R. D. Harcourt University of Melbourne Parkville, Victoria 3052 Australia
Increased-Valence Theory of Valence
There are many molecular systems with one or more sets of four electrons that may be distrihuted among three atomic orbitals (AO's) of three atoms. We shall designate them as 4-el, 3-A0 systems. Some well-known examples are 1 ) The four s electrons of benl Oa and NOs2) ~h~ twosets of four electrons of IinearCO2,NxO,NO.+,
and N23) The four p electrons that participate in the e banding of XeF, and I,-
We are not restricted to triatomic molecules or ions. Very often, polyatomic and intermolecular systems contain 4-el, 3-A0 components. NOa-, SOFz, (PNClz)a, N4S4, and NzO,, have several of them in Figures 1 and 3. I n this paper, we shall describe new types (1, 8) of chemical foimulas for 4 4 , 3-A0 systems. We shall call them "increased-valence" (i.v.) formulas,' because they can have more nearest-neighbor honding than do the simple v.h. formulas, such as (Ia) and (Ib) for 03.
molecules, we can construct MO's that are delocalized over many atoms. Usually, these functions do not easily generate chemical formulas that "illustrate and symbolize the wave-functions" (Id). Linnett proposed the n.p.s.0. formula (111) (8,10,11) which places the four a electrons in different spatial drhitals. Using the double quartet hypothesis (8, 11)' we can derive the sim~lestt v ~ e sof Linnett n.~.s.o. r discussed many of formulas. Recently, ~ i d e (d'has them. The v.h. and n.p.s.0. formulas use bonds that are localized between two atoms. The i.v. formulas of this paper also have localized bonds, and we can derive them from either simple v.h. or n.p.s.0. formulas. For 03,we shall generate the i.v. (IV), (VI), and (V) from the v.b. (I) and (VII), and n.p.s.o. (111).
Types of Chemical Formulas
To describe 4-el, 3-A0 systems, valence-bond (v.b.) (5), molecular orbital (MO) (6, 7), and non-pairing spatial orbital (n.p.s.0) (8, 9) wave functions are often used. For the four a electrons of 03,the chemical formulas that correspond to these functions (10) are (I), (111, and (111). To obtain an i.v. formula for any 4-el, 3-A0 system, with atoms Y, A, and B, we can proceed as follows (1)Write down a formula of type (VIII) (or (IX)) with an electron-pair bond.
Resonance between the two Lewis formulas (Ia) and (Ib) is the simplest type of v.b. representation (5). V.h. descriptions that are more elaborate include longbond formulas, and formulas with incomplete octets (10, 11). The MO formula (11) has two non-localized threecenter bonds. With respect to adjacent oxygen atoms, one is honding, and the other is non-bonding with a node (n) between the terminal atoms. For polyatomic
'
I n references (I), (4),and (4), we used the term "apparent octet4olation." We now refer to s ~ e a of k "increased-valence" because no octet-violation occurs in the wave function, but an increase in valence does occur.
(2) Assume that the AB bond of (VIII) is a doubly occupied bond-orbital. (3) Delocalize one of the Y electrons into the antibonding AB orbital that is vacant in (VIII). delocalization aerier. . This 0 0 X X ates (Xa) and (Xb) (cf., H-H and HexHe or HeoHe for Hz and ~ e (19)). z (4) Should the AO's on Y and A overlap sufficiently well, represent es bonded together the two electrons with opposite spin on Y and A. We thus obtain (Xl), for which the spins are not specified.
Volume 45, Number 7 2, December 7 968
/
779
(5) "Put" the electron of the A .k bond nearer B than A, as it is in ( X I I ) ; the extent of nearest-neighbor banding will then be greater than for (VIII) or (IX). Therefore, formulas of
type (XII) are the i.v. formulas.
I n the three sections of this paper that follow after the next, we shall elaborate and prove the quantum mechanics that pertain to the procedure of (1)-(5).
in which y, g, pas, and +,a are the spin orbitals ya, yg, p ~ a,b 4.s. a and 0 are the spin eigenfunct,ions of S,,and we have represented electrons with these spins by crosses ( x ) and circles (o) in chemical formulas. The wave function for ( X I ) is derived from eqn. (3) by delocdizing one y electron into the antibonding pas* orbital. After this delocalization, yand c,s* are singly occupied. The singlet wave function (i.e., eigenfunction of c' with zero eigenvdlue) for the four electrons is
The Need for Increased-Valence Formulas
There is plenty of evidence that i.v. formulas might often be useful. Ahre than twenty years ago, Samuel (14, 15)) and quite recently, Harcourt (1, 3, 4), Paoloni (16), and Bent (17), advanced theoretical or experimental reasons. Samuel suggested that observed parachors, dipole moments, and bond lengths of many molecules with nitrogen atoms might be accounted for satisfactorily if we assume that sometimes nitrogen can have a valence of five in chemical formulas. As examples to show this, he discussed, as shall we, the bond lengths of NzO and N,H. Pauling's estimates (18) of NN triple and NO double bond lengths are 1.10 and 1.20 i (c.f.,l.lO i and 1.21 for N2 and HNO (I$), with the valence formulas N=N and H-N=O). N20has similar NN and NO lengths, which are 1.13 and 1.19 X (19). Therefore, the pentavalent formula2 N=N=O might reasonably describe the bonds of N20. Similarly, for N8H, a suitable formula could be H-N=N=N. Its HN=N and N=N lengths (19) are 1 . 2 4 i and 1.13i. Pauling's estimate for the length of an NN double bond is also 1.24 i. Some intermolecular systems where valence increase might occur are the transition state of an SNZreaction, c.g., HO---CH,---Br for the reaction of aqueous alkali with CH,Br, and hydrogen bonded systems, e.g., H,O. . . . .H-OH for two water molecules. By using either 3d or 3s AO's, Gillespie (Sf), Preuss (sf?), and Herzberg (23) have suggested how carbon or nitrogen might increase their valence in certain systems. The present work shows that this promotion is not necessary for first-row elements. By using antibonding orbitals, we can increase the valence. I n the appendix, we shall discuss how some other workers have used these orbitals. Because they involve both one- and two-electron bonds, our i.v. formulas differ from those shown in this section. Wove Functions for Increased-Valence Formulas For formula (VIII), let y, a, and b be the three normalized AO's that can accommodate the four electrms. We shall assume that these orbitals are 01,ientated so that the overlap integrals (alb) and (yla) are positive. The orthogonal AB bonding and ant~bondingorbitals of (VIII) are%qns. (1) and (2), with 0 < k 1 or < 1, we position the electron of the AB hond closer to B (as it is in (XU)), or to A, respectively. Similarly, we can generate i.v. formula (XIII) by delocalizing one b eleetro-X) into the antibonding YA orbital, c,.* = 1 - 1 1 . The wave-function for (XIII) is eqn. (6), P,and 2is a bond parameter >l. in which c,, = (a 2y)/d1
.
+
+
+
(lba+,,~l
+ laby+d)/d?
(6)
When Y and B me the same a t o m , as they are for Os or XeF*, (XII) and (XIII) are equivalent formulas that are mirror images. In eqns. ( 5 ) and (B), k = I.
I.v., V.b., and N.p.s.0. Formulas and Wave Functions We shall now demonstrate that a single i.v. formula, such as (XII) or (XIII), "summarizes" the resonance between two canonicd strn~ctures(i.e., v.b. formulas with hond wave functions of the Heitler-London t,ype). We may expand eqn. (5) far (XI11 as lk(lyabbl
+ latbbl) + (ly&al + Ibgadl)l/%"Xl + k*) = (k + + d l v ' G $1
(71
#, and J.z me the normalized bond-eigenfunctions for the canonical structures (XIV) and (XV). Therefore (XII) implies resonance between these structures. Similarly, we may express eqn. (6) for (XI111 as eqn. (a), in which $Z is the bond-eigenfunction for (XVI).
(l$s
+ W4-
(81
When Y and B are the same atoms, k = 2 in eqns. (7) and (8). The resonance between i.v. formulas (XII) and (XIII) is equivalent to resonance between canonical structures (XIV), (XV),
Resonance between three Lewis formulas with quadrivalent (central) nitrogen atoms, NZN-0, N=N=O, and N=N=O, requires a resonance shortening correction to obtain a reasonable account of both bands (SO). a I n eqns. (1) and (2), we have neglected the overlap integrals in the normalization eonstants. When in the determinants of eqns. (3)-(7), all orbitals are assumed to be orthogonal. These simplificrttions are satisfactory for our present purposes. ' These are equivalent to sdding and subtracting two columns or rows of determinrtntal elements that represent orbit& with the same spin functions.
and (XVI). The resrdting wave function is the linear combination of eqns. (7) and (a), which, when normalized, is (A(+,
+ $a] + 2 1 1 . r 1 / d m
(9)
-
light (k # m ) and heavy (k = m) lines, respectively. The 0, and reach its line or hond "intensity" can fade away as k maximum when k = m . Variation in line intensity may occur as a base-displacement reaction proceeds (see below). The sum of N(y,a) and P(a,b) is greater than unity far 1 < k < m, and has a maximum of 1 (4- 1)/2, i.e., 1.207 Because the total bond number (or order) when k = 1 for (XIV) cannot exceed unity, i.v. formulas with 1 < k < m involve mare nearest-neighbor bonding than (XIV), and so we are justified in calling them "increased-valence" formulas. The Y, A, and B atom-charges for (XII) are
+
+ a.
Altogether, there sre six (singlet) canonical structures (X1V)(XIX). Linear combinstions of their wave functions will generate a. "best" wave function (10, 87-30). Linnett and his coworkers have ccalculated these lor the four r electrons of C3Hs-, O., HCOz; and NOz-. They ohtainedslarge coefficients for the un-normalized functions that pertain to our (XIV), (XV), and (XVI). Therefore, the i.v. functions for these systems should he good approximations to their "best" functions. We shall conclude this section by indicating that. i.v. ftinctions are special forms of n.p.s.o. functions. When electrons in adjacent spatial orbitals have opposite spins, the general n.p.s.0. formulas are (XXa) and (XX6).
+
The= functions of eX -( the spatial orbitals y, (ly a)/dl la, (a kb)/dl k2, and b, with k Z 1. If we assume 1 = 0, we have the spatial arbitnls for eqn. ( 5 ) , which is the wave function for i.v. formula (XII). Alternatively, we can obtain the spatial orbitals for the i.v. function of eqn. ( 6 ) by setting k = 0. But when Linnett's n.p.s.o. functions (10, 87, 38,SO) assume either k = I, or k = 1/1 (i.e., when they are oneparameter functions), the i.v. functions cannot be special forms of these.
+
+
+
Bond Orders, Bond Numbers, and Atom Charges for Increased-Valence Formulas As simple criteria of the extent of bonding between two atoms, we shall use bond orders ( P ( ~ , v ) for ) one-electron bonds, and bond numbers (N(a,v)) for tweelectron bonds. 6 bond of (XII), the hond order is calculated For the A as t,he product of the A 0 coefficients in the hond orbital (pd of egn. (51, i.e., as
.
+ ka)
P(a,b) = k/(l The bond-number of the Y-A N(y,a)
=
(10)
bond of (XII) is equal to k21(l
+ k')
(11)
We may deduce this formula as follows. For any k in eqn. 0
0
(2) and (5), the odd-electron charges an A and B of A x B or X ;I,,Bare
+
+
ka/(l k2) and 1/(1 kz). These charges can spm-pair with corresponding fractions of the electron charge on X 0 y or Y . Therefore the YA bond number is given by eqn. (11). We may also obtain these P(a,b) and N(y,a) of eqns. 110) and (11) by an alternative method. B y squaring and integrating eqn. (7), we generate the charge distribution of eqn. (12).
.
lkz($lIJ.l)
+ ( h l b ) + 2k(+tlh)J/(l + kz)
(12)
Since (XII) summarizes the resonance between (XIV) and (XV), YA bonding in (XII) occurs because of the contribution of (XIV) to this resonance. The weight of (XIV) is the coefk'), and this is the ficient of ($,I$,) in eqn. (12), i.e., kV(1 YA bond-nnnlber of eqn. (11). I n eqn. (12), ($9142) = (alb). The bond order for the AB bond k2), as of (XII) is the coefficient of 2(a/b) (SZ), which is k / ( l for eqn. (10). Except when k = m (which generates (XIV)), the YA bond of (XII) is not a "normd" single bond, is., N(y,a) < 1. Therefore, we shall distinguish these two types of YA bonds by using
+
+
and
P(Y,Y) = 1, P(a,a) = 1 P(b,b)
=
1
+ l / ( l + k2)
(13)
+ k"(1 + kZ)
We can deduce these charges from eqn. (5) (or (7)). In eqn. k') and kP/(l kz) to P(a,n) and (5), pas contributes 1/(1
+
+
Pih.h).
- ~ - z - , .
When Y and B are the same atoms, each YA (or AB) bond has a contribution from both N(y,a) and P(y,a). Their total shall be designated as a bond-number, N'(y,a). Using eqn. (9), we can obtain'
N'(y,a) has a maximum of (1
1
+ 4.
+ 4 / 1 4 , i.e.,
0.683 when k
=
Some Examples of Increased-Volence Formulas
I n this and the next section, we shall deduce i.v. formulas for a few of the many systems for which such formulas might be useful. The main purpose here is illustrative, i.e., to indicate what theseformulaslooklike, but later, we shall calculate bond lengths from certain i.v. formulas. We should stress that for some systems, other i.v. formulas apart from those shown exist, and we may not have chosen the best i.v. formulas. Also, for certain purposes, alternative descriptions (e.g., Lewis-Pauling resonance ( 5 ) , MO (6, 7), or n.p.s.0. (8,9)) of these systems might be equally satisfactory or better. To generate i.v. formulas for isolated molecules or ions when atom A has available either one (e.g., hydrogen) or four AO's for bonding, we start with the doublet (if A = H) or octet formulas of Figure 1. I n these, the formal charges are negative on the Y atoms, and (for neutral molecules) positive on one or both of the A and B atoms. These formulas also have the maximum number of two-electron bonds compatible with the octet rule, but usually should be unstable due to the
Overlap has been included in their calculations. Using their two sets of matrix elements for HC02- (67), we have calculated minimum energies (in eV) of -50.6 and -65.9 for the i.v. formulas 0-CH-6 6-CH-0 The energies of other functions (67) are: MO, -49.5, -67.4; n.p.s.o., -50.9, -67.0; "best," -51.4, -67.9. The results of calculations (Sf) indicate that i.v. functions also give low energies for'the r-electrons of Oa, NnO, and NI-. ' The charge distribut,ion for eqn. (9) is Ik'((+tIJ.t)
+ (hlJ.3) + 4 ( h l h ) + 4k((+,l+d
+ ( h l h ) ) + 2kP(hl*31/(2ka + 4)
with
($II$~
=
(alb) and (J.rl#d = Ma)
N(y,a), N(a,b), P(y,a), and P(a,b) are the coefficients of (hI+h), 2(+&), and 2 M J . J .
Volume 45, Number 12, December 1968
(+,I$,) /
781
M -8-q
I
+1
(CEO), 0 +2
:oso:
+l
1
Figure I. Doublet and octet fornulos for genereling i.v. formulas of ikdated nolecvler or ions.
large formal-charge separations. The magnitudes of the formal charges can be decreased so that an increase in bonding occurs. We can achieve this by delocalieing Y electrons into antibonding AB orbitals so that the formal charge on Y is reduced to zero. For neutral molecules, the formal charges on A and B can now be small, i.e., with magnitudes considerably less than unity, and so will conform with the electroneutrality principle (33). The i.v. formulas that result are shown in Figure 2. To demonstrate the procedure, we shall generate i.v. formulas for 03. We could start with the LewisPauling formulas (I), and delocalize one 0- electron into an antibonding 02+ orbital. The i.v. formulas of (IV) result. Alternatively, we could delocalize two 0'- electrons into two antibonding 02+orbitals, and obtain (VI) from (VII). 782
/
Journal o f Chemical Educalion
Figure 2.
1".
formulat generated from thoso given in Figure I.
For the 00 bonds of (IV) and (VI), the maximum totals of bond number bond order are (using k = 1.366) = 1.683; 1 flin eqn. (14)): (IV): '/2(2 (VI): 1/2(1 1.366 1.366) = 1.866. If we want to reduce the amount of resonance, we can start with the Linnett octet formula (111), and delocalize one electron from each 0-'1) into an anti-
+
+
+
+
+
honding 02+'/'orbital. We then ohtain the i.v. formula N\ ,. ,I n this paper, we have generated i.v. formulas from Lewis-Pauling formulas, because these are simpler and hetter known than the Linnett formulas. For some systems, we can assume that the y-electrons are two electrons detached from any atom. When these are delocalized into d i e r e n t antihondiug orhitals, two non-bonding electrons will remain on A in different spatial orhitals. No increase in valence results. But it should he energetically favorable (8) to have the lone-pair electrons spatially separated on A. I n Figure 2, the i.v. formula for SOF2 has this feature. For second (and higher) row elements, i t is possible, hut not always necessary (6, 7) that we may use more than four of their AO's for bonding, i.e., such elements may expand their valence shells. If this occurs, these elements may form five or more honds in the starting formulas that we use to generate i.v. formulas. Such is the case for the sulfur atom of SOaZ-in Figure 1. The value of k that should be used in eqns. (10)-(15) will be different for each system, and we could ohtain it from a variational calculation. However, for many purposes, it is reasonable to assume that we require P(a,b), or N1(y,a) N1(a,b) to he large, but N(y,a) concomitant with satisfactory formal charges on the Y, A, and B atoms. I n Figure 2, we have used k = 1 fior 1 &for YAB or YAY systems. These values generate the maximum amount of honding (in the above sense), hut for some of the systems, e.g., XeF2, the formal charges obtained are not small. When Y = H- or CH3-, k = m which generates (XIV) seems to he satisfactory. Except for molecules where H or CH3 acts as a bridge (e.g., as in B2Hs), A-H or A-CH3 bonds do not show appreciable variations in length for a given atom A in different molecules (19). I n Figure 2, we have displayed the i.v. formula,' together with their atom formal charges, for HF2-, CINO, XeF2, SOFZ,F202,HNa, N20, N3-, NO2+, NOz, NO*-, NO$-, pyrrole, N84, (PNC12)3, M(CO)av- (M = Ni, q = 0; M = Co, q = + l ; M = Fe, q = +2); oxyhemoglobin, and S O P . We have derived these formulas from those of Figure 1. For the YAY systems of Figure 2, the formal charges are for one of the mirror image formulas. When all are taken together, arithmetical averaging of their formal charges is needed. I n Figure 3, some octet and i. v. formulas for NeOaare shown. ~
+
+
+
+
Increased-Valence Formulas for Some Intermolecular Systems
We shall now briefly consider i.v. formulas for some intermolecular systems. The charge-transfer complex of I2with N(Me)3might he reoresented hv the i.v. formula (XXII). We can derive it from ( X ~ I by ) delocalizing 'one e~kctronfrom +
I- into the antibonding NI orbital of I-N(Me)a. One structure instead of the usual two (54) (i.e., those for (Me)3N, 12,and M ~ ~ N - L )is needed to represent the main features of the complex, i.e., that the NI and I1 honds are each rather longer than single honds (55). Similarly, if A and D are any neutral acceptor and donor species, many other charge-transfer complexes
might he represented as A .h
.
This formula could
he a simple alternative to (D,A) * (D-A).
.'.. ...I-me,
:I:
C
- 1
.he, (XXII)
(XXI)
(XXIII)
(XXN)
For two water molecules that are hydrogen-bonded, one simple type of charge-transfer theory involves the (H&H OH) resonance (86). (H$:, H-OH) From (XXIII), we can generate an alternative formula (XXIV), by delocalizing one OH- electron into an antihonding OH orbital of H30+. Because the intermolecular HO hond is much longer than an OH single 0.96 (86)), its experimental bond order hond (1.76 i, is small-Pauling's estimate (86) (he calls it a bond number) is 0.05. Therefore k in eqns. (I), (lo), and (11) is >>I, and so the 0-H hond of 0-H .6 is only slightly weakened. A base displacement reaction8 might he formulated as
..n + R-Y
X
+
[X-R
-+
+ 'J
X-R
When r(XR) (the distance between the Lewis base X and acid RY) is not infinite, the wave functions for x R-Y and X-R (XXV) will interact. A necessary condition for the formation of products is that the energies of these two structures should he equal a t some r(XR) that is rather longer than the X-R hond length of the product. I t is the presence of (XXV) which could be mainly responsible for the breaking and making of bonds. The "intensity" of the X-R hond line in (XXV) will he a function of r(XR) and the overlap between the relevant AO's of X and R. For an organic SN2 reaction such as the hydrolysis of methyl bromide in aqueous alkali, it should be strong. If so, this bond could repel adjacent single honds hetter than does the R .$ hond, and lead to Walden inversion.
.+
' For Fe(CO)&Z-,we have geuerated the i.v. formula by assuming iron can use its 3 p as well aq its 3 4 46, and 4 p AO's for bonding. We have obtained incorrect trends in metal-carbon bond numbers when the 3 p AO's are excluded. To show this, we have ealeulat,ed the following MC bond numbers and CO bond orders for the i.v. formulas of Figure 2. (Experimental estimat,en quoted hy Abel (40) are given in parentheses.) Ni(CO), CO(CO)~Fe(CO)F If the 3 p AO'ti are excluded, the i.v. formula. for Fe(CO)P would be . . Fe (-C-0.) (=C-F).) (=C--i-0 )2 with the MC bond number of 1.71. In our theory, certain penta- and hexacarbanyls also require inclusion of 3 p AO's. In mass spectrometry, a single-barbed arrow is used to indicate a one-electron transfer. We have adopted the same symbolism here.
. ..
.
Volume 45, Number 12, December 1968
.
/
783
An elimination reaction, such as the E2 reaction of potassium ethoxide with isopropyl bromide, could be
.. n f H-CH,CHMe-Br
EtO
Base addition reactions could involve (XXV) as a product. For example, the reaction of HO- with Cop (for which we shall use an i.v. formula) might proceed as
For the reaction of hydrogen with iodine, we could represent the himolecular component of the reaction as
I z 1.v. formula (XXVI) is probably involved in resonance stabilization of the HC08- because reported lengths of the CO bonds (1.28i. 1.32i, 1 3 & ) are all shorter than CO single bonds (e.g., 1.43i for CH,OH(lS)).
-. (XXVI)
For the following nitration of benzene, CsH6N02+ is formed from the addition step. It could have a long CN bond. But, during the subsequent removal of the proton, the electron distribution of the CNOz group should re-organize appreciably. One reason for this could be that CeH5, as distinct from CsHs+, behaves as should the CH, and H groups that were discussed earlier. Therefore, CaHsNOz is shown with a CN single bond. Its length of 1.49 i is similar to 1.47 i for a single bond (as for CH,NH,) (19).
I-H
I n the Pauling resonance description of metallic bonding (57),one of the contributing structures for lithium metal is Li-Li-Li. The central Li atom can form two covalent bonds by using sp hybrids. An alternative structure is Li-Li .Li, which we obtain by delocalizing one electron from ti into an anti-bonding orbital of Li-Li. Conductjon can then proceed by the electron in the A 0 of .Li jumping into the antibonding orbital of an adjacent Li-Li structure. Bond Lengths Calculated from Increased-Valence Formulas
For the relevant systems of Figure 2, we have calculated the NN and NO bond lengths, r(NN) and r(NO), from the i.v. formulas. The relationship between the total bond number bond order, n(NN) and n(NO), and bond length r(n) is assumed to be given by the Pauling-type formula (3, 4)
+
~(n= ) r(1) - 0.8 log n
with r(1) = 1.48; and 1.44 ; for NN and NO bonds. To evaluate n(NN) or n(NO), we have used 1c = 1 &or k = 1 ~ ' 3in eqns. (10) and (11) or (14) for the 4-el, 3-A0 components of the YAB or YAY systems. For t,he single-hond and one-elect,ron bond resonances :Y-A Y: ++ :Y A-Y: and : .iY: u :Y A $: (as in NO,), the maximum contributions to the n(YA) are 0.5 and 1/(2&), i.e., 0.353, respe~tively.~ Thetotaln(NN) and n(N0) are reported in the table, together with the calculated and experimental estimates of the bond lengths. Except for the N-@H bond of N8H, the differ-
+
+
.
.
' If y, a, and b are Y,A, and Y atomic orbitals, then the wave function for t,he one-elect,ron bond of each YAY structure is (w ka)/dland (ka b)/d\/l The normalized linear combination generates a maximum YA bond-order of X d a w h e n k = 1/42.T h e y , a, and b charges are 0.25, 0.5, and 0.25.
+
784
/
Journal o f Chemical Education
+
Total Number
ClNO NOs+ NOS NO,-
+ Oa rndde NN r (n), Calculated and Observed NO Bond Lengths
I+.& I + & 1 & 1+&
2.354 2.366 2.220 1.866
k
n(NN)
+
1.14 1.14 1.16 1.22
1.141.15 1.19, 1.20 1.24
r(NN) (A)
r(NN)
(A)
&
2.5 1.16 1.13 1.866 1.26 1.24 1 43 2.549 1.155 1.15-1.17' NaMILLEN,D. J. and PANNELL, J., J. C h a . Soc., 1322 (1961). PALENIK, G. J., A d a Cryst., 17, 360 (1964). All other lengths are takeken from ref. (19).
+
ences between calculated and observed lengths are not genergreater than 0.03 k For N3H, k = 1 ates formal charges of +0.71 and -0.71 on the two nitrogen atoms of the NNH bond. However, because the nitrogen electmnegativities should be similar, we might expect these charges to be smaller in magnitude. If we use k = fi,they are reduced to +0.5 and -0.5. The calculated NNH length decreases from 1.29; to 136; (expt.1.24;), and the other NN length increases 3 from 1.13; to 1.15; (expt. ~ 1i).
+
Appendix:
..
from eqn. (18). Therefore the central nitrogen atom would be quadrivalent in the N.0 formula, --O. I n his third paper (14), Samuel had implied that eqn. (18) would generate .N=O and hence a. pentavalent nitrogen atom. In replying (16) to Wheland, he gave further justification for this NO formula. Neither Samuel's nor Wheland's formulas obtained from eqn. (17) or (18) are correct. I n retrospect, i t seems both workers held the usual view that vdelence formulas for ~olyatamicd i e magnetic molecules require electron-pair bonds: They attempted to transform eqn. (17) or (18) so that they would obtain such bonds in N.0. But neither knew how to apply the Pauli principle correctly t o electron configurations with antibonding orbitals. The procedure that should be used has been developed for diatomic molecules by Linnett (24, 26). I n this paper, we have described its extension to polyatomic molecules. configuration (eqn. (16)), Linnett and Green (26) From the 0% derived the formula When their theory is applied to eqn. (17) and (la), we ran obtain N-y and.?-?,
:+to:. ..
. ..
. . .
..
and from these formulas N F N = ~ and : N m N - 9 for NnO. The latter is essentially the one shown in Figure 2, hut is obtained using an alternative procedure. By spin pairing the odd-electron charges of two species, we may also obtain i.v. formulas for many other systems. If we start with the Green and Linnett formulas for the ground states of NO and OX,together with oxygen, chlorine, and fluorine atoms, we may generate the i.v. formulas (VIa) and (VIb) for OJ, those of Figure 2 for ClNO and F?OI, and various i.v. formulas for FOa, N202 (e.g., :O=N-N-0: ), 04,and OqNO. Sometimes., we need a formula far an excited state of one (or bobh) of the combining species. This is the case for NO when forming N1O. For more than twenty years, Samuel's work has been virtually neglected. However, usually without attempting to obtain valence formulas, vsrious workers have included antihonding orbit,als in descri~tionsof ~alvatomic molecules. We shall now " list some of these. 1 T o nvromt for metal-ligar,d and lignnd-lrnnd propcrrirs, back donation of drvrrunr frurn metal lo nntilnmdir#gCO, CN, and SO lignnd urliruh har oftelt been invoked (40 .
..
.
Uses of Antibonding Orbitals
Since the inception of MO theory, it has been well-known t,hat ground state M O configurations with three or more electrons have often involved the occupancy of antibanding as well as has t,wo electrons bonding MO's. For example, Hez+ (or Hs-) in the bonding c l s MO, and one electron in the antihonding 8 1 s MO giving the configuration (uls)'(o*ls). The configuration of the 0, molecule,(16), is very familiax, with the two antibonding r*2p orbitals each singly occupied (with parallel spins (38))'
.N-0
(~ls)~(~*ls)~(r2s)'(0.*2s)~(r2p)~(v2p)~(r*2p)~ (16)
The incorporation of twc-center antihonding orbitals into the wave functions of t i - or polyatomie molecules is not so wellknown. Some confusions in method seem to exist in the literature. We shall discuss them here. I n 1964, Samuel (in what should now be considered t o be important papers, and worthy of recognition as such) (14) used antibanding orbitals to increase the valence of certain atoms. For example, he wanted t o show how t,he central nitrogen atom could be pentavalent in the vdence formula C-N=Ofor nitrous oxide. He did this as follows: In his second paper, he wrote down the wave function in eqn. (17) for an excited state of N*.
(rzs)2(n*zs)(rzp)4(.zp)~(~zp)
(17)
From this, he derived the chemical formula :N=$,' with two unpaired electrons. He then used these electrons and the two unpaired p electrons of an oxygen atom, t o form two NO covalent bonds, i.e.,
I n a paper that discussed (39) Samuel's work, Wheland suggested that N 9 0 could be formed by comhinat,ion of the excited NO configuration (~Z~)'(~*Zs)(rZp)'(o~p)~(r*2p)(r*Zp)
(18)
with a nitrogen atom, each with three unpaired electrons. By appeal to the Pauli principle, he obtained the vdence formula
Figure 3. Some incremed-valence formulas for N10,. Pmts Id1 and gensrated fmm (a), If1 from (b), and (91 from (cl. The formulor
(4 ore
. .
involve o n.p.s.o. component 0.N.N. Here, (but not else( 4 and where 138)) we hove assumed, as Linnett hor done 1391, that it contributes 0.5 to the "INN1 of (91. The contribution to eoch n (NO1 of lgl is 0.125.
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785
2) Far some aliphrttic chloro compounds that contain oxygen or fluorine atoms, Lucken (41) has described the interaction of a p A 0 of these atoms with the CCI antibonding o-orbital. 3) Williams (48, 445) has indicated that Lucken's theory in 2) might pertain to many saturated systems-for example to
account far the shortening of the C F bonds and lengthening of the CH bond of CFSH. He has given consideration (48) to the two "local" lobes of antihonding orbitals, and has elaborated Hitckel's assumption that the filling of these orbitals for firstrow elements requires the use of AO's of higher quantum number. 4 ) Sprrttley and Pimentel disoussed the relative extent of interaction of fluorine or hydrogen AO's with antihonding NO or 00 orbitals of FNO, HNO, F2Ox,and H90n(44). 5 ) The long weak NN and SS bands of Ns04 and SZO4'- have been ascribed to deloealisation of oxygen electrons into antibonding N N and SS orbitals (46-47). Short NO bonds of N20a also arise from these deloealizations. 6 ) The Mulliken theory of charge-transfer complexes often assumes that an acceptor orbital is an mtibonding orbital (454). In 1-5, the workers have described their theory as molecular orbital. However, for 2-4, the MO theory given is approximate only. Each of the sysdems of 1 4 constitutes a YAB typesystem with AO's y, a, and h. We mag form MO's in two equivalent ways. The A 0 ?r may be linearlg combined wit,b the AO's a and h either directly, or through the AB bonding and antibonding orbitals, 9.a = a kh and 9.s' = ka - b (eqns. (1) and (2)). Unless 9.s and q.s' belong to different irreducible representations for the symmetry group of YAB, holh 9.s and will combine with y in the same MO (45, 47). Using NzOa, we h m e demonstrated this (47), and have mentioned (48) its relevance to MO descriptions of F,O1 and FNO. The workers of 2-4 have neglected the contribotion of 9,s to their MO's. We should stress that i.7. wave functions with antibonding orbitals, e.g., eqns. (4) and (5), are not MO functions, because they do no1 involve linear combinations of y with either 9.s or
+
(om&*.
Concerning chemical formulas that have been used to designat,e antibonding orbital occupancy, it has been customary to use the simple v.b. resonance notation
?
AB -
r
-3
B,or Y-A-B.
u Y A -
Literature Cited (1) HARCOURT, R.. D., Theorel. Chim. Ada, 6, 131 (1966). (2) HARCOURT, R. D., "Lab Talk," Science Teachers' Association of Victoria No. 11, p. 8 (1967). To be reprinted in Australian Science Teachers' Jovnol (1968). (3) HARCOU~T, R. D., Theoret. Chim. A c h , 2 , 437 (1964); 4, 202 (1968). R. D., Theoref. Chim. Ada, 3, 194, 290 (196.5). (4) HARCOURT, (5) PAULINO,L., '*The Nature of the Chemical Bond," (3rd ed.) Cornell University Press, Ithaca, N. Y., 1960. (6) . . RuNnLlr. R. E., Record of Chemical Progress, 23, 195 (1962j. (7) HUDSON,R. F., Angem Chern. (Int,ernationd Edition), 6, 749 (1967). (8) LINNETT,J. W., "The Electronic Structure of Molecules," me tho en, London, 1964. (9) Luom, W. F., J. CHEM.EDUC.,44, 206, 269 (1967). (10) GOULD,R. D. and LINNETT,J. W., Trans. Faradny S o t , 59, I001 (1963). (11) LINNETT,J. W., J . Amer. Chem. Soe., 83, 2643 (1961). (12) LINNETT, J. W., Ref. (8), p. 15.5. (13) LINNITT,J. W., Ref. (a), p. 27, 34. (14) SIMUEL, R., J. Chem. Phys., 12, 167, 180, 380, 521 (1944). (15) S A M U FR., : ~ J. Chem. Phw., 13, 572 (194.5). (16) PAOIQNI,L., Gazz. chim. Italia, 96, 83 (1966); PAOLONI, L., AND GIUMAN~NI, A. G., Gazz. ehim. Ilalia, 96, 291 (1966). (17) BENT,H. A,, J. CHEM.EDUC.,43, 170 (1966). (18) PAULING, L., Ref. (5), p. 228, 344. (19) SUTTON,L. E., Tables of Interatomic Distances and C o v
.
fieurations in Molecules. Chem. Soc. Special Publieat~ion PAULINO, L., Ref. (6),p. 270. GILLESPIB,R. J., J . Chem. Soc., 1002 (19.52). P~r:uss, H., Angew. Chem. Internal. Edit., 4, 660 (1965). HERZBERO, G., "Molecular Spectra. and Molecular Structure 111." Van Nastrand Co., London, 1966, p. 369. (24) LINNETT.J. W.. J. Chem. Soc.. 275 (1956). (25j POPLE, j . A,, Q U W ~ . Revs., 11; 273 (1957). (26) GREEN,M. L., A N D LINNETT,J. W., J . Chem. Soc., 4959
(20) (21) (22) (23)
flDfiOi~ (27) KIRCHHOFF, W. H., FARRI,:N, J., A N D LINNETT,J. W., J . Chem. Phys., 42, 1410 (196.5) and references therein. (28) HIRST, D. M., A N D LINNETT,J. W., J . Chem. Phys., 43, S74 (1965) and references therein. (29) BOWEN,H. C., AND LINNETT,J. W., J . Chem. Soc., (I.P.T.) lfi7.5 ., 114Rfi> \."-",. (30) LARCHER, J. F., A N D LINNETT, J. W., J . Chem. Soe., (I.P.T.) 1928 (1967). (31) HARCOURT. R . D., A N D SILLITOR, . J.,. calculations to be pub\----,.
-
Thus Psuling (61), Williams (45), and Bent (58) have used the following formulas for 1, 3, and 5 above. 1)
M-C-0
3)
GcF,-H t+ F-CF,
M - C O
H
lished. PARISER.R.. J . Chem. Phus.. 24. 2.50 (1956). PAULINO,L.; Ref. (6), p. i72: 2 7 6 MULLIKTN. R. s...r. CAP+
The i.v. formulas that result from the delocalisation of one electron into an antibonding CO, CH, or NN orbital are
For MCO and CF$H, t,hese formulas summarize the resonance between the v.b. formulas on the R.H.S. of (1) and (3), and the long bond formulas
L-E-1,
and
6-c~~
:H
The resonance that is implied by t,he N,04 i.v. formula is
These are two of seven basis structures that m e needed to describe the distributions of t,he electrons responsible for NN bonding (1, 45).
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Journal of Chemical Education
LUCKBN, E . A. C., J . Chem. Soe., 2954 (19>9). WILLIAM^, J. F. A,, Trans. Farad. Soe., 57, 2089 (1961). WILLIAMS,J. F. A., l'el~ahedmn,18, 1477 (1962). SPRATLEY, R. D., A N D PIMBNTI:L,G. C., J . Amw. Chem. Sac., 88, 2394 (1966). (45) BROWN.R. D.. A N D H . ~ R ~ O U R R.T ,D.. PIOC.Chem. Soe.,
(41) (42) (43) (44)
2 1 6 (i961). (46) BROWN, R . D., A N D HIRCOURT, R . I)., Aud. J . Chem., 16, 737 flQfi3i. ~ ~ ~ - - , (47) BROWN, R. D., A N D H.\ncounT, R . D., Aust. J . Chem., 18, 11lX (1965). (48) TURNER,S. J., A N D HARCOURT, R. D., Chem. Comm., 4 (1967). (49) P.\ULING, L., Ref. (5), p. 331. (50) BENT,H. A,, Inorg. Chem., 2, 747 (1963).
