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Total lattice potential energy of sodium bromide dihydrate NaBr · 2H2O
Solid State Communications, Vol. 51, No. 6, pp. 397-401, 1984. Printed in Great Britain.
0038-1098/84 $3.00 + .00 Pergamon Press Ltd.
TOTAL LATTICE POTENTIAL ENERGY OF SODIUM BROMIDE DIHYDRATE NaBr" 2H20 P. Herzig Institut fiir Physikalische Chemie, Universit~it Wien, W~hringerstrasse 42, A-1090 Wien, Austria and H.D.B. Jenkins and M.S.F. Pritchett Department of Chemistry, University of Warwick, Coventry CV4 7AL, Warwickshire, U.K.
(Received 6 April 1984 by E.F. Bertaut) In addition to presenting comparative calculations by two approaches for the total lattice potential energy of sodium bromide dihydrate, NaBr • 2H20, found to take the value 803.9 kJ mol -~, we investigate the influence of the size and nature of the basis set used to generate multipole moments in a Hartree-Fock calculation which are in turn used to calculate the Madelung constant. The requirement is one of critical size of the basis set and once this is reached the electrostatic energy will be reliable.
1. INTRODUCTION ALMOST 20 years have passed by since Baur [1 ] published his pioneering calculations concerning hydrogen bonding in crystalline hydrates. Although he obtained remarkably good results on the basis of the rather simple assumptions used to represent the charge distribution of the water molecules, not only have the methods available for studying the phenomenon of hydrogen bonding in crystalline solids [2] in general, and methods for calculating lattice energies [3-9] in particular, improved considerably, but also so has our knowledge of the structure of the hydrates themselves [ 10, 11 ]. In addition to presenting an improved and more sophisticated approach to the treatment of a specific hydrate, the present paper should serve two main purposes aside from obtaining the lattice energy for NaBr" 2H20. Firstly, a comparison is made between a point charge model and a model where the electrostatic interactions are described in terms of multipole moments as obtained from Hartree-Fock calculations. Secondly, the influences of the size and nature of the basis set on the electrostatic lattice energy calculated is also examined. Until now no reliable calculated value of the total lattice energy of monoclinic NaBr" 2H20 is to be found in the literature. Ladd and Lee [12] calculated the lattice energy from a Born-Haber cycle which is in fairly good agreement with the value obtained in this work. However, subsequent work by these authors [ 13] is hampered by incorrect structural data [14]. Some years ago, Herzig and Neckel [6] calculated the Madelung energy on the basis of a multipole expansion, but no consideration of the other contributions to the lattice energy was made
at that time. In this paper the total lattice energy of NaBr. 2H20 is computed, and in a subsequent paper [ 15] we shall consider the hydrates of calcium chloride.
2. METHODS OF CALCULATION For comparison both a point charge model and a multipole expansion approach have been employed to calculate the total lattice potential energy of the hydrate, NaBr • 2H20. The total lattice potential energy comprises the following terms u ~ t = U~aec- UR + Udd + Udq,
(i)
where Uelec, UR, Udd and Udq are the electrostatic, repulsion, dipole-dipole and dipole-quadrupole dispersion energy terms.
2.1. Point charge model In this case the water molecules are represented by placing charges qH on the hydrogen atom sites and charges o f - - 2qH on the oxygen atom sites. The electrostatic energy of the lattice is then parameterised as a function ofqH. The computations were performed using the program LATEN [5], which is based on the Bertaut technique [16] for calculating the electrostatic part of the lattice energy and on the Huggins-Mayer parametrization [17] for the respulsion energy. The basic radii of Na ÷ ( F N a + = 0.875 A) and Br- (rBr- = 1.628 A) were taken from [ 18], while the basic radius of oxygen, ?o, representing the water molecule in the repulsion energy calculation, and the charge on the hydrogen
397
398
TOTAL LATTICE POTENTIAL ENERGY OF NaBr" 2H20
Vol. 51, No. 6
Table 1. Components o f the multipole moments o f water (in a.u. ) up to 1 = 4 for 7 different basis sets (taken from Beyer [ 19]), calculated Madelung constants and charges on the hydrogen atoms
Basis set multipole moments
lm
1
2
10 20 22 30 32 40 42 44
7s3p/3s 9s5p2d/4slp 0.4612 0.3761 --0.0422 0.0565 0.7523 1.0449 --1.0271 --1.3799 1.3590 1.8890 --2.5251 --3.0649 2.2871 2.7841 1.5050 2.2139
3
4
5
6
7
11s7pld/6slp 0.4243 --0.0592 0.9405 --1.3215 1.5865 --2.8708 2.5823 1.9829
12s8pld/6slp 0.4266 --0.0649 0.9555 --1.3512 1.6697 --3.0401 2.7362 2.1032
lls7p2d/6s2p 0.3832 --0.0577 0.9600 --1.4190 1.8377 --3.0362 2.6437 2.1513
lls7p3d/6s2p 0.3809 --0.0469 0.9643 --1.4208 1.8514 --3.0532 2.7030 2.1546
lls7p2dlf/6s2p 0.3838 --0.0638 0.9501 --1.4208 1.8494 --3.0638 2.6773 2.1687
Ma
4.4272
4.3430
4.4334
4.4423
4.3281
4.3243
4.3244
qH
0.4431
0.4164
0.4451
0.4480
0.4118
0.4106
0.4106
atom, qH, are determined from the following equilibrium conditions [3]
}o =
/o =
0c Io
(2)
The subscript 0 indicates that the partial derivatives must be calculated for the equilibrium unit cell dimensions. These three equilibrium conditions enable the unknown basic radius of oxygen to be obtained as three different functions of qt4. The combination of these three functions with Upot as a function o f q n and P0 generates three different functional dependencies of Upot on qH. In principle, there should be a value of q n for which the three functions coincide, and in practice three closely adjacent intersection points usually result which, on averaging, give qt4 and hence the basic radius ~o and the lattice potential energy of the salt, U ~ t .
2.2. Multipole expansion approach In this approach the electrostatic energy is expanded in terms of the multipole moments of the water molecules which were obtained from a H a r t r e e - F o c k calculation [19]. In Section 3 the influence of various basis sets on the electrostatic energy is investigated. As was the case for the point charge model, the repulsion energy is derived from the Huggins-Mayer scheme. In this case, however, (and consistent with the method o f treatment of the electrostatic energy), the water molecule is considered as a whole. In the initial calculations apoced radius was assumed for the Br- ion and variable radii were assigned to the water molecule (W) depending on the interaction type W-W, W - N a ÷ or W - B r - . It was found that the radii required in the case of W - W interaction equalled that for W - N a interaction. This arose presumably because of the existence of hydrogen bonds between the bromide ions and the
water molecules and their absence between Na + ions and the water molecules in the crystal. Accordingly two independent basic radii have been assigned to the water molecules in subsequent calculations, depending on the presence ( W - B r - ) or absence (W-W), (W-Na*) of hydrogen bonds between the respective species. These two unknown basic radii, together with the basic radius of the anion Br-, were calculated from the three abovementioned equilibrium conditions. The partial derivatives of Uelee with respect to the lattice parameters were evaluated by a method recently [8] developed. The dispersion energy terms in both approaches nave been calculated by standard methods [4]. 3. MULTIPOLE MOMENTS AND ELECTROSTATIC ENERGY The multipole moments of the gaseous water molecule, calculated by Beyer [ 19] from L C A O - M O - S C F wave functions, have been used to investigate the influence of the basis set on the electrostatic energy calculation for NaBr" 2H20. This approach is justified, since for this particular compound the geometry of the water molecule in the crystal, as determined by Tegenfeldt et al. [20], is very similar to the geometry of the gaseous species. Table 1 displays, for seven different Gaussian basis sets, the multipole moments, the calculated Madelung constants and the charges on the hydrogen atoms that have to be used in a point charge model in order to reproduce the same result as is obtained from the multipole expansion. For the calculation of the multipole moments it was assumed that the water molecules lie in the XZ-plane of the co-ordinate system such that the hydrogen atoms are symmetrical with respect to the Z-axis and the origin is in the centre of mass. The results can be interpreted to indicate that once a critical size of the basis is reached, the Madelung
Vol. 51, No. 6
TOTAL LATTICE POTENTIAL ENERGY OF NaBr. 2H20
Table 2. Results for lattice energy calculations for NaBr- 2H2Ofor the process: NaBr. 2H20(c) -+ Na+(g) + Br-(g) + 2H20(g) using the point charge model Electrostatic energy, Uelee , dispersion energies, Udd , U, d and their derivatives given in the form £~=o Ai qia. ffnergies in kJ mo1-1, lengths in A -x
Uelee (~Ueaecfi)a)o (au~a~/ab)o (aUeae~/aC)o Udd OUad/aa)o (~Udd/ab)o
Ao
A1
606.4 -- 66.9 - 7.1 -- 13.6
823.58 8.68 -91.0 -- 133.5
--~--182.8 12.5 25.8 -
1200
--
I O O C
--
399
800
600
-
I a "-I- . 0
40C
0 I
02
05
04
0.5
06
0.7
0.8
0.9
I
qH
60.3 -- 19.0
Fig. 1.
-- 1 1.4
(aUdd/aC)o
--
Udq (OUdq/aa)o
P21/c) containing four molecules (a = 6.575 A, b =
18.0
5.6 -- 2.4 -1.4 -- 2.2
(~)Udqfi}b)o (aUdq/aC)o
Table 3. Results for lattice energy calculations for NaBr- 2H2Ofor the process: NaBr" 2H20(c) ~ Na+(g) + Br-(g) + 2H20(g) using a point charge model, energies in kJ mo1-1, lengths in ~ Total lattice potential energy, Upot; repulsion energy, UR ; charge on hydrogen atom,
qn and "basic" radius of oxygen atoms, ~o corresponding to intersection of energy minima in Fig. 1 kJ mol -I
10.456 A, c = 6.776 A,/3 = 113.38 °) [20]. The polarisabilities of Na +, Br- and water are taken [21,22] to be 0.179A 3, 4 . 7 7 A 3 and 1.429 A3 and the characteristic energies for Na + and Br- are 82.6 and 15.0 x 10 -9 J molecule -1 . For water the dispersion energy constant 6"6 from [22] corresponds to a characteristic energy of 28.37 x 10 -9 J molecule -x , a value which has been used to calculate the dispersion energy constants for the water-Na + and w a t e r - B r - interactions. It should be noticed that the partial derivatives of the respulsion energy, which are not included in Table 2, can easily be calculated from the equilibrium conditions:
A
Intersection curves
qH
Uelee
(JR
Upot
r0
(8URI \ ai ]o
a and b a and c b and c
0.44 0.43 0.44
934.0 926.2 934.0
195.8 189.0 191.6
804.1 803.1 808.3
1.22 1.20 1.22
where i = a, b, c. Figure 1 shows the plot of the lattice potential energy, Upot for the process:
Average
0.44
931.4
192.1
805.1
1.21
NaBr. 2H20(c) -~ Na+(g) + Br-(g) + 2H20(g),
parameter, or the electrostatic energy, is rather insensitive to the basis actually used, while the individual components of the multipole moment tensor are much more basis-dependent. In the present case at least two d polarisation functions on the oxygen atom and, to a lesser extent, two p functions on the hydrogen atoms, seem to be required. A third d function or an additional f function, however, leave the electrostatic interaction energy almost unaltered. As was the case in [6] only basis set 6 has been used for the calculations of the total lattice energy. 4. RESULTS Tables 2 and 3 give the results obtained for sodium bromide dihydrate on the basis of a point charge model. NaBr" 2H20 has a monoclinic unit cell (space group
=
(i} Ueaee--i \ - - ~ i ]o
(C3Udat + (~q') + \ ai }o
(3) o'
as a function of the charge qu. The intersections lead us to assign: Upot(NaBr" 2H20) = 805.1 kJ mo1-1 qH = 0.44 proton units ?o = 1.21A.
(4) (5) (6)
The total lattice potential energy is made up of the following contributions: 931.4 kJ mo1-1 (U~aee), 192.1 kJ mo1-1 (/JR) and 65.9 kJ mol -I (UD). The value ofqH is close to that (= 0.50) employed by Baur [1 ] in his classic calculation in 1965. Table 4 shows the corresponding results obtained from the multipole expansion approach. The dispersion energy terms are taken to be the same in both calculations. In this case minimization yields the following basic radii:
TOTAL LATTICE POTENTIAL ENERGY OF NaBr" 2H20
400
Vol. 51, No. 6
Taking
Table 4. Electrostatic energy terms of NaBr" 2H20 calculated from the multipole moments. Energies in kJ tool -1, cell lengths inA
Lk/-/~(Na+)(g) = 609.0 kJ mo1-1 [23] z2x//?(Br-)(g) = -- 233.9 kJ tool -~ [23] z_~/?(NaBr- 2H20)(c) = -- 951.9 kJ mo1-1 [23]
Uelec (~Ueaee/~a)o (OUeiec/Ob)o
913.72 -- 70.77 -- 38.13 - - 59.87
(~Uelec/~C)o
2xH?(H20)(I) = -- 241.8 kJ tool -1 [23], we generate Upot(NaBr" 2H20) = 841.0 kJ mo1-1.
Table 5. Lattice energy o f NaBr" 2HzO (kJ mol -~ ). Comparison o f the point charge model and the multipole expansion approach
Uelee
UR Udd
Udq Upot
Point charge
Multipole expansion
931.4 192.1 60.4 5.6 805.1
913.7 177.0 60.4 5.6 802.7
rBr- = 1.705A,
PH2o = 0.743A,
(7) where rH2o is the basic radius of water in the direction of the hydrogen bond. The rather low value ofrn2o seems to be a striking result at first sight. However, it is a feature of the calculation that ?a= o is strongly dependent on the parametrisation of the dispersion energy which cannot be evaluated to a very high degree of accuracy. The repulsive energy itself is not influenced markedly by these variations of ~n=o. Care is therefore required in a physical interpretation of these basic radii. Table 5 compares the different contributions to the lattice energy of NaBr- 2H20 evaluated using the two approaches.
The Born-Haber cycle can be used to estimate the lattice energy of sodium bromide dihydrate:
UP°t+RT ,
Acknowledgement - We are grateful to Professor A. Neckel for many helpful discussions. One of us (PH) wishes to acknowledge grants from the Royal Society under an exchange agreement with the Austrian Academy of Sciences in order to carry out work at the University of Warwick. Mrs Merlin Callaway is thanked for typing the manuscript. REFERENCES 1. 2. 3. 4.
6.
_~ Na(c) + ~Br2(1) + 2H2(g) + O2(g)
whereupon:
7. 8. 9. 10.
Upot(NaBr.2H20) = AH?(Na+)(g) + AH~(Br-)(g) + 2AH~(H20)(/) -
W.H. Baur, Acta Cryst. 19,909 (1965). P. Schuster, G. Zundel & C. Sandorfy (eds.), The Hydrogen Bond, Vols. 1-3, North Holland, Amsterdam (1976). H.D.B. Jenkins & K.F. Pratt, Proc. Roy. Soc. A356, 115 (1977). H.D.B. Jenkins & K.F. Pratt, J. Chem. Soc. Faraday H 74,968 (1978). H.D.B. Jenkins & K.F. Pratt, Comp. Phys. Commun. 13,341 (1978). P. Herzig&A. Neckel, J. Chem. Phys. 71,2131 (1979).
Na+(g) + Br-(g) + 2H20(g)
AH?(NaBr" 2H20)(c)
-
AH~(NaBr-2H20)(c) + RT.
(10)
the electrostatic component of which is also consistent with previous studies [6] made by one of us.
5.
5. DISCUSSION
NaBr" 2H20(c)
This value is, predictively, higher than the value obtained by adopting an ionic model that does not take into account explicitly an energy contribution due to polarisation effects. The difference between the experimental and the theoretical result represents a measure of the covalent element of the lattice bonding. Comparison of our two calculational methods (Table 5) exhibits entirely consistent results which average to generate: Upot(NaBr" 2H20) = 803.9 + 1.2 kJ mo1-1 ,
~ 2 o = 1.346A,
(9)
(8)
11.
AH?(Na+)(g) + AH?(Br-)(g) + 2AH~(HzO)(g)
P. Herzig, J. Phys. Chem. Solids 43,449 (1982). P. Herzig, Solid State Commun. 46,685 (1983). M. Catti, Acta Cryst. A37, 72 (1981). G. Chiari & G. Ferraris, Acta Cryst. B38,2331 (1982). "HBSIO0 Hydrogen Bond Project - Survey 1975
Vol. 51, No. 6
12. 13. 14. 15. 16. 17. 18.
TOTAL LATTICE POTENTIAL ENERGY OF NaBr- 2H20
- UUIC-B19-120" Institute of Chemistry, University of Uppsala, Sweden (1975). M.F.C.Ladd & W.H. Lee, J. Phys. Chem. 69, 1840 (1965). M.F.C.Ladd & W.H. Lee, J. Phys. Chem. 73,2033 (1969). M.F.C.Ladd, Z. Kristallogr. 126, 147 (1968). H.D.B.Jenkins, E. Langadianou, M. Patel & P. Herzig (to be published) (1984). E.F. Bertaut, J. Phys. Radium 13,499 (1952). M.L.Huggins&J.E. Mayer, J. Chem. Phys. 1,643 (1933). H.D.B.Jenkins & K.F. Pratt, Repulsion Energy and Basic Radii for Ions having Spherical Symmetry, unpublished report, University of Warwick (1976).
19.
20. 21. 22. 23.
401
A. Beyer, Doctoral dissertation, University of Vienna (1976); P. Schuster, H. Lischka & A. Beyer, in Progress in Theoretical Organic Chemistry, Vol. 2, p. 89 (Edited by I.G. Czismadia), Elsevier, Amsterdam (1977). J. Tegenfeldt, R. Tellgren, B. Pedersen & I. Olovsson, Acta Cryst. B35, 1679 (1979). K.F. Pratt, Doctoral Thesis, University of Warwick (1978). G.D. Zeiss & W.J. Meath, Mol. Phys. 33,1155 (1977). D.D.Wagman, W.H. Evans, V.B. Parker, R.H. Schumm & R.L. Nuttall, Selected Values of Chemical Thermodynamic Properties, Technical Note 270-8, U.S. Dept. of Commerce, U.S.A., May (1981).
0038-1098/84 $3.00 + .00 Pergamon Press Ltd.
TOTAL LATTICE POTENTIAL ENERGY OF SODIUM BROMIDE DIHYDRATE NaBr" 2H20 P. Herzig Institut fiir Physikalische Chemie, Universit~it Wien, W~hringerstrasse 42, A-1090 Wien, Austria and H.D.B. Jenkins and M.S.F. Pritchett Department of Chemistry, University of Warwick, Coventry CV4 7AL, Warwickshire, U.K.
(Received 6 April 1984 by E.F. Bertaut) In addition to presenting comparative calculations by two approaches for the total lattice potential energy of sodium bromide dihydrate, NaBr • 2H20, found to take the value 803.9 kJ mol -~, we investigate the influence of the size and nature of the basis set used to generate multipole moments in a Hartree-Fock calculation which are in turn used to calculate the Madelung constant. The requirement is one of critical size of the basis set and once this is reached the electrostatic energy will be reliable.
1. INTRODUCTION ALMOST 20 years have passed by since Baur [1 ] published his pioneering calculations concerning hydrogen bonding in crystalline hydrates. Although he obtained remarkably good results on the basis of the rather simple assumptions used to represent the charge distribution of the water molecules, not only have the methods available for studying the phenomenon of hydrogen bonding in crystalline solids [2] in general, and methods for calculating lattice energies [3-9] in particular, improved considerably, but also so has our knowledge of the structure of the hydrates themselves [ 10, 11 ]. In addition to presenting an improved and more sophisticated approach to the treatment of a specific hydrate, the present paper should serve two main purposes aside from obtaining the lattice energy for NaBr" 2H20. Firstly, a comparison is made between a point charge model and a model where the electrostatic interactions are described in terms of multipole moments as obtained from Hartree-Fock calculations. Secondly, the influences of the size and nature of the basis set on the electrostatic lattice energy calculated is also examined. Until now no reliable calculated value of the total lattice energy of monoclinic NaBr" 2H20 is to be found in the literature. Ladd and Lee [12] calculated the lattice energy from a Born-Haber cycle which is in fairly good agreement with the value obtained in this work. However, subsequent work by these authors [ 13] is hampered by incorrect structural data [14]. Some years ago, Herzig and Neckel [6] calculated the Madelung energy on the basis of a multipole expansion, but no consideration of the other contributions to the lattice energy was made
at that time. In this paper the total lattice energy of NaBr. 2H20 is computed, and in a subsequent paper [ 15] we shall consider the hydrates of calcium chloride.
2. METHODS OF CALCULATION For comparison both a point charge model and a multipole expansion approach have been employed to calculate the total lattice potential energy of the hydrate, NaBr • 2H20. The total lattice potential energy comprises the following terms u ~ t = U~aec- UR + Udd + Udq,
(i)
where Uelec, UR, Udd and Udq are the electrostatic, repulsion, dipole-dipole and dipole-quadrupole dispersion energy terms.
2.1. Point charge model In this case the water molecules are represented by placing charges qH on the hydrogen atom sites and charges o f - - 2qH on the oxygen atom sites. The electrostatic energy of the lattice is then parameterised as a function ofqH. The computations were performed using the program LATEN [5], which is based on the Bertaut technique [16] for calculating the electrostatic part of the lattice energy and on the Huggins-Mayer parametrization [17] for the respulsion energy. The basic radii of Na ÷ ( F N a + = 0.875 A) and Br- (rBr- = 1.628 A) were taken from [ 18], while the basic radius of oxygen, ?o, representing the water molecule in the repulsion energy calculation, and the charge on the hydrogen
397
398
TOTAL LATTICE POTENTIAL ENERGY OF NaBr" 2H20
Vol. 51, No. 6
Table 1. Components o f the multipole moments o f water (in a.u. ) up to 1 = 4 for 7 different basis sets (taken from Beyer [ 19]), calculated Madelung constants and charges on the hydrogen atoms
Basis set multipole moments
lm
1
2
10 20 22 30 32 40 42 44
7s3p/3s 9s5p2d/4slp 0.4612 0.3761 --0.0422 0.0565 0.7523 1.0449 --1.0271 --1.3799 1.3590 1.8890 --2.5251 --3.0649 2.2871 2.7841 1.5050 2.2139
3
4
5
6
7
11s7pld/6slp 0.4243 --0.0592 0.9405 --1.3215 1.5865 --2.8708 2.5823 1.9829
12s8pld/6slp 0.4266 --0.0649 0.9555 --1.3512 1.6697 --3.0401 2.7362 2.1032
lls7p2d/6s2p 0.3832 --0.0577 0.9600 --1.4190 1.8377 --3.0362 2.6437 2.1513
lls7p3d/6s2p 0.3809 --0.0469 0.9643 --1.4208 1.8514 --3.0532 2.7030 2.1546
lls7p2dlf/6s2p 0.3838 --0.0638 0.9501 --1.4208 1.8494 --3.0638 2.6773 2.1687
Ma
4.4272
4.3430
4.4334
4.4423
4.3281
4.3243
4.3244
qH
0.4431
0.4164
0.4451
0.4480
0.4118
0.4106
0.4106
atom, qH, are determined from the following equilibrium conditions [3]
}o =
/o =
0c Io
(2)
The subscript 0 indicates that the partial derivatives must be calculated for the equilibrium unit cell dimensions. These three equilibrium conditions enable the unknown basic radius of oxygen to be obtained as three different functions of qt4. The combination of these three functions with Upot as a function o f q n and P0 generates three different functional dependencies of Upot on qH. In principle, there should be a value of q n for which the three functions coincide, and in practice three closely adjacent intersection points usually result which, on averaging, give qt4 and hence the basic radius ~o and the lattice potential energy of the salt, U ~ t .
2.2. Multipole expansion approach In this approach the electrostatic energy is expanded in terms of the multipole moments of the water molecules which were obtained from a H a r t r e e - F o c k calculation [19]. In Section 3 the influence of various basis sets on the electrostatic energy is investigated. As was the case for the point charge model, the repulsion energy is derived from the Huggins-Mayer scheme. In this case, however, (and consistent with the method o f treatment of the electrostatic energy), the water molecule is considered as a whole. In the initial calculations apoced radius was assumed for the Br- ion and variable radii were assigned to the water molecule (W) depending on the interaction type W-W, W - N a ÷ or W - B r - . It was found that the radii required in the case of W - W interaction equalled that for W - N a interaction. This arose presumably because of the existence of hydrogen bonds between the bromide ions and the
water molecules and their absence between Na + ions and the water molecules in the crystal. Accordingly two independent basic radii have been assigned to the water molecules in subsequent calculations, depending on the presence ( W - B r - ) or absence (W-W), (W-Na*) of hydrogen bonds between the respective species. These two unknown basic radii, together with the basic radius of the anion Br-, were calculated from the three abovementioned equilibrium conditions. The partial derivatives of Uelee with respect to the lattice parameters were evaluated by a method recently [8] developed. The dispersion energy terms in both approaches nave been calculated by standard methods [4]. 3. MULTIPOLE MOMENTS AND ELECTROSTATIC ENERGY The multipole moments of the gaseous water molecule, calculated by Beyer [ 19] from L C A O - M O - S C F wave functions, have been used to investigate the influence of the basis set on the electrostatic energy calculation for NaBr" 2H20. This approach is justified, since for this particular compound the geometry of the water molecule in the crystal, as determined by Tegenfeldt et al. [20], is very similar to the geometry of the gaseous species. Table 1 displays, for seven different Gaussian basis sets, the multipole moments, the calculated Madelung constants and the charges on the hydrogen atoms that have to be used in a point charge model in order to reproduce the same result as is obtained from the multipole expansion. For the calculation of the multipole moments it was assumed that the water molecules lie in the XZ-plane of the co-ordinate system such that the hydrogen atoms are symmetrical with respect to the Z-axis and the origin is in the centre of mass. The results can be interpreted to indicate that once a critical size of the basis is reached, the Madelung
Vol. 51, No. 6
TOTAL LATTICE POTENTIAL ENERGY OF NaBr. 2H20
Table 2. Results for lattice energy calculations for NaBr- 2H2Ofor the process: NaBr. 2H20(c) -+ Na+(g) + Br-(g) + 2H20(g) using the point charge model Electrostatic energy, Uelee , dispersion energies, Udd , U, d and their derivatives given in the form £~=o Ai qia. ffnergies in kJ mo1-1, lengths in A -x
Uelee (~Ueaecfi)a)o (au~a~/ab)o (aUeae~/aC)o Udd OUad/aa)o (~Udd/ab)o
Ao
A1
606.4 -- 66.9 - 7.1 -- 13.6
823.58 8.68 -91.0 -- 133.5
--~--182.8 12.5 25.8 -
1200
--
I O O C
--
399
800
600
-
I a "-I- . 0
40C
0 I
02
05
04
0.5
06
0.7
0.8
0.9
I
qH
60.3 -- 19.0
Fig. 1.
-- 1 1.4
(aUdd/aC)o
--
Udq (OUdq/aa)o
P21/c) containing four molecules (a = 6.575 A, b =
18.0
5.6 -- 2.4 -1.4 -- 2.2
(~)Udqfi}b)o (aUdq/aC)o
Table 3. Results for lattice energy calculations for NaBr- 2H2Ofor the process: NaBr" 2H20(c) ~ Na+(g) + Br-(g) + 2H20(g) using a point charge model, energies in kJ mo1-1, lengths in ~ Total lattice potential energy, Upot; repulsion energy, UR ; charge on hydrogen atom,
qn and "basic" radius of oxygen atoms, ~o corresponding to intersection of energy minima in Fig. 1 kJ mol -I
10.456 A, c = 6.776 A,/3 = 113.38 °) [20]. The polarisabilities of Na +, Br- and water are taken [21,22] to be 0.179A 3, 4 . 7 7 A 3 and 1.429 A3 and the characteristic energies for Na + and Br- are 82.6 and 15.0 x 10 -9 J molecule -1 . For water the dispersion energy constant 6"6 from [22] corresponds to a characteristic energy of 28.37 x 10 -9 J molecule -x , a value which has been used to calculate the dispersion energy constants for the water-Na + and w a t e r - B r - interactions. It should be noticed that the partial derivatives of the respulsion energy, which are not included in Table 2, can easily be calculated from the equilibrium conditions:
A
Intersection curves
qH
Uelee
(JR
Upot
r0
(8URI \ ai ]o
a and b a and c b and c
0.44 0.43 0.44
934.0 926.2 934.0
195.8 189.0 191.6
804.1 803.1 808.3
1.22 1.20 1.22
where i = a, b, c. Figure 1 shows the plot of the lattice potential energy, Upot for the process:
Average
0.44
931.4
192.1
805.1
1.21
NaBr. 2H20(c) -~ Na+(g) + Br-(g) + 2H20(g),
parameter, or the electrostatic energy, is rather insensitive to the basis actually used, while the individual components of the multipole moment tensor are much more basis-dependent. In the present case at least two d polarisation functions on the oxygen atom and, to a lesser extent, two p functions on the hydrogen atoms, seem to be required. A third d function or an additional f function, however, leave the electrostatic interaction energy almost unaltered. As was the case in [6] only basis set 6 has been used for the calculations of the total lattice energy. 4. RESULTS Tables 2 and 3 give the results obtained for sodium bromide dihydrate on the basis of a point charge model. NaBr" 2H20 has a monoclinic unit cell (space group
=
(i} Ueaee--i \ - - ~ i ]o
(C3Udat + (~q') + \ ai }o
(3) o'
as a function of the charge qu. The intersections lead us to assign: Upot(NaBr" 2H20) = 805.1 kJ mo1-1 qH = 0.44 proton units ?o = 1.21A.
(4) (5) (6)
The total lattice potential energy is made up of the following contributions: 931.4 kJ mo1-1 (U~aee), 192.1 kJ mo1-1 (/JR) and 65.9 kJ mol -I (UD). The value ofqH is close to that (= 0.50) employed by Baur [1 ] in his classic calculation in 1965. Table 4 shows the corresponding results obtained from the multipole expansion approach. The dispersion energy terms are taken to be the same in both calculations. In this case minimization yields the following basic radii:
TOTAL LATTICE POTENTIAL ENERGY OF NaBr" 2H20
400
Vol. 51, No. 6
Taking
Table 4. Electrostatic energy terms of NaBr" 2H20 calculated from the multipole moments. Energies in kJ tool -1, cell lengths inA
Lk/-/~(Na+)(g) = 609.0 kJ mo1-1 [23] z2x//?(Br-)(g) = -- 233.9 kJ tool -~ [23] z_~/?(NaBr- 2H20)(c) = -- 951.9 kJ mo1-1 [23]
Uelec (~Ueaee/~a)o (OUeiec/Ob)o
913.72 -- 70.77 -- 38.13 - - 59.87
(~Uelec/~C)o
2xH?(H20)(I) = -- 241.8 kJ tool -1 [23], we generate Upot(NaBr" 2H20) = 841.0 kJ mo1-1.
Table 5. Lattice energy o f NaBr" 2HzO (kJ mol -~ ). Comparison o f the point charge model and the multipole expansion approach
Uelee
UR Udd
Udq Upot
Point charge
Multipole expansion
931.4 192.1 60.4 5.6 805.1
913.7 177.0 60.4 5.6 802.7
rBr- = 1.705A,
PH2o = 0.743A,
(7) where rH2o is the basic radius of water in the direction of the hydrogen bond. The rather low value ofrn2o seems to be a striking result at first sight. However, it is a feature of the calculation that ?a= o is strongly dependent on the parametrisation of the dispersion energy which cannot be evaluated to a very high degree of accuracy. The repulsive energy itself is not influenced markedly by these variations of ~n=o. Care is therefore required in a physical interpretation of these basic radii. Table 5 compares the different contributions to the lattice energy of NaBr- 2H20 evaluated using the two approaches.
The Born-Haber cycle can be used to estimate the lattice energy of sodium bromide dihydrate:
UP°t+RT ,
Acknowledgement - We are grateful to Professor A. Neckel for many helpful discussions. One of us (PH) wishes to acknowledge grants from the Royal Society under an exchange agreement with the Austrian Academy of Sciences in order to carry out work at the University of Warwick. Mrs Merlin Callaway is thanked for typing the manuscript. REFERENCES 1. 2. 3. 4.
6.
_~ Na(c) + ~Br2(1) + 2H2(g) + O2(g)
whereupon:
7. 8. 9. 10.
Upot(NaBr.2H20) = AH?(Na+)(g) + AH~(Br-)(g) + 2AH~(H20)(/) -
W.H. Baur, Acta Cryst. 19,909 (1965). P. Schuster, G. Zundel & C. Sandorfy (eds.), The Hydrogen Bond, Vols. 1-3, North Holland, Amsterdam (1976). H.D.B. Jenkins & K.F. Pratt, Proc. Roy. Soc. A356, 115 (1977). H.D.B. Jenkins & K.F. Pratt, J. Chem. Soc. Faraday H 74,968 (1978). H.D.B. Jenkins & K.F. Pratt, Comp. Phys. Commun. 13,341 (1978). P. Herzig&A. Neckel, J. Chem. Phys. 71,2131 (1979).
Na+(g) + Br-(g) + 2H20(g)
AH?(NaBr" 2H20)(c)
-
AH~(NaBr-2H20)(c) + RT.
(10)
the electrostatic component of which is also consistent with previous studies [6] made by one of us.
5.
5. DISCUSSION
NaBr" 2H20(c)
This value is, predictively, higher than the value obtained by adopting an ionic model that does not take into account explicitly an energy contribution due to polarisation effects. The difference between the experimental and the theoretical result represents a measure of the covalent element of the lattice bonding. Comparison of our two calculational methods (Table 5) exhibits entirely consistent results which average to generate: Upot(NaBr" 2H20) = 803.9 + 1.2 kJ mo1-1 ,
~ 2 o = 1.346A,
(9)
(8)
11.
AH?(Na+)(g) + AH?(Br-)(g) + 2AH~(HzO)(g)
P. Herzig, J. Phys. Chem. Solids 43,449 (1982). P. Herzig, Solid State Commun. 46,685 (1983). M. Catti, Acta Cryst. A37, 72 (1981). G. Chiari & G. Ferraris, Acta Cryst. B38,2331 (1982). "HBSIO0 Hydrogen Bond Project - Survey 1975
Vol. 51, No. 6
12. 13. 14. 15. 16. 17. 18.
TOTAL LATTICE POTENTIAL ENERGY OF NaBr- 2H20
- UUIC-B19-120" Institute of Chemistry, University of Uppsala, Sweden (1975). M.F.C.Ladd & W.H. Lee, J. Phys. Chem. 69, 1840 (1965). M.F.C.Ladd & W.H. Lee, J. Phys. Chem. 73,2033 (1969). M.F.C.Ladd, Z. Kristallogr. 126, 147 (1968). H.D.B.Jenkins, E. Langadianou, M. Patel & P. Herzig (to be published) (1984). E.F. Bertaut, J. Phys. Radium 13,499 (1952). M.L.Huggins&J.E. Mayer, J. Chem. Phys. 1,643 (1933). H.D.B.Jenkins & K.F. Pratt, Repulsion Energy and Basic Radii for Ions having Spherical Symmetry, unpublished report, University of Warwick (1976).
19.
20. 21. 22. 23.
401
A. Beyer, Doctoral dissertation, University of Vienna (1976); P. Schuster, H. Lischka & A. Beyer, in Progress in Theoretical Organic Chemistry, Vol. 2, p. 89 (Edited by I.G. Czismadia), Elsevier, Amsterdam (1977). J. Tegenfeldt, R. Tellgren, B. Pedersen & I. Olovsson, Acta Cryst. B35, 1679 (1979). K.F. Pratt, Doctoral Thesis, University of Warwick (1978). G.D. Zeiss & W.J. Meath, Mol. Phys. 33,1155 (1977). D.D.Wagman, W.H. Evans, V.B. Parker, R.H. Schumm & R.L. Nuttall, Selected Values of Chemical Thermodynamic Properties, Technical Note 270-8, U.S. Dept. of Commerce, U.S.A., May (1981).