COVALENT AND IONIC BONDORDERSIN ALKALIHALIDEMOLECULES
85
Covalent and Ionic Bond Orders: Applications to the Alkali Halide Moleculesla
by Owen C. Hofer and Ricardo Ferreiralb Chemistry Department, Indiana University, Bloomington, Indiccna 47406
(Received May 7, 1966)
Energy surfaces E(z,r) for the lithium, sodium, and potassium halides are constructed by considering the energy of the molecules as a function of the ionicity, z, and of the interatomic distance, r. The binding energy is partitioned in a homopolar term, a Madelung term, and a charge-transfer term. The covalent energy is described by a Morse curve with the minimum at ro = r ~ ( c o v ) r~(cov). The Madelung energy is represented by a Born-Mayer function with experimental values for the repulsion constants. The chargetransfer energy is a function of atomic terms. Bond energies, ionicities, bond distances, and vibration frequencies are calculated and found to be in good agreement with the experimental values. The derivative of the primary dipole moment with respect to r is also calculated.
+
Introduction The analysis of the structure of bond energies2a. is certainly important for an understanding of the nature of the chemical bond. Significant and extensive contributions in this area have been made by Mulliien. 2b More recently, Ruedenberg and collabor a t o r ~have ~ made a detailed study of the partitioning of molecular binding energies from the distribution of the electron density and pair density. Within the framework of zeroth-order approximations it is helpful to partition the bond energy in homopolar, electrostatic (Madelung), and charge-transfer energie~,~ and the present paper is an extension of the previous one in that we now included the interatomic distance as a variable in the energy equation. We have restricted the applications of the method to the lithium, sodium, and potassium halides (extension to the rubidium and cesium halides is straightforward). These compounds, because of their high ionicities, are not particularly good systems to deal by ionic-covalent models. However, the reasons for our choice are twofold: first, the alkali halides are the diatomics for which a reasonable potptial function of the ionic structure can be written, namely, the Born-Mayer p06entia1,~and secondly, for these molecules we can assume that the halogen atoms use pure p orbitals and the alkali metals pure s orbitals and assign electronegativity values with some confidence. Neither of these conditions obtains, for example, in
the hydrogen halides or in the interhalogen compounds. It should be pointed out, however, that in principle the method can be applied to all diatomic molecules. The gaseous alkali halides have been described by pure electrostatic models, notably by Rittner,6 who introduced dipole polarizabilities as a correction to the rigid-sphere model and obtained reasonable values for bond energies, dipole moments, etc. The repulsive constants, however, were adjusted from experimental data. We concur with Brewer and Brackett’ that the (1) (a) Work supported by grants from the National Science Foundation and the U.S.A.F. Office of Scientific Research; (b) on leave of absence from the University of Brasilia, Brazil; address correspondence to Department of Chemistry, Columbia University, New York, N. Y. 10027. (2) (a) R.S. Mulliken, “Report of the Shelter Island Conference on Quantum Mechanical Methods in Valence Theory,” 1951, p. 64; (b) R. 8. Mulliken, J. Phys. Chem., 56, 295 (1952); Rec. Chem. Progr., 13, 67 (1952); J . Chem. Phys., 23, 1833, 1841, 2338, 2343 (1955); J . Am. Chem. Soc., 77, 884 (1955); S. Fraga and R. 8. Mulliken, Rev. Mod. Phys., 32, 254 (1960); R. S. Mulliken, J. Chem. Phys., 36, 3428 (1962). (3) K.Ruedenberg, Rev. Mod. Phys., 34, 326 (1962); C. Edmiston and K. Ruedenberg, J. Phys. Chem., 68, 1628 (1964);E.M. Layton, Jr., and K. Ruedenberg, ibid., 68, 1654 (1964); R. R. Rue and R. Ruedenberg, ibid., 68, 1676 (1964). (4) R. Ferreira, ibid., 68, 2240 (1964); R. P. Iczkowski, J . Am. Chem. SOC.,86, 2329 (1964). This partitioning was originally suggested by I. Lindquist, Nova Acta Reg. SOC.Sci. Uppsala, 17, 111 (1960),and many of the ideas contained in our first paper are outlined in C. K. Jgjrgensen, “Orbitals in Atoms and Molecules,” Academic Preas, New York, N. Y., 1962, Chapter 7. (5) M. Born and J. E. Mayer, 2.Physik, 75, 1 (1932). (6) E. S. Rittner, J . Chem. Phys., 19, 1030 (1951).
Volume 70,Number 1 January 1968
OWENC. HOFERAND RICARDO FERREIRA
86
main reason for the good agreement between calculated and experimental values of bond energies by electrostatic models lies in the use of the constants in the exponential repulsion term as variable parameters. It is interesting to note that, at least for the lithium halides, the introduction of higher polarizabilities destroy the agreement.8 On the other hand, quantum mechanical calculationsg show that the bond in LiF(g) has considerable covalent character.
General Formulation Consider a localized two-center bonding A40 in a diatomic molecule -4B *M
=
+
CAPA CB*B
(1)
Let XA(O) and XB(O) be the electronegativities of atoms A and B. If XA(O)> XB(O), then 3 6 1. We will further assume that *A and *B are orthogonal, that is, C A ~ cB2 = 1. In this case the ionicity is defined as = C A ~- c B 2 (0 6 x 6 1). We can also define a covalent bond orderlo AB = ~ n 3 c A 3 c B 1=
ICB~
I C A/
+
2CAcB.
It is easily seen that PAB
= 2cAcB = (1
- x2)'/'
(2)
We shall now write the expression for the energy of the system AB as a function of the interatomic distance T and of the ionicity z. As usual, the coordinates are chosen such that the energy of the separated atoms at infinite distance is zero. The energy equation is
E(x,r)
=
(i -
Do{e-2a(r--ro)
- 2e-a(r-ro) ](I
BOe-")s2 - XA(O)Z
- ,2)1/2 -
+
'/Z(IB
- AB)x2
(3)
where Do is the covalent bond energy between atoms A and B, ro is the covalent bond distance, a is the Morse constant for the covalent structure, I A and IBare the valence-state ionization potentials (VSIP), AA and AB are the valence-state electron affinities (VSEA), and X A (0) and X g (0) are Mullken's electronegativities,ll XA(0) = l / 2 ( : I ~ AA) a n d X ~ ( 0 )= '/z(IB A B ) . The first term in eq. 3 is the homopolar energy; it is the product of the covalent bond energy times the covalent bond order (the unit of covalent bond order is one homopolar bond). The covalent bond energy has a maximum value for T = TO. I n this paper r0 = '/z{re(A-A) ye(B-B)) and DO = (D,(A-A) X De(B-B)]'/'. We also assume that the covalent bond is desscribed by a Morse function12with a constant __a = Z/kw/2Do,where ko is calculated from roby Badger's rule.13
+
+
+
The Journal of Ph&al
Chemistry
(XA@ - XB(O))(X- xZ) ( 3 4 The first term on the right-hand side is the covalent bond energy, the second is the ionic bond energy, and the last term is a function of the electronegativity difference. This last term is always negative (stabilizing) and has a maximum for x = '/z. Since x N (XA(0) - XB(O)), at least for small values of x the last term is essentially proportional to ( X A ( 0 ) - XB(O))~ and represents the contribution of the atomic terms to the ionic-covalent resonance energy. I n the two limiting cases, x = 0 and 2 = 1, E(x,r) reduces to, respectively, Do{e-zu(r-ro) - 2e- a(r (covalent bond energy) and ( - l / r - AA I B ) (ionic bond energy). For z = 0 and r = m , E(x,r) = 0. E($,?) is, in fact, a potential surface,
+
+ ' / ~ ( I A- AA)x' +
XB(0)x
The second term in the energy equation is the Madelung energy, given by the product of the ionic bond order1* QAQB = x2 times the Born-Mayer potential function for the ion pair B+A-. Finally, the last four terms in eq. 3 correspond to the variation in energy due to the charge transfer from B to A.158These terms are independent of r and represent the contribution of the atomic term values to the molecular binding energy. It will be shown in a forthcoming paper that eq. 3 can be derived from SCF-MO theory within the usual approximations of the point charge description.16b It is instructive, however, to consider an alternative form of (3), closely connected with the VB description. Equation 3 can be written E(x,r) = Do(e- 2 a ( r - d - 2e-a(r-ro))(1 - %2)l/2 -
1
+
(7) L. Brewer and E. Brackett, Chem. Reu., 61,425 (1961). (8) W. Klemperer, W. G. Norris, A. Buchler, and A. G. Emslie, J . Chem. Phys., 33, 1534 (1960). The discovery that some aIkaIine earth dihalides are bent (L. Wharton, R. A. Berg, and W.Klemperer, ibid.,39, 2023 (1963))represents a drastic failure of the ionic model. (9) B. J. Ransil, Rev. Mod. Phys., 32,239, 245 (1960); S. Fraga and B. J. Ransil, J. Chem. Phys., 34, 727 (1961); B. J. Ransil, ibid., 36, 1127 (1962); A. D.McLean, ibid., 39, 2653 (1963). (10) C. A. Coulson, Proc. Roy. Soc. (London), A169, 419 (1939); R.S. Mulliken, ref. 2a, p. 67. (11) R. S. Mulliken, J . Chem. Phys., 2, 782 (1934); J . chim. phys., 46, 497 (1949); W. Moffit, Proc. Roy. Soc. (London), A196, 510 (1949); A202, 548 (1950). (12) P. Morse, Phys. Em., 34, 57 (1929). (13) R.M.Badger, J . Chem. Phys., 2, 128 (1934); 3, 710 (1935). (14) R. S. Mulliken, ibid., 23, 1841 (1955). (15) (a) H.0. Pritchard and F. H. Sumner, Proc. Roy. SOC.(London), A235, 136 (1956); R. P. Icekowski and J. L. Margrave, J . Ana. Chem. Soe., 83, 3547 (1961); J. Hinze, M.A. Whitehead, and H. H. Jaff6, ibid., 85, 148 (1963); R. Ferreira, ref. 4. For a just5cation of the parabolic relation between the energy of an atomic orbital and its occupation number see G.Klopman, ibid., 86, 1463 (1964); (b) J. A. Pople, Trans. Faraday Soc., 49, 1375 (1953); G. Klopman, J . Am. Chem. Soc., 86, 4550 (1964).
COVALENT AND IONIC BONDORDERS IN ALKALIHALIDEMOLECULES
the two variable counterparts of the potential curve E(r). The equilibrium values of r and x are obtained by minimizing E(x,r) with respect to the two variables, = 0 and (dE/Bx), = 0. One has
(f - kBOe-kT}x2 = 0
{zr - 2B0e--"}x - XA(O) + ( I A - AA)X+ XB(0) + (IB - AB)x 0 =
(4)
(5)
Equation 4 represents the condition of zero resultant force at equilibrium distance. Equation 5 means that the potential (derivative of the energy with respect to charge) is also zero at equilibrium. The energy, however, is the molecular energy and contains atomic term values plus exchange and ionic terms. The solution of the simultaneous eq. 4 and 5 gives re and x,, and, by substitution in (3), De = -E(xe,re). The force constant k,, hence the vibrational frequency W , is easily obtained
87
Thus, according to (7) the molecule AB would dissociate into Bf2J and AW2J,and this would correspond to an energy E ( x ~ , m )< 0. This is manifestly in error17since AB dissociates into neutral atoms A and B. The ionicity given by (8) is independent of bond distance. This is the result that obtains from the principle of electronegativity equalization.l8 The ionicities obtained from (8) are very As pointed out by J@rgensenZ0and Klopman,21a the difficulty is that in minimizing the energy with respect to x one must realize that the energy in BE/bx is a molecular energy; therefore eq. 8 should include not only the initial atomic term values but also exchange and Madelung terms. This is related to the fundamental difficulty of defining an effective electronegativity of an atom in a molecule. Only if it were also possible to partition the molecular terms unequivocally between A and B would it be meaningful to assign electronegativities for the atoms in the molecule AB.21b Referring to eq. 4,one should note that in general
(9) Since Do(l - x2)"' that za( e - a ( T e - T o ) -
> 0, the
inequality (9) requires < 0, hence re < ro. One does arrive at the well-known result that the equilibrium interatomic distance in a heteropolar molecule AB is shorter than the sum'of the covalent As noted by Schomaker and radii (ro = r A TB). Stevenson,22the shortening is (roughly) proportional to the electronegativity difference. Barte1Iz3gave a rationalization of the Schomaker-Stevenson rule in terms of an arbitrary additional potential superimposed e--2a(~e--ro)
1
+
Rearranging (5)
2
r
+ 2B0e-kr+ (IA- AA) + (IB- AB)
This equation gives the value of x as a function of r along a path of minimum energy. Because the atomic terms in (3) appear as independent of r this path diverges from the actual potential curve of the molecule for distances far from re (see below). For values close to re, however, this path is very close to the actual curve16and it is possible to calculate (dx/dT)T,with some confidence. The primary dipole is (exr), hence (dp,,/ ar)Te = e{ xe ?e@Z/dr)Te ). The limitations of eq. 7 should be pointed out. If r = 03 x according to (7) reduces to
+
)
(16) This i s seen from the good results found in the calculation of the depth of the curve ( D e )and its curvature (he). (17) Mathematically this results from the fact that E(x,T)is square in x and accordingly there are two sets of solutions; thus, for r = a ,x = 0 and x = 223.. Only the first solution has a physical meaning. (18) H. Hinze, M. 8. Whitehead, and H. H. Jaff6, J. Am. Chem. SOC., 85, 148 (1963). The principle of electronegativity equalization was originally formulated by R. T. Sanderson, Science, 114, 670 (1951). See also R. Ferreira, Trans. Faraday Soc., 59, 1064 (1963). (19) R. Ferreira, ref. 4; N. C. Baird and XI. A. Whitehead, Theoret. Chim. Acta, 2, 259 (1964). Figure 2 of the paper by Hinze, et al. (ref. 18), showing a reasonable value for the ionicity of HF is in
error. (20) C. K. J@rgensen,"Orbitals in Atoms and Molecules," Academic Press, New York, N. Y., 1962, p. 80. (21) (a) G. Klopman J. Am. Chem. Soc., 86, 1463 (1964); see also H. 0. Pritchard, ibid., 85, 1876 (1963); R. G. Pearson and H. B. Gray, Inorg. Chem., 2, 358 (1963). (b) This has been pointed out by Professor Mulliken in his 1949 papers in J . chim. phys. (ref. 11). These papers contain a wealth of critically judged informations on MO theory. (22) V. Schomaker and D. P. Stevenson, J . Am. Chem. Soc., 63, 37 (1941). (23) L. s. Bartell, Tetrahedron, 17, 177 (1962).
volume 70,Number 1 January 1966
88
OWENC. HOFERAND RICARDO FERREIRA
Table I
LiF LiCl LiBr LiI
NaF NaCl NaBr NaI KF , KC1 KBr KI
0.943 0.887 0.939 0.848 0.849 0.852 0.881 0.843 0.860 0.873 0.891 0.862
2.045 2.330 2.477 2.669 2.248 2.533 2.680 2.876 2.670 2.955 3.102 3.294
1.658 2.074 2.112 2.436 2.059 2.402 2.458 2.673 2.384 2.776 2.839 3.070
1.5639 2.0207 2.1704 2.3919 1.926“ 2.3606 2.5020 2.7114 2,1714 2.6666 2.8208 3.0478
7.504 8.831 9.519 9.915 8.390 9.822 10.39 10.81 9.841 11.63 12.14 12.69
6.284d 7.119* 6.19“ 6.64“ 8.5’
10.48’ 10.41’
135.5 111.4 100.1 80.4 118.5 100.6 90.2 75.3 116.4 99.9 89.8 76.0
137.5 111.9 100.2 84.6 114.0 97.5 86.7 72.7 117.6 101.3 90.9 76.8
9 62 683 609 549 594 429 348 310 429 301 231 203
964d 641h 563h 49gh 536i 366/ 302/ 258’ 42gi 281f 213’ 186i
-0.302 -0.185 -0.338 -0.307 -0.244 -0.109 -0.213 -0.227 -0.254 -0.080 -0.114 -0.113
2.12 2.42 1.08 0.48 2.66 2.83 1.72
1.13 1.22 3.12 2.72 2.47
’
L. Sutton, Ed., ref. 26. D. R. Lide, P. Cahill, and L. P. Gold, J. Chem. Phys., 40, 156 (1964), and references therein. R. K. L. Wharton, et aE., ref. 39. e A. Honig, M. Mandel, M. L. Stitch, and C. H. Bauer and H. Lew, Can. J. Phys., 41, 146 (1963). See ref. 41. AHOzss values according to L. Brewer and E. Brackett, Chem. Rev., 61,425 (1961). Townes, Phys. Rev., 96,629 (1954). W. Klemperer, W. G. Norris, A. Biichler, and A. G. Emslie, J . C h m . Phys., 33, 1534 (1960). R. K. Ritchie and H. Lew, Cun. J. Phys., 42,43 (1964). j J. R. Rusk and W. Gordy, Phys. Rev., 127,817 (1962).
‘
‘
The zalculated bond energies and vibrational freon the Morse curve of the covalent structure. Bartell’s quencies are in good agreementwith the observed values, interpretation is thus identical with ours, the addiespecially considering that we did not use any adjusttional potential in our case being the Born-Mayer able parameter. The expected bond contraction potential. An interpretation of the bond length is observed. I n spite of the high ionicities the covalent contraction in the hydrogen halides based on the V B method was given some years ago by W a r h u r ~ t , ~ ~bond orders are not very low. The breaking down of the total bond energy according to (3a) is about 15% but his conclusions depend on drastic assumptions homopolar energy, 7501, ionic bond energy, and 10% concerning the derivative of the resonance integral resonance energy. j%~X\k,,,dr. It should be pointed out that other factors besides ionic character (notably the hybridization of the bonding orbitals) may contribute to the observed bond shortening, even in diatomic molecules. (24) E. Warhurst, “Contribution a 1’Etude de la Structure Molecu-
Results and Discussion Table I shows the results of our calculations for the lithium, sodium, and potassium halides. The numerical computations were made at the Indiana University Research Computing Center with a program written by Mr. John P. Chandler.25 The covalent bond distances are from the tables edited by S ~ t t o n . Covalent bond energies are the geometric mean2’ of the bond energies D,(A-A) and D,(B-B) given by CottrelP and Herzberg.29 We also made calculations using the arithmetic mean rule30 but the results differ little from the ones reported here. VSIP are from Pritchard and Skinner31and Hinze and Jaff6.32 VSEA for the halogens are from Berry and re imam^^^ and those for the alkali metals are Clementi’s values.34 The repulsion constants in the Born-Mayer potential were calculated from the experimental data for the noble gases. 35 The Journal of Physical Chemistry
laire,” Volume Commemoratif Victor Henri, Desoer, Liege, 1948, p. 57. See also 0.E. Polansky and G. Derflinger, Theoret. Chimei. Acta, 1, 308 (1963). (25) A copy of the program can be secured through the Quantum Chemistry Program Exchange of the Chemistry Department, Indiana University. (26) “Tables of Interatomic Distances and Configuration in Molecules and Crystals,” L. Sutton, Ed., Special Publication No. 11, The Chemical Society, London, 1958. ~ (27) ~ L. Pauling and J. Sherman, J . Am. Chem. SOC.,59, 1450 (1937). (28) T. L. Cottrell, “The Strengths of the Chemical Bond,” 2nd Ed., Butterworth and Co. Ltd., London, 1958. (29)G. Herzberg, “Spectra of Diatomic Molecules,” 2nd Ed., D, Van Nostrand Go. Inc., New York, N. Y., 1950. (30)L. Pauling and D. M. Yost, Proc. NatE. Acad. Xci. U. X.,18, 414 (1963). (31)H. 0.Pritchard and H. A. Skinner, Chem. Rev.,55, 745 (1955). (32) J. Hinee and H. H. Jaff6, J . Am. Chem. SOC.,84, 540 (1962); J . Phys. Chem., 67,1501 (1963). (33) R. S. Berry and C. W. Reimann, J . Chem. Phys., 38, 1540 (1963). (34) E. Clementi, I B M J . Bes. Develop. Suppl., 9, 2 (1965). (35) E. A. Mason, J . Chem. Phys., 23, 49 11955); E. Whalley and G.Schneider, ibid., 23, 1644 (1955).
COVALENT AND IONIC BONDORDERS IN ALKALIHALIDEMOLECULES
The calculated ionicities are reasonable and show a marked uniformity. As shown b e f ~ r e ,inclusion ~ of overlap would change the numerical values, but not drastically. Sharply drawn trends within homologous groups probably reflect real situations. For example, lithium halides are more ionic than the corresponding sodium and potassium compounds.36 We also found that the bromides are more ionic than the corresponding fluorides (except LiF), chlorides, and iodides but the situation here is not as clear-cut. The results show that the ionicity does not depend solely on the atomic terms and that care should be taken when relating measures of bond ionicity (n.m.r. chemical shifts, n.q.r. coupling constants, thermochemical data, etc.) with differences in atomic electronegativities. We wish to point out the following: although the alkali metals and the halogens greatly differ in atomic orbital energies, the bonds in the alkali halides are very strong, essentially due to the high ionic bond orders. I n the simple LCAO-MO method, molecular orbitals derive from orbitals of the constituent free atoms. It is possible to show that if one approximates the soluOions of the secular equation to the square root, then the energy of the bonding orbital is
where
are the Coulomb integrals of atoms @AB is the resonance integral, and S A B is the overlap integral. It is seen that a strong bond (E
