70
P. Krüger
I q (ω) ≈
m,a,b=±
M
a∗
ilm,q M
b
ilm,q ρ il a m+q,l b m+q (ω + c ) .
(3.16)
We see that the absorption intensity is a weighted sum of a few partial DOS components with angular momentum l a = l ± 1. For high enough symmetry, the interference terms a = b vanish, leaving only the diagonal, usual partial DOS. In some
cases, e.g. for the linear dichroism at the sulphur L 2,3 -edges in MoS 2 , it was found
that interference between p → s and p → d transitions is non-negligible [6]. For the
special case of s-wave core states (K , L 1 , M 1 edges) where l = m = 0, the selection
rules (3.11) give l
= 1, m = q, such that the absorption spectrum for q-polarized
light is directly proportional to the p q -DOS (where q = 0, ±1 or q = x, y, z).
In this section, we have seen that in the independent particle and dipole approximation, the X-ray absorption spectra are approximately given by a weighted sum
of partial DOS with momenta l ± 1, m + q. The weighting factors are local transition matrix elements and reflect the light polarization and orbital symmetry. As
a consequence, XAS can be used to probe the unoccupied DOS of the material in
a site and orbital-resolved way, which gives detailed insight into the local bonding
properties [7].
3.5 Absorption Spectra in Linear Response TDDFT
3.5.1 Time-Dependent Density Functional Theory
DFT is a ground state theory whose application to excited states is ill-founded.
However, a large class of excitations can be computed using the time-dependent
version of DFT. Time-dependent DFT (TDDFT) is the generalization of standard
DFT to time-dependent external potentials V ext (r, t). It was pioneered by Zangwill
and Soven in 1980 [8], who developed a linear response theory for optical absorption
spectroscopy of atoms using a time-dependent version of the LDA. In 1984, Runge
and Gross [9] generalized the Hohenberg–Kohn theorems of DFT to the case of timedependent systems, thus putting TDDFT on a rigorous theoretical ground. TDDFT
has been applied to XAS of solids for the first time in 1998 by Schwitalla and
Ebert [10] and to molecules in 2003 by Stener et al. [11].
The problem at hand is to find the time-dependent electron density n(r, t) of an
interacting electron system subject to a time-dependent external field. In TDDFT,
the exact time-dependent electron density n(r, t) can, in principle, be found from the
knowledge of the external field, the universal energy functional E[n(r, t)] and the
initial density n(r, 0). Linear response functions, including absorption coefficients,
can be expressed as integrals over the electron density change induced by a timedependent external field. Thus, if the exact functional E[n(r, t)] were known, TDDFT
would allow to obtain exact absorption spectra. As in the case of time-independent
P. Krüger
I q (ω) ≈
m,a,b=±
M
a∗
ilm,q M
b
ilm,q ρ il a m+q,l b m+q (ω + c ) .
(3.16)
We see that the absorption intensity is a weighted sum of a few partial DOS components with angular momentum l a = l ± 1. For high enough symmetry, the interference terms a = b vanish, leaving only the diagonal, usual partial DOS. In some
cases, e.g. for the linear dichroism at the sulphur L 2,3 -edges in MoS 2 , it was found
that interference between p → s and p → d transitions is non-negligible [6]. For the
special case of s-wave core states (K , L 1 , M 1 edges) where l = m = 0, the selection
rules (3.11) give l
= 1, m = q, such that the absorption spectrum for q-polarized
light is directly proportional to the p q -DOS (where q = 0, ±1 or q = x, y, z).
In this section, we have seen that in the independent particle and dipole approximation, the X-ray absorption spectra are approximately given by a weighted sum
of partial DOS with momenta l ± 1, m + q. The weighting factors are local transition matrix elements and reflect the light polarization and orbital symmetry. As
a consequence, XAS can be used to probe the unoccupied DOS of the material in
a site and orbital-resolved way, which gives detailed insight into the local bonding
properties [7].
3.5 Absorption Spectra in Linear Response TDDFT
3.5.1 Time-Dependent Density Functional Theory
DFT is a ground state theory whose application to excited states is ill-founded.
However, a large class of excitations can be computed using the time-dependent
version of DFT. Time-dependent DFT (TDDFT) is the generalization of standard
DFT to time-dependent external potentials V ext (r, t). It was pioneered by Zangwill
and Soven in 1980 [8], who developed a linear response theory for optical absorption
spectroscopy of atoms using a time-dependent version of the LDA. In 1984, Runge
and Gross [9] generalized the Hohenberg–Kohn theorems of DFT to the case of timedependent systems, thus putting TDDFT on a rigorous theoretical ground. TDDFT
has been applied to XAS of solids for the first time in 1998 by Schwitalla and
Ebert [10] and to molecules in 2003 by Stener et al. [11].
The problem at hand is to find the time-dependent electron density n(r, t) of an
interacting electron system subject to a time-dependent external field. In TDDFT,
the exact time-dependent electron density n(r, t) can, in principle, be found from the
knowledge of the external field, the universal energy functional E[n(r, t)] and the
initial density n(r, 0). Linear response functions, including absorption coefficients,
can be expressed as integrals over the electron density change induced by a timedependent external field. Thus, if the exact functional E[n(r, t)] were known, TDDFT
would allow to obtain exact absorption spectra. As in the case of time-independent