Comparison of exact-exchange calculations for solids in
  current-spin-density- and spin-density-functional theory
arXiv:0704.0244v2  [cond-mat.mtrl-sci]  4 Jun 2007
Comparison of exact-exchange calculations for solids in current-spin-density- and
spin-density-functional theory
S. Sharma1,2,∗ S. Pittalis2, S. Kurth2, S. Shallcross3, J. K. Dewhurst4, and E. K. U. Gross2
1 Fritz Haber Institute of the Max Planck Society, Faradayweg 4-6, D-14195 Berlin, Germany.
2 Institut für Theoretische Physik, Freie Universität Berlin, Arnimallee 14, D-14195 Berlin, Germany
3 Department of Physics, Technical University of Denmark,
Building 307, DK-2800 Kgs. Lyngby and
4 School of Chemistry, The University of Edinburgh, Edinburgh EH9 3JJ.
The relative merits of current-spin-density- and spin-density-functional theory are investigated
for solids treated within the exact-exchange-only approximation. Spin-orbit splittings and orbital
magnetic moments are determined at zero external magnetic field. We find that for magnetic (Fe,
Co and Ni) and non-magnetic (Si and Ge) solids, the exact-exchange current-spin-density functional
approach does not significantly improve the accuracy of the corresponding spin-density functional
results.
PACS numbers: 71.15 Mb, 71.15 Rf, 75.10 Lp
In the past 30 years, several generalizations of den-
sity functional theory (DFT) have been proposed. In the
early 70’s, DFT was extended to spin-DFT (SDFT) [1] by
including the spin magnetization as basic quantity in ad-
dition to the density. This allows for coupling of the spin
degrees of freedom to external magnetic fields and pro-
duces better results for spontaneously spin-polarized sys-
tems using approximate functionals. Adding yet another
density, the paramagnetic current, leads to the frame-
work of current-SDFT (CSDFT) [2, 3]. CSDFT includes
the coupling of the external magnetic field, through its
corresponding vector potential, to the orbital-degrees of
freedom [3].
SDFT has been enormously successful in predicting
the magnetic properties of materials. This success can
be attributed to the availability of exchange correlation
(xc) functionals which, even though originally designed
for non-magnetic systems, could be systematically ex-
tended to the spin polarized case. The most popular of
these functionals are the local spin density approxima-
tion (LSDA) and the generalized gradient approximation
(GGA). CSDFT, on the other hand, has not enjoyed the
same attention mainly because of problems which arise in
the extension of LSDA and GGA to include the paramag-
netic current density [4–6]. Exposing the homogeneous
electron gas to an external magnetic field leads to the
appearance of Landau levels which, in turn, give rise to
derivative discontinuities in the resulting xc energy den-
sity. Using this quantity to construct (semi-)local func-
tionals then automatically leads to local discontinuities in
the corresponding xc potentials, which are then awkward
to use in practical calculations.
Such problems can be avoided with the use of orbital
functionals and this fact, coupled with the success of
these functionals for SDFT calculations, has led to re-
cent interest in orbital functionals for CSDFT [7–11].
The results from these works have shown mixed suc-
cess. A modified version of the original CSDFT [12] lead
to promising results for spin-orbit induced splittings of
bands in solids, such as Si and Ge [9]. In contrast it was
found that for open-shell atoms and quantum dots the
difference between SDFT and CSDFT results was mini-
mal [7, 8]. Similarly, calculations for solids using a local
vorticity functional [13] and for quantum dots using a
LSDA-type xc functional [14] could not establish the su-
periority of CSDFT over SDFT.
In this work we present a systematic comparison of the
relative merits of CSDFT and SDFT for solids. Since
the Kohn-Sham (KS) system in CSDFT reproduces the
current of the interacting system, one would expect dif-
ferences between SDFT and CSDFT results for orbital
magnetic moments (which can be directly derived from
the current). With this in mind we calculate the orbital
magnetic moment of the spontaneous magnets Fe, Co and
Ni. Since CSDFT is believed to improve the spin-orbit
induced band splitting in the non-magnetic semiconduc-
tors Si and Ge [9] it makes these materials interesting
candidates for a study of the differences between the two
approaches.
Following Vignale and Rasolt [2, 3], the ground state
energy of a (non-relativistic) system of interacting elec-
trons in the presence of an external magnetic field
B0(r) = ∇×A0(r) can be written as functional of three
independent densities the particle density ρ(r), the mag-
netization density m(r) and the paramagnetic current
density jp(r). This functional is given by
http://arxiv.org/abs/0704.0244v2
E[ρ,m, jp] = Ts[ρ,m, jp] + U [ρ] + Exc[ρ,m, jp] +
ρ(r)v0(r) d
3r (1)
m(r) ·B0(r) d
jp(r) ·A0(r) d
ρ(r)A20(r) d
where Ts[ρ,m, jp] is the kinetic energy functional of non-interacting electrons, U [ρ] is the Hartree energy, and
Exc[ρ,m, jp] is the exchange-correlation energy. Minimization of Eq. (1) with respect to the three basic densities
leads to the Kohn-Sham (KS) equation which reads
As(r)
+ vs(r)− µBσ ·Bs(r)
Φj(r) = εjΦj(r) . (2)
Here σ is the vector of Pauli matrices and the Φi are spinor valued wave functions. The effective potentials vs, Bs
and As are such that the ground-state densities ρ, m and jp of the interacting system are reproduced. These effective
potentials are given by
vs(r) = v0(r) + vH(r) + vxc(r) +
0(r)−A
s (r)
, Bs(r) = B0(r) +Bxc(r), As(r) = A0(r) +Axc(r). (3)
Here, v0 is the external electrostatic potential and vH(r) =
ρ(r′)/|r − r′| d3r′ is the Hartree potential. The xc
potentials are given as functional derivatives of the xc energy with respect to the corresponding conjugate densities
which can be obtained from KS wave functions using the following relations
ρ(r) =
i (r)Φi(r), m(r) = −µB
i (r)σΦi(r), jp(r) =
i (r)∇Φi(r)−
i (r)
Φi(r)
where the sum runs over the occupied orbitals. For practical calculations, an approximation for the xc energy functional
Exc[ρ,m, jp] has to be adopted. Here we concentrate on approximations of the xc functional which explicitly depend
on the KS orbitals and therefore only implicitly on the densities. Such orbital functionals are usually treated within
the framework of the so-called Optimized Effective Potential (OEP) method [15–18] where the xc potential is obtained
as solution of the OEP integral equation. Recently, the OEP method has been generalized to non-collinear SDFT [19]
and CSDFT [7, 8]. Another generalization of the OEP method in the context of a spin-current DFT (SCDFT) based
on a different choice of densities has also been put forward [11]. In the present work the formalism of Refs. (7) and
(8) is used and the corresponding OEP equations can be put in a compact form as
(r)Ψk(r) + h.c. = 0, −µB
(r)σΨk(r) + h.c. = 0,
(r)∇Ψk(r) −
Ψk(r)
+ h.c. = 0 ,
where the so-called orbital shifts [17, 20] are defined as Ψk(r) =
∑unocc
Φj(r)Λkj
εk−εj
, here the summation runs over the
unoccupied states and
Λkj =
vxc(r
′)ρkj(r
Axc(r
′) · jpkj(r
′)−Bxc(r
′) ·mkj(r
′)− Φ
where ρkj(r) = Φ
(r)Φk(r), mkj(r) = −µBΦ
(r)σΦk(r) and jpkj(r) =
(r)∇Φk(r−
Φk(r)
Eq. (5) has a structure very similar to the OEP equa-
tions for non-collinear SDFT differing only by the redefi-
nition of the matrix Λ, which now also contains an extra
term depending upon the current density and its con-
jugate field. Due to their similar structure the CSDFT
OEP equations are solved by generalizing the ‘residue al-
gorithm’, successfully applied to solve the non-collinear
SDFT equations [19–21]. The only difference in the case
of CSDFT is introduction of an additional residue coming
from the third OEP equation in Eq. (5). In the present
work we have used the exchange-only exact-exchange
(EXX) functional to solve the OEP equations. The EXX
(gauge invariant) energy functional is the Fock exchange
energy but evaluated with KS spinors
EEXXx [{Φi}] ≡ −
∫ ∫ occ
i (r)Φj(r)Φ
′)Φi(r
|r− r′|
d3r d3r′ .
In order to keep the numerical analysis as accurate
as possible, in the present work all calculations are per-
formed using the state-of-the-art full-potential linearized
augmented plane wave (FPLAPW) method [22], imple-
mented within the EXCITING code [23]. The single-
electron problem is solved using an augmented plane
wave basis without using any shape approximation for
the effective potential. Likewise, the magnetization and
current densities and their conjugate fields are all treated
as unconstrained vector fields throughout space. The
deep lying core states (3 Ha below the Fermi level)
are treated as Dirac spinors and valence states as Pauli
spinors. To obtain the Pauli spinor states, the Hamilto-
nian containing only the scalar fields is diagonalized in
the LAPW basis: this is the first-variational step. The
scalar states thus obtained are then used as a basis to
set up a second-variational Hamiltonian with spinor de-
grees of freedom, which consists of the first-variational
eigenvalues along the diagonal, and the matrix elements
obtained from the external and effective vector fields in
Eq. (2). This is more efficient than simply using spinor
LAPW functions, but care must be taken to ensure there
are a sufficient number of first-variational eigenstates for
convergence of the second-variational problem. Spin-
orbit coupling is also included at this stage.
As was shown above for CSDFT, the magnetic field
couples not only to spin but also to the orbital degrees
of freedom through the vector potential. This makes CS-
DFT specifically important for magnetic materials and
particularly interesting for their orbital properties. By
analogy with SDFT, one might expect that the introduc-
tion of the paramagnetic current density gives an im-
provement in properties such as orbital moments and
spin-orbit induced band splitting, which are related to
this new basic variable. However, within the framework
of existing functionals it is yet to be established conclu-
sively that CSDFT performs better than SDFT for these
properties. The recent development of the OEP method
both for SDFT and CSDFT allows for a direct compar-
ison of these two approaches for the same xc functional,
namely EXX.
The orbital moments of spontaneous magnets Fe, Co
and Ni, in the absence of external magnetic fields and
with spin-orbit coupling included, are presented in Ta-
ble I. For SDFT, the LSDA, GGA and EXX functionals
are used, while for CSDFT the values are obtained us-
ing the EXX functional. It is clear from Table I that
there is no difference between the results obtained us-
ing EXX-CSDFT and EXX-SDFT. Formally, the jp de-
termined from SDFT does not correspond to the true
paramagnetic current density of the fully interacting sys-
SDFT CSDFT
Solid Exp. LSDA GGA EXX EXX
Fe 0.08 0.053 0.051 0.034 0.034
Co 0.14 0.069 0.073 0.013 0.013
Ni 0.05 0.038 0.037 0.029 0.029
36.2 36.7 63.4 63.4
TABLE I: Orbital magnetic moments for bulk Fe, Co and
Ni in µB . The experimental data are taken from Ref. (24).
The final row lists the average percentage deviation of the
numerical results from the experimental value.
tem. Nevertheless, it is standard practice to compute the
orbital magnetic moment L, like those listed in Table I,
which is related to jp from the KS orbitals by the relation
L = 1
r×jp(r)d
3r. The fact the EXX-SDFT and EXX-
CSDFT orbital moments are so close may be viewed as a
post-hoc justification of this practice for magnetic metals.
It should also be noted that in comparison to experiments
the EXX results are significantly worse than their LSDA
and GGA counterparts. One reason, of course, is the fact
that LSDA and GGA also include correlation in an ap-
proximate way which is neglected completely within the
EXX framework.
In a recent work [9] it is shown that the use of the
EXX functional in the framework of SCDFT, improves
the spin-orbit induced splitting of the bands in semicon-
ductors. Unfortunately, it is not clear if this improve-
ment is due to the use of different functionals (going
from LSDA to EXX), or due to the use of an extra den-
sity when going from SDFT to CSDFT. This has moti-
vated us to compare CSDFT and SDFT results for this
quantity using the same functional in both cases. We
have determined the value of this splitting for solid Si
and Ge and the results are presented in Table II. While
the EXX functional significantly improves the agreement
with experimental values, there is almost no change on
going from SDFT to CSDFT. Thus the improvement is
solely due to the orbital based functional. We also note
that the EXX-CSDFT results of Ref. (9) are significantly
different from ours, and in much worse agreement with
experiments. This might be due to the use of pseudopo-
tentials in the previous work. In this respect it is worth
noting that EXX derived KS energy gaps also show sig-
nificant differences depending on whether an all-electron
full-potential or pseudopotential method is used [25].
The paramagnetic current density of Ge for LSDA,
GGA, EXX-SDFT and EXX-CSDFT is plotted in Fig.
1. Ge is chosen as an example since the spin-orbit in-
duced splitting is largest for this system and, unlike in
the case of metallic orbital moments, this quantity does
show some difference on going from SDFT to CSDFT.
We immediately notice that there is no significant qual-
itative difference between the LSDA and GGA currents.
There are, however, pronounced differences in the cur-
Symmetry SDFT CSDFT
point Exp. LSDA GGA EXX EXXp EXXo
Ge Γ7v−8v 297 311 296 291.3 289 258.1
Ge Γ6c−8c 200 229.7 220 201.3 199 173.3
Si Γ25v 44 50 58 42.5 45.5 42.5
9.5 14.0 2.0 2.2 10.5
TABLE II: Spin-orbit induced splittings for bulk Ge and Si in
meV. The experimental data is taken from Ref. (26). EXXp
are results of the present work and EXXo are results from
Ref. (9). The final row lists the average percentage deviation
of numerical results from the experimental value.
rent density between LSDA/GGA and EXX-(C)SDFT:
the current in the latter case being smaller and more
homogeneous than that of the former. This is an inter-
esting finding since it indicates the tendency of (semi-)
local functionals towards higher values of the paramag-
netic current density. It is worthwhile noting previous
EXX-(C)SDFT results for open-shell atoms in which it
was found [7] that this effect was even more pronounced
and lead to vanishing currents.
Even though the EXX-SDFT current is considerably
lower in magnitude than that of EXX-CSDFT and also
has a less symmetric structure, the spin-orbit splittings
for the two cases are almost the same. Similar conclusions
regarding the total energies were also drawn for quantum
dots in external magnetic fields studied using EXX [8]
and other functionals of the current density [14]. From
Fig. 1 it is also clear that one of the major effects of using
the OEP method and of using jp as an extra density is to
change the local structure of the paramagnetic current,
which in turn suggests that quantities depending on local
properties of the currents, such as chemical shifts, might
exhibit larger differences in the two approaches. Such
calculations [27] of chemical shifts, performed using local
functionals, found that for molecules this is not the case.
The effect of the EXX functional on these shifts may be
an interesting subject for future investigations.
To summarize, in this work we have presented EXX-
SDFT and CSDFT calculations for solids. The orbital
magnetic moments of Fe, Co and Ni and the spin-orbit
induced band splitting of Si and Ge are computed. Our
analysis shows only minor differences between EXX- CS-
DFT and SDFT results. The spin-orbit induced band
splittings in EXX calculations are in rather good agree-
ment with experiments, while the results for the orbital
moments are worse than the LSDA or GGA values. This
highlights the importance of proper treatment of corre-
lations for the accurate determination of the orbital mo-
ments.
We acknowledge Deutsche Forschungsgemeinschaft
(SPP-1145) and NoE NANOQUANTA Network (NMP4-
CT-2004-50019) for financial support.
FIG. 1: (Color online)Paramagnetic current density for Ge,
in the [110] plane, calculated using the SDFT and CSDFT.
Arrows indicate the direction and information about the mag-
nitude (in atomic units) is given in the colour bar.
∗ Electronic address: sangeeta.sharma@physik.
fu-berlin.de
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