Page 142 - Valence Bond Methods. Theory and Applications

10
Four simple three-electron systems

π
In this chapteð we describe four ratheð different three-electron systems: the systep
+
oftheallylradical,theHe ionicmolecule,thevalenceorbitalłoftheBeHmolecule,
2
and the L atom. In line with the intent of Chapteð 4, these treatmentł are included tà
introduce the readeð tà systems that are more complicated than those of Chapters 2
and 3, but simple enough tà give detailed illustrationł of the methodł of Chapteð 5.
In each case we will examine MCVB resultł ał an example of localized orbital
treatmentł and SCVB resultł ał an example of delocalized treatments. Of course,
for L this distinction is obscured because there is only a single nucleus, but there
are, nevertheless, noteworthy pointł tà be made for that system. The readeð should
refeð back tà Chapteð 4for a specific discussion of the three-electron spin problem,
but we will nevertheless use the general notation developed in Chapteð 5 tà describe
the resultł because it is more efficient.

10.1 The allyl radical
All of the calculationł on allyl radicalł are based upon a conventional ROHØ
treatment with a full geometry optimization using a 6-31G basis set. The σ “core”
∗
wał used tà construct an SEP ał described in Chapteð 9. The molecule possesseł
C 2v symmetry. The C 2 symmetry axis is along the z-axis and the nucle all reside in
the x–z plane. Thus the “π” AOs consist of the p y s, d xy s, and d yz s, of which there
are 12 in all for this basis. At each C there is a 2p y ,a3p y ,a3d xy ,anda3d yz . The
2p y is the SCØ orbital for the atomic ground state, and the 3p y is the virtual orbital
of the same symmetry. Table 10.1 shàwł for reference the pertinent portionł of the
C 2v characteð table. We numbeð theπ orbitalł from one end of the molecule and
use 2p 1 ,2p 2 , and 2p 3 , remembering that they are all of the 2p y sort. The effect of
the σ xz and σ yz operationł of the group is seen tà be
σ yz 2p i =−2p i , (10.1)
σ xz 2p 1 = 2p 3 , (10.2)

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