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The Generalized Coherent State ansatz: Application to quantum electron-vibrational dynamics
Accepted Manuscript The Generalized Coherent State Ansatz: Application to Quantum Electron-Vibrational Dynamics Raffaele Borrelli, Maxim F. Gelin PII: DOI: Reference:
S0301-0104(16)30271-3 http://dx.doi.org/10.1016/j.chemphys.2016.05.013 CHEMPH 9557
To appear in:
Chemical Physics
Please cite this article as: R. Borrelli, M.F. Gelin, The Generalized Coherent State Ansatz: Application to Quantum Electron-Vibrational Dynamics, Chemical Physics (2016), doi: http://dx.doi.org/10.1016/j.chemphys.2016.05.013
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H(p, q, r, s) = H◦ + V /!"." ?3"" 788"4(6L NB
i o ω 2 2 (pli + qli2 ) + gqli + ωki (rlK + s2lK ) 2 l,i,K X p + (ωqli + g) 2(ki + 1)(rlK+1i rlK + slK+1i slK )
H◦ =
Xnh
l +
l,i,K
+
X
ωpli
l,i,K
V = 2J Re{ O!" (".6F7 6F"3 N,
∂yi H
p
2(ki + 1)(rlK slK+1i − rlK+1i slK )
X
l>m,K,J
C¯lK CmJ hK, αl |J, αm i}.
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(k1 , ..., kd ),
O!" ".&3
hK, αl |J, αm i 788"7.64* 64 =
!" 8' "4 67) "4".*+
V
7."
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1.0
D1 GCS-1 GCS-2 MCTDH
0.8
P (t)
0.6
0.4
0.2
0.0 0
100
200
300
400
500
600
700
800
t (fs) 1.0
D1 GCS-1 GCS-2 MCTDH
0.8
P (t)
0.6
0.4
0.2
0.0 0
100
200
300
400
500
600
700
800
t (fs)
!"#$% &' ()*#+,-!). )/ -0% !.!-!,+ %+%1-$).!1 2-,-% ,2 , /#.1-!). )/ -!3% !. , 3)+%1#+,$ $!." 4!-0 && %+%1-$).!1 2-,-%25 60% *,$,3%-%$2 )/ -0% 3)7%+ ,$% J = 100 13−1 8 ω = 1000 13−1 9 ,: g = 0.2ω8 ;: g = ω5 60% .#3;%$ )/ ;,-0 3)7%2 !2 N = 119 -0% 3,?@& ,.7 =>?@A5 B,20@7)--%7 +!.% C −·: %<1!-,-!). +%D%+2 )/ -0% N %+%1-$).!1 2-,-%2 c = 0 CB&:9 7,20%7 +!.% C@ @: c = 19 /#++ +!.% CE: c = 29 C×: F>6BG $%2#+-5 !
1.0
D1 GCS-1 GCS-2 MCTDH
0.8
P (t)
0.6
0.4
0.2
0.0 0
10
20
30
40
50
60
70
80
t (fs) 1.0
D1 GCS-1 GCS-2 MCTDH
0.8
P (t)
0.6
0.4
0.2
0.0 0
10
20
30
40
50
60
70
80
t (fs)
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R4!YS R4!ZS
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r
hr
kh + 1 K + 1h , αl − 2
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45698
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456A8
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<=7>(?
i
+
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<=7>1?
i
+
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S0301-0104(16)30271-3 http://dx.doi.org/10.1016/j.chemphys.2016.05.013 CHEMPH 9557
To appear in:
Chemical Physics
Please cite this article as: R. Borrelli, M.F. Gelin, The Generalized Coherent State Ansatz: Application to Quantum Electron-Vibrational Dynamics, Chemical Physics (2016), doi: http://dx.doi.org/10.1016/j.chemphys.2016.05.013
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9'& +-#-1 911*%*:9.*-1 91( )'&9.*-1 -B&'9.-'#G 91( α *# 9 )-0B:&H 1?0+&'@ I%&1 n = 0 #.91(9'( :9?+&' )-%&'&1. #.9.&# 9'& -+.9*1&(@57J8 K1:*4& .%& :9..&'G !"# 9'& 1-. &*;&1#.9.&# -, .%& 911*%*:9.*-1 -B&'9.-'@ A. )91 +& &9#*:2 L&'*/&( .%9. √ DMF a n, α = n n − 1, α + α n, α √ ¯ n, α . D6F a† n, α = n + 1 n + 1, α + α !" %9L& .%& )--'(*19.& '&B'&#&1.9.*-157E8 D7F $%&'& .%& .$- '&9: L9'*9+:&#G q, p 9'& '&:9.&( .- .%& )-0B:&H L9'*9+:& α +2 ϕn (x, q, p) = 2−n/2 (n!)−1/2 π −1/4 e−(x−q)
2 /2+ip(x−q/2))
Hn [(x − q)]
q + ip α= √ . 2
N*19::2G *. *# B-##*+:& .- ;*L& .%& 1?0+&' #.9.& &HB91#*-1 -, 9 !" *1 .%& ,-'0 ∞ −|α|2 /2 X n+k 1/2 n−k 2 n−k n, α = e (−1) (k!) L (|α |)(¯ α ) k k (n!)1/2 k=0
DOF
n−k Ln (·) +&*1; .%& 9##-)*9.&( P9;?&''& B-:21-0*9:# -, -'(&' n@ N-' n = 0G 0, α ≡ α G 91( $& '&)-L&' .%& ,-'0?:9 ,-' :9?+&' )-%&'&1. #.9.&#@57J8
I*.%-?. :--#*1; 912 ;&1&'9:*.2 $& )91 $'*.& .%& L*+'-1*) $9L& ,?1).*-1 *1 .%& ,-'0 X Ψ(t) = DQF ClK (t) K, αl (t) l l,K
6
!"#"
l $%"&'("$ ")"&*#+,'& $*-*"$ . l = 1, 2, ..., N /0 K '$ - 12)*'3',4"5 K = (k1 , k2 , ..., kd ) ∈ Nd0 0 -,4 *!" K, αl (t) -#" - $"* +6 !"#$%#&#'%#' 789 4"(,"4 -$ *!" %#+42&* +6 d +,"34'1",$'+,-) 789 d Y ki , αli (t) . K, αl (t) ≡ k1 · · · kd , αl1 (t), · · · , αld (t) =
.:/
i=1
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
Sc
c
!"#" Sc '$ *!" $%-&" $%-,,"4 =D *!" $*-*"$ ', !'&! +,)D c =-$'$ 62,&*'+,$ -#" $'12)*-,"+2$)D "5&'*"40 !(#( !-G" - ,+,3R"#+ G-)2" +6 kiB C6 &+2#$" 6+# -,D %#-&*'&-) %2#%+$"$ *!" =-$'$ '$ *#2,&-*"4 *+ - 1-5'121 H2-,*21 ,21="#B ! 9',&" ', - $D$*"1 '*! d G'=#-*'+,-) 4"A#""$ +6 6#""4+1 *!"#" ')) =" dc 4'$*',&* &+1=',-*'+,$ !-G',A c "5&'*"4 1+4"$0 2$',A $2&! - %-#*'*'+, *!" -G"62,&*'+, +6 OHB ? &-, =" #'**", ', *!" 6+#1 Ψ(x, t) =
N X l=1
(dc) d X X
X Clki1 ...kic (t) ki1 , αli1 (t) . . . kic , αlic (t) l
wi1 ...wic
c=0 i1 ...iC ki1 ...kic =1
wi N h d X X X = Cl (t) αl (t) + Clki (t) ki , αli (t) αl0 (t) l=1
() X wj wi X X d 2
+
i,j=1 ki =1 kj =1
i=1 ki =1
i Clki kj (t) ki , αli (t) kj , αlj (t) αl00 (t) + ... l ,
T
.S/
!"#" " !$%" &$#'('()*"+ '!" αl $, αl = (αl0 , αli) (* '!" -#,' ,.//$'()* $*+ αl = (αl00 , αli, αlj ) (* '!" ,"0)*+ ,.//$'()*1 2* $*$3)4)., &$#'('()* !$, 5""* &"#6)#/"+ 6)# '!" 7.$*'./ *./5"#, K 8 $*+ (*+(0", )6 '!" "9&$*,()* 0)":0("*', ('! %$3." ;"#) $#" )/(''"+< wi #"&#","*' '!" /$9(/./ 7.$*'./ *./5"# )* '!" i='! /)+"1 >,(*4 &$#'('()* ? (' (, &),,(53" ') #"+.0" '!" ,(;" )6 '!" 5$,(, ,"' 5@ 3(/('(*4 (', 0)/5(*$')#($3 "9&3),()*1 A*+""+8 (6 "$0! /)+" (, $33) "+ ') !$%" w 5$,(, 6.*0'()*, '!"* $ 0)/5(*$'()* ('! c ,(/.3'$*").,3@ ! d "90('"+ /)+", (33 (*'#)+.0" wc *" 5$,(, ,'$'",8 $*+ ,(*0" !'!"#" $#" c ,.0! 0)/5(*$'()*, '!" +(/"*,()* )6 '!" Sc ,.5,&$0" (33 5" dc wc1 B!.,8 6)# c = 1 " (33 !$%" &),,(53" 0)/5(*$'()*, $*+ dw 5$,(, ,'$'",8 6)# !d d 2 c = 2 " (33 !$%" 2 w ,'$'", $*+ ,) )*1 B!(, '@&" )6 &$#'('()*(*4 !$, 5""* #"0"*'3@ $&&3("+ 5@ '!" $.'!)#,8 ') '!" ,'.+@ )6 "3"0'#)* ,.&"#90!$*4" (* /)3"0.3$# 0!$(*,1CDEF G./"#(0$3 0)*%"#4"*0" 0$*8 (* &#(*0(&3"8 5" '",'"+ 5@ 3"''(*4 c %$#@ .*'(3 *) ,(4*(-0$*' %$#($'()*, )6 '!" &#)&"#'(", )6 (*'"#",' H !"! "3"0'#)*(0 &)&.3$'()* )# 0)!"#"*0",I $#" )5,"#%"+1 J) "%"#8 ).# /$(* &.#&)," (, ') ,'.+@ '!" #",.3' )6 3) )#+"# $&&#)9(/$'()*, .& ') c = 28 !(0! $#" '!" /),' $&&"$3(*4 6)# '#"$'(*4 ,@,'"/, ('! $ 3$#4" *./5"# )6 +"4#"", )6 6#""+)/1 B!"#"6)#"8 (* '!" 6)33) (*4 " (33 #",'#(0' ).# +(,0.,,()* ') '!" #$%#&' ? '#.*0$'"+ ') '!" "90('$'()* 3"%"3, c = 1, 21 K)#" ,&"0(-0$33@ (* '!" 6)33) (*4 " (33 #"6"# ') '!" "7.$'()*, L $*+ MN 5"3) ΨGCS1 (x, t) =
N h X l=1
ΨGCS2 (x, t) =
N h X l=1
+
wi d X i X Clki (t) ki , αli (t) αl0 (t) l Cl (t) αl (t) +
HLI
i=1 ki =1
wi d X X Clki (t) ki , αli αl0 (t) Cl (t) αl (t) +
(d2) X wj wi X X
i,j=1 ki =1 kj =1
i=1 ki =1
i Clki kj (t) ki , αli (t) kj , αlj (t) αl00 (t) l
HMNI
$, OPQ=M $*+ OPQ=E $&&#)9(/$'()* #",&"0'(%"3@1 A' (, )#'! *)'(0(*4 '!$' '!" ,) 0$33"+ R$%@+)% RM #$%#&' 0)##",&)*+ ') '#.*0$'(*4 '!" $5)%" "9&$*,()* $' '!" "90('$'()* 3"%"3 c = 01 B!" "7.$'()*, )6 /)'()* 6)# '!" &$#$/"'"#, (ClK (t), qli(t), pli(t)) 0$* 5" +"#(%"+ 5@ .,(*4 '!" ,'$*+$#+ /$0!(*"#@ )6 '!" R(#$0=S#"*T"3 '(/"=+"&"*+"*' %$#($'()*$3 U
!"#$" %& '"() ()& *+,!+#,"+#
i ˙ − hΨ|Ψi] ˙ − Ψ H Ψ . L = [hΨ|Ψi 2
-# ()& ./%%/'"#, '& +0/ ( + 1+()&1+("$+% ./!12%+("/# "# ')"$) +%% ()& +3 !+1&(&!4 /. ()& '+5& .2#$("/# +!& !&+% )&#$& '& 4 %"( &+$) $/&6$"&#( "# "(4 !&+% +#0 "1+,"#+!7 +!(8 ClK (t) = rlK (t) + islK (t)9 :)"4 %&+04 (/ ()& 4&( /. 0";&!&#("+% &<2+("/#4=>?8 >>8 >@A X j
ηij y˙j = ∂yi H
BCCD
')&!& y 0&#/(&4 ()& 4&( /. +!+1&(&!4 (rlK , slK , qli , pli )8 ηij "4 + !&+% +#("3 4711&(!"$ (/!8 +#0 H "4 ()& E+1"%(/#"+# .2#$("/#F ηij = i[h∂yi Ψ|∂yj Ψi − h∂yj Ψ|∂yi Ψi] = −2 Im h∂yi Ψ|∂yj Ψi
H = Ψ H Ψ = H(rlK , slK , qli , pli ).
H.(&! $+!!7"#, /2( ()& $+%$2%+("/#4 B4&& + '!"((&# "# ()& ./!1 ηrlK smJ = −2δlm δKJ ηrlK qmh ηslK qmh ηrlK pmh ηslK pmh ηqlj pmh
BCGD BC?D
�"I HD ()& (/! η $+# J&
p p √ = δlm [plh rlK − 2 sl(K−1h ) kh − sl(K+1h ) kh + 1 ] p p √ = δlm [plh slK + 2 rl(K−1h ) kh − rl(K+1h ) kh + 1 ] p p √ = −δlm [qlh rlK + 2 rl(K−1h ) kh + rl(K+1h ) kh + 1 ] p p √ = −δlm [qlh slK + 2 sl(K−1h ) kh + sl(K+1h ) kh + 1 ] X 2 = −δlm δhj (rlK + s2lK ).
BC>+D BC>JD BC>$D BC>0D BC>&D BC>.D
K
K!/1 ()& +J/5& 1&(!"$ (/! ()& L#+% &<2+("/#4 /. 1/("/# (+M& ()& ./!1
N
! "#$ %&'(%)%" *+*"$&
2r˙lK + 2s˙ lK − q˙li
X K
p˙li
2 (rlK + s2lK ) −
X
2 (rlK + s2lK ) +
K
d X
i d X i
ηslK qli q˙li + ηrlK qli q˙li −
i d X i
ηslK pli p˙li = ∂slK H
,-./0
ηrlK pli p˙li = −∂rlK H
,-.10
X
ηrlK pli r˙lK −
X
ηslK pli s˙ lK = ∂pli H
,-.)0
X
ηrlK qli r˙lK +
X
ηslK qli s˙ lK = −∂qli H
,-.20
K
K
3%"#
d X
K
K
K = 1, ..., Kmax
i = 1, ..., d l = 1, ..., N
4$ 5 "%)$ "#/" "#$ &$"6%) "$5* 67 η 7 2 $* 5 " 2%6$)"(+ ) 8'($ "#$ 9/6%/1($* ! "3 2%:$6$5" $($)"6 5%) *"/"$* 1$)/8*$ "#$ $($)"6 5%) 3/9$!85)"% 5 %* 5 " '/6/&$"6%;$27 # 3$9$67 "#$ $<8/"% 5* /6$ ) 8'($2 1+ "#$ =<8/5"8& ! 6)$*> ∂yi H? @5 "#$ "#$6 #/52 "#$ &$"6%) 2%6$)"(+ ) 8'($* "#$ (qli , pli ) 9/6%/1($* ! "#$ ABC 3%"# "#$ PN) $D)%$5"* ! "#$ $E'/5*% 5? F<8/"% 5* -./G-.2 ) 5G *"%"8"$* / *$" ! 2 l (Nl + d) 2%:$6$5"%/( $<8/"% 5* 3#$6$ N %* "#$ 58&1$6 ! $($)"6 5%) *"/"$*7 Nl %* "#$ *%;$ ! "#$ 1/*%* *$" %5 "#$ $($)"6 5%) *"/"$ l7 /52 d %* "#$ 58&1$6 ! 58)($/6 2$H6$$* ! !6$$2 &? I#$ 58&$6%)/( * (8"% 5 ! "#$ *$" ! 2%:$6$5"%/( $<8/"% 5* -. %* 5 " /5 $/*+ "/*J? K52$$27 3#$5 "#$ 58&1$6 ! 9/6%/1($* %* *&/(( / '*$82 G%59$6*$ ! "#$ "$5* 6 η )/5 1$ ) &'8"$27 !"! 1+ *%5H8(/6 9/(8$ 2$) &' *%"% 57 /52 "#$ *+*"$& )/5 1$ * (9$2 8*%5H *"/52/62 &$"# 2* ! 6 62%5/6+ 2%:$6$5"%/( $<8/"% 5*? @5 "#$ "#$6 #/52 ! 6 '6 1($&* 3%"# / (/6H$ 58&1$6 ! 2$H6$$* ! !6$$2 & ) &'8"%5H / '*$82 G%59$6*$ ! η %* 5 " !$/*%1($ /52 &$"# 2* ! 6 %&'(%)%" 2%:$6$5"%/( $<8/"% 5*LMNO &8*" 1$ 8*$2? P$6$ 3$ #/9$ 8*$2 / *'$)%Q)/((+ "/%( 6$2 /''6 /)# 3#%)# ) &1%5$* / R$3" 5GS/'#* 5 &$"# 27 /52 /5 $E'(%)%" ! 86"#G 62$6 S85H$GT8""/ %5"$H6/"% 5? U$"/%(* /1 8" "#$ 58&$6%)/( %&'($&$5"/"% 5 3%(( 1$ H%9$5 $(*$3#$6$?
! "#$%&'()* )++*'(),'-./0 +-*)&-. 12.)$'(/ @5$ ! "#$ *%&'($*" "#$ 6$"%)/( & 2$( 2$*)6%1%5H / *$" ! $($)"6 5%) *"/"$* %5"$6/)"%5H 3%"# $59%6 5&$5"/( '# 5 5* %* )$6"/%5(+ "#$ * G)/(($2 P (*"$%5 ' (/6 5 P/&%(" 5%/5? K"* 3%2$*'6$/2 /''(%)/"% 5* %5 "#$ *"82+ ! )#/6H$ /52 $5$6H+ "6/5*' 6" %5 * (%2* /52 * !" &/""$6 &/J$ %" /5 $E)$(($5" "$*" ! 6 V
!" #$% &" !'(')'*+, -"." /" 0'123 '4 !" 3 2(+ '0 !" '4"5(6&"436'47) -')3 "64 8')7.'4 -7&6) '46749:;< :=>
H=
N X i
i c†i ci − J
N N X † X X (ci+1 ci + H.c.) + ωi a†i ai + g c†i ci (ai + a†i ) i
i
?@AB
i
a†i , ai 74( c†i , ci 7." C'3'4 74( 0".&6'4 1."7 6'4 74( 7446!6)7 6'4 '8".5 7 '.3< g 63 63 !" 3 ."4* ! '0 !" ")"1 .'458!'4'4 1'28)64*< ωi 7." !" C'3'4 0."D2"416"3 74( i 7." !" 36 " "4".*6"3, E" 4' 61" !7 64 !63 173" !" 42&C". '0 ")"1 .'461 3 7 "3 N 63 "D27) ' !" 42&C". '0 F6C.7 6'47) ("*.""3 '0 0.""('& d 74( 7." C' ! )7C"))"( 73 N , G2. !".&'." 64 '2. &'(") 3+3 "& /!"."
!" H.3 74( )73 36 "3 7." 1'28)"(<
!"!
/" 1'436(". 7 &')"12)7. .64*,
I4 !" 0'))'/64*< 0'. !" 37J" '0 36&8)616 +< /" /6)) 7332&" !7 7)) !" C'3'43 !7F" !" 37&" '8 617) 0."D2"41+
ω<
74( 7)) !" 36 "3 !7F" !" 37&"
"4".*+ ?i = 0B, E" /6)) ("31.6C" !" 36 " 8'82)7 6'4 (+47&613 C' ! 64 !" 4'457(67C7 61 ."*6&" ? |J/ω| < 1B 74( 64 !" 7(67C7 61 ."*6&" ? |J/ω| > 1B<
74( 73 7 0241 6'4 '0 !" ")"1 .'458!'4'4 1'28)64* 3 ."4* ! K364*
g,
!" 8.'8". 6"3 '0 #$%3 ("31.6C"( 64 788"4(6L M /" 174 "736)+
1'&82 " !" -7&6) '4674 0241 6'4<
H<
73
H(p, q, r, s) = H◦ + V /!"." ?3"" 788"4(6L NB
i o ω 2 2 (pli + qli2 ) + gqli + ωki (rlK + s2lK ) 2 l,i,K X p + (ωqli + g) 2(ki + 1)(rlK+1i rlK + slK+1i slK )
H◦ =
Xnh
l +
l,i,K
+
X
ωpli
l,i,K
V = 2J Re{ O!" (".6F7 6F"3 N,
∂yi H
p
2(ki + 1)(rlK slK+1i − rlK+1i slK )
X
l>m,K,J
C¯lK CmJ hK, αl |J, αm i}.
174 C" 'C 764"( 747)+ 617))+ 73 ("31.6C"( 64 788"4(6L
O!" P".'5 ! '.(". -7&6) '4674
H◦
641)2("3 7 1)733617)5)6J" "4".*+ 87.
!7 ("8"4(3 "L8)616 )+ '4 !" /7F"871J" 8'36 6'43 74( &'&"4 7
(qli (t), pli (t))<
74( 7 82.")+ D274 2& 1'4 .6C2 6'4 !7 ("8"4(3 '4 !" D274 2& 42&C".3
(k1 , ..., kd ),
O!" ".&3
hK, αl |J, αm i 788"7.64* 64 =
!" 8' "4 67) "4".*+
V
7."
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1.0
D1 GCS-1 GCS-2 MCTDH
0.8
P (t)
0.6
0.4
0.2
0.0 0
100
200
300
400
500
600
700
800
t (fs) 1.0
D1 GCS-1 GCS-2 MCTDH
0.8
P (t)
0.6
0.4
0.2
0.0 0
100
200
300
400
500
600
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t (fs)
!"#$% &' ()*#+,-!). )/ -0% !.!-!,+ %+%1-$).!1 2-,-% ,2 , /#.1-!). )/ -!3% !. , 3)+%1#+,$ $!." 4!-0 && %+%1-$).!1 2-,-%25 60% *,$,3%-%$2 )/ -0% 3)7%+ ,$% J = 100 13−1 8 ω = 1000 13−1 9 ,: g = 0.2ω8 ;: g = ω5 60% .#3;%$ )/ ;,-0 3)7%2 !2 N = 119 -0% 3,?@& ,.7 =>?@A5 B,20@7)--%7 +!.% C −·: %<1!-,-!). +%D%+2 )/ -0% N %+%1-$).!1 2-,-%2 c = 0 CB&:9 7,20%7 +!.% C@ @: c = 19 /#++ +!.% CE: c = 29 C×: F>6BG $%2#+-5 !
1.0
D1 GCS-1 GCS-2 MCTDH
0.8
P (t)
0.6
0.4
0.2
0.0 0
10
20
30
40
50
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70
80
t (fs) 1.0
D1 GCS-1 GCS-2 MCTDH
0.8
P (t)
0.6
0.4
0.2
0.0 0
10
20
30
40
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t (fs)
!"#$% &' ()*#+,-!). )/ -0% !.!-!,+ %+%1-$).!1 2-,-% ,2 , /#.1-!). )/ -!3% !. , 3)+%1#+,$ $!." 4!-0 55 %+%1-$).!1 2-,-%26 70% *,$,3%-%$2 )/ -0% 3)8%+ ,$% J = 1500 13−1 9 ω = 1000 13−1 : ,; g = 0.2ω9 <; g = ω6 70% .#3<%$ )/ <,-0 3)8%2 !2 N = 11: -0% 3,=!3#3 .#3<%$ )/ 2-,-% *%$ 3)8% !2 w = 4 !. <)-0 >?@A5 ,.8 >?@A&6 B,20A8)--%8 +!.% C −·; %=1!-,-!). +%D%+2 )/ -0% N %+%1-$).!1 2-,-%2 c = 0 CB5;: 8,20%8 +!.% CA A; c = 1: /#++ +!.% CE; c = 2: C×; F?7BG $%2#+-6
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α ¯ α ¯ D(α) = D(α)a† + D(α) 2 2 α α D(α) = −D(α)a − D(α) 2 2
R4!
(* *#-G)L 'K.#G& p p (a† − a) (a† − a) ∂ D(α) + i D(α) = D(α) √ − i D(α) D(α) = √ ∂q 2 2 2 2 † † (a + a) (a + a) q q ∂ D(α) = i √ D(α) − i D(α) = iD(α) √ + i D(α) ∂p 2 2 2 2
R4!YS R4!ZS
#&+ ∂ † (a† − a) † ∂ † p D (α) = − √ D(α) D (α) + i D† (α) = ∂q 2 ∂q 2 † † ∂ † ∂ (a + a) † q D (α) = −i √ D(α) D (α) − i D† (α) = ∂p 2 ∂p 2
R4![S R4!\S
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h X plh X ClK K, αl + ClK ∂qlh Ψ = −i 2 K K ∂plh Ψ = i
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r
hr
kh + 1 K + 1h , αl − 2
kh + 1 K + 1h , αl + 2
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X KJ
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2
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456A8
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<=7>(?
i
+
Xp i
2ki [(ωqli + g)slK−1i + ωpli rlK−1i ] + ∂slK V
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<=7>1?
i
+
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∂pli H = ωpli
2ki [(ωqli + g)rlK−1i − ωpli slK−1i ] + ∂rlK V
X
2 (rlK + s2lK ) +
K
∂qli H = (ωqli + g)
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X K
2 (rlK
<=7>+?
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r hr j + 1 i ji pni i ImhK, αl |J+1i , αn i− ImhK, αl |J−1i , αn i− RehK, αl |J, αn i 2 2 2 12,34 !"
∂pni V = −2
X
Vln (rlK rnJ + slK snJ )×
lKJ
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r i hr j + 1 ji qni i RehK, αl |J+1i , αn i+ RehK, αl |J−1i , αn i+ RehK, αl |J, αn i 2 2 2 12,564
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