Physical Chemistry Third Edition

792 19 The Electronic States of Atoms. III. Higher-Order Approximations

conjugates because our function is real. We assume that the nucleus is stationary, so

the correct Hamiltonian operator is:

Ĥ−h ̄

2

2 m

(∇^21 +∇ 22 )+

1

4 πε 0

(

−

Ze^2

r 2

−

Ze^2

r 1

−

e^2

r 12

)

(19.1-4)

where the number of protons in the nucleus is denoted byZ. We will setZequal to 2

at the end of the calculation.

Since the helium-atom Hamiltonian operator is independent of the spin coordinates,

the integral over the spin coordinates can be factored from the space coordinate integra-

tion. Because of the normalization and orthogonality of the spin functions, integration

over the spin coordinates gives a factor of 2, which cancels the normalizing constant

1/2. We could have omitted the spin factor and the spin integration from the beginning.

We now have

W

∫

ψ 100 (1)ψ 100 (2)

(

ĤHL(1)+ĤHL(2)+ e

2

4 πε 0 r 12

)

ψ 100 (1)ψ 100 (2)d^3 r 1 d^3 r 2 (19.1-5)

Exercise 19.1

Show that the spin integration in Eq. (19.1-3) leads to a factor equal to 2.

TheĤHL(1) andĤHL(2) terms in the Hamiltonian operator give ground-state energy eigen-

values for a hydrogen-like atom after operating and integrating.

Exercise 19.2

Show that thêHHL(1) term in Eq. (19.1-5) yields a contribution toWequal toE 1 (HL) and that

thêHHL(2) term yields an equal contribution.

We now have

W 2 E 1 (HL)+

∫

ψ 100 (1)∗ψ 100 (2)∗

(

e^2

4 πε 0 r 12

)

ψ 100 (1)ψ 100 (2)dq (19.1-6)

Evaluation of the integral in this equation is tedious and we give only the result:^2

W 2 E 1 (HL)+

5 Ze^2

8(4πε 0 a)

 2 E 1 (HL)−

5 Z

8

〈V〉H(1s) (19.1-7)

where〈V〉H(1s)is the expectation value of the potential energy for the hydrogen atom

(not hydrogen-like) in its ground state. From Eq. (17.3-19)

W− 2

Z^2 e^2

2(4πε 0 a)

+

5 Ze^2

8(4πε 0 a)

− 108 .8eV+ 34 .0eV− 74 .8 eV (19.1-8)

The numerical values in Eq. (19.1-8) are forZ2. The hydrogen-like orbital

energies are proportional toZ^2 while the term due to electron–electron repulsion is

(^2) I. N. Levine,Quantum Chemistry, 5th ed., Prentice-Hall, Englewood Cliffs, NJ, 2000, p. 254ff.