The total energy of the helium atom is thereforeEtrial 2
8
Z
(^20)
2
h
e
2
(^4)
8 Z
1
8
0
Z
8
e(^20)
4
h
2
Etrial 2 Z^2 8 Z
1
8
0
Z
8
e(^20)
4
h
2
where all terms have had the expression e^4 /8^20 h^2 factored out. In order
for the energy to be a minimum with respect to Z, we must find the value
ofZsuch that
Z
E
0
The only part ofEthat depends on Zis the first parenthetical part, so the
energy can be minimized by determining when the terms in parentheses are
zero. So,Z
E
4 Z 8
1
8
0
0
(2Z^2 8 Z^180 Z)
Z12.7 Variation Theory 397which gives us Z 2 156 . Therefore, the average nuclear charge “felt” by
each electron in He according to this model is ^2176 , or just under 2. Now that
the effective nuclear charge has been determined, the energy can be evaluated
by substituting for all of the constants and Z. One finds that
EHe1.241 10 ^17 J
which, compared to 1.265 10 ^17 J determined experimentally, is high by
1.9%. This is slightly better than the perturbation theory treatment presented
earlier.The above example shows two things. First, variation theory canprovide a
more accurate value for the energy of a system. Second, it comes at a cost: a
cost of effort. However, using computers to do the calculations, the personal
effort can be minimized, so variation theory is particularly well suited for com-
puter applications. In fact, a majority of the effort expended in the modern ap-
plication of quantum mechanics is in the application of computer programs.
Because computers can be programmed with a large number of variables to
change in the course of a calculation, variation problems are almost exclusively
performed on computers.
We have considered the energy of He using several methods. Table 12.2
summarizes the energies using the different methods we have applied in this
chapter, compared with an experimental value. These are not the only meth-
ods possible, and you should understand that the methods used here have been
used in their simplest forms. But Table 12.2 should give you some idea of the
utility of the various tools of quantum mechanics.
Table 12.2 Different energies of the helium atom
E(He) ( 10 ^17 J) Method
1.743 H H approximation
1.198 Using perturbation theory,e^2 /4
0 r 12
1.241 Using variation theory, effective nuclear charge
1.265 Experiment