Molecular Orbital Theory of Bond Order and Valency A. 6. Sannigrahi and Tapas Kar Indian Institute of Technology. Kharagpur-721302, India
The bond order is not observable within the confines of quantum mechanics, and hence it is not possible to define it in a unique manner. Nevertheless, it is a very useful model for chemists, and thus it is worthwhile to see how bond order can be retrieved from MO calculations. The definition of bond order in a polyatomic molecule was first given by Coulson (I)in the context of the Huckel MO theory. A detailed account of this theory is included in almost all standard textbooks on quantum chemistry. The textbook of Pilar (2), however, deserves special mention in this connection because of its elegant matrix formulation of the LCAO-MO theory. Coulson's definition for bond order is applicable when the AO's are assumed to be mutually orthogonal and only m e A 0 per atom is considered. ~ h e s e a p p r o x h a t i o n sare usually invoked in the r-electron theories of Huckel, and of Pariser, Parr, and Pople (3, 4). The definition of bond order beyond the r-electron level was first given by Wiberg (5). who used the term bond index instead of bond order. Based on this pioneering work Armstrong et al. ( 6 ) proposed a definition for valency of an atom in amolecule. These definitions of bond order and valency are, however, restricted to an orthonormal set of AO's. In ab initio calculations (2, 7) or the AO's are those based on the extended Huckel theory (8), not assumed to be orthogonal; that is, the overlap between them is not neglected. For a nonorthogonal basis set the appropriate definition for bond order was proposed by Mayer (9). In subsequent papers (10) Mayer has elaborated the physical significance of bond order and valency a t great length. Since chemical bonding forms an integral part of all undergraduate curricula in chemistry we have considered it descri~tionof the worthwhile to oresent a brief oedaeoeic . MO theory of t h d order and valency as develkped mainly bv. Maver.Theonls ~rereauisitero understand this arricle is . to have some famiii&ty with the MO theory and an elementary knowledge of matrix algebra. Mayer's definitions of bond order and valency are based on the Mulliken population analysis method (11). Hence we shall first give a brief description of the method and then proceed to define bond order and valency. Finally, the results of calculations on some simple molecules are given in order to illustrate the theory. u
The LCAO expansion of an MO is given by
where the summation is taken over all m AO's. In matrix form, where C; is an m X 1 column matrix. The corresponding relation for the entire set of occupied MO's is given by O =xc
where C is an m X (Nl2) matrix. The A 0 overlap matrix is defined as S = X+X
Thus the orthonormality of the MO's implies that OC4 = ctx+xc = CtSC =I,,
where IN12is a unit matrix of order NI2. Since the trace of a unit matrix is equal to its order, eq 7 yields
Mulllken's Population Analysls Let us consider a closed-shell N-electron molecule. We shall approximate its ground state wave functions by a single
Slater determinant (2) of Nl2 lowest occupied MO's. Denoting the nonorthogonal set of m AO's by (x,J and the orthogonal set of N12 occupied MO's by (4;) we can express them by following row matrices.
Journal of Chemical Education
where P = 2CC+ ( C i s an m X (NI2) matrix of the coefficients, and C+ is adjoint of C) anduse ismade of the fact that the trace of a nroduct of matrices remains invariant under their cyclic p.&mutations. For an orthonormal set of AO's, C+C =IN,%and N = tr(P). I t should be noted that the LCAO coefficients of MO's of the same molecule will be different for orthogonal and nonorthogonal sets of AO's. Relation 8 is of central importance in the present context.
I t provides the basis of Mulliken's population analysis (11). and as we shall show shortly the definitions of bond order and valency are also based on this relation. In Mulliken's population analysis scheme, the total number (N) of electrons is partitioned as follows into contribution from atoms and bonds (unlike usual chemical bonds, these hondsinvolve all possible atom pairs in a molecule).
atomic self-charge and active charge, respectively. The valency V Aof atom A is now defined (6,9) by the relation,
N = tr(PS)
it follows from eq 13that an equivalent definition of valency is (16) 1 V A= 26, - BAA
where the quantity in the parentheses is equal to QA, the electron density of atom A. I t is a sum of two terms; the first term is a contribution from all the orbitals centered on A and the second term is a sum of the overlap population (OP) of all bonds formed by A. Thus the overlap population of the AB hond is given by
Deflnltlons of Bond Order and Valency for Nonorthogonal Basis Sets
The definitions of hond order and valency are based on the duodempotency property (a square matrix D is said to he duodempotent, if 0 2 = 20) of the PS matrix, which can he proved as follows. (PSI2= (PS)(PS)
where use has been made of the orthonormality relation (eq 7). Since tr(PS) = N, it follows from eq 11that (12) Xtr(PS)' = N Expanding the left-hand sid'e of eq 12 we get
It is interesting to note that eqs 9 and 13 provide alternative schemes for the oartitionine of N electrons into contributions from atomsand bond; There is, however, a very important difference between the two schemes. In the case of an orthonormal set of AO's for which S6. = doh. the entire second term in eq 9, and those components of the first term for which b # a, vanish. The corresponding terms in eq 13, however, survive. Now the overlap population is zero, hut BABis not. I t is given by BAB=
which is nothing hut Wiberg's bond index. For an orthogonal basis set also VAis defined by eq 15 where for BABeq 17 is used. I t is not obvious a t this stage why BABas defined by eq 14 should be called a bond order. The results of the numerical calculations presented in the next section will show that this quantity is indeed exactly or nearly equal to the multiplicity of a hond as predicted by classical chemical concepts. Bond Orders and Valencles of Some Simple Molecules
We shall first illustrate the definitions of bond order and valency as given by eqs 14 and 15, respectively, taking two simple closed-shell molecules, namely, Hz and HeH+. We shall use the minimal basis set of AO's, which generally consists of the orbitals required to describe the ground state electronic configuration of the constituent atoms. This definition is, however, not quite accurate since one usually considers ls, Zs, and 2p, that is, five orbitals t o constitute a minimal basis set for Li to Ne, although the 2p orbitals are not occupied in the ground state of Li and Be. Similarly, for Na to Ar the minimal basis set consists of is, 29, Zp, 39, and 3p, that is, nine orbitals per atom. Let us first take the case of Hz whose ground state electronic configuration is lo:, where
Here, a and b denote twoH atoms. Since there is only one A 0 per center,a, b, etc., are not distinguished from A,B, etc. The P and S matrices are given by
where the quantity, BAB=
defines (9) the order of the AB bond. The first and the second terms in eq 13 are sometimes (12) referred to as the Volume 65
The SOMI Order and Valencv
oi Some Small Molecules*
Bond order (B,.) CNOOIZ Ab initio
a CND012 Ab initlo
F w d i a m m i c mOlsCYles ~ a k n ~ i eWththeatomsantmsame. ~ol Bondordersforhs nonbondsd atom p i r r are not lncluded in the tlble. his is why relation 15 is not exactly satisfied in many cares.
~ 2.34, ~ ~ ~BHH= 0.22, whichgives QH. = 1.53, QH= 0 . 4 7 , B = B H H=~ 0.72, and VH = VH. = 0.72. The hond order value of 0.72 suggests that He-H hond in HeH+ is not a purely covalent single bond; it has considerable ionic character. I t may be noted that for a purely covalent hond Bas = 1and for a purely ionic bond Bas = 0. Thus Bm is a measure of degree of covalency of the AB bond. In the table are given the bond orders and valencies of some simple molecules calculated (13) from CND012 (14) and minimal basis set ah initio wave functions (15). In the CND012 method onlv the valence electrons and an orthogonal set of valence [email protected]; are considered. The pertinent expressions for bond order and valencv in this case are ~ v e bv n eqs 17 and 15, respectively. As can be seen from the table, the two sets of calculations yield essentially the same results for most of the molecules. The calculated values of Bas and V A are exactly or nearly equal to the values predicted by classical chemical concepts. A slight deviation from the integral valuesstems from the ionic character of the bonds. The bond order in CO is about 2.6, which suggests that this is a polar triple hond in conformity with the Lewis structure, :CEO+:. Similar is the case with the NC bond in HNC. Here N is almost tetravalent, in conformity with the Lewis structure, H-N=C:. The CO bond in H,CO is seen to hare verv feeble triple bond character due to the fact that C is noi exactly sp2hybridized in this molecule. The HF; ion, which happens to be the smallest and strongest H-bonded system, is formed by the interaction of F- and HF. Upon H-bond formation the bond order of H F is considerably reduced, and both the H F bonds in HF; become highly ionic. The results in the table indicate that there is generally a close connection between the delocalized MO theory and the classical Lewis structures. Concluding .Remarks
We have shown how the classical concepts of bond order and valency can be quantitatively defined in the framework of the LCAO-MO theory using general nonorthogonal set of AO's. The results of numerical calculations on several simple molecules have been included to illustrate the theory. Since there is no unique way to define bond order and valency, alternative definitions based on the Lowdin population analysis scheme have been proposed (16) and illustrated with numerical calculations. Within the framework of a given population analysis scheme the calculated values of bond order andvalency depend (13) very much upon the basis sets employed, and for a given basis set they depend (13) upon the method of population analysis especially for molecules with highly polar bonds.
which yields Q. = Qb= 1.0 B,. = B,, = B,, = 1.0
These are expected results indicating that the H-H bond in Hz is a purely covalent single bond. One will get the same values for charge density, bond order, and valency by assuming Sob = 0, that is, for an orthogonal basis set. This is, however, not generally true. The HeH+ is the simplest heteronuclear diatomic molecule. Its ground state is described by the electronic configuration 1c2where for $(la) we shall use the following expression (7):
Literature Clted 1. 2. 3. 4. 5. 6.
The overlap matrix is given (7) hy
7. 8. 9. 10.
Coulaan,C.A.Ploc. Ray.Sor. (London) 1938,A169,413428. Pilar, F. L. Elementary Quantum Chemistry;MeGraw-Hill: New York. 1968. Pariser, R.:Parr, R.G. J. Cham Phys 1953,21.16M71,762-776. Pople. J.A. Tmns.Forodoy Sac. 1953,49. 1375-1385. Wiberg, K. A. Tetrohodron 1968,24.1083-1096. Armsfmng.D.R.;Perkins.P.C.;Stewaut,J.J.P.J.Cham.Soe. iDallan) 1913,83&840. Szabo,A.;Ostlund, N. G. Modern Quantum Ch~mialry,AnIntmductionlo Aduanced Electronic Structure Theory; Mamillan: New York, 1982. Hoffmann, R. J . Chem. Phyr. 1963.39.1397-1412. Mayer, I. Chem. Phys. Lett. 1983.97.270-274. Maver. I.In(. J Quantum Chsm. 1986. 29. 73-84. The references far almast all
Thus, 13. Sennigrshi. A. B.;Kar, T.. unpublished results. 14. Pople, J. A,; Beveridgc, D. L. Appmximote Molecular Orbital Theory; M e G r a w ~ ~ i l l : Now Y a k , 1970. 15. Hehre, W. J.; Radom, R.: Sehleyer, P.v. R.: Pople, J. A. Ab Inilio Moleeulor 01bilol Theory; Wiley-Interscience: New Yark, 1986. 16. Nstiello. M . A : Medrsn0.J. A. Chem. Phys. Let1 t984,105.180-182.