Physical Chemistry Third Edition

18.3 The Ground State of the Helium Atom in Zero Order 771

orthogonality of the spin functions and gives unity for the first and fourth terms by the

normalization of the spin functions. We obtain

(

Probability of finding

electron 1 ind^3 r 1

)



1

2

|ψ 100 ∗(1)|^2 [α∗(1)α(1)+β∗(1)β(1)]d^3 r 1 (18.3-11)

We interpret this to mean that the probability of finding the particle in the volume

elementd^3 r 1 with spin up is the factor multiplying the termα∗(1)α(1):

(

Probability of finding electron 1

ind^3 r 1 with spin up

)



1

2

|ψ 100 ∗(1)|^2 d^3 r 1 (18.3-12)

The probability of finding the particle in this volume elementd^3 r 1 with spin down

is the factor multiplying the termβ∗(1)β(1). It has the same value as the result in

Eq. (18.3-12). The probability of finding either spin irrespective of location is equal

to 1/2. For much of our work with probability densities we can ignore the spins of the

particles.

The energy eigenvalue for our zero-order ground-state wave function is the sum of

two hydrogen-like orbital energies forn1:

E(0) 1 s 1 sE 1 (HL)+E 1 (HL)2(− 13 .60 eV)Z^2 (18.3-13)

where we use a double 1ssubscript to denote that both electrons occupy 1sspace

orbitals. For heliumZ2, so that

E

(0)

1 s 1 s−^108 .8 eV (18.3-14)

This approximate energy eigenvalue is seriously in error, since the experimental value

is− 79 .0 eV. We discuss better approximations in the next chapter.

PROBLEMS

Section 18.3: The Ground State of the Helium Atom

18.5 a.Calculate the expectation value of the kinetic energy of
the electrons in a helium atom in the ground state in the
zero-order approximation.
b.Calculate the expectation value of the potential energy
of a helium atom in the ground state in the zero-order
approximation.
c.Calculate the expectation value of the energy of a
helium atom in the ground state in the zero-order
approximation.
d.Specify which of the values calculated in parts a–c
belong to the predictable case and which belong to the
statistical case as defined in Chapter 16.

18.6 a.Calculate the expectation value of the kinetic energy of
the electrons in a helium atom in the excited state

corresponding to the configuration (1s)(2s)inthe

zero-order approximation.

b. Calculate the expectation value of the potential energy

of the electrons in a helium atom in the excited state

corresponding to the configuration (1s)(2s)inthe

zero-order approximation.

18.7 a. Use the identity

̂S^2 ̂S^21 +̂S 22 +2(̂Sx 1 ̂Sx 2 +̂Sy 1 ̂Sy 2 +̂Sz 1 ̂Sz 2 )

and the facts^1

̂Sxαh ̄

2

β ̂Syα

ih ̄

2

β

̂Sxβ ̄h

2

α ̂Syβ

−ih ̄

2

α

(^1) I. N. Levine,Quantum Chemistry, 6th ed., Prentice-Hall, Upper Saddle River, NJ, 2000, p. 301.