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Semiclassical calculation of thermal rate constants in full Cartesian space:
The benchmark reaction D¿H2\DH¿H
Takeshi Yamamoto and William H. Millera)
Department of Chemistry and Kenneth S. Pitzer Center for Theoretical Chemistry,
University of California, and Chemical Sciences Division, Lawrence Berkeley National Laboratory,
Berkeley, California 94720
?Received 8 October 2002; accepted 6 November 2002?
Semiclassical ?SC? initial-value representation ?IVR? methods are used to calculate the thermal rate
constant for the benchmark gas-phase reaction D?H2→DH?H. In addition to several technical
improvements in the SC-IVR methodology, the most novel aspect of the present work is use of
Cartesian coordinates in the full space ?six degrees of freedom once the overall center-of-mass
translation is removed? to carry out the calculation; i.e., we do not invoke the conservation of total
angular momentum J to reduce the problem to fewer degrees of freedom and solve the problem
separately for each value of J, as is customary in quantum mechanical treatments. With regard to the
SC-IVR methodology, we first present a simple and straightforward derivation of the semiclassical
coherent-state propagator of Herman and Kluk ?HK?. This is achieved by defining an interpolation
operator between the Van Vleck propagators in coordinate and momentum representations in an a
priori manner with the help of the modified Filinov filtering method. In light of this derivation, we
examine the systematic and statistical errors of the HK propagator to fully understand the role of the
coherent-state parameter ?. Second, the Boltzmannized flux operator that appears in the rate
expression is generalized to a form that can be tuned continuously between the traditional half-split
and Kubo forms. In particular, an intermediate form of the Boltzmannized flux operator is shown to
have the desirable features of both the traditional forms; i.e., it is easy to evaluate via path integrals
and at the same time it gives a numerically well-behaved flux correlation function at low
temperatures. Finally, we demonstrate that the normalization integral required in evaluating the rate
constant can be expressed in terms of simple constrained partition functions, which allows the use
of well-established techniques of statistical mechanics.
?DOI: 10.1063/1.1533081?
© 2003 American Institute of Physics.
I. INTRODUCTION
As is well appreciated, the quantum dynamics of a mo-
lecular system is fully determined by the action of the time
evolution operator, exp(?iHˆt/?), onto a given initial wave
function or density matrix. This can now be performed quite
accurately for small polyatomic systems ?e.g., those involv-
ing only three or four atoms? using basis set methods,1,2but
these methods are still difficult to apply to larger systems due
to the exponential growth of the basis set with system size. A
totally different approach for treating quantum dynamics is
that based on real-time path integrals.3The direct application
of real-time path integrals to multidimensional systems or
longer times is, however, still prohibitive because of the no-
torious sign problem;4–7namely, the integrand of a real-time
path integral exhibits large oscillations when the path is far
from the stationary ones, which makes a Monte Carlo evalu-
ation of the integral extremely difficult. ?But note some re-
centinterestingprogressusing
methods8by Rabani et al.9? One approach to greatly improve
the situation is a filtering or smoothing technique10–14such
as the stationary-phase Monte Carlo,6,12which converts an
original integrand to a much smoother one via local averag-
analyticcontinuation
ing operations. A more drastic approximation is to keep only
the contribution from nearby paths around the stationary path
?i.e., classical path?, which results in the semiclassical ?SC?
approximation first obtained by Van Vleck.15,16Though the
original Van Vleck formula is numerically awkward because
of the root-search problem, it can be overcome by employing
the initial-value representation ?IVR?.17,18Semiclassical ap-
proaches, implemented via the initial-value representation
?SC-IVR?, have recently received a rebirth of interest, and a
number of studies have been carried out and have demon-
strated the capability of these approaches to accurately de-
scribe various quantum effects ?for reviews, see Refs. 19–
21?.
This paper is a continuation of our efforts in developing
SC-IVR methods into a practical way for adding quantum
mechanical effects to classical molecular dynamics simula-
tions, here applied specifically to the thermal rate constant of
a chemical reaction. Previous work by us and Wang22
showed how the SC-IVR description of the real-time dynam-
ics could be combined with a fully quantum ?imaginary-time
path integral? description of the Boltzmann operator, exp
(??Hˆ), that appears in the rate expression ?vide infra?. An
application was made to several model problems of chemical
reactions, e.g., a double-well potential coupled to a harmonic
a?Electronic mail: miller@cchem.berkeley.edu
JOURNAL OF CHEMICAL PHYSICSVOLUME 118, NUMBER 51 FEBRUARY 2003
21350021-9606/2003/118(5)/2135/18/$20.00© 2003 American Institute of Physics
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bath, demonstrating the applicability of the overall approach
to systems with many degrees of freedom and its ability to
produce accurate results for the thermal rate constant.
In the present work we apply the SC-IVR approach to
the benchmark gas-phase reaction D?H2→DH?H, which,
though a ‘‘small’’ molecular system, is a ‘‘real’’ system in
three-dimensional space. In addition to several technical im-
provements in the SC-IVR methodology, the most novel as-
pect of the present work is use of Cartesian coordinates in
the full space of six degrees of freedom ?once overall center-
of-mass translation has been removed? to carry out the cal-
culations; i.e., we do not invoke the conservation of total
angular momentum J to reduce the problem to one of four
degrees of freedom and solve it separately for each value of
J, as is performed in standard quantum mechanical
treatments.1,2The reasons for proceeding this way are two-
fold: First, the present calculation is primarily a test of the
methodology; our interest is in applying it to general molecu-
lar systems involving N atoms, in which case reducing the
dimensionality from 3N?3 to 3N?5 degrees of freedom
would be of negligible benefit. Second, the coherent states
involved in the semiclassical propagator are most naturally
defined and straightforward to apply for Cartesian coordi-
nates ?though see the recent work by Kay23on IVR methods
using action-angle variables?. We therefore use the following
Cartesian form of the classical Hamiltonian to describe the
present atom–diatom (A?BC) reaction:
H?P,R,p,r??
P2
2?R?
p2
2?r?V?R,r?,
?1.1?
where r?(x,y,z) is the vector between B and C, R
?(X,Y,Z) is that from A to the center of mass of BC, and
?Rand ?rare the corresponding reduced masses. The quan-
tum Hamiltonian operator is obtained from Eq. ?1.1? via the
usual replacement,
P→?
i“R,p→?
i“r.
?1.2?
In Sec. II we first present an alternate derivation and
analysis of the popular Herman–Kluk ?HK? coherent-state
IVR, which approximates the propagator as follows:
e?iHˆt/??UˆHK?t;??
??2????d? dq0? dp0?qtpt;??
??q0p0;??Ct?q0p0;??eiSt?q0p0?/?.
?1.3?
Here Hˆ is the Hamiltonian for a system with d degrees of
freedom, (q0,p0) are the initial coordinates and momenta for
a classical trajectory, qt?qt(q0,p0) and pt?pt(q0,p0) are
the variables at time t that evolve from these initial condi-
tions, Stis the classical action along this trajectory,
St?q0p0???
0
t
dt??p?t??Tq ˙?t???H„p?t??,q?t??…?,
?1.4?
and Ctis the square root of a determinant that involves the
various monodromy matrices,24
Ct?q0p0;????det?
1
2?Mqq?Mpp???
??Mpq???
i
Mqp
?
i
1/2
,
?1.5?
where
Mqq??qt?q0,p0?
?q0
,Mqp??qt?q0,p0?
?p0
,
?1.6a?
Mpq??pt?q0,p0?
?q0
,Mpp??pt?q0,p0?
?p0
.
?1.6b?
The bra and ket in Eq. ?1.3? are coherent states,25the coor-
dinate representation of which is defined by
?x?qp;????
?
??
d/4
exp???
2?x?q?2?
i
?p"?x?q??.
?1.7?
One notable property of the HK propagator in Eq. ?1.3? is
that it reverts to the Van Vleck propagator in coordinate and
momentum representations as ? goes to ? and 0, respec-
tively. Hence, it represents an intermediate operator between
the two Van Vleck propagators. Historically, Herman and
Kluk derived the form in Eq. ?1.3? by evaluating a time-
evolved wave function via the combined use of the Van
Vleck formula and stationary-phase approximation.26This
derivation is, however, rather complicated because the inte-
gration contour must be distorted into the complex plane,
which makes the stationary-phase trajectory complex. Sub-
sequently, the HK propagator was rederived several other
ways:27–30these derivations include the asymptotic analysis
of a general semiclassical kernel having an integral
expression,27the stationary-phase limit of a coherent-state
path integral,28and, more recently, the application of the
modified Filinovfiltering
element.29In the present paper, we focus on the above prop-
erty of the HK propagator, namely, that it interpolates be-
tween the Van Vleck propagators in coordinate and momen-
tum representations, and construct such an interpolation
operator in an a priori manner. This is accomplished by us-
ing the modified Filinov filtering procedure,11which pro-
vides a convenient way to interpolate between an original
integral and its stationary-phase approximation. We show
that the new operator so defined takes the form in Eq. ?1.3?.
The present derivation is mathematically very simple and
also helpful in fully understanding the role of the coherent-
state parameter ? in Eq. ?1.7?.
Also in Sec. II we consider the issues involved in apply-
ing the SC-IVR to evaluate reactive flux correlation func-
tions for ‘‘real’’ molecular systems, which are more involved
than for the model systems treated previously ?e.g., because
the ‘‘dividing surface’’ that defines the flux operator is, in
general, a nonlinear function of all the coordinates of the
system?. Here we introduce a generalized form of the Boltz-
mannized flux operator that can be tuned continuously be-
tween the traditional ‘‘half-split’’31and ‘‘Kubo’’32forms. It
is then shown that a particular intermediate form has the best
properties of both of these, making it much easier to evaluate
toa coherent-statematrix
2136J. Chem. Phys., Vol. 118, No. 5, 1 February 2003 T. Yamamoto and W. H. Miller
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Page 3
the path integrals for the Boltzmannized flux operator and
also giving a correlation function that is better behaved both
in time and temperature.
In Sec. III we then apply all of this to the calculation of
the reactive flux correlation function ?and thus the rate con-
stant? for the D?H2→DH?H reaction. Particularly impor-
tant is the combined use of the extended ensemble method33
and thermodynamic integration34to evaluate the normaliza-
tion integral required in the Monte Carlo calculations. The
rate constants obtained from T?200–1000 K show excellent
agreement with those obtained from quantum scattering cal-
culations and with the experiment, thus demonstrating the
feasibility and accuracy of the present semiclassical ap-
proach.
II. THEORY
A. Interpolating between the Van Vleck propagators
in coordinate and momentum representations
First, we give a brief sketch of the modified Filinov fil-
tering method,11which will be used to define an interpolation
operator between the Van Vleck propagators in different rep-
resentations. The modified Filinov filtering procedure con-
verts a general multidimensional integral having a highly
oscillatory integrand,
K?? dMzR?z?exp?i??z?/??,
?2.1?
to the one with a less oscillatory integrand as follows:
K?K?c??? dMzR?z?exp?i??z?/??F?z;c?,
?2.2?
where the newly introduced factor F(z;c) is called the modi-
fied Filinov factor and is given by
F?z;c???det?1?
i
?c
?2?
?z?z??
1/2
exp??
c
2?2?
??
?z?
2?.
?2.3?
We note that F(z;c) is essentially a damping factor for the
first derivative of ?(z). That is, it damps the amplitude of
the integrand strongly in the region with a large value of
???/?z?, or, in other words, it filters out the highly oscilla-
tory part of the integrand to generate an approximate but
smoother integrand. Although this operation introduces some
systematic error into the original integral ?and hence K(c) is
not equal to K in general?, the advantage here is that a Monte
Carlo calculation of K(c) becomes much more efficient than
that of K itself. The degree of filtering or smoothing is con-
trolled by the parameter c, and, roughly speaking, the
smoothing effect becomes stronger as c becomes larger. Here
the two limiting cases of c should be noted, i.e., c→0 and
c→?. As c→0 the modified integral reverts to the original
integral since the Filinov factor becomes unity,
lim
c→0
F?z;c??1,lim
c→0
K?c??K,
?2.4?
while in the limit c→? one can show that the modified
integral becomes the stationary-phase approximation to the
original integral, i.e.,
lim
c→?
K?c??KSPA
??
zsp
R?zsp?ei??zsp?/???2?i??M/det?
?2?
?zsp?zsp??
1/2
.
?2.5?
Here the sum is over all the stationary-phase points defined
by ??/?zsp?0. Therefore, the modified Filinov filtering not
only generates a smoother integrand but provides an interpo-
lation formula between the original integral and its
stationary-phase approximation. We note that this latter prop-
erty is the important feature of the modified Filinov
procedure11that is absent from the original Filinov method.10
Next, we describe the Van Vleck propagators in coordi-
nate and momentum representations.15,16Here we consider a
one-dimensional system to make the derivation as straight-
forward as possible. The Van Vleck propagator in the coor-
dinate representation is the stationary-phase approximation
to the path integral of the time evolution operator
exp(?iHˆt/?), which can be written in an operator form as
UˆVVQ?t??? dqf? dqi?qf??qf?
?exp??iHˆt/???qi?VVQ?qi?.
?2.6?
Here the matrix element above is evaluated using classical
mechanical information as
?qf?exp??iHˆt/???qi?VVQ??
q????
1
2?i?Mqp?
1/2
?exp?iSt?qf,qi?/??,
?2.7?
where the summation runs over all classical trajectories q(?)
that satisfy the boundary conditions q(0)?qi and q(t)
?qf, St(qf,qi) is the action integral along these trajectories
given in Eq. ?1.4?, and Mqprepresents an element of the
monodromy matrix in Eq. ?1.6?. ?For simplicity of presenta-
tion, we do not indicate the Maslov index, i.e., the absolute
phase of the square root of Mqp explicitly.? The label
‘‘VVQ’’ is introduced here to distinguish the Van Vleck
propagator in the coordinate representation from that in the
momentum representation.
The Van Vleck propagator in the momentum representa-
tion is defined as
UˆVVP?t??? dpf? dpi?pf??pf?
?exp??iHˆt/???pi?VVP?pi?,
?2.8?
where the matrix element ?pf?exp(?iHˆt/?)?pi?VVPis given by
the general SC expression of Ref. 18, i.e.,
?pf?exp??iHˆt/???pi?VVP??
p????
?1
2?i?Mpq?
1/2
?exp?iSt?pf,pi?/??,
?2.9?
2137J. Chem. Phys., Vol. 118, No. 5, 1 February 2003Thermal rate constants in Cartesian space
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Page 4
where the summation is over all classical trajectories satisfy-
ing the boundary conditions p(0)?piand p(t)?pf, and the
action in the momentum representation, St(pf,pi), is given
by
St?pf,pi???
0
t
dt???q?t??p ˙?t???H„p?t??,q?t??…?;
?2.10a?
it is related to the action in the coordinate representation by
St?pf,pi???pfqf?piqi?St?qf,qi?,
?2.10b?
where (qi,qf) represent the initial and final coordinates of
the trajectories. As discussed before,18the SC approxima-
tions for the coordinate and momentum representations of
the propagator are equivalent in the context of the ‘‘semiclas-
sical algebra’’ that transforms between representations via
the stationary phase approximation.
We now define a new propagator that uses the modified
Filinov procedure to interpolate between the two Van Vleck
propagators above,
UˆVVQP?t;c??? dpf? dpi?pf??pf?
?exp??iHˆt/???pi?VVQ/MF?pi?.
?2.11?
Here the matrix element ?pf?exp(?iHˆt/?)?pi?VVQ/MFis de-
fined as
?pf?exp??iHˆt/???pi?VVQ/MF?? dq1? dq0?pf?q1?
??q1?exp??iHˆt/???q0?VVQ
??q0?pi??F?q1,q0;c?,
?2.12?
where F(q1,q0;c) is the modified Filinov factor of Eq. ?2.3?
with the phase ?(q1,q0) given by St(q1,q0)?pfq1?piq0.
This definition means that we evaluate the integral over
(q1,q0) in Eq. ?2.12? explicitly with the help of the modified
Filinov filtering, rather than using the stationary-phase ap-
proximation. The nature of the new propagator in Eq. ?2.11?
can readily be understood by considering the limit c→0 and
c→?. As c→0 the modified Filinov factor becomes unity
and thus one can remove the integral over (pf,pi) in Eq.
?2.11? to give
lim
c→0
UˆVVQP?t;c??? dq1? dq0?q1??q1?
?exp??iHˆt/???q0?VVQ?q0??UˆVVQ?t?.
?2.13?
In the opposite limit c→?, the integral in Eq. ?2.12? be-
comes the stationary-phase approximation to the original in-
tegral due to the nature of the modified Filinov filtering, i.e.,
lim
c→??pf?exp??iHˆt/???pi?VVQ/MF
???
SPA
dq1dq0?pf?q1??q1?
?exp??iHˆt/???q0?VVQ?q0?pi?
??pf?exp??iHˆt/???pi?VVP,
?2.14?
and hence
lim
c→?
UˆVVQP?t;c??UˆVVP?t?.
?2.15?
UˆVVQP(t;c) therefore represents an interpolation operator be-
tween the Van Vleck propagators in coordinate and momen-
tum representations. This property is similar to that of the
Herman–Kluk propagator in Eq. ?1.3?, and one can, in fact,
show that UˆVVQP(t;c) is identical to the HK propagator, i.e.,
UˆVVQP?t;c???2????1? dq0? dp0?qtpt;??
??q0p0;??Ct?q0p0;??exp?iSt?q0p0?/??
?UˆHK?t;??,
?2.16?
with the coherent-state parameter ? being given by ??1/c
?for details of the derivation, see Appendix A?. Hence, the
HK propagator is identified as a Filinov-smoothed version of
the Van Vleck propagator. This fact also indicates that for
finite values of ? the HK will give a result different from the
Van Vleck propagator, but its Monte Carlo evaluation should
be easier than that of the latter. These numerical properties
will be examined in Sec. IID1.
B. Rate constant calculation via HK-IVR
Here we summarize the previous approach22for the
semiclassical calculation of a thermal rate constant, namely
the double-forward HK-IVR combined with the symmetric
flux–flux correlation function.
The quantum mechanically exact expression for a ther-
mal rate constant k(T) can be written in terms of the flux
correlation function as31
k?T??
1
Qr?T?lim
t→?
Cfs?t?,
?2.17?
where Qr(T) is the reactant partition function per unit vol-
ume and Cfs(t) is the ‘‘flux-side’’ correlation function, de-
fined by
Cfs?t??tr?Fˆ???exp?iHˆt/??h?s ˆ?exp??iHˆt/???. ?2.18?
Fˆ(?) in the above equation is the ‘‘Boltzmannized’’ flux
operator of the form
Fˆ????e??Hˆ/2Fˆe??Hˆ/2,
?2.19?
with the ?bare? flux operator Fˆ, defined by
2138J. Chem. Phys., Vol. 118, No. 5, 1 February 2003T. Yamamoto and W. H. Miller
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Page 5
Fˆ?
i
??Hˆ,h?s ˆ??.
?2.20?
Hˆis the molecular Hamiltonian, h(s) is the Heaviside step
function that depends on the reaction coordinate s, and h(s)
takes the value of 0 ?1? in the reactant ?product? side of the
dividing surface defined by s(q)?0. The rate constant k(T)
can also be expressed as
Qr?T??
0
k?T??
1
?
dt Cff?t?,
?2.21?
where Cff(t) is the ‘‘flux–flux’’ correlation function and is
given by the time derivative of Cfs(t):
Cff?t??
d
dtCfs?t??tr?Fˆ???exp?iHˆt/??Fˆ
?exp??iHˆt/???.
?2.22?
In previous work22it was shown that a slightly different
version of the flux–flux correlation function,
Cff?t??tr?Fˆ??/2?exp?iHˆt/??Fˆ??/2?exp??iHˆt/???,
?2.23?
which can be obtained via cyclic permutation of the Boltz-
mann operator in Eq. ?2.22?, has a distinct advantage over
other flux correlation functions when they are evaluated
semiclassically. This can readily be seen by substituting the
explicit form of the HK propagator in Eq. ?1.3? into the
above correlation function, which yields
Cff?t???2????2d? dq0? dp0? dq0?? dp0?
??q0p0?Fˆ??/2??q0?p0???qt?pt??Fˆ??/2??qtpt?
?Ct?q0p0?Ct*?q0?p0??ei?St?q0p0??St?q0?p0???/?.
?2.24?
An important point here is that the integrand of Eq. ?2.24? is
positive definite at t?0,
Cff?0???2????2d? dq0? dp0? dq0?? dp0???q0p0?
?Fˆ??/2??q0?p0???2,
?2.25?
while the other correlation functions in Eqs. ?2.18? and
?2.22? do not possess this property. We thus expect the inte-
grand of Eq. ?2.24? to be a smoother function of (q0p0;q0?p0?)
than that of the other correlation functions, which leads to a
faster convergence when the integral is evaluated via Monte
Carlo methods. Another advantage in using the symmetric
version of Cff(t) is that one can use the integrand of Cff(0)
in Eq. ?2.25? as a natural weight function for the double
phase-space variables (q0p0;q0?p0?) in a Monte Carlo calcu-
lation, i.e.,
W?q0p0;q0?p0????2????2d??q0p0?Fˆ??/2??q0?p0???2.
?2.26?
With this definition, the flux correlation function can be writ-
ten as the product of ‘‘static’’ and ‘‘dynamical’’ factors as
Cff?t??Cff?0??Rff?t?.
?2.27?
Here Rff(t) is the normalized correlation function that is
evaluated semiclassically as
Cff?0???
?q0?p0??Fˆ??/2??q0p0?
Rff?t??Cff?t?
?qt?pt??Fˆ??/2??qtpt?
?Ct?q0p0?Ct*?q0?p0??
?ei?St?q0p0??St?q0?p0???/??
W
,
?2.28?
with ?¯?Wbeing a Monte Carlo average over W,
?¯?W??dq0?dp0?dq0??dp0? W?q0p0;q0?p0??Ã?¯?
?dq0?dp0?dq0??dp0?W?q0p0;q0?p0??
,
?2.29?
while Cff(0) in Eq. ?2.27? represents the normalization inte-
gral of W:
? dq0? dp0? dq0?? dp0? W?q0p0;q0?p0??
?tr?Fˆ??/2?Fˆ??/2???Cff?0?.
?2.30?
The normalization integral of W is thus related to a quantum
mechanical trace of the Boltzmannized flux operator ?with
no real-time evolution operators? and can therefore be evalu-
ated conveniently using standard imaginary-time path inte-
gral techniques.35,36Due to these attractive features, we will
employ the symmetric version of the flux–flux correlation
function in Eq. ?2.23? in the subsequent calculations.
C. General form of the Boltzmannized flux operator
The simplest combination of the Boltzmann and flux op-
erators that can be used in Eq. ?2.18? ?and originally was
used31?a?? is their product,
Fˆ????e??HˆFˆ.
?2.31?
One can also define several different forms of Fˆ(?) by per-
forming a cyclic permutation of e??Hˆ
mechanical trace in Eq. ?2.18?. One such definition is the
‘‘half-split’’ form31?b?already given in Eq. ?2.19?,
within the quantum
Fˆhalf????e??Hˆ/2Fˆe??Hˆ/2,
?2.32?
which was originally introduced to combine the ‘‘half’’-
Boltzmann operator e??Hˆ/2with the real-time evolution op-
erator e?iHˆt/?to give a single complex-time evolution opera-
tor e?iHˆtc/?
with tc?t?i??/2. Another conventional
definition of Fˆ(?) is the ‘‘Kubo’’ form,32
??
0
FˆKubo????1
?
d? e??????HˆFˆe??Hˆ
?
i
???h?s ˆ?,e??Hˆ?,
?2.33?
2139 J. Chem. Phys., Vol. 118, No. 5, 1 February 2003Thermal rate constants in Cartesian space
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Page 6
which arises in the context of linear response theory. The
second equality in the above equation can be shown by using
the derivative relation
e??????HˆFˆe??Hˆ?i
?
?
??e??????Hˆh?s ˆ?e??Hˆ.
?2.34?
We note that all the different forms of Fˆ(?) above lead to
the same value of the thermal rate constant ?see Appendix B
for a detailed discussion about this?. However, the numerical
behavior of the corresponding flux correlation function or the
ease of the path integral evaluation of Fˆ(?) may be quite
different. For example, the flux correlation function defined
with the Kubo form exhibits much wilder oscillations at low
temperature than that defined with the half-split form, which
will be demonstrated in Sec. IID2, and thus the former is
more difficult to use numerically than the latter. On the other
hand, when one wants to evaluate a matrix element of Fˆ(?)
using path integral techniques, the Kubo form is much more
convenient than the half-split form. This can readily be seen
by considering a coordinate matrix element of FˆKubo(?) in
the second form of Eq. ?2.33?:
?xb?FˆKubo????xa??
i
???h„s?xb?…?h„s?xa?…?
??xb?e??Hˆ?xa?,
?2.35?
which can be evaluated using the standard path integral ex-
pression of the Boltzmann operator,3
?xb?e??Hˆ?xa???
x?0??xa
x?????xbDx???e?S¯???x????/?,
?2.36?
with the imaginary-time action defined by
S¯???x??????
0
??
d??
m
2?
dx???
d??
2
?V„x???…?.
?2.37?
By contrast, the construction of a matrix element of Fˆhalf(?)
is not so straightforward since the bare flux operator Fˆ,
which is essentially a differential operator, appears in the
middle of the Boltzmann operator. ?However, when the reac-
tion coordinate s(x) involves only one of the coordinates
being used ?e.g., s?x1), the matrix element of Fˆhalf(?) can
be constructed rather straightforwardly.? Therefore, the half-
split and Kubo forms of Fˆ(?) have their own merits and
demerits in terms of numerical calculations.
Now we will show that it is possible to define yet an-
other form of Fˆ(?) that possesses the advantages of both
Fˆhalf(?) and FˆKubo(?). First, consider the following general
form of the Boltzmannized flux operator:
Fˆ??;g???
0
?
d? g???e??????HˆFˆe??Hˆ,
?2.38?
where the splitting of the Boltzmann operator is controlled
by the weight function g(?). For Fˆ(?;g) to be Hermitian
and to give the same rate constant via Eqs. ?2.17? and ?2.18?,
we impose the following conditions on g:
?
0
?
d? g????1,g????g?????.
?2.39?
With this definition, the half-split form corresponds to a
delta-function weighting at ???/2,
ghalf?????????
2?,
?2.40?
while the Kubo form arises from an unbiased averaging over
the whole domain:
gKubo????1
?
?0?????.
?2.41?
It is then natural to consider the following g(?) that gives a
partial averaging with respect to ?:
g??????
0otherwise,
1/????
?1????/2????1????/2,
?2.42?
which has an interpolative character as follows:
lim
?→0
g?????ghalf???, lim
?→1
g?????gKubo???.
?2.43?
If we define a general form of Fˆ(?) with g?(?) in Eq.
?2.42?,
Fˆ?????Fˆ??;g??,
?2.44?
Fˆ?(?) interpolates between the half-split and Kubo forms as
a result of Eq. ?2.43?:
lim
?→0
Fˆ?????Fˆhalf???,lim
?→1
Fˆ?????FˆKubo???.
?2.45?
One can also obtain a closed form of Fˆ?(?) by using the
derivative relation in Eq. ?2.34?,
Fˆ?????
i
????e??1????Hˆ/2h?s ˆ?e??1????Hˆ/2
?e??1????Hˆ/2h?s ˆ?e??1????Hˆ/2?.
?2.46?
Here it is interesting to note that the above form reverts to
the second form of FˆKubo(?) in Eq. ?2.33? as ?→1, whereas
in the limit ?→0 the difference of two terms above naturally
generates the bare flux operator Fˆ in the middle of the
Boltzmann operator as Fˆhalf(?).
A clear advantage in using Fˆ?(?) in Eq. ?2.46? is that it
involves only the step function h(s ˆ), and it is thus straight-
forward to construct the corresponding path integral expres-
sion. For instance, a coordinate matrix element of Fˆ?(?) can
be written as
????
x?0??xa
?h„s?x??…?e?S¯???x????/?,
with x??x„(1??)??/2…. This is almost identical to the
path integral expression in Eq. ?2.36?, the only difference
being the presence of the step functions in the integrand
above. Due to this factor the path x(?) contributes to the
?xb?Fˆ?????xa??
i
x?????xbDx????h„s?x??…
?2.47?
2140 J. Chem. Phys., Vol. 118, No. 5, 1 February 2003T. Yamamoto and W. H. Miller
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Page 7
integral only when x?and x?are on opposite sides of the
dividing surface. This fact, in turn, leads to the localization
of the path x(?) around the dividing surface.
The numerical behavior of Fˆ?(?) should depend on the
choice of ?. In Sec. IID2 we will demonstrate that a particu-
lar choice of ?, namely ??1/2, can yield a numerically well-
behaved flux correlation function. With this choice Fˆ?(?)
takes the form
Fˆ??1/2????
i
??/2?e??Hˆ/4h?s ˆ?e?3?Hˆ/4
?e?3?Hˆ/4h?s ˆ?e??Hˆ/4?,
?2.48?
which is in a sense halfway between the half-split (??0)
and Kubo (??1) forms.
D. Numerical example for a 1-D barrier
Here we consider a simple one-dimensional ?1-D? prob-
lem, i.e., the Eckart barrier with parameters that correspond
approximately to the H?H2reaction,
p2
2m?
H?
V0
cosh2?x/a?,
?2.49?
i.e., m?1061.0 a.u., V0?0.425 eV, and a?0.734 a.u., and
?i? examine the dependence of the HK propagator on the
coherent-state parameter ? in light of the analysis given in
Sec. IIA, and ?ii? illustrate the effect of different choices of
the Boltzmannized flux operator discussed in Sec. IIC on the
flux correlation functions.
1. Dependence of the flux correlation function on the
coherent-state parameter
As discussed in Sec. IIA, the HK IVR can be thought of
as a modified Filinov interpolation between the coordinate
and momentum representations of the Van Vleck ?or ‘‘primi-
tive’’ semiclassical? approximation for the propagator. The
coherent-state parameter ? determines how coordinatelike or
momentumlike the HK propagator is. If the SC approxima-
tion for the propagator were equally accurate in coordinate
and momentum representations—as manifest here in the re-
active flux correlation function—then the results would be
independent of ?, but this is, in general, not the case. Here,
therefore, we examine the dependence of the reactive flux
correlation function ?for the example described above, Eq.
?2.49?? on the coherent-state parameter ?.
Figure 1 shows the symmetric flux-flux correlation func-
tion Cff(t) in Eq. ?2.23? for a wide range of ? at a tempera-
ture of T?1000 K. The flux correlation functions were cal-
culated using the double-forward IVR described in Sec. IIB.
Since the related Monte Carlo integrals are fully converged,
the statistical error of Cff(t) in Fig. 1 is negligible. The
coherent-state matrix element of the Boltzmannized flux op-
erator was evaluated exactly using the discrete variable rep-
resentation ?DVR?,37–39so that the deviation of the semiclas-
sical Cff(t) from the exact quantum mechanical one is totally
due to the semiclassical approximation in the HK propagator.
We see from Fig. 1 that the semiclassical Cff(t) is quite
accurate when ? is greater than ?0, while the agreement
becomes worse as ? is decreased. ?Here ?0is defined as ?0
?m?b/?, ?bbeing the harmonic ?imaginary? frequency at
the top of the barrier.? Figure 2 displays the time integral of
Cff(t) in Fig. 1 ?which is proportional to the rate constant? as
a function of ??/?0?solid curve?. The results in Fig. 2 show
that the HK result for the rate can be very much in error for
the smallest values of ? examined here ?i.e., in the momen-
tumlike limit?, while it converges rather quickly to an accu-
rate value as ? is increased ?the coordinatelike direction?.
Also shown in Fig. 2 is the statistical error associated with
the Monte Carlo average over initial conditions of trajecto-
ries in Eq. ?2.24?, which was obtained by sampling a fixed
number of 40000 trajectories and using the block averaging
FIG. 1. The flux–flux correlation function Cff(t) for the one-dimensional
Eckart barrier in Eq. ?2.49? at T?1000 K. The Boltzmannized flux operator
is chosen to be the half-split form. The solid curve is the exact quantum
mechanical result, while other curves are obtained by using the HK propa-
gator, as described in Sec. II B. ? is the coherent-state parameter in Eq.
?1.7?, and ?0is a reference value ?see the text?. The result of HK becomes
increasingly more accurate as ? is increased. In the case of ??4?0, the HK
result reaches the Van Vleck limit and is almost indistinguishable from the
exact quantum result.
FIG. 2. The time integral of the flux–flux correlation function depicted in
Fig. 1. The solid curve is the statistically fully converged result obtained
with the HK propagator, and the dash–dotted line is the exact quantum
mechanical result. The dashed curve is the statistical error of the HK result
when Cff(t) is evaluated with a fixed number of 40 000 trajectories.
2141J. Chem. Phys., Vol. 118, No. 5, 1 February 2003Thermal rate constants in Cartesian space
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Page 8
method.34,40
sampled every five sweeps of the double phase-space vari-
ables (q0p0;q0?p0?) to reduce the correlation among trajecto-
ries. We see from this figure that the statistical error grows
gradually as ? is increased, which is due to the fact that the
HK approaches the Van Vleck propagator in the coordinate
representation that has a highly oscillatory integrand. The
important conclusion of this example is that there is a wide
range of ? values ?provided they are not too small? for which
the statistical error of the IVR calculation is small and also
for which the approximation is quite accurate.
The individual real-time trajectories were
2. Different forms of the Boltzmannized flux operator
Next, we examine the effect of different choices of the
Boltzmannized flux operator on the flux correlation func-
tions. Figure 3 displays the flux–side correlation function
Cfs(t) for the Eckart barrier, which was obtained by integrat-
ing the symmetric flux–flux correlation function in time. The
solid, dashed, and dash-dotted curves in Fig. 3 represent
Cfs(t) defined with the Boltzmannized flux operator Fˆ(?) of
the half-split (??0), intermediate (??1/2), and Kubo
(??1) forms, respectively. Here all the calculations were
performed quantum mechanically with no semiclassical ap-
proximations, and thus the differences among these results
are entirely due to the definition of Fˆ(?). At a high tempera-
ture of T?1000 K ?panel ?a??, where the dynamics involved
is almost classical, the three forms of Fˆ(?) exhibit similar
correlation functions that monotonically increase to the same
plateau value within t???. At a low temperature of T
?200 K ?panel ?b??, where the tunneling effect becomes pro-
found, although Cfs(t) defined with the half-split and inter-
mediate forms of Fˆ(?) show a similar monotonic behavior,
Cfs(t) defined with the Kubo form exhibits wild oscillations
and takes a much longer time to reach the plateau value. The
latter behavior can be qualitatively understood by comparing
the coordinate matrix element of the Boltzmannized flux op-
erator, namely ?x2?Fˆ?(?/2)?x1?. Figure 4 shows the imagi-
nary part of the matrix element, and we can see from this
figure that the matrix elements for the half-split and interme-
diate forms ?panels ?a? and ?b?? are smooth and very similar,
while that for the Kubo form ?panel ?c?? is quite different
from others, exhibiting a discontinuity in the (x1,x2) plane
due to the step function in Eq. ?2.33?. This discontinuity
necessarily leads to the inclusion of very high momenta in
the associated thermal wave packets and causes the oscilla-
tions of Cfs(t) seen in Fig. 3. The Kubo form is thus less
desirable than the other forms in terms of the behavior of the
flux correlation functions.
III. APPLICATION OF THE HK-IVR TO THE D¿H2
REACTION IN FULL DIMENSION
To test the feasibility of the semiclassical approach de-
scribed in Sec. IIB, we apply it to the calculation of a ther-
mal rate constant for the simplest gas-phase reaction,
D?HaHb→?
DHb?Ha.
DHa?Hb,
?3.1?
As discussed in the Introduction, we perform the calculation
in the full 6-D Cartesian space of the atom–diatom system
?see Fig. 5?, with only an overall translation of the center of
mass of the three atoms removed. This is significant in the
following two points: ?i? One can obtain a thermal rate con-
stant corresponding to the bulk experimental condition in the
gas phase directly; i.e., the resulting rate includes all the
contributions from different total angular momenta J. ?ii? The
kinetic energy operator in the Hamiltonian remains a simple
Cartesian form, thus greatly simplifying the numerical
implementation of the SC-IVR and/or path integrals in-
volved. In the present work we make a distinguishable par-
ticle approximation for the two H atoms, which can be ex-
pected to be rather accurate since the reaction ?3.1? is known
to have a tight transition state geometry. Within this approxi-
mation we take into account both arrangement channels in
the reaction ?3.1?.
A. The rate expression
The thermal rate constant k(T) for the above reaction is
defined by the rate equation,
FIG. 3. The flux–side correlation function Cfs(t) for the one-dimensional
Eckart barrier in Eq. ?2.49? at T?1000 K ?panel ?a?? and T?200 K ?panel
?b??. The solid, dashed, and dash–dotted curves correspond to Cfs(t) defined
with the Boltzmannized flux operator of the half-split (??0), intermediate
(??1/2), and Kubo (??1) forms, respectively. At the low temperature of
T?200 K, Cfs(t) defined with the Kubo form exhibits wild oscillations and
takes a much longer time to reach the plateau value.
2142 J. Chem. Phys., Vol. 118, No. 5, 1 February 2003T. Yamamoto and W. H. Miller
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Page 9
d
dt??DHa???DHb???k?T??D??HaHb?,
?3.2?
where ?X? represents the number density of the molecule X.
As described in Sec. IIB, the quantum mechanically exact
expression for k(T) can be written in terms of the flux–flux
correlation function as
Qtrans?T?QH2?T??
k?T??
1
0
?
dt Cff?t?.
?3.3?
Here Qtrans(T) and QH2(T) are the partition functions for
relative translational motion and internal motion of the H2
molecule, respectively, which are given by
Qtrans?T???
?R
2??2??
3/2
,
?3.4?
QH2?T???
v?
j
?2j?1?e???vj.
?We note that for T?200 K the neglect of nuclear spin sym-
metry in the partition function of H2can be justified.41? Fol-
lowing the discussion in Secs. IIB and IIC, the flux–flux
correlation function is taken in the symmetric form
Cff?t??tr?Fˆ??/2?exp?iHˆt/??Fˆ??/2?exp??iHˆt/???,
?3.5?
with the Boltzmannized flux operator chosen to be the inter-
mediate form (??1/2):
Fˆ????
i
??/2?e??Hˆ/4h?s ˆ?e?3?Hˆ/4
?e?3?Hˆ/4h?s ˆ?e??Hˆ/4?.
?3.6?
The reason for choosing the intermediate form of Fˆ(?) is the
good balance of a numerically well-behaved flux correlation
function and the ease of path integral evaluations.
The quantum Hamiltonian operator for describing the
reaction ?3.1? is given by Eqs. ?1.1? and ?1.2?, where ?Rand
?rare the reduced masses defined by
?R?mD•2mH
mD?2mH,
?r?1
2mH,
?3.7?
with mXbeing the mass of the atom X. For simplicity of the
subsequent calculations we introduce a mass-scaled, 6-D
Cartesian coordinate vector x, defined as
x????R
?R,??r
?r?,
?3.8?
with ? being the system-reduced mass given by
???
mD?mH?mH.
mDmHmH
?3.9?
FIG. 4. The coordinate matrix element of the Boltzmannized flux operator.
The imaginary part of ?x2?Fˆ(?/2)?x1? at T?200 K is plotted as a function
of (x1,x2). The panels ?a?, ?b?, and ?c? correspond to the half-split, inter-
mediate, and Kubo forms, respectively. The Kubo form exhibits a drastically
different feature from the other forms, which causes a wild oscillation of the
corresponding flux correlation function shown in Fig. 3?b?.
FIG. 5. The external Jacobi vectors employed in the SC-IVR and path
integral calculations for the D?H2reaction.
2143 J. Chem. Phys., Vol. 118, No. 5, 1 February 2003Thermal rate constants in Cartesian space
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Page 10
With these coordinates the Hamiltonian takes the even sim-
pler form
Hˆ??
?2
2??x
2?V?x?,
?3.10?
where “x
To evaluate the flux correlation functions, it is necessary
to specify the reaction coordinate s(x) that defines the divid-
ing surface that separates the reactant and product region.
Here we define s(x) in such a way that the two product
channels in Eq. ?3.1? are treated equivalently,
2??j?1
6
?2/?xj
2.
s?x??max?sa?x?,sb?x??,
?3.11?
where sa(x) and sb(x) are the reaction coordinates that de-
scribe the individual rearrangement processes,
sa?x??r?Hb?Ha??r?Ha?D?,
?3.12a?
sb?x??r?Ha?Hb??r?Hb?D?,
?3.12b?
with r(X?Y) being the interatomic distance between the
atoms X and Y. The dividing surface corresponding to the
above reaction coordinate is schematically depicted in Fig. 6.
Finally, the calculation of the rate constant can be split
into two steps by rewriting Eq. ?3.3? as
Qtrans?T?QH2?T?Cff?0??
k?T??
1
0
?
dt Rff?t?,
?3.13?
with Rff(t)?Cff(t)/Cff(0). The ‘‘static’’ factor Cff(0) will
be calculated exactly ?in principle? using imaginary-time
path integral techniques, while the ‘‘dynamical’’ factor Rff(t)
will be evaluated approximately with combined use of the
SC-IVR and path integrals.
B. The static factor
The static factor Cff(0) in Eq. ?3.13?, or equivalently the
normalization integral of the weight function for the double
phase-space variables in Eq. ?2.26?, can be obtained using
imaginary-time path integrals as follows. Setting t?0 in Eq.
?3.5? gives a trace formula for Cff(0) as
Cff?0??tr?Fˆ??/2?Fˆ??/2??.
?3.14?
Substituting the explicit form of Fˆ(?/2) in Eq. ?3.6? into the
trace above gives the following:
Cff?0??
32
????2?tr?h?s ˆ?e?3?Hˆ/4h?s ˆ?e??Hˆ/4?
?tr?h?s ˆ?e?2?Hˆ/4h?s ˆ?e?2?Hˆ/4??,
?3.15?
but this expression is numerically inconvenient because each
trace above is divergent for a scattering system ?only their
difference gives a finite value?. A more convenient expres-
sion can be obtained by inserting the identity relation h(s ˆ)
??1?h(s ˆ)??1 in between all the e??Hˆ/4factors and can-
celing the divergent contributions, which gives
Cff?0??
32
????2?Qrrpp?Qrprp?.
?3.16?
Here Qrrppand Qrprprepresent the following constrained
partition functions:
Qrrpp?tr?e??Hˆ/4hr?s ˆ?e??Hˆ/4hr?s ˆ?
?e??Hˆ/4hp?s ˆ?e??Hˆ/4hp?s ˆ??,
?3.17a?
Qrprp?tr?e??Hˆ/4hr?s ˆ?e??Hˆ/4hp?s ˆ?
?e??Hˆ/4hr?s ˆ?e??Hˆ/4hp?s ˆ??,
?3.17b?
where hr(s ˆ)?1?h(s ˆ) and hp(s ˆ)?h(s ˆ) are the projection
operators onto the reactant and product sides of the dividing
surface, respectively. The nature of the above partition func-
tions can be seen by considering their discretized path inte-
gral expressions,
Qrrpp??
?exp????x1¯xP??,
Qrprp??
?exp????x1¯xP??.
?P
2??2??
Pd/2? dx1¯dxPhrrpp?x1¯xP?
?3.18a?
?P
2??2??
Pd/2? dx1¯dxPhrprp?x1¯xP?
?3.18b?
Here d is the mathematical dimension of the system ?i.e., d
?6), P is the number of time slices, and xkrepresents the
discretized path variables for the kth time slice. The dis-
cretized action ? takes the standard form35
??x1¯xP??
?P
2?2??
k?1
P
?xk?xk?1?2??
P?
k?1
P
V?xk?,
?3.19?
while the composite step functions are defined by
FIG. 6. A schematic representation of the three arrangement channels in-
volved in the D?H2reaction. The potential energy surface exhibits a three-
fold symmetry with a high barrier around the triangular geometry of the
DH2complex. The dashed lines indicate a set of dividing surfaces that are
used to calculate the static factor Cff(0). The dividing surfaces with ??1
and ??0 correspond to s(x) in Eq. ?3.11? and s0(x) in Eq. ?3.22?, respec-
tively.
2144J. Chem. Phys., Vol. 118, No. 5, 1 February 2003T. Yamamoto and W. H. Miller
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hrrpp?x1¯xP??hr„s?xP/4?…hr„s?x2P/4?…
?hp„s?x3P/4?…hp„s?x4P/4?…,
hrprp?x1¯xP??hr„s?xP/4?…hp„s?x2P/4?…
?hr„s?x3P/4?…hp„s?x4P/4?….
Therefore, the closed path ?x1,...,xP? gives a nonvanishing
contribution to the integral only when the four ‘‘beads,’’
?xPj/4?j?1,4, are located in the reactant or product side in an
appropriate manner. The situation is illustrated in Fig. 7. This
constraint, in conjunction with the harmonic spiring terms in
the discretized action ?, plays the role to localize the closed
path around the dividing surface.
Equation ?3.16? converts the original problem of com-
puting Cff(0) to that of computing a partition function. This
is advantageous because a number of well-established tech-
niques are available for this purpose.34In the present work
we utilize the extended ensemble method33to examine the
dependence of the partition function on the location of the
dividing surface, and perform a thermodynamic integration34
to estimate the temperature dependence. To accomplish this,
we first introduce a ‘‘variable’’ dividing surface ?or reaction
coordinate? characterized by a parameter ?, which is defined
by
?3.20a?
?3.20b?
s?x;????s1?x???1???s0?x?
?0???1?,
?3.21?
and
s1?x??s?x?,s0?x??R???R?,
?3.22?
with R?being a given constant. Hence, s(x;?) interpolates
between the original reaction coordinate s(x) and the
‘‘asymptotic’’reaction coordinate s0(x) in a continuous man-
ner, which is illustrated in Fig. 6. We note that R?should be
chosen large enough so that the dividing surface for s0(x)
?i.e., ?R??R?) is located well in the reactant valley. Next, we
parametrize the constrained partition function by ? and ? and
evaluate it through the following three steps ?here we restrict
the discussion to Qrrpp).
?1? Calculate the partition function in the asymptotic re-
gion at a given reference temperature, i.e.,
??1??Qrrpp???0;?ref?.
?3.23?
This can be achieved by using a conventional basis set
method such as the DVR,37–39since in this region the inter-
atomic potential between the reactants vanishes and the
quantum mechanical trace in Eq. ?3.17a? splits into those
associated with each reactant molecule ?see Appendix C for
details?.
?2? Evaluate the ratio of the partition function in the
interaction region to that in the asymptotic region at the ref-
erence temperature:
??2??Qrrpp???1;?ref?
Qrrpp???0;?ref?.
?3.24?
In the present work we employ the extended ensemble
method33to compute the above ratio. This method was origi-
nally developed to calculate the ratio of partition functions
having different temperatures, where the ‘‘extended’’ degree
of freedom was chosen to be the discretized temperatures. In
the present case we choose the location of the dividing sur-
face ?or more precisely, the parameter ? that specifies the
variable reaction coordinate? as the extended degree of free-
dom. Specifically, we discretize ? as
?m?m
M
?m?0,1,...,M?,
?3.25?
and consider an ‘‘extended’’ space ?(X,m)? that is the direct
product of the path space ?X?(x1¯xP)? and the set of the
different reaction coordinates ??m?m?0,M. Then, we define
an unnormalized distribution function ? for this space by
??X,m??hrrpp?X;?m?exp????X;?ref??.
?3.26?
With this definition, ?(2)can be expressed as the ratio of two
volume integrals:
??2??? dX??X,M??? dX??X,0??PM/P0.
?3.27?
Here Pmrepresents the marginal probability to find the sys-
tem in a particular value of m:
Pm?? dX??X,m??? ?
m??0
M? dX??X,m???.
?3.28?
Hence, ?(2)can be obtained by running a Monte Carlo tra-
jectory in the extended space according to the distribution
function ?(X,m) and counting the number of states that be-
longs to each value of m.
?3? Perform a thermodynamic integration34to compute
the ratio
??3??
Qrrpp???1;??
Qrrpp???1;?ref?.
?3.29?
FIG. 7. A schematic representation of the discretized path integral for the
constrained partition functions Qrrpp?panel ?a?? and Qrprp?panel ?b?? in Eq.
?3.18?. The circles represent the discretized path variables ?xk? while the
springs represent the kinetic energy terms in the discretized action in Eq.
?3.19?. The four ‘‘beads,’’ ?xPj/4?j?1,4, need to be located in the reactant or
product side of the dividing surface according to the constraints in Eq.
?3.20?. For example, in panel ?a? the two beads xP/4and x2P/4need to be in
the reactant side while x3P/4and x4P/4have to be in the product side.
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Page 12
This can be achieved by taking the log derivative of Eq.
?3.18a? with respect to ?:
?
??lnQrrpp??????dXhrrpp?X?exp????X;???E?X;??
?dXhrrpp?X?exp????X;???
???E?x;???,
?3.30?
where ??1 is assumed. E(X;?) is the thermodynamic esti-
mator of internal energy,35,36
E?X;???Pd
2??
?P
2?2?2 ?
k?1
P
?xk?xk?1?2?1
P?
k?1
P
V?xk?.
?3.31?
Integrating Eq. ?3.30? with respect to ? gives the ratio as
??3??exp???
?ref
?
d???E?x;?????.
?3.32?
Steps ?2? and ?3? above can be performed using standard path
integral Monte Carlo techniques.35,36Finally, combining
?(1), ?(2), and ?(3)gives the desired value of the constrained
partition function Qrrpp(?;??1) in the interaction region
?and subsequently the static factor Cff(0)].
C. The dynamical factor
The dynamical factor Rff(t) ?i.e., the normalized flux–
flux correlation function? will be calculated semiclassically
according to the prescription given in Sec. IIB, where the
exact time evolution operator is replaced by the HK propa-
gator. Here we evaluate the coherent-state matrix element of
the Boltzmannized flux operator in Eq. ?2.24? using dis-
cretized path integrals as follows. First, we rewrite the matrix
element as
?qbpb?Fˆ????qapa?
?? dxb? dxa?qbpb?xb??xb?Fˆ????xa??xa?qapa?. ?3.33?
The coordinate matrix element of Fˆ(?) can be evaluated
using the discretized version of Eq. ?2.47? with ??1/2 as
??/2?
??h„s?x3P/4?…?h„s?xP/4?…?
?exp??
wkV?xk??,
?xb?Fˆ????xa??
i
?P
2??2??
Pd/2? dx1¯dxP?1
?P
2?2??
k?1
P
?xk?xk?1?2
??
P?
k?0
P
?3.34?
with x0?xa, xP?xb, and wk?1 for 1?k?P?1 and 1/2
otherwise. Substituting Eq. ?3.34? and the coordinate repre-
sentation of the coherent state into Eq. ?3.33? yields a dis-
cretized path integral of the coherent-state matrix element
as42
?qbpb?Fˆ????qapa?
?C? dx0dx1¯dxP?h„s?x3P/4?…?h„s?xP/4?…?
?exp???
2??qb?xP?2??x0?qa?2?
?
i
???pb"?xP?qb??pa"?x0?qa??
?
?P
2?2??
k?1
P
?xk?xk?1?2??
P?
k?0
P
wkV?xk??,
?3.35?
where C is an overall constant. This integral can be evaluated
conveniently by choosing the weight function for the path
variables as
W?x0¯xP??exp???
2??qb?xP?2??x0?qa?2?
?xk?xk?1?2?.
?
?P
2?2??
k?1
P
?3.36?
With this weight function the coherent-state matrix element
can be written in a Monte Carlo form as
?qbpb?Fˆ????qapa??C???h?s?x3P/4???h„s?xP/4?…?
?exp???
?exp?
?pa"?x0?qa????
where ?¯?Wrepresents an average over W,
?¯?W??dx0¯dxPW?x0¯xP???¯?
?dx0¯dxPW?x0¯xP?
and C? is another constant. Since W(x0¯xP) is a multidi-
mensional Gaussian, we can sample it straightforwardly
through the normal-mode sampling36or by the staging
algorithm.43
Now we summarize the computational procedure of the
SC-IVR and path integral calculations. The initial conditions
of real-time trajectories (q0,p0) and (q0? ,p0?) were sampled
using the standard Metropolis method44with the weight
function
P?
k?0
P
wkV?xk??
i
???pb"?xP?qb?
W
,
?3.37?
,
?3.38?
W?q0p0;q0?p0?????q0p0?Fˆ??/2??q0?p0???2,
?3.39?
and a pair of trajectories ran every five sweeps of
(q0p0;q0?p0?). A fixed number of 105trajectories were propa-
gated to obtain the correlation function at a given tempera-
ture, and the calculation repeated for different temperatures
of T?200, 300, 500, and 1000 K. The sampling of the path
2146J. Chem. Phys., Vol. 118, No. 5, 1 February 2003 T. Yamamoto and W. H. Miller
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Page 13
variables (x0¯xP) in Eq. ?3.35? was performed via normal-
mode sampling.22,36The number of time slices P for com-
puting a matrix element of Fˆ(?/2) was chosen to be 4 and 20
at T?1000 and 200 K, respectively, which correspond to 8
and 40 time slices for e??Hˆ. The number of imaginary-time
paths sampled was chosen as 8?104and 1.2?106for T
?1000 and 200 K, respectively. ?We note in passing that the
number of paths above could probably be reduced by using
the bisection Monte Carlo36or the staging algorithm,43
namely by discarding a set of paths that do not contribute to
the integral in an early stage of the path construction, but we
did not pursue this idea in the present work.? One simple but
effective technique used to reduce the computational effort is
to store precalculated fluctuation paths ?i.e., the fluctuation
from the straight-line path connecting qaand qbin Eq.
?3.35?? in fast memory and reuse them to evaluate different
matrix elements. This is possible because the fluctuation
paths do not depend on the phase-space variables. The
coherent-state parameter ? in Eq. ?1.7? was chosen to be ?
??s???s/?, where ?srepresents the frequency of the
symmetric stretching mode at the transition state. We con-
firmed that the correlation function at T?1000 K does not
change significantly, even if we set ? to 2?sor ?s/2. Finally,
we applied the first-order Filinov smoothing to the
exp?i?St(q0p0)?St(q0?p0?)?/?? factor in Eq. ?2.24?, which
yields a damping factor for the monodromy matrix elements
and was useful in accelerating the Monte Carlo convergence
with respect to the number of trajectories.
Figure 8 shows the normalized correlation function
Rff(t) thus obtained for T?1000 K ?panel ?a?? and 200 K
?panel ?b??. At the high temperature of T?1000 K, the cor-
relation function is statistically well converged and exhibits a
smooth, monotonic decay without any indication of recur-
rences. At the lowest temperature examined (T?200 K),
though the statistical error of Rff(t) becomes larger, it can
still be seen that Rff(t) decays almost monotonically to zero
with only a slight negative lobe. Hence, the overall behavior
of the flux correlation function is characteristic of a ‘‘direct’’
reaction and is very similar to that of the one-dimensional
Eckart barrier studied in Sec. IID. Figure 9 displays the ther-
mal rate constant k(T) obtained by combining the static fac-
tor Cff(0) with the time integral of Rff(t). The agreement of
the present semiclassical result with that given by quantum
scattering calculations45is very good at high temperatures
?the deviation is ?10% at T?1000 K) and also good, even
at the lowest temperature ??20% at T?200 K). We also plot
a ‘‘crude’’ quantum transition-state-theory ?QTST? estimate
of the rate in Fig. 9, which is defined here as
kQTST?T??
1
Qtrans?T?QH2?T?Cff?0???
2,
?3.40?
that is, Rff(t) is assumed to decay with the ‘‘thermal time’’46
t??? and the time integral of Rff(t) is approximated by
??/2. As expected, the approximate rate gives accurate re-
sults at high temperatures while it becomes poor at low tem-
peratures. This is because the true flux correlation function
decays much faster than the thermal time at low tempera-
tures, which is also evident in Fig. 8. Overall, it is encour-
aging to see that the SC-IVR can evaluate a thermal rate
constant directly using the Cartesian coordinates of the sys-
tem. This is especially so because the use of such coordinates
will be required for simulating more complex chemical reac-
tions.
Although encouraging, we should point out a numerical
difficulty associated with the present approach; i.e., the
Monte Carlo evaluation of the coherent-state matrix element
using the path integral in Eq. ?3.35? is computationally quite
demanding. This is because it involves a plane-wave term
arising from the momentum factor of the coherent state,
exp?
i
???pb"?xP?qb??pa"?x0?qa???,
?3.41?
which makes the integrand more oscillatory as the momenta
(pb,pa) are increased. Due to this factor the Monte Carlo
evaluation of Eq. ?3.35? is much more time-consuming than
that of the coordinate matrix element, i.e., ?xb?Fˆ(?)?xa?. In
the present work we alleviated this problem by using several
ad hoc techniques. ?For instance, when the coherent-state
matrix element was dominated by statistical error we ap-
proximated its value by zero. Also see the discussion in Ref.
FIG. 8. Normalized flux–flux correlation function Rff(t)?Cff(t)/Cff(0) for
the D?H2reaction; ?a? T?1000 K and ?b? T?200 K. Error bars indicate
one standard deviation obtained with 105pairs of trajectories. The ‘‘thermal
time’’ ?? is 7.6 fs for T?1000 K and 38 fs for T?200 K. Rff(t) decays
much faster than the thermal time at the low temperature of T?200 K.
2147J. Chem. Phys., Vol. 118, No. 5, 1 February 2003Thermal rate constants in Cartesian space
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Page 14
22.? However, it is obvious that one needs a more systematic
solution to it if one wants to apply the present approach to
larger systems.
IV. CONCLUDING REMARKS
In the present paper we have first presented a simple and
straightforward derivation of the HK propagator, which re-
lies on the useful property of the modified Filinov filtering
method. In this derivation, the HK propagator is defined a
priori as an interpolation operator between the Van Vleck
propagators in coordinate and momentum representations.
The fact that the HK is a Filinov-smoothed version of the
Van Vleck propagator suggests that the former may give a
different result from the latter, since the Filinov procedure
introduces some systematic error into the original integral.
Although this has been confirmed by numerical examples, it
was also demonstrated that one can reach the accurate Van
Vleck limit with reasonable computational effort by gradu-
ally increasing the coherent-state parameter ? and monitor-
ing the systematic convergence to the Van Vleck limit.
Second, we have applied the HK propagator to a thermal
rate constant calculation of the simplest gas-phase reaction.
The important point here is that all the calculations were
carried out using Cartesian coordinates of the system, and
thus the rate constant corresponding to the experimental con-
dition was directly obtained. This was made possible by in-
troducing the intermediate form of the Boltzmannized flux
operator, which has a good balance of the well-behaved flux
correlation functions and the simplicity of path integral ex-
pressions. Also, this form allows the use of an arbitrary re-
action coordinate, which is prerequisite for describing a
chemical reaction in terms of Cartesian coordinates.
Although the present semiclassical estimates of the rate
are very accurate and are thus encouraging, the application of
the present method to larger systems may be hampered by
the oscillatory nature of the path integral expression for the
coherent-state matrix element. This may be circumvented by
the following strategies. First, one can apply the Filinov
smoothing technique to the integrand of Eq. ?3.35?. Second,
the coherent-state matrix element may be approximated as
?qbpb?Fˆ????qapa?
?? dxb? dxa?qbpb?xb??xb?Fˆ????xa??xa?qapa?
??qb?Fˆ????qa?? dxb? dxa?qbpb?xb??xa?qapa?.
?4.1?
Here the variation of ?xb?Fˆ(?)?xa? as a function of (xb,xa)
is assumed to be slow compared to that of the coherent
states, which becomes valid when ? is chosen sufficiently
large. Then, the integral over (xb,xa) can be performed ana-
lytically, which results in the following ‘‘zeroth-order’’ ap-
proximation to the coherent-state matrix element:
?qbpb?Fˆ????qapa???qb?Fˆ????qa?
??
4?
??
d/2
exp??
1
2??2?pb
2?pa
2??.
?4.2?
The advantage of this approximation is that one needs only
the evaluation of a coordinate matrix element, i.e.,
?qb?Fˆ(?)?qa?. Higher-order approximations are also pos-
sible by considering the derivative of ?xb?Fˆ(?)?xa? with re-
spect to (xb,xa). Another possible strategy is to sample real-
time trajectories and imaginary-time paths in one Monte
Carlo integral, rather than in two independent Monte Carlo
integrals as in the present work. One can then avoid using a
weight function that is ‘‘contaminated’’ with statistical noise,
thus resulting in a more stable calculation. The feasibility
and performance of these approaches remain to be explored.
ACKNOWLEDGMENTS
The authors would like to thank the generous allocation
of supercomputer time from the National Energy Research
Scientific Computing Center ?NERSC?. This work was sup-
ported by the Director, Office of Science, Office of Basic
Energy Sciences, Chemical Sciences, Geosciences, and Bio-
sciences Division, U.S. Department of Energy under Con-
tract No. DE AC03-76SF00098 and by the National Science
Foundation Grant No. CHE-0096576. T.Y. gratefully ac-
knowledges the support of JSPS Postdoctoral Fellowships
for Research Abroad.
APPENDIX A: DERIVATION OF THE HK PROPAGATOR
Here we explicitly calculate the momentum matrix ele-
ment ?pf?exp(?iHˆt/?)?pi?VVQ/MFin Eq. ?2.12?, which defines
the interpolation operator UˆVVQP(t;c). Substituting the Van
Vleck formula in Eq. ?2.7? into Eq. ?2.12? gives
FIG. 9. An Arrhenius plot of the thermal rate constant k(T) for the D
?H2reaction. The solid curve is the result obtained from quantum scatter-
ing calculations ?Ref. 45? for the LSTH potential surface ?Ref. 47?. The
diamonds are the present result obtained with the SC-IVR and path inte-
grals. The dashed curve is a ‘‘crude’’QTST approximation to the rate, where
the time integral of the normalized flux–flux correlation function is replaced
by ??/2 ?see the text?. The dotted curve is the experimental result ?Ref. 48?.
2148J. Chem. Phys., Vol. 118, No. 5, 1 February 2003 T. Yamamoto and W. H. Miller
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Page 15
Kfi??pf?exp??iHˆt/???pi?VVQ/MF
?? dq1? dq0
1
2???
q????
1
2?i?Mqp?
1/2
?exp?i?St?q1,q0??pfq1?piq0?/??
?F?q1,q0;c?,
?A1?
which is rewritten in a form suitable for the Filinov proce-
dure as
Kfi?? dq1? dq0?
q???R?q1,q0?exp?i??q1,q0?/??
?F?q1,q0;c?.
?A2?
Here the amplitude R and the phase ? are naturally chosen as
2???
R?q1,q0??
1
1
2?i?Mqp?
1/2
,
?A3a?
??q1,q0??St?q1,q0??pfq1?piq0.
?A3b?
The evaluation of the modified Filinov factor F(q1,q0;c) in
Eq. ?2.3? requires the first and second derivatives of
?(q1,q0). The derivative relation24of the action St,
?St?q1,q0?
?q1
?p1?q1,q0?,
?St?q1,q0?
?q0
??p0?q1,q0?,
?A4?
gives the first derivatives of ? as
??
?q1?p1?q1,q0??pf,
??
?q0??p0?q1,q0??pi,
?A5?
and further differentiation of the first derivatives gives the
following second derivatives:
?2?
?q1?q1??p1?q1,q0?
?q1
?Mpp
Mqp,
?A6a?
?2?
?q0?q1??p1?q1,q0?
?q0
??
1
Mqp,
?A6b?
?2?
?q0?q0???p0?q1,q0?
?q0
?Mqq
Mqp.
?A6c?
The second equalities in the above equations can easily be
obtained by first writing the definition of the monodromy
matrix,
dq1?Mqqdq0?Mqpdp0, dp1?Mpqdq0?Mppdp0,
?A7?
and rearranging terms as
dp1?Mpp
Mqpdq1?
1
Mqpdq0,
?A8?
dp0?
1
Mqpdq1?Mqq
Mqpdq0,
where we have used the fact that MqqMpp?MqpMpq?1.
Substituting these derivatives into the definition of the modi-
fied Filinov factor in Eq. ?2.3? yields
F?q1,q0;c???
2ic
?Mqp?
?exp??
1/2
Ct?q0p0?
2?2??pf?p1?2??pi?p0?2??,
c
?A9?
where Ctrepresents the following ‘‘Herman–Kluk prefac-
tor:’’
Ct?q0p0???
1
2?Mqq?Mpp??
icMqp?ic
?Mpq??
1/2
.
?A10?
Combining Eqs. ?A1? and ?A9? leads to the following ex-
plicit form of the momentum matrix element:
Kfi??2????1? dq1? dq0?
q???
1
?Mqp??
c
??2?
1/2
?Ct?q0p0?exp?iSt?q1q0?/??
?exp??
?exp??
c
2?2?pf?p1?2?
i
?pfq1?
?piq0?.
c
2?2?pi?p0?2?
i
?A11?
By noting the coordinate and momentum representations of
the coherent state,
?q??qp????
?
??
1/4
exp???
2?q??q?2?
i
?p?q??q??,
?A12a?
?p?q?,
?A12b?
?p??qp????
1
???2?
1/4
exp??
1
2??2?p??p?2?
i
and invoking the ‘‘IVR trick,’’17i.e., changing the integra-
tion variable from the final coordinate q1to the initial mo-
mentum p0,
? dq1?
q????? dp0?Mqp?,
?A13?
one can simplify Eq. ?A11? as
Kfi??2????1? dq0? dp0Ct?q0p0?
?exp?iSt?q0p0?/???pf?qtpt???q0p0??pi?,
?A14?
with ??1/c. It is therefore seen that the interpolation opera-
tor UˆVVQP(t;c) in Eq. ?2.11? is identical with the Herman–
Kluk propagator. In this derivation it is evident that the HK
propagator becomes exact for a quadratic potential regardless
of the value of ?. This is because in such a case ?i? the
underlying Van Vleck formula is exact, ?ii? the amplitude
R(q1,q0) in Eq. ?A3a? becomes a constant, and ?iii? the
modified Filinov procedure involves no approximations.11It
is also interesting to note that one can define the Filinov
parameter to be a diagonal matrix:
c?diag?c1,c0?,
?A15?
2149J. Chem. Phys., Vol. 118, No. 5, 1 February 2003Thermal rate constants in Cartesian space
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Page 16
which leads to a ‘‘generalized’’form of the HK propagator as
Kfi??2????1? dq0? dp0Ct?q0p0;?1?0?
?exp?iSt?q0p0?/???pf?qtpt?1??q0p0?0?pi?,
?A16?
where the generalized HK prefactor is defined by
Ct?q0p0;?1?0???
1
2??1
1/2Mqq?0
?1/2??1
?1/2Mpp?0
1/2
??
i?1
1/2Mqp?0
1/2?
i
??1
?1/2Mpq?0
?1/2??
1/2
,
?A17?
with ?1?1/c1and ?0?1/c0. This form has been derived
previously by Kay through the asymptotic analysis of a gen-
eral semiclassical kernel.27
APPENDIX B: FLUX CORRELATION FUNCTIONS
DEFINED WITH THE GENERAL FORM OF
THE BOLTZMANNIZED FLUX OPERATOR
Here we show that the flux correlation function can be
defined in several different manners in conjunction with the
general form of the Boltzmannized flux operator in Eq.
?2.38?.
The quantum mechanically exact expression of the ther-
mal rate constant for a gas-phase reaction can be written as31
k?T?Qr?T??tr?e??HˆFˆPˆ??Fr.
Here Fˆis the flux operator in Eq. ?2.20? and Pˆrepresents a
projection operator onto the product region in the infinite
future,
?B1?
Pˆ?lim
t→?
exp?iHˆt/??h?s ˆ?exp??iHˆt/??.
?B2?
This can also be written as31
n? dE???n
Pˆ??
??
??E?????n
??E??,
?B3?
where ???n
total energy E that flows into the product arrangement chan-
nel ? with internal quantum numbers n in the infinite future.
It is evident from Eq. ?B3? that Pˆcommutes with the total
Hamiltonian, i.e.,
?(E)? represents a scattering eigenstate of Hˆwith
?Hˆ,Pˆ??0,
?B4?
which leaves us some freedom concerning the way in which
the Botlzmann operator appears in Eq. ?B1?. For example,
we can split the Boltzmann operator as
Fr?tr?e???/2???HˆFˆe???/2???HˆPˆ?,
where Fris the reactive flux defined in Eq. ?B1? and ? is an
arbitrary constant. Substituting the projection operator in Eq.
?B2? into the above equation yields
?B5?
Fr?lim
t→?
tr?e???/2???HˆFˆe???/2???HˆeiHˆt/?h?s ˆ?e?iHˆt/??
?B6a?
?lim
t→?
Cfs?t;??,
?B6b?
where we have defined a flux–side correlation function that
depends parametrically on ?. One can then show the follow-
ing symmetry of Cfs(t;?):
Cfs?t;??*?Cfs?t;???,Cfs??t;?????Cfs?t;??.
?B7?
Now we define another flux–side correlation function by
taking a weighted average of Cfs(t;?) with respect to ?:
I?t???
??/2
Cfs
?/2
d? g ˜???Cfs?t;??.
?B8?
Substituting the definition of Cfs(t;?) into Cfs
compact form as
I(t) gives a
Cfs
I?t??tr?Fˆ???eiHˆt/?h?s ˆ?e?iHˆt/??.
?B9?
Here Fˆ(?) represents the general form of the Boltzmannized
flux operator in Eq. ?2.38?, which is expressed in a slightly
different fashion as
Fˆ?????
??/2
?/2
d? g ˜???e???/2???HˆFˆe???/2???Hˆ,
?B10?
with
?
??/2
?/2
d? g ˜????1,g ˜????g ˜????.
?B11?
Using the symmetry of Cfs(t;?) and g ˜(?) in Eqs. ?B7? and
?B11? gives the following property of Cfs
I(t):
Cfs
I?t??Cfs
I?t?*,Cfs
I??t???Cfs
I?t?.
?B12?
One can also express the reactive flux in the time-integral
form,
I?0???
Fr?lim
t→?
Cfs
I?t??Cfs
0
?
dtCff
I?t?,
?B13?
where the flux–flux correlation function is defined by
Cff
I?t??
d
dtCfs
I?t??tr?Fˆ???eiHˆt/?Fˆe?iHˆt/??.
?B14?
Note that Cfs
function of t.
Another useful form of the flux–side correlation func-
tion arises by taking an average of Cfs(t;?) as
II?t???
??/4
??/4
I(0) in Eq. ?B13? vanishes since Cfs
I(t) is an odd
Cfs
?/4
d???
?/4
d? g ˜????g ˜???Cfs?t;?????, ?B15?
which has the same symmetry as Eq. ?B12?. Substituting the
following expression:
Cfs?t;??????tr?e???/4????HˆFˆe???/4????HˆeiHˆt/?
?e???/4???Hˆh?s ˆ?e???/4???Hˆe?iHˆt/??, ?B16?
2150 J. Chem. Phys., Vol. 118, No. 5, 1 February 2003T. Yamamoto and W. H. Miller
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Page 17
into Eq. ?B15? yields
Cfs
II?t??tr?Fˆ??/2?eiHˆt/?hˆ??/2?e?iHˆt/??,
?B17?
where hˆ(?/2) is defined similarly as Eq. ?B10?. The corre-
sponding flux–flux correlation function becomes
Cff
II?t??
d
dtCfs
II?t??tr?Fˆ??/2?eiHˆt/?Fˆ??/2?e?iHˆt/??,
?B18?
with which the reactive flux can be written as
II?t???
0
Fr?lim
t→?
Cfs
?
dt Cff
II?t?.
?B19?
Here we should emphasize that different choices of g ˜(?) or
different averaging of Cfs(t;?) with respect to ? provide
numerically different flux–side ?or flux–flux? correlation
functions, although the long time limit of Cfs(t) ?or the time
integral of Cff(t)] gives a unique value of the reactive flux
Fr. For example, in the case of a free particle, Cff(t) defined
with g ˜(?)??(?) ?which corresponds to the half-split form
of Fˆ(?)] becomes
Cff
I?t??Cff
II?t??
1
2???
???/2?2
?t2????/2?2?3/2,
?B20?
while that defined with g ˜(?)?1/? ?corresponding to the
Kubo form? becomes
Cff
I?t??
1
2?????2
??t2?????2?t?3/2
?2t?t2?????2
,
?B21?
and
Cff
II?t??
4
?????3??t2????/2?2
??t/2??t2?????2?t?1/2?.
?B22?
Notice that the Cff
comes singular at t?0.
I(t) for the Kubo form in Eq. ?B21? be-
APPENDIX C: CONSTRAINED PARTITION FUNCTION
IN THE ASYMPTOTIC REGION
In the limit ?→0, where the dividing surface is located
in the reactant valley, the constrained partition function in
Eq. ?3.17? can be evaluated conveniently using basis set
methods such as the discrete variable representations
?DVR?.37–39This is because the intermolecular potential be-
tween the reactants vanishes and thus the total Hamiltonian
separates into two parts as
lim
?→0
Hˆ?TˆR?Hˆr.
?C1?
Here TˆRand Hˆrdescribe the relative translational motion
and internal motion of the H2molecule, respectively,
TˆR??
?2
2?R“R
2, Hˆr??
?2
2?r?r
2?VH2?r?.
?C2?
Since the reaction coordinate for ??0 depends only on R,
the trace in Eq. ?3.17? factors into a trace over R and one
over r,
Qrrpp???0??Qrrpp
R
?QH2,
?C3?
where Qrrpp
lational motion,
R
represents the contribution from relative trans-
Qrrpp
R
?trR?e??TˆR/4hr?s ˆ0?e??TˆR/4hr?s ˆ0?
?e??TˆR/4hp?s ˆ0?e??TˆR/4hp?s ˆ0??,
?C4?
while QH2is the partition function of the H2molecule,
QH2?trr?e??Hˆr?.
?C5?
Qrrpp
normal basis set into the R space:
R
can be evaluated by introducing the following ortho-
?R?nlm??R?1?n?R?Yl
m??,??,
?C6?
where (R,?,?) are the polar coordinates of R, Yl
spherical harmonics, and ?n(R) is a radial DVR basis func-
tion satisfying ??n???n?R??n?n, with the inner product de-
fined as
??b??a?R??
0
mis the
?
dR ?b?R?*?a?R?.
?C7?
?In practice, one may use the sinc-DVR basis39for ?n(R),
although it satisfies Eq. ?C7? only approximately.? With this
basis set, the matrix element of TˆRbecomes
?n?l?m??TˆR?nlm????n??TˆR
l??n?R?l?l?m?m,
?C8?
with
TˆR
l??
?2
2?R
d2
dR2?l?l?1??2
2?RR2 .
?C9?
Now inserting the resolution of identity,
Iˆ??
n?
l?
m
?nlm??nlm?,
?C10?
in between all the neighboring hr, hp, and e??TˆR/4and in-
tegrating out the angular basis yields
Qrrpp
R
??
l?0?2l?1?tr
??e??TR
l/4hre??TR
l/4hre??TR
l/4hpe??TR
l/4hp?,
?C11?
where various DVR matrices are defined as
?TR
l?n?n???n??TˆR
l??n?R,
?hp?n?n??n?nh?R??Rn?,
hr?1?hp.
?C12?
Here we have used the usual DVR ?i.e., diagonal? approxi-
mation for a position-dependent operator, and Rnis the DVR
grid point corresponding to ?n(R). Combining Eqs. ?C5?
and ?C11? gives the desired value of the constrained partition
function in the asymptotic region.
2151 J. Chem. Phys., Vol. 118, No. 5, 1 February 2003Thermal rate constants in Cartesian space
Downloaded 17 May 2005 to 169.229.129.16. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
Page 18
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