- Email: [email protected]
The electronic properties of water molecules in water clusters and liquid water
20 October 2000
Chemical Physics Letters 329 (2000) 283±288
www.elsevier.nl/locate/cplett
The electronic properties of water molecules in water clusters and liquid water Yaoquan Tu, Aatto Laaksonen * Physical Chemistry Division, Arrhenius Laboratory, Stockholm University, S-10691 Stockholm, Sweden Received 7 August 2000; received in ®nal form 4 September 2000
Abstract A novel, self-consistent approach, applicable both for ground and excited electronic states, is introduced to calculate molecular properties in clusters and liquids. Using the method, carried out here at the second-order Mùller±Plesset perturbation theory (MP2) level, we obtain an average dipole moment of 2.65 D for water in liquid. Signi®cant changes in quadrupole moment and polarizability, due to surrounding molecules, are also found along the water plane in the direction perpendicular to the axis bisecting the H±O±H bond angle. Ó 2000 Elsevier Science B.V.
1. Introduction Water molecules exist as distinctly structured entities in both gaseous state and condensed phases. Their electronic properties play a crucial role in determining the anomalous behavior of water and ice. An understanding of the properties of bulk liquid and ice requires a good quantitative description of a single water molecule in these systems. Coulombic interactions are dominant between the water molecules in the condensed phases, and the behavior of the water molecules is mainly determined by their electrostatic properties and responses to the microelectric ®elds of the surroundings. Therefore, studies focused on the molecular electronic properties such as the multipole moments and polarizabilities, carried out in condensed state conditions, should give valuable information about intermolecular interactions
*
Corresponding author. Fax: +46-8-152-187. E-mail address: [email protected] (A. Laaksonen).
0009-2614/00/$ - see front matter Ó 2000 Elsevier Science B.V. PII: S 0 0 0 9 - 2 6 1 4 ( 0 0 ) 0 1 0 2 6 - 5
when compared to the corresponding quantities for isolated molecules. The dipole moment of the water molecule has been among the most frequently studied properties during the last few years. In the gas phase, the experimental dipole moment of a water molecule is 1.855 D [1]. This value can be con®rmed by various high-level ab initio quantum chemistry calculations. In the condensed phases, the dipole moment will increase due to the polarization effects imposed by the surroundings. The studies of water clusters by Gregory et al. [2] show that the average dipole moment of a water molecule increases from 2.1 to 2.7 D as the size of the cluster increases from a dimer to a hexamer. In liquid water, it seems that the dipole moment of a single water molecule is more dicult to determine. The often cited `experimental' value (2.6 D) of Coulson and Eisenberg [3] is, in fact, a theoretical calculation on ice Ih. A recent study by Batista et al. on ice Ih by using an induction model suggests that the average dipole moment would reach a value as high as 3.09 D and that the smaller value of
284
Y. Tu, A. Laaksonen / Chemical Physics Letters 329 (2000) 283±288
Coulson and Eisenberg is due to the less accurate parameters used in the calculation [4]. In order to reproduce the experimental static dielectric constant 78:3, the water dipole moment is estimated to be about 2.6 D [5,6]. An average dipole moment exceeding 2.6 D leads to a signi®cant overestimation of [7]. Theoretical calculations based on varying models give very dierent dipole moments. The value obtained from the classical induction model is around 2.5±3.2 D, depending on the empirical parameters used [8±13]. Using the generalized self-consistent reaction ®eld approach with con®guration interaction calculation (CISD), Jansen et al. [14] report a value of 2.62 D. Recently, the average water dipole moment in liquid water has been calculated from ab initio Car± Parrinello molecular dynamics (CPMD) [15] simulations. However, the values obtained from CPMD depend very much on the way the electron density is partitioned. By integrating the electron density to the spherical cut-o around each atom in the water molecule, Laasonen et al. [16] obtain an average water dipole moment of 2.66 D. Based on the ¯ux of the electron density gradient to identify the electron distribution of individual molecules, a smaller value of 2.47 D is obtained [17]. More recently, by using the maximally localized Wannier functions to de®ne molecules, Silvestrelli and Parrinello [18] ®nd that the average dipole moment of a water molecule in the liquid should be about 3.0 D, a value much larger than those obtained from other calculations. The large uncertainty in the reported values of the dipole moment of liquid water re¯ects the fact that our basic understanding of the water molecule in condensed phases is still fairly limited. While the ultimate average water dipole moment in the liquid state remains as a challenge, the other molecular properties, such as geometry, quadrupole moment and polarizability, are also important measures behind the intermolecular interactions and have been the subject of recent studies [17,19±21]. Obviously, the polarizabilities cannot be calculated through the classical induction model, and one has to use quantum calculations to determine them. However, in view of the current uncertainty in determining the water dipole moment caused by the diculty in the parti-
tion of the charge density in ab initio molecular dynamics (MD) simulation wavefunctions, there is no reason to believe that the related quadrupole moments and polarizabilities would be easier to determine. 2. Methods In this Letter, we present a novel approach to calculate the molecular properties of single molecules in condensed states. It has been used here to determine the dipole moments, quadrupole moments and polarizabilities of a single water molecule in water clusters and liquid water. The basic idea of our approach is to carry out the quantum calculation only on one constituent molecule at a time, avoiding the prohibitively expensive quantum calculation of the whole system. The interactions between the molecules are simpli®ed to pure Coulombic interactions. According to our method, we can write the Schr odinger equation for a chosen molecule A as: ! X V^BÿA jWA i EA jWA i; 1 H^A B6A
where H^A is the Hamiltonian of the isolated molecule A. V^BÿA is the Coulomb interaction between molecules A and B. The interaction term can be represented as (in atomic units): X X /B ~ ri Za /B ~ Ra ; 2 V^BÿA ÿ i2A
a2A
where i and a denote the electrons and the nuclei, respectively. Za is the charge of the nucleus a. ri is the molecular electrostatic potential /B ~ (MEP) of molecule B at the position ~ ri : Z X qB 1 Zb ds1 ri ÿ : 3 /B ~ ~ r1 j j~ ri ÿ~ j~ r ÿ Rb j i b2B Each molecule in the system is calculated according to an equation similar to Eq. (1). The calculation of the whole system is carried out until selfconsistency is reached. Obviously, the approach used here accounts only for the major interactions
Y. Tu, A. Laaksonen / Chemical Physics Letters 329 (2000) 283±288
(the electrostatic and polarization interactions) between the molecules and neglects the other interactions. In view of the major interaction in liquid water being the Coulomb interaction, this is assumed to be a reasonable approximation. In a sense, the approach is a compromise between the classical induction model and quantum chemical super-molecule calculations. Compared to the classical induction model, the approach gives more reliable results because no parameters are needed. Also, it allows us to calculate some properties that are dicult or impossible to be treated with the classical induction model. At the same time, the diculties in partitioning the electron density in the quantum chemical super-molecular calculations can be avoided. Through the multipolar expansion of the charge density of a molecule according to the distributed multipole analysis (DMA) of the electrostatic potential by Stone and Alderton [22], the MEP of the molecule can be perfectly reproduced. It has been shown that, when the atomic sites and bond middle points are used as expansion centers, the multipolar expansion to third order (octupole) is accurate enough to be used in the calculations of the electrostatic potential (ESP) of a molecule [23]. We ®nd from our water dimer calculations that, for the calculation of the dipole and quadrupole moments, even the ESP charges derived by ®tting the MEP in some area according to the Merz±Singh±Kollman scheme [24,25] can reasonably be used to reproduce the MEP. The relative errors in the monomer dipole and quadrupole moments are less than 1% when the distributed multipoles are simply replaced by ESP charges in reproducing the electrostatic potentials. Therefore, in this Letter, each water molecule is calculated quantum mechanically in the presence of the other water molecules represented by ESP charges centered on atomic sites. The self-consistent calculations continue until the ESP charges on each water molecule converge to 10ÿ5 jej unit. The quantum calculations and the ESP ®ttings are carried out by GA U S S I A N 98 [26]. In order to take into account the electron correlation eects, the second-order Mùller±Plesset perturbation theory (MP2) is used throughout this Letter.
285
3. Results and discussion In the present approach, each molecule only `feels' the electric ®elds from the surroundings and not the exchange repulsion. It is likely that the results thus obtained depend strongly on the basis set since the exchange repulsion is missing. In order to study the basis set eects, two sets of basis set are chosen to calculate the monomer dipole moment in several small water clusters (from dimer to tetramer). One is Dunning's correlation consistent basis set aug-cc-pvdz [27] and the other is Sadlej's polarizability basis set [28]. In the calculation, the water clusters are ®xed in their MP2optimized geometries. In Fig. 1, we compare the average monomer dipole moments from the two basis sets with those obtained in the work of Gregory et al. [2]. For a gas state water molecule, the calculated dipole moments from both basis sets are in good agreement with that from the experiment. The relative errors are within 1%. In the clusters, the water dipole moment increases signi®cantly because of the intermolecular interactions. From the ®gure, we can see that, in water clusters, the monomer dipole moments from Sadlej's basis set are slightly larger than those from the aug-cc-pvdz basis set. It is clear that the monomer dipole moments from Sadlej's basis set are in better agreement with those from the work of Gregory et al. It seems that these dipole moments are also quite similar to those obtained by the classical induction model using accurate parameters [30]. As we pointed out before, our approach is in a sense similar to the classical induction model. Sadlej's basis set is larger than Dunning's aug-cc-pvdz basis set and thus gives a larger polarizability for a gas state water molecule. In fact, for the water molecule, the molecular multipole moments and polarizability calculated from Sadlej's basis set [28] are very close to those from the very large basis set (aug-cc-pvqz) [4]. Therefore, we believe that the results calculated here with Sadlej's basis set would be close to those with very large basis set, and in the following study, only the results from Sadlej's basis set are presented and discussed. Table 1 lists the calculated average monomer dipole moments, quadrupole moments and
286
Y. Tu, A. Laaksonen / Chemical Physics Letters 329 (2000) 283±288
Fig. 1. The monomer dipole moments (in Debye) calculated in this work at the MP2 level using Sadlej's polarizability basis set (as stars) and aug-cc-pvdz basis set (as diamonds) together with those from the DMA study by Gregory et al. [2] (as squares).
Table 1 Calculated average monomer dipole moments, quadrupole moments and polarizabilities in water clusters. All clusters are ®xed in their MP2 optimized geometries. The x- and z-direction lie on the plane of the water molecule with the origin on the oxygen and z-axis bisecting the H±O±H bond angle
a
No.a
lz (D)
l (D)
Qxx (DA)
Qyy (DA)
Qzz (DA)
1 2 3 4
1.88 2.07 2.26 2.47
1.88 2.08 2.28 2.50
)4.55 )4.45 )4.29 )4.23
)7.95 )7.98 )8.05 )8.05
)6.08 )6.03 )6.01 )5.94
axx (a.u.) 10.26 10.10 9.95 9.87
ayy (a.u.)
azz (a.u.)
9.62 9.62 9.76 9.62
9.91 9.83 9.82 9.76
The number of water molecules in the cluster.
polarizabilities in dierent water clusters using Sadlej's basis set. From the table we can see that, from gas state to cluster, the absolute value of the monomer quadrupole moments decreases signi®cantly in the plane of the molecule and along the direction perpendicular to the axis bisecting the HOH bond angle. For example, the quadrupole moment in the in the gas state direction changes from )4.55 DA in the tetramer. In the direction to )4.23 DA perpendicular to the molecular plane, it changes very little. Similar changes also happen to the water polarizabilities. From Table 1, we can see that the major change in polarizability is also
along the same direction and little change is found in the direction perpendicular to the molecular plane. Obviously, the changes in quadrupole moments and polarizabilities are dierent from those in dipole moments where the signi®cant change happens only in the direction along the axis bisecting the HOH bond angle. Because each water molecule in the clusters is almost symmetrical with respect to the axis bisecting the HOH bond angle, the dipole moments in the directions other than the axis cannot give us more information about the polarization of the molecule. However, the changes of quadrupole moments and polarizabilities can contribute to our understanding of the
Y. Tu, A. Laaksonen / Chemical Physics Letters 329 (2000) 283±288
polarization of the water molecule: the less negative values of the quadrupole moments in the molecular plane re¯ect a ¯ow of electrons from the two hydrogen atoms to the oxygen, and the direction perpendicular to the axis bisecting the HOH angle is more sensitive to the surroundings because of the signi®cant changes of the quadrupole moment and polarizability in the direction. In extending the calculations to the monomer properties in liquid water, the con®gurations from classical MD simulation are used. The potential model for water molecule used in the MD simulation is Jorgensen's TIP3P model [29]. In this model, the water molecule is ®xed in its experimental gas state equilibrium geometry RO±H and \ H±O±H 104:52°). 0:9572 A, The MD simulation has been performed in a canonical NVT ensemble. The number of water molecules was 256, the density 0:997 g=cm3 and the temperature 298.15 K. The time step was 1 fs. Twenty con®gurations were selected from the MD simulation at 1 ps intervals after an equilibration of 200 ps. In the calculation of the monomer properties in each con®guration, the water molecule which is calculated quantum mechanically is always in the center of the box with length of and the periodic boundary condition is 19:7 A, for the interused. The cut-o distance of 9.85 A molecular interactions between the quantum molecule and other molecules (represented as ESP charges on atomic sites) is also used. The monomer dipole moments, quadrupole moments, and polarizabilities averaged over 20 con®gurations (totally 5120 water molecules) are listed in Table 2. The average monomer dipole moment in liquid water from this Letter is 2.65 D. This value is in reasonable agreement with those estimated from the experimental static dielectric constants (2.6 D) using the classical induction model [5,6]. However, it is much smaller than that from Sil-
287
vestrelli and Parrinello. In our approach, only the Coulomb interaction between the water molecules is considered and other interactions, such as exchange, charge transfer and dispersion interactions, are neglected. Among the interactions neglected, the charge transfer interactions could enhance the dipole moment. However, in view of the monomer dipole moments in water clusters from this Letter being very close to the accurate ones of Gregory et al. [2], the accurate averaged dipole moment of a water molecule in liquid water would be also close to 2.65 D. In this Letter, the water molecule is ®xed in its gas state equilibrium geometry. In reality, it deforms in the condensed states. However, our previous quantum mechanical MD simulations at similar ambient conditions indicate that, although the ¯uctuations are significant, the average deformation from the equilibrium gas phase geometry is fairly small [21]. Therefore, we expect that the average dipole, calculated here with the gas phase equilibrium geometry, should be reasonably close to that when calculated with real ¯uctuating geometries. As found in the water clusters, the signi®cant changes in quadrupole moments and polarizabilities are found only in the plane of the water molecule and along the direction perpendicular to the axis bisecting the HOH bond angle. The water quadrupole moment in that direction changes in the gas state to )4.27 DA in from )4.53 DA liquid water and the polarizability from 10.05 to 9.39 a.u. This corresponds to a 6% change in quadrupole moment and 7% in polarizability, respectively. In the direction perpendicular to the plane of the water molecule, both the quadrupole moment and polarizability change very little. This again shows that the polarization of the water molecule is in the molecular plane, and the direction perpendicular to the axis bisecting the HOH angle is more sensitive to the surroundings.
Table 2 Calculated average monomer dipole moments, quadrupole moments and polarizabilities in liquid water. The x- and z-direction lie on the plane of the water molecule with the origin on the oxygen and z-axis bisecting the H±O±H bond angle lz (D)
l (D)
Qxx (DA)
Qyy (DA)
Qzz (DA)
axx (a.u.)
ayy (a.u.)
azz (a.u.)
2:64 0:17
2:65 0:17
ÿ4:27 0:07
ÿ7:99 0:07
ÿ5:94 0:06
9:39 0:17
9:52 0:32
9:51 0:20
288
Y. Tu, A. Laaksonen / Chemical Physics Letters 329 (2000) 283±288
Acknowledgements This work has been supported by the Swedish Council for Natural Sciences (NFR). The authors want to thank Dr. Lars Ojam ae for supplying the cluster con®gurations and Dr. Andrzej J. Sadlej for sending us his polarizability basis sets. References [1] F.J. Lovas, J. Phys. Chem. Ref. Data 7 (1978) 1445. [2] J.K. Gregory, D.C. Clary, K. Liu, M.G. Brown, R.J. Saykally, Science 275 (1997) 814. [3] C.A. Coulson, D. Eisenberg, Proc. Roy. Soc. London A 291 (1966) 445. [4] E.R. Batista, S.S. Xantheas, H. J onsson, J. Chem. Phys. 109 (1998) 4546. [5] S.L. Carnie, G.N. Patey, Mol. Phys. 47 (1982) 1129. [6] K. Watanabe, M.L. Klein, Chem. Phys. 131 (1989) 157. [7] M. Sprik, J. Chem. Phys. 95 (1991) 6762. [8] P. Barnes, J.L. Finney, J.D. Nicholas, J.E. Quinn, Nature (London) 282 (1979) 459. [9] M. Sprik, M.L. Klein, J. Chem. Phys. 89 (1988) 7556. [10] P. Ahlstr om, A. Wallqvist, S. Engstr om, B. J onsson, Mol. Phys. 68 (1989) 563. [11] J. Caldwell, L.X. Dang, P.A. Kollman, J. Am. Chem. Soc. 112 (1990) 9144. [12] U. Niesar, G. Corongiu, E. Clementi, G.R. Kneller, D.K. Bhattacharya, J. Phys. Chem. 94 (1990) 7949.
[13] J. Brodholt, M. Sampoli, R. Vallauri, Mol. Phys. 86 (1995) 149. [14] G. Jansen, F. Colonna, J.G. Angy an, Int. J. Quant. Chem. 58 (1996) 251. [15] R. Car, M. Parrinello, Phys. Rev. Lett. 55 (1985) 2471. [16] K. Laasonen, M. Sprik, M. Parrinello, R. Car, J. Chem. Phys. 99 (1993) 9080. [17] L.D. Site, A. Alavi, R.M. Lynden-Bell, Mol. Phys. 96 (1999) 1683. [18] P.L. Silvestrelli, M. Parrinello, Phys. Rev. Lett. 82 (1999) 3308. [19] N.W. Moriarty, G. Karlstr om, J. Phys. Chem. 100 (1996) 17791. [20] N.W. Moriarty, G. Karlstr om, J. Chem. Phys. 106 (1997) 6470. [21] F. Hedman, A. Laaksonen, Comput. Phys. Commun. 128 (2000) 284. [22] A.J. Stone, M. Alderton, Mol. Phys. 56 (1985) 1047. [23] P.N. Day, J.H. Jensen, M.S. Gordon, S.P. Webb, W.J. Stevens, M. Krauss, D. Garmer, H. Basch, D. Cohen, J. Chem. Phys. 105 (1996) 1968. [24] U.C. Singh, P.A. Kollman, J. Comput. Chem. 5 (1984) 129. [25] B.H. Besler, K.M. Merz Jr., P.A. Kollman, J. Comput. Chem. 11 (1990) 431. [26] M.J. Frisch et al., GA U S S I A N 98, Rev. A.3, Gaussian, Inc., Pittsburgh, PA, 1998. [27] R.A. Kendall, T.H. Dunning Jr., R.J. Harrison, J. Chem. Phys. 96 (1992) 6796. [28] A.J. Sadlej, Collect. Czech. Chem. Commun. 53 (1988) 1995. [29] W.L. Jorgensen, J. Chandrasekhar, J.D. Madura, R.W. Impey, M.L. Klein, J. Chem. Phys. 79 (1983) 926. [30] E.R. Batista, S.S. Xantheas, H. J onsson, J. Chem. Phys. 111 (1999) 6011.
Chemical Physics Letters 329 (2000) 283±288
www.elsevier.nl/locate/cplett
The electronic properties of water molecules in water clusters and liquid water Yaoquan Tu, Aatto Laaksonen * Physical Chemistry Division, Arrhenius Laboratory, Stockholm University, S-10691 Stockholm, Sweden Received 7 August 2000; received in ®nal form 4 September 2000
Abstract A novel, self-consistent approach, applicable both for ground and excited electronic states, is introduced to calculate molecular properties in clusters and liquids. Using the method, carried out here at the second-order Mùller±Plesset perturbation theory (MP2) level, we obtain an average dipole moment of 2.65 D for water in liquid. Signi®cant changes in quadrupole moment and polarizability, due to surrounding molecules, are also found along the water plane in the direction perpendicular to the axis bisecting the H±O±H bond angle. Ó 2000 Elsevier Science B.V.
1. Introduction Water molecules exist as distinctly structured entities in both gaseous state and condensed phases. Their electronic properties play a crucial role in determining the anomalous behavior of water and ice. An understanding of the properties of bulk liquid and ice requires a good quantitative description of a single water molecule in these systems. Coulombic interactions are dominant between the water molecules in the condensed phases, and the behavior of the water molecules is mainly determined by their electrostatic properties and responses to the microelectric ®elds of the surroundings. Therefore, studies focused on the molecular electronic properties such as the multipole moments and polarizabilities, carried out in condensed state conditions, should give valuable information about intermolecular interactions
*
Corresponding author. Fax: +46-8-152-187. E-mail address: [email protected] (A. Laaksonen).
0009-2614/00/$ - see front matter Ó 2000 Elsevier Science B.V. PII: S 0 0 0 9 - 2 6 1 4 ( 0 0 ) 0 1 0 2 6 - 5
when compared to the corresponding quantities for isolated molecules. The dipole moment of the water molecule has been among the most frequently studied properties during the last few years. In the gas phase, the experimental dipole moment of a water molecule is 1.855 D [1]. This value can be con®rmed by various high-level ab initio quantum chemistry calculations. In the condensed phases, the dipole moment will increase due to the polarization effects imposed by the surroundings. The studies of water clusters by Gregory et al. [2] show that the average dipole moment of a water molecule increases from 2.1 to 2.7 D as the size of the cluster increases from a dimer to a hexamer. In liquid water, it seems that the dipole moment of a single water molecule is more dicult to determine. The often cited `experimental' value (2.6 D) of Coulson and Eisenberg [3] is, in fact, a theoretical calculation on ice Ih. A recent study by Batista et al. on ice Ih by using an induction model suggests that the average dipole moment would reach a value as high as 3.09 D and that the smaller value of
284
Y. Tu, A. Laaksonen / Chemical Physics Letters 329 (2000) 283±288
Coulson and Eisenberg is due to the less accurate parameters used in the calculation [4]. In order to reproduce the experimental static dielectric constant 78:3, the water dipole moment is estimated to be about 2.6 D [5,6]. An average dipole moment exceeding 2.6 D leads to a signi®cant overestimation of [7]. Theoretical calculations based on varying models give very dierent dipole moments. The value obtained from the classical induction model is around 2.5±3.2 D, depending on the empirical parameters used [8±13]. Using the generalized self-consistent reaction ®eld approach with con®guration interaction calculation (CISD), Jansen et al. [14] report a value of 2.62 D. Recently, the average water dipole moment in liquid water has been calculated from ab initio Car± Parrinello molecular dynamics (CPMD) [15] simulations. However, the values obtained from CPMD depend very much on the way the electron density is partitioned. By integrating the electron density to the spherical cut-o around each atom in the water molecule, Laasonen et al. [16] obtain an average water dipole moment of 2.66 D. Based on the ¯ux of the electron density gradient to identify the electron distribution of individual molecules, a smaller value of 2.47 D is obtained [17]. More recently, by using the maximally localized Wannier functions to de®ne molecules, Silvestrelli and Parrinello [18] ®nd that the average dipole moment of a water molecule in the liquid should be about 3.0 D, a value much larger than those obtained from other calculations. The large uncertainty in the reported values of the dipole moment of liquid water re¯ects the fact that our basic understanding of the water molecule in condensed phases is still fairly limited. While the ultimate average water dipole moment in the liquid state remains as a challenge, the other molecular properties, such as geometry, quadrupole moment and polarizability, are also important measures behind the intermolecular interactions and have been the subject of recent studies [17,19±21]. Obviously, the polarizabilities cannot be calculated through the classical induction model, and one has to use quantum calculations to determine them. However, in view of the current uncertainty in determining the water dipole moment caused by the diculty in the parti-
tion of the charge density in ab initio molecular dynamics (MD) simulation wavefunctions, there is no reason to believe that the related quadrupole moments and polarizabilities would be easier to determine. 2. Methods In this Letter, we present a novel approach to calculate the molecular properties of single molecules in condensed states. It has been used here to determine the dipole moments, quadrupole moments and polarizabilities of a single water molecule in water clusters and liquid water. The basic idea of our approach is to carry out the quantum calculation only on one constituent molecule at a time, avoiding the prohibitively expensive quantum calculation of the whole system. The interactions between the molecules are simpli®ed to pure Coulombic interactions. According to our method, we can write the Schr odinger equation for a chosen molecule A as: ! X V^BÿA jWA i EA jWA i; 1 H^A B6A
where H^A is the Hamiltonian of the isolated molecule A. V^BÿA is the Coulomb interaction between molecules A and B. The interaction term can be represented as (in atomic units): X X /B ~ ri Za /B ~ Ra ; 2 V^BÿA ÿ i2A
a2A
where i and a denote the electrons and the nuclei, respectively. Za is the charge of the nucleus a. ri is the molecular electrostatic potential /B ~ (MEP) of molecule B at the position ~ ri : Z X qB 1 Zb ds1 ri ÿ : 3 /B ~ ~ r1 j j~ ri ÿ~ j~ r ÿ Rb j i b2B Each molecule in the system is calculated according to an equation similar to Eq. (1). The calculation of the whole system is carried out until selfconsistency is reached. Obviously, the approach used here accounts only for the major interactions
Y. Tu, A. Laaksonen / Chemical Physics Letters 329 (2000) 283±288
(the electrostatic and polarization interactions) between the molecules and neglects the other interactions. In view of the major interaction in liquid water being the Coulomb interaction, this is assumed to be a reasonable approximation. In a sense, the approach is a compromise between the classical induction model and quantum chemical super-molecule calculations. Compared to the classical induction model, the approach gives more reliable results because no parameters are needed. Also, it allows us to calculate some properties that are dicult or impossible to be treated with the classical induction model. At the same time, the diculties in partitioning the electron density in the quantum chemical super-molecular calculations can be avoided. Through the multipolar expansion of the charge density of a molecule according to the distributed multipole analysis (DMA) of the electrostatic potential by Stone and Alderton [22], the MEP of the molecule can be perfectly reproduced. It has been shown that, when the atomic sites and bond middle points are used as expansion centers, the multipolar expansion to third order (octupole) is accurate enough to be used in the calculations of the electrostatic potential (ESP) of a molecule [23]. We ®nd from our water dimer calculations that, for the calculation of the dipole and quadrupole moments, even the ESP charges derived by ®tting the MEP in some area according to the Merz±Singh±Kollman scheme [24,25] can reasonably be used to reproduce the MEP. The relative errors in the monomer dipole and quadrupole moments are less than 1% when the distributed multipoles are simply replaced by ESP charges in reproducing the electrostatic potentials. Therefore, in this Letter, each water molecule is calculated quantum mechanically in the presence of the other water molecules represented by ESP charges centered on atomic sites. The self-consistent calculations continue until the ESP charges on each water molecule converge to 10ÿ5 jej unit. The quantum calculations and the ESP ®ttings are carried out by GA U S S I A N 98 [26]. In order to take into account the electron correlation eects, the second-order Mùller±Plesset perturbation theory (MP2) is used throughout this Letter.
285
3. Results and discussion In the present approach, each molecule only `feels' the electric ®elds from the surroundings and not the exchange repulsion. It is likely that the results thus obtained depend strongly on the basis set since the exchange repulsion is missing. In order to study the basis set eects, two sets of basis set are chosen to calculate the monomer dipole moment in several small water clusters (from dimer to tetramer). One is Dunning's correlation consistent basis set aug-cc-pvdz [27] and the other is Sadlej's polarizability basis set [28]. In the calculation, the water clusters are ®xed in their MP2optimized geometries. In Fig. 1, we compare the average monomer dipole moments from the two basis sets with those obtained in the work of Gregory et al. [2]. For a gas state water molecule, the calculated dipole moments from both basis sets are in good agreement with that from the experiment. The relative errors are within 1%. In the clusters, the water dipole moment increases signi®cantly because of the intermolecular interactions. From the ®gure, we can see that, in water clusters, the monomer dipole moments from Sadlej's basis set are slightly larger than those from the aug-cc-pvdz basis set. It is clear that the monomer dipole moments from Sadlej's basis set are in better agreement with those from the work of Gregory et al. It seems that these dipole moments are also quite similar to those obtained by the classical induction model using accurate parameters [30]. As we pointed out before, our approach is in a sense similar to the classical induction model. Sadlej's basis set is larger than Dunning's aug-cc-pvdz basis set and thus gives a larger polarizability for a gas state water molecule. In fact, for the water molecule, the molecular multipole moments and polarizability calculated from Sadlej's basis set [28] are very close to those from the very large basis set (aug-cc-pvqz) [4]. Therefore, we believe that the results calculated here with Sadlej's basis set would be close to those with very large basis set, and in the following study, only the results from Sadlej's basis set are presented and discussed. Table 1 lists the calculated average monomer dipole moments, quadrupole moments and
286
Y. Tu, A. Laaksonen / Chemical Physics Letters 329 (2000) 283±288
Fig. 1. The monomer dipole moments (in Debye) calculated in this work at the MP2 level using Sadlej's polarizability basis set (as stars) and aug-cc-pvdz basis set (as diamonds) together with those from the DMA study by Gregory et al. [2] (as squares).
Table 1 Calculated average monomer dipole moments, quadrupole moments and polarizabilities in water clusters. All clusters are ®xed in their MP2 optimized geometries. The x- and z-direction lie on the plane of the water molecule with the origin on the oxygen and z-axis bisecting the H±O±H bond angle
a
No.a
lz (D)
l (D)
Qxx (DA)
Qyy (DA)
Qzz (DA)
1 2 3 4
1.88 2.07 2.26 2.47
1.88 2.08 2.28 2.50
)4.55 )4.45 )4.29 )4.23
)7.95 )7.98 )8.05 )8.05
)6.08 )6.03 )6.01 )5.94
axx (a.u.) 10.26 10.10 9.95 9.87
ayy (a.u.)
azz (a.u.)
9.62 9.62 9.76 9.62
9.91 9.83 9.82 9.76
The number of water molecules in the cluster.
polarizabilities in dierent water clusters using Sadlej's basis set. From the table we can see that, from gas state to cluster, the absolute value of the monomer quadrupole moments decreases signi®cantly in the plane of the molecule and along the direction perpendicular to the axis bisecting the HOH bond angle. For example, the quadrupole moment in the in the gas state direction changes from )4.55 DA in the tetramer. In the direction to )4.23 DA perpendicular to the molecular plane, it changes very little. Similar changes also happen to the water polarizabilities. From Table 1, we can see that the major change in polarizability is also
along the same direction and little change is found in the direction perpendicular to the molecular plane. Obviously, the changes in quadrupole moments and polarizabilities are dierent from those in dipole moments where the signi®cant change happens only in the direction along the axis bisecting the HOH bond angle. Because each water molecule in the clusters is almost symmetrical with respect to the axis bisecting the HOH bond angle, the dipole moments in the directions other than the axis cannot give us more information about the polarization of the molecule. However, the changes of quadrupole moments and polarizabilities can contribute to our understanding of the
Y. Tu, A. Laaksonen / Chemical Physics Letters 329 (2000) 283±288
polarization of the water molecule: the less negative values of the quadrupole moments in the molecular plane re¯ect a ¯ow of electrons from the two hydrogen atoms to the oxygen, and the direction perpendicular to the axis bisecting the HOH angle is more sensitive to the surroundings because of the signi®cant changes of the quadrupole moment and polarizability in the direction. In extending the calculations to the monomer properties in liquid water, the con®gurations from classical MD simulation are used. The potential model for water molecule used in the MD simulation is Jorgensen's TIP3P model [29]. In this model, the water molecule is ®xed in its experimental gas state equilibrium geometry RO±H and \ H±O±H 104:52°). 0:9572 A, The MD simulation has been performed in a canonical NVT ensemble. The number of water molecules was 256, the density 0:997 g=cm3 and the temperature 298.15 K. The time step was 1 fs. Twenty con®gurations were selected from the MD simulation at 1 ps intervals after an equilibration of 200 ps. In the calculation of the monomer properties in each con®guration, the water molecule which is calculated quantum mechanically is always in the center of the box with length of and the periodic boundary condition is 19:7 A, for the interused. The cut-o distance of 9.85 A molecular interactions between the quantum molecule and other molecules (represented as ESP charges on atomic sites) is also used. The monomer dipole moments, quadrupole moments, and polarizabilities averaged over 20 con®gurations (totally 5120 water molecules) are listed in Table 2. The average monomer dipole moment in liquid water from this Letter is 2.65 D. This value is in reasonable agreement with those estimated from the experimental static dielectric constants (2.6 D) using the classical induction model [5,6]. However, it is much smaller than that from Sil-
287
vestrelli and Parrinello. In our approach, only the Coulomb interaction between the water molecules is considered and other interactions, such as exchange, charge transfer and dispersion interactions, are neglected. Among the interactions neglected, the charge transfer interactions could enhance the dipole moment. However, in view of the monomer dipole moments in water clusters from this Letter being very close to the accurate ones of Gregory et al. [2], the accurate averaged dipole moment of a water molecule in liquid water would be also close to 2.65 D. In this Letter, the water molecule is ®xed in its gas state equilibrium geometry. In reality, it deforms in the condensed states. However, our previous quantum mechanical MD simulations at similar ambient conditions indicate that, although the ¯uctuations are significant, the average deformation from the equilibrium gas phase geometry is fairly small [21]. Therefore, we expect that the average dipole, calculated here with the gas phase equilibrium geometry, should be reasonably close to that when calculated with real ¯uctuating geometries. As found in the water clusters, the signi®cant changes in quadrupole moments and polarizabilities are found only in the plane of the water molecule and along the direction perpendicular to the axis bisecting the HOH bond angle. The water quadrupole moment in that direction changes in the gas state to )4.27 DA in from )4.53 DA liquid water and the polarizability from 10.05 to 9.39 a.u. This corresponds to a 6% change in quadrupole moment and 7% in polarizability, respectively. In the direction perpendicular to the plane of the water molecule, both the quadrupole moment and polarizability change very little. This again shows that the polarization of the water molecule is in the molecular plane, and the direction perpendicular to the axis bisecting the HOH angle is more sensitive to the surroundings.
Table 2 Calculated average monomer dipole moments, quadrupole moments and polarizabilities in liquid water. The x- and z-direction lie on the plane of the water molecule with the origin on the oxygen and z-axis bisecting the H±O±H bond angle lz (D)
l (D)
Qxx (DA)
Qyy (DA)
Qzz (DA)
axx (a.u.)
ayy (a.u.)
azz (a.u.)
2:64 0:17
2:65 0:17
ÿ4:27 0:07
ÿ7:99 0:07
ÿ5:94 0:06
9:39 0:17
9:52 0:32
9:51 0:20
288
Y. Tu, A. Laaksonen / Chemical Physics Letters 329 (2000) 283±288
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