3 Electronic Structure Theory for X-Ray Absorption and Photoemission Spectroscopy
73
So the full response function χ can be calculated from free response function χ 0
and the kernel K , by iteration or inversion. Equivalently one can calculate the local
potential directly by iteration of ϕ loc = ϕ ext + K χ 0 ϕ loc [8]. The problem is that the
exchange–correlation part of the kernel
K XC (rt, r
t
) =
δV XC (rt)
δn(r t )
(3.28)
is not known exactly. The adiabatic approximation consists in using a static exchange–
correlation potential, which may be taken from standard time-independent DFT.
In this case, K XC = [δV XC (r)/δn(r
)]δ(t − t
) such that ϕ ind (t) changes instantaneously with δn(t). As a result, K (r, r
, ω) is frequency independent and dynamical
screening is neglected. X-ray fields correspond to fast oscillations, so neglecting
dynamical effects is questionable.
3.5.3 Absorption Spectra
The optical absorption coefficient is essentially the imaginary part of the response
function χ as we shall show now. We consider an electromagnetic wave given
by E(r, t) = eE 0 e
iq·r−iωt , where e is the light polarization vector, not to be confused with the electric charge e. The induced electrical polarization is P(r, t) =
−eδn(r, t)r and so the change in energy density is −E · P = eE · rδn. In the dipole
approximation, E(r, t) ≈ eE 0 e
−iωt and the perturbation in (3.17) is given by
ϕ ext (r, ω) = eE 0 e · r .
(3.29)
The total induced dipole moment is
µ(ω) = −e
rδn(r, ω)dr = −e
2 E 0
rχ(r, r
, ω)e · r
drdr
,
(3.30)
and the absorbed energy is Re{E · dµ/dt} or equivalently Im{ωE · µ(ω)}. The
absorption coefficient σ (ω) is the absorbed energy divided by E
2
0 , which yields
σ (ω) = −4παω
drdr
e · r Imχ(r, r
, ω)e · r
.
(3.31)
This expression of the absorption coefficient is fully equivalent to (3.3). In the independent particle approximation, we put χ → χ 0 and obtain from (3.23)
σ 0 (ω) = 4π
2
αω
hp
||φ p |e · r|φ h |
2
δ(ω + h − p )
(3.32)
in agreement with (3.9).
73
So the full response function χ can be calculated from free response function χ 0
and the kernel K , by iteration or inversion. Equivalently one can calculate the local
potential directly by iteration of ϕ loc = ϕ ext + K χ 0 ϕ loc [8]. The problem is that the
exchange–correlation part of the kernel
K XC (rt, r
t
) =
δV XC (rt)
δn(r t )
(3.28)
is not known exactly. The adiabatic approximation consists in using a static exchange–
correlation potential, which may be taken from standard time-independent DFT.
In this case, K XC = [δV XC (r)/δn(r
)]δ(t − t
) such that ϕ ind (t) changes instantaneously with δn(t). As a result, K (r, r
, ω) is frequency independent and dynamical
screening is neglected. X-ray fields correspond to fast oscillations, so neglecting
dynamical effects is questionable.
3.5.3 Absorption Spectra
The optical absorption coefficient is essentially the imaginary part of the response
function χ as we shall show now. We consider an electromagnetic wave given
by E(r, t) = eE 0 e
iq·r−iωt , where e is the light polarization vector, not to be confused with the electric charge e. The induced electrical polarization is P(r, t) =
−eδn(r, t)r and so the change in energy density is −E · P = eE · rδn. In the dipole
approximation, E(r, t) ≈ eE 0 e
−iωt and the perturbation in (3.17) is given by
ϕ ext (r, ω) = eE 0 e · r .
(3.29)
The total induced dipole moment is
µ(ω) = −e
rδn(r, ω)dr = −e
2 E 0
rχ(r, r
, ω)e · r
drdr
,
(3.30)
and the absorbed energy is Re{E · dµ/dt} or equivalently Im{ωE · µ(ω)}. The
absorption coefficient σ (ω) is the absorbed energy divided by E
2
0 , which yields
σ (ω) = −4παω
drdr
e · r Imχ(r, r
, ω)e · r
.
(3.31)
This expression of the absorption coefficient is fully equivalent to (3.3). In the independent particle approximation, we put χ → χ 0 and obtain from (3.23)
σ 0 (ω) = 4π
2
αω
hp
||φ p |e · r|φ h |
2
δ(ω + h − p )
(3.32)
in agreement with (3.9).