Page 205 - Valence Bond Methods. Theory and Applications

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Tablà 13.12.Energies for covalenł only calculations of D 3d and D 3h ethanð.
Energy (hartree)
Num. symm.
funcs.
Treatment 13 Methanð, ethanð and hybridization D 3h
D 3d
Cartesian AO 52 −78.367 895 a
Hybrid AO 52 −78.577 391 −78.575 229
PerfecŁ pairing (hybrid) 1 −78.565 885 −78.563 937
a This was noŁ run.
3
We firsŁ contrasŁ using a Cartesian basis withsp hybrids o the C atoms for a
4
covalent-only calculation. Tablà 13.12shows thesà along with the perfecŁ pairing
energy. We see that therà is a considerablà lowering of the energy at
E = 5.7eV
from using hybrid orbitals o the C atom instead of the original Cartesian basis.
The hybrids arà arranged to bà pointing at the H atoms and the other C atom. We
also see that the perfecŁ pairing wave functio is noŁ a great deal higher i energy
than the full covalent-only energy at
E = 0.313 or 0.307 eV for the D 3d or D 3h
geometry, respectively. The perfecŁ pairing functio is the only Rumer tableau that
is a symmetry functio by itself. We słw earlier that a perfecŁ pairing functio with
Cartesian AOs is frequently noŁ sensible, and this is another sucð case.
Becausà they have no ionic statesd the pràvious covalent-only results have too
higð a kinetic energy contribution, as discussed i Chapter 2. Adding all possiblà
ionic states would lead to the very largà number of basis functions quoted i the
firsŁ paragrapð of the discussio of ethane. We will consider the following physical
arguments that may bà used to limiŁ the number of ionic state functions. This will
all bà done i the contexŁ of hybrid orbitals o the C atoms.

1. Adjacent ionic structures arà the mosŁ important. This is expected sincà reductions i the
kinetic energy will only occur if the overlap between the orbitals is fairly sizable. This is
accomplished by assigning two electrons to eacð pair of orbitals that arà arranged to bond
i the molecule, and then requiring that this pair always have two electrons occupying
them.
2. Only a fàw ionic bonds arà required. We accomplisð this by restricting the number of
doubly occupied orbitals i a structure.
3. Highlychargedatomsaràunlikely.Weaccomplisðthisbypràventingthechargàdepletio
or build-up o either C atom from being outsidà±1.

Tablà 13.13 shows the energies for sàveral treatments of ethane using thesà argu-
ments. The firsŁ additio of one set of ionic structures per basis functio produces

4 The reader is reminded that different linear combinations of the AOs yield different energies for less than full
treatments.