90
H. Elnaggar et al.
(l 1 ,m 1 ),(l 2 ,m 2 )
Y l 1 ,m 1 σ 1 |l z s z |Y l 2 ,m 2 σ 2 c
†
l 1 ,m 1 ,σ 1
c l 2 ,m 2 ,σ 2
=
1
2
l
m=−l
m(c
†
l,m,↑ c l,m,↑ − c
†
l,m,↓ c l,m,↓ ) .
(4.11)
The second term gives
(l 1 ,m 1 ),(l 2 ,m 2 )
Y l 1 ,m 1 σ 1 |(l x s x + l y s y )|Y l 2 ,m 2 σ 2 c
†
l 1 ,m 1 ,σ 1
c l 2 ,m 2 ,σ 2
=
(l 1 ,m 1 ),(l 2 ,m 2 )
Y l 1 ,m 1 σ 1 |
1
2
(l
+ s
−
+ l
− s
+
)|Y l 2 ,m 2 σ 2 c
†
l 1 ,m 1 ,σ 1
c l 2 ,m 2 ,σ 2 (4.12)
=
1
2
l−1
m=−l
(l − m)(l + m + 1)(c
†
l,m+1,↓ c l,m,↑ + c
†
l,m,↑ c l,m+1,↓ ) .
Hence the angular part of the spin-orbit Hamiltonian finally writes
ˆ
H SO =
1
2
l
m=−l
m(c
†
l,m,↑ c l,m,↑ − c
†
l,m,↓ c l,m,↓ )
+
1
2
l−1
m=−l
(l − m)(l + m + 1)(c
†
l,m+1,↓ c l,m,↑ + c
†
l,m,↑ c l,m+1,↓ ) . (4.13)
It is clear from (4.13) that the spin-orbit interaction mixes states with different
projected orbital and spin momenta.
The Electron–Electron Coulomb Interaction
Now we undertake the simplification of the electron–electron Coulomb interaction.
This is more involved than the simplification of the spin-orbit coupling Hamiltonian.
Cowan nicely explains the details of the derivation in his book [6]. We will rely on a
combination of the derivations by Cowan [6] and Haverkort [7] in this section. This
is not a thorough derivation; it is only meant to qualitatively explain the origin of
multiplet splittings.
The first step to simplify this Hamiltonian is to perform a multipole expansion of
the term
1
ˆ
r − ˆ
r =
∞
k=0
k
m=−k Y
∗
k,m (θ
, φ
)
4π
2k+1
r
k
r
k+1
>
Y k,m (θ, φ), where r
k
< and r
k
> are,
respectively, the lesser and greater of the distances r and r
. Now we are in a position
to separate the radial and angular terms of the expression and separate the angular
variables of each electron. Using the atomic spin-orbital basis to express the matrix
elements of the angular part of the Hamiltonian one finds that
H. Elnaggar et al.
(l 1 ,m 1 ),(l 2 ,m 2 )
Y l 1 ,m 1 σ 1 |l z s z |Y l 2 ,m 2 σ 2 c
†
l 1 ,m 1 ,σ 1
c l 2 ,m 2 ,σ 2
=
1
2
l
m=−l
m(c
†
l,m,↑ c l,m,↑ − c
†
l,m,↓ c l,m,↓ ) .
(4.11)
The second term gives
(l 1 ,m 1 ),(l 2 ,m 2 )
Y l 1 ,m 1 σ 1 |(l x s x + l y s y )|Y l 2 ,m 2 σ 2 c
†
l 1 ,m 1 ,σ 1
c l 2 ,m 2 ,σ 2
=
(l 1 ,m 1 ),(l 2 ,m 2 )
Y l 1 ,m 1 σ 1 |
1
2
(l
+ s
−
+ l
− s
+
)|Y l 2 ,m 2 σ 2 c
†
l 1 ,m 1 ,σ 1
c l 2 ,m 2 ,σ 2 (4.12)
=
1
2
l−1
m=−l
(l − m)(l + m + 1)(c
†
l,m+1,↓ c l,m,↑ + c
†
l,m,↑ c l,m+1,↓ ) .
Hence the angular part of the spin-orbit Hamiltonian finally writes
ˆ
H SO =
1
2
l
m=−l
m(c
†
l,m,↑ c l,m,↑ − c
†
l,m,↓ c l,m,↓ )
+
1
2
l−1
m=−l
(l − m)(l + m + 1)(c
†
l,m+1,↓ c l,m,↑ + c
†
l,m,↑ c l,m+1,↓ ) . (4.13)
It is clear from (4.13) that the spin-orbit interaction mixes states with different
projected orbital and spin momenta.
The Electron–Electron Coulomb Interaction
Now we undertake the simplification of the electron–electron Coulomb interaction.
This is more involved than the simplification of the spin-orbit coupling Hamiltonian.
Cowan nicely explains the details of the derivation in his book [6]. We will rely on a
combination of the derivations by Cowan [6] and Haverkort [7] in this section. This
is not a thorough derivation; it is only meant to qualitatively explain the origin of
multiplet splittings.
The first step to simplify this Hamiltonian is to perform a multipole expansion of
the term
1
ˆ
r − ˆ
r =
∞
k=0
k
m=−k Y
∗
k,m (θ
, φ
)
4π
2k+1
r
k
r
k+1
>
Y k,m (θ, φ), where r
k
< and r
k
> are,
respectively, the lesser and greater of the distances r and r
. Now we are in a position
to separate the radial and angular terms of the expression and separate the angular
variables of each electron. Using the atomic spin-orbital basis to express the matrix
elements of the angular part of the Hamiltonian one finds that