Concise Physical Chemistry

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

272 THE VARIATIONAL METHOD: ATOMS

This amounts to finding the wave functionφ(r)=e−αrthat has a specificαselected

from among all possible values ofα. The minimization gives

α=

mee^2


^2

From this, the minimum energy can be found to be [Problem 7.11, McQuarrie (1983)]

E=−

1

2

(

mee^4


^2

)

=− 2. 180 × 10 −^18 J

which is the ground state of H originally found by Bohr (Fig 15.2). The spectrum of

allowed energies can be calculated from the quantum numbersn:

εH=−

1

2

(

mee^4


^2

)(

1

n^2

)

, n= 1 , 2 , 3 ,...

The energies are negative for a stable system. The resulting energy level differences

can be used to determine the lines in the hydrogen spectrum in agreement with

Fig. 15.1. This should come as no surprise because we started out with the exact

ground state orbital.

At this point it is reasonable to define an atomic unit of energy, thehartree

Eh=

(

mee^4 /
^2

)

, which is twice the energy of the ground state of the hydrogen atom

εH. (Please do not confuse the measured energyεHwith theunitof energyEH.)

εH≡ 2 Eh=2(2. 180 × 10 −^18 J)= 4. 359 × 10 −^18 J=2625 kJ mol−^1

= 627 .51 kcal mol−^1

The process of searching out energy minima in this way is frequently referred to

asminimizationfor obvious reasons. It is also calledoptimizationbecause in cases

where the orbital is not known, the optimum energy and the best possible values of any

parameters in the orbital expression are found. These parameters are sometimes called

optimization parametersto distinguish them from known constants. One arrives at

different optimization parameters by different optimization procedures, but universal

constants like Planck’s constanthdo not change.

17.3.1 Optimizing the Gaussian Function

Let us try a function that is similar to, but not identical to, the exact function. A

variational calculation can be carried out with the approximateGaussianfunction

φ(r)=e−αr

2

. It is of the same form as the lowest hydrogen orbital except that the
numerator has anr^2 , wherershould be. This function can be optimized by calculus