847
J. Phys. Chem. 1082, 86, 847-857
Emplrlcal Potentlal Parameters for Alkall Halide Molecules and Crystals, Hydrogen Hallde Molecules, Alkall Metal Dimers, and Hydrogen and Halogen Molecules B. Thlmmo Gowda and Sidney W. Bonron’ Department of Chamlshy, Hydrocarbon Research Instffute, Universky of Southern Callk?mia. Los Angeles, Callfornla 90007 (Received: September 8, 1981; I n Flnal Form: November 4, 1981)
Empirical potential parameters were computed for alkali halide diatomic molecules by using an ionic Rittner potential with dispersion (Rd) and exponential repulsion contributions: V(R) = -(e2/R) - (a1+ a2)e2/(2R4) - 2ala2e2/R7 A exp(-R/p) - C/R6. To estimate the repulsion and dispersion constants A , p, and C, we used the experimental data on the bond energies, the equilibrium distances, and the spectroscopic frequencies (force constants). The polarizabilities used were computed from the experimentally determined dipole moments. Significant contributions from the dispersion terms are found to the dissociation energy in contrast to the models for alkali halide crystals where they can be neglected. It is found on comparison with the homonuclear diatomics H2, X2, and alkali metal dimers (M2) that pij = (pii + pjj)/2. In the latter only dispersion terms are used in V(R) and for the HX, M2, and H2molecules it is found empirically that pij = 0.135(Ri,). Values for the halogens fall close to this line. Empirical methods of computing the “C” parameters are shown not to be very accurate. The potential parameters of alkali halide molecules have been used to predict the interionic distances and lattice energies for alkali halide crystals. The agreement between the predicted and experimental values is reasonably good. The repulsive and dispersive parameters have been employed to describe gas-phase hydration energies of alkali metal cations and halide anions with good results. Despite the differences in parameters used, both gas and crystal potential function parameters appear to describe hydration energies equally well, suggesting that experimental origin and consistency are more important than presumed physical significance.
+
Introduction Physical chemistry may be defined as that branch of chemistry which seeks to understand and represent the behavior of chemical systems in terms of mechanical and, ultimately, atomic and molecular models. In the past 2 decades physical chemists have achieved remarkable success in being able to obtain thermochemical data for ionic species in the gas phase.1*2 This has opened the prospect of making quantitative analysis and models of reactions involving ions both in solvents and in the gas phase. The importance of ion-pair processes in chemistry has provided incentives for us to consider the possibility of constructing molecular models for the process of ion solvation. While the ion-solvent interactions seem predominantly of the ion-dipole type, the polarization, repulsion, and dispersion interactions also play a large role. Thus to understand ion-solvent interaction or in general the interaction of ions with atoms and molecules we need to have reliable methods to estimate repulsive and dispersive interactions. The lack of good values for the potential parameters to estimate the repulsion and dispersion energy contributions has hindered the physical chemists’ successful use of electrostatic type calculations. In the past, the most common calculation of ion-molecule interaction used minimization of the bond energy function with respect to the radius to determine a one-parameter repulsive potential. The bond energy was then computed by using the equilibrium bond distance. Some workers3 have calculated the potential coefficient ( A ) for the repulsions between Ne-Ne, Ar-Ar, etc. from the 6-12 po(1) (a) Kebarle, P. Annu. Rev. Phys. Chem. 1977,28,445, and references therein. (b) Sunner, J.; Kebarle, P. J. Phys. Chem. 1981,85, 327. (c) Sunner, J.; Kazushige, K.; Kebarle, P. Ibid. 1981, 85, 1814. (2) (a) Castleman,Jr., A. W. Chem. Phys. Lett. 1978,53,560. (b) Lee, N.; Keesee, R. G.; Castleman, Jr., A. W. J. Chem. Phys. 1980,72,1089. (c) Mark, T. D.; Peterson, K. I.; Castleman, Jr., A. W. Nature (London) 1980,285,392. (d) Keeaee, R. G.; Castleman,Jr., A. W. Chem. Phys. Lett. 1980,74,139. (e) Keesee, R. G.; Lee, N.; Castleman, Jr., A. W. J. Chem. Phys. 1980, 73, 2195. (3) (a) Dzidic, I.; Kebarle, P. J. Phys. Chem. 1970, 74, 1466. (b) Arshadi, M.; Yamdagni, R.; Kebarle, P. Ibid. 1970, 74, 1475.
0022-365418212086-0847$01.25/0
0.38-
CI-
1.3
1.5
1.7 I
1-
Br-
I 1.9 l l
2.1I
I
2.1
Radii of halide ions
Figure 1. Plot of the repulsive parameter, p, vs.
hallde Ions for alkali halide molecules.
radii (Pauling’s)of the
tential function of Lennard-Jones using the relation Ai = 4EiR:2 where Ei is the depth of the potential well and Ri is the interatomic distance at zero potential energy. A , values for the interactions between two nonidentical noble gas atoms like Ne-Ar were taken as the geometric means of the A values for Ne-Ne, Ar-Ar (ANeAr= Plots of log Ai for the inert gases vs. their respective van der Waals’ radii gave a straight line which was then used to select Ai value for any species for which an appropriate radius is available. From this method, repulsive potentials for the systems (M+)-0 and (X-)-H were computed. Unfortunately, the ion-atom potentials determined in the above fashion have been unsuccessful in accounting for the hydration energies of ions. The basic problem appears to be that a general one-parameter potential of the form V(R) = AR-12 is inadequate for comparing several
ANeNekAr).
0 1982 American Chemical Society
848
The Journal of Physical Chemistry, Vol. 86,No. 5, 7982
Gowda and Benson
ions because the radial dependences of ion-molecule repulsive potentials are quite dependent on the ion. For the quantitative representation of the ion-molecule or ionatom or ion-ion interactions one needs at least a two-parameter repulsive potential of the form V ( R ) = A exp(Rlp) or AR-N with the parameters A and p or N determined independently. Recent practice is to use the potentials for the inert gas pair and ion-inert gas potential determined from scattering experiments4to estimate the repulsive potentials and hence the repulsive energies for the ions. Though there is extensive literature, both experimental (scattering and m ~ b i l i t yand ) ~ theoretical5 on this subject, it still requires further refinements before one can use this type of data for our purpose. Empirical formulae to calculate dispersion energies have also been found to be not too successful. Although, recently, there has been extensive theoretical work in predicting dispersion energies,6we found nothing useful for our work, as these efforts are still at the preliminary stages. Thus we had to find other methods to compute repulsion and dispersion interactions. Since the bond dissociation energies, ionization potentials, and electron affinities are known to good accuracy, we thought it worthwhile to use this data for alkali halide molecules to compute their potential parameters in the Rittner potential function' and employ the latter parameters of alkali fluoride to compute the repulsive and dispersive interactions between the corresponding alkali positive ion and the water molecule (4) (a) Mason, E. A.; Vanderslice, J. T. In "Atomic and Molecular Processes", Bates, D. R., Ed.; Academic Press: New York, 1962. (b) Olson, R. E.; Smith, F. T.; Mueller, C. R. Phys. Reu. A 1970, 1, 27. (c) Boerboom, A. J. H.; Van Dop, H.; Los, J. Physica 1970, 46, 458. (d) Hogervorst, W. Physica 1971,51,59,77, and 90. (e) Smith, F. T. 'Physics of Electronic and Atomic Collisions", VI1 ICPEAC, 1971;North Holland: Amsterdam, 1972; p 1. (f) Inouye, H.; Kita, S. J. Chem. Phys. 1972,56, 4877. 1972,57, 1301. J . Phys. SOC.Jpn. 1973, 34, 1588. (g) Amdur, I.; Jordan, J. E.; Chien, K. R.; Fung, L. W. M.; Hance, R. L.; Hulpke, E.; Johnson, S. E. J. Chem. Phys. 1972,57,2117. (h) Amdur, I.; Jordan, J. E.; Fung, L. W. M.; Hermans, L. J. F.; Johnson, S. E.; Hance, R. L. Ibid. 1973, 59, 5329. (i) Leonas, V. B. Sou. Phys. Usp. 1973, 15, 266. (j) Gengenbach, R.; Hahn, C.; Toennies, J. P. Phys. Reu. A 1973, 7,98. (k) Powers, T. R.; Cross, Jr., R. J. J. Chem. Phys. 1973, 58, 626. (1) Chen, C. H.; Siska, P. E.; Lee, Y. T. Ibid. 1973,59,601. (m)Sielanko, J.; Van Dop, H.; Los, J.; Kistemaker, J. Physica, 1973, 70, 591. (n) Bickes, Jr., R. W.; Lantzsch, B.; Toennies, J. P.; Walaschewski, K. Faraday Discuss. Chem. SOC.1973,55,167. (0)Weise, H. P. Ber. Bunsenges. Phys. Chem. 1973,77,578. (p) Ng, C. Y.; Lee, Y. T.; Barker, J. A. J. Chem. Phys. 1974, 61, 1996. (9) Boyle, J. F.; Smith, F. J. Physica, 1974, 75, 351. (r) Buck, U. Adu. Chem. Phys. 1975, 30, 313. ( 8 ) Chigin, V. I. Sou. Phys. Tech. Phys. 1975, 20, 429. (t) Budenholzer, F. E.; Galante, J. J.; Gislason, E. A.; Jorgensen, A. D. Chem. Phys. Lett. 1975,33,245. (u) Kita, S.; Noda, K.; Inouye, H. J . Chem. Phys. 1975,63,4930. (v) Foreman, P. B.; Lees, A. B.; Ro, P. K. Chem. Phys. 1976, 12, 213. (w)Champion, R. L.; Doverspike, L. D. Phys. Reu. A 1976,13,609. (x)Gatland, I. R.; Morrison, W. F.; Ellis, H. W.; Thackston, M. G.; McDaniel, E. W.; Alexander, M. H.; Viehland, L. A.; Mason, E. A. J.Chem. Phys. 1977,66,5121. 01)Este, G. 0.;Knight, G.; Scoles, G. Chem. Phys. 1978,35, 421. (z) Toennies, J. P.; Welz, W.; Wolf, G. J . Chem. Phys. 1979, 71, 614. (5) (a) Gordon, R. G.; Kim, Y. S. J . Chem. Phys. 1972,56, 3122. (b) Maitland, G. C.; Smith, E. B. Chem. Phys. Lett. 1973,22,443. (c) Kim, Y. S.; Gordon, R. G. J . Chem. Phys. 1974,60,4323, 4332. (d) Wagner, A. F.; Das, G.; Wahl, A. C. Ibid. 1974,60, 1885. (e) Tang, K. T.; Toennies, J. P. Ibid. 1977,66, 1496. (f) Dunker, A. M.; Gordon, R. G. Ibid. 1978, 68, 700. (g) Mingelgrin, U.; Gordon, R. G. Ibid. 1979, 70, 3828. (h) Waldman, M.; Gordon, R. G.Ibid. 1979, 71, 1325, 1340, 1353. (6) (a) Scoles, G. Annu. Rev. Phys. Chem. 1980,31,81, and references therein. (b) Gordon, R. G. J . Chem. Phys. 1968, 48,3929. (c) Davison, W. D. J. Phys. B 1968,1,597. (d) Kramer, H. L.; Herschbach, D. R. J . Chem. Phys. 1970,53,2792. (e) Starkschall, G.; Gordon, R. G. Ibid. 1971, 54,663. (0 Ahlberg, R.; Goscinski, 0. J.Phys. E 1974, 7, 1194. (g) Tang, K. T.; Norbeck, J. M.; Certain, P. R. J. Chem. Phys. 1976,64, 3063. (h) Magnasco, V.; Battezzati, M.; Austi, R. Chem. Phys. Lett. 1977,51,375. (i) Manakov, N. L.; Ovsiannikov, V. D. J. Phys. B 1977,10,569. 6) Zeisa, G. D.; Meath, W. J. Mol. Phys. 1977, 33, 1155. (k) Margoliash, D. J.; Meath, W. J. J. Chem. Phys. 1978,68,1426. (1) Yoffe, J. A. Theor. Chim. Acta (Berl.) 1979,52,155. (m) Maeder, F.; Kutzelnigg, W. Chem. Phys. 1979, 42, 95. (n) Pena, M. D.; Pando, C.; Renuncio, J. A. R. J . Chem. Phys. 1980, 72, 5269. (7) Rittner. E. S. J . Chem. Phys. 1951, 19, 1030.
5
e C.j
m
E
t-: 01
J
2
._
k k
0
sx
The Journal of Physical Chemistry, Vol. 86, No. 5, 1982 849
Empirical Potential Parameters
TABLE 11: Data Used for the Estimation of Potential Parameters for Alkali Halide Molecules -
~~
1O'l6K,
M , D ~
MX Li F
A
Duo,' eV
I,d eV
E,d eV
u , cm-' ~
eV cm-*
hv/2, e V
-V,e eV
6.325 7.120 7.268 7.429
1.564 2.021 2.170 2.392
5.945 4.818 4.341 3.604
5.392
LiCl Li Br Li I
3.400 3.61 3 3.363 3.063
910.3 643.3 563.5 490.0
15.586 8.878 7.509 5.858
0.056 0.040 0.035 0.030
7.994 6.637 6.405 5.963
Na F NaCl Na Br NaI
8.156 9.001 9.118 9.236
1.926 2.361 2.502 2.711
4.948 4.211 3.799 3.1 60
5.1 39
3.400 3.61 3 3.363 3.063
536.1 364.6 298.5 259.2
10.997 6.785 5.837 4.811
0.033 0.023 0.019 0.016
6.720 5.759 5.593 5.252
2.172 2.667 2.821 3.048
5.117 4.384 3.929 3.387
4.341
KI
8.593 10.269 10.628 10.820
3.400 3.613 3.363 3.063
426.0 279.8 219.2 186.5
8.528 5.311 4.61 2 3.818
0.026 0.017 0.014 0.012
6.084 5.130 4.921 4.677
Rb F RbCl RbBr RbI
8.547 10.510 10.860 11.480
2.270 2.787 2.945 3.177
5.078 4.380 3.994 3.430
4.177
3.400 3.61 3 3.363 3.063
373.3 233.3 169.5 138.5
7.957 4.964 4.324 3.593
0.023 0.014 0.011 0.009
5.878 4.959 4.818 4.553
CsF
7.884 10.387 10.820 11.690
2.345 2.906 3.072 3.31 5
5.290 4.51 4 4.081 3.474
3.894
3.400 3.613 3.363 3.063
352.6 214.2 149.5 119.2
7.599 4.676 4.074 3.395
0.022 0.013 0.009 0.007
5.806 4.809 4.621 4.312
KF KCl KBr
csc1 csBr csI
Reference 35.
Reference 40.
' References 36, 38, and 39.
as F- and H 2 0 have the same number of electrons and almost the same radius. In the present paper we shall describe the computation of potential parameters for alkali halide molecules and crystals, hydrogen halide molecules, alkali metal dimers, and hydrogen and halogen molecules. We shall also check the consistency of the computed potential parameters of the alkali halide molecules by using them to predict the interionic distances and lattice energies for the alkali halide crystals. Potential Parameters for Ion-Ion Interactions in Alkali Halide Molecules A number of attempts have been made to compute potential parameters for alkali halide molecules and cryst a l ~ . ~Thirty - ~ ~ years ago Rittner7presented a theory of the alkali halide molecules in the spirit of the Born-Mayer lattice theory. This theory has been found to be reasonably successful and most frequently used for the potential curves and dipole moment functions for the electronic ground states of the alkali halides. It has also been useful (8) Born, M.; Mayer, J. E. 2. Phys. 1932, 75, 1. (9) Honig, A,; Mandel, M.; Stitch, M. L.; Townes, C. H. Phys. Reu. 1954,96, 629. (10) Pauling, L. Proc. Natl, Acad. Sci, India 1956, A%, 1. (11) (a) Varshni, Y. P. Trans. Faraday SOC.1957,53,132. (b) Varshni, London 1960, 76, 794. J. Chem. Y. P.; Shukla, R. C. Proc. Phys. SOC. Phys. 1961, 35, 582. J.Mol. Spectrosc. 1965, 16, 63. (12) (a) KlemDerer. W.: Rice. S. A. J. Chem. Phvs. 1957.26.618. (b) " \-, - -~ Rice, S. A.; Klemperer, W. Zbid. 1957, 27, 573. (13) Frost, A. A.; Woodson, J. H. J. Am. Chem. SOC. 1958,80, 2615. (14) (a) Milne, T. A.; Cubicciotti, D. J. Chem. Phys. 1959, 30, 1418, 1625. (b) Cubicciotti, D. J. Phys. Chem. 1961, 65, 1058. (15) (a) Fumi, F. G.; Tosi, M. P. J. Phys. Chem. Solids 1964.25.31. (h) M. P.Solid State Phvs. 1964. 16. 1. -, Tnsi. - - -., . - -. (16) Spears, K. G. j.Chem."Phys. 1972; 57, 1842. (17) Brumer, P.; Karplus, M. J. Chem. Phys. 1973, 58, 3903. (18) Kim, Y. S.; Gordon, R. G. Phys. Reu. % 1974,9, 3548. (19) Matcha, R. L.; King, Jr., S. C. J.Am. Chem. Soc. 1976, 98, 3415, 3420. (20) Catlow, C. R. A.; Diller, K. M.; Norgett, M. J. J. Phys. Chem. 1977, IO, 1395. (21)(a) Mohammad, S. N. Indian J. Pure Appl. Phys. 1978,16,646. (b) Shanker, J.; Gupta, A. P.; Sharma, 0. P. Phil. Mag. 1978,37,329. (c) Shanker, J.; Agrawal, H. B.; Agrawal, G. G. J . Chem. Phys. 1980,73,4056. (22) Solomonik, V. G. J. Struct. Chem. 1978, 19, 860. (23) Davidovits, P.; McFadden, D. L., Ed. "Alkali Halide Vapors"; Academic Press: New York, 1979. (24) Berkowitz, J.; Bataon, C. H.; Goodman, G. L. J. Chim. Phys. 1980, 77, 631.
.--,
r -
I
References 36 a n d 37. e V = -D,,'
-- I
+ E - hvI2.
TABLE 111: Polarizabilities ( A ) of the Ions in Alkali Halide Molecules Computed from the Experimental Dipole Moments recommended ac1BrF(anion) (0.446) (2.156) (3.154)
Li'
Nata K' Rb'
Cs' a
0.103 0.312 1.129 1.756 2.901
0.019 0.312 1.040 1.681 2.860
-0.024 0.312 1.002 1.726 2.951
1a(5.070) (cation)
-0.078 0.312 1.218 1.605 2.706
0.103 0.312 1.10 1.69 2.86
a ( N a + )= 0.312 A 3 has been assigned from other data.
in finding reasonable values for the potential parameters. Although a number of workers have made calculations with the Rittner potential to estimate the potential parameters for alkali halide molecules, these calculations were based on the older data or else used London formula to compute the van der Waals' constant. As will be seen later, the potential parameters computed vary with the polarizabilities used and the polarizabilities computed by different g r o ~ p sdo~ not ~ *agree ~ ~with ~ ~each other. Also the free ion polarizabilities for the halide anions are now considered to have large errors%and their use in dense media is open to question. Table I lists the polarizabilities of alkali positive and halide negative ions computed by different groups. So we have used the experimentally determined dipole moments (pd)36(Table 11) of the alkali halides to compute the polarizabilities of alkali cations and halide anions from the expression7
~
(25) (a) Fajans, K.; Joos, G. 2. Phys. 1924,23,1. (b) Fajans, K. "Encyclopediaof Chemistry"; Clark and Hawley, Ed.; Reinhold New York, 1957; p 764. (26) Born, M.; Heisenberg, W. Z. Phys. 1924, 23, 388. London, Ser. A 1927, 114, 191. (27) Pauling, L. Proc. R. SOC. (28) Mayer, J. E.; Mayer, M. G. Phys. Reu. 1933,43,605. (29) Tessman, J. R.; Kahn, A. H.; Shockley, W. Phys. Reu. 1953, 92, 890. (30) Ruffa, A. R. Phys. Reu. 1963, 130, 1412. (31) Pirenne, J.; Kartheuser, E. Physica 1965, 31, 284. (32) Michael, A. J. J. Chem. Phys. 1969, 51, 5730. (33) Wilson, J. N.; Curtis, R. M. J. Phys. Chem. 1970, 74, 187. (34) Sadlej, A. J. J. Phys. Chem. 1979, 83, 1653. (35) Lovas, F. J.; Tiemann, E. J. Phys. Chem. Ref. Data 1974,3,609,
and references therein.
850
The Journal of Physical Chemistry, Vol. 86, No. 5, 1982
pd
= eRe -
(
)
R,4e(al + a2)+ 4Reealaz R,6 - 4alaz
Gowda and Benson I
0 LI'
(1)
where e is the electronic charge, Re is the equilibrium interionic distance (Table II), and a1 and a2are the polarizabilities of the alkali (+) and halide (-) ions, respectively. While doing so we have to fix the polarizability of one of the ions to get the polarizabilities of other positive and negative ions. We have computed the various sets of polarizabilities by furing the polarizabilities of the ions from Li+ to I- and used them to compute the potential parameters. The fixing of polarizability has little effect on the overall potentials and still less will be its effect on the computed hydration energies using the parameters obtained. However, it is advisable to fix the cation polarizability as the fixing of anion polarizability requires the use of an arbitrary value as the anion polarizabilities are relatively uncertain. A typical set of polarizabilities computed by fixing the polarizability of Na+ ion as 0.312 A3 (from Pirenne and Kartheuser31) is shown in Table 111. Brumer and Karplus17 have argued from an exchange perturbation theoretical analysis that the Rittner expression~~ for the molecular dipole moment and the potential are unbalanced and hence the 4ala2/R6term in the dipole moment function and the corresponding 2ala2/R7 term in the potential function be neglected to make them consistent. Their arguments led to the truncated expressions 2 and 4, which do not include the induced dipole
-7.5 A
Rb'
i
2.3 R a d i i of halide ions
Flgure 2. Behavior of the potential for alkali halide molecules.
In using the experimental data on dissociation energies, corrections should be made for kinetic energy terms representing the difference in translational, rotational, and vibrational energies between the molecule and the free ions from which it is composed, including zero point energy
AHoexpt = V ( R )- f/,kT+ 1/hV Q,
+ k P ( d In Q,/aT)
= ( 1 - e-hv/kr)-l
the rotational partition function. V(Re)= -(De I - E )
+
contributions. We used the latter expressions to compute the polarizabilities from the experimental dipole moments and hence the potential parameters. When we employed these parameters of alkali fluorides to compute the hydration energies of alkali cations, the discrepancy between the computed and experimental values increased and also the fixing of the polarizabilities has considerable effect on the values of the potential parameters, if we use the truncated expressions. We thus decided to employ the Rittner expressions. In the present invesitgations we take a simple, direct approach to compute the repulsive parameters and the van der Waals' constants from the Rittner potential. Since the bond dissociation energies and spectroscopic data for the alkali halide molecules, ionization potentials for the alkali atoms, and electron affinities for the halogen atoms are well-known ( f l kcal), we used the Rittner potential and its first and second derivatives to obtain the parameters A , p, and C. The Rittner potential considers the electrostatic attraction between two polarizable spheres of charges * l . It also includes a repulsion term of the exponential form and a van der Waals' attraction term. The resulting Rittner model potential is V(R)=
where the successive terms represent, respectively, the Coulombic attraction, the ion-induced dipole interaction, the short-range repulsion, and the long-range van der Waals' attraction term. The truncated Rittner potential17 is
(aV/aR)Re= 0
(5) (6)
(7)
(d2V/dR2)Re = K , vibrational force constant for the diatomic molecule with frequency u and reduced mass p , where U
= (1/2T)(K/p)'/'
(8)
Here e, Re, al,and a2have the same meaning as before; A and p are the repulsive potential coefficient and parameter, respectively; C is the van der Waals' constant; h and It are Planck's and Boltzmann's constants, respectively; De is the bond dissociation energy to atoms; I is the ionization potential of the alkali atom; and E is the electron affinity of the halogen atom. The data employed in the calculations are listed in Table 11. Bond dissociation energies, ionization potentials, and electron affinities were taken from various source^.^^^^ Vibrational frequencies to compute the force constants and the interionic distances were taken from Gordy et al.40 Polarizabilities used were from Table I. To examine the effect of polarizabilities on the values of potential parameters, we have used the polarizabilities.computedby various groups, ranging from Pauling's free ion polarizabilities to crystal ion polarizabilities. The results of the calculations for the repulsion and van der Waals' constants are shown in Tables IV and V. The choice of polarizabilities makes little difference in most cases with p, but it has (36) "Handbook of Chemistry and Physics";CRC Press: Boca Raton; 1980-81, 61st ed. (37) Rosenstock, H. M.; Draxl, K.; Steiner, B. W.; Herron, J. T. J. Phys. Chem. Ref. Data 1977,6, Supplement No. 1. (38) Gaydon, A. G. "Dissociation Energies and Spectra of Diatomic Molecules";Chapman and Hall London, 1968; 3rd ed. (39) Herzberg, G. "Molecular Spectra and Molecular Structure, I. Spectra of Diatomic Molecules";Van Nostrand: New York, 1950; 2nd ed. (40) (a) Rusk, J. R.; Gordy, W. Phys. Reu. 1962,127,817. (b) Clouser, P.; Gordy, W . Ibid. 1964, 134, A863. ( c ) Veazy, S. E.; Gordy, W . Ibid. 1965, 138, A1303. (d) Pearson, E.; Gordy, W . Ibid. 1969, 177, 52.
The Journal of Physical Chemistry, Vol. 86, No. 5, 7982 851
Empirical Potential Parameters
TABLE IV: Potential Parameters for Alkali Halide Moleculesa Obtained b y Using Various Polarizabilities
P au ling
-
TKS'
A,eV
p,A
1048c, MX
A,eV
p, A
eVcm6
Wilson and Curtisd
1048c, eVcm6
Pirenne and Kartheusef
low,
1048~. A,eV
p,A
eVcm6
A,eV
8.95 392.8 0.3107 -13.94 958.2 -0.59 1216.7 0.2553 0.2720 11.42 2052.3 88.97 1164.9 0.3333 0.3184 54.13 2121.4 0.3096 0.3299 195.33 3009.5 0.3256 233.23 2004.4 0.3405 222.05 2942.8 0.3598 422.98 3299.8 0.3565 475.21 2307.1 0,3703 284.86 3265.7 30.70 1173.0 0.2921 -13.43 1967.3 NaF 0.2792 23.04 2106.3 0.2764 96.94 2828.3 NaCl 0.3358 167.45 2853.4 0.3325 198.89 2071.7 0.3416 NaBr 0.3556 435.17 3507.5 0.3535 461.72 2772.9 0.3606 327.28 3478.2 NaI 0.3840 711.21 3530.9 0.3820 747.39 2759.7 0.3899 517.04 3515.0 5.56 2450.5 KF 54.56 2045.7 0.3044 0.3007 60.01 2449.0 0.3013 KCI 0.3566 267.37 3234.0 0.3555 283.08 2770.5 0.3584 168.78 3247.1 KBr 0.3780 507.57 3436.5 0.3770 511.76 2975.2 0.3800 357.55 3437.9 KI 0.4106 1160.89 3838.4 0.4097 1170.06 3329.2 0.4131 895.55 3841.1 RbF 0.3071 57.40 2605.0 0.3083 48.81 2388.7 0.3085 -3.35 2639.0 RbCl 0.3634 353.91 3698.1 0.3629 367.51 3350.2 0.3641 246.51 3718.4 RbBr 0.3902 922.33 4196.8 0.3897 921.92 3843.5 0.3913 759.44 4205.4 RbI 0.4212 1661.95 4396.3 0.4206 1666.18 3980.8 0.4226 1372.10 4403.0 CsF 20.15 2885.7 0.3164 88.85 2772.6 0.3189 55.65 2909.9 0.3163 CsCl 0.3701 421.96 4132.4 0.3699 394.88 3960.8 0.3699 304.25 4174.4 CsBr 0.3940 862.75 4392.9 0.3937 813.29 4175.0 0.3941 682.62 4418.0 CsI 0.4189 121.25 4495.8 0.4184 1161.00 4189.4 0.4187 983.51 4510.5 Present calculation employing Rittner potential. Pauling's free ion polarizability. ' Reference 29. e Reference 31. LiF LiCl LiBr LiI
788.5 1661.9 2651.0 3015.9 1887.0 2543.7 3285.9 3335.6 2522.7 3099.3 3337.0 3737.3 2744.9 3613.3 4135.0 4323.2 3126.4 4150.6 4400.6 4466.1
p,A
eVcm6
0.2640 0.3107 0.3264 0.3569 0.2780 0.3329 0.3539 0.3823 0.3011 0.3556 0.3772 0.4098
3.49 83.93 226.30 469.00 24.34 201.35 463.94 754.50 51.36 297.11 526.99 1194.65
0.3078 0.3630 0.3898 0.4207 0.3177 0.3700 0.3938 0.4186
45.80 384.76 940.67 1695.56 57.95 426.40 848.23 1210.67
Reference 33.
TABLE V : Potential Parameters for Alkali Halide Molecules present calculation truncated Rittner potential
MX
A,eV
p,A
1048C, eVcm6
Rittner potential
low, A,eV
p,A
eVcm6
Brumer and Karplusa
_____
A,eV
0.2531 1303.1 11.18 1401.8 0.2504 11.49 782.8 0.3046 2479.1 116.07 2674.0 0.3020 123.23 998.3 0.3220 3380.9 273.52 3585.5 0.3199 285.48 1031.7 0.3530 3661.7 543.39 3851.8 0.3512 564.52 1116.5 47.13 1372.9 NaF 0.2755 2257.7 44.84 2404.3 0.2738 NaCl 0.3311 3100.9 262.44 3298.3 0.3293 273.71 1551.7 NaBr 0.3523 3743.8 547.08 3936.4 0.3507 563.34 1547.3 NaI 0.3807 3769.7 879.03 3952.0 0.3791 904.12 1595.5 KF 0.3039 2279.7 87.54 1778.4 62.96 2732.7 0.2996 KC1 0.3572 3198.5 363.39 3559.9 0.3545 403.14 2313.6 KBr 0.3786 3421.6 625.35 3750.6 0.3760 670.48 2385.2 KI 0.4108 3859.2 1350.68 4139.6 0.4085 1405.16 2485.2 RbF 0.3121 2289.6 89.88 2099.9 48.92 2875.9 0.3071 RbCl 0.3654 3538.1 447.65 3977.9 0.3627 508.12 2912.4 RbBr 0.3916 4079.3 1036.86 4464.5 0.3892 1105.81 RbI 0.4222 4326.7 1858.99 4656.1 0.4200 1935.52 CsF 43.57 3096.1 0.3175 117.71 2282.1 0.3249 2258.6 CsCl 0.3732 3779.8 474.15 4382.7 0.3705 580.99 3612.8 CsBr 0.3967 4099.9 935.89 4631.2 0.3940 1052.17 CsI 0.4212 4248.0 1369.77 4718.6 0.4190 1501.27 4311.4 a Reference 17. Reference 22. Reference 16. Reference 11.
LiF LiCl LiBr LiI
significant effect on the values of A and C. So the magnitudes of these parameters depend on the polarizabilities used. Hence, we computed the polarizabilities of the alkali cations and halide anions from the experimental dipole moments of alkali halide molecules as described before and used them in the calculation of A , p, and C. The resulting values are shown in Table V (and Figure 1). Our values are also compared with those of Brumer and Karplus,17 Solomonik,22Spears,16and Varshni and Shukla." Contributions of the different terms to the total potential are shown in Table VI and the plot of the potential vs. ionic radii of the halide ions is shown in Figure 2. All the three constants, A , p , and C, increase as we go from fluoride to iodide or from lithium to cesium (Tables IV and V). Also,
p,
A
Solo moni k A,eV
801.4 1087.3 1133.8 1240.8 1610.3 2008.5 1999.8 2074.7 2127.7 3086.4 3191.9 3367.3 2550.9 3930.2 4125.6 4427.1 0.3148 2931.6 0.3527 4975.7 5352.7 0.3864 5985.6
0.2725 0.3310 0.3499 0.3766 0.2821 0.3367 0.3553 0.3816 0.3003 0.3474 0.3635 0.3881 0.3059 0.3491
p,A
Spears' A,eV
p,A
0.262 840.8 0.2625 0.314 1031.2 0.3245 0.333 1143.0 0.3452 0.359 1052.5 0.3786 0.273 1692.8 0.2769 0.319 1962.3 0.3293 0.337 1917.4 0.3496 0.362 1913.6 0.3770 0.293 2182.9 0.2999 0.332 2873.3 0.3469 0.348 2882.2 0.3651 0.371 2909.1 0.3913 0.299 2522.3 0.3073 0.335 3488.2 0.3518 0.350 3563.7 0.3691 0.371 3570.5 0.3946 0.306 2773.2 0.3189 0.340 4118.6 0.3614 0.354 4237.1 0.3782 0.373 4399.0 0.4024
Varshni and Shuklad
0.2674 0.3253 0.3441 0.3702 0.2770 0.3290 0.3488 0.3744 0.2964 0.3432 0.3607 0.3844 0.3076 0.3466 0.3627 0.3849 0.3050 0.3540 0.3687 0.3886
the computed repulsion and dispersion energies for alkali halide molecules (Table VI) generally increase as we go from fluoride to iodide. The contributions of different terms will also be affected by the choice of polarizabilities. We have also checked the consistency of the repulsive potentials by computing the ratio of the parameters p and A for various alkali halide molecules (Table VII). The agreement is reasonable with some systematic deviations. The present calculation not only gives the consistent repulsive potentials for alkali halide molecules, but also the van der Waals' constant. Although we wanted to separate alkali halide potentials into individual ionic potentials we could not do so as it requires at least one accurate M+-M+ or X--X- potential and an exact combining rule. The
852
The Journal of Physical Chemistry, Vol. 86, No. 5, 7982
Gowda and Benson
TABLE VI: Contribution of Different Termsa t o the Total Potential for Alkali Halide Moleculesa this work, TKS,
MX
Pauling’s,c
this work, TKS,b Pauling’s,c
-
C/Rpb IId IIId
A exp(-R,lp) Id IId IIId
-v
-e2/R,
ct1,a2
O , , O L ~
ol,02
ff1,ffZ
ffI,ff2
011,
LiF LiCl LiBr LiI
9.206 7.124 6.635 6.01 9
0.661 0.975 1.057 1.138
0.810 1.290 1.359 1.4 21
1.298 1.605 1.571 1.581
0.058 0.046 0.041 0.034
0.024 0.018 0.015 0.012
0.038 0.022 0.018 0.013
0.785 1.809 2.734 3.014
0.612 1.306 2.234 2,537
0.041 0.794 1.871 2.258
2.716 3.317 4.063 4.241
2.657 3.101 3.838 4.025
2.508 2.909 3.690 3.908
7.994 6.636 6.405 5.963
NaF NaCl NaBr NaI
7.475 6.098 5.755 5.311
0.397 0.572 0.637 0.717
0.550 0.780 0.839 0.912
0.644 0.897 0.917 0.978
0.041 0.047 0.046 0.042
0.077 0.085 0.080 0.070
0.056 0.047 0.041 0.035
0.923 1.580 2.296 2.278
0.602 1.148 1.882 1.883
0.451 0.967 1.774 1.792
2.116 2.538 3.140 3.096
1.984 2.353 2.962 2.923
1.906 2.249 2.893 2.863
6.720 5.759 5.594 5.252
KF KC 1 KBr
KI
6.629 5.399 5.104 4.724
0.499 0.463 0.483 0.514
0.640 0.611 0.6 24 0.648
0.611 0.64 5 0.642 0.667
0.062 0.071 0.070 0.066
0.109 0.119 0.112 0.101
0.111 0.093 0.082 0.071
0.834 1.120 1.330 1.752
0.520 0.787 1.015 1.459
0.572 0.743 1.007 1.448
1.939 1.923 2.067 2.379
1.812 1.785 1.935 2.255
1.839 1.750 1.914 2.233
6.084 5.130 4.920 4.677
RbF RbCl RbBr RbI
6.343 5.166 4.889 4.53 2
0.580 0.459 0.464 0.478
0.711 0.589 0.587 0.594
0.670 0.610 0.596 0.606
0.070 0.080 0.080 0.076
0.118 0.129 0.123 0.112
0.138 0.115 0.102 0.089
0.657 1.084 1.695 1.882
0.357 0.784 1.413 1.620
0.420 0.755 1.414 1.616
1.711 1.831 2.310 2.415
1.651 1.710 2.194 2.306
1.692 1.688 2.183 2.292
5.878 4.959 4.818 4,553
CsF CSCl CsBr CSI
6.140 4.955 4.687 4.343
0.786 0.506 0.4 86 0.472
0.947 0.635 0.606 0.582
0.831 0.619 0.586 0.572
0.094 0.101 0.100 0.095
0.159 0.162 0.155 0.140
0.189 0.148 0.130 0.114
0.708 0.965 1.252 1.131
0.335 0.656 0.968 0.875
0.534 0.701 1.027 0.916
1.921 1.718 1.904 1.730
1.774 1.600 1.794 1.629
1.888 1.614 1.809 1.634
5.806 4.809 4.621 4.312
ff2
Id
As a function of polarizabilities of alkali cations, c y I , and halide anions, 0 1 ~(all values in e V ) . Reference 29. Reference 27. C, A , and p used were obtained from the Rittner potential using ( I ) polarizabilities computed from the experimental dipole moments, (11) TKS p ~ l a r i z a b i l i t i e sand , ~ ~ (111) Pauling’s p o l a r i z a b i l i t i e ~ . ~ ~ TABLE VII: Consistency of Repulsive Potentials of Alkali Halide Molecules ratio o f repulsive potentials
XF-
c1Rr IFCI Br
I
KX/
RbX/
RbX
CSX
0.91 0.9 3 0.9 3 0.93
0 98 0.98 0.97 0.97
0.97 0.98 0.99 1.00
A,IA, 0.88 0.9 3 1.05 0.95
0.95 0.89 0.84 0.89
0.93 0.91 0.96 0.99
LiX/ NaX
NaX/
0.91 0.92 0.91 0.93 0.58 0.81 0.91 0.97
Kx
P,lP*
M’
MF/MCI
Li‘ Na
Rb’ CS’
0.83 0.83 0.8 5 0.85 0.86
Li‘ Na’ K+ Rb’ Cs’
0.52 0.7 3 0.7 7 0.7 2 0.71
+
K+
MCl/MBr
MBr/MI
PIIPI
0.94 0.94 0.94 0.93 0.94
0.91 0.93 0.92 0.93 0.94
0.75 0.84 0.95 0.89 0.95
0.93 1.00 0.91 0.96 0.98
A,/A,
repulsive potential of M+-M+ or X--X- obtained from the scattering experiments is ~ n c e r t a i n . ~ If we compare our repulsion parameters to those of Brumer and Karplus,” Spears,16Solomonik,22and Varshni and Shukla” we see that the values of p are generally in close agreement while the A parameters differ considerably. The present calculations are most interesting in suggesting fairly significant dispersion contributions to the binding energies of all the alkali halide molecules. This
is contrary to the expectations from crystal data which suggest very small dispersion contributions to the binding energy. However, the two results are compatible with the fact that the gas-phase molecules show an equilibrium anion-cation separation smaller by some 10-20% than that in the crystal. Since the dispersion terms are very sensitive to Rij varying as Rij* while the Coulombic term varies as Riyl this is what we might expect. Note that in all cases the repulsion term (Table VI) exceeds the dispersion term. While the Coulombic term dominates the potential, the value of the equilibrium separation becomes more sensitive to the higher order terms [Riy4,Rij*, and exp(-Ri,/pij)]. The force constant is most sensitive of all to these higher order terms and least sensitive to the Coulombic term. The current analysis shows that if one is using a Coulombic term to portray the alkali halide binding then the other parameters for dispersion must be included in order to yield good values for the force constants. To check the consistency of the parameters determined by using the polarizabilities computed from the experimental dipole moments, we used them to predict the lattice energies and the interionic distances in alkali halide crystals. Equilibrium interionic distances (R,) were determined by minimizing the lattice energy (v) equation for NaC1-type structure
(aU/aR),, = 0 where AMis the Madelung constant, a factor of 6 appears because each ion has six nearest neighbors. The lattice energies were then computed. The resulting values for Ro and U are shown in Table VIII, along with the experimenta136*41p42 and Kim and Gordon’s18calculated values. (41) Smith, C. S.; Cain, L. S. J. Phys. Chem. Solids 1975, 36, 206. (42) Pauling, L. T h e Nature of the Chemical Bond”, 3rd Ed; Cornell University Press: Ithaca, NY; 1960.
Empirical Potential Parameters
The Journal of Physical Chemistry, Vol. 86, No. 5, 1982 853
TABLE VIII: Equilibrium Interionic Distances and Lattice Energies for Alkali Halide Crystals, Predicted by Using Potential Parameters from t h e Alkali Halide Gas Molecules
R,,, a MX
this work
expt
LiF LiCl LiBr LiI
2.081 2.633 2.784 3.064
2.013 2.570 2.751 3.006
Na F NaCl Na Br NaI
2.409 2.927 3.077 3.346
KF KCl KBr KI RbF RbCl RbBr RbI CsF CSCl c s Br CSI a Reference 18.
-U, eV Kim and Gordon"
Kim and Gordona
this workb
expt
1.93 2.47 2.66
10.683 9.042 9.062 8.499
10.736 8.840 8.363 7.845
11.282 8.936 8.338
2.317 2.820 2.989 3.236
2.31 2.86 3.04
9.590 8.394 8.445 7.864
9.565 8.145 7.741 7.296
9.639 7.922 7.484
2.718 3.264 3.440 3.687
2.674 3.146 3.300 3.533
2.60 3.05 3.20
8.592 7.507 7.307 7.159
8.508 7.410 7.068 6.726
8.850 7.601 7.250
2.852 3.390 3.540 3.799
2.826 3.291 3.445 3.671
2.77 3.19 3.32
8.170 7.303 7.440 7.164
8.135 7.140 6.840 6.529
8.412 7.345 7.020
2.986 3.542 3.710 3.992
3.005 3.47 3.62 3.83
7.891 6.977 6.902 6.433
7.669 6.829 6.539 6.259
These values have been corrected for zero point energy.
The agreement is reasonably good. The calculated lattice energies have been corrected for zero point energies by using the frequencies computed from the experimental Debye characteristic temperatures (6, = hv,,/k) of alkali halide crystals.43
TABLE IX: Data Used for t h e Calculation of Potential Parameters for Hydrogen, Alkali Metal Dimers, and Halogen and Hydrogen Halide Molecules
Potential Parameters for Atom-Atom Interactions The potential parameters for alkali metal dimers, hydrogen, halogen, and hydrogen halide molecules were computed by using a potential without ionic interaction terms. The function (V? used was of the form
where R is the interatomic distance. The remaining terms have the same meaning as before. This potential and its first and second derivatives
(aV'/aR),@= 0
A
D,,n,b eV
cm-'
0.742
4.480
4395.2
35.801
0.272
4.752
2.673 3.079 3.923 4.22e 4.48
1.104 0.759 0.556 0.434 0.394
351.4 159.2 92.6 57.3 42.0
1.589 1.072 0.615 0.513 0.431
0.022 0.010 0.006 0.004 0.003
1.125 0.769 0.561 0.438 0.396
1.418 1.988 2.284 2.667
1.587 2.475 1.971 1.542
892.1 564.9 325.4 214.6
27.803 20.537 15.381 10.754
0.055 0.035 0.020 0.013
1.643 2.510 1.991 1.556
0.917 5.854 1.275 4.436 1.414 3.751 1.605 3.057
41 38.5 2991 .O 2649.7 2309.5
60.284 32.231 25.699 19.614
0.257 0.185 0.164 0.143
6.111 4.622 3.915 3.201
molecule
(12)
1 0 I b K , hv/2, e V c m - * eV
u,'
eV
References 45-47. References 36, 38, and 39. Reference 44. V' = D,,' - hu/2. e As calculated from covalent radii g i v v in ref 48.
-
where K is the vibrational force constant, were solved for C, A , and p. The vibrational frequencies to compute the force constants were taken from Nakamoto.44 Bond dissociation energies and interatomic distances were taken
0.6-
0.5-
(43) (a) Barron, T. H. K.; Berg, W. T.; Morrison, J. A. Proc. R. SOC. London, Ser. A 1957,242,478. (b) Martin, D. L. Phil. Mag. 1955,46,751. Proc. Phys. SOC. London 1964, 83, 99. (c) Briscoe, C. V.; Squire, C. F. Phys. Rev. 1957,106, 1175. (d) Alers, G. A.; Neighbors, J. R. Rev. Mod. Phys. 1959,31,675. (e) Lewis, J. T.; Lehoczky, A.; Briscoe, C. V. Phys. Rev. 1967,161,877, and references therein. (f') Harrison, J. P. Rev. Sci. Instrum. 1968, 39, 145. (9) Marshall, B. J.; Cleavelin, C. R. J. Phys. Chem. Solids 1969,30,1905. (h) Marshall, B. J.; Kunkel, J. R. J. Appl. Phys. 1969,40,5191. (i) Robbins, R. A.; Marshall, B. J. Ibid. 1971,42, 2562. (j)Konti, A.; Varshni, Y. P. Can. J. Phys. 1971, 49, 3115. (k) Rollefson, R. J.; Peressini, P. P. J.Appl. Phys. 1972,43,727. (1) Varshni, Y.P.; Konti, Y. J. Phyc. C 1972,5,2562. (m) Cleavelin, C. R.; Pederson, D. 0.;Marshall, B. J. Phys. Rev. B. 1972,5, 3193. (n) Pathak, P. D.; Trivedi, J. M. Acta Crystallogr., Sect. A 1973,29,45. 1974,30, 321. (0) Berg, W. T. Phys. Rev. B 1976,13,2641. (p) Birch, J. A,; Collins, J. G.; White, G. K. Aust. J. Phys. 1979,32,463. (44) Nakamoto, K. "Infrared and Raman Spectra of Inorganic and Coordination Compounds"; Wiley: New York, 1978; 3rd ed.
0.4-
(dl 0.3
-
0.5
1.0
1.5
2.0
2.5
Interatomic
3.0
distance
3.5
(A)
4.0
4.5
Figure 3. Plot of the repulsion parameter, p , vs. the interatomic distances for alkali metal dimers, hydrogen, and halogen and hydrogen halide molecules.
854
Gowda and Benson
The Journal of Physical Chemistry, Vol. 86, No. 5, 1982
TABLE X: Potential Parameters and Contributions of Different Terms to t h e Total Potential for Alkali Metal Dimers, Hydrogen, and Halogen and Hydrogen Halide Molecules A molecule
H,
llP,
A, eV 31438.9
/I-'
9.5792
0.1044
Li Na 1 K, Rb 2 cs,
7687.5 5427.3 4187.1 3568.0 3317.2
2.7308 2.4105 1.9343 1.8495 1.7550
0.3662 0.4149 0.5170 0.5407 0.5698
F2
23479.3 32484.6 44133.1 76624.8
6.2118 4.3518 4.1394 3.9452
0.1610 0.2298 0.2416 0.2535
41189.5 31934.0 27596.8 23435.9
7.8912 5.7618 5.2462 4.6956
0.1267 0.1736 0.1906 0.2130
c4 Br2 1,
HF HCl H Br HI
' V' =
--C/Reb+ A exp(-Re/p).
5.089
30.493
25.741
4.752
6.322 4.015 2.682 1.893 1.673
5.196 3.246 2.120 1.455 1.277
1.125 0.769 0.561 0.438 0.396
41.89 505.7 773.5 1302.7
5.153 8.191 5.449 3.620
3.510 5.681 3.458 2 .O 64
1.643 2.510 1.991 1.556
21.27 108.34 163.70 268.37
35.769 25.219 20.481 15.699
29.658 20.597 16.566 12.499
6.111 4.622 3.91 5 3.200
2305.9 3421 .O 9775.1 10689.7 13529.3
(0.1327)b (0.1671)b (0.17 3 0 ) b (0.1790)b
Values in parentheses are the arithmetic means of the
p
values f o r H: and X,.
801
i
701
60-
!
v
-5
(eV) -4-
40L
I
%,
I
1.0
1
I
I
I
1.5 2.0 2.5 3.0 I n t e r a t o m i c dislcnce
I
I
3.5
4.0
I
1
",5
(81
Flgure 4. Plot of the repulsion coefficient, A , of alkali metal dimers, hydrogen, and halogen and hydrogen halide molecules vs. their interatomlc distances.
from the various sources.36,38,39,45-48 The data employed and the potential parameters obtained are shown in Tables IX and X, respectively. Contributions of the attraction and repulsion terms to the total potential are also shown in Table X. The potential parameters and total potential energies have been plotted vs. interatomic distances for alkali metal dimers, hydrogen, and halogen and hydrogen halide molecules (Figures 3-5). They (with the exception of F2) fall either on straight lines or on the smooth curves indicating the consistency of the calculation. It is found on comparison that the arithmetic mean of the p values for homonuclear molecules is approximately equal to the p values for the heteronuclear molecules [that is pij = (pii + p i j ) / 2 ] . p values for alkali halide molecules obtained in this manner are shown in Table XI. These values agree with the values obtained directly from the Rittner and truncated Rittner potentials to h0.04 A. With the hydrogen halides, the maximum deviation is about (45) JANAF Thermochemical Tables, National Bureau of Standards, US., 1971, No. 37. (46) 'Tables of Interatomic Distances and Configuration in Molecules and Ions"; The Chemical Society, London; Spec. Publ. 1958, No. 11. 1965, No. 18. (47) Barrow, G. M. "Physical Chemistry"; McGraw-Hill: New York, 1979; 4th ed. (48) Coulson, C. A. "Valence"; Oxford University Press: New York, 1956; p 180.
Flgure 5. Behavior of the potential for alkali metal dimers, hydrogen, and halogen and hydrogen halide molecules.
TABLE XI:
Comparison of Potential Parameters
LiI
0.2531 0.3046 0.3220 0.3530
0.2504 0.3020 0.3199 0.3512
0.2636 0.2980 0.3039 0.3099
0.2111 0.2728 0.2930 0.3229
0.2640 0.2722 0.2742 0.2802
NaF NaCl NaBr NaI
0.2755 0.3311 0.3523 0.3807
0.2738 0.3293 0.3507 0.3791
0.2880 0.3224 0.3283 0.3342
0.2600 0.3187 0.3378 0.3660
0.2783 0.2954 0.3129 0.3131
KF KCI KBr
KI
0.3039 0.3572 0.3786 0.4108
0.2996 0.3545 0.3760 0.4085
0.3390 0.3734 0.3793 0.3853
0.2932 0.3600 0.3808 0.4115
0.3096 0.3324 0.3453 0.3489
RbF RbCl RbBr RbI
0.3121 0.3654 0.3916 0.4222
0.3071 0.3627 0.3892 0.4200
0.3509 0.3853 0.3912 0.3971
0.3065 0.3762 0.3976 0.4289
0.3257 0.3532 0.3587 0.3623
CsF CsCl CsBr CsI
0.3249 0.3732 0.3967 0.4212
0.3175 0.3705 0.3940 0.4190
LiF LiCl LiBr
0.3654 0.3166 0.3453 0.3998 0.3923 0.3517 0.4057 0.4147 0.3105 0.4117 0.4475 0.3187 ' From T Rittner potential using polarizabilities computed from the experimental dipole moments. From Rittner potential using polarizabilities computed from the experimental dipole moments. Arithmetic mean of t h e P values for M, and X, (see Table X). From the plot of P values of H2, Liz, Na,, K2, Rb,, and Cs2 vs. their interatomic distances (see Figure 3). E Smith.4P
The Journal of Physical Chemistty, Vol. 86, No. 5, 1982 855
Empirical Potential Parameters
TABLE XII: Data Used and Repulsive Parameter Computed for Alkali Halide Crystals
MX
Ro,a
-U,e V
LiF LiCl Li Br LiI
2.013 2.570 2.751 3.006
10.7362 8.8398 8.3631 7.8449
Na F NaCl Na Br Na I
2.317 2.820 2.989 3.236
9.5652 8.1454 7.7413 7.2957
KF KCl KBr KI
2.674 3.146 3.300 3.533
Rb F RbCl RbBr RbI CsF CSCl c s Br c sI
1048c,Q e V cm6
.,b A
0.655 10.67 45.38 8 3.01
u,c
a
P,d
a
u,e
a
u,f
a
0.2842 0.2535 0.2498 0.2120
0.2839 0.2495 0.2355 0.1886
0.2504 0.3020 0.3199 0.3512
0.299 0.342 0.353 0.430
0.2614 0.2786 0.2882 0.2617
12.23 88.00 234.05 272.13
0.2807 0.2692 0.2891 0.2521
0.2761 0.2455 0.2402 0.1996
0.2738 0.3293 0.3507 0.3791
0.330 0.317 0.340 0.386
0.2555 0.2903 0.3014 0.3099
8.5082 7.4097 7.0677 6.7257
90.50 273.38 506.18 780.18
0.2820 0.2825 0.3138 0.2956
0,2561 0.2313 0.2410 0.1964
0.2996 0.3545 0.3760 0.4085
0.334 0.337 0.335 0.355
0.2770 0.3048 0.3295 0.3264
2.826 3.291 3.445 3.671
8.1351 7.1402 6.8397 6.5288
229.69 390.09 504.31 1092.26
0.2968 0.2817 0 2894 0.2971
0.2438 0.2174 0.2188 0.1741
0.3071 0.3627 0.3892 0.4200
0.328 0.31 8 0.335 0.337
0.2945 0.3233 0.3321 0.3431
3.005 3.47 3.62 3.83
7.6687 6.8293 6.5392 6.2594
667.84
0.3505
0.2528 0.2018 0.2142 0.1807
0.31 7 5 0.3705 0.3940 0.4190
0.282
From ref 52. Considering dispersion energy. gas molecules. e Smith.4e f Catlow e t aLzo
Ignoring dispersion energy.
Repulsive parameter for alkali halide
TABLE XIII: Dispersion Energies ( - C / R e 6 )for Alkali Halide and Hydrogen and Halogen Molecules (in e V ) Slater-Kirkwood formula London formula
MX or
present work
true
IP
N as the outer subshell electrons
eff IP
N as the effective electron no.
P
x 2
I"
IIQ
IQ
IIQ
IQ
IIQ
IQ
IIQ
LiF LiCl LiBr LiI
0.785 1.809 2.734 3.014
0.612 1.306 2.234 2.537
0.015 0.01 7 0.01 5 0.01 2
0.006 0.007 0.006 0.004
0.033 0.036 0.033 0.027
0.013 0.014 0.012 0.010
0.099 0.062 0.051 0.038
0.045 0.024 0.019 0.014
0.110 0.088 0.081 0.069
0.052 0.037 0.033 0.027
Na F NaCl Na Br Na I
0.923 1.580 2.296 2.278
0.602 1.148 1.882 1.883
0.01 3 0.020 0.019 0.017
0.025 0.035 0.033 0.029
0.029 0.044 0.043 0.039
0 055 0.079 0.073 0.064
0.086 0.074 0.066 0.055
0.137 0.114 0.099 0.080
0.105 0.11 3 0.112 0.105
0.168 0.174 0.168 0.155
KF KCl KBr KI
0.834 1.120 1.330 1.752
0.520 0,787 1.015 1.459
0.022 0.032 0.031 0.029
0.038 0.053 0.050 0.045
0.048 0.072 0.070 0.065
0.085 0.120 0.11 3 0.101
0.105 0.101 0.094 0.081
0.161 0.147 0.133 0.112
0.1 54 0.169 0.171 0.165
0.235 0.247 0.242 0.229
RbF RbCl RbBr RbI
0.657 1.084 1.695 1.882
0 357 0.784 1.413 1.620
0.025 0.037 0.037 0.035
0.042 0.060 0.057 0.051
0.056 0.084 0.083 0.078
0.095 0.135 0.127 0.115
0.108 0.108 0.102 0.090
0.162 0.154 0.141 0.121
0.173 0.193 0.196 0.191
0.258 0.274 0.270 0.257
CsF CSCl Cs Br c sI
0.708 0.965 1.252 1.131
0.335 0.656 0.968 0.875
0.034 0.048 0.048 0.045
0.058 0.078 0.073 0.066
0.077 0.109 0.107 0.101
0.1 31 0.175 0.165 0.149
0.125 0.125 0.119 0.107
0.189 0.178 0.164 0.143
0.228 0.243 0.246 0.240
0.341 0.346 0.340 0.324
9.535 0.658 0.837 0.696 0.569
9.535 0.658 0.837 0.696 0.569
10.965 1.481 1883 1.567 1.280
10.965 1.481 1.883 1.567 1.280
16.525 1.564 1.408 1.O68 0.797
16.525 1.564 1.408 1.068 0.797
16.525 1.399 1.781 1.548 1.381
16.525 1.399 1.781 1.548 1.381
H2
F* c 1 2
Br2 12
IIQ
Polarizabilities used for alkali cations and halide anions were I, computed from the experimental dipole moments o f alkali halide molecules (this work), and 11, from Tessman et aLZ9
rt0.03 A (Table X). Also, the p values for hydrogen halides fall on the smooth curve obtained by the plot of p values vs. interatomic distances for alkali metal dimers and hydrogen molecule (Figure 3). This encouraged us to obtain the p values for all the alkali halide molecules from this plot, using their interatomic distances. The values obtained are also shown in Table XI. These values agree with the values obtained from the Rittner potential to about h0.03 A. So this plot (Figure 3) can be used to obtain the
values for other molecules whose internuclear distances are known. It is interesting to note that the p values for halogen molecules fall below this curve. Also, the plot of p values vs. the interatomic distances for the HX, Mz, and Hz molecules is approximately a straight line with slope equal to 0.135. Thus it was found empirically that pij = 0.135Rjj. Using spectroscopic data for the alkali halide monomers Gilbert49has shown that approximate additivity rules of p
858
Gowda and Benson
The Journal of Physical Chemistry, Vol. 86, No. 5, 1982
the form Rij = Ri + Rj and pij = pi + pj hold for the radii and hardness parameters in the Born-Mayer repulsive potential. Smith,5oBrumer,5l and others6ahave also discussed the combining rules for p . Potential Parameters for Ion-Ion Interactions in Crystals Independently, we have also determined the repulsive potentials for alkali halide crystals with NaCl structure by using recently available values for the lattice energies but neglecting dispersion terms
AMe2 u=-R + 6B exp( --:)
(au/aR),,
=
o
TABLE XIV: Energies for the Hydration of Alkali (+ ) and Halide (-) Ions (kcal mol-')
M'. (OH,),, calcda calcdb calcd' calcdd calcde calcdf calcdg calcdh calcdf calcdJ exptk
Li'
Na'
K+
Rb'
Cs+
36.1 36.4 36.4 36.0 36.0 34.6
23.4 24.4 24.3 22.1 24.2 24.6 25.6 35.8 27.9 42.9 24.0
17.4 18.0 17.9 15.8 16.9 17.6 17.6 23.3 18.8 31.2 17.9
15.3 15.8 15.7 13.6 14.5 15.0 15.5 22.1 15.9 28.1 15.9
13.8 14.3 14.1 11.8 12.4 12.8 13.4 21.7 15.0 23.1 13.7
58.4 53.3 -34.0
(15)
U , AM,e , and R have the same meaning as before; B and u are
the repulsion potential coefficient and parameters, respectively. We felt that dispersion energy was small enough in crystals to omit in our calculations of B and Q using the above equations. So we made two calculations: (1) ignoring the dispersion energy, and ( 2 ) considering dispersion energy as computed by Hajj from optical data.52 Lattice energies were taken from the "Handbook of Chemistry and The interionic distances used were from Smith and Cain41and P a ~ l i n g .The ~ ~ determined parameters along with the data employed are shown in Table XII. The results are also compared with the gas-phase parameters and other calculations. The p values for gas molecules are generally higher than the u values for crystals. Comparison of Dispersion Energies We have also estimated the van der Waals' constants, C,and hence the dispersion energies for all the alkali halide molecules, hydrogen, and halogen molecules using London formulaMwith true and effective ionization potentials" and by Slater-Kirkwood formula55with N as outer subshell electrons and effective number of electrons,"@ and compared them with the values computed in this work. London Formula with True Ionization Potentials.
Here a1and a2are the polarizabilities of the alkali cations and halide anions, respectively, I 2 is the second ionization potential of the alkali atom, and E is the electron affinity of the halogen atom. For the homonuclear molecules the above equation becomes
where I is the first ionization potential of the atom and CY is the atom polarizability. London Formula with Effective Ionization Potentials.54
F' C1
EW
I-
calcd' calcdm exptl" calcd' calcdm exptln expto calcd' calcdm exptn calcd' dcdm exptn expto
17.5 22.9 23.3 13.3 14.2 13.1 14.9 12.4 12.5 12.6 11.3 10.6 10.2 11.1
16.6 21.0 16.6 12.8 13.6 12.7 12.6 12.0 12.1 12.3 10.9 10.4 9.8 9.9
14.7 18.0 13.7 11.5 12.1 11.7 11.5 10.8 10.9 11.5 9.9 9.5 9.4 9.3
13.3 16.0 13.5 10.6 11.2 11.1 10 9 10 0 10.2 10.9 9.2 9.0
11.0 13.7 13.2 8.9 10.0
8.4 9.2 7.8 8.2
Calculated by employing potential parameters computed from the Rittner potential using polarizabilities computed from the experimentally determined dipole moments by fixing t h e polarizability o f (a) Lit (0.029 A'), ( b ) Na' (0.312 A'), and ( c ) K' (1.136 A3). d-f Calculated by employing potential parameters computed from t h e truncated Rittner potential using polarizabilities computed from the experimentally determined dipole moments by fixing the polarizability of ( d ) Li' (0.029 A'), ( e ) Na' (0.312 A 3 ) , and ( f ) K+ (1.136 A'). P Calculated by using crystal parameters. Calculated by using parameters o f 0-He, 0-Ne, 0-Ar, 0-Kr, and 0-Xe, determined from scattering experiments (Foreman et al.4v) for computing repulsive interactions of O-Li', 0-Na+, 0-K', 0-Rb', and 0-Cs', respectively. While using these parameters SlaterKirkwood formula was used for computing dispersion energies. ' Calculated by using parameters from Eliezer and Krindel"' for repulsive interactions and Slater-Kirkwood formula for computing dispersion energies. Dzidic and Kebarle's c a l ~ u l a t i o nby~ ~ using Lennard-Jones potentials. Experimental values f r o m Dzidic and Kebarle.3a Calculated with Pauling's radii." Calculated with Ladd's radii.'8 Fxperimental values from Arshadi e t al.3b ' Experimental values from Keesee and Castleman.2d
systems and 2.25 for systems with closed sp shells. For homonuclear molecules
where U = j I . Slater-Kirkwood Formula. where U1 = jE, U , = jI,, and j = 1.15 for two electron (49) Gilbert, T. L. J . Chem. Phys. 1968,49, 2640. (50) Smith, F. T. Phys. Reu. A 1972, 5, 1708. (51) Brumer, P. Phys. Reu. A 1974, 10, 1. (52) (a) Hajj, F. J. Chem. Phys. 1966, 44, 4618. (b) Narayan, R. J. Phys. Chem. Solids 1977, 38, 1097. (53) London, F. Z.Phys. Chem. 1930, B l l , 222. (54) Pitzer, K. S. Adu. Chem. Phys. 1959, 2, 59. (55) Slater, J. C.; Kirkwood, J . G. Phys. Rev. 1931, 37, 682. (56) Scott, R. A.; Scheraga, H. A. J. Chem. Phys. 1965, 42, 2209.
Edisp
3e h ala2 1 - (20) 2m1I2 ( c Y ~ / N+~(aZ/N2)lI2 ) ~ / ~ R,6
= --
where h = h / 2 n , e and m are the electronic charge and mass,respectively, and h is the Planck's constant. a1and cyz are the polarizabilities of the alkali and halide ions, respectively. Nl and N2are the number of electrons in the outer subshell or the effective number of electrons for alkali and halide ions, respectively." For the homonuclear molecules, the above equation takes the form
Empirical Potential Parameters
The effective N values exceed the actual number of outer subshell electron^.^^^^ The data used for the calculation of dispersion energies by these formulae were taken from various so~ces.36~37~40~47~M~~ Polarizabilities for alkali cations and halide anions used were from (i) present studies, which have been computed from the experimental dipole moments (see Table I) and (ii) Tessman et al.29(Table I). Atom polarizabilities were taken from P i t ~ e rand ~~ Hirschfelder et al.57 The estimated dispersion energies are shown in Table XIII. The results are also compared with the present calculation. The dispersion energies computed from the empirical formulae are far lower than the values computed in the present calculation.
Test of Consistency of the Potential Parameters Obtained Finally, to check the consistency of the potential parameters determined for alkali halide gas molecules and crystals, we used the parameters of alkali fluorides to estimate the repulsion and dispersion energy contributions in the calculation of ion hydration energies. The details of our electrostatic calculation has been discussed in a separate paper.58 A typical set of results for both alkali (+) and halide (-) ions obtained by using both gas-phase and crystal potential (57) Hirschfelder, J. 0.; Curtiss, C. F.; Bird, R. B. "Molecular Theory of Gases and Liquids"; Wiley: New York, 1954; p 950. (58)Gowda, B. T.; Benson, S. W. J. Phys. Chem. In press.
The Journal of Physical Chemistry, Voi. 86, No. 5, 1982
857
parameters are given in Table XIV. The results are also compared with the experimental value^.^,^ The ion hydration energies obtained by these two sets (gas phase and crystal) of parameters agree very well with each other and also with the experimental values thus indicating the consistency of the potential parameters employed. It appears that the parameters computed from Rittner potential are more successful in describing the gas-phase hydration energies than those computed from the truncated Rittner potential. We have also used the parameters of 0-He, 0-Ne, 0-Ar, 0-Kr, and 0-Xe, recently determined from the scattering experiments? for computing the repulsive interactions of 0-Li+, 0-Na+, 0-K+, 0-Rb+, and 0-Cs+, respectively, and Eliezer and K ~ i n d e parameters l~~ computed from the repulsive potentials for 0-0 and the noble gas atom pairs. While using these parameters, the Slater-Kirkwood formula was used for computing dispersion energies. The computed hydration energies obtained by using these parameters along with Dzidic and Kebarle's calculated values by using parameters computed from Lennard-Jones potentials are also shown in Table XIV. A comparison of these results with the experiment indicates that the parameters computed from the scattering experiments are not successful in describing the gas-phase hydration energies.
Acknowledgment. B.T.G. thanks the Government of India, New Delhi, for the award of a National Scholarship for post-doctoral research abroad. ~~
~~~
(59) Eliezer, I.; Krindel, P. J. Chen. Phys. 1972,57,1884.
