Subscriber access provided by RYERSON UNIV
Article
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.
Page 1 of 40
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 $ § , 1 ∗
†
ACS Paragon Plus Environment
Journal of Chemical Theory and Computation
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 2 of 40
! ! ) $%&' ! $%&' ! * + # , " $%&' , - #
$%&' , * ) ) ! ! *
!
" # $ % # # &' ' # # #
(
# # $ ) ) *
ACS Paragon Plus Environment
Page 3 of 40
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 ! ./
,
$ # # ./) # 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- ./) ;
ACS Paragon Plus Environment
Journal of Chemical Theory and Computation
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 4 of 40
# ./ < 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
Page 5 of 40
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
&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
- # ' @
ACS Paragon Plus Environment
Journal of Chemical Theory and Computation
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 6 of 40
# 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
ACS Paragon Plus Environment
Page 7 of 40
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
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
ACS Paragon Plus Environment
Page 8 of 40
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
Journal of Chemical Theory and Computation
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
+Ω
q22 2
.
A
./) 0 γ = 4 × 10−4 ω1
)! ω = 2ω1 δt = 0.05ω1−1 ;E " 107 Ω E E@HE C c3 c4 EH25 × 10−4 EH40 × 10−4 , kB T = 0.03 ω1 T ∼ 60 > ω1 = 2π × 40 $5 # # 5 ) C
ACS Paragon Plus Environment
Page 9 of 40
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
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
ACS Paragon Plus Environment
Journal of Chemical Theory and Computation
-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
Page 10 of 40
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
ACS Paragon Plus Environment
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
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
average energy
Page 11 of 40
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 -
ACS Paragon Plus Environment
Journal of Chemical Theory and Computation
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
Page 12 of 40
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 # # ( *
ACS Paragon Plus Environment
Page 13 of 40
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
( ) 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 # %)$ ;
ACS Paragon Plus Environment
Journal of Chemical Theory and Computation
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 14 of 40
! 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 !
?
ACS Paragon Plus Environment
Page 15 of 40
2500 effective temperature (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
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- # @
ACS Paragon Plus Environment
Journal of Chemical Theory and Computation
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
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
ACS Paragon Plus Environment
Page 17 of 40
(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
ACS Paragon Plus Environment
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
Page 18 of 40
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
ACS Paragon Plus Environment
Page 19 of 40
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 E D 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
ACS Paragon Plus Environment
Journal of Chemical Theory and Computation
(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
ACS Paragon Plus Environment
Page 21 of 40
5 4 # ./ #
./)
> *BE >
- # / /% # ! $ , ) ) $ 0 γ E @ A $5 )! ν @ $5 $ γ ) +)% %) )F 1 *
0.03
reduced polarization
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
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 # # ./)
γ = 16 $5 1 ./) = +)%))F &'
# P = 16 = AE >
DE > *@@ > # # &' A; >
DC > *@C >
, *
γ ./) 1 γ # # *
ACS Paragon Plus Environment
Journal of Chemical Theory and Computation
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 22 of 40
' γ ./ = ! 6&7- #
( 5 ) 2 6&7- ./) # ./ ! ( 6&7- # # # $ γ , 2 2 # /% # # ./ ! # ./
# < ) ./ ) ) 6&7- ) ) = # # # # ./ ' # ./ # # γ # # ! #
$ !
6&7- ./) γ -
**
ACS Paragon Plus Environment
Page 23 of 40
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
) F ! # # , # ./) # # 2 # ' # ./) # # γ γ # 2 %)$ # )# γ % # # ! # ./ 0 0 6&7- # ' ./ = #
# 2 # ./)
( 2 # 2 # : O : /% # ( 2 8 )- / -8 # P / 2 , F +O 3Q) )1 ' % 2 $&F LO L F 4 O # F 4 O FJ+4
*;
ACS Paragon Plus Environment
Journal of Chemical Theory and Computation
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 24 of 40
< + # # # 2
. #
# . , $ : R(t) =
R(t) = 0 R(t)R(t + τ ) =
** +∞
−∞
IR (ω, T )−iωτ
ω 2π
*;
( )> R(t) IR (ω, T ) ' ./) IR # * tn = nδt : # # : # Xj Wj +∞
Rn ≡ R(tn ) =
*?
Wj Xj+n
j=−∞
# Xj 5 1 *; *? Wj # =
+∞
Wj Wj−l =
j=−∞
+∞ −∞
IR (ω, T )−iωtl
ω 2π
*@
˜ (ω) , 1 W (t)= tl = lδt W Wj ≡ W (tj ) =
+∞
−∞
*?
˜ (ω)−iωtj ω W 2π
ACS Paragon Plus Environment
*A
Page 25 of 40
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
M *A δt → 0= +∞
Wj Wj−l
j=−∞
∞
˜ (ω)W ˜ (−ω)−iωtl ω W 2π −∞
1 = δt
*B
˜ (ω) # F W (t) W *@ *B=
˜ (ω) = W
=
√ Wj = δt
δtIR (ω, T )
ω IR (ω, T )−iωtj 2π
+∞
−∞
*C
*D
' 5 1
1 Wj = √ N 1 Xj = √ N
N/2
˜ l −i2πjl/N W
;E
˜ l −i2πjl/N X
;
l=−N/2+1 N/2 l=−N/2+1
N # / *D ;E=
1 Wj = √ Nδω
+∞ −∞
˜ (ω)−iωtj ω W
;*
Nδtδω = 2π #
˜l = √ 1 W Nδt ωl = lδω 1 ;E ;
Rn =
IR (ωl )
;;
˜ −l X ˜ l −i2πnl/N W
;?
Rn N/2 l=−N/2+1
*@
ACS Paragon Plus Environment
Journal of Chemical Theory and Computation
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 26 of 40
;; Rn , = N/2 1 ˜ l −i2πnl/N Rn = √ IR (ωl )X Nδt l=−N/2+1
;@
˜ l - , 1 R 1 Rn = √ N
N/2
˜ l −i2πnl/N R
;A
l=−N/2+1
#
˜l = R
IR (ωl ) ˜ Xl δt
;B
˜ l # : # X ˜ + iN ˜l ˜ l = Ml √ X 2
;C
˜ l N ˜l : # 5 M ˜ l M ˜l = M ˜ −l # X ˜l = −N˜−l 1 = N ˜l = R
IR (ωl ) ˜ ˜l Ml + iN 2δt
;D
' Rn # =
˜ l N ˜l l = 1, .., N/2 − 1 : : # M ˜ l ;D * F R ˜l = R ˜l = R ˜ N −l l = N/2 + 1, .., N − 1 ; 4 5 R ˜ l = 0 l = 0 N/2 ? F R @ F Rn ;A *A
ACS Paragon Plus Environment
Page 27 of 40
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
!" #
- # , /% - )# # # # . & # # ' ) 2 # i m - # =
dpi i − pG pi − 1 . T r(pG ) pi = fi − γpi + R dt Wg Nf Wg
?E
i u Φ(u1 , ..., uN ) −γpi R fi = −∇ i - ./ pi ui #
pi pG dui ui = + dt m Wg
?
h " pG
pG h dh = dt Wg
?*
dpG 1 p2i = V (t)(X − Pext Id) + Id − γG pG + LG dt Nf i m
?;
V (t) Wg L L # Nf # Pext Id X ' # 7 ?; 5 −γG pG γG 0 # LG
; × ; 2γG Wg kB T # 5 δt # *B
ACS Paragon Plus Environment
Journal of Chemical Theory and Computation
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 28 of 40
F
. . . #' *BD
* 9 R $ ; 2 7
&. ./ .
,, + B
0 / 1 0 9 %
M & J P 29 ? F 8 9 I :
&. 0. . ,", A AC &. 0. . ,""
@ EA @ F ' + 9 7
&. 0. . , , ;;AC
A /
/ 8 9 7
&. 0. . , *? E@
B 2 $ 9 F P 9 - 9 $
9 : ! 8 )8
. . .
,"2 DEAE
C F 9 / : 9 &
. . . ,"2 E;EAE;
D F 1 9 I )% J 9 & & 9 1 F
. 0. 0. .
,%
E@E;) E@EA E 2 $ 9 2 7 $
9 1 1 &. ./
.
% ?;@?E*
4 I 9 > P 9 F * / 9 / 9 I +
. . 3
-# EA?;E@
. 0. 0. . , *@D
; / P 9 & 9 1 1 9 4 .
. 3
- * ? E
? / P 9 & 9 / - 7 9 :
+ 9 4 9 1 1
. 3
2 E*? E? *C
ACS Paragon Plus Environment
.
Page 29 of 40
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
@ 2 $ 9 $
9 F P 9 : ! 8 )8
. . .
,"'
DCDE* A /)J 9 - +
&. 0. . ,", CBAC)CBC;
B $ # 4 9 7
&. 0. . ,2, *??@ C
C F 9 / : 9 &
&. 0 1 0 . #
D : 4 9 + 5 + 9 1 5)4 I *E / ) 5 % J 9 / 8 )- 9 + * $ 5)+" 8 9 F 1 9 J 7 :
. . 3
. . 3
-' ;?*EB
BE) CE
- E ?;E;
&. 0. 1 0 .
,,
CA )CBE ** > # +
. 4. . *@@
*; F $ / 9 (
. . -2 ;?
*? 9 9 : + 9 O 5 7 +
+. . "2
*@@);EB *@ F P 9 2 $ 9 - 9 $
9 : ! 8 8 . .
3
-% **?;E *A / 8 )- 9 +
&. . . ,%% ABD)ACD
*B & F 9 > 9 R 9 / > *C 8 & :
&. ./ .
. . *
"
. . 3
!
*D
'# E*ABEA
," **? )**AE
*D : : 9 2 $ 9 $
9 . ;E IG2 : 9 R >
!
'# ED? E@
ACS Paragon Plus Environment
. 3
-' E ? ;
Journal of Chemical Theory and Computation
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
; - 7 9 :
+ 5 !
9 F % 9 ! ;* 6 ( 9 I #
. . 3
;; 6 ( 9 I # 9 + # >
2 @E?B
. . 3
;? . 9 & # ' 8
&. 0. .
;@ . 9 & # ' 8
0 . . 00.
;E
, "
ACS Paragon Plus Environment
A;E ?;*
,# ;**
Page 30 of 40
Page 31 of 40
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
$ %
;
ACS Paragon Plus Environment
Journal of Chemical Theory and Computation
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
ACS Paragon Plus Environment
Page 32 of 40
Page 33 of 40
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
Journal of Chemical Theory and Computation
ACS Paragon Plus Environment
Journal of Chemical Theory and ComputationPage 34 of 40
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
ACS Paragon Plus Environment
Page 35 of 40
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
Journal of Chemical Theory and Computation
ACS Paragon Plus Environment
Journal of Chemical Theory and ComputationPage 36 of 40
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
ACS Paragon Plus Environment
Page 37 of 40Journal of Chemical Theory and Computation
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
ACS Paragon Plus Environment
Journal of Chemical Theory and ComputationPage 38 of 40
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
ACS Paragon Plus Environment
Page Journal 39 of of 40Chemical Theory and Computation
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
ACS Paragon Plus Environment
Journal of Chemical Theory and ComputationPage 40 of 40
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
ACS Paragon Plus Environment
