Abinitio theory of strongly correlated systems
We work on the abinitio description of strongly correlated systems using four fundamentally different approaches, namely Reduced Density Matrix Functional Theory (RDMFT), Dynamical Mean Field Theory (DMFT), SelfInteraction Corrected (SIC) Density Functionals, and LDA+U.
Recently, reduced density matrix functional theory has appeared as an
alternative approach to handle complex systems. It has shown great
potential for improving upon DFT results for finite systems
^{
[1]
}
as well as extended systems
^{
[2] ,
[3]
}.
RDMFT uses the onebody reduced density matrix (1RDM),
γ, as the basic variable.
^{12, 2111 (1975)"">[4]}.
This quantity, for the ground state, is determined through the minimization
of the total energy functional, under the constraint that
γ is ensemble Nrepresentable.
The total energy as a functional of
γ
can be expressed as (atomic units are used throughout)
where, ρ(r) the electron density is the diagonal of the 1RDM
and v(r) is the external potential. The first two terms in Eq.(1) are
the kinetic and external potential energies. The electronelectron interaction
can be cast in the last two terms, the first being the Coulomb repulsion energy
and E_{xc} the
exchangecorrelation (xc) energy functional.
In principle, Gilbert's
^{[4]
} generalization of the
HohenbergKohn theorem to the 1RDM guarantees the existence of a functional
E_{v}[γ] whose minimum yields
the exact γ and the exact groundstate energy of the systems characterized
by the external potential v(r). In practice,
however, the xc energy is an unknown functional
of the 1RDM and needs to be approximated. In the past years, a plethora of approximate
functionals, based on the orignal functional of Müller ^{
[5]
},
have been introduced. An assessment of the relative performance ^{
[6],
[7],[8] } of these functionals for a
large set of atoms and molecules reveals that the so called BBC3 ^{[9]} and PNOF0 ^{[10]} and power functionals
^{[3]} yield
results for molecular systems, with an accuracy comparable to the MP2 method
(as demostrated in Fig. 1).
 Fig. 1: Percentage deviation of the correlation energy, obtained using various functionals with in RDMFT, from the exact CCSD(T) results ^{ [11] }
As for the extended systems the real challange for any abinitio theory is the prediction of
an insulating state for the strongly correlated materials in the absence of any magnetic order
one of the most dramatic failures of the usual local/semilocal density approximations
to the exchangecorrelation (xc) functional of density functional theory (DFT) is the incorrect
prediction of a metallic ground state for the strongly correlated Mott insulators, of which
transition metal oxides (TMOs) may be considered as prototypical.
For some TMOs (NiO and MnO) spin polarized calculations do show a very small band gap
(up to 95% smaller than experiments) but only as the result of AFM ordering,
however all TMOs are found to be metallic in a spin unpolarized treatment. On the
other hand, it is well known experimentally that these materials are insulating
in nature even at elevated temperatures (much above the Néel
temperature). In our recent work we have shown that RDMFT not only pridicts accurate gaps for
conventional semiconductors, but also demonstrates insulating behavior for Motttype insulators
in the abscence of any long range magnetic order^{[3]
}.
This clearly points towards the ability of RDMFT to capture physics well beyond the reach of
most modern day groundstate methods.
Despite this success the effectiveness of RDMFT as a ground state theory
has been seriously hampered by the absence of a
technique for the determination of spectral information. The reason for this lies in the fact
that unlike DFT, RDMFT does not result in a set of single particle KohnSham like equations.
Although the value of the band gap itself can be determined from the chemical potential,
which shows a discontinuity at integer particle numbers
^{
[12],
[13]
}
, no other spectral information
is directly accessible for comparision with experiments like (inverse)photo emission spectrascopy.
However, recently this impediment has been remove, in our recent work we propose a method
for calculating the spectrum. Using this approach the spectra of
a set of TMOs are determined (see Fig. 2). In all cases we find good agreement with
experimental data^{[14]}.
 Fig. 2: Spectral density for the TMOs in presence of AFM order. Site and angular momentum projected spectral density are also presented for transition metal e_{g} and t_{2}g states and Oxygenp states. In addition XPS and BIS spectra (shifted up for clarity) are presented for comparison.
In Fig. 2 is also present the site and angular momentum projected spectral density.
The electronic gap, as expected, always occurs between lower and upper Hubbard bands
dominated
by transition metal
dstates. As a validation of our method for
the calculation of spectra we may compare the features
of the projected spectral density, and in particular the
ordering in energy of the t
_{2}g and
e
_{g} states, with well established
< span class="romani">abinitio manybody techniques such as
dynamical mean field theory and the GW method
^{[15.
16]}.
In all cases we find excellent
agreement, signaling that the method we present here
yields an accurate description of the detailed features of
the spectral density.
This opens up many future possibilities, such as the
study of High T_{C} superconductors in their underdoped Mott insulating
phase. It should also, hopefully, stimulate efforts to develop the theory
formally, including a temperature dependent and magnetic extension of RDMFT
and functionals
within RDMFT.
External coworkers on this project: N. N. Lathiotakis and N. Helbig.
To top
Recently, reduced density matrix functional theory
(RDMFT) has appeared as an
alternative approach to handle complex systems. It has shown great
potential for improving upon DFT results for finite systems
^{
[1]}
as well as extended systems
^{
[2],
[3]}.
RDMFT uses the onebody reduced density matrix (1RDM),
γ, as the basic variable
^{[4]}.
This quantity, for the ground state, is determined through the minimization
of the total energy functional, under the constraint that
γ is ensemble Nrepresentable.
The total energy as a functional of
γ
can be expressed as (atomic units are used throughout)
where, ρ(r) —the electron density— is the diagonal of the 1RDM
and v(r)
< next line I add explicit space between Eq. and (1): Eq. (1) >
is the external potential. The first two terms in Eq. (1) are
the kinetic and external potential energies. The electronelectron interaction
can be cast in the last two terms, the first being the Coulomb repulsion energy
and E_{xc} the
exchangecorrelation (xc) energy functional.
In principle, Gilbert's^{
[4]
} generalization of the
HohenbergKohn theorem to the 1RDM guarantees the existence of a functional
E_{v}[γ]
whose minimum yields the exact γ and the exact groundstate energy of the systems
characterized by the external potential v(r). In practice,
however, the xc energy is an unknown functional
of the 1RDM and needs to be approximated. In the past years, a plethora of approximate
functionals, based on the orignal functional of Müller^{
[5]},
have been introduced. An assessment of the relative performance of these
functionals^{[6,
7,
8]} for a
large set of atoms and molecules reveals that the so called BBC3^{[9]}
and PNOF0^{[10]} and power functionals^{[3]} yield
results for molecular systems, with an accuracy comparable to the MP2 method
(as demostrated in Fig. 1).
 Fig. 1: Percentage deviation of the correlation energy, obtained using various functionals within RDMFT, from the exact CCSD(T) results ^{ [11] }
As for the extended systems the real challenge for any abinitio theory is the prediction of
an insulating state for the strongly correlated materials in the absence of any magnetic order
—one of the most dramatic failures of the usual local/semilocal density approximations
to the exchangecorrelation (xc) functional of density functional theory (DFT) is the incorrect
prediction of a metallic ground state for the strongly correlated Mott insulators, of which
transition metal oxides (TMOs) may be considered as prototypical.
For some TMOs (NiO and MnO) spin polarized calculations do show a very small band gap
(up to 95% smaller than experiments) but only as the result of AFM ordering,
however all TMOs are found to be metallic in a spin unpolarized treatment. On the
other hand, it is well known experimentally that these materials are insulating
in nature even at elevated temperatures (much above the Néel
temperature). In our recent work we have shown that RDMFT not only predicts accurate gaps for
conventional semiconductors, but also demonstrates insulating behavior for Motttype insulators
in the abscence of any long range magnetic order^{[3]}.
This clearly points towards the ability of RDMFT to capture physics well beyond the reach of
most modern day groundstate methods.
Despite this success the effectiveness of RDMFT as a ground state theory
has been seriously hampered by the absence of a
technique for the determination of spectral information. The reason for this lies in the fact
that unlike DFT, RDMFT does not result in a set of single particle KohnSham like equations.
Although the value of the band gap itself can be determined from the chemical potential,
which shows a discontinuity at integer particle numbers^{[12,
13]},
no other spectral information
is directly accessible for comparision with experiments like (inverse)photo emission spectroscopy.
However, recently this impediment has been removed, in our recent work we propose a method
for calculating the spectrum. Using this approach the spectra of
a set of TMOs are determined (see Fig. 2).
In all cases we find good agreement with
experimental data^{[14]}.
 Fig. 2: Spectral density for the TMOs in presence of AFM order. Site and angular momentum projected spectral density are also presented for transition metal e_{g} and t_{2g} states and Oxygenp states. In addition XPS and BIS spectra (shifted up for clarity) are presented for comparison.
In Fig. 2 is also present the site and angular momentum projected spectral density.
The electronic gap, as expected, always occurs between lower and upper Hubbard bands
dominated
by transition metal dstates. As a validation of our method for
the calculation of spectra we may compare the features
of the projected spectral density, and in particular the
ordering in energy of the t_{2g} and
e_{g} states, with wellestablished
abinitio manybody techniques such as
dynamical mean field theory and the GW method^{[15,
16]}.
In all cases we find excellent
agreement, signaling that the method we present here
yields an accurate description of the detailed features of
the spectral density.
This opens up many future possibilities, such as the
study of High T_{C} superconductors in their underdoped Mott insulating
phase. It should also, hopefully, stimulate efforts to develop the theory
formally, including a temperature dependent and magnetic extension of RDMFT
and functionals
within RDMFT.
External coworkers on this project: N.N. Lathiotakis and N. Helbig.
References
 [1]

N. N. Lathiotakis,
N. Helbig, and
E. K. U. Gross,
Phys. Rev. A 72,
030501 (2005).
 [2]

N. N. Lathiotakis,
N. Helbig, and
E. K. U. Gross,
Phys. Rev. B 75,
195120 (2007).
 [3]

S. Sharma,
J. K. Dewhurst,
N. N. Lathiotakis,
and E. K. U.
Gross, Phys. Rev. B
78, 201103
(2008).
 [4]

T. L. Gilbert,
Phys. Rev. B 12,
2111 (1975).
 [5]

A. M. K. Müller,
Phys. Lett. 105A,
446 (1984).
 [6]

N. N. Lathiotakis,
S. Sharma,
J. K. Dewhurst,
F. G. Eich,
M. A. L. Marques,
and E. Gross,
Phys. Rev. A 79,
040501 (2009a).
 [7]

N. Lathiotakis,
S. Sharma,
N. Helbig,
J. K. Dewhurst,
M. Marques,
F. Eich,
T. Baldsiefen,
A. Zacarias, and
E. K. U. Gross,
Zeitschrift fur Physikalische Chemie.
224, 467 (2010).
 [8]

N. N. Lathiotakis,
N. Helbig,
A. Zacarias, and
E. K. U. Gross,
J. Chem. Phys. 130,
064109 (2009b).
 [9]

O. Gritsenko,
K. Pernal, and
E. J. Baerends,
J. Chem. Phys. 122,
204102 (2005).
 [10]

M. Piris, Int.
J. Quant. Chem 106, 1093
(2006).
 [11]

J. A. Pople,
M. HeadGordon,
and
K. Raghavachari,
J. Chem. Phys. p. 5968
(1987).
 [12]

N. Helbig,
N. N. Lathiotakis,
and E. K. U.
Gross, Phys. Rev. A
79, 022504
(2009).
 [13]

N. Helbig,
N. N. Lathiotakis,
M. Albrecht, and
E. K. U. Gross,
Europhys. Lett. 77,
67003 (2007).
 [14]

S. Sharma,
S. Shallcross,
J. Dewhurst, and
E. K. U. Gross,
arxiv:0912.1118.

 [15]

J. Kunes et al., Nat. Mat. 7. 198 (2008).
 [16]

C. Rödl, F. Fuchs, J. Furthmüller, and F Bechsted, Phys. Rev. B 79, 235114 (2009).
To top
The Kondo effect is one of the most intriguing manybody phenomena ^{[1]} : It arises when a local magnetic moment is weakly coupled to a sea of noninteracting conduction electrons. Such a situation can be realized for example by a transition metal impurity in a metal host where the strongly interacting electrons within the open d or fshell of the impurity give rise to the formation of a local magnetic moment. Another possible realization is a molecule with an unpaired electron coupled to metallic leads.
At low temperatures the effective coupling between the magnetic moment and the conduction electrons becomes antiferromagnetic giving rise to the formation of a singlet state where the total spin S of the system is zero. Due to the formation of this socalled Kondo singlet state the magnetic moment of the impurity or molecule is effectively screened by the interaction with the conduction electrons. Hence the behavior becomes very different from that of a free magnetic moment. For example the magnetic susceptibility as a function of temperature does not obey the CurieWeiss law anymore.
Another important consequence of the formation of the Kondo singlet state is the appearance of a sharp temperature dependent resonance in the spectral function right at the Fermi level. In the case of magnetic impurities in metal hosts the appearance of this Kondo peak leads to an increase in the resistance of the metal at low temperatures. Thus the Kondo effect solved the longstanding puzzle of the resistance minima at very low temperatures, first observed in the 1930s in Au metal samples ^{[2]} which were later traced back to the presence of magnetic impurities in the metal host ^{[3]}.
Here we study theoretically the Kondo effect that occurs in atomic or molecular devices such as single magnetic atoms or molecules on metal surfaces or in nanocontacts^{ [4]}. The challenge is to understand in detail how and under what circumstances the Kondo effect can emerge in a given system, and how the Kondo effect is altered by adding or changing the constituents. The ultimate goal is to being able to actually predict theoretically whether a given system will show the Kondo effect or not, and what type of Kondo effect.
To this end we have developed a novel approach for calculating the electronic structure and transport properties of a molcular device that explicitly takes into account the strong electronic correlations originating from the strongly interacting electrons within the open d or fshells of transition metal atoms that are ultimately responsible for the Kondo effect^{[5], [9]}. This approach combines ab initio electronic structure calculations on the level of Density Functional Theory with sophisticated many body methods such as the OneCrossing Approximation (OCA)^{[5]} and the Dynamical MeanField Theory (DMFT)^{[9]} that account for the strong correlations of the d or felectrons.
 Orbital Kondo effect in CoBz_{2} sandwich molecule ^{[7]} . (a) Geometry of CoBz_{2} sandwich molecule in Cu nanocontact. (b) Hybridization function calculated from the LDA electronic structure of the sandwich molecule in a Cu nanocontact as shown in (a) for different distances d between the tip atoms of the Cu nanocontact and the central Co atom of the sandwich molecule. (c) At low temperatures a sharp Kondo resonance appears in the spectral function of the Co 3dorbitals right at the Fermi level due to an orbital Kondo effect in the doubly degenerate E_{2}orbitals. (d) Corresponding transmission function showing the typical Fanolineshape resulting from the appearance of the Kondo resonance in the Co 3d spectral function.
Fig. 1 shows results of an LDA+OCA calculation of the electronic structure and
transport properties of CoBenzene sandwich molecule (CoBz_{2})
trapped between the tips of a Cu nanocontact ^{
[7]
}.
Our calculation predict that the strong correlations in the co
3dshell give rise to a socalled orbital Kondo effects
which stems from the orbital degeneracy of the doubly degenerate
E_{2}levels in the Co 3dshell. Here the pseudospin
labeling the two degenerate E_{2}levels is screened instead of the
normal electron spin as in the usual Kondo effect. Using our LDA+OCA methodology
we have also studied the Kondo effect of Co atoms on graphene sheets
^{
[6]
}, and the Kondo effect of metallic nanocontacts hosting magnetic
impurities in the contact region
^{
[5]}.
When a nanoscale device contains several transition metal atoms that exhibit strong electronic correlations, the direct application of the above mentioned LDA+OCA methodology^{[5]} is computationally not feasible anymore. Therefore we have recently adapted the socalled Dynamical MeanField Theory (DMFT) originally conceived to describe strong electronic correlations in bulk materials^{[8]} to the case of nanoscopic conductors such as nanocontacts and molecules attached to electrodes^{[9]}. The basic assumption of DMFT is that nonlocal electron correlations, i.e. correlations between electrons on different atoms are small and hence can be neglected. In this case it is easy to show that the problem of many interacting electrons on a lattice can be mapped on an Anderson impurity model, that is an interacting site coupled to a "bath" of noninteracting electrons. This problem can then be solved with an impurity solver, for example the above mentioned OCA impurity solver. However, since the "bath" depends on the electronic structure of each site and thus on the local correlations on each site, the problem has to be solved selfconsistently. This is the socalled DMFT selfconsistency condition.
 Figure 2: Dynamical MeanField Theory for nanoscale devices (NanoDMFT). (a) Illustration of NanoDMFT selfconsistency cycle for a molecular conductor^{[9]} and schematic drawing of a molecular conductor showing the division of the system into left (L) and right (R) electrodes and device region (D) that hosts the strongly correlated subspace (C) consisting of the dorbitals of the magnetic atoms (red circles). (b) Application of NanoDMFT method to a Ni dimer hosted in a Cu nanocontact: Spectral function calculated with NanoDMFT method near the Fermi level for different temperatures. The inset shows the geomtry of the system.
In the case of a molecular conductor (i.e. a nanocontact or a molecule connected to
bulk electrodes) we are dealing with a finite region (the molecule or nanocontact) and
hence a finite set of correlated atomic sites in contrast to a solid for which DMFT was
originally developed. Also in contrast to a crystalline solid, each of the correlated atoms
of the molecular conductor in principle has a different environment. Hence for each correlated
atom in the conductor we have to solve an individual Anderson impurity problem in each step
of the NanoDMFT selfconsistency. This is illustrated in Fig. 2a. Fig. 2b shows the application
of the NanoDMFT selfconsistency procedure to the case of a Ni dimer suspendend between
the tips of a Cu nanocontact
^{
{9]
}
. The strong correlations of the 3delectrons of the two
Ni atoms give rise to a Kondo effect signaled by a sharp Kondo peak in the 3dspectral function
at the Fermi level and the concomittant Fano lineshape in the lowbias conductance
characteristics of the nanocontact. Such a system has recently been realized experimentally
^{
[10]
}.
References
 [1]
 J. Kondo, Prog. Theor. Phys. 32, 37 (1964); A. C. Hewson, The Kondo problem to Heavy Fermions
(Cambridge University Press, Cambridge, 1993).
 [2]
 W. J. de Haas et al., Physica 3, 440 (1936).
 [3]
 M. Sarachik et al., Phys. Rev 135, 1041 (1964).
 [4]
 Madhavan et al., Science 280, 567 (1998);
P. Wahl et al., PRL 95, 166601 (2005);
Iancu et al., Nano Lett. 6, 820 (2006);
N. Néel et al., PRL 98, 016801(2007).
 [5]
 D. Jacob, K. Haule and G. Kotliar, Phys. Rev. Lett. 103, 016803 (2009).
 [6]
 D. Jacob and G. Kotliar, Phys. Rev. B 82, 085423 (2010).
 [7]
 M. Karolak, D. Jacob and A. I. Lichtenstein, Phys. Rev. Lett. 107, 146604 (2011).
 [8]
 See e.g. G. Kotliar et al., Rev. Mod. Phys. 78, 865 (2006), and references therein.
 [9]
 D. Jacob, K. Haule and G. Kotliar, Phys. Rev. B 82, 195115 (2010).
 [10]
 J. Bork et al. Nature Physics (2011).
To top
In the last years there has been a general strong interest in finding materials with speciﬁc
or even parametrisable magnetic properties. A lot of the promising candidates are strongly
correlated electronic systems which in many ways are still a challenge to be properly described
theoretically regarding their electronic groundstate properties. One important example of
such materials are transition metal monoxides (TMO), speciﬁcally MnO, FeO, CoO, and NiO.
They are chargetransfer insulators, well known for strong correlation effects associated with
the TM 3d electrons. Originating from the Andersontype superexchange, their equlibrium
magnetic structures are of the antiferromagnetic II (AFII) order, characterized by planes of
opposite magnetization which are stacked in (111)direction (see Fig. 1).
 (Fig. 1)
A firstprinciples description of these systems is rather difficult since the conventional
local spin density approximation (LSDA) fails to reproduce correctly their insulating
ground state or predicts much too small band gaps and magnetic moments. This is
associated with an unphysical selfinteraction of an electron with itself, occurring in the
Hartree term of the LSDA energy functional on account of the local approximation applied
to the exchange–correlation energy functional. This selfinteraction becomes important for
localized electrons like d electrons of TM elements in their monoxides. In the latter, the
selfinteractions push the localized electron orbitals into the valence band, usually resulting
in too strong a hybridization with the other valence electrons (see Fig. 2, upper panel).
This problem was recognized many years ago and a remedy was proposed by Perdew
and Zunger ^{[1]} to simply subtract the spurious selfinteractions
from the LSDA functional,
orbital by orbital, for all the localized states. The resulting SIC–LSDA approach treats both
localized and itinerant electrons on equal footing, leading to split d and fmanifolds and
describing the dual character of an electron (see Fig. 2, lower panel). We implemented the
selfinteraction corrections method within the the multiple scattering theory and applied it to
study electronic magnetic properties of TMO^{ [2],
[3],
[4]}.
 (Fig. 2)
 (Fig. 3)
To characterize the magnetic properties of TMO we investigated thermally induced
magnetic fluctuations, which are treated using a meanfield disordered local moment
(DLM) picture of magnetism
^{[2]}.
This involves the assignment of a local spinpolarization
axis to all lattice sites. The orientations vary slowly on the timescale of electronic motion.
To determine the magnetic properties of TMO we investigated the spin fluctuations that
characterize the paramagnetic state. In Fig. 3,
we present the results of our paramagnetic spin susceptibility calculations for NiO.
These show the paramagnetic state to be dominated by spin fluctuations with wave vector
q_{max} = (0.5; 0.5; 0.5) (in units of 2pi/a), which corresponds to the symmetry
point L in the Brillouin zone. This indicates that the system will order into the AF II structure,
that concurs with the experimentally observed ground state of this system and also calculations
at T = 0 K, where the most stable structure was determined by comparing the total energies
of different magnetic configurations. We examined the temperature dependence of the static
spin susceptibility in particular looking for a divergence which would signify that paramagnetic
states become unstable with respect to the formation of a spin density wave, characterized
by the wave vector q_{max}. Our paramagnetic susceptibility calculations indicate,
that like NiO, the other members of the TMO series have a tendency to order into the AF II
structure. The temperatures at which we predict this ordering to occur are shown in Fig. 4.
We find a good agreement with the experiment, with the exception of NiO where we
underestimated the temperature by about of a third. This suggests that some additional
physics, not at work in the other TMOs, may be of relevance to the determination of
the ordering temperature of NiO.
 (Fig. 4)
To top
References
 [1]
 P. Perdew and A. Zunger, Phys. Rev. B 23 , 5048 (1981)
 [2]
 M. Lüders, A. Ernst, M. Däne, Z. Szotek, A. Svane, D. Ködderitzsch, W. Hergert, B. L. Györffy,
and W. M. Temmerman, Phys. Rev. B 71 , 205109 (2005)
 [3]
 M. Däne, M. Lüders, A. Ernst, D. Ködderitzsch, W. M. Temmerman, Z. Szotek, and W. Hergert,
Journal of Physics: Condensed Matter. 21, 045604 (2009)
 [4]
 G. Fischer, M. Däne, A. Ernst, P. Bruno, M. Lüders, Z. Szotek, W. Temmerman, and W. Hergert,
Phys. Rev. B 80, 014408 (2009)
To top
The LSDA+U Method
The description of strongly correlated systems is a difficult task.
Conventional density functional theory (DFT) calculations, using local/semilocal
exchangecorrelation functionals, cannot really capture strong Mott
localizations
^{[1]}
; Mott insulators when
treated with these exchangecorrelationfunctionals and without any
long range magnetic order, show a metallic ground state.
Over the years many ideas were developed to treat strongly correlated
systems. Among them, most prominent within the framework of DFT are
selfinteractioncorrected local spin density approximation (SICLSDA)
^{ [2]}
hybrid functionals
^{[3]}
and LSDA+U. Out of these methods LSDA+U is particularly useful
as it is computationally less demanding and hence can easily be used
for larger systems of modern day interest. Furthermore, with LSDA+U
it is possible to systematically increase the strength of the onsite
repulsion U in order to gain insight into the effect of Coulomb
correlations. The total energy in the LSDA+U method consists of
the LSDA total energy plus a `Hubbardlike' term
^{[4]}
where
I,mσ ⟩ are localized atomic orbital at lattice
site
I, angular momentum m and spin component σ.
The occupation number matrix elements, n
^{Iσ}_{m1m3
}, are defined as:
A major problem within the LSDA+
U method is that the electronic
interactions are already partially included in the LSDA energy thus
a simple addition of the `Hubbardlike' term to the LSDA energy would
lead to double counting (DC) errors. Hence, a extra "DC term"
is subtracted from the LSDA+
U total energy to avoid this error. An
ideal DC term should subtract the mean field part from the `Hubbardlike'
term; leaving only an orbital dependent correction to the orbital
independent LSDA potential
^{[5]}
There is no rigorous way to construct a DC term, the usual approaches
include the fully localized limit (FLL), around mean field (AMF) and
interpolations between these two
^{[5]}
The FLLDC term is derived from Eq. (1) by taking the limit of fully occupied orbitals
and approximating the matrix elements by averaged values U^{I}
and J^{I}, leading to the following energy correction
One could use the bare Coulomb interaction in the evaluation of the matrix elements in Eq. (1)
and for the determination of U
^{I} and J
^{I}.
But this would totally neglect screening
effects, which are important in solid states. For the Coulomb interaction
the matrix elements can be written as a product of real prefactors
a
_{k} and Slater integrals F
_{k}^{I}:
This means, that only the Slater integrals F
_{k}^{I} with even
k are needed in Eq. (4). The Slater integrals
represent the radial part of the Coulomb interaction, which is mostly
affected by screening effects. Hence, they are replaced by "screened
Slater integrals" { S
_{0}^{I}, S
_{2}
^{I}, ..., S
_{2l}^{I}} .
These parameters are chosen in such a way that they allow for many
body (screening) effects. In practice these screened Slater integrals
are usually reexpressed in terms of only two parameters:
 U^{I} the screened averaged Coulomb on site repulsion
 J^{I} the screened exchange interaction.
Note, that the DC term in Eq. (3) is already expressed
in terms of
U^{I} and
J^{I}. If the orbital quantum number l
is two or greater, additional conditions are needed to ensure a unique
map between { S
_{0}^{I},S
_{2}^{I
},...,S
_{2l}^{I}}
and { U
^{I},J
^{I}} . The TMOs are such a case, because
the transition metals have partially filled dshells (l=2).
For isolated transition metal atoms the ratio of the Slater integral
is constant with good accuracy (between
0.62 and 0.63)
^{[6]}.
Screening should effect
F
_{4}^{I} and F
_{
2}^{I} in equal measure, hence the ration
is used to obtain the relations:
and
.
The two main approaches for determining
the values of
U^{I} and
J^{I} are:
 To chose the parameter in such a way to reproduce as many experimental
observables as possible. This approach is frequently applied to large
systems, where calculation of U^{I} and J^{I} is difficult.
 To calculate the parameters U^{I} and J^{I}
abinitio.
This brings the `first principle' character back to the LSDA+U
method. (But the construction of a DC term is still not unique)
For such a calculation two schemes are used: (1) Originally the value
of
U^{I} was chosen based on a constrained LSDA calculation
^{[7]}
and (2). A newer method is the linear response approach, which leads
to smaller values of
U^{I} compared to constrained LSDA
^{[8]}.
The LSDA+U method has proved to be able to reproduce the correct
band structure for the Mott insulators
^{[7]}
These materials have a partially filled d (or f) shell,
and a d  d (or f  f)
band gap. The gap is caused by a large onsite Coulomb
repulsion which splits the d (or f) bands in a lower (occupied)
and upper (unoccupied) Hubbard band. This mechanism is captured by
the LSDA+U method. Bands with mainly d (or f) character are
shifted down in energy if n_{mm}^{Iσ}
> 0.5 and shifted up if n_{mm}^{Iσ} < 0.5.
The magnitude of the energy shift is proportional
to the value of U. By increasing the Coulomb repulsion the d (or f)
states around the Fermi level are shifted either up or down
and a gap opens. This can be seen in Fig. (1) for the
four TMOs, which are prototypical Mott insulators.
Not only ground state properties like the gap, magnetic moment or
charge distribution are effected by the U, also excitations like
magnons or phonons change
^{[9]}.
In Fig. (1) the magnon spectrum for three TMOs is shown. The magnon energies are
suppressed with increasing U, which is due to enhanced charge localization
around the transition metal atoms
^{[14]}.
Figure 1: (Top) Opening of a gap due to onsite Coulomb repulsion in the TMOs.
(Bottom) Magnon spectrum for NiO, CoO and MnO. LSDA results are shown with red
circles, the two different values of U with green squares and blue diamonds, experiment
^{[10],
[11],
[12]}
with
^{[13]} black triangles and the J
_{1
}J
_{2} results with orange line.
References
 [1]
 A. J. MoriSánchez, P. Cohen and W. Yang, Phys. Rev. Lett. 100, 146401 (2008).
 [2]
 Z. Szotek, W. M. Temmerman, and H. Winter, Phys. Rev. B 47, 4029 (1993).
 [3]
 C. Rödl, F. Fuchs, J. Furthmüller, and F. Bechstedt, Phys. Rev. B 79, 235114 (2009).
 [4]
 A. I. Liechtenstein, V. I. Anisimov, and J. Zaanen, Phys. Rev. B 52, R5467 (1995).
 [5]
 F. Bultmark, F. Cricchio, O. Grå näs, and L. Nordström, Phys. Rev. B 80, 035121 (2009).
 [6]
 F. M. F. de Groot, J. C. Fuggle, B. T. Thole, and G. A. Sawatzky, Phys. Rev. B 42, 5459 (1990).
 [7]
 V. I. Anisimov, J. Zaanen, and O. K. Andersen, Phys. Rev. B 44, 943 (1991).
 [8]
 M. Cococcioni and S. de Gironcoli, Phys. Rev. B 71, 035105 (2005).
 [9]
 A. Floris, S. de Gironcoli, E. K. U. Gross, and M. Cococcioni, Phys. Rev. B 84, 161102 (2011).
 [10]
 M. T. Hutchings and E. J. Samuelsen, Phys. Rev. B 6, 3447 (1972).
 [11]
 S. Tomizasu, K. Itoh, JPSJ 75, 084708 (2006).
 [12]
 G. Pepy, J. Phys. Chem. Solids 35, 433 (1973).
 [13]
 G. Fischer, M. Däne, A. Ernst, P. Bruno, M. Lüders, Z. Szotek, W. Temmerman, and W. Herger, arXiv:0905.0391 (2009).
 [14]
 F. Essenberger, S. Sharma, J. K. Dewhurst,C. Bersier, F. Cricchio, L. Nordström E. K. U. Gross Phys. Rev. B (accepted 2011).
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