Physical Chemistry , 1st ed.

meters, and then the expression is minimized with respect to those parameters.

In calculus terms, if the energy is some expression in terms of a single variable

E(a), then the minimum energy occurs when the slope of a plot ofEversus a

is zero:‡



E



(

a

a)

(^) at aamin^0
and the energy evaluated at this point is the minimum energy:
E(amin) Emin
This minimum energy is the “best” energy that this trial wavefunction can pro-
vide. When there are multiple variables in the trial wavefunction, then the ab-
solute minimum with respect to all variables simultaneously is the lowest en-
ergy that such a trial wavefunction produces. Although variation theory does
provide more complicated expressions for the energies of excited states, the
above relatively simple expressions apply only to the ground state of a system.
The trial wavefunctions can have any number of variable parameters, limited
mostly by the efficiency in determining the energy minimum. Variation theory
is best illustrated by example. We will start by using a trial wavefunction with-
out parameters to show that equation 12.23 is satisfied. For the particle-in-a-
box of length a, assume that instead of a sine function, the ground-state wave-
function is instead an upside-down parabola at the center of the box,a/2:
axx^2
This trial wavefunction is shown in Figure 12.8. As you can see, it meets all of
the requirements of a wavefunction for a particle-in-a-box system: it is single-
valued, continuous, integrable, and goes to zero at the boundaries. To calculate
the energy for this trial wavefunction, we need to evaluate

a
0
(axx^2 )*Hˆ(axx^2 ) dx
where the particle-in-a-box Hamiltonian is (^2 /2m)(d^2 /dx^2 ). The second de-
rivative of the trial wavefunction is 2, which reduces the integral to




m

2



a

0

(axx^2 ) dx

The expression inside the integral can be integrated, then evaluated between

the limits 0 to a. One gets





6

2

m

a^3



This trial wavefunction is not normalized, but one can determine the normal-

ization constant to be 30/a^5. This makes the predicted energy of the ground

state (adjusted by the square of the normalization constant)

Etrial

m

5 

a

2

 2  4

5

2

h

m

2

a^2



This compares with a true energy for the ground state of the particle-in-a-box

ofh^2 /8ma^2 , or a difference of 1.32%. The approximated energy is higherthan

the true energy by 1.32%.

12.7 Variation Theory 395

‡Maximum energies also meet this criterion, so it is important to verify that the energy

so determined is a minimum, not a maximum.

x  0 a
Figure 12.8 Trial wavefunctions for a varia-
tion-theory treatment of the ground state of the
particle-in-a-box. The solid line is the trial para-
bolic wavefunction, and the dotted line is the true
wavefunction.