Electronic structure of lithium-based antiperovskite hydrides

Journal of Alloys and Compounds, 209 (1994) 159-165 JALCOM 1095

159

Electronic structure of lithium-based antiperovskite hydrides Emilio Orgaz and Mich~le Gupta lnstitut de Science des Matdriaux, URA 446, b(tt. 415, Universitd de Paris-Sud, 91405 Orsay C~dex (France) (Received November 11, 1993)

Abstract We have studied theoretically the electronic structure of EuLiH3, SrLiH3 and BaLiH 3 by means of the augmented plane wave method. The energy bands and electronic density of states and their partial wave analysis are presented. These hydrides crystallize within the inverse cubic perovskite structure (Pm3m). Our investigation shows that these hydrides are semiconductors with relatively large energy gaps. The energy gaps are of indirect type between the X symmetry point, at the top of the valence band (H s states), and the M symmetry point, at the bottom of the conduction band (D d, s states; D=-Eu, Sr or Ba). Our results are in general agreement with optical spectroscopy data. They account for the three absorption peaks observed experimentally in EuLiH3 where the f electrons of the Eu atom are involved.

1. Introduction

Since 1964 to date, several ternary light metal hydrides such as LiBeH3 [1], the perovskites KMgH3 [2] and CsCaH3 [3] and the antiperovskite phases SrLiH3 [4, 5], BaLiH3 [4, 6] and EuLiH3 [7] have been synthesized. The physicochemical properties of these systems have not yet been completely investigated owing to their low chemical stability which makes them difficult to handle and measure. However, recently the electronic properties of these ternary compounds have received some attention in view of the possibility of metallic behaviour and perhaps superconductivity. In 1987, Overhauser [8] analysed the X-ray diffraction data of Bell and Coates [1] of the LiBeH3 phase and proposed a complex cubic perovskite structure for this material. From these results, he estimated the density and the electron density parameter which he concluded to be in the range of the values assumed for metallic hydrogen. Thus Overhauser suggested that the LiBeH3 phase, if metallic, could be superconducting. The electronic structure of the cubic perovskite LiBeH3 has been largely investigated. The first ab initio band structure calculations [9, 10] showed that this compound is a semiconductor with a small energy gap (0.21 eV) if one assumes the structure proposed by Overhauser. The energy gap appears between the X and R symmetry points of the simple cubic irreducible Brillouin zone (SCIBZ). The effect of isostatic pressure was simulated by decreasing by 3.4% the crystal parameter. Gupta and PercheronGuegan found that this perovskite becomes metallic with a low density of states (DOS) at Fermi energy.

Pseudopotential local density formalism calculations by Yu and Lam [11] and Martins [12], and molecular Hartree-Fock calculations by Press et al. [13] and Seel et al. [14] confirmed these results. Nevertheless, the electronic parameters resulting from these calculations do not match with those assumed by Overhauser: the calculated electronic density is twice as large as the estimation, and the optimal lattice parameter obtained by total energy calculations [12, 14] is twice as large as the value derived from the X-ray experimental data. From these results it appears that the LiBeH3 system is not a good candidate for metallic behaviour or for superconductivity. In addition, Martins argues, from a crystal chemistry viewpoint, that LiBeH3 should have a silicate-like structure and not a perovskite structure. Recently, experimental investigation confirmed this suggestion. Cantrell et al. [15, 16] indexed the X-ray pattern of LiBeH3 in the FeTiO3-ilmenite type of structure (which is rhombohedral). Among the lithium-based ternary hydrides, EuLiH3, SrLiH3 and BaLiH3 have been found to crystallize in the inverse perovskite structure [4-7]. The electric, magnetic and optical absorption properties of SrLiH3

0925-8388/94/$07.00 © 1994 Elsevier Science S.A. All rights reserved SSDI 0925-8388(93)01095-L

TABLE 1. Structural parameters in fingstroms, band widths to and energy gaps Ag in electronvolts a EuLiH3 [7] SrLiH3 [4] BaLiH3 [6]

d(Li-H) d(D-H) d(D-H) in DH2 to

3.796 1.898 3.834 1.917 4.023 2.012

2.684 2.711 2.845

2.45 2.49 2.67

Ag

4.94 4.16 5.10 3.54 3.82 3.93

160

E. Orgaz, M. Gupta / Electronic structure of DLiH3 (D=-Eu, Sr, Ba) E n e r g y B a n d s of EuLiH 3

N u m b e r of Electrons

0

10

5 '

'

'

i

i

'

'

0.4 /

I

I

/

0.2 I I I

Ef

o

r t I I

o

r~ I

-0.2

-0.4

J

-0.6

0

(o)

X

M

F

R

,

,

I

~

,

I

20

M

I

,

,

40

I

I

,

60

80

Total DOS N u m b e r of E l e c t r o n s

Energy Bands of SrLiH 3

5

10

0.4

/

0.2

I

0

v

tw

O r~

-0.2

-0.4

I

-0.6

(b) Fig. i .

20

F

X

M

r

R

40

60

I

L

i

80

Total DOS

(con~nued)

E. Orgaz, M. Gupta / Electronic structure of DLiH3 (D =-Eu, St, Ba) Energy Bands of BaLiH3

161

Number of Electrons 0

5

10 '

q

I

,

,

I

"R ~.~

i Ef

0

I I

-0.2

~

-

~

-

i

-0.4

t

~

--0.6

L

0

(c)

r

x

M

r

R

M

~

,

l

20

~

,

1

I

,

40

,

,

I

60

I

,

,

80

Total DOS

Fig. 1. Energy bands along some high symmetry directions of the irreducible Brillouin zone of the simple cubic structure and total DOS for (a) EuLiH3, (b) SrLiH3 and (c) BaLiH3. Energy in rydbergs and DOS in states of both spins per rydberg per unit cell.

and EuLiH3 were investigated by Greedan in 1971 [17, 18]. The electric conductivity from room temperature to 400 °C of SrLiH3 and EuLiH3, determined by the van der Pauw technique on single crystals, is of the order of 10-7 (11 cm)-I at room temperature for both compounds. The energy gap derived from the activation energy for electric conduction seems to be out of range compared with the values determined by optical measurements. This could indicate that the intrinsic electric conduction is not being observed as a result, perhaps, of impurity effects. Diffuse reflectance and optical absorption spectra exhibit peaks at 4 eV for S r L i H 3 and 2.0, 4.0 and 5.15 eV for EuLiH3. In the same investigation, Greedan [17] found that E u L i H 3 is a ferromagnetic insulator with Tc = 38 K and Oc = 40 K, while SrLiH3, and presumably BaLiH3, do not have magnetic properties. We have investigated theoretically the electronic structure of EuLiH3, SrLiH3 and BaLiH3 by means of an ab initio technique and analysed our results in the light of the available experimental data. 2. Results and discussion

In the inverse perovskite structure hydrides DLiH3, the D atom (D = Eu, Sr and Ba) is located at the (0,0,0)

position, the Li atom at the centre of the cubic unit cell, (-~, ~, -~) position, and the H atoms at the 1 0) positions. In these compounds the H atoms form octahedral cages around the Li atom with Li-H distances slightly shorter than in LiH (NaCI structure), while the D - H distance is around 8% larger than in the DH2 dihydrides (Table 1). The electronic properties of these hydrides have been determined by means of the augmented plane wave (APW) method [19, 20]. The non-self-consistent crystal potential was calculated in the non-overlapping muffin tin (MT) approximation where the local exchange potential was modelled by means of the X,, Slater method (a = 1). In fact, from previous investigations of several hydrides, we concluded that our results of non-selfconsistent APW calculations with Slater exchange potential are quite similar to those obtained with self-consistent methods using more elaborate exchange potentials in the local density approximation (LDA) framework. Because of the short Li-H distance, the MT spheres occupy less than 60% of the unit cell volume. The departure of the potential outside the MT spheres from a constant value was taken into account (warped MT corrections). The energy eigenvalues and wavefunctions

E. Orgaz, M. Gupta / Electronic structure of DLiH3 (D~Eu, Sr, Ba)

162

have been calculated ab initio at 56 points in the SCIBZ. The ab initio energy bands were expanded into a set of 30 symmetrized plane waves giving a maximum r.m.s. error of 3 mRy. The DOSs have been calculated by the linear energy tetrahedron method [21]. The energy bands due to the localized f electrons of the rare earth based compound EuLiH3 are not plotted in the energy vs. wavevector k dispersion curves. It is well known that these bound f states are very sensitive to correlation effects which are not fully included in the LDA. In our calculation of the crystal potential we have of course taken into account the full electronic density of the rare earth atom, including f electrons. The calculated energy of the resonance corresponding to the electrons gives us the general position of the f electrons. These results will be used below in our discussion of the experimental optical absorption spectrum. However, we decided not to plot the f bands since the width of these states is very sensitive to nonEu

LDA corrections. In the case of BaLiH3 and SrLiH3, the 4f electrons are located at higher energies than in the europium compound, they have also been treated in the same manner, and are not plotted in the E(k) dispersion curves. In Fig. 1 we plot the energy bands and total DOS and in Fig. 2 the partial DOS for the DLiH3 (D- Eu, Sr and Ba) hydrides. The analysis of the wavefunction is shown in Tables 2--4. For the three hydrides, at the centre of the SCIBZ, point F, the lowest energy state has a strong H s character; it is hybridized with D s and Li s states. Next, at higher energy, a doubly degenerate state is mainly composed of H s with a bonding contribution from the D d~,orbitals. The first three bands are filled by the six valence electrons of the compounds; since they are separated from the higher energy bands, the hydrides are found to be semiconducting. From the partial wave analysis of the DOS inside the MT spheres (shown in Fig. 2), it appears

1=0

1 Ef i

0

__~b

_

Eu

1=1

1 Ef l Eu

I=2

20

°I

10 0

Ef i ____+__

_

Li 1=0

Li 1=0

]

2

2

1

1 Ef i

o

0 Li 1= 1

Li 1= 1

Z

2

1

1 Ef I

0

.

28

'

H

l=0

14 0

L

-0.4

(o) Fig. 2.

H

3[

-0.2 0 Energy (Ry)

~

0.2

-i ° -~ 0.4

1=0

1

0

(b)

-0.4

-0.2

0 Energy (Ry)

0.2

0.4

(contmued)

E. Orgaz, M. Gupta /Electronic structure of DLiHs (D=--Eu, Sr, Ba)

.

10

TABLE 2. Wavefunction analysis for EuLiH3 at the F (0,0,0), X (1,0,0), M (1,1,0) and R (1,1,1) symmetry points (in units of lr/a) (the Fermi energy is 0.384 Ry) E (Ry)

0

Et I

0

24 120

E

~

Li 1=0

2

Et

0 IA 1=1

4 0 2

Ef

0

H

0 -0.4

-0.2

Eus

Eup

Eud

Lis

Lip

H s

- 0.132 0.174 0.775 0.777 0.864 1.188

1 2 3 2 3 1

0.10 0.00 0.00 0.00 0.00 0.40

0.00 0.00 0.08 0.00 0.00 0.00

0.00 0.11 0.00 0.60 0.82 0.00

0.06 0.00 0.00 0.00 0.00 0.03

0.00 0.00 0.23 0.00 0.00 0.00

0.47 0.62 0.00 0.18 0.00 0.24

X (1,0,0) 0.011 0.021 0.233 0.584 0.603 0.890

1 1 1 1 2 2

0.07 0.00 0.00 0.00 0.00 0.16

0.00 0.10 0.00 0.00 0.00 0.00

0.03 0.00 0.00 0.63 0.43 0.30

0.00 0.06 0.00 0.00 0.00 0.00

0.05 0.00 0.00 0.00 0.10 0.00

0.50 0.53 0.77 0.00 0.00 0.25

M (1,1,0) 0.032 0.091 0.534 0.712 0.878

1 2 1 2 1

0.00 0.00 0.13 0.00 0.00

0.00 0.07 0.00 0.00 0.00

0.11 0.00 0.35 0.64 0.27

0.04 0.00 0.00 0.00 0.08

0.00 0.06 0.00 0.00 0.00

0.52 0.59 0.00 0.00 0.27

R (1,1,1) 0.033 3 0.702 1 0.850 3

0.00 0.00 0.00

0.00 0.00 0.28

0.11 0.00 0.00

0.00 0.27 0.00

0.05 0.00 0.00

0.54 0.00 0.00

1=0

20

( C)

Degeneracy

r (o,o,o)

Ba 1=1

2

163

0

02

0.4

TABLE 3. Wavefunction analysis for SrLiH3 at the F (0,0,0), X (1,0,0), M (1,1,0) and R (1,1,1) symmetry points (in units of 7r/ a) (the Fermi energy is 0.364 Ry)

Energy (Ry)

Fig. 2. Partial wave analysis of the DOS inside each MT, for (a) EuLiH3, (b) SrLiH3 and (c) BaLiH3. DOS in states of both spins per rydberg per total number of atoms in the unit cell (per D atom, per Li atom or per three H atoms).

clearly that the valence bands are dominated by a strong H s character. An important bonding contribution from the D d orbitals is also present. In this energy range a hybridization with Li s, p and D s, p states is also observed. The top of the valence band (third band) occurs at the X symmetry point; it is formed by nonbonding H-s states as shown in the wavefunction analysis of Tables 2-4. The valence band widths to of the three hydrides listed in Table 1 are 4.94 eV, 5.10 eV and 3.82 eV for EuLiH3, SrLiH3 and BaLiH3 respectively. The smaller width is obtained for the Ba compound which has a lattice constant about 6% larger than that of the Eu and Sr hydrides. This large increase in the metal-H and H-H distances leads to a decrease in the energetic separation between bonding states at the bottom of the valence band and non-bonding states at

E (Ry)

Degeneracy

Sr s

Sr p

Sr d

Li s

Li p

H s

-0.141 0.181 0.729 0.738 0.813 1.072

1 2 3 2 3 1

0.13 0.00 0.00 0.00 0.00 0.46

0.00 0.00 0.11 0.00 0.00 0.00

0.00 0.13 0.00 0.64 0.87 0.00

0.05 0.00 0.00 0.00 0.00 0.03

0.00 0.00 0.18 0.00 0.00 0.00

0.44 0.61 0.00 0.18 0.00 0.25

X (1,0,0) 0.001 0.007 0.235 0.560 0.596

1 1 1 1 2

0.00 0.10 0.00 0.00 0.00

0.10 0.00 0.00 0.00 0.00

0.00 0.04 0.00 0.68 0.50

0.50 0.00 0.00 0.00 0.00

0.00 0.05 0.00 0.00 0.08

0.51 0.47 0.73 0.00 0.00

M (1,1,0) 0.043 0.080 0.490 0.681

1 2 1 2

0.00 0.00 0.19 0.00

0.00 0.07 0.00 0.00

0.12 0.00 0.35 0.71

0.04 0.00 0.00 0.00

0.00 0.05 0.00 0.00

0.51 0.56 0.00 0.00

R (1,1,1) 0.045 3 0.674 1 0.748 3

0.00 0.00 0.00

0.00 0.00 0.33

0.12 0.00 0.00

0.00 0.24 0.00

0.05 0.00 0.00

0.53 0.00 0.00

r (o,o,o)

164

E. Orgaz, M. Gupta /Electronic

structure

of DLiH3 (D~Eu, Sr, Ba)

TABLE 4. Wavefunction analysis for BaLiH3 at the F (0,0,0), X (1,0,0), M (1,1,0) and R (1,1,1) symmetry points (in units of ~r/a) (the Fermi energy is 0.316 Ry)

Eu-s/H-s/Li-s ""-. u-deg/H-s Eu.dt2g .....

0.0 E (Ry)

Degeneracy

Bas

Bap

Bad

Li s

Li p

H s

-g F (1,0,0) -0.111 1 0.130 2 0.641 2 0.710 3 0.731 3 1.085 1

0.14 0.00 0.00 0.00 0.00 0.46

0.00 0.00 0.00 0.11 0.00 0.00

0.00 0.16 0.63 0.00 0.88 0.00

0.06 0.00 0.00 0.00 0.00 0.02

0.00 0.00 0.00 0.23 0.00 0.00

0.47 0.59 0.20 0.00 0.00 0.25

X (1,0,0) 0.011 0.033 0.172 0.466 0.518

1 1 1 1 2

0.09 0.00 0.00 0.00 0.00

0.00 0.14 0.00 0.00 0.00

0.05 0.00 0.00 0.72 0.55

0.00 0.06 0.00 0.00 0.00

0.05 0.00 0.00 0.00 0.07

0.51 0.52 0.76 0.00 0.00

M (1,1,0) 0.008 0.083 0.456 0.594

1 2 1 2

0.00 0.00 0.13 0.00

0.00 0.09 0.00 0.00

0.16 0.00 0.49 0.72

0.04 0.00 0.00 0.00

0.00 0.05 0.00 0.00

0.50 0.60 0.00 0.00

R (1,1,1) 0.013 3 0.599 1 0.805 3

0.00 0.00 0.00

0.00 0.00 0.37

0.15 0.00 0.00

0.00 0.28 0.00

0.05 0.00 0.00

0.53 0.00 0.00

J

''o-.

Li-pAl-p/Sr-p"" ' ' " i ~ ~Li-pAl-p/Ba-p

l

Ba-deg/H-s Eu-f H-s/Eu-deg

H-s/Sr-deg

H-s/Ba-deg

H-s/Eu-s/Li-s

H-s/Sr-s/Li-s

H-s/Ba-s/Li-s

EuLiH 3

SrLiH 3

tO

BaLiH 3

Fig. 3. Energy diagram of the F point eigenvalues (in rydbergs) for the EuLiH3, SrLiH3 and BaLiH3 hydrides.

--~

~

i 5.95 " ~ - ~ q l ~

At the Brillouin zone centre F point, the conduction band in these hydrides appears at 7-8 eV higher in energy above the top of the valence band (Fig. 3). For EuLiH3 and SrLiH3 the first states above the Fermi energy are, in order of increasing energy, a triply degenerate Li p-H p-Eu(Sr) p state very close in energy to a doubly degenerate Eu(Sr) de,-H s antibonding state. The next state is triply degenerate of Eu(Sr) dr2, non-bonding character and, at higher energies, the Eu(Sr) s-H s-Li s antibonding state appears. The case of BaLiH3 is equivalent with just an inversion in the order of the Ba de,-H s and Li p--H p-Ba p states and the presence of the empty Ba f states close to the triply degenerate Ba dr2, states. By inspection of Fig. 1, we observe that the three compounds are semiconductors with an indirect gap between the top of the valence band at X and the

Sr-degAl-s Sr-dt2g/ Ba-f "~ . . . . . . . . , Ba-dt2~g~

"ollaz~.

I

5.0 e V ) .

Ba-s/H-s/Li-s

Ll•_pal . p/Eu_p

-1.0

the top and thus to a reduction in the total width of the occupied states. Similarly in the perovskite structure hydride LiBeH3 the valence band width was found to decrease with increasing lattice parameters [9, 10]. In the case of EuLiH3, the Eu f states are located in the gap at around 1.4 eV from the top of the valence band at the F point. For SrLiH 3 the unoccupied f states are very high in energy (about 15.5 eV, above the top of the valence band) while, for BaLiH3, the Ba f states are close to the bottom of the conduction band (about

Sr-sAl-s/Li-s

Eu 5d

at

M

Sr 4d

at

M

/

fig= 4.16

Sr 4d,s at M

Eu 5d,s at M Eu 4f

!061

Ag= 3,54

at × n-satX

H-s

H-s

(a) EuLiH 3

(b) SrLiH3

~ V

I

a 5d

at

M

Ba 4f Ba 5d,s at M

. ....

x

H-s

(c) BaLiH 3

Fig. 4. Schematic energy diagram showing the optical transitions (a) for EuLiH3, (b) for SrLiH3 and (c) for BaLiH3. The energy transitions and energy gaps are indicated in electronvolts.

bottom of the conduction band at M. At X, the third band is formed by the non-bonding H s state while the first unoccupied states at the M symmetry point (fourth band) is formed by D d-D s hybrid states. It

E. Orgaz, M. Gupta /Electronic structure of DLiH3 (D=-Eu, Sr, Ba)

should be noted that the Li charge is strongly delocalized in these systems. The energy gap values Ag listed in Table 1 are 4.16 eV, 3.54 eV and 3.93 eV for EuLiH3, SrLiH3 and B a L i H 3 respectively. The band widths for these compounds are clearly smaller than for the hypothetical metallic BeLiH3 (to= 17 eV). This is merely a direct consequence of the larger metal-hydrogen distances in DLiH3 compared with BeLiH3. The Ag values are also larger, characterizing these materials as semiconductors of relatively large energy gap. The trends in to and Ag for these hydrides as a function of the Li-H distance (or lattice parameter) exhibit the expected behaviour in the cases of SrLiH3 and BaLiH3. We observe a slight decrease (increase) in to (Ag) as the Li-H distance increases. For EuLiH3, the presence of the interacting f electron makes it more complicated to interpret. In our calculation the localized f electrons were found to lie in the gap, yielding a significant increase in the energy gap and a decrease in the bandwidth. If we consider the Eu f electrons in the energy eigenvalue calculation, some mixing is obtained between Eu f and H s states at the top of the valence band and hence the energy gap is increased by 0.5 eV. The partial DOS analysis plotted in Fig. 2 shows that the bottom of the conduction bands is largely dominated by the contribution of the D (Sr, Ba, Eu) d states which hybridize with Li and Sr p and to a lesser extent with the s states. The comparison of the calculated electronic structure with the optical data is given schematically in Fig. 4, as proposed by Greedan [18]. In this diagram, we included the values of the energies that resulted from our calculations. The electronic transitions observed by Greedan appear at 2.0 eV, 4.0 eV and 5.15 eV and are attributed to Eu f~conduction band, valence band --->conduction band and Eu f ~ E u d respectively. From our calculations it appears that the Eu f bound states are located at 0.6 eV above the top of the valence band, which yields different energies for these transitions (3.5 eV, 4.16 eV and 5.95 eV respectively). The corresponding energy diagram for SrLiH3 and BaLiH3 (Figs. 4(b) and 4(c)) is simpler than for E u L i H 3. Only one transition at 4 eV was observed in SrLiH3 which corresponds to the calculated value: 3.54 eV. The good agreement between experiment and theory in the case of SrLiH3 indicates that the correlation effects in EuLiH3 are more important, yielding larger energy values for the observed electronic transitions. We recall that, in the LDA, we neglect some of the correlation energy which is frequently large in the case of localized f electrons. Thus, the relatively large calculated energy gap of EuLiH3 should be considered

165

as an upper limit, in the light of the approximations introduced in the LDA calculations. 3. Conclusions

In summary, we have calculated the electronic structure of the cubic antiperovskite DLiH3 (D ---Eu, Sr and Ba) hydrides. We observe that these compounds are semiconductors characterized by rather large indirect energy gaps between the X and M symmetry points of the SCIBZ. The top of the valence band is mainly composed of H s non-bonding states, while the bottom of the conduction band is formed by the hybrid D dD s states. Our calculations confirm the assignation of the electronic transitions observed by Greedan. We have explained the trends observed in the valence band width to and energy gaps Ag by taking into account the variation in the Li-H distances and the presence of the f electrons in the energy gap of the E u L i H 3 compound. References 1 N.A. Bell and G.E. Coates, J. Chem. Soc. A, (1968) 628. 2 E.C. Ashby, R. Kovar and R. Arnott, J. Am. Chem. Soc., 92 (1970) 2182. 3 H.H. Park, M. Pezat and B. Darriet, Rev. Chim. Mindr., 23 (1986) 323. 4 C.E. Messer, J.C. Eastman, R.G. Mers and A.J. Maeland, Inorg. Chem., 3 (1964) 776. 5 C.E. Messer and I.S. Levy, lnorg. Chem., 4 (1965) 543. 6 A.J. Maeland and A.F. Andresen, J. Chem. Phys., 48 (1968) 4660. 7 C.E. Messer and K. Hardcastle, lnorg. Chem., 3 (1964) 1327. 8 A.W. Overhauser, Phys. Rev. B, 35 (1987) 411. 9 M. Gupta and A. Percheron-Guegan, J. Phys. F, 17 (1987) L201. 10 M. Gupta and A. Percheron-Guegan, Chem. Scr., 28 (1988) 117. 11 R. Yu and P.K. Lam, Phys. Rev B, 38 (1988) 3576. 12 J.L. Martins, Phys. Rev B, 38 (1988) 12776. 13 M.R. Press, B.K. Rao and P. Jena, Phys. Rev B, 38 (1988) 2380. 14 M. Seel, A.B. Kunz and S. Hill, Phys. Rev B, 39 (1989) 7949. 15 J.S. Cantrell and T.A. Beiter, Z. Phys. Chem., N.F., 163 (1989) 233. 16 J.S. Cantrell, T.A. Beiter, P.C. Souers and P. Barry, J. LessCommon Met., 172-174 (1991) 213. 17 J.E. Greedan, J. Phys. Chem. Solids, 32 (1971) 819. 18 J.E. Greedan, J. Phys. Chem. Solids, 32 (1971) 1039. 19 T. Loucks, Augmented Plane Wave Method, Benjamin, New York, 1967. 20 L.C. Mattheiss, J.H. Wood and A.C. Switendick, in B. Adler, S. Fernbach and M. Rotenberg (eds.), Methods in Computational Physics, Vol. 8, Academic Press, New York, 1968. 21 G. Lehmann and M. Taut, Phys. Status Solidi (B), 54 (1972) 469.