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Structure factor calculations for molecular liquids: Beyond the rism approximation
Physica B 156 & 157 (1989) 161-163 North-Holland, Amsterdam
STRUCTURE FACTOR CALCULATIONS FOR MOLECULAR LIQUIDS: BEYOND THE RISM APPROXIMATION B. SALAMITO CEN-Grenoble,
and P.H. FRIES
DRF-Laboratoires
de Chime’,
85 X, 38041 Grenoble
Cedex, France
A simple model of acetonitrile molecules approximated as dipolar rods is developed. The pair distribution function is calculated using an HNC type approximation and tested by comparison with diffraction data. The dominant molecular arrangements are presented.
1. Introduction Liquid acetonitrile (CH,CN) is a polar solvent which is widely used in chemistry. It has received much experimental and theoretical attention since it provides a good example of a molecule having a large dipole moment of 3.92 D in the gas phase, no important hydrogen bonding, and a strongly anisotropic shape, together with a high symmetry which simplifies the theoretical description. Detailed diffraction experiments aimed at the study of the intermolecular distribution of the molecules have been carried out and data from neutron scattering using different isotopes and from X-rays are available [l, 21. Realistic site-site potential models have also been devised from ab initio molecular orbital calculations [35] and have been used in simulations and sitesite integral equation theories [6] to represent the interactions between actual molecules. This work is concerned with the dominant features of the molecular pair distribution obtained from an integral equation of the hypernetted chain (HNC) type [7] using the simplest site-site intermolecular potential which incorporates the principle features of the shape and Coulomb interactions.
where the label i = 1,2 denotes the absolute position and orientation of the ith molecule. U,, is a short range anisotropic repulsion which corresponds to the intuitive molecular shape and U,, is the dipole-dipole long range interaction [8]. In order that U,, contains the essential features of the intermolecular repulsive forces it is sufficient that at the molecular contact, for a given relative orientation of the molecules, it increases to values of a few k,T when the intercentre distance R decreases by a few percent. The potential UP, is adapted to the geometry of collision between two molecules with its softness increasing with the contact intercentre distance. This property allows its accurate expansion in a reduced basis of angular functions Qm”’ which are invariant under a global rotation of the molecules. Such a molecular potential is obtained as a superposition of isotropic site-site interactions using suitably parametrized error functions as described in ref. [8]. A test of the validity of UP, for correctly describing the geometry of the molecules is provided by the Barker-Henderson contact intercentre distance [7] which for a given relative orientation of the molecules is defined by d,,
2. The potential model
= I(1
- exp[-pU,,]}
dR .
(2)
0
The potential U(12) between two acetonitrile molecules can be written in the form V12) = Gv(12)
+ Q&12),
* Equipe Chimie de Coordination.
(1)
The repulsion U,,, has been chosen so that d,, correspond to the minimum distance of approach between the centres of the molecules approximated by nearly sphero-cylindrical rods which mimic the rigid hard-core shape designed by Hsu
0921-4526/89/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
B. Salamito and P.H. Fries I Liquid acetonitrile
162
and Chandler [9]. The rod diameter is chosen equal to the value 3.4 A of the Van der Waals diameter for the nitrile carbon and its length is 5.8 A.
scattering factors [l, 21 a”‘(q) and of FourierHankel transforms [lo] h”““‘(q) of the coefficifunction ents hmn’ (R) of the total correlation h(12) with respect to the basis of angular functions Qmn’. More precisely,
“I?,/ f,
3. Theory The integral equation theory used to calculate the pair distribution function g(12) consists of the Ornstein-Zernike equation [7, lo] coupled with a closure HNCAR which is of the HNC type at long distances and which Amplifies the Repulsive effects of the soft shape intermolecuso that it lar potential U,, at short distances efficiently prevents the overlap of the molecules, even for those orientations where the dipoledipole interaction is strongly attractive. Let h(12) = g(12) - 1 be the total pair correlation function which is the sum of a direct contribution ~(12) and of an indirect part ~(12). Denoting by w( 12) = -In g( 12) the dimensionless potential of mean force, the closure HNCAR is z ~(12)
= P&&2)
- 1 exp[-P
&&‘2’)1
R
x
&
[pU,,(1’2’)
- 77(1’2’)] dR’ , (3)
where R and R’ are distances between the centres of the molecules having the same relative orientation. The molecular polarizability is also taken into account with the help of the selfconsistent mean field theory introduced by Patey et al. [ll]. The solution g( 12) of the HNCAR theory is computed by iterative methods [8, lo]. In order to assess the value of the present potential model and of the HNCAR integral equation we should compare all possible experimental structural and thermodynamical data to their theoretical counterparts obtained from the pair distribution function. Here, we are mainly concerned with the intermolecular coherent differential cross section per molecule (dgl employed in the neutron and X-ray dan)inter diffraction studies. For a given length q of the scattering vector, this cross section can be expressed as a series of products of molecular
PC rnnl (2m + I)(2n X (y
+ I)
‘fi i] i”-“a’“(q)a”(q)h”““‘(q).
(4)
p is the molecular number density, the f’;l”” are normalization constants and the (z ;; :,) are the usual 3-j symbols. The factors a”(q) are determined by the location of the diffraction centres on a molecule and depend on q because of the comparable magnitude of the radiation wavelength and of the molecular size. 4. Structural
study
4.1. Comparison with diffraction experiments Neutron and X-ray scattering intensities have been measured for liquid acetonitrile by Bertagnolli et al. [l, 21. The experimental intermolecular scattering contributions are shown in fig. 1 and compared with our theoretical results obtained at 291 K and for a molar volume V= 52.18cm’ (ref. [3]). The curves exhibit oscillations with a good agreement of phase showing that the characteristic intermolecular distances in the model fluid are near to those of real acetonitrile. The theoretical amplitudes of the oscillations are somewhat too small, but this discrepancy can be partly explained by the uncertainty about the determination of the experimental intermolecular cross section which is obtained as the small difference of two large quantities
[31. A further test of the theory is provided by the study of the dielectric permittivity. Assuming a spherically symmetric polarizability [12] CY= 3.789 A3, the calculated dielectric constant increases from 25 for a non-polarizable model to 36 for a polarizable molecule carrying an effective dipole moment of 4.89 D. This result is in good agreement with the experimental value [13] E = 38.5.
B. Salamito and P.H. Fries I Liquid acetonitrile
400.
163
a 46.
a.1
4.5
4.2
3.0
Fig. 2. Geometry of the dominant pair arrangements the associated g( 12) values.
with
and antiparallel arrangements. An other preferred configuration corresponds to parallel molecules shifted so that the methyl group is brought near to the N atom. Other important arrangements are shown in fig. 2.
300.
5. Conclusion
b
I
1
2.
4. Q
I 6.
The main result of this work is that a liquid of simple dipolar short rods shows the principal structural features of acetonitrile through the molecular pair distribution function g( 12) calculated within an HNC type approximation. In particular, knowledge of g(12) for all positions and orientations of the molecules allows a direct and systematic search of the dominant molecular arrangements which help to clarify the -local structure.
G-7
Fig. 1. Comparison of experimental (dots) and theoretical (solid line) intermolecular coherent cross section (da/da) of diffraction studies for liquid acetonitrile. (a) The neutron cross section for CD,C14N. (b) The reduced X-ray cross section q(duldO),,,,,f,( q)-’ where f,(q) is an average scattering length per electron [2].
4.2. The pair distribution function The well-resolved structures of the intermolecular radial atom-atom distribution functions calculated from site-site integral equation (RISM) theories [6,9] can be combined to determine the preferred neighbouring configurations of two molecules. However, the procedure is not straightforward and can lead to ambiguous results. On the other hand, the dominant molecular arrangement and their relative importance are obtained by direct inspection of the molecular pair distribution function g( 12). In spite of the crudeness of our potential model the major structural features appearing in the simulations [3] have been found [14]. The first coordination sphere contains roughly ten to eleven molecules. The three closest are in side by side
References [l] (a) H. Bertagnolh, P. Chieux and M.D. Zeidler, Mol. Phys. 32 (1976) 759; (b) ibid. p. 1731. [2] H. Bertagnolli and M.D. Zeidler, Mol. Phys. 35 (1978) 177. (31 H.J. BBhm, I.R. McDonald and P. Madden, Mol. Phys. 49 (1983) 347. [4] D.M.F. Edwards, P.A. Madden and I.R. McDonald, Mol. Phys. 51 (1984) 1141. (51 W.L. Jorgensen and J.M. Briggs, Mol. Phys. 63 (1988) 547. [6] K.J. Fraser, L.A. Dunn and G.P. Morriss, Mol. Phys. 61 (1987) 775. [7] J.P. Hansen and I.R. McDonald, Theory of Simple Liquids (Academic Press, London, 1986). [8] P.H. Fries and B. Salamito, J. Phys. France 49 (1988) 1397. [9] C.S. Hsu and D. Chandler, Mol. Phys. 36 (1978) 215. [lo] P.H. Fries and G.N. Patey, J. Chem. Phys. 82 (1985) 429. [ll] J.M. Caillol, D. Levesque, J.J. Weis, J.S. Perkyns and G.N. Patey, Mol. Phys. 62 (1987) 1225. [12] M.T. Khimenko and N.N. Gritsenko, Russ. J. Phys. Chem. 54 (1980) 106. [13] N.A. Lange, Handbook of Chemistry (McGraw-Hill, New York, 1961) p. 1222. [14] B. Salamito and P.H. Fries, to be published.
STRUCTURE FACTOR CALCULATIONS FOR MOLECULAR LIQUIDS: BEYOND THE RISM APPROXIMATION B. SALAMITO CEN-Grenoble,
and P.H. FRIES
DRF-Laboratoires
de Chime’,
85 X, 38041 Grenoble
Cedex, France
A simple model of acetonitrile molecules approximated as dipolar rods is developed. The pair distribution function is calculated using an HNC type approximation and tested by comparison with diffraction data. The dominant molecular arrangements are presented.
1. Introduction Liquid acetonitrile (CH,CN) is a polar solvent which is widely used in chemistry. It has received much experimental and theoretical attention since it provides a good example of a molecule having a large dipole moment of 3.92 D in the gas phase, no important hydrogen bonding, and a strongly anisotropic shape, together with a high symmetry which simplifies the theoretical description. Detailed diffraction experiments aimed at the study of the intermolecular distribution of the molecules have been carried out and data from neutron scattering using different isotopes and from X-rays are available [l, 21. Realistic site-site potential models have also been devised from ab initio molecular orbital calculations [35] and have been used in simulations and sitesite integral equation theories [6] to represent the interactions between actual molecules. This work is concerned with the dominant features of the molecular pair distribution obtained from an integral equation of the hypernetted chain (HNC) type [7] using the simplest site-site intermolecular potential which incorporates the principle features of the shape and Coulomb interactions.
where the label i = 1,2 denotes the absolute position and orientation of the ith molecule. U,, is a short range anisotropic repulsion which corresponds to the intuitive molecular shape and U,, is the dipole-dipole long range interaction [8]. In order that U,, contains the essential features of the intermolecular repulsive forces it is sufficient that at the molecular contact, for a given relative orientation of the molecules, it increases to values of a few k,T when the intercentre distance R decreases by a few percent. The potential UP, is adapted to the geometry of collision between two molecules with its softness increasing with the contact intercentre distance. This property allows its accurate expansion in a reduced basis of angular functions Qm”’ which are invariant under a global rotation of the molecules. Such a molecular potential is obtained as a superposition of isotropic site-site interactions using suitably parametrized error functions as described in ref. [8]. A test of the validity of UP, for correctly describing the geometry of the molecules is provided by the Barker-Henderson contact intercentre distance [7] which for a given relative orientation of the molecules is defined by d,,
2. The potential model
= I(1
- exp[-pU,,]}
dR .
(2)
0
The potential U(12) between two acetonitrile molecules can be written in the form V12) = Gv(12)
+ Q&12),
* Equipe Chimie de Coordination.
(1)
The repulsion U,,, has been chosen so that d,, correspond to the minimum distance of approach between the centres of the molecules approximated by nearly sphero-cylindrical rods which mimic the rigid hard-core shape designed by Hsu
0921-4526/89/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
B. Salamito and P.H. Fries I Liquid acetonitrile
162
and Chandler [9]. The rod diameter is chosen equal to the value 3.4 A of the Van der Waals diameter for the nitrile carbon and its length is 5.8 A.
scattering factors [l, 21 a”‘(q) and of FourierHankel transforms [lo] h”““‘(q) of the coefficifunction ents hmn’ (R) of the total correlation h(12) with respect to the basis of angular functions Qmn’. More precisely,
“I?,/ f,
3. Theory The integral equation theory used to calculate the pair distribution function g(12) consists of the Ornstein-Zernike equation [7, lo] coupled with a closure HNCAR which is of the HNC type at long distances and which Amplifies the Repulsive effects of the soft shape intermolecuso that it lar potential U,, at short distances efficiently prevents the overlap of the molecules, even for those orientations where the dipoledipole interaction is strongly attractive. Let h(12) = g(12) - 1 be the total pair correlation function which is the sum of a direct contribution ~(12) and of an indirect part ~(12). Denoting by w( 12) = -In g( 12) the dimensionless potential of mean force, the closure HNCAR is z ~(12)
= P&&2)
- 1 exp[-P
&&‘2’)1
R
x
&
[pU,,(1’2’)
- 77(1’2’)] dR’ , (3)
where R and R’ are distances between the centres of the molecules having the same relative orientation. The molecular polarizability is also taken into account with the help of the selfconsistent mean field theory introduced by Patey et al. [ll]. The solution g( 12) of the HNCAR theory is computed by iterative methods [8, lo]. In order to assess the value of the present potential model and of the HNCAR integral equation we should compare all possible experimental structural and thermodynamical data to their theoretical counterparts obtained from the pair distribution function. Here, we are mainly concerned with the intermolecular coherent differential cross section per molecule (dgl employed in the neutron and X-ray dan)inter diffraction studies. For a given length q of the scattering vector, this cross section can be expressed as a series of products of molecular
PC rnnl (2m + I)(2n X (y
+ I)
‘fi i] i”-“a’“(q)a”(q)h”““‘(q).
(4)
p is the molecular number density, the f’;l”” are normalization constants and the (z ;; :,) are the usual 3-j symbols. The factors a”(q) are determined by the location of the diffraction centres on a molecule and depend on q because of the comparable magnitude of the radiation wavelength and of the molecular size. 4. Structural
study
4.1. Comparison with diffraction experiments Neutron and X-ray scattering intensities have been measured for liquid acetonitrile by Bertagnolli et al. [l, 21. The experimental intermolecular scattering contributions are shown in fig. 1 and compared with our theoretical results obtained at 291 K and for a molar volume V= 52.18cm’ (ref. [3]). The curves exhibit oscillations with a good agreement of phase showing that the characteristic intermolecular distances in the model fluid are near to those of real acetonitrile. The theoretical amplitudes of the oscillations are somewhat too small, but this discrepancy can be partly explained by the uncertainty about the determination of the experimental intermolecular cross section which is obtained as the small difference of two large quantities
[31. A further test of the theory is provided by the study of the dielectric permittivity. Assuming a spherically symmetric polarizability [12] CY= 3.789 A3, the calculated dielectric constant increases from 25 for a non-polarizable model to 36 for a polarizable molecule carrying an effective dipole moment of 4.89 D. This result is in good agreement with the experimental value [13] E = 38.5.
B. Salamito and P.H. Fries I Liquid acetonitrile
400.
163
a 46.
a.1
4.5
4.2
3.0
Fig. 2. Geometry of the dominant pair arrangements the associated g( 12) values.
with
and antiparallel arrangements. An other preferred configuration corresponds to parallel molecules shifted so that the methyl group is brought near to the N atom. Other important arrangements are shown in fig. 2.
300.
5. Conclusion
b
I
1
2.
4. Q
I 6.
The main result of this work is that a liquid of simple dipolar short rods shows the principal structural features of acetonitrile through the molecular pair distribution function g( 12) calculated within an HNC type approximation. In particular, knowledge of g(12) for all positions and orientations of the molecules allows a direct and systematic search of the dominant molecular arrangements which help to clarify the -local structure.
G-7
Fig. 1. Comparison of experimental (dots) and theoretical (solid line) intermolecular coherent cross section (da/da) of diffraction studies for liquid acetonitrile. (a) The neutron cross section for CD,C14N. (b) The reduced X-ray cross section q(duldO),,,,,f,( q)-’ where f,(q) is an average scattering length per electron [2].
4.2. The pair distribution function The well-resolved structures of the intermolecular radial atom-atom distribution functions calculated from site-site integral equation (RISM) theories [6,9] can be combined to determine the preferred neighbouring configurations of two molecules. However, the procedure is not straightforward and can lead to ambiguous results. On the other hand, the dominant molecular arrangement and their relative importance are obtained by direct inspection of the molecular pair distribution function g( 12). In spite of the crudeness of our potential model the major structural features appearing in the simulations [3] have been found [14]. The first coordination sphere contains roughly ten to eleven molecules. The three closest are in side by side
References [l] (a) H. Bertagnolh, P. Chieux and M.D. Zeidler, Mol. Phys. 32 (1976) 759; (b) ibid. p. 1731. [2] H. Bertagnolli and M.D. Zeidler, Mol. Phys. 35 (1978) 177. (31 H.J. BBhm, I.R. McDonald and P. Madden, Mol. Phys. 49 (1983) 347. [4] D.M.F. Edwards, P.A. Madden and I.R. McDonald, Mol. Phys. 51 (1984) 1141. (51 W.L. Jorgensen and J.M. Briggs, Mol. Phys. 63 (1988) 547. [6] K.J. Fraser, L.A. Dunn and G.P. Morriss, Mol. Phys. 61 (1987) 775. [7] J.P. Hansen and I.R. McDonald, Theory of Simple Liquids (Academic Press, London, 1986). [8] P.H. Fries and B. Salamito, J. Phys. France 49 (1988) 1397. [9] C.S. Hsu and D. Chandler, Mol. Phys. 36 (1978) 215. [lo] P.H. Fries and G.N. Patey, J. Chem. Phys. 82 (1985) 429. [ll] J.M. Caillol, D. Levesque, J.J. Weis, J.S. Perkyns and G.N. Patey, Mol. Phys. 62 (1987) 1225. [12] M.T. Khimenko and N.N. Gritsenko, Russ. J. Phys. Chem. 54 (1980) 106. [13] N.A. Lange, Handbook of Chemistry (McGraw-Hill, New York, 1961) p. 1222. [14] B. Salamito and P.H. Fries, to be published.