- Email: [email protected]
Spin-polarized electronic structures of La2CuO4
Communications, .",'~_~Pr~nted / State •in Great SBritain. o Vol.66,No.6, l i pp.629-632, d
1988.
0038-1098/88 $3.00 + .00 Pergamon Press plc
SPIN-POLARIZED ELECTRONIC STRUCTURES OF La=CuO, Ken~ Shiraishi, Ateushi Oshiyama', Nobuyuki Sh~,-a, Takashi Nakayama" and Hircehi Kamhnura Department of Physics, University of Tokyo, Bunkyo-ku, Tokyo 113 ;Japan Fundamental Research Labor&tories, NEC Corporation, Miymmhi, Miyamae-ku, K a w u a ~ ~la" and Department of Physics, Cldba University, Chiba 280 Japan" ( Received 22 February 1988 by J. Kanamori)
We have performed the spin-polarised total-energy band-structure calculation for L~CuO 4 within the local spin density functional formalism using the semirel&tivktic normconserving pseudopotentials and the gaumian-orbitals bask set. An antiferroma~netic innlating state k found to be stable a p ~ t a parama4paetic metallic state.
The discovery of oxide superconductor| with tra~dtion temperature T¢ of 30--40K* for singlelayered perovskJte structure and thcee with Tc---90KU for triple-layered structure has triggered great expansion of research on high temperature superconductivity. In particular, considerable efforts'~' to clarify the mechanisms of the superconductivity have been made so far. The spinunpolarised enerp/-band calculation7~', on the other hand, have revealed that Cu 3a and O 9p orbitak constitute two dimensional bands in each layer of the materials. The resulting metallic bands, however, are incompatible with the experimental observation of insulating states. Since correlation between electron spins seems to be important in the materials, we here attempt a first-principles ~ n - p ~ / J e d ~mev~ql bandstructure calculation for La2CuO4 within local-spindensity-functlonal formalism °J° using the normconserving pseudopotential method n. Among the results from the present calculation are (i) that in La~C'~O4 an antiferromagnetic state is stable against a paramagnetic state, (ii) that the magnetic moment is mainly localized at Cu =+ sites with the magnitude of 0.43/~n, (iii) that La~CuO 4 is an insulator with the energy gap of 0.6 eV (iv) and that upon doping of divalent Sr, Ba or Ca atoms, the holes are created in the bands consisting of Cu d and 0 p orbitals. These theoretical findings are consistent with the observed antiferromagnetic state *~'*s and with the behavior of Hall coefficient14. Reliability of the results is also discussed at the end.
In the present calcuJation, we fn'st construct the normconserving psoudopotentialsIs by solving all-electron atomic Dirac equations and then fitting the results for the valence states: ( ~ , 48, 4p ) of
Ou, ( 2., 2p, a~ ) of o, ( ~, e., ep ) of LL ~nd ( S., 5p, 4d ) of Sr. The resulting pseudopotentials are placed at each atomic sites in the.crystal, and then the valence electron spin density k determined selfcona~tentiy with the mean difference between input and output cryiP,al potential of less than 104 a.u. We use the exchange-correlation functional form by Ceperley and Alder 1' parametrised by Perdew and Zunger .7. The basis set employed here consists of atomic gn,_,~!an orbitais. We have used 2 exponents for s and p orbitals of La and Cu, 3 exponents for d orbital of Cu and also for s and p orbitals of O.*s The values of the exponents are determined by fitting the numerically obtalne~d atomic orbitals: They are 0.23 and 0.70 for a and p orbltals of La and Cu, 0~S12, 1.0996 and 4.4573 for s orbital of O, 0.2213, 1.0694 and 5.50 for p orbital of O and finally 0.4151, 1.4595 and 5.4206 for d orbital of Cu in atomic units. We have calculated total energy and band structure for both paramagnetic ( spinunpolarhmd ) and antiferromagnetic spin configurations. In the antiferromagnetic confis~u'ation the spins of neighboring Cu ~+ sites are opposite so that a tetragonal body centered structure 1° is divided into two sublattices a and b, as shown in Fig. 1. In this fi~qL,'e the unit cell contains 4 La atoms, 2 Cu atoms and 8 0 atoms. It is noted that the present paramagnetic state is different from the state in 629
630
SPIN-POLARIZED ELECTRONIC STRUCTURES OF La2CuO 4
Vol.
66, No. 6
:u(U)
Cu(a)
c~ Fig. 1: Unit cell of (LasCuO,)s ~ for calculation of paramzsnetic and ant[ferromagnetic states, as, a= and a s are the primitive lattice vectors, u d Cu(a) and Cu(b) represent two sublattices.
/
)
, , ....
Fig. 2: Spin demdty of the antiferromzgnetic stato of La=CuO 4 in the Cu-O layer with the contour spscinz of 0.1 electrou/Ca.u.)8.
lO-
which only the interlayer spin ordering is destroyed. T h e calculated total energy for the antiferromagnetic state is lower than that of the paramagnetic state by 0.1 eV per unit ceil: The valence-total energy per unit cell is -468.164 Ry for the paramagnetic state and -468.173 Ry for the antiferromagnetic state. The energy gain comes from the exchange-correlation energy: The kinetic and electrcetatic parts of the total energy in the antiferromagnetic state is larger by 0.2 eV, whereas the exchange-correlation part is smaller by 0~ eV, compared with the corresponding parts of the parzmagnetic state. The 9alculatod spin density is shown in Fig. 2. It is clear from the figure that the spin-density is localized at Cu sites and constructed mainly from Cu d(xS-y=) orbital. The integrated spin over a cube around ~ ( i.e. the spin per CuO~ ) is 0.43,s, which is consistent with t h e observed values. ~=,~ The energy-band structures of the paramagnetic and the antlferromagnetic states are shown, respectively, in Fig. 3 and Fig. 4, together with the corresponding BrUlouine zone. Our results for the paramagnetic state reproduce well the previous results by Mattheiss 7 and Yu et al 8. It is found that the Cu d(x2-y~) - O p band which is a single metallic band in the spin-unpolarim~d calculation ( Fig. 3 ) splits into two bands in the
e,
0 r
$
R
Y
r
z
T
Y
Fig. 3: Energy band structure for the paramagnetic state of LzzOuO,. The inset shows the Brlllouine zone. b v b 2 and b s are the primitive reciprocal lattice vectors. antiferromagnet~c state, and that La=CuO, becomes an insulator with the ener~/-gap of 0.6 eV. Both the valence-band-top and the conduction-band. bottom consist mainly of the d(x2-y2) orbital. This splitting comes from the antfferromagnetic spin density which doubles the period of the crystal potential through exchange-correlation interaction. This instability is triggered by the Fermi-surface nesting of the Cu d(x~-y~) - 0 p band for the paramagnetic state with the two-dimenslonal spanning vector parallel to the layers. This implies
Vol. 66, No. 6
10 ~
SPIN-POLARIZED ELECTRONIC STRUCTURESOF La2CuO 4
--......___._../- f
" ~
0
r
s
R
Y
r
z
T
Y
Fig. 4: Energy band structure for the antiferromagnetic state of La=CuO4.
that the antiferromagnetic state obtained here is essentially two-dimensional i n its character, and is stabilized by gain in the band energy through exchange-correlation interaction. We now consider the effect of substitution of the La atom by the divalent atom M ( M=Sr, Ba or Ca ) based on the above spin-polarked energy bands. Upon substitution the Fermi-level position decreases. When the Fermi level crosses the lower d(x2-y2) ban:d, the holes are created, and give rise to positive Hall coefficient. This is consistent with experimental results. Further, with Increasing the doping concentration, two types of holes appear; one in the Cu d(x2-y2) band and the other in the Cu d(z2) band, as already shown by our local-densltyfunctional calculation. 21 Since the obtained antiferromagnetic state is energetically favorable due to Fermi-surface nesting of the ¢/(x2-y=) band in the paramagnetic state, the crossover from the antiferromagnetic state to the paramagnetic state with increasing the M content x is expected. At this critical value of x, the energy gap in the antiferromagnetic state disappears, and the relevant carriers may change their characters from hole to electron. This will cause the sudden change in the sign of the Hall coefficient. like
Next, based on our band structures, we would to predict the optical properties of
631
antiferromzgnetic L a ~ O u O 4. Since the energy gap of antiferromagnetic I ~ C ~ O t k 0.6 eV, we expect the fundamental ab~rption edge around 0.6 eV. However, the characters of the hlghest occupied band ( HOB ) and the lowest unoccupied band ( LUB ) are mainly both d(xS-yj). Bemuse of the same parity for HOB and LUB, the oecUlator strength for transition between HOB and LUB is very small, that is the order of 104..-10"4, compared with that of allowed tramdtion, and thus the transition between HOB and LUB is expected to be very small. On the other hand, tran~tions from 10th and further lower bands below HOB which have p character to LUB are strong. Thus we expect that optical spectrum with relatively strong intensity starts from the energy region around 4 eV, but has a tail of weak inteusity over a wide energy region from 0.6 eV to the absorption edge. Very recently, zeverzl authors have attempted the local spin density functional calculation for La=CuO 4 using Linear Muffin-Tin Orbital (LMTO) method ~ and Linear Combination of Atomic Orbital (LCAO) method=4. Hateugal and Fu~wura u has shown by the LMTO method that the ground state of La=CuO4 is the antlferromagnetic semimetalllc state, but the other LMTO calculation has obtained the Jrpin-unpolarised ground state. The discrepancy in the conclusion obtained by two different methods of pseudopotential and LMTO may be due to the following fact; in LMTO method the potential is usually averaged spherically in a Muffin-tin sphere so that the method may not be suitable for expressing anisotropic spin density in layered materials, while In the. pzeudopotential method the frozen core approximation may give some effect on spin densities. In this context it is still an open question whether the exact solution for the ground state of La=CuO, within the local spin density functional formalism is an antiferromagnetic state. Aeknowledg,mm, ts: AO acknowledgss useful discussion with J. Misuki. This work was supported in part by a Grant-in-Aid from the Ministry of Education, Science and Culture. Computation were performed with NEC SX2 at NEC Laboratories, HITAC S-810 st the computer center of the Universlty of Tokyo and HITAC S-S10 at the Institute for Molecular Science.
632
SPIN-POLARIZEDELECTRONICSTRUCTURESOF La2Cu04
Vole 66, No. 6
Rofm,me~ 1. J.G. Bednor| and KeA. Muller, Z. Phys. B04, 189
C19m). 2. M.K. Wu st al, Phys. Rev. Left. 18, 908 C1987). 3. S. Hlkaml, T. Hiral and S. Ka~zdzhna, Jpn. J. Apple Phys. N, L314 (198T). 4. P. W. Andemon, Science ~ | , 1196 (1987); PeW. Anderson, G. Buh&ran, Z. Zou and T. Hsu, Phys. ~ . Le~. u , 27oo ( 1 ~ ) . 5. H. Ksmlmum, in Pro¢. of Adr/at/m P,~mrch C'onfarz~ on H~h Tm,qv,e ' a ~ $ ~ , ~ ' s ( Trimte 1987, World Scientific ) ed. by S. Lundqvkt et al; H. Kamimurn, Jpn. J. Appl. Phys. X, L627 (1987); H. Aokl and H. Kamimura, Solid St. Commun. IS, ~ S (19S7). 6. See, for example, Proc. of 18th Int. Conf. on
Low Temperature Phy~lc~ (Kyoto lm~7) ed. by Y.Na4;aoka, Jpn. 3. Appl. Phys. N, Supplement and Ref.S. 7. L~. M&ttheks, Phys. Rev. Left. 5S, 1028 (1987). 8. J. Yu, AJ. Freemu and J.H. Xu, Phys. Rev. L ~ . u , 10SS (l~rr). 9. P. Hohenberg and W. Kohn, Phys. Rev. B IN, 864 (1965); W. Kohn and L. J. Sham, Phys. Rev. AldO, 1188 (1965). I0. For a review, see T ~ z ~ of Thz InAcm~geneo~ EI~ Gas ed. by S. Lundqvkt and N.H. March ( Plenum P r e l , New York 1988 ). 11. MeT. Yin and M.L. Cohen, Phys. Rev. B20, 5668 (19~): A. Oehiymna and M. Salto, Phys. Rev. B ~ , 6156 (1087) and J. Phys. Soc. Jpn ~ , H04 (1987). 12. D. Vaknin et al, Phys. Rev. Left. ~ , 2802 CXOST).
13. G. Shirane et al, Phys. Rev. Left. $9, 1613 (1987). 14. N.P. Ong et al, Phys. Rev. B3|, 8807 (1987); M.F. Hundley, A. Zettl, A. Stacy and M.L. Cohen, Phys. Rev. BSS, mOO (1987). 15. D.R. Hamann, M. Schiuter and C. Chiang, Phys. Rev. Lett. U, 1404 (1979): G.B. Bachelet, D.R. Hamann and M. Schluter, Phys. Rev. B•, 4199 C1982). 16. D.Me C~perley and B.J. Alder, Phys. Rev. Left. 4,, s ~ C19~). 17. J~Perdew and A. Zunger, Phys. Rev. B~, 5048 (1981). 18. To ex&min the completenmm of the bask set, we have also performed the calculation with the extended bask set in which 4 exponents are used for d orbihtl of Cu and p orbital of O. The re|ulte are e~entially identical to those presented in the text. 19. J~D. Jorgensen et al., Phys. Rev. Left. U, 1024 (1987) 20. Y. J. Uemurn et al, Phys. Rev. Left, 19, 1045 (19ST). 91. K .qhirakhi, Dr. Thuk, Un|v of Tokyo 1987. 22. P.A. Sterne and C~. Wang, preprint. 23. Y. Hatlm_gai and T. Fujlwarn, to be publkhed in Solid St. Commun. 24. T.C. Leung, X.W. Wang and B.N. Harmon, to be publkhed in Phys. Rev.
1988.
0038-1098/88 $3.00 + .00 Pergamon Press plc
SPIN-POLARIZED ELECTRONIC STRUCTURES OF La=CuO, Ken~ Shiraishi, Ateushi Oshiyama', Nobuyuki Sh~,-a, Takashi Nakayama" and Hircehi Kamhnura Department of Physics, University of Tokyo, Bunkyo-ku, Tokyo 113 ;Japan Fundamental Research Labor&tories, NEC Corporation, Miymmhi, Miyamae-ku, K a w u a ~ ~la" and Department of Physics, Cldba University, Chiba 280 Japan" ( Received 22 February 1988 by J. Kanamori)
We have performed the spin-polarised total-energy band-structure calculation for L~CuO 4 within the local spin density functional formalism using the semirel&tivktic normconserving pseudopotentials and the gaumian-orbitals bask set. An antiferroma~netic innlating state k found to be stable a p ~ t a parama4paetic metallic state.
The discovery of oxide superconductor| with tra~dtion temperature T¢ of 30--40K* for singlelayered perovskJte structure and thcee with Tc---90KU for triple-layered structure has triggered great expansion of research on high temperature superconductivity. In particular, considerable efforts'~' to clarify the mechanisms of the superconductivity have been made so far. The spinunpolarised enerp/-band calculation7~', on the other hand, have revealed that Cu 3a and O 9p orbitak constitute two dimensional bands in each layer of the materials. The resulting metallic bands, however, are incompatible with the experimental observation of insulating states. Since correlation between electron spins seems to be important in the materials, we here attempt a first-principles ~ n - p ~ / J e d ~mev~ql bandstructure calculation for La2CuO4 within local-spindensity-functlonal formalism °J° using the normconserving pseudopotential method n. Among the results from the present calculation are (i) that in La~C'~O4 an antiferromagnetic state is stable against a paramagnetic state, (ii) that the magnetic moment is mainly localized at Cu =+ sites with the magnitude of 0.43/~n, (iii) that La~CuO 4 is an insulator with the energy gap of 0.6 eV (iv) and that upon doping of divalent Sr, Ba or Ca atoms, the holes are created in the bands consisting of Cu d and 0 p orbitals. These theoretical findings are consistent with the observed antiferromagnetic state *~'*s and with the behavior of Hall coefficient14. Reliability of the results is also discussed at the end.
In the present calcuJation, we fn'st construct the normconserving psoudopotentialsIs by solving all-electron atomic Dirac equations and then fitting the results for the valence states: ( ~ , 48, 4p ) of
Ou, ( 2., 2p, a~ ) of o, ( ~, e., ep ) of LL ~nd ( S., 5p, 4d ) of Sr. The resulting pseudopotentials are placed at each atomic sites in the.crystal, and then the valence electron spin density k determined selfcona~tentiy with the mean difference between input and output cryiP,al potential of less than 104 a.u. We use the exchange-correlation functional form by Ceperley and Alder 1' parametrised by Perdew and Zunger .7. The basis set employed here consists of atomic gn,_,~!an orbitais. We have used 2 exponents for s and p orbitals of La and Cu, 3 exponents for d orbital of Cu and also for s and p orbitals of O.*s The values of the exponents are determined by fitting the numerically obtalne~d atomic orbitals: They are 0.23 and 0.70 for a and p orbltals of La and Cu, 0~S12, 1.0996 and 4.4573 for s orbital of O, 0.2213, 1.0694 and 5.50 for p orbital of O and finally 0.4151, 1.4595 and 5.4206 for d orbital of Cu in atomic units. We have calculated total energy and band structure for both paramagnetic ( spinunpolarhmd ) and antiferromagnetic spin configurations. In the antiferromagnetic confis~u'ation the spins of neighboring Cu ~+ sites are opposite so that a tetragonal body centered structure 1° is divided into two sublattices a and b, as shown in Fig. 1. In this fi~qL,'e the unit cell contains 4 La atoms, 2 Cu atoms and 8 0 atoms. It is noted that the present paramagnetic state is different from the state in 629
630
SPIN-POLARIZED ELECTRONIC STRUCTURES OF La2CuO 4
Vol.
66, No. 6
:u(U)
Cu(a)
c~ Fig. 1: Unit cell of (LasCuO,)s ~ for calculation of paramzsnetic and ant[ferromagnetic states, as, a= and a s are the primitive lattice vectors, u d Cu(a) and Cu(b) represent two sublattices.
/
)
, , ....
Fig. 2: Spin demdty of the antiferromzgnetic stato of La=CuO 4 in the Cu-O layer with the contour spscinz of 0.1 electrou/Ca.u.)8.
lO-
which only the interlayer spin ordering is destroyed. T h e calculated total energy for the antiferromagnetic state is lower than that of the paramagnetic state by 0.1 eV per unit ceil: The valence-total energy per unit cell is -468.164 Ry for the paramagnetic state and -468.173 Ry for the antiferromagnetic state. The energy gain comes from the exchange-correlation energy: The kinetic and electrcetatic parts of the total energy in the antiferromagnetic state is larger by 0.2 eV, whereas the exchange-correlation part is smaller by 0~ eV, compared with the corresponding parts of the parzmagnetic state. The 9alculatod spin density is shown in Fig. 2. It is clear from the figure that the spin-density is localized at Cu sites and constructed mainly from Cu d(xS-y=) orbital. The integrated spin over a cube around ~ ( i.e. the spin per CuO~ ) is 0.43,s, which is consistent with t h e observed values. ~=,~ The energy-band structures of the paramagnetic and the antlferromagnetic states are shown, respectively, in Fig. 3 and Fig. 4, together with the corresponding BrUlouine zone. Our results for the paramagnetic state reproduce well the previous results by Mattheiss 7 and Yu et al 8. It is found that the Cu d(x2-y~) - O p band which is a single metallic band in the spin-unpolarim~d calculation ( Fig. 3 ) splits into two bands in the
e,
0 r
$
R
Y
r
z
T
Y
Fig. 3: Energy band structure for the paramagnetic state of LzzOuO,. The inset shows the Brlllouine zone. b v b 2 and b s are the primitive reciprocal lattice vectors. antiferromagnet~c state, and that La=CuO, becomes an insulator with the ener~/-gap of 0.6 eV. Both the valence-band-top and the conduction-band. bottom consist mainly of the d(x2-y2) orbital. This splitting comes from the antfferromagnetic spin density which doubles the period of the crystal potential through exchange-correlation interaction. This instability is triggered by the Fermi-surface nesting of the Cu d(x~-y~) - 0 p band for the paramagnetic state with the two-dimenslonal spanning vector parallel to the layers. This implies
Vol. 66, No. 6
10 ~
SPIN-POLARIZED ELECTRONIC STRUCTURESOF La2CuO 4
--......___._../- f
" ~
0
r
s
R
Y
r
z
T
Y
Fig. 4: Energy band structure for the antiferromagnetic state of La=CuO4.
that the antiferromagnetic state obtained here is essentially two-dimensional i n its character, and is stabilized by gain in the band energy through exchange-correlation interaction. We now consider the effect of substitution of the La atom by the divalent atom M ( M=Sr, Ba or Ca ) based on the above spin-polarked energy bands. Upon substitution the Fermi-level position decreases. When the Fermi level crosses the lower d(x2-y2) ban:d, the holes are created, and give rise to positive Hall coefficient. This is consistent with experimental results. Further, with Increasing the doping concentration, two types of holes appear; one in the Cu d(x2-y2) band and the other in the Cu d(z2) band, as already shown by our local-densltyfunctional calculation. 21 Since the obtained antiferromagnetic state is energetically favorable due to Fermi-surface nesting of the ¢/(x2-y=) band in the paramagnetic state, the crossover from the antiferromagnetic state to the paramagnetic state with increasing the M content x is expected. At this critical value of x, the energy gap in the antiferromagnetic state disappears, and the relevant carriers may change their characters from hole to electron. This will cause the sudden change in the sign of the Hall coefficient. like
Next, based on our band structures, we would to predict the optical properties of
631
antiferromzgnetic L a ~ O u O 4. Since the energy gap of antiferromagnetic I ~ C ~ O t k 0.6 eV, we expect the fundamental ab~rption edge around 0.6 eV. However, the characters of the hlghest occupied band ( HOB ) and the lowest unoccupied band ( LUB ) are mainly both d(xS-yj). Bemuse of the same parity for HOB and LUB, the oecUlator strength for transition between HOB and LUB is very small, that is the order of 104..-10"4, compared with that of allowed tramdtion, and thus the transition between HOB and LUB is expected to be very small. On the other hand, tran~tions from 10th and further lower bands below HOB which have p character to LUB are strong. Thus we expect that optical spectrum with relatively strong intensity starts from the energy region around 4 eV, but has a tail of weak inteusity over a wide energy region from 0.6 eV to the absorption edge. Very recently, zeverzl authors have attempted the local spin density functional calculation for La=CuO 4 using Linear Muffin-Tin Orbital (LMTO) method ~ and Linear Combination of Atomic Orbital (LCAO) method=4. Hateugal and Fu~wura u has shown by the LMTO method that the ground state of La=CuO4 is the antlferromagnetic semimetalllc state, but the other LMTO calculation has obtained the Jrpin-unpolarised ground state. The discrepancy in the conclusion obtained by two different methods of pseudopotential and LMTO may be due to the following fact; in LMTO method the potential is usually averaged spherically in a Muffin-tin sphere so that the method may not be suitable for expressing anisotropic spin density in layered materials, while In the. pzeudopotential method the frozen core approximation may give some effect on spin densities. In this context it is still an open question whether the exact solution for the ground state of La=CuO, within the local spin density functional formalism is an antiferromagnetic state. Aeknowledg,mm, ts: AO acknowledgss useful discussion with J. Misuki. This work was supported in part by a Grant-in-Aid from the Ministry of Education, Science and Culture. Computation were performed with NEC SX2 at NEC Laboratories, HITAC S-810 st the computer center of the Universlty of Tokyo and HITAC S-S10 at the Institute for Molecular Science.
632
SPIN-POLARIZEDELECTRONICSTRUCTURESOF La2Cu04
Vole 66, No. 6
Rofm,me~ 1. J.G. Bednor| and KeA. Muller, Z. Phys. B04, 189
C19m). 2. M.K. Wu st al, Phys. Rev. Left. 18, 908 C1987). 3. S. Hlkaml, T. Hiral and S. Ka~zdzhna, Jpn. J. Apple Phys. N, L314 (198T). 4. P. W. Andemon, Science ~ | , 1196 (1987); PeW. Anderson, G. Buh&ran, Z. Zou and T. Hsu, Phys. ~ . Le~. u , 27oo ( 1 ~ ) . 5. H. Ksmlmum, in Pro¢. of Adr/at/m P,~mrch C'onfarz~ on H~h Tm,qv,e ' a ~ $ ~ , ~ ' s ( Trimte 1987, World Scientific ) ed. by S. Lundqvkt et al; H. Kamimurn, Jpn. J. Appl. Phys. X, L627 (1987); H. Aokl and H. Kamimura, Solid St. Commun. IS, ~ S (19S7). 6. See, for example, Proc. of 18th Int. Conf. on
Low Temperature Phy~lc~ (Kyoto lm~7) ed. by Y.Na4;aoka, Jpn. 3. Appl. Phys. N, Supplement and Ref.S. 7. L~. M&ttheks, Phys. Rev. Left. 5S, 1028 (1987). 8. J. Yu, AJ. Freemu and J.H. Xu, Phys. Rev. L ~ . u , 10SS (l~rr). 9. P. Hohenberg and W. Kohn, Phys. Rev. B IN, 864 (1965); W. Kohn and L. J. Sham, Phys. Rev. AldO, 1188 (1965). I0. For a review, see T ~ z ~ of Thz InAcm~geneo~ EI~ Gas ed. by S. Lundqvkt and N.H. March ( Plenum P r e l , New York 1988 ). 11. MeT. Yin and M.L. Cohen, Phys. Rev. B20, 5668 (19~): A. Oehiymna and M. Salto, Phys. Rev. B ~ , 6156 (1087) and J. Phys. Soc. Jpn ~ , H04 (1987). 12. D. Vaknin et al, Phys. Rev. Left. ~ , 2802 CXOST).
13. G. Shirane et al, Phys. Rev. Left. $9, 1613 (1987). 14. N.P. Ong et al, Phys. Rev. B3|, 8807 (1987); M.F. Hundley, A. Zettl, A. Stacy and M.L. Cohen, Phys. Rev. BSS, mOO (1987). 15. D.R. Hamann, M. Schiuter and C. Chiang, Phys. Rev. Lett. U, 1404 (1979): G.B. Bachelet, D.R. Hamann and M. Schluter, Phys. Rev. B•, 4199 C1982). 16. D.Me C~perley and B.J. Alder, Phys. Rev. Left. 4,, s ~ C19~). 17. J~Perdew and A. Zunger, Phys. Rev. B~, 5048 (1981). 18. To ex&min the completenmm of the bask set, we have also performed the calculation with the extended bask set in which 4 exponents are used for d orbihtl of Cu and p orbital of O. The re|ulte are e~entially identical to those presented in the text. 19. J~D. Jorgensen et al., Phys. Rev. Left. U, 1024 (1987) 20. Y. J. Uemurn et al, Phys. Rev. Left, 19, 1045 (19ST). 91. K .qhirakhi, Dr. Thuk, Un|v of Tokyo 1987. 22. P.A. Sterne and C~. Wang, preprint. 23. Y. Hatlm_gai and T. Fujlwarn, to be publkhed in Solid St. Commun. 24. T.C. Leung, X.W. Wang and B.N. Harmon, to be publkhed in Phys. Rev.