Physical Chemistry , 1st ed.

the order in which orbitals are filled by electrons. Perhaps the most common

is shown in Figure 12.6. This ordering of orbitals, and the idea that each sub-

shell is completely filled before electrons occupy the next orbital, is strictly ap-

plicable to the electron configurations of 85 of the first 103 elements. (There

is little or no experimental verification of the electron configurations of ele-

ments 104 or larger to date.) Figure 12.7 shows the relationship between the

aufbau principle and the structure of the periodic table. Table 12.1 lists the

electron configurations of the elements in their lowest electronic states.

12.6 Perturbation Theory

In a previous section, we presumed that the wavefunctions of multielectron

atoms can be approximated as products of hydrogen-like orbitals:

Z H,1H,2H,3H,Z (12.13)

where Zis the wavefunction for an atom having a nuclear charge ofZand

H,1,....H,Zare the hydrogen-like wavefunctions for each of the Zelec-

trons. Generally speaking, this is a very useful qualitativedescription of the

electrons in larger atoms. However, we did see that in the case of He, it is not

a good system for making a quantitativeprediction of the total electronic en-

ergy of the system. As noted, there is no known exact solution to the differen-

tial equation that is the Schrödinger equation for the helium atom; it has no

analyticsolution. There is no known simple (or complicated, for that matter!)

expression for that we can substitute into the Schrödinger equation as given

in equation 12.5 and have it satisfied so that an eigenvalue Eis produced.

386 CHAPTER 12 Atoms and Molecules

7 s 7 p

6 s 6 p 6 d 6 f 6 g

5 s 5 p 5 d 5 f 5 g

4 s 4 p 4 d 4 f

3 s 3 p 3 d

2 s 2 p

1 s

and so on

6 d

5 d

**

*

4 d

5 f

4 f

**

*

3 d

7 s

6 s

5 s

4 s

3 s

2 s

7 p

6 p

5 p

4 p

3 p

2 p

1 s

Figure 12.6 A convenient way to

remember the order of filling of the

subshells in most atoms (1–85).

Simply follow the order of subshells

crossed by the arrows.

Figure 12.7 The aufbau principle rationalizes

the structure of the periodic table. Compare the

order of filling of the subshells in Figure 12.6 with

the labels in the periodic table here. (Note where

the 4fand 5fsubshells are filled.)