IMP LE M EN TA Ç Ã O N O C O N TE X T O D A

  N o me d o A u to r

  M A RS A L E D U A R D O B EN TO |F U N C

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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 conhecido 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 DISSERTAđấO DE MESTRADO

  INVESTIGAđấO DE ESTRATÉGIAS DE

  2014 UNIVERSIDADE DO ESTADO DE SANTA CATARINA – UDESC CENTRO DE CIÊNCIAS TECNOLÓGICAS – CCT PROGRAMA DE PốS-GRADUAđấO EM FễSICA

IMPLEMENTAđấO NO CONTEXTO DA FORMULAđấO KOHN-SHAM DA TEORIA DO FUNCIONAL DA DENSIDADE

  JOINVILLE, 2014

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  ˆ HΨ(r 1 , r 2 , .., r N ; R 1 , R 2 , .., R M ) = EΨ(r 1 , r 2 , .., r N ; R 1 , R 2 , .., R M ),

  HJ= I H , r , .., r ; R , R , .., R )

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  1 Z Z i j ˆ

  H = − ∇ + R − ∇ r i i i=1 i=1 i j>i 2m 2m 4πǫ |R − R | Z i e i j X N M N N X X X 2 56-67

  1 Z j e 1 e

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    • X N N M N N ~ 2 X X X X 2 2

      1 Z e j 1 e ˆ r

      H = − ∇ + − i i=1 i j i j>i 2m 4πǫ |r − R | 4πǫ |r − r | e i j i j 56-U7

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      ˆ H = ˆ T + ˆ V + ˆ ee ext .@AB/ V ,

      T

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      V V %8* 5)'7,*',#/0 ˆ ext,1 # ˆ ext,2 0 ,*- C%#? (1) (1) (1) (1) E = hΨ | ˆ H |Ψ i < hΨ | ˆ H |Ψ i, 1 1 2 2 .@AL/

      E = hΨ 1 | ˆ T + ˆ V ee + ˆ V ext,1 |Ψ 1 i < hΨ 2 | ˆ T + ˆ V ee + ˆ V ext,1 |Ψ 2 i, .@AM/

      #8 C%# Ψ 1 # Ψ 2 7<) *7 H%'DN#7 (# )'(* () #7,*() H%'(*8#',*- *77)5&6

      V V 6= Ψ

    • (*7 *)7 ;),#'5&*&7 ˆ ext,1 # ˆ ext,2 0 +#7;#5,&=*8#',#0 5)8 Ψ (1)
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        E < E d r n + 2 (r) V ext,1 (r) − ˆ V ext,2 (r) .

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        E + < E d r n (r) 1 ext,2 ext,1 .@A @/ V (r) − ˆ V (r) .

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        E + E < E + E =⇒ 2 (r) :#&49%';,<#= . /0>1 1 (r) 1 ?#94%&4#2 n &<# @#') ()9 ;8+%A % n :%(# Ψ ()B% ';C)D 2

        9)&4) ') Ψ / E+ ()B%5 Ψ 6= Ψ =⇒ (r) 6= n (r), 1 2 )&4<# =⇒ n 1 2 . /0F1 n(r)

        ')4)9$;&% Ψ(r) → Ψ(r) G C+&:;#&%A ') n(r) / H;&%A$)&4)2 :I)8%$#( %5 Ψ(r , ..., r ) = Ψ[n(r)]. 1 N . /0J1

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        ˆ HΨ = ( ˆ T + ˆ V + ˆ ee ext . /0T1 V )Ψ = E Ψ ,

        ( ˆ T + ˆ V )Ψ [n] ee V = E − , ext . /0U1

        Ψ [n] #+ ()B%2 Ψ ')4)9$;&% V ext % $)&#( ') +$% :#&(4%&4)/ K&4<#2 () Ψ G C+&:;#&%A ') n(r)2 V ext 4%$3G$ ()9V/ W)(+$;&'#2 4)$#( n(r) −→ Ψ (r 1 , ..., r N ) −→ V ext −→ ˆ H −→ {Ψ, E} d ,

        . /0X1 :#$ d 9#4+A%&'# *+%;(*+)9 )(4%'#( )A)49Y&;:#(/ H;&%A$)&4)2 @#')$#( )(:9)Q)9 % )&)98;% :#$# +$ C+&:;#&%A '% ')&(;'%')5

        E[n] = hΨ[n]| ˆ T + ˆ V + ˆ ee ext . / !1 V |Ψ[n]i.

        " #$%&$'( )$ *%+$,-$&. $ /%+, )$'%,01&( 23$ 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%=

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        03-'$15)% ( 3' 4%1$,:5(7 $?$15;% v s,σ = Q $23(BC% )$ K:+&R)5,.$& 0$&<S 2

        ∇ N I N I N I − + v [n](r) ψ (r) = ε ψ (r), s,σ L = !M kσ kσ kσ

        2 $ N I N I X X N σ 2 N I X n (r) = f kσ |ψ (r)| = n (r), σ σ k kσ σ L = M

        ≤ 1 $' 23$ 0 ≤ f kσ 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 N I X Z 3 N I E [n] = T s [n] + d r n (r)v s,σ [n](r). σ σ L = VM

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

        δ(E [n] − µ R d r n (r )) σ σ = 0, N I L = WM

        δn (r) σ $' 23$ X Z 3 N I ′ ′ X N = d r n (r ) = N . σ σ σ σ L = GM

        "3 0$X(S δT [n] s N I + v [n](r) = µ. s,σ L = FM

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        E[n] = T [n] + E [n] + E [n] + V [n], s H xc ext @ - AB [n]

        1$/&%;&#+" $% #)"# /&,+(*5)*34$# 1$ $,$('*" /*,C+*/" T s 1$ )% #*#D

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        [n] [n] " )% ;&+$,/*"= $H+$(,& V ext - I=C% 1*##&0 E xc ($;($#$,+" " $,$('*" 1$ $&")' % )"&&%*'+!" @JKB0 2)$ *,/=)* +&1"# "# /&(($34$# ,$/$##6(*"# <#

        [n] ";(&H*%"34$# $L$+)"1"# ,&# +$(%&# 1$ T [n] $ V ee 0 &) #$M"0

        E [n] = (T [n] − T [n]) + (V [n] − E [n]) , xc s ee H @ - NB [n]

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

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      • $%&# 2)$?

        δT [n] δE [n] δE [n] s H xc

      • v [n](r) = µ. ext

        @ - QB δn (r) n (r) n (r) σ σ σ

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        $% 2)$ δE [n] H v [n](r) = =⇒ H,σ !&+$,/*"= 1$ G"(+($$; @ -T B

        δn σ (r) δE [n] xc v xc,σ [n](r) = =⇒

        !&+$,/*"= 1$ 9(&/" $ K&(($="3F&. @ -TTB δn (r) σ

        I##*%0 " ;"(+*( 1" $2)"3F& ,F& *,+$("'$,+$ @ - WB0 #&%&# /";"D :$# 1$ &5+$( " %$#%" 1$,#*1"1$ 1& $#+"1& L),1"%$,+"= 2)$ &5+$(X"%&# /"#& $#+*EC##$%&# ($#&=E$,1& " $2)"3F& 1$ O/P(Y1*,'$( *,+$("'$,+$ 1$

        [n](r) "#$%&' (&)*&'+ ,-)- $''&. /010"&' #%$2$3-) & *&%04($-2 050%$1& v s,σ /-/& *02- 06#-78& 9 +!:;+ <'*0($=(-"04%0. %0"&'> 2

        ∇ − + v [n](r) ψ kσ (r) = ε kσ ψ kσ (r),

        !,σ 9 +!?;

        2 v [n](r) = v [n](r) + v [n](r) + v [n](r) =⇒

        !,σ H,σ xc,σ ext,σ

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      • #. -A /4 O"P> "% 41, $/&1, 8%&,+A (%1/) C4% Z hom
      • 3

          !" hom E [n] = e [n] d r, xc xc n→n(r) L >QRM

          [n]

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          ##" GGA 3 E = e [n, ∇n] d r, xc xc n→n(r) L >QWM

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          $##" M GGA 2 3 E = e [n, ∇n, ∇ n, τ ]d r, xc xc L >QGM 2 n

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          1 2 2 τ (r) = |∇ ψ k (r)| .

          L >!aM

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        • $(#%)$@

          E [n] = E [n] + E [n], xc x c ; 3AB= &'+ E x 0#("' ' ?#)+' "# ?)'&$ # E c $ #(#)>%$ "# &'))#2$.1'3 C00%+7 ' :*$)?' "#>)$* "$ &2$00%5&$.1' &'(&#%?*$2 8)'8'0?$ 8') 4#)"#D %(&2*% $ "#(0%"$"# "# #(#)>%$ "# ?)'&$ e x 7 ($ %"#%$ "# E%8#)6FFC@ Z HGGA 2 3

          !!" E [n] = e [n, ∇n, ∇ n, τ, e ]d r. xc xc x ; 3A =

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          1

          #"

          V [n] = 1 − E [n], ee H ; 3AZ= N

          0#("' #00$ $ &'))#.1' "# $*?'6%(?#)$.1' 0#>*("' Y#)+% # C+$2"% ;YC6 ee = 0 TUW= ; [=7 >$)$(?%("' :*# V 8$)$ N = 13 W$I# 'I0#)9$) :*# $ YCTUW -'% 8)'8'0?$ &'+ ' %(?*%?' "# &'))%>%) ' +'"#2' "# \E'+$06 Y#)+% ; <7 ]7Z^=7 :*#7 8') 0*$ 9#O7 &'(0%"#)$ $8#($0 ' ?#)+' "# _$)?)## &'+' $ #(#)>%$ "# %(?#)$.1' ;'* 0#J$7 (1' 2#9$ #+ &'(?$ ?)'&$ # &')6 )#2$.1'=3 4)'8'0?' #+ B] [ # JK ?'+$("' $ "#(0%"$"# &'+' 9$)%K9#2 &E$9#7 ' +'"#2' "# \E'+$06Y#)+% 8'"# 0#) 9%0?' &'+' *+$ $I')"$>#+ $(&#0?)$2 "$ RY\3

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          1

          !

          V [n + n ] = 1 − E [n + n ] ee A B H A B N + N A B

          1 = 1 − (E [n ] + E [n ]) H A H B A ;BDC

          N + N A B

          1

          1 6= 1 − E [n ] + 1 − E [n ], H A H B

          N A N B ,(%,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 min min

          E [n] = hψ | ˆ x ee H A ;B!C n n V |ψ i − E [n], & min min min min

          E c [n] = hΨ | ˆ T + ˆ min min n n n n A ;BOC V ee |Ψ i − hψ | ˆ T + ˆ V ee |ψ i, 8#. ψ & Ψ ,(%,8)(%# #$E,-),9 Q#:(4R:). & I@(1S&9 %& #(%) %& n n .@,-#9 8#$'#9/ ?@& %2# #$,*&. T %&(9,%)%& n(r); U,)(-& %& @. &67-$#(/

          ˆ V ≡ 0 ee / 8#(%@+,(%#4(#9 ) (1) (1)

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