RICARDO FERREIRA
2240
A Zeroth-Order Approximation for Bond Energies, Hybridization States,
and Bond Ionicities. I.
Diatomic Molecules and A'-B' Crystals'"
by Ricardo Ferreiralb Chemistry Department, Indiccna University, Bloomington, Indiana
(Receiwed .March 18,1964)
In the first part of the paper bond energies and ionicities of 33 diatomic molecules and crystals of the type AI-B' are calculated by writing the bond energy D(A-B) as a function of the ionicity II: and maximizing D(A-B) with respect to x. This is made possible by partitioning the bond energy into a sum of a homopolar term (a decreasing function of z) and a heteropolar tern? (the sum of an electrostatic term and a charge-transfer term). It is found that crystals are inore ionic than the corresponding diatomic molecules, and also that lithium halides are more ionic than the corresponding sodium and potassium halides. In the second part of the paper, the bond energy is expressed as a function of two variables, the ionicity and the degree of s-character (a') of the more electronegative atom. D(A-B), x, and aa for ten binary compounds are calculated by maximizing D(A-B) with respect to x and a. It is shown that the fluorine atom has a higher degree of s-character than the other halogens in corresponding compounds.
Introduction A zeroth-order approximation in quantum chemistry can be defined as one in which the molecular properties of interest to the chemist (bond energies, force constants, ionicities, etc.) are calculated from data of the corresponding isolated atoms (ionization potentials, electron affinities, etc.).2 Considering on one hand the great difficulties inherent in ab initio calculations of fair sized nzolecules and, on the other hand, the fact that many of the so-called electron correlation effects are implicitly included in the free atom data, it is possible that zeroth-order calculations will remain useful for some time to come. The most general physical principle operative in the chemical bonding is that the bond energy must correspond to a minimum in the total energy of the system with respect to the separated atoms. The bond energya of a heteroatomic molecule, D(A-B), can be given by the sum of a homopolar term and a heteropolar term. The latter is given by the sum of an electrostatic term plus the energy variation due to the process of charge transfer from the less electronegative atom to the more electronegative As pointed out by L i n d q ~ i s tin , ~ this partitioning of the bond energy the heteropolar term includes only the energy connected The Journal of Phgsical Chemistry
with charge transfer; the classical Coulomb iiiteraction between atoms, without electron transfer, is part of the homopolar bond energy. Lindqvist6 also noted that as we increase the ionicity of a bond, the energy due to the heteropolarity increases but simultaneously the homopolar energy decreases. We have done some work along these lines4bbased on the principle of electronegativity We realize now, however, that this generalization in its original form is not strictly valid (except if the bonded atoms are separated by an infinite distance!) and this led us to the decision of making a more detailed study of this subject. (1) (a) This work was supported by grants from the U.S.A.F. Office of Scientific Research and the National Science Foundation; (h) on leave of absence from the University of Brasilia, B r a d (2) We also make use of the experimental dissociation energies of homopolar molecules. (3) Since the energy minimum calculated from quantum mechanics includes the zero-point energy we will use, whenever it is possible, bond energies, De, and not bond dissociation energies, DO. These two quantities are related by the expression De = Do 4- '/zhvo. (4) (a) R. P. Iczkowski and J. L. Margrave, J. A m . Chem. Soc., 83, 3547 (1961); (b) R. Ferreira, Trans. Faraday Soc., 5 9 , 1064, 1075 (1963). (5) I. Lindqvist, Nova ActaReg. SOC.Sci. Uppsala, 17, No. 11 (1960). (6) R. T. Sanderson, Science, 114, 670 (1951).
ZEROTH-ORDER APPROXIMATION FOR BONDENERGIES
The One-Parameter 'Treatment Shull' has pointed out the difficulties that one encounters when trying to define the ionic character, or ionicity, of a chemical bond. In this paper we will define ionicity in the following way : consider a localized two-center bonding RIO in a diatomic molecule AB *M
=
CA+A
+
(1)
C B ~ B
2241
between two different atoms and, in fact, the concept of a normal covalent bond itself breaks down at a more sophisticated level. Pauling and Yostg first proposed an arithmetic mean rule
D(A-R)o
==
l/z(D(A-A)
+ D(B-B))
(5) This equation gives meaningless results when applied to the alkali metal hydrides'l and Pauling and Sherrnanl2later suggested a geometrical mean rule
+ + ZCACBSAB D(A-B)o (D(A-A) *D(B-B))"% (6) SAB= 0, so that + - The theoretical basis for eq. 6 is the criterion of bond the normalization condition reduces to The MO is normalized, hence
C A ~
C B ~
= 1. We will further assume that
=
C A ~
C B ~
1. If XA(O) and XIt(0) are the electronegativities of the neutral atoms A. and B and if XA(O) > XB(O)? we can write 1 2 2 l C B [ . We define the ionicity of a bond as
] C AI
x
= cA2
-
cB2
(0 5 x 15 1)
(2)
Considering the two extreme values, if CB = 0, x = 1, = # A ; we can say in this case that the bond is an ionic bond. If CB = CA = ('/z)"*, then z = 0, +M = I/-V'~(+A $B), and we can say that the bond A-E is a normal covalent bond. This definition, of course, does not remove the difficulties discussed by Shull' in relation to the VB treatment since +A and $B are not orthogonal. It has the advantage, however, that it leads to a simple formulation of the function p(x) (see below) in terms of the concept of bond order.a In this first approximation we will consider the variation of bond energy with the single parameter x. I n a second treatment (two-parameter treatment) we will discuss how the bond energy varies with both the ionicity anld the hybridizstion state of the bonded atoms. For two atoms at the internuclear equilibrium distance ( r e )we can write in a general way J/M
+
D(A-B)
=
D('A-B)ok(x)
+ *(z) + 6(x)
(3)
where D(A-B)o is the normal covalent bond energyjg p(x) is a decreasing function of x, @(x) represents the electrostatic energy, and a($) is another function of x that gives the variation in energy due to the charge transfer. If one knows the forms of the functions p(x), cP(x), and 6(x), and the value of D(A-B),, we can maximizelo D(A-B) with respect to x d(D(A-B))/dx
=
0
(4)
and thus calculate x and, by substituting in (3), obtain D(A-B). T h e Normal Covalent Bond Energy, D(A-B)o. At the present time there is apparently no rigorous way of calculating the energy of a normal covalent bond
strength defined as the product of the angular parts of the bond orbitals (Slater atomic orbitals) along the bond direction. There is now sufficient evidence showing that this criterion of bond strength is less satisfactory than the one based on the numerical value of the overlap integral.13-16 On the other hand, a somewhat rough MO c a l ~ u l a t i o nfavors ~~ the arithmetic mean rule. Thus, it was possible to show that
+
PAB = '/z(PAA PBB)
(7)
+
where @AB = S#A*H$BdT - '/z[.f$A*H+AdT J # B * H J / B ~ ~ ] S A Since" B. D(A-B)o = --@AB, it iS reasonable to suppose that eq. 5 holds within the limits of validity of ey. 7. In this report we will consistently use the arithmetic mean rule to calculate D(A-B)o. Two words of caution: first, as Mulliken17 pointed out, the arithmetic mean rule, or the more fundamental eq. 7 , does not show the dependence of D(A-B)o on the bond distance. As is always the case in this type of approximation, we will consider the atoms at their equilibrium distance, re. Second, the arithmetic mean rule implies that the hybridization states of atoms A and B are the same in the homoatomic molecules Az (7) H. Shull, J . A p p l . Phys., 33, 290 (1962). (8) C. A. Coulson, Proc. Roy. SOC.(London), A169, 419 (1939). (9) L. Pauling and D. M. Yost, Proc. Natl. Acad. Soc., 18, 414 (1932). (10) The maximum value of D(A-B), considered as a positive quantity, corresponds to the minimum in the energy of the molecule as compared with the separated atoms. (11) It seems that in the alkali metal hydrides the concept of electronpair bonds begins to break down even at this level. (12) L. Pauling and J. Sherman, J . Am. Chem. Soc., 59, 1450 (1937). (13) A. Maccoll, Trans. Faradag Soc., 46, 369 (1950). (14) R. S. -Mulliken, J . Phus. Chem., 5 6 , 295 (1952). (15) C. A. Coulson, "Valence," Oxford University Press, London, 1953, pp. 198-201. (16) F. A. Cotton, J . Chem. Phys., 35, 228 (1961). (17) R. S. Mulliken, J . Chim. Phgs., 46, 535 (1949). (18) The equality B(A-B)o = -2PAB has a reasonable MO justification. The same cannot be said for the relation B(A-B) = - 2 D A B , since the ionic contribution to the total bond energy depends on the difference of the diagonal matrix elements.
Volume 68,Number 8 August, 1964
RICARDO FERREIRA
2242
and Bz and in the heteroatomic system AB. We will discuss the limitations of this point in detail later (in the two-parameter treatment). T h e Function p(x). By a simple extension of the concept of bond order,8 it is possible to define what may be called a covalent bond order. In Coulson’s treatment the mobile bond order between two atoms is given by
pAB
njcAjCBj
=
(8)
j
where nj is the number of electrons in the jth orbital. Since we are considering a single localized a-MO with two electrons, the covalent bond order is simply ~ C A C B . From eq. 2 it is easily seen that
pAB
=
=
(1 - x2)”’
= p(x)
(9)
The hoinopolar bond energy is given by
D(A-B)op(z) = ‘/Z[D(A-A)
+ D(B-B)](1 - x*)’”
(10)
If C A = C B = (l/z)l’z, p(x) = 1 and the covalent bond order is equal to 1. I n this case the honiopolar bond energy js equal to D(A-B),,. The homopolar bond energy decreases with increasing values of x and if x = 1 it goes to zero.19 It should be pointed out again that the function p(x) = P A B is, strictly speaking, dependent on the bond length, but in this treatment we will consider only the equilibrium distance. T h e Electrostatic Function, @(x), Although many potential functions have been proposed for the essentially ionic compounds,2oa it is not a straightforward matter to calculate the electrostatic energy in molecules like HC1, BrF, etc. Wallzobused the original Born-Land6 function
where A is the Madelung constant (for diatomic molecules, A = 1) and n is a repulsive coefficient.2’ In crystals n is determined from compressibility data but for the gaseous compounds only reasonable guesses can be made. Wall attributed to n values ranging from 2 to 6 in HF, 3 to 7 in HC1, 3.5 to 7.5 in HBr, and 4.5 to 8.5 in HI. His results, however, are not strictly valid because he was gauging them by a comparison, unwarranted as we know now, between the experimental dipole moments and the primary dipole moments. Warhurst22 in his treatment of the hydrogen halides used a Born-hlayer potentialz3with 1 / p = 3.00. For the alkali metal halides, Born and Rlayer had shown that the best value of 1 / p is 2.90. Warhurst’s value was thus chosen because the repulsive The Journal o f Physical Chemistry
term in the hydrogen halide molecules is smaller than the one in the alkali metal halides. Since for the extremely ionic compounds the BornLand6 functionz4gives results within h57, of the experimental values and since for the essentially covalent compounds, where the Born-Land6 function is probably a poorer approximation, the term @(x) is relatively unimportant, we will use eq. 11 in the calculation of the electrostatic energy. With the exception of the hydrogen halides we will use the repulsive coefficients given by Sherman.26 For the hydrogen halides we will make two sets of calculations, the first one with Sherman’s coefficieiits for the halogen anions and n = 5 for hydrogen, the second one with n = 9 for the four hydrogen halides. The value n = 9 was chosen to take into account the smaller repulsive terms in these molecules. T h e Functzon 6 (x), We will a ~ s u m e ~ * that ~ ~ ~the -*~ energy of an electron in an atomic orbital is a continuous and digerentiable function of the charge in the atom, x, in the interval +1 2 x 2 -1. Furthermore, this function E ( X ) is single-valued for each valence state. This simple electrostatic approach is justified because in one given orbital the electrons must have opposite spins and the exchange integral vanishes. From the experimental values of ~ ( r for ) the three integral values of x, 1, 0, - 1 (corresponding, respectively, to Ivs, 0, and -Evs), an infinite number of curves can be drawn. The function adopted is the simplest one, namely, the parabolaz9
+
(19) The concept of covalent bond order was firstly discussed by Professor Mulliken at the Shelter Island Conference on Valence Theory (Conference Report, 1951, p. 67). It would also be possible to use the generalized bond order definition of Chirgwin and Coulson [Proc. Roy. SOC.(London), A201, 196 (1950)], but the inclusion of overlap would change little the numerical results. (20) (a) See the complete analysis of P.-0. Ldwdin, Phil. M a g . Suppl., 5 , No. 17 (1956); (b) F. T. Wall, J . Am Chem. Soc., 61, 1051 (1939). (21) In this paper 2 is a pure number (cf. eq. 2) which measures the fractional (excess or decrement) electron content of the bonded atoms. (22) E. Warhurst, Victor Henri Memorial Volume, Liege, Desoer, 1948, p. 57. (23) M. Born and J. E. Mayer, 2.P h y s i k , 75, 1 (1932). (24) M . Born and A. LandB, Verhandel. Deut. Physik. Ges., 20, 210 (1918). (25) J. Sherman, Chem. Ret;., 11, 93 (1932). (26) H. 0. Pritchard and F. H. Sumner, Proc. Rou. SOC. (London), A235, 136 (1966). (27) C. K. Jgkgensen, “Orbitals in Atoms and Molecules,” Academic Press, New York, N. Y . , 1962, p. 80. A. ‘I Whitehead, . and H . H. JaffB, J. Am. Chem. Soc., (28) J. Hinze, > 85, 148 (1963).
ZEROTH-ORDER APPROXIMATION FOR BOXDENERGIES
2243
6(%)
= (l/Z(IA
+
(l/Z(IB
+
-
'/2(1A
+
EB)x
6(Z) =
.....-.A
P I
I
.#-
2245
Pexp
ere
is due to the fact that the total electric inomeiit is given by the vector sum of th primary, overlap, and
z e r g , D.
x, e.u. 0.909 0.862 0,852 0.817 0.766 0.800 0.801 0.772 0,752 0.831 0.839 0.823 0.986 0.968 0.972 0.971 0.966 0.971 0.972 0,979 0.969 0.977 0.979 0.615 (0.718)” 0.251 (0.273)” 0.153 (0.162)‘ 0,069(0,070)” 0.184 0.241 0.278 0.067 0.128 0,063
liexp,
D.
6.83 8.26
6.28 5.9
9.07
8.5
10.02 9.18 10.6 11.4 12.0
4.9 7.33 10.5 10.4 11.1
2 . 7 1 ( 3 . 1 6 ) ” 1.74-1.91 1.53(1.67)“ 1.08 1.02(1.10)a 0.80 0.53(0.54)‘ 0.42 1.44 0.88 2.02 1.29 0.69 1.43
0.57 0.65
a These values are calculated with n = 9 in the Born-Land6 function.
hybridization moments. Two general conclusions, however, should be pointed out. 1. Crystals are more ionic than the corresponding diatomic molecules. This results from the fact that the larger Madelung constant makes the electrostatic terms more important in the case of the crystals. This is also expected from the classical (polarizability) models, since the polarization of the anion by the cation is larger for diatomic molecules. 2. Lithium compounds are more ionic than the corresponding sodium compounds and, in most cases, than the corresponding potassium compounds. This effect, experimentally found by Dailey and T o w n e ~ , ~ ~ is due to the large Coulomb energy connected with the short interatomic distances in the lithium compounds. (34) B. P. Dailey and C . H. Townes, J . Chem. Phys., 23, 118 (1955).
Volume 68, Sumber 8 August, 1964
RICARDO FERREIRA
2246
Bond Energies. The bond energies calculated from eq. 21 are shown in Table V. The experiniental values for D(,4-B) are given in the second coluniii of this table; they were obtained in the following way: for the gaseous alkali metal halides the values shown are the dissociation enthalpies35 Values given for the hydrogen halides and interhalogen compounds are bond energies, De, obtained from the bond dissociation energies and zero-point frequencies.36,37 For the ionic crystals, the experimental bond energies are atomization energies calculated from the relatioii
D(A-B)
=
GAB
+ EA -
(30)
Ig
where U ~ B is the lattice energy obtained from the Born-Haber cycle. For comparison, the last column of Table V shows the values of D(A-B) calculated from the classical Born-Land6 function. Table V : Calculated and Experimental Bond Energies“ D(A-B)
D(A-B) (oalod.)
(exptl.)
137.5 111.9 100.2 84.6 114.0 97.5 86.7 72.7 117.6 101.3 90.9 76.8 201.5 163.9 148.2 128.3 151.9 138.5 120,5 174.5 154.5 141,9 125.4 139.9 106.5 90.3 73.8 61.6 (55.9) (46) 52.7
50.1 42.3 a
n
The Journal of Physical Chemistry
Land6)
140.0 113.7 96.7
80.3 113.5 103.1 88.9 74.3 111.0 106.5 92.8 78.4 199,2 158.3 140.7 120.9 150.4 134.7 116.2 173,O 152.2 138.6 122.0 111.4 (120.1)’ 90.2(90.7)’ 80.2 (80.3)h 7 2 . 7 (72.7)b 53.7 51.9 50.6 52.5 49.0 41.4
Bond energies in kcal. mole-’. 9 in the Born-Land6 function.
=
D(A-13) (Born-
197,6 154.3 139.1 119.4 147,2 133.5 115,3 171.9 149.4 138,8 121.7
Values calculated with
It is seen that the agreement between calculated and experimental values is reasonably good for the extremely ionic and extremely covaleiit compounds but poorer for the molecules of an intermediate character (HF, HCI, CIF, etc.). For the 23 essentially ionic conipounds the average over-all accuracy is f3y0. It should be pointed out that for the ionic crystals, the calculated bond energies tend t o be lowerethan the experimental values; this probably is due to the choice of the Born-Land6 function, since this classical niodel gives still lower values. It is possible that our results could be improved by the use of a better potential function @ ( . E ) like the Born-Mayer function, 2 3 but we do not think this is worthwhile considering the approxiniate character of this treatment. We will show next that the larger discrepancies found for HF, HC1, etc., are in part due to our neglect of the hybridization of the bonding orbitals.
The Two-Parameter Treatment Thus far we have considered the bond energy as a function of a single parameter, the ionicity IC. A more complete treatment, however, should include as a second variable the hybridization state of the bonded atoms. Mulliken’s well-known saying that “a little hybridization goes a long way” reminds us of the dangers of many arbitrary assignments of hybrid wave functions. For example, there is now sufficient evidence, both from SCF-?\IO and VB c a l c u 1 a t i o n ~showing , ~ ~ ~ ~ ~that in the halogen molecules the bonded atoms use almost pure p-orbitals. We hope to show, nevertheless, that in heteroatomzc molecules the more electronegative atom uses s, p hybrid orbitals, because in this case the s-p proniotion energy is compensated not only by an increase in the overlap integral but also by an augmented electrostatic term @(IC),due to an increase in X h ( 0 ) . This analysis indicates also that we can safely neglect s-p hybridization of the less electronegative atom, a fact that has been recognized for some time now. 40 Let us assume that atom A (X,(O) > XB(O))uses a normalized hybrid u-A0 of the form y5.4
=
+ (1 - a2)”jbp
(31 1
(35) L. Brewer and E. Brackett, Chem. Rea., 61, 425 (1961). (36) T. L. Cottrell, “The Strengths of the Chemical Bond,” 2nd Ed., Butterworths, London, 1958. (37) G. Hersberg, “Spectra of Diatomic Molecules,” 2nd Ed., D. Van Nostrand, New York, N. Y., 1950. (38) B. J. Ransill, Rev. M o d . Phya., 32, 245 (1960). (39) G. L. Caldow and C. A . Couleon, Trans. Faraday SOC.,5 8 , 633 (1962). (40) H. A. Bent, J . Chem. F’h:,?., 33, 304, 1259 (1960).
ZEROTH-ORDER APPFLOXIMATION FOR BONDER'ERGIES
All the energy terms of this A 0 are linear functions of the amount of s-character, a2. For instance =
IA(LY2)
aZI*(s)
+ (1 - a2)I*(p) ==
+ a2(IA(s) - IA(P))
IA(P)
(32)
I n the same way we can write
+ - EA(p)) xA(p) + a'(XA(s) - xA(p))
-@*(a2) = EA(p) xA(a') =
a2(EA(S)
(33) (34)
The promotion energy is also a linear function of the amount of s-character, that is, PA(& = C Y ~ P A . Mulliken, et have shown that hybrid A 0 overlap integrals are given by the linear combination of noli-. hybrid overlap integrals. Since we have assumed that only atom A hybridizes, eq. 77 of the article by Rlulliken, ef al., reduces to
S ( a ) = alS(1)
+ (1 - a""'S(0)
(36)
2247
Equation 38 is the two-variable counterpart of eq. 21; it is easily seen that when cy2 = 0, (38) reduces to (21). Calculation of Bond Energies, Ionicities, and Degrees of s-Character. Assuming that D(d-B) as given in eq. 38 is a continuous and differentiable function of x and a i n theintervalso 2; x 5 1and0 5 a 5 1, the necessary conditions for D(A-B) to be a maximum42are that (bD(A-B)/bx), = 0 and (bD(A-B)/da)z = 0. Hence (dD (A-B) /dx) a
X(0))
= -D (A-B) OX
+ (1 -
a2)'/7
(1 - 5 ') -
+ [2d(;r;
"1
[ (8(1)/
x
+
and
where X(0) is the overlap integral if atom A uses a pure p-orbital and S(1) is the overlap integral if atom A uses a pure s-orbital. Assuming that the normal covalent bond energy is directly proportional to the overlap integral, eq. 36 applies.
-
(421
D (A-B) 0,ar = rI, (A-B)o(S(a) /S(O))
(36)
Hence D(A-B)o,a = D(A-B)o[a(S(l)/S(O))
+ (1 - a2)''']
(37)
It is known that the length of a bond A-B will vary with the hybridization state of the bonded atoms. However, for the sma,ll amount of s-character we will discuss in this paper, it can be assunied that re is independent of the hybridization state of A and we can now write
+
D(A-B)O(l - ~ ~ ) ~ ' ~ [ a ( S ( l ) / X ( 0 ) ) Ax2 (n - 1) (1 - a2)1/7 [xA(P) re n a'(X,(s) - XA(P))]Z - '/Z[[lA(P) aZ(I*(s) -
D(A-B)
=
+
IA(p)) 1 - [EA(p) X B X
-
~
+
~
+
+
+
a'(EA(S)
'/z(IB
- EA(p)) IlZ'
-
- E B ) ~-' a 2 P ~(38)
This system of simultaneous equations, eq. 39 and 40, can be solved by an iterative method to give x aiid CY, and by substitution in (38) we can calculate D(A-B). From (39) we obtain (41), and likewise from (40) we get (42). It is easi1,y seen that if a = 0, (41) reduces to (26). Let us call xo the value of z calculated from (25). We can now substitute xo in (42) and obtain a value of a that we will call 011. Substituting CY' in (41) we obtain a value 21; this value is substituted in (42) and we obtain a', and so on by iteration until self-consistency is obtained in both x aiid a . Substituting the self-consistent 5 and a in eq. 38 we calculate the bond energy, D(A-B), Results and Discussion We have made the calculations for LiF(g), LiCl(g), KaF(g), LiF(c), LiCl(c), HF(g), HCl(g), HBr(g), (41) R. S. Mulliken, C A. Rieke, D. Orloff, and H. Orloff, J . Chem. Phys., 17, 1248 (1949). (42) The sufficient condition (dgf/drz)(d2f/da*) - d*f/dzdo > 0 is also shown by the function D(A-B) = f .
Volume 68, ,\'umber
8
August, 1964
RICARDO FERREIRA
2248
Table VI : Ionicities, Hybridization States, and Bond Energies % Ionicity
0,910 0.860 0.768 0.985 0.965 0 . 6 2 3 (0.724)" 0.274 (0.297)" 0 . 1 7 1 (0.180)a 0.079 (0.080)" 0.187 a
8-
character
a
0.121 0.226 0.068 0.250 0.388 0.210 ( 0 , 2 4 2 ) " 0.200 (0.203)" 0 . 1 5 5 (0,156)' 0.124 ( 0 , 1 2 4 ) " 0.062
1.48 5.09 0.47 6.23 15.1 4.4 (5,8)" 4.0 (4.1)" 2.4(2.4)" 1 . 5(1.5)" 0.38
D(A-B) (calcd.), kcal. mole-'
D(A-B) (exptl.), kcal. mole-'
140.8 116.1 114.0 200.0 160.4 1 2 0 . 8 (129.9)" 99.1 ( 99.7)" 8 6 , 2 ( 86.3)" 76.6 ( 76.6)" 55.3
137.5 111.9 114.0 201.5 163.9 139.9 106.5 90.3 73.8 61.6
These values are calculated with n = 9 in the Born-Land6 function.
HI(g), and ClF(g). With the exception of hydrogen bromide, these are the systems for which numerical values for the overlap integrals are reported in Mulliken's tables.41 The overlap integrals for HBr(g) were obtained by interpolation. The s-p promotion energies for the halogen atoms are43: F, 20.89 e.v.; C1, 10.76 e.v.; Br, 10.95 e.v.; I, 10.17 e.v. All the other necessary data are shown in Tables I, 11, and 111. The calculations were made with the IBM 709 computer of the R.C.C. of Indiana University. The results are shown in Table VI. It is seen that for the extremely ionic compounds the calculated ionicities and bond energies did not change appreciably from the values calculated by the simpler treatment. There was, however, a definite improvement in the results for the hydrogen halides and ClF (for the hydrogen halides the value n = 9 gave better results, particularly in the case of HF). It is interesting to note that the amount of s-character of the halogen atoms in the hydrogen halides is small and that it is largest for fluorine (4.4y0 or, alternatively, 5.8%) , decreasing monotoiiically to iodine (1,5y0).This trend is observed in spite of the considerably larger s-p promotion energy of fluorine compared with the other halogens, and it is due to the great increase of the term ( X F ( S )- X,(p)) in eq. 38. XOW,
The Journal of Physical Chemislry
it is well known that the bond angle data in homologous series of molecules like HzO, H2S, H2Se, and H2Te, or, YH3, PH3, AsH3, and SbH3, when analyzed within the assumption that the orthogonality condition applies to the localized bonding orbitals of these molecules, indicate that the amount of s-character is a niaxinium for the second period atoms (0, X, etc.). Because the lighter atoms have in general larger s-p promotion energies than their higher homologs the resulting apparent contradiction has been held against the assumption of strict orthogonality, or against the criterion of bond angle data as a measurement of hybridization state. Our analysis, however, shows that a 2 is indeed maximum for the second period atoms and the stabilization factor is the gain in ionic energy brought by the increased electronegativity of the more electronegative atom. Aclcnowledgmenis. We are greatly indebted to Dr. Keith Howell, nh. Gene Barnett, and Mr. Hollace Cox for their collaboration in the numerical calculations, and to Professor Harrison Shull for his comments and hospitality. Thanks are also due to Dr. G. Klopman for many helpful discussions and to Professor Coulson for the benefit of correspondence. (43) J. Hinze and H. H. JaffQ,J . Am. Chem. SOC.,84, 540 (1962); J . P h ~ s Chem., . 67, 1501 (1963).
