Geoscience Reference
In-Depth Information
The standard DFT has limitations in applying
to Fe-bearing oxides and silicates in the fol-
lowing case. One-electron approximation with
the standard DFT approaches fails to describe
the interactions between localized electrons with
strongly correlated behavior, such as 3
d
electrons
of Fe in the Fe-O system. Conventional mean-
field type treatments on the exchange-correlation
potential including both LDA and GGA usually
produce (semi)metallic bands or considerable un-
derestimations of the band gap for Fe-O bonding
in oxides and silicates (Fang
et al
., 1998; Co-
coccioni & de Gironcoli, 2005; Tsuchiya
et al
.,
2006a; Metsue & Tsuchiya, 2011) and also fail to
reproduce their phase stability (Fang
et al
., 1998).
Since they also do not provide the correct crystal
field effects that break the d-orbital degeneracy,
more sophisticated classes of technique, such as
LDA
n
I
m
are the
m
-th atomic orbital occupations
with spin
σ
for the atom
I
experiencing the
Hubbard correction (iron in the present case), and
n
Iσ
=
m
n
I
m
.LDA
+
U
correctly opens the band
gaps with reasonable orbital occupations in tran-
sition metal oxides (Cococcioni & de Gironcoli,
2005; Tsuchiya
et al
., 2006a; Metsue & Tsuchiya,
2011) and successfully reproduces their phase
stability (Fang
et al
., 1998). The main problem of
applying LDA
U
to materials under pressure is
the determination of the Hubbard
U
parameter.
Tsuchiya
et al
. (2006b); Hsu
et al
. (2010, 2011),
Metsue and Tsuchiya (2011) and Metsue and
Tsuchiya (2012) computed the effective
U
in
ferropericlase (Mg
1
−
x
Fe
x
)O and (Mg
1
−
x
Fe
x
)SiO
3
(post-)perovskite nonempirically based on an
internally consistent linear response approach
(Cococcioni & de Gironcoli, 2005). Presently, the
ab initio
LDA
+
DMFT (dynamical mean-
field theory), etc., are needed to treat the many-
body effect of electrons more accurately and to
investigate geophysically important iron-bearing
systems. Among these schemes, the LDA
+
U
,LDA
+
U
+
U
technique is a quite powerful
technique to investigate the physical properties
of iron-bearing systems relevant to the Earth's
deep interior.
Elastic constant tensor for a single crystal can
be calculated with high precision by a method
described briefly below. Combined with the
ab
initio
planewave pseudopotential method, pio-
neering works by Karki
et al
. (1997a,b,c. 1998)
applied the technique to determine the elasticity
of major mantle minerals of MgO, SiO
2
,MgSiO
3
perovskite, and CaSiO
3
perovskite as a function
of pressure. Direct comparison between the
ab
initio
and seismologically inferred elastic proper-
ties of Earth's interior now spawns a new line of
geophysical research. Determination of the elas-
tic constants by means of the DFT simulation
proceeds as follows: (1) at a given pressure (or
volume) the crystal structure is first fully opti-
mized using the damped molecular dynamics or
similar algorithms; (2) the lattice is slightly de-
formed by applying a small strain. The stress in
the strained configuration is calculated, and the
values of the elastic constants follow from the
linear stress-strain relation
+
U
method (Anisimov
et al
., 1991) is for now the
most practical method for minerals, where the
correction, described using a parameter
U
,isap-
plied to the onsite Coulomb interaction between
Fe
d
states. In the LDA
+
U
method, it is replaced
by the Hubbard-like screened Coulomb interac-
tion
E
Hub
calculated using the localized basis, and
then the total energy is modified as
E
LDA
+
U
[
n
(
r
)]
+
E
LDA
[
n
(
r
)]
E
Hub
[
n
I
m
}
=
+
{
]
−
E
DC
[
{
n
Iσ
], (7.3)
where
E
DC
is the energy doubly counted in
E
LDA
and
E
Hub
. Within the rotationally invariant for-
mulation (Liechtenstein
et al
., 1995; Dudarev
et al
., 1998), the Hubbard correction to the energy
functional is greatly simplified to
}
E
Hub
[
n
I
mm
}
E
DC
[
n
Iσ
{
]
−
{
}
]
n
I
mm
−
m
2
I
U
n
I
mm
n
I
m
m
=
m
,
σ
2
I
,
σ
U
Tr
[
n
Iσ
(1
n
Iσ
)]
=
−
(7.4)
σ
ij
=
c
ijkl
ε
kl
(7.5)
