AmericanMineralogist,Volume70,pages601407, 1985
Thermodynamicpropertiesof Nacl obtainedby computercalculation
CH.nnr,nsJ. Hosrnrr,nn
Battelle Northwest
Richland, W ashington 99352
Abotrect
Someof the thermodynamicpropertiesof the NaCl systemhave beencalculatedfrom first
principles using computer simulation techniques.Pairwise-additiveinteraction potentialsfor
the NaCl system were calculated using SCF-LC\O ab initio methods,These interaction
potentials were used in conjunction with the Monte Carlo techniqueto calculatevolumes,
third law entropies,and enthalpiesof NaCl at four temperaturesand 1 bar. The calculated
properties were mostly within five percent of their correspondingexperimentalvalues.The
melting temperaturewas calculatedto be 950 K (the experimentalvalue is 1074K) reflecting
the difficulty inherent in predicting phase change information from first principles. The
volume of crystallineNaCl as a function of pressurewas also calculatedusing a free cnergy
minimization technique.The calculatedvolumescomparedwell with experimentalresultsto
pressuresof 120 kbar. In generalthe calculatedthermodynamicpropertiesdid not increase
rapidly enough with temperature;the interaction potential functions obtained from the ab
initio calcrilationswere too negativecomparedto Born-Huggins-Mayer effectivepair potentials. This work demonstratesthe possibility of determiningthermodynamicdata using computer simulation techniques,as well as some of the fundamental difficulties in applying
Hartree-Fock pair potentialsto predict the thermodynamicpropertiesof condensedmatter.
Introduction
have been examined(e.g.,Krogh-Moe et al., 1969;Woodcock and Singer, 1971; Larson et al., 1973; Adams and
Some recent attempts to understand the nature of
various geochemicalsystemsapplied interaction potential McDonald' 1974;Michielsenet al', 1975;Woodcock' 1975,
functions in conjuncti,onwith computer simulati,ontech- p. 12-23).aswell as silicateglasses(Woodcock et al., 1975t
1979;Soulesand Busby,1981;Angell et al', 19E1;
niquesto predictequilibriumprop.riir.. One particularap- Soule.s,
elceltgt al', 1982,Hostetler,1982)'
proach employed interparticle potentials calculated wiih
work I presentthe resultsfrom a first principks
modified ilectron-gas theory or Hartree-Fock self.In.this
simulation
of the Nacl system.The interaction potential
consistent-fieldtheoiy. Combined with free-energy-irritt:t^':tt for the systemhave beendeterminedusing SCFmization techniques,Lnairrg energies,uona aistan?s, aio
elasticpropertiesof crystalshave beencalculatedr.. urturi :CA9 ab initio quantum calculations' These potential
halide irystals (e.g.,Cohen and Gordon, L975; Zhad,.anov functionshavebeenusedin Monte Carlo computersimulaand Pdyakov,l97l;Mackrodt and Stewart,1979)as well tions to calculatethe enthalpy,entropy, and volume of
both crystallineand molten NaCl as a function of temperas various crystalline oxides-Mgo, Mno, c"6, n"o,
ature
at one bar' The potential functions were also used
FeO, ZnO, Al2O3, SiOr, and TiO, (e.g.,Mackrodt ani
Stewart, 1979; Cohen and Gordon, 1976; Dienes et al., with ener.gyminimization methodsto calculatethe volume
1g75;Tossell,1gg0).Empirical potentialshave also been of crystallineNaclasafunctionof pressure.Thesecalculausedwith energyminimizationtechniquesto predict lattice tions.provide a sensitivetest of the accuracyof the comenergiesfor seviral silicatecrystal structures(Catlow et al., puted interactionpotential functions.
1982).
Procedure
In addition, Monte Carlo (MC) and moleculardynamics
(MD) techniqueshave been used to predict equilibrium
Interaction potentialfunctions
and rransporrpropertiesof various systems.Til;;;:
The total potential energy of a systemof N interacting
eling efforts have started with potentials (usually of the
Born-Huggins-Mayer form, see Tosi and Fumi, 1964; particlesisgivenby:
Woodcock, 1975p. 4-12 for reviewsof empirical potential
I N N
1 N N N
f u n c t i o n s ) c o n t a i n i n g p a r a m e t e r s w h i c h w e r e d e t e r m i n e dU : ;
(1)
I
I
d
,
,
+
;
I
I Id,t"+...
't i=t i= I
Jr i=l j=l
k=l
by fits to propertiesof crystalline
materialat 298 K. MC
i+i
i+ j+k
and MD calculationsemploy the interactionpotentialsto
predict the properties of the system as a function of the relative to a standardstate of infinitely dispersedparticles.
state variables. Both crystalline and liquid salt systems Each index runs over all the particles,and the summands
0003-o04x/85/0506-{60
I $02.00
601
ffi2
HOSTETLER: THERMODYNAMIC PROPERTIESOF NACI
are the interactionpotential functionsrepresentingthe contribution to the potential energyfrom the particlesindexed
in the function. The interaction potentials (and the potential energy)are functionsof the particle positions.In lieu of
adequatetheoreticalknowledgeofthe effectsofmany-body
interactions,defining a pairwiseadditive potential function
which reproducesthe potential energyofthe systemis convenlent:
i=r
j=l
zrzrez . Aii
r
r"
/
Bii
t'
(U' - U\ + P(V' - V) -NkTrn(V'lV)
(5)
where k is Boltzmann'sconstant and N, P, and T are the
where the angular brackets denote an ensembleaverage.
The degreeto which this effectivepair potential fails to
reproduceother properties of the systemis a measureof
the importanceof many-bodyinteractions.Stillinger (1964,
p. a-fi) proposedtheoreticalarguementsthat polarization
forces(which accountfor approximatelyone percentof the
potential energyfor the NaCl system)are the only important many-bodyinteractionsin molten salts.
The effectivepotentialsusedin this study havethe form:
d"(r) :
w:
of particles,the pressure,and the temperatureof
(2)number
the system.If W is negative for the transition the new
,:(+i irr')
\!
tion a Markov chain of configurationsis generatedby displacing a randomly chosenparticle by a random amount.
The volume of the systemis also randomly changed.The
pseudo-Boltzmannweighting factor, W, from a state with
potential energy u and volume V to a state U', V' is calculated:
cii
r'
where the Z, are the formal ionic charges,e is the fundamental unit of electroniccharge,and A,r, B,,, and C,, are
adjustableparameters.The potential is a function of the
particle types and the distancebetweenthem, r. The first
term representsthe coulombic interaction between two
ions and the latter terms representa combination of the
dispersionforcesand repulsiveforces.This particular form
for the potential functions is similar to the Pauling(1928)
form, and was chosenfor convenientuse with the (N,P,T)
Monte Carlo technique(discussed
below).
In order to determine the parametersin the effective
pair-potential, a data set was generatedconsistingof the
energiesof a pair of particles as a function of separation.
The energiesof eachparticle pair were calculatedusing ab
quantum
initio restrictedHartree-Fock self-consistent-field
techniqueswhich involve computing all necessaryone- and
two-electronintegrals.After initial assumptionsare made
(discussedbelow) no input data are neededto evaluatethe
(Popleet al., 1978)
energyof the system.The Gaussian-76
program was used to calculate energies.The parameters
were fit with the least squarestechnique using the equation:
configuration is accepted for inclusion in the ensemble
average. Otherwise the configuration is accepted with
probability which is exp(-W/kT). If the new configuration
is rejectedthe old configurationis retainedand includedin
the ensembleaverage.This algorithm ensuresthat each
configuration shows up in the ensembleaveragewith a
frequencyproportional to its weighting factor. The process
is repeatedfor the numberofconfigurationsdesired.
Two parameters govern the elliciency of the Monte
Theseare the maximum amount
Carlo samplingprocesses.
any particle can be displaced,and the maximum changein
volume for a transition. Theseparameterswere chosenby
trial and error such that the acceptanceratio was closeto
0.5.
The ensembleaverageof any configuration dependent
property correspondsto the value of the property at the
givenP and 7. In particular:
E:
(6)
<U + 3/2NkT)
H:<E+PV>
(7\
v: <v>
(8)
where E is tbe internal energy,.EIthe enthalpy, and V the
volume of the system.The third law entropy of the system
was computed using (N,I/,D Monte Carlo runs at each
calculated volume with the cell approximation method
(Goslingand Singer,1970):
(d
, ' 2_k T L:\4 . " ) r . s {+l h f r " 4' )l .i
I
+;
lnC + lnf,
(9)
where C: 1 for solids, NN/t{! for liquids, f, is the free
volume (Kirkwood, 1950), and @-,n is the minimum value
(4) of the potential in the free volume. The free volume is equal
I I d,d.): E. - I Er
ijK
to a(Y)V where a(V) is the acceptance ratio as a function of
whereE. is the energyof the clusterand E* is the reference volume. The MC jump parameter is varied and the maxistateenergyofparticlek.
mum calculated free volume is the MC estimate of the true
free volume. Although the free volume can be estimated
Monte Carlo calculations
rather precisely using this method, the treatment of the
The Monte Carlo method is based on the theory of minimum potential energy term is much less accurate. GosMarkov processes
and was first describedby Metropolis et ling and Singer (1970) found that entropies computed in
al. (1953).A modification of the basic method to allow this manner were systematically lower than corresponding
calculation in the isobaric-isothermal(N,P,T) ensemble experimental values for liquid argon. In spite of this bias,
was describedby Adams and McDonald (1974)and was we chose the free volume method for computing entropies
usedfor this work. Starting from a given initial configura- in order to conserve computational resources; calculation
HOSTETLER: THERMODYNAMIC PROPERTIESOF NaCI
of entropiesusing p dV integration along isothermswould
haverequiredabout ten MC runs for eachtemperature.
To include the effectsof the long-rangedcoulomb potential and to minimize surfaceeffectsin the MC simulations
periodic boundary conditions are enforcedon the system.
Two computational techniqueswere used in conjunction
with the periodic boundary conditions. The minimum
image convention allows each particle in the system to
interact only with its N-l nearestneighborsin the system;
eachneighbor is either a particle in the systemor an image
of the particle produced by the periodic boundary conditions. The Ewald technique(Ewald, 1921;Brush et al.,
1966)was also used to evaluatethe long range coulomb
potential.
Free energy minimization
The free energyminimization method was usedto calculate the volumesand freeenergiesofcrystalline phasesas a
function of pressure.The B1 and 82 phasesof NaCl were
studiedat 0 K. (The phasechangebehaviorfor NaCl seems
to be fairly insensitiveto temperature,Cohen and Gordon,
1975)Using this simplifying assumption,the free energyis
just:
603
Table 1. Basisset sizeand balance
Baeis
Set
Nurber
Funclions
of
fo!
Nuober
Functlons
Na
of
for
Cl
Na Functlon6
Per Electron
C1 Functlons
Per Electron
sto-3c'
IE
l8
t.8
t.0
sto-3clzb
24
36
2.4
2.0
(12,e)c
42
7a
4.2
4.3
1 6, 4 1 d
t2
l8
1.2
t.0
sto-3c'
30
30
3.0
aThree
Gsueslan
bmrbl.
2"."
cfrelve
fits
STo-3C
to
functlons
eussian
(Pople
slat€r-type-ofbltels
(tso
basls
sith
per
functlons
6 type
syNtry,
orbttsl)
et
al.,
1978)
(Cle@ntr,
1964)
(vetuard,
9 p funetlons
r968).
dslx
b.Bts
Gausslan
s
(vetllsrd,
ePolellratlon
Eype functlons,
1968).
(d)
functions
4 p functlons;
added
to
the
contrscEed
STo-3G basls
fron
(Pople
rhe
et
aI.,
(f2,9)
1978)
constraint on electronpopulation distribution, which is determined in part by mathematicalflexibility. The lack of
balance is a potential source of error in determining
Hartree-Fockenergies.
In Table 2 energiesfor the Na+-Cl- pair separatedat
(10)
G : U ( V \+ P v
3.0A and for the referencestatesare comparedfor several
Using the interactionpotential model describedabove,at a basis sets.The energy obtained with the Born-Hugginsgiven pressurethe free energy dependson only one vari- Mayer (BHM) model at this distanceis 7.55x 10-1e J
able, the volume. Minimization of G with respectto the (Adams and McDonald, 1974).Even for the large (12,9)
volume gives the free energy of the crystalline phase.The basisset the agreementis rather poor. The STO-3G basis
set was also used to calculate interaction energiesas a
conditionfor the B1-82 phasechangeis G(Bl) : G(B2).
functionof distancefor the Na+-Cl- ion pair. In Figure I
the ab initio potential is plotted along with the coulomb
Results
energy
and the BHM potential. The STO-3G potential is
The energiesof Na+-Na+, Na*{l-,
and Cl-{lpairs were calculatedas a function of separationdistance approximatelytwenty percenttoo negativeat distancesless
usingthe Gaussian-76(Popleet al., 1978)program.This ab than 3A.
Negative deviations in computed potentials can be
inirio method useslinear combinations of atomic orbitals
(LCAO) to form fhe molecular orbitals. The number of causedby counterpoiseerror. This error is a spuriouslowatomic orbitil/ fdnctions, their mathernatical form, and ering of the energy of a cluster relative to the reference
their adjustableparameterscomprise a basis set for that energies.Counterpoiseerror is a mathematicalartifact of
atom. Larggr'basissetsare more flexible and possiblyallow the orbital approximation resulting from extra degreesof
better descriptionolthe chargedistribution in a molecular freedom in the cluster relative to the reference state
environment. Calculations were performed using several (Carsky, 1980).In order to decreasethe counterpoiseerror
"ghost" atomic centerswere added to the calculationsof
publishedbasissets(describedbriefly in Table l).
Closed-shellions were chosedas the referencestate for the referencestate energy.These ghost centerscarry the
the quantum calculations.The SCF equationsassumeeach orbitals of the other nuclear center in the cluster but no
of the electrons in a system move in a coulombic field nuclear or electroniccharge.Thus the electronsin the refgeneratedby the nuclei and the time averagedpositions of erencestate ion can interact with the orbitals of the other
the other electrons.In reality the electronmotions are correlated. The correlation energy can be quite large. Using Table 2. Calculated energies for Na+{lions separated by 3.0A
closed-shellions as the referencestatefor calculationson a
cation-anion cluster approximately conservescorrelation
Basls sel
E*.+ (1.u.)
ECl- (A,U.)
ENa+cl- (A.u.)
AE (lo-19 J)
error. When the energy of the referencestate ions is subtracted from the ion pair energy, the correlation energy
-159.1a462
-454.4aO42
-614.46194
-8.58
sT0-3G
approximatelycancels.
-62t.41244
-16r.57095
-459.56094
-7.87
(12,9)
It should be kept in mind that all of the basissetsexcept
4
5
9
.
5
1
5
9
1
-62r.36484
1
6
r
,
6
6
0
r
r
-8.23
I6,41
for the (12,9)basisset usedin this calculation were unbal-454.48042
-614.51582
-r0.93
-t59.78462
sTo-3c*
anced (i.e. contained different numbers of basis functions
per electronfor Na and Cl). This lack of balanceacts as a
HOSTETLER: THERMODYNAMIC PROPERTIESOF NaCI
5 x 10-18
Table 4. Interaction potentials for the NaCl system
STO.3G POTENTIAL
4
o (ro-19.1)
BHM POTENTIAL
a
;3
separatlon
2
2
u
(A)
N"*cr-
2.60
3r
2.80
2.85
2.90
? os
3.00
3.25
3.50
3, 7 0
z
r-o
o
4
-1
.2
12345
N.+ . ct' D|STANGE(i)
ion in the cluster, so the counterpoisecorrection is approximately the samein the referencestateand the cluster.
Table 3 containsthe resultsfor four basissetsfor Na+{lpair at 3A as corrected for counterpoiseerror. Counterpoise error generallyincreaseswith both atomic number
and negative charge (Carsky, 1980)and in these caleulations most of the counterpoisecorrection does come from
the Cl--ghost term. Also note that the counterpoise
error
is inverselyproportional to basissetsize.A large portion of
the negativedeviationsin the calculatedNa+{lpotentials can be correctedin this way.
Given the resultsin Table 3 and the constraintson computer time, the STO-3G* basiswas chosenfor calculation
of the detailed interparticle potential functions. The computed interaction energiesfor the three pairwise interactions at 28 interionic distancesare listed in Table 4. These
results were all correctedfor counterpoiseerror using the
method describedabove.
Expansion coefflcientswere fit to the data in Table 4
using equations(3) and (4). The expansioncoeflicientsfor
each potential are presentedin Table 5. A small constant
was also neededto fit the data. The deviations from the
Adams and McDonald (1974)BHM potentialsare plotted
in Figure 2. The total deviation is negativenear the equilibrium geometry,becomingsmaller with increasingseparation (i.e. the Hartree-Fock potentials are too soft). The
effects of the deviations on computed thermodynamic
propertieswill be examinednext.
ENa+ (A.U.)
sT0-3G
-159.7E483
ECI- (A.U,)
-454.49319
E*.+ar-
(A.u.)
-6t4.46t94
STo-3GDZ
-t60,52267
-456.47274
-6t7.r7922
-E,01
[6,41
-161,66014
-459.52043
-621.164a4
-8.03
sT0-3G*
-159.7E599
-457.91274
-617.87281
o.uo/
-5.796
5.9L2
5 . 8 37
5.764
5,789
4.05
4.10
4.15
4.20
4,30
-
5.694
5.625
5.590
5.527
_
-
5.490
5.353
).cu)
4,40
4.s0
4.70
4.90
5.00
-s.146
_
-4.626
5.24r
5,125
4 '907
4.107
4.613
4 ,584
4,011
6 .3 0
6.50
_
-3.666
-3. 553
4.OO2
5.655
5,067
The Monte Carlo calculations were performed for a
systemcontaining 64 ions (i.e.,32 of each type) in a cube
with periodic boundary conditions.For runs below the experimentallydeterminedmelting point (1074K) the starting configuration was a simple cubic lattice with edge
length of 8.4A. For the runs above 1074 K, a random
initial configuration was prepared by heating the crystalline starting configuration at 2000 K for 105configurations. The initial 105 configurations were also discarded
from eachofthe high temperatureruns. Three independent
runs were made at each temperatureof 2 x lDs configurations, and the resultsaveraged.The maximum particle displacementand volume changeparameterswere chosenfor
eachtemperatureso that the acceptanceratio was near 0.5.
The pressurewas one bar for all of the MC runs.
Thermodynamicpropertiescalculatedwith the MC technique at 4 temperaturesare listed in Table 6. Exceptfor the
Table 5. Expansion coelficients for the interaction potential
functions (short-rangc terms)
crr
L j ( s h o r t- ) ! t - + + - + + - + +
1lr
r
r
Na-cl
ds (fO-I9.r)
-7.99
cl-cl-
6.007
-
4,00
Table 3. Calculated energies for Na*{lions separated by 3.0A
corrected for counterpose error
Ba61s set
-5.184
3 . 75
3,80
3.90
Fig. l. Comparison
of the ab initio STO-3GNa+-Cl- potential with the empiricalBorn-Huggins-Mayer
potential(Adams
andMcDonald,1974).
Na*Na*
-8.278
-8.061
- 7 . 9 74
-7.883
-7 .788
-7 .69r
-7 .592
-7.091
-6.616
arl
(J A')
bri1 (J A')
.r,
t:
i8)
Na Ne
-1,082 x I0 '"
a.520x r0
(3)
^" (3)
-3.205 x 10-16 (3)
K = 1,521 x to-21 (,1)
K
- 2 . 7 5 6x t o - 1 8 ( r )
o
2 . 2 4 8x r o - 1 7( r )
cl cl
2 . 1 4 4x 1 0 - 1 6( l )
- 2 . 3 2 2x t o - 1 5 ( r )
s . r 2 5x l o - r 5 ( l )
605
HOSTETLER: THERMODYNAMIC PROPERTIESOF NACI
.A
120
o
d
:
ioo
g'
+
Fig. 2. Deviationsof the
correctedSTO-3G*popotential
tentials(this work) from "ounr.rp"or."
the Born-Huggins-Mayer
(AdamsandMcDonald,1974).
ttoo ttoo
,."rJ"ollr^.,*,
Fig. 3. Gibb'sfunction( - (G, - H""")lf) asa functionof temperature.Heavyline from Robieet al. (1978).
Opentrianglesthis
meltingpoint of NaCl, solid
work. Solid circleis experimental
trianglerepresents
meltingpointderivedthiswork.
(The actualmelting temperatureis 1073K.) The freeenergy
results are illustrated graphically in Figure 3. This figure
enthalpy function, the valuesare within five percentof the points out the diffrculty ofaccurately predictingphaserelaavailable thermodynamic data. The enthalpiesare about tions using this method due to the similarity of slopesof
the Gibb's function for the liquid and the solid. A better
ten percent too low. The volumesfound for all of the liquids were all lower than the volume at the melting point, result might have been obtained by interpolating over a
which is 37.6 crn3. The entropies were computed using smallertemperaturerange.
Using the free energy minimization technique, the
(N,I/,f) MC runs at the volume determinedby the (N,P,T)
result; the low volumestending to produce low entropies. volume of the 81 and B2 phasesof crystallineNaCl were
Theseresultsindicate that the potentialsare too soft com- determinedat 0 K as a function ofpressure.Thesedata are
pared to the Adams and McDonald (1974)BHM poten- presentedin Figure 4. The heavy solid line is the negative
tials, which more a@uratelyreproduce the enthalpy and of the differencebetweenthe more stable phase and the
volume of the Bl phaseat zero pressure(Y") divided by
volume at 1073K. This result is consistentwith the data in
Figure 2 which show decreasingdeviation with increasing Vor.Theupper row of dots are the samequantity for the B1
phase,and the lower row ofdots are for the 82 phase.The
distance.
A common thermodynamictechniqueusesinterpolation phase transition was found to be at 160 kbar versusthe
on the Gibb's function to find the temperatureat which experimentalvalue of 300 kbar (Bassettet al., 1968).This
two phasesare in equilibrium. Linear interpolation for the result is quite similar to the value of 107 kbar determined
low and high temperaturedata showsliquid and crystalline by Cohen and Gordon (1975)using the ionic model. Both
NaCl have equal free energiesat 950 K. If the sameprocedute were used with the experimentallydeterminedfree
energydata the equilibrium temperaturewould be 1040K.
AVr2 lcms) 1.014
26.50.
Vor lch3l
a TH|S WOR(
b - BASSEn.t.r
Table 6. Monte Carlo results for the NaCl system at
four temperatures
zgexf
fl(r)''b
uc:
EX-
0
0
toooxf
ttooxs
35.45 (s0)
39.88
60.34 (s0)
67.91
tzoorS
72.29 <50) 135.73 (100)
72,t2
L39.37
164.45 (100)
171.89
r94,42 (100)
201.30
c(T)''c
ltc
EX
72,29
72.t2
99.28
99.49
104.11
103.96
128.60
133.s2
MC
H
21.12 <5)
27 .02
28.69 (4)
unlts
d
of
J/ml
30.35 (4)
(
E(r) - (\-u2e8)/r
c(T) - -(cr-H298)/r
bnte
Carlo
results
expertuental
te€ult8
shple
cublc
lattlce
randon
start
Adans
start
and UcDomld
.
*a-A-x,
(1974),
Robte
P r -r t k b . r ,
1@'
{3oo)b
u''"'"',13:33",0
'''
.-::-a-a-a-.-,
,....;:_::_:
aa
.
Mc
EX
b
02
65.82 (40)
67.78
s"
__, 3.
vr lcm3l la.7aa
tr7.36,b
o1
tl.ootb
l26.9Slb
t1968)
{
THIS WORK
"",on"on
rr*.,
100
120
3s.r-3 (3)
Fig. 4. Volume changesof the 81 and 82 phasesof NaCl as a
function of pressure.Vo,-volume of BI phaseat zero pressure;
Vr, Vr-volumes of 81 and 82 phasesat the phasechangepressure; Vr, : Vr - Vzi p12- pressureof the phase transition.
Upper dots and crossesfor B1 phase,lower dots for 82 phase.
Heavy line represents volume change for thermodynamically
stablephase.
606
HOSTETLER: THERMODYNAMIC PROPERTIESOF NaCl
models provide a good prediction of the Bl phase up to
aroundl00kbar.
Discussion
Computation of accurateinteraction potential functions
using ab initio scF-LcAomethodsrequirescare in basisset
selection.Errors in computed potentials can arise from a
variety of sources:incomplete cancellationof correlation
error, incomplete correction for counterpoiseerror, and
lack of basis set balance.The basissets used in this work
were optimized for isolated neutral atoms, and no reoptimization for isolated ions or for ion-pairs was performed.
Also, only ion-pair clusterswere used to develop the effective pair potentials in this work. Another possible approach would employ solid state energy calculations to
develop pseudopotentialsfrom a geometry variation. Despite theselimitations, the computedinteraction potentials
are quite closeto empiricallydeterminedpotentials(Adams
and McDonald, 1974)and to potentials determinedusing
an ionic model (Gordon and Kim, 1972).
Monte Carlo calculationsprovide a sensitivetest of interaction potentials.The equilibrium propertiescomputed
in this study deviatefrom the experimentalvaluesbecause
the potential functions are too negative.Furthermore, the
amount by which the Hartree-Fockpotentialsdeviatefrom
an effectivepair potential changeswith interparticle separation (Fig. 2). The deviationsin the enthalpy function are
the largest,approximately ten percent.Ten percent of the
enthalpy function representsa difference of 3 kJ in the
enthalpy at 1000K. The lattice energy of the Bl phaseof
NaCl was calculated to be -764 kJ. This agreementto
within one-half percent(relativeto the ion referencestate)
is consistentwith the deviationsin the potential functions
from the BHM model (Fig. 2) and illustratesthe difliculty
in obtaining thermodynamicdata from potential energybased techniques. The thermodynamic quantities are
derivedfrom quite small diflerencesbetweenstatescontaining large potential energies.(The potential energiesthemselves are computed as small diflerences between the
Hartree-Fock energiesof the cluster and referencestate,
seeequation (4),Tables2 and 3.) This diffrculty is similarly
manifestedin computing phaserelationships.For both the
one atmosphereMC simulationsand the 0 K energyminimizations, the predictedphasechangepoints were not as
accurateas the predictedfreeenergiesand volumes.
This work demonstratesthe possibility of determining
thermodynamicdata (for a simple system)from first principles using computer simulation techniques.This work
also points out some of the fundamentaldifliculties in applying Hartree-Fock pair potentials to condensedmatter.
Future work, refining the interaction potential functions,
should lead to a great improvement in the accuracy of
computedequilibrium properties,and an increasedunderstanding of the interparticle interactions in the NaCl
system.
Acknowledgments
This work was supportedby NASA Grant NSG 7576.The
author would like to thank M. Drake, A. Woronow, and J. Jones
for helpful discussions.Two reviewsby M. Bukowinski were also
appreciated.
References
Adams,D. J. and McDonald, I. R. (1974)Rigid-ion modelsof the
interionic potential in the alkali halides.Journal of PhysicsC:
SolidStatePhysics,7,2761-2775.
Angell, C. A., Boehm, L., Cheeseman,P. A., and Tamaddon, S.
(1981)Far IR spectraand electricalconductivity of Li and Na
glassesby laboratory and conputer simulation experiments.
Solid Statelonics, 5, 659-462.
Angell, C. A., Cheeseman,P. A., and Tamaddon, S. (1982)Pressure enhancementof ion mobilities in liquid silicatesfrom computer simulationstudiesto 800kilobars.Science,2fE, E85-E87.
Bassett,W. A., Takahashi,T., Mao, H., and Weaver,J. S. (196E)
Pressure-inducedphase transformations in NaCl. Journal of
Applied Physics,39, 319-325.
Bridgeman,P. W. (1945)Polymorphic transitions and geological
phenomena.AmericanJournal of Scienoe,2434,90.
Brush, S. G., Sahlin, H. L., and Teller, E. (1966)A Monte Carlo
study ofa one-componentplasma.Journal ofChemical Physics,
&.2102-2tt8.
Carsky, P. (1980) Ab lnitio Calculations: Methods and Applicationsin Chemistry.Springer-Verlag,New York.
Catlow, C. R. A., Thomas,J. M., Parker,S. C., and Jefferson,D. A.
(1982)Simulating silicatestructureand the structural chemistry
of the pyroxenoids.Nature, 295,658-662.
Clementi, E. (1964)Simple basis set for molecular wavefunctions
containing first- and second-rowatoms. Journal of Chernical
Physics,40,194-1945.
Cohen, A. J. and Gordon, R. G. (1975) Theory of the lattice
energy, equilibrium structure, elastic constants,and pressureinduced phase transitions in alkali-halide crystals. Physical
Review,812,322V3241.
Cohen, A. J. and Gordon, R. G. (1976) Modified electron-gas
study of the stability, elastic properties,and high-pressurebehavior of MgO and CaO crystals.PhysicalReview,Bl4 45934ffi5.
Dienes,G. J., Welch, D. O. Fischer,C. R., Hatcher, R. D., Lazareth O., and Samberg, M. (1975) Shell-model calculation of
some point-defectpropertiesof c-AlrOr. PhysicalReview,811,
3060-3070.
Ewald, P. P. (1928)Die berechnungoptischer und elektrostatischer gitterpotentiale.Annalender Physik, 64,253-287.
Gordon, R. G. and Kim, Y. S. (1972)Theory for the forces between closed-shellatoms and molecules.Journal of Chemical
Physics,56,3122-3133.
Gosling, E. M. and Singer, K. (1970)Determination of the free
volume and the entropy by a Monte Carlo method. Pure and
Applied Chemistry, 22, 303-309.
Hostetler, C. J. (1982)Theoretical investigationsof geochemical
: MgO-AlrO.-SiOr. EOS,63,ll4l.
systems
Kirkwood, J. G. (1950)Critique of the free volume theory of the
liquid state.Journal of ChemicalPhysics,18,380-382.
Krogh-Moe, J., Ostvold, T., and Forland, T. (1969)Monte Carlo
studieson fusedsalts II. Calculationson a model of fusedlithium chloride at 1073K. Acta Chemica Scandinavia,23,24212429.
Larson, B., Forland, T., and Singer, K. (1973)A Monte Carlo
calculation of thermodynamic properties for the liquid
NaCl + KCI mixture.MolecularPhysics,26,152l-1532.
Mackrodt, W. C. and Stewart, R. F. (1979)Defect propertiesof
ionic solids: II. Point defect energies based on modified
electron-gaspotentials.Journal of PhysicsC: Solid State Physics, 12,431-49.
HOSTETLER: THERMODYNAMIC PROPERTIESOF NACI
607
Tosi, M. P. and Fumi, F. G. (1964)Ionic sizesand Born repulsive
parametersin the NaCl-type alkali halides-Il. The generalized
Huggins-Mayer form. Journal of Physics and Chemistry of
rw2.
Solids,25,45-52.
Michielsen,J., Woerlee,P., Graaf, F. V. D., and Ketelaar,J. A. A. Tossell,J. A. (1980)Calculation of bond distancesand heats of
(1975) Pair potential for alkali metal halides with rock salt
formation for BeO, MgO, SiOr, TiOr, FeO, and ZnO using the
ionic model.AmericanMineralogist,63, 163-173.
structure. Transactions of the Faraday Society II, 71, l73{.l_Veillard, A. (1968)Gaussianbasisset for molecularwavefunctions
t'!41.
containing second-row atoms. Theoretica Chimica Acta, 12'
Pauling L. (1928) The influence of relative ionic sizes on the
405411.
propertiesof ionic compounds.Journal of the AmericanChemi
Woodcock, L. V. (19?5)Molecular dynamicscalculationson the
cal Society,50, 1036-1045.
molton ionic salts. In J. Braunstein,G. Mamantov, and G. P.
Pople, J. A., Binkley, J. S. Whitehead,R. A., Harihan, P. C., and
Smith. Eds.. Advancesin Molten Salt Chemistry,Volume 3' p.
Seeger,R. (1978)GAUSSIAN-76: an ab initio molecularorbital
1-74, PlenumPress,New York.
calculation.Quantum ChemistryProgram Exchange,11, 368.
Robie, R. A., Hemingway,B. S., and Fisher, J. R. (1978)Thermo- Woodcock,L. V. and Singer,K. (1971)Thermodynamicand structural propertiesof liquid ionic salts obtained by Monte Carlo
dynamic propertiesofminerals and relatedsubstancesat298.15
computation.Transactionsof the FaradaySociety,6'1,12-29K and 1 bar pressureand at higher temp€ratures.United States
P. (1976)MolecWoodcock, L. V., Angell, C. A., and Cheeseman,
GeologicalSurveyBulletin 1452.
ular dynamicsstudiesof the vitreousstate: Simpleionic systems
Soules,T. F. (1979)A moleculardynamic calculationof the strucand silica.Journal of ChemicalPhysics,65, 1565-1577.
ture of sodium silicateglasses.Journal of ChemicalPhysics,71,
calcula457H.578.
Zhadanov,V. A. and Pdyakov,V. V. (1977)Parameterless
tion of the bond forcesin ionic crystals.Soviet PhysicsCrystalSoules,T. F. and Busbey,R. F. (1981)Sodium diffusion in alkali
lography,22,488490.
silicateglassby moleculardynamics.Journal of ChemicalPhysics, 75,969-975.
Stillinger, F. H. (1964)Equilibrium theory of pure fused salts.In
Manuscript receioeil, Nooember 19, 1981:
M. Blander,Ed., Molten Salt Chemistry,p. 1-105. Interscience
accepteilfor publication, Jantnry 14, 1985.
Publishers.New York.
Metropolis, N., Rosenbluth,A. W., Rosenbluth,M. N., Teller, A.
H., and Teller, E. (1953)Equation of state calculationsby fast
computing machines.Journal of Chemical Physics,21, 1087-