88
H. Elnaggar et al.
due to the CF and the 3d orbitals are split into two groups: the e g orbitals pointing
towards the ligands and the t 2g orbitals pointing between the ligands. The number of
possible electronic states is given by the number of allowed arrangements of three
electrons into ten spin orbitals, i.e., C
3
10 = 120 microstates. The energy separation
between these states arises due to the combined effect of: (i) CF splitting, (ii) electronic repulsions, and (iii) spin-orbit coupling, i.e., the multiplet effects. Electrons
occupying closed shells do not actually contribute to the energy splitting of the electronic levels; there is only one way to completely fill a shell giving a single average
energy of the configuration.
All these multiplet states can be further grouped in so-called term symbols (or
spectroscopic terms) according to their energy, spin and orbital moments. The relative
energy positions of these spectroscopic terms for a 3d
n transition metal ion in O h CF
(and neglecting 3d spin-orbit coupling) were calculated by Tanabe and Sugano and
are available in several references (e.g., [3, 4]). Similar diagrams are available in [5]
for symmetries lower than O h , such as trigonal or tetragonal. The relative energies
of the electronic states depend on the CF parameters as well as the Racah parameters
that relate to the electronic repulsions. The determination of the spectroscopic terms
becomes very complex when the spin-orbit coupling and Zeeman terms are included
in the Hamiltonian and/or if lower symmetries are considered. LFM theory takes
these effects into account and has been realized in several computer codes.
Key Ideas of Ligand-Field Multiplet Theory
Atomic multiplet theory, crystal field multiplet theory, and LFM theory (sometimes
collectively referred to as the multiplet theory) are based on concepts that were
developed in atomic physics and make use of group theory. One has to solve the
Schrödinger equation for the ion with its N electrons in a given configuration
ˆ
H |g = E|g ,
(4.4)
where ˆ
H is the Hamiltonian of the system for the chosen configuration, E and |g are
the eigenvalue and eigenstate, respectively. The different eigenstates are functions
of N electrons, hence they are called many-body (or multi-electronic) states. The
Hamiltonian is expressed as
ˆ
H = ˆ
T + ˆ
V + ˆ
V ee + ˆ
H SO + ˆ
H C F ,
(4.5)
where ˆ
T is the kinetic energy of the electrons, ˆ
V the Coulomb attraction between electrons and the nucleus, ˆ
V ee the electron–electron Coulomb interaction, ˆ
H SO the spinorbit coupling interaction, and ˆ
H C F the CF Hamiltonian, which takes into account
the local environment of the absorbing atom.
These interactions will now be expressed in second quantization formalism. In
this notation, any operator can be expressed in terms of creation (c
†
τ ) and annihilation
(c τ ) operators. The operator c
†
τ creates a state characterized by the quantum numbers
τ (for example, if we choose to express the states as spin-orbitals, τ will be the set
H. Elnaggar et al.
due to the CF and the 3d orbitals are split into two groups: the e g orbitals pointing
towards the ligands and the t 2g orbitals pointing between the ligands. The number of
possible electronic states is given by the number of allowed arrangements of three
electrons into ten spin orbitals, i.e., C
3
10 = 120 microstates. The energy separation
between these states arises due to the combined effect of: (i) CF splitting, (ii) electronic repulsions, and (iii) spin-orbit coupling, i.e., the multiplet effects. Electrons
occupying closed shells do not actually contribute to the energy splitting of the electronic levels; there is only one way to completely fill a shell giving a single average
energy of the configuration.
All these multiplet states can be further grouped in so-called term symbols (or
spectroscopic terms) according to their energy, spin and orbital moments. The relative
energy positions of these spectroscopic terms for a 3d
n transition metal ion in O h CF
(and neglecting 3d spin-orbit coupling) were calculated by Tanabe and Sugano and
are available in several references (e.g., [3, 4]). Similar diagrams are available in [5]
for symmetries lower than O h , such as trigonal or tetragonal. The relative energies
of the electronic states depend on the CF parameters as well as the Racah parameters
that relate to the electronic repulsions. The determination of the spectroscopic terms
becomes very complex when the spin-orbit coupling and Zeeman terms are included
in the Hamiltonian and/or if lower symmetries are considered. LFM theory takes
these effects into account and has been realized in several computer codes.
Key Ideas of Ligand-Field Multiplet Theory
Atomic multiplet theory, crystal field multiplet theory, and LFM theory (sometimes
collectively referred to as the multiplet theory) are based on concepts that were
developed in atomic physics and make use of group theory. One has to solve the
Schrödinger equation for the ion with its N electrons in a given configuration
ˆ
H |g = E|g ,
(4.4)
where ˆ
H is the Hamiltonian of the system for the chosen configuration, E and |g are
the eigenvalue and eigenstate, respectively. The different eigenstates are functions
of N electrons, hence they are called many-body (or multi-electronic) states. The
Hamiltonian is expressed as
ˆ
H = ˆ
T + ˆ
V + ˆ
V ee + ˆ
H SO + ˆ
H C F ,
(4.5)
where ˆ
T is the kinetic energy of the electrons, ˆ
V the Coulomb attraction between electrons and the nucleus, ˆ
V ee the electron–electron Coulomb interaction, ˆ
H SO the spinorbit coupling interaction, and ˆ
H C F the CF Hamiltonian, which takes into account
the local environment of the absorbing atom.
These interactions will now be expressed in second quantization formalism. In
this notation, any operator can be expressed in terms of creation (c
†
τ ) and annihilation
(c τ ) operators. The operator c
†
τ creates a state characterized by the quantum numbers
τ (for example, if we choose to express the states as spin-orbitals, τ will be the set