Concise Physical Chemistry

c17 JWBS043-Rogers September 13, 2010 11:28 Printer Name: Yet to Come

A VARIATIONAL TREATMENT FOR THE HYDROGEN ATOM 271

The same problem for a sum ofNterms demands that the general secular deter-

minant be zero. This leads toNroots:

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

H 11 −ES 11 H 12 −ES 12 ... H 1 N−ES 1 N

H 21 −ES 21 H 22 −ES 22 ... H 2 N−ES 2 N

..

.

..

. ...

..

.

HN 1 −ESN 1 HN 2 −ESN 2 ... HNN−ESNN

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

= 0

Many readers will be familiar with the use of this mathematical formalism in Huckel ̈

molecular orbital theory.

17.3 A VARIATIONAL TREATMENT FOR THE HYDROGEN ATOM:

THE ENERGY SPECTRUM

Starting with the radial Hamiltonian

Hˆ =

(

−^12 ∇ 12 −

2

r 1

)

=−

1

2 r^2

d

dr

r^2

d

dr

−

2

r

and the exact wave function for the hydrogen atomφ(r)=e−αr, the variational

method leads to

E=

h^2 α^2

8 π^2 me

−αe^2

wheremeis the mass of the electron (McQuarrie, 1983). This result is similar to

the exact solution for the particle in a cubic boxE=h^2 n^2 / 8 ml^2 except that there

is a potential energy term−αe^2. We can expect some similarities between the two

systems. One similarity is that in each there is aspectrumof specific energy levels,

each corresponding to a specific quantum number. Another is that there is no zero

energy because the lowest quantum number is 1, not 0.

We would like to carry out a systematic search for the minimum energy, so we

set the first derivative ofEwith respect toαequal to zero. The derivative goes to

zero at a minimum, maximum, or inflection point. If our trial function is reasonably

good,^1 only the minimum is found at this level of calculation. (For more complicated

systems, maxima, saddle points, and the like may be found.)

The derivative is

dE

dα

=

2 
^2 α

me

−e^2 = 0

(^1) As chemists, we bring centuries of empirical evidence to bear on the decision of what is or is not a good
atomic or molecular wave function. A negative exponential is reasonable for the hydrogen atom.