Physical Chemistry Third Edition

18.5 Angular Momentum in the Helium Atom 779

The three values ofMLcorrespond toL1 with no states left over, so that onlyPterms

occur. Each symmetric spin factor combines with each one of the three antisymmetric space

factors to give the nine states of the^3 Pterm, and the antisymmetric spin factor combines

with each one of the symmetric space factors to give the three states of the^1 Pterm.

The states in the previous example can be counted up more simply by listing all the

possible combinations ofm 1 ,m 2 ,ms 1 , andms 2 that can occur, and then marking off

the states for each possible term, starting with the largest possible value ofL. For each

value ofLwe then work through the possible values ofS, starting with the largest.

These entries are shown in Table 18.1. Some of the actual states are linear combinations

of the wave functions corresponding to the entries in this list, but the number of them is

correctly counted. For example, for each set of values ofm 1 andm 2 there is an entry for

ms 1  1 /2 andms 2 − 1 /2 and another entry forms 1 − 1 /2 andms 2  1 /2. The

actual states correspond to the symmetric spin factor and the antisymmetric spin factor,

but the two states are correctly counted. It does not matter which of the two lines in the

table we assign to the triplet and to the singlet if we are just counting up the states.

The process of assigning the terms used in Example 18.3 can be summarized as

follows: The largest value ofMLis identified, which must be equal to the largest value

ofL. The largest value ofMSoccurring with this value ofMLis identified, which must

be equal to the largest value ofSoccurring with the largest value ofL. The states with

the appropriate values ofMLranging from+Lto−Lfor the largest value ofLare

found, as are the states with the appropriate values ofMSfrom+Sto−S. For this

example, the maximum value ofLis 1 and the maximum value ofSoccurring with it

is 1, corresponding to a^3 Pterm. For each value ofML, three states withMSequal to

1, 0, and−1 are found. The nine states with the appropriate values ofMLandMSfor

this term are marked in the table. After the states for the largest value ofLare found,

the largest remaining value ofMLis identified and the next value ofLis assigned to be

equal to this value. The largest remaining value ofMSoccurring with this value ofML

is identified and this value is set equal to the value ofSthat corresponds to this value

ofL. The states with the appropriate values ofMLandMSare found. In the example,

the largest remaining value ofMLis 1 and the only value ofMSoccurring with it is 0,

Table 18.1 Terms for the (1s)(2p) Configuration of the He Atom for

Example 18.4

Values Terms

m 1 m 2 ms 1 ms 2 ML MS^3 P^1 P

01 + 1 / 2 + 1 /21 1 x

01 + 1 / 2 − 1 /21 0 x

01 − 1 / 2 + 1 /21 0 x

01 − 1 / 2 − 1 / 21 −1x

00 + 1 / 2 + 1 /20 1 x

00 + 1 / 2 − 1 /20 0 x

00 − 1 / 2 + 1 /20 0 x

00 − 1 / 2 − 1 / 20 −1x

0 − 1 + 1 / 2 + 1 / 2 −11x

0 − 1 + 1 / 2 − 1 / 2 −10x

0 − 1 − 1 / 2 + 1 / 2 − 10 x

0 − 1 − 1 / 2 − 1 / 2 − 1 −1x