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O desenvolvimento da Teoria do Funcional da Densidade

(DFT) tem se concentrado, sobretudo, em dois pilares

fundamentais: (1) a busca por funcionais de troca e

correlação (XC) mais precisos; (2) a viabilidade de

implementação computacional diante de sistemas com

muitos elétrons. Nesse contexto, o objetivo principal

deste trabalho consiste em utilizar sistemas quânticos

unidimensionais, mais simples de serem tratados

numericamente, como laboratórios teóricos para o

desenvolvimento de alternativas de implementação

numérica de funcionais orbitais (OFs) da densidade. Por

definição, OFs são todos aqueles que dependem apenas

implicitamente da densidade, via formulação explícita em

termos dos orbitais Kohn-Sham. Exemplos típicos são os

funcionais XC advindos da correção de auto-interação de

Perdew e Zunger (PZSIC). Formalmente, via equações de

Kohn-Sham, a implementação de OFs deve ser procedida

por meio do método do potencial efetivo otimizado (OEP)

que, no contexto computacional, é con

hecido por se

tornar demasiadamente custoso, inclusive para sistemas

com poucos elétrons (N ̴10). Sendo assim, investigamos,

de forma sistemática, alternativas de simplificar ou evitar

o procedimento OEP, tomando como referência a

implementação da correção PZSIC aplicada a cadeias de

Hubbard unidimensionais.

Joinville, 2014

DISSERTAÇÃO DE MESTRADO

FUNCIONAIS ORBITAIS:

INVESTIGAÇÃO DE ESTRATÉGIAS DE

IMPLEMENTAÇÃO NO CONTEXTO DA

FORMULAÇÃO KOHN-SHAM DA

TEORIA DO FUNCIONAL DA

DENSIDADE

ANO

2014

UNIVERSIDADE DO ESTADO DE SANTA CATARINA

UDESC

CENTRO DE CIÊNCIAS TECNOLÓGICAS

CCT

PROGRAMA DE PÓS-GRADUAÇÃO EM FÍSICA

MARSAL EDUARDO BENTO

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!" # $%&'(&%# )&'$*&+&,#-./

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ˆ

HΨ(r1,r2, ..,rN;R1,R2, ..,RM) =EΨ(r1,r2, ..,rN;R1,R2, ..,RM),

HJ= I $- 80$Hˆ3E$Ψ(r1,r2, ..,rN;R1,R2, ..,RM)(5#(&.- + 6.-(2,+5(.5+3

(19)

!

"#$%#&'()*+#,'#- . /*+(0'1,(*,1 &1+%0#'1 21 $($'#+* 3 2*21 %1"4

ˆ

H =−

M X

i=1

~2

2mZi

∇2Ri

N X i=1

~2

2me

∇2ri+

1 4πǫ0

M X i M X j>i

ZiZj

|RiRj|

− 1

4πǫ0

N X i M X j

Zje

|riRj| +

1 4πǫ0

N X i N X j>i e2

|rirj|,

56-67

#+ 89# mZ: Z # R $;1 * +*$$*: &*"<* # %1$(=;1 21$ ,>&0#1$: me:

−e #r $;1 * +*$$*: &*"<* # %1$(=;1 21$ #03'"1,$- .$ 21($ %"(+#("1$

'#"+1$ 2* #89*=;1 56-67 (,2(&*+ * #,#"<(* &(,3'(&* 21$ ,>&0#1$ # #03? '"1,$: "#$%#&'()*+#,'#- .$ '#"+1$ "#$'*,'#$ "#%"#$#,'*+ *$ #,#"<(*$ 2# (,'#"*=;1 5"#%90$;1 ,>&0#1?,>&0#1: *'"*=;1 ,>&0#1?#03'"1, # "#%90$;1 #03'"1,?#03'"1,7- @#$10)#" * #89*=;1 2# A&/"B2(,<#" &1+ #$$# /*+(0? '1,(*,1 3 (,)(C)#04 1 %"1D0#+* 89E,'(&1 2# +9('1$ &1"%1$ &1,$($'# ,* '*"#F* 2# #,&1,'"*" D1*$ *%"1G(+*=H#$ %*"* 1 /*+(0'1,(*,1 56-67 #I19 %*"* * F9,=;1 2# 1,2*Ψ

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

ˆ

H =−

N X i=1

~2

2me

∇2ri 1

4πǫ0

N X i M X j

Zje

|riRj|+

1 4πǫ0

N X i N X j>i e2

|rirj|

56-U7 # 9+ /*+(0'1,(*,1 ,9&0#*" 2#$&"#)#,21 1 +1)(+#,'1 21$ ,>&0#1$: 89# ,;1 $#"C &1,$(2#"*21 ,#$'# '"*D*0/1- @#$10)#" * #89*=;1 2# A&/"B? 2(,<#" &1+ 1 /*+(0'1,(*,1 *,'#"(1" *(,2* 3 9+* '*"#F* #G'"#+*+#,'# &1+%0#G*- J %*"'(" 2# *<1"* *21'*"#+1$ 9,(2*2#$ *'Q+(&*$: #$&10/(2*$ 2# +121 89#4

(20)

! "#$%#&' (# )$)#*+#%,-.$)*

"#$%&'() * &(#&* () *+,&$) )+&$&'*- . /0 1)2#'3#+$ # 4)2' 5)'6 7&(#+*+*8 %8 7&7,#8* 5)8 9 #-:,+)'7 (#75+&,)7 ;#-) 2*8&-,)'&*') Hˆ

'<) +#-*,&=>7,&5)?

ˆ

H = ˆT+ ˆVee+ ˆVext, .@AB/

#8 C%# Tˆ : ) );#+*()+ #'#+$&* 5&':,&5*0 Vˆee : ) );#+*()+ #'#+$&* (#

&',#+*D<) #Vˆext : ) ;),#'5&*- #E,#+')A F) ;+&'5>;&) =*+&*5&)'*-0 ,#8)7

C%#?

E0=hΨ1|Hˆ|Ψ1i<hΨ2|Hˆ|Ψ2i −→Ψ16= Ψ2, .@AG/

5)8 E0 7#'() * #'#+$&* () #7,*() H%'(*8#',*-A

I J8 (# (#8)'7,+*+ C%# %8 7&7,#8* ;)(# 7#+ 5)8;-#,*8#',# #7;#5&J5*() ;)+ 8#&) (* (&7,+&3%&D<) (# (#'7&(*(# n(r)0 1)2#'3#+$ #

4)2' ;+);%7#+*8 * #E&7,K'5&* (# ()&7 ;),#'5&*&7 (&7,&',)7 .;)+ 8*&7 (# %8* 5)'7,*',#/0Vˆext,1#Vˆext,20 ,*- C%#?

E0(1)=hΨ1|Hˆ(1)|Ψ1i<hΨ2|Hˆ(1)|Ψ2i, .@AL/

E0(1)=hΨ1|Tˆ+ ˆVee+ ˆVext,1|Ψ1i<hΨ2|Tˆ+ ˆVee+ ˆVext,1|Ψ2i, .@AM/

#8 C%#Ψ1#Ψ2 7<) *7 H%'DN#7 (# )'(* () #7,*() H%'(*8#',*- *77)5&6

*(*7 *)7 ;),#'5&*&7Vˆext,1#Vˆext,20 +#7;#5,&=*8#',#0 5)8Ψ16= Ψ2A

E0(1)<hΨ2|Tˆ+ ˆVee+ ˆVext,1+ ˆVext,2−Vˆext,2|Ψ2i,

E0(1)<hΨ2|Hˆ(2)+ ˆVext,1−Vˆext,2|Ψ2i,

E0(1)< E (2)

0 +hΨ2|Vˆext,1−Vˆext,2|Ψ2i,

.@A!/

E0(1)< E (2)

0 +

Z

d3r n2(r)

h

ˆ

Vext,1(r)−Vˆext,2(r)

i

. .@A O/

F* 8#78* H)+8*?

E0(2)=hΨ2|Hˆ(2)|Ψ2i<hΨ1|Hˆ(2)|Ψ1i, .@A /

# #',<)?

E0(2)< E0(1)+

Z

d3r n1(r)

h

ˆ

Vext,2(r)−Vˆext,1(r)

i

(21)

!

"#$%&'# %( )*+%,-)( . /0!1 ) . /0 12 #34)$#(5

E0(1)+E (2)

0 <

Z

d3r[n2(r)−n1(r)]

h

ˆ

Vext,1(r)−Vˆext,2(r)

i

+E0(1)+E (2)

0 .

. /061 78#9%2 ()n2(r) =n1(r)2 4)9)$#( *+)5

E0(1)+E (2) 0 < E

(1)

0 +E

(2)

0 =⇒:#&49%';,<#= . /0>1

?#94%&4#2 n2(r) &<# @#') ()9 ;8+%A % n1(r):%(# Ψ1 ()B% ';C)D

9)&4) ')Ψ2/ E+ ()B%5

Ψ16= Ψ2=⇒)&4<#=⇒n1(r)6=n2(r), . /0F1

n(r)')4)9$;&%Ψ(r)Ψ(r)G C+&:;#&%A ')n(r)/

H;&%A$)&4)2 :I)8%$#( %5

Ψ(r1, ...,rN) = Ψ[n(r)]. . /0J1

K(4% ')$#&(49%,<# '# L)#9)$% ') M#I)&3)98DN#I& G :I%$%'% ') !" #$%&' (' ()*$ #' .01/

O @#((PQ)A @9#Q%9 % @#((;3;A;'%') ') ')(:9)Q)9 *+%;(*+)9 )(4%'#( )R:;4%'#( % @%94;9 '% ')&(;'%') '# )(4%'# C+&'%$)&4%A n(r)/ ?%9%

4%&4#2 3%(4% $#(49%9 *+) Vext = Vext[n(r)]/ ?%94;&'# '% )*+%,<# ')

":I9S';&8)9 @%9% # )(4%'# C+&'%$)&4%A5

ˆ

HΨ0= ( ˆT+ ˆVee+ ˆVext)Ψ0=E0Ψ0, . /0T1

Vext=E0−( ˆT+ ˆVee)Ψ0[n]

Ψ0[n]

, . /0U1

#+ ()B%2Ψ0 ')4)9$;&%Vext % $)&#( ') +$% :#&(4%&4)/ K&4<#2 ()Ψ0G

C+&:;#&%A ')n(r)2Vext4%$3G$ ()9V/ W)(+$;&'#2 4)$#(

n(r)−→Ψ0(r1, ...,rN)−→Vext−→Hˆ −→ {Ψ, E}d, . /0X1

:#$ d 9#4+A%&'# *+%;(*+)9 )(4%'#( )A)49Y&;:#(/ H;&%A$)&4)2 @#')$#(

)(:9)Q)9 % )&)98;% :#$# +$ C+&:;#&%A '% ')&(;'%')5

(22)

" #$%&$'( )$ *%+$,-$&. $ /%+, )$'%,01&( 23$ 4%)$'%0 3156 758(& ( )$,05)()$ $7$1&9,5:( :%'% ;(&5<;$7 :+(;$ ,% 1&(1('$,1% )$ 0506 1$'(0 )$ '351%0 :%&4%0= >%,?%&'$ )50:315&$'%0 ,( 4&@A5'( 0$BC%D $' !EFG H(71$& /%+, $ I3 J$3 K+(' L M 4&%430$&(' ( 03-015135BC% )$ 3' 0501$'( )$ 4(&1N:37(0 5,1$&(.$,1$0 4%& 3' 0501$'( )$ 4(&1N:36 7(0 ,C% 5,1$&(.$,1$0 03-'$15)(0 ( 3' 4%1$,:5(7 $?$15;% −?3,:5%,(7 )(

)$,05)()$−23$ 5,:%&4%&( 1%)%0 %0 $?$51%0 )( 5,1$&(BC%=

!" #$%&'(#) *# +,-./)-&0

>%,05)$&$'%0 3' 0501$'( )$ 4(&1N:37(0 ,C% 5,1$&(.$,1$0 LOPM 03-'$15)% ( 3' 4%1$,:5(7 $?$15;%vs,σ[n](r)= Q $23(BC% )$ K:+&R)5,.$&

0$&<S

−∇

2

2 +vs,σ[n](r)

ψN Ikσ(r) =εkσN IψN Ikσ(r), L = !M

$

nN I(r) =X

σ Nσ

X k

fkσ|ψkσN I(r)|2= X

σ

nN Iσ (r), L = M

$' 23$ 0 ≤ fkσ ≤ 1 T % ,U'$&% )$ %:34(BC% )% %&-51(7 kσD :%' σ

5,)5:(,)% (0 )3(0 4%005-575)()$0 )$ 045, {↑,↓}= Q $,$&.5( 1%1(7 0$&<

)()( 4%&S

EN I[n] =Ts[n] + X

σ Z

d3r nN Iσ (r)vs,σ[n](r). L = VM

Q )501&5-35BC% )$ )$,05)()$ )% $01()% ?3,)('$,1(7 T %-15)( 4%& '$5% )( '5,5'58(BC%

δ(EN I[n]µP σ

R

d3rnN I σ (r′)) δnN I

σ (r)

= 0, L = WM

$' 23$

N =X

σ Z

d3r′ nN Iσ (r′) = X

σ

Nσ. L = GM

"3 0$X(S

δTs[n] δnN I

σ (r)

(23)

!"##$%&# "'&(" " )% #*#+$%" *,+$("'$,+$- .$#+$ /"#&0 )%" 1"# $2)"34$# 56#*/"# 1" 789 1*: ($#;$*+& < $,$('*" +&+"= 1$ )% #*#+$%" $=$+(>,*/&?

E[n] =Ts[n] +EH[n] +Exc[n] +Vext[n], @ - AB

1$/&%;&#+" $% #)"# /&,+(*5)*34$# 1$ $,$('*" /*,C+*/" Ts[n]1$ )% #*#D

+$%" !" # $%&'(% $%0 $% 2)$ &# $=C+(&,# #$ %&E$% #$% #$,+*( " ;($D #$,3" 1&# 1$%"*#0 *,+$("3F& /=6##*/" 1$ G"(+($$ EH[n]$ $,$('*" 1$E*1&

" )% ;&+$,/*"= $H+$(,&Vext[n]- I=C% 1*##&0Exc[n]($;($#$,+" " $,$('*"

1$ $&")' % )"&&%*'+!" @JKB0 2)$ *,/=)* +&1"# "# /&(($34$# ,$/$##6(*"# <# ";(&H*%"34$# $L$+)"1"# ,&# +$(%&# 1$T[n]$Vee[n]0 &) #$M"0

Exc[n] = (T[n]−Ts[n]) + (Vee[n]−EH[n]), @ - NB

/&%T[n]$Vee[n]#$,1& "# $,$('*"# /*,C+*/" $ 1$ *,+$("3F&- O$,1& "##*%0

)%" E$: /&,P$/*1& & L),/*&,"=Exc[n]$H"+&0 " $,$('*" 1& ;(&5=$%" 1$

%)*+&# /&(;&# C $H"+"%$,+$ 1$+$(%*,"1"- !"(" & #*#+$%" *,+$("'$,+$0 +$%&# 2)$?

δTs[n] δnσ(r)+

δEH[n] nσ(r) +

δExc[n]

nσ(r) +vext[n](

r) =µ. @ - QB

R&P, $ OP"% *%;)#$("% 2)$ "# %*,*%*:"34$# @ - SB $ @ - QB =$E"##$% < %$#%" 1*#+(*5)*3F& 1$ 1$,#*1"1$ 1& $#+"1& L),1"%$,+"=0 &) #$M"0

nσ(r)≡nN Iσ (r). @ -TUB

V')"=",1& "# $2)"34$# @ - SB $ @ - QB0 +$%&#?

vs,σ[n](r) =vH,σ[n](r) +vxc,σ[n](r) +vext,σ[n](r), @ -TWB

$% 2)$

vH,σ[n](r) = δEH[n] δnσ(r)

=⇒!&+$,/*"= 1$ G"(+($$; @ -T B

vxc,σ[n](r) = δExc[n] δnσ(r)

=⇒!&+$,/*"= 1$ 9(&/" $ K&(($="3F&. @ -TTB

(24)

"#$%&' (&)*&'+ ,-)- $''&. /010"&' #%$2$3-) & *&%04($-2 050%$1& vs,σ[n](r)

/-/& *02- 06#-78& 9 +!:;+ <'*0($=(-"04%0. %0"&'>

−∇

2

2 +v !,σ[n](r)

ψkσ(r) =εkσψkσ(r), 9 +!?; v !,σ[n](r) =vH,σ[n](r) +vxc,σ[n](r) +vext,σ[n](r) =⇒

,&%04($-2 @&A4BCA-",

9 +!D; 0

n(r) =X

σ Nσ

X k

fkσ|ψkσ(r)|2= X

σ

nσ(r). 9 +!E;

F' 06#-7G0' 9 +!?;B9 +!E; )0(0H0" & 4&"0 /0 !"#$% & ' ()*+,-*#. 9@C;+ F 06#-78& /0 @&A4BCA-" I )0'&21$/- /0 5&)"- -#%&(&4'$'%04%0. (&45&)"0 0'6#0"-%$3-/& -H-$J&>

K ($(2& I )0'&21$/& /- '0L#$4%0 5&)"-> (&"07-"&' (&" #" 1-2&) %0'%0 *-)- - /04'$/-/0M - '0L#$) (-2(#2-"&' & *&%04($-2 050%$1& -%)-1I' /- 06#-78& 9 +!D; 0 '#H'%$%#N"&' & )0'#2%-/& 0" 9 +!?;. )0(-2(#2-4/& -''$"n(r)- *-)%$) /- 06#-78& 9 +!E;+ ,&) =". (&"*-)-"&' (&" & 1-2&)

(25)

!

!" # $%&'(#&)* +, -.#') , '#./

.,*)01# 23'4

"#$%&%'(%) *+,))%) -% ,.&/0#1,23%) .,&, / $4'*#/',+ 56 )% )%7 84#&,1 -4&,'(% , 9#)(:&#, -/ -%)%';/+;#1%'(/ -, "<=> ?%1 -@;#-,A , ,.&/0#1,2B/ 56 1,#) )#1.+%) C4% ./-%1/) #1,8#',&A .&/./)(, DE %1 FGH! ./& I/9%'J%&8 % K/9' LFMA N , *9,1,-, !"#$%& '(# ) )*+,%) )* -#. -A /4 O"P> "% 41, $/&1, 8%&,+A (%1/) C4%

Exc !"[n] = Z

ehomxc [n]n→n(r)d3r, L >QRM

*/1 ehom

xc [n] #'-#*,'-/ , %'%&8#, -% (&/*, % */&&%+,2B/ ./& ;/+41% -%

41 )#)(%1, 9/1/8S'%/ */1 -%')#-,-% n> P.%),& -, )#1.+#*#-,-%A ,

,.&/0#1,2B/ -, -%')#-,-% +/*,+ $%T 9#)(:&#, $/&'%*%'-/ J/') &%)4+(,-/) .,&, 14#(/) )#)(%1,) 'B/ 9/1/8S'%/)>

U )%84'-/ .,))/ %1 -#&%2B/ ,/ ,.&#1/&,1%'(/ -/) $4'*#/',#) -% %'%&8#, 56 '/) +%;, ,/ */'*%#(/ -% ,.&/0#1,2B/ 8%'%&,+#T,-, -% 8&,-#%'(%)A /4 VVPA

Exc##"= Z

eGGAxc [n,∇n]n→n(r)d3r, L >QWM

C4% #'(&/-4T / 8&,-#%'(% -, -%')#-,-% ∇n*/1/ #'8&%-#%'(% ,-#*#/',+

X ,.&/0#1,2B/ +/*,+>

?%84#'-/ , .&/./)(, -% Y%&-%Z L[,*/J\) +,--%&M L]MA *9%8,1/) ,/ (%&*%#&/ .,))/ */'*%#(4,+^ , *9,1,-, 1%(,7VVPA -%_'#-, */1/

E$##"xc =

Z

eM GGA

xc [n,∇n,∇2n, τ]d3r, L >QGM

%1 C4% #'*+4`1/) /) O,.+,*#,'/)∇2n-, -%')#-,-%A ,+N1 -, -%')#-,-%

/&J#(,+ -% %'%&8#, *#'N(#*,A

τ(r) = 1

2

X k

|∇2ψ

k(r)|2. L >!aM

P O"P N 41 $4'*#/',+ +/*,+ %'C4,'(/ C4% , VVP 41 $4'*#/',+ )%1#+/7 *,+ -, -%')#-,-%> P ,.&/0#1,2B/ 1%(,7VVP N 41 $4'*#/',+ )%1#+/*,+ -, -%')#-,-% % -/) /&J#(,#) ψkσ &%)4+(,'(%) -/ *E+*4+/ K/9'7?9,1> U

(26)

"# $"%&%'($) *+ +$%') (,+#)' "# %(-')+$./#0 (1' 2'&$%03 4') "#5(%6 .1'7 8'"#+'0 )##0&)#9#) Exc "#5(%"' 8#2$ #:*$.1' ; 3 <= "$ 0#>*%(?#

+$(#%)$@

Exc[n] =Ex[n] +Ec[n], ; 3AB=

&'+ Ex0#("' ' ?#)+' "# ?)'&$ # Ec $ #(#)>%$ "# &'))#2$.1'3 C00%+7 '

:*$)?' "#>)$* "$ &2$00%5&$.1' &'(&#%?*$2 8)'8'0?$ 8') 4#)"#D %(&2*% $ "#(0%"$"# "# #(#)>%$ "# ?)'&$ ex7 ($ %"#%$ "# E%8#)6FFC@

Exc !!"[n] = Z

eHGGA

xc [n,∇n,∇2n, τ, ex]d3r. ; 3A =

G+ ?'"'0 '0 "#>)$*0 "$ &2$00%5&$.1' 8)'8'0?$ 8') 4#)"#D7 H &'+*+ 'I0#)9$)+'0 ' &E$+$"' #))' "# $*?'6%(?#)$.1'7 &*J$ "#5(%.1' 0#)K $I')"$"$ ($ 0#:*L(&%$3

!" #$%%&'(&) *& +,-$./0-&%+'1$

4$)$ 0%0?#+$0 &'(?#("' $8#($0 *+ #2H?)'( ;N = 1=7 0$I#+'0

:*# $ %(?#)$.1' #2#?)M(%&$ Vee "#9# 0#) (*2$3 N' #(?$(?'7 $ 9$(?$>#+

# &'(9#(%L(&%$ "# 0# *?%2%O$) $ "#(0%"$"# #+ 9#O "$0 -*(./#0 "# '("$ -$O &'+ :*# #00$ 8)'8)%#"$"# 0#J$ *+$ "$0 +$%0 "%-P&#%0 "# 0#)#+ 0$6 ?%0-#%?$0 8') :*$2:*#) -*(&%'($2 "$ "#(0%"$"#7 %(&2*%("' $ QRC7 FFC # $?H +#0+' $2>*+$0 "$0 +$%0 +'"#)($0 E%8#)6FFC03 S ' &E$+$"' #))' "# $*?'6%(?#)$.1' ;TUG=7 "$("' ')%>#+ V (#&#00%"$"# "# !""#$%#& '# ()*!+,-*#"($.! ;TUW=7 *+ 9P(&*2' #X$?' $ 0#) 0$?%0-#%?' # :*# 8'"# 0#) $82%&$"' $ &$"$ *+ "'0 "#>)$*0 &'(&#%?*$%0 $8)#0#(?$"'0 ($ 0#.1' $(?#)%')3

Y#)+% # C+$2"% 8)'8*0#)$+ $ 8)%+#%)$ TUW7 #0&)%?$ &'+'@

Vee#"[n] =

1− 1

N

EH[n], ; 3AZ=

0#("' #00$ $ &'))#.1' "# $*?'6%(?#)$.1' 0#>*("' Y#)+% # C+$2"% ;YC6 TUW= ; [=7 >$)$(?%("' :*# Vee = 0 8$)$ N = 13 W$I# 'I0#)9$) :*#

(27)

!

"#$ %&'&(%&$ %) *$)(%&+) ,(-&*$)%) N (# %&(#.,()%#$/ )

)'$#0,.)12# %& 3&$.,45.)6%, (2# 7 8#(9,9-&(-& &. -).)(:#; ")$) %#,9 9,9-&.)9/ A & B/ (# 6,.,-& %& 9&')$)12# ,(<(,-)/ %&=&$>).#9 -&$

?@&

Vee[nA+nB] =Vee[nA] +Vee[nB]. A ;BBC

"#$7./ -&.#9

Vee ![nA+nB] =

1− 1

NA+NB

EH[nA+nB]

=

1− 1

NA+NB

(EH[nA] +EH[nB])

6

=

1− 1

NA

EH[nA] +

1− 1

NB

EH[nB],

A ;BDC

,(%,8)(%# @. '$#E6&.) 8#(8&,-@)6 %# .#%&6#; 5'&9)$ %) ,.'#$-F(8,) :,9-G$,8)/ ) 8#$$&12# %& 3&$.,45.)6%, (2# 9&$H 8#(9,%&$)%) 8#. .),9 %&-)6:&9 (# %&8#$$&$ %&9-& -$)E)6:#;

5 '$#'#9-) I&,-) &. JKLJ '#$ "&$%&M & N@(*&$ AOC -#$(#@49& ) 8#$$&12# %& )@-#4,(-&$)12# .),9 8#(:&8,%) & ,.'6&.&(-)%); ")$-,(%# %) !"#$%&' %#9 -&$.#9 %& -$#8) & 8#$$&6)12#/ -&.#9P

Ex[n] =hψnmin|Vˆee|ψminn i −EH[n], A ;B!C

&

Ec[n] =hΨnmin|Tˆ+ ˆVee|Ψnmini − hψnmin|Tˆ+ ˆVee|ψminn i, A ;BOC

8#.ψmin

n &Ψminn ,(%,8)(%# #$E,-),9 Q#:(4R:). & I@(1S&9 %& #(%) %&

.@,-#9 8#$'#9/ ?@& %2# #$,*&. T %&(9,%)%&n(r); U,)(-& %& @. &67-$#(/

ˆ

Vee ≡0/ 8#(%@+,(%#4(#9 )

Ex[n(1)] =−EH[n(1)], A ;BLC

&

Ec[n(1)] = 0, A ;BKC

@.) =&+ ?@&

(28)

"#$ n(1)(r) %&'%"(&'# ( ')&*%'(') )+),-.&%"( ') /$ *%*,)$( ') /$

)+0,-#&1

2#-,(&,#3 /$ 4%5#,0,%"# 6/&"%#&(+ 78 )9(,# :Exc[n] =Ex[n] + Ec[n]; ')<) *(,%*6(=)- ( *)>/%&,) %>/(+'(')?

EH[n(1)] +Exc[n(1)] = 0. : 1@A;

B# )&,(&,#3 5(-( ( $(%#-%( '#* 6/&"%#&(%* (5-#9%$('#* #* <C&D "/+#* : 1EF; ) : 1EG; #/ : 1@A; &H# *H# *(,%*6)%,#*1 I(*)('#* &)**) 6(,#3 2)-')J ) K/&>)- :!; 5-#5/*)-($ L/)?

Exc !"#$[n]≡Exc !"#$[n↑, n↓]

=Exc%&'()[n↑, n↓]−

X σ

X k

(EH[nkσ] +Exc%&'()[nkσ,0]),

: 1@ ; L/) 5#') *)- (5+%"('( ( L/(+L/)- 6/&"%#&(+ ') )&)->%( (5-#9%$('# :MNO3 PPO3 ),"1;3 "#$?

nkσ(r) =fkσ|ψkσ(r)|2, : 1@Q;

*)&'# ')R&%'( "#$# ( !"#$ % ! &'($)%*1 S 6/&"%#&(+ 2KTU8 */V,-(% #-D V%,(+ 5#- #-V%,(+ ( "#&,-%V/%WH# ') (/,#D%&,)-(WH# -)6)-)&,) X ')&*%'(') #-V%,(+nkσ(r)3 >(-(&,%&'# L/) #* <C&"/+#* : 1EF; ) : 1EG; *)Y($ *(,%*6)%D

,#* 5(-( /$( '%*,-%V/%WH# ') ')&*%'(') ') /$ )+0,-#&1 S &Z$)-# 0)$

Exc%&'()[nkσ,0] %&'%"( L/) ( "#--)WH# 0 6)%,( *)5(-('($)&,) 5(-( "('(

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

B)**) *)&,%'#3 # -)")&,) "#&")%,# ') `)--# ') ')+#"(+%=(WH#a ,)$ *%'# /,%+%=('# "#$# /$( >)&)-(+%=(WH# '# ADTU_3 +)<(&'# X %')%( ') `)--# ') (/,#D%&,)-(WH# ') $/%,#* )+0,-#&*a3 #/ BDTU_ :Q@;1

(29)

!

!" # $%&#'# #() ( *+,* *-$).-/-0

1,23(*

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

vxc,σ[n](r) = δEorb

xc [{ψmτ}] δnσ(r)

. I EMNJ

O80 /0)3(/()&0 (@4#+/+2'%()2( ' &(4()&P)/+' &0 -.)/+0)'# &( ()(1D+' Q= /0% ' &()*+&'&( nσ(r), 0 410/(&+%()20 -01%'#%()2( /011(20 &(

21'2'1 (**( 2+40 &( *+2.'780 $ 0 /3'%'&0 RS9 IR42+%+?(& ST(/2+C( 902()2+'#J 0. R9U IR42+%+?(& 902()2+'# U(230&J I!, V, WXJ, L.( D(1' .% 402()/+'# %.#2+4#+/'2+C0 #0/'# /0%.% ' 20&0* 0* 01>+2'+* B;E S% *A)2(*(, 2(%0* L.(

vxc,σ[n](r) = δEorb

xc [{ψmτ}] δnσ(r)

=X α Nα X k Z

δEorbxc [{ψmτ}] δψkα(r′)

δψkα(r′) δnσ(r)

d3r′+c.c.

I EMMJ "4G* %')+4.#'780 %'2(%K2+/' &(*/1+2' )0 '4P)&+/( ", ' (L.'780 RS9 40&( *(1 (*/1+2' /0%0Y

X k

Z ψ∗

kσ(r′) [fkσvxc,σ[n](r′)−uxc,kσ[n](r′)]GKS,kσ(r′,r)ψkσ(r)d3r′

+c.c.= 0,

I EMZJ /0%0≤fkσ≤1,

uxc,kσ[n](r′) = 1

ψ∗

kσ(r′) δEorb

xc [{ψmτ}] δψkσ(r′)

(30)

"

GKS,kσ(r′,r) =

X m(m6=k)

ψ∗

mσ(r)ψmσ(r′) εkσ−εmσ

. # $%&'

()*) +"*),- ./ "01/23) # $%4' ) 5)6".78/9 9)7/9 *19685987/68+)

vxc,σ[n](r′).3) /5/:"7" 8,)9/;) ;" <):*/ "=59>786/- ;"+".;) ,": 86":/;)

? /16)@7).,8,6A.78/ ;".6:) ;" 7/;/ 7879) /16)@7).,8,6".6" ;/, "01/2B", CD # $EF'@# $E4'$ G/9 5:)7";8*".6) 6:/H 7).,8I) 1* 7).,8;":J+"9 71,6) 7)*516/78)./9$ K",," ,".68;)- 1*/ ;/, 5:8*"8:/, /5:)=8*/2B", 5/:/ / "01/23) LMN <)8 5:)5),6/ 5): C:8"I":- O8 " P/<:/6" #FQ'- /5:)=8*/.;) / 7R/*/;/ <1.23) ;" S:"". CD "=5:",,/ ./ "01/23) # $%&' 7)*)T

GKS,kσ(r′,r) =

X m(m6=k)

ψ∗

mσ(r)ψmσ(r′) εkσ−εmσ

≈ 1

∆ε

X m(m6=k)

ψ∗

mσ(r)ψmσ(r′)

≈ 1

∆ε[δ(r−r

)ψ

kσ(r)ψkσ(r′)].

# $%!'

O"+/.;) ) :",196/;) ;/ "01/23) # $%!' ./ "01/23) LMN # $%4'-)U6"*), / /5:)=8*/23) COPT

vxc,σKLI[n](r) = Nσ

X k

|ψkσ(r)|2 2nσ(r)

uxc,kσ[n](r) +fkσ¯vKLIxc,kσ[n]−u¯xc,kσ[n]

+c.c.

# $4Q' 7)*

¯

vxc,kσ[n] = Z

ψ∗kσ(r′)vxc,σ[n](r′)ψkσ(r′)d3r′, # $4V' ¯

uxc,kσ[n] = Z

ψ∗

kσ(r′)uxc,kσ[n](r′)ψkσ(r′)d3r′. # $4 '

W5",/: ;" /8.;/ .3) 98."/:- 7)* vKLI

xc,σ[n](r) /5/:"7".;) "* /*U), ),

(31)

!

"#$% &'()* +, $,+-.,/&.)%+ % +,01#$% 2,.)% $& 3'24)& '4#5& $& ,61&78% 9:;<=> , +1?+24214.)%+ % .,+1'2&$% #& ,61&78% 9:;<@>* %?2,)%+ & &-.%A4)&78% $, B'&2,. 9C* DE>F

vxc,σSlater[n](r) = Nσ

X k

|ψkσ(r)|2 2nσ(r)

uxc,kσ[n](r) +c.c., 9:;@ >

+,01#$% & 61&'* $4G,.,#2, $&+ &?%.$&0,#+ HIJ , KL"* #8% )&4+ +, G&/ #,M,++N.4% 1) MN'M1'% &12%OM%#+4+2,#2, 4#2,.#% $,#2.% $% -.%M,$4),#2% $&+ ,61&7P,+ KB;

QN &4#$& )&4+ 1)& G%.)& $, +4)-'4RM&. & ,61&78% HIJ* M%#O +4$,.&#$% % +,014#2,F

|ψkσ(r)|2 nσ(r)

= |ψkσ(r)|

2

P

kfkσ|ψkσ(r)|2

≈ P1

kfkσ = 1

. 9:;@D>

B1?+24214#$% 9:;@D> ,) 9:;@ >* %?2,)%+ & &-.%A4)&78% $, !"#$ %&'($& 9STU> 9DE>F

vxc,σGAM[n](r) = 1 2Nσ

X k

uxc,kσ[n](r) +c.c. 9:;@<>

H+ -.%M,$4),#2%+ HIJ* KL"* B'&2,. , STU +8% M%)-',2&),#2, 0,.&4+* -%$,#$% +,. &-'4M&$%+ & )*$&)*+, G1#M4%#&' $& $,#+4$&$,; I) #,#51) )%),#2% &$)424)%+ 1)& G%.)& ,+-,MVRM& -&.& Eorb

xc [{ψmτ}];

T'() $4++%* ,) 2%$%+ %+ M&+%+ &2( &614 $4+M124$%+ % MN'M1'% $& $,.4W&$& G1#M4%#&' uxc,kσ[n](r) ,) .,'&78% &%+ %.?42&4+ +, G&/ #,M,++N.4%; X%

,#2&#2%* -&.& G1#M4%#&4+ )&4+ )%$,.#%+ , +%R+24M&$%+ M%)% &+ 54-,.O SST+* 4#M'1+4W, % MN'M1'% $,+2, 2,.)% -%$, &-.,+,#2&. .,'&24W& M%)O -',A4$&$,; J%. ,++& .&/8%* &'() $, -# .&#/0$, % HIJ* ( 4#2,.,++&#2, $4+M124.)%+ &'2,.#&24W&+ $, +1#234&' -%. M%)-',2%;

!" #$%&'&()*)%+('*# ,)'#-&(#.#

I) :!!<* Y&R,.% , S%#/&',+ 9D:> %?+,.W&.&) 61, 1) G1#M4%O #&' $, $,#+4$&$, +4)-',+ -%$,.4& +,. 124'4/&$% M%)% &-.%A4)&78% -&.& & 4)-',),#2&78% $, 1) G1#M4%#&' )&4+ +%R+24M&$%F 2.&2&O+, $& $*2'4 0'5-#-2650#$ +-0$&'5$"$ 9BBY>; H -&++% 4#4M4&' M%#+4+2, ,) 4$,#24RM&. & +,014#2, 401&'$&$,F

ExcB[n] = EB

xc[n] EA

xc[n]

(32)

"#$ EA

xc[n] %ExcB[n] &'(&")'(# *+'"&#')&, (% -.#") % "#..%/)01# 2+)&,3

2+%.4A%B5 67#.)4 -#$)'(# ) (%.&8)() *+'"&#')/ (% 9:5;;< -%$#,= vBxc,σ[n](r) =

1

EA xc[n]2

δEB

xc[n] δnσ(r)

ExcA[n]−ExcB[n] δEA

xc[n] δnσ(r)

EAxc[n]

+F[n]δE A xc[n] δnσ(r)

= v B

xc,σ[n](r)ExcA[n]−ExcB[n]vAxc,σ[n](r) EA

xc[n]

+F[n]vAxc,σ[n](r).

9:5;>< ?#'*#.$% )@#.()(# ') /&-%.)-+.) 9A <4 B#(%$#, )($&-&. 2+%=

vxc,σB [n](r)≈vxc,σGSSC[n](r) =F[n]vAxc,σ[n](r), 9:5;C<

"#$ DEE? &'(&")'(# # 2+% (%F'&$#, "#$# !"#$%&#' &(% )$%*#$4 8&,-# 2+% # *)-#. F[n]&'(%B%'(% () B#,&01# ,#@.% ) 2+)/ )-+)5

G+ ,%H)4 )($&-&'(# 2+% I ").)"-%.&J% +$ *+'"&#')/ "#$B/%K#4 # B#-%'"&)/ (% -.#") % "#..%/)01#vB

xc,σ[n](r)B#(% ,%. (&.%-)$%'-% #@-&(#

) B).-&. (% +$ #+-.# B#-%'"&)/4 vA

xc,σ[n](r)5 I),-) +-&/&J). +$ *)-#.

(% %,")/) F[n] ")B)J (% +'&. )$@), ), )@#.()7%',5 ?#$# '1# L#+8%

'%'L+$) %,B%"&F")01# &'&"&)/4 vA

xc,σ[n](r) B#(% ,% .%*%.&. ) 2+)/2+%.

B#-%'"&)/ B.#8%'&%'-% (% +$ *+'"&#')/ %KB/M"&-# () (%',&()(%4 (% *N"&/ #@-%'01#5 O,,% P # 7.)'(% $P.&-# () &(%&) EE?4 B%.$&-&. ) )B.#K&$)01# (% *+'"&#')&, "#$B/%K#, ) B).-&. (% )@#.()7%', "#',&(%.)8%/$%'-% $)&, ,&$B/&F")(),5

Q$ *)-#. (% %,")/#')$%'-# 7/#@)/F[n]B#(% ,%. &'"#$B/%-# ')

-).%*) (% (%,".%8%. B#'-#3)3B#'-# ), (&*%.%'0), (%vA

xc%vxcB4 ,%'(# &'-%3

.%,,)'-% &'-.#(+J&.$#, +$ *)-#. (% %,")/#')$%'-# /#")/4 f[n]4 (%F'&(#

"#$#

f[n](r) = e

B[n](r)

eA[n](r). 9:5;R<

6,,&$4 ) %K%$B/# () %2+)01# 9:5;C<4 B#(%$#, %,".%8%. 2+%

vBxc,σ[n](r)≈vLSSCxc,σ [n](r) =f[n](r)vxc,σA [n](r), 9:5>S<

(33)
(34)

!"#$%" #$

&'(()*#

'+,#,!$+-,"+)%

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

!")+),A@."#. 1$%1%B#% .@ #$QB &)@."BR.B8 '@, &,B G.$BR.B @,)B .B#'&,&,B . ,1A)+,&,B &% @%&.A% &. 4'55,$& &)( $.B1.)#% ,% +,B% '")&)N @."B)%",A8 +'S% T,@)A#%"),"%8 .@ A)"P',P.@ &. B.P'"&, F',"#)(,CD%8 - .B+$)#% +%@%U

ˆ

H =−tX ijσ

c†iσcjσ+H.c.+U X

i

c†ici↑c†i↓ci↓+

X i,σ

Vic†iσciσ, 6 37?

.@ F'. c† .ciσ &.B+$.G.@8 $.B1.+#)G,@."#.8 , +$),CD% . , &.B#$')CD%

&. '@, 1,$#*+'A, +%@ B1)" σ ∈ {↑,↓}"% B*#)% i3 V 1$)@.)$% #.$@% &%

T,@)A#%"),"% $.1$.B."#, , .".$P), +)"-#)+,8 ,#$,G-B &% !""#$% ."#$. %B B*#)%B . &% 1,$W@.#$% t3 V B.P'"&% #.$@% +%$$.B1%"&. X $.1'ABD%

Y%'A%@5),", A%+,A U ."#$. %B .A-#$%"B3 2I % ZA#)@% #.$@% +,$,+#.$)(,

1%BB*G.)B 1%#."+),)B .[#.$"%BVi3 H@, +%"LP'$,CD% .A.#$O")+, 1.$@)#)&,

1,$, % 7\4] - ,1$.B."#,&, ", LP'$, 37U +,&, B*#)% &, $.&. 1%&. .B#,$ G,()%8 '")+,@."#. %+'1,&% 1%$ '@ .A-#$%" +%@ B1)" &" %' '!($8 %' &'1A,@."#. %+'1,&% 1%$ .A-#$%"B +%@ B1)"B %1%B#%B3

(35)

!

,

t

,

U

,

U

,

U

!"#$% &'( ) *+,$+-+./%012 3+ #4% 52.6"#$%012 ,+$4!/!3% ,%$% 2 423+72 3+ 8#99%$3 #.!3!4+.-!2.%7'

"#$%&' (% %)*+&'", % ' +$".' (' ,/,+%$0 )*$ (% 1',,/2/)/+& -3-&/-45' 6'"+7"8- (' 1-&9$%+&' (% /"+%&-45' U0 -2&/"(' %,1-4'

1-&-- /"3%,+/:1-&--45' (% ,/,+%$1-&--, ;&1-&--61-&-- % ;'&+%$%"+% /"+%&1-&--:%"+%,< =1-&--$2*$0 3-)% &%,,-)+-& >8% - /"6)8,5' (% 1'+%"6/-/, %?+%&"',0 6'$' /$18&%@-,0 * ;-6/)$%"+% /$1)%$%"+A3%)<

B-&- ' 6-,' 8"/(/$%",/'"-) % 8"/;'&$%0 ' $'(%)' (% C822-&( * %?-+-$%"+% ,')#3%) %$ +%&$', (' !" #$ %& '&#(&0 (-"(' '&/:%$ D 6'".%6/(- ,')845' (% E/%2 % F8 G!!H0 6'";'&$% (/,68+/&%$', - ,%:8/&<

!" #$%&'($ )* %+*, * -&

I 1-&+/& (- ,')845' (% E/%2 % F80 1'(%$', (%,+-6-& +&J, ,/+8K -4L%, 1-&+/68)-&%,M

N< I %"%&:/- +'+-) (' ,/,+%$-0 "' %,+-(' ;8"(-$%"+-)0 6'$ 2-"(-,%$/K1&%%"6./(- %U >8-)>8%& * (%,6&/+- 1'&M

E(n= 1, U)

L t =−4

Z ∞

0

J0(ω)J1(ω)

ω(1 +eωU/2t) dω, !"#$

%& '(% J0(ω) % J1(ω) )*+ ,) -(./0%) 1% 2%))%3 1% +41%& 0 %15

4%)6%789:,&%.8%" ; <4,.1%=, n = N/L 4%64%)%.8, , 1%.)91,1%

&>19, 1% 6,48?7(3,)5 7+& N )%.1+ + .@&%4+ 1% %3>84+.) % L +

.@&%4+ 1% )?89+)" #" U → ∞

E(n, U → ∞)

L t =−

2

πsen(π n), !"!$

!" U = 0

E(n, U = 0)

L t =−

4

πsen

π n

2

(36)

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

!" #$%&'#()* )+, #* -*).%* ).

/011#2) 0(&)&-.(3&*(#%

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

ˆ

HKS=−t X

ijσ

c†cjσ+H.c.+ X

i

Vef f,i c†iσciσ, L ;!O

/,2

Vef f,i=ViH[ni] +Vixc[ni] +Vi[ni], L ;NO

&'4&'%'-*#-+, #% /,-*&$8($56'% +,% 4,*'-/$#$% +' H#&*&''0 *&,/# ' /,&3 &')#51, ' '>*'&-,0 &'%4'/*$B#2'-*';

!"!# $%&'%()*+)&, -./

",2#-+, /,2, 8#%' ,% *&?% )$2$*'% '>#*,% L ;JO3L ;MO0 C$2# '* #) LM!0MNO 4&,4(%'&#2 (2# 4#&#2'*&$:#51, 4#&# # '-'&@$# *,*#)E0 /(P#

B#)$+#+' %' '%*'-+' +' 9,&2# #4&,>$2#+# 4#&# *,+,% ,% B#),&'% +' n' U; "#) '7(#51, 4#&#2'*&$:#+# = '%/&$*# /,2,X

E !"#$%&(n, U)

Lt =−

2β(U/t)

π sen

πn β(U/t)

(37)

!

" #$%&'()%*β + ,()(%'-.$,* #*%/ β(U/t)sen

π

β(U/t)

= 2π

Z ∞

0

J0(ω)J1(ω)

ω 1 +eωU/2tdω, 0 123

4(%-56$.,* 78( .*9 6$9*9 (:)%('*9 ,( n= 1;U = 0(U → ∞*9 )%<9

=-'-)(9 (:$)*9 9>* %(68#(%$,*9; 6*' β(U = 0) = 2 (β(U → ∞) = 11

?78-; @? %(A(%(B9( $ C@()D( ?.9$)EF ( GH"I J9 -.-6-$-9 ,*9 $8)*%(91 K'$ 4(E 6*.D(6-,$ $ (.(%L-$ )*)$= E(n, U),* 9-9)('$ D*'*B

L<.(* ( -.5.-)*; #*,('*9 (:)%$-% $ (.(%L-$ ,( )%*6$ ( 6*%%(=$M>* 98NB )%$-.,* $9 6*.)%-N8-MO(9 6*.D(6-,$9 ,( (.(%L-$ 6-.+)-6$ .>* -.)(%$L(.)( ( P$%)%((/

Exc !"#$%&(n, U) =E !"#$%&(n, U)−E !"#$%&(n, U = 0)−EH(n, U).

0 1Q3 H$N(.,* 78(;

EH(n, U)

L =

U

4n

2, 0 1RS3

+ #*99T4(= (96%(4(% $ (.(%L-$ ,( )%*6$ ( 6*%%(=$M>* #*% 9T)-* ,( 8' 9-9B )('$ 8.-A*%'(/

Exc !"#$%&(n, U)

L =−

2tβ(U/t)

π sen

β(U/t)

+4t

πsen π 2n −U 4n 2. 0 1RR3 ? $#%*:-'$M>* GU? #$%$ 8' 9-9)('$ .>* D*'*L<.(* #*,( (.)>* 9(% 6*.9)%8T,$1 I*.A*%'( VW ,-968)-,*; * #%*6(99* 6*.9-9)( (' $#%*:-'$% $ (.(%L-$ Exc (' 6$,$ 9T)-* #(=$ 6*%%(9#*.,(.)( (.(%L-$ ,*

9-9)('$ -.5.-)* ( 8.-A*%'(/

Exc !"#'!(#$%&[n, U] = L X i=1

E !"#$%&xc (ni, U)

L , 0 1RX3

4W=-,$; (.)%()$.)*; $#(.$9 #$%$ 0 ≤ ni ≤ 11 Y$%$ 6*.9-,(%$%'*9 $

9-)8$M>* (' 78(1< ni≤2; + .(6(99W%-* 78( A$M$'*9 8'$ )%$.9A*%'$B

M>* #$%)T68=$BN8%$6* (ciσ → (−1)ic†iσ) .* P$'-=)*.-$.* ,( P8NN$%,

*%-L-.$= 06$9* 78( .>* 9(%W 6*.9-,(%$,* .(9)( )%$N$=D*31

Z-.$='(.)(; #$%$ 78( )$= $#%*:-'$M>* 9(V$ ('#%(L$,$ .* P$B '-=)*.-$.* [*D.BHD$' (:#*9)* .$ (78$M>* 0 1\3; ,(4('*9 (A()8$% $ '-.-'-E$M>*; 6*.,8E-.,*B.*9 $* #*)(.6-$= ]I $)8$.,* 9*N%( * -B+9-'* 9T)-*/

Vixc[ni, U] =−2t cos

πni β(U/t)

+ 2t cosπni

2

−U ni

(38)

"#$ %&' ()*+,( ( -(*&./+$0 ,& *1(/$ & /(11&0$23(4 )$5*$ %(06 *$1#(5 .$ &78$23( 9 :;< & &# 5&=8+,$ 58)5*+*8+1 ( 1&580*$,( .( >$6 #+0*(.+$.( ?(@.6A@$# 9 :B<: C -$1*+1 ,$D4 1&5(0%&.,( ( -1()0&#$ ,& E(1#$ $8*(/(.5+5*&.*&4 ()*&#(5 (5 (1)+*$+5 & &.&1=+$5 #(.(&0&*1F.+/$54 /(#-0&*$.,( ( /+/0(:

G(5*&1+(1#&.*&4 8# #H*(,( $0*&1.$*+%( I &78$23( -$1$#&*1+6 '$,$ E(+ *$#)H# ,&5&.%(0%+,(: J# %&' ,& -1(-(1 8#$ -$1$#&*1+'$23( ,$ &.&1=+$ &# E8.23( ,(5 0+#+*&5 /(.@&/+,(54 K+%$0,( L: M$#-( N1: &* $0 9O!< ,&5&.%(0%&1$# 8#$ $)(1,$=&# .8#H1+/$ -$1$ $ Exc !4 1&5(06

%&.,( $5 &78$2P&5 ,& L+&) & Q8 -$1$ ,+%&15(5 %$0(1&5 ,& U4 #(.*$.,(

&# /$,$ /$5(4 8#$ #$0@$ R.$ ,& ,&.5+,$,&5 /(# $5 1&5-&/*+%$5 &.&16 =+$5 & -(*&./+$+5 ,& *1(/$ & /(11&0$23(: C55+#4 &# /$,$ 5D*+(4 $ &.&1=+$ ,& *1(/$ & /(11&0$23( H 0(/$0#&.*& $-1(S+#$,$ -(1T

Exc !"#$!%&'(n, U)

L =

L X i=1

E !(n

i, U)

L +

4t πsen

π

2ni

−U

4n

2

i

.

9 :UO< -$1$ 0≤ni ≤1: C78+4E !(ni, U) +.,+/$ $ &.&1=+$ &S$*$ ,( 5+5*&#$

+.R.+*( & 8.+E(1#& /(# ,&.5+,$,& ni4 ()*+,$ .8#&1+/$#&.*& %+$ -1(6

/&,+#&.*( #&./+(.$,(: G$1$ 8#$ ,$,$ ,&.5+,$,& +.*&1#&,+V1+$ n.3(

(39)
(40)

!"#$%&#'%(

&)*%+'%(, & -&./0&

./ 1"**'). $&-&

0'*&)'+2)%&

+/2)%$&

"#$%&' ($)* +,%-#.'* '* /#%-)'%$)* 01 )%-'23'2$(4 ($)* -'(5 3.67$ &6%86 $ *62 $ *#$ )(3.6(6%&$9:' -'(3#&$-)'%$.4 -'(' '-'226 %' -$*' 8'* /#%-)'%$)* '2;)&$)* 8$ 86%*)8$86< = 32';.6($ *6 6%-'%&2$ %$ 862)+$9:' 6 )(3.6(6%&$9:' 8' 3'&6%-)$. 01 67326**' 36.$ 6>#$9:' ?@<ABC< D($ +6E >#6 %:' -'%F6-6('* $ 8636%8G%-)$ 673.,-)&$ 8' /#%5 -)'%$. 86 6%62H)$ -'( 26.$9:' I 86%*)8$864 ' 32'-68)(6%&' =JK 86+6 *62 6(326H$8'4 .6+$%8' $ #( 6.6+$8' -#*&' -'(3#&$-)'%$. LM 3$2$ *)*5 &6($* -'( 3'#-'* 6.N&2'%* ?N ∼10C< O ($)'2)$ 8$* )(3.6(6%&$9P6*

$#&'-'%*)*&6%&6* 8'* /#%-)'%$)* '2;)&$)* $)%8$ N /6)&$ +)$ ()%)()E$9:' '2;)&$.4 '# *6L$4 6( +6E 8$ 6>#$9:' ?@<ABC4 *#;*&)&#)5*6 $ 862)+$8$ /#%5 -)'%$. 6( 26.$9:' $'* '2;)&$)*4 6>#$9:' ?@<AQC4 %' 3'&6%-)$. 6/6&)+' 8$ 6>#$9:' R'F%5SF$(<

(41)

!

!" #$%&'$'()*+,- .' /0(1#-(*#2

-34#)*#2

"#$# % &#'% ()'&$*+% (#' &#(*)#' (* ,-..#$(/ # *0-#12% 34" #56)&#(# #% 7-8&)%8#6 "9:;< 5%(* '*$ *'&$)+# &%=%>

Nσ X k L X i=1

ψkσ∗ (i)fkσ[vxc,σ[n](i)−uxc,kσ[n](i)]GKS,kσ(i, j)ψkσ(j)

+c.c.= 0,

? @AB &%=

uxc,kσ[n](i) = δE !"#$ xc [n] δnkσ(i)

, ? @CB GKS,kσ(i, j) =

L X m(m6=k)

ψ∗

mσ(j)ψmσ(i) εkσ−εmσ

, ? @DB

*0≤fkσ≤1@ E0-)/LF % 8G=*$% (* 'H+)%' * Exc !"#$[n] F (*I8)(% (*

#&%$(% &%= # *0-#12% ?C@JCB@

4= 'H8+*'*/ +#=.F= $**'&$)+#' 5#$# % &#'% (#' &#(*)#' ()'&$*+#' (* ,-..#$(/ #' #.%$(#K*8' 0-* !"#$!%&'" # *0-#12% 34" #56)&#(# (% 7-8&)%8#6 "9:;< '2%>

A@ LM;

vxc,σKLI[n](i) = Nσ

X k

fkσ

|ψkσ(i)|2 2nσ(i)

uxc,kσ[n](i) + ¯vxc,kσKLI[n]−u¯xc,kσ[n]

+c.c.

? @ B &%=

¯

vxc,kσ[n] = L X i=1

ψ∗

kσ(i)vKLIxc,σ[n](i)ψkσ(i), ? @JB

*

¯

uxc,kσ[n] = L X i=1

ψ∗

(42)

"# $%&'()

vxc,σSlater[n](i) = Nσ

X k

fkσ

|ψkσ(i)|2 2nσ(i)

uxc,kσ[n](i) +c.c. * #+,

-# ./0

vxc,σGAM[n](i) = 1 2Nσ

X k

fkσuxc,kσ[n](i) +c.c. * #1,

/2 &34)5&6(72 89( (:;'&< & (89&=>4 ?@AB C4) 29& :(D !# .$$E

vGSSC

xc,σ [n](i) = EB

xc[n] EA

xc[n] vA

xc,σ[n](i); * #F,

"# G$$E

vLSSCxc,σ [n](i) = eB

xc[n](i) eA

xc[n](i)

vAxc,σ[n](i). * #!H,

I& &C%;J&=>4 5&2 &34)5&6(72 .$$E ( G$$E &4 K97J;47&% AL$ME +N/O

GP/QRIB (< :(D 5( &C)4S;<&)<42 4 C4'(7J;&% TE AL$ME & C&)';) 54 C4'(7J;&% TE GP/B &C)4S;<&)(<42 4 C4'(7J;&% 5( ;7'()&=>4 AL$ME '4'&%B ;7J%9;754 4 '()<4 5( U&)')((# $4<(7'( &22;< V C422W:(% 6&)&7';) 89( 4 C4'(7J;&% 5( ;7'()&=>4 2(X& 79%4 C&)& 2;2'(<&2 5( 9< (%V')47#

I& C)YS;<& 2(=>4 &C)(2(7'&)(<42 42 )(29%'&542 5( 9<& ;7:(2O ';6&=>4 89( (76%4349 4 JZ%J9%4 5( (7()6;&2 ( 5(72;5&5(2 54 (2'&54 K975&<(7'&% (< J&5(;&2 5( U933&)5 &3()'&2B J4< ( 2(< ;<C9)(D&2B (2'&3(%(J(754 & J4<C&)&=>4 542 )(29%'&542 ?@A J4< &2 &%'()7&';:&2 89( 4 2;<C%;[J&< 49 (:;'&<# \<& :(D 89( (2J4%](<42 ')&3&%]&) J4< 9<& GP/B 7>4 ;7J%9;)(<42 89&%89() C422;3;%;5&5( 5( <&67(';D&=>4B 49 2(X&B J472;5()&)(<42 N↑ = N↓ ( vext,↑[n](r) = vext,↓[n](r) (< '4542

42 J&242# A4) (S(<C%4B 2( N = 5B '()(<42 N↑ =N↓ = 2,5# /22;<B

(43)

! "#$%&'()

vHxc[n](r) =

δ(EH[n] +Exc[n]) δn(r)

=X σ

Z δ(E

H[n] +Exc[n]) δnσ(r′)

δnσ(r′) δn(r) d

3r

= 1 2

X σ

Z

vHxc,σ[n](r′) δn(r′)

δn(r)d

3r

=vHxc,σ[n](r).

*

+,,-."/" "' 0"1(&"' 1&'0/(2"' 1( 3455"/16 78/ '4" 9(:)

vHxc[n](i) =vHxc,σ[n](i). *

+,!-;4 '(<"6 42&%&:"/(=8' 8 >+,!-;4#0&8#"% ?@ ABCDEBFGH 08#>8/=( 1&'042&18 #8 @"7I24%8 J6 08#'2/4I18 " 7"/2&/ 1" =&#&=&:"KL8 (= /(%"KL8 M 1(#C '&1"1( 282"% n(i)+ B &#0%4'L8 1( ="N#(2&:"KL8 '(/&" 78''I9(% 0"'8 (=C

7/(N$''(=8' 4=" "7/8O&="KL8 1( 1(#'&1"1( %80"% 1(7(#1(#2( 1( '7&# *DPEB-6 (= Q4(6 78/ (O(=7%86 4= 0"'8 08= N↑ = 3 (N↓ = 2 '(/&"

"1=&''I9(%6 2"% Q4(vHxc,↑[n](i)=6 vHxc,↓[n](i)+

!" #$%&'(')*$ +,-(+ $+& + ./'.

'0-+(,'-12'.

!"!# $%&'()*& &() %)+,-(.*&

./&=(&/86 &#9('2&N"/(=8' 8 1('(=7(#R8 1( 0"1" "%2(/#"2&9" 9&" "#$%&'( 1"' (#(/N&"' ( 1(#'&1"1(' 18 ('2"18 >4#1"=(#2"% 7"/" 0"1(&"' "5(/2"' '(= Q4"&'Q4(/ 782(#0&"&' (O2(/#8'+ H8''" "=8'2/" S 08#'2&C 24I1" 78/Ns= 250"1(&"' "5(/2"' 1( 1&>(/(#2(' 2"="#R8'6 #T=(/8' 1(

7"/2I04%"' ( 1(#'&1"1(' =S1&"' *L = 5,6,7, ...,14U N = 4,6, ..106 08=

N/L < 0,85-+ H('2( 0"7I24%86 8 2(/=8 V0"1(&" "5(/2"W '&N#&X0" 4=

'&'2(=" 4#&1&=(#'&8#"% 1(L'I2&8' ( 08= 08#1&KL8 1( 08#28/#8 "5(/2"6

(44)

"#$%&'($)*+ &)' ',-./+$ $+%'%0+%/1' 1*) 2'+$ ,*+ ($+&.%'3*+ 3*+ 45 +/+%$)'+ $+%&3'3*+6 $) 7&$ * 3$+8/* '1&)&.'3* $,%($ *+ ($+&.9 %'3*+ :"; $ +&'+ '<(*=/)'>?$+ @ )$3/3* ' <'(%/( 3*+ $((*+ )@3/*+ <$(1$,%&'/+6 3$A,/3*+ 1*)*B

AE= 100

PNS

j=1

E

j

approx−E j OEP

PNS

j=1 E j OEP

, C DE!F

AEn= 100

PNS

j=1

PLj

i=1

n

j

approx(i)−n j OEP(i)

PNS

j=1

PLj

i=1

n

j OEP(i)

, C DE F

$) 7&$ EOEPj CnjOEPF +$ ($#$($) G+ $,$(H/'+ C3$,+/3'3$+F *2%/3'+ 8/'

<(*1$3/)$,%* :";D Lj (*%&.' * ,I)$(* %*%'. 3$ +0%/*+ i 7&$ 1*)<?$

1'3' +/+%$)'j$NS @ * ,I)$(* %*%'. 3$ +/+%$)'+ ,* 1*,J&,%* 3$ 3'3*+D

K<./1',3* ' 1*(($>L* ;MNOP '* #&,1/*,'. QK9RSKTUV6 *+ ($+&.%'3*+ +L* '<($+$,%'3*+ ,' AH&(' DED

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

!"!" #$%&'()% *+( $(,-.'/)%

P*)* <(X=/)* /%$)6 /,8$+%/H'($)*+ +/+%$)'+ 1*) /)<&($]'+ .*1'./]'3'+6 %*)',3* '+ +$H&/,%$+ ')*+%('+B L = 156 N = 4 $ Vi =

−1,0tC/)<&($]' '%('%/8'F *&Vi= 1,0tC/)<&($]' ($<&.+/8'FD K.@) 3*+

3$+8/*+ ,'+ $,$(H/'+ %*%'/+6 ','./+')*+ *+ 3$+8/*+ ,* +0%/* 3' /)<&($]'6 '7&/ 3$A,/3* 1*)*B

AEni= 100|napprox(imp)−nOEP(imp)|

|nOEP(imp)|

(45)

! " # $

% %& %!

!"#

!

"

$%& '()

*(+,-. /+0 /**1 (**1

! " # $

% ! &

!" #

!

"

#

$%& #'()

#*(+,-. #/+0 #/**1 #(**1

(46)

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G #%4&-$: 7+2#-*%$)$%.+# ) ,$%#%2<) *% &. ,+(%27-)' %S(%$2+ *+ (-,+ T#+;(JB+&'+.PU: %#7$-(+ 7+.+=

Vi =−

t p

a2+ (ii

0)2

, C 5VWE

7+.a#%2*+ &. ,)$X.%($+ *% #&)0-/)<1+ % i0+ #Q(-+ 7%2($)' *) 7)*%-)5

B)P% +P#%$0)$ R&% ) 0)$-)<1+ *+ ,)$X.%($+ a,%$.-(% #-.&')$ *-;%$%2J

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