Zero-Point Energy Leakage in Quantum Thermal ... - ACS Publications

Oct 21, 2016 - Sorbonne Universités, UPMC Université Paris 06, CNRS, Institut des Nanosciences de Paris (INSP), 4 Place Jussieu, 75252 Paris,. Franc...
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Zero-point Energy Leakage in Quantum Thermal Bath Molecular Dynamics Simulations Fabien Brieuc, Yael Bronstein, Hichem Dammak, Philippe Depondt, Fabio Finocchi, and Marc Hayoun J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.6b00684 • Publication Date (Web): 21 Oct 2016 Downloaded from http://pubs.acs.org on October 24, 2016

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

Journal of Chemical Theory and Computation is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

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Journal of Chemical Theory and Computation

                      † §   ‡ §    ∗ † ¶    ‡ 



 

 

 ∗ ‡    ¶ 

                         !           "#   $          %   &  '

  !      $  (   )    *+       ,,-    !                

                                                          !          "    # $%&'           ! #!           (  !                

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



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

   $%&' ,             *               )              )              !              !            *                     

                                                       !   

 "    #      $                              %            #                  #                      &' '               # #         #         



 (

  #  #                         $                          )          )  *

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Journal of Chemical Theory and Computation

             #  #        

     +      #    , -      #  )             #  ./        0          !                         ./           

    



 ,             

           $         #       # ./)           #            1          )        2     3 #          #    $  ./      # 2 1            

   



4  ./       5 )    2 

6&7- 2    #       " 



 6&7-  2  #           )           )                  "   6&7-   #     !              - ) 8   



9           ./  6&7-   # 

   # /  ) 5   2  $          6&7-   ./   2   #     4      6&7-  #  ./)       #    /  ) 5   2                #       #  $      )  

   2   2     :     2   

         6&7-          2    #              



$            6&7-  ./)     ;

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          #             ./                     <   6&7-         3     #       ./               1        =          )                  - )8              / %3  '              



     

 ./   #    ,    -                !      /         -        ./               x    m   #        f (x)  = 

m¨ x = f (x) − mγ x˙ + R(t)

 

                      )          # #       R(t)   #   :    5   =

< R(t) > = 0  < R(t)R(t + τ ) > =

* +∞ −∞

IR (ω, T ) −iωτ

ω . 2π

;

7  ;   ( )>           
            &4 IR (ω, T )     T         #    ?

ACS Paragon Plus Environment

      

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&4     &4    Ix  #     <  )              = 

χ˜ (ω) =

ω Ix (ω, T ) 2kB T

?



˜ (ω)         # χ(ω) ˜     1      χ ˜    x ˜(ω)   1           R(ω)      =

˜ x˜(ω) = χ(ω) ˜ R(ω)

@

1      #     #  &4    Ix   &4       IR = 2 Ix (ω, T ) = |χ(ω)| ˜ IR (ω, T )

A

  <  )       #    = 

2kB T χ˜ (ω) IR (ω, T ) = . 2 ω |χ(ω)| ˜ '                 ω0  

B

 @ 

 #  

χ(ω) ˜ =

m [ω02

1 . − ω 2 + iγω]

C

/         B  &4        #           =

IR (ω, T ) = 2mγ kB T

∀ω.

/       ( )>    ;  

D   

-     #   '                @

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   #              kB T              ω  '             #       #





1 1 ⎦   θ(ω, T ) = ω ⎣ + 2 exp ω − 1 kB T

 E

              ./       &4          D #           #    E       #          <  )       # F   (    # > #  = 

χ ˜ (ω) =

ω Ix (ω, T ) 2θ(ω, T )





     ( )>       &4      R(t)     ./  =

IR (ω, T ) = 2mγ θ(ω, T )

 *

'     -     IR  ω         R(t)  #          #               )! ω               #                  )     ./          =    0 γ         )! ω              #     (   -                0 γ    #                    ,    #                # /   +  

ω    #                #             )              A

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 ω   γ                '             2        #     ω      2ω  ω #            '           ./)           0 γ                 D  *



 

      '      #   ./)          )         2      #                ./)             )   4 G           # #  $  

H= 1 1 1 1 H = mx˙ 21 + mω12 x21 + mx˙ 22 + mω22 x22 + C3 (x1 − x2 )3 + C4 (x1 − x2 )4 2 2 2 2

 ;

  x1  x2          ω1  ω2      )   m      C3  C4       $   H   #

˜ = H/ω1   =       H 2 2 2 2 ˜ = q˙1 + q1 + q˙2 + Ω2 q2 + c3 (q1 − q2 )3 + c4 (q1 − q2 )4 H 2 2 2 2

 ?

      #   =

ω2 Ω= , ω1

ξ=

 , mω1

xi qi = , ξ

C3 ξ 3 c3 = , ω1

C4 ξ 4 c4 = , ω1

t∗ = ω1 t,

q˙i =

dqi .  @ dt∗

q1  q2             Ω                 ω1 > ω2   )          # B

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Ω=0.5

0.5 0.4 0.3 0.2 0

2

4

6

8 104 x c3

0.6 average energy

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

average energy

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10

12

14

Ω=0.25

0.4 0.2 0 0

5

10

QTB

15

20 4 10 x c4

QTB

ε1

25

30

QTB

ε2

εc

35

Exact

1  =     1  2              c   # ./)             c3  c4    =  #   c3 = 0 c4 = 0  Ω = 0.5 -  =     c4 = 0 c3 = 0  Ω = 0.25 /   c = 0    #                         6&7-   ./                  '         <    #                 1  2   =

1 =

q˙12 2



+

q12 2



,

2 =

q˙22 2



2



q22 2

.

 A

 ./)            0 γ = 4 × 10−4 ω1

 )!    ω  = 2ω1    δt = 0.05ω1−1             ;E     "     107        Ω      EE@HEC       c3  c4      EH25 × 10−4  EH40 × 10−4                ,          kB T = 0.03 ω1  T ∼ 60 >  ω1 = 2π × 40 $5           #            #     5 )                          C

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1 0.8

ζ

(a)

4

10 x c3 = 2.4 9.7 14.6 19.4

0.6 0.4 0.2 0 0.48

0.5

0.52

0.54

0.56

0.58

0.6

Ω 1

0.6

ζ

(b)

104 x c4 = 3.9 7.7 15.4 30.8

0.8

0.4 0.2 0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Ω

1  *= 6 )    2   ,         ζ  B 

         Ω   @   =  #   c3 = 0 c4 = 0 -  =     c4 = 0 c3 = 0                              5 )  9      1 = 0.5  2 = Ω/2 1 

      #    ./      =

Ω = 0.5    #    Ω = 0.25                c3 = 0  c4 = 0  ./                     5 )   '         c3  c4      ./)           =         

       *  6&7-  

'        6&7-          Ω c3 c4    ,  $   1          ζ        6&7-=

Δ   − Δ  (   −   ) − (  −  ) 1 2 1 2 ζ= =       Δ 1 − 2

 B

(  ,   2       ζ = 1          D

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-4

(a)

arbitrary units 0

ω1-ω2

2ω2

0.5

1

γ=4x10 ω1

ω1+ω2

1.5

2

0

-4

(b)

-3

(c)

γ=4x10 ω1

0.5

1

2 -3

(d)

γ=4x10 ω1

1.5

γ=4x10 ω1

arbitrary units

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0

0.5

1

1.5

2

0

0.5

ω/ω1 oscillator 1

1

ω/ω1

1.5

2

oscillator 2

1  ;= I#              * #  # ./)           #       Ω = 0.5  c3 = 2.4 × 10−4

              Ω = 0.2  c4 = 15.4 × 10−4                 0= γ = 4 × 10−4 ω1  γ = 4 × 10−3 ω1 

        =  =   '        2   ζ = 0 1 2      1,2 = 1,2  ' ,  *    #   ζ     Ω  ! 

    c3  c4     %       6&7-                         Ω '   #        Ω = 0.5 ,  *  '   #        2  # #      #          2ω    ,  #  #               ./)    #   ,  ; =    2ω2 ω1 − ω2  ω1 + ω2  #     Ω = 0.5      #       (ω1 ; 2ω2)  (ω2 ; ω1 − ω2 ) 4        #            3ω 9 6&7   #       Ω = 1/3 ,  *# (        ,  6&7-           Ω 1  ;#       Ω = 0.2  c4 = 15.4 × 10−4          ω1  ω2    E

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Ω=0.5

0.5

2x10-2 1x10-2

0.4

4x10

-3

4x10

-4

4x10-3

0.3

1x10-2 -2 2x10

0

average energy

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average energy

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2

4

6

8 4 10 x c3

10

12

14

Ω=0.25

0.5 0.4 0.3 0.2 0.1 0

5

10

15

20 4 10 x c4

25

30

35

1  ?= 7!    0 γ     ω1             A            =  #   c3 = 0 c4 = 0  Ω = 0.5 -  =     c4 = 0 c3 = 0  Ω = 0.25      #      #    ./)                 $      2      ,  6&7-      Ω < 1/3          γ

  0      

4 × 10−4 ω1  2 × 10−2 ω1  (            6&7-    = Ω = 0.5   #    Ω = 0.25       ,  * 1  ?     6&7-      γ  '   γ      2                #   '     c3 = 2.4 × 10−4

    γ    4 × 10−3 ω1   0    6&7- ζ = 0.08 1  ;   #    #          γ =   6&7-   #      2        2ω2 ω1 − ω2  ω1 + ω2                 #        6&7-      γ     #      2                               

     γ/2π  -             

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                      γ  6&7-       ,  ? 1  ;        γ     

     Ω = 0.2  c4 = 15.5 × 10−4    0          #  !    '           ttr       #          JI7         

     ttr

   #     #              *       1  @      ttr  Ω = 0.5       #     c3                            6&7-         γ                  ω = 1/ttr  1       

c3 = 4 × 10−4  ,   ttr ∼ 400ω1−1    ω ∼ 2.5 × 10−3ω1    ,  ?        γ = 10−2 ω1         2    6&7-    γ  ω  Ω=0.5 1500

-1

ttr (ω1 )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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1000

500 1

1.5

2

2.5

3

3.5

4

-4

10 x c3

1  @= 7     #      ttr    ω1−1       #     c3  Ω = 0.5     JI7               $  c4 = 0 '              =     

  #       ,     !   6&7- #    γ #              #    (  *

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          (   )         3        3        #       # 

   # #      %      %H$     )      # 8    )#   =

V (r) =

−b(r−r0 ) 

a(r−r0 )  u0 a e − 1 + b e − 1 − u0 a + bea(r−r0 )

 C

  r   %H$   u0        #  r0   #   %H$   a  b                   = r0 = 0.96 K            #   %$−  a  7.11 K−1

b  2.00 K−1  u0 = 2.73 I     %H$      ν     

       V      100 $5 %      %H%     # #      =

2 V (R) = C0 1 − e−α0 (R−R0 ) − C0

 D

  C0  α0              R0  %H%  #           = C0 = 3.81 I R0 = 2.88 K 

α0            %H%    ν   # 10  60 $5  ./)         0.1      #          #   *     "   3                # V (r) + V (R − r)  

 #)   (       ,   L L %H$ #  ∼ 1 K

  L  # L ∼ 1.9 K                            5 #   %)$       ;

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             !  6&7-     )#                    )   ν2      %H%           )   ν1      %H$     '           %H$       

    )       %H%                 '      <      Ω = ν2 /ν1     0 γ   6&7-  T = 600 >    

ν2          α0   D  ν  ,  100 $5       ν1      ./)          &'        # P = 20                       1   ./)     2                2        

          # &' '         2      2       ,     )                  )        !    T  T  $  %   ,    2  = N 1  kB T (i) = E 2 N i=1 k

N 1  (i) kB T = E , 2 N i=1 k

*E

(i)

  N = 3  N = 3    #   $  %    Ek 

  2     i '              2       #         =        !               = )         2     !                   ./    &'         1  ,  A         2         !          

?

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2500 effective temperature (K)

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Journal of Chemical Theory and Computation

H - PIMD ZPEL

2000

H - QTB-MD O - QTB-MD

ZPEL

1500

O - PIMD

1000 0

1000

2000 time (ps)

3000

4000

1  A= 7!    T  T             ./)  &'  T = 600K  $  Ω = 0.5  γ = 0.2 $5              6&7-       '     6&7-   #  ,         =

ζ=

(T − T )() − (T − T )() (T − T )()

.

* 

ζ = 0       2   0 < ζ < 1   2         Ω    ,  B 4          3    ; 6&7-     Ω ∼ 1/2 1  B      6&7-   #  #      #    γ         %        6&7-  #   Ω < 0.2=                           !               Ω

          

1  C    #    

  d  d     Ω = 0.5     ./) &'    )     ' ,  C        d  #     ! #  6&7- %      d  #        6&7-=  ./)  #    #         2              6&7- $   6&7-     # @

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1

Page 16 of 40

γ = 0.2 THz γ = 2 THz γ = 5 THz γ = 10 THz γ = 20 THz

0.8 0.6 ζ

0.4 0.2 0 0.2

0.3

0.4

0.5

0.6

0.7

Ω

1  B= 6&7-  , #      ζ   ,   *  

        Ω    γ  ./) d  #      &'            

(            

          γ       #        9       2  #      2             γ       ,  ; 1  D   #       )        Ω = 0.5    !      γ 0.2  10 $5 1 γ = 0.2 $5 6&7-     γ = 10 $5  6&7      ,  B (            0    #    2    $      2           , #        γ  7        #             #           

'        !      2    #    #         6&7-                6&7-          # #     F         0     6&7- '                   9          A

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(a) : = 0.5

-1

probability density (Å )

2 Standard-MD PIMD

1.5

QTB-MD (0.2 THz) QTB-MD (20 THz)

1 0.5 0 0.8

1 d

1.2

OH

-1

10

1.4

(Å)

12 probability density (Å )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Journal of Chemical Theory and Computation

: = 0.5

(b)

8 6 4 2 0 2.7

2.8 d

2.9 (Å)

3

3.1

OO

1  C= & # #  #    %H$     %H% #         ) &'  ./)          0 γ 0.2  20 $5   Ω = 0.5 #      #   #            



      

(     !     0 γ   6&7-            

      M ./)    /  ) N5  .      6&7-  T = 10 >           # - )8    /kB = 1450.6 > σ = 2.54 K  ! = 1.37σ = 3.49 K '                  )      )    #   ./    #    B

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Journal of Chemical Theory and Computation

0.7

γ=2 THz γ=10 THz

0.6 0.5 arbitrary units

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0.4 0.3 0.2 0.1 0 20

40

60

80

100

120

frequency (THz)

1  D= I#    #  # ./)     Ω = 0.5          0= γ = 0.2 $5 #     γ = 10 $5        #     2  (     ./)     1   

 !      γ  ( ,     γ = 0.9 $5 ./)              #      #       ,  E '      #       #             #  $       γ  E $5     #    ./)           #  θ(ν, T ) 

E   #

     ,  E           6&7-    5 #  ./    ,  E               #    5 #         #      γ        0    6&7-   γ = 9 $5            1       γ     2              #  #  # ./)                               ./   -       # #     #    2   #       )            %4 1       γ   γ                Δω     2   %4    γ   ,    #    F        

γ  #     #           Δν = Δω/2π = γ/2π   C

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J = 0.9 THz J = 10 THz

200

150 1

slope / QD slope

K

B

2 E /k (K)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Journal of Chemical Theory and Computation

100

50

0.8 0.6 0.4 0.2 0

0

2

0

4

2

4

6

6

J (THz)

8

10

8

frequency (THz)

1  E= 4  2    #       T = 10 >             #    ./)     γ     ED  E $5       )! 20 $5                            #  . θ(ν, T )  E  ν = ω/2π  '=       γ         #    5         #       , 

 '         -      %4

 #  #   5  1             #

kB T  '     ./) kB T   #   # θ(ω, T )  E 1 

  

    6&7-    %4 #    ./)  "             ) '   , 

#     6&7-   

%4   # #     ./)  "     #  )      )               

      / %3 /%              5 #             !   #   <          '            )    )      =   #  +    # %       # F    7                                     ,  

                      D

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(a)

Standard-MD (10 THz) QTB-MD (10 THz)

(b)

Standard-MD (2 THz) QTB-MD (2 THz)

DOS (ps)

500 400 300 200 100 0

500

DOS (ps)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 40

400 300 200 100 0

0

5

10

15

20

frequency (THz)

1  = I#       %4          !      γ     #       1        )          5 # kB T      -    # θ(ν, T )  E      ./)         0 γ   =   10 $5

 # 2 $5

*E

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 5  4  #       ./    #     

      ./)               

>  *BE > 

-  #           /         /%     #  ! $         , )   )                 $                    0 γ       E@  A $5    )!    ν                    @ $5 $           γ           )     +)% %) )F 1  * 

    0.03

reduced polarization

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Journal of Chemical Theory and Computation

PIMD QTB (16 THz)

0.02 R

C

C

200

T O

T (K)

0.01

T

O

300

100

R 0

0 100

1

J(THz)

150

200

10

250

300

T (K)

1  *=           5       )      / %3  #  # ./) γ = 16 $5  &' P = 16    I            #    +)% %)  )F                       0 γ   ./     5           #  # &'  5              #  # ./) 

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Page 22 of 40

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Journal of Chemical Theory and Computation

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Page 24 of 40

       

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Journal of Chemical Theory and Computation

M   *A       δt → 0= +∞ 

Wj Wj−l

j=−∞





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   ;; Rn ,  = N/2   1 ˜ l −i2πnl/N Rn = √ IR (ωl )X Nδt l=−N/2+1

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Journal of Chemical Theory and Computation



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; × ;                     2γG Wg kB T         #      5         δt   #              *B

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