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2 HiLAPW 1 first-principles or ab initio calculation HiLAPW 2 HiLAPW (Hiroshima Linear-Augmented-Plane-Wave) 1, 2, 3, 4, 5 ( ) 6 7 1

3 4, 5 8 ( ) ( ) (Fortran90) SX-6i( 8GFLOPS) 4GFLOPS 2.1 (LAPW: Linear Augmented Plane Wave) LAPW LAPW Andersen2 Koelling-Arbman3 Soler- Williams 4, 5 Soler-Williams 2.2 Dirac Kohn-Sham Anderson SCF 2.4 APW LAPW Weinert 7 2

4 2.5 Hellman-Feynman Car-Parrinello FLAPW Soler-Williams4, 5 FLAPW 2.6 Blöchl HiLAPW 1 J.C. Slater, Phys. Rev. 51, 392 (1937). 2 O.K. Andersen, Phys. Rev. B 12, 3060 (1975). 3 D.D. Koelling and G. Arbman: J. Phys. F: Metal Phys. 5, 2041 (1975). 4 J.M. Soler and A.R. Williams, Phys. Rev. B 40, 1560 (1989). 5 J.M. Soler and A.R. Williams, Phys. Rev. B 42, 9728 (1990). 6 D.D. Koelling and B.N. Harmon: J. Phys. C: Solid State Phys. 10, 3107 (1977). 7 M. Weinert, J. Math. Phys. 22, 2433 (1981). 8 P.E. Blöchl, O. Jepsen and O.K. Andersen, Phys. Rev. B 49, (1994). 3

5 HiLAPW 1 HiLAPW xsets xlapw makefile 1 2 Tru64 F90 Alpha Compaq Alpha Linux F90 Alpha ( ) Absoft Pro Fortran F90 PowerMac (Mac OS X) Intel ifc PC F90 LAPACK BLAS.cshrc set path set path = (. $home/hilapw/bin $path) ( csh ) hilapw.tar.gz # cd ~ # gunzip hilapw.tar # tar xvf hilapw.tar # cd hilapw/sources # make clean # make -f makefile_tru64 all # make -f makefile_tru64 install ps lib make # cd ~/hilapw/ps # make clean # make -f makefile_tru64 all # make -f makefile_tru64 install # cd ~/hilapw/lib # make clean 1

6 1: HiLAPW xsets atomdata wavout atomdens sets.out spgrdata sets.in xlapw wavin wavout lapw.in ekn dis foa tau ten lapw.out xdoss wavin pdos doss.in doss.out xnewa k wavin wavout newa.out xwbox 3D wavin wbox.list wbox.in wbox.out xpbox 3D wavin pbox.list pbox.in pbox.out xspin wavin wavout spin.out xwcon wav wavin wavout wavin.frm wavout.frm wcon.in wcon.out xsymm wavin ekl cml symm.out xrept ekl eig1 cml eig2 rept.out 2

7 # make -f makefile_tru64 all # make -f makefile_tru64 install # source ~/.cshrc hilapw1 # cd ~ # mkdir hilapw1 3 fcc Cu 3.1 SCF hilapw1 Cu getdata 3 atomdata atomdens spgrdata ~/hilapw/data # cd hilapw1 # mkdir Cu # cd Cu # getdata 11 ~/hilapw/data # tar xvf ~/hilapw/data/cu.tar SCF # JOB-SCF Alpha JOB-SCF 3.2 # xdoss # mv pdos pdosa2 # getfermi outa2 > fermia2 # PSP < psp_tdos > tdos.ps # ps2pdf tdos.ps (ps) tdos.ps 1 getfermi PSP psp_tdos ps PSP ~/hilapw/ps/psplot.f xdoss pdosa2 3

8 4.0 fcc Cu TOTAL DOS(/eV) ENERGY(eV) 1: fcc Cu # PSP < psp_pdos > pdos.ps # ps2pdf pdos.ps s p e g t 2g fcc Cu 0.8 DOS(/eV-atom-spin) ENERGY(eV) 2: fcc Cu s p e g t 2g 3.3 # JOB-EK # JOB-SYM JOB-EK k JOB-SYM ps 4

9 # PSP < psp_ek > ek.ps # ps2pdf ek.ps 3 ENERGY(eV) W Q fcc Cu L Λ Γ X S K Σ Γ 3: fcc Cu 3.4 sets.in xsets sets.in 01: 01:fcc Cu 02:-----nspin 03:1 04:-----space group 05:Fm-3m 06: : :-----atoms 09:1 10:Cu 1 11: :-----k points 13:0 14: :fcc Cu

10 02:-----nspin 03: NSPIN :-----space group 05:Fm-3m 06: : spgrdata 06 a, b, c Å 07 α, β, γ a,b,c α, β, γ a b b a 08:-----atoms 09:1 10:Cu 1 11: NTTP a, b, c 12:-----k points 13:0 14: k 0 1 k k 14 k 3 k k 2π/a, 2π/b, 2π/c k 4 Si 4.1 SCF fcc Cu # cd ~/hilapw1 # mkdir Si # cd Si # getdata # tar xvf ~/hilapw/data/si.tar 6

11 # JOB-TEN a = 5.20Å a = 5.60Å 11 SCF GET-TEN Murnaghan Murnaghan # GET-TEN > TEN # xefitm < TEN > fit_ten xefitm Murnaghan ~/hilapw/lib/efitm.f fit_ten Coefficients for Murnaghan fitting c1 = D+07 c2 = D-02 E = V0 = E0 = B = B0 = GPa ( 2 ) a.u. 98GPa 4.02 # tail -103 fit_ten > TEN2 # PSP < psp_ten > TEN.ps # ps2pdf TEN.ps 4 TOTAL ENERGY(Ry) Si VOLUME(a.u.) 4: Si Murnaghan 7

12 LAPW 1 Soler-Williams LAPW Andersen Koelling-Arbman Weinert α MT v α (r) Kohn-Sham Rydberg h α ψ αlm (r; E) 2 + v α (r) ψ αlm (r; E) =Eψ αlm (r; E). (1) ψ αlm (r; E) ψ αlm (r; E) =R αl (r; E)i l Y lm (ˆr). (2) Y lm (ˆr) R αl (r; E) (1) h αl R αl (r; E) d2 dr 2 2 d l(l +1) + r dr r 2 + v α (r) R αl (r; E) =ER αl (r; E). (3) (3) P αl (r; E) =rr αl (r; E) d2 l(l +1) + dr2 r 2 + rv α (r) P αl (r; E) =EP αl (r; E). (4) E sα 0 (R αl (r; E)) 2 r 2 dr = sα (3) E 0 (P αl (r; E)) 2 dr =1. (5) h αl Ṙ αl (r; E) =EṘαl(r; E)+R αl (r; E), (6) Ṙαl(r; E)( dr αl (r; E)/dr) Ṙαl(r; E) (2 sα ) 2 N αl = (Ṙαl (r; E) r 2 dr. (7) 0 1

13 ψ αlm (r; E) =Ṙαl(r; E)i l Y lm (ˆr). (8) 2.2 MT α < > α <ψ αl m ψ αlm > α = δ l lδ m m, (9) < ψ αl m ψ αlm > α =0, <ψ αl m ψ αlm > α =0, (10) (11) < ψ αl m ψ αlm > α = δ l lδ m mn αl, (12) (10, 11) (5) d de sα 0 (R αl (r; E)) 2 r 2 dr =2 sα αmt h α 0 R αl (r; E)Ṙαl(r; E)r 2 dr =0. (13) <ψ αl m h α ψ αlm > α = δ l lδ m me, (14) < ψ αl m h α ψ αlm > α =0, (15) <ψ αl m h α ψ αlm > α = δ l lδ m m, (16) < ψ αl m h α ψ αlm > α = δ l lδ m mn αl E, (17) (16 (6) 2.3 MT (3) R αc (r; E αc ) P αc (r; E αc )= rr αc (r; E αc ) d2 l(l +1) + dr2 r 2 + rv α (r) P αc (r; E αc )=EP αc (r; E αc ). (18) (4) P αc r (18) P αl r sα d P 2 αl dr 2 P d 2 sα αc P αc dr 2 P αl dr =E E αc P αc P αl dr, (19) 0 sα d d P αl dr dr P d αc P αc dr P αl dr = 0 0 sα d P αl dr P d αc P αc dr P αl 0 r =0 P αc P αl 0 P αc d dr P αc (19) E E αc (20) 2

14 3 3.1 LAPW φ n (r) = φ n (r)+ α Θ( r α ) l max lm φ αlmn (r α ) φ αlmn (r α ) r α = r R α R α α lm l max l max l = (21) lm l m= l φ n (r) =Ω 1/2 exp(ik n r). (22) k n = k K n K n n (21) Θ(x) φ αlmn (r α ) φ αlmn (r α )= A αlmn R αl (r α ; E αl )+B αlmn Ṙ αl (r α ; E αl ) i l Y lm (ˆr), (23) φ αlmn (r α ) φ αlmn (r α )=4πΩ 1/2 j l (k n r α )Y lm(ˆk n )exp(ik n R α )i l Y lm (ˆr α ) (24) j l (x) (23) A αlmn B αlmn A αlmn =4πΩ 1/2 a αln Y lm (ˆk n ) exp(ik n R α ), (25) B αlmn =4πΩ 1/2 b αln Ylm(ˆk n )exp(ik n R α ), (26) a αln = s 2 α k n j lṙαl j l Ṙ αl, (27) b αln = s 2 α j l R αl k lj l R αl. (28) j l =(dj l(z)/dz) z=kn R αl = R αl ( ; E αl ) LAPW MT MT lm l max l l MT MT Soler-Williams (21) LAPW Andersen Soler-Williams MT s (21) 3

15 3.2 k i ψi k (r) (21) LAPW ψ k i (r) = n φ n (r)c ni. (29) C ni (21) LAPW ψ k i (r) = ψ k i (r)+ α Θ( r α ) l max lm ψαlmi k (r α) ψ αlmi k (r α), (30) ψ k i (r) =Ω 1/2 n exp(ik n r)c ni, (31) ψ k αlmi(r α )= PαlmiR k αl (r α ; E αl )+Q k αlmiṙαl(r α ; E αl ) i l Y lm (ˆr), (32) ψ k αlmi (r α)=j k αlmi (r α)i l Y lm (ˆr α ), (33) P k αlmi Qk αlmi J k αlmi (r α) P k αlmi = n A αlmn C ni, (34) Q k αlmi = n B αlmn C ni, (35) J k αlmi (r α)= n φ αlmn (r α )C ni. (36) Jαlmi k (r α) (36) Kαlmi k (r α) sα Ñαlmi k = Jαlmi k (r α) 2 r 2 dr, (37) 0 K k αlmi (r α)=4πω 1/2 n k 2 n j l(k n r α )Y lm (ˆk n ) exp(ik n R α )C ni. (38) (30) k i ψi k (r) 2 = ψ i k (r) 2 + Θ( r α ) α ( )( lmax ) ψ αlmi k (r α) ψαl k m i (r α) + c.c. lm l m ( )( lmax ) ψ αlmi k (r α) ψk αl m i (r α) c.c. lm l m 4

16 + ( lmax lm ( lmax lm ψ k αlmi (r α) ψ k αlmi(r α ) )( lmax ) ( lmax ψαl k m i (r α) + l m lm ) ψk αl m i(r α ) c.c. l m )( lmax ψ k αlmi (r α) = ψ i k (r) 2 + Θ( r α ) α lmax l max ( ψαlm(r k α )ψαl k m (r k α) ψ αlm(r α ) ψ ) αl k m (r α) + lm l m l max lm>l max l m )( lmax l m ψk αl m i (r α) ( ψk αlmi(r α ) ψαl k m i(r α ) ψ ) αl k m i(r α ) + c.c.. (39) (39) ) ψ k i (r) 2 = ψ k i (r) 2 + α Θ( r α ) l max lm ψαlmi k (r α)ψαlmi k (r k α) ψ αlmi (r α) ψ αlmi k (r α) = Ω 1 nn exp i(k n K n ) r C nic n i + 1 4π Θ( r α ) α l max lm P k αlmir αli (r α )+Q k αlmiṙαli(r α ) 2 J k αlmi(r α ) 2. (40) MT (40) (39) MT MT MT 4.2 (40) k i n v (r) = ωi k ψi k (r) 2 k,i = K = K ñ v (K)exp(iK r)+ α ñ v (K)exp(iK r)+ α Θ( r α )Y 00 n v,α00 (r α ) ñ v,α00 (r α ) Θ( r α )Y 00 n v,α00 (r α ). (41) ωi k k i ñ v (K) =Ω 1 ωi k Cni C n iδ(k + K n K n ), (42) k,i nn n v,α00 (r α )=(4π) 1/2 k,i ω k i l max lm P k αlmi R αli(r α )+Q k αlmiṙαli(r α ) 2, (43) 5

17 ñ v,α00 (r α )=(4π) 1/2 k,i ω k i l max lm J k αlmi (r α) 2. (44) 4.3 α n c,α (r α ) n c,α (r α ) MT MT MT n c (r) = α = α n c,α (r α ) ñ c,α (r α )+ α n c,α (r α ) ñ c,α (r α ) = ñ c (r)+ α Θ( r α )Y 00 n c,α00 (r α ) ñ c,α00 (r α ). (45) ñ c,α (r α ) MT n c,α (r α ) (45) ñ c (r) = K exp(ik r)ñ c (K), (46) ñ c (K) ñ c (K) = Ω 1 ñ c,α (r α )exp( ik r)d 3 r α α = 4πΩ 1 α sα exp( ik R α ) ñ c,α (r α )j 0 (Kr α )rαdr 2 α, (47) (41) (45) n e (r) n e (r) = n v (r)+n c (r) = ñ e (K) exp(ik r)+ K α Θ( r α )Y 00 n e,α00 (r α ) ñ e,α00 (r α ), (48) ñ e (K) =ñ v (K)+ñ c (K), (49) n e,α00 (r α )=n v,α00 (r α )+n c,α00 (r α ), (50) ñ e,α00 (r α )=ñ v,α00 (r α )+ñ c,α00 (r α ). (51) N e N e = Ωñ e (K =0)+ α k,i ω k i l max lm Pαlmi k 2 + Q k αlmi 2 N αl Ñ αlmi k 6

18 +(4π) 1/2 n c,α00 (r α ) ñ c,α00 (r α ) α = Ñe + N e,α Ñe,α. (52) α Ñ e =Ωñ e (K =0), N e,α = N v,α + N c,α, Ñ e,α = Ñv,α + Ñc,α, (53) (54) (55) N v,α = k,i Ñ v,α = k,i ω k i ω k i l max lm l max lm P k αlmi 2 + Q k αlmi 2 N αl, (56) Ñ k αlmi. (57) N c,α =(4π) 1/2 n c,α00 (r α )d 3 r α. (58) Ñ c,α =(4π) 1/2 ñ c,α00 (r α )d 3 r α. (59) ρ(r α ) q αlm q αlm = Ylm rl ρ(r α )d 3 r α, (60) α q α00 q α00 =(4π) 1/2 ( N v,α Ñv,α + N c,α Ñc,α Z α ). (61) 4.5 Weinert Weinert 4π ñ ps (K) = (2ν + 3)!! j ν+1(k ) Ω (K ) ν+1 q α00 exp(ik R α ). (62) α ν K MT K max Weinert ñ(r) =ñ e (r)+ñ ps (r). (63) MT MT 7

19 5 E T U E xc E = T + U + E xc. (64) 5.1 (30) T T = ωi k ψi k (r)( 2 )ψi k (r)d3 r k,i = ωi k ψ i k (r)( 2 ) ψ i k (r)d 3 r k,i + k,i ω k i α l max lm ψαlmi(r k α )( 2 )ψαlmi(r k k α ) ψ αlmi(r α )( 2 ) ψ αlmi(r k α ) d 3 r α. (65) (65) lm 5.2 U (48) MT U int int MT MT U MT int MT U MT MT U = U int int + U MT int + U MT MT. (66) U int int + U MT int (63) MT ñ(r)ñ(r ) U intint + U MT int = r r d 3 rd 3 r ñ(r)ñ(r ) α r r d 3 rd 3 r. (67) U U = Ũ + U α Ũ α. (68) α Ũ = 1 2 Ṽ (r)ñ(r)d 3 r, (69) U α = Vion(r)n α e (r)d 3 r + 1 Ve α (r)n e (r)d 3 r, 2 (70) Ũ α = 1 Ṽ α ñ(r)d 3 r. 2 (71) 8

20 ñ(r ) Ṽ (r) =2 r r d3 r. (72) V α ion (r) = 2Z α r, Ve α n e (r ) r r d3 r, (74) Ṽ α ñ(r ) (r) =2 r r d3 r. (75) Vion α V e α (r) Ṽ α MT α (72) Ṽ (r) =8π K 0 ñ(k) exp(ik r). (76) K2 K =0 Ũ Ũ = 4π Ω K 0 ñ(k) 2 K 2. U α MT α (73, 74) V (r) =V α ion (r)+v α e (r)+v α out (r) (r ). (78) Vout α MT α α (78) (72) V α out(r) =Ṽ (r) V α ion(r) V α e (r) ( r = ), (79) α V (r) =V α ion (r)+v α e (r)+ṽ () V α ion () V α e (). (80) (80) U α U α = 1 V (r)n e (r)d 3 r + 1 Vion α 2 2 (r)n e(r)d 3 r 1 2 = 1 V (r)+v 2 ion(r) α n e (r)d 3 r 1 2 Ṽ ( )+ 2Z α 2N e,α Ṽ (sα ) V α ion () V α (73) (77) e () N e,α N e,α. (81) α N e,α = n e (r)d 3 r. (82) 9

21 N e,α (54) N e,α N e,α = ñ e (r)d 3 r + N e,α Ñe,α. (83) Ũ α U α α ñ(r ) (75) Ṽ (r) =Ṽ α (r)+ṽ α out (r) (r ). (84) Ṽ α out (r) α ñ(r ) Ṽ α out(r) Ṽ α out(r) =Ṽ (r) Ṽ α (r) ( r = ). (85) (71) Ũ α = 1 Ṽ (r)d 3 r Ṽ (sα ) Ṽ α ( ) Ñ α. (86) Ñ α = ñ(r)d 3 r. (87) 5.3 E xc (48) E xc = Ẽxc + Exc α Ẽα xc. (88) α Ẽ xc = ε xc ñ e (r)ñ e (r)d 3 r. (89) Exc α = ε xc n e (r)n e (r)d 3 r. (90) Ẽxc α = ε xc ñ e (r)ñ e (r)d 3 r. (91) 10

22 1 1.1 n e (r) = ñ e (r) + α Θ( r α ) n e,α (r α ) = ñ e (r)θ(r I) + α Θ( r α )n e,α (r α ) (1) ñ e (r) = K ñ e (K) exp(ik r) (2) n e,α (r α ) = lm Y lm (ˆr α ) n e,αlm (r α ) (3) n e,α (r α ) = lm Y lm (ˆr α )n e,αlm (r α ) (4) n e,αlm (r α ) = n e,αlm (r α ) + ñ e,αlm (r α ) (5) ñ e,αlm (r α ) = 4πi l K ñ e (K)j l (Kr α )Y lm( ˆK) exp(ik R α ) (6) 1.2 U MT U int int MT MT U MT int MT U MT MT U = U int int + U MT int + U MT MT. (7) U int int + U MT int ñ(r) = ñ e (r) + ñ ps (r) = K ñ(k) exp(ik r) (8) 1

23 MT ñ(r)ñ(r ) U int int + U MT int = r r d 3 rd 3 r ñ(r)ñ(r ) α r r d 3 rd 3 r. (9) U U = Ũ + U α Ũ α. (10) α Ũ = 1 2 Ṽ (r)ñ(r)d 3 r, (11) U α = Vion(r)n α e (r)d 3 r + 1 Ve α (r)n e (r)d 3 r, 2 (12) Ũ α = 1 Ṽ α ñ(r)d 3 r. 2 (13) ñ(r ) Ṽ (r) = 2 r r d3 r. (14) Vion(r) α = 2Z α r, (15) Ve α n e (r ) (r) = 2 r r d3 r, (16) Ṽ α ñ(r ) (r) = 2 r r d3 r. (17) V α ion V α e (r) Ṽ α MT α 1.3 Ũ (14) Ṽ (r) = 8π K 0 ñ(k) exp(ik r) K2 = K 0 Ṽ (K) exp(ik r) (18) K = 0 Ũ Ũ = 4πΩ ñ(k) 2 K 2 K 0 = Ω Ṽ (K)ñ(K) (19) 2 K 0 2

24 1.4 U α U α MT α (15) (16) V (r) = V α ion(r) + V α e (r) + V α out(r) (r ) (20) MT α Vout α (16) (1) Ve α n e (r ) (r) = 2 r r d3 r = 8π sα 1 2l + 1 lm 0 r > ( r< r > ) l n e,αlm(r )r 2 dr Y lm (ˆr) V e α = lm V e,αlm (r)y lm (ˆr) (21) V e,αlm (r) = 8π sα 2l r > ( r< r > ) l n e,αlm(r )r 2 dr (22) 1 r r = l 4π 1 2l + 1 r > ( r< r > ) l m Y lm (ˆr)Y lm(ˆr ) (23) α (20) (14) V α out(r) = Ṽ (r) V α ion(r) V α e (r) ( r = ), (24) Ṽ MT Ṽ (ŝ α ) = lm Ṽ αlm ( )Y lm (ŝ α ) (25) Ṽ αlm (r) = 32π 2 K K 2 ñ(k)j l (Kr)i l Y lm( ˆK) exp(ik R α ) = 4π K Ṽ (K)j l (Kr)i l Y lm( ˆK) exp(ik R α ) (26) r = r ŝ α MT V (r) = V α ion(r) + V α = 2Z α ( 1 1 r e (r) + Ṽ (ŝ α) Vion(ŝ α α ) Ve α (ŝ α ) ) + lm V e,αlm (r) V e,αlm ( ) + Ṽαlm( ) Y lm (ˆr) (27) 1.5 Ũ α (8) α ñ(r) = lm ñ αlm (r α )Y lm (ˆr α ) (28) 3

25 ñ αlm (r α ) = 4πi l K ñ(k)j l (Kr α )Y lm(k) exp(ik R α ) (29) (14) Ṽ (r) = 8π sα ( ) l 1 r< ñ αlm(r)r 2 dr Y lm (ˆr) 2l + 1 lm 0 r > r > = Ṽlm(r)Y α lm (ˆr) (30) lm Ũ α Ũ α = 1 sα Ṽ α 2 lm(r)ñ αlm (r)r 2 dr (31) lm 0 4

26 Soler-Williams 1 ψ i (r) = Ω 1/2 K exp i(k + K) r ψ ik = K φ k+k (r)ψ ik (1) MT ψ i (r) = ψ i (r) + Θ( r α ) ψ iαlm (r α ) ψ iαlm (r α ) Y lm (ˆr α ) α lm = K φ k+k (r)ψ ik (2) φ k+k (r) φ k+k (r) = Ω 1/2 exp i(k + K) r + Θ( r α ) φ αlm (r α ) φ αlm (r α ) Y lm (ˆr α ) (3) α lm r α = r R α α R α 2 R α MT ψ iαlm (r α ) ψ i (r) {ψ ik } {R α } E = E {ψ ik }, {R α } (4) 3 R α {ψ ik } F α = de dr α 1

27 = E R α {ψik } + i,k E ψ ik dψ ik dr α + E ψ ik dψik dr α F (1) α + F (2) α (5) (5) 1 F (1) α 2 F (2) α δr α δr α ψ(r) R T (R) T (R)ψ(r) = ψ(r R) (6) δr α δψ i = T ( δr α )ψ i (r) ψ i (r) = ψ i (r + δr α ) ψ i (r) = δr α ψ i (r) (7) 4 F (1) α 4.1 T = w i ψi (r)( 2 )ψ i (r)dr i = w i ψ i 2 ψi dr + i β s β { ψi 2 ψ i + ψ } i 2 ψi dr (8) δr α δt δt = w i { δψi 2 ψ i ψi 2 δψ i + δ ψ i 2 ψi + ψ i 2 δ ψ } i i = 2 w i Re { δψi 2 ψ i + δ ψ } i 2 ψi dr i (9) (9) n(r) MT ñ(r) ñ(r) MT n(r) ñ(r) n(r) 2

28 n(r) ñ(r) V (r) Ṽ (r) MT V (r) = Ṽ (r) MT I U = U I + U I MT + U MT MT + U MT (10) U I U I MT MT U MT MT MT U MT MT ñ(r) MT (10) 1 3 ñ(r) ñ(r) Ũ ñ(r) MT Ũ MT U = Ũ ŨMT + U MT = Ũ + ( U β Ũ β) (11) β (11) Ũ = 1 Ṽ (r)ñ(r)dr (12) 2 Ṽ = 2 U β = ñ(r ) r r dr V β ion (r)n e(r)dr (13) V β e (r)n e (r)dr (14) V β ion (r) = 2Z β r β Ve β ne (r ) (r) = 2 r r dr Ũ β = 1 Ṽ β (r)ñ(r)dr 2 s β Ṽ β ñ(r ) = 2 s β r r dr (15) (16) (17) (18) Ũ ñ(r) ñ(r) = ñ e (r) + β ñ β (r) (19) ñ e (r) = i w i ψ i (r) 2 (20) 3

29 δr α ñ(r) δñ(r) ñ α (r) δr α T ( δr α ) ñ(r) ñ α (r) ñ(r) = ñ e (r + δr α ) + β ñ β (r + δr α ) ñ α (r + δr α ) ñ e (r) β ñ β (r) ñ α (r) δ ñ α (r) = δr α ñ(r) ñ α (r) (21) δñ(r) = δr α ñ(r) ñ α (r) + δ ñ α (r) (22) Ṽ (r)δ ñ α dr (23) δñ(r) (12) δũ = Ṽ (r)δñ(r)dr = Ṽ (r)δr α ñ(r)dr Ṽ (r)δr α ñ α (r)dr + (23) 1 ñ(r) ñ(r)ñ(r ) r r dr 0 (23) 2 Ṽ (r)δr α ñ α (r)dr = Ṽ (r) ñ α (r)δr α ds + δr α Ṽ (r) ñα (r)dr (25) 0 δũ = δr α Ṽ (r) ñα (r)dr + Ṽ (r)δ ñ α dr (26) (24) U α MT α U β U α δu α = Vion(r) α + Ve α (r) δn e (r)dr s α = V (r)δn e (r)dr Vout(r)δn α e (r)dr (27) α out(r) α V α V (r) = V α ion(r) + V α e (r) + V α out(r) (28) 4

30 4.2.4 Ũ α Ũ α α ñ(r) δũ α = Ṽ α (r)δñ(r)dr s α = Ṽ (r)δñ(r)dr Ṽout(r)δñ(r)dr α (29) out α ñ(r) out ñ δñ(r) α (22) Ṽ α δñ(r) = δñ e (r) + δ ñ α (r) = δr α ñ e (r) + δ ñ α (r) (30) δũ α = Ṽ (r)δñ e (r)dr + Ṽ (r)δ ñ α (r)dr Vout(r)δñ(r)dr α (31) V α δu α δu = δũ + δu α δũ α = δṽ (r) ñα (r)dr + V (r)δn e (r) Ṽ (r)δñ e(r) dr (32) δṽ (r) = δr α Ṽ (r) (33) δn e (r) = δr α n e (r) = 2 i w i Re δψ i (r)ψ i (r) (34) δñ e (r) = δr α ñ e (r) = 2 i w i Re δ ψ i (r) ψ i (r) (35) δñ(r) δn(r) δn e (r) α 4.3 E xc = ε xc (ñ e (r)) ñ e (r)dr + ε xc (n e (r)) n e (r) ε xc (ñ e (r)) ñ e (r) dr (36) β s β δñ e (r) δn e (r) δe xc = µ xc (n e (r)) δn e (r) µ xc (ñ e (r)) δñ e (r) dr (37) β s β µ xc (n) = d (ε(n)n) (38) dn 5

31 4.4 {ψ ik } α δe (1) δe (1) = δṽ (r) ñα (r)dr +2 { w i Re δψi (r) 2 + V (r) + µ xc (r) ψi (r) i δ ψ } i (r) 2 + xc(r) Ṽ (r) + µ ψ i (r) (39) F (1) µ xc (r) = µ xc (n e (r)) (40) µ xc (r) = µ xc (ñ e (r)) (41) 5 F (2) α < ψ i S ψ j > ψi (r)ψ j (r)dr = ψi (r)ψ j (r) ψ i (r) ψ j (r) dr = δ ij ψ i (r) ψ j (r)dr + β s β (42) K > K > ψ i >= K K > ψ ik (43) ψ i >= K K > ψ ik (44) α α δ < ψ i ψ j > = δψi (r)ψ j (r) δ ψ i (r) ψ j (r) + ψi (r)δψ j (r) ψ i (r)δ ψ j (r) dr { = 2Re δψi (r)ψ j (r) δ ψ i (r) ψ } j (r) dr δ < ψ i ψ j >= δ < ψ j ψ i > δψ ik (45) δ < ψ i ψ j > + K δψ ik < K ψ j > + < ψ i K > δψ jk = 0 (46) δψ ik δψ ik = j δψ ij ψ jk (47) 6

32 δψ ij = δψji δψ ij = 1 2 δ < ψ i ψ j > δψ ik δψik = { ψjkre j F (2) α δψi (r)ψ j (r) δ ψ i (r) ψ } j (r) dr δe (2) = δψ E ik ψ + E δψ ik i,k ik ψ ik E ψ ik = w i < K H ψ i > δe (2) = ij = 2 i { (w i + w j )Re < ψ j H ψ i > δψi (r)ψ j (r) δ ψ i (r) ψ } j (r) dr { w i ε i Re δψi (r)ψ i (r) δ ψ i (r) ψ i (r) } dr (48) (49) (50) (51) (52) H < ψ j H ψ i >= ε i δ ji 6 ( δe (1) + δe (2)) F α = lim δr α 0 δr α = Ṽ (r) ñα (r)dr 2 { ψi w i Re (H ε i )ψ i (r) ψ } i ( i R α R H ε i ) ψ i (r) dr α = Ṽ (r) ñα (r)dr 2 i w i Re < ψ i R α (H ε i ) ψ i > α < ψ i R α ( H ε i ) ψ i > α (53) α ψi / R α frozenaugmentation exp i(k + K) R α R α Re < ψ i R α (H ε i ) ψ i > α = Im K Kψ ik < K (H ε i ) ψ i > α (54) 7

33 R α ik F α = Im KṼ K ñ α K K 2 i w i KψiK Im K < K (H ε i ) ψ i > α < K ( H ε i ) ψ i > α (55) Ṽ K ñ α K Ṽ (r) ñ α (r) (9) (ψ1 ψ 2 ) dr = ψ1 ψ 2 ds (56) V V ψ1 2 ψ 2 dr = S S ψ1 ψ 2 ds ψ1 ψ 2 dr (57) V ψ 1 = ψ i ψ 1 = ψ i ψ 2 = δψ i ψ 2 = δ ψ i ψi 2 δψ i + ψ i 2 δ ψ i dr = ψi δψ i + ψ i δ ψ i ds S ψi δψ i + ψ i δ ψ i dr = δψ i ψi + δ ψ i ψ i ds S δψ i ψi + δ ψ i ψ i dr = { δψ i 2 ψ i + δ ψ i 2 ψi dr} (58) ψ i = ψ i ψ i = ψ i 8

34 SCF SCF 1 SCF v(r) Rydberg Hψ i (r) 2 + v(r) ψ i (r) = ε i ψ i (r). (1) n(r) n(r) = i ψ i (r) 2 (2) v(r) v(r) = v ext (r) + 2 dr n(r ) r r + µ xc(n(r)) (3) (1) (2) (3) n(r) = F n(r) (4) (4) SCF self-consistent field SCF n (i+1) = F n (i) F (i) (5) (5) n (i) n (i 1) F (i) F (i 1) 1

35 2 2.1 n (i) F (i) n (i+1) n (i+1) = (1 α)n (i) + αf (i) (6) Dederichs Zeller 1 n ( ) α < α crit 2/µ max µ max ɛ = I δf δn µ min α opt 2/ (µ max + µ min ) α crit Si (7) 2.2 Anderson 2 Anderson 2 n (i+1) ñ (i) F (i) n (i+1) = (1 α )ñ (i) + α F (i) (8) ñ (i) F (i) 2 ñ (i) = (1 β)n (i) + βn (i 1) (9) F (i) = (1 β)f (i) + βf (i 1) (10) β ñ (i) F (i) F (i) ñ (i) 2 = drw(r) F (i) ñ (i) 2 < F (i) ñ (i) F (i) ñ (i) > (11) w(r) (11) (9) (10) β 0 β β = < R(i) R (i 1) > < R (i) R (i) > (12) R (i) = F (i) n (i) R (i) = R (i) R (i 1) (13) (14) 2

36 TOTAL ENERGY (Ry) simple α =0.1 simple α =0.2 Anderson α' =0.1 Anderson α' = EXTERNAL ENERGY (Ry) ITERATIONS : fcc Cu Anderson α α Anderson 1 Anderson E fcc Cu E (i) = dr v (i) (r) v (i 1) (r) n (i) (r) (15) 1 P.H. Dederichs and R. Zeller, Phys. Rev. B 28, 5462 (1983). 2 D.G. Anderson, J. Assoc. Comput. Math. 12, 547 (1965). 3

37 1 Anderson Anderson 1 Anderson n (i+1) ñ (i) F (i) n (i+1) = (1 α)ñ (i) + α F (i) (1) ñ (i) F (i) M + 1 ñ (i) = β 0 n (i) + β 1 n (i 1) + + β M n (i M) (2) F (i) = β 0 F (i) + β 1 F (i 1) + + β M n (i M) (3) β i ñ (i) F (i) D =< F (i) ñ (i) F (i) ñ (i) > (4) (4) (2) (3) M M D = β j β k < R (i j) R (i k) > (5) j=0 k=0 R (i) = F (i) n (i) i β i β 0 M β 0 = 1 β j j=1 (6) (6) (5) D = M M β j β k < R (i j) R (i) R (i k) R (i) > j=1 k=1 M +2 β j < R (i) R (i j) R (i) > + < R (i) R (i) > (7) j=1 β k 0 M < R (ij) R (ik) > β k =< R (i) R (ij) > (8) k=1 1

38 fcc Cu Emax=25Ry 10-1 CHARGE DISTANCE α=0.1, M=0 α=0.1, M=1 α=0.1, M=2 α=0.1, M=3 α=0.1, M=4 α=0.1, M=5 α=0.2, M=0 α=0.2, M=1 α=0.2, M=2 α=0.2, M=3 α=0.2, M=4 α=0.2, M= ITERATIONS : fcc Cu R (ij) = R (i) R (i j) (9) 1 fcccu M M = 4 M = 5 1 V. Eyert, J. Comp. Phys. 124, 271 (1996). 2

39 1 {α τ α }r αr + τ α (1) α 3 3 τ α non-symorphic τ α non-primitive {α τ α }ψ(r) ψ({α τ α } 1 r) (2) {α τ α } 1 {α τ α } 1 = {α 1 α 1 τ α } (3) 2 k {E R} {E R}ψ k (r) = ψ k ({E R} 1 r) = ψ k (r R) = e ik R ψ k (r) (4) αk {E R}ψ αk (r) = e iαk R ψ αk (r) (5) {E R}{α a}r = {E R}(αr + a) = αr + a + R = α(r + α 1 R) + a = {α a}{e α 1 R}r (6) {E R}{α a}ψ k (r) = {α a}{e α 1 R}{α a}r = {α a}ψ k ({E α 1 R} 1 r) = {α a}ψ k (r α 1 R) = e iαk R {α a}ψ k (r) (7) 1

40 ψ αk (r) {α a}ψ k (r) {E R} e iαk R ψ αk (r) = λ {α a} {α a}ψ k (r) (8) λ {α a} 2 = 1 (9) 3 Y lm (θ, φ) (θ, φ) (θ, φ ) (α, β, γ) (α, β, γ) D l m m Arfken Mathematical Methods for Physisists Y lm (θ, φ ) = m Y lm (θ, φ)d l m m(α, β, γ) (10) D l m m(α, β, γ) = l+m (l + m)!(l m)!(l + ( 1) k m )!(l m )! k!(l m k)!(l + m k)!(m m + k)! k=0 ( e imγ cos β ) 2l+m m 2k ( sin β ) m m+2k e im α 2 2 (11) 4 ψn(r) k ψ k n(r) = νlm φ k νlm(r)c k νlm,n (12) φ k νlm (r) R νl(r) φ k νlm(r) = R φ νlm (r τ ν R)e ik (τ ν+r) (13) φ νlm (r) = R νl (r)y lm (ˆr) (14) φ k νlm (r) {α τ α} {α τ α }φ k νlm(r) = φ k νlm({α τ α } 1 r) = R φ νlm (α 1 r α 1 τ α τ ν R)e ik (τnu+r) = R φ νlm (α 1 {r τ α α(τ ν R)})e iαk α(τν+r) (15) (τ ν + R) {α τ α } {α τ α }(τ ν + R) = α(τ ν + R) + τ α = τ ν + R (16) ν R ν {α τ α }φ k νlm(r) = R φ ν lm(α 1 (r τ ν R ))e iαk (τ ν +R ) e iαk τ α (17) 2

41 φ νlm {α τ α }φ k νlm(r) = φ νlm (r τ ν R )Dm l m(α 1 ) e iαk (τ ν +R ) e iαk τα R m = m R φ νlm (r τ ν R )e iαk (τ ν +R ) D l m m(α 1 )e iαk τα = m φ αk νlm (r)dl m m(α 1 )e iαk τα (18) (r) ψ n(r) k ψn αk ψ αk n (r) = λ {α τ α} {α τ α }ψ k n(r) = λ {α τα} νlm φ k νlm({α τ α } 1 r)c k νlm,n = λ {α τα} ν lm = λ {α τ α } = ν lm φ αk C αk ν lm,n = λ {α τ α} m m ν lm φ αk φ αk ν lm (r)dl m m(α 1 )e iαk τα C k νlm,n ν lm (r) m D l m m(α 1 )C k νlm,n ν lm (r)cαk ν lm,n (19) D l m m(α 1 )C k νlm,n (20) λ {α τ α} = λ {α τα} e iαk τα (21) 5 ψn(r) k ψ k n(r) = K φ k+k (r)c k+k n (22) φ k+k (r) φ k+k (r) = Ω 1/2 expi(k + K) r (23) {α τ α }φ k+k (r) = φ k+k ({α τ α } 1 r) = φ k+k (α 1 r α 1 τ α ) = φ α(k+k) (r)e iα(k+k) τα (24) (r) ψ n(r) k ψn αk ψ αk n (r) = λ {α τ α} {α τ α }ψ k n(r) = λ {α τ α} K {α τ α }φ k+k (r)c k+k n = λ {α τ α} K φ α(k+k) (r)e iα(k+k) τ α C k+k n 3

42 = λ {α τ α} K φ αk+k (r)e i(αk+k ) τ α C k+α 1 K n = K φ αk+k (r)c αk+k n (25) C αk+k n = λ {α τ α} e i(αk+k ) τ α C k+α 1 K n (26) 4

43 X IT 1 1 IT r = (x, y, z) a b c r = ax + by + cz (1) x, y, z IT x, y, z (1) r x r y r z = S 11 S 12 S 13 x S 21 S 22 S 23 y S 31 S 32 S 33 z S = a x b x c x a y b y c y a z b z c z (2) (3) S lattice system a b c a a 0 0 S cubic = 0 a 0 (4) 0 0 a a c b y c z S hex = 3 2 a a a c (5) lattice type T R 1, R 2, R 3 R R = R x 1 R x 2 R x 3 R y 1 R y 2 R y 3 R z 1 R z 2 R z 3 = ST (6) 1

44 T 6 P F I A A B B C C P = (7) I = F = /2 1/2 1/2 0 1/2 1/2 1/2 0 1/2 1/2 1/2 1/2 1/2 1/2 (8) (9) 1/2 1/2 1/ A = 0 1/2 1/2 (10) B = C = 0 1/2 1/2 1/2 0 1/ /2 0 1/2 1/2 1/2 0 1/2 1/ (11) (12) 2 IT r = (x, y, z) r = (x, y, z ) I4/mcm x, y, z; x, ȳ, z; x, ȳ, z; x, y, z; x, y, z; x, ȳ, z; x, ȳ, 1 2 z; x, y, 1 2 z; ȳ, x, z; y, x, z; y, x, z; ȳ, x, z; ȳ, x, z; y, x, z; y, x, 1 2 z; ȳ, x, 1 2 z. P 6 3 /mmc x, y, z; ȳ, x y, z; y x, x, z; ȳ, x, z; x, x y, z; y x, y, z; x, ȳ, z; y, y x. z; x y, x, z; y, x, z; x, y x, z; x y, ȳ, z; x, ȳ, z; y, y x z; x y, x, z; x, y, 1 2 z; ȳ, x y. 1 2 z; y x, x, 1 2 z; y, x, z; x, y x, z; x y, ȳ, z; ȳ, x, 1 2 z; x, x y, 1 2 z; y x, y, 1 2 z. 1 2 D Sr = DSr (13) 2

45 r = S 1 DSr (14) r = Wr (15) D D = SWS 1 (16) P 6 3 /mmc 3 (y x, x, z) W y x x z = x y z (17) W (5) (16) hex = S hex D (3) S 1 hex = (18) W D 1 N.F.M. Henry and K. Lonsdale (eds.), International Tables for X-ray Crystallography, 3rd ed., (The Kynoch Press, England, 1969). 3

46 Murnaghan 1 Murnaghan Murnaghan Ω p 1 ( p = B ) B 0 Ω B 1 Ω 0 (1) p = 0 Ω 0 (1) B(Ω) Ω dp ( ) B Ω dω = B 0 Ω 0 ( B 0 = Ω dp ) dω 0 B 0 (2) (2) (3) Ω db(ω) ( ) B Ω dω = B 0B Ω 0 (4) B B = 1 ( Ω db ) = 1 ( ) Ω 2 d2 p B 0 dω B 0 dω (1) E(Ω) = B ( ) B 0Ω 1 Ω B B E 1 Ω 0 (5) = B 0 Ω B 0 B (B 1) Ω1 B + B 0 B Ω + E (6) E 2 (6) E(Ω) = c 1 Ω 1 B + c 2 Ω + c 3 (7) 1

47 c 1, c 2, c 3, B ( c1 (B 1) Ω 0 = B 0 = c 2 B c 2 ) 1/B (8) (9) 3 1 1Ry/Bohr 3 Pa 1 a.u. = 1 Ry 1 Bohr 3 = J ( m) 3 = Pa = GPa (10) 1 F.D. Murnaghan, Proc. Natl. Acad. Sci. U.S.A. 30 (1944)