Theory Department
Max Planck Institute of Microstructure Physics

Theory Department > Research > Ab-initio theory of strongly-correlated systems

Ab-initio theory of strongly correlated systems

We work on the ab-initio description of strongly correlated systems using four fundamentally different approaches, namely Reduced Density Matrix Functional Theory (RDMFT), Dynamical Mean Field Theory (DMFT), Self-Interaction Corrected (SIC) Density Functionals, and LDA+U.

Research topics

RDMFT S. Sharma, K. Dewhurst, T. Baldsiefen, F. Eich, A. Zacarias

NanoDMFT D. Jacob, S. Frank, S. Motahari

SIC A. Ernst

LDA+U F. Essenberger, S. Sharma

Reduced Density Matrix Functional Theory (RDMFT)

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 one-body reduced density matrix (1-RDM), 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 N-representable. The total energy as a functional of γ can be expressed as (atomic units are used throughout)

(1)

where, ρ(r) -the electron density- is the diagonal of the 1-RDM and v(r) is the external potential. The first two terms in Eq.(1) are the kinetic and external potential energies. The electron-electron interaction can be cast in the last two terms, the first being the Coulomb repulsion energy and E_xc the exchange-correlation (xc) energy functional. In principle, Gilbert's ^[4] generalization of the Hohenberg-Kohn theorem to the 1-RDM guarantees the existence of a functional E_v[γ] whose minimum yields the exact γ and the exact ground-state energy of the systems characterized by the external potential v(r). In practice, however, the xc energy is an unknown functional of the 1-RDM 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 ab-initio 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/semi-local density approximations to the exchange-correlation (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 Mott-type 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 ground-state 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 Kohn-Sham 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 t₂g states and Oxygen-p 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 d-states. 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 t₂g and e_g states, with well established < span class="romani">ab-initio many-body 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 under-doped 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 co-workers on this project: N. N. Lathiotakis and N. Helbig.

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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 one-body reduced density matrix (1-RDM), 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 N-representable. The total energy as a functional of γ can be expressed as (atomic units are used throughout)

(1)

where, ρ(r) —the electron density— is the diagonal of the 1-RDM 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 electron-electron interaction can be cast in the last two terms, the first being the Coulomb repulsion energy and E_xc the exchange-correlation (xc) energy functional. In principle, Gilbert's^[4] generalization of the Hohenberg-Kohn theorem to the 1-RDM guarantees the existence of a functional E_v[γ] whose minimum yields the exact γ and the exact ground-state energy of the systems characterized by the external potential v(r). In practice, however, the xc energy is an unknown functional of the 1-RDM 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 ab-initio 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/semi-local density approximations to the exchange-correlation (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 Mott-type 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 ground-state 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 Kohn-Sham 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 Oxygen-p 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 d-states. 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 well-established ab-initio many-body 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.

External co-workers 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. Head-Gordon, 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).

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NanoDMFT: Dynamical Mean-Field Theory for strongly correlated electrons at the nanoscale

Kondo effect in molecular devices from first principles

The Kondo effect is one of the most intriguing many-body phenomena ^[1] : It arises when a local magnetic moment is weakly coupled to a sea of non-interacting 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 f-shell 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 so-called 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 Curie-Weiss 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 long-standing 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 f-shells 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 One-Crossing Approximation (OCA)^[5] and the Dynamical Mean-Field Theory (DMFT)^[9] that account for the strong correlations of the d- or f-electrons.

: Orbital Kondo effect in CoBz₂ sandwich molecule ^[7] . (a) Geometry of CoBz₂ 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 3d-orbitals right at the Fermi level due to an orbital Kondo effect in the doubly degenerate E₂-orbitals. (d) Corresponding transmission function showing the typical Fano-lineshape 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 Co-Benzene sandwich molecule (CoBz₂) trapped between the tips of a Cu nanocontact ^[7]. Our calculation predict that the strong correlations in the co 3d-shell give rise to a so-called orbital Kondo effects which stems from the orbital degeneracy of the doubly degenerate E₂-levels in the Co 3d-shell. Here the pseudospin labeling the two degenerate E₂-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].

Dynamical mean-field theory for nanoscale conductors

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 so-called Dynamical Mean-Field 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 non-local 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 non-interacting 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 self-consistently. This is the so-called DMFT self-consistency condition.

: Figure 2: Dynamical Mean-Field Theory for nanoscale devices (NanoDMFT). (a) Illustration of NanoDMFT self-consistency 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 d-orbitals 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 self-consistency. This is illustrated in Fig. 2a. Fig. 2b shows the application of the NanoDMFT self-consistency procedure to the case of a Ni dimer suspendend between the tips of a Cu nanocontact ^{9] . The strong correlations of the 3d-electrons of the two Ni atoms give rise to a Kondo effect signaled by a sharp Kondo peak in the 3d-spectral function at the Fermi level and the concomittant Fano lineshape in the low-bias 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).

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Exchange interactions in magnetic oxides

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 ground-state properties. One important example of such materials are transition metal monoxides (TMO), speciﬁcally MnO, FeO, CoO, and NiO. They are charge-transfer insulators, well known for strong correlation effects associated with the TM 3d electrons. Originating from the Anderson-type 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 first-principles 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 self-interaction 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 self-interaction becomes important for localized electrons like d electrons of TM elements in their monoxides. In the latter, the self-interactions 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 self-interactions 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 f-manifolds and describing the dual character of an electron (see Fig. 2, lower panel). We implemented the self-interaction 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 mean-field disordered local moment (DLM) picture of magnetism ^[2]. This involves the assignment of a local spin-polarization axis to all lattice sites. The orientations vary slowly on the time-scale 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)

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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)

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The LSDA+U Method

The description of strongly correlated systems is a difficult task. Conventional density functional theory (DFT) calculations, using local/semi-local exchange-correlation functionals, cannot really capture strong Mott localizations ^[1] ; Mott insulators when treated with these exchange-correlation-functionals 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 frame-work of DFT are self-interaction-corrected local spin density approximation (SIC-LSDA) ^[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 on-site 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 `Hubbard-like' term ^[4]

(1)

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:

(2)

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 `Hubbard-like' 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 `Hubbard-like' 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 FLL-DC 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

(3)

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:

(4)

(5)

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" { S₀^I, S₂ ^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 re-expressed 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 { S₀^I,S₂^I,...,S_2l^I} and { U^I,J^I} . The TMOs are such a case, because the transition metals have partially filled d-shells (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 F₄^I and F₂^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 ab-initio. 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 on-site 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 on-site 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 J₁J₂ results with orange line.

References

[1]: A. J. Mori-Sánchez, P. Cohen and W. Yang, Phys. Rev. Lett. 100, 146401 (2008).

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