Physical Chemistry Third Edition

19.1 The Variation Method and Its Application to the Helium Atom 791

function. The variation energyWis calculated as a function of the parameters, and the

minimum value ofWis found by the methods of calculus.

EXAMPLE19.1

Calculate the variational energy of a particle in a hard one-dimensional box of lengtha, using

the variation functionφ(x)Ax(a−x).

Solution

To normalize the wave function:

1 A^2

∫a

0

x^2 (a−x)^2 dxA^2

∫a

0

(a^2 x^2 − 2 ax^3 +x^4 )dx

A^2

(

a^5

3

−

2 a^5

4

+

a^5

5

)

A^2

a^5

30

A

(

30

a^5

) 1 / 2

Since there are no variable parameters, the variational energy will not be minimized.

W−A^2

̄h^2

2 m

∫a

0

x(a−x)

d^2

dx^2

[x(a−x)]dx−A^2

h ̄^2

2 m

∫a

0

x(a−x)(−2)dx

A^2

h ̄^2

m

∫a

0

(ax−x^2 )dxA^2

h ̄^2

m

(

a^3

2

−

a^3

3

)



5 h ̄^2

ma^2



5 h^2

4 π^2 ma^2

 0. 12665

h^2

ma^2

This variational energy is only 1.32% higher than the correct ground-state energy, in which

the factor 1/ 8  0 .125 occurs in place of the factor 0.12665.

Application of the Variation Method to the Helium Atom^1

Let us first use the zero-order orbital wave function of Eq. (18.3-2) as a variation func-

tion. This is a single function, so no energy minimization can be done, but the procedure

illustrates some things about the method. The zero-order function is normalized so that

the variation energy is

W

1

2

∫

ψ 100 (1)ψ 100 (2)[α∗(1)β∗(2)−β∗(1)α∗(2)]Hψ̂ 100 (1)ψ 100 (2)

×[α(1)β(2)−β(1)α(2)]dq′ 1 dq′ 2 (19.1-3)

wheredq′ 1 anddq 2 ′indicate integration over space and spin coordinates and where the

space orbitalψ 100 is a hydrogen-like 1sorbital. We omit the symbols for the complex

(^1) Our treatment follows that in J. C. Davis, Jr.,Advanced Physical Chemistry, The Ronald Press,
New York, 1965, p. 221ff.