PHYSICAL REVIEW B 69, 085104 共2004兲
Weighted-density-approximation description of rare-earth trihydrides
Zhigang Wu and R. E. Cohen
Geophysical Laboratory, Carnegie Institution of Washington, 5251 Broad Branch Road NW, Washington, DC 20015, USA
D. J. Singh
Center for Computational Materials Science, Naval Research Laboratory, Washington, DC 20375, USA
R. Gupta
Commissariat a l’Energie Atomique, Centre d’Etudes de Saclay, Service de Recherches de Métallurgie Physique Commissariat
à l’Energie Atomique, 91191 Gif Sur Yvette Cedex, France
M. Gupta
Institut des Sciences des Materiaux, Batiment 415, Universite Paris-Sud, 91405 Orsay, France
共Received 24 October 2003; published 19 February 2004兲
Density-functional calculations within the weighted density approximation 共WDA兲 are presented for YH3
and LaH3 . We investigate some commonly used pair-distribution functions G. These calculations show that
within a consistent density-functional framework a substantial insulating gap can be obtained while at the same
time retaining structural properties in accord with experimental data. Our WDA band structures agree with
those of GW approximation very well. It shows that there is no strong correlation in H 1s states, and the
absence of self-interaction in H 1s is crucial to obtain correct band structures of these rare-earth trihydrites.
But the calculated band gaps are still 1.0–2.0 eV smaller than experimental findings.
DOI: 10.1103/PhysRevB.69.085104
PACS number共s兲: 71.15.Mb, 71.30.⫹h, 78.20.Bh
I. INTRODUCTION
The rare-earth trihydrides, M H3 , where M ⫽Y, La–Lu,
are the subject of considerable practical and theoretical interests. The rare earths absorb hydrogen readily and in the trihydride form have among the largest hydrogen to metal ratios of the elemental hydrides.1 The fully hydrogenated
materials are insulating, while those with less hydrogen are
metallic, and in thin film form they can be used as switchable
mirrors.2 It is important to demonstrate a correct description
of these trihydrides in first-principles approaches in order to
validate calculations for more complex metal hydrogen systems. However, standard first-principles calculations based
on density-functional theory 共DFT兲 within the local-density
共LDA兲 and generalized gradient 共GGA兲 approximations incorrectly predict metallic behavior for the rare-earth trihydrides. Here we explore application of the weighted density
approximation 共WDA兲. We find a WDA that works for these
materials 共and others兲, indicating that the gap problem here
is not due to unusual correlations or quasiparticle corrections, but is simply a problem with the LDA and GGA exchange correlation 共xc兲 functionals.
The rare-earth dihydrides, M H2 , are metallic. The trihydrides on the other hand are insulating with optical gaps of
the order of 2 eV, with a metal insulator transition at a hydrogen content correlated with rare-earth ionic radius.2 This
insulating gap of the trihydrides may be understood in terms
of simple picture, in which H forms a negative ion, H⫺ , fully
ionizing the metal atoms, thus forming a gap between H
bands and higher lying metal sd conduction bands. This
simple picture is supported by infrared and photoemission
measurements.3,4 It is consistent with the fact that all the
rare-earth trihydrides are insulators despite differences in the
0163-1829/2004/69共8兲/085104共6兲/$22.50
details of their crystal structures. These range from the f cc
derived structure of LaH3 to the not fully solved hexagonal
structure of YH3 . 5– 8
Computations within the LDA predict a metallic band
structure with substantial overlap between H and metal derived bands.8 –11 This severe gap problem could be due to
several possible causes: 共1兲 use of an incorrect crystal structure; 共2兲 an extreme case of LDA error in predicting gaps; 共3兲
unusual correlation effects; or 共4兲 self-interaction errors in
LDA. Early attempts at resolving this discrepancy focused
on crystal structure. Indeed it was shown that with optimized
crystal structures, DFT calculations could yield insulating
band structures for YH3 . 12,13 Unfortunately, these relaxed
structures are not consistent with diffraction measurements,
and such an explanation would not explain the similar insulating properties of the other M H3 compounds.
DFT is a theory of ground-state properties, and the eigenvalues of the Kohn-Sham equations cannot be equated to
quasiparticle excitations.14 Nevertheless, any nonpathological accurate approximation to the exact DFT should predict
an insulating Kohn-Sham band structure, because static responses and other ground-state properties depend on the
presence or absence of a gap. For example, LO-TO splitting
would be present in the phonon spectrum of ionic insulator
like YH3 but not in a metal.
Recent studies on rare-earth trihydrides have considered
electron correlation effects,15–19 particularly electron correlations on the H sites. However, since there are no partially
occupied d or f orbitals in LaH3 and YH3 , nor is there evidence that these materials are near a quantum critical point,
these hydrides represent a novel class of strong correlated
materials. Calculations done within the GW approximation
showing band gaps that are reconcilable with experiment are
69 085104-1
©2004 The American Physical Society
PHYSICAL REVIEW B 69, 085104 共2004兲
WU, COHEN, SINGH, GUPTA, AND GUPTA
obtained, both with the cubic BiF3 structure (LaH3 ) and the
hexagonal HoD3 structure 共a simplified YH3 structure兲.17–19
However in the GW calculations, the dielectric function is
inherently wrong since it comes from the LDA, where the
ground state is metallic and not insulating. Also, the GW
calculations are very cumbersome, and it would be advantageous to a simple and fast method that gives reasonable results.
The GW method approaches the problem as being most
simply viewed as due to the treatment of excited states, but if
the problem is due to self-interaction, it is fundamentally a
ground-state problem. The LDA makes large errors in description of the one-electron hydrogen atom because of incomplete cancellation of the Hartree and xc self-interactions.
One way to remove the spurious self-interactions in LDA
and GGA calculations is the so-called self-interaction
correction.20 In this approach an orbital dependent selfinteraction is calculated at each step and used to create an
orbital dependent single-particle Hamiltonian. This method
suffers from basis set dependence, e.g., the computed correction is zero for a Bloch state representation. We explore
whether one can remain within conventional ground-state
DFT and obtain the proper electronic structure of rare-earth
trihydrides.
of G,27 and another G-J type for k⫽4 (G-J4), which was
tested by Rushton et al. for silicon.28
One subtle issue in WDA implementations is shell
partitioning.23 Although the WDA has the exact form of xc
energy, it does not describe the exchange energy between
core and valence states very well due to the approximate
model G function. Outside core regions, the core density
vanishes, so does the core-valence exchange energy. However in the WDA scheme, because the range of integration of
G is similar to the size of atoms, and there is no distinction
between core and valence electrons, core and valence electrons would dynamically screen valence electrons equally,
and it could cause the core-valence exchange energy to be
exaggerated. If the exact G function for an arbitrary electron
system is known, it will naturally describe intershell interactions correctly. On the other hand, the LDA can give correct
intershell contributions, since the LDA depends only on local
density. Based on this observation, a shell partitioning approach was proposed23,30 to guarantee the correct behavior of
core-valence xc interaction, in which the valence-valence interactions are treated with the WDA, while core-core and
core-valence with the LDA.
If n v(r) and n c(r) denote valence and core electron density, respectively, the xc energy
II. WEIGHTED DENSITY APPROXIMATION
LDA
WDA
LDA
E xc关 n 兴 ⫽E xc
关 n 兴 ⫹E xc
关 n v ,n 兴 ⫺E xc
关 n v ,n 兴 ,
The weighted density approximation 共WDA兲 共Refs. 21–
23兲 has the exact nonlocal form of DFT expression of xc
energy,
E xc关 n 兴 ⫽
冕冕
n 共 r兲 n 共 r⬘ 兲
兩 r⫺r⬘ 兩
关 ḡ xc共 r,r⬘ ,n 兲 ⫺1 兴 drdr⬘ .
where
WDA
E xc
关 n v ,n 兴 ⫽
共2兲
where the weighted density n̄(r) can be determined from the
xc hole condition:
冕
n 共 r⬘ 兲 G 关 兩 r⫺r⬘ 兩 ,n̄ 共 r兲兴 dr⬘ ⫽⫺1.
冕冕
n v共 r兲 n v共 r⬘ 兲
兩 r⫺r⬘ 兩
G 关 兩 r⫺r⬘ 兩 ,n̄ 共 r兲兴 drdr⬘ ,
共5兲
共1兲
Since the exact pair-distribution function ḡ xc(r,r⬘ ,n) is elusive, a model WDA function G is made,
ḡ xc共 r,r⬘ ,n 兲 ⫺1⫽G 关 兩 r⫺r⬘ 兩 ,n̄ 共 r兲兴 ,
LDA
E xc
关 n v ,n 兴 ⫽
冕冕
n v共 r兲 n v共 r兲
兩 r⫺r⬘ 兩
G 关 兩 r⫺r⬘ 兩 ,n 共 r兲兴 drdr⬘ .
共6兲
The weighted density n̄(r) can be chosen from the following
sum rule
冕兵
共3兲
This assures no self-interaction in a one-electron system. The
WDA also gives correct results for the uniform electron gas
because it reduces to the LDA in this case. However because
WDA
is incorrect,
G WDA(r,r⬘ )⫽G WDA(r⬘ ,r), the form of V xc
and it causes wrong asymptotic behavior at large distance
away from nucleus.23 Fortunately this error is not substantial
when calculations are done at experimental structure.
In the WDA, an approximation is made for the function
G, a natural choice of G would be of a uniform system, but
Mazin and Singh25 emphasized, based on the xc kernel, that
G of the uniform electron gas is not optimal. Several model
forms of G have been suggested, such as the GunnarssonJones
共G-J兲
共Ref.
24兲
ansatz,
G G-J(r,n)⫽c 兵 1
⫺k
⫺exp(⫺关r/␭兴 )其, with k⫽5; and the Gritsenko et al.
共GRBA兲 共Ref. 26兲 ansatz, G GRBA(r,n)⫽c•exp(⫺关r/␭兴k),
with k⫽1.5. In this paper, we also used a homogeneous type
共4兲
n v共 r⬘ 兲 G 关 兩 r⫺r⬘ 兩 ,n̄ 共 r兲兴 ⫹n c共 r兲 G 关 兩 r⫺r⬘ 兩 ,n 共 r兲兴 其 dr⬘
⫽⫺1.
共7兲
The xc potential for core and valence states are different in
that the xc energy depends on core and valence electron density explicitly. The derivation is attached in the appendix. An
accurate potential makes the calculations fully consistent, so
that atomic forces can be determined by the HellmannFeynmann theory. We found that the calculated forces within
the WDA agree with numerical results of finite energy difference very well.
III. RESULTS AND DISCUSSION
A. Calculation details
The calculations were done using a plane-wave basis with
pseudopotential method. Pseudopotentials were generated by
Troullier and Martins29 scheme. We treated semicore states
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PHYSICAL REVIEW B 69, 085104 共2004兲
WEIGHTED-DENSITY-APPROXIMATION DESCRIPTION . . .
TABLE I. With the calculated atomic positions and fixed c/a
ratio, the optimized equilibrium volumes 共in Å3 ) of hexagonal YH3
(Y2 H6 ). Here a, b, c, and d denote uniform, GRBA, G-J, and G-J4
forms of G, respectively. Numbers in parentheses are the percentage
deviations from experiment.
LDA
WDA 共a兲
WDA 共b兲
WDA 共c兲
WDA 共d兲
Expt.
72.8
(⫺6.2)
75.0
(⫺3.4)
77.4
(⫺0.3)
75.6
(⫺2.6)
79.4
(⫹2.3)
77.6
of Y and La with LDA, higher states and H 1s states with
WDA, and lower states are pseudized. The ionic configurations and cutoff radii are 4s 2 4p 6 4d 2 and r s,p,d
c
⫽1.55,1.55,1.67 Bohr for Y, 5s 2 5p 6 5d 0.5 and r s,p,d
c
⫽1.77,1.77,1.77 Bohr for La, and 1s 0.92p 0.1 and r s,p
c
⫽1.15,1.15 Bohr for H. Well converged plane-wave basis
sets were used; the energy cutoffs are 156 Ry for hexagonal
YH3 , 121 Ry for cubic YH3 , and LaH3 . We used a 48⫻48
⫻84 real space grid and 12 k-point mesh for hexagonal
YH3 , and a 48 real space grid and 12⫻12⫻12 k-point mesh
for cubic YH3 and LaH3 to obtain the band structure.31
B. Hexagonal YH3
As mentioned before, although the exact structure of hexagonal YH3 is still under investigation, its insulating property is believed to be an electronic property, i.e., not sensitive
to the crystal structure.32,33 We studied a simplified hexagonal LaF3 structure with a Y2 H6 unit cell,11 in which Y atoms
form a hexagonal close-packed 共hcp兲 structure. The measured lattice constants are a⫽3.672 Å and c/a⫽1.81. We
optimized the H positions using LDA and WDA 共with four
types of G), and all the results are consistent with the previous LDA result.11 We also optimized the equilibrium volume with fixed c/a ratio. Table I shows that the WDA volumes deviate by only about 2–3 % from experimental data.
The calculated bands of the LDA and WDA 共with the
G-J4 type of G) are shown in Fig. 1. The LDA predicts a
direct band gap of 1.0 eV at ⌫ and an overlap of 0.93 eV
between ⌫ and K, which leads to a semimetal. Our present
LDA results are a little different from previous results 共0.6
eV for the ⌫ gap and 1.3 eV for the overlap11兲 because we
used the Hedin-Lundqvist xc function, whereas the Wigner
ansatz was used previously. The WDA 共with the G-J4 type of
FIG. 1. LDA and WDA 共with the G-J4 type of G) electron
bands of the hexagonal YH3 .
TABLE II. Calculated LDA and WDA 共a, b, c, and d as seen in
Table I兲 fundamental and ⌫ gaps 共in eV兲, compared with GW
results.17 A negative band gap means band overlap.
LDA WDA 共a兲 WDA 共b兲 WDA 共c兲 WDA 共d兲 GW
fd. gap ⫺0.93
⌫ gap
1.0
⫺0.74
1.2
⫺0.35
1.5
⫺0.72
1.2
0.41
2.2
0.6
2.9
G) predicts a direct ⌫ gap of 2.2 eV, and opens up a fundamental gap of 0.41 eV between ⌫ and K. Thus, the WDA
predicts a substantial insulating gap. Our WDA band gaps
are close to the GW calculations17 of 0.6 eV for the fundamental gap and 2.9 eV for the direct ⌫ gap. In the Brillouin
zone of the more complicated tripled HoD3 structure of
YH3 , this indirect ⌫-K gap will be folded to form a direct
gap at ⌫ point, and they are roughly of the same order of
magnitude.11,17 Thus, a small gap between ⌫ and K in the
simple hexagonal structure makes YH3 a semiconductor
even in the complicated HoD3 structure.
We also calculated the WDA bands with the other three G
functions, and the band gaps are shown in Table II. We found
that the WDA with all these G’s gives better band gaps than
the LDA. Except for the G-J4 form, the GRBA 共Ref. 26兲
form is superior to the other two, because it predicts a larger
direct ⌫ gap and smaller overlap between ⌫ and K. These
results illustrate that the WDA band structure is sensitive to
the choice of G. We compare these pair-distribution functions by drawing them together in Fig. 2, where n
⫽3/(4 ␲ r s3 ), and in the WDA calculations of YH3 , 0.8⬍r s
⬍2.3. We found the G-J4 form has the shortest range, and
the range of GRBA ansatz is shorter than the other two. Our
calculations indicate that a shorter range G gives larger band
gaps. Interestingly, such functions, particularly GRBA, yield
static responses closer to Monte Carlo results for the uniform
electron gas.25
C. Cubic YH3 and LaH3
We performed LDA and WDA 共with G-J4 only兲 calculations on cubic BiF3 structure of YH3 and LaH3 , in which the
Y or La atoms constitute a face-centered cubic 共fcc兲 lattice
and H atoms are located at the tetrahedral and octahedral
sites. Recently the insulating fcc YH3 was stabilized by existence of MgH2 , 33 but there are large discrepancies in the
FIG. 2. Four types of the pair-distribution function G(r,n) with
r s ⫽0.8 and r s ⫽2.3.
085104-3
PHYSICAL REVIEW B 69, 085104 共2004兲
WU, COHEN, SINGH, GUPTA, AND GUPTA
Although our DFT-WDA band gaps of YH3 and LaH3
agree with GW quasiparticle calculations very well, the measured optical gap is ⬃2.8 eV for YH3 and ⬃1.8 eV for
LaH3 , which are still 1–2 eV larger. Gelderen et al.17 argued
that the fundamental band gap of hexagonal YH3 is not the
measured optical gap due to the forbidden transition between
the highest valence and the lowest conduction bands. They
concluded that the measured value corresponds to the direct
gap at ⌫ of the hexagonal LaF3 structure. However the real
hexagonal YH3 is more complex and it may have low symmetry so that the dipole matrix element could may well not
be zero. Our DFT-WDA method may underestimate band
gaps due to the DFT gap discontinuity at the Fermi level.36
IV. SUMMARY
FIG. 3. LDA and WDA 共with the G-J4 type of G) electron
bands of the cubic YH3 and LaH3 .
measured volumes of fcc YH3 . Ahuja et al.13 expected
32.2 Å3 , while van der Molen et al.33 extrapolated 36.9 Å3
and Gogh et al.32 obtained 38.0 Å3 . Our LDA and WDA
equilibrium volumes are 32.4 Å3 and 35.2 Å3 , respectively.
We calculated band structure at experimental volume of
36.9 Å3 . For LaH3 the measured volume 44.2 Å3 is close to
both LDA and WDA results, and we used this volume to
calculate band structure.
In Fig. 3 are the calculated band structures of cubic YH3
and LaH3 . The LDA predicts that both of them are metals
with direct overlaps of 1.0 eV for YH3 and 0.46 eV for LaH3
at ⌫ point, which agree with previous calculations.19,34 Contrary to the LDA, the WDA predicts that both trihydrides are
semiconductors with direct ⌫ gaps of 0.19 eV and 0.80 eV
for YH3 and LaH3 , respectively. However the fundamental
gaps are about 0.1 eV smaller, because the energy of lowest
conduction band at L is slightly lower than that at ⌫. Our
WDA band gaps are in good agreement with previous GW
calculations, e.g., Alford et al.34 predicted a ⌫ gap of 0.2–0.3
eV for YH3 and 0.8 –0.9 eV for LaH3 , and Chang et al.35
obtained 0.8 eV for LaH3 . We found that the hexagonal and
cubic YH3 have similar small gaps, suggesting that the insulating properties are of electronic origin rather than structural.
Another interesting feature of the WDA band structures is
that compared with the LDA, the overall valence bandwidth
hardly changes. It supports a view that rare earth hydrites are
not strong correlated systems. This is consistent with previous GW calculations,17,19,34 but in contrast to the many-body
model Hamiltonian calculations which show a large valence
bandwidth decrease.15 Thus, our results support the previous
conclusion17,34 that rare-earth trihydrides are just band insulators, and that strong correlations are not required.
In conclusion, we have studied the band structures of the
hexagonal YH3 and cubic YH3 and LaH3 using the firstprinciples WDA method. The calculations showed that YH3
and LaH3 are both semiconductors, and this insulating property does not rely on the detailed crystal structure. The calculated band gaps agree with GW results very well, and they
are also consistent with experiment. Our WDA calculations
predicted good ground-state structures as well. The success
of WDA may be attributed to the absence of self-interaction
in H 1s states.
ACKNOWLEDGMENTS
We are grateful for helpful discussions with I. Geibels, P.
Vajda, and K. Yvon. This work was supported by the Office
of Naval Research under ONR Grant Nos. N000149710052,
N00014-02-1-0506, and N0001403WX20028. Calculations
were done on the Center for Piezoelectrics by Design 共CPD兲
computer facilities and on the Cray SV1 at the Geophysical
Laboratory supported by NSF EAR-9975753, and the W. M.
Keck Foundation. We also thank the Institut du Developpement et des Ressources en Informatique Scientifique 共IDRIS兲
for a grant of computer time. D.J.S. is grateful for the hospitality of the University of Paris Orsay during part of this
work.
APPENDIX:
XC POTENTIAL OF THE WDA
The xc potential V xc(r) in the DFT is the functional derivative of the xc energy,
V xc共 r兲 ⫽
␦ E xc关 n 兴
.
␦ n 共 r兲
共A1兲
In the shell-partitioning scheme as described in Sec. II, the
xc potential of core states
xc
V core
共 r兲 ⫽
␦ E xc关 n 兴
LDA
⫽V xc
共 r兲 ⫹V 1 共 r兲 ⫺V 2 共 r兲 ,
␦ n c共 r兲
共A2兲
where the first term is the ordinary LDA potential, and
085104-4
V 1 共 r兲 ⫽
冕
n v共 r兲 n v共 r⬘ 兲 ␦ G 关 兩 r⫺r⬘ 兩 ,n̄ 共 r兲兴
dr⬘ , 共A3兲
␦ n c共 r兲
兩 r⫺r⬘ 兩
PHYSICAL REVIEW B 69, 085104 共2004兲
WEIGHTED-DENSITY-APPROXIMATION DESCRIPTION . . .
V 2 共 r兲 ⫽
冕
n 2v 共 r兲 ␦ G 关 兩 r⫺r⬘ 兩 ,n 共 r兲兴
dr⬘ .
␦ n 共 r兲
兩 r⫺r⬘ 兩
V 5 共 r兲 ⫽
共A4兲
In order to evaluate V 1 (r), we need the expression for
␦ n̄(r)/ ␦ n c(r). This can be done by considering the variation
of sum rule Eq. 共7兲 with respect to n c(r),
␦ n̄ 共 r兲
⫽
␦ n c共 r兲
n 2 共 r兲
冕
n v共 r兲
n v共 r⬘ 兲
␦ G 关 兩 r⫺r⬘ 兩 ,n̄ 共 r兲兴
␦ n̄ 共 r兲
冕冕
V 6 共 r兲 ⫽
冕
n v共 r⬘ 兲 ␦ G 关 兩 r⫺r⬘ 兩 ,n̄ 共 r兲兴
冕
h 2 共 r兲 ⫽
␦ n̄ 共 r兲
兩 r⫺r⬘ 兩
n v共 r⬘ 兲
␦ G 关 兩 r⫺r⬘ 兩 ,n̄ 共 r兲兴
␦ n̄ 共 r兲
冕
dr⬘
V 7 共 r兲 ⫽2
dr⬘ ,
dr⬘ ,
共A6兲
冕
n 2v 共 r兲 h 1 共 r兲
.
n 2 共 r兲 h 2 共 r兲
共A8兲
␦ n̄ 共 r兲
⫽⫺
␦ n v共 r兲
␦ E xc关 n 兴
xc
LDA
⫽V xc
V vln
共 r兲 ⫽
共 r兲 ⫹V 3 共 r兲 ⫹V 4 共 r兲 ⫹V 5 共 r兲
␦ n v共 r兲
共A9兲
V 3 共 r兲 ⫽
V 4 共 r兲 ⫽
1
冕
冕
兩 r⫺r⬘ 兩
n v共 r⬘ 兲
兩 r⫺r⬘ 兩
V 5 共 r兲 ⫽
G 关 兩 r⫺r⬘ 兩 ,n̄ 共 r兲兴 dr⬘ ,
共A10兲
G 关 兩 r⫺r⬘ 兩 ,n̄ 共 r⬘ 兲兴 dr⬘ ,
共A11兲
G. G. Libowitz and A. J. Mealand, in Handbook on the Physics
and Chemistry of Rare Earths, edited by K. A. Gschneidner, Jr.
and L. Eyring 共North-Holland, Amsterdam, 1979兲, Vol. 3, p.
299.
2
J.N. Huiberts, R. Griessen, J.H. Rector, R.J. Wijngaarden, J.P.
Dekker, D.G. de Groot, and N.J. Koeman, Nature 共London兲 380,
231 共1996兲; J.N. Huiberts, R. Griessen, R.J. Wijngaarden, M.
Kremers, and C. Van Haesendonck, Phys. Rev. Lett. 79, 3724
共1997兲.
3
M. Rode, A. Borgschulte, A. Jacob, C. Stellmach, U. Barkow, and
J. Schoenes, Phys. Rev. Lett. 87, 235502 共2001兲.
4
J. Osterwalder, Z. Phys. B: Condens. Matter 61, 113 共1985兲.
5
N.F. Miron, V.I. Shcherbak, V.N. Bykov, and V.A. Levdik, Kristallografiya 17, 404 共1972兲 关Sov. Phys. Crystallogr. 17, 342
共1972兲兴.
共A14兲
冕
G 关 兩 r⫺r⬘ 兩 ,n̄ 共 r⬘ 兲兴
n v共 r⬙ 兲
n 共 r兲
2
冕
␦ G 关 兩 r⬘ ⫺r⬙ 兩 ,n̄ 共 r⬘ 兲兴
␦ n̄ 共 r⬘ 兲
,
dr⬙
共A15兲
n c共 r兲
n v共 r⬘ 兲
␦ G 关 兩 r⫺r⬘ 兩 ,n̄ 共 r兲兴
␦ n̄ 共 r兲
.
dr⬘
共A16兲
And the final expressions for V 5 and V 6 are
where
n v共 r⬘ 兲
G 关 兩 r⫺r⬘ 兩 ,n 共 r兲兴 dr⬘ ⫹V 2 共 r兲 .
兩 r⫺r⬘ 兩
␦ n̄ 共 r⬘ 兲
⫽⫺
␦ n v共 r兲
共A7兲
Similarly the xc potential of valence states is
⫹V 6 共 r兲 ⫺V 7 共 r兲 ,
n v共 r兲
共A13兲
Again we can obtain ␦ n̄(r⬘ )/ ␦ n v(r) and ␦ n̄(r)/ ␦ n v(r) from
variation of Eq. 共7兲,
we can simplify
V 1 共 r兲 ⫽
n v共 r兲 n v共 r⬘ 兲 ␦ G 关 兩 r⫺r⬘ 兩 ,n̄ 共 r兲兴
dr⬘ ,
␦ n v共 r兲
兩 r⫺r⬘ 兩
. 共A5兲
If we denote h 1 (r) and h 2 (r) as
h 1 共 r兲 ⫽
n v共 r⬘ 兲 n v共 r⬙ 兲 ␦ G 关 兩 r⬘ ⫺r⬙ 兩 ,n̄ 共 r⬘ 兲兴
dr⬘ dr⬙ ,
␦ n v共 r兲
兩 r⬘ ⫺r⬙ 兩
共A12兲
冕
n v共 r⬘ 兲
h 1 共 r⬘ 兲 ␦ G 关 兩 r⫺r⬘ 兩 ,n̄ 共 r⬘ 兲兴
h 2 共 r⬘ 兲
V 6 共 r兲 ⫽⫺
␦ n̄ 共 r⬘ 兲
n v共 r兲 n c共 r兲 h 1 共 r兲
.
n 2 共 r兲 h 2 共 r兲
dr⬘ ,
共A17兲
共A18兲
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