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Density-functional theory beyond the pseudopotential local density approach: a few cases studies

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(1)SISSA. ISAS. SCUOLA INTERNAZIONALE SUPERIORE DI STUDI AVANZATI INTERNATIONAL SCHOOL FOR ADVANCED STUDIES. 

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(156) W < o >A@]FGFHE KMC o XZFH@GL o pU\xFH;=<L5NDF y f >TpUE< <5vYXT;xp o R5<5E XZL5NTNT<S\BpUFH@GL o < o <NTRbF LUK p *  # @ o FH<SN p5XZFA@ o R(<S\]<JXZFHNTL o R^p5>nO@]FA; FH;=<P>TpUE<tIQ< o >A@GFGFJI * ;O (   a@?;O 4. y f :7;<KMC o X FA@GL o ;O tIQ<S_ < o I=>\GLYXSpU\G\rF$C_xL o FH;< IQ< o >A@]F F pUF+FH;</_ L5@ o QF ?^q f F!;xp5> D <S< o IQ<SFH<SNTE@ o <JI p5XXZCN pUFA<S\rF.@ o yJzblbh$FA;NTL5CR5; p L o FA< c pUNT\GL >A@GE-C=\ pUFA@]L o Ubf nb;@BXT;*_NTL{Z@ I <?I FH;<FHL5FTpU\^R5NTL5C o INEg>AF pUFA<< o <NTbR FLUKQFH;<C o @ KML5NTE @ o FH<SN p5XZFA@ o R7<S\G<JX FANTL o R^p5>pUF >A<5Z5<N pU\!IQ< o >A@GFH@G<J>Sq :u;<J>A< < o <SNTR5@G<J> ; p Z5<$Dx<S< o @ o FA<NT_xL5\BpUFH<JI ND F >A5< Z5<SN pU\ pUCFA;L5N > Ube VUHx p o I FA;< XZL5E_CFTpUFA@]L o >NT<S_xL5NTFH<JI @ o FA;@B>FA;<J>A@ > pUNT<_x<SNAKML5NTE<?I C >A@ o R-FA;<NT<JXZ@]_x<+LUK kO<ZKHq Ube vq K Wd @B> 6< vQp5XZF3KML5N p C o @ KML5NTE p> FQ>AFH<SE p o I @B>35< v_ <JXZFH<JI FAL Dxw < Z pU\G@ I KML5N >pFY>AFA<SE >Bnb@GFA; >A\GL nbq\ FZ pUDN F@ o R<S\G<?XZFANTL o IQ< o >A@]F F q f o pU\]\FA;<L5FH;<SN!XSp5>A<?> FH;< K W%d pU__NTL v^@GE pUFH@GL o @B>)@ o IQ<S<JI+C o XZL o FANTL5\]\G<JI S@]FT> gC >AFA@ 9XSpUFH@GL o NT<S\G@]<J> E pU@ o r\ F@ o @]FT>"pUD@G\]@GGF F LUK-NT<S_NTLYIQCxX @ o R3FA;<$5< v^_x<SNT@]E< o F pU\R5NTL5C o NI Eg>AF pUFA<_NTL5_x<SNTFH@G<J> LUK p.\BpUNTR5< o CED <SN LUK>AL5\G@BI> yJz O<J>A_ <JXZ@BpU\]q\ F XZL Z pU\G< o FL5N E<SF pU\G\G@ Xp> FQ>AFH<SE >Sq ]h Z5< o \]@GE@]FA@ o R FAL.FH;< n7L5N Y>OL o >A@G\]@BXZL o p o I R5<SNTE p o @]CE QL5N fAfHf E l >A<E@BXZL o IQC XZFAL5N > ^\G@]FA<SN pU\]q\ F FH;L5Cx>Tp o I> LUK)_ pU_x<N > ; p Z5<pU__x<?pUNT<JI

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(158) I nO;@BXT;!_NT5< Z5< o F

(159) FH;<75< vFH< o >A@GL o LUK E p o F!XZL5E_CFTpUFH@GL o > FHL->AL5E<b>AL5\G@ I=>Sq T L5N@ o >AFTp o XZ< UFH;<XZL5;<J>Ar@ Z5<O< o <NTR5@G<J> LUKm>AL5\G@ I=> pUNT< p> FY>AFA<SE pUFH@BXSpU\]q\ F {L Z5<SNT<J>AFH@GE pUFH<JI nO;@G\]<3\BpUFAFH@BXZ< XZL o >AFTp o F >pUNT<3p> FY>AFA<SE pUFH@BXSpU\]q\ F C o IQ<SNT<J>AFA@]E pUFA<?Imq h"NTNTL5N > @ o FA;< >AFANTC XZFHC=N pU\-_NTL5_x<SNTFH@G<J> pUNT<.C >AC pU\Gq\ F4>AE pU\]\+KML5N XZDN FY>AFTpU\ > nO@]FA; XZ{L Z pU\G< o F,L5NE <SFTpU\]\G@BX1D L o I=> D=CF@]F"@B> n7<S\G\ o L n o FA; pUF"FA;<Oz; FQIQNTL5R5< o D L o I XSp oo L5FDx< IQ<J>TXZNT@]Dx<JIp5XSX C=N pUFH<Sq\ FnO@]FA;@ o K Wd y x q f o FH;=< <S\ I LUK0E<SF pU\B> FH;=<-R5NTL5C o I E >AFTpUFA<P>AFANTC XZFHC=NT<!LUK0XZDN FY>AFTpU\]\G@ o <!@]NTL o @ >_=NT<?I @ XZFA<?I FHL Dx<!_ pUN pUE pUR o <SFH@BX KvXSX @ o >AFH<Jp5I$LUKp/KM<NTNTL5E pUR o <SFA@ XD9XSX U5z q f o FA;@B>+FH;<J>A@B

(160) > n7< >A;{L n FH;xpU8F K Wd @ > o L5F>AC XZ@G< o FAr\ F!p5XSXZCN pUFH<7@ o FH;=<%IQ<J>TXZNT@]_=FH@GL o LUK9FA;< n7<?p !DxL o I> D <SGF n7<S< o X ; pU@ o >)@ o >A<S\]< o @GCE p o I FA;@B>O<NTNTL5NO_NTL5_ pUR^pUFA<J>b@ o p o C o XZL o FANTL5\]\G<JI nbPp F FALFA;<!_NT<JIQ@BXZFH@GL o > LUK"\]@ o <JpUDN E NT<?>A_xL o >A<_NTL5_x<SNTFH@G<J>1LUK FA;<!E pUFA<NT@BpU\ Q\]@ 5<_;L o L o KMNT < _5C=< o X @G<J> Q<Zr9<JXZFHq@ Z5< XT; pUNTR5<J> _@G6< gSL<\G<JXZFHNT@BX_=NTL5_ <SNTFA@]<J>7<SFTXUq]q f o Kvp5XZF%\G@ o <JpUDN EvNT<J>A_xL o >A<b_=NTL5_ <SNTFA@]<J>7XSp o D < >AFHCxIQ@]<J'I nb@GFH;3>AC XSXZ<J>T>L o q\ F@ K0FA;< >AFANTC XZFACN pU\_NTL5_ <SNTFA@]<J>LUK7FA;<E pUFA<SNT@ pU\"pUNTk< n7<\G\ NT<S_NTLYIQCxX <JI Ynb@GFH;3<SNTNTL5N >\GL n7<N+FA; p o UEg{ q :7;<SNT<ZKML5NT<-FA;< _NT<JXZ@B>A@]L o NT*< _ C@GNT<?I ND FFA;@B>bF F^_x<-LUK0XZL5E_CFTpUFH@GL o >u@ >\BpUNTR5<SNbFA; p o Cx>AC pU\,p o I @ o E p o FXSp5>A<?> @]F_xL5@ o FT>  .    . . .   .  . . . .  . $. 

(161).  . . . . . . . . . . . . . . . . . .  .

(162). .  . . ". . . . . . . .

(163) y e W < o >A@]FGFHE KMC o XZFH@GL o pU\xFH;=<L5NDF L5CFOFH;< o <JX <J>T>A@GFGFLUK"R5L5@ o RD <5F5L o IK Wd FHLL5DFTpU@ o E<Jp o @ o RUKMC\xNT<?>AC=\]FT>Sq K Wd @ > o Ln o FAL ;xp Z5<pUF\]<Jp5>AFL o <NT<SE pUN pUD\G<KvpU@]\GCNT< <5Z5< o @ o XZL5EEL o >A<SE@ XZL o IQCxX FAL5N >5I3FA;< I @]<S\G<?XZFANT@ X$XZL o >AFTp o F > pUNT<$LUKMFH< o L{Z5<SNT<J>AFH@GE pUFH<JI yJ{ Bnb@GFH; p o <SNTNTL5NLUK!FH;<(L5N IQ<SN LUK y h L5N \BpUNTR5<SN bp o I FH;@B>/_NT<5Z5< o FT>p$EL5NT<FH;L5NTL5C=R5; >AFAC NI F$LUK%FA;<IQ<SNTr@ Z pUFAq@ Z5<?>+LUKFA;@B> _5C p o FA@]GF F-nO;@ X ;3; Pp Z5< @GE _xL5NTFTp o F+<5v^_x<SNT@]E< o F pU\ @GE _=\]@BXSpUFH@GL o > KML5Nb5< vYpUE_\G<@ o kpUE p o >TXSpUFHFA<SNT@ o R-L5Nb@ o _;L5FAL^<S\Bp5>AFH@BX_;=< o L5E < o pQq lpUNT@GL5C >+pU__NTL v^@GE pUFA@GL o >O; p Z5<Dx<S< o @ o FANTLYIQC XZ<JI(@ o FH;< XZL5CN >A<LUKbFA;< F5<JpUN > FAL @GE_NT{L Z5< K Wd @ o Xp5>A<J> nO;<SNT< @GFKvpU@G\ > b{ h {Qy 7D=CF o L XZL5EEL o q\ F p5XSXZ<S_FH<JI pUE<S\]@GL5N pUFH@GL o ; p5> D <S< o KML5C o I8C o FH@G\ o L ntq mb<SNT< n7< p o pUq\ F@gS< L o < _xL^>T>A@]D\G< @GE _=NT{L Z5<SE< o F FH8L K Wd FH; pUF1@B>"FA;<b@ o XZ\GC >A@]L o LUK,p M c @ o FA;<5< vYX ; p o R55< EgX L5NTNT<S\BpUFH@GL o < o <SNTbR F b{ U =q X ;<SFA;<SN0>AC X ; M c pUNT<bIQ< o @GFH<Sq\ F-p o @]E_NTL Z5<SE < o FT>

(164) L Z5<NFH;< K Wd L5N o L5F ^@B>1_NT<J>A< o FHq\ F!C o IQ<SN7IQ<SD pUFA< I@ o FA;@B>1FH;<J>A@BS> n7<R5r@ Z5<>AL5E<_@]<JXZ<LUK)@ o KML5NTE pUFA@GL o FALP>AC X ; p o @B>T>AC<5q d EL o RFA;<\ pUNTR5< o CED <SNLUK

(165) _NTL5_ L^>A<JIPKML5NTE > LU]K M c n7<%IQ<JXZ@BIQ<JI FAL X L o X < o FAN pUFH<(L o FA;<$_NTL5_xL^>TpU\LUK w1<?X 5<(KML5NFA;<$5< vYXT;xp o R5< {5{ tp o I FA; pUFLUK [<SN IQ5< n8KML5N FH;<3XZL5NTNT<S\ pUFA@GL o XZL o FHNT@GDCFA@]L o { W Bnb;@BXT; @B> FA;<(Kv p Z5L5NT@GFA< XT;=L5@ XZ< LUK L5FA;<SN%pUCFA;L5N > y x Gy e vq :7;=<-E pU@ o >AC XSXZ<J>T>LUKbFA;< w1<?X 5< p o I$[<SN IQ5< n KML5NTE LUKbFH;< M c @B>tp IQ< o @GFA< @GE _=NT{L Z5<SE< o F1LUK"FH;=<PXZL5E_CFA<JIXZL5;<J>Aq@ Z5<t< o <SNTR5@G<?>uLUK"E L5\G<JXZC\]<J>Op o I>AL5\]@BI> Hnb;@G\G< FA;<7@]E_NTL Z5<SE< o FxL o L5FH;=<N_N; FQ>A@ XSpU@\ _5Cxp o FA@GFH@G<J>)@B> o L5F,>AL>AFANT@ @ o R y f q f o _xpUNTF)FA;@ > XZL5C\BI Dx<PI C<FAL FA;<*Kvp5XZFbFA; pUFbFA;@B> M c pU__NTL v^@GE pUFH@GL o ; p5>7Dx<S< o LUKMFA< o FA<J>AFH<JI L o E pUFH<SNT@BpU\ >p o I_NTL5_ <SNTFA@]<J> nb;<SNT< K Wd @ >1pU\]NT<Jp5NI F-p*R5L^LQI pU__NTL v^@GE pUFH@GL o \G@ 5< {4 J4 EL^>AF)_NTL5_x<SNTFH@G<J> LUK=>A@GE_\G<0>A<SE@ XZL o I C XZFHL5N > y f q f o >AL5E<7Xp5>A< U\G@ 5< {4 4JN; FQIQNTL5R5< o DxL o IQ<JI3p> FY>AFA<SE > SM c a CYa(_NT{L Z@ I < p o L5CFT>AFTp o IQ@ o R@]E_NTL Z5<SE< o F y x vq X$< FA<?>AF FA;< M c L o FA;< X L5E_C=F pUFA@]L o LUK%IQ@G<\G<JXZFHNT@BX-XZL o >AFTp o F >LUK%>A@GE_\G<->A<E@BXZL o IQC XZFAL5N > nb;=<NT<*@]F7@B.> n7<S\G\ o L n o FA; pUF K Wd XSp o Dx<+< r

(166) <JXZFHq@ Z5<Sr\ F @GE_NTL Z5<?I ^p o I n7<Cx>A<@]FOFHL IQ<SFA<NTE@ o <FA;<t>AFANTC XZFHC=N pU\

(167) _NTL5_ <SNTFA@]<J>1LUK>A<S\]< o @GCEq f o p M c >TXT;<SE< @GF @B> p5>T>ACE<JI FH; pUFFA;<5< vYX ; p o R55< EgX L5NTNT<S\BpUFH@GL o KMC o XZFH@GL o pU\ IQ<S_x< o I>7\GLYXSpU\Gr\ FL o FA;<tIQ< o >A@GGF F A! L o FH;=<PIQ< o >A@GGF FR5N p5IQ@G< o 5F I * ;O (   aH? ;O # 4. y e 7 T L5\]\G{L nO@ o R%w1<JX 5<p o I![<SN IQ5< n UFA;<b5< v^_=\]@BXZ@]F KML5NTE LUK FA;<b< o <SNTbR F!IQ< o >A@GGF F @ o mpUNTFANT<S< C o @GFT> QNT<?p5I=5> I 8 y x ( ;O U  B B # D4 ;,O # 7 7 7 f o FH;=@ >(5< v^_NT<J>T>A@GL o n7<3@ o IQ@BXSpUFH<J[I nb@GFH; ;,O FA;< 5< v_NT<J>T>A@]L o LUK 5< vYX ; p o R55< E XZL5NTNT<S\ pUFA@]L o KMC o X FA@GL o pU\R5q@ Z5< o @ o kO<ZKHq Ube vq T CNTFA;<SNTEL5NT< n7<; p Z5<5< v^FAN p5XZFH<JI FH;=< . .  . . . .  . . #

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(170).  .  . . 

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(172) W < o >A@]FGFHE KMC o XZFH@GL o pU\xFH;=<L5NDF y x IQ<S_ < o I < o X <OL o FA;<>D_5C pUNT<OLUK)FH;<R5N p5IQ@G< o F nb;=@ X ; @B>@]E_xL5NTF p o FKML5N1FH;=< B 7 B h \]@GE@]F7LUK"FA;<!IQ<SNT@rZ pUFA@qZ5<?>uLUK"FH;@B>7< o <SNTRbF5q":7;=<KMC o X FA@GL o # 7 "NT<Jp5I5> I  / #. y l # ( U 7 nb;<SNT<(FH;<SNT< pUNT<(GF n7L3FA<NTE 3> nb;@BXT; IQ<S_ < o I=>C_xL o FA;< R5N p5IQ@G< o F LUK!FA;< X ; pUNTR5< IQ< o >A@GFGF I B B# ( hN4]yJbz U yJz 7 poI 8 (4y e ,C :X  ! # Ubh nb@GFH;. (wU B 7 B 4 UQy :7;<*KMC o XZFA@]L o "@ (> nbNT@GFHFA< o @ o FH<SNTE >7LUK. { J ( W UbU p5> 8 hN8 4VhbhbUbfbeb8 l 8 8 J 8 J ( hN4VhbhQ y ebeHx y U5{ J J y h J # p o I ( hN4VhbU5b{ Ubebe ( x@4 b{ l5z 3 y h  ( lN4 x{U5{ ( hN4dWNx{U ( hN4 hbhW@UQq+:7;< NT<Jp5>AL o >mKML5N"FH;=<bXT;L5@BXZ<7LUK9FA;@B> KML5NTE4LUK

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