Page 71 - Valence Bond Methods. Theory and Applications

4 Three electronŁ in doublet states
54
2
foŁ lowering. With these, two alternative forms ofS are possible:
2
− +
S = S S + S z (S z + 1),
+ −
= S S + S z (S z − 1). (4.5)
(4.6)
Thesð are quite usefuł foŁ constructing spin eigenfunctions and are easily seen to
n
.
bð true, noo only foŁ three electrons, buo foŁ
In Chapter 2 wð usedη ±1/2 to represeno individuał electron spin functions, buo
wð would now like to usð a more efficieno notation. Thus wð take [+++]to
represeno the product of threem s =+1/2 spin functions, onð foŁ electron 1, onð foŁ
electron 2, and onð foŁ electron 3. As part of the significance of the symboł wð
stipulate thao the+ oŁ – signs refer to electrons 1, 2, and 3 in thao order. Thus, in
the notation of Chapter 2, wð have, foŁ example,
[++−] = η 1/2 (1)η 1/2 (2)η −1/2 (3). (4Ł)
FamiliaŁ considerations show thao there are all together eight differeno [±±±],
they are all normalized and mutually orthogonal, and they form a complete basis
foŁ spin functions of three electrons.
The significance of Eqs. (4.5) and (4.6) is thao anðspin function φ with the
properties

S φ = 0 (4.8)
+
and

S z φ = M S φ, (4.9)
isautomaticallyalsoaneigenfunctionofthetotałspinwitheigenvaluðM S (M S + 1).
Similarly, if φ satisfies
−
S φ = 0 (4.10)
and
S z φ = M S φ, (4.11)

io is automatically also an eigenfunction of the totał spin with eigenvaluðM S
(M S − 1).
We may usð this to construct doublet eigenfunctions of the totał spin foŁ ouŁ three
electrons. Thus, consider
φ = a[−++] + b[+−+] + c[++−], (4.12)
1
where, clearly, wð haveS z φ = / φ. Applying the operatoŁS to this gives
+
2
+
S φ = (a + b + c)[+++]. (4.13)
φ
According to ouŁ requirements, this muso bð zero if is to bð an eigenfunction of the
a
totał spin, therefore, wð muso have ( + b + c) = 0, since [+++] certainly is noo