The time-convolutionless (TCL) quantum master equation provides a powerful tool to simulate reduced dynamics of a quantum system coupled to a bath. The key quantity in the TCL master equation is the so-called kernel or generator, which describes effects of the bath degrees of freedom. Since the exact TCL generators are usually hard to calculate analytically, most applications of the TCL generalized master equation have relied on approximate generators using second and fourth order perturbative expansions. By using the hierarchical equation of motion (HEOM) and extended HEOM methods, we present a new approach to calculating the exact TCL generator and its high order perturbative expansions. The new approach is applied to the spin-boson model with different sets of parameters, to investigate the convergence of the high order expansions of the TCL generator. We also discuss circumstances where the exact TCL generator becomes singular for the spin-boson model, and a model of excitation energy transfer in the Fenna-Matthews-Olson complex.

[1]
A.
Nitzan
,
Chemical Dynamics in Condensed Phases: Relaxation, Transfer and Reactions in Condensed Colecular Systems
,
New York
:
Oxford University Press
, (
2006
).
[2]
U.
Weiss
,
Quantum Dissipative Systems
, 4th Edn.
New Jersey
:
World Scientific
, (
2012
).
[3]
B. J.
Berne
,
G.
Ciccotti
, and
D. F.
Coker
, Eds.,
Classical and Quantum Dynamics in Condesed Phase Simulations
,
New Jersey
:
World Scientific
, (
1998
).
[4]
V.
May
and
O.
Kühn
,
Charge and Energy Transfer Dynamics in Molecular Systems
, 3rd Edn.
Weinheim
:
Wiley-VCH
, (
2011
).
[5]
D. E.
Makarov
and
N.
Makri
,
Chem. Phys. Lett.
221
,
482
(
1994
).
[6]
N.
Makri
and
D. E.
Makarov
,
J. Chem. Phys.
102
,
4600
(
1995a
).
[7]
N.
Makri
and
D. E.
Makarov
,
J. Chem. Phys.
102
,
4611
(
1995b
).
[8]
H.
Wang
,
M.
Thoss
, and
W. H.
Miller
,
J. Chem. Phys.
115
,
2979
(
2001
).
[9]
H.
Wang
and
M.
Thoss
,
J. Chem. Phys.
119
,
1289
(
2003
).
[10]
Y.
Tanimura
and
R.
Kubo
,
J. Phys. Soc. Jpn.
58
,
101
(
1989
).
[11]
A.
Ishizaki
and
G. R.
Fleming
,
J. Chem. Phys.
130
,
234111
(
2009
).
[12]
J. C.
Tully
,
Faraday Discuss.
110
,
407
(
1998
).
[13]
J. C.
Tully
,
J. Chem. Phys.
137
,
22A301
(
2012
).
[14]
M.
Thoss
,
H. B.
Wang
, and
W. H.
Miller
,
J. Chem. Phys.
115
,
2991
(
2001
).
[15]
T. C.
Berkelbach
,
D. R.
Reichman
, and
T. E.
Markland
,
J. Chem. Phys.
136
,
034113
(
2012
).
[16]
A.
Montoya-Castillo
,
T. C.
Berkelbach
, and
D. R.
Reichman
,
J. Chem. Phys.
143
,
194108
(
2015
).
[17]
W. T.
Pollard
,
A. K.
Felts
, and
R. A.
Friesner
,
Adv. Chem. Phys.
93
,
77
(
1996
).
[18]
H. P.
Breuer
and
F.
Petruccione
,
The Theory of Open Quantum Systems
,
Oxford University Press on Demand
, (
2002
).
[19]
Y.
Yan
and
R.
Xu
,
Annu. Rev. Phys. Chem.
56
,
187
(
2005
).
[20]
K.
Blum
,
Density Matrix Theory and Applications
,
Springer Science & Business Media
, (
2013
).
[21]
S.
Nakajima
,
Prog. Theor. Phys.
20
,
948
(
1958
).
[22]
R.
Zwanzig
,
J. Chem. Phys.
33
,
1338
(
1960
).
[24]
H.
Mori
,
Prog. Theor. Phys.
33
,
423
(
1965
).
[25]
A. J.
Leggett
,
S.
Chakravarty
,
A. T.
Dorsey
,
M. P. A.
Fisher
,
A.
Garg
, and
W.
Zwerger
,
Rev. Mod. Phys.
59
,
1
(
1987
).
[26]
G. D.
Mahan
,
Many-Particle Physics
,
New York
:
Springer Science & Business Media
, (
2013
).
[27]
E.
Gull
,
A. J.
Millis
,
A. I.
Lichtenstein
,
A. N.
Rubtsov
,
M.
Troyer
, and
P.
Werner
,
Rev. Mod. Phys.
83
,
349
(
2011
).
[29]
D. R.
Reichman
,
F. L. H.
Brown
, and
P.
Neu
,
Phys. Rev. E
55
,
2328
(
1997
).
[30]
M.
Aihara
,
H. M.
Sevian
, and
J. L.
Skinner
,
Phys. Rev. A
41
,
6596
(
1990
).
[31]
Q.
Shi
and
E.
Geva
,
J. Chem. Phys.
119
,
12063
(
2003
).
[32]
E. Y.
Wilner
,
H.
Wang
,
M.
Thoss
, and
E.
Rabani
,
Phys. Rev. B
89
,
205129
(
2014
).
[33]
J.
Cerrillo
and
J.
Cao
,
Phys. Rev. Lett.
112
,
110401
(
2014
).
[34]
M.
Xu
,
Y.
Yan
,
Y.
Liu
, and
Q.
Shi
,
J. Chem. Phys.
148
,
164101
(
2018
).
[35]
M.
Tokuyama
and
H.
Mori
,
Prog. Theor. Phys.
56
,
1073
(
1976
).
[36]
N.
Hashitsumae
,
F.
Shibata
,
M.
Shingū
,
J. Stat. Phys.
17
,
155
(
1977
).
[37]
F.
Shibata
,
Y.
Takahashi
, and
N.
Hashitsume
,
J. Stat. Phys.
17
,
171
(
1977
).
[38]
S.
Chaturvedi
and
F.
Shibata
,
Z. Phys. B
35
,
297
(
1979
).
[39]
H.
Grabert
,
P.
Schramm
, and
G. L.
Ingold
,
Phys. Rep.
168
,
115
(
1988
).
[40]
B.
Garraway
,
Phys. Rev. A
55
,
2290
(
1997
).
[41]
M.
Murao
and
F.
Shibata
,
J. Phys. Soc. Jpn.
64
,
2394
(
1995
).
[42]
A.
Smirne
,
H. P.
Breuer
,
J.
Piilo
, and
B.
Vacchini
,
Phys. Rev. A
82
,
062114
(
2010
).
[43]
W. M.
Zhang
,
P. Y.
Lo
,
H. N.
Xiong
,
M. W. Y.
Tu
, and
F.
Nori
,
Phys. Rev. Lett.
109
,
170402
(
2012
).
[44]
G.
Nan
,
Q.
Shi
, and
Z.
Shuai
,
J. Chem. Phys.
130
,
134106
(
2009
).
[45]
L.
Kidon
,
E. Y.
Wilner
, and
E.
Rabani
,
J. Chem. Phys.
143
,
234110
(
2015
).
[46]
S.
Jang
,
J.
Cao
, and
R. J.
Silbey
,
J. Chem. Phys.
116
,
2705
(
2002
).
[47]
G.
Gasbarri
and
L.
Ferialdi
,
Phys. Rev. A
97
,
022114
(
2018
).
[48]
D.
Chruściński
and
A.
Kossakowski
,
Phys. Rev. Lett.
104
,
070406
(
2010
).
[49]
D.
Egorova
,
M.
Thoss
,
W.
Domcke
, and
H.
Wang
,
J. Chem. Phys.
119
,
2761
(
2003
).
[50]
M.
Schröder
,
U.
Kleinekathöfer
, and
M.
Schreiber
,
J. Chem. Phys.
124
,
084903
(
2006
).
[51]
S.
Mukamel
,
Principles of Nonlinear Optical Spectroscopy
,
New York
:
Oxford
, (
1995
).
[52]
R.
Doll
,
D.
Zueco
,
M.
Wubs
,
S.
Kohler
, and
P.
Hänggi
,
Chem. Phys.
347
,
243
(
2008
).
[53]
A.
Ishizaki
and
Y.
Tanimura
,
J. Phys. Soc. Jpn.
74
,
3131
(
2005
).
[54]
Q.
Shi
,
L. P.
Chen
,
G. J.
Nan
,
R. X.
Xu
, and
Y. J.
Yan
,
J. Chem. Phys.
130
,
084105
(
2009
).
[55]
M.
Xu
,
L.
Song
,
K.
Song
, and
Q.
Shi
,
J. Chem. Phys.
146
,
064102
(
2017
).
[56]
M.
Sparpaglione
and
S.
Mukamel
,
J. Chem. Phys.
88
,
3263
(
1988
).
[57]
M.
Sparpaglione
and
S.
Mukamel
,
J. Chem. Phys.
88
,
4300
(
1988
).
[58]
H. T.
Chen
,
T. C.
Berkelbach
, and
D. R.
Reichman
,
J. Chem. Phys.
144
,
154106
(
2016
).
[59]
M. G.
Mavros
and
T.
Van Voorhis
,
J. Chem. Phys.
141
,
054112
(
2014
).
[60]
[61]
S.
Chakravarty
and
A. J.
Leggett
,
Phys. Rev. Lett.
52
,
5
(
1984
).
[62]
Q.
Shi
,
L. P.
Chen
,
G. J.
Nan
,
R. X.
Xu
, and
Y. J.
Yan
,
J. Chem. Phys.
130
,
164518
(
2009
).
[63]
[64]
Y.
Tanimura
,
J. Phys. Soc. Jpn.
75
,
082001
(
2006
).
[65]
R.
Xu
and
Y.
Yan
,
Phys. Rev. E
75
,
031107
(
2007
).
[66]
Y.
Tanimura
and
S.
Mukamel
,
J. Phys. Soc. Jpn.
63
,
66
(
1994
).
[67]
A.
Ishizaki
and
Y.
Tanimura
,
Chem. Phys.
347
,
185
(
2008
).
[68]
Y. A.
Yan
,
F.
Yang
,
Y.
Liu
, and
J. S.
Shao
,
Chem. Phys. Lett.
395
,
216
(
2004
).
[69]
Y.
Zhou
,
Y. A.
Yan
, and
J. S.
Shao
,
Europhys. Lett.
72
,
334
(
2005
).
[70]
Y.
Zhou
and
J. S.
Shao
,
J. Chem. Phys.
128
,
034106
(
2008
).
[71]
S.
Jang
,
J. Chem. Phys.
131
,
164101
(
2009
).
[72]
D. P.
McCutcheon
and
A.
Nazir
,
New. J. Phys.
12
,
113042
(
2010
).
[73]
H. T.
Chang
,
P. P.
Zhang
, and
Y. C.
Cheng
,
J. Chem. Phys.
139
,
224112
(
2013
).
[74]
M.
Cho
,
H. M.
Vaswani
,
T.
Brixner
,
J.
Stenger
, and
G. R.
Fleming
,
J. Phys. Chem. B
109
,
10542
(
2005
).
[75]
S. I.
Vulto
,
M. A.
de Baat
,
R. J.
Louwe
,
H. P.
Permentier
,
T.
Neef
,
M.
Miller
,
H.
van Amerongen
, and
T. J.
Aartsma
,
J. Phys. Chem. B
102
,
9577
(
1998
).
This content is only available via PDF.
You do not currently have access to this content.