A density functional (DF) approach to high-Z systems [4-9] has to reflect their relativistic character: The corresponding auxiliary single-particle equations are of Dirac form, including a multiplicative (four component) xc-potential, in which also the retardation corrections (Breit+...) to the Coulomb interaction show up (for a review see [9]). This relativistic DF scheme is usually introduced within the framework of quantum electrodynamics (QED) [4-9], and thus necessarily reflects the various subleties of QED. This becomes particularly obvious for the existence theorem of relativistic DF theory (RDFT), which guarantees the equivalence of the DF single-particle description with the conventional many-body treatment. As the proof of the existence theorem relies on the finiteness of ground state four currents and energies, it is essential to work with renormalized quantities. One can verify [7,9], however, that the proof is in fact compatible with the standard renormalization procedure of QED, thus providing a firm basis for RDFT.
The most simple version of this DF scheme is the Dirac-Fock-Slater (DFS) approach, in which only the relativistic kinematics of the auxiliary particles is taken into account, while the xc-potential is approximated by the nonrelativistic local density approximation (LDA). All nonlocal (gradient etc) and relativistic contributions to the xc-potential (as well as radiative corrections) are neglected. In order to deal with the nonadditivity of relativistic and xc effects, relativistic corrections to the functional dependence of the xc-energy on the density are required, which, until recently, have only been known within the relativistic LDA (RLDA) [4-6]. In the RLDA the density dependence of the xc-energy density of the relativistic homogeneous electron gas is employed with the actual inhomogeneous density of the system of interest. The RLDA, however, was found to be rather inaccurate, in particular for very heavy atoms [6].
The RLDA has two obvious conceptual shortcomings:
In order to address the first issue we have developed a fully 'nonlocal' RDFT scheme, the relativistic optimized-potential-method (ROPM) [6] (note that 'nonlocal' in this context refers to the kernel of the xc-functional - the resulting xc-potential nevertheless remains a local operator). In the (R)OPM one allows the xc-energy to depend explicitly on the Kohn-Sham single-particle orbitals, which themselves are implicit functionals of the density (or, more generally, the four current). The (R)OPM can thus be viewed as an extension of the conventional DF approach: Not only the kinetic energy but also the xc-energy is represented in terms of the auxiliary single-particle orbitals, rather than the density or four current. As a consequence, the local xc-potential is determined by an integral equation, which has to be solved selfconsistently together with the Kohn-Sham equations.
The concept of orbital-dependent functionals is most easily introduced in the case of the exchange-only energy, which is simply defined by the (relativistic) Fock expression, written in terms of Kohn-Sham orbitals. For the correlation energy, on the other hand, approximations are unavoidable. A simple semiempirical orbital-dependent representation of the correlation energy has been suggested by Colle and Salvetti [9]. In a more systematic fashion orbital-dependent xc-functionals can be derived by a perturbation expansion on the basis of the Kohn-Sham Hamiltonian [11-13]. A subsequent expansion in powers of e^2 directly leads to the exact exchange-only energy (in first order) and provides a Moller-Plesset type approximation for the correlation energy (in second order). This correlation functional explicitly contains dispersion contributions, so that it allows an DF description of the van-der-Waals interaction [13,14]. It should be emphasized that, while these functionals are derived by perturbation theory, they are applied selfconsistently within the (R)OPM, i.e. an integral equation has to be solved for each order in e^2.
The extension of nonrelativistic many-body methods to the relativistic domain quite generally leads to the question of how to deal with the negative energy continuum states. In this respect the no-(virtual-)pair approximation, in which these states are projected out completely, represents the standard in conventional many-body approaches as the relativistic Hartree-Fock (RHF) approximation. It is well known, however, that this projection leads to gauge dependent results for the xc-energy, as soon as the transverse electron-electron interaction is taken into account [15]. In addition to the no-pair approximation there exists a second possible source of gauge dependence: Individual energy contributions (like the exchange energy) may become gauge dependent if they are evaluated with single-particle orbitals which experience a nonlocal potential [16] (as e.g. in the RHF scheme). However, due to the multiplicative nature of the DFT single-particle potential this second source of gauge dependence is not present in ROPM calculations [13] (note that gauge invariant transition matrix elements can also be obtained with RHF-orbitals, if a suitable resummation of the perturbation series is used [17]). In particular, one can show that the exact DFT treatment of the no-pair exchange via the ROPM ensures gauge invariance even if the transverse interaction is used selfconsistently [13], in contrast to the RHF scheme [16]. On the other hand, approximations for the DFT correlation energy are not gauge invariant, as soon as the no-pair approximation is used in their derivation. As a first analysis of the importance of gauge effects for the DFT correlation energy we have examined the no-pair approximation for the RLDA [18]. The results demonstrate that at the present level of sophistication of relativistic density functionals this gauge dependence is quantitatively irrelevant.
As the ROPM provides a direct way to include the exact exchange in DF
calculations, ROPM-results are clearly superior to the corresponding
RLDA-data [6,8,13].
However, the solution of the (R)OPM integral equation for the
xc-potential is computationally rather involved, so that for
applications to complex systems an approximate treatment of the
xc-energy functional is still of interest.
For approximations three different routes are available:
Within the framework of orbital-dependent xc-functionals Krieger, Li
and Iafrate (KLI) have suggested to resort to an approximate form of
the OPM integral equation [19], which can be solved much more easily.
This approximation has also been extended to the relativistic domain
[20,13], leading to highly accurate results for atoms, at least at the
exchange-only level.
Alternatively, one can use ROPM results to construct explicit density
functionals like the relativistic generalized
gradient approximation (RGGA) [21,22], in which the RLDA is
systematically extended by utilizing the information contained in
the first density gradient (illustrative plots
of the characteristic gradient for atoms, molecules and solids).
The RGGA significantly improves atomic xc-energies over the RLDA, thus
making the RGGA a promising tool for electronic structure calculations
for molecules and solids with very heavy constituents.
On the other hand, an application of this RGGA to metallic Gold and
Platinum within the linearized-augmented-plane-wave (LAPW) scheme [23],
showed that the inclusion of relativistic corrections in the GGA does
not improve the somewhat inaccurate results obtained for the cohesive
properties for these systems with the nonrelativistic GGA [24] (as the
relativistic contributions to the xc-energy cancel out in the energy
surface).
Similarly, while the dissociation energies of heavy diatomic molecules
are improved, the RGGA overcorrects the errors in the RLDA bond lengths
[25,26].
The problems of the GGA with these systems have to be attributed to
the limited 'nonlocality' of gradient corrected xc-functionals, which
is also inherent to the RGGA.
As a third route for utilizing the information contained in atomic ROPM
results the pseudopotential concept, in which
the electronic degrees of freedom which are irrelevant for molecular
or solid state properties are projected out, offers itself.
We have thus constructed norm-conserving pseudopotentials from ROPM
all-electron potentials [27-30], both within the framework of the
inherently relativistic Bachelet-Hamann-Schlüter scheme [31]
as well as on the basis of the relativistic extension [32] of the
Troullier-Martins scheme [33].
In this way an optimum 'valence Fock space' is generated for the actual
molecular or band structure calculation, which may then rely on the KLI
procedure (or even the LDA/GGA).
First results [29,30] for atoms, molecules and solids indicate that,
at least on average, ROPM pseudopotential results reproduce the
all-electron ROPM data more accurately than their RLDA counterparts
reproduce the corresponding all-electron RLDA data.
The fact that the RLDA only depends on the density, rather than on the full four current, has stimulated the search for alternative forms of RDFT which allow an explicit treatment of magnetization effects and, in particular, of the spin-degree of freedom. A formalism in which the role of the spatial components of the four current is played by the magnetization density m has been suggested rather early [5,34]. Within the no-pair approximation one can give rigorous arguments that this m-dependent RDFT approach is completely legitimate as long as there is no external magnetic field present [9]. For non-vanishing external magnetic field it still represents a reasonable approximation. A corresponding m-dependent extension of the RLDA has been derived for the exchange part of the xc-functional [34-37].
In general, the direction of m can vary with position. Ground states for which the direction of m either changes from site to site (inter-atomic non-collinear magnetism [38]) or even on the atomic scale (intra-atomic non-collinear magnetism [39]) were found for a number of solids. On the other hand, the non-collinearity of m turned out to be only of very limited importance for open-subshell atoms [40], i.e. the single site problem. Moreover, the structure of the single-particle equations with non-collinear m is numerically rather involved. For this reason the collinear form of m-dependent RDFT, in which the oriention of m is globally fixed as m=(0,0,m), represents the standard in calculations for magnetic materials. In the nonrelativistic limit the latter form approaches the usual spin-density functional theory. For that reason, the m-dependent RDFT is subsequently termed relativistic spin-density functional theory (RSDFT), in spite of the fact that the notion of spin is not well-defined in the fully relativistic context. Accordingly, nonrelativistic spin-density functionals are often used for the 'collinear' xc-functional.
Even the collinear variant of RSDFT suffers from the intricate structure of its single-particle equations. Until recently, the corresponding equations have only been solved for atomic ions with their very attractive Kohn-Sham potential [41-44]. First results for neutral atoms were obtained by a basis set expansion [45]. A robust finite differences shooting method, which is easily implemented in standard LAPW codes, has been developed in our group [46]. This scheme not only allows the application of collinear RSDFT to arbitrary neutral atoms within the (R)LDA, but also permits (R)GGA calculations with their much more structured potentials.
The importance of non-collinearity for molecular binding properties has been studied [47,48] utilizing a basis set expansion of the molecular Kohn-Sham states in terms of the atomic Kohn-Sham orbitals of the constituents. A comparison with the corresponding collinear results seems to indicate that non-collinearity does not necessarily show up in the spectroscopic constants, even if both the molecule and its constituents exhibit unpaired spins. On the other hand, on the GGA level the results obtained with the non-collinear scheme so far excellently agree with the experimental data.
Last modified: April 14, 2004
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