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Models for strongly polar liquids. The influence of molecular polarizability
MODELS FOR STRONGLY THE INFLUENCE
15 July 1980
CHEMICAL PHYSICS LETTERS
Volume 73, number 2
POLAR LIQUtDS.
OF MOLECULAR
POLARIZABILITY
G.N. PATEY, M-L,. KLEIN Dwlaon of Chemistry, National Research Councd of Gmada, Ottawa, C&a& KIA OR6
and 1.R MCDONALD Department of Physxal Chemlsv. Gzmbndge CB2 I EP. UK Recewed
26 March 1980.
Unwenity
of Cambridge,,
in final form 19 April 1980
A recently proposed thermodynarmc perturbation approach IS used to estimate the can*abution of abdity to the average dipole moments and thermodynamic properties of model HF and HCIliquids
In two recent artrcles [ 1,2], a thermodynamic pertheory for polarizable dipolar hard spheres has been developed. This treatment used a rigid dipolar fluid as a reference system and has been termed the “rigid-dipole-reference perturbation theory” (RDRPT). The method has a disadvantage in that Monte Carlo (MC) or molecular dynamics (MD) calculations on the reference system are required. It does, however, yield free-energy results in good agreement with complete MC calculations [2] for polarizable dipolar hard spheres at a small fraction of the computational effort requrred to obtam the latter. F’revrous work [l-3] has shown that polarizabrlrty does make a significant contribution to the thermodynamic properties of polar liquids and that, furthermore, the magrutude of this contnbution is strongly influenced by the amsotropy of the polarizabrlrty tensor. In the present note we apply the RDRPT method to estimate the effects of polarization on the thermodynamic properties of two associated liquids, HF and HCl, for which extensive MD calculations have recently been carried out [4,5]. The models used in the MD calculations consist of appropriately tailored atomatom potentials supplemented by interacting charge turbation
molecti
polariz-
distributions chosen so as to reproduce the known molecular dipole moment 21,and quadrupole moment Q. These rigid models constitute our reference system Polarizable models are constructed by associating with these rigid molecules the appropriate polarizability tensor a. The induced interactions are then treated as
a perturbation. From the configurations generated during the course of a MD computation, we select a representative sample suitable for use in a RDRPT calculation. We replace the charge distribution on each molecule with point dipole and quadrupole moments located at the molecular centres of mass and the polarization (perturbation) energy Ufor instantaneous configurations is obtained by iteration [1,2]. The only significant departure from the procedure followed in previous work [ 1,2] is that quadrupole-induced-dipole (QID) as well as dipole-induced.-
dipole (DID) interactions are now included in Um ‘Ihe mean value (Ur,& is obtained by averaging over the sample of reference-system configurations generated by MD calculations. K$,& is a fii-order approximation to the difference in free energy between the reference (i.e. rigid models) and polarizable fluids. In the present case, however,
from the best avarIable values, Q,, = 3.01 A3 and (YI = 2.70 A [I 11. However, the mean polarizabllity is essentAIy unchanged. We estimate that the difference in anisotropy will decrease (Up,> by no more than =S% [I], which is insufficient to affect our basic conclusions. Our results are in table 2. The average total and electrostatic energies, tv> and t&d respectively, for the ngid reference fluid are included. In addition we calculate the electrostatrc energy of the reference system assuming that the charge distribution can be replaced by point dipole and quadrupole moments and the dqmle-dipole (DD), dipole-quadrupole (DQ) and quadrupole-quadrupole (QQ) contributrons are reported. Also grven are the magnitude of the mean total (permanent + induced) dipole moment , On>, associated with a singIe particle (as measured in the molecular coordinate system), and CUP,>, which IS broken down into DID and QID contrrbutrons. Furthermore, for each of the terms in ( Upen), values resulting from a pairwrse additivity assumptron are meluded. These numbers are obtained by assuming independent pair interactions, by which we mean that the mteractron between a particuIar pair of particles is assumed to depend only upon then permanent and mutuafiy induced dtpole moments. The mutually mduced dipole moments must still be found by rteration, but only a pair of particles is involved. The true induced interactions are, of course, many-body m character. It can be seen from table 2 that the enhancement of (G&l reIative to the permanent moment P is much
Table 1 State condlhons and molecular propeties System HF HCI
15 July 1980
CHEMICAL PHYSICS LETTERS
Volume 73, number 2
T (K)
V (cm3 mol-l)
!D,
278 201
19.96 30.74
1.82 1.04
0.96 0.72 2.6 3.37 3.13 2.39
Upert IS not calculated exactly but IS obtarned by replacmg the discrete charge drstnbutrons of the reference system with point dipoles and quadrupoles. (Ur,& 1s also a rough estimate of the contnbution made by the induced mteractions to the internal energy of the polar&able fluid. The complete firstorder approximation to the internal energy [6] consists of (U ) plus a fluctuation term which we rgnore, but p” or dense liquids (Up& is the dornmant contnbution. The state condrtions considered are given in table 1, together with a summary of the molecuIar properties, mcluding the components of the polarizabdity tensor parallel and perpendicular to the dipole (on and LYE,respectively). The values of y and Q quoted m table 1 are those of the pomt-charge models used in refs. [4,5]. They doffer somewhat from the best expenmental values appropriate to the equilibrium internucIear distances (1.2745 A for HCI and 0.9170 A for HF) namely c(~a = 1.0933 D [7], /.+r~ = 1.7965 D [8], QHa = 3.53 B, and QHF = 2.21 B [9]. The components of the polanzabihty tensor were taken from ref. [IO]- For HCl these differ considerabIy since
Table 2 RDRPT results for HF and HCI System
-tUjINkT
-lUe~ecl/h%T
Itm)l CD)
-WpeaVNkT
(ko 10) a) true
point dipole + quadrupole approxunation (*0.15) a) DD
DQ
QQ
total
-0.16
10 57
HF
9.71
11.31
6.56
4.17
HCI
9.89
2.37
1.25
1.22
0 61
a) These are estimated upper bounds on the statistical error. b) The quantities in brackets are obtained m the pairwse adltive
376
(to.011 a)
3.08
approximation.
DID b)
QID b)
total b)
1.76 (1.07)
0.50 (0.06)
2.26 (1.13)
2.15
0.61 (0.42)
0.24 (0.03)
0.85 (0.45)
1.29
Volume 73, number 2
CHEMICAL
PHYSICS
the same for both liquids studied (18% for HF and 24% for HCl). We note that the two polanzabdity anisotropies are nearly equal, and that the effect of the smaller permanent dipole moment of HCl IS more than offset by its larger polanzabihty. The polarization effects, however, are more significant for HF, where
LETTERS
15 July 1980
12% of ( CJk This result is in good accord with a recent quantum-mechanical estimate [4] and highlights the danger of relying too heatiy on the “effective” pair potential concept.
References [ 11 G.N. Patey and J-P. Vakau, Chem. Phys Letters 42 (1976) 407; 58 (1978) 157. [ 21 G.N. Patey, GM_ Torrie and J.F. Vakau, 1. Chem. Phys 71 (1979) 96. 131 MS. Wertheim, hfoL Phys 37 (1979) 83. t41 M.L. Klein and I.R. McDonald, J. Chem Phys. 7l
377
15 July 1980
CHEMICAL PHYSICS LETTERS
Volume 73, number 2
POLAR LIQUtDS.
OF MOLECULAR
POLARIZABILITY
G.N. PATEY, M-L,. KLEIN Dwlaon of Chemistry, National Research Councd of Gmada, Ottawa, C&a& KIA OR6
and 1.R MCDONALD Department of Physxal Chemlsv. Gzmbndge CB2 I EP. UK Recewed
26 March 1980.
Unwenity
of Cambridge,,
in final form 19 April 1980
A recently proposed thermodynarmc perturbation approach IS used to estimate the can*abution of abdity to the average dipole moments and thermodynamic properties of model HF and HCIliquids
In two recent artrcles [ 1,2], a thermodynamic pertheory for polarizable dipolar hard spheres has been developed. This treatment used a rigid dipolar fluid as a reference system and has been termed the “rigid-dipole-reference perturbation theory” (RDRPT). The method has a disadvantage in that Monte Carlo (MC) or molecular dynamics (MD) calculations on the reference system are required. It does, however, yield free-energy results in good agreement with complete MC calculations [2] for polarizable dipolar hard spheres at a small fraction of the computational effort requrred to obtam the latter. F’revrous work [l-3] has shown that polarizabrlrty does make a significant contribution to the thermodynamic properties of polar liquids and that, furthermore, the magrutude of this contnbution is strongly influenced by the amsotropy of the polarizabrlrty tensor. In the present note we apply the RDRPT method to estimate the effects of polarization on the thermodynamic properties of two associated liquids, HF and HCl, for which extensive MD calculations have recently been carried out [4,5]. The models used in the MD calculations consist of appropriately tailored atomatom potentials supplemented by interacting charge turbation
molecti
polariz-
distributions chosen so as to reproduce the known molecular dipole moment 21,and quadrupole moment Q. These rigid models constitute our reference system Polarizable models are constructed by associating with these rigid molecules the appropriate polarizability tensor a. The induced interactions are then treated as
a perturbation. From the configurations generated during the course of a MD computation, we select a representative sample suitable for use in a RDRPT calculation. We replace the charge distribution on each molecule with point dipole and quadrupole moments located at the molecular centres of mass and the polarization (perturbation) energy Ufor instantaneous configurations is obtained by iteration [1,2]. The only significant departure from the procedure followed in previous work [ 1,2] is that quadrupole-induced-dipole (QID) as well as dipole-induced.-
dipole (DID) interactions are now included in Um ‘Ihe mean value (Ur,& is obtained by averaging over the sample of reference-system configurations generated by MD calculations. K$,& is a fii-order approximation to the difference in free energy between the reference (i.e. rigid models) and polarizable fluids. In the present case, however,
from the best avarIable values, Q,, = 3.01 A3 and (YI = 2.70 A [I 11. However, the mean polarizabllity is essentAIy unchanged. We estimate that the difference in anisotropy will decrease (Up,> by no more than =S% [I], which is insufficient to affect our basic conclusions. Our results are in table 2. The average total and electrostatic energies, tv> and t&d respectively, for the ngid reference fluid are included. In addition we calculate the electrostatrc energy of the reference system assuming that the charge distribution can be replaced by point dipole and quadrupole moments and the dqmle-dipole (DD), dipole-quadrupole (DQ) and quadrupole-quadrupole (QQ) contributrons are reported. Also grven are the magnitude of the mean total (permanent + induced) dipole moment , On>, associated with a singIe particle (as measured in the molecular coordinate system), and CUP,>, which IS broken down into DID and QID contrrbutrons. Furthermore, for each of the terms in ( Upen), values resulting from a pairwrse additivity assumptron are meluded. These numbers are obtained by assuming independent pair interactions, by which we mean that the mteractron between a particuIar pair of particles is assumed to depend only upon then permanent and mutuafiy induced dtpole moments. The mutually mduced dipole moments must still be found by rteration, but only a pair of particles is involved. The true induced interactions are, of course, many-body m character. It can be seen from table 2 that the enhancement of (G&l reIative to the permanent moment P is much
Table 1 State condlhons and molecular propeties System HF HCI
15 July 1980
CHEMICAL PHYSICS LETTERS
Volume 73, number 2
T (K)
V (cm3 mol-l)
!D,
278 201
19.96 30.74
1.82 1.04
0.96 0.72 2.6 3.37 3.13 2.39
Upert IS not calculated exactly but IS obtarned by replacmg the discrete charge drstnbutrons of the reference system with point dipoles and quadrupoles. (Ur,& 1s also a rough estimate of the contnbution made by the induced mteractions to the internal energy of the polar&able fluid. The complete firstorder approximation to the internal energy [6] consists of (U ) plus a fluctuation term which we rgnore, but p” or dense liquids (Up& is the dornmant contnbution. The state condrtions considered are given in table 1, together with a summary of the molecuIar properties, mcluding the components of the polarizabdity tensor parallel and perpendicular to the dipole (on and LYE,respectively). The values of y and Q quoted m table 1 are those of the pomt-charge models used in refs. [4,5]. They doffer somewhat from the best expenmental values appropriate to the equilibrium internucIear distances (1.2745 A for HCI and 0.9170 A for HF) namely c(~a = 1.0933 D [7], /.+r~ = 1.7965 D [8], QHa = 3.53 B, and QHF = 2.21 B [9]. The components of the polanzabihty tensor were taken from ref. [IO]- For HCl these differ considerabIy since
Table 2 RDRPT results for HF and HCI System
-tUjINkT
-lUe~ecl/h%T
Itm)l CD)
-WpeaVNkT
(ko 10) a) true
point dipole + quadrupole approxunation (*0.15) a) DD
DQ
total
-0.16
10 57
HF
9.71
11.31
6.56
4.17
HCI
9.89
2.37
1.25
1.22
0 61
a) These are estimated upper bounds on the statistical error. b) The quantities in brackets are obtained m the pairwse adltive
376
(to.011 a)
3.08
approximation.
DID b)
QID b)
total b)
1.76 (1.07)
0.50 (0.06)
2.26 (1.13)
2.15
0.61 (0.42)
0.24 (0.03)
0.85 (0.45)
1.29
Volume 73, number 2
CHEMICAL
PHYSICS
the same for both liquids studied (18% for HF and 24% for HCl). We note that the two polanzabdity anisotropies are nearly equal, and that the effect of the smaller permanent dipole moment of HCl IS more than offset by its larger polanzabihty. The polarization effects, however, are more significant for HF, where
LETTERS
15 July 1980
12% of ( CJk This result is in good accord with a recent quantum-mechanical estimate [4] and highlights the danger of relying too heatiy on the “effective” pair potential concept.
References [ 11 G.N. Patey and J-P. Vakau, Chem. Phys Letters 42 (1976) 407; 58 (1978) 157. [ 21 G.N. Patey, GM_ Torrie and J.F. Vakau, 1. Chem. Phys 71 (1979) 96. 131 MS. Wertheim, hfoL Phys 37 (1979) 83. t41 M.L. Klein and I.R. McDonald, J. Chem Phys. 7l
377