We present a scheme to express a bath correlation function (BCF) corresponding to a given spectral density (SD) as a sum of damped harmonic oscillations. Such a representation is needed, for example, in many open quantum system approaches. To this end we introduce a class of fit functions that enables us to model ohmic as well as superohmic behavior. We show that these functions allow for an analytic calculation of the BCF using pole expansions of the temperature dependent hyperbolic cotangent. We demonstrate how to use these functions to fit spectral densities exemplarily for cases encountered in the description of photosynthetic light harvesting complexes. Finally, we compare absorption spectra obtained for different fits with exact spectra and show that it is crucial to take properly into account the behavior at small frequencies when fitting a given SD.

1.
U.
Weiss
,
Quantum Dissipative Systems
, 3rd ed. (
World Scientific Publishing Company
,
2008
).
2.
V.
May
and
O.
Kühn
,
Charge and Energy Transfer Dynamics in Molecular Systems
(
Wiley-VCH
,
2000
).
3.
H.-P.
Breuer
and
F.
Petruccione
,
The Theory of Open Quantum Systems
(
Oxford University Press
,
2002
).
4.
C.
Meier
and
D. J.
Tannor
,
J. Chem. Phys.
111
,
3365
(
1999
).
5.
U.
Kleinekathöfer
,
J. Chem. Phys.
121
,
2505
(
2004
).
6.
A.
Ishizaki
and
Y.
Tanimura
,
J. Phys. Soc. Jpn.
74
,
3131
(
2005
).
7.
N. S.
Dattani
,
F. A.
Pollock
, and
D. M.
Wilkins
,
Quantum Phys. Lett.
1
,
35
(
2012
).
8.
N.
Singh
and
P.
Brumer
,
Mol. Phys.
110
,
1815
(
2012
).
9.
C.
Kreisbeck
,
T.
Kramer
,
M.
Rodríguez
, and
B.
Hein
,
J. Chem. Theory Comput.
7
,
2166
(
2011
).
10.
D.
Süß
,
A.
Eisfeld
, and
W. T.
Strunz
, “
Hierarchy of stochastic pure states for open quantum system dynamics
,” e-print arXiv:1402.4647 [quant-ph].
11.
M. R.
Wall
and
D.
Neuhauser
,
J. Chem. Phys.
102
,
8011
(
1995
).
12.
V. A.
Mandelshtam
and
H. S.
Taylor
,
J. Chem. Phys.
107
,
6756
(
1997
).
13.
N. S.
Dattani
,
D. M.
Wilkins
, and
F. A.
Pollock
, e-print arXiv:1205.4651 [quant-ph].
14.
S.
Mostame
,
P.
Rebentrost
,
A.
Eisfeld
,
A. J.
Kerman
,
D. I.
Tsomokos
, and
A.
Aspuru-Guzik
,
New J. Phys.
14
,
105013
(
2012
).
15.
I.
Burghardt
,
R.
Martinazzo
, and
K. H.
Hughes
,
J. Chem. Phys.
137
,
144107
(
2012
).
16.
Y.
Tanimura
and
R.
Kubo
,
J. Phys. Soc. Jpn.
58
,
101
(
1989
).
17.
V.
Chernyak
and
S.
Mukamel
,
J. Chem. Phys.
105
,
4565
(
1996
).
18.
V.
Barsegov
,
V.
Chernyak
, and
S.
Mukamel
,
Isr. J. Chem.
42
,
143
(
2002
).
19.
M. M.
Toutounji
and
G. J.
Small
,
J. Chem. Phys.
117
,
3848
(
2002
).
20.
M.
Toutounji
,
J. Chem. Phys.
130
,
094501
(
2009
).
21.
M.
Wendling
,
T.
Pullerits
,
M. A.
Przyjalgowski
,
S. I. E.
Vulto
,
T. J.
Aartsma
,
R.
van Grondelle
, and
H.
van Amerongen
,
J. Phys. Chem. B
104
,
5825
(
2000
).
22.
M.
Rätsep
and
A.
Freiberg
,
J. Lumin.
127
,
251
(
2007
).
23.
A.
Damjanović
,
I.
Kosztin
,
U.
Kleinekathöfer
, and
K.
Schulten
,
Phys. Rev. E
65
,
031919
(
2002
).
24.
C.
Olbrich
and
U.
Kleinekathöfer
,
J. Phys. Chem. B
114
,
12427
(
2010
).
25.
S.
Shim
,
P.
Rebentrost
,
S.
Valleau
, and
A.
Aspuru-Guzik
,
Biophys. J.
102
,
649
(
2012
).
26.
S.
Valleau
,
A.
Eisfeld
, and
A.
Aspuru-Guzik
,
J. Chem. Phys.
137
,
224103
(
2012
).
27.
A. O.
Caldeira
and
A. J.
Leggett
,
Ann. Phys.
149
,
374
(
1983
).
28.
R. S.
Knox
,
G. J.
Small
, and
S.
Mukamel
,
Chem. Phys.
281
,
1
(
2002
).
29.
A.
Kell
,
X.
Feng
,
M.
Reppert
, and
R.
Jankowiak
,
J. Phys. Chem. B
117
,
7317
(
2013
).
30.
J.
Hu
,
R.-X.
Xu
, and
Y.
Yan
,
J. Chem. Phys.
133
,
101106
(
2010
).
31.
J.
Hu
,
M.
Luo
,
F.
Jiang
,
R.-X.
Xu
, and
Y.
Yan
,
J. Chem. Phys.
134
,
244106
(
2011
).
32.
S.
Jang
,
J.
Cao
, and
R. J.
Silbey
,
J. Phys. Chem. B
106
,
8313
(
2002
).
33.
G.
Ritschel
,
J.
Roden
,
W. T.
Strunz
, and
A.
Eisfeld
,
New J. Phys.
13
,
113034
(
2011
).
34.
S.
Mukamel
,
Nonlinear Optical Spectroscopy
(
Oxford University Press, Inc.
,
1995
).
35.
J.
Adolphs
and
T.
Renger
,
Biophys. J.
91
,
2778
(
2006
).
36.
See supplementary material at http://dx.doi.org/10.1063/1.4893931 for the explicit fit parameters and additional plots.
37.
J.
Roden
,
W. T.
Strunz
,
K. B.
Whaley
, and
A.
Eisfeld
,
J. Chem. Phys.
137
,
204110
(
2012
).
38.
A.
Croy
and
U.
Saalmann
,
Phys. Rev. B
80
,
073102
(
2009
).
39.
A.
Croy
and
U.
Saalmann
,
Phys. Rev. B
82
,
159904
(
2010
).
40.
Handbook of Mathematical Functions
, edited by
M.
Abramowitz
and
I. A.
Stegun
(
N.B.S.
,
1964
).
41.
Our choice corresponds to the Fourier transform of the energy-gap correlation function.34 
42.
We use the same symbol J(ω) to denote a given spectral density as well as our fit functions Eq. (3), which we use to approximate the former.
43.
For the superohmic case n > 2 one can solve the integral efficiently using Fourier transformation methods.
44.
Higher order terms lead in general to time-dependent coefficients.
45.
For poles of first order one has
$\protect \mathrm{Res}_{\omega _{i}}{[f(\omega )]}=\protect \qopname{}{m}{lim}_{\omega \rightarrow \omega _{i}} (\omega -\omega _{i})f(\omega )$
Res ωi[f(ω)]=limωωi(ωωi)f(ω)
.
46.
We were also not able to anti-symmetrize Eq. (D4) such that it is still meromorphic.
47.
At the moment we can make no statement for n not belonging to the set of natural numbers.

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