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Makri, N.

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JOURNAL OF CHEMICAL PHYSICS VOLUME 111, NUMBER 14 8 OCTOBER 1999Iterative evaluation of the path integral for a system coupled
to an anharmonic bath
Nancy Makri
School of Chemical Sciences, University of Illinois, 601 S. Goodwin Avenue, Urbana, Illinois 61801a)
and Theoretical & Physical Chemistry Institute, National Hellenic Research Foundation, 48 Vassileos
Constantinou Avenue, Athens, Greece 11635
~Received 18 June 1999; accepted 4 August 1999!
An iterative algorithm is presented for evaluating the path integral expression for the reduced
density matrix of a quantum system interacting with an anharmonic dissipative bath whose influence
functional is obtained via numerical methods. The method allows calculation of the reduced density
matrix over very long time periods. © 1999 American Institute of Physics.
@S0021-9606~99!03238-9#In spite of persistent efforts, the problem of calculating
the quantum time evolution of a wave-function or density
matrix in a multidimensional Hamiltonian remains unsolved.
Recent work has revolved around methods based on mean
field, quantum-classical, or semiclassical ideas. The most
rigorous of these approaches, semiclassical evolution with
the Van Vleck propagator,1,2 is often highly accurate,3–8 yet
extremely demanding numerically because it requires multi-
dimensional integration of oscillatory functions as well as
evaluation of a prefactor that scales nonlinearly with the
number of particles. Current efforts to make it practical ex-
ploit filtering techniques,9–11 the self-cancellation achieved
via combined forward–backward propagation,12–16 or formu-
lations which avoid calculation of the prefactor.11,17–19 While
these approaches appear promising, they are bound to fail at
long propagation times or if tunneling effects are
prominent.20 Treatment at a higher level becomes necessary
in such situations.
In a series of papers by our group, we have argued that
the path integral-influence functional formulation of quan-
tum dynamics21,22 offers significant advantages when dealing
with large-dimensional problems. One begins by identifying
the observable ‘‘system’’ @the degree~s! of freedom s being
probed in the calculation# and the remaining ‘‘bath’’ degrees
of freedom x which interact with the system and thus affect
its dynamics but whose precise state is not followed. Thus,
the Hamiltonian is split into two terms,
H[H0~s ,ps!1Hb~x,p,s !. ~1!
Expressing the full propagator as a path integral, and collect-
ing all bath variables into an influence functional, one arrives
at a formal path integral representation where only paths of
the low-dimensional system are summed over. For example,
the reduced density matrix of the system takes the form
a!Address to which corrspondence should be sent. Electronic mail:
nancy@makri.scs.uiuc.edu6160021-9606/99/111(14)/6164/4/$15.00
Downloaded 08 Dec 2008 to 194.177.215.121. Redistribution subject p˜ ~s8,s9;t ![Trb^s9ur~ t !us8&
5E Ds1E Ds2 expS i\ S0@s1# D
3expS 2 i\ S0@s2# DF@s1 ,s2# . ~2!
Here, s1 , s2 are forward and backward paths of the observ-
able system respectively, S0 is the corresponding action, and
the influence functional is given by the expression
F@s1 ,s2#5Trb~Ub@s1#r~0 !Ub
21@s2# !, ~3!
where Ub is the time evolution operator of the bath along a
chosen system path. Note that the time parametrization of
these paths makes the bath Hamiltonian time-dependent.
There are numerous advantages of this representation, as
well as severe obstacles. The explicit form of the influence
functional—an intrinsically quantum mechanical quantity
not obtainable by classical molecular dynamics methods—is
not available except in very restrictive situations, the most
notable of which is the case of a harmonic bath.22 Yet the
simple structure of the influence functional, where the only
operators appearing are the time propagators and the initial
density matrix, implies that its evaluation may require less
work than that required to obtain the full quantum dynamics
for the same number of particles. Further, the consistency of
the bath in terms of spectator coordinates implies that its
dynamics may be treated at a less rigorous level than that of
the observable system, i.e., errors due to approximate treat-
ment of the influence functional are expected to affect the
result in a less significant way compared to the error that
would arise if a similar approximation were used to propa-
gate the observable system. The semiclassical approximation
lends itself naturally to this goal: it keeps track of quantum
mechanical phases rather faithfully and its forward–
backward version12,13 is ideally suited. Thus, assuming that
the influence functional from a sufficiently large environ-
ment can be computed numerically, one is confronted with
the problem of evaluating the remaining path integral with
respect to the system. Monte Carlo sampling of paths fails4 © 1999 American Institute of Physics
to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp
6165J. Chem. Phys., Vol. 111, No. 14, 8 October 1999 Path integral for an anharmonic bathas the integrand is a highly oscillatory multidimensional
function, while explicit enumeration of paths is feasible only
for short times. The only remaining avenue seems to be an
iterative formulation. The iterative evaluation of the path in-
tegral has proved a very fruitful method in the case of a
harmonic dissipative bath;23–26 its extension to general an-
harmonic environments is undertaken in the present letter.
Since we aim at a numerical propagation method, we
switch to the discretized path integral representation, in
which the reduced density matrix takes the form
r˜ ~sN
2
,sN
1 ;t !
5E ds06E ds16flE dsN216 ^sN1ue2iH0Dt/\usN211 &fl
3^s1
1ue2iH0Dt/\us0
1&^s0
2ueiH0Dt/\us1
2&fl
3^sN21
2 ueiH0Dt/\usN
2&F~s0
6
. . . ,sN
6!. ~4!
If the bath is harmonic, the influence functional is an expo-
nential of a quadratic form.22 In the case of a general anhar-
monic environment, F may contain multiple-site interactions,
as described in the cumulant expansion of the influence
functional,27
F~s0
6s1
6
. . .sN
6!5expH 2 (
k150
N
~ak1
1 sk1
1 1ak1
2 sk1
2 !
2 (
k150
N
(
k250
N
~bk1k2
11 sk1
1 sk2
1 1bk1k2
12 sk1
1 sk2
2 1fl !
2 (
k150
N
(
k250
N
(
k350
N
~gk1k2k3
111 sk1
1 sk2
1 sk3
1
1gk1k2k3
112 sk1
1 sk2
1 sk3
2 1fl !2flJ . ~5!
As shown in Ref. 27, the coefficients in this expansion are
given by multitime correlation functions of the force exertedDownloaded 08 Dec 2008 to 194.177.215.121. Redistribution subject on the system due to its interaction with the bath. Such cor-
relation functions decay irreversibly if the environment is
macroscopic. As a consequence, the nonlocal interactions in
the influence functional arising from condensed media have
finite range. If the expansion coefficients are known, it is a
straightforward matter of extending the harmonic bath meth-
odology developed in our group to devise an iterative proce-
dure for evaluating the evolution of the reduced density ma-
trix.
However, the expansion coefficients in Eq. ~5! are not
known in general. In a classical treatment of the bath, low-
order terms in the series can be obtained by molecular dy-
namics simulations. The latter is costly and the evaluation of
terms beyond the two-time correlation function is extremely
demanding. Further difficulties are the required determina-
tion of imaginary parts and the inability to test the accuracy
of the classical approximation. An alternative approach
which we have considered is to evaluate the entire influence
functional for each set of system paths numerically using
forward-backward semiclassical dynamics ~FBSD!.12,13 This
approach is superior because it is not restricted to low-order
terms in the cumulant expansion, is capable of describing the
quantum behavior of the bath, and yields the real and imagi-
nary parts of the influence functional in a single calculation.
Although a numerically constructed influence functional
does not assume the form of a systematic series expansion, it
is shown below that it can still be cast in a form suitable for
iterative propagation.
As in the case of a harmonic bath,26 the scheme proceeds
via multiplication of an array R of path segments that span
the memory length by a propagator matrix T. The memory
length tm[DkmaxDt ~where Dkmax is an integer!, given
roughly by the decay time of the bath correlation functions,
plays the role of a convergence parameter in the calculation.
We define path segments
l i
6[$s1
6
,s2
6
, . . . ,sDkmax
6 % i , ~6!
and the propagator matrix between the ‘‘old’’ path segment i
and the ‘‘new’’ path segment j,Ti j~nDkmax![T~~snDkmax
6
,snDkmax11
6
, . . . ,s ~n11 !Dkmax21
6 ! i ,~s ~n11 !Dkmax
6
, . . . ,s ~n12 !Dkmax21
6 ! j!
5 )
k5nDkmax11
~n11 !Dkmax
^sk21
1 ue2iH0Dt/\usk
1&^sk
2ueiH0Dt/\usk21
2 &
3
F˜ ~snDkmax
6
, . . . ,s ~n11 !Dkmax21
6
,s ~n11 !Dkmax
6
, . . . ,s ~n12 !Dkmax21
6 !
F˜ ~0,...,0,s ~n11 !Dkmax
6
, . . . ,s ~n12 !Dkmax21
6 !
. ~7!Here, the tilde indicates that the influence functionals must
be evaluated with the proper boundary conditions: In order to
include all the proper interactions for continued propagation,
the old–new path pair is extended in one or both directions,
i.e.,F˜ @ l i ,l j#[Trb~U ~0 !U@ l j
1#U@ l i
1#
3r~0 !U21@ l i
2#U21@ l j
2#U ~0 !
21
!, ~8!
where U (0) signifies time evolution with the bath Hamil-to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp
6166 J. Chem. Phys., Vol. 111, No. 14, 8 October 1999 Nancy Makritonian in the absence of system–bath interactions. According
to Eq. ~8!, the influence functional entering the midtime
propagator is given by evolving the initial density matrix by
one memory length under the influence of the force along the
system path segment l i
1
, subsequently evolving under the
force due to l i
1
, followed by propagation with the isolated
bath Hamiltonian by one more memory length and by the
entire series of operations in reverse order in the backward
time direction. The time lengths and forces involved in the
calculation of F˜ are shown in Fig. 1. Finally, the denomina-
tor in Eq. ~7! involves the same sequence of operations but
with U@ l i
6# replaced by U (0). It is straightforward to show
that successive multiplication with the propagator defined in
Eq. ~7! reproduces the exact dynamics of Eq. ~4! subject to
the assumption of finite memory length.
After the first propagation step, the propagator matrix
can be stored and used for successive iterations. This is so
because the influence functional is translationally invariant,
i.e., the F˜ calculated from a given pair of path segments
remains unchanged if additional zero-force segments are at-
tached to either or both ends of Eq. ~8!, as long as l i
6 are not
too close to endpoints. This property is a consequence of
dephasing and can be exploited to achieve dramatic compu-
tational savings.
The details of the algorithm will be presented in a future
publication.28 Here, we illustrate the scheme by presenting
the long-time dynamics of a two-level system ~TLS! coupled
to a nonlinear bath of 50 two-level systems according to the
Hamiltonian
H52\Vsx
02(
i51
n 1
2 \v isx
i 2sz
0(
i51
n
ciA \2v i szi . ~9!
In the last equation, sx and sz are the usual Pauli spin ma-
trices, the tunneling splittings are 2\V and \v i for the bare
system and the bath spins, respectively, and the parameters
of the bath are specified from the spectral density function of
the Ohmic form,29
J~v!5
p
2 (i51
ci
2
v i
d~v2v i!5
p
2 \jve
2v/vc,
with vc56V . Further, we assume that the interaction be-
tween system and bath is turned on at t50, at which time the
system is in the ‘‘up’’ state and that the bath is prepared at
the temperature 1/kBb . The discrete character of H0 turns
FIG. 1. Schematic representation of two system path segments and addi-
tional memory erasing segments for constructing the influence functionals
required for the ~a! numerator and ~b! denominator of the midtime propaga-
tor. The system has two states indicated as up- or down-arrows which de-
termine the force exerted on the bath. The circles correspond to propagation
with the isolated bath Hamiltonian.Downloaded 08 Dec 2008 to 194.177.215.121. Redistribution subject the integrals in Eq. ~4! into sums. In the present situation, the
influence functional factorizes and is computed exactly by
the matrix multiplication method.30
Figure 2 shows the evolution of the average position of
the observable TLS as calculated by iterative evaluation of
the path integral for \Vb52, j50.1 and b50.05, j50.5.
The path integral time step is Dt50.25 V21 and the memory
length is tm53Dt . The total propagation time equals 60
elementary time steps. At these long times, full evaluation of
the path integral by global summation is not feasible, but at
short times the present results are in good agreement with
those reported in Ref. 31.
In summary, it is possible to formulate an iterative
scheme for evaluating the path integral of a low-dimensional
system coupled to a general dissipative bath if the influence
functional of the latter can be calculated at some level of
approximation. Use of this idea in conjunction with the
FBSD methodology for evaluating anharmonic influence
functionals will lead to a powerful approach to the dynamics
of polyatomic chemical systems. The combined path
integral-semiclassical treatment offers two distinct advan-
tages compared to fully semiclassical schemes: it avoids
treating the important, highly quantum mechanical system of
interest via the semiclassical approximation; and the semi-
classical evaluation of the influence functional requires only
short-time dynamics, over which FBSD is likely to be ex-
FIG. 2. Expectation value of the TLS position as a function of time as
obtained with the iterative scheme described in this letter. ~a! \Vb52,
j50.1 and \Vb50.05, j50.5.to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp
6167J. Chem. Phys., Vol. 111, No. 14, 8 October 1999 Path integral for an anharmonic bathtremely accurate. Future publications will exploit these ideas
in the context of solvation dynamics.
This work has been supported by the National Science
Foundation under Grant No. 1-5-32189. I thank the Director
and the researchers of the Theoretical and Physical Chemis-
try Institute for their hospitality during my sabbatical leave
from the University of Illinois.
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Iterative evaluation of the path integral for a system coupled to an anharmonic bath

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