For a cyanide species where the mass is me,k 1 10 ^7 kgm/s^2 , and
a 1.15 Å (that is, 1.15 10 ^10 m), we can evaluate the expression above
and get
1,real 1,PIAB0.15162,PIAB
A more complete treatment includes contributions from 3 , 4 , and so on,
but their contributions become less and less important as the difference be-
tween the ideal energies increases. Finally, recall that a 2 for 2,real(or any
other real ) will be different from the a 2 calculated above for 1,real.12.7 Variation Theory
The second major approximation theory used in quantum mechanics is called
variation theory.Variation theory is based on the fact that any test wavefunc-
tion for a system has an average energy that is equal to or greater than the true
ground-state energy of that system. Therefore, the general idea is that the lower
the energy, the better the approximated energy(and therefore,the better the
wavefunction). What one does is to suppose a trial wavefunction that has some
variable parameter in it, determine the expression for the energy of the system
(using the Schrödinger equation or the definition of average energy,E), and
then determine what value the variable must have in order to yield the lowest
possible energy. Since the wavefunction should also provide average values for
other observables, those other values can be determined once a minimum en-
ergy is determined for that trial wavefunction.†One of the strengths of varia-
tion theory is that the trial wavefunctions can be anyfunction, as long as the
function meets the standards of wavefunctions in general (that is, continuous,
integrable, single-valued, and so on) and satisfies any inherent requirement
of the system (such as approaching zero as xapproaches or the system
barriers).
One way of stating the basic idea behind variation theory is the following:
for a system having a Hamiltonian operator Hˆ, true wavefunctions true, and
some lowest-energy eigenvalue E 1 , the variation theoremstates that for any
normalized trial wavefunction :*HˆdE 1 (12.23)
Ifis identically equal to truefor the ground state, then equation 12.23 is an
equality. Ifis not exactlythe ground-state wavefunction, then equation 12.23
is an inequality and the energy produced by the integral is always greaterthan
the true ground-state energy of the system. Therefore, the lower the predicted
energy, the closer it is to the true ground-state energy and the “better” an
energy eigenvalue it is. Proof of variation theory is left as an exercise at the
end of this chapter. For an unnormalized wavefunction, equation 12.23 is
written as*H
*
ˆ
dd
E 1 (12.24)
Usually, the trial wavefunctions have some set of adjustable parameters
(a,b,c,.. .). The energy is calculated as an expression in terms of those para-394 CHAPTER 12 Atoms and Molecules
†However, there is no guarantee that this trial wavefunction will yield accurate values for
other observables.