3 Electronic Structure Theory for X-Ray Absorption and Photoemission Spectroscopy
77
3.7.2 GW Approximation
The GW approximation was invented by Hedin in 1965 [17] and owes its name from
the form of this self-energy, which is = i GW , i.e. the product (or convolution) of
Green’s function (G) and the screened Coulomb interaction (W ). The latter is given
by [18]
W (r, r
, ω) =
e
2
4ππ 0
dr
−1
(r, r
, ω)
|r − r |
,
(3.38)
where (r, r
, ω)
−1 is the inverse dielectric function. This expression may be understood by analogy to the electrostatic energy between two electrons in a polarizable
medium, which is given by e
2
/[4ππ 0 r |r − r
|], where r is the relative dielectric permittivity. The dielectric function (r, r
, ω) generalizes r to inhomogeneous media
and dynamic screening effects. The GW approximation is most often used in a nonself-consistent way, i.e. as = i G 0 W 0 with the free Green’s function G 0 instead
of the full Green’s function G. The GW approximation has been very successful
for correcting band energies of weakly correlated systems. In particular, bandgaps
of semi-conductors are very well reproduced in the GW approximation, while the
values obtained in DFT (except for DFT-HF hybrid functionals) are systematically
too small [19].
3.7.3 Bethe–Salpeter Equation
At this point, we briefly switch back to the problem of absorption spectroscopy. Since
light absorption creates an electron–hole pair, absorption spectra are described with
an electron–hole (i.e. a two-particle) Green’s function G eh . If the excited electron
and the hole do not interact, G eh is just the product of the one-particle removal (hole)
Green’s function G h and the addition (electron) Green’s function G e . Electron–hole
interaction leads to coupling of these two Green’s functions, which may be expressed
in a Dyson-type equation as [20]
G eh (1, 2; 1
, 2
) = G e (1, 1
)G h (2, 2
)
(3.39)
+
G e (1, 3)G h (2, 4)K (3, 4; 5, 6)G eh (5, 6; 1
, 2
)d3d4d5d6 ,
where 1 stands for all coordinates of particle 1 and K is the interaction kernel.
In the Bethe–Salpeter equation (BSE) approach, (3.39) is solved with K given by
the screened Coulomb interaction in (3.38) and the bare exchange interaction. The
electron and hole Green’s functions, G e and G h , are commonly computed in the
GW approximation. The BSE approach is arguably the most accurate first-principles
method for absorption spectroscopy in solids, but it is computationally very demanding. It was first applied to X-ray spectra by Shirley in 1998 [21]. It accounts well
77
3.7.2 GW Approximation
The GW approximation was invented by Hedin in 1965 [17] and owes its name from
the form of this self-energy, which is = i GW , i.e. the product (or convolution) of
Green’s function (G) and the screened Coulomb interaction (W ). The latter is given
by [18]
W (r, r
, ω) =
e
2
4ππ 0
dr
−1
(r, r
, ω)
|r − r |
,
(3.38)
where (r, r
, ω)
−1 is the inverse dielectric function. This expression may be understood by analogy to the electrostatic energy between two electrons in a polarizable
medium, which is given by e
2
/[4ππ 0 r |r − r
|], where r is the relative dielectric permittivity. The dielectric function (r, r
, ω) generalizes r to inhomogeneous media
and dynamic screening effects. The GW approximation is most often used in a nonself-consistent way, i.e. as = i G 0 W 0 with the free Green’s function G 0 instead
of the full Green’s function G. The GW approximation has been very successful
for correcting band energies of weakly correlated systems. In particular, bandgaps
of semi-conductors are very well reproduced in the GW approximation, while the
values obtained in DFT (except for DFT-HF hybrid functionals) are systematically
too small [19].
3.7.3 Bethe–Salpeter Equation
At this point, we briefly switch back to the problem of absorption spectroscopy. Since
light absorption creates an electron–hole pair, absorption spectra are described with
an electron–hole (i.e. a two-particle) Green’s function G eh . If the excited electron
and the hole do not interact, G eh is just the product of the one-particle removal (hole)
Green’s function G h and the addition (electron) Green’s function G e . Electron–hole
interaction leads to coupling of these two Green’s functions, which may be expressed
in a Dyson-type equation as [20]
G eh (1, 2; 1
, 2
) = G e (1, 1
)G h (2, 2
)
(3.39)
+
G e (1, 3)G h (2, 4)K (3, 4; 5, 6)G eh (5, 6; 1
, 2
)d3d4d5d6 ,
where 1 stands for all coordinates of particle 1 and K is the interaction kernel.
In the Bethe–Salpeter equation (BSE) approach, (3.39) is solved with K given by
the screened Coulomb interaction in (3.38) and the bare exchange interaction. The
electron and hole Green’s functions, G e and G h , are commonly computed in the
GW approximation. The BSE approach is arguably the most accurate first-principles
method for absorption spectroscopy in solids, but it is computationally very demanding. It was first applied to X-ray spectra by Shirley in 1998 [21]. It accounts well