Physical Chemistry Third Edition

776 18 The Electronic States of Atoms. II. The Zero-Order Approximation for Multielectron Atoms

z

z z

z

z

s 2

s 1

z

s 2

I 2 I 2

I (^1) I
1
S
L
L
J
S
L
s 1
s 1
z
S
s 2
I 1
I 2
(a) (b) (c)
(d) (e)
(f) (g)
Figure 18.3 Examples of Vector Addition of Angular Momenta.(a) Two electron spins:
ms 1  1 /2,ms 2 − 1 /2,Ms0,S0. (b) Two electron spins:ms 1  1 /2,ms 2 − 1 /2,
Ms0,S1. (c) Two electron spins:ms 1  1 /2,ms 2  1 /2,Ms1,S1. (d) Two
pelectrons’ orbital angular momenta:l 1 1,ml1,l 2 1,m 2 −1,ML0,L1.
(e) Twopelectrons’ orbital angular momenta:l 1 1,m 1 0,l 2 1,m 2 0,ML0,
L2. (f) Twopelectrons’ orbital angular momenta:m 1 0,l 1 1,m 2 0,l 2 1,ML0,
L0. (g) Total orbital and spin angular momenta:L1,ML1,S 1 /2,MS 1 /2,
MJ 3 /2,J 3 /2.
For two electrons, we have the operators
̂L^2 (̂lx 1 +̂lx 2 )^2 +(̂ly 1 +̂ly 2 )^2 +(̂lz 1 +̂lz 2 )^2 (18.5-10)
̂Lẑlz 1 +̂lz 2 (18.5-11)
̂S^2 (̂sx 1 +̂sx 2 )^2 +(̂sy 1 +̂sy 2 )^2 +(̂sz 1 +̂sz 2 )^2 (18.5-12)
̂Sẑsz 1 +̂sz 2 (18.5-13)