780 18 The Electronic States of Atoms. II. The Zero-Order Approximation for Multielectron Atoms
corresponding to a^1 Pterm. If there were more states remaining, the process would be
continued until all states have been assigned to terms.Exercise 18.4
By explicit operation witĥLẑLz 1 +̂Lz 2 , show that the eigenvalues of the two space factors
in Eq. (18.5-23) and Eq. (18.5-24) both equalh ̄, corresponding toML1.If we wish to include the values ofJ, we can determine the possible values from
the values ofMJ, the quantum number for itszcomponent. SinceJis the vector sum
ofLandS, thezcomponents add algebraically:
̂JẑLz+̂Sz (18.5-25)Therefore,
MJML+MS (18.5-26)The possible values ofJcan be deduced by using the rule that for each value ofJ, the
values ofMJrange from+Jto−J. Since the largest value ofMJequals the largest
value ofMLplus the largest value ofMS, the largest value ofJis
JmaxL+S (18.5-27)The smallest value ofJis
Jmin|L−S| (18.5-28)
Jmust be non-negative.EXAMPLE18.4
Assign theMJvalues to the 12 states of Table 18.1. Show that the following terms occur:(^1) P
1 ,
(^3) P
2 ,
(^3) P
1 ,
(^3) P
0
Solution
We adopt the notation that the left factor in each product is for electron 1 and the other factor
is for electron 2. The terms are counted by assigning the proper set of eigenvalues for each
term.
ML MS MJ term
(ψsψp 1 +ψp 1 ψs)(αβ−βα)101^1 P 1
(ψsψp 1 −ψp 1 ψs)αα 1123 P 2
(ψsψp 1 −ψp 1 ψs)(αβ+βα)101^3 P 2
(ψsψp 1 −ψp 1 ψs)ββ 1 − 103 P 0
(ψsψp 0 +ψp 0 ψs)(αβ−βα)000^1 P 1
(ψsψp 0 −ψp 0 ψs)αα 0113 P 1
(ψsψp 0 −ψp 0 ψs)(αβ+βα)000^3 P 1
(ψsψp 0 −ψp 0 ψs)ββ 0 − 1 − 1 3 P 1
(ψsψp− 1 +ψp− 1 ψs)(αβ−βα) − 10 − 1 1 P 1
(ψsψp− 1 −ψp− 1 ψs)αα − 1103 P 2
(ψsψp− 1 −ψp− 1 ψs)(αβ+βα) − 10 − 1 3 P 2
(ψsψp− 1 −ψp− 1 ψs)ββ − 1 − 1 − 2 3 P 2