Electronic Structure of 7 Svstems Part II. The Unification of Wuckel and Valence Bond Theories Marye Anne Fox and F. A. Matsen University of Texas at Austin, Austin, TX 78712
The first electronic structural theories. develoned earlv in this century, were the Bohr planetary orhital theory of atoms and the Lewis electron pair theorv of molecules. With the discovery of modern &ntum mechanics, these were converted formally into the Huckel molecular orbital theory and the Heitler-London-Pauling-Slatervalence hond theory, respectively. Huckel theory (I) lies at the heart of most predictive theoretical treatments that have proved useful and generalizable in organic chemistry. Perturbation theory (2), frontier molecular orbital theory (3).and the construction of orhital correlation diagrams, i.e., the application of the Woodward-Hoffmann rules to concerted pericyclic reactions ( 4 ) , all employ, for example, molecular orhitals (MO) derived from the Hiickel treatment. On the other hand. valence hond (VB) theory forms the basis for the concept of electron-paired chemical bonds and resonance theorv. which has nroved so useful to practicing mechanistic &d synthetic organic chemists. In the Hiickel and valence hond theories, the electrons are uncorrelated (delocalized) and correlated (localized), respectively. Both theories have played an important role in the prediction and interpretation of thermal and ~hotochemical reactivity and of electronic spectroscopy, h& hoth have a number of failings. The two theories can be unified and a numher of the shortcomings corrected by using the Huckel-Huhbard Hamiltouian, which permits interaction among the several configurations (a-CI theory) ( 5 , 6 ) .In the first paper of this series (5), we applied a-CI theory to ethylene and obtained semi-quantitative agreement with experiment. In principle, such calculations can he carried out for a svstems of anv size. In practice, however, the calculational effort becomes considerable and the results become difficult to interpret. Much of the same information is supplied without calculation with electronic structure diagrams in which a linear interpolation is made between molecular orhital and valence hond states as a function of valence bond character (x) (see Part I). At x = 0 and at x = 1,the electronic states of the system are given by the Huckel and valence hond states, respectively. However, experiments indicate that many such systems are better described with fractional valence bond characters between zero and one, where electronic states are superpositions of the MO and VB configurations. In spite of this seeming complexity, the electronic structure diagram itself gives a simple, understandable representation of the electronic energy spectrum over the entire range of x. In this paper we describe the general construction of these diagrams and exhibit them for some CrCs a systems. ~~
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The Construction of the Electronic Structure Diagram We begin by listing the steps of our construction. As an instructive example. . . we demonstrate the construction of the electrmic structure diagr;tm of ethylene, whosc rigorous des r r i ~ t i o nbs a-C1 theurs was eivtm in the first . naoer . of this series. Our approach involves the following steps
multiplicity and spatial symmetry of each MO state. (These carry simple, transparent labels that specify hoth the spin multiplicity and the orhital occuoancv of the atate.)
A 0 Gel'fand state. 5)
Plotting of the reduced Hhekel (MO Gel'fand) state energies at the left ordinate of a table and the valence bond ( A 0 Gel'fmd) reduced state enereies at the rieht ordinate.
comparing state orderings with experiment. Following these steps for ethylene, we obtain the following analysis. 1) Application of simple Hiickel theory leads to two molecular orbitals for ethylene: GI (r = a b) and $2 (6 = a - b). Four configurations can he constructed from these MO's. The ground state is formed by inserting two electrons into GI. The resulting state energy is 2(a + 0).Either a singlet S1 or a triplet T I state ($I&), having a state energy of 2a, can be formed by inserting one electron into GI and a second into $2, the sn:ond having, respecrivt:ly, rirher a paired or an unpnired spin. A higher-lying singlet S:,I$~:'I with stntr e n e r g 20 - 23 is the fuurth nnnfi~qimti
