92
H. Elnaggar et al.
G
(k)
=
min(|2l 1 ,2l 2 |)
k=0
R n 1 ,l 1 (r )R n 2 ,l 2 (r
)|
r
k
<
r
k+1
>
|R n 2 ,l 2 (r )R n 1 ,l 1 (r
) .
(4.18)
The F
(k) and G
(k) are called Slater integrals. The magnitude of the multiplet splitting depends on the magnitude of the Slater integrals. For an atomic calculation
(corresponding to the case of a free ion in spherical symmetry) radial integrals are
calculated self-consistently using a Hartree–Fock model, with values typically of the
order of a few eV (not more than tens of eV). In the point group symmetry, where
the absorber is considered with its environment, these values are reduced empirically
in order to take into account the effect of the chemical bond which delocalizes the
electrons. This reduction factor is an adjustable parameter (typically, 60–80% for a
iono-covalent bond, 100% being the ionic limit case of a free ion).
The Crystal Field Hamiltonian
Let us now extend our theoretical framework to include the effect of the CF potential
on the absorbing ion. This is done by considering the N nearest neighbours as point
charges (Z i e) at positions R i . The electrostatic potential due to these point charges
at position r, V C F (r), is expressed as
V C F (r) =
N
i=1
Z i e
2
|r − R i |
.
(4.19)
A multipole expansion of the potential can be used to expand (4.19), leading to the
expression
V C F (r, θ, φ) = e
2
∞
k=0
r
k
k
m=−k
C k,m (θ, φ)Q k,m ,
(4.20)
with Q k,m ≡
4π
2k+1
1/2 N
i=1 Z i (
1
R i
)
k+1 Y
∗
k,m (θ i , φ i ). Y k,m is the spherical harmonic,
C k,m is the renormalized spherical harmonic, and
∗ is the complex conjugate. Note
here that one assumes that the radial extent of the 3d orbitals is smaller than the
distance between the absorbing ion and its first neighbours (i.e., r R i ). The CF
Hamiltonian can be developed using the single-particle basis (atomic spin-orbitals
as discussed before for the Coulomb interaction). Separating the radial and angular
parts leads to
ˆ
H C F =
τ 1 ,τ 2
k,m
A k,m Y 1 ,m 1 σ 1 |C k,m |Y 2 ,m 2 σ 2 c
†
τ 1
c τ 2 .
(4.21)
Here, the A k,m combine all the radial parts of (4.20). They are related to the usual
CF parameters (10Dq, Ds, Dt, …) which are usually not known precisely. The CF
parameters (or A k,m ) are either fitted parameters or taken from experiments (optical
absorption, electron paramagnetic resonance, …). It is important to warn the reader
against the temptation to fit the calculation with an unreasonable number of CF
H. Elnaggar et al.
G
(k)
=
min(|2l 1 ,2l 2 |)
k=0
R n 1 ,l 1 (r )R n 2 ,l 2 (r
)|
r
k
<
r
k+1
>
|R n 2 ,l 2 (r )R n 1 ,l 1 (r
) .
(4.18)
The F
(k) and G
(k) are called Slater integrals. The magnitude of the multiplet splitting depends on the magnitude of the Slater integrals. For an atomic calculation
(corresponding to the case of a free ion in spherical symmetry) radial integrals are
calculated self-consistently using a Hartree–Fock model, with values typically of the
order of a few eV (not more than tens of eV). In the point group symmetry, where
the absorber is considered with its environment, these values are reduced empirically
in order to take into account the effect of the chemical bond which delocalizes the
electrons. This reduction factor is an adjustable parameter (typically, 60–80% for a
iono-covalent bond, 100% being the ionic limit case of a free ion).
The Crystal Field Hamiltonian
Let us now extend our theoretical framework to include the effect of the CF potential
on the absorbing ion. This is done by considering the N nearest neighbours as point
charges (Z i e) at positions R i . The electrostatic potential due to these point charges
at position r, V C F (r), is expressed as
V C F (r) =
N
i=1
Z i e
2
|r − R i |
.
(4.19)
A multipole expansion of the potential can be used to expand (4.19), leading to the
expression
V C F (r, θ, φ) = e
2
∞
k=0
r
k
k
m=−k
C k,m (θ, φ)Q k,m ,
(4.20)
with Q k,m ≡
4π
2k+1
1/2 N
i=1 Z i (
1
R i
)
k+1 Y
∗
k,m (θ i , φ i ). Y k,m is the spherical harmonic,
C k,m is the renormalized spherical harmonic, and
∗ is the complex conjugate. Note
here that one assumes that the radial extent of the 3d orbitals is smaller than the
distance between the absorbing ion and its first neighbours (i.e., r R i ). The CF
Hamiltonian can be developed using the single-particle basis (atomic spin-orbitals
as discussed before for the Coulomb interaction). Separating the radial and angular
parts leads to
ˆ
H C F =
τ 1 ,τ 2
k,m
A k,m Y 1 ,m 1 σ 1 |C k,m |Y 2 ,m 2 σ 2 c
†
τ 1
c τ 2 .
(4.21)
Here, the A k,m combine all the radial parts of (4.20). They are related to the usual
CF parameters (10Dq, Ds, Dt, …) which are usually not known precisely. The CF
parameters (or A k,m ) are either fitted parameters or taken from experiments (optical
absorption, electron paramagnetic resonance, …). It is important to warn the reader
against the temptation to fit the calculation with an unreasonable number of CF