Walter England' Ames Laboratow Energy Research and Development Admmistrotion Iowa State University Ames, 50010 and Department of Phvs~cs Colorado State University Fort Collins, 80521
I The LMO Description of Multiple Bonding and Multiple Lone Pairs
I
Energy localized orbitals (LMO's) are commonly accepted as quantum mechanical counterparts of classical localized valence. Hence, what they reveal about the localization of electrons is of particular interest and they, therefore, receive considerable attention in the literature (I). One of the many questions to which LMO analyses have been addressed concerns multiple bonds and multiple lone pairs. For our purposes, the multiple bonds can be classified as: (1) double bonds such as in ethylene, (2) triple bonds such as in acetylene, and (3) aromatic bonds such as in one of the benzene resonance structures (2); while the lone pain may occur in (1) doublets such as found on oxygen in formaldehyde, and (2) triplets such as found on each F in Fz. The question is basically the same in each of these situations: Are the localized orbitals sigma and pi orbitals or are they mixtures of sigma and pi orbitals? The traditional valence description of multiple bonds and multiple lone pairs classifies them as sigma and pi orbitals. Each set of multiple bonds or lone pairs is said to consist of one pure sigma orbital and one or two pure pi orbitals. However, the LMO valence description fails to support this. There, each set is found to consist of orbitals that are neither sigma nor pi orbitals. In bonding situations these are highly bent off of the bond skeleton and are called banana bonds (3). The LMO lone pairs are also directed off of the bond skeleton, hut away from the neighboring atom (3). It is clear that the local hybridization of the classical valence orbitals is very different from that of the LMO's. This difference will be important for applications that depend on the shape of the orbital. Moreover, since LMO's are rigorous quantum mechanical analogs of classical valence ideas, i t is of basic interest to understand the disparity. Localized Molecular Orbitals There is an inherent flexibility associated with any single determinant closed-shell molecular orbital wave function, namely, each such wave function is unchanged by a unitary transformation of the occupied molecular orbitals (a detailed discussion is given in Paper 3 of Reference ( I ) ) . Conseauentlv. there are infinitelv manv sets of molecular orbitals associated with each clo"sed-shell system, and for example, obsewables can be calculated with sets of localized or delocalized molecular orbitals. Energy localized molecular orbitals are a particular set of closed-shell molecular orbitals. They are defined by mathematical criteria (4) whicb we will review in the next paragraph. We will also give the physical significance of these criteria. The electron repulsion energy for any single determinant closed-shell molecular orbital (MO) wave function is (4) R = D + C - X (1) where D is the intmorbitol Coulombic repulsion energy, or orbital self-repulsion energy 'Present address: Chemistry Division, Argonne National Lahoratory, Argonne, Illinois 60439.
D
=
z(illi2)
Cis the interorbital Coulombic repulsion energy c = ezzy;rl,~)
(2) (3)
and X is the interorbital Exchange repulsion energy X = XZ'(ij1ijj (4) All sums are over the occupied MO's &, and the primes mean that the i = j terms are omitted from the sum. The integral notation is defined by
rijhl)
JJdV,dV,rn,(l~,(l)rn~(2)+,(2)/r,~ (5) Energy localized MO's (LMO's) are the MO's that simultaneously maximize D and minimize C and X against unitary transformations (4). The total MO energy and total electron repulsion energy R are unchanged by these transformations (4). Maximization of D means that, on the average, the LMO short-range Coulombic interactions are as large as possible. Minimization of C means that they are also the MO's whose average long-range Coulombic interactions are as small as possible. Finally, minimization of X means that their average short-range exchange interactions are likewise as small as possible. Succinctly, enerev localization accomnlishes three thines: concentration'cf the MO's, long-run& separation of kfferent MO'n, and short-rance separation of different MO's. If i t were poisibie to reduce X to zero, one would have =
R = D + C (6) This is the repulsion energy for a Hartree product wave function (5). The Hartree product for a closed-shell takes the Pauli principle into account by doubly occupying each MO, but does not include the part of the principle which requires antisymmetry of the total wave function. Since it is generally impossible to reduce X to zero, the LMO's are those MO's for whicb R most nearly approaches the Hartree form, i.e. they are proper quantum mechanical orbitals that identify with specific electrons as closely as is possible. Although their original definition was given more than 20 years ago by Lennard-Jones and Pople (6). LMO's remained largely conceptual-even after the advent of computers-until Edmiston and Ruedenberg (3, 4) gave practical methods for calculating them. Listing a b initio results for diatomic and small polyatomic molecules in two papers (3, 7), the latter authors convincingly demonstrated that LMO's were quantum mechanical analogs of the intuitive notions of localized valence. Since tbis work, LMO's have occurred frequently in the literature (I). Perhaps one of the most stimulating papers was given by Trindle and Sinanoglu (8) who applied the method to CNDO (9), an approximate self-consistent-field (SCF) MO theory. In many cases, their results were the same as the ab initio. Because CNDO calculations are simple and fast, whereas, ab initio calculations are very time consuming, tbis opened new areas of problems to energy localization. Volume 52, Number 7, July 1975 / 427
The Description of Multiple Bonds and Lone Pairs
The LMO's will be assumed to have the form
A,
=
(a
+ a , s , ) / ( l+ a,l)'12
with ",?
= a
2
=
a2
=
integer
>0
(7) (8)
and a and rt are molecular orbitals satisfying
The lone pairs in the fluorine molecule have previously been written in the form of eqn. ( 7 ) (10). We shall use ethylene as an example for unsaturated bonding orbitals. The orbital a is a CC sigma bond consisting mainly of sp2 hybrids on each carbon that lie along the CC axis. The ri orbital, with lobes above and below the molecular plane, is made from the carbon 2 p orbitals whose nodes are in the plane. One equivalent Xi orbital can be formed from these by taking ai = 1. The other results when a , = - 1 . The eqns. ( 7 ) and (8) give
while eqn. (9) shows that the XI and a , rt sets both satisfy localization criteria (4) and hence qualify as possible LMO's. Furthermore, the degree of localization of these orbital sets can be assessed from their contributions to eqn. ( 2 ) , i.e. we need not consider the other LMO's in the molecule. In what follows, we can therefore restrict our attention to just one of the Xi and supress the i. The localization sum for the X's is a')-'[(a21a2) aYr'ln2) 2ay(a4n2) + D(X) ( 1 2 ( m l o s ) ) l (11) and for the a , n is
-
+
+
D(0.n)
=
(a21a1)
+
+
aYn21P)
The difference is A = D(X) - D(a,a)
which will he positive whenever
D is typically an order of magnitude greater than X, so the A's are almost always more localized then the a, r . In general, however, A1 may be positive, negative, or zero. This can be understood in terms of the relative spacial proximities of the a and n orbitals. First, suppose a and r overlap very little in an absolute sense. Then
where 1 arl denotes the absolute overlap of a and a , i.e.
In this case, the a and r are most localized. This would be the situation if they came from different valence shells, an example of which has been discussed by Edmiston and Ruedenberg (7). Now suppose that a and r overlap strongly in an absolute sense, as would he the case if both came from the same valence shell. The overlap of A* with itself is then much greater than with A,, and hence
so that the X's are most localized. The situation described by eqn. (20) obtains for unsaturated molecules because the sigma and pi orbitals are from the same valence shell. If we imagine each doubly occupied localized orbital to be an electron pair, then electrostatic interactions between pairs of electrons in a set of banana orbitals are smaller than those in a set of sigma and pi orbitals. Alternatively, electrostatic interactions between paired electrons are greater in a set of banana orbitals than in a set of sigma and pi orbitals. Of course, we cannot have perfect pairing, but the localized orbital description identifies electrons with specific orbitals as closely as is quantum mechanically possible. Exactly parallel remarks apply to multiple lone pairs. Acknowledgment
When A > 0, the X's are most localized and when A < 0, the a , r a r e mas$ localized. Note that the sign of A is independent of a2, which allows us to perform the analysis with a2 = 1 in our equations. The A's are the usual digonal orbitals
and we find
The author is grateful to Dr. M. S. Gordon for reading and criticizing the manuscript. Literature Cited Ill Sevenrt reviem of lacalized orbitals have appeared: Bcnt. H. A,. Forachr. Chrm. Forseh., 22, 1 119701: Weinstein, H.,Paunez, R.. and Cohen. M., "Advances in Atomic and Molecular Phydes," Vol. 7, Academic Press. New Yark. 1971: and England. W., Sslmon, L. 8.. and Ruedenberg, K.. Fortaehr. Chem Forach.. 23. 31 119711. 121 England, W., and Gordon, M. S.,J.Amer. Cbem Soc., 91,6861 119691. 131 Edmirton. C..andRuedonhe~,K , J Chrm.Phy8, 43,897 119651. 141 Edmisfon, C., and Ruedenberg, K..RPU.Mod. Phy. .. 35. 457 11961). 151 Bethe. H.A,. and Jaekiw, R.. "Intermediate Quantum Mechanics: 2nd Ed., W. A . Benjamin. Ine.. New York. 19M,p.52. 161 LennedJonos. J . E.. and Paple. J. A . Pmc. Rw. Sm. Ihndonl, A 198. 166 14%
.
