Coupled-Cluster-R12-Methoden mit Auxiliarbasisfunktionen German

Heike Fliegl Coupled-Cluster-R12-Methoden mit Auxiliarbasisfunktionen Coupled-Cluster-R12-Methoden mit Auxiliarbasisf...

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Heike Fliegl Coupled-Cluster-R12-Methoden mit Auxiliarbasisfunktionen

Coupled-Cluster-R12-Methoden mit Auxiliarbasisfunktionen von Heike Fliegl

Dissertation, Universität Karlsruhe (TH) Fakultät für Chemie und Biowissenschaften, 2006

Impressum Universitätsverlag Karlsruhe c/o Universitätsbibliothek Straße am Forum 2 D-76131 Karlsruhe www.uvka.de

Dieses Werk ist unter folgender Creative Commons-Lizenz lizenziert: http://creativecommons.org/licenses/by-nc-nd/2.0/de/

Universitätsverlag Karlsruhe 2006 Print on Demand ISBN-13: 978-3-86644-061-6 ISBN-10: 3-86644-061-8

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# 2E,+3 %."*+,"

0.30 0.29 0.28 0.27

Ψ(Θ12)

0.26 0.25 0.24 0.23 0.22 0.21 0.20 -π

-0.5π

0 Θ12

0.5π

π

EAS %." %."*+,"

21!3 - n U G : 4 7

4# # 9# 2GE,+3

,E

+ # , ( &

4# *+," 1 ?0 !0"- " 4 " ' # # ( MEFN $ & ?0 . # # S 4 4 "( 4# 4

& ' H ; I"2*.3"1& M+FEN ' *+," MG,' BGN 4# !

r12' . # ; M+FGN *+,"4 -6":' H- I' f12 = r12

N

2 ≈ bi 1 − exp(−γi r12 )

2E,+3

i=1

# # 4 ' . r12 & ' 9 1 !( !

r12 = 0 0 $ # M+FGN $ 0

$ ' % *; # M+FA"++FN' 9# -6": # # 1"$ #

Ψ = Ki=1 ci φi # φi # ' & : φi =

exp(−βi rp2i qi )

N

exp(−αik |rk − Aik |2 )

2E,,3

k=1

K

?& $# N !; (

$ $7 βi αik 4' & # ' Aik 9# 8 -6" :' # . :

' & φi !& #

( & !; M+FBN / $ ' ;# : φi #' # - M+FB'+FDN 1# " 4 " # 5 # 4# 7 ( -6": *+,"' ,G

( )

* )

#& 6 . # 6 !; 0 $ ,FFA # # ( *+,"

4# f12 &6 f12 = exp(−γr12 ) ≈

N

2 ck exp(−αk r12 )

2E,E3

2 ck r12 exp(−αk r12 )

2E,G3

k=1

f12 = r12 exp(−γr12 ) ≈

N k=1

4 r12 M+++N # " ' - 2E,E3 M++,N

#6 H- I

*+," % "? 4 *+," ' # : - 2E,G3 - 2E,E3' # ! ' M+++N 1 $7 γ # ) # ' 5 γ 0 = & #

! γ = 1

4

*+," 1

#&' 7

0 ' ( *+," 4 " 4# *."1& * ( " # # ,' %%! %%! ( # . 8 ? * ()*. MEF'E,'G,'GD'++E'++GN ,(*+,"

#( ' *."1& 4# 7 & ( 0 ' 5 MGBN

# & * %%,"*+,"

,A

,J

& % . # & % "% "*+,"

( 6 # & %%(*+," & ( % "% "- %%! (*+," %%,(*+,"-( . & # %%,(*+,"- % ( :(

) "7 * # &

7 : *+,"& %%,(*+,"-( &' (

%%,(*+," # $ $ 7 "1& #

%%,"& # # ' ? M,'+,'+EN $ R %%,(*+," @ * M++AN

' ( % "% "*+,"

#' # , (

% "% "

' 7

|CC = exp(Tˆ)|HF 2G+3 @' # % "5 2G,3 Tˆ = Tˆ1 + Tˆ2 + ... + Tˆ2 , 5 Tˆ2 & #'

' ( ()*/ !!' # 2E3 2EE,3 ' ()*,-& !! ;( & M+F,N ukl (1, 2) = w ˆ 12 |ϕk (1)ϕl (2) , 2GE3 & {ϕα}

# 5 w ˆ 12 @ & + , - 2E+D3 2E+K3 $ @ , @ 5 wˆ 12' 4 MGGN 4 M++JN ' &6 ˆ 1 )(1 − O ˆ 2 )(1 − Vˆ1 Vˆ2 )f12 w ˆ 12 = (1 − O 2GG3

,B

+

& 4# 9 Vˆ1 =

2GA3

|ϕa (1)ϕa (1)| ,

a

":"5 {ϕa} 9 8 (1 − Vˆ1Vˆ2) - 2GG3

' ()* &

5 '

#' ( 4# @ + * &6 - 2GG3 4 @ - 2E+K3 # 4 8 # *+,"&

' (

8 # ?0 % "% "- " #( 0 # ' # /' *+," #' /

":"5 & # - 2H "

I3 ":"*"8 ' *+," &6 Tˆ2 =

1 2

ijkl

cij kl

kl wαβ Eαi Eβj =

tμ2 τˆμ2

2GJ3

μ2

αβ

# 2E3 ()*

+ cijkl .

2GB3 5 Eαi $ MGN kl wαβ = ϕα (1)ϕβ (2)|w ˆ 12 |ϕk (1)ϕl (2) ,

tμ # &6

2

α Eαi |HF = i

2GD3 *"8 2E3 . i' j ' ( 5 ' #& 5 ' $ & / ' α' β ' #

' )*( . / - 2G,3 # *+," % "% " ! "" " ME' +,' +EN %%,"

M,N $# % ( Tˆ = Tˆ1 + Tˆ2 + Tˆ2 , 2GK3 8# - 2H "

I3 ":"*"8 ' 0

,D

+ ,!- -.

$" 8# &6 Tˆ1 =

tia Eai =

Tˆ2 =

tμ1 τˆμ1

μ1

ai 1 2

tij ab Eai Ebj

=

2G+F3 2G++3

tμ2 τˆμ2

μ2

abij

# - 2GJ3

5

*+," Tˆ2 & % " tμ ( τˆμ ,

# $ 1 0 $" 8#"% " tia tijab # 5 Eai !( $ MGN # - 2GD3 *"8 - # 2E3 ; . a' b' 2

3 5

i

i

' )*( *( 4# ?0 & !0" - 2,G3 % "% "*+,"

(

% "% "

9' # ( , # % "% "- $ % "5 Tˆ - 2,A3 ˆ 1 Tˆ2 + Tˆ2 + Tˆ2 )|HF , ECC−R12 = EHF + HF|H( 2 1

2G+,3

% "% "- 2H "

I3 %%! ( *+,

MGFN &6 2G+E3 2G+G3 ˆ exp(Tˆ)|HF = 0 Ωμ = μ2 |[fˆ, Tˆ2 + Tˆ2 ] + exp(−Tˆ )Φ 2G+A3

#

' M++EN

(

" 2fαi = fiα = 03

- 2G+E3 7 [fˆ, Tˆ2 ]

7 # 7 μ2|[fˆ, Tˆ2 ]|HF μ2 |[fˆ, Tˆ2 ]|HF +

' #& , ' # - ( 2G+G3 2G+A3 8# & "5 ˆ = fˆ + Φ ˆ + hK 2G+J3 H # :"5 fˆ : $ Φˆ ( " " # hK # ˜ˆ ˜ˆ ˆ Ωμ1 = μ1 |[fˆ, Tˆ1 ] + Φ + [Φ, T2 + Tˆ2 ]|HF = 0 , ˆ exp(Tˆ)|HF = 0 , Ωμ2 = μ2 |[fˆ, Tˆ2 + Tˆ2 ] + exp(−Tˆ)Φ

2

,K

+

":"$ ! & 9 ' # 7 # 1 Φ˜ˆ # #' Tˆ1"V 2G+B3 : ; 1 % " - 0 : Φ˜ˆ # Φˆ 8# 9

" # "8& # 1 ' )Q 5 MGN ˜ˆ ˆ exp(Tˆ1 ) Φ = exp(−Tˆ1 )Φ

a 1

= HF|E † , ai i 2 ab 1

= HF|(2E † E † + E † E † ) , ai bj aj bi ij 6

2G+D3 2G+K3

@ : 9 *+," μ2 | # @ kl † αβ αβ

= (w )kl ij

2G,F3 ij

αβ

- #' # 9 8 αβ ij 2G+K3 @ # # . a b α β

@ 9 ( ' # -

S a b

τˆ |HF = δab δij , i j ab cd

τˆ |HF = Pˆijab δac δbd δik δjl , ij kl kl k l †

τˆ |HF = Pˆijkl ϕk (1)ϕl (2)|w ˆ 12 w ˆ 12 |ϕk (1)ϕl (2)δii δjj ij i j

2G,+3 2G,,3

= Pˆijkl δii δjj Xkl,k l ,

2G,E3

= Pˆijab δii δjj (w† )ab kl

2G,G3 2G,A3

kl ab †

τˆi j |HF = Pˆijab δii δjj ϕk (1)ϕl (2)|w ˆ 12 |ϕa (1)ϕb (2) ij

ab i j kl

τˆ |HF = Pˆijab δii δjj wab . ij

kl

5 Pˆijkl !; $ # &6 2G,J3

rs rs sr Pˆpq Apq = Ars pq + Aqp

4706 Arspq 7 klij τˆiabj |HF abij τˆiklj |HF + - 2GG3

@ : ,

EF

++ -

' #&

, &6 - 2E+K3

$& Tˆ 7 8# &' %%! (*+,"% "% "

@ ' 4# (

" 1& # %%,"

M,N :

%%! ( *+,"% "% "- 2G+G3 2G+A3 tμ tμ 8# & # %%,(*+,"

$& ( ' V : Φˆ Tˆ1 & # ' exp(−Tˆ)Φˆ exp(Tˆ) - 2G+G3 2G+A3 Φ˜ˆ ' - 2G+B3 @ #' # %%! (*+," - tμ $ & ( & 0 %%,(*+,"% "% "- M++AN 2

2

1

2G,B3 2G,D3 2G,K3 # %%," ,"

7 # %%,( *+," ,(*+,"

$ ( tμ

' & %%,(*+,"

,(*+,"

. ?0( % "% "%%,(*+,"- :

: ,(*+,"

&

- 2G,B3

& #' #& - 2G,D3 2G,K3 Tˆ1" : Φˆ ' = ' tμ = 0 ' ,(*+,"

' 7 ( μ2 |[fˆ, Tˆ2 ]|HF μ2 |[fˆ, Tˆ2 ]|HF +

' #& , # ˜ˆ ˜ˆ ˆ ΩCCSD−R12 = μ1 |[fˆ, Tˆ1 ] + Φ + [Φ, T2 + Tˆ2 ]|HF = 0 , μ1 ˆ ˆ ˆ ˆ˜ ΩCC2 μ2 = μ2 |[f , T2 + T2 ] + Φ|HF = 0 , ˜ˆ ˆ ˆ ˆ ΩCC2 μ2 = μ2 |[f , T2 + T2 ] + Φ|HF = 0

1

1

'' *( 8 % "% " ω ' # , '

$# = ωSR AR 2GEF3 0 # ) "7 A &6 - 2,E+3 (

% "% "- Ωμ % ( tν @ S

Sμ ν = μi |ˆ τν |HF , 2GE+3 i

j

i j

j

E+

+

#

/ 0 S %%,(*+," %%! (*+,"

# .7 - 2G,+3 2G,A3 # %%,(*+," %%! ( *+,"

7 ⎞ ⎛ Sμi νj

1 0 0 ⎝ 1 Sμ2 ν2 ⎠ , = 0 0 Sμ2 ν2 Sμ2 ν2

2GE,3

#' # / 0 Sμ ν Sμ ν + , - 2GG3

, - 2E+K3' &

# /

$# - 2GEF3 0' SR # *+," ij ij Sikjl,i mj n Rmn = Xkl,mn Rmn 2GEE3 2 2

2 2

i j mn

mn

4# - 2G,E3 7

2GEG3 # , & 6 0 ij kl ij Siajb,i kj l Rkl = wab Rkl 2GEA3 † Xkl,mn = ϕk (1)ϕl (2)|w ˆ 12 w ˆ 12 |ϕm (1)ϕn (2)

i kj l

kl

i j Sijkl,i aj b Rab

i aj b

=

2GEJ3

ij (w† )ab kl Rab .

ab

SR # 4 # *+," Rkli j

8# ij Rab ij *+," cijkl % " % " tab

: & % "% "

) "7

ˆ τˆν ] exp(Tˆ)|HF Aμ ν = μi | exp (−Tˆ)[H, 2GEB3

# MGN . # %%,(*+,"

4#( Tˆ1"& "5 ˜ˆ ˆ exp(Tˆ1 ) H = exp (−Tˆ1 )H 2GED3 ˜ˆ 8 # H¯ = [H, τˆν ]' ˜ˆ H tν '

Ωμ tν &6 i j

j

1

1

Aμi νj

E,

1

1

⎞ ˜ˆ ˜ˆ ¯ ¯ ˆ ˆ μ |H + [H, T2 + T2 ]|HF μ1 |[Φ, τˆν2 ]|HF μ1 |[Φ, τˆν2 ]|HF ⎟ ⎜ 1 ˜ˆ ⎜ = ⎝ εν2 δμ2 ν2 μ2 |[fˆ, τˆν2 ]|HF ⎟ μ2 |[Φ, τˆν1 ]|HF ⎠ ˜ ˆ ˆ ˆ μ2 |[f , τˆν2 ]|HF μ2 |[f , τˆν2 ]|HF μ2 |[Φ, τˆν1 ]|HF ⎛

2GEK3

+/ 0 " ' 1 -.

εν # $< 0 5 ' # - 2GEK3 ":" 5 # 7 μ2 |[fˆ, τˆν ]|HF μ2 |[fˆ, τˆν ]|HF +

' #& , # 2

2

2

'+ , ( % - *( : &

%%," %%! " % "% "- :

#

0 ) " ? M,' +,' +EN @ &( %%,"-

2+3 8

%%,"& # *+," . : # &

*+,"& ( $ R *+,"& %%,(*+,"4 * & + , @

2,3 *+,"& "% "% "- ' #

"- Ωμ *+," "- Ωμ 8# # ' &6 2

2

(ij) ij ab

[fˆ, Tˆ2 ]|HF = Cab,kl ckl ij

2GGF3

kl

(ij) ij kl

[fˆ, Tˆ2 ]||HF = Ckl,ab tab , ij

2GG+3

ab

4# 7 2GG,3

(ij) Cab,kl = ϕa (1)ϕb (2)|(fˆ1 + fˆ2 − εi − εj )w ˆ 12 |ϕk (1)ϕl (2) ,

":"5 5 i' j # εi εj ' #& fˆ1 fˆ2 $ ":"5 ; : + C

' 7 # % "% "- 8# Ωμ Ωμ ' ! "- 2G,B3 2

2

*+,"& "% "% "- # *+," "- Ωμ 7 ( [fˆ, Tˆ2 ] (ij) kl

[fˆ, Tˆ2 ]|HF = Bkl,mn cij 2GGE3 mn , ij 2

mn

EE

+

4# 7 † Bkl,mn = ϕk (1)ϕl (2)|w ˆ 12 (fˆ1 + fˆ2 − εi − εj )w ˆ 12 |ϕm (1)ϕn (2) (ij)

2GGG3

# B C ,(*+," MGBN 9 # : & @ # !& $ 1& * 9# @ * MGBN $ B C * , &6 - 2GG3 * MGG' ++JN # . # 2GA,3 *+," "%%,"- Ωμ 7 2

V˜klij =

kl ˜ 1 †

Φ|HF ˆ = ϕk (1)ϕl (2)|w ˆ 12 exp(−Tˆ1 ) exp(Tˆ1 )|ϕi (1)ϕj (2) ij r12

2GGA3

# . / V"7' ,(*+," MGBN ' # . Tˆ1"& ( : Φ˜ˆ # 5 Tˆ1"V V - 2GGA3 V". ,(*+," *+,"& ! "% "% "- Ωμ &6 1

a ˜ mi ˆ Tˆ2 ]|HF = ˆ 1 w

[Φ, (2cim ˆ 12 |ϕk (1)ϕl (2) kl − ckl )ϕa (1)ϕm (2)|(1 − T1 ) i r 12 klm 1 kl (2clk (1 + Tˆ1 )w ˆ 23 |ϕi (1)ϕl (2)ϕk (3) − mn − cmn )ϕm (1)ϕn (2)ϕa (3)| r 12 mnkl ˜ˆ ki (2cik ˆ 12 |ϕm (1)ϕn (2) 2GGJ3 + mn − cmn )ϕa (1)ϕk (2)|f2 w kmn

˜ˆ

5 ∗ f :"5' @ ( 7 ρ(1, 2) = 2 i ϕi (1)ϕ˜i(2) # & ϕ˜i # &6 ϕ˜i = ϕi +

ϕa tia ,

2GGB3

ϕi tia

2GGD3

a

ϕ˜a = ϕa −

i

! #' Tˆ1"V .( - 2GGA3 2GGJ3 - ( 2GGJ3

+

*+,"& % " % "%%,(*+,"- EG

+/ 0 " ' 1 -.

/ ( H """;I"2*.3"1& M+FE'++B'++DN 4( # 7 ' ## 5 ( # $ 7 ' !"1& ' 4 ! * MGBN # 2GA+3 +- ) . ,

8 4 #'

& H7 ; 7I' !"1& '

$ !" 1& & - 2GGA3 2GGJ3 &6 4 ! MGBN ' # & V". M++AN' 7 : ' $ !"1& C B - 2GG,3 2GGG3 ' # 2GA,3 & & $ !"1& # & + 0 2GGK3

w ˆ 12 = (1 − Pˆ1 − Pˆ2 + Pˆ1 Pˆ2 )r12

# Pˆ1 → Pˆ1 Pˆ2 Pˆ2 → Pˆ2 Pˆ1

2GAF3

# # 9 Pˆ1 Pˆ2 &6

Pˆ2 =

|φp (2)φp (2)|

2GA+3

p

@ $

&6 H """;I" 2*.3"1& 7 {φp } & # wˆ 12† r1

12

† w ˆ 12

1 ABS 1 1 1 ≈ 1 − r12 Pˆ1 Pˆ2 − r12 Pˆ2 Pˆ1 + r12 Pˆ1 Pˆ2 r12 r12 r12 r12

2GA,3

. , / ' Pˆ1 Pˆ2 Oˆ1 Oˆ2 # 7 # , &6 - 2E+K3 2GG3 ' : , @ : &6 - EA

+

2GG3 ' , - 2E+K3 wˆ 12† r1 & 4# !"1& 12

† w ˆ 12

1 ABS ˆ 1 Pˆ2 1 − r12 O ˆ 2 Pˆ1 1 + r12 O ˆ1 O ˆ 2 1 − r12 Vˆ1 Vˆ2 1 ≈ 1 − r12 O r12 r12 r12 r12 r12

2GAE3

. - ,(*+," % "% "*+," Tˆ" Tˆ1"& 5' # # V˜ " . - 2GGA3 ? 7

1 1 † † w ˆ 12 exp(−Tˆ1 ) exp(Tˆ1 ) = w ˆ 12 exp − Tˆ1 (1) − Tˆ1 (2) exp Tˆ1 (1) + Tˆ1 (2) r12 r12

2GAG3

@ : : # #' Tˆ1 # 9 $ ( ˆ ˆ ! &6 T1 = i T1(i)

!; $ i # # $ Tˆ1(i) Tˆ1 $ i # ' Tˆ1 (i) $ i

5 7 &6 - 2GAG3 5 $7 exp Tˆ1 (1) + Tˆ1 (2) |φi (1)φj (2) = |φ˜i (1)φ˜j (2) 2GAA3 ' 0 "8 Tˆ1 "& - 2GGB3 2GGD3 # # : # & # # ! - 2GAG3 $ #

5 # ' (Tˆ1(i))2

0 Tˆ1 (i)

1 1 † † w ˆ 12 exp(−Tˆ1 (1) − Tˆ1 (2)) =w ˆ 12 (1 − Tˆ1 (1))(1 − Tˆ1 (2)) . r12 r12

2GAJ3

(1 − Pˆi)Tˆ1 (i) = 0' (1 − Pˆi)

& ( 9' $ i #

":"5 @ + .& 1 † † 1 w ˆ 12 exp(−Tˆ1 (1) − Tˆ1 (2)) =w ˆ 12 , r12 r12

2GAB3

!"1& # - 2GA,3 #( , wˆ 12 9 9 ' 5 wˆ 12 - 2GG3 , † ˆ 1 )(1 − O ˆ 2 ) − Vˆ1 Vˆ2 w ˆ 12 = r12 (1 − O 2GAD3 EJ

+/ 0 " ' 1 -.

# ' & - 2GAJ3 † w ˆ 12 (1− Tˆ1 (1))(1− Tˆ1 (2))

1 ˆ 1 )(1− O ˆ 2 )− Vˆ1 Vˆ2 (1− Tˆ1 (1))(1− Tˆ1 (2)) 1 = r12 (1− O r12 r12

2GAK3

' # ' 1 ˜ ˜ 1 r12 Vˆ1 Vˆ2 1 − Tˆ1 (1) 1 − Tˆ1 (2) = r12 Vˆ1 Vˆ2 , r12 r12

Vˆ1 Vˆ2

2GJF3

4# @ 9 ˜ Vˆ1 = |ϕa (1)ϕ˜a (1)| ,

2GJ+3

a

# 9 V˜ˆ1 # 8 ϕ˜a - 2GGD3 @ Tˆ1(i)Oˆi = Tˆ1 (i) Oˆi Tˆ1(i) = 0' ( & - 2GAK3 ˜ˆ ˜ˆ 1 ˆ 1 )(1 − O ˆ 2 ) 1 − Tˆ1 (1) 1 − Tˆ1 (2) 1 = r12 (1 − O r12 (1 − O , 1 )(1 − O2 ) r12 r12

2GJ,3

4# @ 9 ˜ˆ O |ϕ˜j (1)ϕj (1)| , 1 =

2GJE3

j

# 9 O˜ˆ1 & ( ϕ˜j - 2GGB3 @ 1 !"1& - 2GAE3 &6 † w ˆ 12 (1

1 ABS ˜ˆ ˆ ˜ˆ ˆ ˜ˆ ˜ˆ ˜ˆ ˜ˆ 1 ˆ ˆ − T1 (1))(1 − T1 (2)) ≈ r12 [1 − O1 P2 − O2 P1 + O1 O2 − V1 V2 r12 r12

2GJG3

# 4# !"1& - 2GGJ3 # Tˆ1 "(

"8 4# ( ϕ˜a &6 - 2GGD3 # & # # - 2GGJ3 !"1& #

Tˆ1"' 5 ( ϕi(1) "8' 4# ϕ˜i &6 - 2GGB3 7 # EB

+

- 2GGJ3 4# wˆ 12 , - 2GG3 ϕm (1)ϕn (2)ϕa (3)|

1 ˆ 2 )(1 − O ˆ 3 )(1 − Vˆ2 Vˆ3 )]r23 |ϕ˜i (1)ϕl (2)ϕk (3) [(1 − O r12

2GJA3

# 5 ϕa(3) "8

' 9( (1 − Oˆ3) & # 6 :' Vˆ3 #' ϕa(3) ' (

# Vˆ3 = 1 #' 9 &6 ˆ 2 )(1 − Vˆ2 Vˆ3 ) = 1 − O ˆ 2 − Vˆ2 = 1 − Pˆ2 (1 − O

2GJJ3

# !"1& # !

2GJB3

ABS 1 − Pˆ2 ≈ Pˆ2 − Pˆ2 ,

Vˆ2Vˆ3 , &6 - 2E+K3 !"1& :

2GJD3

ˆ 2 ABS ˆ2 . 1−O ≈ Pˆ2 − O

) # , &6 - 2E+K3 2GG3 # !"5 M++KN

' 7 ( {φp } ":"5 {φj } # 5 {φp} & # *+," ! "%%,(*+,"-

−ϕm (1)ϕn (2)ϕa (3)|

˜ ˜ip ABS 1 ip w ˆ 23 |ϕ˜i (1)ϕl (2)ϕk (3) ≈ − rplk a gmn − gmn Spp r12 p p

2GJK3

4# R 7 Spp = ϕp |ϕp 0 : ,

2,3 #& $( !"1& 2Pˆ −Pˆ 3 4# !"5

# . 4# H ; I"2%!3"1& M++JN ' 7 ( & 7 2%!3 #' 5

!"1& *+," - 2GGJ3 R &6 - 2GJB3 1 !"1& # :

" 2-%3 2fβk = fkβ = 03 # # !"5

' 7 ' 9 # ,' 5 - & *+," ED

+/ 0 " ' 1 -.

! "% "% "*+,"- ABS ˜ ϕa (1)ϕk (2)|fˆ2 w ˆ 12 |ϕm (1)ϕn (2) ≈ GBC pc p c p c l pc l mn rap (2gkl − glk )tc − (2gkl − glk )tc Spp p

cl

p

2GBF3

cl

0 : ,

2,3

+-

! /& %

$ !"1& : *+,"& ! " *+," "%%,"- ( & . # B C ' : *+,"& (

" *+," "% "% "- # 7 :

& C B * ( & @

2E3 / 1 & 0 ' # & :"5 2GB+3 fˆ12 = fˆ1 + fˆ2 , ! # $ ":"5 2GB,3 fˆ1 = tˆ1 + ˆj1 − kˆ1 , ' 9# $ $ tˆ1 (ij) $ kˆ1 # % "5 ˆj1 @ Bkl,mn &6 - 2GGG3 7 1 † ˆ 1 † ˆ † ˆ 12 [f12 , w ˆ , f12 ]w w ˆ 12 (fˆ12 − εi − εj )w ˆ 12 = w ˆ 12 ] + [w ˆ 12 2 2 12 1 † 1 † ˆ 12 w + w ˆ 12 (fˆ12 − εi − εj ) + (fˆ12 − εi − εj )w ˆ 12 w ˆ 12 2 2

2GBE3

@

; - 2GBE3 #&

& - ' 5 $ :"5 '

" 2-%3

' & ( # 7 wˆ 12† wˆ 12(fˆ12 − εi − εj ) &6 GBC † ϕk (1)ϕl (2)|w ˆ 12 w ˆ 12 (fˆ12 − εi − εj )|ϕm (1)ϕn (2) ≈ (εm + εn − εi − εj )Xkl,mn ,

2GBG3 EK

+

4# 7 Xkl,mn' - 2G,E3 ' (

Xkl,mn 9 # :

2E3

$ 7 & 7 X (

2GA+3 4# : + # !"1& # &6 ABS † † w ˆ 12 w ˆ 12 = w ˆ 12 r12 ≈ r12 (1 − Pˆ1 Pˆ2 − Pˆ1 Pˆ2 + Pˆ1 Pˆ2 )r12 2GBA3 8 & + ) ' '

H I

" 2-%3 #( H7I

" 2$%3

' 5 :"5 # ( EBC [fˆ12 , (1 − Pˆ1 )(1 − Pˆ2 )] ≈ 0 , 2GBJ3

7 # & # - ( 2GBE3

2GBB3 @ - 2GB+3 2GB,3 $ !"1& # 2GA+3 ' +

EBC † † ˆ † ˆ † ˆ w ˆ 12 [f12 , w ˆ 12 ] ≈ w ˆ 12 [fˆ12 , r12 ] = w ˆ 12 [t1 + tˆ2 , r12 ] − w ˆ 12 [k1 + kˆ2 , r12 ] ,

2GBD3 † ˆ w ˆ 12 [k1 + kˆ2 , r12 ] ≈ r12 (1 − Pˆ1 Pˆ2 − Pˆ1 Pˆ2 + Pˆ1 Pˆ2 )[kˆ1 + kˆ2 , r12 ] , 2GBK3 # & . [tˆ1 + tˆ2 , r12] ( # 0' . ( ' ' # 1 - 2GBK3

!"1& # & # 9( ' .

0 : † ˆ w ˆ 12 [k1 + kˆ2 , r12 ] - 2GBK3 ' 7 † ˆ † w ˆ 12 k1 r12 w ˆ 12 r12 kˆ1 & kˆ2 0 # # 7 # & 9 0 wˆ 12† kˆ1r12 wˆ 12† r12kˆ1 ABS † ˆ w ˆ 12 [t1 + tˆ2 , r12 ] ≈ r12 (1 − Pˆ1 Pˆ2 − Pˆ1 Pˆ2 + Pˆ1 Pˆ2 )[tˆ1 + tˆ2 , r12 ] , ABS

ABS † ˆ w ˆ 12 k1 r12 ≈ r12 (1 − Pˆ1 Pˆ2 − Pˆ1 Pˆ2 + Pˆ1 Pˆ2 )kˆ1 r12 ,

2GDF3

ABS

≈ r12 Pˆ1 Pˆ2 kˆ1 Pˆ1 r12 − r12 Pˆ1 Pˆ2 kˆ1 Pˆ1 r12 − r12 Pˆ1 Pˆ2 kˆ1 Pˆ1 r12 + r12 Pˆ1 Pˆ2 kˆ1 Pˆ1 r12

ABS † w ˆ 12 r12 kˆ1 ≈ r12 (1 − Pˆ1 Pˆ2 − Pˆ1 Pˆ2 + Pˆ1 Pˆ2 )r12 kˆ1 ,

2GD+3 2GD,3

ABS

≈ r12 Pˆ1 Pˆ2 r12 Pˆ1 kˆ1 − r12 Pˆ1 Pˆ2 r12 Pˆ1 kˆ1 − r12 Pˆ1 Pˆ2 r12 Pˆ1 kˆ1 + r12 Pˆ1 Pˆ2 r12 Pˆ1 kˆ1

GF

2GDE3

+/ 0 " ' 1 -.

7 #

# & 7 B * +

@

2E3 . - + # , -%"1& # 1& ' 2GDG3

7 & # wˆ 12† [fˆ12, wˆ 12] - 2GBB3 &6 ˆ 1 )(1 − O ˆ 2 )] GBC [fˆ12 , (1 − O ≈ 0

2GDA3

!"1& # , &6 - 2E+K3 GBC † † ˆ † ˆ † ˆ w ˆ 12 [f12 , w ˆ 12 ] ≈ w ˆ 12 [fˆ12 , r12 ] = w ˆ 12 [t1 + tˆ2 , r12 ] − w ˆ 12 [k1 + kˆ2 , r12 ]

2GDJ3 ABS † ˆ ˆ 1 Pˆ2 − Pˆ1 O ˆ1 O ˆ2 + O ˆ 2 )[kˆ1 + kˆ2 , r12 ] w ˆ 12 [k1 + kˆ2 , r12 ] ≈ r12 (1 − O 2GDB3 ( & 9 # . $ !" 1& - 2GD+3 2GDE3 + wˆ 12† kˆ1r12 wˆ 12† r12kˆ1 , ABS

† ˆ ˆ2 + O ˆ 2 )[tˆ1 + tˆ2 , r12 ] , ˆ 1 Pˆ2 − Pˆ1 O ˆ1O w ˆ 12 [t1 + tˆ2 , r12 ] ≈ r12 (1 − O

ABS † ˆ ˆ 1 Pˆ2 − Pˆ1 O ˆ1 O ˆ2 + O ˆ 2 )kˆ1 r12 w ˆ 12 k1 r12 ≈ r12 (1 − O

2GDD3

ABS

ˆ 1 Pˆ2 kˆ1 Pˆ1 r12 − r12 Pˆ1 O ˆ 2 kˆ1 Pˆ1 r12 ≈ r12 Pˆ1 Pˆ2 kˆ1 Pˆ1 r12 − r12 O ˆ1 O ˆ 2 kˆ1 Pˆ1 r12 + r12 O

2GDK3

ABS † ˆ 1 Pˆ2 − Pˆ1 O ˆ1 O ˆ2 + O ˆ 2 )r12 kˆ1 , w ˆ 12 r12 kˆ1 ≈ r12 (1 − O

2GKF3

ABS

ˆ 1 Pˆ2 r12 Pˆ1 kˆ1 − r12 Pˆ1 O ˆ 2 r12 Pˆ1 kˆ1 ≈ r12 Pˆ1 Pˆ2 r12 Pˆ1 kˆ1 − r12 O ˆ1 O ˆ 2 r12 Pˆ1 kˆ1 + r12 O

2GK+3 & B * , @

2E3 / & B * @ , &6 - 2GG3

' # & &6 w ˆ 12 = w ˜ 12 − Vˆ1 Vˆ2 r12 , 2GK,3 ˆ 1 )(1 − O ˆ 2 )r12 , w ˜ 12 = (1 − O 2GKE3 G+

+

5 w˜ 12 (

, - 2E+K3 B C + , : &6 - 2E+K3 ,(*+," '

' ( & @ , & H I &

' # & & # / 4# w˜ 12 - 2GBE3 † † † w ˆ 12 (fˆ12 − εi − εj )w ˆ 12 = w ˜ 12 (fˆ12 − εi − εj )w ˜ 12 − w ˜ 12 (fˆ12 − εi − εj )Vˆ1 Vˆ2 r12 2GKG3 − r12 Vˆ1 Vˆ2 (fˆ12 − εi − εj )w ˜ 12 + r12 Vˆ1 Vˆ2 (fˆ12 − εi − εj )Vˆ1 Vˆ2 r12 , (ij) (ij) # 7 Bkl,mn 4# [2] Bkl,mn (ij) [2]Cab,kl @ , (ij)

Bkl,mn = +

[2]

(ij)

Bkl,mn −

[2]

(ij)

ab Ckl,ab rmn −

ab ab rkl (εa

+ εb − εi −

[2]

(ij)

ab Cab,mn rkl

2GKA3

ab mn εj )rab

ab (ij) (ij)

# [2]Bkl,mn [2]Ckl,ab @ - ( 2GGG3 2GG,3 4# w˜ 12 † [2] (ij) Bkl,mn = ϕk (1)ϕl (2)|w ˜ 12 (fˆ12 − εi − εj )w ˜ 12 |ϕm (1)ϕn (2) , 2GKJ3 (ij) † [2] Ckl,ab = ϕk (1)ϕl (2)|w ˜ 12 (fˆ12 − εi − εj )|ϕa (1)ϕb (2) . 2GKB3

# . B * ,

$ : 7 C @

† † † † w ˆ 12 (fˆ12 − εi − εj ) = [w ˆ 12 , fˆ12 ] − (εi + εj )w ˆ 12 + fˆ12 w ˆ 12 , 2GKD3 4# -%" &6 - 2GDG3 GBC † [w ˆ 12 , fˆ12 ] ≈ [r12 , fˆ12 ] = [r12 , tˆ1 + tˆ2 ] − [r12 , kˆ1 + kˆ2 ] 2GKK3 :"5

# $ !"1& . r12kˆ1 kˆ1 r12 kˆ2' 7 #' &6 ABS r12 kˆ1 ≈ r12 Pˆ1 kˆ1 ABS kˆ1 r12 ≈ kˆ1 Pˆ1 r12 .

G,

2G+FF3 2G+F+3

+/ 0 " ' 1 -.

$ - 2GKD3 wˆ 12 (ij) w˜ 12 &6 , - 2E+K3' [2] Ckl,ab 2G+F,3 # - 4# - 2GK,3 : ( (ij) Ckl,ab * @ , &6 - 2GG3 ' (ij) 7 Ckl,ab @ &6 [2]

(ij) ab Ckl,ab = ϕk (1)ϕl (2)|[r12 , fˆ12 ]|ϕa (1)ϕb (2) + (εk + εl − εi − εj )rkl

2G+FE3 4# # . B 7 C (ij) . , # Ckl,ab (ij) (ij) ab Ckl,ab = [2] Ckl,ab − (εa + εb − εi − εj )rkl . 2G+FG3 7 : & C @

2E3 † † w ˆ 12 (fˆ12 − εi − εj ) = w ˜ 12 (fˆ12 − εi − εj ) − r12 Vˆ1 Vˆ2 (fˆ12 − εi − εj )

+- # 0 (ij) # & @ ,( 7 Bkl,mn *+," %%(*+,"(ij) 9# 0 1 ! 4#( # Bkl,mn & $ [tˆ1 + tˆ2, r12]' 1 ! + : &' - 2GBE3 ' ' 1 ! +2 ( (ij)

& Bkl,mn ' 1 ! % 1&

& !& ( #' 9 # # 7

-%"' $%" !"1& # $ 0< ,(*+,"! MAEN : ' -%"' $%" !"1& # ' : & # 0 ' 7 B @ 7

' $# 1&

5 ( *+," ' # " :"5 @ ' $# B 06

@ ' 7 X' R 7 $ * ' # # 5 " W 7 # # # *+," 1& 2-%' $%' !3 . :

'

GE

+

$# B #' # 1 ;( & - ' ( - ' 7 B & @ '

' # - # 7 B @ 7 C 0 1 ! &' C # # ( & # &' [tˆ1 +tˆ2, r12] ' 1 ! + : &'

- 2GKD3 ' 1 ! +2' #&

& 1 ! % #

'. , ( % - *( 8 $# - 2GEF3 0 # 4# H ( I' 7 A S M,' +E' +,F' +,+N

# AR SR ( - # : & AR & SR 2GG3 - 2GEE3 2GEJ3

# *+,"& AR 0 4# @ A - ( 2,E+3 *+,"& % "% "-( % " # & AR '

%%,"

' 4

& (

2G3 ' #& 7 *+,"& AR * # &

2A3 @ 4 *+," & %%,(*+,"-

2,3 AR

2A3' # '

*+,"& AR '

*+," &' # 0' 0 %%,(*+," -

tijab ij *+," cij kl 4 Rab Rklij *+,"& AR ' $ Rai 4 &' & %%,(*+,"- Tˆ1 "V ( : ϕ˜i ϕ˜a - 2GGB3 GG

+2 0 " ' 1 -

2GGD3 ' % "% "$( tia * # $ $ 4 Rai 5 ( ϕ¯i =

a

ϕ¯a = −

ϕa Rai

ϕi Rai .

2G+FA3 2G+FJ3

i

Rai " @

2A3 . ¯.7 Rai ' # < . # ip (V † )kl ˆlk tia" . -( a ¯m g (V †)kla˜m gˆ˜lkip / ϕ˜i' ϕ˜a ϕ¯i ϕ¯a

V *+,"& ( %%,(*+,"- ' *+," *+,"& % " % "-'

%%," %%! "

' . 6 # (

GA

+

GJ

' (

)(* +,-!

. # # $ G ( & %%! 2*+,3"

M+,,N

6 # *+,"& 4# 7 & 7 $# %%! 2*+,3"

%%! 232*+,3"

M+,EN $ & " (

%%! 23"

M+,GN

+ )/( 0 " %%! 2*+,3"- ' & # %%,(*+,"

' 2H "

I3 %%! (*+,"

( 0 %%! (*+,"% "% "- # G - 2G+E3 2G+A3

- %%! 2*+,3"1&( 5 & *+," "

" # # + @ , - 2GG3 :

$ # # 4 ' *+,"8# # ' (

# %%! 2*+,3"1& # 5 H5I '

0 $

2A+3

5 Φˆ Tˆ2 ' - ' # H5I #' #& ( 5' #' 6 ( ' H5I

# . %%! (*+,"- 4# *+," ' O( *+," ' 5 Tˆ2 # & # MGFN 0' # ' O "ζ & / . %%! 2*+,3"

' (

M+,AN 8 '

' *+," ' &' 9 - $ 6 $ 6

GB

/ ! ,!3451

ˆ = fˆ H

ˆ + Φ

Tˆ = Tˆ1 + Tˆ2 + ...+

Tˆ2

- 6 6 H5I F + F +

A+S 4# 5 # H5I / %%! 2*+,3"% "% "- ' # & %%! (*+,"- 8# ˜ˆ ˜ˆ ˆ Ωμ2 = μ2 |[fˆ, Tˆ2 + Tˆ2 ] + Φ + [Φ, T2 + Tˆ2 ] 1 ˜ˆ ˆ ˆ ˜ˆ ˆ ˆ ˜ˆ ˆ ˆ ˜ˆ ˆ ˆ T2 ], T2 ] + [[Φ, + [[Φ, T2 ], T2 ] + [[Φ, T2 ], T2 ] + [[Φ, T2 ], T2 ] |HF = 0 , 2 ˜ˆ ˜ˆ ˆ Ωμ2 = μ2 |[fˆ, Tˆ2 + Tˆ2 ] + Φ + [Φ, T2 + Tˆ2 ] 1 ˜ˆ ˆ ˆ ˜ˆ ˆ ˆ ˜ˆ ˆ ˆ ˜ˆ ˆ ˆ T2 ], T2 ] + [[Φ, + [[Φ, T2 ], T2 ] + [[Φ, T2 ], T2 ] + [[Φ, T2 ], T2 ] |HF = 0 2

2A+3 2A,3

Tˆ1 "& : Φ˜ˆ ( - 2G+B3 # Φˆ H5I %%,(*+,"

# %%! 2*+,3"

1& ! "%%! (*+,"- . * %%! 2*+,3"1& #( -

" & ( ' 7 Tˆ2 & ˜ˆ ˆ ˆ +

' 7 μ2 |[[Φ, T2 ], T2 ]|HF

' ( & {ϕα} 5 {ϕp} : , :

' 7 &

*+," "- Ωμ # ' 5 & ' #

' 5 ' *+," "- & *+," "- %%! 2*+,3"1& &6 2

ˆ˜ + [Φ, ˆ˜ Tˆ2 ], Tˆ2 ]|HF = 0 , ˆ˜ Tˆ2 ] + 1 [[Φ, Ωμ2 = μ2 |[fˆ, Tˆ2 + Tˆ2 ] + Φ 2

2AE3

˜ˆ ˆ ˜ˆ ˆ ˆ # # [Φ, T2 ] [[Φ, T2 ], T2 ] # O(Tˆ22 ) O(Tˆ23 ) 5 Tˆ2 # 4 &

& ( $ -06 : + ( ˜ˆ ˆ ˆ 7 μ2 |[[Φ, T2 ], T2 ]|HF

' & {ϕα } 5 {ϕp} , - 2GG3' 5 & #

*+," #

GD

/ ! ,!34 .

-

'

%%! 2*+,3"1& & 4# *."1& 28 $3 # 0( 0 9 : ' - %%! (*+,"

# %%! 2*+,3"

H I 1& ' # & %%! "*+,"

#' H I *."1& # 6 %%! " %%! 2*+,3" # %%! (*+,"

0 4 1 4 MGFN *+," "

O & #' Tˆ1 Tˆ2 & # * MGFN 1& X 4 & Tˆ1" & & 0

& ( $ # ' #& %%! 2*+,3"

# 5 Tˆ2

-

! ( % . 0#1( 2/

. * # %%! 2*+,3"

+ & : & + 7 : &'

%%! "

MG' +,N' # !

. : # *+,"& %%! 2*+,3"

* + $ 4 G %%,(*+,"

' *+,"& " *+," "% "% "- ˆ˜ Tˆ ] [Φ, ˆ˜ Tˆ ] " 7 [Φ, 2 2 *+," "- &6

ij ab ˆ ˜ Tˆ2 ]|HF =

[Φ, c (V˜ † )kl ab , kl

ij

2AG3

kl

ij kl ˆ ˜ Tˆ2 ]|HF =

[Φ, tab V˜klab ij

2AA3

ab

4# . ˆ 1 exp(Tˆ1 )w (V˜ † )kl ˆ 12 |ϕk (1)ϕl (2) ab = ϕa (1)ϕb (2)| exp(−T1 ) r12

2AJ3 GK

/ ! ,!3451

(V˜ †)klab & Tˆ1"V (

.' exp(Tˆ1 )w ˆ 12 |ϕk (1)ϕl (2) = w ˆ 12 |ϕk (1)ϕl (2)

2AB3

1 wˆ 12 |ϕk (1)ϕl (2) & ( ' Tˆ1"V # &

"%%! "- ˆ˜ Tˆ ], Tˆ ] [[Φ, ˆ˜ Tˆ ], Tˆ ] ' 7 [[Φ, 2 2 2 2 &6

ab ˆ il kj jk

[[Φ, ˜ Tˆ2 ], Tˆ2 ]|HF = −Pˆijab ˜ † )mn ˜ † mn ij t (2c − c )( V + tkl ab mn mn kl ab (V )kl cmn ij klmn

2AD3

klmn

4# . ˆ 1 exp(Tˆ1 )w (V˜ † )mn ˆ 12 |ϕm (1)ϕn (2) , kl = ϕk (1)ϕl (2)| exp(−T1 ) r12

2AK3

# $ - 2AB3' (V˜ †)mn kl ˆ . k l T1 "& %%,(*+,"

# Tˆ1 " V˜ . 4# ϕ˜a ϕ˜i ' - 2GGB3 2GGD3 @ ' 1 $ 7 &' GA+ #' 0 V˜ . &6 ˆ ab (V˜ † )kl ˜k S˜bl − Pkl ab = Sa

pq

V˜klab

= Sak Sbl −

Pˆklab

kl pq rpq g ˜ + a ˜b

pq

2A+F3

pq ab rkl gpq ,

2A++3

pq

pq ab rkl gpq

+

pq

ˆ mn (V˜ † )mn kl = δkm δln − Pkl

kl pq rpq ga˜˜b ,

pq

mn pq rpq g˜˜ + kl

mn pq rpq gk˜˜l

2A+,3

pq

# - 2A++3' & Tˆ1"V 5 r1 Tˆ1 " &

V˜klab '

12

*+,"& % "% "%%! 2*+,3" - * +

AF

/ ,!34 ,!3

4346

+ )/( 0 )/10/( 02 .

%%! "

&# $ " " ' # # %%! 23"

M+,GN' %%! 2*+,3"

$# % "% "- %%! 23(*+,"

ˆ Tˆ2 ] + [Φ, ˆ Tˆ2 ]|HF = 0 , μ3 |[fˆ, Tˆ3 ] + [Φ,

2A+E3

ˆ Tˆ2 ] + 5( [Φ, & & {ϕα} 5

' ijk%%! 232*+,3"

# 1& ( tabc 9

%%! 23"

+ %%! 232*+,3"- &6

E = E + ΔE

2A+G3

# $ & # %%! 232*+,3"1&

& %%! 23(*+,"

%%! 23"

M+A'+,GN ΔE =

ai

ij a

ab

ˆ Tˆ3 ]|HF +

ˆ ˆ 2tia i [Φ, (2tab − tji ab ) ij [Φ, T3 ]|HF .

2A+A3

abij

# /

%%! 23"

' " %%! 232*+,3" %%! 23(*+,"

( %%! 2*+,3" %%! (*+,"$" 8# tia tijab # #'

%%! "

A+

/ ! ,!3451

A,

. ! . *+,"&

2,3 2G3 * & + , %%,(*+,"

4( % "% "% M,'+,'+EN ?51 M+,JN

. 3 ! # : ( n

V 06 " 7 # N N # - # N

! % "% " ( ?0 % "% "- 4# ..!"2H I3" ' # ; M+,B' +,DN ' & - Ωai Ωaibj

! " "- # # !

*+,"& #' #& *+," "4 Ωkilj ( #

& %%,(*+,"- # 9 . $" 8#" *+," tia ' tijab cijkl # ' ( Ωμ = 0

/ & # # % "% "*+,"* & ,(*+,"* ( &

. wˆ 12' *+," ! cijkl

' 7 B , 7 C # i

"

% "% " ?51

!' &' : #' 7 -06 O(n2N 2) M+,N : *+," # ! ( '

7 #' # 7 -06 %%(*+, O(n2N (N + N )) AE

2 " " -6

. : ' # # . # δ . #' # # Rαk,l ( δα *+,". rβγ #& ?0 % "% "- $# # & 0 !

.7 δ # # !

# #( ' . *+,"& % "% "

7"7"

: H ( I"2?!3"* #

?51" :

4 %%,(*+,"- & . V˜klij (V †)kla˜m & V˜klij 9 % " % ". #' (V †)kla˜m ,(*+,"! . (V †)klαm &6 † kl (V † )kl tia (V † )kl Λpαa (V † )kl 2J+3 a ˜m = (V )am − im = αm , α

i

α ;( 4# 7 Λp # .7

' . V˜klij 9( . # $ 4# #& αβ ˜ % "% ". . : Vkl ( 6' #& % "% ".' (

α β 7 Λh 5 9 # ' 06 . ' ( N 2 >> n2 ∗ nIter ' *# ' V˜klij #& 9 . # . !

0 . . # (V †)klαm Tˆ1 ( # #

L7 C 4#

7

t1 =

0 0 , tia 0

2J,3

H

" I" Tˆ1" tia ' ( Λh' Λp Λp = C(1 − t1 T ) , 2JE3 h Λ = C(1 + t1 ) , 2JG3 0 Tˆ1" ϕ˜i ϕ˜a - 2GGB3 2GGD3 O ( # M+,KN & Λ¯ p Λ¯ h AG

2 " 7 # " )

ϕ¯a ϕ¯i - 2G+FJ3 2G+FA3 # *+," & AR # $

Λ" @

2J3

8 7 ! & 0 ' & *+," #

%%,"& # ( (V †)klαm # N 6 ! % "% " . # : + &6 O(n3N 2(N +N )) , # O(n3N (N 2 + nN ))

# ( *+,"& * M+,N H. "I ! # 0 # ' 5". # ' ! - # # 0 ( % ". ' # V˜klij 0 ' - . : δ δ Rαk,l Rαk,l # *+," % ". ' ( #' #& *+,". : # ij ˜ 4 ' Vkl 0 ' . ! V˜klij O(n4N (N + N )) + O(n4(N 2 + nN )) , #

.

% "% " R 7 S $( 7 8 SR # *+,"& 9 # # ! ( ρ = AR *+,"& ! " "$ ρai ρaibj ' #& *+," "$ ρkilj # ?0 $# AR = ωSR 4#

" M+,F'+,+N' 4 #' (A − ωS)R = 0

8 R ( # &

' ) "7 % "% "4 % " <( O

: df (x) f (x + h) − f (x − h) dx 2h

2JA3

# 6 # $# AR = ωSR 0 ω . 4 . %%,(*+, # ! ' *# O(n3N 3) . . AA

2 " " -6

7 O(n4N (N + N ))' #& &' # ( B C ' # O(n6) O(n4N 2) *# %%,(*+,"* (

0

%%,"*' ! O(nN 4 ) # "1"4 9 & # 5 # , %%, #' # ' %%(*+,"* % "% "

0 5' # # %%! %%E ! ( 2N 6 N 7 3 # # 9 $# %%,(*+, % "% "

0 5 ' "1"4& #

. ( % - ( 4 . * + # % "% ". . : δ Rα,kl =

pq Cδq Cαp rkl ,

2JJ3

pq

δ Rα,kl =

pq Cδ q Cαp rkl

2JB3

pq

' #pq ". p q . ( rkl rklpq ,(*+,"! : # # 0 % " βδ #

% "% ( . gαγ "% !

' %%(*+," 0 #'

/ 4# *+,". (V †)klαm &6 ( 2%+3

! # * +

. ' %%,(*+,"-( # ' ?0 % "% "- # . 2%,3 # ( *+,"& %%,(*+,"- (

:

%%,"& @

2+3 V # %%,"- - AR .' ?0 $# #'

: + & AJ

2 ' 1 " "

# ) "7 0 .( V¯klij V¯klij = Pˆklij

2JD3

¯ h V μ˜j , Λ μi kl

μ

4# 7 Λ¯ h '

2J3 @ 8 V¯klij - . Vklμ˜j # # ?0 % "% "- ( # (V †)klαm : L7 2 2%+33 Λ"7 &6 * (˜ a˜i|˜b˜j) = Λpαa Λhβi Λpγb Λhδj (αβ|γδ) 2JK3 αβγδ

# ' Vklμ˜j . ?0 ' . # O(n4 N 2 ) ! AR # AR 0 . (V †)kla¯m 4# 7 Λ¯ p ¯ p (V † )kl , (V † )kl = 2J+F3 Λ a ¯m αa αm α

# 4# -( (V †)klαm / - 2J+F3 9 &6 j p † kl (V † )kl = − R Λ (V ) = − Raj (V † )kl 2J++3 a ¯m a αj αm jm j

j

# (V †)kljm . ?0 AR O(n3V N ) !' - 2J+F3 9 . # #&' O(n4V ) ! # * +

6 *+,"& AR

2%E3 (

' # 7 :

& #'

2G3 2A3 $ # V". & ' ( & / - -

* + # . 7 7 δ ?0 AR # 4 ' % "% "- - 0 ' # *# *+," ?0( AR 7 O(n6) 8 *+," AB

2 " " -6

. ( % - ( 4

4 *+,"& * , & +' # . : # , &6 - 2GG3 ' 8 & , - 2E+K3 # + , - . % "% ". # ,(*+,"! # ( . rklαβ ' rklαβ # % " αβ αβ . gmn ' gmn : ' & 9 # # # * *+,". ' # + # #' $ δ ab Rα,kl = Cαa Cδb rkl , 2J+,3

ab

@ : , #' / (

! V˜ ". ' # *+,". - 2JJ3 ' # ' . ' .7 7

&' &6 αp αβ δ Rα,kl = Cp δ rkl = Cp δ Cp β rkl 2J+E3

p

p β

& & . (V †)klαm ! 2%G3 ' #

2

2,33 4# ( . - 2J+,3 # % "% ". % ". ' & :"7 # : . H " I "

2,3 2A3 # &6 δˇ αδ αδ Iα,kl = gkl − gkl Sδδ , 2J+G3

Sδδ =

δ

2J+A3

Cδp Spδ ,

p

# ' ! Sδδ ( , &

. %%,(*+,"-( Ωμ i

AD

2 ' 1 " " (ij) 2%A3 2%B3 . * , # 7 Bkl,mn ' ,(*+,"!(ij) : #' 7 Ckl,ab 0 #

. ' # (ij) 4 7 Bkl,mn ! 2.5 n6 S .5 n4 V 23 0' (ij) # # ,(*+,"! # 7 Ckl,ab 9 % "% ". 2H =;I3 O(n4V 2) ! 2%/3 *+,"

" (ij) 4# & 9# 1& Ckl,ab 2%J3

- # + , . V¯klij 4# Vklμ˜j #

V¯klij , / + &6 V¯klij = Pˆklij

¯ hμi V μ˜j − Pˆ ij Λ kl kl

mq ¯ ˜i˜ j ˆ ij rkl gmq + Pkl

mq

μ

˜˜

m˜ ¯ n ij rkl gmn −

mn

ab ij rkl g¯ab ,

2J+J3

ab

# % ". g¯abij 2J+B3

¯˜ ij ij ˜i˜ g¯ab = Pˆab (ga¯j˜b + ga˜ij˜b ) ,

%%," "& ρ ( 0 # 21ρFaibj 3 4 + # , . Vklμ˜j ˜

Vklμj = Sμk S˜jl −

mq

˜

mq μj rkl gmq −

mq

˜

mq jμ rlk gmq +

mn

˜

mn μj rkl gmn −

˜

ab μj rkl gab ,

2J+D3

ab

?0 % "% "- : . (V †)klαm 2%G3 /' !

L7 7 Λh # : (V †)kla¯m # - 2J+F3 2 J++3 ( , . (V †)klim 0' 4# (V †)klαm ( 7 α # $ #' +

! &' , (V †)klαm . # # +

#' & ( AR ' 2%D3

, . ?0 AR *+," . 7 7 δ # . - AK

2 " " -6

+ 7 *# *+," & 9 O(n6)' O(n4V 2) ' ?0 %%,(*+,"- *+," AR #(

' +

JF

/ ! )(* +,-! & . *+," & A * + %%! 2*+,3"

4 % "% "% M,' +,' +EN ?51 M+,JN - . "& M+,GN $( # %%! 2*+,3 %%! 232*+,3"

# %%E"% M+G'+A'+EFN ?51

5 ) & %%! 2*+,3"! "- Ωai %%,(*+,"

' # !

# ' J # - # %%,(*+,"

# - % "% " & *+,"& # & %%! " "-( Ωaibj ' * M+,N ' ' #& *+," "- Ωkilj %%,(*+, #' %%! 2*+,3"

- 2AA3 # *+,"& %%! 2*+,3"

- 2AG " A+F3 # ' # V". # . - %%,(*+,"

# V". ' #

.' # # (V †)klab 0 - # % "% " . . : kl γδ kl γδ ˆ kl (V † )kl = S S − P r g + rγδ gαβ 2B+3 αk βl αβ αβ γδ αβ

γδ

γδ

: ' α' β 9 5

# 0 *# (V †)klαβ 7 O(n2 N 3 (N + N ))' # n N J+

8 " " ,!3451

-06 " W 7 0 O(n2 V 2 N (N + N )) 5 % "% ". # ' (V † )klαβ

% "% " . # # . (V † )klαm' & %%,(*+," 0 #' Vklab' - 2A++3 V˜klab ' : * . O(n3N 2) O(n2V N (N + V )' # V

# # 4 ' & 5 V 2 #& % "% ". # 0 . % ". *+,"& % "% "- Ωai # %%,(*+,"

# # V˜klij ( Vklαβ ϕ˜i &6 V˜klij =

2B,3

Λhαi Λhβj Vklαβ

αβ

O(n3N (N +n)) 5 (V †)klαm ,( *+,"! # %%! 2*+,3"4 ' . (V †)klαβ # !

# #' & αβ Vkl ". %%,(*+,"

+ 0 ! # 9 0 * 06 !;' # N 2 >> n2 ∗ nIter ' #

5 6 ( % - . 2B+3 2B,3 # %%! 2*+,3"-( & Ωμ @ # * M+,'+EN J * %%,(*+,"

# *+," - 2AG3 # ! i

ΩB αiβj =

† kl cij kl (V )αβ

2BE3

kl

& # 6 # ΩBαiβj

%%! " ΩBF αiβj

BF + BF − ΩBF αiβj = Ωαβ,ij + Ωαβ,ij , 1 h h ± δ± δ± ΩBF (Λ Λ + Mγ,ij )Jαβ,γ αβ,ij = 2 γ≥δ γi δj

J,

2BG3 2BA3

8 " " ' 1

* M+,N ' # 0 : δ± @ .

" Mγ,ij δ± % ". Jαβ,γ # * M+,N # ' . α ≥ β γ ≥ δ # i ≥ j .

# ΩBF αiβj & &6 ΩBF aibj =

p + BF − p (ΩBF αβ,ij + Ωαβ,ij )Λαa Λβb

2BJ3

αβ

# % "% " ( < # #' ΩAaibj (

%%! "

M+,N &6 p p ΩA tkl Λαk Λβl ΩBF 2BB3 aibj = ab αiβj kl

αβ

& 7 - ( 2AD3 #' ( # - 2AD3 ( # #

%%! M+EN Elj2 # %%! ". Elj2 6 #

# %%! 2*+,3". *+," # . ' (

& 7 O(n6) ' 7 *# O(n4N 2) : ; % "% "*+,"* ' 7 " W 5 #( 06 N >> n ! 9 %%! 2*+,3". (

%%! ' . O(n2N 4) O(n3N 3) ' # ' & *+,"& & $ &

& %%! (*+,"

MGFN 0(

'

*# ' %%! 2*+,3"

( # # . .*%%*+,(5! M+E+N %%! (*+,". . ' O(n3N 4) 1 4 ' . %%! (*+,". ' ! O(n3N 3) ( %%! (*+,"

-06 #

%%! "

4 %%! 2*+,3"1& 4

& %%! (*+,"

* ' # 1& * 8 $ 2*.3 JE

8 " " ,!3451

$ 7 #' %%! (*+,"

*."1& # %%! 2*+,3 # &

# %%! 2*+,3" %%! (*+,"

' 4# 7 ' 5 5 #& # ' ?

' @ 8 0

7 &' # # * *.(1& M+FE' ++B' ++DN % ". /*55?$ M+E,"+EJN ' #' 4 ˆ Tˆ3 ]|HF μ2 |[Φ, ˆ Tˆ3 ]|HF - 8 "& μ1 |[Φ, 2A+A3 %%! 232*+,3"

*+,"& #( . %%! 2*+,3"* # ( . 23 ' " ( # . . # & & ˜ˆ ˆ ˆ Tˆ3 ]|HF μ2 |[H, μ1 |[H, T3 ]|HF %%E"! " "- Ωμ Ωμ M+G'+A'+EFN' # Ωμ

%%E". Tˆ1"

' !

( ˆ Tˆ3 ]|HF μ2 |[Φ, ˆ Tˆ3 ]|HF & 0 23" μ1 |[Φ, 0 &6 - 2A+A3 # # . 23" %%! 2*+,3"

%%! 23W%%E" M+G'+A'+EFN %%! 2*+,3" * # 1

2

JG

3

8 " " ' 1

• • • • •

. Ωai = 0' Ωaibj = 0 Ωkilj = 0 BF + BF − ΩBF αiβj = Ωαβ,ij + Ωαβ,ij ΩCaibj ΩDaibj Ωaibj = Ωaibj + ΩCaibj + ΩDaibj ΩGai ΩHai Ωai = Ωai + ΩGai + ΩHai ΩBαiβj

? (V ) *+," c Ω = c (V ) → O(n N ) Ω = Ω ± Ω † kl αβ

ij kl

B αiβj

#

• • •

† kl αβ

ij kl kl

B± αβ,ij ± ΩB αβ,ij α

B αi,βj

≥β

4

B βi,αj

2

i ≥ j ΩBαi,βj αi ≥ βj

B± BF ΩBF αiβj = Ωαiβj + Ωαβ,ij p p BF ΩBF aibj = αβ Λαa Λβb Ωαiβj

ΩFkilj ? (V †)klαβ ΩFkilj = V˜klij = αβ ΛhαiΛhβj (V †)klαβ

•

Ωkilj = Ωkilj + ΩFkilj

•

? Vklab Ωkilj = Ωkilj +

•

ΩEkilj =

•

Ωkilj = Ωkilj + ΩEkilj

→ O(n3 N (N + n))

mn

ij ab ab tab Vkl

→ O(n4 N 2 )

(ij)

6 Bkl,mn cij mn → O(n )

B+S !

( *+,"& %%! 2*+,3"- Ωμ * + i

JA

8 " " ,!3451

•

ΩGai ? (V †)klαm (V †)kla˜m = α Λpαa(V †)klαm → O(n3V N ) mi † kl 4 ΩGai = klm(2cim ˜m → O(n V ) kl − ckl )(V )a

• • • • •

Ωai = Ωai + ΩGai ΩJai Ωai = Ωai + ΩJai ΩAaibj = kl tklab αβ Λpαk Λpβl ΩBF αiβj Ωaibj = Ωaibj + ΩAaibj *+," ΩEaibj

? (V ) (V ) E = (2c − c E = E + E † mn αl

† mn kl

2 lj

kmn

kj mn

2 lj

2 lj

2 lj

• • •

=

α

Λpαk (V † )mn → O(n4 N ) αl

jk † mn mn )(V )kl

→ O(n5 )

ΩEaibj = −Pˆabij l tilabElj2 Ωaibj = Ωaibj + ΩEaibj ΩIai Ωai = Ωai + ΩIai

B,S : 2B+3S !

*+,"& %%! 2*+,3"-( Ωμ * + i

JJ

0 #! . # # %%,(*+,"

( # # 1 # ' 12' %5 : #

7 3

%%,(*+,"* # GAE !&( '

. B C # 4# # ?51 M+,JN # ((4T8 &( M,F' D+' D,N 9# 5 ' %%,(*+,"* , &6 - 2GG3

( , - 2E+K3' #

%%,( *+,"* , &6 - 2GG3 # $ 4 ,

%%,(*+,"* # 4( # !"1& 7 # +K+GDJGE, 2KJGE, 3 * MGBN # # +"5 H I"1& #' #& 1 & *

# # ! * # 7

& 2re 3 ' %5' 12 : * M+EBN : * # ,F+B+G++DA # 5 " 7 # # ' # < ": # +D E M,FN' 6 # ' $7 ηl : 2l + 3 ηl = ηs 2D+3 5 #' ,F+B+G++DA # # .7 l O ηs # $7 0 ! JB

9 -6

< ": # : 1 # ,F+G++KBAE 5 " E,,G+D+A+,KJ 7 #' * MGG' GBN 8 & # $7 #' : MDGN (X 3 EX − Y 3 EY ) E∞ = 2D,3 (X 3 − Y 3 ) EX EY " - ' & X Y # 7 O Lmax + 1 # & 1 # - 2D,3 - #& MDG'+EDN 8 & # # 9' : # # ' # - * ; (

' & " #' ( ' # $ #

: - 2D,3 # ' # 0 & & %%," $7( # X = 5 Y = 6 ' # #' ":" 9# # 6 &

7 . # %%,(*+,"$ 1

( # 1 2, ← ,3 1 1 2E ← ,3 R

$ ' 7 ' @

2 +3 2 G3 8& # %%,(*+," -" 8 6 # &

%%,(*+,"-" # ( 1

9# 1 8& 1 + , 4

%%,"

@

2 +3 " 2 G3 -" 8 1 ( 2D+3 2D,3 4

%%, # & ' - - 2D,3 7 #'

& -( : 2D,3 ' JD

9 "

*+,"- # ( :

- 0 # ' *

# #' ": ( # $< $ # < # ,(*+," %%,(*+,"-( 06 # ' ? ,"? # ? ,"? # M+EK"+GEN' # 2(BJEAD $h3 * M+GGN ( # #' &6 ! M+GAN

# %%,(*+,"$ * & + , -" ( 8 ' #

& + 2D+3 5 wˆ 12 - 2GG3 2E+D3 0' # 5 " 7 2Pˆ1 = Pˆ1 3 ' &6 ˆ2 + O ˆ 2 − Vˆ1 Vˆ2 ˆ 1 Pˆ2 − Pˆ1 O ˆ1O 1 − Pˆ1 Pˆ2 − Pˆ1 Pˆ2 + Pˆ1 Pˆ2 = 1 − Pˆ1 Pˆ2 = 1 − O 2DE3 #

$ ( + , # ' #

2 + " ,3 ' 0

' . !& + ,

: %%,(*+,"-

& %%,(*+,"-

' # 8 4 - &

' *+,"& 4 (

%%, 8 ( %%,(*+," 8 # 9 *+,"& 1 # %%,(*+,"$ # -" 8 ,' & # ,(*+,"- MGBN' +

← %%,( 1

*+," 1 2, ← ,3 R # %%,"4 # @

2D+3 %%,(*+,"$ + , ' # $ ( +

:

& *+," # 06

%%, 1 *+,"$

%%, &6 # ' < # & %%,(*+, 06 FF,+ 4 ' #& < *+," FF+ 4 & ? JK

9 -6

D+S %%," %%,(*+,"$ -" 1 8

D,S %%," %%,(*+,"$ -" 1 8 1

! ← %%,( 1

*+," 1 2E ← ,3 R 1

2D,3 (

. :

# 6 7 #'

'

BF

9 "

: ' $ !"1& ' 0 & V #

*+," 06

%%, 6 ' ' , #' 06 + & ' # ,

+ #( 06 < # %%,(*+," + , 0.004 0.006 4' #&

%%,"

0.022 4 ' (

4# *+,"& # ' '

%%, # 6 # *+,"&

D+S 1 2, ← ,3 4 , 1& $ # ,F+B+G++DA 5 " 7 # : + # $ ' %%, %%,(*+, AAG+ AJKJ A,FB A,JB A+GA A+BG A+,J A+GE A++K A+EF & ? A+FK

D,S 1 2E ← ,3 1 & + , 1& 4 4# # ,F+G++KBAE 5 " E,,G+D+A+,KJ 7 %%, + , +ADFD +JJDE +JJBE +JFEF +J,E, +JEA+ +J+EB +J,+K +J,K, +J,+A +J,AD +J,DD +J,GG +J,BA +J,DJ +J,AB +J,DE +J,DA & ? +J,BK B+

9 -6

7 2 . # %%,(*+,"* ' :' %5 12

$ ' 7 ' @

1

Σ+

"

1

# #(

Π

1Σ+" 1Π" R & . - & %%,(*+,"

( 1Σ+"8

%%," ' %%,(*+," 06

%%, : 1Π" R

( ( R &' &

' %%,(*+," 1Σ+"8 4

%%,

' #& ! 1Π"R # 8& 4 -' 2DE3 2DG3

' & 1Σ+"8 < %%,(*+, & 06 0.01 4 4 1' < # -06 #' $ 9 ' & ? -

R & 9 ' %%,(*+,"$ & 06

%%,

DES 4 & + , 1& 7 # # 1 Σ+ 1Π %%, + , %%, + , ((4 8 JEKE JGKG JAFA ,DJJ ,KDA EFEE ((48 JGGG JGK+ JGKF ,DEG ,DDD ,DKK ((4C8 JGJA JGK+ JGK, ,D,E ,DA, ,DAJ ((4A8 JGJK JGDJ JGDJ ,D+K ,DED ,DEK ((4J8 JGB, JGDJ JGDE ,D+B ,DEG ,DE+ & ? JGBJ ,D+G

B,

9 6 )

DES %%," %%,(*+,"$ -" 1Σ+"8

DGS %%," %%,(*+,"$ -" 1Π"8

BE

9 -6

1

Σ+

$ %

1

1

Σ+

$ %

1

(

1

$

1

Σ+

" Π & 1

$ %%,(*+,"! R & 1Σ+' % 1Σ+ 1Π : @

2 ,F3' #&

2 D3 2 ++3 2 +3 2 E3

V #

: %%,(*+," & + , ( ' # $ , # # 06 +

& 1Π"R $

%%,"

( ' ! 1 + Σ "R & # %%, # &6' # 9# 0 0# # #

2 ,F3 ( # . *+,"

%%, - 4 9# 8' 9 8 ( & ' # , #

+ Σ+

" Π %' 1

%%,(*+," 8& Σ ' % 1Σ+ 1Π @ (

2 ,+3 8& # -(

2 +,3 2 +A3 2 G3 2 J3

- # :

/ -" 4 8( ) 8 9# & 8& # -

%%," "ζ "

R &' #& *+,"$ "ζ "

#( $ 06 4 & R & 1Σ+' % 1Σ+ 1Π 0.08 4' 0.05 4 0.01 4' #&(

%%, 0.06 4' 0.03 4 0.003 4 4 6 # %%,( *+, 1Σ+"R &

%%, 4 ' 8&

*; "% Σ− u

1

Πg

+

)

1

Δu

2

$ %%,(*+," 12 @

2 ,,3 0

2 +J3 2 +K3 ( 2 B3 2 K3

. :

(

%%,"

R &

: < %%,"* 7 "ζ " & 7 0.001 4 ' # BG

9 6 )

& # # . # %%,"$ *+,"& # ( & & * ' # , # 06 + . %%,(*+," #

%%, . 06 # %%,(*+, & # FF+ FF, 4 . 4

%%,"$ ' %%,(*+, 7 "ζ " ( 8& 4 -' 9 8 ( ' # %%,(*+,

%%, $ # 4 - ( 8& ' # # - , #

+

10.80 10.75

CC2 Ansatz 1 Ansatz 2 Limit

Energie in eV

10.70 10.65 10.60 10.55 10.50 10.45 10.40 10.35 D

T Q 5 Kardinalzahl der aug-cc-pVXZ Basis

6

DAS %%, %%,(*+, X R 12 4

1

Σ− u"

BA

9 -6

9.85 9.80

CC2 Ansatz 1 Ansatz 2 Limit

Energie in eV

9.75 9.70 9.65 9.60 9.55 9.50 9.45 D

T Q 5 Kardinalzahl der aug-cc-pVXZ Basis

6

DJS %%, %%,(*+, R 12 4

1

Πg "

11.30 11.25

CC2 Ansatz 1 Ansatz 2 Limit

Energie in eV

11.20 11.15 11.10 11.05 11.00 10.95 10.90 10.85 D

T Q 5 Kardinalzahl der aug-cc-pVXZ Basis

6

DBS %%, %%,(*+, # R 12 4 BJ

1

Δu "

9 6 )

. # 1 # # *+," 4

%%, # -" 9# 8' *+," #

& $ *+,"& ' *+," ,

( + # . : # ; ' / %%,(*+," @ 4& ( #

'

?0# *+,"

BB

Kapitel 8. Anwendungen des CC2-R12–Modells

8.4. Analyse der R12–Beitr¨ age zu den berechneten Anregungsenergien Der R12–Ansatz ist bekannt f¨ ur seine Verbesserung des Konvergenzverhaltens von Grundzustandsenergien zum Basissatzlimit, wenn die dominante Fehlerquelle auf dynamische Korrelation zur¨ uckzuf¨ uhren ist. Inwieweit sich die Verwendung des R12–Ansatzes auf die Berechnung von Anregungsenergien auswirkt, ist jedoch bisher noch nicht untersucht worden. Ziel dieses Abschnitts ist das unerwartet langsame Konvergenzverhalten des CC2-R12 f¨ ur Anregungsenergien zu verstehen. Daf¨ ur ist eine detaillierte Untersuchung der Korrelationsbeitr¨age zu letzteren notwendig. Die Einteilung dieser Korrelationsbeitr¨age in einen korrelierten und einen unkorrelierten Anteil ist allerdings schwierig, da es im Gegensatz zum Grundzustand f¨ ur den angeregten Zustand kein unumstrittenes nicht–korreliertes Referenzmodell wie beispielsweise das Hartree–Fock–Modell gibt. Außerdem kann sich aufgrund der Ber¨ ucksichtigung der dynamischen Korrelation der Charakter der Wellenfunktion, also der wichtigsten Slater– Determinanten, deutlich ¨andern. F¨ ur die Analyse der Beitr¨age zu den CC2-R12–Anregungsenergien wird folgende Zerlegung vorgenommen, die sich an der Struktur der Jacobi–Matrix aus Gleichung (4.39) orientiert. Sowohl f¨ ur das CC2– als auch f¨ ur das CCSD–Modell kann die Anregungsenergie ω als ω =

1 † SR R

1† A12 R 2† A21 R 2† A22 R 1 + R 2 + R 1 + R 2 1† A11 R R

(8.4)

dargestellt werden. Der Einfachheit halber ist in Gleichung (8.4) keine Unterteilung in konventionelle Doubles– und R12–Doubles–Beitr¨age vorgenommen worden. Der Nenner † SR ist im Prinzip nur eine Normierungskonstante und kann bei entsprechender NorR gleich 1 gesetzt werden. Deshalb wird er im Folgenden mierung der Versuchsvektoren R auch nicht weiter ber¨ ucksichtigt. Bei den in dieser Studie untersuchten einfachanregungs¨ dominierten Uberg¨angen gilt f¨ ur die entsprechenden Singles– und Doubles–Anteile des 2 || ||R 1 ||. Dies hat zur Folge, dass sich die letzten beiden Terme Versuchsvektors ||R aus Gleichung (8.4) nahezu gegenseitig aufheben, da n¨aherungsweise † (A21 R † S22 R 2 ||2 ) ≈ 0 1 + A22 R 2) = ωR 2 = O(||R R 2 2

(8.5)

gilt. Es ist also ausreichend, sich bei der Untersuchung der Beitr¨age zu den CC2-R12– Anregungsenergien auf die ersten beiden Terme aus Gleichung (8.4) zu beschr¨anken, da diese als die dominanten Beitr¨age angesehen werden k¨onnen. Unter Einf¨ uhrung der Kurzschreibweise Ri | = μi μi | (mit i = 1, 2) kann der erste Term aus Gleichung (8.4) in zwei Beitr¨age aufgeteilt werden. Der erste Beitrag ist das Matrix˜ˆ ˆ element R1 |[H, R1 ]|HF und wird im Folgenden als singles only“–Term bezeichnet, da ” er nur von Einfachanregungsbeitr¨agen dominiert ist. Der zweite Beitrag hingegen ist das

78

9+ : ' 1 #

ˆ Tˆ2 ], R ˆ 1 ]|HF $< ; ( 7 R1|[[H,

-

˜ˆ ˆ # - 2DG3 R1|[H, R2 ]|HF

#

$ <" < # - 8 # ( ˆ Tˆ2 ], R ˆ 1 ]|HF -' # $ ( -06 # R1|[[H, @ # ! #' # # ; &

-" 8 (

8 %%," %%! " . % "% "*+,"

& *+,"&

"& ˜ˆ ˆ ˆ Tˆ2 + Tˆ2 ], R ˆ 1 ]|HF R1 |[H, ˆ 2 ]|HF 4 &6 R1|[[H, R2 + R

"*

P ← 7 %%,(*+,"& 1P 2, ← ,3 R @ 4

%%,"

2 ,E3 2 EJ3 # ˜ˆ ˆ H ;I" R1|[H, R1 ]|HF # %%,(*+, (

%%, #

# # ' *+," & 2Tˆ2 ' R 2 3 7 &' ; - ' 9 # # FB 4 FK 4 ' # 2DD3

' %%,( *+,"

23 ' #& (

%%, & ' $ ' &

' *+,"$

%%, 4 & & 1 /

2 EJ3 *+," ˜ˆ ˆ ˆ 2 ]|HF #

& 7 R1|[H, R2 + R ' - & %%, %%,(*+, ' # *+,"$ &

1 P 2, ← ,3 R 6 & # *+," # '

5 wˆ 12 # : -

@ +2,2 $ :

0 ' *+," !; ' R 1P 2, ← ,3 0 '

%%,"

2D+3 &

: & ˜ˆ ˆ ˆ Tˆ2 ], R ˆ 1 ]|HF R1 |[H, R1|[[H, R2 ]|HF

& 1

BK

9 -6

Energie in meV

900

850

800

CC2 CC2-R12

750

2

3

4

5

6

Kardinalzahl der Basis ˆ Tˆ2 + Tˆ2 ], R ˆ 1 ]|HF DDS %%," %%,(*+, & R1|[[H, 1 P 2, ← ,3 R

"*

1

P

! ←

& 1

2 ,G3 2 EB3 ( *+," # # ' ; #6 R '

*+," ' & #

%%, &

: ( $ #

' :

1 : ' *+,"&

7 *+,"& %%,"$ 06 & ˜ˆ ˆ ˆ 2 ]|HF

%%, R1 |[H, R2 + R ' *+,"& 7

' #

2 EB3 ( ˆ Tˆ2 + Tˆ2 ], R ˆ 1 ]|HF & - R1|[[H, 2DK3' # ' ,

+ +

%%, *+,"$< 6

DF

9+ : ' 1 #

2600 2400 Energie in meV

2200 2000 1800

CC2 CC2-R12 Ansatz 1 CC2-R12 Ansatz 2

1600 1400 1200 1000 2

3

4

5

6

7

Kardinalzahl der Basis ˆ Tˆ2 + Tˆ2 ], R ˆ 1 ]|HF DKS %%," %%,(*+, & R1|[[H, 1 P 2E ← ,3 R 1

"*

1

Σ+

"

1

# &

Π

;

2 ,A3 2 ,J3 ( 7 *+," "& 7 ˜ˆ ˆ ˆ Tˆ2 + Tˆ2 ], R ˆ 1 ]|HF R1 |[H, ˆ 2 ]|HF @

2 ED3 R1 |[[H, R2 + R 1 + 2 EK3 Σ "R ' &

' &' " $ ' *+,"

- # ' H ;I" ' # ( ˆ Tˆ2 + Tˆ2 ], R ˆ 1 ]|HF - R1|[[H,

%%, H ;I" # ˆ Tˆ2 ], R ˆ 1 ]|HF" ' 7 6 # R1|[[H, : 2D,3 :

#

2DE3 1 # 9# & 7 *+,"& 4

%%,

: 1Π"R # & ! # 1P 2, ← ,3 R 8 H ;I" *+,"

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Berechnete Energie in Hartree

-128.720

CCSD CCSD(R12) CCSD-R12

-128.740 -128.760 -128.780 -128.800 -128.820 -128.840 -128.860 D

T

Q 5 Kardinalzahl der aug-cc-pVXZ Basis

6

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4

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Berechnete Energie in milli Hartree

100 MP2-R12 CCSD(R12) CCSD-R12

10

D

T Q Kardinalzahl der aug-cc-pVXZ Basis

5

6

K,S Δ*+,"& 1 Eh'

(

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Berechnete Energie in milli Hartree

100 MP2-R12 CCSD(R12) CCSD-R12

10

D

T Q Kardinalzahl der aug-cc-pVXZ Basis

5

6

KES Δ*+,"& : Eh'

(

!

MP2-R12 CCSD(R12) CCSD-R12

Berechnete Energie in milli Hartree

100

10

D

T Q Kardinalzahl der aug-cc-pVXZ Basis

5

6

KGS Δ*+,"& 12 Eh'

(

! K+

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MP2-R12/aux MP2-R12 CCSD(R12)/aux CCSD(R12) CCSD-R12

Berechnete Energie in milli Hartree

100

10

D

T Q Kardinalzahl der aug-cc-pVXZ Basis

5

6

KAS Δ*+,"& 1 +K+GDJGE, ( 7 Eh'

( !

Berechnete Energie in milli Hartree

100 MP2-R12/aux MP2-R12 CCSD(R12)/aux CCSD(R12) CCSD-R12

10

D

T Q Kardinalzahl der aug-cc-pVXZ Basis

5

6

KJS Δ*+,"& : +K+GDJGE, 2KJGE, 3 7 Eh'

(

! KG

< ' ,!34.

Berechnete Energie in milli Hartree

100

MP2-R12/aux MP2-R12 CCSD(R12)/aux CCSD(R12) CCSD-R12

10

D

T Q Kardinalzahl der aug-cc-pVXZ Basis

5

6

KBS Δ*+,"& 12 +K+GDJGE, ( 7 Eh'

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2

3

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Fehler in kJ/mol

60 40 20 0 -20 -40 -60 -80 -100 1

2

3

4

5

6 7 8 9 10 11 12 13 14 15 Reaktionsnummer

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2

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4

5

6

7

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Reaktionsnummer

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( $7 # M+++N . * ! # ' 6 %%! 232*+,3": ((42P38 & ":"-

":" %%! 232*+,3"- 4# & ( ! * # 0( ((42CP38 ":"- - (( 42P38W%%! 232*+,3"* 4# (

+FJ

< ' ) ' ,=3 ·>2 = ! "

: % " " %

);3·<2; . # !53· 25 7 ' # #' # *

!# & & 1 (

," % "% "4 # $ ( *+," # ,(*+," %%! 232*+,3" M+GBN 3 /'

-

& 0 * *( # & !# & & 0# M+BJ"+BKN SO3 + H2 O SO3 · H2 O SO3 · H2 O + H2 O SO3 · (H2 O)2 SO3 · (H2 O)2 H2 SO4 · H2 O H2 SO4 · H2 O H2 SO4 + H2 O .

2+3 2+ 3 2+3 2+3

8 ! ' / 5 M+DFN :(

' # ' / # ' 2!54 7 !53· 25 # *

. $7 M+BJ'+BD'+BKN #' -#( 2!54 5 !53 # 5 25 & -# kobs ( & kobs Ea(obs) ≈ −52 )W ≈ −13 W M+BD'+BKN C ! M+BB'+D+N / $"*( @ * 2+3 2+3 ' * 2+3 #( ! ' * 2+3

*( 2−+3 & 8 ! $7 # );

M+BKN -& 0 * 2+3 2+3 SO3 · (H2 O)2 → H2 SO4 + H2 O

2KK3 +FB

< ,!3451

# -# !# & ( &6 d[H2 SO4 ] = k∗1c [SO3 · (H2 O)2 ] 2K+F3 dt # ! SO3 · (H2O)2 7 C( & M+D,N ' d[SO3 · (H2 O)2 ] ≈ 0. 2K++3 dt ' SO3 · (H2O)2 # 8# ' ( 4 * - # ' - # 2+3 - # 2+ 3 = #' SO3 · (H2O)2 7 [SO3 · (H2 O)2 ] =

K1a k∗1b [SO3 ][H2 O]2 k∗−1b + k∗1c

2K+,3

#' # K1a - # * 2+3 ( # ! ' k∗−1b >> k∗1c ' $ - 2K+,3 - 2K+F3 ' -# !#( & &6 d[H2 SO4 ] k∗ = K1a ∗1b k∗1c [H2 O]2 [SO3 ] = kobs [H2 O]2 [SO3 ] dt k−1b

2K+E3

# 7 % * 2+ 3 2+3 # ' *# # 0 -#( 2∗3 - 2K+E3 7

*# kobs Ea(obs) &6 Ea(obs) ≡ RT2

d ln kobs = ΔH01a + E∗a(1b) − E∗a(−1b) + E∗a(−1c) dT

2K+G3

# ΔH01a ! * 2+3 E∗a(i) # * i $ ' *' ' E∗a(i) # M+DEN / 4 # 7

Ea(obs) ΔH01a ' # De = −ΔE(0K) &6 −ΔH01a (T) = De + ΔΔU(0K → T) + ΔZPVE + RT 2K+A3 # " 1

# 2K,3 ΔΔU ΔZPVE # +FD

< ' ) ' ,=3 ·>2 = ! "

!53· 2 5 ( - 2K+A3 & * ΔH01a ( # 0 ; - 2K+G3 4 kobs $ & 4 ΔH01a & 4& & kobs 4@ * 3 4

! # 4# & ( !53· 2 5 M+DGN " $ #

1 - #'

9# HI HI # - !53 25 ( : - !53· 25 7 # HI # # H:7I #' # - !53· 2 5 7 - $&' #' # HI"2%3" M+DA"+DDN #' H I"2!!$3": M+DA"+DDN 0 ' < # $ %" @ H-I ' : H-I * %" # #' 1 H-I - HI !53 H-I' ? ( # HI # / 4# 1 !53· 2 5 ( 7 # &6 [a] [b] [a] [b] A(A) + B(B) → {A(AB) + B(AB) } 2,3 [ag] [gb] [ab] {A(AB) + B(AB) } → AB(AB) 2, 3 # # .7" - .7 $< * 2,3 2, 3 [a] [b] [a] [b] ΔE1 = E(A(AB) ) + E(B(AB) ) − E(A(A) ) − E(B(B) ) 2K+J3 [ab] [ag] [gb] ΔE2 = E(AB(AB) ) − E(A(AB) ) − E(B(AB) ) 2K+B3 De = −ΔE1 − ΔE2 2K+D3 +FK

< ,!3451

%" - De !53· 25 7 ΔE1 $' # # ' 25 !53 : - # - ΔE2 # H( I # ! $< # 7 :' # 7 - # ( 7 ! 25 !53 # !53· 25 # *.(,"1 M+EE' +EGN 4# ,(C84 M+DKN /*55?$ M+E,N # 4 $ * 2+ 5 +,, !3 HI :

: # H I"1& # . :

' # 7 7 # - !53· 25 7 @

2$,J3' 2K+F3

K+FS !53· 25 0 ! @ (

2$,J3 ++F

< ' ) ' ,=3 ·>2 = ! "

$ /*55?$"4 # *.(,(*+,"-( M+KFN * + 1& # #

5 " 7 ' 2:3 ( # - @

2$,B3 . *.(,(*+," # ; *."1& # .( ,(*+,"

' # ! MGBN ?51" #' 4# 7(

0 ( *."1& . *.(,(*+," # !

5 7 *."1& # : ( H*+,"@I *."1& . *.(,(*+," # ( & *."1& 8# 5 r1 ' r12 [r12, t1 ] M+FE' +EE' +EG' +KFN *."1& % ".

," ( M+FE'+EE'+EGN

7 & M+K+N / H*+,"@I *."1&'

7 ( & ' # *."1& ' # ; MGA'+K,N 4# < H; @I"2 :3 *.": *.(,(*+, # 25 !53 ,(*+,"4 MGBN' # ?51" M+,JN #'

2$,B3 # 5 # *.(,(*+,"* # 12

%%! 232*+,3"- M+,,' +,EN # *( + 1& MGBN ?51" M+,JN !53 ' 4# # ((42CP38 5 M,F' +AKN

( & ((42AP38 7 MGBN 8 2 5 # =& ;;

M+KEN # #

%%! 23"- M+,GN 5?*5" M+KG"+KDN 4# # ( # ((42CP38 ((%48

M,F'+AK'+KKN %%! 23"* ((%48 #( #' # # H I"1& ' & $<' * $ #' $ @

2$,K3 2$EF3 8& # & $< # #" # MGN ":"1

+++

< ,!3451

3 )

25 !53 # F,G GB, )W *.(,2:%3W,(C84 1 *.( ," 7

0 " 1 4 # #( - # # &# ( ' *.(,"1 *.(, ( 25' 4# ; )," =& ;;

M+KEN F,G )W #' *.( ," !53 ; =& ' 0 ( # 8& # %%! 232*+,3"1 RSO U +G,,F #' 4# !5" &' - ' (

2$,D3 @' @ ( !5" # *.(,"1 &( - %%! 232*+,3"* < %%! 232*+,3"$ # 2(J,E,FD JJJ Eh3 ! 2(J,E,FJ DA, Eh3 GBJ )W %%! 23"$ ' ((%48 ( * #

2$EF3 ' # * # $< # FFB )W & !53 GJK )W - ΔE1 U GKE )W . - ΔE1 H I

# ΔE2 # ΔEFix #' 2 Fix

ΔE2 ΔE2 - 2K+D3 @ /( ΔE2 ' ΔEFix :7 2 !53· 25 7 ( & ΔE2 ΔEFix

2K,3 2 06 %%! 23W((42CP38 *

2$,K3 7 :' # 4 #( # < # ,W((42CP38 *.(,(*+,"$ ' 4# H I"1& ( #' & & # ? ++,

< ' ) ' ,=3 ·>2 = ! "

,(?( M+AGN * # ( # %%! 23W((%48 -' H I"1& #' 0 @ & ":"1

2$EF3 # < # #" ( # 0 H I # ΔE2 ΔEFix −45.74 )W −37.08 )W # 2

K,S $ & !53· 25 7 )W ΔE2 -# #

& 7( ! ΔE2Fix # ( !' # # :7 - # ΔE2Fix ΔE2 4 %%! 232:%3 ((42CP38 (EAD, (GGFB *.(,(*+,2:%3

2 :3 (+A+ (,FJ * %%! 232:

3 ((%48 (FFE (FF+ ": ((%48 PF,D PFGF - (EBFD (GABG <

' %" ,W((42CP38 1 ,, )W #' #& *.(,(*+,"1 F,, )W # ' H I"2!!$3": 4# 7

-06 # 8 ΔE1 ΔE2 # "" ; =& 25 # 6 # ΔE1 ' : 2 5 !53 *.( ,W,(C84 ' ! ΔE1 ΔE2 # H I De = −ΔE ( # 4# * M+DGN # $< ΔE Fix x + ΔE1 x2 2K+K3 ΔE(x) = ΔEFix 2 + ΔE2 − ΔE2 : x

# ' # $7&

x U F' x U + xopt ' ΔE(x) # & ΔE : :

x U F ' - # # ΔE(x) = ΔE2Fix : :

x U + - 25·!53 7

& 7 ΔE(x) = ΔE1 +ΔE2 / 4# ΔE1 = 4.93 )W ' ΔE2Fix = −37.08 )W ΔE2 = −45.74 )W ' xopt = 0.88 ++E

< ,!3451

# ΔE = −40.88 )W 8 4 # ΔE $7

x U + −40.81 )W ' :

!53· 25 7 5 x 6 $< De GFDD )W

# :(

# -"* 7< 9 *.( ,W,(C84 # & ! 9( %%! 23W((42CP38 6 # ΔE2 ΔEFix 2 &

2K,3 4 ( : O

' ' & & ( # . ΔE2 ' " FB )W , W ' #& FG )W # # & + )W # : # !53· 25 7 De = (40.9 ± 1) )W # 3+ !!

. ! # !53· 25

7 4# ,"' *.(,"' *.(,(*+,"' %%! 23" %%! 232*+,3"4 De = (40.9 ± 1) )W 5 " 1

#( * ΔH01a(298K) −34.7 ± G )W 2−8.3 ± + W 3 M+GBN $ & 4' - # 2+3 6 !# & & 0 ΔH01a #' 06 Ea(obs) ' ≈ −13 W &6 - 2K+G3 # # A W * 2+ 3 2+3 $ 0 ! ; #

++G

2 3 ! . # 1&

%%! (*+," - 4# 7 * + # M+,,N & O "ζ "C &

%%! 2*+,3"1& - ( C & #

& %%! (*+,"

1 " %%! 23"

/ *( #' %%! 232*+,3"1& * M+,EN ( " M++,N # M+++',FFN #' %%(*+,"

4# ' # # r12 exp(−γr12)

*+,"' 4 ( # %%," 4# 7

* &( + , 7 0 M++AN #' 4# *+," %%,(*+," # ( ; & %%,(*+," #' 4

%%,"

# " & -" 8 ! 8 *+,"& -( $ #' π ← σ"R & (

# # :

' -@ σ "5 ' *+," # # !; 0 *+," R &( $ ! ' . ( *+,"' #

5 ' 4 (

' & # M+GJN ! 4 # - ; $# %%,(*+,"

0 $( # - ; $ 6 %%" 0 %%"

* %%(*+," %%(*+,"#" $ 7( $ %%! 2*+,3" M,F+N ++A

? ;"" # )

++J

"!

++B

#)

5 !!$ %% %%!' %%! ' %%! %%! (*+,' %%! (*+, %%! 2*+,3 %%! 23 %%! 23(*+, %%! 232*+,3 %%! MN %%! MN(*+, %%! MN2*+,3 %%,' %%E %%,(*+, %. %.!' %.! :%. %.(*+, % : -5 : 5 , ,(*+, *. *.(,(*+, !%:

++D

% "% % "% "

$" 2! 3' 8#" 2 3 # 2 3 % %%! " %%! "

7

1&

%%! (*+, %%! "

0 " ' # %%! " ! " " " " # # %%! (*+,"

0 " &6 %%! 23"1& 1&

%%! 23"*+, %%! "

0 " ' # %%! " " " # # %%! (*+,"

0 " &6 %%! MN"1& 1&

%%! MN(*+, !0 1& %%! %%! %%,"

7

%@ 2 @# #3 @# # * (: # # ! :

"%. %."

7

-6 ; ": Q

" !0 # 5 ,"

7

; ,(*+,"

*."1& @

4 " "

++K

' ( ) -6

+S $7 %%,"- Ωai = H I J E F ΩG ai + Ωai + Ωai + Ωai Ωaijb = Ωaijb + Ωaijb . α' β ' ( % ". # grspq grspq = φr (1)φs(2)| r1 |φp (1)φq (2) @ Tˆ1 & φ˜i φ˜a - 2GGB3 2GGD3 @ 12

+,F

Ωai ΩG ai ΩH ai ΩIai ΩJai

= cdk (2tik − tki )g dc cd kl cd lk k˜a d˜i = − dkl (2tad − tad )glk ki ˆ = ck (2tik ac − tac )fkc = fˆai

Ωaijb ΩE aijb ΩFaijb

fˆkc fˆai

cb cb j = fkc + bj (2gkj − gjk )tb ˜ ˜ i = fai + (a − i )ta + kc (2ga˜ick − ga˜cik )tkc

= (a + b − i − j )tij ab ˜i˜ j = ga˜˜b

,S $7 *+,"& %%,(*+,"-( H I CC2 E Ωai = ΩCC2 + ΩG ai ai + Ωai + Ωai Ωaibj = Ωaibj + Ωaibj E F Ωkilj = ΩE kilj + Ωkilj + Ωkilj & . # rrspq pq rrs = φr (1)φs (2)|r12 |φp (1)φq (2) @ : @ % " . grspq

2+3 R # Spq ( Tˆ1 & φ˜i φ˜a - 2GGB3 2GGD3 @ B C - 2GGG3 2GG,3 @ + , ( . 7 @

2E3 +S wˆ 12 = (1 − Pˆ1)(1 − Pˆ2)r12

Ωai ΩG ai ΩH ai ΩIai im ykl V˜klij (V † )kl a ˜m

= =0 =0

im † kl ˜m klm ykl (V )a

Ωai ΩG ai ΩH ai ΩIai

im = klm ykl (V † )kl a ˜m lk mn ˜ip = − mnkl ymn p rap ˆlk g mn ik = kmn ymn ˆp k p rap g

Ωaibj ΩE aibj

=0

Ωkilj ΩE kilj ΩE kilj ΩFkilj

(ij) = mn Bkl,mn cij mn =0 = V˜klij

Ωkilj ΩE kilj ΩE kilj ΩFkilj

(ij) = mn Bkl,mn cij mn (ij) ij = ab Ckl,ab tab = V˜klij

pq ˜i˜j pq ˜i˜ = δik δjl − Pˆklij pq rkl gpqj + pq rkl gpq kl pq kl pq lk pq = Sa˜k δlm − pq rpq ga˜m − pq rpq gm˜a + pq rpq ga˜m ˆ 1 )(1 − O ˆ 2 )r12 w ˆ 12 = (1 − O

˜

mi U 2cim kl − ckl

,S

ip gˆlk gˆp k V˜klij (V † )kl a ˜m

Ωaibj ΩE aibj

=

(ij) ij kl Cab,kl ckl

˜ij ˜ip = glk − j glk Sjp p c p c l jc jc l = cl (2gkl − glk )tc − j cl (2gkl − glk )tc Sjp ˜ ˜ ˜ ij mq ˜ i j m˜ ˜ n i˜ j = δik δjl − Pˆkl mq rkl gmq + mn rkl gmn mq mq kl lk kl mn = Sa˜k δlm − mq rmq ga ˜m ˜m − a + mq rmq gm˜ mn rmn ga ˆ 1 )(1 − O ˆ 2 )(1 − Vˆ1 Vˆ2 )r12 w ˆ 12 = (1 − O

˜

ip gˆlk gˆp k V˜ ij

kl (V † )kl a ˜m

˜ip ˜ip = glk − p glk Spp p c p c l pc pc l = cl (2gkl − glk )tc − p cl (2gkl − glk )tc Spp ab ˜i˜j ˜i˜ ˜ ˜ j ij mq ˜ i j m˜ ˜ n = δik δjl − Pˆkl mq rkl gmq + mn rkl gmn − ab rkl ga˜˜b kl cd mq mq kl lk kl mn = Sa˜k δlm − mq rmq r g + r g − ga mn a ˜ m ˜m ˜m − m˜ a mq mq mn cd rcd ga

+,+

' ( ) -6

ES $7 X' C B * & +S wˆ 12 = (1 − Pˆ1)(1 − Pˆ2)r12 Xkl,mn (ij) Ckl,ab (ij) Bkl,mn

pq mn pq mn pq mn = smn kl − pq (rkl rpq + rlk rpq ) + pq rkl rpq =0 = 12 (Tkl,mn + Tmn,kl ) + 12 (εk + εl + εm + εn − 2εi − 2εj )Xkl,mn + 12 (Qkl,mn + Qmn,kl ) − 12 (Pkl,mn + Pmn,kl )

smn kl trs pq Kqp pmn pq

2 = ϕk (1)ϕl (2)|r12 |ϕm (1)ϕn (2) ˆ = ϕp (1)ϕq (2)|[t1 + tˆ2 , r12 ]|ϕr (1)ϕs (2) = ϕq (1)|kˆ1 |ϕp (1) r mn = r (Kpr rrmn q + Kq rpr )

Tkl,mn Qkl,mn Pkl,mn

pq mn pq mn pq nm = δkm δnl − pq (rkl tpq + rlk tpq ) + pq rkl tpq m n = r (Xkl,r n Kr + Xkl,mr Kr ) pq mn p q mn pq mn p q mn = p q rkl pp q − pq rkl ppq − p q rkl pp q + pq rkl ppq

,S

ˆ 1 )(1 − O ˆ 2 )r12 w ˆ 12 = (1 − O

[2]

Xkl,mn [2] (ij) Ckl,ab

[2]

(ij)

Bkl,mn

Tkl,mn Qkl,mn Pkl,mn

ij mn iq mn iq mn = smn − (r r + r r ) + r r kl kl iq lk iq iq ij klp ijab p b a ab ab = −tkl + (εk + εl − εi − εj )rkl + p (Kk rp l − rkl Kp ) q ab aq b + q (Kl rkq − rkl Kq ) = 12 (Tkl,mn + Tmn,kl ) + 12 (εk + εl + εm + εn − 2εi − 2εj )Xkl,mn + 12 (Qkl,mn + Qmn,kl ) − 12 (Pkl,mn + Pmn,kl ) ij mn iq mn iq nm = δkm δnl − iq (rkl tiq + rlk tiq ) + ij rkl tij = r (Xkl,r n Krm + Xkl,mr Krn ) iq mn ij mn p q mn p j mn = p q rkl pp q − iq rkl piq − p j rkl pp j + ij rkl pij ˆ 1 )(1 − O ˆ 2 )(1 − Vˆ1 Vˆ2 )r12 w ˆ 12 = (1 − O

Xkl,mn (ij) Ckl,ab (ij) Bkl,mn

+,,

= = =

ab mn Xkl,mn − ab rkl rab [2] (ij) ab Ckl,ab − (εa + εb − εi − εj )rkl (ij) (ij) [2] ab B − ab [2] Ckl,ab rmn − ab kl,mnab mn + ab rkl (εa + εb − εi − εj )rab [2]

[2]

(ij)

ab Cab,mn rkl

GS $7 ρ = AR %%,"

( # ρ ρai = 1ρGai + 1 H 2 H 2 I 1 F 2 E ρai + 1 ρIai + 1 ρJai + 2 ρG ai + ρai + ρai ρaibj = ρaibj + ρaibj Tˆ1 ϕ˜i ϕ˜a % "% "$ tμ # ( ϕ¯i ϕ¯a' - 2G+FA3 2G+FJ3 @ 1

1 G ρai 1 H ρai 1 I ρai 1 J ρai 1 F ρaibj

= cdk (2tik − tki )g dc cd kl cd lk k¯a d¯i = − dkl (2tad − tad )glk ki ¯ = ck (2tik ac − tac )fkc = f¯ai ¯˜ ˜¯ ˜˜ ˜˜ = ga˜ij˜b + ga˜ij˜b + ga¯ij˜b + ga˜ij¯b

f¯kc f¯ai fˆab fˆji

cb = jb 2gkj − g cb )Rj ˆ i jk ˆ b i ˜ ˜ = b fab Rb − j fji Ra + kc (2ga˜ick − ga˜cik )Rck bc k = εa δab + kc (2ga˜bck − gk˜ a )tc ˜ib ˜ib k = εi δji + kb (2gjk − gkj )tb

2 G ρai 2 H ρai 2 I ρai 2 E ρaibj

ik ki dc = cdk (2Rcd − Rcd )gk˜a kl lk d˜i = − dkl (2Rad − Rad )glk ik ki ˆ = ck (2Rac − Rac )fkc ij = Rab (εa + εb − εi − εj )

+,E

' ( ) -6

AS $7 *+,"& ρ = AR %%,(*+,"

# ρ ( 1 H 2 H ρai = ρCC2 + 1 ρG ρai + 1 ρIai + 2 ρG ρai + 2 ρIai ai ai + ai + 2 E 2 E ρaibj = ρCC2 ρaibj ρkilj = 1 ρFkilj + 2 ρE ρkilj ( aibj + kilj + ˆ T1 ϕ˜i ϕ˜a % " % "$ tμ # ϕ¯i ϕ¯a ' - 2G+FA3 2G+FJ3 @ ( @ . (V † )kla˜m ' gˆ˜lkip gˆp k @

2 ,3 +S wˆ 12 = (1 − Pˆ1)(1 − Pˆ2)r12

1

1 G ρai 1 F ρkilj

† kl = klm c˜im ¯m kl (V )a = V¯klij

c˜im kl ¯ Vklij

mi im mi ˜ im R = 2cim = 2Rkl − Rkl kl − ckl kl ¯i˜ ˜i¯ ˜i¯ pq pq ¯i˜ j j j j = −Pˆklij pq rkl (gpq + gpq ) + pq rkl (gpq + gpq )

,S

ˆ 1 )(1 − O ˆ 2 )r12 w ˆ 12 = (1 − O

1 G ρai 1 H ρai 1 I ρai 1 F ρkilj

= klm c˜im (V † )kl a ¯m kl lk mn ¯ip = − mnkl c˜mn p rap ˆlk g mn = kmn c˜ik ¯p k mn p rap g ij = V¯

V¯klij g¯p k

˜i¯ j j j mq ¯ ˜i˜ mq ˜ ¯i˜ j j ¯ n ˜i˜ m˜ ˜ n ˜i¯ ˆ kl (rm˜ = −Pˆijkl mq [rkl gmq + rkl (gmq + gmq )] + Pij mn kl gmn + rkl gmn ) pc pc jc jc = cl (2gkl − glk )Rcl − j cl (2gkl − glk )Rcl Sjp

kl

2 G ρai 2 E ρkilj

2 G ρai 2 H ρai 2 I ρai 2 E ρaibj 2 E ρkilj 2 E ρkilj

˜ im (V † )kl = klm R a ˜m kl (ij) ij = mn Bkl,mn Rmn

˜ im (V † )kl = klm R a ˜m kl ˜ lk mn ˜ip = − mnkl Rmn p rap ˆlk g mn ik ˜ mn = kmn R ˆp k p rap g (ij) ij = kl Cab,kl Rkl (ij) ij = ab Ckl,ab Rab (ij) ij = mn Bkl,mn Rmn

ˆ 1 )(1 − O ˆ 2 )(1 − Vˆ1 Vˆ2 )r12 w ˆ 12 = (1 − O V¯klij g¯p k

+,G

˜i¯ j j j mq ¯ ˜i˜ mq ˜ ¯i˜ j j ¯ n ˜i˜ m˜ ˜ n ˜i¯ ˆ kl (rm˜ = −Pˆijkl mq [rkl gmq + rkl (gmq + gmq )] + Pij mn kl gmn + rkl gmn ) ¯ ˜ ˜ ˜ ab ij (ga˜˜b + ga¯ij˜b ) −Pˆijkl ab rkl pc p c pc pc = cl (2gkl − glk )Rcl − p cl (2gkl − glk )Rcl Spp

JS @ Tˆ1 C 7 L t1 R1 $#

5 @ .7 α # # Λp Λpαi Λpαa ¯p Λ ¯p Λ αi ¯p Λ αa

= C(1 − tT1 ) = Cαi = Cαa − j Cαj tja = −CR1T =0 = − j Cαj Raj = − j Λpαj Raj

Λh Λhαi Λhαa ¯h Λ ¯ hαi Λ ¯h Λ αa

= C(1 + t1 ) = Cαi + b Cαb tib = Cαa = CR 1 = b Cαb Rbi = b Λhαb Rbi =0

+,A

' ( ) -6

+,J

! 5 .

+,B

## 2

• •

. (V †)klαm = Sαk δlm ! δ δ % ". gαγβδ = (αβ|γδ) gαγβδ = (αβ|γδ) (αβ|mδ) = γ Cγm(αβ|γδ) δ ! (αβ|mδ) Iβ,αm δ δ ∗ ? Rβ,kl δ † kl δ 3 3 ∗ (V † )kl αm = (V )αm + β Iβ,αm Rβ,kl → O(n N ) δ ∗ (αβ|mδ ) = γ Cγm (αβ|γδ ) δ ∗ ! (αβ|mδ ) Iβ,αm δ ∗ ? Rβ,kl δ † kl δ 3 2 ∗ (V † )kl αm = (V )αm − β Iβ,αm Rβ,kl → O(n N N ) ∗ (mβ|γδ ) = α Cαm (αβ|γδ ) δ ∗ ! (mβ|γδ ) Iβ,γm δ δ ∗ ! Rβ,kl Rβ,lk δ † kl δ 3 2 ∗ (V † )kl γm = (V )γm − β Iβ,γm Rβ,lk → O(n N N ) $ ! δ δ

•

%+S !

(V †)klαm * +

+,D

• •

. Ωai = 0' Ωaibj = 0 Ωkilj = V˜klij = δik δlj ! δ % ". gαδβγ Iα,δ ˜i˜j = βγ ΛhβiΛhγj gαδβγ δ δ δ ? Rα,kl Ωkilj = Ωkilj + α Rα,kl Iα,˜i˜j → O(n4 N 2 ) ΩFaibj = Λpδb α ΛpαaIα,δ ˜i˜j Ωaibj = Ωaibj + ΩFaibj

Ω Ω Ω G ai

• •

H ai

ai

H = Ωai + ΩG ai + Ωai

$ ! δ ! δ % ". gαδβγ Iα,δ ˜i˜j = βγ ΛhβiΛhγj gαδβγ δ ? Rα,kl

Ωkilj = Ωkilj − 2

• • • •

α

δ δ Rα,kl Iα, → O(n4 N N ) ˜i˜ j

$ ! δ ΩJai ΩIai Ωai = Ωai + ΩJai + ΩIai ΩEaibj Ωaibj = Ωaibj + ΩEaibj

(ij) 6 ΩEkilj = mn Bkl,mn cij mn → O(n ) Ωkilj = Ωkilj + ΩEkilj ΩGai ? (V †)klαm (V †)kla˜m = α Λpαa(V †)klαm → O(n3V N ) mi † kl 4 ΩGai = klm(2cim ˜m → O(n V ) kl − ckl )(V )a

•

•

Ωai = Ωai + ΩGai

%,S !

*+,"& %%,(*+,"- Ωμ *( +

& Ωμ @

2+3 2,3 i

i

+,K

## 2

•

! 4 1ρFaibj ρaibj = ρaibj + 1ρFaibj 2ρGai 2ρHai ρai = ρai + 2ρGai + 2ρHai 2ρEaibj ρaibj = ρaibj + 2ρEaibj ? (V †)klim 1ρGai = − i Ria(V †)klim → O(n4V ) ρai = ρai + 1ρGai 1ρHai 2ρIai ρai = ρai + 1ρHai + 2ρIai ? Vklμ˜j 1ρFkilj = Pˆklij μ Λ¯ hμiVklμ˜j → O(n4N ) ρkilj = ρkilj + 1ρFkilj (ij) ij 2 ρEkilj = mn Bkl,mn Rmn → O(n6 ) ρkilj = ρkilj + 2 ρEkilj 2 ρGai = klm R˜klim(V †)kla˜m → O(n4V ) ρai = ρai + 2 ρGai 1ρIai ρai = ρai + 1ρIai $ ! 4

•

%ES !

*+,"& ρ = AR * +

%%,(*+,"& ρ @

2G3 2A3

+EF

•

. (V †)klαm = Sαk δlm ! δ δ δ ! (αβ|mδ) Iβ,αm δ δ ab ∗ ? Rβ,kl = ab Cαa Cγb rkl δ † kl δ 3 3 ∗ (V † )kl αm = (V )αm − β Iβ,αm Rβ,kl → O(n N ) δ δ ∗ ! (αβ|mδ ) Iβ,αm δ δ ∗ Ii,αm = β Cβi Iβ,αm → O(n2 N 2 N ) δ δ δ ∗ ? Rβ,kl Ri,kl = β Cβi Rβ,kl → O(n3 N N ) δ † kl δ 4 ∗ (V † )kl αm = (V )αm − i Ii,αm Ri,kl → O(n N N ) δ ∗ ! (mβ|γδ ) Iβ,γm δ δ ∗ Ii,γm = β Cβi Iβ,γm → O(n2 N 2 N ) δ δ ∗ ! Ri,kl Ri,lk δ † kl δ 4 ∗ (V † )kl γm = (V )γm − i Ii,γm Ri,lk → O(n N N ) $ ! δ δ ? (V †)klαm kl kl ? rαβ rijkl = αβ CαiCβj rαβ

•

ij ij ? gαβ gαm =

•

(V †)klαm = (V †)klαm +

• •

• •

β

ij

ij Cβm gαβ

kl ij rij gαm → O(n5 N )

%GS !

(V †)klαm * ,

+E+

## 2

• •

. Ωai = 0' Ωaibj = 0 Ωkilj = V˜klij = δik δlj ! δ % ". gαδβγ Iα,δ ˜i˜j = βγ ΛhβiΛhγj gαδβγ ΩFaibj = Λpδb α ΛpαaIα,δ ˜i˜j Ωaibj = Ωaibj + ΩFaibj

Ω Ω Ω G ai

H ai

ai

H = Ωai + ΩG ai + Ωai

•

$ ! δ ? rklab Ωkilj = Ωkilj − ab rklabg˜a˜i˜j˜b

•

˜n ? rklαβ rklm˜ =

•

αβ i˜ j ? gmn g˜mn =

•

• •

αβ

αβ

→ O(n4 V 2 )

αβ Λhαm Λhβn rkl → O(n3 N (N + n)) αβ Λhαi Λhβj gmn → O(n3 N (N + n))

˜ n ˜i˜ j Ωkilj = Ωkilj + mn rklm˜ gmn → O(n6 ) ! δ % ". gαδβγ Iα,δ ˜i˜j = βγ ΛhβiΛhγj gαδβγ → O(nN 2N (N + n)) Im,δ ˜i˜j = α CαmIα,δ ˜i˜j → O(n3N N ) δ δ δ ? Rα,kl Rm,kl = α Λhαm Rα,kl → O(n3 N N ) ˜ δ δ Ωkilj = Ωkilj − 2 m Rm,kl → O(n5 N ) ˜ Im,˜i˜ j $ ! δ ΩJai ΩIai Ωai = Ωai + ΩJai + ΩIai ΩEaibj Ωaibj = Ωaibj + ΩEaibj

• • •

%AS !

%%,(*+,"-( * ,

& Ωμ @

2+3 2,3 i

+E,

•

ΩEkilj & 9# 1& C 1& S ab ∗ ? . kkl ab ij 4 2 ∗ ΩE kilj (A) = − ab kkl tab → O(n V ) 1& X S ab ∗ ? . kkl ab ∗ ? 5 εμ . rkl ab ab ab ∗ Xkl = (εa + εb − εk − εl )rkl + kkl

i

∗ ΩE kilj (A ) = −

ab

ab ij Xkl tab → O(n4 V 2 )

1& S ∗ ∗ ∗ ∗ • •

? . kklab Kklab Mklab = Kklab − kklab ? 5 εμ . rklab Xklab = (εa + εb − εk − εl )rklab + Mklab ΩEkilj (B) = − ab Xklabtijab → O(n4V 2) i

Ωkilj = Ωkilj + ΩEkilj ΩEaibj & 9# 1& C 1& S ab ∗ ? . kkl *+," cklij ab kl 4 2 ∗ ΩE aibj (A) = − kl kkl cij → O(n V ) 1& X S ab ∗ ? kkl ' rklab' *+," cklij 5 εμ ab ab ab ∗ Xkl = (εa + εb − εk − εl )rkl + kkl ab kl 4 2 ∗ ΩE aibj (A ) = − kl Xkl cij → O(n V ) 1& S ab ∗ ? . kkl Kklab Mklab = Kklab − kklab ab ∗ ? 5 εμ ' rkl *+," cklij ab ab ∗ Xkl = (εa + εb − εk − εl )rkl + Mklab ab kl 4 2 ∗ ΩE aibj (B) = − kl Xkl cij → O(n V )

i

i

•

Ωaibj = Ωaibj + ΩEaibj

%JS : 2%A3S . kklab Kklab p ab ab ab kkl = ϕk (1)ϕl (2)|[tˆ1 + tˆ2 , r12 ]|ϕa (1)ϕb (2) Kkl = p (Kk rp l − pb a aq ab Kp ) + q (Klq rkq − rkl Kqb ) @ @ # rkl . 7 C @

2E3

+EE

## 2

•

ΩHai ? clkmn c˜lkmn = 2clkmn − cklmn ! δ δˇ δ ∗ ? Iα,kl Rα,mn δˇ δˇ ∗ Ii,kl = α Λhαi Iα,kl → O(n3 N N ) δˇ δi 5 = kl c˜lk ∗ Mmn mn Ii,kl → O(n N ) δi δ Rα,mn → O(n3 N N ) ∗ Ωαi = − mn Mmn $ ! δ ΩHai = α CαaΩαi Ωai = Ωai + ΩHai ΩIai tlα = c Λhαctlc ! δ δˇ δˇ δˇ ∗ ? Iα,kl Fδ k = α tlα(2Iα,kl − Iα,lk )

• •

→ O(n2 N N ) δ αk δ ? Rα,mn Mmn = Rα,mn Fδ k $ ! δ ? cikmn c˜ikmn = 2cikmn − ckimn ak αk Mmn = α Cαa Mmn → O(n3 N V ) ak ΩIai = kmn c˜ikmnMmn → O(n3 N V ) Ωai = Ωai + ΩIai ΩEkilj ΩGai +

∗

→ O(n3 N N )

•

%BS : 2%A3 2%J3S !

%%,(*+,"- * ,

+EG

! 4 1ρFaibj = g¯abij ρaibj = ρaibj + 1ρFaibj 2ρGai 2ρHai ρai = ρai + 2ρGai + 2ρHai ? rklab ρkilj = − ab rklabg¯abij → O(n4V 2) 2ρEaibj ρaibj = ρaibj + 2ρEaibj ? (V †)klim 1ρGai = − i Ria(V †)klim → O(n4V ) ρai = ρai + 1ρGai 1ρHai 2ρIai ρai = ρai + 1ρHai + 2ρIai ¯ n ˜i˜ j ρkilj = ρkilj + mn rklm˜ gmn → O(n6 ) i˜ j ρkilj = ρkilj − Pˆklij mδ rklmδ¯ g˜mδ → O(n5 N ) ? Vklμ˜j ρkilj = ρkilj + Pˆklij μ Λ¯ hμiVklμ˜j → O(n4N ) ρkilj = 1ρFkilj = V¯klij ij ij 2ρEkilj = ab Ckl,ab Rab → O(n4 V 2 ) ρkilj = ρkilj + 2 ρEkilj ij ij 2 ρEaibj = kl Ckl,ab Rkl → O(n4 V 2 ) ρaibj = ρaibj + 2 ρEaibj (ij) ij 2 ρEkilj = mn Bkl,mn Rmn → O(n6 ) ρkilj = ρkilj + 2 ρEkilj 2 ρGai = klm R˜klim(V †)kla˜m → O(n4V ) ρai = ρai + 2 ρGai mn ˜ip lk 2 ρHai = − mnkl R˜mn ˆlk → O(n3 N N + n5 N ) p rap g ρai = ρai + 2 ρHai mn ik 2 ρIai = kmn R˜mn ˆp k → O(n3 N V ) p rap g ρai = ρai + 2 ρIai 1ρHai = − mnkl c˜lkmn p rapmngˆ¯lkip → O(n3N N + n5N ) ρai = ρai + 1ρHai 1ρIai = kmn c˜ikmn p rapmng¯p k → O(n3N V ) ρai = ρai + 1ρIai 1ρIai ρai = ρai + 1ρIai • $ ! 4 %DS !

*+,"& ρ = AR * ,

%%,(*+,"& ρ @

2G3 2A3 •

+EA

## 2

+EJ

( % ! 5 0

+EB

!

# ## 9

+S %%,(*+,"- & + , 1& $ # ,F+B+G++DA 5 " 7 # %%, + , (+GJEBJEK (+GJGDDJF (+GJGDDJJ (+GJGAGK, (+GJGKJAE (+GJGKJJG (+GJGBBJB (+GJGKBGK (+GJGKBJ+ (+GJGDGFG (+GJGKBB, (+GJGKBBK (+GJGDJFJ (+GJGKBBA (+GJGKBD, & ? (+GJGKD

,S %%,(*+,"- 8 1 2, ← ,3 + , 1& $ # ,F+B+G++DA 5 " 7 # %%, + , (+GGEEKKD (+GGEKAGB (+GGEKAAF (+GGAG+,A (+GGAJFDK (+GGAJFKE (+GGADJD+ (+GGAKAKK (+GGAKJFE (+GGJFF,F (+GGJFBD+ (+GGJFBBD (+GGJFGDD (+GGJ+,G, (+GGJ+,EK & ? (+GGJ++

ES %%,(*+,"- 1 & + , 1& $ # ,F+G++KBAE 5 " E,,G+D+A+,KJ 7 # %%, + , (+,DBG+BKK (+,DDBDDBA (+,DDDF,,B (+,DDB+GDB (+,DKFBFGA (+,DKE+,DB (+,DKFDBDD (+,DK,GFKD (+,DKEJ+B+ (+,DK,,K,F (+,DKE+JD, (+,DKEJJJE (+,DK,D,F+ (+,DKEGJJB (+,DKEJBA+ (+,DKEFEB+ (+,DKEJ,ED (+,DKEJBB+ & ? (+,DKEG+ +ED

GS %%,(*+,"- 8 1 2E ← ,3 1 + , 1& $ # ( ,F+G++KBAE 5 " E,,G+D+A+,KJ 7 # %%, + , (+,D+JFDBD (+,D,JABDD (+,D,JBA,+ (+,D,D,EKD (+,DE+FAGD (+,DEEFGFF (+,DE+ABGD (+,DE,DFAK (+,DEEBGGA (+,DE,BFEF (+,DEEG,FA (+,DEED+FE (+,DEE+,EE (+,DEEJADG (+,DEED,J, (+,DEE,KEB (+,DEEBDEG (+,DEED,KK & ? (+,DEEAD

AS %%,(*+,"- & + , 1& $ # 7 ( # %%, + , ((4 8 (,A+DKFJF (,A+KD+KF (,A,F+JBA ((48 (,A,FGK,J (,A,FKF+, (,A,FKK++ ((4C8 (,A,+FEKG (,A,+,ADJ (,A,+,DEJ ((4A8 (,A,+,,+D (,A,+EJFK (,A,+EJBJ ((4J8 (,A,+,KBE (,A,+G,AE (,A,+EKG, & ? (,A,+GF

JS %%,(*+," 1Σ+"8 & + , 1& $ # 7 # %%, + , ((4 8 (,GKAG+E+ (,GKAKAED (,GKJ,,KE ((48 (,GKJD+,F (,GKBFGBJ (,GKB+G++ ((4C8 (,GKB,BKG (,GKBGFGE (,GKBG,JB ((4A8 (,GKBGGDA (,GKBA,JD (,GKBAEFK ((4J8 (,GKBA+GD (,GKBADKE (,GKBAJD+ & ? (,GKBJ+ +EK

!

# ## 9

BS %%,(*+," 1Π"8 & + , 1& $ # 7 # %%, + , ((4 8 (,AFDEBGB (,AFDDAFB (,AFKF+GG ((48 (,A+FFBDE (,A+F,DBD (,A+FEEDJ ((4C8 (,A+FJJGK (,A+FBBB+ (,A+FBDKJ ((4A8 (,A+FDJ,G (,A+FKE,J (,A+FKEG+ ((4J8 (,A+FKGGA (,A++F++J (,A+FKK+A & ? (,A++FJ

DS %%,(*+,"- : & + , 1& $ # 7 ( # %%, + , ((4 8 (+,GEDGJ,A (+,GG,KEJD (+,GGAGJ++ ((48 (+,GGK+JGA (+,GA++A+F (+,GA,,BEK ((4C8 (+,GA,A,,F (+,GAEAABF (+,GAG++E+ ((4A8 (+,GAEBABB (+,GAGGD,G (+,GAGJDGF ((4J8 (+,GAG,EDK (+,GAGDA++ & ? (+,GAGK

KS %%,(*+," 1Σ+"8 : & + , 1& $ # 7 ( # %%, + , ((4 8 (+,GFD,D,K (+,G+,GFEJ (+,G+GDFG, ((48 (+,G+DBBDK (+,G,FAKKJ (+,G,+B+,K ((4C8 (+,G,,++GK (+,G,EFABF (+,G,EJ+EJ ((4A8 (+,G,EEJEF (+,G,GF,JG (+,G,G,,D+ ((4J8 (+,G,EKEFE (+,G,GGKG, & ? (+,G,GB +GF

+FS %%,(*+," % 1Σ+"8 : ( & + , 1& $ # 7 # %%, + , ((4 8 (+,GFJBK+B (+,G+FKE+B (+,G+EEAGA ((48 (+,G+BEABF (+,G+K+BJB (+,G,F,KBG ((4C8 (+,G,FBJGE (+,G,+BFGA (+,G,,,JFF ((4A8 (+,G,,FEKB (+,G,,BF+D (+,G,,KF+D ((4J8 (+,G,,J,+K (+,G,E+DEB & ? (+,G,EG

++S %%,(*+," 1Π"8 : & + , 1& $ # 7 ( # %%, + , ((4 8 (+,G+GGKDJ (+,G+DBFGF (+,G,FKDK+ ((48 (+,G,AGEGE (+,G,BE,+F (+,G,DEBDE ((4C8 (+,G,DDJJ+ (+,G,KDGE, (+,GEFEDDE ((4A8 (+,GEF+EBF (+,GEFD,J, (+,GE+F,EA ((4J8 (+,GEFJE,J (+,GE+,,E, & ? (+,GE+E

+,S %%,(*+,"- %5 & + , 1& $ # 7 # %%, + , ((4 8 (++EFJFKK+ (++E+FFD+B (++E+,GKB+ ((48 (++E+AFFDG (++E+JDAAG (++E+BD,ED ((4C8 (++E+DFBJG (++E+KFE,B (++E+KAE+A ((4A8 (++E+K+DEA (++E+KDADK (++E,FFEGK ((4J8 (++E+KJ,GD (++E,F+KFK & ? (++E,F, +G+

!

# ## 9

+ES %%,(*+," 1Σ+"8 %5 ( & + , 1& $ # 7 # %%, + , ((4 8 (++,JAEAKD (++,JDDBA, (++,B+FDJE ((48 (++,BG,FAK (++,BADE+G (++,BJBGGE ((4C8 (++,BBGFA, (++,BD,G,E (++,BDB+F, ((4A8 (++,BDBE,A (++,BKE,GE (++,BKGDK+ ((4J8 (++,BKE,DE (++,BKD,AK & ? (++,DF+

+GS %%,(*+," % 1Σ+"8 %5 ( & + , 1& $ # 7 # %%, + , ((4 8 (++,JEEDFF (++,JJDDED (++,JK+FDA ((48 (++,B,,KDK (++,BEK+,A (++,BGD,K+ ((4C8 (++,BAGJJD (++,BJ,KJB (++,BJBJG+ ((4A8 (++,BJBADB (++,BBEGJB (++,BBA+FB ((4J8 (++,BB,KE+ (++,BBBDBB & ? (++,BDF

+AS %%,(*+," 1Π"8 %5 & + , 1& $ # 7 ( # %%, + , ((4 8 (++,BEDJEG (++,BBGKJG (++,BKAEGA ((48 (++,DEFDGG (++,DGBK++ (++,DAJ,JG ((4C8 (++,DJ,,FB (++,DB+FBD (++,DBAJGE ((4A8 (++,DBEGKB (++,DBKBKD (++,DD+GF+ ((4J8 (++,DBBKKE (++,DDEE,J & ? (++,DDG +G,

+JS %%,(*+,"- 12 & + , 1& $ # 7 # %%, + , ((4 8 (+FK,DE++K (+FKE,++JA (+FKEGAGJJ ((48 (+FKEBFBJJ (+FKEDDEGF (+FKEKBAJK ((4C8 (+FKGFF+,E (+FKGFKFF+ (+FKG+EK,B ((4A8 (+FKG+FJJB (+FKG+BFBF (+FKG+DDF+ ((4J8 (+FKG+GDJK (+FKG,FE,+ (+FKG,F+J, & ? (+FKG,F

+BS %%,(*+," 1Πg "8 12 & + , 1& $ # 7 ( # %%, + , ((4 8 (+FDKE+GGB (+FDKJAFJK (+FDKDGAF+ ((48 (+FKF,,EEB (+FKFED+AK (+FKFGAJ+G ((4C8 (+FKFA,,,J (+FKFJF,DA (+FKFJGAJF ((4A8 (+FKFJ,D,, (+FKFJDJAG (+FKFBF+E, ((4J8 (+FKFJBFA, (+FKFB,FGK (+FKFB+DED & ? (+FKFBE

+DS %%,(*+," X 1Σ−u "8 12 ( & + , 1& $ # 7 # %%, + , ((4 8 (+FDDKBBDE (+FDKEFGJB (+FDKAFFDA ((48 (+FDKDKG,G (+FKFFGDGF (+FKF+,,FD ((4C8 (+FKF+DKD, (+FKF,JBBB (+FKFE+F+K ((4A8 (+FKF,KGBJ (+FKFEAFBF (+FKFEJAJ+ ((4J8 (+FKFEEJGA (+FKFEDGA, (+FKFED,DJ & ? (+FKFEK +GE

!

# ## 9

+KS %%,(*+," # 1Δu"8 12 ( & + , 1& $ # 7 # %%, + , ((4 8 (+FDDBDEF, (+FDK+,BKA (+FDKEFKKB ((48 (+FDKJKA,J (+FDKDAD,D (+FDKK,JKG ((4C8 (+FDKKK,ED (+FKFFBJEK (+FKF++AAG ((4A8 (+FKFFKD,G (+FKF+ABAG (+FKF+B+FD ((4J8 (+FKF+GFGK (+FKF+KFDD (+FKF+DDGF & ? (+FKF,F

+S %%," %%,(*+,"$ -" ( 1Σ+"8 :

+GG

,S %%," %%,(*+,"$ -" ( % 1Σ+"8 :

ES %%," %%,(*+,"$ -" ( 1Π"8 : +GA

!

# ## 9

GS %%," %%,(*+,"$ -" ( 1Σ+"8 %5

AS %%," %%,(*+,"$ -" ( % 1Σ+"8 %5 +GJ

JS %%," %%,(*+,"$ -" ( 1Π"8 %5

BS %%," %%,(*+,"$ -" ( 1Πg "8 12 +GB

!

# ## 9

DS %%," %%,(*+,"$ -" ( X 1Σ−u "8 12

KS %%," %%,(*+,"$ -" ( # 1Δu"8 12 +GD

,FS : 4 & + , 1& 7 # # 1 Σ+ % 1 Σ+ 1Π %%, + , %%, + , %%, + , ((4 8 D,+, DEFK DEG, DJ+D DBFK DBEB JA,+ JAKG JJAK ((48 D,JD DE+E DE+J DJAA DBF+ DBF+ JGAB JGDG JAF, ((4C8 D,BG D,KK D,KK DJG, DJJD DJJD JGEB JGAE JGAJ ((4A8 D,B+ D,DD D,DB DJE+ DJGD DJGD JG,D JGEB JGED ((4J8 D,GB D,J+ DJFE DJ+B JG,G JG,K & ? JG+K

,+S %5 4 & + , 1& 7 # # 1 Σ+ % 1 Σ+ 1Π %%, + , %%, + , %%, + , ((4 8 ++FDJ ++,+E ++,JD ++J,G ++BAA ++DFB DBB, DDJB DKBF ((48 +++FE +++JE +++BD ++J,, ++JDA ++JKK DJDB DB,A DBJ+ ((4C8 ++FJB +++FF +++FD ++AKA ++J,K ++JED DJJD DJDB DJKK ((4A8 ++FFB ++FEF ++FEE ++AGG ++AJD ++AB+ DJJ, DJBA DJBK ((4J8 +FKJA +FKDG ++A+K ++AEK DJJF DJJK & ? +FKFB ++GDA DJAB +GK

+AF

# ## 9 !

,,S 12 4 & + , 1& 7 # # X 1Σ−u 1Πg # 1 Δu %%, + , %%, + , %%, + , ((4 8 +FGDJ +FJEF +FBAK KAJK KJKF KD,, ++F+J +++FD ++,BD ((48 +FEBB +FGEB +FGDJ KGD+ KA,K KABB +FK+D +FKAA ++F+B ((4C8 +FEB+ +FEKK +FG+K KGJB KGDK KAFB +FKFK +FK,, +FKGK ((4A8 +FEBE +FEKA +FGF+ KGJA KGD+ KGDD +FKFB +FK,F +FKE+ ((4J8 +FEBG +FEK+ +FEK+ KGJA KGBB KGBD +FKFB +FK+D +FK,+ & ? +FEBA KGJA +FKFB

5.80 5.70

Energie in eV

5.60

CC2 Ansatz 2 Limit

5.50 5.40 5.30 5.20 5.10 5.00 2

3

4

5

6

Kardinalzahl der Basis

+FS %%, %%,(*+, 4

16.80 CC2 Ansatz 1 Ansatz 2 Limit

Energie in eV

16.60 16.40 16.20 16.00 15.80 2

3

4

5

6

7

Kardinalzahl der Basis

++S %%, %%,(*+, 1 4 +A+

!

# ## 9

6.52 6.50 Energie in eV

6.48 6.46 6.44

CC2 Ansatz 1 Ansatz 2 Limit

6.42 6.40 6.38 D

T

Q

5

6

Kardinalzahl der aug-cc-pVXZ Basis

+,S %%, %%,(*+, R 4

1

Σ+ "

3.05

Energie in eV

3.00

CC2 Ansatz 1 Ansatz 2 Limit

2.95 2.90 2.85 2.80 D

T

Q

5

6

Kardinalzahl der aug-cc-pVXZ Basis

+ES %%, %%,(*+, R 4 +A,

1

Π"

8.36 CC2 Ansatz 1 Ansatz 2 "Limit"

8.34

Energie in eV

8.32 8.30 8.28 8.26 8.24 8.22 8.20 D

T

Q

5

6

Kardinalzahl der aug-cc-pVXZ Basis

+GS %%, %%,(*+, R : 4

1

Σ+ "

1

Σ+ "

8.74 CC2 Ansatz 1 Ansatz 2 "Limit"

Energie in eV

8.72 8.70 8.68 8.66 8.64 8.62 8.60 8.58 D

T

Q

5

6

Kardinalzahl der aug-cc-pVXZ Basis

+AS %%, %%,(*+, % R : 4

+AE

!

# ## 9

6.70 CC2 Ansatz 1 Ansatz 2 "Limit"

Energie in eV

6.65 6.60 6.55 6.50 6.45 6.40 D

T

Q

5

6

Kardinalzahl der aug-cc-pVXZ Basis

+JS %%, %%,(*+, R : 4

1

Π"

11.30 CC2 Ansatz 1 Ansatz 2 "Limit"

11.25

Energie in eV

11.20 11.15 11.10 11.05 11.00 10.95 10.90 10.85 D

T

Q

5

6

Kardinalzahl der aug-cc-pVXZ Basis

+BS %%, %%,(*+, R %5 4 +AG

1

Σ+ "

11.85 11.80

CC2 Ansatz 1 Ansatz 2 "Limit"

Energie in eV

11.75 11.70 11.65 11.60 11.55 11.50 11.45 D

T

Q

5

6

Kardinalzahl der aug-cc-pVXZ Basis

+DS %%, %%,(*+, % R %5 4

1

Σ+ "

9.00 8.95 Energie in eV

8.90

CC2 Ansatz 1 Ansatz 2 Limit

8.85 8.80 8.75 8.70 8.65 D

T

Q

5

6

Kardinalzahl der aug-cc-pVXZ Basis

+KS %%, %%,(*+, R %5 4

1

Π"

+AA

!

# ## 9

,ES ; %%,(*+,"& 4 4

%%,

˜ˆ ˆ R1 |[H, R1 ]|HF

%%, %%,(*+, GKD+ GKBD GK,E GK,+ GK,E GK,, GK,G GK,G GK,A GK,A

ˆ Tˆ2 + Tˆ2 ], R ˆ 1 ]|HF R1 |[[H,

%%, BEA DEG DJA DBD DDE

%%,(*+, DK, DKG DKA DKA DKA

˜ˆ ˆ ˆ 2 ]|HF R1 |[H, R2 + R

%%, (,,G (JB+ (BBE (DFB (D,+

%%,(*+, (,,A (JB, (BBE (DFB (D,+

EJS *+,"& 4 7

2 ,E3 (

; %%,(*+," ˜ˆ ˆ ˆ Tˆ2 ], R ˆ 1 ]|HF R1|[[H, R1 |[H, R2 ]|HF BEJ (,,A DEG (JB, DJJ (BBE DBD (DFB DDE (D,+ ˜ ˆ Tˆ2 ], R ˆ 1 ]|HF ˆ R ˆ 2 ]|HF R1|[[H, R1 |[H, +AJ F JF F ,K F +J F ++ F

+AJ

,GS ; %%,(*+,"& 4 1 4

%%,

˜ˆ ˆ R1 |[H, R1 ]|HF

%%, +BKAG +BBB+ +BB,B +BB++ +BBFJ +BBFA

+ +BDG+ +BBGF +BB+G +BBFG +BBFF +BJKK

, +BDJ+ +BB,E +BBFG +BBFF +BJKK +BJKD

ˆ Tˆ2 + Tˆ2 ], R ˆ 1 ]|HF R1 |[[H,

%%, ++EB ,FAG ,,BG ,EJD ,GF, ,G+B

+ ,FDK ,,BB ,EJA ,G+J ,GEB ,GGB

, ,FBJ ,G+F ,GGA ,GGD ,GGK ,GAF

˜ˆ ˆ ˆ 2 ]|HF R1 |[H, R2 + R

%%, (EKJ, (GAE, (GJ+, (GJ+J (GJ+D (GJ+K

+ (EKBF (GAEE (GJ+, (GJ+B (GJ+K (GJ+D

, (EKK, (GAEA (GJ+E (GJ+B (GJ+K (GJ+D

+AB

((4 8 ((48 ((4C8 ((4A8 ((4J8

˜ˆ ˆ R1 |[H, R1 ]|HF

%%, J,KA J,+G J+DD J+B+ J+JG

+ J,D+ J,FB J+DG J+JK J+J,

, J,JD J,F+ J+D, J+JD J+J,

ˆ Tˆ2 + Tˆ2 ], R ˆ 1 ]|HF R1 |[[H,

%%, DDG +FGG +FKK ++,F ++EF

+ +FF+ +FKK ++,K ++EK ++GJ

, +FGF ++FB ++E, ++GF ++GG

˜ˆ ˆ ˆ 2 ]|HF R1 |[H, R2 + R

%%, (KBF (+FFG (+F+G (+F+A (+F+A

+ (KBJ (+FFJ (+F+A (+F+J (+F+J

, (KDE (+FFK (+F+J (+F+J (+F+J

,JS ; %%,(*+,"& 4 1Π" 4

%%, ((4 8 ((48 ((4C8 ((4A8 ((4J8

˜ˆ ˆ R1 |[H, R1 ]|HF

%%, ,DFF ,DFE ,DFE ,DF, ,DF,

+ ,BKD ,DF, ,DF, ,DF, ,DF,

, ,BKK ,DF, ,DF, ,DF+ ,DF,

ˆ Tˆ2 + Tˆ2 ], R ˆ 1 ]|HF R1 |[[H,

%%, +F+, ++D+ +,EK +,J+ +,B+

+ ++EE +,EB +,JD +,DF +,DB

, ++D+ +,GB +,B, +,D, +,DA

˜ˆ ˆ ˆ 2 ]|HF R1 |[H, R2 + R

%%, (KBF (+,+J (+,DG (+E+F (+E,,

+ (+FFK (+,+B (+,DA (+E+F (+E,,

, (+FFK (+,+B (+,DA (+E+F (+E,,

+AD

# ## 9 !

,AS ; %%,(*+,"& 4 1Σ+" 4

%%,

,BS ; %%,(*+,"& 4 1Σ+" : 4

%%, ((4 8 ((48 ((4C8 ((4A8 ((4J8

˜ˆ ˆ R1 |[H, R1 ]|HF

%%, D,EJ D+BF D+,, DFKD DFGD

+ D,,K D+JJ D+,F DFKJ DFGB

, D,+G D+J+ D++D DFKJ

ˆ Tˆ2 + Tˆ2 ], R ˆ 1 ]|HF R1 |[[H,

%%, +F,F ++DE +,EK +,J+ +,JB

+ ++,B +,EA +,JD +,BK +,D+

, ++BE +,G+ +,JK +,DF

˜ˆ ˆ ˆ 2 ]|HF R1 |[H, R2 + R

%%, (+EFB (+EAD (+EJ+ (+EAD (+EGF

+ (+E+G (+EJE (+EJG (+EJ+ (+EG,

, (+E+G (+EJF (+EJE (+EJF

,DS ; %%,(*+,"& 4 % 1Σ+" : 4

%%, ((4 8 ((48 ((4C8 ((4A8 ((4J8

˜ˆ ˆ R1 |[H, R1 ]|HF

%%, DGJF DEA, D,BJ D,EA D+KB

+ DGGA DEGA D,BE D,EE D+KJ

, DG,D DEGF D,B+ D,E,

ˆ Tˆ2 + Tˆ2 ], R ˆ 1 ]|HF R1 |[[H,

%%, KKF ++A+ +,FB +,,K +,JB

+ +FKJ +,FE +,EJ +,GD +,AA

, ++GE +,FK +,EB +,GD

˜ˆ ˆ ˆ 2 ]|HF R1 |[H, R2 + R

%%, (+FGG (+FBJ (+FBD (+FBD (+FDA

+ (+FGJ (+FBB (+FBK (+FBK (+FDG

, (+FA, (+FBB (+FBD (+FBK

,KS ; %%,(*+,"& 4 1Π" : 4

%%,

+AK

((4 8 ((48 ((4C8 ((4A8 ((4J8

˜ˆ ˆ R1 |[H, R1 ]|HF

%%, JEDE JEBG JEJD JEJA JEJA

+ JEBB JEBF JEJJ JEJE JEJG

, JEBK JEB, JEJB JEJA

ˆ Tˆ2 + Tˆ2 ], R ˆ 1 ]|HF R1 |[[H,

%%, +ED, +AAD +J,F +JGA +JAJ

+ +GKF +J++ +JAF +JJE +JB+

, +AGB +J,F +JA, +JJA

˜ˆ ˆ ˆ 2 ]|HF R1 |[H, R2 + R

%%, (+G,G (+JJA (+BG, (+BBG (+BDD

+ (+GAK (+JDK (+BAA (+BDE (+BKD

, (+GA, (+JDF (+BAG (+BDE

((4 8 ((48 ((4C8 ((4A8 ((4J8

˜ˆ ˆ R1 |[H, R1 ]|HF

%%, ++BE, ++AGA ++G+K ++EFJ ++,EJ

+ ++B+F ++AEA ++G+G ++EFE ++,EG

, ++JDG ++A,G ++G+F ++EF,

ˆ Tˆ2 + Tˆ2 ], R ˆ 1 ]|HF R1 |[[H,

%%, +EG+ +ADK +JJA +JDD +JKA

+ +GKA +JJ, +BFE +B+G +B+J

, +AB+ +JDG +B+A +B+D

˜ˆ ˆ ˆ 2 ]|HF R1 |[H, R2 + R

%%, (,A+J (,AB, (,AAJ (,A,+ (,GKK

+ (,A,B (,ABD (,AJF (,A,E (,AFF

, (,A,J (,ABG (,AAK (,A,,

E+S ; %%,(*+,"& 4 % 1Σ+" %5 4

%%, ((4 8 ((48 ((4C8 ((4A8 ((4J8

˜ˆ ˆ R1 |[H, R1 ]|HF

%%, ++KDA ++D+K ++BEG ++JJK ++JGE

+ ++KJ+ ++DFK ++BEF ++JJJ ++JGF

, ++KEE ++BKB ++B,A ++JJA

ˆ Tˆ2 + Tˆ2 ], R ˆ 1 ]|HF R1 |[[H,

%%, +,BJ +A+A +AK, +J+K +JE,

+ +G,B +ADB +JEF +JGJ +JAE

, +AFE +JFK +JG+ +JGK

˜ˆ ˆ ˆ 2 ]|HF R1 |[H, R2 + R

%%, (,FDE (,+K, (,,E, (,,J, (,,D+

+ (,FDJ (,+KE (,,EE (,,JE (,,D+

, (,FDG (,+DK (,,E+ (,,J,

E,S ; %%,(*+,"& 4 1Π" %5 4

%%, ((4 8 ((48 ((4C8 ((4A8 ((4J8

˜ˆ ˆ R1 |[H, R1 ]|HF

%%, DB+E DJDE DJBE DJBF DJJK

+ DBFD DJD+ DJB+ DJJD DJJD

, DBFJ DJD+ DJB, DJJK

ˆ Tˆ2 + Tˆ2 ], R ˆ 1 ]|HF R1 |[[H,

%%, ,EAG ,JAJ ,BAJ ,BKG ,D++

+ ,A+K ,BEG ,BKJ ,D,, ,DEG

, ,J,, ,BJA ,D+F ,D,B

˜ˆ ˆ ˆ 2 ]|HF R1 |[H, R2 + R

%%, (,JEJ (EFF+ (E++F (E+A+ (E+JK

+ (,BFD (EFG+ (E+E+ (E+JB (E+DE

, (,B++ (EFEB (E+EA (E+JB

+JF

# ## 9 !

EFS ; %%,(*+,"& 4 1Σ+" %5 4

%%,

EES ; %%,(*+,"& 4 1Πg " 12 4

%%, ((4 8 ((48 ((4C8 ((4A8 ((4J8

˜ˆ ˆ R1 |[H, R1 ]|HF

%%, KJJK KJ,G KJ+E KJ++ KJ+F

+ KJJJ KJ,E KJ++ KJ+F KJ+F

, KJJJ KJ,G KJ+, KJ++ KJ++

ˆ Tˆ2 + Tˆ2 ], R ˆ 1 ]|HF R1 |[[H,

%%, ,DJ, E,GF EEJ+ EG+F EGEF

+ EFJB EEE, EGFD EGG, EGAD

, E,FJ EED, EGE+ EGA+ EGAB

˜ˆ ˆ ˆ 2 ]|HF R1 |[H, R2 + R

%%, (EGFA (EDEJ (EKJE (GF+, (GFEE

+ (EGKD (EDDG (EKDK (GFEF (GFGK

, (EA+A (EDK+ (EKKB (GFEE (GFGD

EGS ; %%,(*+,"& 4 X 1Σ−u " 12 4

%%, ((4 8 ((48 ((4C8 ((4A8 ((4J8

˜ˆ ˆ R1 |[H, R1 ]|HF

%%, DEEE D,BK D,JA D,J+ D,JF

+ DE,E D,BA D,JE D,AK D,AD

, DE+G D,B, D,J, D,AK D,AK

ˆ Tˆ2 + Tˆ2 ], R ˆ 1 ]|HF R1 |[[H,

%%, ED,B G,+B GEGD GEKK GG,,

+ GFEE GEFK GEK+ GGEE GGAF

, G,FB GEB, GG,+ GGGE GGGK

˜ˆ ˆ ˆ 2 ]|HF R1 |[H, R2 + R

%%, (+DAB (,E,, (,GGB (,GKG (,A+A

+ (+K+E (,EAG (,GJ+ (,AFA (,A,A

, (+KAJ (,EJG (,GB+ (,AFK (,A,A

EAS ; %%,(*+,"& 4 # 1Δu" 12 4

%%,

+J+

((4 8 ((48 ((4C8 ((4A8 ((4J8

˜ˆ ˆ R1 |[H, R1 ]|HF

%%, DDJD DDFJ DBKF DBDJ DBDG

+ DDAG DDF+ DBDD DBDE DBDE

, DDGJ DBKB DBDB DBDG DBDE

ˆ Tˆ2 + Tˆ2 ], R ˆ 1 ]|HF R1 |[[H,

%%, EDJD G,A+ GED+ GGEE GGAA

+ GFBJ GEGG GG,A GGJJ GGDE

, G,GE GGFA GGAG GGBB GGDE

˜ˆ ˆ ˆ 2 ]|HF R1 |[H, R2 + R

%%, (+KAA (,EKF (,A+B (,AJJ (,ADD

+ (,FAK (,GGA (,AGJ (,ADJ (,JFA

, (,FAB (,GGF (,AGD (,ADJ (,JF,

!

# ## 9

EBS *+,"& 4 7

2 ,G3 (

; %%,(*+," 1

ˆ Tˆ2 ], R ˆ 1 ]|HF R1 |[[H,

˜ˆ ˆ R1 |[H, R2 ]|HF

ˆ Tˆ2 ], R ˆ 1 ]|HF R1 |[[H,

˜ˆ ˆ R1 |[H, R2 ]|HF

+ ++EF ,FA, ,,BE ,EJB ,GF, ,G+J + KAK ,,A K, GK EA E+

, ++E, ,FA, ,,BE ,EJB ,GF, ,G+B

, KGG EAD +B, D+ GB EE

+ (EKJA (GAE+ (GJ++ (GJ+J (GJ+D (GJ+D

, (EKBB (GAE, (GJ++ (GJ+J (GJ+D (GJ+D

+ (A (, (+ (+ (+ F

, (+A (E (, (+ (+ F

EDS *+,"& 4 7

2 ,A3 (

; 1Σ+"%%,(*+," ((4 8 ((48 ((4C8 ((4A8 ((4J8 ((4 8 ((48 ((4C8 ((4A8 ((4J8 +J,

ˆ Tˆ2 ], R ˆ 1 ]|HF R1 |[[H,

+ DDG +FGG +FKK ++,F ++EF

, DK+ +FGJ ++FF ++,F ++EF

ˆ Tˆ2 ], R ˆ 1 ]|HF R1 |[[H,

+ ++D AA EF +K +J

, +GK J+ E, ,F +G

˜ˆ ˆ R1 |[H, R2 ]|HF

+ (KB+ (+FFG (+F+G (+F+A (+F+J

, (KB, (+FFG (+F+G (+F+A (+F+A

+ (G (, (+ (+ F

, ++ (A (+ (+ F

˜ˆ ˆ R1 |[H, R2 ]|HF

EKS *+,"& 4 7

2 ,J3 (

; 1Π"%%,(*+," ((4 8 ((48 ((4C8 ((4A8 ((4J8 ((4 8 ((48 ((4C8 ((4A8 ((4J8

ˆ Tˆ2 ], R ˆ 1 ]|HF R1 |[[H,

+ +F++ ++D+ +,EK +,J+ +,B+

, +F,+ ++DE +,EK +,J+ +,B+

ˆ Tˆ2 ], R ˆ 1 ]|HF R1 |[[H,

+ +,+ AA EF +K +B

, +J+ JG EE ,+ +G

˜ˆ ˆ R1 |[H, R2 ]|HF

+ (+FFK (+,+B (+,DA (+E+F (+E,,

, (+FFK (+,+B (+,DA (+E+F (+E,,

+ F F F F F

, F F F F F

˜ˆ ˆ R1 |[H, R2 ]|HF

GFS *+,"& 4 7

2 ,B3 (

; 1Σ+"%%,(*+," : ((4 8 ((48 ((4C8 ((4A8 ((4J8 ((4 8 ((48 ((4C8 ((4A8 ((4J8

ˆ Tˆ2 ], R ˆ 1 ]|HF R1 |[[H,

+ +F+B ++D, +,ED +,JF +,JJ

, +F,B ++D+ +,EB +,JF

ˆ Tˆ2 ], R ˆ 1 ]|HF R1 |[[H,

+ ++F AE EF +K +A

, +GJ JF E, ,F

˜ˆ ˆ R1 |[H, R2 ]|HF

+ (+EFJ (+EAD (+EJ+ (+EAK (+EGF

, (+EFG (+EAA (+EJF (+EAD

+ (D (A (E (, (,

, (+F (A (E (,

˜ˆ ˆ R1 |[H, R2 ]|HF

+JE

!

# ## 9

G+S *+,"& 4 7

2 ,D3 (

; % 1Σ+"%%,(*+," : ((4 8 ((48 ((4C8 ((4A8 ((4J8 ((4 8 ((48 ((4C8 ((4A8 ((4J8

ˆ Tˆ2 ], R ˆ 1 ]|HF R1 |[[H,

+ KDB ++AF +,FB +,,K +,GF

, KKD ++AF +,FJ +,,D

ˆ Tˆ2 ], R ˆ 1 ]|HF R1 |[[H,

+ +FK AE ,K +K +A

, +GA AK E+ ,F

˜ˆ ˆ R1 |[H, R2 ]|HF

+ (+FEK (+FBA (+FBD (+FBD (+FDE

, (+FED (+FBE (+FBB (+FBD

+ (B (, (+ (+ (+

, (+G (G (+ (+

˜ˆ ˆ R1 |[H, R2 ]|HF

G,S *+,"& 4 7

2 ,K3 (

; 1Π"%%,(*+," : ((4 8 ((48 ((4C8 ((4A8 ((4J8 ((4 8 ((48 ((4C8 ((4A8 ((4J8 +JG

ˆ Tˆ2 ], R ˆ 1 ]|HF R1 |[[H,

+ +EBG +AAA +J+K +JGE +JAA

, +EDJ +AAG +J+D +JGE

ˆ Tˆ2 ], R ˆ 1 ]|HF R1 |[[H,

+ ++J AJ E+ ,F +J

, +J+ JJ EG ,,

˜ˆ ˆ R1 |[H, R2 ]|HF

+ (+G+D (+JJ, (+BG+ (+BBE (+BDD

, (+G+A (+JJF (+BGF (+BBE

+ (G+ (,B (+G (+F (+F

, (EB (,F (+G (+F

˜ˆ ˆ R1 |[H, R2 ]|HF

GES *+,"& 4 7

2 EF3 (

; 1Σ+"%%,(*+," %5 ((4 8 ((48 ((4C8 ((4A8 ((4J8 ((4 8 ((48 ((4C8 ((4A8 ((4J8

ˆ Tˆ2 ], R ˆ 1 ]|HF R1 |[[H,

+ +EED +ADD +JJG +JDB +JKG

, +EGJ +ADD +JJG +JDB

ˆ Tˆ2 ], R ˆ 1 ]|HF R1 |[[H,

+ +AB BG EK ,B ,,

, ,,A KJ A+ E+

˜ˆ ˆ R1 |[H, R2 ]|HF

+ (,A+A (,AB, (,AAB (,A,+ (,GKK

, (,AFK (,AJB (,AAA (,A,F

+ (+, (J (E (, (+

, (+B (B (G (,

˜ˆ ˆ R1 |[H, R2 ]|HF

GGS *+,"& 4 7

2 E+3 (

; % 1Σ+"%%,(*+," %5 ((4 8 ((48 ((4C8 ((4A8 ((4J8 ((4 8 ((48 ((4C8 ((4A8 ((4J8

ˆ Tˆ2 ], R ˆ 1 ]|HF R1 |[[H,

+ +,B+ +A+E +AK+ +J+K +JE+

, +,BK +A+G +AK+ +J+D

ˆ Tˆ2 ], R ˆ 1 ]|HF R1 |[[H,

+ +AJ BG EK ,B ,,

, ,,G KA AF E+

˜ˆ ˆ R1 |[H, R2 ]|HF

+ (,FD+ (,+K+ (,,E, (,,J, (,,D+

, (,FBE (,+DJ (,,EF (,,J+

+ (A (, (+ (+ F

, (++ (E (+ (+

˜ˆ ˆ R1 |[H, R2 ]|HF

+JA

!

# ## 9

GAS *+,"& 4 7

2 E,3 (

; 1Π"%%,(*+," %5 ((4 8 ((48 ((4C8 ((4A8 ((4J8 ((4 8 ((48 ((4C8 ((4A8 ((4J8

ˆ Tˆ2 ], R ˆ 1 ]|HF R1 |[[H,

+ ,EGJ ,JAE ,BAG ,BKE ,D+F

, ,EJ+ ,JAE ,BAE ,BK,

ˆ Tˆ2 ], R ˆ 1 ]|HF R1 |[[H,

+ +BE D+ G, ,K ,G

, ,J+ ++, AB EA

˜ˆ ˆ R1 |[H, R2 ]|HF

+ (,JEF (,KKD (E+FD (E+AF (E+JD

, (,J,K (,KKA (E+FB (E+GK

+ (BD (GE (,E (+B (+A

, (D, (G, (,D (+D

˜ˆ ˆ R1 |[H, R2 ]|HF

GJS *+,"& 4 7

2 EE3 (

; 1Πg "%%,(*+," 12 ((4 8 ((48 ((4C8 ((4A8 ((4J8 ((4 8 ((48 ((4C8 ((4A8 ((4J8 +JJ

ˆ Tˆ2 ], R ˆ 1 ]|HF R1 |[[H,

+ ,DJF E,EK EEJ+ EGFK EGEF

, ,DDF E,GA EEJ, EGFK EGEF

ˆ Tˆ2 ], R ˆ 1 ]|HF R1 |[[H,

+ ,FB KE GB EE ,D

, E,J +EB JK G, ,B

˜ˆ ˆ R1 |[H, R2 ]|HF

+ (EG+B (EDG+ (EKJA (GF+E (GFEG

, (EG,B (EDGJ (EKJD (GF+G (GFEA

+ (D+ (GE (,G (+B (+A

, (DD (GA (,K (+K (+E

˜ˆ ˆ R1 |[H, R2 ]|HF

GBS *+,"& 4 7

2 EG3 (

; X 1Σ−u "%%,(*+," 12 ((4 8 ((48 ((4C8 ((4A8 ((4J8 ((4 8 ((48 ((4C8 ((4A8 ((4J8

ˆ Tˆ2 ], R ˆ 1 ]|HF R1 |[[H,

+ ED,D G,+D GEGD GEKK GG,+

, EDJG G,,K GEA+ GGF+ GG,,

ˆ Tˆ2 ], R ˆ 1 ]|HF R1 |[[H,

+ ,FA K+ GE EG ,K

, EGE +GE BF G, ,B

˜ˆ ˆ R1 |[H, R2 ]|HF

+ (+DAB (,E,E (,GGD (,GKG (,A+A

, (+DAK (,E,G (,GGD (,GKA (,A+A

+ (AJ (E+ (+E (++ (+F

, (KB (GF (,E (+G (+F

˜ˆ ˆ R1 |[H, R2 ]|HF

GDS *+,"& 4 7

2 EA3 (

; # 1Δu"%%,(*+," 12 ((4 8 ((48 ((4C8 ((4A8 ((4J8 ((4 8 ((48 ((4C8 ((4A8 ((4J8

ˆ Tˆ2 ], R ˆ 1 ]|HF R1 |[[H,

+ EDB, G,AE GED, GGEE GGAA

, EKFJ G,JE GEDA GGEA GGAJ

ˆ Tˆ2 ], R ˆ 1 ]|HF R1 |[[H,

+ ,FG K+ GE EE ,D

, EEB +G, JK G, ,B

˜ˆ ˆ R1 |[H, R2 ]|HF

+ (+KAA (,EKF (,A+B (,AJJ (,ADD

, (+KAK (,EK+ (,A+B (,AJJ (,ADD

+ (+FG (AA (,K (,F (+B

, (KD (GK (E+ (,F (+G

˜ˆ ˆ R1 |[H, R2 ]|HF

+JB

!

+JD

# ## 9

% ! 5 1

+JK

+BF

# ## <

$+S - Eh $< Δ*+, Eh !%: , Δ*+,a %%! Δ2*+,3b Δ*+,c 1 ((4 8 (+,DGKJEAF (+,DBFE,,E (KGB+G (+,DBFJAFG (DGKBG (++F,AF ((48 (+,DAEE,BE (+,DDFABK, (E,KEG (+,DDFBEJD (,B+GB (EEBG+ ((4C8 (+,DAGEBAJ (+,DDGFKKK (+A,BE (+,DDG+A+G (++AAB (+,JBA ((4A8 (+,DAGJBDJ (+,DDAGBAA (DJAB (+,DDAEADG (J+K+ (JG+B ((4J8 (+,DAGBFJ, (+,DDAKKEA (AJJA (+,DDABJJK (EK+G (EDDJ − : ((4 8 (KKG,D,D, (KKJJAKGD (BFJFD (KKJJ,JKF (J+EAF (BBJBE ((48 (KKGAFDFB (KKBGADBK (,EAFF (KKBEKEF+ (+DGEB (,+KKD ((4C8 (KKGABGJ, (KKBBGGFB (+FG,B (KKBJAD+K (B,KJ (DJJJ ((4A8 (KKGAK,J+ (KKBDAAAB (J,KF (KKBBGDDK (GFD, (GGAA ((4J8 (KKGAKGEF (KKBDKKB+ (EBKD (KKBBDFJJ (,EJE (,AGE a Δ*+, U Δ*+,2,3 U ,(*+, " , b Δ2*+,3 U Δ*+,2%%! 2*+,33 U %%! 2*+,3 " %%! c Δ*+, U Δ*+,2%%! (*+,3 U %%! (*+, " %%!

$,S - Eh $< Δ*+, Eh !%: , Δ*+,a %%! Δ2*+,3b Δ*+,c ((4 8 (,A+,JG,B (,A+DDDFF (++DD, (,A,+J,GF (BDFK (+F+EF ((48 (,A+EF,F+ (,A,FGAKF (G+JA (,A,,KED+ (,+GE (,AK, ((4C8 (,A+E+EB+ (,A,+FFFG (,FFG (,A,EE+FE (FBDD (FD,A ((4A8 (,A+E+AKB (,A,++DFK (+,,G (,A,EGFAG (FG,J (FG,E ((4J8 (,A+E+JEF (,A,+,AAB (FD+A (,A,EGEJA (F,JA (F,AD : ((4 8 (+FFFEEGJJ (+FF,AABEB (BJJ+, (+FF,AKGGJ (JJGDA (DJEKF ((48 (+FFFJ+FJK (+FFEGFDJA (,JDKE (+FFEG,FGE (,FKJJ (,AE+A ((4C8 (+FFFJDAAK (+FFEJKBAD (+,EFD (+FFEJK+F+ (DJBJ (KGDF ((4A8 (+FFFBFABE (+FFEDFAAB (BFFG (+FFEBDFGF (GJGG (GBAG ((4J8 (+FFFBFBBF (+FFEDGBF+ (GAAF (+FFED+FDB (,KFB (,DDF (JFDD, (D,E+J 12 ((4 8 (+FDKJFJAF (+FK,+G,K+ (+EJ,DF (+FK,DFGJF ((48 (+FDKDGJKD (+FKEJG+JE (,GF+B (+FKEJ+ABG (+BEFJ (,,JGG ((4C8 (+FDKK+AAD (+FKEKE,DD (++,GG (+FKEDJBKE (JKAF (BJD, ((4A8 (+FDKK,KAK (+FKGFEBEK (JA,G (+FKEKGADJ (EBG+ (EDF+ ((4J8 (+FDKKE+EA (+FKGFBK+F (GGE, (+FKEKB,KB (,G,+ (,EKF %5 ((4 8 (++,BAGJK+ (++EFAEKJD (BG+A, (++EFJFKBK (J,EF+ (D+DKE ((48 (++,BD+GEE (++E+G,+B, (,J+,A (++E+GGA,F (+K+JK (,EEB, ((4C8 (++,BDKFEJ (++E+B,D,G (+,+BB (++E+B+J+F (BBB, (DEJ+ ((4A8 (++,BKFJAD (++E+DEDBB (BFAK (++E+DFFJF (G+BJ (G,FB ((4J8 (++,BKFDGD (++E+DD,KB (GJEF (++E+DEFFD (,J+G (,ADF a Δ*+, U Δ*+,2,3 U ,(*+, " , b Δ2*+,3 U Δ*+,2%%! 2*+,33 U %%! 2*+,3 " %%! c Δ*+, U Δ*+,2%%! (*+,3 U %%! (*+, " %%! +B+

+B,

# ## <

$ES - Eh $< Δ*+, Eh !%: , Δ*+,a %%! Δ2*+,3b Δ*+,c (BJFG+G,D (BJ,JFBJA (JFGKE (BJ,JDAEG (AFAAK (JAFAD 2 5 ((4 8 ((48 (BJFJFJ+E (BJE,DKAD (,FKK+ (BJEEEJJA (+AEJ, (+BDBB ((4C8 (BJFJJFF+ (BJEA+K+E (KJA+ (BJEAG,+G (J,BD (JJBF ((4A8 (BJFJBE,+ (BJEJF,,A (AG+, (BJEJFJGD (E,KA (EE+D ((4J8 (BJFJBGJ, (BJEJEG+B (EA+G (BJEJ,D+E (,FAD (,FE, 1 3 ((4 8 (AJ,FAGGF (AJGFGJKA (GJG+D (AJG+KB++ (EBEAB (GD,KG ((48 (AJ,,FEDG (AJGJFAEB (+AEJB (AJGB,,FJ (+FGGG (++DJG ((4C8 (AJ,,GFEA (AJGBBBB, (B+AJ (AJGDJBD+ (G,GJ (GGFA ((4A8 (AJ,,GK+B (AJGDED,A (GF,F (AJGKFE+G (EFG, (EFE, ((4J8 (AJ,,AF,, (AJGDJ+BA (,JGG (AJGK,FGG (+KAF (+K,, (GF+KKAKD (GFEJBE,B (EAAGB (GFEKFJ,J (,BGBB (EAKDF % 4 ((4 8 ((48 (GF,+EJJJ (GFG+GGAK (++,FJ (GFGEGEAB (BFA+ (D++K ((4C8 (GF,+J,KD (GFG,BGF, (A+GA (GFGGGB,, (,BDB (,KFD ((4A8 (GF,+BFFA (GFGE+DKJ (,KAJ (GFGGBBJA (+GDK (+GBG a,b,c

$+

$GS & - Tˆ1' Tˆ2 Tˆ2 Eh %%! 2*+,3 %%! (*+, ˆ ˆ ˆ ˆ ˆ E(T1 , T2 ) E(T2 ) Δ*+, E(T1 , T2 ) E(Tˆ2 ) Δ*+, ECCSD " ESCF 1 ((4 8 (,FGJKE (KFGEA (DGKBG (,F,K+F (++BGKG (++F,AF (,+F+AG ((48 (,B++FE (EF+ED (,B+GB (,BFEGJ (EBGDK (EEBG+ (,BGFKA ((4C8 (,KAK,G (+EEK+ (++AAB (,KABA, (+GJD+ (+,JBA (,KBBAD ((4A8 (EFAJ+G (BEBJ (J+K+ (EFAABE (BJGG (JG+B (EFJBKK ((4J8 (EFKBBB (GBGG (EK+G (EFKBDG (GBFK (EDDJ (E+FJFB − : ((4 8 (,,KF+K (JJBEK (J+EAF (,,BGGG (DGJEJ (BBJBE (,EGGFD ((48 (,DADE, (,+FKK (+DGEB (,DA,DG (,A,FK (,+KKD (,DDGKG ((4C8 (EFJD+B (DDEJ (B,KJ (EFJAEG (+FGKF (DJJJ (EFDEAB ((4A8 (E+GABA (A+EG (GFD, (E+GGD+ (AJF, (GGAA (E+AJ,D ((4J8 (E+BKJJ (EFE, (,EJE (E+BK+A (E,JG (,AGE (E+DJEJ

+BE

+BG

# ## <

$AS & - Tˆ1' Tˆ2 Tˆ2 Eh %%! 2*+,3 %%! (*+, ˆ ˆ ˆ ˆ ˆ E(T1 , T2 ) E(T2 ) Δ*+, E(T1 , T2 ) E(Tˆ2 ) Δ*+, ECCSD " ESCF ((4 8 (DBJBD (KKGG (BDFK (DB+FB (+,DEJ (+F+EF (DKD+E ((48 (KD,J, (EFJF (,+GE (KDFBJ (EJKJ (,AK, (KK+DF ((4C8 (+F+,EJ (+,DG (FBDD (+F+,+J (+EG+ (FD,A (+F+BEE ((4A8 (+F,+GJ (FBEB (FG,J (+F,+GD (FBE, (FG,E (+F,GAB ((4J8 (+F,,AD (FGBG (F,JA (+F,AE, (FGJ+ (F,AD (+F,BEA : ((4 8 (,,F,JG (B,,F+ (JJGDA (,+DEDE (KEKDB (DJEKF (,,AKDF ((48 (,BBK,K (,GF+, (,FKJJ (,BB,BJ (,KF+E (,AE+A (,DFKBG ((4C8 (,KDBBE (+FGGJ (DJBJ (,KDJ+G (++GFD (KGDF (EFFAG, ((4A8 (EFJEGD (ABJE (GJGG (EFJE,G (ADKB (GBAG (EFBGJB ((4J8 (EFKAGK (EJBG (,KFB (EFKAAB (EJEK (,DDF (E+FE+J %5 ((4 8 (,KK,EJ (JKEAE (J,EF+ (,KBF++ (K++BF (D+DKE (EFJ,DD ((48 (EAKGDB (,,BJK (+K+JK (EADB,G (,BBEA (,EEB, (EJEFDB ((4C8 (EDFGJA (KDD+ (BBB, (EDFE+B (+FJ+D (DEJ+ (ED,ABE ((4A8 (EDDFJD (AA+F (G+BJ (EDDFJ, (AAGB (G,FB (EDKGF, ((4J8 (EK+,GG (EA,K (,J+G (EK+,AK (EGDF (,ADF (EK,+AK 12 ((4 8 (E+,AGJ (JD+GJ (JFDD, (E+FFFB (K,++K (D,E+J (E+KD+F ((48 (EBEEJF (,FD,, (+BEFJ (EB,,DJ (,B,EG (,,JGG (EBJDBJ ((4C8 (EKE+JA (KF,F (JKAF (EK,KA, (KKJA (BJD, (EKA,EA ((4A8 (GFFEE+ (AFEB (EBG+ (GFFE+E (A++A (EDF+ (GF+J,B ((4J8 (GFE,GE (EEEK (,G,+ (GFE,AB (E,KG (,EKF (GFG+J+

$JS & - Tˆ1' Tˆ2 Tˆ2 Eh %%! 2*+,3 %%! (*+, ˆ ˆ ˆ ˆ ˆ E(T1 , T2 ) E(T2 ) Δ*+, E(T1 , T2 ) E(Tˆ2 ) Δ*+, ECCSD " ESCF (,,+GD+ (AJ+DG (AFAAK (,+KBEA (B,GEF (JAFAD (,,B+FJ 2 5 ((4 8 ((48 (,BF+K, (+D,,, (+AEJ, (,JKB+G (,+,+A (+BDBB (,BEFA, ((4C8 (,DJAKB (BDKA (J,BD (,DJAFF (DEDE (JJBF (,DD,+E ((4A8 (,K,EG+ (G,D+ (E,KA (,K,EEJ (GEFK (EE+D (,KEE,D ((4J8 (,KGJDE (,B,J (,FAD (,KGJK, (,JK+ (,FE, (,KAEAF (+DJAGD (E+KAB (,BGBB (+DA+,, (G+DDJ (EAKDF (+K+F,D % 4 ((4 8 ((48 (,+DJDG (KFAB (BFA+ (,+DEBA (+FGEA (D++K (,,FJK+ ((4C8 (,,BEAD (EDAE (,BDB (,,BE+A (GF+D (,KFD (,,DG,A ((4A8 (,EF++G (,+EG (+GDK (,EF+,, (,++, (+GBG (,EFBJF 1 3 ((4 8 (,FK+AG (G,GBG (EBEAB (,FBAJA (AGKKK (GD,KG (,+G,B+ ((48 (,GKEBF (+,DKA (+FGGG (,GKFEJ (+GJAF (++DJG (,A+D,, ((4C8 (,J+EDJ (AJFB (G,GJ (,J+EEK (AD+E (GGFA (,J,BGJ ((4A8 (,JAEKB (EFG, (EFG, (,JAGF, (EF,J (EFE, (,JAEKB ((4J8 (,JBF++ (+KJ+ (+KAF (,JBF,, (+K,, (+K,, (,JBF,,

+BA

# ## <

$BS - $h $< Eh $ # +K+GDJGE, 2KJGE, 3 5 " 7(

# !%: , Δ*+,a %%! Δ2*+,3b Δ*+,c : (+FFFBFBBK (+FFEDA,J+ (A,FD (+FFED+AEK (E,K+ (E,,J 5 (BJFJBEK+ (BJEJEAE+ (GEEF (BJEJ,KEJ (,A+A (,GAB 2 % 2 (+FFFBFBBK (EKFGKJBE (,+BB (EKFBFJEJ (FKJD (FKEK :2 (+KDBBEDFJ (+KKEBAGFA (KJ+B (+KKEJDB,B (JFBJ (AKJD (+FDKK,KEG (+FKGFDJDE (A,BK (+FKEKBBE, (,DBA (,DFG 12 % 4 (+KDBBEDFJ (GFGEE,+A (,DF, (GFGGDABA (+EJA (+E,E %1 (K,K+AABK (KE,KBA,+ (GBKG (KE,K,BF, (,GDG (,G+, 1% (K,KFF+AJ (KE,JK,,E (GBBF (KE,JKJFK (,GBA (,GFG 1 (K,K+AABK (AJGDAK+J (EGDF (AJGK,GBJ (+DGK (+DF, %5 (++,BKFJEJ (++E+DK+,G (AA+A (++E+DEAFF (EFK, (EF+B %52 (+DBB,AFGK (+DDGF+J+, (KAEB (+DDEBDB,+ (AAAB (AG,K (++,BKFJEJ (BB+KJEK+ (G,AF (BB,FFFKD (,FBG (,FFD %2 2 a Δ*+, U Δ*+,2,3 U ,(*+, " , b Δ2*+,3 U Δ*+,2%%! 2*+,33 U %%! 2*+,3 " %%! c Δ*+, U Δ*+,2%%! (*+,3 U %%! (*+, " %%!

$DS - $h $< Eh $ # +K+GDJGE, 2KJGE, 3 5 " 7 # %%! MN Δ2*+,3a Δ*+,b %%! 23 Δ2*+,3c Δ*+,d : (+FFEKFJFA (E,AE (E+DK (+FFEKF,JB (E,AK (E+KG (BJEB,KKD (,GBB (,G,+ (BJEB,BFE (,GDF (,G,G 25 % 2 (EKFBJEJA (FKGB (FK+K (EKFBJ,AA (FKGB (FK+K :2 (+KKEK,+KB (AKKF (ADDG (+KKEK+EEE (AKKJ (ADKF 12 (+FKG+KBKK (,D+F (,BG+ (+FKG+DDKA (,DFK (,BGF (GFGAJFFB (+EGG (+EFG (GFGAABJE (+EG, (+EF, % 4 %1 (KEE+EBDK (,G,F (,EA+ (KEE+,K,D (,G,F (,EA+ 1% (KE,KF+GB (,G+, (,EGE (KE,DK,+E (,G+G (,EGA 1 (AJAF+KJ+ (+D+K (+BBE (AJAF+B+, (+D+D (+BB, %5 (++E,FGEJG (EF,, (,KGK (++E,F,KJA (EFE+ (,KAD %52 (+DDG+ED,+ (AGGF (AE+J (+DDG++J+E (AGAE (AE,D (BB,+K+B+ (,F+D (+KAG (BB,+DAFB (,F+B (+KAE %2 2 a Δ2*+,3 U Δ*+,2%%! MN2*+,33 U %%! MN2*+,3 " %%! MN b Δ*+, U Δ*+,2%%! MN(*+,3 U %%! MN(*+, " %%! MN c Δ2*+,3 U Δ*+,2%%! 232*+,33 U %%! 232*+,3 " %%! 23 d Δ*+, U Δ*+,2%%! 23(*+,3 U %%! 23(*+, " %%! 23 +BJ

$KS Δ*+,"& Eh 4# # +K+GDJGE, 2KJGE, 3 7 Δ*+,2,3a Δ*+,2%%! 2*+,33b 1 ((4 8 (GABAE (EDJ+D ((48 (,FG,E (+ABDG ((4C8 (+FFA+ (BE,J ((4A8 (BAA+ (AEFA ((4J8 (JEJE (GEBJ :− ((4 8 (EAFFD (,BE,G ((48 (+GJJA (+F,K+ ((4C8 (D,,E (AG+J ((4A8 (AAK+ (EAED ((4J8 (GG++ (,B,G a,b

$+

+BB

# ## <

$+FS Δ*+,"& Eh 4# # +K+GDJGE, 2KJGE, 3 7 Δ*+,2,3a Δ*+,2%%! 2*+,33b ((4 8 (K+G+ (G,A+ ((48 (GFBB (+AJE ((4C8 (,+DE (FBGD ((4A8 (+EDA (FGAE ((4J8 (+,BK (FG+J : ((4 8 (EDGFK (EFD+A ((48 (+B,,K (+,AFE ((4C8 (DJEF (ADAE ((4A8 (J,GK (GFJJ ((4J8 (AGGJ (EGEJ 12 ((4 8 (+F+A,F (,BDFK ((48 (+BA,B (++,GG ((4C8 (DDBA (A,FB ((4A8 (JGFF (EAKJ ((4J8 (AGAA (,KK+ %5 ((4 8 (GF,JG (,KBKA ((48 (+DJ,B (+,,,E ((4C8 (KJAB (ADEE ((4A8 (JD,F (EKGB ((4J8 (ABGJ (E,GD (E+JDD (,EKJD 2 5 ((4 8 ((48 (+G+BA (KAA+ ((4C8 (B,EB (GA,+ ((4A8 (AFFF (,KDD ((4J8 (GAAF (,JFD (,JGAK (+DBGB 1 3 ((4 8 ((48 (++GJG (BFDA ((4C8 (ADAG (EEE+ (,EFAB (+A,GD % 4 ((4 8 ((48 (KEFD (A,G+ ((4C8 (GBA+ (,GJF a,b

$+

+BD

Berechnete Energie in milli Hartree

100 MP2-R12 CCSD(R12) CCSD-R12

10

D

T Q Kardinalzahl der aug-cc-pVXZ Basis

5

6

Berechnete Energie in milli Hartree

$+S Δ*+,"& :− Eh'

(

!

MP2-R12 CCSD(R12) CCSD-R12

10

1

0.1 D

T Q Kardinalzahl der aug-cc-pVXZ Basis

5

6

$,S Δ*+,"& Eh'

(

! +BK

# ## <

Berechnete Energie in milli Hartree

100 MP2-R12 CCSD(R12) CCSD-R12

10

D

T Q Kardinalzahl der aug-cc-pVXZ Basis

5

6

$ES Δ*+,"& %5 Eh'

(

!

Berechnete Energie in milli Hartree

MP2-R12 CCSD(R12) CCSD-R12

10

1 D

T Q Kardinalzahl der aug-cc-pVXZ Basis

$GS Δ*+,"& 1 ( ! +DF

3

5

6

Eh'

Berechnete Energie in milli Hartree

MP2-R12 CCSD(R12) CCSD-R12

10

1 D

T Kardinalzahl der aug-cc-pVXZ Basis

$AS Δ*+,"& % ( !

4

Q

5

Eh'

100

Berechnete Energie in milli Hartree

MP2-R12 CCSD(R12) CCSD-R12

10

D

T Q Kardinalzahl der aug-cc-pVXZ Basis

$JS Δ*+,"& ( !

2

5

6

5 Eh'

+D+

MP2-R12/aux MP2-R12 CCSD(R12)/aux CCSD(R12) CCSD-R12

100 Berechnete Energie in milli Hartree

# ## <

10

D

T Q Kardinalzahl der aug-cc-pVXZ Basis

5

6

$BS Δ*+,"& :− +K+GDJGE, ( 7 Eh'

( !

Berechnete Energie in milli Hartree

100 MP2-R12/aux MP2-R12 CCSD(R12)/aux CCSD(R12) CCSD-R12

10

D

T Q Kardinalzahl der aug-cc-pVXZ Basis

5

6

$DS Δ*+,"& %5 +K+GDJGE, ( 7 Eh'

( ! +D,

Berechnete Energie in milli Hartree

MP2-R12/aux MP2-R12 CCSD(R12)/aux CCSD(R12) CCSD-R12

10

1

D

T Q Kardinalzahl der aug-cc-pVXZ Basis

5

6

Berechnete Energie in milli Hartree

$KS Δ*+,"& +K+GDJGE, 2KJGE, 3 7 Eh'

(

!

MP2-R12/aux MP2-R12 CCSD(R12)/aux CCSD(R12) CCSD-R12

10

1

D

T Q Kardinalzahl der aug-cc-pVXZ Basis

5

6

$+FS Δ*+,"& 25 +K+GDJGE, 2KJGE, 3 7 Eh'

(

! +DE

+DG

# ## <

$++S !%:"' ,(*+,"' %%! 2*+,3" %%! 232*+,3"- + 1&( Eh 4

,"' %%! " %%! 23"* 5 # ((42P38 7 0

((42AP38 # !%: , ,(*+, %%! %%! 2*+,3 %%! 23 %%! 232*+,3 (++EEFGK (++JAF,E (++JJJBB (++B,J+A (++BEEJF (++B,J+A (++BEEJF 2 (BJFJF,D, (BJE,DKK, (BJEG+D,E (BJEEEJAB (BJEG,,JJ (BJEG,E,J (BJEAFDEJ 25 %5 (++,BBKDDK (++E+G,G++ (++E+ADK+J (++E+GG,DK (++E+AAFGF (++E+J,+BA (++E+B,B,K % 4 (GF,+EJAD (GFG+GGAK (GFG,EGFB (GFGEGEJ+ (GFGEKEDD (GFGGFDKA (GFGGADGB (+DBBFJDBA (+DDE,+JG+ (+DDEAFGKB (+DDE+F,GA (+DDE,KDF+ (+DDEGFAAG (+DDEAKBJB %52 !52 (AGBEF,,DJ (AGBKBKDFE (AGDF++EED (AGBKJKEEE (AGBKKF+EB (AGDFFJ++K (AGDF,JGD+ (+AFDEDDFA (+A+EGBKDB (+A+EB+G+E (+A+EAGFG+ (+A+EJKBDB (+A+EBG,J+ (+A+EDKBD, 2 52 !53 (J,,+JJDJ, (J,EFDE,,D (J,E+,B+KG (J,EFJAGF, (J,EFKA+JE (J,E++,+,D (J,E+G+E,K (+JKJGBKAB (+JKJABFF, (+JKJBA+JJ %51 2 (+JKFFGAGB (+JKJ,FBBF (+JKJGK+,K (+JKJ,KGD+ % 2%5 (+A+BDAGBF (+A,EEGJFE (+A,EADKED (+A,EG+AEA (+A,EABFDG (+A,EJKFDB (+A,EDGEEJ %2 45 (+A,K,D+JG (+AEA+GFGA (+AEAGF,DD (+AEAE++F, (+AEAGBJAJ (+AEAAJAD, (+AEAB,DJE (++AFK,FEK (++AA,KFFD (++AAGKAK+ (++AAGJGDB (++AAAKJAG (++AAJ,EAB (++AABAEGB % 35

Tabelle E.12.: Berechnete SCF–, MP2-R12–, CCSD(R12)– und CCSD(T)(R12)–Grundzustandsenergien mit Ansatz 1 und N¨aherung B in in Eh im Vergleich zu konventionellen MP2–, CCSD– und CCSD(T)–Rechnungen. Als Orbitalbasis wurde eine aug-cc-pV(T+d)Z Basis und als Auxiliarbasis eine v¨ollig dekontrahierte aug-cc-pV(5+d)Z Basis verwendet. Molek¨ ul H2 S NH3 H2 CO CS2 HNO3 C2 H2 C2 H4 C2 H6 COCl2 Cl2 CH3 CHO

SCF MP2 MP2-R12 CCSD -398.715993 -398.911014 -398.921152 -398.934920 -56.220324 -56.460541 -56.471485 -56.472209 -113.913144 -114.316410 -114.334927 -114.325357 -832.971927 -833.475430 -833.499278 -833.486658 -279.561077 -280.514329 -280.556591 -280.496843 -76.849103 -77.164057 -77.176963 -77.175090 -78.064694 -78.404529 -78.418577 -78.428217 -79.260379 -79.635368 -79.651677 -79.665951 -1031.804169 -1032.585458 -1032.628427 -1032.607359 -919.002357 -919.390544 -919.416517 -919.423204 -152.974287 -153.552446 -153.578813 -153.571232

CCSD(R12) CCSD(T) -398.940526 -398.943591 -56.478963 -56.480543 -114.337294 -114.342922 -833.500464 -833.523291 -280.525555 -280.545011 -77.182581 -77.192183 -78.436299 -78.443714 -79.675291 -79.679886 -1032.634851 -1032.648634 -919.439204 -919.442727 -153.587855 -153.596596

CCSD(T)(R12) -398.949085 -56.487197 -114.354665 -833.536636 -280.573207 -77.199483 -78.451637 -79.689073 -1032.675566 -919.458393 -153.612936

185

Molek¨ ul H2 H2 O CO CH4 CO2 SO2 H2 O2 SO3 HCONH2 CH2 CO C2 H4 O CH3 OH

SCF -1.133499 -76.065649 -112.787390 -40.216289 -187.719442 -547.316925 -150.848728 -622.187562 -169.015402 -151.795017 -152.937493 -115.099534

MP2 -1.166739 -76.351907 -113.172893 -40.427399 -188.373789 -548.039627 -151.391022 -623.166383 -169.669316 -152.377023 -153.557910 -115.563094

MP2-R12 -1.167521 -76.359404 -113.182859 -40.432036 -188.390969 -548.059340 -151.405089 -623.193694 -169.685621 -152.391138 -153.572843 -115.574704

CCSD -1.173854 -76.354162 -113.171183 -40.444724 -188.357091 -548.023161 -151.392474 -623.141078 -169.671923 -152.378274 -153.568710 -115.576062

CCSD(R12) -1.174175 -76.358869 -113.177217 -40.447132 -188.367922 -548.035233 -151.401312 -623.158230 -169.681784 -152.386651 -153.577498 -115.583007

CCSD(T) -1.173854 -76.363576 -113.190262 -40.451718 -188.389399 -548.062663 -151.414289 -623.191338 -169.701352 -152.407591 -153.596009 -115.593170

CCSD(T)(R12) -1.174175 -76.368223 -113.196179 -40.454088 -188.400028 -548.074456 -151.422991 -623.208138 -169.711034 -152.415796 -153.604641 -115.600014

186

Anhang E. Tabellen und Abbildungen zu Kapitel 9

Tabelle E.13.: Berechnete SCF–, MP2-R12–, CCSD(R12)– und CCSD(T)(R12)–Grundzustandsenergien mit Ansatz 1 und N¨aherung B in in Eh im Vergleich zu konventionellen MP2–, CCSD– und CCSD(T)–Rechnungen. Als Orbitalbasis wurde eine aug-cc-pV(Q+d)Z Basis und als Auxiliarbasis eine v¨ollig dekontrahierte aug-cc-pV(5+d)Z Basis verwendet.

Tabelle E.14.: Berechnete SCF–, MP2-R12–, CCSD(R12)– und CCSD(T)(R12)–Grundzustandsenergien mit Ansatz 1 und N¨aherung B in in Eh im Vergleich zu konventionellen MP2–, CCSD– und CCSD(T)–Rechnungen. Als Orbitalbasis wurde eine aug-cc-pV(Q+d)Z Basis und als Auxiliarbasis eine v¨ollig dekontrahierte aug-cc-pV(5+d)Z Basis verwendet. Molek¨ ul H2 S NH3 H2 CO CS2 HNO3 C2 H2 C2 H4 C2 H6 COCl2 Cl2 CH3 CHO

SCF MP2 MP2-R12 CCSD -398.719393 -398.927952 -398.933604 -398.949371 -56.223971 -56.477766 -56.483660 -56.486775 -113.920514 -114.348647 -114.359480 -114.353525 -832.980361 -833.517090 -833.530956 -833.522171 -279.579953 -280.592867 -280.618256 -280.567141 -76.853475 -77.185720 -77.193077 -77.192837 -78.069387 -78.427679 -78.435454 -78.447094 -79.265305 -79.659733 -79.668387 -79.685633 -1031.818254 -1032.657463 -1032.683039 -1032.672632 -919.009096 -919.431713 -919.447480 -919.461168 -152.983904 -153.596179 -153.611149 -153.608807

CCSD(R12) CCSD(T) -398.952267 -398.958971 -56.490152 -56.495730 -114.360044 -114.372338 -833.529550 -833.561389 -280.583240 -280.618440 -77.196744 -77.210940 -78.451223 -78.463579 -79.690227 -79.700523 -1032.687764 -1032.717768 -919.470130 -919.483312 -153.617610 -153.635922

CCSD(T)(R12) -398.961804 -56.499053 -114.378742 -833.568510 -280.634232 -77.214740 -78.467618 -79.705036 -1032.732570 -919.492071 -153.644564

187

Anhang E. Tabellen und Abbildungen zu Kapitel 9

Tabelle E.15.: Berechnete SCF–, MP2–, CCSD– und CCSD(T)–Grundzustandsenergien in einer aug-cc-pV(5+d)Z Basis in in Eh . Molek¨ ul H2 H2 O CO CH4 CO2 SO2 H2 O2 SO3 HCONH2 CH2 CO C2 H4 O CH3 OH H2 S NH3 H2 CO CS2 HNO3 C2 H2 C2 H4 C2 H6 COCl2 Cl2 CH3 CHO

188

SCF MP2 CCSD CCSD(T) -1.133637 -1.167366 -1.174241 -1.174241 -76.066965 -76.360211 -76.360587 -76.370281 -112.789001 -113.183909 -113.179589 -113.199138 -40.216996 -40.431893 -40.447765 -40.454897 -187.722229 -188.392734 -188.371839 -188.404923 -547.320762 -548.062805 -548.041211 -548.081757 -150.851223 -151.406905 -151.404812 -151.427239 -622.193063 -623.198320 -623.166405 -623.218034 -169.017847 -169.686647 -169.684933 -169.715051 -151.797166 -152.392105 -152.389476 -152.419427 -152.939741 -153.573642 -153.580322 -153.608257 -115.101404 -115.575327 -115.585209 -115.602751 -398.720057 -398.934533 -398.953753 -398.963687 -56.224852 -56.483816 -56.491120 -56.500282 -113.922239 -114.360286 -114.362299 -114.381574 -832.981847 -833.533400 -833.533309 -833.573505 -279.584155 -280.621247 -280.589243 -280.641734 -76.854438 -77.193201 -77.197955 -77.216398 -78.070486 -78.435578 -78.452501 -78.469301 -79.266545 -79.668200 -79.691414 -79.706596 -1031.820947 -1032.684673 -1032.692657 -1032.739238 -919.010376 -919.448117 -919.472969 -919.496110 -152.986149 -153.611786 -153.620326 -153.648056

$+JS 4 # 2ΔECalc − ΔEExp 3 # 2((42P383 $7 )W 1 Δ, Δ,(*+, Δ%%! Δ%%! 2*+,3 Δ%%! 23 Δ%%! 232*+,3 + +KB ,FB (,,F (,+E (AG (GK , (,,G (,EB +FB DD FJ (+E E (JGE (JKF +GJ KE DA EE G GJ JF (DB (DE (GA (G, A (,D (EB (+G (,J (FJ (+B J (+B+ (+DB (AD (BB +D (F, B (,, (EE (GB (JF (F, (+A D E+ EE (+, (++ ++ +, K (DB (+FE (JK (D, (,D (G+ +F (BE+ (D+G (FD (K, +,J G+ ++ (+BE (,,+ AB FK KK A+ +, (,DE (E,K +F, AD BK EJ +E FA FD (BA (BF (,K (,A +G (BD (+F, GK ,G +A (+F +A GK E, (EE (GJ (,, (EJ

$+BS : ; ((42P38 1

)W , %%! %%! 23 ,(*+, %%! 2*+,3 %%! 232*+,3 : (+G+ (++ +B (+J+ (EE (FA : ,J+ B, G, ,FJ JK ,D ! # ,AE KE AA ,BB DF EE 7 : BE+ ,,F +,J D+G ,+E A+ +DK

$+KS : ; ((42CP38 1

)W , %%! %%! 23 ,(*+, %%! 2*+,3 %%! 232*+,3 : (+J, (GF (++ (+JD (GD (+K : ,+J JJ +B ,,A JB ,E ! # ,KJ BB +B EFB BB ,F 7 : DAF +KK G+ DDE +KA AE +KF

# ## <

$+DS 4 # 2ΔECalc − ΔEExp3 # 2((42CP383 $7 )W 1 Δ, Δ,(*+, Δ%%! Δ%%! 2*+,3 Δ%%! 23 Δ%%! 232*+,3 + ,EA ,GA (+KK (+KA (,D (,G , (,EG (,EF KJ KE (FJ (+F E (BG+ (BJF ,B FE (EF (AE G A, J, (DG (D+ (G+ (EK A (,K (E+ (+A (+K (FD (+, J (,FF (,FE (KE (+FF (+A (,, B (,J (,A (A+ (AG (FJ (+F D EJ G+ (FB (FG +B ,F K (BG (BJ (AB (J+ (+A (+K +F (DAF (DDE (+JF (+KG (+J (A+ ++ (,AB (,DG (GJ (B+ (FE (,D +, (E,G (EGF GK E+ ,B +F +E +E +A (JA (JE (+K (+B +G (+FF (++, ,E +, (+E (,A +A B+ JD (+A (,F (FG (FK

$,FS 4 # 2ΔECalc − ΔEExp3 # %%! 23" ' ((42TP38 & # $7 )W 1 8 C8 8WC8 A8 C8WA8 82*+,3 C82*+,3 + (AG (,D (FA (+E FE (GK (,G , FJ (FJ (+D (F+ (F+ (+E (+F E DA (EF (JD (G, (AB EE (AE G (GA (G+ (G+ (ED (EA (G, (EK A (FJ (FD (,+ (FB (FB (+B (+, J +D (+A (E+ (+B (+K (F, (,, B (F, (FJ (+J (FA (FG (+A (+F D ++ +B +K ,F ,E +, ,F K (,D (+A (++ (+G (+E (G+ (+K +F +,J (+J (BE (EB (AD G+ (A+ ++ KK (FE (GK (EK (JE A+ (,D +, BK ,B (F+ +B FA EJ +F (,, (,A (+B +E (,K (+K (+, (,F +G +A (+E (EK (,F (,K (+F (,A +A (,, (FG FB (FE (FG (EJ (FK

$,+S : ; 2*+,3"& %%! 23"1 ( & )W # 7

*# # Δ¯ U : ' Δ¯ abs : ' Δstd U ! #' Δmax U 7 : ¯ ¯ abs %%! 23 Δstd Δmax Δ Δ ((42P38 +B G, AA +,J ((42CP38 (++ +B +B G+ $7 28WC83 (,G ,B ,J BE ((42AP38 (+A +K +K G, $7 2C8WA83 (+K ,E ,A JE ((42P38 2*+,3 (FA ,D EE A+ ((42CP38 2*+,3 (+K ,E ,F AE +K+

# ## <

$,,S 4 # 2ΔECalc −ΔERef 3 # %%! 23" ((42TP38 & 2*+,3"& 7 2C8WA83 )W 1 8 C8 8WC8 A8 82*+,3 C82*+,3 + (AB (E+ (FD (+J (A, (,B , FB (FA (+B FF (+, (FK E +G, ,B (++ +A KF FG G (+F (FJ (FJ (FE (FB (FG A F+ (F+ (+G FF (+F (FA J EB FG (+, F, +B (FE B F, (F, (+, (F+ (++ (FJ D (+, (FJ (FG (FE (++ (FE K (+A (F, F, (F+ (,D (FJ +F +DG G, (+A ,+ KK FB ++ +J, JF +G ,G ++G EA +, BG ,, (FJ +, E+ FA +E (FB FE +F F, (FE FA +G GG +J (+F FK +K FG ++ F+ (E, (FA +A (+D FF

$,ES : ; 2*+,3"& %%! 23"1 ( & )W # 7 2C8WA83 *# # Δ¯ U : ' Δ¯ abs : ' Δstd U ! #' Δmax U 7 : ¯ ¯ abs Δ Δ %%! 23 Δstd Δmax ((42P38 EJ A+ BE +DG ((42CP38 FD +A ,, JF $7 28WC83 (FA +F +F +B ((42AP38 FG FB +F ,G ((42P38 2*+,3 +E EJ AF ++G ((42CP38 2*+,3 (F+ FK +E EA +K,

$,GS 4 # 2ΔECalc − ΔERef 3 # %%! 23" ((42TP38 & 2*+,3" & 9# 4# ((42AP38 ":" - 7 2C8WA83 )W 1 8 C8 8WC8 A8 82*+,3 C82*+,3 + (J, (EF (FD (+J (AB (,B , +J FF (+, FF (FE (FE E D, EF (FK +A ,K FJ G (FB (FJ (FJ (FE (FE (FG A +B FF (+E FF FJ (FA J ,B FG (+, F, FB (FE B +F (FE (+, (F+ (FE (FJ D (+F (FB (FA (FE (FK (FG K (FK (F, F, (F+ (,E (FB +F +,F G, (+A ,+ EA FB ++ ++F GB F, ,G J, ,, +, J+ ,, (FJ +, +D FA +E (FG FG +F F, FF FA +G AE +B (FK FK ,D FA +A (+G F+ ++ F+ (,D (FG

$,AS : ; 2*+,3"& %%! 23"1 ( & 9# 4# ((42AP38 ":"- )W # 7 2C8WA83 *# # Δ¯ U : ' Δ¯ abs : ' Δstd U ! #' Δmax U 7 : ¯ ¯ abs Δ Δ %%! 23 Δstd Δmax ((42P38 ,J GF AF +,F ((42CP38 FD +G ,F GB $7 28WC83 (FA FK FD +A ((42AP38 FG FB +F ,G ((42P38 2*+,3 FG ,+ ,K J, ((42CP38 2*+,3 (F+ FD ++ ,B +KE

# ## <

100 80

MP2 CCSD CCSD(T)

Fehler in kJ/mol

60 40 20 0 -20 -40 -60 -80 -100 1

2

3

4

5

6

7

8

9 10 11 12 13 14 15

Reaktionsnummer 100 80

MP2-R12 CCSD(R12) CCSD(T)(R12)

Fehler in kJ/mol

60 40 20 0 -20 -40 -60 -80 -100 1

2

3

4

5

6

7

8

9 10 11 12 13 14 15

Reaktionsnummer

$++S # 2ΔECalc − ΔEExp 3 ((42P38 1 # 7

*( )W +KG

$,JS *.(,2:%3W,(C84 - !53· 2 5 1 23

& 7 - ' #& 2 :73 - # 8 1 @ 2K+F3 & "# *.( ,2:%3W,(C84 25 2 C2v !;3 KADF +FG,,◦ : *.(,2:%3W,(C84 !53 2 D3h !;3 & "# +G,BJ +,FFF◦ 23 2 :73 5A" B KJ,J KADF 5A" J KJ,J KADF B"5A" J +FAA, +FG,, 5A"!+ ,E,GJ ,GAJB !+"5E +G,BA +G,BJ !+"5, +G,E, +G,BJ !+"5G +G,BA +G,BJ 5E"!+"5, ++KDD +,FFF 5,"!+"5G ++KDD +,FFF 5G"!+"5E ++KGA +,FFF 5G"!+"5E"5, +JKBK +DFFF B"5A"!+"5G ++ABK ++JFK

$,BS *.(,(*+,"$ Eh 4# # + 1& # H I"1& 2:%3 $ EÊÁ¹ÅÈ¾¹Ê½¾ = EÊÁ¹ÅÈ¾ + EÊÁ¹ΔÊ½¾ *.": Eh # δÊÁ¹ÅÈ¾ = EÊÁ¹ÅÈ¾ − EÅÈ¾ δÊÁ¹ΔÊ½¾ = EÊÁ¹ΔÊ½¾ − EΔÊ½¾ @ EÊÁ¹ÅÈ¾¹Ê½¾ EÊÁ¹ΔÊ½¾ δÊÁ¹ÅÈ¾ δÊÁ¹ΔÊ½¾ (BJEJB D+, (FF+F BDG FF+D (FFE+ 2 5 23 (BJEJB BGF (FF+F BKF FF+D (FFE+ 2 5 23 (BJEJB BKJ (FF+F JBJ 2 5 2-3 !53 23 (J,E,,K FD+ (FFGE FKB FF+D (FFDJ !53 23 (J,E,,B E+, (FFGE FKJ FF+K (FFDJ (J,E,,B EG+ (FFG, BKG !53 2-3 !53· 25 23 (JKKJ+, JAE (FFAG ++G !53· 25 2 :73 (JKKJ++ F,+ (FFAG FGF 5 2- :73 (BJEJB DJ, (FF+F JK, 2 (J,E,,K ++F (FFG, DA+ !53 2- :73 +KA

# ## <

$,DS ((42CP38W%%! 232*+,3"- !53 Eh * !"5 W !%: ,"*+, %%! 2*+,3 %%! 232*+,3 +G+ −622.195986 −623.192672 −623.160309 −623.208112 +G, −622.192980 −623.193633 −623.159973 −623.208651 +GE −622.189193 −623.193817 −623.158851 −623.208417

$,KS ((42CP38 $ Eh $ # 4 ( !%: , %%! %%! 23 (BJFJAK,, (BJEA+K+B (BJEAG,FB (BJEJEADA 2 5 23 5 23 (BJFJAAKD (BJEA+DGJ (BJEAG+FG (BJEJEA,+ 2 (BJFJAKJ, (BJEA,+GB (BJEAGEDB (BJEJEBBD 2 5 2-3 !53 23 (J,,+KF+A, (J,E+JJGDF (J,E+G+KBJ (J,E+K+JD+ !53 23 (J,,+DDJKJ (J,E+JGJBG (J,E+GFE+B (J,E+DKKAE !53 2-3 (J,,+DDDDE (J,E+JA,,, (J,E+GFBDF (J,E+KFGGB !53· 2 5 23 (JKD,JDG+F (JKKAEGFKK (JKKA++JAK (JKKAB+F+F !53· 2 5 2 :73 (JKD,JJG+, (JKKAE,AB+ (JKKAFKKA+ (JKKAJKA,D (BJFJAKJ, (BJEA,+GB (BJEAGEDB (BJEJEBBD 2 5 2- :73 !53 2- :73 (J,,+KFE,, (J,E+JJKA+ (J,E+G,EBB (J,E+K,+FD

+KJ

$EFS %%! 232:%3W((%48 2 4 # 3 %%! 232:

3W( (%48 2

$ # 3 $ Eh $ " #" 2 +3 "-# 243 Eh # !%:W((%48 1 !%: + 4 %%! 232:%3 %%! 232:

3 (BJFJF BJE F+KK DDG (F,A+ GDA (BJEGA GDB (BJEKK JGG 2 5 23 (BJFJF GJF F+KK DBD (F,A+ GBA (BJEGA GAD (BJEKK JFB 2 5 23 5 2-3 (BJFJF ADG F+KK DBG (F,A+ GJK (BJEGJ FFF (BJGFF +B+ 2 !53 23 (J,,+JD AKK G++J BFD (AEGJ B,A (J,E+,+ KK, (J,EJ+A BJD !53 23 (J,,+JB FD+ G++J BFA (AEGJ BEF (J,E+,F +BG (J,EJ+E KD, (J,,+JB EAA G++J BF, (AEGJ B,E (J,E+,+ ,FE (J,EJ+A +B+ !53 2-3 !53· 25 23 (JKD,G+ ADB GE+J EEK (AAKB DF+ (JKKGD, DEG (BFFFEF KBJ !53· 25 2 :73 (JKD,EK JJE GE+J G+F (AAKB K+B (JKKGD+ J++ (BFFF,K BF+ (BJFJF DBB F+KK DD+ (F,A+ GDF (BJEGA KAD (BJGFF +EA 2 5 2- :73 !53 2- :73 (J,,+JD DA+ G++J BFJ (AEGJ B+K (J,E+,, DK, (J,EJ+J BKA

+KB

+KD

# ## <

6 5 4 " 7 !

1$

+KK

0 ) ' # <

, ; )" 5 !# # ,(C84 M+DKN' /*55?$" M+K+N # 8( & # # s":' 5 :( '

& # p"!' 5 : ' ' # : -6": 2-5 H-6 ; I3 # 6 # < : $7 FFG,D 2s3' FFE+B 2p3' FFB,, 2d3' F+G 2f 3' F,KB 2g3' # : $7 +,DFF 2s3' BFF' +BF 2d3' ,,A 2f 3' +DG 2g3' ((42CP38 M,F'+AKN #' ? ! +Es+,pBdGf Eg ' ,,s+ApBdGf Eg :

, ; ) = 4# 5 !< 4# !# & # ,(C84 #& 6 # # s":' # : '

& . # ! p":' : ' 6 # < : &' ((4C8 M,F' D+N # 0 $7 = : FFJKAK 2s3' FFAEGD 2p3' F+AG 2d3' FE,G 2f 3' FB+G 2g3 : EFFF 2s3' E,F ++F 2d3' D,B 2f 3' JF 2g3 !< +FsKpJdGf Eg ' +BsKpJdGf Eg :

, ; = 5 < ,(C84 6 # s":' : '

& # < : $7 FF,EJE 2s3' FFDGD 2p3' F+K 2d3' FEJ 2f 3 # : $7 JE 2p3 & DsApEd,f -6": ,FF

0+ ( #

,' & 7 !# ' !<

< # ,(C84 7 & M,F,N # # 6 < : & : !# # : $7 +,BFFF 2s3 FFAAD 2s3' F+E+ 2p3' KBEGAA 2d3' ,KJJG 2d3'FFD+BB 2d3' ,KKK 2f 3' F+BD+ 2f 3' +F+FD 2g3' F,DF++ 2g3' ,FEJ 2h3' FAFK 2h3 +Gs++p+,dDf AgEh 7 !< # : $7 ,,FDF 2s3' F++A 2s3' F,,J, 2p3' +DFAE 2d3' JDFA 2d3' F++J 2d3' ,GBK 2f 3' FG+E, 2f 3' KJEJB 2g3' FJ,GA 2g3' AAE+ 2h3 +ED,D 2h3 ' # :

++sKp+FdBf AgEh 7 : < # : $7 F+FDA 2s3' DGDGD 2p3' FFKBJ 2p3' F,BEA 2d3' F,ED+ 2f 3' +F 2g3' BsJpGdEf ,g 7

,F+

0 ) ' # <

,F,

% A+ D+ D, DE K+ K, + ,

4# 5 # H5I GD 1 2, ← ,3 %%,(*+," B+ 1 2E ← ,3 %%,(*+," 1 B+ B, $7

* )W KB $ & !53· 25 7 )W ++E $7 %%,"- +,F $7 *+,"& %%,(*+,"-( +,+ E $7 X' C B * ( & +,, G $7 ρ = AR %%,"

+,E A $7 *+,"& ρ = AR %%,(*+,"

+,G J @ Tˆ1 +,A + %%,(*+,"- +ED , %%,(*+,"- 8 1 2, ← ,3 +ED E %%,(*+,"- 1 +ED G %%,(*+,"- 8 1 2E ← ,3 1 +EK A %%,(*+,"- +EK J %%,(*+," 1Σ+"8 +EK B %%,(*+," 1Π"8 +GF D %%,(*+,"- : +GF K %%,(*+," 1Σ+"8 : +GF +F %%,(*+," % 1Σ+"8 : +G+ ++ %%,(*+," 1Π"8 : +G+ +, %%,(*+,"- %5 +G+ +E %%,(*+," 1Σ+"8 %5 +G, +G %%,(*+," % 1Σ+"8 %5 +G, ,FE

#

+A %%,(*+," 1Π"8 %5 +G, +J %%,(*+,"- 12 +GE +B %%,(*+," 1Πg "8 12 +GE +D %%,(*+," X 1Σ−u "8 12 +GE +K %%,(*+," # 1Δu"8 12 +GG ,F : +GK ,+ %5 +GK ,, 12 +AF ,E ; %%,(*+," +AJ EJ*+,"& 4 7

2 ,E3 (

; %%,(*+," +AJ ,G ; %%,(*+," 1 +AB ,A ; 1Σ+"R +AD ,J ; 1Π"R +AD ,B ; 1Σ+"R : +AK ,D ; % 1Σ+"R : +AK ,K ; 1Π"R : +AK EF ; 1Σ+"R %5 +JF E+ ; % 1Σ+"R %5 +JF E, ; 1Π"R %5 +JF EE ; 1Πg "R 12 +J+ EG ; X 1Σ−u "R 12 +J+ EA ; # 1Δu"R 12 +J+ EB*+,"& 4 7

2 ,G3 (

; %%,(*+," 1 +J, ED*+,"& 4 7

2 ,A3 (

; 1Σ+"%%,(*+," +J, EK*+,"& 4 7

2 ,J3 (

; 1Π"%%,(*+," +JE GF*+,"& 4 7

2 ,B3 (

; 1Σ+"%%,(*+," : +JE G+*+,"& 4 7

2 ,D3 (

; % 1Σ+"%%,(*+," : +JG G,*+,"& 4 7

2 ,K3 (

; 1Π"%%,(*+," : +JG GE*+,"& 4 7

2 EF3 (

; 1Σ+"%%,(*+," %5 +JA GG*+,"& 4 7

2 E+3 (

; % 1Σ+"%%,(*+," %5 +JA GA*+,"& 4 7

2 E,3 (

; 1Π"%%,(*+," %5 +JJ GJ*+,"& 4 7

2 EE3 (

; 1Πg "%%,(*+," 12 +JJ ,FG

#

GB*+,"& 4 7

2 EG3 (

; X 1Σ−u "%%,(*+," 12 GD*+,"& 4 7

2 EA3 (

; # 1Δu"%%,(*+," 12 $+ - Eh $< Δ*+, Eh $, - Eh $< Δ*+, Eh $E - Eh $< Δ*+, Eh $G & - Tˆ1 ' Tˆ2 Tˆ2 Eh $A & - Tˆ1 ' Tˆ2 Tˆ2 Eh $J & - Tˆ1 ' Tˆ2 Tˆ2 Eh $B - $h $< Eh $ # +K+GDJGE, 2KJGE, 3 5 " 7 # $D - $h $< Eh $ # +K+GDJGE, 2KJGE, 3 5 " 7 # $K Δ*+,"& Eh 4# # +K+GDJGE, 2KJGE, 3 7 $+F Δ*+,"& Eh 4# # +K+GDJGE, 2KJGE, 3 7 $++4 # %%! 232*+,3" %%! 23"- ((42P38 $+,4 # %%! 232*+,3" %%! 23"- ((42P38 $+E4 # %%! 232*+,3" %%! 23"- ((42CP38 $+G4 # %%! 232*+,3" %%! 23"- ((42CP38 $+A !%:"' ,"' %%! " %%! 23"- ((42AP38 Eh $+J4 # ((42P38 1 $7( $+B: ; ((42P38 1 $+D4 # # $7 (( 42CP38 1 $+K: ; ((42CP38 1 $,F4 # %%! 23"1 $7 $,+: ; %%! 23"1 7

*# $,,4 # %%! 23"1 $,E: ; %%! 23"1 7 2C8WA83 *( #

+JB +JB +BF +B+ +B, +BE +BG +BA +BJ +BJ +BB +BD +DG +DA +DJ +DB +DD +DK +DK +KF +KF +K+ +K+ +K, +K, ,FA

#

$,G4 # %%! 23"1 +KE $,A: ; %%! 23"1 7 2C8WA83 *( # +KE $,J*.(,2:%3W,(C84 - !53· 25 +KA $,B*.(,(*+,"$ Eh +KA $,D ((42CP38W%%! 232*+,3"- !53 Eh +KJ $,K ((42CP38 $ Eh +KJ $EF %%! 232:%3W((%48 $ Eh +KB

,FJ

! E+ E, EE EG EA B+ B, D+ D, DE DG DA DJ DB DD DK K+ K, KE KG

% "? 21!3 - %."

. α' β ' %." %."*+,"

21!3 - n U B n U E 4 7

%." %."*+,"

21!3 - n U G : 4 7

!

*+,"& %%! 2*+,3"- : 2B+3 %%,(*+," -" 1 8( %%,(*+," -" 1 8( 1 %%," %%,(*+,"$ -" 1Σ+"8 %%," %%,(*+,"$ -" 1Π"8 %%, %%,(*+, X 1Σ−u "R 12 4 %%, %%,(*+, 1Πg "R 12 4 %%, %%,(*+, # 1Δu"R 12 4 ˆ Tˆ2 + Tˆ2 ], R ˆ 1 ]|HF %%," %%,(*+, & R1|[[H, 1 P 2, ← ,3 R ˆ Tˆ2 + Tˆ2 ], R ˆ 1 ]|HF %%," %%,(*+, & R1|[[H, 1P 2E ← ,3 R 1 - 1 Δ*+,"& 1 Eh Δ*+,"& : Eh Δ*+,"& 12 Eh

+A +D ,, ,E ,E JA JJ BF BF BE BE BA BJ BJ DF D+ DK KF K+ K+ ,FB

##

KA KJ KB KD KK

Δ*+,"& 1 7 KG Δ*+,"& : 7 KG Δ*+,"& 12 7 KA *< ((42CP38 1 +FF *< ((42CP38 ( (42AP38 1 +FG K+F !53· 2 5 ++F %+ !

(V †)klαm * + +,D %, !

*+,"& %%,(*+,"- +,K %E !

*+,"& ρ = AR * + +EF %G !

(V †)klαm * , +E+ %A !

%%,(*+,"-( * , +E, %J : 2%A3 +EE %B : 2%A3 2%J3 +EG %D !

*+,"& ρ = AR * , +EA + %%," %%,(*+,"$ -" 1Σ+"8 : +GG , %%," %%,(*+,"$ -" % 1Σ+"8 : +GA E %%," %%,(*+,"$ -" 1Π"8 : +GA G %%," %%,(*+,"$ -" 1Σ+"8 %5 +GJ A %%," %%,(*+,"$ -" % 1Σ+"8 %5 +GJ J %%," %%,(*+,"$ -" 1Π"8 %5 +GB B %%," %%,(*+,"$ -" 1Πg "8 12 +GB D %%," %%,(*+,"$ -" X 1Σ−u "8 12 +GD K %%," %%,(*+,"$ -" # 1Δu"8 12 +GD +F %%, %%,(*+, 4 +A+ ++ %%, %%,(*+, 1 4 +A+ +, %%, %%,(*+, 1Σ+"R 4 +A, ,FD

##

+E %%, %%,(*+, 1Π"R 4 +G %%, %%,(*+, 1Σ+"R : 4 +A %%, %%,(*+, % 1Σ+"R : 4 +J %%, %%,(*+, 1Π"R : 4 +B %%, %%,(*+, 1Σ+"R %5 4 +D %%, %%,(*+, % 1Σ+"R %5 4 +K %%, %%,(*+, 1Π"R %5 4 $+ Δ*+,"& :− Eh $, Δ*+,"& Eh $E Δ*+,"& %5 Eh $G Δ*+,"& 1 3 Eh $A Δ*+,"& % 4 Eh $J Δ*+,"& 25 Eh $B Δ*+,"& :− 7 $D Δ*+,"& %5 7 $K Δ*+,"& 7 $+F Δ*+,"& 25 7 $++ # ((42P38 1

+A, +AE +AE +AG +AG +AA +AA +BK +BK +DF +DF +D+ +D+ +D, +D, +DE +DE +KG

,FK

##

,+F

8 M+N * ) ' "3 + 44- & 4 ' . 0 5 "' ! +FGB' 2 S * >;3' !@ ! 2+KKA3 M,N 5 %' )Q' % ; ?' +!' GFK 2+KKA3 MEN - * ) ' ) % ;' ,-' +K+F 2+KD,3 MGN ' )Q ) 5 ' . 0 5 "'

; % 2,FFF3 MAN * ' % !' +' DK 2+K,D3 MJN 4 :' 8 ;' -.' +,J 2+KEF3 MBN 4 :' 8 ;' -' BKA 2+KEF3 MDN 5 ?0#' % ;' ' ,FB 2+KAK3 MKN ' 5 %' )Q ) 5 ' % ; ?' ++' BA 2+KKA3 M+FN 5 %' ' )Q ) 5 ' % ; ?' /-' +DA 2+KKJ3 M++N * - >' / " 5 " & + . ' 57( /; ' 57 2+KDK3 M+,N ' ! Z' 5 %' ) % ;' .0+' G+AB 2+KKA3 M+EN 5 %' ' ' )Q' ! Z' ) % ;' .0/' JK,+ 2+KKJ3 M+GN 5 %' )Q' ) % ;' .0!' BG,K 2+KKA3 M+AN ' 5 %' )Q' ! [ ' ) % ;' .0-' +DFD 2+KKB3 ,++

$

M+JN ! 1 ! 5 ' . 6 "' 54$*' 1# > 2+KKJ3 M+BN $ !0' ;' ,1' EJ+ 2+K,K3 M+DN $ !0' ;' ,1' GDK 2+K,K3 M+KN * 5' ;' 2+' GAB 2+K,B3 M,FN $7 % %; $ ! ' 4 +FW,+WFE M,+N ' ) % ;' .02' A+KG 2+KKD3 M,,N 5 %' )Q' ) % ;' .0!' BG,K 2+KKA3 M,EN )Q' ) % ;' 1!' EEEE 2+KKF3 M,GN ' ) % ;' .0-' DFAK 2+KKB3 M,AN % !#' ; *' .-' +F+A 2+KJ,3 M,JN % !#' % ;' ' ,G+ 2+KJE3 M,BN %

' ) ! * ' ) % ;' ,.' G+G, 2+KBK3 M,DN * 1

' ) % ;' 2!' ++BE 2+KDA3 M,KN $ ;

' 8 ;' /+' EGB 2+K,K3 MEFN ' % ' -2' GGA 2+KDA3 ME+N ? Q

! ' ; *' +-' J+D 2+KEG3 ME,N ' % ; ?' .!+' +B 2+KDB3 MEEN ) ' 1 % ;' * $ ' % ' ,1' EJ+ 2+KK+3 MEGN ' % ; ?' .2-' ADE 2+KK+3 MEAN $ : 4 : ! ...' ) % ;' ..!' EKKF 2,FFF3 MEJN $ : 4 '

' : ! ... - %\\' ) % ;' ..+' ,DBA 2,FF+3 MEBN $ : 4 '

' * ' % !

: ! ...' ) % ;' ..2' DAKG 2,FFE3 MEDN ) 1' ' % ; ?' .11' GKB 2+KK,3 ,+,

$

MEKN ) 1 ' ) % ;' .0.' BBED 2+KKG3 MGFN ) 1 4 ' % ; ?' !+' +JJ 2,FFF3 MG+N ) 1 4 ' ' " ()* ' "3 (4- & 5' B' ! +E+' 2 ( S ) ?;3' !@' ! 2,FF,3 MG,N ) 1' ' ()*3 + ' " 5 "' ( +4 "' E' ! +' 2 S * ) 3' !@' ! 2+KKB3 MGEN ' ) % ;' .0' +FDKF 2,FFG3 MGGN ' ' % ' .0,' +BE 2,FF,3 MGAN : * ;' ) % ;' ..1' GJFB 2,FFE3 MGJN ! ( : * ;' ) % ;' ..1' AEAD 2,FFE3 MGBN % % !' ) % ;' ..-' JEKB 2,FF,3 MGDN ! ˘' ) 1' . ) C %' .0/' K,K 2,FFA3 MGKN $ ' % 4

' ) % ;' .' ,+GEFJ 2,FFA3 MAFN ! (' ) % ;' ..' ++B 2,FFG3 MA+N ) ; : * ;' ) % ;' ..' GGBK 2,FFG3 MA,N % % !' ' % ; %' .+1' + 2,FF,3 MAEN ) ;' $ 4 ' *

; : * ;' ; % % ;' ,' ,B+F 2,FFA3 MAGN ) 1' ()* "' ' " " "' ! +GK' 2 S ) *; #( 3' # ' 2,FFE3 MAAN ) 1' 4 ' ) % ;' ../' ,F,, 2,FF+3 MAJN ) 1' % ; %' +' E, 2,FFE3 MABN 5 %' )Q % &' . ) C %' -2' + 2+KKD3 MADN % &' . % ! ! & !! , 1 4 . ' ' /& 2 3 2,FFA3 MAKN : 4' $ ! ) ) ' ) % ;' 2' EBAE 2+KDA3 ,+E

$

$ ! *9' ; * ' !!' ,K,D 2+KDJ3 ) ' ) % ;' ..2' EFFJ 2,FF,3 ' = ) ' ; % % ;' -' ,FAK 2,FFG3 0' + " !! , 1 % ! 7 ! 0" ( . (8*' ( ' /& 2 3 2,FFE3 MJGN ) 5 )Q' 5/ ( 5 " - +

MJFN MJ+N MJ,N MJEN

0 & . 9! 0 & : 4 ' . 0 5 "' ! DAB' 2 S * >;3'

MJAN MJJN MJBN MJDN MJKN MBFN MB+N MB,N MBEN MBGN MBAN MBJN MBBN ,+G

!@' ! 2+KKA3

' % ' 2!' ,JE 2+KK,3 !' : * .' ) % ;' 11' EBED 2+KKE3 ) 5 )Q' ) % ;' 2' E,EA 2+KDA3 % &' 5 % )Q' ) % ;' .02' DEE+ 2+KKD3 #' ' ) % $' 40( 2,FFJ3

' 5 " & ' ' " : 4

" "' ! E' 2 S ) *; #3' # ( 2,FFE3

' ()* . ; ' . . + ! & 6 "' ! +D+' 2 S ) -3' % . 2,FFF3

' 12/ : 4 ' ' $0 8 2+KKJ3 % % !' <! " & "

! ()* ' =' ' /& /( 2,FFG3

' 12/ : 4& ' " & "' ! ,EA+' 2 S 4 * ! ; 3' ;' % 2+KKD3

' : * ;' ! ( $ : 4 ' . * ; %' 2,FFJ3 $ ' . " 5 " + ' ;' 1# > 2+KJF3 ) ! 5 !˘ ' ) % ;' ++' +DKK 2+KJJ3

$

MBDN MBKN MDFN MD+N MD,N MDEN MDGN MDAN MDJN MDBN MDDN MDKN MKFN MK+N MK,N MKEN MKGN MKAN MKJN MKBN MKDN MKKN M+FFN M+F+N

) ! 5 !˘ ' ) % ;' ++' EJFD 2+KJJ3 ) ! 5 !˘ ' ) % ;' +-' DAG 2+KJB3 ' % ' .0' +A+ 2+KAB3 *

' * ) ' ) % ;' 1-' JBKJ 2+KK,3 ' )' ' !22' EEK 2+KKB3 %." %."*+,"

# # # 4

0 2,FFA3 ' ' ) 1' ) % ;' .0-' KJEK 2+KKB3

) ...' ) % ;' 1-' GGDG 2+KK+3 ) % ! ' ; *' !.' EEE 2+K,D3 $ ;

' 8 ;' -/' +G 2+KEF3 % ? ' ; *' ..' +JGK 2+KAD3 % ? ' ; *' ../' +,+J 2+KAK3 % ? ' ; *' .-' +GBF 2+KJ,3 - ' ; * ' /.' AF+K 2+KKA3 ?# ' . ) C %' +.' B+K 2+KK,3 ?# ' . ) C %' +/' GGA 2+KKE3 ?#' . ) C %' +/' DB 2+KKE3 ' - ?#' . ) C %' /!' ABA 2+KKA3

? ) * 1 ' . 6 . & 0 ' " & "' ! +BEA' 2 S 4 * ! ; 3' ;' % 2+KKD3 ?# ) ' ) % ;' .0/' GJEJ 2+KKJ3 ! ?#' ) % ;' ../' AEJ, 2,FF+3 ?# ) ' * ; %' /.' AF+ 2,FFF3 ) ! ! ! ' ; * ' +' KFD 2+KB+3 ) ! ! ! ' ; * ' ..' G+D 2+KBG3 ,+A

$

M+F,N ' '

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

; * ! ' ) % ;' ,.' EEKJ 2+KBK3 M++KN - ' ( < . ' 4 2+KKA3 M+,FN $ * ' ) % ;' .,' DB 2+KBA3 M+,+N $ * ' ) ; ' .!' ?+BK 2+KDF3 M+,,N : ' % &' ) % ;' .' DG+FB 2,FFA3 M+,EN : ' % & ' . ) C %' .0-' ,EFJ 2,FFJ3 M+,GN *' - ' ) (-' % ; ?' ./,' GBK 2+KDK3 ,+J

$

M+,AN ' 5 ' " *()* 0/?()*@' <! + " & 6 /" ' ! D' 2 ( S ' ) ;' 3' %

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

' % ; ?' 2-' ,GE 2+KKD3 M+EKN ) 0' ) % ;' 11' A+JB 2+KKE3 M+GFN 4 ' ' ) % ;' 1+' ,FF, 2+KK+3 M+G+N ! 7' ) ! #' ) % ;' 21' EAA 2+KDD3 ,+B

$

M+G,N M+GEN M+GGN M+GAN M+GJN M+GBN M+GDN M+GKN M+AFN M+A+N M+A,N M+AEN M+AGN M+AAN M+AJN M+ABN M+ADN M+AKN M+JFN M+J+N M+J,N

,+D

#' )#' ) ; ' .' ,KJA 2+KBK3 ! ! ^' ; * ' +.' GJBF 2+KKF3 * #' ) ! #' ) % ;' ..0' G+JA 2+KKK3 ' + ! ' " - 4 & ' ( ' /& 5 2,FFG3 % 1' % & ' ) % ;' 2,FFJ3 : ' - 06' 5 ' 5 ' ) % ;' ( 40< 2,FFJ3 * ;' .

- !;' ;' +' +,JA 2+KB,3 ) ? .

' ! ' 0' A,E 2+KJG3 ? -; - * ' ;' !,' +KF+ 2+KBK3 ' [

;' #' ) -' ) % ;' 2,FFJ3

' 0 2,FFG3 ' [

; ) -' ;' .0!' ,+FK 2,FFA3

?' ;' 1-' AAK 2+KKK3 ) ! ? ! > ' ) % ;' ..' +FBGJ 2,FFF3 - ' ' ) % ;' 2!' A+FA 2+KDA3

' ? ' )Q' ) 5 ' ) ; ' !' *+FE 2+KKK3 / ' ) 1' ! ) % * ) ' ) % ;' 2!' GFG+ 2+KDA3 )' ' ) % ;' ..+' K,GG 2,FF+3 () ' 0 2,FFA3

7

! # 1.!" ?( 2,FFA3 ! %!2' 25' %52' 2!' %5' %5% 2' 1 3 ' 252' 153' %2 2' %2 4' % 4' !52' !53 %2 45 %' )' 1.!()1: ' =' ) ; % * ' K' %! . 2+KKD3

$

M+JEN ! 2%5 : - ' :; ! --' BKG 2+KBF3 M+JGN ! % 35 ) '

% ! ) +1' EFDK 2+KBJ3 M+JAN ! %2 6 - ') % ! :; + -2' ,,,G 2+KB,3 M+JJN ! % 2%5 * ? 1

' ? 4 ;' ) % ; !' +JB 2+KB+3 M+JBN ! %51 2 -' % +.' JBF 2+KAD3 M+JDN ! % 3% 5 ' ? ! %( ' ) % ! ..!' EGGB 2+KK+3 M+JKN % ?' > * - ' ; * ' !,' BDA 2+KDD3 M+BFN !&' % * ' ) % ;' .00' AD,K 2+KKG3 M+B+N & M+B,N & * ' ) % %' .0' +FG 2+KDK3 M+BEN * ' ) % %' .1' +BGJ 2+KKD3 M+BGN ' : : * ' % ; ?' !-' A++ 2,FF,3 M+BAN .' +' %; 0 5 " ' 5 "3 & . 5 " ; +0 0" 0;' 2 S . ) :3'

M+BJN M+BBN M+BDN M+BKN M+DFN M+D+N M+D,N

% !; 2+KKD3 % $ ' ) );' * ' ) ' * : 4' ) % !' ..-' +FE+G 2+KKG3 % ' ) % !' ..-' +FE+J 2+KKG3 $ * ?9;' * ? - ;' ) ; %' .00' +KK++ 2+KKJ3 ) );' / 0 > %' ) ; % ' .0.' +FFFF 2+KKB3 5 ' A <! & 0A3·<2A 0" 2,FFF3 ? ) ?' : ' ) % ;' ..' DDEF 2,FFF3 - ' " ' ; 4% 2+KKB3

,+K

$

M+DEN :' 5 " & B ( ' ' 1# > 2+KBE3 M+DGN ' ) - % 9 ( *9 : 9 ' ; % % ;' ' ,,,B 2,FFF3 M+DAN ? ?' ) % ;' /1' GAAB 2+KBE3 M+DJN ? ?' ) % ;' 1.' ,EGD 2+KDK3 M+DBN ! : ; : ' ;' .1' AAE 2+KBF3 M+DDN : 9 ' ) - % 9 ( *9 ) ?' % *' 1+' +DBE 2+KKG3 M+DKN : * ' ; % % ;' ,' E,KB 2,FFA3 M+KFN - 06' % 4

' ) % %' 4 2,FFJ3 M+K+N 25 &3 27 &3 & M+K,N () : * ;' ) % ;' .+' FAG++G 2,FFJ3 M+KEN 5 ? ;;' ) ) ;' ) % ;' .0/' JGKF 2+KKJ3 M+KGN () ' ) # ' * ?' !' % ' ' : * ( ;' - *' * ' ' ' ? %' ) 5 ' ) ;' : $' % ' - ' ? ;' ! ) 1( ' ;' $ ' 1 ' ' * ' / !' !

' ) !' * ' .A(A; ! & ! M+KAN * ?' / *; ?' ) % ;' 1/' ADDK 2+KK+3 M+KJN % ' () ' % ; ?' .10' + 2+KK,3 M+KBN ) 5 ) # ' % ; ?' ,' E,+ 2+KKG3 M+KDN !' * ? () ' ;' 1-' B+K 2+KKK3 M+KKN ' )' ) % ;' ..,' +FAGD 2,FF,3 M,FFN # ' ) % ;' 8 40< 2,FFJ3 M,F+N % 1' 0 2,FFJ3 M,F,N % &' ; % % ;' ,' AK 2,FFA3 ,,F

. : ' 0' % & * ' + 8 > 0 & +' ) % !' ./' KD,+' 2,FFE3 : ' % &' " - 9 3 5 0/?()*@ ' ) % ;' .' DG+FB' 2,FFA3 ! : ' % & ' " - 3 5 *()* & ' !' ) % ;' .+' 12

12

FGG++,' 2,FFJ3 + : ' % & ' 8 & ?5@ 5 12 . 0/?()*@' . ) C %' .0-' ,EFJ' 2,FFJ3 / : ' - 06' 5 ' 5 ' + & ! !" & 0A3 ·<2 A '' ) % ;' 40< ' 2,FFJ3