3 Electronic Structure Theory for X-Ray Absorption and Photoemission Spectroscopy
71
DFT, however, the exact functional is unknown. Moreover, finding good approximate
functionals is even more difficult in TDDFT than in standard DFT.
3.5.2 Linear Response Theory
Here we shall outline the theory of absorption spectroscopy in linear response following Zangwill and Soven [8]. We consider an interacting electron system as described
by the unperturbed Hamiltonian H in (3.4), and try to find its response to a timedependent applied field ϕ ext (r, t) such as the electromagnetic field of an X-ray beam.
The perturbation Hamiltonian is written as
H
(t) =
ϕ ext (r, t)n(r, t)dr ,
(3.17)
where n(r, t) is the electron density. It differs from the density of the unperturbed
system n
0
(r) by the induced density
δn(r, t) = n(r, t) − n
0
(r) .
(3.18)
The fundamental assumption of linear response theory is that the response of the
system, δn, is proportional to the applied field ϕ ext , i.e.
δn(r, t) =
dr
dt
χ(r, r
, t − t
)ϕ ext (r
, t
) ,
(3.19)
where χ , the response function, is an intrinsic property of the unperturbed system.
In the frequency domain, this relation reads
δn(r, ω) =
dr
χ(r, r
, ω)ϕ ext (r
, ω) .
(3.20)
It can be shown that χ is given by the retarded density–density Green’s function
χ(r, r
, t − t
) = −iθ(t − t
)0|[ ˆ
n(r, t), ˆ
n(r
, t
)]|0 ,
(3.21)
where ˆ
n(t) = e
i Ht
ˆ
ne
−i Ht is the density operator in Heisenberg representation, |0 is
the exact ground state of H with energy E 0 , [ , ] denotes the commutator and θ(x) is
the Heaviside step function [θ(x) = 1 for x > 0 and θ(x) = 0 for x < 0]. By inserting a complete set of excited states
m |mm| and performing a time-frequency
Fourier transformation, we obtain the following exact (‘Lehmann’) representation:
χ(r, r
, ω) =
m
0| ˆ
n(r)|mm| ˆ
n(r
)|0
ω − E m + E 0 + iη
−
m
0| ˆ
n(r
)|mm| ˆ
n(r)|0
ω + E m − E 0 + iη
, (3.22)
71
DFT, however, the exact functional is unknown. Moreover, finding good approximate
functionals is even more difficult in TDDFT than in standard DFT.
3.5.2 Linear Response Theory
Here we shall outline the theory of absorption spectroscopy in linear response following Zangwill and Soven [8]. We consider an interacting electron system as described
by the unperturbed Hamiltonian H in (3.4), and try to find its response to a timedependent applied field ϕ ext (r, t) such as the electromagnetic field of an X-ray beam.
The perturbation Hamiltonian is written as
H
(t) =
ϕ ext (r, t)n(r, t)dr ,
(3.17)
where n(r, t) is the electron density. It differs from the density of the unperturbed
system n
0
(r) by the induced density
δn(r, t) = n(r, t) − n
0
(r) .
(3.18)
The fundamental assumption of linear response theory is that the response of the
system, δn, is proportional to the applied field ϕ ext , i.e.
δn(r, t) =
dr
dt
χ(r, r
, t − t
)ϕ ext (r
, t
) ,
(3.19)
where χ , the response function, is an intrinsic property of the unperturbed system.
In the frequency domain, this relation reads
δn(r, ω) =
dr
χ(r, r
, ω)ϕ ext (r
, ω) .
(3.20)
It can be shown that χ is given by the retarded density–density Green’s function
χ(r, r
, t − t
) = −iθ(t − t
)0|[ ˆ
n(r, t), ˆ
n(r
, t
)]|0 ,
(3.21)
where ˆ
n(t) = e
i Ht
ˆ
ne
−i Ht is the density operator in Heisenberg representation, |0 is
the exact ground state of H with energy E 0 , [ , ] denotes the commutator and θ(x) is
the Heaviside step function [θ(x) = 1 for x > 0 and θ(x) = 0 for x < 0]. By inserting a complete set of excited states
m |mm| and performing a time-frequency
Fourier transformation, we obtain the following exact (‘Lehmann’) representation:
χ(r, r
, ω) =
m
0| ˆ
n(r)|mm| ˆ
n(r
)|0
ω − E m + E 0 + iη
−
m
0| ˆ
n(r
)|mm| ˆ
n(r)|0
ω + E m − E 0 + iη
, (3.22)