Ab initio electronic structure calculations are dominated by density functional theory (DFT). They have revolutionized that field. The subject is reviewed in this volume by Jones (see Chap. 8), one of the pioneers in the field. But DFT is a ground-state theory and therefore it is of little surprise that it fails, in particular for strongly correlated systems, when low-energy excitations are calculated from it. The failure is inherent and independent of any approximations which are made for the potential in the Kohn-Sham equation. This is seen by considering the simplest possible system of strongly correlated electrons, i.e., two electrons in two orbitals. As seen below, this model also shows a characteristic feature of strong correlations, namely the appearance of new low-energy scales. In weakly correlated systems the characteristic energy scale is given by the Fermi energy eF, or alternatively, by the electronic hopping matrix elements tj between neighboring sites i and j. Strong correlations cause additional, much lower energy scales.

The two orbitals are denoted by L (for ligand) and F (for 4 f, for example) and we assume the corresponding orbital energies to be e; and ef with ef < e;. Two electrons in the F orbital are expected to repel each other with an energy U Ae with Ae = e; — ef. When both electrons are in the L orbital, or when one electron is in the L and the other in the F orbital, we neglect their Coulomb interaction.

This is justified, if the ligand orbital has a large spatial extent. It applies when, for example, the ligand orbital is that of a large molecule or when it is a Bloch state. We assume that the hybridization t between the two orbitals is small, i.e., t ^ Ae. The Hamiltonian of the system is

H = ei X l++ ef X f+ fa + 1 X (l+f* + f+l*) + Un\n#. (1.2)

The l + (la), fa+(fa) create (annihilate) electrons with spin a in the L and F orbitals, respectively; furthermore nf = f + fa. The Hamiltonian is so simple that it can be easily diagonalized. When t = 0, the ground state of the system has energy E0 = el + ef and is fourfold degenerate. One electron is in the F orbital, while the other is in the L orbital. The four states are eigenstates of the total spin S and consist of a singlet \ &S=0) and a triplet \ &S=i). The system has one excited state of the form

The energy of that state is Eex = 2el. The state f+f+ \0) is excluded from further consideration, since its energy is of order U and we assume U

When the hybridization is turned on, the singlets \&S =0) and |0ex) are coupled, while the S = 1 states \&S=1) remain unchanged. The coupling leads to the two eigenvalues

For small values of t there is a low-lying triplet excitation above the singlet ground state. One can attach a characteristic temperature T* = 2t2/(kB Ae) to the energy gain associated with the singlet formation. It is by a factor t/Ae smaller than the energy scale set by the hopping matrix element t and is an example of the new low-energy scales, which are generated by strong correlations. The same system can be treated using the density functional theory. In fact, that is quite interesting to do so, because one can derive explicitly the exact exchange-correlation potential vxc[p] as function of the density. When the 2 x 2 Kohn-Sham equation is solved, one finds that the energy difference between the two eigenvalues should not be interpreted as the excitation energy, since it is of order t instead of T *. These findings hold irrespective of approximations to the functional [3].

With the above pointed out, it is clear that treatments of low-energy excitations by Kohn-Sham equations have no sound basis. Nevertheless, semi-empirical methods based on density functional theory have had numerous successes. An excellent example is the renormalized band-structure method [4]. It is the only one which has been able to make detailed predictions for the effective mass anisotropies at the Fermi surface of heavy-quasiparticle systems. The idea hereby is to formulate the quasiparticle dispersions in terms of phase shifts like in an independent electron approach. For these phase shifts, one is using the ones obtained from an LDA, i.e., a local approximation to the density functional, except for those of the strongly correlated electrons. For Ce intermetallic compounds like CeRu2Si2, CeSn3, CeCoIn5, etc., these are the f electron phase shifts at the Ce sites. For the latter, a phenomenological ansatz is made. It has the effect of reducing the bare bandwidth to a renormalized one of order kBT*. Take CeRu2Si2 as an example. The phase shifts near the Fermi energy qf are irt(Q)} = {^Ce00, ^(q)} , vI M = 1,2, (1.5)

where l, l', l" are angular momenta and the indices v, m count different atoms within the unit cell. Except for rCD3(e) all other phase shifts are assumed to be given by the LDA. Regarding the f phase shifts at a Ce site, only the one with j = 5/2 (Hund's rule multiplet) is relevant and, more specifically, with the symmetry of the crystalline field ground-state doublet, i.e., rCe(e) with x = 1,2. It can be parameterized by the resonant form

with two parameters r and Q. One of them is fixed by the f -electron number at a Ce site, while the remaining one is fixed by requiring that the large y coefficient of the low temperature specific heat C = yT is reproduced. As mentioned earlier, with these phase shifts not only a Fermi surface but also the strongly anisotropic quasiparticle masses can be determined. They agree well with experiments.

Another extension of density functional theory which is often used is the LDA+U method. Here the LDA is supplemented by adding an on-site Coulomb interaction U and exchange interaction J term to the LDA energy. For example, for d electrons it is

E = Elda + — 2_&nia(l)&nj-a(l) C--- ^ Snia(l)Snja0(l), (1.7)

lija li (^j)a where l is a site index while i and j are d orbital indices. Furthermore, Snia(l) = nia(l) — n0(l) where n0(l) = nd(l)/10 and nd(l) is the total d electron number at site l. The LDA is an orbital-independent molecular-field approximation and therefore an inclusion of the deviations Snia(l) allows for an improved treatment of correlations. The potential entering the Kohn-Sham equation is obtained from 8E/8nia(l) as

Vif(l) = Vlda c Uj^lnj —a(l) C (U — J)Yi1nja(l), (1.8) j j &

which shows that results different from LDA are obtained if the spin orbitals are differently populated, i.e., when Snia(l) ^ 0. An unequal population is favored by a large Coulomb interaction U like in any unrestricted mean-field approximation. As explained earlier, this way charge fluctuations are suppressed and correlation energies (not wavefunctions) are improved. There has also been an approach developed in which the LDA is used for computing Wannier functions and Coulomb parameters as input for a multiband Hubbard Hamiltonian. The latter is treated by a generalized tight-binding method, i.e., one which combines the exact diagonalization of an isolated cluster, i.e., a unit cell, with a perturbation treatment of the intercluster hopping and interactions [5] (see Chap. 4.4). The aim has been to find, e.g., the size of the gap in La2CuO4, which in LDA is absent.

Another hybrid method is the LDA+DMFT [6]. The dynamical mean-field theory (DMFT) [7-9] is a dynamical coherent potential approximation (DCPA) [10] which was stimulated by work on the Hubbard model in infinite dimensions [11,12] (see Chap. 6.5). While the standard coherent potential approximation (CPA) introduced by Hubbard in connection with his Hamiltonian (see below) reduces at temperature T = 0 to a self-consistent field (SCA), i.e., Hartree-Fock theory, the DMFT (or DCPA) contain correlation effects in that limit. A site of an infinite lattice with, e.g., d electrons is treated as an impurity in a medium for which an LDA calculation has been done. The electronic self-energy S(a>) at the impurity site is computed and the medium is self-consistently modified until the self-energy of the impurity coincides with that of the medium. A shortcoming of the DMFT and DCPA is that only on-site correlations are treated, i.e., any k dependence of the self-energy is neglected. From quantum chemical calculations it is well known that intersite correlations are important and must be treated if one is interested in quantitative results.

Extensions of the original DMFT to a cluster DMFT [13] go in the right direction, but the clusters one can treat are rather small. Therefore, one cannot distinguish, e.g., between short-range AF correlations and long-range AF order. One possible way of including properly the short-ranged correlations is by using the method of increments in connection with a self-consistent projection operator method. This allows S(k,a>) to be calculated with rather high accuracy. These approaches are still at their infancy as far as realistic calculations for specific materials are concerned, but some promising results have been obtained.

A rather different approach to strongly correlated electron systems is based on wavefunction methods [2]. They are combined with quantum chemical techniques and provide a rigorous theoretical framework for addressing the correlation problem which avoids any uncontrolled approximations. Many-body wavefunctions can be explicitly constructed at levels of increasing sophistication and accuracy. But in order to ensure size extensivity of the modifications induced by correlations the wavefunctions have to be formulated using cumulants. Standard quantum chemical methods thus offer a systematic path to converged results. They provide the right framework for coping with issues like a rigorous treatment of the ubiquitous short-range correlations and of a realistic representation of the crystalline environment. The way to proceed is to cut out a finite atomic cluster C from the infinite solid which is large enough to describe the crucial short-range correlations. Partially filled d -electron shells require a multiconfiguration representation of the correlated many-electron wavefunction, which is achieved using the complete-active-space (CAS) self-consistent field (CASSCF) method. This way, strong correlations can be very well accounted for. The crystalline environment is described by an effective one-electron potential which is extracted from prior Hartree-Fock calculations for the periodic system. Remember that Hartree-Fock calculations describe charge distributions quite well. They are robust against correlation corrections, even when the latter are strong. With this approach the ground state of such strongly correlated systems as LaCoO3 and LiFeAs was determined as well as the Zhang-Rice-like electron removal band for CuO2 planes in La2CuO4 [14].

Wavefunction-based quantum chemical calculations for strongly correlated electrons are a field with high potential for the future. They have been somewhat neglected, because they require investments in program development and time, features not particularly favored by research funding. However, concerning actual calculations, they are the best one can do to obtain insight into the most important microscopic processes.

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