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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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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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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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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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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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Page 11

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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