5 Enabling Materials By Dimensionality: From 0D to 3D Carbon-Based. . .
179
χ(r, t, r
, t
) = χ KS (r, t, r
, t
)
dτ
d
3 x
dτ
d
3 x
χ KS (r, t, x, τ )
δ(τ − τ )
|x − x |
+ f xc (x, τ, x
, τ
)
χ(x
, τ
, r
, t
)
(5.20)
where f xc (r, t, r , t ) =
δv xc (r,t)
δn(r ,t )
n gs (r,t)
is the energy-dependent exchangecorrelation kernel. The independent particle response function χ KS is calculated
usually by solving the standard Kohn-Sham equations. As for static DFT, the
time-dependent exchange-correlation potential is unknown. Thus, calculations of
χ(r, t, r , t ) usually rely on the so-called adiabatic local-density approximation
(ALDA) in which the time dependence of the functional is neglected [110].
However, in systems where excitonic effects are expected to have a strong influence
on spectral features due to an ineffective electronic screening, e.g. in insulators such
as diamond, the use of a bootstrap kernel [125] that includes effects beyond the
RPA is necessary.
Assuming translational invariance, the ELF can be computed inserting Eq. 5.13
into Eq. (5.15). The inversion procedure can be cumbersome for large basis sets and
large k-point grids. Thus, wherever possible the most viable option is to assess the
microscopic dielectric matrix by inverting only the head of the matrix, which means
to neglect the off-diagonal elements ( G,G (q, W ), G, G = 0) for all q [115]. These
off-diagonal terms include the fluctuations of the fields on atomic scale, called the
local field effects (LFE). Nevertheless, for highly inhomogeneous or strongly locally
polarizable systems, such as in the case of diamond and graphite, strong microscopic
local fields can exist, and thus LFE can play a significant role in the description
of the dielectric properties [126], particularly at small wavelengths, to the point of
invalidating even qualitative results. This is the case, for example, of our 3D carbonbased materials, and, thus, we will include LFE in our analysis. With the inclusion
of the LFE, one can show that the dielectric function in reciprocal space is [127]:
−1
G,G (q, W ) = δ G,G + ν
s
G,G (q)χ G,G (q, W )
(5.21)
where ν s
G,G (q) =
4πe 2
|q+G||q+G | is the Fourier transform of the Coulomb potential
and χ G,G (q, W ) is the microscopic polarizability.
ELFs and Related Observables of Diamond and Graphite
While for computational details we refer to Ref. [64], using LR-TDDFT we
calculated the ELFs of diamond and graphite according to Eq. 5.15. In the top
panels of Fig. 5.29, we report the ELFs of diamond (left) and of graphite (right)
in comparison to the Drude-Lorentz approach (black lines).
179
χ(r, t, r
, t
) = χ KS (r, t, r
, t
)
dτ
d
3 x
dτ
d
3 x
χ KS (r, t, x, τ )
δ(τ − τ )
|x − x |
+ f xc (x, τ, x
, τ
)
χ(x
, τ
, r
, t
)
(5.20)
where f xc (r, t, r , t ) =
δv xc (r,t)
δn(r ,t )
n gs (r,t)
is the energy-dependent exchangecorrelation kernel. The independent particle response function χ KS is calculated
usually by solving the standard Kohn-Sham equations. As for static DFT, the
time-dependent exchange-correlation potential is unknown. Thus, calculations of
χ(r, t, r , t ) usually rely on the so-called adiabatic local-density approximation
(ALDA) in which the time dependence of the functional is neglected [110].
However, in systems where excitonic effects are expected to have a strong influence
on spectral features due to an ineffective electronic screening, e.g. in insulators such
as diamond, the use of a bootstrap kernel [125] that includes effects beyond the
RPA is necessary.
Assuming translational invariance, the ELF can be computed inserting Eq. 5.13
into Eq. (5.15). The inversion procedure can be cumbersome for large basis sets and
large k-point grids. Thus, wherever possible the most viable option is to assess the
microscopic dielectric matrix by inverting only the head of the matrix, which means
to neglect the off-diagonal elements ( G,G (q, W ), G, G = 0) for all q [115]. These
off-diagonal terms include the fluctuations of the fields on atomic scale, called the
local field effects (LFE). Nevertheless, for highly inhomogeneous or strongly locally
polarizable systems, such as in the case of diamond and graphite, strong microscopic
local fields can exist, and thus LFE can play a significant role in the description
of the dielectric properties [126], particularly at small wavelengths, to the point of
invalidating even qualitative results. This is the case, for example, of our 3D carbonbased materials, and, thus, we will include LFE in our analysis. With the inclusion
of the LFE, one can show that the dielectric function in reciprocal space is [127]:
−1
G,G (q, W ) = δ G,G + ν
s
G,G (q)χ G,G (q, W )
(5.21)
where ν s
G,G (q) =
4πe 2
|q+G||q+G | is the Fourier transform of the Coulomb potential
and χ G,G (q, W ) is the microscopic polarizability.
ELFs and Related Observables of Diamond and Graphite
While for computational details we refer to Ref. [64], using LR-TDDFT we
calculated the ELFs of diamond and graphite according to Eq. 5.15. In the top
panels of Fig. 5.29, we report the ELFs of diamond (left) and of graphite (right)
in comparison to the Drude-Lorentz approach (black lines).