Page 61 - Mechanism and Theory in Organic Chemistry

Appendix 2

MOLECULAR

ORBITAL THEORYa

Tlie electronic state functions of a molecule are those functions Y which satisfy
the Schrodinger equation :

Here E, is the total electronic energy and X, is the electronic Hamiltonian
~perator.~ The state function Y is a function of the coordinates of all the elec-
trons. X, is a prescription for carrying out on YP. a sequence of mathematical
operations (partial differentiation with respect to the various coordinates and
division by electron-electron and electron-nuclear separations) that is deter-
mined from the laws of mechanics and from the properties (number of electrons,
number and positions of nuclei) of the particular molecule being considered. The
Hamiltonian operator, though it may be quite complicated for a large molecule,
can be written relatively easily; the unknown quantities in the equation are Y
and E,. In general, there will be many possible functions Y that are solutions for a
given X,. Each of them represents a different possible state of the molecule, and
each has its associated energy. (From this point on we shall consider the electronic
ground state only.) Because the complexity of the many-particle molecular
systems is so great, it is impossible to solve A2.1 for Y and E. Approximate
methods must be used if there is to be any hope of progress.

.
" For further information consult the sources cited in footnote 2, p. 10.
Equation A2.1 has already been simplified by making the assumption, known as the Born-Oppen-
heimer approximation, that nuclear and electron motions can be considered separately. Equation
A2.1 applies only to electron motions; the nuclei are considered to be in fixed locations.