Reply to the Comment on ''Monte Carlo studies of ...

Reply to the Comment on ’’Monte Carlo studies of the fluid–solid phase transition in the Lennard‐Jones system’’ Harold J. Raveché, Raymond D. Mountain, and William B. Streett Citation: The Journal of Chemical Physics 62, 4582 (1975); doi: 10.1063/1.430380 View online: http://dx.doi.org/10.1063/1.430380 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/62/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Freezing line of the Lennard-Jones fluid: A phase switch Monte Carlo study J. Chem. Phys. 124, 064504 (2006); 10.1063/1.2166395 Determination of fluid-phase behavior using transition-matrix Monte Carlo: Binary Lennard-Jones mixtures J. Chem. Phys. 122, 064508 (2005); 10.1063/1.1844372 Solid–liquid phase coexistence of the Lennard-Jones system through phase-switch Monte Carlo simulation J. Chem. Phys. 120, 3130 (2004); 10.1063/1.1642591 Comment on ’’Monte Carlo studies of the fluid–solid phase transition in the Lennard‐Jones system’’ J. Chem. Phys. 62, 4581 (1975); 10.1063/1.430379 Monte Carlo studies of the fluid‐solid phase transition in the Lennard‐Jones system J. Chem. Phys. 61, 1960 (1974); 10.1063/1.1682197

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Letters to the Editor

of-state: Aex(p) =Aex(po) NkT

fP (~_ ) !!:Ii

NkT + Po

p'kT

1

p"

(1)

We chose pri =1. 2 and used the corresponding value of the free energy quoted in HV. The free energies for p* '" 1. 2 are listed in Table I and compared to the predictions of quasiharmonic theory. The comparison shows that the free energies computed from (1) converge smoothly toward their harmonic values as p* increases; at p* =2. 4 the difference is less than the uncertainties on the MC calculations. Note that the thermal fraction of the internal energy also tends towards its harmonic limit t and that the ratio 17J7)/d decreases rapidly as p* increases, thus ensuring the validity of harmonic theory. Since the extension of the HV data to high densities yields the expected quasiharmonic behavior, the HV free energies along the T* =2.74 isotherm must be very accurate. The preceding arguments and calculations support the validity of the HV melting parameters both at low (arguments a and b) and at high temperatures (arguments c and d). Consequently it remains to understand why the results of SRM differ so significantly. Since their equation-of-state data agree with the HV results, the differences can only stem from the difficulty of obtaining a van der Waals loop in the coexistence region. The work of Alder and Wainwright 8 and WoodS has shown that the size of the simulated systems is too small to allow coexistence of the two phases in three dimensions. However Alder and Wainwright were able to obtain coexistence in two dimensions (hard disks); their van der Waals loop in the pressure versus density curve, which corresponds to a reversible gradual transition from the fluid to the solid phase, is very different from the "loop" obtained by SRM. In particular the former loop is rather flat,

whereas the latter contains a vertical part which corresponds to an irreversible pressure jump between the solid and fluid branches. Because SRM were unable to obtain true coexistence between the fluid and the SOlid, their "loop" is not a proper van der Waals loop, and the equal area construction does not apply. Consequently, the melting parameters which they obtained from such a construction cannot be correct. The authors acknowledge useful suggestions made by L. Verlet. Most arguments presented here have already been raised by B. J. Alder, W. G. Hoover, and L. Verlet during a discussion following an oral report of the SRM work. *Equipe associee N" 453 au Centre National de la Recherche Scientifique. tIBM postdoctoral fellow. tW. B. Streett, H. J. Raveche, and R. D. Mountain, J. Chern. Phys. 61, 1960 (1974). 2N. A. Metropolis, M. N. Rosenbluth, A. W. Rosenbluth, A. H. Teller, and E. Teller, J. Chern. Phys. 21, 1087 (1953). 3J. P. Hansen and L. Verlet, Phys. Rev. 184, 151 (1969). 4W. G. Hoover and F. H. Ree, J. Chern. Phys. 47, 4873 (1967); and 49, 3609 (1968). 'A. Michels, H. Wijker, and H. K. Wijker, Physica 15, 627 (1949). 6 L• Verlet, Phys. Rev. 159, 98 (1967); W. W. Wood, in Physics of Simple Liquids, edited by H. N. V. Ternperley, J. S. Rowlinson, and G. S. Rushbrooke (Wiley, New York, 1968). 7J. D. Weeks, D. Chandler, and H. C. Andersen, J. Chern. Phys. 54, 5237 (1971). BB. J. Alder and T. E. Wainwright, Phys. Rev. 127, 359 (1962); and J. Chern. Phys. 33, 1439 (1968). 9J • J. Weis, Mol. Phys. 28, 187 (1974). tOL. Verlet and J. J. Weis, Phys. Rev. A 5, 939 (1972). l1J. P. Hansen, Phys. Rev. A 2,221 (1970). 12W. G. Hoover, M. Ross, K. W. Johnson, D. Henderson, and J. A. Barker, J. Chern. Phys. 52, 4931 (1970).

Reply to the Comment on "Monte Carlo studies of the fluid-solid phase transition in the Lennard-Jones system" Harold J. Raveche and Raymond D. Mountain National Bureau of Standards, Washington, D. C. 20234

William 8. Streett Science Research Laboratory, U,S. Military Academy, West Point, New York 10996 (Received 3 March 1975)

We agree that the thermodynamic properties of the stable Lennard-Jones fluid and solid phase reported by Street, Raveche, and Mountain (SRM)l and those reported by Hansen and Verlet (HV)2 are identical to within 1 %. Beyond this statement we give the following reply to the preceding comment. 3 To minimize confusion and subterfuge we will make our remarks brief. (1) It does not follow that since a "flat" loop was observed for a particular hard-disk system, that every finite system must have the same loop. The qualitative The Journal of Chemical Physics, Vol. 62, No. 11, 1 June 1975

features of the loop determined by SRM are, in fact, similar to those that were observed with hard disks. As described in SRM, the curve connecting the fluid and solid branches was schematically illustrated in Fig. 9 as a vertical line, because the branches were observed to be connected by a very steep curve. (2) It is specious to point to differences in transition pressures. The emphasis should be on the volume (density) differences, since the pressure depends on this, and small variations in volume can, on some isotherms, Copyright © 1975 American Institute of Physics


Letters to the Editor

cause large differences in pressure. (3) As was shown in detail in Table I, Figs. 1,2, and 4 of SRM, there are no appreciable size effects on the isotherms. This was verified in computations involving more than 106 configuration.

(4) The real point at issue is the uncertainty in the coexisting densities determined by the HV method and those determined by the SRM method. No amount of intercomparison with calculations on Hamiltonians other than the Lennard-Jones function can determine this uncertainty. Rather, it is our opinion that the error bars in the HV and SRM coexisting densities overlap. For

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example: If in the vicinity of a loop we assign a somewhat large uncertainty of 1.5% to our Monte Carlo pressures, then the largest uncertainty in the coexisting densities determined by SRM is approximately 3%. Now, if the uncertainty in HV coexisting densities for the Lennard-Jones caSe is between 1% and 2%, then the results of both methods are in agreement, and there is no need for circumlocution. lW. B. Streett, H. J. Raveche, and R. D. Mountain, J. Chern. Phys. 61, 1960 (1974). 2J • P. Hansen and L. Verlet, Phys. Rev. 184, 151 (1969). 3J . P. Hansen and E. L. Pollock, J. Chern. Phys. 62, xxx (1975), preceding comment.

The affinity of oxygen for two electrons M. F. C, Ladd Department of Chemical Physics, University of Surrey, Gui/dford, Surrey, England (Received 4 April 1974)

In past calculations of the C and D constants of the van der Waals potential for ionic crystals by Mayer's method! [see Eqs. (1)-(4)], the values of ~i and ~J have been multiplied by arbitrarily chosen constants in the ranges 0.75-0.9 and 2.4-2.8, respectively.I,2 This unsatisfactory situation was resolved 3 in a model which used polarizabilities, aj and ai' calculated from static dielectric constants. C = 6. 5952clJ + O. 90335(cjj + c n ) , D =6. 1457dlJ +0. 40005(d u +dn ),

(1)

(2)

c lJ =1. 5~1 ~JaiQJ/(~i + ~J)'

(3)

d iJ = 2. 25 e-2ciJ[(~iQJPi) + (~JQ/PJ)] .

(4)

In this model, ~j is the ionization potential of the positive ion, ~J is the single electron affinity of the negative ion, and Pi and Pi are the "effective" numbers of polarizable electrons in the ions, calculated following Mayer. 1 A recent calculation by Cantor, 4 following Ladd's equation,3 of the cohesive energies of MgO, CaO, SrO, and BaO has allowed, through the use of new data, a revision of the electron affinity of oxygen for two electrons, E(02-). In these calculations, the C and D constants were evaluated on three bases, none of which is entirely satisfactory. The Huggins and Sakamoto basis5 calculates ciJ by comparison with the values for the alkali-metal halides. In Boswarva's method,6 the values for both a and ~ are semiempirical, and, in the view of the present author, 3 the use of the electronic polariza,bility incorrectly estimates the values of c iJ ' and, hence, of diJ. Equating ~J to the electron affinity, 0- 02-, is ambiguous, vide Eq. (2), although the estimate of 8 eV for ~i is very close to the best value in the work reported here. Cantor reduces the value of (:l(SrO) by its experimental uncertainty, which is probably justifiable, but neglects The Journal of Chemical PhYSics, Vol. 62, No. 11, 1 June 1975

the value of 153 kcal/mole for E(02-) obtained through BaO, and weights, by an arbitrary method, the remaining values to obtain the result of 149 ± 8 kcal/mole. These procedures do not seem to be wholly satisfactory. Indeed, if the highest electron affinity is taken to be the most reliable, vide Huggins and Sakamoto,5 then the value of 153 (from BaO) should not necessarily have been eliminated. Indeed, it is not outside the significance limits of the result quoted. The results reported herein have been calculated along the lines given by Ladd. 3 Data have been taken from Cantor, 4 but with certain significant differences. The polarizabilities have been re-evaluated. 3 BeO has been brought into the calculations, taking data from Gaffney and Ahrens,7 except that tJ(BeO) was taken from So gaS and ~HJ(BeO) from Parker, 9 but, again, reevaluating a(BeO). The value of ~i' used in Eq. (3), was taken, initially, as 9 eV, from the values of E(02-)IO and E(O-). 11 The calculations led, through a BornFajans-Haber (isothermal) thermochemical cycle, 3 to an average value for E(02-), From this result, a new value for ~i was determined and re-cycled through the calculations. This procedure converged rapidly with E(02-) = 145 kcal/mole (~J = 7.72 eV). Table r lists the important quantities from the final iterative cycle. The present value of 145 kcal/mole is in good agreement with that of Cantor, 4 but arises from a single, straightforward calculation of the constants, C and D. The variations in the five values for E(02-), obtained from the oxides, represent uncertainties in the data and are communicated to the result. It is not possible to assess accurately the standard deviation E(02-), because the thermochemical cycle can be completed only when this quantity is known. This situation could be alleviated by the experimental measurement of either E(02-) or E(02- - 0-), but so far this Copyright © 1975 American Institute of Physics
