Physical Chemistry , 1st ed.

Spin is unaffected by any other property or observable of the electron, and the

spin component of a one-electron wavefunction is separable from the spatial

part of the wavefunction. Like the three parts of the hydrogen atom’s electronic

wavefunction, the spin function multiplies the rest of. So for example, the

complete wavefunctions for an electron in a hydrogen atom are

Rn,

,mm

for an electron having msof^12 . A similar wavefunction, in terms of , can be

written for an electron having ms^12 .

12.3 The Helium Atom

In the previous chapter, it was shown how quantum mechanics provides an ex-

act, analytic solution to the Schrödinger equation when applied to the hydro-

gen atom. Even the existence of spin, discussed in the last section, does not al-

ter this solution (it only adds a little more complexity to the solution, a

complexity we will not consider further here). The next largest atom is the he-

lium atom, He. It has a nuclear charge of 2, and it has two electrons about

the nucleus. The helium atom is illustrated in Figure 12.3, along with some of

the coordinates used to describe the positions of the subatomic particles.

Implicit in the following discussion is the idea that both electrons of helium

will occupy the lowest possible energy state.

In order to properly write the complete form of the Schrödinger equation

for helium, it is important to understand the sources of the kinetic and po-

tential energy in the atom. Assuming only electronic motion with respect to a

motionless nucleus, kinetic energy comes from the motion of the two elec-

trons. It is assumed that the kinetic energy part of the Hamiltonian operator

is the same for the two electrons and that the total kinetic energy is the sum

of the two individual parts. To simplify the Hamiltonian, we will use the sym-

bol ^2 , called del-squared, to indicate the three-dimensional second derivative

operator:

^2 





x

2

 2 



y

2

 2 



z

2

 2 (12.3)

This definition makes the Schrödinger equation look less complicated.^2 is

also called the Laplacian operator.It is important to remember, however, that

del-squared represents a sum of three separate derivatives. The kinetic energy

part of the Hamiltonian can be written as



2





2

^21 

2





2

^22

where ^21 is the three-dimensional second derivative for electron 1, and ^22 is

the three-dimensional second derivative for electron 2.

The potential energy of the helium atom has three parts, all coulombic in

nature: there is an attraction between electron 1 and the nucleus, an attraction

between electron 2 and the nucleus, and a repulsionbetween electron 1 and

electron 2 (since they are both negatively charged). Each part depends on the

distance between the particles involved; the distances are labeled r 1 ,r 2 , and r 12

as illustrated in Figure 12.3. Respectively, the potential energy part of the

Hamiltonian is thus

Vˆ

4

2



e

0

2

r 1



4

2



e

0

2

r 2



4

e



2

0 r 12



374 CHAPTER 12 Atoms and Molecules

r 12

r 2

r 1

2 +

e 1

e 2

Figure 12.3 Definitions of the radial coordi-
nates for the helium atom.