- Email: [email protected]
An extended rism equation for molecular polar fluids
Volume 83, number 2
AN EXTENDED
CHJ35cAL
PHYSICS LE-ITERS
RISM EQUATION FOR MOLECULAR
15 uctobes
1981
POLAR FLUIDS
Fumio I-IIIUTA and Peter J. ROSSKY Deprtment
of Ciiemls~,
Unwernty
of T-sat
Austin, Austm, Texas 78712, USA
Received 22 June 1981
The REM integral equation is extended to molecules wth charged ties via a renormabzation of the Coulomb potent%& and the introduction of appropriate closure relations For a fluid of diatom& with atomic cIk%gesof 50.2 e the equation yields tie-site correlation funchons 111qualitatwe agreement with those from computer simulation.
1. Introduction
A frurtful approach to the model&g of intermolecular interactions in molecular fluids has been the use of so-called interaction site models (EMS) [I-4] _ In such models the interactions between molecular entrtres is represented by a sum of pair-wise addrtme spherically symmetric potent&s which act between sites located in the molecules. Potentials of this form are reasonable representations for both nonpolar [5-7] and highly polar molecules [8,9]. One mqor motivation for an ISM representation is that the mathematical aspects of the statistical mechanics of molecular systems is simplified, since no explicit orientatron dependence is necessary in the descnption of the interacttons The correspondmg graphical expansion formulation for these models has been developed in detail by Chandler and coworkers [2-4, 10, Ii]. An approximate integral equatron formulation known as the RISM equation [23, which leads to the site-site pair correlation functions has been developed and applied to the study of molecular liquids whose structure is dominated by short-range intermolecular repulsrve forces. The results of these studies have demonstrated that the RISM equation yields a reliable picture of the short-range structure of such liquids [7] - The quality of the results is particularly interesting in light of the fact that the nature of the approxrmations involved in the REM: equation is somewhat different from that familiar in the corresponding equa-
tions for atomic fluids. If analyzed in terms of the appropriate graphical formalism [IO], one finds that not only are a set of graph&d contributions omitted, but also a new class of diagrams with topologies not present in the exact expansion are introduced into the calculation. It 1s natural to ask whether the same type of integral equation can be productively applied to polar systerns. As for liquids composed of non-polar molecules, one can consider separately the theoretical predictions for the long and short-range structure of the fItrid [7,12]. For polar molecules the long-wavelength property of special interest is the static dielectric constant It is readiIy shown that for a broad class of closure relations the RISM integral equation leads only to the low-density ideal gas result [ I3]_ Nevertheiess, the question of the short-range behavior of the pair correlation functions remains open, and we investigate this aspect of the theory here. The present article is devoted to the development of the appropriate extension of the REM equation to the case in which moIecuIar sites bear charges, and an appIication to the simplest relevant modeI, namely, a fluid composed of neutraI diatomic molecules in which each of the pair of atoms is associated with a charge of equal magnitude. We test the predictions by comparison with molecular dynamics computer simulation results. In section 2, we present the formuIation of the integral equation in&ding the choice of closure. The resuits obtained for the dIatomIc model are presented in section 3. The conchrsions are given In section 4. 329
Volume 83, number 2
CHEMICAL PHYSICS LETTERS
1S October 1981
2. F o r m u l a t i o n o f the integral e q u a t i o n
%,(0 =f~v(r)[l + t~.(r)],
In this section we outline the extension o f the RISM integral e q u a t i o n t o a f o r m appropriate for the present purposes. We assume a familiarity with the elements o f the standard t h e o r y o f a t o m i c fluids [14] as well as the corresponding t h e o r y for an ISM representation and for the RISM e q u a t i o n [2--4, 10]. We make use o f the renormalization techniques and n o t a t i o n o f Rossky and Dale [15] and refer the reader there for detailed clef'tuitions and derivations o f the results employed. The RISM integral e q u a t i o n for intermolecular site--site correlations has a f o r m analogous to the usual O r n s t e i n - Z e r n i k e equation [14] for atomic fluids, n a m e l y (for a o n e - c o m p o n e n t molecular fluid o f density p),
where
p 2 h ( r ) = ~ * c * o a ( r ) + p2 co*c*h(r),
(1)
where h, t , . and c are matrices with elements labelled by paixs o f molecular sites, which we will d e n o t e by try, and * denotes convolution. Here hcr t is an intermolecular s i t e - s i t e pair correlation function, ca. r the c o r r e s p o n d i n g direct correlation function, and coa.~ is given b y
coa~ = pa a~a (r) + psa. t (r).
(2)
sc,~(r) is the intramolecular s i t e - s i t e pair distribution f u n c t i o n for distinct sites in a molecule, and vanishes for ot = T. The integral e q u a t i o n given in eq. ( I ) can be written c o m p a c t l y as a chain sum on hypervertices as (see ref. [ 1 5 ] )
p2h=p2 e [ c l
¢~]
= sum o f all simple chains o f one or more c b o n d s on black to hypervertices between terminal white p circles.
(3)
The generalization for mixtures o f molecular species is immediate. In i t s original c o n t e x t , for molecules c o m p o s e d o f fused b2xd spheres, the needed closure c o n d i t i o n was imposed b y direct analogy to the hard-sphere P e r c u s Yevick (PY) t h e o r y [ 2 ] . The generalization to continuous potentials has been given [10] and applied t o molecules c o m p o s e d o f fused L e n n a r d - J o n e s spheres [ 12]. The PY-like closure takes the f o r m 330
(4)
f~.(r) = exp[-~u~,,(O] - 1
and t~(r)
= h~(r)
- c~(r).
Here. u t ~ ( r ) is the site--site potential and [3 = l l k B T , where k B is B o l t z m a n n ' s c o n s t a n t and T is the temperature. Here, we wish to consider molecules with charged sites. Hence, as in the case o f a t o m i c electrolytes [ 16], it is advantageous to renormalize the integral e q u a t i o n so as to remove any explicit reference to the longranged potential. D e n o t i n g this b y Ulr and defining u • = u - u Ix, the rearrangement can be carried o u t directly using the techniques o f refi [ 15]. F r o m eq. (3), we can obtain immediately
h= e[c--Ol~+Q]
+Q,
(5)
where (~ = --JfUlr and 0 = C [~1 o)]. T h e renormaliTation accomplished in eq. (5) is closely analogous to the D e b y e - H ~ c k e l r e s u m m a t i o n in electrolyte t h e o r y [ 17], where Q, here, corresponds to the screened Debye-Hfmkel potential. T h e chain sum Q has been considered as a leading a p p r o x i m a t i o n to h 0 s e w h e r e [1 1]. It is, however, i m p o r t a n t to n o t e that the presence o f the intramolecular correlation functions cause Q t o d e c a y as r - 6 rather than exponentially as it does in electrolytes [ 1 1 ] . In addition t o eq. (5) [which is equivalent to eq. (3)] a closure relation, analogous to that given in eq. (4), is necessary to f o r m a closed set o f equations. Here, we take the view that a reasonable choice is one analogous to that which is reasonably successful in the t h e o r y o f a t o m i c electrolytes [ 18]. This leads immediately to two alternatives. First, one can consider a closure analogous to the h y p e r n e t t e d chain (HNC) closure, n a m e l y ctz~' -- ~ ' 7 = exp(--lffu*a'r + Qa~, + T~b) - - 1 -- r ~ .
-
Q~,-r,
(6)
where
(7)
Alternatively, one can consider a closure analogous to Alnatt’s modrfication [ 191 of the Percus-Yevick closure (PYA), that is, c Qy - &
= expG&&
- 1--a7-
+ Q,Xl
+
~~1 (8)
Qa,-
The latter choice has the apparent advantage that in the hmit of zero site charges, it reduces to eq. (4), the PY-lrke closure. The latter is already established as a good description for the uncharged case [6,12]. We note here that for liquid molecular densities, a(r) is rapidly decaying, and eq. (8) gives results wluch drffer negligibly from the lmearized form, c a7 =
exp(-flu&Xl + Q,,W
- 1 - Q,
- T,~.
15 October 1981
CJXEMICAL PHYSICS LETTERS
Volume 83, number 2
+ r,J (9)
The solution of eq. (5) with either eq. (6) or eq. (8) as closure can be obtained usmg iterative schemes completely analogous to that used m earlier stu&es of atomic electrolyte solutions [ 16,18,20] _ The closures mtroduced above all lead tc approximate integral equations in the same sense as that mentroned in section 1. That IS, the site-site pair correlation functions which follow from a solution to eq. (5) usmg these includes only a subset of terms present m the exact graphical expansron for the ISM, and, in addrtion, mcludes a set of terms which are absent in the exact prescription. For the HNC closure [eq. (6)], both of these sets are larger than for the PYA closure [eq. (8)], and rt is only by trial calculatrons that the predrctive capacity of these equations can be tested.
3. Application
the atoms are separated by a rigid bond of length Z= l- 1 A (The difference between a rigid bond and onewith a reasonable standarddcviaticminbond length h negligible for the present purposes. as argued elsewhere 161.) A charge of +Ze is placed on one nucleus and a charge of -2e is pIaced on the other, where e is the magnitude of the electronic charge. Hence, the site-site interaction potential is us,(f)
=
~E[(cT/)~*- (c~/r)~]+ Z&e*/r_
For the present model, we use e/kB = 44 K, 0 = 3.341 A, and 2 = 3.2, which corresponds to a dipole moment of 1.06 D, a value comparable, for example, to that of CO. All calculations have been carried out at liquid-mtrogen conditrons *: T= 72 K and p = 0.01867 molecules/A3. 3.2. Simulationresults To test the validrty of the integral equation we have carried out molecular dynamics computer simulations of the charged and uncharged models_ The sample used consists of 216 diatomic molecules. All
calculations have been carried out using periodic boundary condrtions [2 11. For the system with charge,
* The temperature is the mean obtained from the simuIation results and is used in the integral equation caIcuIafioions.
2
P
In this sectron we describe results obtained from an apphcation of the extended RJSM equation described above, and compare to computer simulation data. 3. I. ikfolecular model The model considered here corresponds, except for site charges, to that employed in earlier studies [S, 123 as a model for liquid molecular nitrogen; each N atom is represented by a Lennard-Jones sphere, and
4
6
8
r rEi1 Fig. 1. Effect of bcundaxy conditions on site-site coneIaticn iimcti~w k++(r) and h+_(r). &O h, spherical cut off (0. ~1, fidl minimumimage(10.69 A, cnbic cut off) (-).
331
rolume83. number 2
15 October 1981
CHEMICAL PHYSICS LETTERS
rparate smurlahons were carried out, using different runcation schemes for the potential, m order to enkure that the simulation results for ante-sate correlatron ktncttons were not sensittve to the boundary condrions. From these results, shown in fig. 1, it is evident that the results are not sensitive to the boundary coniittons, as expected from earlier studres [22]. In fig. 2 we give computer simulation results for 10th the uncharged case hNN(r) and for the charged :ase, b_(r) and k++(r). These results show a clear iplitting of h,, into the two charged components. For the uncharged case the geometrical interpretatron of &,I&) has already been given by several authors [S. 131. Upon charging the atomic centers, the small --T part sf the first peak in h&r) evolves in the major peak m k+_(r), a result of the coulombic attraction between dtes, the peak corresponds to the We-site contact distance. Correspondingly, the shoulder regon at ~4.3 A, correspondmg to stormc sues wrth an mtervenmg bonded atomrc partner, 1s depressed m h+_(r). For the ++ pair, the contact interaction has a repulsrve coulombic component, and the first peak m h,(r) slufts to larger r. The shoulder in h&r) evolves into the mam peak, corresponding to an mtervening bonded partner with a site charge which is opposite to that of both sites mvolved m the correlation fimctlon.
3.3. R&M results In fig. 3 we present the REM equation results and m fig. 4, we compare with those from the simulation (solid lure). The PYA results [eq. (8)] are indicated by fiied crrcles and the HNC results [eq. (6)J by triangles. As 1s evtdent from figs. 3 and 4, the integral equation provides a qualitatively correct description of the effect of the site charges. In partrcular, the splitting of the fint peak in h&r) accordmg to the charge type of the pair is properly accounted for. A discrepancy remams, some of which must be attributed to the underlying errors present in the integral equation in the absence of charge, as manifest by the comparison shown m fig. 2 (fdled circles). In this hght, the result for h+_(r) (fig. 4a) appears to be very satisfactory. For h++(r) the discrepancy is clearly larger, the amplitude being somewhat too small, and the sluft to larger r being underestrmated. It 1s clear from the results shown m fig. 4 that the altematrve closures [eq. (6) versus eq. (S)] yield quite snmlar results, and there 1s kttle evidence here to recommend one over the other. In the region just beyond the first maxunum, the PYA-like ciosure ap pears slightly more accurate, but the difference is not compelhng It IS interesting to note that the curves for the charged ste correlation func’ilons which follow from
2 2
a
P I
I
4
Fw 2. Computer nmulation results for site-site correlation functions k&r), k-(r) are shown as sohd lines, m(r) (un&arged) by a dashed line. The filled circles show the RISM result with PY-like closwe [eq. (4)] for m(r)_
332
6
6
r(i)
Fw 3. RISM equation prediction of correlation functions with PYA-like closure [eq. (8)]. The dashedcurve showy
bmm.
CHEMICAL
Volume 83. number 2
PHYSICS LETl-JZRS
1s October L981
charged sites- The renormahzed equation, combined with closures analogous to those used in electrolyte theory, has been applied to a simple model of a Quid composed of diatomic molecules similar to N,, but with partial charges of %.2c located at the nuclear centers. The results of this application have shown that the equation yields a qualitatively correct descrip tion of the intermoleculaf site-site correlations in the liquid when cornbared to computer simulation remits for the same model. These results are encouraging and indicate that the approach warrants further study, including, for example, the possibility for the PYA-like closure of employing a modified electrostatic potential which is optirmzed [2] in the region of strong T_enna.rd-Jones repulsion. A more detailed study of the integral equation and its predictions for a variety of models wiil be presented io a later paper.
Acknowledgement
P I
Support of the work reported here by grants from the Robert A. Welch Foundation (Grant No. F-761) and the Research Corporation is gratefully acknowledged 6 (a)
4
rI Xl
Fie 4- Comptin of simulatronand integralequation pre dictions Srmulauon(-1, WA &sure fe4. U-01(.I, HNc dosure [eq. (6j] (A). HNCresultsare omitted where they
aould
obscure
those
from
PYA.
the integral equation are symmetrically displaced to either side of the uncharged (N-N) function (see fig. 3) and that this symmetry is approximately reflected by the molecular dynamics results (fig. 2). It can be demonstrated [23] that the integral equation should exhibit this property to a high degree for the present case, in which the densrty IS high, and where, in the absence of charge, the sites are equivalent_
4. conclusions We have presented an extension of the REM integral equatron to a form appropriate for models with
References [l]
J.R.
Sweet
and
WA.
Steele,
J. Chem
Phys
47
(19673
3029. [2] D. Chandler and H.C. Anderson, J. Chera Phys 57 (1972) 1930. [ 3J B.M. Ladanyi and D. Chandler, J. Chem Phyr 62 (1975) 4308 [4] L-R Pratt and D. C%andler. J. Chem Fhyr 65 (1977) 2925. [S] J. Bamjas, D. Levxsque and B. Quentxec, Phys Rev. A7 (1973) 1092. [6] W. Lowden and D. Chandkr, J. Chen~ Phys 61<1974) 5228. [7] D. Chandler, Ann- Rev. Phys Chea 29 (1978) 441. [S] W.L. Jorge.nszn,J. Chea Phys 70 (1979) 5888. [9] 0. Matsuoka, E Clementi and M. Yoshimine, I- themPhys 64 (1976) 1351. [lOI D-chandler. Mel Phys 31(197Q 1213. ill] D. chandh, J. Chem. Fhyr 67 (1977) 1113. 1121 K. Kojh ad K. Arakawa, Bull- Chem. Sac. Japan 51 (1978) 1977. 1131 D-E. S* and C-G- Gray, MoL phys, to be published.
333
Volume 83, number 2
CHEMJCAL
[ 141 G. St&
in: The equilibnum theory of classical fluids, eds. ELL Fnsch and J.L. Lebomtz (BenJamm, New
York, 1964). [ 151 P.J. Rossky and W.D.T. Dale, J. Chem. Phys 73 (1980) 2457. 1161 P.J. Rossky, LB. Dudowm, B.L. Tembe and H.L. Friedman, J. Chea Phys 73 (1980) 3372 1171 J-E Mayer, J. Chem. Phyr 18 (1950) 1426. [ 181 J.C. Rasaiah, J. Chem. Phys 56 (1972) 3071. J.C. Rasuah and HA. Friedman, J. Chem Phys 48 (1968) 2742
334
PHYSICS LETlERS
15 October 1981
[19] A.R Allnatt, Mol. Phys 8 (1964) 533. [20] P-J. Rossky and HA. Friedman, J. Chem Phys 72 (1980) 5694. 1211 J-P. Vrdleau and SG. ~~0~ 111:Modem theoretical chsnustry, Vol. 5, ed. B.J. Berne (Plenum Press, New York, 1977). [22] C Pangah, hi, Rao and B.J. Bern&, MoL Phyr 40 (1980) 661. [23] F. Himta, B.ht Pettitt and P.J. Rosky, to be published.
AN EXTENDED
CHJ35cAL
PHYSICS LE-ITERS
RISM EQUATION FOR MOLECULAR
15 uctobes
1981
POLAR FLUIDS
Fumio I-IIIUTA and Peter J. ROSSKY Deprtment
of Ciiemls~,
Unwernty
of T-sat
Austin, Austm, Texas 78712, USA
Received 22 June 1981
The REM integral equation is extended to molecules wth charged ties via a renormabzation of the Coulomb potent%& and the introduction of appropriate closure relations For a fluid of diatom& with atomic cIk%gesof 50.2 e the equation yields tie-site correlation funchons 111qualitatwe agreement with those from computer simulation.
1. Introduction
A frurtful approach to the model&g of intermolecular interactions in molecular fluids has been the use of so-called interaction site models (EMS) [I-4] _ In such models the interactions between molecular entrtres is represented by a sum of pair-wise addrtme spherically symmetric potent&s which act between sites located in the molecules. Potentials of this form are reasonable representations for both nonpolar [5-7] and highly polar molecules [8,9]. One mqor motivation for an ISM representation is that the mathematical aspects of the statistical mechanics of molecular systems is simplified, since no explicit orientatron dependence is necessary in the descnption of the interacttons The correspondmg graphical expansion formulation for these models has been developed in detail by Chandler and coworkers [2-4, 10, Ii]. An approximate integral equatron formulation known as the RISM equation [23, which leads to the site-site pair correlation functions has been developed and applied to the study of molecular liquids whose structure is dominated by short-range intermolecular repulsrve forces. The results of these studies have demonstrated that the RISM equation yields a reliable picture of the short-range structure of such liquids [7] - The quality of the results is particularly interesting in light of the fact that the nature of the approxrmations involved in the REM: equation is somewhat different from that familiar in the corresponding equa-
tions for atomic fluids. If analyzed in terms of the appropriate graphical formalism [IO], one finds that not only are a set of graph&d contributions omitted, but also a new class of diagrams with topologies not present in the exact expansion are introduced into the calculation. It 1s natural to ask whether the same type of integral equation can be productively applied to polar systerns. As for liquids composed of non-polar molecules, one can consider separately the theoretical predictions for the long and short-range structure of the fItrid [7,12]. For polar molecules the long-wavelength property of special interest is the static dielectric constant It is readiIy shown that for a broad class of closure relations the RISM integral equation leads only to the low-density ideal gas result [ I3]_ Nevertheiess, the question of the short-range behavior of the pair correlation functions remains open, and we investigate this aspect of the theory here. The present article is devoted to the development of the appropriate extension of the REM equation to the case in which moIecuIar sites bear charges, and an appIication to the simplest relevant modeI, namely, a fluid composed of neutraI diatomic molecules in which each of the pair of atoms is associated with a charge of equal magnitude. We test the predictions by comparison with molecular dynamics computer simulation results. In section 2, we present the formuIation of the integral equation in&ding the choice of closure. The resuits obtained for the dIatomIc model are presented in section 3. The conchrsions are given In section 4. 329
Volume 83, number 2
CHEMICAL PHYSICS LETTERS
1S October 1981
2. F o r m u l a t i o n o f the integral e q u a t i o n
%,(0 =f~v(r)[l + t~.(r)],
In this section we outline the extension o f the RISM integral e q u a t i o n t o a f o r m appropriate for the present purposes. We assume a familiarity with the elements o f the standard t h e o r y o f a t o m i c fluids [14] as well as the corresponding t h e o r y for an ISM representation and for the RISM e q u a t i o n [2--4, 10]. We make use o f the renormalization techniques and n o t a t i o n o f Rossky and Dale [15] and refer the reader there for detailed clef'tuitions and derivations o f the results employed. The RISM integral e q u a t i o n for intermolecular site--site correlations has a f o r m analogous to the usual O r n s t e i n - Z e r n i k e equation [14] for atomic fluids, n a m e l y (for a o n e - c o m p o n e n t molecular fluid o f density p),
where
p 2 h ( r ) = ~ * c * o a ( r ) + p2 co*c*h(r),
(1)
where h, t , . and c are matrices with elements labelled by paixs o f molecular sites, which we will d e n o t e by try, and * denotes convolution. Here hcr t is an intermolecular s i t e - s i t e pair correlation function, ca. r the c o r r e s p o n d i n g direct correlation function, and coa.~ is given b y
coa~ = pa a~a (r) + psa. t (r).
(2)
sc,~(r) is the intramolecular s i t e - s i t e pair distribution f u n c t i o n for distinct sites in a molecule, and vanishes for ot = T. The integral e q u a t i o n given in eq. ( I ) can be written c o m p a c t l y as a chain sum on hypervertices as (see ref. [ 1 5 ] )
p2h=p2 e [ c l
¢~]
= sum o f all simple chains o f one or more c b o n d s on black to hypervertices between terminal white p circles.
(3)
The generalization for mixtures o f molecular species is immediate. In i t s original c o n t e x t , for molecules c o m p o s e d o f fused b2xd spheres, the needed closure c o n d i t i o n was imposed b y direct analogy to the hard-sphere P e r c u s Yevick (PY) t h e o r y [ 2 ] . The generalization to continuous potentials has been given [10] and applied t o molecules c o m p o s e d o f fused L e n n a r d - J o n e s spheres [ 12]. The PY-like closure takes the f o r m 330
(4)
f~.(r) = exp[-~u~,,(O] - 1
and t~(r)
= h~(r)
- c~(r).
Here. u t ~ ( r ) is the site--site potential and [3 = l l k B T , where k B is B o l t z m a n n ' s c o n s t a n t and T is the temperature. Here, we wish to consider molecules with charged sites. Hence, as in the case o f a t o m i c electrolytes [ 16], it is advantageous to renormalize the integral e q u a t i o n so as to remove any explicit reference to the longranged potential. D e n o t i n g this b y Ulr and defining u • = u - u Ix, the rearrangement can be carried o u t directly using the techniques o f refi [ 15]. F r o m eq. (3), we can obtain immediately
h= e[c--Ol~+Q]
+Q,
(5)
where (~ = --JfUlr and 0 = C [~1 o)]. T h e renormaliTation accomplished in eq. (5) is closely analogous to the D e b y e - H ~ c k e l r e s u m m a t i o n in electrolyte t h e o r y [ 17], where Q, here, corresponds to the screened Debye-Hfmkel potential. T h e chain sum Q has been considered as a leading a p p r o x i m a t i o n to h 0 s e w h e r e [1 1]. It is, however, i m p o r t a n t to n o t e that the presence o f the intramolecular correlation functions cause Q t o d e c a y as r - 6 rather than exponentially as it does in electrolytes [ 1 1 ] . In addition t o eq. (5) [which is equivalent to eq. (3)] a closure relation, analogous to that given in eq. (4), is necessary to f o r m a closed set o f equations. Here, we take the view that a reasonable choice is one analogous to that which is reasonably successful in the t h e o r y o f a t o m i c electrolytes [ 18]. This leads immediately to two alternatives. First, one can consider a closure analogous to the h y p e r n e t t e d chain (HNC) closure, n a m e l y ctz~' -- ~ ' 7 = exp(--lffu*a'r + Qa~, + T~b) - - 1 -- r ~ .
-
Q~,-r,
(6)
where
(7)
Alternatively, one can consider a closure analogous to Alnatt’s modrfication [ 191 of the Percus-Yevick closure (PYA), that is, c Qy - &
= expG&&
- 1--a7-
+ Q,Xl
+
~~1 (8)
Qa,-
The latter choice has the apparent advantage that in the hmit of zero site charges, it reduces to eq. (4), the PY-lrke closure. The latter is already established as a good description for the uncharged case [6,12]. We note here that for liquid molecular densities, a(r) is rapidly decaying, and eq. (8) gives results wluch drffer negligibly from the lmearized form, c a7 =
exp(-flu&Xl + Q,,W
- 1 - Q,
- T,~.
15 October 1981
CJXEMICAL PHYSICS LETTERS
Volume 83, number 2
+ r,J (9)
The solution of eq. (5) with either eq. (6) or eq. (8) as closure can be obtained usmg iterative schemes completely analogous to that used m earlier stu&es of atomic electrolyte solutions [ 16,18,20] _ The closures mtroduced above all lead tc approximate integral equations in the same sense as that mentroned in section 1. That IS, the site-site pair correlation functions which follow from a solution to eq. (5) usmg these includes only a subset of terms present m the exact graphical expansron for the ISM, and, in addrtion, mcludes a set of terms which are absent in the exact prescription. For the HNC closure [eq. (6)], both of these sets are larger than for the PYA closure [eq. (8)], and rt is only by trial calculatrons that the predrctive capacity of these equations can be tested.
3. Application
the atoms are separated by a rigid bond of length Z= l- 1 A (The difference between a rigid bond and onewith a reasonable standarddcviaticminbond length h negligible for the present purposes. as argued elsewhere 161.) A charge of +Ze is placed on one nucleus and a charge of -2e is pIaced on the other, where e is the magnitude of the electronic charge. Hence, the site-site interaction potential is us,(f)
=
~E[(cT/)~*- (c~/r)~]+ Z&e*/r_
For the present model, we use e/kB = 44 K, 0 = 3.341 A, and 2 = 3.2, which corresponds to a dipole moment of 1.06 D, a value comparable, for example, to that of CO. All calculations have been carried out at liquid-mtrogen conditrons *: T= 72 K and p = 0.01867 molecules/A3. 3.2. Simulationresults To test the validrty of the integral equation we have carried out molecular dynamics computer simulations of the charged and uncharged models_ The sample used consists of 216 diatomic molecules. All
calculations have been carried out using periodic boundary condrtions [2 11. For the system with charge,
* The temperature is the mean obtained from the simuIation results and is used in the integral equation caIcuIafioions.
2
P
In this sectron we describe results obtained from an apphcation of the extended RJSM equation described above, and compare to computer simulation data. 3. I. ikfolecular model The model considered here corresponds, except for site charges, to that employed in earlier studies [S, 123 as a model for liquid molecular nitrogen; each N atom is represented by a Lennard-Jones sphere, and
4
6
8
r rEi1 Fig. 1. Effect of bcundaxy conditions on site-site coneIaticn iimcti~w k++(r) and h+_(r). &O h, spherical cut off (0. ~1, fidl minimumimage(10.69 A, cnbic cut off) (-).
331
rolume83. number 2
15 October 1981
CHEMICAL PHYSICS LETTERS
rparate smurlahons were carried out, using different runcation schemes for the potential, m order to enkure that the simulation results for ante-sate correlatron ktncttons were not sensittve to the boundary condrions. From these results, shown in fig. 1, it is evident that the results are not sensitive to the boundary coniittons, as expected from earlier studres [22]. In fig. 2 we give computer simulation results for 10th the uncharged case hNN(r) and for the charged :ase, b_(r) and k++(r). These results show a clear iplitting of h,, into the two charged components. For the uncharged case the geometrical interpretatron of &,I&) has already been given by several authors [S. 131. Upon charging the atomic centers, the small --T part sf the first peak in h&r) evolves in the major peak m k+_(r), a result of the coulombic attraction between dtes, the peak corresponds to the We-site contact distance. Correspondingly, the shoulder regon at ~4.3 A, correspondmg to stormc sues wrth an mtervenmg bonded atomrc partner, 1s depressed m h+_(r). For the ++ pair, the contact interaction has a repulsrve coulombic component, and the first peak m h,(r) slufts to larger r. The shoulder in h&r) evolves into the mam peak, corresponding to an mtervening bonded partner with a site charge which is opposite to that of both sites mvolved m the correlation fimctlon.
3.3. R&M results In fig. 3 we present the REM equation results and m fig. 4, we compare with those from the simulation (solid lure). The PYA results [eq. (8)] are indicated by fiied crrcles and the HNC results [eq. (6)J by triangles. As 1s evtdent from figs. 3 and 4, the integral equation provides a qualitatively correct description of the effect of the site charges. In partrcular, the splitting of the fint peak in h&r) accordmg to the charge type of the pair is properly accounted for. A discrepancy remams, some of which must be attributed to the underlying errors present in the integral equation in the absence of charge, as manifest by the comparison shown m fig. 2 (fdled circles). In this hght, the result for h+_(r) (fig. 4a) appears to be very satisfactory. For h++(r) the discrepancy is clearly larger, the amplitude being somewhat too small, and the sluft to larger r being underestrmated. It 1s clear from the results shown m fig. 4 that the altematrve closures [eq. (6) versus eq. (S)] yield quite snmlar results, and there 1s kttle evidence here to recommend one over the other. In the region just beyond the first maxunum, the PYA-like ciosure ap pears slightly more accurate, but the difference is not compelhng It IS interesting to note that the curves for the charged ste correlation func’ilons which follow from
2 2
a
P I
I
4
Fw 2. Computer nmulation results for site-site correlation functions k&r), k-(r) are shown as sohd lines, m(r) (un&arged) by a dashed line. The filled circles show the RISM result with PY-like closwe [eq. (4)] for m(r)_
332
6
6
r(i)
Fw 3. RISM equation prediction of correlation functions with PYA-like closure [eq. (8)]. The dashedcurve showy
bmm.
CHEMICAL
Volume 83. number 2
PHYSICS LETl-JZRS
1s October L981
charged sites- The renormahzed equation, combined with closures analogous to those used in electrolyte theory, has been applied to a simple model of a Quid composed of diatomic molecules similar to N,, but with partial charges of %.2c located at the nuclear centers. The results of this application have shown that the equation yields a qualitatively correct descrip tion of the intermoleculaf site-site correlations in the liquid when cornbared to computer simulation remits for the same model. These results are encouraging and indicate that the approach warrants further study, including, for example, the possibility for the PYA-like closure of employing a modified electrostatic potential which is optirmzed [2] in the region of strong T_enna.rd-Jones repulsion. A more detailed study of the integral equation and its predictions for a variety of models wiil be presented io a later paper.
Acknowledgement
P I
Support of the work reported here by grants from the Robert A. Welch Foundation (Grant No. F-761) and the Research Corporation is gratefully acknowledged 6 (a)
4
rI Xl
Fie 4- Comptin of simulatronand integralequation pre dictions Srmulauon(-1, WA &sure fe4. U-01(.I, HNc dosure [eq. (6j] (A). HNCresultsare omitted where they
aould
obscure
those
from
PYA.
the integral equation are symmetrically displaced to either side of the uncharged (N-N) function (see fig. 3) and that this symmetry is approximately reflected by the molecular dynamics results (fig. 2). It can be demonstrated [23] that the integral equation should exhibit this property to a high degree for the present case, in which the densrty IS high, and where, in the absence of charge, the sites are equivalent_
4. conclusions We have presented an extension of the REM integral equatron to a form appropriate for models with
References [l]
J.R.
Sweet
and
WA.
Steele,
J. Chem
Phys
47
(19673
3029. [2] D. Chandler and H.C. Anderson, J. Chera Phys 57 (1972) 1930. [ 3J B.M. Ladanyi and D. Chandler, J. Chem Phyr 62 (1975) 4308 [4] L-R Pratt and D. C%andler. J. Chem Fhyr 65 (1977) 2925. [S] J. Bamjas, D. Levxsque and B. Quentxec, Phys Rev. A7 (1973) 1092. [6] W. Lowden and D. Chandkr, J. Chen~ Phys 61<1974) 5228. [7] D. Chandler, Ann- Rev. Phys Chea 29 (1978) 441. [S] W.L. Jorge.nszn,J. Chea Phys 70 (1979) 5888. [9] 0. Matsuoka, E Clementi and M. Yoshimine, I- themPhys 64 (1976) 1351. [lOI D-chandler. Mel Phys 31(197Q 1213. ill] D. chandh, J. Chem. Fhyr 67 (1977) 1113. 1121 K. Kojh ad K. Arakawa, Bull- Chem. Sac. Japan 51 (1978) 1977. 1131 D-E. S* and C-G- Gray, MoL phys, to be published.
333
Volume 83, number 2
CHEMJCAL
[ 141 G. St&
in: The equilibnum theory of classical fluids, eds. ELL Fnsch and J.L. Lebomtz (BenJamm, New
York, 1964). [ 151 P.J. Rossky and W.D.T. Dale, J. Chem. Phys 73 (1980) 2457. 1161 P.J. Rossky, LB. Dudowm, B.L. Tembe and H.L. Friedman, J. Chea Phys 73 (1980) 3372 1171 J-E Mayer, J. Chem. Phyr 18 (1950) 1426. [ 181 J.C. Rasaiah, J. Chem. Phys 56 (1972) 3071. J.C. Rasuah and HA. Friedman, J. Chem Phys 48 (1968) 2742
334
PHYSICS LETlERS
15 October 1981
[19] A.R Allnatt, Mol. Phys 8 (1964) 533. [20] P-J. Rossky and HA. Friedman, J. Chem Phys 72 (1980) 5694. 1211 J-P. Vrdleau and SG. ~~0~ 111:Modem theoretical chsnustry, Vol. 5, ed. B.J. Berne (Plenum Press, New York, 1977). [22] C Pangah, hi, Rao and B.J. Bern&, MoL Phyr 40 (1980) 661. [23] F. Himta, B.ht Pettitt and P.J. Rosky, to be published.