Valence-bond studies of 4-electron 3-centre bonding units. I. The π-electrons of O 3 , NO 2− ,...

Valence-bond studies of 4-electron 3-centre bonding units. I. The welectrons of O,, NO,-, and CH2N2

Received May 26, 1977

RICHARD D. HARCOURT and WALTER ROSO. Can. J. Chem. 56,1093 (1978). Some nb-i~l ir io valence-bond wave-functions are reported for the n-electrons of the ground-

states of 0 3 , NOz-, and CHzNz. Examination of these wave-functions provides further support for the hypothesis that, for the ground-states of many electron-excess molecules, important valence-bond structures are those that are compatible with the electroneutrality principle, i.e. they carry either small or zero formal charges on each of the atoms. For O3 and CHzNz, the important valence-bond structures with zero atomic formal charges are

Each of these structures has a 'long-bond' between non-adjacent atoms. The significance of 'long-bond' (or spin-paired diradical) structures for the electronic mechanism of 1,3-dipolar cycloaddition reactions is discussed and 'increased-valence' descriptions of the electronic structure of each molecule are presented. Some comments on the utility of 'increased-valence' structures are provided.

RICHARD D. HARCOURT et WALTER ROSO. Can. J. Chem. 56, 1093 (1978). On rapporte quelques fonctions d'onde valence-liaison calculCes par des mCthodes nb b ~ i t i o

pour des Clectrons n des Ctats fondamentaux de 0 3 , NO2- et CH2N2. Un examen de ces fonctions d'onde fournit un support additionnel l'hypothese selon laquelle pour les Ctats fondamentaux de plusieurs molCcules posskdant des Clectrons en excks, les structures importantes de liaison de valence sont celles qui sont compatibles avec le principe de I'Clectro- neutralit6 c'est-a-dire celles oh les charges formelles parties par chaque a to~ne sont ou tri-s faibles ou Cgales A zCro. Pour O3 et CHZNZ, les structures importantes de liaison de valence avec des charges formelles atomiques de zero sont

Chacune de ces structures possident une liaison longue entre deux ato~nes non-adjacents. O n discute de la signification des structures a liaisons longues (ou du diradical a spin pairk) pour le mecanisme Clectronique des rCactions de cycloadditions dipolaire-1,3 et on prCsente des descriptions d'augmentation de valence de la structure Clectronique de chaque molecule. O n fournit quelques commentaires sur 17utilitC des structures d'augmentation de valence.

[Traduit par le journal]

Introduction tures for thousands of molecular systems in which For electron-excess molecular systems, a very im- the valence-bond arrangements of 1 and 2 pertain for

portant type of triatomic bonding unit involves four one or more sets of four electrons (1, 3, 4). Further, electrons distributed amongst three overlapping these bonding units inay be either intra- or interatomic orbitals located on three atoms (1-4). The molecular. Larger n-centre bonding units (for exam- resulting 4-electron 3-centre bonding is usually described qualitatively by means of either delocalized 3-centre molecular orbital theory or by valence-bond resonance theory (1-4). The simplest type of valence- bond description involves resonance between the standard valence-bond structures 1 and 2, each of which has an electron-pair bond between adjacent atoms and a lone-pair of electrons on the third atom. This valence-bond representation shows perhaps more immediately than does a molecular orbital description that 4-electron 3-centre bonding must be an extremely widespread phenomenon; it is possible

ple, 6-electron 4-centre) with additional atoms, overlapping atomic orbitals and electrons, represent elaborations of the basic 4-electron 3-centre bonding unit. The purpose of this and subsequent papers is to report the results of some non-empirical valence- bond calculations for a number of molecules that involve either or both of these types of bonding units.

f ---.-: --> Y A-B Y-A B Y A B

to write down Lewis-Langmuir valence-bond struc- 4 5 6

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1094 C A N . J. CHEM. VOL. 56. 1978

In contrast to what is the case for molecular orbital theory, few valence-bond wave-functions have been reported for the ground-states of systems with 4-electron 3-centre bonding units. Of those that have been reported (4-1 9) (and they range from semi-empirical to ab-initio), most show up what we consider to be an important (but much-ignored) result, namely that the bond-eigenfunction coefficient for the 'long-bond' structure 3 can have appreciable magnitude. Indeed, its magnitude is sometimes calculated to be larger than those for either of the standard valence-bond structures 1 and 2. This is particularly the case when 1 and 2 carry atomic formal charges as in

i-, if, :.> : , Y A-B and Y-A B

and 3 does not. The ranlifications of this result for qualitative valence-bond theories of structure and reactivity are very considerable (see for example, refs. 14, 20, 21). Very often, it is not sufficient to write down valence-bond structures of types 1 and 2, and then hope to obtain a realistic representation for the electronic structure of the molecule. Where appropriate, 3 must be included also. In nulnerous papers, this has been done (1, 3, 4, 14, 20, 21), and new types of valence formulae (namely, the 'increased-valence' formulae of the next paragraph) have been generated that summarize resonance between the standard and 'long-bond' (or spin-paired diradical) structures 1, 2, and 3. (Of course, valence- bond structures of types 4-6 are also needed to form a complete valence-bond basis set for a singlet-spin state involving 4-electron 3-centre bonding.) One purpose of the present series of papers is to provide further quantitative evidence that important valence- bond structures for the ground-states of electron- excess n~olecules carry either small or zero atomic formal charges, and that such structures may often be of type 3. The formal charge requirement is compatible with the electroneutrality principle (22).

It may be noted that resonance between all valence- bond structures that do not involve 'long-bonds' may be sumnlarized by using ,-centre' bond orbitals in- stead of Heitler-London wave-functions as the wave- functions for the electron-pair bonds that occur in the standard valence-bond structures 1 and 2 (1, 3b, 4-10). To include the 'long-bond' structure also, another type of valence formula is needed. For this purpose, the 'increased-valence' formulae 7 and 8 have been devised (1, 3,4, 14,20,21). These formulae have one more bonding electron than have any of 1-3 and may be easily generated from the standard valence-bond structures 1 and 2 by delocalizing a non-bonding Y or B electron into a bonding YA or AB orbital, as is shown in 9 and 10. Because the 1-electron bond A.B summarizes resonance between

the valence formulae A h and A B, it is easy to demonstrate that 7 and 8 summarize resonance between 1 and 3, and 2 and 3, respectively if Heitler- London type wave-functions are used to describe the A-B and Y-A bonds of 7 and 8. Alternatively, if localized molecular orbitals (or bond-orbitals) are used to describe these bonds in 7 and 8, resonance between these 'increased-valence' formulae is equivalent to invoking resonance between all of the valence- bond structures 1-6 when Heitler-London type wave-functions are used to describe the electron-pair bonds of 1-3 (1, 3b, 4, 14c). Linnett's non-paired spatial orbital formulae 11 (2, 5-10) may also be used to summarize resonance between the standard and 'long-bond' valence-bond structures 1-3, as well as 4. Both 'increased-valence' and non-paired spatial orbital formulae will be used in the discussions for this series of papers.

Y . A-B Y-A - B

.? Y A-B y - ~ p i 9 . A .

9 10 11

Because of limitations to our computing facilities (we have used a CYBER 72, CDC 6400 computer, with a store capacity of 64K words of central mem- ory), it has not been possible to perform all-electron calculations on many of the triatomic and polyatomic systems that we have selected to study. Therefore, we have restricted our attention t o a consideration of the different types of valence-bond structures that arise from variations in the distributions of only some of the electrons, namely, those that could delocalize appreciably. For non-linear planar molecules, these electrons were usually either some or all of (a) the n-electrons, and (6) the lone-pair n'-electrons oc- cupying atomic orbitals. that overlap appreciably with the orbitals of.0-bonds. On occasions, we have also allowed for the polarization of certain localized a- or n-bonds, and when this was done, additional valence-bond structures were introduced into the calculations. For a calculation, it was possible to take account of 6 atoms, 20 atomic orbitals, 20 electrons, and 25 valence-bond structures. When the number of structures that we could write down exceeded 25, we have used chemical intuition and the electroneutrality principle to decide which structures should be included in the treatment.

In the present paper, we shall report the results of valence-bond calculations for the n-electrons of O,, NO2-, and CH2N2. We are currently preparing for publication other papers that describe the results of similar types of valence-bond calculations for the

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HARCOURT AND ROSO 1095

o-electrons of HF2-, and the n- and n'-electrons for by writing wi = ai'C2. Equation 3 may then be ex- a number of nitroso and nitro compounds, and the pressed as eq. 4. To avoid using spherical harmonics, oxalate anion. the CGF's in 141 were redaced bv the siin~ler lobe

Method of Calculation

. ., Gaussian functions using R = 0 . 0 3 ~ ~ - " ~ , in which R is the distance of the lobe centre from the nucleus .

Since completing this work, Gerratt (23) has pub- (30). The resulting S T 0 is given by [5]. lished a review of valence-bond theory. We shall N

briefly summarize the procedure that we have used [3] $sTo(C = 1, I.) = C aiyiCG(ui, r ) to construct the Hamiltonian .and overlap matrix i = l

elements for the valence-bond secular equations of N

[2]. These equations pertain to the linear combination C41 $STO(~, 1.1 = C ai~iCG(u;C2, 1'1 of [ I 1, in which the $i represent the normalized bond- i = 1

eigenfunctions for the different canonical valence- N

bond structures. C ~ I = C aiyiLG(ai1C2, R, I * )

N i = 1

(a) As basis atomic orbitals, we have used single- zeta Slater-type orbitals (STO's) with the 'best atom' orbital exponents of Clementi and Raimondi (24). For the hydrogen 1s exponent, Hehre and Pople's value of 1.3 (25) was used. The optimization of exponents in valence-bond calculations makes heavy demands on computer time, and we have not at- tempted to do this for the polyatomic systems.

(b) The bond-eigenfunction for a valence-bond structure involves a linear combination of Slater determinants. Numerous procedures are available for the construction of the appropriate spin eigen- states (23, 26). Because we have been concerned only with singlet (S = 0) and doublet (S = i) spin-states, it is possible to use the simple rule described by Kauzmann (27) to construct the spin eigenfunctions. The branching diagram indicates the existence of spin-degeneracy for systems that involve three or more singly-occupied orbitals (23,26). The procedure that we have used generates the spin-eigenfunction that is appropriate for the electron-pairing scheme of a given valence-bond structure. The ground-state eigenvectors for [ l ] that we shall report pertain when both Y and the $i bond-eigenfunctions are normalized to unity.

(c) To evaluate the Hamiltonian and overlap matrix elements (Hij and Sij) of [2], we have used Lowdin's cofactor method (28).

(d) To calculate the 1-electron and 2-electron integrals over atomic orbitals for the Hij and Sij, the STO's were expressed as linear combinations of cartesian Gaussian functions (CGF's), yiCG, as in [3] (29). The optimum linear combination of CGF's was determined by Stewart (29) for S T 0 exponents (5) of unity. Non-unity values for the exponents are used by scaling (30,31) the CGF exponents of [3], i.e.

The number of Gaussian components, N, used in all calculations was five (30). When testing the valence-bond procedure, we found that this number is the minimum required to give valei~ce-bond wave- functions that agree closely with some published ST0 calculations for LiH (3 1).

(e)With the current valence-bond program, which was prepared by one of us (W.R.), it is possible to perform one or more of the following types of calculations. (i) All-electron calculations: For certain diatomic and small polyatomic molecules, the program permits all of the electrons t o be included specifically in the construction of the Slater determinants to form the bond-eigenfunctions. Minimal atomic orbital basis sets were used in calculations of this type for LiH, HF, CO, and HNO. (ii) All- valence-electron calculations: For larger molecules, we were unable to treat explicitly all of the electrons, and therefore we have invoked the approximations described here and (iii) below. For all-valence electron calculations, only the valence-shell electroils and orbitals are included in the construction of the bond- eigenfunctions. Inner-shell electrons are treated as part of the core, whose charge is equal to the nuclear charge minus the number of inner-shell electrons. We have tested the validity of this approximation by comparing the results of valence-bond calculations for HNO when procedures (i) and (ii) are both used. These calculations will be described in the second paper of this series. (iii) Har?zilton's approxir?zation: Many of the molecules that we have examined have rather more than 20 valence-shell electrons and orbitals. For them, it was necessary to invoke Hamil- ton's approximation (1 1, 32), and include only some of the valence-shell orbitals in the Slater deterini- nants. The remaining electrons are assigned to the core, and are considered only in the calculation of penetration integrals. When this is done, Hamilton's approximation involves the neglect of all electron exchange, except for that which occurs between the

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1096 CAN. J. C H E M .

electrons explicitly included in the Slater determinants. In Part 11 of this series, we shall compare calculations for FNO and HF2- using Hamilton's ap- proxi~nation and either all-valence electron or all- electron procedures.

(f) In our discussions of the valence-bond wave- functions, we shall follow Maclagan and co-workers (33) by assuming that the bond-eigenfunction 41i whose coefficient has the largest magnitude in the ground-state valence-bond wave-function of [1 ] represents the most important valence-bond structure. Because the $i are not orthogonal, the weights for the valence-bond structures are not equal to Ci2. However, an inspection of the structural weights and coefficients for an ab-it7itio study of C6H6 (34) indicates that the magnitudes for these two quantities correlate approximately (but 1701 exactly) with each other, in so far as if a coefficient Ci has a large magnitude, then the valence-bond weight (or occupation number (34)) is also large.

To ensure correct functioning of the program, valence-bond wave-functions were calculated for a number of systems for which such wave-functions have been reported previously by other workers. Our results for an all-electron calculation for LiH are reported in Table 1, where they are compared with those of Miller et al. (3 1). The calculation of Linnett and co-workers (9) for the forinate anion, with Hamilton's approximation, was repeated. Exact

TAULE 1. Valence-bond wave-functions for LiH

LJ Li

Structure configuration This work Ref. 31

Li-H Is22s1 0.6665 0.6916 Li-H ls22p' 0.3913 0.4251 Li+ H - 1 s2 0.2390 0.2248 Li- H + ls22s2 - 0.0554 - 0.0696 Li- H + ls22p2 -0.0169 -0.0317 Li- H + 1 ~ ~ 2 ~ ~ 2 ~ ~ - 0.0347 -0.0583

Electronic energy (eV) -216.946 -217.01 Dipole moment (D) 6.039 6.04

Value (au) Randomly selected

integrals* This work Ref. 31

[nib] 0.1658 0.1663 0.5546 0.5545

[al$"Ia] -4.7824 - 4.7869 [alf?rOlbl -0.7174 -0.7169 [clHOP] -0.8225 - 0.8226 [aalaal 1 ,6875 1.6875 [ah I ah1 0.021 2 0.0212 [aalabl 0.1658 0.1656 ~bd I ccn 0.1117 0.1117

agreement between our calculations and those of Linnett et al. was not expected, because the latter workers approximated some of the molecular integrals, whereas we have evaluated them accurately. The two wave-functions are the following, for which the bond-eigenfunctions are defined in ref. 9. Linnett and co-workers (9)

This work

Experimental bond-lengths (35) have been assumed for all calculations. The bond-angles for 0, and NO2- were approximated t o 120" and 116" respectively.

Results ( a ) Ozone and Nitrite

The ground-states of 0, and NO2- each have four n-electrons distributed amongst three overlapping atomic orbitals. Linnett and co-workers (6, 10) have calculated valence-bond wave-functions for these systems, using Hamilton's approximation. Our results for this type of calculation, and also those for all-valence electron calculations, are reported in Tables 2 and 3. For comparison, eigenvectors obtained by other workers (4, 6, 10) are also reported in these tables.

Each calculation indicates that the bond-eigenfunction $, for the valence-bond structure with a 'long' 00 n-bond has a coefficient with a large magnitude. This is particularly the case for O,, and is in accord with the electroneutrality principle in so far as the valence-bond structure carries zero formal charges on all atoms. On the basis of their generalized valence-bond calculations, Goddard and co-workers (17) have concluded that O, is primarily a diradical rather than a zwitterion, and that the 'long' 00 n-bond structure best represents the electronic structure of this molecule. They have used this valence- bond structure to discuss the Criegee mechanism of ozonolysis of olefins (17).

For 0, and NO2-, the standard valence-bond structures that are most usually displayed are the Lewis-Langmuir octet structures 1 and 2 of Tables 2 and 3. However, because the bond-eigenfunction for the 'long-bond' octet structure 3 of either table is also calculated to be important, it is necessary to include this type of structure in the valence-bond representation for the ground-state. This may be done either by writing down the 'long-bond' structure together with the other structures, or by devising valence formulae that summarize resonance between all of these structures. These latter formulae may be

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TABLE 2. n-Electron valence-bond structures and bond-eigen- by delocalizing an in-plane nf-electron as well as a function coefficients (Ci) for 0 3 ground-state n-electron from the (-ve) oxygen atom. In the cal-

culations of the present paper, no delocalization of Ci the n'-electrons has been taken account of for either

Type Structure A* B? Ref. 4* O3 or NO,-. For each of these systems, and for CH,N,, we have also not allowed for any polarization of 00, NO, CN, or N N o-bonds. Therefore,

0.337 valence-bond structures such as 19, 20, and 21 have been omitted from the calculations, even though the absence of formal charges for them might suggest that they may be important structures. However, these structures are not required to interpret reac-

0.337 tions of these molecules that involve only the break- ing of their n-bonds. The '1,3-dipolar' cycloadditions that are to be discussed in the next section form one

3 0.8619 0.7934 0.859 such class of reactions. Electronic mechanisms for reactions such as

(++) X + 0, + XO + 0,, with X = H or NO, are described in ref. 36. They utilize 'increased-valence'

4 :.0. . /O\ . .O: 0.0391 0.0670 0.110 structures 17 and 18 for 0, and its diradical character.

(-) (-1

'Hamilton's approximation for the four n-electrons. ?All valence electrons in Slater determinants. Could and Linnett (6) have

calculated Y = 0.351(v, ' - I -v i ' )+0.390v~'+0.124yr, '+0.0 in which the v,' are non-normallzed bond-e~genfilnctlons.

either of the 'increased-valence' (1, 3, 4, 14, 20, 21) or non-paired spatial orbital type (2, 5-10). For O,, these valence formulae are either 12 + 13 (30, 4, 14b) or 14 (3, 4, 6). We may generate the 'increased- valence' formulae 12 and 13 from the standard valence-bond formulae by delocalizing a non- bonding n-electron from a terminal oxygen atom into a bonding 00 n-orbital, as is shown in 15 and 16 (1, 4, 20, 21).

TABLE 3. n-Electron valence-bond structures and bond-eigenfunction coefficients (Ci) for NO2- gound-state

Type Structure A* B? Ref. 4*

16 (+) (-)

15 *Hamilton's approximation for the four n-electrons. ?All valence electrons in Slater determinants. (For this calculation, the For '3, 'increased-va1ence' l7 and 'best atom' exponents were adjusted for residual formal charges.) Hirst a n d

may also be generated which carry zero formal Linnett (10) have calculated Y = O0.306(v,' +, v,,) + o . ~ ~ I v , , +,0.185~,' +0.070(v5' + VG'), in which the v,' are non-normallzed bond-e~genfunc-

charges on all atoins (30,20b, 36). These are obtained tlons.

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1098 C A N . J . CHEM. VOL. 56. 1978

TABLE 4. n-Electron valence-bond structures and bond-eigenfunction coefficients for CH2N, ground-state

Coefficients Coefficients

TY pe Structure 6 electrons 10 electrons Type Structure 6 electrons 10 electrons

1 : +L

H ,C -N=N N: 0.3068 0.2822 7 H ~ C ~ N - N : 0.2746 0.3140 - lt'

(b) Dinzoinethane Diazomethane has two sets of n-electrons. The

standard valence-bond structures are 1 and 2 of Table 4. They each involve four n- and two nf-electrons. Huisgen (37) has classified both 0, and CH2N2 as 1,3-dipolar molecules, and has assumed that the sextet valence-bond structures 5 and 6 of Table 2, and 4 and 5 of Table 4 are the important structures for a concerted mechanism for 1,3-dipolar cycloaddition reactions. An alternative concerted mechanism has been proposed (38) for this class of reaction, which utilizes the 'long-bond' (or spin- paired diradical) structure of type 3 in both Tables 2 and 4 rather than the sextet structures. More recently, Goddard and Walch (18) have also concluded that the spin-paired diradical structure for CH2N, is important for a description of the electronic structure and cycloaddition reactions of this molecule. The results of our valence-bond calculations provide further support for this hypothesis.

We shall report here the results of two valence- bond calculations, each of which utilizes the Hamil- ton approximation. The first calculation includes explicitly only the six x-electrons in the Slater determinants, whereas the second also includes two CN

o-electrons and two CC o-electrons. Twelve valence- bond structures were considered that differ in the positions of the six n-electrons. These structures are displayed in Table 4, together with the eigenvectors for the two sets of calculations. The hybridizations for the central nitrogen atom, and the carbon atom, were assumed to be sp and sp2 respectively.

In Table 4, the valence-bond wave-functions have similar sets of coefficients. They indicate that the most important of the bond-eigenfunctions is that which pertains to the 'long-bond' structure 3, with zero formal charges on all atoms. The bond-eigenfunction coefficients for the zwitterionic structures 1 and 2 are smaller than are those for 3. As is the case for O,, the absence of formal charges in the 'long- bond' structure 3 may be responsible for the impor- tance of its bond-eigenfunction. Structure 7 also carries zero formal charges on all atoms, and the coefficient for its bond-eigenfunction also has appreciable magnitude. Each of these results is in accord with what we would deduce from the electroneutrality principle.

The important valence-bond structures for the Huisgen mechanism of 1,3-dipolar cycloaddition are structures 4 and 5 of Table 4. Fo r the six-electron valence-bond calculation, the coefficients for the bond-eigenfunctions for these structures are 0.1044 and 0.0473. These are much smaller than the 0.3068, 0.2297, and 0.3770 for the bond-eigenfunctions that pertain to the zwitterionic and 'long-bond' structures 1,2, and 3. Although it is possible that the im- portance of the sextet structures increases as a 1,3-

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Hz?.-N=N : H2C/R%N: formal charges on the carbon atom and the central X- -.-.- -7. ,\ /, nitrogen atom. A comparison of these formal

/C-C >?=&/ \ charges with those of 22 and 23 might lead to the \ conclusion that 22 is the more important of the two

. . 'increased-valence' formulae. ( - t ) ( + A ) (+A) (-9 -

H ~ C T N ~ N : H2C=NTN: ( h ) H ~ C / ~ ' \ N:

Some Comments on the Utility of \ I - \'J/ + Ac -& Increased-valence Formulae

I / >LC \ \ \ ,C=C I , / \ / ' When Heitler-London procedures are used to construct wave-functions for the A-B and Y-A

(-4) (+) (+9 bonds of 'increased-valence' formulae 7 and 8, H~c-;-N=N: these structures provide an economical (and easily-

L O U / ' & derived) method for su~nmarizing resonance be- \\ // /c=C \ tween the standard and 'long-bond' structures 1-3.2

\

FIG. 1. Electronic mechanisms for 1,3-dipolar cycloaddition reactions.

dipolar cycloaddition reaction proceeds, it may be more favourable energetically to assume that the 'reactive' structures are those that are important for the valence-bond representation of the ground-state of the 1,3-dipolar molecule. Of these latter structures, 3 is calculated to be the most important. If we use structure 3 alone, we may devise a simple mechanism for cycloaddition, namely that of a of Fig. 1. This mechanism has also been postulated recently by Goddard and Walch (1 8). I t does not involve a reor- ganization of the atomic formal charge distribution as the reaction proceeds, and therefore it is compatible with the observed lack of solvent dependency on the rate of cycloaddition (37, 39).

The 'increased-valence'formulae 22 and 23 for CH2N2 (38) may be generated from the zwitterionic structures 1 and 2 of Table 4, by delocalizing a carbon and a (terminal) nitrogen non-bonding n-electron into a bonding CN or N N n-orbital. Resonance between these 'increased-valence' structures summarizes resonance between the valence-bond structures 1, 2, and 3 of Table 4. If we use 22 and 23 to represent the electronic structure of CH,N2, the mechanism for 1,3-dipolar cycloaddition becomes that of b of Fig. 1 (38). Alternatively, use of the Lin- nett non-paired spatial orbital formula 24 gives the cycloaddition mechanism (c) of Fig. 1 .' Each of these electronic mechanisms depends on the contribution of the spin-paired diradical structure 3 of Table 4 to the description of the electronic diazomethane.

The wave-functions of Table 4 generate the atomic formal charges of 25 and 26. Both calculations locate a small formal positive charge on the terminal nitrogen atom, and rather larger negative and positive

'Firestone has formulated mechanism (c) as a two-step mechanism (39). See ref. 38 for some comments on the electronic aspects of this procedure.

For 0, and CH2N2, the economy gained by using 'increased-valence' formulae 12, 13, 22, and 23 in- stead of the standard and 'long-bond' structures of Tables 2 and 4 is obviously not great. But if a molecule has two or more sets of 4-electron 3-centre bonding units, as has 0, with standard valence-bond structures 15 and 16, then the economy that is obtained by using 'increased-valence' formulae 17 and 18 is rather more considerable. Resonance between these latter structures is equivalent to invoking resonance between the standard structures 15 and 16 and the 'long-bond' structures 27-31.

30 31

2The S = 0 wave-function for 7 is JyW,.al;J + Iv,,JbCl, with vr. = k y + n and k > 0 (1, 3, 4, 146). This wave-function may be expressed as k(lyJn5) + lyjjbd)) + lyda6) + nybiil, which is a linear combination of the bond-eigenfunctions for the standard and 'long-bond' str~~ctures 1 and 3.

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1100 C A N . J. CHEM. VOL. 56. 1978

If at least some of the 'long-bond' structures are required to provide an elementary valence-bond description of certain moIecules, it then does not be- come easy to make deductions about bond properties when these structures are written dew; together with the standard valence-bond structures. For example by writing down the 'long-bond' structures 27-29 with zero formal charges as well as the standard valence-bond structures 15 and 16, one does not obtain an obvious understanding of why the 00 bond-lengths of 1.28 A for 0, are closer to the 00 double-bond length (1.21 A) than to the single-bond length (1.48 A). This is because although 15 and 16 alone imply some double-bond character for the 00 bonds, each of 27-29 has only 00 single bonds. Although inclusio~l of the 'long-bond' structures in the resonance description must lower the energy, the nearest-neighbour bond-orders (or bond-num- bers) seem to be reduced in value. Inclusion of additional 'long-bond' structures such as 30 and 31 must lead to a further lowering of the energy. However, because each of these latter structures has only one (nearest-neighbour) 00 single-bond, a further re- duction in the (nearest-neighbour) 00 bond-order might be deduced. Therefore we see177 to obtain a paradoxical situation when 'long-bond' structures as well as standard valence-bond structures are used to represent the electronic structure of a molecule, namely, that inclusion of the 'long-bond' structures lowers the energy but seans to reduce the nearest neighbour bond-orders. This paradox arises because simply writing down the standard and 'long-bond' structures does not provide an obvious way to show the effect of resonance on nearest-neighbour bond orders. One way to resolve this difficulty is to use 'increased-valence' formulae such as 17 and 18 for 0, to represent the electronic structure. We have al- ready indicated that 17 and 18 summarize resonance between the standard and 'long-bond' structures 15, 16, and 27-31. But by inspection of 17 and 18, we can easily make the appropriate deductions about the 00 bond properties. Each of these structures has an 0, ground-state type double-bond ( O t O ) and a weakened 0, excited-state type double-bond (0=0). Therefore by using 'increased-valence' formulae 17 and 18 as a method to summarize resonance between the standard and 'long-bond' structures 15, 16, and 27-31, we ensure that inclusion of the 'long-bond' structures both lowers the energy and increases the 00 bond-orders relative to the values that pertain for resonance between the standard structures 15 and 16 alone. The 'increased-valence' formulae take account of the effect of resonance on nearest-neighbour bond-order in a manner which is identical to that which pertains when the valence structure A-B

for a 1-electron bond is ysed to sumparize resonance between the structures A B and A B. Neither of the latter two structures alone has an AB bonding electron, but resonance between them generates AB bonding, which is indicated quite clearly by means of the valence structure A-B.

Acknowledgments One of us (W.R.) acknowledges the award of a

Commonwealth Postgraduate Scholarship, which was held throughout the tenure of this work (1971- 1973). The manuscript for this paper was completed during study leave by one of us (R.D.H.) at Cornell and Gottingen Universities. Professors R. Hoffinann and W. A. Binge1 are thanked for the use of the facilities of their departments at Cornell and Got- tingen. We are indebted to Drs. F. R. Burden and B. T. Hart, and the Monash University Chemistry Department for making available their integral evaluation program.

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Valence-bond studies of 4-electron 3-centre bonding units. I. The π-electrons of O 3 , NO 2− ,...

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Transcript of Valence-bond studies of 4-electron 3-centre bonding units. I. The π-electrons of O 3 , NO 2− ,...