86
H. Elnaggar et al.
XMχ D and NRLD signals are related to moments that are more complex, the anapole
orbital moment and other higher order moments.
4.1.3 The Many-Body Problem in Spectra Calculations
The calculation of an absorption spectrum is a formidable task: it requires the calculation of the ground state of the system, the excited states of the system, and the
interaction of the system with the electromagnetic field (X-ray beam). This means
that the theoretical approach required to calculate XAS has to be suitable for calculating the electronic structure in addition to properly considering the interaction
with the electromagnetic field. Approximations have often to be made to calculate
the absorption (or the scattering Kramers–Heisenberg) cross section and the spectroscopist therefore has to choose which theoretical approach is the most suitable for
the problem at hand.
In principle, the ground and excited states can be determined by solving the
Dirac equation which accounts for all relativistic effects and includes all possible
interactions in the Hamiltonian. This full treatment provides a relativistic, manybody, extended description of the electronic states. Unfortunately, in practice, it is
not possible to perform such a calculation as it is computationally very consuming.
In most cases one solves instead the Schrödinger equation and introduces relativistic
effects as perturbations (e.g., the spin-orbit interaction). Furthermore, one can make
use of the Born–Oppenheimer approximation to separate the electronic properties
of the system from the dynamics of the nuclei. In order to describe the electronic
part of the wave function, various theoretical approaches can be used such as (i) the
single-particle extended picture (DFT-based approaches), (ii) the many-body atomic
picture (multiplet theory), and (iii) the many-body extended picture (beyond DFT
methods).
4.1.3.1 The Single-Particle Extended Picture of Electronic States
DFT-based methods can be used to describe the electronic states using a singleparticle extended picture. Although DFT methods should formally only apply to
ground state calculations, they are often used for the calculation of excited states
probed in core level spectroscopies. DFT methods simplify the ground state wave
function of N electrons by replacing them with a fictitious, non-interacting system of
independent electrons, that have the same electronic density as the real system. The
correct charge density minimizes the total energy of the system. The Schrödinger
equation is transformed into a system of equations (called the “Kohn–Sham equations”) with an effective Hamiltonian and wave functions, which are functions
of only one space variable. This implies that DFT is essentially a single-particle
approach, although some many-particle (many-body) interactions are contained in
the exchange and correlation term of the electronic effective potential. The exact
H. Elnaggar et al.
XMχ D and NRLD signals are related to moments that are more complex, the anapole
orbital moment and other higher order moments.
4.1.3 The Many-Body Problem in Spectra Calculations
The calculation of an absorption spectrum is a formidable task: it requires the calculation of the ground state of the system, the excited states of the system, and the
interaction of the system with the electromagnetic field (X-ray beam). This means
that the theoretical approach required to calculate XAS has to be suitable for calculating the electronic structure in addition to properly considering the interaction
with the electromagnetic field. Approximations have often to be made to calculate
the absorption (or the scattering Kramers–Heisenberg) cross section and the spectroscopist therefore has to choose which theoretical approach is the most suitable for
the problem at hand.
In principle, the ground and excited states can be determined by solving the
Dirac equation which accounts for all relativistic effects and includes all possible
interactions in the Hamiltonian. This full treatment provides a relativistic, manybody, extended description of the electronic states. Unfortunately, in practice, it is
not possible to perform such a calculation as it is computationally very consuming.
In most cases one solves instead the Schrödinger equation and introduces relativistic
effects as perturbations (e.g., the spin-orbit interaction). Furthermore, one can make
use of the Born–Oppenheimer approximation to separate the electronic properties
of the system from the dynamics of the nuclei. In order to describe the electronic
part of the wave function, various theoretical approaches can be used such as (i) the
single-particle extended picture (DFT-based approaches), (ii) the many-body atomic
picture (multiplet theory), and (iii) the many-body extended picture (beyond DFT
methods).
4.1.3.1 The Single-Particle Extended Picture of Electronic States
DFT-based methods can be used to describe the electronic states using a singleparticle extended picture. Although DFT methods should formally only apply to
ground state calculations, they are often used for the calculation of excited states
probed in core level spectroscopies. DFT methods simplify the ground state wave
function of N electrons by replacing them with a fictitious, non-interacting system of
independent electrons, that have the same electronic density as the real system. The
correct charge density minimizes the total energy of the system. The Schrödinger
equation is transformed into a system of equations (called the “Kohn–Sham equations”) with an effective Hamiltonian and wave functions, which are functions
of only one space variable. This implies that DFT is essentially a single-particle
approach, although some many-particle (many-body) interactions are contained in
the exchange and correlation term of the electronic effective potential. The exact