4 X-ray Dichroisms in Spherical Tensor and Green’s Function Formalism
87
analytical expression of this term is not known. This means that DFT can only be
applied in an approximate form for example using the local density approximation
(LDA) or the generalized gradient approximation (GGA)].
One can determine the ground state wave function by solving the Kohn–Sham
equations which are constructed from the single-particle wave functions. The ground
state wave function is therefore formed by a single Slater determinant. This is an
important point because it limits the ability to treat a many-body response of the
system described through a linear combination of Slater determinants. To illustrate
this let us take an example with two electrons. Coupling s = 1/2 to s = 1/2 yields
S = 1 or 0. This corresponds to four |S, Ms wave functions: |S = 1, Ms = 1 ,
|S = 1, Ms = −1 |S = 1, Ms = 0 , and |S = 0, Ms = 0 . These functions need
to meet the property of being anti-symmetric under particle exchange and it can
be shown that only |S = 1, Ms = 1 , |S = 1, Ms = −1 can each be expressed as
a single Slater determinant. However, the |S = 1, Ms = 0 and |S = 0, Ms = 0
states can only be expressed as the combination of two Slater determinants and thus
cannot be calculated in DFT.
One can group the various DFT-based methods according to their characteristics:
– Cluster or periodic: The Kohn–Sham equations can be solved either for a cluster
centred around the absorbing atom (direct or real space methods) or starting from
a unit cell of the crystal (or a multiple unit cell, which is called supercell) in order
to take advantage of the 3D periodicity (reciprocal space methods).
– Self-consistency or not: The Kohn–Sham equations can be solved without or
(preferably) with self-consistency, i.e., using an iterative cycle where two successive steps are mixed until a convergence criterion is reached to determine the
charge density.
– Type of basis functions used to expand the orbital solutions of the Kohn–
Sham equations: either localized functions [linear combination of atomic orbitals
(LCAO), linear muffin-tin orbitals (LMTO)], or delocalized functions [plane waves
(PW), full-potential linearized augmented plane waves (FLAPW)].
– Approximation made on the shape of the electronic potential: For example in
LMTO or multiple scattering theory, the potential is approximated to be spherically
symmetric in the atoms, and constant between them (muffin-tin). In full-potential
methods [FLAPW, or projector augmented wave (PAW)-pseudopotentials], no
approximation is made, which is generally preferable, even though it makes the
calculations more consuming.
4.1.3.2 The Many-Body Atomic Picture of Electronic States
A Simple Introduction to the Many-Body Atomic Picture
Let us consider as an example a Cr
3+ ion in an octahedral (O h ) environment. Here
the solid is reduced to an atom embedded in a mean field known as the crystal field
(CF) that mimics the effect of the inter-atomic interactions. The atomic electronic
configuration is 1s
2 2s
2 2 p
6 3s
2 3 p
6 3d
3 . The degeneracy of the Cr 3d levels is lifted
87
analytical expression of this term is not known. This means that DFT can only be
applied in an approximate form for example using the local density approximation
(LDA) or the generalized gradient approximation (GGA)].
One can determine the ground state wave function by solving the Kohn–Sham
equations which are constructed from the single-particle wave functions. The ground
state wave function is therefore formed by a single Slater determinant. This is an
important point because it limits the ability to treat a many-body response of the
system described through a linear combination of Slater determinants. To illustrate
this let us take an example with two electrons. Coupling s = 1/2 to s = 1/2 yields
S = 1 or 0. This corresponds to four |S, Ms wave functions: |S = 1, Ms = 1 ,
|S = 1, Ms = −1 |S = 1, Ms = 0 , and |S = 0, Ms = 0 . These functions need
to meet the property of being anti-symmetric under particle exchange and it can
be shown that only |S = 1, Ms = 1 , |S = 1, Ms = −1 can each be expressed as
a single Slater determinant. However, the |S = 1, Ms = 0 and |S = 0, Ms = 0
states can only be expressed as the combination of two Slater determinants and thus
cannot be calculated in DFT.
One can group the various DFT-based methods according to their characteristics:
– Cluster or periodic: The Kohn–Sham equations can be solved either for a cluster
centred around the absorbing atom (direct or real space methods) or starting from
a unit cell of the crystal (or a multiple unit cell, which is called supercell) in order
to take advantage of the 3D periodicity (reciprocal space methods).
– Self-consistency or not: The Kohn–Sham equations can be solved without or
(preferably) with self-consistency, i.e., using an iterative cycle where two successive steps are mixed until a convergence criterion is reached to determine the
charge density.
– Type of basis functions used to expand the orbital solutions of the Kohn–
Sham equations: either localized functions [linear combination of atomic orbitals
(LCAO), linear muffin-tin orbitals (LMTO)], or delocalized functions [plane waves
(PW), full-potential linearized augmented plane waves (FLAPW)].
– Approximation made on the shape of the electronic potential: For example in
LMTO or multiple scattering theory, the potential is approximated to be spherically
symmetric in the atoms, and constant between them (muffin-tin). In full-potential
methods [FLAPW, or projector augmented wave (PAW)-pseudopotentials], no
approximation is made, which is generally preferable, even though it makes the
calculations more consuming.
4.1.3.2 The Many-Body Atomic Picture of Electronic States
A Simple Introduction to the Many-Body Atomic Picture
Let us consider as an example a Cr
3+ ion in an octahedral (O h ) environment. Here
the solid is reduced to an atom embedded in a mean field known as the crystal field
(CF) that mimics the effect of the inter-atomic interactions. The atomic electronic
configuration is 1s
2 2s
2 2 p
6 3s
2 3 p
6 3d
3 . The degeneracy of the Cr 3d levels is lifted