• Core-hole $ valence-electron Coulomb and exchange interactions: F and G.
These two-electron interactions are described by so-called Slater integrals. The
“Coulomb interactions” are the ordinary repulsions between electrons in
overlapping regions of space and are quantified in Slater parameters F. The
“exchange interaction” captures the effect of electron-electron correlation that
occurs between electrons with the same spin, and it is captured in Slater
G parameters.
• Ligand field interactions: 10 Dq, Ds, and Dt. In the ligand field description, the
effect of the ligands is to split the atomic multiplets, and this perturbation depends
on the symmetry of the ligands around the central ion. In octahedral symmetry the
ligand field is described by 10 Dq. In square planar symmetry, additional parameters Ds and Dt are required.
There are also weaker valence electron interactions to consider:
• Valence electron spin-orbit coupling: ξ 3d . This is usually on the order of 10’s of
meV. The final-state effects are usually small, but the ground-state ξ 3d can be
important for causing splitting of the available ground-state energies that leads to
temperature-dependent spectra.
• Valence-electron $ valence-electron Coulomb and exchange interactions: F dd
and G dd . There can also be electron-electron interactions in the valence shell.
The Hamiltonian for the ground-state atomic multiplets can be written as:
H gs ¼ H 3d þ H mu þ H ls
ð7:4Þ
where H 3d is the average energy of the 3d states, H mu includes all two-electron
integrals, and H 1s is the 3d spin-orbit coupling. The Hamiltonian for the final excited
state includes extra terms:
H es ¼ H 3d þ H mu þ H ls þ H c þ H c:ls þ H c:3d
ð7:5Þ
where H c is for the core-hole energy, H c:ls for the core-hole spin-orbit interaction,
and H c:3d for the core-valence Coulomb and exchange interactions.
In Fig. 7.9, we illustrate the sequential turning on of different terms in the
Hamiltonian. We use a hypothetical Ti
4+ L-edge because there are no d-electrons
to complicate the ground state. In the absence of any additional interactions, the 2p
L-edge is described as a 2p
6 3d
0
! 2p
5 3d
1 transition, and this yields a single line:
1 S!
1 P
ð7:5Þ
7.6.2 Core-Hole Spin-Orbit Splitting
ξ 2p . A 2p
5 configuration can be treated as a single hole with L ¼ 1 and S ¼ 1/2. The
coupling of these angular momenta leads to two possible terms: J ¼ L + S ¼ 3/2 and
7.6 Charge-Transfer Multiplet Theory
175
These two-electron interactions are described by so-called Slater integrals. The
“Coulomb interactions” are the ordinary repulsions between electrons in
overlapping regions of space and are quantified in Slater parameters F. The
“exchange interaction” captures the effect of electron-electron correlation that
occurs between electrons with the same spin, and it is captured in Slater
G parameters.
• Ligand field interactions: 10 Dq, Ds, and Dt. In the ligand field description, the
effect of the ligands is to split the atomic multiplets, and this perturbation depends
on the symmetry of the ligands around the central ion. In octahedral symmetry the
ligand field is described by 10 Dq. In square planar symmetry, additional parameters Ds and Dt are required.
There are also weaker valence electron interactions to consider:
• Valence electron spin-orbit coupling: ξ 3d . This is usually on the order of 10’s of
meV. The final-state effects are usually small, but the ground-state ξ 3d can be
important for causing splitting of the available ground-state energies that leads to
temperature-dependent spectra.
• Valence-electron $ valence-electron Coulomb and exchange interactions: F dd
and G dd . There can also be electron-electron interactions in the valence shell.
The Hamiltonian for the ground-state atomic multiplets can be written as:
H gs ¼ H 3d þ H mu þ H ls
ð7:4Þ
where H 3d is the average energy of the 3d states, H mu includes all two-electron
integrals, and H 1s is the 3d spin-orbit coupling. The Hamiltonian for the final excited
state includes extra terms:
H es ¼ H 3d þ H mu þ H ls þ H c þ H c:ls þ H c:3d
ð7:5Þ
where H c is for the core-hole energy, H c:ls for the core-hole spin-orbit interaction,
and H c:3d for the core-valence Coulomb and exchange interactions.
In Fig. 7.9, we illustrate the sequential turning on of different terms in the
Hamiltonian. We use a hypothetical Ti
4+ L-edge because there are no d-electrons
to complicate the ground state. In the absence of any additional interactions, the 2p
L-edge is described as a 2p
6 3d
0
! 2p
5 3d
1 transition, and this yields a single line:
1 S!
1 P
ð7:5Þ
7.6.2 Core-Hole Spin-Orbit Splitting
ξ 2p . A 2p
5 configuration can be treated as a single hole with L ¼ 1 and S ¼ 1/2. The
coupling of these angular momenta leads to two possible terms: J ¼ L + S ¼ 3/2 and
7.6 Charge-Transfer Multiplet Theory
175