Journal of Molecular Structure (Theochem), 149 (1987) 91-96 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

BOND ORDERS AND VALENCE INDICES: RELATIONS TO MULLIKEN’S POPULATION ANALYSIS AND COVALENT CHEMICAL REACTIVITY*

0. P. SINGH and J. S. YADAV

Department of Physics, Banaras Hindu University, Varanasi-221005 (India)

(Received 28 August 1985)

ABSTRACT

Ab initio SCF theory is applied to calculate the bond orders and valence number of an atom in a molecule following Mayer’s suggestions. The ab initio bond orders are compared with Mulliken’s overlap population and with semi-empirical results. The reactivities of different atoms, in terms of Jug’s normal-, hyper- and sub-valences, are also discussed in terms of their affinities for covalent bond formation. The traditional view, that valency is directly related to the atomic charge, is found to be invalid. The valency of an atom is found to be determined by a function of the orbital occupancies corrected for intra- atomic terms.

INTRODUCTION

In an earlier paper [ 11, the authors applied Mayer’s suggestions [2, 31 to calculate the bond orders and valence numbers of atoms in some fluoro- substituted molecules and discussed the same in the light of M&liken’s popu- lation analysis and its affinity for covalent bond formation, respectively. According to Gopinathan and Jug [4, 51, in chemical reactions, a subvalent atom in a molecule may form further covalent bond(s) with other reagents whereas a hypervalent atom may either break or weaken the existing bond(s) so as to convert its sub- or hyper-valency to its normal valency. As the affinity of atoms for covalent bond formation changes from molecule to molecule and covalent chemical reactivity has direct relation with the valency, it was thought adequate to apply the same method to calculate the above indices of some other substituted benzenes and to see how the dif- ferent atoms of the molecules behave towards their tendencies for covalent bond formation. It should be mentioned that we are dealing with reactivity as a tendency for covalent bond formation; nucleophilic or electrophilic reactivity is thus excluded.

The present paper deals with the results of such investigations for toluene, phenol and aniline and these are discussed in terms of the relative strength of chemical bonds and their reactivities.

*Dedicated to Professor Gerhard Herzberg.

0166-1280/87/$03.50 0 1987 Elsevier Science Publishers B.V.

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METHOD OF CALCULATION

Mayer [2, 31 defined the bond order, BAB, between atoms A and B and the actual total valence, VA, of an atom A in the molecule for a closed shell system as,

where the notations XeA and weB indicate that the summations have to be carried out for all the basis orbit& centred on atoms A and B, respectively. P and S are the density and overlap matrices, respectively.

VA = c (2p, --pi) - 1 (pS),, (ps),, (2) WA ~c.ve.4

(Ic#V)

where p,, = (PS),,,, is the gross orbital population on the pth basis orbital. The first and second sums represent the actual bonding power of the dif- ferent basis orbitals and the intra-atomic bond orders of the atom A, respec- tively.

The percentage “excess valence ” of different atoms in the molecules were also calculated by using the STO-3G method. Following Mayer’s suggestion [ 41, the term “excess valence” was used instead of Gopinathan’s and Jug’s “free valence”, in order to avoid the use of the latter in different meanings. The definitions of normal-, hyper- and sub-valences are given in the paper by Gopinathan and Jug [4]. In order to incorporate the above suggestions of Mayer and Jug, the Gaussian 76 program of Pople et al. [7] was extended. All the molecules, except toluene, were taken as planar. All bond angles are 120”, and C-C and C-H bond lengths are 1.397 and 1.084 IL, respectively, except those involving the substituents. The C-C(CHs), C-G(OH), C-N(NH2), C(CHs)-H, G-H and N-H bond lengths are 1.540,1.380,1.420, 1.100, 0.940 and 1.040 A, respectively. LC-G-H (phenol), LC-N-H (aniline) and LC-C-H (toluene) are 105, 126.5 and 109.5”, respectively, provided that the atoms H 13 and Hi5 in toluene are above and below the plane of the ring, respectively.

RESULTS

The calculated bond orders, overlap populations and valence numbers are summarized in Table 1 and a qualitative correlation between the “excess valence” of atoms in molecules and their affinity for covalent bond formation with other reagents is given in Table 2. The following features emerge from these tables.

The -CHs, --OH and -NH2 substituents are more electronegative than hydrogen and, therefore, the electronic charge flows from C-H bond regions to the attractive C-C, C-C(CH3), C-N(NH2) and C-G(OH) bond regions, making the C-H bond orders slightly less than unity. Furthermore, similarly

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TABLE 1

Calculated bond orders, overlap populations and valence numbers for some molecules using the STO-3G and 4-31G basis sets

Bond order and overlap populations*

Bond STO-3G 4-31G

Valence number

Atom STO-3G 4-31G

1.401(0.508) ‘1.443(0.509) 1.425(0.513) 1.446(0.523) 1.427(0.470) 1.376(0.475) 0.782(0.195) 0.940(0.382) 0.947(0.383) 0.951(0.385) 0.947(0.382) 0.946(0.385)

0.784(0.272)

Phenol

C--c, 1.378(0.498)

C--c, 1.444(0.509)

C&--C, 1.426(0.506)

C,-C, 1.433(0.507)

C,-C, 1.436(0.507)

C,-C, 1.385(0.499)

C-G, 1.035(0.297)

C,--H, 0.970(0.393)

C,--H, 0.972(0.394)

Cd--H,, 0.972(0.395)

C,--HI, 0.972(0.394)

C,---H,, 0.970(0.393)

0,--H,, 0.925(0.268)

Aniline

C-C* 1.372(0.501)

C,-C, 1.444(0.510)

C,-C, 1.427(0.506)

C,--C 1.427(0.506)

C,-C, 1.444(0.510)

C,--C, 1.372(0.501)

C--N, 1.047(0.389)

C---H, 0.972(0.394)

C,--H, 0.972(0.394)

C,--H,, 0.973(0.395)

C,--HI 1 0.972(0.394)

G--H,, 0.972(0.394)

NT---Ha 0.932(0.353)

N,--H,, 0.932(0.353)

Toluene

C1-C 1.416(0.506)

C,-C, 1.442(0.507)

C,-C, 1.430(0.506)

C,--C 1.435(0.507)

C*-C, 1.432(0.506)

C,-C 1.421(0.507)

C--c, 0.996(0.372)

C,--H, 0.968(0.394)

C,---H, 0.972(0.394)

%---HI, 0.972(0.394)

C,--H, 1 0.972(0.394)

G-H,, 0.971(0.394)

CT--He 0.978(0.380)

CT--H,, 0.983(0.383)

‘G--H,, 0.978(0.380)

aOverlap population values are in parentheses.

1.386(0.524) 1.449(0.518) 1.434(0.520) 1.431(0.520) 1.449(0.518 1.386(0.524)

0.858(0.211) 0.951(0.387) 0.950(0.382) 0.960(0.386) 0.950(0.382) 0.951(0.387) 0.847(0.334) 0.847(0.334)

1.418(0.511) 1.445(0.515) 1.431(0.516) 1.443(0.519) 1.430(0.509) 1.438(0.524) 0.954(0.284) 0.952(0.386) 0.951(0.384) 0.952(0.385) 0.951(0.383) 0.950(0.385) 0.951(0.388) 0.960(0.393) 0.951(0.388)

3.9402 3.5930 3.9731 3.8389 3.9768 3.8468 3.9763 3.8470 3.9769 3.8612 3.9715 3.8119

2.0825 1.6197

0.9950 0.9294

0.9956 0.9351 0.9964 0.9384

0.9958 0.9355 0.9972 0.9362

0.9519 0.8089

3.9465 3.6414

3.9697 3.8473 3.9771 3.8549 3.9750 3.8468 3.9771 3.8633 3.9697 3.8473 3.0327 2.6048 0.9968 0.9387 0.9961 0.9375 0.9969 0.9346 0.9961 0.9375 0.9968 0.9387 0.9689 0.8475 0.9689 0.8475

3.9761 3.7814 3.9774 3.8451 3.9777 3.8560 3.9776 3.8525 3.9777 3.8526 3.9768 3.8696 3.9635 3.7932 0.9965 0.9391 0.9962 0.9377 0.9964 0.9387 0.9962 0.9378 0.9965 0.9392 0.9959 0.9295 0.9963 0.9372 0.9959 0.9295

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TABLE 2

Valency and covalent chemical reactivitya

Atoms “Excess valence” or “valence defect” (%)

Hyper Normal Sub -

Predicted affinity for covalent bond formationb

Phenol

C, C* C3 Cd C, C6 07 %

Aniline

C, C* C, C, C, C6 N, HI3 I-L

Toluene

C, C.2 C3 C, C, C6 NV HI, HI, HIS

- - - - - -

- - - - -

- - - - - - - - - -

- - - - - - - - -

- - - - - - - - - -

1.50 unreactive 0.75 unreactive 0.50 unreactive 0.50 unreactive 0.50 unreactive 0.75 unreactive

- slightly antireactive 5.00 slightly reactive

1.25 unreactive 0.75 unreactive 0.50 unreactive 0.50 unreactive 0.50 unreactive 0.75 unreactive

- slightly antireactive 3.00 slightly reactive 3.00 slightly reactive

0.50 0.50 0.50 0.50 0.50 0.50 1.00 0.40 0.40 0.40

unreactive unreactive unreactive unreactive unreactive unreactive unreactive unreactive unreactive unreactive

aAll hydrogen atoms attached to ring carbons are found to have small subvalence. Their subvalence values in different molecules have not been listed in order to reduce the size of the table. bThe terminology of Gopinathan and Jug is used [4,5].

to our previous study [ 11, the C-H bond orders depend upon their relative positions with respect to the substituent. Owing to their polarity, the N-H and O-H bonds have smaller bond-order values in aniline and phenol mol- ecules, respectively, than the C-H bond orders of the benzene ring, while this order is reversed for the C(CH3)-H bonds. It seems that because N and 0 are more electronegative than carbon, the flow of electronic charge from N-H and O-H bond regions takes place more readily towards the more attractive C-N and C-O bond regions in aniline and phenol, respectively,

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than the flow of charge from the C(CH3)-H bond regions to the less attrac- tive C-C(CH3) bond region in toluene. This is also supported by the fact that the STO-3G C-C(CH3) bond order is less than the corresponding C-N and C-O bond orders in their respective molecules. Furthermore, the C-C(CHB), C-N(NH2) and C-O(OH) bonds also attract charges from their adjacent C-C bonds. This makes the bond orders of the adjacent bonds less than those of the C-C bonds which are farther from the substituent. How- ever, the attraction of charges from these adjacent bonds also depends upon the nature of the substituent group. There is much less migration of charge from the C1--Cz and C6-C1 bond regions to the Ci-C, region in toluene than the migration of charge from the corresponding bonds to the C&-N, and C1-O, bonds in aniline and phenol, respectively.

In most normal compounds, the valencies of C, N and 0 have been taken as 4, 3 and 2, respectively, when computing excess valencies. The oxygen atom in phenol and the nitrogen atom in aniline are found to have very little hypervalence character and can be treated as slightly antireactive, whereas C(CH3) has a small subvalence character. An atom in a molecule having high hypervalence, i.e., large negative excess valence, is said to be “antireactive” in the sense that it would not only oppose the formation of further covalent bonds but would also tend to reduce its hypervalency by breaking or weaken- ing the existing bonds.

The hydrogen atoms attached to oxygen and nitrogen are found to have small subvalence values and, therefore, are more reactive than the other hydrogen atoms in the molecule. Furthermore, the carbon atoms attached to the substituent have higher subvalence values than other carbon atoms in the molecule. However, none of the atoms in the present study can be treated as highly reactive or antireactive so far as their affinities for covalent bond formation are concerned; this is quite clear because they show a very little hyper- or subvalence character. Such an observation is in accordance with the chemical structure requirements in this set of molecules.

The traditional view that the valency of an atom in a molecule is related to the atomic charge is found to be invalid in this study. It is found that the valency is a function of the orbital occupancies, corrected for the intra- atomic terms, rather than by the mere sum of these orbital occupancies. It is to be noted that for the chemical significance these intra-atomic terms represent losses in the overall bonding ability of the atom.

The examination of the net atomic charges (table not given) on each atomic site, reveals that an electrophilic substitution will take place preferen- tially at the or-ho and pm positions. The ortho-, Pam-directing properties of the substituents were also observed in our earlier semi-empirical studies [ 81 and experimental observations [ 91.

CONCLUDING REMARKS

As in the work by Mayer [2], a parallelism between the bond-order index, B AB, and the Mulliken overlap population was observed for the chemically

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bonded atoms. However, for non-bonded atoms, a different situation some- times occurs. Both positive and negative values have been found for bond- order indices, whereas the corresponding values of Mulliken overlap popula- tions, generally, but not necessarily, carry the same sign. The latter fact perhaps may be considered as an expression of the repulsive character of the interaction. Thus, like the Wiberg index, WAB, in the case of an orthonormal basis set, the bond-order index does not appropriately distinguish between the non-bonding and anti-bonding situations occuring for chemically non- bonded atoms [6]. The difference in the behaviour of these two indices may be due to the fact that the overlap population is connected with the charge- density distribution and the electrostatics of the molecule. On the other hand the bond-order index is related to the exchange effects in bonding.

The “free valence” defined by Gopinathan and Jug [4, 53 may be a pos- sible measure of covalent chemical reactivity, but to call it a “free valence” is a misnomer because it is not the difference of the actual valence index and the sum of the bond orders. Note that Mayer’s sum of the bond-order indi- ces and Gopinathan and Jug’s valence index are the same in semi-empirical theory for closed shell systems. The difference between the classical valence (reference valence index) and the sum of bond-order indices, or actual valence index, can be attributed to the delocalization and polarization effects, or, in some cases, to the inadequacy of the basis sets. Therefore, as suggested by Mayer [6], this difference might better be named as “excess valence” and “valence defect” for hyper- and sub-valence cases, respectively, rather than “free valence”.

ACKNOWLEDGEMENTS

One of us (OPS) is grateful to UGC (New Delhi) for the award of a teacher fellowship and to Dr. P. P. Singh, Principal, M.L.K.(P..G.) College, Balrampur for academic leave and encouragement.

REFERENCES

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M. D. Newton, Quantum Chemistry Program Exchange (QCPE), Program No. 368, Department of Chemistry, Indiana University, Bloomington, IN 47405, U.S.A.

8 J. S. Yadav, P. C. Mishra and D. K. Rai, Mol. Phys., 26 (1973) 193. 9 E. Baciocchi and G. Illumminati, in A. Streitwieser and R. W. Tafh (Eds.), Progress in

Physical Organic Chemistry, Interscience, New York, 1967, p. 5.