynamically generated even in systems whose quantities. underlying interactions are short-ranged. Some discussion of the rate of approach to the thermo At first sight, dynamic properties approach dynamic limit has taken place in the context of the thermodynamic limit more rapidly. Cavity the long-time tail phenomenon.' Hydrodynamic back- calculations, which can be obtained from the theory flow effects make the diffusion constant approach of Titulaer and Oeutch,1 indicate that the correla its true value very slowly if drop ooundary condi tion function of electric field fluctuations on tions are used. In stochastic simulations, rather a fixed ion (which should be related to the long- than being dynamically generated, hydrodynamic range force contribution to the drag coefficient) farces must be introduced directly into the model. and field gradient fluctuations approach the limit Ermack and McCammon^ have shown how to do this, more rapidly, differing by terms proportional but the question of boundary condition prob'ems to Rc"^ and Rc"^ respectively. One must take care, remains to be addressed. -32- Long-range forces are of course present in 4. E. Gosling, and K. Singer, Chem. Phys. Lett. aqueous biopolymer systems. In this case the large 39, 361 (1978). size of the molecules makes it necessary to use large systems even when taking into account only 5. R. L. Fulton, J. Chem. Phys. 68, 3089 (1978). the short-range forces. In the classic simulation of Rossky et al.9 the peptide is surrounded by 6. I. R. McDonald, and J. C. Rasaiah, Chem. Phys. only two layers of water. Approaches similar to Lett. 34, 382 (1975); G. Patey, and J. Valleau, those used for the long-range force problem in J. Chem. Phys. 63, 2334 (1975); D. J. Adams, simple systems may be helpful here. and J. C. Rasaiah, Faraday Discussion No. 64 (1978). REFERENCES 7. T. Keyes, B. Ladanyi, J. Chem. Phys. 62, 4787 (1975); W. W. Wood, Fundamental Problems in 1. U. M. Titulaer, and J. M. Deutch, J. Chem. Statistical Mechanics III, North Holland Phys. 50, 2303 (1974). Publ. (1975). 2. E. L. Pollock, and B. J. Alder, Phys. Rev. 8. D. Ermak, J. A. McCammon, J. Chem. Phys. 69, Lett. 41, 903 (1978). 1352 (1978). 3. G. Ciccotti, and G. Jacucci, Phys. Rev. Lett. 9. P. J. Rossky, M. Karplus, A. Rahman, 35, 789 (1975). Biopolymers 18, 825 (1979). -33- LONG-RANGE FORCES AND ION TRANSPORT ACROSS MEMBRANES Kent R. Wilson Department of Chemistry University of California, San Diego La Jolla, CA 92093 We are involved in several simulation studies representing the rest of the helix) interacting in which long-range forces are important, includ with waters and a Li+ ion, in a molecular dynamic ing the computation of vibrational spectra in study of the water and ion motions in the pore. polar solvents and of the molecular dynamics of A computer animated film has been prepared which reactions in polar solutions. A particular ex shows these atomic motions. Tests of calculated ample is the question of how ions are selectively versus experimental ion mobility in pure water are transported through membranes. The nature of the encouraging. microscopic machinery which accomplishes this is an intriguing puzzle in chemical physics whose so With Arnie Hagler of Weizmann Institute, we lution is of considerable biological importance. are working now on treating all the atoms of the Cells and cellular subsystems are surrounded by whole dimer (30 amino-acid residues). We are membranes, so that the molecules and ions which close to modelling ion + water + rigid gramicidin. pass in and out must be transported from the aque The next step will be to allow all the protein ous, high-dielectric exterior, through a low- atoms to move as well in order to look at the de dielectric membrane medium, and again into an tailed process of transport. aqueous medium. One mechanism is by pores, pro teins which form passages through the membrane. By using a set of linked processors, including These pores are key elements, for example, in the an array processor, we are able to handle thousand- transmission of nerve impulses along axons, as atom molecular dynamics for runs of tens *:o hun well as in the passage of signals across synapses dreds of picoseconds. However, it seems doubtful between nerve cells. that we can model enough atoms explicitly to pron- erly take into account the long-range forces in He are using molecular dynamics to model the any brute-force manner. Thus the proper solution transport of ions through a simple polypeptide of this problem, as well as of many others, is pore, gramicidin. Its primary structure is known, tied to progress in the general area of simulation and there is considerable evidence as to its three- of systems with long-range forces. dimensional structure. Urryl has proposed a head- to-head dimer of two helices, a structure which REFERENCES has passed a number of chemical, nmr and x-ray diffraction tests. Part of his model involves 1. D. W. Urry, Proc. Nat. Acad. Sci. USA 68, 672 the ability of the he Veal pore to distort, to (1971). — allow the carbonyl oxyjens to rotate into the pore to partially solvate the positive ions, re 2. R. 0. Watts, Chem. Phys. 2ji, 367 (1977). placing some of the water molecules, the helices perhaps resembling a python swallowing a pig as 3. E. Clementi and H. Pookie, J. Chem. Phys. 57, the ion passes. 1077 (1972). ~ We have, in initial tests, used the Watts2 4. H. Kistenmacher, H. Popkie and E. Clementi, water-water potential and dementi's quantum ibid. 58, 1689 and 5627 (1973). potential surfaces for ion-water3'4 and water amino-acid5 to model a "bare-bones" protein 5. E. Clementi, F. Cavallone and R. Scordamaglia, (consisting only of the carbonyls plus a soft wall J. Amer. Chem. Soc. 99, 5531 (1977). -34- ION TRIPLES IN 2-i ELECTROLYTES P. J. Rossky and J. B. Dudowicz and H. L. Friedman Department of Chemistry Department of Chemistry University of Texas State University of New York Austin, TX 78712 Stony Brook, NY 11794 INTRODUCTION In the following, we will use this correspondence to identify the ionic size in the soft sphere A mutual comparison of results obtained from model. computer simulation and approximate analytical theories for the same models provides i ^urce for HNC AND MC RESULTS the evaluation of the shortcomings of .jch ap proach. Here, we report some aspects of a compari Fig. 1 shows the like charge pair correlation son ot Monte-Carlo simulation (MC) and the function obtained for the model from the HNC equa hypernetted chain (HNC) equation for models of 2-2 tion at a series of representative salt concentra electrolyte solutions, with the primary goal of tions. The dashed lines shew the corresponding evaluating the accuracy )£ the latter. The param results for the hard sphere model with R = A.2 A. eterization of the model corresponds to an aqueous The two models give corresponding structure as 2-2 electrolyte at 25°C, but corresponding results mentioned above. The amplitude of the local maxi are expected for appropriately reduced charge and mum in g|| (r) is, not surprisingly, sensitive to dielectric constant. Earlier comparisons** have ionic size, as shown in Fig. 2; the curves are shown that for 1-1 electrolytes HNC and MC agree labeled by the corresponding charge hard sphere very closely, but for 2-2 systems the const s ten.-y diameter, as described above. is less clear. Of particular interest, related earlier studies ot 2-2 systems have found The MC calculations are carried out using evidence for substantial triple ion formation in standard methods, using a minimum image trunca the wide concentration range 10~u M i Cg.^ <. 0.2 M tion of the potential. The truncation point in (CCT is the stoichiometric salt concentration), units of the Debye Kappa (K) is, for 216 parti cles, 7.7 K"1 (c = 0.005 Ml and 16.9 K~> this being most clearly manifest by a local maxi ST mum in the like charge pair correlation function g.pj.tr) near twice the ionic diameter. One may infer that this follows from the formation of linear trimers, H—K The results presented below indicate that this distinct structural feature is a figment of the HNC approximation, and hence that caution should be used when applying the HNC equa tion to dilute systems with high charge/dielectric constant ratio. MODEL A continuum dielectric solvent model and equal diameter soft spheres ions are used. The pair potential is ,:> u. .(r) = B. . - + Z.Z.e2/tr where e is the magnitude of the electronic charge, Z^ is the ionic charge in units of e of a particle of species i, z is the solvent bulk dielectric .vr.it'int, and i is the nominal ion diameter. Further ve put5 : vhfc: re -. = ; /k3T, ?-o is 3oltzmann s constant, T is t'-r- >.-JS•'•', He terr.pe r •> t^re , and -. is expressed in Ar.t-str .~ uniTs. ': -r the 2-2 electrolyte solution Q stud/ v~ take -..r= 7H.VJH, 1 = 12, T = 29S.16 K, and 7 = I • '"'- 2-rs A • The si; parage ters correspond to tho'^c; •i-et'i in f-fl-jier studies *' on charged hard sphere models with diameter F: = i.2 A; with the above choice of J the un 1 i ke r.fwr^e soft sphere Figure 1. Like-charge soft sphere pair correlation mode] ha? a mini nun at the sa"*; distance (h .2 A), functions from HNC. Curves are labeled by salt and the two mode 1 h are f <\ Figure 3. HNC and MC results for g++(r) at CST = 0.005 M. MC results from 64 particle system, 640 K single particle noves. r(A) Figure 2. Ionic size dependence of g++(r) from HNC (cSj = 0.005 M). Curves are labeled by the corresponding hard sphere diameter. (cST = 0.5625 M); for 64 particles (cST = 0-005 M) we find 5.1 <""J. Hence, boundary effects should be small in the cases considered below, r/ (or»iVj= 2.1 A comparison of internal potential energies R I) obtained from MC (216 particles) and HNC indicate likely errors in HNC of no more than about 27-. at cST = 0.2 M and 0.5625 M, and about 7'£ at Figure 4. Unnormaiized like-charge pair correlation = CgT 0.005 M. However, for concentrations functions from constrained Konte Carlo simulation CST ^ 0-20 M there are serious discrepancies in the for various ionic sizes (CST = 0.005 H). The structure shown by the like-charge pair correla dijtance axis is divided in sections of length tions, while the unlike-charge correlations a; pear 2.1 A; the two curves to the right are shifted qualitatively consistent.4 At such low concentra by 4.2 A and 8.4 A, respectively, with respect tions It is difficult to obtain accurate evalua to that labeled 3.8 A. tions . < g_f_f_(r) in the region r = 2R. A system of 216 inns is predicted by HN'C to include on the average only about 0.4 -H- pairs with separatum less than 10.5 A at c = 0.005 M. A typical gT Tae pair correlation functi nod result Is shown in Fig. 3. To enhance sampling in particular pair Jij_4.fr> ') is i on .1! to the region of interest, we have carried out con [in. true pair correlation funct r) f. r strained Monte-Carlo calculations using a system Ion r- < 10. "•> X; tiic propprtio ens rant of bb ions. The procedure follows similar me thuds n.- 1 i t v however, nor easily evn 1 ua tod. ' K.> - r . t. descrihed elsewhere.7 An artificial potential u' the veracity of the structural Nat in t is introduced only between two particular positive Mr) near r = 2K appareni. in the HNC cal it ions, say 1 and 2, -111 ns. ve can examine ^^.(r) for a series of ion i c r adi i . and compare w:th the trend* in Fie., 2. Thie c a leu lated !0 r,. £ in.5 A functions v.'\ , f r) .ire -shown in Fie,, i. It is e v i - u'(r,,) dent that nu local maximum aeoears in r) for r,, :> 10.5 A anv of tne diameters considered. -36- 0R1GTN OF THE DISCREPANCY Une can argue that the differences in g+.j.tr) can be assigned to insufficient cross correlation among counter-ions which are strongly attracted to CL8 the -H- (or —) ion pair. A quantitative correc —^HNC tion to the HNC result may be evaluated by calculating the leading bridge diagram correction tu ] r. { fi++(r ) J namely, in standard notation, F ^Tcr' ^C"' s++( o.i- "•' • l+<^>' / ^ I / BHNC where for the honds we use the HNC result for !i_(_^{rj and h+_(r). The new approximation *~o g_H_(r) denoted BHNC is n—^ " i i . 1 i BHNC, , + S^r+ ) HNC ,> (r) = e gj_^ (r) l>b S^tr) is evaluated by ordinary MC integration yielding the representative result shown by circles in Fig. 5 . Error bars (one standard deviation in Figure 5. HNC and BHNC theories (see text) for S++) are indicated. For cg-r .C 0.2 M, the correc 9++(r) (cST = 0.005 H). tion becomes negligible. It is clear that the local maximum in gt l(r) is removed upon the addi tion of the leading correction to the HNC result. Current work is aimed at obtaining an accurate assessment of the thermodynamic consequences of the failures in the HNC predictions and a quantitative 2. J. C. Rasaiah, J. Chem. Phys. 5_», 3071 (1972). evaluation of the accuracy of the BHNC results and re lated app roximat ions. 3. H. L. Friedman and B. Larsen, .1. Chem. Phvs. _7C_* 92 (1979). ACKSiiWLEDf'.MKST 4. P. .1. Ros.sky, J. D. Dudowicz, B. Tembe , and II. L. Friedman (to be published). We gr ttt-fu I ly urknuw ledge support of this work through a grant 1 r<>m the National Science 5. P. S. Ramanathan and H. L. Friedman, .1. Chem. Foundation and the granti ug of an NSF National Phys. 54_, 1086 (1971). Needs Postdoctoral Fellowship to PJR. n. B. Larsen and J. C. Rasaiah (to be published). REFERENCES 7. C. Pangali, M. Ran, and R. .1. Berne, J. Chem. I. 0. N. Card and J. P. Valleau, .1. Chem. Phvs. Phys. 7J_, 2975 (1979); G. N. Patev and 62 32 '1970). J. P. Valleau, .1. Chem. Phvs. 63, 23 j4 (1975). -37- Hsir.. Id ;.. F-ioin! r,er-,.r,_r.'irit :' ~.t : f :i.--v Y rk, . i . ' Endsrby •! at — — \~• 4 4M NiCI2 '• / • \ " Hard Sptierss Quirk! S Sopar —' 1 L- k 0 I 2 3l Figure 1. Partial structure factor from neutron diffraction^-'' and hard sphere model.*? -3H- 0 5 " .1/3 (molarity) 7' ii;<~>' ••'. Position of first peak as a function •:' -tir,! it-• fyl' • for neutron diffraction'''' and for •v.r.i sphere mode'.'' Bl denutes the Braqq's law ' •> "e ' iiq. ''!. V' '• V. c*,-jV<^vv-'" ". gve ?. =.il 3' ,'istr 'but-on 'unction for Ni-Ni nai-"S ' rnr Mole' A fir severs' concentrations. -39- Figure 4. Partial structure factors for Ni-Ni pairs for Model A. 1/3 [molarity] Figure 5. Molarity dependence of krj for Models A, B, and C. BL denotes the Bragg's law line (Eq. ?). • i, •«•'.•• i.. W. Ni-ils..i: ,in,l I. K. KinUTbv, I. 1'hv !I. I.dj'. (]y?H). -40- Lxtric effect in th*-*' n->Iu'jil ity. The neutron 11. R. H. Stokes, 3. Pharu?, and P. Mills, ,T. di l"fr:ic:l Lor: exrerimen t ~. involve DpO solut ions Solution Chem., 8_, I489 (1979). at ."--.°~. " ' 12. N. Quirke and A. K. Soper, J. Phys. C. , 10_, '•!. T. Font ana , Col id St. Commun. 1_8_, 765 1802 (1977). 'l i-V). 13. D. Elkoubie, Ph.!"'. Thesis, Universite' Pierre '. '••jbi;-: ; i , 1. Vai^anr, F. Mirliirdo, and et Marie Curie, Paris, 1978. '-. 'A^iiorl; •.,-»!, -". Fhyn. f, 1£, 1*689 (1977). I1* • H. L. Friedman, unpublished results. •:. H. Mar:-:, and M. I. To-i, Phys. Letts 50ft, 15- The g(r) determined by an 11=6^ simulation agrees with Verlet's ll=P.6h result for the F*. A. F.-'r i nr. •:: and ?. H. ftokes, "Electrolyte same model vithin the thickness of the line ."-il-j'.L^:::-.," ruf.^rv-rthn, London, 1955. of the p;raph, in the r-range up to L/2 for •". F. '.'• '~n:•:,-or.' , : r i vatf eommunicat ion . the 11-6^ simulation. -41- SUMMARY OF SESSION III P. J. Rossky Department of Chemistry University of Texas at Austin Austin, TX 78712 The session centered around what might best expansion of the equivalent conductance, as do be termed long-range force problems in solutions. short-range forces. As pointed out by wolynes, the model and real systems of primary interest here are, in general, In discussion, A. Hagler pointed out that less well standardized and less well studied than the problem of periodicity in boundary conditions those of interest in earlier sessions. was of much less concern when studying solvation in crystals where substantial water of crystalliza Wolynes drew attention to problems related tion is present, and the "solute" is naturally to both static and dynamic properties. In periodic. H. Andersen emphasized the advantages particular, the fact that the approach to the of considering the passage to long-range forces thermodynamic limit may be more rapid for some after that to the thermodynamic limit, analogous dynamical properties than for static ones to the technique considered by B. Larson (see emphasizes the fact that focusing on a particular Session I). system property may mislead one as to the amount of valid informatio . which may be obtained from The fact that, in most systems exhibiting simulations on fini:e systems within present long-range forces, short-range forces are also limitations. A related point, reiterated in a very important formed a theme in the three short different context by Kent Wilson, is the general contributions. Rossky presented simulation results question of the relative role of long- and short- indicating that at very low concentration the range forces in determining the basic structure hypernetted chain integral equation is not capable and dynamics of systems which include strong short- of accurately representing the short-range structure range ordering. It seems this question is one of highly charged ionic systems, although the that requires i .rther comparison using controlled lonq-ranqe part is treated well. The alt -native studies which treat the long-range forces at possibility of treating the short-range clustering different levels of accuracy. aspect via "chemical" models (e.g., physical cluster theory) was raised by J. Barker. Wilson A further general point of particular emphasized the possible role of forces at short importance raised in this session is the fact range between ions and polypeptides in the important that certain long-range forces can contribute to problem of ion transport through membrane pores. dynamical quantities and have no influence on time- In Friedman's contribution, a study was made of averaged quantities. Two primary examples raised models for very concentrated NiCl? solutions by Wolynes are that of the hydrodynamical interac (approaching fused salt densities) where packing tion, which contributes even in systems with only forces are important and where, it was pointed short-range static forces, and instantaneous dipolar out, the combination of long-range Coulombic and forces between distorted ionic "atmospheres" short-range forces collaborate, leading to corre surrounding moving charges in ionic fluids. The lation lengths even Tonger than either alone would latter contributes to linear terms in the density produce. SESSION IV SPECIAL TECHNIQUES FOR LONG-RANGE FORCE SIMULATION -45- REACTION FIELD METHOD FOR POUR FLUIDS 0. A. Barker IBM Research Laboratory San Jose, CA 95193 The long-range dipolar interaction plays an a) = £w + I .W "i j W important role in determining the structure of i^ fluids composed of molecules with permanent dipole moments (of which water is an example of great chemical and biological importance). The dipole where interaction affects particularly the angular correlations of the dipoles, but these in turn u = -Li J ZjE-1} (5) must affect the spatial arrangement of molecules T ii 2c+ 1 (in the model* of Pople for water, first-neighbor angular correlations determine second-neighbor and positions; on the other hand for "hard-spheres and central dipoles" the dipolar interactions appear to have relatively little effect on the angle averaged radial distribution function at -a'3 ^ Hi-Wj. *ij- <*> least at high densities). The dielectric constant depends on the angular correlations through the %- 0 , R >a Kirkwood g-factor. ij g = ( m, . Im. )/u (1) Thus Wi is a self-energy and W^j is an effective short-range pair potential which The first computer simulations for water 2,3 describes the effect of the long-range interactions gave a value for this factor smaller by a factor on the short-range correlations. It is readily of about 20 than its experimental value; this was verified that Wi gives the correct long-range due to neglecting the effect of the long-range correction to the second virial coefficient. It part of the dipolar interaction which tends to is also possible to check the validity of Wij by favor fluctuations of polarizations. If the dipole comparison with density expansions of the angular potential is truncated as finite distance, these correlation functions, and work on this is in fluctuations are suppressed. If one imagines a progress. Note that the potential Wij is expected set of molecules in the fluid contained in a to correct only the short-range angular correla spherical volume of radius a, volume V and in a tions. In the spirit of this method, one should configuration with total dipole moment m, then calculate the long-range angular correlations from the surrounding molecules will behave lTke a the dielectric constant by electrostatics. dielectric with the macroscopic dielectric constant e and will produce a field (the Onsager reaction The reaction field method was used for dipolar field) in the sphere which is given by lattice systems by Adams and McDonald* who found that it gave results similar to the Ewald method (see below). Useful discussions are given by BerendsenS and Friedman^. Levesque et al.7 conclude - -3 2£+l - \3V/ 2E+1 (2) that the reaction field method is unsatisfactory. However this conclusion is based on the assumption that the LHNC equation deals correctly with the This is exactly the field at the center of the long-range interaction. This seams at least as sphere; if the moment lis a point dipole at the questionable as the reaction fie'd method, which center of the sphere or a uniform distribution has a sound physical basis. of polarization, it is exact everywhere in the sphere. This field produces a contribution to The other method which has been used is Ewald the (free) energy given by summation (for discussion and references see E. R. Smith, this proceedings). This method sums the interaction to infinite range, but it does this l(4ir\2(e-l) 2 for highly artificial configurations in which the 2 \3v) 2c+l HI (3) polarization of the unit computational cell is replicated periodically. These are not the configurations which are important in fluids; in Thus we expect that configurations of large m should a fluid the dipole moments of different regions be favored by the weighting factor exp(-oi/kTj. are largely uncorrected, the mean dipolar energy of two different regions is zero, and the reaction Barker and Watts3 proposed that this effect field corrections come from the second moment of should be included in simulations by truncating the dipolar energy, not from the mean. However, the potential at distance a and including an the Ewald method as customarily applied gives a additional potential given by mean dipolar intercell energy equal to -46- 4l T Some of the paradoxes concerning Ewald summa 1 I \ J- tion which arose in the discussion are resolved in papers by Stuart8 and Redlack and Grindlay.9 and this happens to be identical with the result given by the reaction field Eq. (3) when the dielectric constant is large [so that 2(e-l)/(2c+l) REFERENCES approaches 1]. Thus one would expect the Ewald method to work well for large dielectric constant, 1. J. A. Pople, Proc. Roy. Soc. (London) A221, but to over-estimate the short-range angular corre 498 (1954). lations for •„- near 1. This appears to be consistent with the results of Adams and McDonald'1 and of 2. A. Rahman and F. H. Stillinger, J. Chem. Phys. Pollock and Alder (see E. R. Pollock, this 55_, 3336 (1971); F. H. Stillinger and A. Rahman, proceedings). One could say that the Ewald method J. Chem. Phys. 57, 1282 (1972) and 60, 1545 is justified for large E by the fact that it (1974). _ approximates to the reaction field. 3. J. A. Barker and R. 0. Watts, Molec. Phys. 26, Smith (this proceedings) has proposed adding 789 (1973); R. 0. Watts, Molec. Phys. 28, 1069 the te. ii (1974); 0. A. Barker, Molecular Dynamics and Monte Carlo Calculations on Water, CECAM Report (1972) p. 21. 1 CM J 4. D. J. Adams and I. R. McDonald, Molec. Phys. 32, 931 (1976). to the Ewald energy to cancel the corresponding 5. H. J. C. Berendsen, Molecular Dynamics and negative term mentioned above. If the arguments Monte Carlo Calculations on Water, CECAM given here are valid this would be roughly equiva Report (1972), p. 29. lent to truncating the potential without reaction field. More reasonable would be to add 6. H. Friedman, Molec. Phys. 29, 1533 (1975). 7. D. Levesque, &. N. Patey, and 0. J. Weis, l/4-\ 2 r, Z(c-l) 1 Molec. Phys. 34, 1077 (1977). 8. S. N. Stuart, J. Comp. Phys. 29, 127 (197E). to correct from infinite to finite dielectric 9. A. Redlack and J. Grindlay, J. Phys. Chem. constant. Solids 36, 73 (1979). -47- A MONTE CARLO STUDY OF ELECTRIC POLARISATION IN WATER R. 0. Watts, The Australian National University Canberra, Australia The Monte Carlo method developed by Barker and The next problem concerns the calculation of the Watts I 1] to study the structure and thermo dielectric constant from the Monte Carlo calculat dynamic properties of liquid water has been ion. Standard dielectric theory ( 5! shows that modified to include the effect of strong electric fields. As most models of the water pair 4TT the periodic image convention. Thus the field is calculated for a series of values of cR until acting on the molecules is treated as being the two dielectric constants agree. At this point external, rather than a cavity field. This point the calculation is self-consistent - the reaction is important, for if the field is a cavity field, field used to account for long range effects all dielectric constants reported in table L need corresponds to the model value. In th,e case of the to be divided by 3c_/(2£n+l) , approximately 1.5. Rowlinson potential truncated at 6.2 A (a 64 -48- Table 1. Summary of dielectric properties for water. Results from cube Results from sphere 2 2 No. Calc. E E* P E M /V P E R M /V 500 K 1.0 0.0 0.86 - 2.56 0.52 _ 6.17 500 1.0 3.25 3.83 7.51 2.98 3.60 5.97 250 2.0 6.01 3.62 23.4 6.06 3.64 5.88 250 5.0 15.0 3.61 129.0 15.0 3.61 5.81 250 10.0 26.4 3.30 24.2 3.11 4.37 376 20.0 32.7 2.42 32.3 2.41 2.81 500 50.0 34.7 1.60 678.0 34.5 1.60 2.65 500 16.0 0.0 4.17 - 18.0 0.25 - 8.11 500 1.0 12.1 11.54 101.0 12.9 12.24 9.54 250 2.0 21.5 10.37 270.0 23.1 11.06 9.59 250 5.0 29.4 6.12 31.2 6.44 7.77 250 10. 32.3 3.81 33.7 3.94 6.04 500 30.0 1.0 18.5 17.12 202.0 19.0 17.55 7.93 500 50.0 1.0 17.7 16.96 190.0 18.6 17.20 8.48 1740 80.0 0.0 6.67 - 31.8 3.33 _ 19.20 500 1.0 18.2 16.86 64.4 18.0 16.68 4.30 Notes (i) P in Debye £-3 * io3 (ii) M2/V in Debye2 8-3 x 102. particle system) it is found that z - 8, in poor A full report of the study, together with agreement with the experimental value of 78.5 thermodynamic and structural data, will be for water. published elsewhere. In addition to studying the effect of the applied field on the polarisation of the system, the atom-atom distribution functions were also REFERENCES calculated. It is found that the position of the first peak in g00(r), and its magnitude, increase 1. J.A. Barker and R.O. Watts, Chem. Phys. Letts. as the field strength is increased. The oxygen- _3, 144 (1969). hydrogen radial distribution function shows a marked decrease in the height of its first two 2. J.S. Rovlinson, Trans. Farad. Soc. 47^ 120 peaks, suggesting that the 0-H correlation is (1951). reduced by the applied field. Changes in gnjj(r) 3. J.A. Barker and R.O. Watts, Molec. Phys. 2£, are particularly marked and at high field strengths 789, (1973). most of the structure in this function vanishes. Finally, the dipole-dipole correlation function 4. R.O. Watts, Molec. Phys. 28, 1069 (1974). g (r) tends towards zero at long distances, for 5. H. Frohlich, Theory of Dielectrics, Oxford zero field strength, as has been demonstrated in University Press (1958). earlier work [ 3l . At high field strengths this function is large at long distances demonstrating 6. A. Rahman and F.H. Stillinger, J. Chem. Phys. that the system is strongly polarised. 55, 3336 (1971); 57, 1281 (1972). -49- LATTICE SUMS FOR PERIODIC BOUNDARY CONDITIONS E. R. Smith Mathematics Department, University of Melbourne Parkville, Victoria 3052 Australia We may write the energy of a configura tice sum over lattice vectors with compon tion of N particles of charge q± at r^ ents -M SUHHARY OF SESSION IV R. L. Fulton Department of Chemistry Florida State University Tallahasee, FL 32306 The presentation of the techniques used in The techniques used include: Minimum image; the simulations involving long-range forces was Reaction fields; and Ewald sums. The use of not restricted to this session. Because of this minimum image (crudely, keep only the most important 1 will include in this summary aspects of the forces from the nearest cells) is generally agreed simulation of long-range dipolar forces which were to give poor results when used to treat dipolar presented in other sessions "and discussions. To forces. This technique will not be discussed. provide some orientation I consider briefly the The other techniques use, either implicitly or general techniques used in the simulations to explicitly, an infinite number of replicated cells. minimize the effects of surfaces, then consider the prescriptions used in dealing with dipolar The use of reaction fields was suggested some forces and finally enumerate the difficulties and time ago in an attempt to mimic the back reaction the problems still outstanding. the long range polarization fluctuations have on the molecules in the basic cell. To implement Because the machine simulations use a limited this scheme in Monte Carlo simulations, the center number of particles {usually of the order of 256 of the basic cell is located at the particle being or 512 particles) with the resulting large ratio considered for the next move. The energy is cal of surface to volume, replication techniques are culated as the sum of two parts: the first consists used in an attempt to minimize surface effects of the contribution from the particles contained and hopefully to provide more accurate calculations in a sphere inscribed in the basic cell; the second of the bulk properties of the system. The basic consists of the contribution from the reaction cell in which the simulations are carried out and field calculated by treating the medium external its contents, that is the positions and orientations to the sphere as a continuum having the "proper" of the molecules, is replicated. The extent of dielectric constant. The particle is then movea, replication depends upon the range of the forces a similar calculation made of the new energy and involved, a judicious guess on the part of tv= the move accepted or rejected according to estab investigator and the technique used in tht infla lished criteria. An a priori calculation must tions. For potentials which decay rapidly, for therefore be done iteratively to find the dielectric example as the inverse sixth power of the inter- constant. molecular distance, it seems to be adequate to replicate nearest neighbor cells, or at least to Some two particle correlation functions, such consider only the effect of potentials extending as the angular correlation function which is long- into nearest neighbor cells. Problems arise with ranged, seem to have a discontinuity at the sphere such a truncation when dealing with dipolar forces. radius in this technique. Such discontinuities may disappear for large cell size. Nonetheless, The motion (either in a dynamical or in a this is disconcerting. It is to be noted that Monte Carlo sense) is governed by the forces between the basic cell used for replication is cubic, while the particles in the basic cell and neighboring the reaction field is that appropriate to a sphere. cells. But dipolar forces are long-ranged, and Some particles in the basic cell are therefore this brings about difficulties. For example, replaced by a continuum. Are there consistency the long-range polarization fluctuations in non- problems arising from this? This could probably replicated, infinite, isotropic media are given by be answered if a statistical model having the features of the Monte Carlo simulation were established. The use of Ewald sums provoked a spirited discussion which lasted the entire workshop. Ewald sums arise in the following way. The basic cell and its contents are infinitely replicated. The forces and torques on the particles in the basic cell are those due to the particles in all These fluctuations fall off as the inverse cube the cells. Because of the imposed periodicity of the distance |r-r'|. In another sense these these involve Ewald sums. The molecule- dynamic fluctuations represent polarization correlations. simulations use these forces and torques in the The long-range forces are responsible for long- equations of motion; the Monte Carlo procedure range correlations. In the replication technique uses the corresponding energies. The sums involved this correlation is lost--rather a periodicity are conditionally convergent, the value of the is imposed on the correlations. Can these two sum depending on the prescription jsed for the different prescriptions be reconciled? Perhaps, summation. The differing prescriptions can be but first a brief description of techniques related to different bourdary conditions. See peculiar to dipolar systems. for example, E. Smith's abstract. This raises the following two problems. (A third is raised temperature for the replicated system the same later.) as the dielectric constant for an unreplicated system at the same temperature and density? The 1) If the dielectric constant is related consistency mentioned above leads to optimism. to the polarization fluctuations, what But is must be recognized that replication enforces is that formula? e particular order on the system which is lacking in the non-replicated system. Does this constraint 2) The computed mean square polarization change e from the case of no constraint? (This fluctuations depend on the sum. The constraint differs from that imposed by finite formulae must be different for the boundaries.) This quesMon does not seem to have different summation procedures if they been answered definitively. If the dielectric are to give the same dielectric constant constants are the same, a further question arises. for given temperature and given density. Do we know how to scale other properties, such We raise the question, "Are the formulae as two body correlation functions, so that they consistent?" Another way to put the are applicable to non-replicated systems? (In question is, "Is the dielectric constant addition, I urge authors to use care in identifying independent of the procedure used to carry which Ewald sum is being used.) out the sum?" Smith answers the second question in the affirmative and gives Problems related to the dynamical behavior various (consistent) possibilities for of quantities when using replicated systems were the connection between the dielectric not considered. It is well known in the literature constant and the fluctuations. I note of dielectrics of non-replicated systems what that '.he same program can be carried out pitfalls beset the unwary. Similar pitfalls must for anisotropic media, and that there exist when dealing with replicated systems, tind are, in fact, an infinite number of we must avoid them. fluctuation formulae. To get consistency seems to require (in principle) a large R. 0. Watts raised the question of how to basic cell size. How large was not interpret the applied field when using replicated specified, [it is possible (again in systems in simulations. This is important when p-inciple) to make some estimate of the dealing with non-linear (e.g., saturation) size using the fourth cumulant of the phenomena. This seems not to be a trivial problem. cell polarization because the first correction to the relations between the A qeneral question of whether or not long- various fluctuation formulae involves range forces could be ignored when calculating 1/15 <&MZ AM2> -1/3 PLASMAS: ESPECIALLY THOSE ACCURATE RESULTS THAT MAY BE USED FOR TESTING VARIOUS SIMULATION METHODS -55- REV1EW TALK SIMULATION OF PLASMAS Ian R. McDonald Department of Physical Chemistry University of Cambridge Cambridge CB 1EP, U.K. INTRODUCTION "quanticity" of the electrons, on the other hand, car be discussed in terms of the Fermi temperature. The plasmas to be discussed in this talk are intended as models of matter under conditions of 2/3 5 2 density and temperature which are sufficiently Tp = ykB = h^3p/i) /2mekB - (6*10 )rs" K extreme for it to be considered fully ionized. As such, they are relevant to a number of important astrophysical problems (planetary interiors, white where fp is the Fermi degeneracy energy. Assuming dwarf stars, neutron star crusts, etc.), and in classical behavior for the ions, three different the physics of laser implosion. Apart from these electronic regimes may be distinguished. practical applications, a further reason for study ing idealized models of this type is for the con 1. T--Tp. In this case the electron gas tribution it makes to a deeper understanding of is highly degenerate. If, in addition, the Thomas- the properties of strongly coupled Coulomb fluids. Fermi screening length ^TF is greater than a. The often unusual behavior of such systems is, of course, a direct consequence of the long range if of the Coulomb potential. -1/2 All the simulations to which we shall refer /a = W12Z ,1/3 have been based on the standard Monte Carlo (MC) TF and molecular dynamics (MD) methods used in the study of liquids. The earliest published papers which, in effect, means that rs • 1, then the polari in this field were those of Barker^ and of Brush, zation of the electron gas by the ions is suffici Sahlin, and Teller,? but we shall be primarily ently weak for the electrons to be regarded as concerned with the work described in a series of forming a rigid, uniform background. The ions may papers by Hansen and others, the first^ of which then be treated as a classical one-component plasma appeared in 1973. Most of the MC and MD calcula or OCP. tions have been based on samples of 250 ions, with the forces and energies calculated by the Ewald 2. T • Tp. Under these conditions the elec method described by Brush et al.^ Despite its trons also behave classically, resulting in a obvious limitations, we hope to show that this genuine two-component plasma or TCP, in which the approach does do adequate justice to the long-range ions and electrons are treated on an equal footing. force problem with which this meeting is concerned. (This is a more realistic model for the process of laser implosion, at least in its later stages). Of course, a fully classical system of this type SOME DEFINITIONS would not be stable, since the ion-electron attrac tion would lead to collapse. The problem can be Consider a system of N ions of charge z|e! overcome within a classical framework by working and mass mi and NZ electrons of mass me, all in a with effective pair potentials which incorporate volume V. The i on number density is n = N/V and short-range quantum diffraction effects. The tern a convenient uni t of length is the ion-sphere perature should also be sufficiently high to avoid radius a * (3/4 n)l/3_ A dimensionless quantity the occurrence of bound states. of similar signi ficance for the electrons is the 3/4-rp)l/3/a = Zl^a/ao, where parameter rs = ( 0 3. T ^ Tp. This is a much more complicated ;• = Zn is the ch arge density and a0 = h'/4-?mee? situation. The electron gas is now partially de is the electroni c Bohr radius. Some typical values generate and neither the OCP nor TCP are valid of r5 are models. rs - 3 metallic conductor THE OCP rs < 1 - laser imploded pellet The Coulomb Coup 1ing Parameter. The main feature of the OCP is that the electron background rs =• 10"2 - white dwarf star is assumed to be rigid; it therefore plays no role in the dynamics, though it does contribute to the Since m; »me, it is reasonable to hope that thermodynamics of the system. Corrections to in many circumstances it will prove possible to static properties arising from polarization of the use classical statistical mechanics to describe background can be made, however, by perturbation the ions even when the electrons require a quantum methods. In the absence of screening, the interac mechanical treatment. A necessary condition for tion between ions is the bare Coulomb potential, this to be true is that the thermal deBroglie wave Z^e'/r. Thermodynamic properties therefore obey length for the ions be much less than a. The certain scaling relations and the equilibrium state -56- o' 'he system is completely characterized, not by more surprising is that the effect of the plasma the density, n, ana temperature, T, separately, oscillations are also apparent in the motion of but bv the single quantity '•' = Z^e'/akgT. The single ions.7.8 In particular, it is evident that Coulomb coupling parameter . is, roughly, the mean for sufficientl" large ~ the ions become trapped •nterarr i.jn enerqv 1 i i ded by the mean kinetic in a plasma wave, and the velocity autocorrelation energy. For 1 the system can be described4 function Z(t) develops strong oscillations at a in te'riis of et trier Llebye-H'ucke I theory (for static frequency very close to ^n. This behavior is Quantities: or V'WSJV tneor^ (for dynamic proper illustrative of the overwhelmingly collective ties':. !n principle this offers the possibility nature of the ionic motion in a strongly coupled of testing i-esjlts of computer simulations based plasma. For T ' 1, on the other hand, Z(t) decays :r a particular treatment of the long-range forces monotonically to zero in a quasiexponential fashion, aqa'ns* analytical '"esu'ts which are exact in a the change to an oscillatory decay occurring at „ = i 1-def ined 'i-it. !n practice, however, simula- : "•• 10 (cf. Static Properties). It is also t or proves tc be an efficient technique only for notable that both the shear viscosity and thermal Three regions of ' are therefore of special conductivity of the OCP have a minimum at T ^ 10. Clearly, therefore, '.' -10 represents a boundary between two regimes of different character.9,10 '.' - '. - weak coupling ("hot") Collective effects can be seen more directly : • 10 - intermediate coupling ("warm") in the behavior of S(k, J), the spectrum of density 100 - strong coupling ("cold") fluctuations. (Note that in the OCP, charge and number density fluctuations are equivalent.) For small k (ka • 1), MD calculations show that the A natural scale of time is provided by the spectrum is dominated by a sharp "plasmon" peak, plasma frequency ..p, since all theories of the centered close to nip, which shifts and broadens dynamic properties are agreed in predicting that as k increases." Simultaneously, a low-frequency a'_ long wavelengths the dominant phenomenon is wing develops, but at small k there is no sign of a collective plasma oscillation at a frequency the central (Rayleigh) peak characteristic of nor mal fluids." A hydrodynamic analysis'-^ shows ? ? .n gi ven bv ."" = 4--. e /1m .. that a (thermal) diffusive mode jj, present, but p ^ ' P its intensity is insignificant in comparison with that of the (mechanical) "plasmon" mode. This Static Properties. Computer "experiments last result is directly traceable to the long range (MC.i hTve-sHown^ that the OCP undergoes a phase of the Coulomb interaction.13 transition to a body centered cubic structure at ~ ^ 155. However, as in other fluids, short range For small r, the dispersion of the "plasmon" order becomes evident well below the transition peak is positive, in agreement with Vlasov theory, density.^ For 7 ^ 1, the radial distribution func and at large r it is negative, as suggested by tion g(r) increases monotonically to reach g(r) = 1 hydrodynamic arguments;" the change takes place, as r - =. This is qualitatively in agreement with once again at r = 10. At all values of r, however, r Oebye-Huckel theory. At = 100, by contrast, g(r) the computer results strongly suggest that as k - 0 has a strongly oscillatory behavior similar to that the spectrum reduces to a delta-function at the of a dense liquid. The onset of these oscillations frequency »„, as is expected on theoretical grounds. occurs at T s 10, or somewhat lower, suggesting that it is in this range that collective effects begin to dominate. For r ; 10, Brush et al.2 found Ionic Mixtures. A number of investigations^"^ that minimum image and Ewald methods of summation have been made of the properties of mixtures of of the pair energies led to essentially identical OCPs, particularly of the system H+ + He++, but also results for g(r), but at higher charge densities of more asymmetric mixtures. When account is taken the minimum image method led to spurious oscilla of polarization effects, these systems are found to tions indicative of a premature crystallization undergo a phase separation!4*"' below a certain into a simple cubic structure. Largely because of critical temperature, and the results are of inter these results, it has been generally assumed that est for studies of the composition of the deep the minimum image method can be safely used for interior of planets such as Jupiter and Saturn. - 10, but this is now known to be untrue, as «c sha 11 see below. The dynamical properties of H+ - He++ mixtures have also been studied by MD methods.17,18 The Giv°n gi'r , the stat". structjre factor S(k) most interesting feature of such systems is that an t>e obtained straigntf V"«a*"dly by Fourier there is no unique plasma frequency.!' The charac t'ans'ormat'on. For th- jCE, tne conditions of teristic frequency which arises in the Vlasov jt'l linger and Lovett0 show that 5(k) • 0 as {' • 0) limit differs from that arising in the k • 0 and, further, that Sfk) /anishes as k' with hydrodynamic approach (valid for sufficiently large a coefficient related to the 3ebye length. The T) by a term proportional to [l\/m\ - 12/^2)' > simulation results dre found to satisfy these where the subscripts 1 and 2 refer to the two ionic e>act conditions, at least when the Ewald method species. However, the MD results suggest that the or summation is used. hydrodynamic limit is, in practice, never reached. The form of the charge fluctuation spectrum differs Dynamical Properties. All expectations4 are significantly from that observed for the OCP. At that the collective dynamics of the OCP will be large r there is both a "plasmon" peak and a narrow, dominated by the "plasmon" phenomenon, i.e., a central peak, the latter arising from interdiffu- propagating charge density fluctuation. What is sion of the two species. In addition, the "plasmon" -57- peak 's found to have a certain width even in the HO Calculations. By adopting the effective limit k • 0, i.e., the plasma oscillation has a pair potentials introduced above, a number of MD finite lifetime even at infinitely long wavelength. simulations of a hydrogen plasma1^, 23 have been From the Doint of view of this meeting, the most carried out for values of !' and rs of interest for important fact to emerge from these calculations laser implosion experiments {? 2, rs ; 1). The is that use of the minimum image and Ewald methods large proton-electron mass ratio did not create leads to different results^ for the spectrum of any special problems, and no serious difficulties charge density fluctuations, even for values of ' were encountered in either reaching or maintaining lying in what had previously been regarded as the thermal equilibrium. In this context it should be "safe'' region for use of the minimum image method. added that it would also be of interest to study In particular, it has been found that the miminum of one in which the temperature of the electrons imaae result for the frequency of the "plasmon" is much higher than that of the protons. Deafc is shifted by up to 15% with respect to the Ewald result, a large effect for such a well-defined The structure of the plasma is unusual insofar collective mode. All available evidence suggests r nsv as both gje(r) and gee( ) e non-zero values that it is the Ewaid result which is correct (or, at r - 0\ in fact, gje(r) has its maximum value at least, more nearly correct). there. The form of the function gee(r) is suggestive of an almost uniform background of THE TCP electrons and is the only one of the three partial distribution functions which is strongly influenced Effective Potentials. Simulation of a TCP by symmetry effects. In every case the tomputed requires the use of effect r.t pai'" potentials that values of 9lf..(r) a. c in excellent agreement with incorporate quantum diffraction effects, i.e., HNC predictions. The diffusion coefficient of the uncertainty principle. Symmetry effects can the ions turns out to be close to that expected also be taken into account, but these are generally on the basis of calculations for the OCP. Apart of 'ess importance. The significance "v the un from an obvious scaling of the time scale by a certainty principle is that the particles (ions factor (mj/me)l''2 - 43, the velocity autocorrela and electrons! cannot be correctly treated as point tion functions for ions and electrons are almost tharqes, but rather as smeared out charge distri identical. The electrical current autocorrelation butions. If these distributions are Gaussian in function decays roughly on the time scale of the shape, the electrostatic interaction is exponential- electronic motion, but there is an enhancement like at small separations r. More formally, the of the electrical conductivity compared to that many-body Slater sum predicted on the basis of a naive Nernst-Einstein relation by a factor of two or more. There is no immediate explanation for this unexpected result. W(lril) -- n.n* e*P (--En)-n The major effort in this work has been devoted to the study of the spectrum of charge density where -n and En are, respectively, the eigenfunc- fluctuations over a wide range of wavenumber. tions and eigenvalues of the full Hamiltonian, may Qualitatively the spectra resemble those obtained be rewritten in a form reminiscent of the classical for the ionic mixtures discussed in the Ionic Boltzmann expression, namely Mixtures section. Vlasov theory turns out to be poor at al1 densities and temperatures for which calculations have been made, but a memory function W - exp{ - .-^ j w(i,j) } approach which takes account of the coupling of the partial densities of the two charged species gives excellent results. If this effect is not where w(i,j) is an effective pair potential. In treated correctly, neither the strength of the particular,20,21 neglecting density dependence, central peak nor this damping of the plasmon the two-body Slater sum can be rewritten in such oscillations can be satisfactorily explained. a form, with CONCLUSION w^(r) = (l.l^lr) ll - exp (-r/-.1;.) } The main conclusion to be drawn from this work is that despite the obvious limitations of playing the role of a temperature-dependent effec the Ewald treatment, useful results :ar. and have tive Dair potential for particles of species >,r; been obtained for a variety of model systems in the quantity ".,; is given by which long-range interactions are dominant. There is no evidence that use of the Ewald method has in any situation led to results which are qualitatively Sr =h/ (8-' -^k T)1/Z • incorrect, and quantitatively the errors appear B to be very small. where i-,; is the reduced mass of an -»,.; pair. The REFERENCES potentials w.,r(r) remain finite at r = 0 and behave _1 as r for r '< -l£. Note that the properties of 1. A. A. Barker, Australian J. Phys. 18, 119 the TCP are separately dependent on temperature (1962). — and density or on two independent combinations of these. The latter are conveniently taken to 2. S. G. Brush, H. L. Sahlin and E. Teller, be the parameters r and r5 already introduced. J. Chem. Phys. 45, 2102 (1965). -58- J. P. Hansen, Phys. Rev. A 8, 3096 (1973). 13. M. Baus, Physica 79A, 377 (1975). See, e.g., S. Ichimaru, "Basic Principles 14. J. P. Hansen and P. Vieillefosse, Phys. Rev. of Plasma Physics," (Benjamin, Reading, Lett. 37, 391 (1976). Mass., 1973). 15. J. P. He isen, G. M. Torrie and P. Vieillefosse, E. L. Pollock and J. P. Hansen, Phys. Rev. Phys. Rev. A 16, 2153 (1977). A 3, 3110 (1973). f. H. Stillinger and R. Lovett, J. Chem. Phys. 16. B. Brami, J. P. Hansen and F. Joly, Physica 49, 1991 (1968). 95A, 505 (1979). J. P. Hansen, E. L. Pollock and I. R. McDonald, 17. I. R. McDonald, P. Vieillefosse, and J. P. Phys. Rev. Lett. 32, 277 (1974). Hansen, Phys. Rev. Lett. 39, 271 (1977). 0. P. Hansen, I. R. McDonald and E. L. Pollock, Phys. Rev. A U, 1025 (1975). 18. J. P. Hansen, 1. R. McDonald and P. Vieillefosse, Phys. Rev. A, in pre:. P. Vieiilefosse and J. P. Hansen, Phys. Rev. A 12, 1105 (1975). 19. M. Baus, Phys. Rev. Lett. 40, 793 (1978). B. Bernu and P. Vieillefosse, Phys. Rev. A 20. A. A. Barker, J. Chem. Phys. 55_, 1751 (1971). 18, 2345 (1978). 21. C. Deutsch, Phys. Lett. 60A, 317 (1977). J. P. Hansen and I. R. McDonald, Theory of Simple Liquids, (Academic Press, New York, 22. J. P. Hansen and I. R. McDonald, Phys. Rev. 1976). Lett. 41, 1379 (1978). M. Baus and J. P. Hansen, Phys. Reports, 23. J. P. Hansen and I. R. McDonald, to be 59, 1980. published. -59- SUMMARY OF SESSION V H. C. Andersen Department of Chemistry Stanford University Stanford, CA 9120b The review topic for the session was the mathematical procedures necessary to calculate simulation of plasmas, but most of the discussion the sum of the electrostatic interactions in the concerned the more general problem of how, in limit that the radius of the spherical array principle, to perform simulations on systems with approaches infinity, keeping the size of the basic long-range forces. cube-fixed. Three apparently different points of view The third is that of Roy Pollock,2 who dis were presented. The conceptual frameworks of cussed various contributions to the electrostatic those participating in the discussion were energy of an elliptically shaped array of cubically sufficiently different and the terminologies were replicated systems. sufficiently different that precise discussion was difficult. No clear consensus emerged from the discu:sion. One approach is that of John Barker (see his paper for the session on special techniques). REFERENCES It can be described as a physically motivated way of justifying use, a certain type of reaction field 1. S. de Leeuw, J. M. Perram, and E. R. Smith in simulations. "Simulation of Electrostatic Systems in Periodic Boundary Conditions," Proc. Roy. Soc. The second is that of E. R. Smith.1 He (in press). considered a spherical array of cubically repli cated systems of charged particles surrounded by E. L. Pollock, (See Session II of this a homogeneous dielectric, and he discussed the Proceedings). SESSION VI LONG-RANGE FORCE PROBLEMS IN SIMULATION OF SURFACE PHENOMENA -63- SURFACE SIMULATIONS AND LONG-RANGE FORCES* J. D. Doll Los Alamos Scientific Laboratory, University of California Los Alamos, New Mexico 87545 Experimental surface results^-"? indicate that REFERENCES AKJ FOOTNOTE both the structure and dynamics of adsorbates are strongly influenced by long-range interactions. Structurally we find that adsorbate overlayers are quite frequently ordered even when the overlayer Work performed under the auspices of the U.S. to substrate mesh sizes of four to six (and even Department of Energy. larger) are observed experimentally in low energy electron diffraction data. Dynamical coupling of adsorbates separated by such distances has been 1. J. J. Lander, Surf. Sci. 1, 125 (1964). suggested by surface diffusion studies.3 2. G. Somorjai, Principles of Surface Chemistry, A major problem in surface simulations is Prentice Hall, Englewood Cliffs, NJ (1972). that the nature of these adsorbate/substrate and adsorbate/adsorbate interactions is poorly 3. G. Ehrlich, CRC Critical Review in Solid State understood. Preliminary analysis^>5 has suggested Sciences, February, 205 (1974). qualitative features of these interactions, and certain more detailed model studies are becoming 4. T. L. Einstein and J. R. Schrieffer, Phys. available, but chemisorption theory remains a Rev. B7, 3629 (1973). significant bottleneck with regard to surface simulation. 5. P. J. Feibelman, Surf. Sci. 27, 438 (1971). -64- SOME EXACT RESULTS AND SOME HYPERNETTED CHAIN APPROXIMATION RESULTS FOR AN ELECTRIC DOUBLE LAYER Lesser Blum Douglas Henderson Department of Physics IBM Research Laboratory and University of Puerto Rico San Jose, California 95193 Rio Piedras, Puerto Rico 00931 Consider a system of charged hard spheres 2 4 6 8 10 12 11 of charge 'ze and diameter a in a medium of uniform 0.08 1 I / 1 1 1 I , - -oil. We assume that image forces can be neglected. If we let P +(X) and P _(x) be the W w MGC / HNC concentrations of cations and anions, respectively, / at a distance x from the charged wall, p(x) = / / rw+(x) + .-'W_U) and q(x) = ze[Pw+(x) - Pw-(x)3, / / / / where ..(x) and q(x) are the density and charge 0.06 / / profiles, respectively. It is convenient to define / / D x x P X / / 9ws(x) = »'(x)/ and 9wd( ) = I>w-( ) - w+( )]/P' / / where . is the density of the hard spheres in the '/ bulk. // 11 If c (x) and c j(x) are the direct corre ws wt | 0.04 1M - lation functions which are related to gws(x) 1 1 and gwd(x) through the usual Ornstein-Zernike i relations, then wel can establish the following - I exact asymptotic relations 0.02 cwd(x) • -SzeEx , as x -• • (U) cws(x) - -R(3p/3r-)e , as x - (lb) «W" • ^/t^UJdt , . x-- (10 ' i i 1 1 ! .. ,., 1 1 J- S 0.2 0.4 0.6 0.8 Charge density (coul-m ) Another exact result,1.2 which is of interest, Figure 1. Potential difference across the diffuse layer involves the value of gWs(x) at x = 0: in a 1M model ionic fluid (o=2.76 A, z=l , T=298 K, t=78.4, near a hard planar wall as a function of the charge density on the wall. ^ -^^* {¥) (2) which is satisfied by the HNC results shown in Fig. 1. Despite the fact that the non-monotonic where t is the pressure of the hard spheres in behavior of the HNC results cannot be excluded the bulk tid K = (4n6z2e2p/c)1'2 is the usual ; by any general considerations, it appears from electrostatic screening parameter. recent computer simulations5 that this non monotonic behavior is not present. Recently, we have calculated^ gWc(x) and gwcj(x) using the hypernetted chain (HNC) approximation. The diffuse layer potential difference, calculated REFERENCES AND FOOTNOTE from Supported in part by the University of Puerto Rico, Center of Energy and Environment (CEER) and NSF Grant No. CHE77-14611. t) dt (3) -/ 9wd 1. D. Henderson and L. Blum, J. Chem. Phys. 59, 5441 (1978). is plotted in Fig. 1. The most interesting feature 2. D. Henderson, L. Blum, and J. L. Lebowitz, of the HNC potential is that it is not monotonic. J. Electroanalytical Chem. 102, 315 (1979). This is a surprising result. Despite some effort we have not been able to exclude such behavior by 3. D. Henderson, L. Blum, and W. R. Smith, Chem. any general argument. The strongest exact result Phys. Lett. 63, 381 (1979). we have been able to establish is 4. L. Blum, J. L. Lebowitz, and D. Henderson, J. Chem. Phys. (in press). 3* (4) 3E 5. G. Torrie and J. Valleau, unpublished work. -65- TREATMENT OF LONG RANGE FORCES IN MONTE CARLO CALCUIATIONS ON ELECTRICAL DOUBLE LAYERS J.P. Valleau and G.M. Torrie Department of Chemistry Department of Mathematics University of Toronto Royal Military College Toronto, Ontario Kingston, Ontario Canada Canada Recently, we have reported1 some preliminary results of a Monte Carlo study of a system of charged hard spheres next to a planar charged wall i.e. an electrical double layer. The central Monte Carlo cell is a rectangular prism WxWxL with impene trable walls at 1=0 and 1=1 but periodic boundary mage box conditions in the X. and tj directions. The wall at 1=0 carries a uniform charge density a = - AWe/h'~ where AW is the fixed excess of positively charged spheres in the cell. This Monte Carlo system differs in two respects from the semi-infinite system whose properties we seek. First, the bulk electrolyte does not extend an infinite distance perpendicular to the charged surface but terminates instead at the impenetrable wall at Z=L. Secondly, in the two directions parallel to the surface, there is the usual periodicity whose effects must central box be minimized. These problems have arisen in the simulation of dense systems of neutral particles, but the situation for dilute systems of charged particles is qualitatively different. Close to the wall at Z=L formation of a com plete ionic atmosphere about each ion is imposs ible. At low concentrations (Z0.10M) this is found to be the dominant effect of the wall at l~i mage box leading to depletion of both cations and anions by a few per cent within a couple of ionic diameters of the wall. At higher concentrations the packing effects of the short range excluded volume potential predominate leading to a small peak in the singlet ion densities at Z-L. For equal size ions either the depletion or enrichment near 1=1 affects cations and anions equally. Thus, there is no charge separation and hence no resultant field acting on the region of interest near 1=0. Figure 1: Long-range forces have a non-zero average for surface problems. The diagram For the simulation of bulk coulombic systems shows a two-dimensional representation of the under conditions corresponding to dilute aqueous central Monte Carlo cell and its periodic electrolytes the nearest image convention is replicas. Each cell contains an excess of ordinarily used. (In fact, the Ewald summation counterions (•) over coions (0) which is procedure gives identical results under these balanced by the surface charge density 0. conditions;2 presumably both methods are reliable The force felt in the central box due to this for such systems.) For the double layer systems charge separation in the image boxes is application of the nearest image convention would primarily due to the dipolar component of this mean including the interaction of each ion with charge separation. only those ions and that part of -3 uniform sur face charge within the rectangular irism WxWxL centered (in the X and tj directi-^) on the the system by means of Ewald-type summations particle in question. In contract to the simula for the two-dimensional lattice of cells, on tion of the bulk system, the inhomogeneity in the the other hand, would also introduce explicitly Z direction of the double layer system means that into the Hamiltonian the effect of the spurious the charge distribution outside the nearest image lon>3 range correlations due to the periodicity box exerts a net force on each ion perpendicular of the system, just as it does for three to the wall, neglect of which would distort the dimensional systems. One consequence of such a charge distribution in the central cell. [Fig, 1] procedure in the double layer case is that (A similar problem occurs in simulations of the fluctuations of the net dipole roment of the liquid vapour interface of a fluid of particles system would be inhibited and, indeed, the 3 interacting via short-ranged potentials. ) To asymmetry of the system makes it likely that include this contribution from the remainder of even the mean charge distribution would, in -66- principle, be affected. The procedure we have adopted instead is to replace the charge distribu tion of each replicated box with the average charge distribution of the central box (as measured over all preceding configurations) rather than its instantaneous charqe distribution. The nearest image contribution to the energy is then sufplemented by adding to it the interaction of each ion with this average "external" charge 0 0 0 0 0 distribution. We have tested two ways of implementing such a procedure in practice. The first is a variation on a technique due to Ladd,1* originally designed for the calculation of Ewald sums, in which the mean charge distribution of the cell is expanded in a multipole series. Each ion then experiences the field generated by the infinite square lattice of identical multipoles. For the double layer croblem the dipole term has by far the dominant effect and is the only one that need be considered if the 2 coordinate of the origin is chosen so as 9+-S. to make the quadrupole moment vanish. Alternatively, the charge distribution normal to tne surface can be approximated by a set of infinite Sheets at closely spaced values of Z, each sheet carrying a different uniform charge density. An ion then interacts with each such sheet of charge, less the square "hole" corres ponding to the central box within which all Figure 2: The mean charge density normal to the interactions are calculated explicitly as usual. charged surface can be represented by a set of [Fig. 2] Both methods have given identical closely spaced infinite planes at Z^t each carry results for the double layer systems we have ing a uniform charge per unit area equal to the studied though the second method is more general mean charge per unit ^irea between Z/-AZ and 2.-+AZ since it implicitly includes all orders in the multipole expansion. Such higher order terms in the central cell. The charge distribution will become important, for example, when charge external to its central cell seen by an ion is oscillation occurs in the double layer at high represented by the whole of each plane which lies fields. outside that central cell, i.e. by planes of charge with holes in them corresponding to the position of the central cell, as shown. REFERENCES 1. G.M. Torrie and J.P. Valleau, Chem. Phys. 3. J.K. Lee, J.A. Barker and J.M. Pound, Lett. 65, 343 (1979). J. Chem. Phys. 60, 1976 (1974). 2. C.5. Hoskins and E.R. Smith, unpublished. 4. A.J.C. Ladd, Mol. Phys. 3_3_, 1039 (1977). -67- SUHHARY OF SESSION VI D. Henderson IBM Research Laboratory San Jose, CA 95193 Computer simulations of surface problems will tation of a reaction-field method for a surface involve difficulties (and opportunities) even under simulation is not even partially formulated. the most favorable of circumstances. Long-range Other problems discussed, usually without any forces will pose particularly severe problems. consensus as to a solution, were surface reconstruc The reaction-field method appears to be the tion, non-local energies, surface potentials, image promising method Of including long-range potential fortes in electrode problems, and the role of long- contributions in bulk simulations. The implemen range forces in surface tension calculations.