Page 246 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

9.2 Linear variation functions 237
X
2
E ÿ E k ˆ ja n j (E n ÿ E k )
n(>k)
from which it follows that
E > E k (9:7)
Thus, the quantity E is an upper bound to the energy E k corresponding to the
state ø k . For situations in which ö can be made orthogonal to each exact
eigenfunction ø 0 , ø 1 , ... , ø kÿ1 , the coef®cients a 0 , a 1 , ... , a kÿ1 vanish
according to equation (9.4) and the inequality (9.7) applies.
An example is a one-dimensional system for which the potential energy V(x)
is an even function of the position variable x. The eigenfunction ø 0 with the
lowest eigenvalue E 0 has no nodes and therefore must be an even function of x.
The eigenfunction ø 1 has one node, located at the origin, and therefore must
be an odd function of x. If we select for ö any odd function of x, then ö is
orthogonal to any even function of x, including ø 0 , and the coef®cient a 0
^
vanishes. Thus, the integral E ˆhöjHjöi gives an upper bound to E 1 even
though the ground-state eigenfunction ø 0 may not be known.
When the exact eigenfunctions ø 0 , ø 1 , ... , ø kÿ1 are not known, they may
be approximated by trial functions ö 0 , ö 1 , ... , ö kÿ1 which successively give
upper bounds for E 0 , E 1 , ... , E kÿ1 , respectively. In this case, the function ö 1
is constructed to be orthogonal to ö 0 , ö 2 constructed orthogonal to both ö 0 and
ö 1 , and so forth. In general, this method is dif®cult to apply and gives
increasingly less accurate results with increasing n.

9.2 Linear variation functions
A convenient and widely used form for the trial function ö is the linear
variation function
N
X
ö ˆ c i ÷ i (9:8)
iˆ1
where ÷ 1 , ÷ 2 , ... , ÷ N are an incomplete set of linearly independent functions
which have the same variables and which satisfy the same boundary conditions
as the exact eigenfunctions ø n of equation (9.1). The functions ÷ i are selected
to be real and are not necessarily orthogonal to one another. Thus, the overlap
integral S ij , de®ned as
S ij h÷ i j÷ j i (9:9)
is not generally equal to ä ij . The coef®cients c i are also restricted to real values
and are variation parameters to be determined by the minimization of the
variation integral E .