A Continuous QuantitativeRelationship between Bond Length, Bond Order, and Electronegativity for Homo- and Heteronuclear Bonds Lyle Peter Seattle Pacific University, Seattle. WA 98119 Since the earliest days of quantitative molecular structure determination, attempts have been made to apply, in a simple manner, the empirically determined structural parameters to the prediction of structures of other species and to correlate the structures to chemical bonding theories. Pauling and Huggins ( I ) suggested that molecular hond lengths could be adequately estimated by simple addition of the single, double, or triple (2) covalent radii. Schomaker and Stevenson (3) pointed out deficiencies in simple addition when applied to single honds, and proposed the equation ~ A I S is I the singlr bond internudear distance, rA1 and rel are single bond covalent radii for anv elrments A and 13., , Ar ,ru is the absolute value of the elect;onegative difference (Pauling scale) between elements A and B. All distances are in picometers. This equation, known as the Schomaker-Stevenson Rule (or S-S Rule), accounts for hond shortenine due to partial ionic character hy including the elertronegativity differrnre term. Although it has heen criticized fim its lark of sophistication (41, its simple, universal applicability bas made it a popular tool in making first approximation hond length estimates. The author has plotted observed versus calculated single bond distances for more than 90 different heteronuclear comhinations for p-block elements, using both simple addition of covalent radii and an ionic character correction term similar to that of Schomaker and Stevenson. Ideally, the curves produced should he straight lines with slopes of unity and intercepts of zero. Table 1summarizes a least-squares treatment of the data points. Slopes, intercepts, straight line currelation (:oefficie~~& and ~ t a n d a r ddevi&ons arelisted. \'nI~~esa~erecal~~~llated with and without tluorine-containina honds, because fluorine bond data introduce significant de: viations from the ideal in both ionic character corrected and uncorrected plots, and there is experimental evidence (5) that indicates that the fluorine covalent radius may be artificially large because of unusually large electron-electron repulsions in the F2 molecule. The best lines were obtained when the electronegativity difference term was included, with the line omitting fluorine comhinations being statistically equivalent to the ideal. This indicates that the Schomaker-Stevenson Rule is on the right track, a t least for pblock elements. However, it is not applicable in multiple bonding situations. This paper reports the development of an equation that relates quantitatively and continuously bond order and bond length for any homo- and heteronuclear bond.
Table 1. Linear Least Squares Slopes, Intercepts, and Correlation Coetflcleds for Observed Versus Calculated Slngle Bond Lengths for pBlock Elements r*
k+a NO.01 Data Pts.
Slope Slope Std. Dev. Intercept Resid. Std. Dev. Corr. Coelf.
+ re -
r*
+ rs -
(with F)
lOlAxl (with F)
(without F)
(without F)
Ideal
95 0.870 0.024 33.79 7.75 0.967
95 0.961 0.013 7.48 4.39 0.991
80 0.937 0.021 18.13 5.75 0.981
80 0.987 0.015 1.62 4.07 0.991
...
A
lqAd
1 0 0 0 1
,...-
-
Equation Derlvatlon A review of the literature of single and multiple honds between p-block elements reveals that heteronuclear multiple honds are also shorter, in general, than the simple sums of homonuclear derived, double and triple bond, covalent radii, hut this shortening diminishes with increasing hond
order. A Schomaker-Stevenson-like relationship can he written for multiple honds dae.
+%, - kd A x d
=r ~ ,
(2)
where n is the hond order, and k , is a bond-order-dependent term. Pauling pointed out (2), that for carbon-carbon bonds, there is the quantitative and continuous relationship between bond order and length d, = dl - (71) log n
(3)
The form of thi.iequntion can be extended to other elements 16'1,andwhengeneralired and divided by two.. weaet, - . for any element E, covalent radii where CF is a unitlrss multiple bond pnrometer. A strength of eqs 3 nnd 4 is that the\ handle snioothls nonintearal - as well as integral hond orde;s. As it turns out, k , in eq 2 can he expressed also as a continuous function of hond order k, = 10 - (17) log n
(5)
Substituting the right side of eqs 4 and 5 into eq 2, and rearranging, we get
Dlscusslon Equation 6 gives the formula for estimating a hond length of any homo- or heteronuclear hond of any order, integral or nonintegral, simply, from tabulated single bond radii, electronegativities, and the multiple bond parameters. When rearranged, hond orders can he estimated from experimental internuclear distances. Table 2 lists values of Q,, CE, and X E for several p-block elements and rhenium. The covalent single hond radii are, in general, averages obtained from several reported homonuclear bond distances. Since hond lengths can change with Volume 63 Number 2
February 1986
123
Table 2. Bonding Parameters For Selected Elements Single Bond
Elemem
Radi~s.~h
Multiple Bond Parameter. O
Table 3. A Comparison of Calculated and Observed Bond Lengths (in picometers) tor Some Multiple Bonds
Electroneg: XE
Bond
Order
Calculated
Observed
Referencee
SN SeSe TeTe
CIO 00 CN
co
N0 SN S0 SeS PP ASAS ASP CN
00 N0 SN PS ASS SIN CN
co
coordination numher, the rEl listed are for coordination numher of four for Group IV, three for Group V, two for Groun VI. and one for Grouo VII. The multinle hond oarameters'for all but B, Ge, Sn, and the halogenswere calculated from exnerimental homonuclear multiple bond distances, frequently from diatomic molecule spe&oscopic data (7). The CE'S for the others were estimated from heteronuclear bonds or are extrapolations. The Allred-Rochow electronegativity scale (6) is preferred by the author, hut any scale could he used with minor modifications. Perhaps a better electronegativity scale (6a) would eliminate the fluorine problem. Equation 6 extends and combines the hond order-bond leneth continuitv of Pauline's (ea.3). - eauation . .. and the nartialionic character correction of ~chomakerand stevenion. More snecificallv. - . it extends the Paulina eauation, which was designed for homonuclear bonds,-to -heteronuclear bonds, i t extends the S-S Rule to multiple bonding situations, and combines them in one equation. I t does this, in part, by inclusion of the term identified in eq 5. This term accounts for the observed continuous inverse relationship between the magnitudes of partial ionic character and bond order. ~quarion-6 has the advnnrngr ot continuity in relating both hond lrngth and partial ionic character to bond order. I t therrfirre deals handilv u,ith bonds ot noninteeral order. and it eliminates the need for tabulated multiplebond dis: tances and extranolations from them. When n = 1, the last term ineq 6vnnishes, and theequation red~~ceit
