March, 1952
STRUCTURE OF BONDENERGIES AND ISOVALENT HYBRIDIZATION
295
MAGIC .FORMULA, STRUCTURE OF BOND ENERGIES AND ISOTTI4LEiST BYROBERTS. MULLIKEN Department of Physics, The University of Chicago, Chicago 87, Illinois Received December 96. 1961
In order to give approximate quantitative form to the Slater-Pauling “criterion of maximum overlapping,” a “magic formula” is presented. It gives the energy of atomization DOof any molecule as a sum of terms of which the principal ones are functions of overlap integrals (theoretical), of atomic ionization potentials (experimental), and also of degree of hybridization where the latter can occur without change of valence (isovalent hybridization). A preliminary fitting to three molecules whose Dovalues are sufficiently well known yields a magic formula suitable for molecules containing first-row atoms and hydrogen, ahd which fits the molecules CH, Nz, 0 2 , Fz,CH,, C~HB, C?H+CzHz! Hz+,Hez+, within about f0.3 e.v. per bond. For molecules with isovalent hybridization, the actual degree of hybridization 1s obtained as a by-product. Especially interesting is what the magic formula says about the structure of bond energies; that is, about the wa in which the numerous theoretical terms, some bonding, some repulsive, add to give DO. As compared with VB theory maryses by Van Vleck, Penney and others, it indicates much larger bonding and repulsive exchange terms. It indicates strikingly large exchange repulsions by inner shells, here in agreement with earlier conclusions of James and of Pitser. The magic formula indicates that the x bonding terms in multiple ‘bonds are much larger than is generally believed. Another striking conclusion is that isovalent hybridization is often very important for molecular stability. This conclusion, agreeing with indications from several other types of evidence for molecules such as CH, OH, HCI, HzO, 0 2 and Nz, is also in harmony with recent theoretical work of Moffitt and of Kotani, using conventional VB methods. The terms in the magic formula for Do can be collected into groups, one for each bond, corresponding to the usual concept of bond energies. Gross bond energies (a category introduced by Van Vleck in terms of VB theory) and net bond energies are tabulated for several molecules. Section IV includes an analysis and clas~ificationof types of hybridization and promotion which occur for atoms in valence states. Suggestions for estension, further applications, and (Section XIII) improvement of the magic formula are given.
I. Introduction It should be emphasized that the magic formula Quantum mechanics gives a satisfactory qualita- in its present form and with the present values of its tive explanation of the major facts of chemical coefficients is still preliminary. And further, as valence, and is capable “in principle” of predicting compared with the hoped-for future development of all energy relations quantitatively. But because really quantitative calculations of chemical binding of mathematical difficulties, no such quantitative energies, the attempt t o construct a magic formula accounting has yet been attained. Slater and may probably be looked on as a stop-gap effort. Pauling, in 1931, proposed as a rough measure of 11. LCAO MO Background of Magic Formula the strength of any covalent bond, formed by two It is instructive to begin with a survey of LCAO electrons on adjacent atoms, the quantum-mechanical criterion of maximum overlapping of the orbitals MO energy expressions for the simple molecules (one-electron wave functions) occupied by these H2+, Hz, He.+, He2, for states involving occupation electrons. With the idea of implementing this of the lowest-energy MO’s lag and la,; in LCAO criterion somewhat quantitatively, C. A. Rieke approximation and the writer before the war sought semi-empirical (18, + 18b)/(2 + 28)a; 1 U u =(Is. - 1sb)/(2 - 2S)t 1Ug relations between overlap integrals and bond Here Is, and 1 s b are 1s AO’s (atomic orbitals) on the energies.3 Recently the writer proposed a pre- two atoms a and b, and S is their overlap integral liminary equation for this p u r p o ~ e . ~An improved “magic is described below. S I 1S.lSb dv I n the history of valence theory, two main quantum-mechanical methods for describing molecular For H2+, the energies of the ground state (one electronic structures have proved useful, namely, the electron in la,) and the first excited state (one elecVB (valence-bond) method, and the MO (molec- tron in la,) can be, respectively, expressed in ular orbital) method in its LCAO form. The the forms magic formula has been built in a general pattern E = E= - ( C + + E‘) f P/(l f 8 ) (1) indicated by the VB method, and in this respect Here EH is the energy of a normal hydrogen atom, is by no means entirely novel, but the specific Further forms of its details have been suggested largely by C+ = - e z / R - w LCAO MO theory. The structure of the formula p = T - S W and the numerical values of the terms which appear (2) in it are such, it is believed, as to afford increased w =SUblSa2dtJ; 7 = U b l S s l S b dv insights into the nature of chemical binding and to provide helpful suggestions for further work. R is the internuclear distance, and Ub is the potential due to
s
S
(1) This work was asskted hy the ONR under Task Order IX of Contract NBori with The University of Chicago. (2) Presented at Symposium on Bond Energies of Division of Phyaical and Inorganio Chemistry, American Chemical Society Meeting at New York, September 6, 1951. (3) Some of the material in Sections I1 and I11 below was presented at B sympoeium in 1942, but only an Abstract was published: R. 8. Mulliken and C. A. Rieke, Rev. Modern. Phys.. 14, 159 (1942). (4) R. 9. Mulliken, J . A m . Chem. Soc., 7 2 , 4493 (1950),in particular Eq. (8) and (O), footnote 14a, and Table X. (5) J . Chem. P h ~ s .19, , 900 (1951). Note added in proof.
nucleus b. It should be noted that w , T and 6 are all negative quantities. -E’ (which is included merely for bookkeeping purposes) is whatever energy is needed to correct the error of the LCAO approximation so as to make Eq. (1) exact. Resonance energy expressions of the type p are of central importance in ordinary LCAO theory.6
(6) Combinations of the form of i3 of Ea. ( 2 4 were introduced with thesymbol Y by Mulliken, J . Chem. Phya., 3, 573 (1935). Eq. (16), and identified with Hilckel’s semi-empirical LCAO quantity j3 hy Rliilli!,en and Rieke, J . A m . Chem. SOC.,63,44, 1770 (1941).
ROBERTS. MULLIKEN
296 Equation (1) is obtained as follows.’
As is well known,
E = e2/R - E’ + ( a 4 r)/(l st S)
where (Y
=JlSahlSa
dv,
S
T sz
h being the electronic Hamiltonian. On substituting
p=r-s(Y
(24
and h=h.$.Ub
ha being the Hamiltonian for atom a alone, one readily obtains a Ex + w , p = T - S W , and Eq. (1) 5
For the ground state of Hz+,Eq. (1) gives for the dissociation energy D e , equal to EH- E De (C+ E’) 8/(1 S) (3) Now it would be very pleasant if C+ 3. E’ in Eq. (3) could be neglected and D e put equal t o the dominant resonance term --@/(l S) alone. Actually, this procedure gives rather good results (De= 2.79 e.v., -@/(l S) = 2.22 e.v., by direct theoretical c o m p ~ t a t i o n ) . ~ ~ ~ Similarly for Hz, where7 De = ( C E’) 2/3/(1 8) (4) the theoretically computed resonance term -28/ (1 S) agrees rather well (4.07 e.v.) with the observed D e (4.76 e.v.).*Pi0 The results on Ht and H2+, reinforced by similar (though less reliable) evidence on ?r bonds in CsHz and CZH4,’ suggest that the resonance terms of LCAO theory may be used as a basis for a semi-empirical systematics of bond energies. One might now try to use D expressions of the form A/?/(1 S), computing p theoretically for each type of bond. However, a different path will be followed here, on which the next step is to replace @ by a function of S. The simplest reasonable choice is to put
+
+ -
3
+
+
+
+ -
+
+
-20
(5)
ASI
where I is the ionization energy for the appropriate valence AO. A is then a factor to be determined empirically so as to satisfy equations like (3) and (4)omitting terms like C E’, as well as may be possible for a number of molecules simultaneously. This procedure gives for a one-electron bond and an electron-pair bond, respectively
+
D = *AZS/(l D = AZS/(l
+ S) + 8)
&3(u + s u a l s . 2 do)
is involved. Also by direct computation in two instances, p has been found approximately proportional to S. Namely, for 1s-1s binding in H2+ or Hz, in e.v., - fl = 7 5 for R > 1 A. (but at the equilibrium distance for Hz, -/3 FJ 4.78); and for 2 p n - 2 ~binding ~ as in C Z Hor ~ C2Hn, -fl SJ 1 0 s (this is valid over a range of R extending to both sides of the
equilibrium distances for these molecules) .I1 These results suggest the following conclusions, confirmed later in this aper: (1) -6 may usually be taken proportional to S, u t with a larger proportionality constant for r than for u bonds; (2) but if R is unusually small (relative to A 0 size), 0 may become smaller (cf. ref. 56 below). Another instructive approach to Eq. (6) and (7) is the following. Consider the char e density p for an electron in the MO la, in H2+. I n LCA8 approximation (cf. Eq. ( l ) ) , this is (in units of -e) 8) (1% lSb)/(l s) P (lug)* !d18a8 f lsbs)/(l As compared with the charge distribution %(lsa’ which would exist if the electron were distributed with equal probability between 1s AO’s on the two nuclei, the above distribution represents a shift of a fraction Jlss 18b dv/(l 8 ) = S/(1 S ) of the charge into the region of overlap between the-two nuclei. Since, aside from the Coulomb term, which would correspond to the unshifted distribution f(ls.2 I@), it is precisely this partial shift into the increased field in the overla region which is primarily responsible for the stability of $+? it is to be expect,ed that D. for the latter should be approximately proportional to S/(1 8). The assumption of approximate proportionality of B to Z is made plausible by the following reasoning. Moleculeformation viewed according to the VB method is a phenomenon which causes perturbations in the energies of the valence electrons of the participating atoms. Bond energy is then a measure of the extent of these perturbations. Other things being equal, the perturbation energy should be more or less proportional to (perhaps a nearly constant fraction of) the original energy of the valence electrons, for any one of which its binding energy -I is a measure. (See also ref. 39 below.) s k1
-
+ +
.+
+
+
+
+
+
-
In the ground states of Hez+ and Hez, there are two lug, and, respectively, one or two luu electrons. The resonance contributions to D are --@/(l 4S) for each lug and +@/(l- S) for each lau electron (cf. Eq. (l),(4), (5)). Putting -28 = ASI, one obtains for Hez+and Hez, respectively 2 D = -p(- 1 + S
- -) 1 -1 s
=
(6) (7)
(7) For further details. see R. 8. Mulliken, 1. chim. phys., 46, 497 see Eqs. (38)-(40) for I, wand @; Eq. (43) and Table (1949). For Hp+, I1 for dissociation energy equation and data. For HI,see Eqs. (65) and (69) for w and @ ( @is slightly different for HI than for Ha+, but this is neglected in Eq. ( 5 ) ) ; Eq. (77)and Table I11 for dissociation energy equation and data. For similar material on CxHn and C1H4,see Tables V and VI. (8) The De value given ia the experimental value corrected to no zero-point energy. Later in the paper, uncorrected D values (D@ values) will be used. For HI, Do 4.48 e.v. (9) The theoretical C + is -0.74 e.v., requiring E’ 1.31 e.v. It in seen that a much better result is obtained by ignoring the Coulomb term C+ of LCAO theory than by including it. (IO) In Eq. (4)
-
c = -es/R
Rough proportionality of b to S is seen to be plausible if one studies the forms of T, w and p in Eq. (2). Among other things, the rough relation
1sbhha dv
y
Vol. 56
- 2~ - a(Ja f Jab)
where Jasand J.b are as in Eq. (18). Here, just as for Hi+, inalumionof C (computed value -1.55 e.v.1 would only have worsened the agreement (E’ = 2.24 e.v.).
Equation (9) predicts repulsion for two He atoms, in agreement with VB theory. Equation (8) for the three-electron bond indicates weaker bonding than for the one-electron bond. In the process of generglizing Eq. (7)-(9) to obtain a magic formula, as described later in this paper, it was found necessary empirically to introduce (cf. Eq. (23)) a correction factor Y to temper somewhat repulsions like those of Eq. (9). Because of the largely empirical character of the final formulas, Eq. (9) may as well also be simpli(11) For further details, see R. 5. Mulliken, J . chim. p h y s . , 46, 675 (19491, Section 28.
I ,
STRUCTURE OF BONDENERGIES AND ISOVALENT HYBRIDIZATION
March, 1952
fied by dropping the factor 1/(1 - S2), which usually is not far from 1.12 The result is ’
Hez: D
= AI
s (iTs
= -2vAZSa
-
+ +2 4 1 -
= l(1 S M l SI1 = 1 2S(1 S)(l
-
-
- + ...
(11)
Y)
If this factor
p is empirically needed in (lo), use of the same factor in the antibonding term in (8) is indicated. Equation (8) then becomes Hez+: D = A I S - ips =
(-l + s -) 1 - 8
+
MIS [(2 - PI - (2 P)SI (12) This predicts a somewhat stronger three-electron bond than Eq. (8) does. Although Eq. (6)-(9) above have been obtained by LCAO MO theory, it is well known that VB theory gives identically the same wave functions as LCAO theory for H2+, Hez+ and He2 in their ground states, the two theories differing only for Hz.la Hence, Eq. (6), Eq. (8) or (12), and Eq. (9) or (lo), may be considered to be based on either the VB or the LCAO approximation, only (7) being LCAO only. Since VB theory is tolerable for all values of R, all the equations but (7) should likewise be acceptable for all R values. Although the LCAO formula (7) will be used below as a basis for dealing with bonded attractions by the magic formula, it must be recognized that Eq. (7) cannot be valid outside a limited range of smaller R values near equilibrium. For larger R values, the corresponding VB formula must be given preference. As has been shown previ~usly,~ Eq. (9) with A = 0.65 fits the curve of repulsion between two He atoms over a broad range of R. Eq. (10) with v = 0.7 and an increased A almgives a good fit. In using Eq. (6)-(12), the necessary S values can be looked up in existing tables.14
111. Valence-Bond Theory Background of Magic Formula In the VB method, each electron is first assigned to an A 0 on one atom. Electrons with unpaired opposite spins on adjacent atoms may then form electron-pair bonds. In the simplest example, Hz,the approximate energy of the lZ+g ground state according to VB theory16is usually given in the form (12) At first, a different empirical modification of Eq. (9) was tried, consisting in replacing the terms S/(1 s) and S/(1 8) by S/ (1 OS)and S/(1 b Z ) , respectively, and trying t o determine a and b to give as good fits as possible to observed D values for Hs+, HI, Her+. Hez. A number of trial values of a and b were tested, mostly with a > b > 0,but finally it was ooncluded that the use of a form like Eq. (10) has better possibilities for fitting observed D’s, beside# being simpler. (13) This is readily verified by setting up the complete antisymmetrized VB and LCAO wave functions for each case in detail, including spins, and comparing. The statement is true, of course, only if Is hydrogenic AO’s with a single 2 value are used in each case. (14) (a) It. 6. Mulliken, C. A. Rieke, D. Orloff and H. Orloff, J. O h m . Phya., 17, 1248 (1949); (b) R. 8. Mulliken, ibid., 19, 900 (1951). The symbolism and notation of ref. 14b has been used in the present paper. (15) See for example L. Pauling and E. Bright Wilson, Jr., “Introduction t o Quantum Mechanics,” McGraw-Hill Book Co., Inc., New York, N. Y., 1935, Section 42a.
+
-
+
-
+
E e (Hi1 Hiz)/(l Sa) (13) In Eq. (13), Hll and H12 can be written in the form
(10)
with Y near 0.7 (see later in this paper). Introduction of the factor p into the antibonding term in the first form of Eq. (10) makes the two forms of Eq. (10) agree if P
+
207
Hii where
Hji
3
s
$j(
Ha,
+ Hb2 + Hint)$i dol
do2
18s (1) h b (2); $2 3 18b (1) 18, (2) $1 HalE --haVia/87r% - ez/r,l; Hbz E -hzA2Z/8n2m ea/%* and the interaction operator Hilltis Hint = e*/R ea/rlz - e2/Tbl - ee/r.2 Equation (13) can be r e c a ~ t to ’~~~~ E Hn q / ( l S2) where (14) q Hlz - S2Hii If a bookkeeping term -E‘ is now added to the right of the first Eq. (14), such as to convert it from an approximate to an exact equality, and if then E is subtracted from ~ E Hone , obtains for the dissociation or bond energy De = (C E’) - 0/(1 5’) where (15) C ~ EH H11 = - $1 Hint$l dv
-
+
+
FJ
+
+
1
+
S
1
I
is the relatively small1J “Coulomb” term. l8 Equation (15) may conveniently be written De (C E’) X ; X = -q/(l + S 2 ) (16) the exchange term18 X being the main term responsible for bonding. It is instructive to compare the VB expression (16) for De of Hz with the corresponding LCAO expression of Eq. (4). Quantitative theoretical computations have of course been made15 by both equations, with the result that (omit,ting the bookkeeping terms E’) the VB expression gives a somewhat better computed D e than does the LCAO expression. However, if the primary bondS) ing terms alone are used, that is, -28/(1 of Eq. (4) and X = -v/(l S2) of Eq. (16), the former gives a surpfisingly good D value, but the latter of a poor D value.19 This points toward adoption of the procedure proposed in Section 11, of approximating D semi-empirically by -28/(1 8) with -20 replaced by AS1 (cf. Eq. (5), (7)). In view of the strongly empirical character of the search for a magic formula, a definite decision to use expressions of the LCAO-suggested form ASI/(1 S) for the energies of the bonding electrons of electron-pair bonds seems justified, and will now be made, for the case of molecules with the distances
+
+
+
+
+
+
(18) The definition of 7 is constructed in analogy t o t h a t of the LCAO MO parameter 13 (cl. footnote 6). The quantity here called q (earlier-aee footnote 3 - c a l l e d a) has also been used recently by van Dranen and Ketelaar, J . Chem. Phya., 18, 1125L (1950)-their a‘. (17) The energy of the *E+”repulsive state is also given by Eq. (13) and (14) if the plus signs are changed to minus signs. (18) I n the customary terminology of atomic structure theory, intratomic repulsion integrals analogous in form to the interatomic (two-center) integrals J a b and Ksb of Eq. (18) are called Coulomb and exchange integrals, respectively, and it seems appropriate to use the 8ame terminology for Jab and Kab. In VB theory, on the other hand, integrals $$i Hint $i dr and $$i Hint $1 do are commonly called Coulomb and exchange integrals. I n the following, expressions of the type v/(l SI) will be called “exchange terms” t o signalize the fact that their dominant components a m “exchange integrals” in the VB sense. (19) This happens because of the terms C, which have opposite effects in the two cases, making the computed De somewhat better in Eq. (16). but much worse (cf. footnote 10) in Eq. (4). See sentence containing Eq. (4) for actual values of De and -28/(1 8).
+
+
ROBERT S. MULLIKEN
298
Vol. 56
between bonded atoms near their equilibrium .D is now the total energy required t o dissociate the molecule (devoid of thermal energy, as at 0°K.) values. A study of the detailed structure16of X of Eq. (16) shows completely into atoms in their ground states (energy of atomization). that it can be written in the form (20) contains one term of the type x = - [258 ( K s b - S’Jab)] /( 1 8’) (17) X i Equation j for each electron-pair bond in the predominant Here 8 is the familiar resonance energy quantity of LCAO VB structure, and one term - 3 X k l for every pair theory (cf. Eq. (2), (as)), andIBKah and J a b are of electrons not o n Ihe Same atom and not bonded to each other. For X i j ’ s , generalizing from the case of Hz, expressions of the form A S I / ( l K a b ~ S S l s . ( l ) l s b ( l ) e z / r l 1S,‘2)lSb(2’d?ll * dvz 8)will be adopted for the magic formula. The X k l ’ S may be Jab 1Sa(’) ISs( ‘)e2/r121Sb(2) 1Sb c2) dol dvz classified in various ways. For present purposes, 1 they may first be divided into the following two The convenient and rather accurate approximationza types : (1) homogeneous: those involving (as do also the X i j ’ s ) two AO’s of the same kind (both Q, &b tS2(Jna Jab) where J , is an intratomic Coulomb integral defined analo- or both T ) ; (2) heterogeneous: those involving gously to J a b in Eq. (18), may now be introduced into Eq. orbitals of different kinds (one Q, one T ; or one (17). Oae then has T + , one T-, or one rx,one T ~ ) .For ~ ~type (2) XkI’S, S = 0, and Eq. (17) shows that these reduce x * -[25@ +8’(Jas - Jab)]/(1 Sa) (19) to the relatively simple exchange integrals K k l Usually21 H > 0, as a result of strong predominance of the positive term -258 (cf. Section 11)over the negative term (cf. Eq. (18)); the notation K k l will therefore be used hereafter for the type (2) Xkl)S. -!S2(Jan - J a b ) . (Note that J,. > J a b for R > 0.)
+
+
1
sss
+
t
’
+
+
t is of interest to compare the primary bonding terms of LCAO and VB theory as given in Eq. (4) and (19). It is seen that these both t.ake the form of a resonance energy expression22 -28, modified by a factor which is somewhat lus an added term in the different in the two cases-and 8 ) ; in case of Eq. (19). I n Eq. (4), t f e factor is 1/(1 %q. (19) it is S / ( 1 4-52). For a strong bond as in Hz, the difference is not great (for H ~ , . l / ( l 8 ) = 0.57, S / ( l 82) = 0.48), but for bonds with smaller S the VB factor leads to the prediction of much weaker bonding than does the LCAO factor; in the VB case, the added term is*(Jan- J a b ) acts to cut down the predicted D somewhat further. All in all, the slower variation of D with S indicated by LCAO theory seems to be in much better general agreement with observed bond energies than the more rapid variation suggested by VB theory.23
+
+
+
The preceding discussion must now be generalized to the many-electron, polyatomic, case. Here no exact .general VB theory expression is available, but for molecules with all electrons paired, at least in rough a p p r o x i m a t i ~ n conyentional ,~~ VB theory yields D = ( C + E’) + ZXij - 4 2 X ~- P + RE (20) bonds non-bonded pairs
In Eq. (20), each exchange term X is defined as in Eq. (16)’ P denotes promotion energy (see below), and RE refers to resonance energy if present. (20) R. 8. Mulliken, ref. 7, Eq. (63). (21) Heisenberg’s theory of ferrbmagnetism assumes X < 0 for 3d-3d bonds between iron or similar atoms a t metallic R values. Eq. (19) shows how X < 0 might be possibIe in special cases. (22) By this is here meant (although the terminology is not very satisfactory) a term such a s occurs for Hz+,where removal of (electron-exchange) degeneracy is not involved. The fact t h a t the “exchange energy” of VB theory owes its negative sign to a predominant term of “resonance” character was pointed out by the writer some time ago (CAena. Revs., 9,364 (1931)),b u t probably is not generally realized. (23) By reasoping similar t o that used in Section 11, the VB equation (19) with -28 put equal to cSI suggests the use of a semi-empirical form ASV/(1 82) for bond energies. , Indeed, this fits the observed D values on the series Hs, Liz, Naz,. . ,better than the form A S I / 8)-but not very well. I n other cases, the fit is not good. For (1 example, it appears impossible with this form to escape from a computed D < 0 for F:, after necessary antibonding terms are inoluded as desoribed below (Eq.(20) et eeq.). (24) Cy., e.p., ref. 15, p. 376, reoast in t e r m of quantities 9 and X instead of Hia. In Eq. (ZO), multiple exohange integrals and certain ether complications have been ignored. I t is here aasumed t h a t their effects can be taken care of sufficiently well, along with those of the term8 C E‘, by empirical adjustment of the coeffioients in Eq. (21).
+
+
+
The type (1) HkI’S, on the other hand, should according to VB theory be of the same structure (cf. E (17)) as the Xij’s. However, comparison with LCAO M% theory suggests a diflerent structure for the xij’s and Xkl’s, as can be seen by considering the cases of HZand He2 as special cases of Eq. (21), and comparing with Eq. (4) and (9), respectively: H?(VB): D = (C E’)vB X I ~ J ~ ; H2(LCAO): D = (C E’)LCAO28/(1 AS) Hez(VB): D = (C E’)vB - ~XI~,I,,; Hez(LCA0): D = ( C E‘)LCAO488 If the terms C E’ are dropped, then in the one case, X corresponds to -2p/(1 S), in the other to -288, a difference of a factor S(1 8 ) . Since i t has alread been deit will cided to Pollow the lead of LCAO theory for the z’s, inake for clarity to adopt at this point a distinctive notation for the two kinds of X’s. Accordingly, the Xij’s will still S hereafter be called Ykl’s. be called Xij’s but the X ~ I ’will
+
+
+
+
-
+
+
+ +
++
IV. Magic Formula Equation (20) is ngw ready to be recast to form a basis for the magic formula. For this purpose, (a) the term (C E’) will be considered as absorbed into the other terms ( rimarily into the X i j ’ s ) ; (b) the homogeneous kl’s will be called Ykl’s; (c) the heterogeneous XkI’S, times -1, will be called Kmn’s; (d) for D, the value Do uncorrected for zero-point vibrational energy will be used, since this will be mol‘e convenient than a corrected value De, and in view of the rough and largely empirical character of the magic formula2e; (e) the X’s and Y’s will from here on be considered as semi-empirical functions of the S’s, t o be so adjusted as to reproduce as nearly as possible the exact DOvalues of actual molecules. Equation (20) now takes the form DO = ZXij - $ZYkl + 3 Z K m n - P + RE (21)
+
H
bonds
non-bonded pairs
The quantities on the right of Eq. (21) have been so defined that every letter symbol now stands for an (25) Among u orbitals are n.9, npo, nds, and hybrids of these. The moat usual n orbitals, and the only ones considered in the present paper, are nm. (26) Fitted to Do values, Eq. (21) can of course reproduce Do for only one isotope of a molecule, but the variations of Da between isotopes are alwaya less than the uncertainty involved in the rouphnem of the formula a s a t present constituted.
.t‘
STRUCTURE OF BONDENERGIES AND ISOVALENT HYBRIDIZATIOX
March, 1952
intrinsically positive quantity, with possible rare exceptions for the X k 2 l For molecules in their ground states with all electrons paired, Eq. (21), taken together with semi-empirical “magic” expressions for the X and Y terms, now constitutes the magic formula. For the X i j ’ s as already decided above, LCAO-based expressions Xij = L4,Sij7iJ/( 1
+
Sij)
(22)
of the form of Eq. (7), are to be used. Here the A’s are coefficients to be adjusted empirically, and the I’s are suitable mean ionization energies (see below). The S i j ’ s are to be computed theoretically. Before deciding on a magic form for the Ykl’S, it may be noted that they are of three kinds: (a) lone-pair :lone-pair Ykl’S; (b) lone-pair :bondedelectron Ykl’S; (e) Yk?S between electrons in different bonds.27 For the first of these kinds, a t least for like lone pairs in the homopolar case,28LCAO and VB theory coincide (cf. Sect’ion 11), and the LCAO Eq. (9) may reasonably be adopt’ed as prototype for a suitable magic form. I t will now be assumed, this time following VB theory, that the same type of expression is equally valid for all kinds of Ykl’S.29 However, trial and error studies suggest one modification (already discussed in connection with Eq. (9)-(12) in Section II), leading to the magic form Ykl
=
VAkSk12~kI
(23)
This form will be adopted and used from here on.‘* The effect of the factor v, for which values somewhat less than 1 were found to give the best fit to observed Do’s, is to diminish somewhat the importance of the non-bonded repulsion^.^^ Further explanation is now needed as to how to obtain values for certain quantities appearing in Eq. (21)-(23). The K’s (cf. Eq. (18)) of Eq. (21) are to be obtained theoretically. Fortunately, they are relatively small, so that rough estimates are adequate for the present discu~sion.~‘ (27) It may be noted in passing t h a t the K-’s can be classified in the same way into three kinds. (28) With reference to heteropolar lone-pair:lone-pair repulsions in LCAO theory, if one sets u p a n LCAO wave function for the artificial case of the interaction of a 1s)z shell on one atom with a 2s)l shell on another, one finds a n expression for the interaction energy which is roughly proportional to 8 3 , just as (cf, Eq. (19)) by VB theory. (29) In general, non-bonded repulsions appear in a very different guise in LCAO theory from that in VB theory: see R. 8. Mulliken, J . Chem. Phys., 19, 912 (1951), where it is pointed out that direct antibonding effects and “forced hybridization” effects in LCAO theory apparently constitute the respective analogs of the lone-pair: lone-pair and the other kinds of non-bonded repulsions of VB theory. (30) This seems not unreasonable. The flexibility introduced by the adjustable A’s takes care roughly of the various errors involved in the use of the simple approximation (22) for the bonding terms; in particular, the effect of the bonding parts of the omitted terms C E’ of Eq. (20). The factor Y then gives in the simplest way some further flexibility in representing the non-bonded repulsions. Earlier workers (cf., e.g., H. M. James, J . Chem. Phys., 2, 79: (1934)) have also felt t h a t VB theory predicts somewhat too large non-bonded repulsions as well as too small bonded attractions. (31) Formulas for computing the K’s involving Is, 28, and 2 p Slntertype AO’s are given in a paper by K. Riidenberg, ibid., 19, 1459 (1951). For the present paper, a rough procedure was used, whereby SCF-A0 K ‘ s for Nm, On, FP,and CH were estimated from related Slater-A0 K’s given for C=O bonds in COz b y J. F. Mulligan, ibid., 19, 347 (1951). Taking Ksosc and Kacso as an example, their average (0.37 e.v.) may be taken as characteristic for C=O and used as a basis to estimate
+
.
299
The promotion energy P in Eq. (21) occurs because without it Eq. (21) would in general be valid only for dissociation into certain hypothetical atomic states, with the valence-electron spins completely unpaired, called “valence Since valence states in general have higher energies than atomic ground states, the quantity P , equal to the sum B P n of the valence-state promotional energies taken over all the atoms involved, must be subtracted in Eq. (21) in order to give the desired Do, which has been defined to correspond to dissociation into atoms in their ground states. For the special case of s-univalent atoms (H, Li, etc.), ground and valence states are identical (P, = 0). Often Pn may involve no configurational excitation. For example, to obtain trivalent nitrogen, one must “promote” the atom from its s2p3, 4S, ground state, where the spins of the three p electrons are all parallel, to a trivalent valence state VT~of the same s2p3 configuration. There is required a promotion energy of 1.70 e.v. This kind of promotion might be called advalent promotion, that is, intraconfigzirational promotion from ground state to a valence state. Trivalent nitrogen gives stronger bonding, however, if there is partial further promotion toward the trivalent valence state of the configuration sp4; that is, s2p3, 4Xu -t s2p3, V3 -+ (partially) sp4, V3. This second kind of promotion, extraconfigurational but without change of valence, and only partial, corresponds to a kind of s,p hybridization which Moffitt3S has called second-order hybridization. These effects might more descriptively be called isovalent promotion and isovalent hybridization. Other cases occur in which extraconfigurational and complete promotion is accompanied by increase in valence : say, pluvalent promotion and pluvalent hybridization. Thus for carbon, only advalent promotion ( s 2 p 2 , 3Pgto s2p2, Tiz) is required for bivalency, but for tetravalent carbon, pluvalent promotion to sp3, V4 or to a related hybrid V4 state is required (these tetravalent valence states, incidentally, are very considerably higher in energy than the lowest state, 5Xu, of sp3). Similarly for bivalent beryllium, pluvalent promotion is required from nullvalent s2, V o (same as s2, lSg) to sp, V 2 (lying between sp, 3Puand sp, ‘PU). Valence-state promotion energies are obtainable from spectroscopic data on atomses4 Table VI11 in Appendix I contains illustrative values for several first-row atoms in valence states suitable for use in K s r for Nz, On and Fz. I n a similar way, values for K s w (0.56 e.v.) and K s s , (0.041 e.v.) for C-0 were obtained, whereas K s , l e was found negligible. To estimate K values for Nn, Oz and Fz,it was assumed that they differ from those for C=O in the ratios of the squares of the overlap integntls Sam, S.r and Ssr for K r s , K s c a n d K r r , respectively. Tho integral K.r,mr was treated similarly. Since in the present paper, X’s,Y‘s and K’s corresponding to SCF-AO’s are desired, the squares of ratios of estimated (cf. footnote 42) SCF-A0 S’sfor Nz, On and Ft t o Slater-A0 8’s for C-0 were used. The same sort of procedure was used also for C-C, C=C and C G C bonds. To estimate the CH integral K r h (between 2 p i c and ISH), the Mulligan K r s for C-0 to (0.37 e.v.) was multiplied by the square of the ratio of 87gL1~~,, sslate? [z10,aaO), giving 0.68 e.v. ‘(32) (33) further (34)
See footnote 14 of reference given in footnote 4. W. Moffitt, Proc. Roy. SOC.(London), ‘2028, 534 (1950); and reference to Moffitt in Appendix I1 below. R. 8. Mulliken, J . Chem. Phva., 2, 782 (1934).
ROBERT S. MULLIKEN
300
linear molecules.s6 Table I X in Appendix I contains additional valence-state promotion energies, suitable for tetravalent carbon in molecules of several types of symmetry. Use will be made of Tables VI11 and I X later in this paper. The resonance energy RE in Eq. (21) refers first of all to resonance energy in the usual sense, as for example in benzeneeS6 But it may sometimes include further terms. In particular, when one uses Eq. (21) for carbon compounds in which tetravalent carbon is assumed, with Pnvalues from Table IX, one underestimates D. As Voge has methane is about 1.3 e.v. more stable because the carbon is very considerably more in the s2p2 condition than if it were purely tetravalent with tetrahedral AO’s. This difference may be regarded with Eq. (21) as demotional resonance energy, to be added as part of the RE term after the bond energy has been obtained for pure tetrahedral valence. For any molecule with polar bonds, a polar RE term must also be included in Eq. (21). These polar RE values would presumably be of the same orders of magnitude as the observed deviations from additivity of bond energies for heteropolar bonds, used by Pauling in setting up his electronegativity scale.as A procedure is now needed for obtaining the mean 1 values in Eq. (22) and (23). On the basis of rough theoretical c o n s i d e r a t i o n ~ sand ~ ~ ~ ~of simplicity and empirical acceptability, each f is (35) See also footnote 33. (36) If the term “resonance energy” is used in a strict sense, some resonance energy is present for nearly all molecules, since in general numerous “excited” and “ionic” VB structures mix to some extent into the usual single predominant ground-state structure. However, the corresponding numerous small “normal” resonance terms are covered (as part of the E’ term in Eq. (20)) in the empirical adjustment of the A’s and perhaps especially Y in Eq. (22) and (23). Hence, R E in Eq. (21) corresponds only to “excess“ resonance energy above normal, which is really what is usually meant by the term “resonance energy.”?‘ (37) H. H. Voge, J . Ckem. Phys., 4, 581 (1936); 16,984 (1948). See also Kotani and Siga, Proc. Phys. Math. SOC.Japan, 19, 471 (1937). (38) See, e.&, L. Pauling, “The Nature of the Chemical Bond,” Cornel1 University Press, Ithaca, N. Y., 1940. (39) Referring to Eq. (2) for the case of homopolar bonding, it is seen that insofar as I is its major term, the quantity @ is the potential energy of a charge of magnitude S, located in the overlap region, in a field of potential ub. If then - 2 8 is put equal to A S 1 (Eq. ( 5 ) ) , the I used should be so chosen as to be a measure of the magnitude of ub in the overlap region. But since ub in this region depends on all the electrons in the outer shell, a n I averaged over the latter is suitable (Rule (1) for 7). A similar analysis for the heteropolar case (cf. Ref. 7, Sections 15-16), leads again t o Rule (l), pEus Rule (2). The foregoing arguments are directly applicable t o the bonding terms Xij in Eq. (21) and t o Ykl repulsions between like lone pairs (cf. Eq. (8), (9)). It is here assumed t h a t the same kind of argument based on the form of 8 in Eq. (2) can be extended to alE Yki terms. (40) The application of Rule (2) to A 0 pairs differing greatly in I deserves further discussion beyond that in footnote 39. Consider for example Yls.za between the l a lone pair of carbon and the 28 lone pair of fluorine in CF,. The overlap integral is small but not negligible. Most of the overlap occurs in the region where l a of carbon ia strong. The potential in which the overlap charge finds itself is then nearly the same as for a l a carbon AO, suggesting that I for l a carbon should be used. Rule (2), which calls for the arithmetic mean of this and I of the outer shell of fluorine, is a compromise between this and the use of a smaller I such as for example a geometric mean. Considerable support for Rule (2) for inner-shel1:outer-shell Ykl’s is afforded by the magnitudes of Kmn’s as estimated theoretically (cf. J. F. Mulligan, J . Chem. Phya., 19, 347 (1951)). That inner-shel1:outer-shell repulsions are very large is shown also by the work of H. M. James, ibid., 8 , 794 (1934), on Lir. Also, a direct estimate by the m i t e r of Y I . , for ~~ Lii gives a result in close agreement with the magic formula expression 0.7. Eq. (231 with A = 1.16, Y
-
Vol. 56
to be obtained by aGeraging (1)intratomically, for each AO, over (valence-state) I values, obtained from spectroscopic data,S4for all electrons in the same valence shell as the given AO; (2) then interatomically.40141 The S’s in Eq. (22)-(23) may be taken from theoretically computed tables. l 4 Extensive trials were made first using 8’s based on Slater AO’s, and then on SCF AO’s. The latter should of course be more accurate, and after numerous trials it was tentatively concluded that they must be used, and they have been used below,42although it had been hoped at first that the much simpler Slater S’s would give acceptable results. Since most of the terms in Eq. (21), likewise the S’s in Eq. (22)-(23), depend on R values, it needs to be emphasized that Eq. (21)-(23) are intended here to be used primarily for molecules with their bonds a t equilibrium lengths; and the S values are to be computed, and the A values and Y determined, to fit this situation. However, the possibility that Eq. (21)-(23) can be extended or adapted to reproduce interatomic potential curves over considerable ranges of bond distances may well deserve exploration.4 8 * 4 4 Moreover, it is to be noted that the non-bonded repulsion ,(Eq. (23)) and attraction terms, the Y’s and K’s, are not limited in their validity to any particular range of R values. Thus they should be capable as they stand of representing closedshell interactions and the repulsion potentials of steric hindrance for varying configurations of nonbonded atoms (here see last paragraphs of Section 11, and ref. 4).46 V. Magic Formula for Special Cases The magic formula Eq. (21)-(23) is applicable for molecules in their ground states with all electrons paired. It can, however, easily be modified to include several other cases. (41) What is meant by Rule (1) should be clear from fo_otnotesd of Table VI11 and b of Table IX. As an example of Rule (21, IiS.zab would ?(n-r)b). be $(lla. (42) Actually, S C F 8’s have been published only for C-C and C-H bonds (cf, footnote 14b). I n the work below, they were estimated in the following way for N=N, 0=0 and F-F bonds. From the Slater p value for N, 0 or F, together with the equilibrium R value, a Slater p value (cf. ref. 14a, Table I and Eq. (12)) was determined. The S value corresponding to this p was then looked u p in a table of S C F S’s for carbon-carbon bonda. This was done for S(ls,2po) using Table I X of ref. 14b, and for S(2a,2pa), 8(2pr,2po) and S(2pr,2pr) using Table VI of ref. 14b. Hybrid S a , S(le,@),S(8,B) and S(8,oB) were then computed using Eqs. (Z), (4), (6) of ref. 14b. [Accurate analytical expressions for l a , 2.3, and 2p SCF AO’s of C, N, 0, and (by extrapolation) F have recently been determined by Dr. P.-0. L6wdin, and will Boon be published. Direct computation of 8’s for Nt and On using these expressions, by Mr. C. W. Soherr in this Laboratory, gives values in satisfactory agreement with those obtained in the manner described above. Details will be published later.] (43) Note that in general Eq. (21)-(23) are capable of giving a minimum of energy (maximum Do) as a function of the bond distances, for example, for though only if important Y k l t e r m are present-ot, Hz. To make the single term A S 1 / ( 1 8)give a maximum D for HI. one would have to make A a suitable function of distance. (44) I n fact, calculatibns with a preliminary form of the magic formula, as applied to N=N, N=N, N-N, &., show that it gives maxinium values for D. calculated and plotted as a function of R, a t R values very roughly equal to the observed equilibrium values. (45) Related computations on the variations of exchange energies with bond angle, and their effect on bending vibration frequencies, have been made by various authors using VB theory: 6.0.. J. H. Van Vleck and P. C. Cross, J . Chem. Phys., 1, 357 (1934): references in footnote 55 of the present paper: and other more recent articles.
+
+
STRUCTURE OF BONDENERGIES AND ISOVALENT HYBRIDIZATION
March, 1952
Odd-electron molecules involving what Pauling calls one-electron or three-electron bonds are easily included by using Eq. (6) for the former and Eq. (11)-(12) for the latter instead of Eq. (22), just as in the prototype cases of HI+ and Hez+ for which these equations were developed. Various moleculeions (e.g., Nz+, Oa+) fall under this case. For radicals containing unused valence electrons, e.g., CH or NH, Eq. (21)-(23) can be used directly, although some care may be needed in determining what valence states and P values are required. The casewf 0 2 is somewhat special, but is most easily treated by computing Do for the first excited (‘Ag) state, to which Eq. (21)-(23) are directly applicable, then correcting to the ‘2, ground state. For excited states, LCAO theory (Section 11) may often be used as a guide in setting up a magic formula. VI. Determination of A’s and v in Magic Formula ; and of Degrees of Hybridization In order to give empirical reality to the magic formula, Eq. (21)-(23), values of A and v must first be determined so as to make it fit a few molecules whose Do values are reliably known. Predicted Dovalues obtained from the resulting formula can then be checked against empirical Do values for additional molecules. After considerable preliminary exploration, the following plan was adopted for the first step in this procedure. First it was decided to try to get along with only three empirical parameters: A , (for all u bonds and u-u non-bonded repulsions), A, (for all ?r bonds and r-r non-bonded repulsions), and v.25 This plan required a fitting to a t least three representative molecules whose DO values are reliably known. For this purpose, the molecules CH, 0 2 , N2 and Fz were selected, the first two to be fitted exactly, the last two as well as possible. For the CH radical, the value DO = 3.47 e.v. is known with a high degree of probability, although there may be a very slight possibility that it is 0.1-0.3 e.v. higher (not more).46 For 02, the value DO = 5.08 e.v. (within about 0.01 e.v.) is certain.47 For Fz, there has been some uncertainty about Do,but there is scarcely any doubt that it is in or close to the range 1.6 0.3 For Nz,there is perhaps some doubt as to whether Do is 7.37 or 9.76 e.v., but the evidence is decidedly in favor of the second of these values, and all other values seem to be definitely excluded.49
*
(46) ’The value 3.47 e.v. for the 2 I I ground state of CH (and 3.52 e.v. for CD) was determined from predissociation in the u = 0 and 1 levels of the excited state by T. Shidei, J a p . J . Phps., 11, 23 (1936), and confirmed by others. This value is accepted by Herzberg (cf. footnote 47) and by Gaydon. On the other hand, by a Birge-Sponer extrapolation of the vibrational levels of the 2 2 - state, one obtains 3.70 e.v. as a probable upper limit to Do. (That the ’U is the ground state is shown by the occurrence of absorption from it in interstellar space.) (47) Cf. G. Herzberg, “Spectra of Diatomic Molecules,” Second Edition, D. Van Nostrand Co.. Inc., New York, N. Y.,1950. (48) R.N. Doesoher, J . Chem. Phu8.. 19, 1070 (1961), and referenoea given there. Evans, Warhurst and Whittle, J . Chem. SOC..1524 (1950). H. J. Schumacher, C. A . , 46, 2300‘ (1951). (49) Cj. 0.B. Kistiakowsky, J . A m . Chem. SOC.,73, 2972 (1951); A. E. Douglas pnd G . Herzberg, Con. J . P h w . . as. 294 (1951). The latter work definitely eliminates all values but 7.38 and 9.76 e.v.: the former work seems to be incompatible with the smaller value. Con-
30 1
The magic formula for CH involves only A, and v ; for the other three molecules, it also involves A,. It is convenient then to begin with CH, determining A , for each of several assumed v values; that is, A , as a function of v. Then assuming the same A,(v) for 0 2 , Ar(v) is determined. Using A,(v) and A,(v), Dois computed as a function of v for Naand for Fz,23and compared with the empirical Do values to obtain a bestcompromise v. A very interesting by-product of the process of determining A,, A,, and v is that at the same time the degree of s,pu hybridization a2 in each of the molecules used must and (to the extent that the magic formula is correct in structure) can be determined (see Sections VII-IX, in particular, Fig. 1 for CH, and its caption).
VII. The Fitting for the CH Radical The VB electron configuration for CH may be written’4b it12
hodPhg*h)T )
where all the AO’s but h are carbon atom AO’s; hp and hop are mutually orthogonal 2s,2pu hybrids of the forms ha =
ots
+ BPU;
hop = Bs
-CV~U
(24)
where a! > 0, p > 0 (thus making hp overlap h more strongly than if p = 0 ) , with a2 p2 = 1. The magic formula Eq. (21) becomes Do = Xgh - Ykh - Y0g.h )Km, - p (25) if RE is assumed negligible.6o In each subscript, the first symbol refers to the carbon AO, the second to ISH. It can be shown (see Appendix 11) that P is given by Po a2AP,with Po and AP having the meaning and numerical values given in Table VIII. Next making use of Eq. (22)-(23), Eq. (25) becomes 3.47 = Do = Au(14.24 S p h / ( l S p h ) - 15l~S’kh14.24~s 2 0 p , h ) - a2Ap i(Krh - Po) (26) The numbers 14.24 and 151 are average I values as required, being, respectively, ~ ( T L IH)and ~ ( I K IH),where and IKrefer to carbon and are taken from Table VIII, and IH= 13.60 e.v. Since v cannot yet be determined, trials were made for each of the four assumed Y values 1.0, 0.85, 0.70 and 0.55. The right side of Eq. (26) is a sum of three groups of terms: first, A, 1, which, with v assumed fixed, depends on two adjustable quantities, namely, A , and a 2 ; second, -a2AP, which is a constant times cy2; third, ( i K r h - PO),a constant. For each assumed value of v, values of A , and a! satisfying Eq. (26) were simultaneously determined by a graphical procedure described in the caption of Fig. 1.
+
+
+
+
+
+
+
--
siderations on the nature of active nitrogen advanced by G. Cario and L. H. Reinecke, Abh. Braunschw. Wiss. Uesell., 1, 8-13 (1949), also strongly favor 9.76 8.v. But in favor of 7.38 e.v., see H. D. Hagstrum, Rev. Modern Phss., 23, 185 (1951). (50) If the C-H bond were strongly polar, there would be an appreciable R E . To estimate the polarity, one needs the relatfve electronegativity of C and H for an hg-1s bond. As was pointed out some time ago (cf. footnote 34), the electronegativity of carbon.dependa strongly on the degree of 8 , p hybridization a2 in the carbon bond orbital. As it happens, a* as deduced below for hg in CH (cf. Table I) is such that carbon h a and hydrogen 1 8 are almost equally electronegative. Thus the assumption R E = 0 is justified.
ROBEHT S. MULLTKEN
302
3.47
3
-
2
$
d
v
4 1
0
0.0
0.2
0.1 CY2
Fig. l.-Do
+.
0.3
0.4
for CH computed using the right side of Eq.
(26), and plotted as function of degree of hybridization asin the carbon bonding A 0 ha (cf. Eq. (24)), for each of two assumed values of v (0.7 and 1.0). For each graph, the ordi-
nate scale has been adjusted (by adjusting the value of A , ) until the maximuin ordinate of the graph is equal t o the empirical DOof 3.47 e.v. (left side of Eq. (26)). In this way, for any assumed value of Y , a simultaneous determination is made of those values of A , and a2 which can reproduoe the empirical DO. I n this procedure, the final a2 has the character of an eigenvalue. For the correct value of v, the 012 value so determined should be the actual degree of hybridization, if the magic formula is essentially correct in form. (To make the curves have their maxima exact1 a t 3.47 e.v., A , should be slight1 increased for Y = 0.7, sligttly decreased for v = 1.0; but d e r e is no point in determining A s t o more than two decimals.)
The necessary S values, for any a! value, were first looked up in published tables,51 Po and AP were taken from Table VIII, and K , h was taken as 0.8 (51) L.c., ref. 14b, Table X I (SCF values) for Sph and 806,h (positive hybrids for E&, negative hybrids for So0,h);ref. 14a, Table IV, for Skh Values. (62) J. R. Stehn, J . C h s m Phys., 6, l$6 (1937), and G,W. Sing, ibid., 6, 378 (1938), applied VB theory in instructive studies of CH, NH. OH and FH. Using atomic and molecular spevtroscopic data, they obtained empirical values for various exchange and Coulomb energies. Ho*ever, they made assumptions (inaluding peglect of ianershell non-bonded repulsion@,and to hybridization), which in thg light of the present work render very questionable the meanings of the numerical values they obtained. However, their value of 0.8 e.v. for Xrh ( J r in their notation) appears free from serious objection. It agrees well with the value 0.68 e.v. estimated semi-theoretically a t the end of footnote 31 above.
Vol. 56
The graphical eigenvalue procedure used in Fig. 1 needs justification. First of all, it should be noted that it is not necessary that the procedure be valid except nem the correct value of v (which, moreover, need not a t first be known), and the following reasoning is to be understood as applying only to that value of Y, Suppose, then, that one i s working with a magic formula, including correct values of v and A,, such that when the degree of hybridigation existing in the true molecular wave function is assumed, the formula will correctly reproduce the true bond energy. Under these circumstances, an exact formula for the energy of the true wave function and of others differing from it only in, degree of hybridization would show a maximum Do (minimum total energy) for the true degree of hybridization. If the structure of the magic formula represents approximately correctly the various elements which combine to give the final Do,it should show essentially the same property. If the magic formula were really bad, it might give no maximum a t all for Do as function of a, and the fact that it actually gives well-defined maxima (cf. Fig. 1) is reassuring. The writer feels that if coefficients for the magic formula can be obtained by the procedure desaribed, such that observed Dovalues for a reasonably large and varied group of molecules can subsequently be reproduced, then both the formula and the procedure are probably valid to a reasonable degree of approximation. The results obtained for CH are summarized in Table I. In Table 11, details of the individual terms in Eq. (25) and (26) for DO of CH are given, for the parameter values ( v = 0.7, A , = 1.16) later adjudged to be about correct. This tabulation gives a vivid picture of how the various bonding, repulsive, and promotion terms combine to give the final resultant Do.
A,
TABLE I CH AS FUNCTION OF ASSUMED VALUE OF v 1.0 0.85 0.70 0.55 1.33 1.16 1.07 1.25
a2
0.22
VALUES OF A,
AND
Y
a2 FOR
0.20
0.155
0.10
TABLE I1 STRUCTURE OF DOFOR CH IF v = 0.7, A , = 1.16" CY30
0.509 0 071 0.553
sBh
S kh SOP,h
a2 =
0.155
0.686
0.071 0.308
5.57 6.73 -0.62 -0.62 - Ykli -3.54 -1.09 - h -a2AP 0.00 -1.46 0.40 0.40 3KTh -0.49 -Fa -0.49 1.32 3.47 Do a See Table VI11 for Po and A9, ref. 51 for S values, ref. 52 for K n h , Eq. (25)-(26) for magic formula. XBh
(ev.)
A comparison, for v = 0.7, between the results computed for the case of no hybridization (a! = 0) and for the probable actual hybridization (az =
1(
'
STRUCTURE OF BOND ENERGIES AND ISOVALENT HYBRIDIZATION
March, 1952
0.155) likewise shows vividly the importance of the relatively small amount of isovalent hybridization in stabilizing the molecule (see Fig. 1 and , ~ ~ VB theory with semiTable 11). M ~ f f i t tusing empirical values of the necessary exchange integrals (instead of Eq. (22) and (23)), and maximizing the bond energy with respect to degree of hybridization in essentially the same way as here, has obtained similar results (a2about 0.1 for CH; he has also discussed NH and OH). Noteworthy in Table I1 is the fact that hybridization is effective in two ways, namely, by increasing the size of bonding term X a h and by decreasing the magnitude of the non-bonded repulsion term - Yop,h.
VIII. The Fitting for O2
303
after inserting the value of A,, (as determined in Section VI11 for CH) corresponding to any one of the v values 1.0, 0.85, 0.7, 0.55, G becomes a quantity which depends only on a and v, To obtain the hybrid SCF 8’s required in computing the Y’s by Eq. (23), in C and G, the method of ref. 42 and 51 was used. Finally H, after inserting FL for the oxygen atom from Table VIII, depends only on v and on the as yet undetermined AT. For a n y assumed value of v, the complete expression on the right of Eq. (30) now depends (aside from known constants) only on A , and a. It is then possible, for any particular assumed v value, to determine A , and a2 by the method that was used to determine A , and a2 for CH. The results are included in Table IV.
Since the *2-, ground state of 0 2 is a somewhat special case in terms of VB theory, it will be more convenient to make a fitting to the first excited, lAg, electronic state, whose VB electron configurat i ~ n ’is~ ~
IX. Definite Determination of Empirical Parameters for the Magic Formula Determination of Degrees of Hybridization For the final determination of A,, A , and v, it is k)’ k)2 ha@)’hop)’ hp.ha) a ~ a’)’ ) a’)Z convenient next to compute DO for several mole(27) with one ha-ha bond and one T-T bond5a; ha and cules by the magic formula, for each of the v values hopare as defined in Eq. (24). For the lAg state, 1.0, 0.85, 0.70, 0.55, using the corresponding apvalues of A , and A , as determined in Do is less by its known excitation-energy (0.98 propriate Sections VI1 and VIII. By comparing these come . ~ . ) ~than 7 Do (5.08 e.v.) of the 32-gstate. Eq. puted Do(v) values with empirical Dovalues, one (21) now takes the form can determine what v value gives the best average 4.10 = Do = (Xpg + xnr) - (2Yg,o@+ fit. With v thus fixed, the values of A,, A,, and 2YOP,O@$. 2yks f 4Yk,Oa 2Yr,)+ (3Kpr (28) of a2for the molecules considered are determined. 6Koa.n 4-6Kks 2Krs’) - PO + CY’AP) Selecting first the ground states of N2 and Fz, it may be with Po and AP as given for the V Zvalence states noted that their electron configurations are formally identical with (27) for the lAg state of 02,except for the r elecof the oxygen atom listed in Table VIIIn54 In Eq. (28), Kk, is negligible and can be dropped. trons. The respective P partial configurations are Further, it is convenient to rewrite some of the other terms as follows (the relations stated are easily proved) 2(yk@ y k , D @ ) = 2 ( y k s f YkU); 3(Ki39r f
+
+
+
+ Kur)
Equations for NZand Fz corresponding to _Eq. (28)-(31)are easily written; values of PO,AP, ZIC and 11,for use in conThe terms in Eq. (28) can then be usefully re- nection with these equations are found in Table VIII, and the necessary 8’s and K ’ s are obtained in the manner degrouped as scribed for 02 following Eq. (31). Regarding A , and A, ~ . ~ O = D ~ = C + E ( Y ) + F ( C ~ ) + G ( C Y , Y ) + H(30) ( A ~ , Y now ) as known (for any given Y ) , and DOas a quantity to bc computed as a function of v , the analog of Eq. (39)is where K0a.r) =
‘
1
+
+
C = -2Po (3Km 3Kur E(V) = -2( Yka Yku)
H ( A , , v ) = X,,
+
- 2Ymr
3(Km
(29)
+ 2Krr)
=
A,IZ[Sn/(l
+ S),
- 2vSZrj
I
Using Eq. (23) for the Y’s and the method of ref. 31 for the K’s, C becomes a known constant and E ( v ) a known function of v. Similarly, F ( a ) becomes a known function of CY. G is an expression which (using Eq. (22) and (23)) is seen to be proportional to A,, and also to depend on a and v, but (53) One has either 9 r + . ~ + ) ~ - ) 2r-)* or T+)Z r+)l, the two cases corresponding t o the M L = -2 and + 2 sub-states of the doubly degenerate electronic state ‘Ag. (54) For a = 0 , 6 = 1 in Eq. (24), the state of each oxygen atom would be szucr’2, Vz; for a = 1, p = 0, it would be au~mr’z,Vz. The proof t h a t P for each oxygen a t o m is Pa a * A P is indicated in Appendix 11. A factor two is then required in Eq. (28) since bolh oxygen atonis are promoted. ,-a,-)
+
Where a choice is given in Eq. (34), the first alternative refers to Nz, the second to Fz. The next step, for either Nzor Fz, for each assumed Y value, is to plot the computed Do against oca, as in Fig. 1. The maximum of each such curve then gives the desired value of DO(also of d )for the given Y value. If the electron configuration of CHd is written using tetrahedral carbon o r b i t a l ~ “ ~ ~ ~ 6 k)’ te,&) teb.hb)le,.h,) tedhd) (55) For some disoiifiRioiis of hydrocarbous using VB methods, see for example, J. H. Van Vleck and A. Sherman, Rev. Modern Phya., 7, 167 (1935)-and references there cited: R. Serber, J . Chem. Phys., 17, 1022 (1035): W.G. Penney, I’rans. Faraday Soc., 51, 734 (1935)
ROBERTS. MULLIKEN
304
$01, 5G
TABLE I11 COEFFICIENTS OF TERMS IN DOFORMULAS (EQ.(21)) FOR n = 2 n = 4 n-6
XBh
x88
2 4 6
1 1 1
-Yog,h
t
2 1 0
4 t
-Y'sh
2 3
1 2 3
-P$b
vc
1
n = 2 n = 4 n = 6
--'ah
-Y.B
XTr
1 2 3 %u
-Yns
1 1 1
-Ykh
-Y'kh
2 4 6 Ksr
-Yas
a
t1
2
+YBP
-Yxr
t
0
-1 2 -21
t 1
- Yka 2 2 2
yks
2 2 2
-Y"hh
-yhh
-y'hh
2 4 6
0 1 3
0 1 312
1 3
Kss'
Kksd
Ksh
K'sh
B
2 2 2 2 1 4 2 2 n = 2 4 4 2 2 2 2 1 4 n = 4 2 2 n = 6 1 . 4 6 6 2 2 a Unprimed exchange terms refer to A 0 pairs on nearest, singly primed to those on more distant, and doubly primed to pairs on most distant neighbor atoms. The subscripts @ refer to d i , tr, te for n = 2 , 4 , 6 respectively, OB to the corresponding orthohybrids (cf. footnote 14b). For CZHS,the coefficients given above were worked out on the basis of the opposed form See Table I X for the Pt values (cases B', C D for n = 2, 4, 6, resbectively). For the computations of (symmetry &h). Table IV, it is assumed that the R E correction analogous to the Voge deniotional RE correction (cf. footnote 37) for CHI = 1.28 e.v.) is the same per carbon atom in C2Hn. This must be considered n guesstimate. Polar RE corrections for the Negligible. -H bonds have been neglected.
bV
the magic formula becomes
Do
4Xte,h
- 4Ykh - 2YOte,h f 4 K s h - 3Yhh P(te4,Vd
+V
(35)
The terms -2YOte.h - 4 & , come from transformation of a term -6Yte'h corresponding to the 12 non-bonded exchange terms between H atoms and the te AO's to which they are not bonded; KBh is taken as in ref. 52, except for a slight correction for the smaller C-H distance. P is Voge's value for case D of Table I X and V is Voge's demotional RE correction*' (see Section IV). A polar R E term, though probably appreciable, has been neglected in E (35). I n computing X and the Y's using Eq. (22)-(23), %e S values are obtained as for CH," the? values are I H = 13.60 e.v., I K = 288 (Table VIII), and IL = 13.68 (Table IX). For CzH. (n = 2 , 4 , or 6), the non-bonded exchange terms are very numerous, and the D Oformulas are best presented in tabular form (Table III).66 The electron configuration@ are
.
CzHB: k)Z k ) z te.4,)
teb'hb) te,.h,) tea.ted*) tee* .h, *) teb *-hb*)tea *.ha*)
C2Hd: k)' k)'
lrb'hb)
tra'ha)
tr,'tTc*)
t?b*'hb*)
tra*.ha*) 7rx.1Tx*) CZHZ: k)2 k)' di'.h) di.di*) di'**h*)TT*) x ' * x ' * )
1
(36)
Many of the non-bonded exchange integrals involving hybrid AO's occur in groups which (by using relations similar to Eq. (29) above) can conveniently and rigorously be transformed into Y's and K's which are largely non-hybrid, and this transformation has been done in preparing Table 111. For Hz and He*, Eq. (21)-(23) reduce to Eq. (7) and(10), respectively. For Hz+ and Hez+, Eq. (6) and (11)-(12) are respectively applicable.
The results of computations on all the molecules mentioned above, carried out in accordance with the first paragraph of this Section, are presented in Table IV. As will be seen from this Table, the computed Do varies rapidly, and in opposite ways for N z and Fz, with v, so that the best v is rather sharply determined, On the whole, agreement of observed and computed D ovalues is best for about v = 0.7, with A , = 1.16, A, = 1.53. The fact that a compromise v can be determined a t a11 to fit (even though roughly) such different molecules as N p and FZ gives important support to the validity of the general structure of the magic formula.
Excluding Hp and Lip as e x ~ e p t i o n a lthe , ~ ~average difference per electron-pair bond between observed and computed Do values for v = 0.7 is about 1 0 . 3 e . ~ . ~In ' view of the presence of only three adjustable parameters in the magic formula, the agreement found is all that one could reasonably hope for, and much too good to be reasonably attributable to accident. As a corollary, this argument indicates that the hybridization coefficients in the v = 0.7 column in Table IV are probably fairly near the truth. The following further point is worth emphasizing. Although the specific values for the coefficients in the magic formula were obtained by procedures which some readers may question, the formula with these coefficients now stands on its own merits, independent of these procedures. Its degree of quantitative validity should be judged by its empirical success in representing observed bond energies. One should also bear in mind that the present magic formula is a first edition, in need of further test by application to more molecules, and undoubtedly capable of further improvement in various ways (see Section XIII).
X. Validity and Uses of the Magic Formula Procedure The magic formula has now been implemented with specific values for its coefficients. One may ask: How good is it? What can it be used for? (50) The figures on Hz and Lia suggest t h a t a distinct coefficient A,, with value about 0.7. may be needed for pure s-8 bonds, as was proposed in Ref. 4. Or possibly A s is anomalous for molecules which have unusually small € values, as is true of Ha and Liz (cf. second column in Table IV); the fact that proportionality of the theoretiaally computed LCAO quantity - B to S fails when g gets small (see Section 11: sentence with Ref. 11) suggests this possibility. The discrepancy for Fn might then possibly he attributed to its exceptionally lares E value. For other possible ways of dealing with Ha, Lir and Fi, Bee also footnote 23 and Section XIII. (57) If the hydrocarbons, for which both the observed Do values and the assumed resonance energies (Table 111, footnote c) are rather uncertain, were omitted, a somewhat improved fit for the remaining molecules could be obtained with (say) Y = 0.75 (or perhaps 0.85, if As is made somewhat smaller than for Y = 0.85 in Table I V , and exact agreement for Oa is sacrificed)
.
STRUCTURE OF BOND ENERGIES AND ISOVALENT HYBRIDIZATION 305
March, 1952
TAB^ IV COMPUTED DOVALUESFROM MAGICFORMULA Y
Molecule
Ed
A, A,
1.0 1.33 3.36
Computed Do Values (e.v.)%b*c 0.85 0.70 1.25 1.16 2.28 1.53
0.55 1.07 0.975
Observed DB (e.v.1.
3.47 3.47 3.47 (a2 = 0.10) 0.155) 4.10 4.10 4.10 4.10 1.3 4.10 02 (a' = 0.12) ( a 2 = 0.14) (a2 = 0.09) (a2 = 0.06) 4.91 ' 9.76 or 7.37 8.32 27.27 13.70 1 .o Nz (a2 = 0.21) ( a 2 = 0.31) ( a 2 0.26) ( a a = 0.15) 1.97 -0.46 3.82 3.12' 1.8 1 . 6 i0 . 2 Fa (a* = 0.024) ( a 2 = 0.031) (a*= 0.018) (a' = 0.014) 2.89 3.12 2.66 1.0 3.32 2.65 H2 7.79 6.78 0.7 7.32 6.24 4.48 Ha 1.46 1.7 He2 + 2 . 1 f 1.0h -4.30 1.7 -3.66 -2.37 -3.01 He2@ (-2.38)' 2.06 0.8 1.03 Liz 17.31 17.64 0.9 17.61 ' 17.36 15.02or 17.30 CH, 1.2 25.03 27.62 22.05 26.72 24.90or29.46 C& 24.23 21.47 25.86 1.0 19.11 or 23.67 22.79 C2H4 23.83 14.16 31.81 0.9 12.90 or 17.46 18.36 CiHz See text of this Section for methods of computation. The a ' values given were determinedby maximizing the computed Doin the manner. illustrated in Fig. 1. For the hydrocarbons, aavalues were assumed corresponding to the electron configurations in (36), but demotional R E corrections were then included in the computed D i s t o allow for deviations from the assumed states of hybridization. This correction was 1.28 e.v. for C H , 2.56 e.v. for all CzH, cf. Table 111, note c). Polar RE corrections have been neglected. For deiinition of E., see Eq. (3) of ref. 4. * For CH, re 46; for Fz and Nz, ref. 48, 49; for Hea, ref. 4, Table X; for the other diatomic molecules, ref. 47. The values for CH4and C2H, are based on heats of formation AH: for 0°K. taken from the American Petroleum Institute Project No. 47 Tables (15.99 kcal. per mole for CHI, 16.52 for CzHO, -14.52 for CaHn, and -54.33 for CZHZ), combined with DO= 4.478 e.v. = 103.2 kcal. for HZ and, for the heat of sublimation L of graphite, either (a) 124 kcal. or (b) 175 kcal. (see footnote 58). 1 e.v. = 23.06 kcal./mole is assumed throughout. f 'Ag state. 0 At 1.06 A. Band-spectrum value, 3.1 e.v., not very reliable ref. 47); theoretical value, 2.2 e.v. (S. Weinbaum, J. Chem. Phys., 3, 547 51935)); electron-impact value, between 1.2 an 2.1 e.v. (J. A. Hornbeck and J. P. Molnar, Phys. Rev., 84, 621 (1951)). Theoretical.
CH
0.9
3.47
(a2 = 0.22)
3.47
(a2
= 0.20)
( a a=
-
+
Q
I.
6
The first question is discussed at the end of Section 11; Eq. (35) and Tables 111,VI), and in view of the I X and in Section XIII. fact that the only possibly needed further correcI n answer to the second question: the magic tions to the computed D oof CH4 would probably be formula, used in connection with the maximizing polar RE corrections which would increase Do, the procedure described above, may be expected to be definite indication given by the magic formula in valuable principally in three ways. They are: (1) favor of the high heat of sublimation of carbon may as a tool for the approximate computation or esti- have real significance. mation of bond energies; (2) as an expression The magic formula should be applicable to radiwhich shows roughly quantitatively how the total cals, and so, by the use of differences between heats energy of atomization results from the addition and of atomization for molecules and radicals, should subtraction of significant individual terms; (3) in permit the calculation of energies of dissociation determining approximate degrees of isovalent hy- into radicals. bridization. The first of these applications will be The magic formula may be helpful also in the preconsidered now, the second in Section XI, and the diction of bond energies for excited and ionized third in Section XII. molecules. As can be seen from Table IV, the magic formula The formula, either in its present form or after is not as yet reliable enough to yield conclusive de- improvement, may also be useful in dealing with cisions between controversial alternative observed non-bonded repulsions (see the last paragraph of Dovalues, such as occur in several cases (see last Section IV). column of Table IV). However, it is striking that, no matter what value-pairs A,, Y that fit CH are XI. The Theoretical Structure of Chemical Bond Energies taken, the magic formula always predicts for CHI Any system of dividing the total energy of atomia value of Dowhich agrees closely with the highest of the "observed" Dovalues corresponding to the zation of a molecule into a sum of individual terms several competing values58for the heat of sublima- is more or less arbitrary unless these terms corretion of graphite. (Note that A , is not involved for spond to realizable physical processes-as, for exeither CH or CH,.) I n view of the nature of the ample, if Dois expressed as the sum of energies resimilarities and differences in the structure of the quired to remove the atoms one by one in a specified C-H bond in CH and CH4 (I$.Eq. (25) and Table order.sg Even then, there is no unique way of analyzing Do into a sum of terms. And even in a (58) The extreme values are 124 and 175 kcal. For recent reviews, see H. D . Springall, Research, 3, 260 (1950): H. D. Hagstrum. Rm. Modern Phua., 23, 185 (1951).
(59) For a recent analysis of these problems, see M. Szwarc and M. 0.Evans, J. Chem. Phus., 18, 618 (1950).
ROBERTS. MULLIKEN
306 TERMS IN
TABLE V Nz BONDENERGY‘ USING THE MAGlC FORMULAb Al. Constant Terms (Total C) 2Yks - Z Y k s - 2Po $. 2& f 2 K s r f &w’ = c 1.04 - 3 . 2 4 - 3.40 0.89 1.68 0.10 = 9.96
COMPUTED
-
2x~r
14.97
Vol. 56
+
+
+
A2. Terms Including Those Depending on Hybridization C - 2alAP f 2K0g.n + xgg 2Yg0,3 - 2Y0&0,3 2 Y ~ , ~ ,=3 Do a’ = 0: 9.96 0.00 0.89 3.26 - 7.15 - 5.96 - 1.04 = -0.04 aa 0.21: 9.96 - 5.44 0.03 8.22 4.38 - 0.05 - 0 . 0 2 +8.32
+ +
+
+ +
+
+
-
-
B. Distribution of Terms Between u and
+ +
-
T
Bonds“ a2 = 0
-
a2 = 0 . 2 1
-
bond: Xgg $ZK,, - 2P0/3 2a2AP ZY -14.57 - 5.78 ?r bonds: 2XTT K,,? tZK,, - 4&/3 +14.53 +14.10 Total (Do) - 0.04 8.32 For certain terms where the 0 Cj. Eq. (33)-(34). b Eq. (21)-(23) with A, = 1.16, A , = 1.53, Y = 0.7 (cf. Section IX). distribution between the u and ?r bonds is of necessity rather arbitrary it has been effected as follows. Of the promotion energy, the amount 2Po is necessary for the trivalent valence state, and it is here assigned one-third to the one u and twothirds to the two 7r bonds. The additionaLpromotion energy 2a2AP is assigned exclusively to the u bond, since hybridization affects almost exclusively this bond, The mixed terms Krr are divided equally between the u and ?r bonds. fl
+
+
contributions acComparing the regarded as merely a matter of convenience for calculated results for az = 0 and a2 = 0.21, it further computation or understanding. I n spite of all indicates that without isovalent hybridization these considerations, everyone recognizes the use- (which is equivalent-see Table VIII-to partial fulness of writing Do as a sum of terms of more or promotion from s2p3 to sp4 without change of the less theoretical character. Most often, Dois cal- formal valence three), the bond strength would be culated as a sum of standard contributions (“bond very small. energies”) one for each chemical bond-plus corA notable feature of Tables V and VI is the large rections if necessary for LLresonance.” The stand- size of the non-bonded repulsion terms Yka, Yko, ard bond energies are determined empirically to fit Yk,o,3, etc., involving the inner-shell (Is) electrons.m observed thermal data.38 The valence-shell repulsion terms and Y o ~ , p The present magic formula likewise is a sum of would be large if there were no hybridization (az= terms, including corrections for “resonance,”86 0), but are made much smaller by hybridization. and contains coefficients adjusted to make it fit ob- Another point of interest is the very considerable served thermal data. However, it has (like the positive contributions made by the non-bonded VB theory formulass5from which it is adapted) a attractions ( K terms). The breakdowns of the total energies of atomizamuch richer structure than the ordinary additive bond energy formula, and, if soundly based, should tion into individual terms in Tables 11, V and VI afford a much deeper and more detailed theoretical are similar to those presented by Van Vleck, Peninsight into the various factors which determine the ney and others some years ago,e1guided by theoretitotal energies of chemical binding, and greater cal computations and by spectroscopic and other possibilities of prediction. The possibilities of the empirical data. However, the individual terms are magic formula for steric hindrance effects have al- now in many cases much larger. There are two main reasons for this difference: (1) the terms corready been mentioned. Finally, whereas the use of a table of bond ener- responding to inner-shell: outer-shell repulsions gies involves a different empirical bond energy for earlier were almost always neglected, whereas aceach kind of bond, the magic formula has a more tually they are probably fairly large40; (2) the universal character in that, for all molecules built possibility of hybridization was usually neglected from first-row atoms, or first-row atoms and hy- earlier except for CHI and C2Hn,and except for drogen, it contains only three empirical coefficients. CH, NH and OH in a recent paper by MoffitLZ2 (To cover the whole periodic system, more will When these two effects are considered, one arrives doubtless be needed.) To be sure, the magic form- a t much larger bonding terms than when they are ula involves also certain other quantities which ignored. By making suitable combinations of individual must be determined: the overlap integrals 8, theoretically; the ionization and promotion energies, terms in the magic formula, theoretical expressions best from empirical spectroscopic data on atoms. (60) This can be understood as follows. As two nitrogen atoms apThe way in which the magic formula gives insight proach each other, c bonding first sets in, with little hybridization and moderate non-bonded repulsions; z bonding is weak because the into the structure of chemical bond energies can ronly overlap integral is small. At closer approach, both hybridization and best be appreciated in terms of examples. For non-bonded e repulsions increase. Meantime the T bonding increases this purpose, reference may be made to Table I1 so rapidly with increasing w overlap that equilibrium is not established and Figure I above, for the CH radical, to Table V until long after the interaction sum has turned negative. (61) See J. H.Van Vleck and A. Sherman, Rev. Modern P h y a . , 1 , 167 below, for the Nzmolecule, and to Table VI. Ta- (1935), for a review; also ref. 37, 52, 55. Reference should also be made ble VB indicates, contrary to the usual idea that T t o the important work of M . Kotani and collaborators in Japanese bonds are weak, that the ?r contribution to the bond- journals. purely theoretical approach, any attempt to break
ing is very large, the sum of the
Q
Do down into a sum of terms must ultimately be tually being strongly negative.
,
5
L
STRUCTURE OF BONDENERGIES AND ISOVALENT HYBRIDIZATION
March, 1952
307
TABLE VI STRUCTURE OF ATOMIZATION ENERGIES~ A. HOMOPOLAR DIATOMIC MOLECULES Terms-(Bk)’
-up
P--
-Pd
X
Fz
-0.80
0 2
-4.31 -8.84
6.25 7.95 8.22
N2
Bond snergiesc
Terms in Atomization Energy (e.v.) b C , D Term? T,?Fand r , ~ Terms ‘ BY ZK X ZY ZK
(-0.25) (-1.60) (-4.30)
-1.91 -5.50 -8.73
0.83
...
2.21
6.29 14.97
2.60
-1.29 -2.62
....
0.04 .08 .10
Gross
Net
3.92 9.3gb 17.16
3.12 5.08’ 8.32
B. HYDROCARBON MOLECULES~ 1. CH, Intragroup Terms (e.v. per C-H Bond) u,h Term8 (Zyk)”
X
6.73 6.66 6.63 6.77 6.91
CH CH4 CzHs CzHi CzHz
‘ ZY
( -0.62) ( - .70) ( - .68) ( - .74) ( - .79)
-1.71 -1.00 -0.97 .90
-
- .so
h,h
T,h
ZY
ZK
.... -0.65 - .44
- .18 ....
0.40 .82 .83 .86 .89
2. Intergroup Terms (e.v. per Molecule) and C-H Terms-
PH-H
h,h
ZY
(ZYk)’
ZY
ZK
-0.56 - .21 - .01
(-0.07) ( - .07) ( - .02)
-2.86 -2.26 -0.67
0.36 .30 .09
CH CHI C1Hs CZH~
Terms (ZYk)e
a&
c.
X
6.20 6.72 7.22
(-1.38) (-2.54) (-4.23)
T,T
ZY
-2.16 -3.43 -4.80
zK 1.86 2.48 2.96
and
r,x’
X
ZY
ZK
...
-1.20 -0.96
0.06 .09 .13
5.56 12.45
...
HYDROCARBON BONDENERGIES (EV.)h
- P 4- RE& -1.95 -1.42 -1.42 -2.845
1
C-C Terms
I
uh
C-H Gross
Bonds Net
5.42 5.82 5.75
3.47 4.40 4.33
6.25
4.79
C-C,
C=C, o r C=C Net
Gross
3.51
0.67
9.48 3.62 -1.63 6.85 5.22 CzHz -9.765 17.67 7.91 Computed by magic formula with coefficients as in u = 0.7 column of Table IV. The terms are grouped by type (X, Y K P ) and by category (a,.; U , T ; etc.), in a way which will be made clear by comparison with the detailed table of terms f i r ”,(Table V, with or2 = 0.21). * The nature of the terms in detail can be seen in Eq. (28)-(31) for 0 2 , Eq. (33)-(34) for N2 and F2(also Table V for N2). The net bond energy is the sum of all the terms listed (except the Y k k (cf. footnote e), which are included in the ZY’s). The gross bond energy is the sum obtained excluding - P and so corresponds to the energy P is 2p0 2 d A P with Po and AP from Table VI11 and a2 from Table of formation from the promoted valence state. I V ( V = 0.7 column). e These sums represent those portzons of the immediately-following ZY’s which result from non-bonded repulsions between the inner shell 6.h: shell, Iss or k 2 ) electrons of one atom and the valence electrons of its neighbor. They are separately listed to show their frequently large magnitude. f Here for the aZ-, ground state of 02,an additional term +0.98 e.v. (determined empirically from the molecular spectrum) has been mcluded, equal to the extra bond energy of the 3 2 - state as compared with the ’As state for which the magic formula computation was made. In CH4, and within each CH: group in C2H, (m = n/2), there are C-H terms (intragroup a,h and ~ , in h Table VIB) and H-H terms (intragroup h,h in Table VIB). I n addition, there are in CZH,also intergroup terms of three kinds: H-H terms, between one H, group and the other; C-H terms (a,h and ~ , hbetween ) H , in one group and carbon atom electrons of the other; and carboncarbon terms of various categories. (See E% (25) and Table I1 for CH, Eq. (35) for CHI, and Table I11 for C2H,, for a complete listing of all these terms; cf. also footnote 55.) Each C-H bond energy is taken (cf. footnote 55) as one mthof the sum of the following terms: (a) all intragroup C-H and H-H terms in CH,; (b) all intergroup H-H terms; (c) one-half the sum of the intergroup C-H terms. Each C-C or C=C or C=C bond energy is taken (cf. also footnote 55) as the sum of (a) one-half the sum of the intergroup C-H terms; (b). the total of all C-C terms. For each net bond energy, a term (cf. footnote k below) - P R E from the second column is mcluded; for the gross bond energies, these promotion terms are omitted. k The values in this column are all per bond (C-H or C-C or C = C or CEC). For CH, RE = 0 and - P = -Po - @ZAP,with Poand AP as in Table VIII, and or2 = 0.155 (cf. Table I). For CH4, - P RE is one-fourth of (-6.97 1.28),where 6.97 e.v. is the tetrahedral promotion energ Pt of carbon (Table IX, case D),and 1.28 e.v. is the Voge resonance energy ( R E ) correction V (s:~ Section IV and.Ref. 373: Similar1 for CzH, (somewhat arbitrarily), the amount - ( P t V)/4 is assigned to each individual electron-paw-bond terminus enJing on carbon; hence -(Pt V)/4 to each C--H bond and -(a/2)(Pt.- V)to the carbon-carbon bond, with a = 1, 2 , 3 for CzHs, C2H4, and CzHz, respectively; note that Pt differs somewhat in the three molecules (see Table 111, especially footnotes b, c); cf. also footnote 55. ~~~~~
5
+
+
+
+
-
or numerical values can be obtained which are counterparts of the ordinary “bond energies.” I n the case of diatomic molecules, the bond energy is the same as the atomization energy, hence is the sum of all terms in the magic formula. For polyby the hydrocarmolecules, bolls C*H, in Table VIC, there is some arbitrariness
(especially for the promotion energy corrections) in dividing the total atomization energy among the carbon-hydrogen and carbon-carbon bonds. 62 The ordinary assumptions3*of approximate constancy and additivity of standard bond energies (02) For earlier VB theory discussions of the division of Do among bond energies, see ref. 56, especially the paper by Serber.
308
ROBERTS. MULLIKEN
have been criticized by Serber62and others.59 It will be noted that the magic formula yields definite bond energies which are not necessarily constant or additive; but in particular cases where they actually are so-as for example in a series of molecules such as the normal paraffins-it should be able to show the reasons why; and in cases where they are not, it should make clear the reasons. Here the magic formula (preferably in a future improved edition, including if possible a systematic formulation of the RE term) should give interesting insights. As an example, using the present magic formula-and probably significant in spite of tfie latter’s limitations-Table VIC shows approximate equality of C-H bond strengths for CH4 and c2H6a relation which presumably would extend also to the higher paraffins-but somewhat varying C-H bond strengths for other types of hydrocarbons. As Van Vleck clearly pointed out in the case of hybrid carbon bonds16*it is useful to distinguish between gross bond energies, referred to atoms in suitable valence states, and net bond energies, representing what is left after promotion energy deductions have been made.@ Gross bond energies are truer measures of real or intrinsic bond strengths, and it is with these rather than with net bond energies that equilibrium distances and force constants should tend to be correlated. (This idea should, however, be applied with caution, since degrees of hybridization or promotion may often vary appreciably with interatomic distances even near equilibrium; for sufficiently large distances, of course, they must change.) Table VI lists both gross and net bond energies for a number of bonds, as computed using the magic formula based on the v = 0.7 column of Table IV. Interesting, and probably significant in spite of the preliminary character of the present magic formula, is the slowly increasing C-H bond strength from CH to CHI, CaHa, C2H4 and C2H2, in spite of a nearly constant primary bonding term X h g . 6 4 For the carbon-carbon bonds,62 the computed variation from C-C to C=C to C=C in Table VIC seems to be too rapid, and this impression is confirmed by reference to Table IV, where the computed atomization energy (for v = 0.7) is relatively too small for C2H6 and too large for C2H2, as compared with the observed values.s6 An interesting problem is the possible resolution of the (net or gross) bond energy of a double or triple bond into a u bond energy plus a ?r bond energy or bond energies. Unfortunately there is no uniquely justifiable logical basis for doing so. Nevertheless, it can be done in one or another way (63) Cf.ref. 61, p. 195; also ref. 37. (64) For an earlier VB calculation, based on similar consideration of
varying hybrid character in the carbon bonding A 0 in the C-H bond, and in the bond energy of the latter, see T. Forster, 2. physik. Chem.. 4SB, 68 (1939), in particular the Fig. I n a recent semi-empirical analysis relating bond energies to bond distances, G. Glockler ( J . Chem. Phvs., 16, 842 (1948)) also obtains similar results. (65) Possibly hyperoonjugation (R.S. Mulliken, C. A. Rieke and W. G. Brown, J . Am. Chem. Soc., 63, 41 (1941)) is partly responsible. However, the authors mentioned estimated the hyperconjugation energy per mole as 2.5 kcal. for CzHs, and 5.5 kcal. for CZHI,not nearly large enough to account for the discrepancies under discussion. It seems not impossible t h a t a revision of the calculations (cf. footnote 30 of ref. 74) might give considerably larger computed hyperoonjugatioo energies.
Vol. 56
if one is willing to accept a considerable degree of arbitrariness (particularly so for net bond energies) in making fractional assignments of some of the terms in Dopartly to the u and partly to the ?r bonds. The terms in question include the promotion energy, the u,?r K terms, and in C2H, the H-H and C-H intergroup terms of Table VIB.62 One way of making such a division is illustrated by Table VB for the nitrogen molecule. This makes the u net bond energy negative in the Na triple bond, and the ?r net bond energy very large and positive.60 Applying the same method also to other molecules listed in Table VI, one would obtain as a general result that u net bond energies are positive for single bonds, near zero in double bonds, negative in triple bonds. The u gross bond energies appear to be always positive. A very arbitrary but simple way to obtain separate u and ?r bond energies is to use the assumption of constancy of bond energies, including equality of energies of u bonds in single and multiple bonds. Although this procedure cannot be logically justified, and is seriously a t variance with the results of the magic formula approach, a few results obtained by it may be of interest for comparative purposes (see Table VII; the assumption of constancy of u bond energies is embodied in footnote b of the Table). TABLE VI1 u
AND
a BONDENERGIES
Assuming Constancy of C-H and C-C u Bond Energies4 Bond
H-H C-H C-C
in Hz in CHI in CaH6
Bond energies B (e.v.) Net6 Gross0
Type
4.48 4.48 3.78 or 4.35 5.20 or 5.77 u 2.41 or 3.55 5.26 or 6.40 u ~ 2 . 4 1or 3 . 5 5 1 ~ ‘IH4 ?r 1.69 or 2.83 4.62 or 5.76 u [2.41 or 3.55Id C s C in CzHz n 1.48 or 2.62 4.74 or 5.88 This material was first presented at a meeting in 1942 (footnote 3), and was later published in part in Ref. 7 (see footnotes** to Tables V and VI1 there). See also footnote Obtained as follows (and see Table IV), footnote e): 62. B(C,H,) = AH;)(CmHn) mL +nDa(Hz) B( C-H) = $Do(CHI); B( C-C) = Do(CzHa) - 6B(C-H) B,( C=C) = Do(CzII4) 4B( C-H) B( C-C) B(C-C)I B,(CzC) = +[Da(CzH2! - 2B(C-H) c &(grossj B(gross) = B(nei) B(net) +-lm(Pt am(Pt 1 V ) ,with m = 1 For for CHI, 2 for CzH,, C2H,, and V = 1.28 e.v. (cf. Table VI, footnote IC), with Ptvalues from I’able IX. Assumed. u u
i
+
-
+
+
-
-
XII. Degrees of Hybridization and Bond Properties Recently evidence has appeared from a number of sources indicating the essential importance of isovalent hybridization (Le,, partial hybridization without increase of formal valence-see Section IV) for the energy and other properties of many molecules. One of the most interesting aspects of the magic formula, taken in connection with the procedure for maximizing Doas a function of degree of isovalent hybridization, is its ability to yield information about the latter quantity (see Table IV, v = 0.7 column, for results on CH, N2, 0 2 and Fz). Moffittaahas already applied a similar procedure to CH, NH and OH, using a conventional VB formula with smaller values than here for the X’s and Y’s, and neglecting the Yk1s;61 he obtained calculated
J
March, 1952
STRUCTURE OF BONDENERGIES AND ISOVALENT HYBRIDIZATION
309
degrees of isovalent hybridization somewhat smaller siderable amount of s,p hybridization in the bondthan here. ing AO. Regardless of the quantitative soundness The interpretation of information on the follow- of the conclusion in this particular case, it is clear ing properties of molecules has also yielded rough that isovalent hybridization must be of essential values for degrees of isovalent hybridization: importance in determining the actual electronegamolecular dipole moments, atomic electronega- tivities of atoms like N, 0 and C1.68 The effect, on electronegativities, bond strengths, t i v i t i e ~molecular , ~ ~ ~ ~ ~quadrupole moments,69 coupling constants of molecular force-fields with nuclear bond lengths and dipole moments, of varying dequadrupole moments, 70 absolute intensities of mo- grees of s,pu hybridization in u bonds formed by lecular electronic spectra.’l The degrees of isoval- carbon atoms with pluvalent hybridization-in parent hybridization indicated by all these methods ticular, of G-H bonds in CH4 and CZH~,CzH4, agree roughly with those by the present method. CzHz-has been discussed by various authors. 72 A brief discussion will be given here only of the effect of isovalent hybridization on dipole moments XIII. Criticisms and Possible Improvements on the Magic Formula and on electronegativities. Robinsonee has shown that if HC1 were covalent, it would have, according As has been emphasized above, the present magic formula to a VB theory calculation, a considerable dipole (cf. Section IV, in particular ref. 30), both as to its precise and as to the choice of numerical values of its coeffimoment of polarity H-C1+ if there were no s,p form cients, is still preliminary. I n this Section. some criticisms hybridization, while about 12% s,p hybridization of its possible shortcomings and some possible leads for its in the chlorine bonding A 0 would give a dipole future improvement will be sketched. First of all, it may be an oversimplification to lump the moment of the observed magnitude and with sign “Coulomb” terms C of VB theory into the other terms H+CI-. Actually, there is no reasonable doubt (compare Eq. (21) with Eq. (19), (20)). This procedure that HC1 has primary heteropolar character of po- was adopted because in the VB theory for Hz,the theoreticlarity H+C1-, so that less than 12% hybridization ally computed Coulomb term C is relatively small, espewould be sufficient to account for the observed mo- cially at the equilibrium R; also because in LCAO theory for Hz-which was used as a basis for the adopted form of ment. However, the VB calculation is not reliable the X’s (Eq. (22)) in the magic formula-it was found emenough to permit quantitative conclusions. The pirically that the sum of Coulomb plus error terms ( C + main point to be emphasized here is that dipole E‘ in Eq. (4)) is small. However, theoretically computed moments are extremely sensitive to small amounts Coulomb energies indicateS7,bb that in C-H bonds (and in general whenever pu and hybrid u bonds are present) the of isovalent hybridization. Coulomb terms of VB theory are much larger than in H2.73 In a rough LCAO c a l c ~ l a t i o n the , ~ ~ writer con- If so, the fact that the ma ic formula, using Eq. (22) for cluded that as much as 20% s , hybridization ~ in the primary bonding terms of VB theory, works as well a s the bonding A 0 of C1 in HCl would be needed to ex- it does may be a result of the likely possibility that in general terms, if actually important, may be more or plain: (a) the observed dipole moment; (b) a the Coulomb proportional to the overlap integral S. The fact that sufficient electronegativity of the chlorine atom to less the value of A,, determined by fitting CH and other moleaccount for the latter. The second point is quite cules involving pu and hybrid-A0 bonds, is toQ large to fit distinct from the first, and is also of interest for it- H2and Liz might then be explainable as a result of the preslarger Coulomb terms in the former. Howself. Some time in setting up a semi-theoreti- ence ofallmuch this is as yet hypothetical. cal scale of electronegativities, the writer pointed ever, Another rather uncertain feature in the magic formula lies out that the electronegativity of an atom should in the method of calculating the non-bonded repulsions bevary greatly with the type of bonding A 0 it was tween inner (here ls)_and outer electrons. As computed using, and in the case of a hybrid A 0 should depend using Eq. (23) with I values taken as averages of Is and outerelectron I values, these particular non-bonded repulstrongly on the degree of hybridization. As ap- sions rather large (cf. Z Y k columns in Table VI). plied to the chlorine atom, the writer’s c o n c l ~ s i o n ~Although ~ are often there are rather good reasons89 for giving credence was that, for it to have the degree of electronegativ- to these large values, a more thorough study-theoretical ity necessary to account for a strong enough H+C1- or empirical-would be desirable. (It seems possible even heteropolarity in the H-Cl bond to reproduce the that such a study might point to somewhat largemather than values.) observed HC1 dipole moment, there must be a con- smaller When a study of the effect of Coulomb terms is made, 66p67
3
(66) D. Z. Robinson, J . Chem. Phys., 17, 1022 (1949), theoretical calculations on HCI. (67) Reference 7. p. 541. (68) The effect of isovalent hybridization on the electronegativities of N and 0 was discussed in ref. 34 (p. 787 and Table I), and it was pointed out there that the observed (Pauling) electronegativities of N end 0 aould be explained (aside from an improbable explanation using “second-stage electronegativities”) only by assuming fairly large amounts of isovalent 8 , p hybridization. For the trivalent N atom, the result was 44%. which, however, now seems rather too large to be credible. In the same discussion, Pauling’s electronegativity for the carbon atom was found to agree fairly well with that calculated by the writer for a tetrahedral hybrid carbon AO. (69) C. Greenhow and W. V. Smith, J . Chem. Phys., 19, 1298 (1951), explanation of molecular quadrupole moments deduced from microwave line-broadening. Theoretical computations indicated that 20% s , p hybridization in the u bond of Na. and 5-10% in that of 0 2 , could explain the magnitudes of these quadrupole moments. (70) C. H. Townes and B. P. Dailey, ibid., 17,782 (1949): ”Hybridization of the normal covalent bonds of N, C1, and As with at least 15% 8 aharaoter is clearly shown.” (71) H. Shull: intensities in C; and N;+ spectra (to be published
soon).
possible effects corresponding to the “multiple exchange integrals” of VB theory24 should also be looked into. With explicit inclusion of Coulomb terms in the magic formula (and possibly smaller Y k terms), leading to smaller A , and revised A , and Y values, the resulting revised Table IV might perhaps show improved agreements between computed and observed DOvalues, in particular for Hf, Liz and Fz,66 and for the molecules CzH.. (For CH4 and CZH,, polar RE correclions for the CH bonds, neglected in Table IV, should be’included.) With smaller Aa values, the X’s and Y’s would become smaller, and thereby closer to the much smaller effective exchange integrals indicated by the earlier work of Van Vleck and others.6’ The magic formula should of course be tested and adjusted by fitting to more molecules, including molecules containing atoms higher in the periodic system. For this purpose, i t will be necessary to obtain self-consistent-field S
(72) Reference 64; ref. 4 (Fig. 2 and p. 4500, and references in footnote 24 there); A. Maocoll, Trans. Faraday Soc., 46, 359 (1950); and see Table VI above. (73) Also in Li:, the Coulomb energy seems to be relatively somewhat larger than in Ht (of. H. M. James, J . Chem. Phys., 2,794 (1934)).
ROBERT S. MULLIKEN
3 10
values for overlaps between such atoms. The hybridization situation will a h be more complicated. The usefulness of the present magic formula is to some degree limited by the fact that corrections under the eneral heading of resonance energy are often needed (qf. 8ection IV); these corrections, although usually of moderate size, must be obtained by outside considerations. They are needed in the following situations: (1) where strongly polar bonds are present; (2) where there is unusual stabilization by conjugation or aromatic re~onance’~;(3) where atoms are present in a part,ially demoted valence state. With regard to polar bonds, reference may be made to Section V above. I n extreme ionic bonds, the procedure indicated there fails, but the magic formula should now be valid with the molecular structural units taken as ions instead of atoms, and an ionic attraction term added. I n addition to RE terms like those used in VB theory for aromatic or conjugated molecules, a small second-order hyerconjugation RE termBJ4is probably needed in the magic formula for C2H4, C2H6, and higher olefins and paraffins. The magic formula is particularly good in cases where there is only isovalent hybridization (cf. Section IV). Here
Vol. 56
the use of hybrid-A0 S values, together with a subtractive correction for the corresponding excess promotion energy, completely takes care of the effect of hybridization on DO. But when, as in CH, and CsH, (cf. Section IV), pZuyaEent hybridization is modified by partial demotion, no simple way is apparent for taking care of the demotion energy except by an RE correction like the Voge correction in Tables I11 and VI. Like other RE terms, this one must be computed or estimated by special methods. Finally, it may be recalled that the use of the magic formula is a t present limited to molecules with bond lengths at their equilibrium values, except with respect to nonbonded interactions (see last paragraph of Section IV). It seems possible that, when perfected, the magic formula may be capable of reproducing D as a function of all the interatomic co6rdinates for large ranges of these. I n relation to this possibility, the ,magic formula, taken in connection with the standard procedure adopted above of always maximizing the computed Dowith respect to degree of hybridization, has the very good property that it permits needed adjustment of hybridization with varying R without extra complications.
w
APPENDIX I. VALENCE-STATE ENERGY DATA TABLE VI11 SOMEVALENCE-STATE ENERGY DATAAPPLICABLE TO FIRST-ROW ATOMSWHENIN LINEARMOLECULES” Ground state
Ionization energies (e.v.)
Promotion energies’ PO (e.v.) Stateb
r
AP(e.v.) Id ?Le 0.00 ...... ..... (66) 5.39 s2p2, 3 P o SZUP, VI 0.49 SU%r, vz 9.45 288 14.88 s2p3, 4 s S2UPP’) v3 1.70 su2PP’J Va (12.60) (398) 18.11 0 s y , SPl s~u7rTr’2,vi 0.67 SU~P“’~, 16.49 530 21.22 F s2p6, 2Ps/2 sZua2r’2, v1 0.02 su2?r2r’Z, VI [20.92] 686 124.88 ] 0 All energies are in e.v., assuming 13.60 e.v. as the ionization energy of the H atom. Po denotes energy above the ground state, AP the additional promotion energy to reach the state listed. The POvalues are from Ref. 34, and the A P values lestimated uncertainties a few tenths e.v.) are from data in ref. 34, except the value for fluorine which is extrapolated. (The values for carbon are sli htly inconsistent with Voge’s GI, GZand Pt values in Table IX, but since the present values had already been used in thefater computations, it did not seem worth while to readjust them. Voge’s parameters would lead to Po = 0.32 and Po A P = 10.04 e.v.) The detailed configurations given corresljond to quantization in a force-field of cylindric symmetry, as in diatomic or other linear molecules; the symbols s, U , r,P , respectively, mean 29, 2pu, 2 p ~ and + 2 p ~ - . a The IK values are 1s 1 ’ s of Holweck quoted by Dauvillier, J . phys. radium, [6] 8, 1 (1927) (see also J. Thibaud, ibid., 8,447‘(1927)), corrected slightly to agree with footnote a. (Actually, slightly higher estimated values-291, 401, 542, 696, for C, N, 0, F-were used in the calculations in this paper.) The IL vahes are L shell valence-state values from footm electrons in the S ~ U Tvalence ~ states above. For note 34 or-from, data given there, obtained by averaging over the 3 example, ILfor nitrogen is one-fifth of 21.(S2UPP‘,VS +S U T P ’ , V 4 ) IO(S*U7r7r’,V3 +S 2 U P , V 2 ) 2I1(S2UT7r’,VS +S2U7rJ V2)
Atom
Li C N
9,
2s
StslteC
8, v1
v,
+
+
+
+
TABLE IX VALENCE-STATE ENERGY DATAFOR TETRAVALENT CARBON^ Case
A A’
State
Pt(e.v.)
.
P~(e.u.)b
8.47 13. 68 sulrn’, v 4 8.78 . B‘ didi‘m’, V4 7.79 trtr’tr’’u, V. 7.14 C 6.97 D tete‘te”te’’’, Vd a The A 0 symbols are in Part as in Table VI& note c; in addition, d i , tr, te refer to digonal, trigonal and tetrahedral 2s,2p hybrids, respectively. Pt denotes promotion energy to a tetravalent valence state. The value 6.97 e.v. for Case D is from H. H. Voge J. Chem. Phys., 4, 581 (1936); 16, 984 (1948). The vahes for the other unprimed cases are obtained from this by using Voge’s 1948 values of GZ (0.21 e.v.) and GI (2.24 e.v.) and a formula. of J. H. Van Vleck, J . Chem. Phys., 2, 20 (1934), Eq. (7), noting that Cases D , C, B (didi‘rxry), A, correspond to p = 3, 1/3, $, and 1, respectively, in Van Vleck’s formula, The cases A , B’, are obtained from A and B, respectively, by adding 1iG2 (cf. Tables in ref. 34). b One-fourth of 2s plus threefourths of 2p ionization energy for State A . Since the whole magic formula procedure is rough, this I for State A can serve for all tetravalent states. suPxTY, v 4
(74) For a theoretical analysis and some theoretical computations of conjugation and resonance energies, see R. 8. Mulliken and R. G. Parr, J . Chem. Plays., 19, 1271 (1951). Serber (cf. footnote 55) has also discussed the problem. Mulliken and Parr also give some anslyais of hyperaohjugation energies.
Appendix 11. Energies of Isovalent Hybrid Valence States For the carbon atom in the state k)2 hog)2 ha) T), V2, first consider the antisymmetrized wave function written in de) hg(5) terminant form, (6!)-* Det k ( 1 ) k’(2) h o ~ ( 3hog’(4) the unprimed symbols refer to AO’s with posi~ ( 6 )where , tive spin (ms = $1, the primed ones to AO’s with negative spin (m. = -4). Writing out hog and hg as per Eqs. (24), the determinant may be expanded (cj., e.g., Margenau and Murphy, “The Mathematics of Physics and Chemistry,” D. Van Nostrand Co Inc 1943, p. 289) into a linear combination of eight dete&inaAts of which four vanish, and the remaining four combine (using 012 p2 = 1) to give the result lJ(k2hoa2hpr) = O19(k2spu2,) &b(k*s2pun) (37) This derivation is for the case of a carbon atom with m, = +1/z for the spins of hg and T. An analogous derivation evidently holds for each of the other three possible combinations of m. values for these two AO’s, hence also for the k2hop2hg?r,V ZvaZence state, since each VZvalence state wave function is a linear combination, of like form in all cases, of wave functions for the four different ma rombinations mentioned. W. Moffitt,” in a similar discussion of hybrid valence states, arrives a t a relation the same as Eq. (37) exce t that, apparently erroneously, he gives 012 and p2 as coekcients instead of O1 and 8. a later paper,3s he makes computations whose results, when graphed (his Fig. I), are in agree-
+
+
+
(75) W. Moffitt, Proc. Roy. Soc. (London), 968, 524 (1948), top of P. 527.
f
March, 1952
BONDLENGTHIN
C O N J U G l T E D AND
ment with Eq. (38) and Table VI11 of the present paper. B y computing E = fIC.*HIC.dv with $ taken as a VZ valence-state function given by an equation corresponding to Eq. (37), one obtains, after noting that the two terms on the right of Eq. (37) are mutually orthogonal if orthogonal are used AO's are used or in general if true exact
+ + +
1
E(kzhop2hj3a,Vz)= p2E(k2s2pun, V2) d E ( k a ~ p u 2V2) ~, = E(k2s2pZ, aP) Po d A P where Po and AP hate the meanings and the numerical values given in the carbon atom entry in Table VIII. (The derivation of Eq. (38)based on determinant wave functions, t,houghstrict,lynot exactly valid for true exact wave functions,
AROMATIC MOLECULES
311
should be very nearly so, and entirely sat>isfactoryfor present purposes. The values of POand AP listed in Table VIII, being based on spectroscopic data, corresponding to accurate wave functions.) For hybrid valence states of other atoms, like that of carbon in having the configuration k ) Z hs) . . ., but containing additional T electrons, equations analogous to ( 3 7 ) (38) . and (38) can be proved by the same kind of procedure.
Acknowledgment.-The author is greatly indebted to Mr. Tracy J. Kinyon for extensive assistsnce in carrying out the numerical computations for this paper.
FACTORS AFFECTING THE BOND LENGTHS IN CONJUGATED AND *4ROMATIC MOLECULES BY C. A. COULSON Wheatstone Physics Dept., King's College, London, England Received December 26, 1061
A critical discussion is given of some of the factors which influence the bond lengths of conjugated and aromatic molecules. The three quantities (1) bond order, (2)hybridization type and (3) formal charge distribution are particularly important, usually, though not always, in the sequence (1) more than (2)more than (3). A critical survey of published bond lengths in aromatic hydrocarbons vindicates the concept of fractional bond order, by which bond lengths may be estimated to within about 0.015 1. When hetero-atoms such as N and 0 are present the accuracy is much less.
Recent very accurate three-dimensional Fourier analyses of the crystal structure of organic molecules such as naphthalene' or the pyrimidines2 make it desirable that we should consider as carefully as possible the accuracy with which theoretical predictions of the bond lengths in these molecules may be made. In many cases the experimental uncertainty is no more than 0.015 A. and with more refined electrical methods for measuring the intensity of the various scattered rays in a Laue diagram, it may even improve. We are led, therefore, to ask first, what factors affect these bond lengths, and second, how closely may we reasonably hope to calculate them? We are not concerned here with simple molecules, for which, as WarhurstS and others have shown, a wider variety of technique is available. Our concern is with large molecules where some simplifying assumptions have to be made. The most important of these is the assumption that resonating bonds can be described in terms of a fractional bond order, and that this bond order effectively determines the corresponding bond length. There are however other factors of importance, as we shall see. The present account will be concerned with (1) bond order, (2) hybridization type and (3) formal charges, all of which are significant in this connection. since the pioneer work (1) Bond Order.-Ever of Pauling, efforts have been made to improve and refine the definition of fractional bond order.4 The crucial question is this: is there a genuine correla(1) 9. C. Abrahams, J. M. Robertson and J. a. White, Acta CryalalE. 2, 233, 238 (1949).
(2) W. Cochran, dbid.. 4, 81 (1951). (3) E. Warhurst, Proc. Roy. SOC.(London), 8207, 32 (1951). and earlier papers. (4) For a review see C. A. Coulson, e'bid., 8207, 91 (1951), and C.A. Conlson, R. Daudel and J. M. Robertson, dbid., 8207, 306 (1951).
tion between calculated order and observed length? Thanks to the greatly improved accuracy of recent experimental work. it is possible to answer this question unambiguously. Figure 1 shows the conventional curve for C-C bonds, drawn to pass through the basic points corresponding to diamond (C-C), ethylene (C=C) and acetylene (C=C). ?\diamond 1.5 h
04 3M 1.4
8
I
V
& 1.3 1.2 1.0
Fig. 1.-An
2.0 3.0 Fractional bond order. order length curve for carbon-carbon bonds.
These all have integral bond orders, 1, 2 and 3, respectively. If the concept of bond order is to be useful, then the points for other molecules where the orders are no longer integral, should lie on or near this curve. Figure Z6 shows all the presently available data. The legend underneath defines the molecules to which the various points apply. The solid curve represents about the best curve through the points. This curve follows extremely closely the shape predicted theoretically by the writer,o of the form (5) Taken, by permission, from ref. (4a). ( 6 ) C. A. Coulson, Proc. Roy. Soc. (London), 8 1 6 9 , 413 (1939).
