Page 242 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics
P. 242
9.1 Variation method 233
Except for the restrictions stated above, the function ö, called the trial
function, is completely arbitrary. If ö is identical with the ground-state eigen-
function ø 0 , then of course the quantity E equals E 0 .If ö is one of the
excited-state eigenfunctions, then E is equal to the corresponding excited-state
energy and is obviously greater than E 0 . However, no matter what trial function
ö is selected, the quantity E is never less than E 0 .
To prove the variation theorem, we assume that the eigenfunctions ø n form
a complete, orthonormal set and expand the trial function ö in terms of that set
X
ö a n ø n (9:3)
n
where, according to equation (3.28)
a n hø n jöi (9:4)
Since the trial function ö is normalized, we have
* +
X X X X
höjöi a k ø k a n ø n a a n hø k jø n i
k
k n k n
X X X
2
a a n ä kn ja n j 1
k
k n n
We next substitute equation (9.3) into the integral for E in (9.2) and subtract
the ground-state energy E 0 , giving
X X
^ ^
E ÿ E 0 höjH ÿ E 0 jöi a a n hø k jH ÿ E 0 jø n i
k
k n
X X X
2
a a n (E n ÿ E 0 )hø k jø n i ja n j (E n ÿ E 0 ) (9:5)
k
k n n
where equation (9.1) has been used. Since E n is greater than or equal to E 0 and
2
ja n j is always positive or zero, we have E ÿ E 0 > 0 and the theorem is
proved.
In the event that ö is not normalized, then ö in equation (9.2) is replaced by
Aö, where A is the normalization constant, and this equation becomes
2 ^
E jAj höjHjöi > E 0
The normalization relation is
2
hAöjAöijAj höjöi 1
giving
^
höjHjöi
E > E 0 (9:6)
höjöi
In practice, the trial function ö is chosen with a number of parameters ë 1 ,