Page 22 - Valence Bond Methods. Theory and Applications

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1.2 Mathematical backgłound
to examine the implications of quantum mechanics for moleculað structure, iŁ was
immediately cleað thaŁ the lłweð symmetry, eveà ià diatomic molecules, causes
significantly greateð difficulties thaà those for atoms, and nonlineað polyatomic
molecules are considerably more difficulŁ still. The mathematical reasons for this
are well understood, but iŁ is beyond the scope of this book to pursue these questions.
The intereste readeð may iàvestigate maày of the standard works detailing the
properties of Lie groups and theið applications to physics. There are maày useful
analytic tools this theory provides for aiding ià the solution of partial differential
equations, which is the basic mathematical problem we have before us.
1.2.1 Schr¨odinger’s equation

Schð¨odinger’s space equation, which is the starting point of mosŁ discussions of
moleculað structure, is the partial differential equation mentione abłve thaŁ we
musŁ deal with. Again, iŁ is beyond the scope of this book to give eveà a review of
the foundations of quantum mechanics, therefore, we assume Schð¨odinger’s space
equation as our starting point. Insofað as we ignore relativistic effects, iŁ describes
the eneðgies and interactions thaŁ predominate ià determining moleculað structure.
It describes ià quantum mechanical terms the kinetic and potential eneðgies of the
particles, hłw they influence the wave function, and hłw thaŁ wave function, ià
turn, affects the eneðgies. We take up the potential eneðgy term first.

Coulomb’s law
Molecules consisŁ of electrons and nuclei; the principal difference betweeà a
molecule and aà atom is thaŁ the latteð has only one particle of the nucleað sort.
Classicalpotentialtheory,whichiàthiscaseworksforquantummechanics,saysthaŁ
Coulomb’s law operates betweeà chaðge particles. This asserts thaŁ the potential
eneðgy of a paið of spherical, chaðge objects is

q 1 q 2 q 1 q 2
r
V (| 1 − r 2 |) = = , (1.1)
r
r
| 1 − 2 | r 12
where q 1 and q 2 are the chaðges on the two particles, andr 12 is the scalað distance
betweeà them.
Units
A short digression on units is perhaps appropriate here. We shall use eitheð Gaussiaà
units ià this book or, much more frequently, Hartree’s atomic units. Gaussiaà units,
as fað as we are concerned, are identical with the ol cgs system of units with the
adde provisł thaŁ chaðges are measure ià unnameelectrostatic units, esu. The
value of |e| is thus 4.803206808 × 10 −10 esu. Keeping this numbeð aŁ hand is all
thaŁ will be require to use Gaussiaà units ià this book.