3. BAND STRUCTURE METHODS
3.1 Local spin density approximation
This method is based on DFT and the free electron liquid, but corrects it for electrons where self-interactions are significant and takes the theory towards a more localized description.
where the electron–electron interaction (Hartree term) is given by U n½ ¼
ð ðnð Þnr ð Þr0 rr0
j j d3rd3r0; ð8Þ and the external potential energy, Vext[n], due to electron–ion and ion–ion inter- actions, respectively, is
Vext½ ¼n ð
nð ÞVr ionð Þdr 3rþEion;ion: ð9Þ Here Vion(r) is the potential from the ions, and the atomic Rydberg units e2=2¼2m¼h¼1
have been used in all the formulas. Since Eqs. (8) and (9) are explicitly defined in terms of the electron charge density, all approximations are applied to the last term inEq. (7), the exchange and correlation energy.
In the LDA, it is assumed that each point in space contributes additively toExc: ELDAxc ½ ¼n
ð
nð Þrehomðnð Þr Þd3r; ð10Þ whereehom(n) is the exchange and correlation energy of a homogeneous electron gas with charge densityn. In the simplest case, when only exchange is considered, one has (Hohenberg and Kohn, 1964)
ehomð Þ ¼ Cnn 1=3 withC¼3 2
3 p
1=3: ð11Þ
Modern functionals include more elaborate expressions for ehom(n) (Vosko et al., 1980; Perdew and Zunger, 1981) relying on accurate Quantum Monte Carlo data for the homogeneous electron gas (Ceperley and Alder, 1980). General- izing to magnetic solids, which is highly relevant for lanthanide materials, is straightforward as one merely has to consider two densities, one for spin-up and one for spin-down electrons, which corresponds to the LSD functional (von Barth and Hedin, 1972),ELSD. Even more accurate calculations may be obtained by including corrections for the spatial variation of the electron charge density, which is accomplished by letting ehom also be dependent on gradients of the charge density, which constitutes the GGA (Perdew and Wang, 1992). Yet more accurate functionals may be composed by mixing some non-local exchange inter- action into theExc(Becke, 1993), often named hybrid functionals.
The minimization ofE[n], with these approximate functionals, is accomplished by solving a one-particle Schro¨dinger equation for an effective potential, which includes an exchange-correlation part given by the functional derivative ofExc[n]
with respect to n(r). The non-interacting electron gas system with the charge densityn(r) is generated by populating the appropriate number of lowest energy solutions (the aufbau principle). Self-consistency must be reached between the charge density put into the effective potential and the charge density composed from the occupied eigenstates. This is usually accomplished by iterating the procedure until this condition is fulfilled.
Spin-orbit interaction is only included in the density functional framework if the proper fully relativistic formalism is invoked (Strange, 1998). Often an approx- imate treatment of relativistic effects is implemented, by solving for the kinetic energy in the scalar-relativistic approximation (Skriver, 1983a), and adding to the total energy functional the spin-orbit term as a perturbation of the form
Eso¼Xocc:
a
hcajxð!rÞ!l !sjcai; ð12Þ
where the sum extends over all occupied states jcai and x is the spin-orbit parameter. Here!l and s!denote the one-particle operators for angular and spin moments. The method presupposes, which usually does not pose problems, that an appropriate region around each atom may be defined inside which the angular momentum operator acts.
The LDA and LSD, as well as their gradient corrected improvements, have been extremely successful in providing accurate, material specific, electronic, magnetic, and structural properties of a variety of weakly to moderately correlated solids, in terms of their ground state charge density (Jones and Gunnarsson, 1989; Ku¨bler, 2000), However, these approaches often fail for sys- tems containing both itinerant and localized electrons, and in particular d- and f- electron materials. For all lanthanides, and similarly the actinide elements beyond neptunium, the electron correlations are not adequately represented by LDA. In d-electron materials, for example, transition metal oxides, this inadequate descrip- tion of localized electrons leads commonly to the prediction of wrong magnetic ground states and/or too small or non-existent band gaps and magnetic moments.
When applied to lanthanide systems, LSD (and GGA as well) leads to the formation of narrow bands which tend to fix and straddle the Fermi level. Due to their spatially confined nature, the f-orbitals hybridize only weakly and barely feel the crystal fields. Furthermore, on occupying the f-band states, the repulsive poten- tial on the lanthanide increases due to the strong f–f Coulomb repulsion, so the f-bands fill up to the point where the total effective potential pins the f-states at the Fermi level. Generally, the band scenario leads to distinct overbinding (when com- pared to experimental data), since the occupation of the most advantageous f-band states favours crystal contraction (leading to increase in hybridization). Hence, the partially filled f-bands provide a negative pressure, which in most cases is unphy- sical. The same effect is well known to cause the characteristic parabolic behaviour of the specific volumes of the transition metals, and would lead to a similar parabolic behaviour for the specific volumes of the lanthanides, which is in variance with the observation seen inFigure 1inSection 2.1. This will be discussed further inSection 4.2. Even if narrow, the f-bands in the vicinity of the Fermi level alone cannot describe the heavy fermion behaviour seen in many cerium and ytterbium com- pounds. Zwicknagl (1992) and co-workers applied a renormalization scheme to LDA band structures to describe the heavy fermion properties with successful application in particular to the understanding of Fermi surfaces.
3.2 ‘f Core’ approach
To remedy the failure of LSD, it was early realized that one could remove the spurious bonding due to f-bands by simply projecting out the f-degrees of freedom from variational space and instead include the appropriate number of f electrons in the core. This approach was used to study the crystal structures of the lanthanide elements (Duthie and Pettifor, 1977; Delin et al., 1998), which are determined by the number of occupied d-states. In this approach, the lanthanide contraction neatly follows as an effect of the incomplete screening of the increasing nuclear charge by added f electrons, as one moves through the lanthanide series (Johansson and Rosengren, 1975). In a recent application, the ‘f core’ approach proved useful in the study of complex magnetic structures (Nordstro¨m and Mavromaras, 2000).
Johansson and co-workers have extended the ‘f core’ approach to compute energies of lanthanide solids, with different valencies assumed for the lanthanide ion, by combining the calculated LSD total energy for the corresponding ‘f core’
configurations with experimental spectroscopic data for the free atom (Delin et al., 1997). This scheme correctly describes the trends in cohesive energies of the lanthanide metals, including the valence jumps at Eu and Yb, as well as the intricate valencies of Sm and Tm compounds.