Steven L. Mielke and Donald G. Truhlar- A new Fourier path integral method, a more general scheme for extrapolation, and comparison of eight path integral methods for the quantum mechanical

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A new Fourier path integral method, a more general schemefor extrapolation, and comparison of eight path integral methodsfor the quantum mechanical calculation of free energies

Steven L. MielkeDepartment of Chemistry, The Johns Hopkins University, Baltimore, Maryland 21218

Donald G. TruhlarDepartment of Chemistry, Chemical Physics Program, and Supercomputer Institute, University ofMinnesota, Minneapolis, Minnesota 55455-0431

Received 23 June 2000; accepted 12 July 2000

Using an isomorphism of Coalson, we transform five different discretized path integral DPI

methods into Fourier path integral FPI schemes. This allows an even-handed comparison of these

methods to the conventional and partially averaged FPI methods as well as a new FPI method. It

also allows us to apply to DPI methods a simple and highly effective perturbative correction scheme

previously presented for FPI methods to account for the error due to retaining only a finite number

of terms in the numerical evaluation of the propagator. We find that in all cases the perturbative

corrections can be extrapolated to the convergence limit with high accuracy by using a correlated

sequence of affordable calculations. The Monte Carlo sampling variances of all eight methods

studied are very similar, but the variance of the perturbative corrections varies markedly with

method. The efficiencies of the new FPI method called rescaled fluctuation FPI and one of Fourier

analog methods compare favorably with that of the original FPI method. The rescaled fluctuation

method not only proves practically successful, but it also gives insight into the origin of the

dominant error in the conventional FPI scheme. 2001 American Institute of Physics.

DOI: 10.1063/1.1290476

I. INTRODUCTION

Path integral methods,1 3 especially when coupled with

Monte Carlo integration, provide a powerful means of calcu-

lating accurate quantal partition functions and hence absolute

free energies. Two different families of techniques have

evolved that are distinguished by the method used to repre-

sent the paths. Discretized path integral DPI methods1,2,46

represent a given path using a finite number of discrete

coordinate-space points that are equidistant in imaginary

time and are usually referred to as beads. Fourier path inte-

gral FPI methods1,2,721 represent the deviations of the

paths from free-particle paths by a Fourier expansion typi-

cally a Fourier sine expansion. The Fourier expansion may

simply be truncated to a finite number of terms, which is

called conventional FPI C-FPI, or one may use an approach

called partial averaging10,11 FPI PA-FPI in which the po-

tential is replaced by an effective potential that approxi-

mately includes the higher-order Fourier components that arenot retained after truncation.

Coalson9 has shown that the DPI formulations can be

mapped isomorphically onto Fourier-like formulations that

can be implemented with only slight differences from the

conventional FPI method. In the present paper, we use this

approach to transform five DPI methods into analog FPI

schemes, and we compare these methods to the C-FPI and

PA-FPI methods as well as to a new method introduced be-

low. The relative efficiency of various DPI and FPI methods

has already been widely studied9,14,2225 and debated!. By

using the Fourier analogs of the DPI methods we can employ

essentially the same Monte Carlo sampling scheme for all

eight methods, and this permits a more even-handed com-

parison of relative efficiencies than has been available previ-

ously.

As we have recently shown,20 one major advantage of

FPI calculations is that the paths Fourier expansion length

can be truncated at a moderate number of terms, and the

effect of additional terms can be considered as a perturba-

tion. A perturbative correction for the contribution from a

specific number of additional terms can be explicitly calcu-

lated in another Monte Carlo calculation that is substantially

less expensive than the primary calculation. Several pertur-

bative corrections, for various numbers of additional retained

terms, can be calculated simultaneously and in a correlated

fashion such that the infinite expansion limit can be obtained

via extrapolation without significant distortion due to statis-

tical sampling error.20

A major goal of the present article is to show that thesetechniques for perturbative corrections and extrapolation of

correlated calculations can be adapted with minor modifica-

tions for use with the Fourier analogs of DPI methods. An-

other goal is to examine the effectiveness of these techniques

as a function of which path integral method is employed. We

will also examine whether some of the Fourier analogs of the

DPI methods can provide performance better than that of the

original FPI method.

All specific applications in the present paper are for

vibrational-rotational partition functions of molecules. For-

JOURNAL OF CHEMICAL PHYSICS VOLUME 114, NUMBER 2 8 JANUARY 2001

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mal results about convergence rates are fairly general in the

following discussion we will assume that the potential is

bounded from below and possesses four continuous deriva-

tives, although most of results hold for less restrictive con-

ditions, but conclusions drawn from specific applications

may need to be retested if the methods are applied to other

problems because different algorithms may be favorable for

different kinds of problems.

II. THEORY

We begin by outlining the eight FPI Monte Carlo meth-

ods that we will compare. We then discuss the methods for

obtaining perturbative corrections and accurately extrapolat-

ing them.

II.A. Conventional and partially averaged FPI methods

The internal i.e., vibrational-rotational partition func-

tion of a molecule in its ground electronic state may be ob-

tained by calculating the trace of the canonical density op-

erator, i.e.,

Q T 1sym

dxx,x;, 1

where sym is a symmetry number, x is an N-dimensional

point in mass-scaled Jacobi coordinates N is 3NA3, where

NA is the number of atoms, is 1/kBT, where kB is Boltz-

manns constant,

x,x;xexpHx 2

is the coordinate representation of the density operator, and

H is the Hamiltonian operator. Elements of the density op-

erator may be expressed as path integrals

x,x;xx

Dx s exp1

0

dsHx s , 3

where is Plancks constant divided by 2, and xxDx(s)

denotes the summation over all paths parameterized by

imaginary time s and beginning at x and ending at x.

In the C-FPI method we set x equal to x and expand the

resulting closed paths in Eq. 3 in a Fourier series,

xj s xjk1

K

ajk sin ks , 4where K is the length of the Fourier expansion. After some

simplification one obtains the expression

Q K TJ T

sym

j1

N

dxj j1

N

k1

K

dajk

expj1

N

k1

Kajk

2

2k2Sx,a , 5

where the k are the fluctuation parameters given by

k2

22

2k2, 6

is the scaling mass of the mass-scaled Jacobi coordinates,

S(x,a) is the contribution of the potential energy to the ac-

tion integral for a given path and is calculated by

Sx,a0

dsV x s , 7

V(x) is the potential energy, and J(T) is the Jacobian1 of the

transformation from the integral over paths to the integral

over Fourier coefficients. Note that the kinetic energys con-

tribution to the action has been explicitly integrated. Equa-

tion 5 can be put into an expression more appropriate for

Monte Carlo integration by multiplying and dividing by the

free-particle partition function and restricting the configura-tion space to a finite domain D. We then obtain

Q K TQ fp T

sym

D

dx

da expj1

N

k1

Kajk

2

2k2 expSx,a

D

dx

da expj1

N

k1

Kajk

2

2k2

, 8

where the free particle partition function is given by

Q fp TVD 22 N/2

9

and VD is the volume of the domain D. Partition functions

calculated with the C-FPI method converge asymptotically

as O(1/K). 26

Instead of simply ignoring contributions from Fourier

components with kK, one can use the approach of partial

averaging.10,11 The effects of kK are approximated by in-

voking Gibbs inequality.2 The PA-FPI method may then be

implemented exactly like the C-FPI method except that thepotential is replaced by an effective potential defined by11

VeffPA

x s 22 s N/2dp

expi1

N

p i2/22 s V x s p, 10

where

2 s kK1

k2 sin2 ks/, 11

or more conveniently for computation,

622 J. Chem. Phys., Vol. 114, No. 2, 8 January 2001 S. L. Mielke and D. G. Truhlar



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2 s 1

s s

k1

K

k2 sin2 ks/. 12

The PA-FPI method gives a rigorous lower bound on the

partition function and converges asymptotically as

O(1/K2). 26 Analytic integration of the N-dimensional Gauss-

ian transform of Eq. 10 is rarely possible for realistic mo-

lecular potential energy functions, so a gradient expansion is

usually employed

VeffPA

x s V x s 2 s

2 i1

N2V x s

x i2 . 13

Even more rapidly convergent procedures can be derived us-

ing cumulant methods11,14 but these have not been widely

used for numerical work due to their increased complexity.

II.B. Discretized path integral methods

The DPI methods can be derived by starting with the

identity

x,x;x,x;/P P, 14

where P is the number of equidistant time slices. Using Eq.

14 in Eq. 1 and inserting the resolution of the identity Ptimes gives

Q T1

sym dx1 dxP

i1

P

xiexp P H xi115

with the requirement that xP1x1 . At this point the stan-

dard Trotter approximation27

exp P

Hexp P

T exp P

V , 16or symmetric Trotter approximation28

exp P

Hexp 2 P

V exp P

T

exp 2 P

V , 17where T and V denote the kinetic and potential energy op-

erators, is sometimes invoked. Partition functions calculated

using either of these approximations can be shown28 to con-

verge as O(1/P 2). Either approximation, together with Eq.

15, yields rigorous upper bounds on the exact partition

function; in particular, it can be shown29,30 that

Q T; P2p1 Q T; P2p2Q T; P; p 2p 1 .

18

Equation 15, together with one of the Trotter or

Trotter-type approximations, is still not in a form that is easy

to evaluate, and an additional approximation is required. Ei-

ther the midpoint Trotter MT,

xiexp TP expV

P xi1

fpxi ,xi1 ;/P expV xixi1/2, 19

or trapezoidal Trotter TT approximation,

xiexp V2 P expT

P expV

2 P xi1

fpxi ,xi1 ;/P expVxiVxi1/2 , 20

where fp(xi ,xi1 ;/P) is the free particle density, is com-

monly used to reduce Eq. 15 to a form that can be readily

implemented. It is quite common to see claims made in the

literature for calculations using the MT or TT approxima-

tions that have been proven only for the Trotter approxima-

tion. Formal proofs of asymptotic convergence rates with

these more approximate schemes are apparently not avail-

able, but as we will discuss below, it is possible to show that

partition functions calculated with the MT and TT schemes

converge asymptotically as O(1/P) and O(1/P 2), respec-

tively.Considerable attention has been given to the calcula-

tion of higher order corrections to the Trotter

approximation.22,28,3135 One of the most widely used

expressions,22,32,33 which we will simply refer to as the

TakahashiImada TI approximation, can be implemented

by replacing the potential in Eq. 17 with an effective po-

tential

VeffTI

xVx1

24

P

2

VV. 21

Partition functions calculated with this expression converge

as O(1/P 4). 28,32 One could use Eq. 21 in conjunction with

either Eq. 19 or Eq. 20, but we will only consider the

latter option, which converts the trapezoidal Trotter approxi-

mation into the trapezoidal TakahashiImada TTI approxi-

mation. In the following discussion we will assume that the

TTI approximation has the same asymptotic convergence

rate as the TI approximation.

Another approach to treating Eq. 15 is to expand the

step propagator in a power series.3638 This yields

xiexp HP xi1fpxi ,xi1 ;/P

expi

n2

i/P n

1Wnx .22

The first two terms are given by38

W2x 0

1

d Vxixi1xi 23

and

W3xi

2

0

1

d 12Vxxxixi1xi ;

24

623J. Chem. Phys., Vol. 114, No. 2, 8 January 2001 A new Fourier path integral method



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higher order terms are available but are prohibitively expen-

sive to evaluate numerically. We follow the notation of

Makri and Miller37,38 by referring to the expansion through

the terms W3(x) as the first order propagator FOP, and we

will refer to the expansion through the term W2(x) as the

zero order propagator ZOP. We note that these expansions

are accurate to O(1/P 2) and O(1/P), respectively.36,38 The

MT scheme can be considered a midpoint-rule integration

approximation of the ZOP scheme; since this integration isaccurate to O(1/P 2), the MT scheme converges at the same

asymptotic rate as the ZOP schemeO(1/P).

II.C. Fourier analogs of DPI methods

Discretized path integral methods, including the five

MT-DPI, TT-DPI, TTI-DPI, ZOP-DPI, and FOP-DPI that

we have detailed above, can be transformed into FPI meth-

ods using an isomorphism established by Coalson.9 Coalson

also showed that a P-point TT-DPI scheme is equivalent to

the C-FPI scheme with an infinite number of Fourier coeffi-

cients but with the action integrals integrated using a P-point

trapezoidal rule.9

One can see from this that the TT-DPIscheme converges at the same rate as trapezoidal-rule inte-

gration, i.e., O(1/P 2); thus, we see that partition functions

calculated with Eq. 20 have the same asymptotic conver-

gence rate as those obtained with Eq. 17.

Coalson further showed9 that if we change the fluctua-

tion parameters in the C-FPI method from those of Eq. 6 to

k;K2

22

1

2K1sin k/ 2K1 2, 25

that a Fourier expansion of length K produces a path for

which the P equidistant-time points with PK1 are dis-

tributed as in an infinite Fourier expansion. Using this rela-

tionship, each DPI method can be transformed into an FPI

method; this leads to five Fourier analog methods that we

will label MT-FPI, TT-FPI, TTI-FPI, ZOP-FPI, and FOP-

FPI. Specifically the MT-FPI method is implemented by re-

placing the action integral in Eq. 7 by a P-point midpoint

rule,

S MT-FPIx,a1

P i1

P

Vxixi1/2 , 26

and the TT-FPI method is implemented by replacing the ac-

tion integral with a P-point trapezoidal rule integration,

S TT-FPIx,a1

2 P i1

P

VxiVxi1 , 27

where, as usual, xP1x1 . The TTI-FPI method is obtained

by replacing the potential in Eq. 27 by the effective poten-

tial in Eq. 21. The ZOP-FPI method involves accurate in-

tegration of V(x) over a path obtained by connecting the

adjacent pairs of discretized points with straight line seg-

ments rather than using the actual Fourier path; the FOP-FPI

method can be implemented by integrating an effective po-

tential over the same path. The effective potential, as ob-

tained from Eqs. 23 and 24, is

VeffFOP

xVx 12V, 28

where

xxi/xi1xi . 29

It is worthwhile to explicitly note that the convergence

order in K of the Fourier analog methods is the same as the

convergence order in P of the DPI methods from which they

are derived.

II.D. A new FPI method

At first glance only slight differences distinguish the

C-FPI and TT-FPI methods, and Coalson argued9 that the

two methods are essentially the same. He noted however

that the TT-FPI scheme converged somewhat faster than the

C-FPI scheme in some limited numerical tests while the MT-

FPI scheme converged somewhat more slowly than the

C-FPI scheme. Apart from the initial studies,9,11 there have

apparently been no calculations that utilize Fourier analogs

of DPI schemes. As noted above, the C-FPI scheme con-

verges as O(1/K) which is undesirably slow, especially at

low temperatures where large values of K are often required

to yield accurate results. The TT-FPI scheme converges asO(1/K2) and thus should provide a considerable advantage

over the C-FPI scheme, provided that the Monte Carlo sam-

pling variance is not greatly different between the two meth-

ods. This difference in convergence rates must be predomi-

nately due to the different fluctuation parameters; to

emphasize this we consider yet another FPI method which

we will refer to as rescaled fluctuation FPI RF-FPI and

which differs from the C-FPI method only by the use of Eq.

25 instead of Eq. 6. The RF-FPI scheme reduces to the

TT-FPI scheme if quadratures over the paths Eq. 7 are

integrated with a P-point trapezoidal rule, but we do not

restrict ourselves to equidistant-time quadrature nodes in the

RF-FPI method as explained in Sec. III.The points on the Fourier path determined by the fluc-

tuation parameters given by Eq. 25 are ideally distributed

only at the P equidistant-time points, so extension of the

integration in the RF-FPI method to the entire Fourier path

introduces a slight deviation from the quadrature result that

would be obtained on a K Fourier path. Since the TT-FPI

result is a trapezoidal-rule approximation of the RF-FPI re-

sult, this deviation can be seen to be of order O(1/P 2) and

thus the RF-FPI method converges as O(1/K2). The TTI-FPI

method cannot profitably be similarly generalized since the

O(1/P 2) error from the finite expansion of the paths would

reduce the asymptotic convergence rate to O(1/K2) from

O(1/K4). The RF-FPI method is expected to be useful insituations where we desire a continuous specification of the

path, or where we can use the extra flexibility in quadrature

choice to integrate some or all of the paths with fewer than P

quadrature points.

If we consider Eq. 25 in detail, we see that the pa-

rameters depend explicitly on the maximum expansion

length K or equivalently P and are enlarged compared to

the parameters of Eq. 6. The high-k parameters differ the

most, and the expression

k1

K

k2 sin2 ks n /

k1

k2 sin2 ks n / 30




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holds exactly for the points

s n n1

K1, n1,2,...,K1. 31

One can view the higher convergence rate of the TT-FPI

scheme or the RF-FPI scheme as compared to the C-FPI

scheme as resulting from this k-dependent magnification of

the fluctuation parameters to compensate for the truncation

of the path expansion to finite K. This situation can be com-pared with that for the PA-FPI scheme, which also converges

as O(1/K2), and where the fluctuation parameters for kK

determine via Eq. 11 the spatial extent over which the

potential energy is Gaussian averaged. A notable difference

though is that the TT-FPI and RF-FPI schemes achieve the

faster O(1/K2) rate of convergence without requiring La-

placian evaluations.

II.E. Perturbative corrections and their extrapolation

The perturbative correction approach20 can be defined

via the elementary identity

Q K TQ K TQ corr,K,K T, 32

where

Q corr,K,K TQ K TQ K T, 33

and where K is less than K. We consider K small enough

that the first term on the right-hand side of Eq. 32 can be

affordably calculated and well converged; the second term

on the right hand side of Eq. 32 is expensive to calculate

but small in magnitude for sufficiently large K) . We

have previously shown20 that in the C-FPI method, for a

given number of Monte Carlo samples, we can calculateQ K(T) and Q corr,K,K(T) with a similar relative sampling

variance. Thus to achieve a given absolute accuracy we need

substantially fewer samples for the expensive Q corr,K,K(T)

term than we need for the inexpensive Q K(T) term.

In order to calculate Q corr,K,K(T), we perform simulta-

neous calculations for several values of K, the lowest of

which is taken to equal the K of the previous paragraph. In

the C-FPI method, for each Monte Carlo sample we form

paths for each value of Kfrom a single set of random Fourier

coefficients. Each of these paths begins and ends at the same

configuration space sample point, x, and the lower-order

paths have Fourier expansions that are truncated versions of

the highest-order path. We can then accumulate statistics on

Q K(T), and the various Q K(T), and Q corr,K,K(T) terms

in a single run. Except for statistical errors, the perturbative

corrections are calculated exactly by this treatment. We then

calculate Q K(T) using a substantially larger number of

samples than we use for the Q corr,K,K(T) run.

This procedure is also used for the PA-FPI method, but

we must modify the approach for the other six methods as

these have fluctuation parameters that vary with K. For these

cases we calculate the Fourier expansion coefficients at each

configuration space point using a single set of random num-

bers but letting the parameters vary with K, i.e., we use

ajk;Kjkk;K 34

instead of

ajkjkk 35

in the NK-dimensional Monte Carlo average over the a

space in Eq. 8. This generates a family of paths that have

Fourier expansions that are as similar as possible while still

yielding sequences of partition functions that have the cor-

rect asymptotic convergence rates. As we will see in the

example calculations that follow, the Q corr,K,K(T) terms for

these methods have higher variances than we obtained for

correction terms in the C-FPI scheme, but we can still

achieve substantial savings by using these techniques.

Another possible extrapolation strategy exists for the

DPI schemes. Since each of the P discretization points is

distributed as in an infinite Fourier expansion we can select

subsets of these points to calculate lower order partition

function approximants. In the case of the TT scheme this

amounts to extrapolation of different trapezoidal-rule inte-

gration approximations of the same path. An attractive fea-

ture of this approach is that we can obtain additional lowerorder results without the need to perform any additional po-

tential evaluations if we use either the TT or the TTI scheme.

If we choose the largest desired value of P as a power of two,

this same-path extrapolation scheme saves nearly a factor

of two in the cost of potential evaluations as compared to the

scheme of Eq. 34. Unfortunately, we found numerically

that the same-path extrapolation approach yields pertur-

bative corrections that have a much higher variance than

those of the similar-Fourier-expansion approach of Eq.

34 and thus the latter approach is preferable.

We have two calculations of Q K(T) from the proce-

dure above, one using a large number of samples and a less

accurate result obtained during the calculation of

Q corr,K,K(T); we distinguish these two results with super-

scripts of L and S, respectively, to denote large and

small samples. The statistical errors in Q corr,K,K(T) and

Q K ,S(T) are highly correlated so we can enhance the ac-

curacy of the final results via

Qcorr,K,K TQ K ,L T

Q K ,S TQ corr,K,K T. 36

We perform calculations of Q corr,K,K(T) at three or more

values of K, and we extract Q corr,,K(T) by fitting to the

functional form

Q corr,K,K TQ corr, ,K TA

Kn

B

Kn1, 37

where n is the leading order of the asymptotic convergence

rate i.e., n1 for the C-FPI, MT-FPI, and ZOP-FPI meth-

ods, n2 for the PA-FPI, TT-FPI, FOP-FPI, and RF-FPI

methods, and n4 for the TTI-FPI method. Partition func-

tions calculated with the Trotter approximation can be

shown39 to be even functions of P; thus, one might expect

that an expansion in even powers of K might be better than

the form of Eq. 37 for the TT scheme. It seems likely

however that this result holds only for the Trotter approxi-




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mation proper rather than the TT approximation; fits of the

form of Eq. 37 perform as well as or better than a form

involving only even powers for all of the methods considered

here.

III. COMPUTATIONAL DETAILS

In order to illustrate the various path integral methodsand compare their relative efficiency, we performed partition

function calculations at a temperature of 300 K for HCl us-

ing a potential that has been used previously16,40 to illustrate

methods for calculating quantal free energies.

Monte Carlo sampling, as implemented in our

algorithm,16,17,19,20 involves sampling in two distinct

spacesthe configuration space x and the Fourier space

aeach of which is sampled in an uncorrelated fashion.

Given a sampling of the Fourier space, we construct a rela-

tive Fourier path; when this is added to a configuration

space sample we obtain an absolute Fourier path. A large

amount of computational effort is required to actually form

the relative Fourier path. If we sample the x and a space atthe same rate then the formation of the relative paths domi-

nates the computational cost for typical problems. Instead we

choose to sample the x space much more frequently than the

a space typically 101000 times as often; thus, a given

relative path is reused many times.19 In particular, for the

present study we reuse relative paths 100 times. This means

that the sequence of absolute paths has some short-term cor-

relation, but numerical tests19 indicate that Monte Carlo vari-

ances for this type of sampling can be very accurately esti-

mated using formulas appropriate for uncorrelated sampling.

If we reuse the relative paths a large number of times,

the computational expense is strongly dominated by the cost

of the potential evaluations. Our algorithm15,19 for the C-FPI

scheme uses Gaussian quadrature to evaluate the action inte-

grals in an effort to minimize the number of potential evalu-

ations required for accurate integration. In the present paper

we also use Gaussian quadrature for the PA-FPI and RF-FPI

calculations. Since Gaussian quadrature uses irregularly

spaced quadrature nodes we must form the relative paths

using matrix multiplication.19 For the FPI analogs of the five

DPI methods we only need to determine the path at the K

1 discretized points that are evenly spaced in imaginary

time, i.e., the set of points given in Eq. 31. Equation 4

then becomes

xj s nxjk1

K

ajk sin n1 kK1 , 38and we implement this via a fast Fourier sine transform. This

is substantially faster than matrix multiplication generating

the entire path via matrix multiplication requires O(2NK2)

operations, whereas the FFT procedure only requires

O(NKlog K) operations, but the path generation phase of

the algorithm still presents a computational bottleneck; there-

fore, we still reuse relative paths to increase efficiency.

We restrict our configuration space domain D to a hy-

perannulus defined by 1u where the hyper-radius is

given by

j1

N

xj2. 39

We subdivide D into several concentric hyperannulii and

sample these via an adaptively optimized stratified sampling

scheme15,19 AOSS. We also sometimes employ importance

sampling in the configuration space using functions of the

atomatom distances.19,20 In many cases, particularly for

systems of high dimensionality and low temperature, a largefraction of absolute paths that we sample contribute negligi-

bly to the partition function. We have implemented a number

of geometric and energetic screening criteria that permit us

to identify such cases early in the action integral evaluation

phase, and we can then save substantial computational effort

by early termination of the evaluation of contributions from

these unimportant paths.19

Extensive details on the implementation of our algo-

rithms have been presented previously, and we refer the in-

terested reader to these sources for additional details.15,17,19,20

The calculations presented here used an adaptively opti-

mized stratified sampling scheme with a sampling domain

that is defined by 150 a 0 and u150 a 0 where the scal-ing mass is equal to the mass of an electron and which is

subdivided into 20 equal volume strata. In each case a total

of 2107 samples was calculated; 10% of these are distrib-

uted uniformly in an initial probe phase and the remain-

der are distributed in 20 AOSS phases as explained

previously.19 Masses of 1.007 825 and 34.968 852 amu are

used for H and Cl, respectively. In the present study the same

number of samples (2107) was used to calculate both the

Q K(T) results and the perturbative correction results; this

facilitates comparisons of the statistical errors. In actual ap-

plications we would refine the results by performing a calcu-

lation with a large number of samples at a single moderate

value of K and apply the correction procedure outlined in

Sec. II.E.

IV. RESULTS

Accurate variational calculations are available16 for HCl

at 300 K for the same potential as used here, and they can be

used as benchmarks for the present results. Table I lists

Q K(T) and its associated statistical errors for various K for

the eight methods studied. In particular we tabulate the 2

statistical error, which is calculated via

i1Nstrata

varfi

Ni , 40

where var(fi) denotes the variance of the Monte Carlo sam-

pling of the integrand in strata i, Nstrata is the number of

strata, and Ni is the number of samples in strata i. We also

tabulate the 2 relative statistical error given by 2/Q K

(T) which we express as a percentage. Table I also gives

partition functions extrapolated to K obtained by fitting

the last several points to the functional form of Eq. 37.

Figure 1 displays the unsigned truncation error vs K for vari-

ous unextrapolated calculations.

Table II displays perturbative corrections and associated

errors for various FPI methods and values of K and K.




7/10

Table III compares TT-FPI partition functions and perturba-

tive corrections obtained using the path sequence of Eq. 34

to calculations using the same-path extrapolation ap-

proach.

V. DISCUSSION

Several factors must be considered in evaluating the ef-

ficiency of the FPI methods considered here. These include

the value of K that is required to get reasonable results, the

variance of the calculation of Q K(T), the variance of the

perturbative corrections, the quadrature costs of evaluatingthe action integrals, and the additional costs incurred in some

of the methods for evaluating gradients or Laplacians.

As indicated in Fig. 1, the MT-FPI and ZOP-FPI meth-

ods have the largest errors for a given value of K and the

slowest convergence rates. The poor performance of these

methods is in marked contrast to the seemingly similar TT-

FPI scheme. Makri and Miller37 concluded that trapezoidal

Trotter calculations are superior to midpoint Trotter calcula-

tions because the former is a two-point approximation to the

exact integration of the straight-line-segment path used in the

propagator power series schemes while the later is only a

one-point integration approximation. In contrast, in the

present study we find that the TT scheme is not only more

accurate than the MT scheme but also substantially more

accurate than the ZOP method. To correctly explain these

trends we must consider the way that the paths enter into the

DPI schemes.

Equation 15 is an exact expression for Q(T); for a

specific set of P discrete points, we still must integrate over

all possible paths that pass through these points. Thus the P

distinguished points in the DPI schemes can be thought of as

representing a set of paths which we will denote

C(x1

,x2

, . . . ,xP

). The trapezoidal Trotter approximation in-

volves an operator approximation as given in Eq. 17 as

well as a P-point trapezoidal rule integration scheme to ap-

proximate the contribution of each member of the set

C(x1 ,x2 , . . . ,xP). The MT scheme involves a similar opera-

tor approximation but differs from the TT scheme in that its

calculation involves potential evaluations at points that do

notlie on each member of C(x1 ,x2 , . . . ,xP). In particular, the

statistical distribution of a midpoint between two adjacent

discretized points is narrower than the distribution from

which the discretized points themselves are chosen, and thus

the MT scheme yields partition functions that are biased to-

ward higher values.

TABLE I. Partition functions and 2 statistical errors for various methods. The variational result Ref. 16 is 1.651102.

K MT-FPI ZOP-FPI C-FPI TT-FPI RF-FPI FOP-FPI PA-FPI TTI-FPI

Q K(T)

1 1.379 5.959101 6.231101 3.732101 5.715101 2.986104 9.369104 7.269102

2 4.800101 2.870101 2.508101 1.637101 2.041101 1.548103 4.550103 3.654102

4 2.352101 1.319101 1.088101 6.347102 7.760102 5.344103 1.139102 2.173102

8 1.046101 6.414102 5.079102 2.978102 3.351102 1.052102 1.531102 1.744102

16 5.242102 3.713102 3.022102 2.009102 2.104102 1.413102 1.628102 1.659102

24 3.833

10

2

2.947

10

2

2.490

10

2

1.813

10

2

1.857

10

2

1.524

10

2

1.641

10

2

1.650

10

2

32 3.204102 2.593102 2.252102 1.742102 1.768102 1.572102 1.645102 1.648102

64 2.362102 2.097102 1.930102 1.671102 1.679102 1.625102 1.647102 1.647102

96 2.109102 1.942102 1.831102 1.657102 1.662102 1.637102 1.647102 1.646102

128 1.988102 1.866102 1.784102 1.653102 1.656102 1.641102 1.648102 1.647102

1.648102 1.647102 1.649102 1.647102 1.649102 1.647102 1.648102 1.646102

2 statistical error

1 1.7103 4.8104 5.3104 3.6104 5.2104 2.6107 8.0107 1.7104

2 4.5104 3.2104 3.1104 2.2104 2.8104 1.8106 5.7106 1.0104

4 3.1104 1.9104 1.8104 1.2104 1.5104 7.9106 1.9105 6.4105

8 1.7104 1.1104 9.7105 6.2105 7.4105 1.8105 2.9105 4.5105

16 9.7105 7.1105 6.1105 4.3105 4.6105 2.7105 3.3105 3.7105

24 7.4105 5.8105 5.0105 3.9105 3.9105 3.0105 3.3105 3.6105

32 6.3105 5.2105 4.6105 3.7105 3.7105 3.1105 3.4105 3.5105

64 4.7105 4.3105 3.9105 3.5105 3.4105 3.3105 3.4105 3.4105

96 4.310

5 4.010

5 3.710

5 3.510

5 3.410

5 3.310

5 3.410

5 3.410

5

128 4.0105 3.8105 3.6105 3.5105 3.4105 3.3105 3.4105 3.4105

2 relative % error

1 0.12 0.08 0.08 0.10 0.09 0.09 0.09 0.23

2 0.09 0.11 0.12 0.14 0.14 0.12 0.12 0.28

4 0.13 0.14 0.16 0.18 0.19 0.15 0.16 0.29

8 0.16 0.17 0.19 0.21 0.22 0.18 0.19 0.26

16 0.19 0.19 0.20 0.21 0.22 0.19 0.20 0.23

24 0.19 0.20 0.20 0.21 0.21 0.20 0.20 0.22

32 0.20 0.20 0.20 0.21 0.21 0.20 0.20 0.21

64 0.20 0.20 0.20 0.21 0.20 0.20 0.20 0.21

96 0.20 0.20 0.20 0.21 0.20 0.20 0.20 0.21

128 0.20 0.20 0.20 0.21 0.20 0.20 0.20 0.21




8/10

The ZOP scheme can be thought of as an operator ap-

proximation together with a rule for replacing each member

ofC(x1 ,x2 , . . . ,xP) with a single path obtained by connecting

the P discretization points with straight line segments. TheZOP scheme is identical to the pure-bead limit i.e., pure

discretized limit of the mixed bead-Fourier scheme of

Vorontsov-Velyaminov et al.41 They argue that replacing the

potential evaluations at the P points with integration over the

straight lines between beads gives a better representation of

the potential energy. This argument neglects the consider-

ation that points on the straight-line-segment path have a

narrower statistical distribution than the full set of properly

weighted points on members of the set C(x1 ,x2 , . . . ,xP), and

that using only this path introduces a bias towards partition

functions that are too large.

The mixed bead-Fourier method of Vorontsov-

Velyaminov et al.41 augments a discretized path representa-tion with Fourier expansions between adjacent beads.

Vorontsov-Velyaminov et al.41 state that neither the pure-

bead nor the pure-Fourier limits of their mixed bead-

Fourier scheme are optimal. This result must be understood

as a consequence of their choice of implementation such that

the pure-bead limit reduces to the ZOP scheme instead of

the much more rapidly converging TT scheme; if their algo-

rithm is suitably modified, a TT pure-bead limit would

probably be optimal. A better mixed-discretized-Fourier ap-

proach could be devised using TT-DPI and TT-FPI schemes,

and this mixed method would converge as the inverse square

of the number of path variables for any partitioning of the

work. Such a mixed representation might be useful as it

would permit both extrapolation in the Fourier space and

importance sampling in configuration space for multiple

points along the path. A mixed scheme such as this might

also facilitate use of more advanced stratified sampling strat-egies than those we currently employ.

The FOP-FPI scheme also involves accurate integration

over a path where the P discretization points are connected

by straight lines. This method is equivalent to the mixed

bead-Fourier approach of Vorontsov-Velyaminov et al.41 in

the pure-bead limit with gradient partial averaging over

the trivial Fourier paths consisting of straight lines connect-

ing adjacent beads. It tends to converge monotonically from

below and yields good accuracy and an O(1/K2) conver-

gence rate. It performs somewhat better than the TT-FPI

scheme at low K, but at higher K the accuracy of the TT-FPI

and FOP-FPI schemes are very similar. As a numerical

FIG. 1. Unsigned truncation error, Q (T)Q K(T) , as a function of K forvarious FPI methods.

TABLE II. Perturbative corrections and statistical errors for various meth-

ods, K and K.

Method K K Qcorr,K,K(T) 2 error 2 rel. % error

MT-FPI 64 96 2.53 103 5.0 106 0.20

64 128 3.73 103 7.4 106 0.20

ZOP-FPI 128 192 7.45 104 1.5 106 0.20

128 256 1.11 103 2.2 106 0.20

128 384 1.48 103 3.0 106 0.20

C-FPI 64 96 9.83 104

2.1 106

0.2264 128 1.46 103 3.1 106 0.21

128 192 4.65 104 9.8 107 0.21

128 256 6.93 104 1.4 106 0.21

TT-FPI 32 64 7.16 104 3.8 106 0.53

32 128 9.01 104 4.1 106 0.45

32 192 9.37 104 4.1 106 0.44

32 256 9.49 104 4.2 106 0.44

64 128 1.85 104 1.3 106 0.70

64 192 2.21 104 1.4 106 0.62

64 256 2.33 104 1.4 106 0.59

128 192 3.60 105 4.2 107 1.15

128 256 4.79 105 4.5 107 0.95

RF-FPI 64 96 1.70 104 1.2 106 0.72

64 128 2.29 104 1.4 106 0.59

FOP-FPI 64 96 1.16

10

4

6.3

10

7

0.5464 128 1.59 104 8.2 107 0.52

64 256 2.04 104 1.1 106 0.52

PA-FPI 8 16 1.10 103 1.1 105 1.02

8 24 1.14 103 1.1 105 1.01

8 32 1.16 103 1.2 105 1.00

8 64 1.17 103 1.2 105 1.00

8 128 1.17 103 1.2 105 1.00

16 24 1.31 104 4.0 106 3.10

16 32 1.66 104 4.6 106 2.75

16 64 1.90 104 4.9 106 2.59

16 96 1.93 104 5.0 106 2.56

16 128 1.95 104 5.0 106 2.54

TTI-FPI 8 16 9.42 104 2.4 105 2.51

8 24 9.57 104 2.4 105 2.49

8 32 9.73 104 2.4 105 2.48

8 64 9.74 104 2.4 105 2.48

8 128 9.73 104 2.4 105 2.48

16 16 9.20 105 8.4 106 9.17

16 24 1.07 104 9.2 106 8.59

16 32 1.21 104 9.7 106 8.02

16 64 1.21 104 9.8 106 8.04

16 128 1.20 104 9.8 106 8.17




9/10

method though, the FOP-FPI scheme is extremely expensive

since it requires costly Laplacian evaluations and because

accurate integration of the straight-line-segment paths re-

quires additional functional evaluations as compared to the

number required for the TT-FPI scheme.

Three of the eight methods we have considered here in-

volve accurate integration of the full Fourier path. The con-

ventional FPI scheme displays poor accuracy and exhibits

slow convergence. The RF-FPI scheme performs very simi-

larly to the TT-FPI scheme. The PA-FPI scheme shows rapid

O(1/K2) convergence from below and yields superior accu-

racy. Unfortunately the PA-FPI scheme requires expensive

Laplacian evaluations. In many situations the cost of these

Laplacian calculations increases approximately linearly with

the dimensionality of the system and thus we expect that the

increased performance from partial averaging will not typi-

cally be sufficiently compelling to offset the increased cost.

The MT-FPI, TT-FPI, and TTI-FPI schemes all require

P potential evaluations to integrate each path. The C-FPI,

RF-FPI, and PA-FPI schemes can occasionally produce ac-curate results with fewer than P-point quadratures, but typi-

cally somewhat more than P points are required for accurate

integration; for small K values, substantially more than P

points can be required. For the present system the quadrature

costs are rather modest, but there have been reports23 of sys-

tems which require about three times as many quadrature

points as path variables to achieve accuracy of better than

1.5% in the PA-FPI scheme.

Among the methods considered here, the TTI-FPI

scheme yields the most accurate results for a given value of

K and possesses the fastest asymptotic convergence rate

O(1/K4). Unfortunately the asymptotic rate in not realized

until fairly high K, and thus the performance is not as dra-matic as one might initially expect. Also the gradient calcu-

lations needed for the effective potential of Eq. 21 make

the method quite expensive. Still, the methods cost is com-

parable to that of the PA-FPI scheme while yielding some-

what greater accuracy except at very small K.

One of the most important aspects to consider in evalu-

ating the efficiency of a Monte Carlo method is the variance

of the sampling. One must consider the magnitude of the

variance both as a function of method and as a function of K.

It is a common problem in DPI calculations for the statistical

errors to increase as P increases, and a number of methods

have been proposed to alleviate this problem.4244 Topper18

has stated that the sampling variance typically decreases

rapidly as a function of K in FPI schemes. We observe

neither trend in the results given in Table I; the sampling

variance does decrease as a function of K, but only at the

same often sluggish rate as the partition function. The 2

relative error is essentially independent of the path integral

method employed, and rapidly approaches a value of about0.20%0.21% as K is increased. Thus we may conclude that

any reports of differences in sampling variances are likely

consequences of the Monte Carlo sampling strategy em-

ployed rather than a property of the path integral methods

themselves.

The variance of the perturbative corrections varies sig-

nificantly depending on the path integral method used. The

C-FPI scheme yields correction terms with two-standard-

deviation relative errors that are comparable to those of the

underlying Q K(T) calculations about 0.21% for widely

varying values of K and K. The 2 relative errors for the

perturbative corrections of the MT-FPI and ZOP-FPI scheme

are also fairly small for the useful ranges of K and K. Forthe TT-FPI, FOP-FPI, and RF-FPI methods the 2 relative

errors for the perturbative corrections are all significantly

larger than the sampling errors of the underlying Q K(T)

calculations. The relative variance of the perturbative correc-

tions increases as the magnitudes of the corrections decrease,

as K increases, and as K decreases. The PA-FPI and TTI-

FPI schemes show the largest relative statistical errors for the

perturbative corrections; for the TTI-FPI method with K

16, the relative statistical error is over 40 times larger than

the relative statistical error in the Q K(T) calculations. The

PA-FPI and TTI-FPI path integral methods also have rela-

tively large absolute statistical errors, and this lessens theirperformance advantages as we must either use more samples

in calculating the perturbative term or use a higher value of

K. The TT-FPI, RF-FPI, and FOP-FPI methods have rela-

tively small absolute statistical errors in their perturbative

corrections, which is a critical advantage for practical calcu-

lations.

The statistical error of the perturbative corrections also

varies depending on the sequence of paths used to obtain the

results. In Table III we see that the statistical error for per-

turbative corrections in the TT scheme is a factor of about

2.4 larger if we use the same path approach instead of the

similar Fourier expansion sequence of Eq. 34. The

TABLE III. Q K(T), Q corr,K, K(T), and statistical errors calculated using two different path sequence ap-

proaches and the TT-FPI scheme.

P Q P1(T) 2 error rel. % error Qcorr,P1,31(T) 2 error rel. % error

Same-path approach with K127 path

128 1.656 102 3.4 105 0.20 9.50 104 1.0 105 1.08

64 1.676 102 3.4 105 0.20 7.55 104 9.2 106 1.21

32 1.751 102 3.6 105 0.21

Sequence of Eq. 34

128 1.656 102 3.4 105 0.20 9.48 104 4.3 106 0.45

64 1.675 102 3.4 105 0.20 7.56 104 3.9 106 0.52

32 1.751 102 3.6 105 0.21




10/10

same path approach would thus require nearly 6 times as

many samples to achieve the same accuracy as the use of Eq.

34, and this additional cost greatly outweighs the savings

from reusing potential evaluations.

The extrapolated values in Table I are all very similar

because results for each of the methods were calculated with

the same random number sequence. We also performed one

calculation with the TT-FPI method, K32, and 5108

samples; this gave a result of (1.74770.0007)102

,where again the quoted error is 2. Using this result and a

K perturbative correction derived from the data in Table

I, we obtain a corrected partition function of (1.652

0.001)102 which is in excellent agreement with the

variational result16 of 1.651102.

We have expressed no opinion on whether the trapezoi-

dal Trotter scheme or any of the other DPI methods is

better implemented in the discretized or Fourier formthe

optimal choice may vary with the problem and may depend

strongly on sampling strategies. Even if the discretized rep-

resentation proves more efficient for a particular problem,

one can still make use of the Fourier representation to afford-

ably calculate a perturbative term to correct for the trunca-tion to a finite number of beads. Furthermore, we have

pointed out that there may be some advantages to a mixed

discretized-Fourier method obtained by combining the TT-

DPI and TT-FPI schemes.

VI. CONCLUDING REMARKS

We have compared eight different Fourier path integral

methods including the conventional and partial averaging

versions, a new Fourier method based on rescaled fluctua-

tions, and five discretized schemes that have been trans-

formed into Fourier schemes by using the isomorphism of

Coalson.9

This isomorphism allows us to apply Monte Carlosampling in the Fourier space as well as to adapt our pertur-

bative correction and correlated extrapolation schemes to

DPI methods. The Monte Carlo relative sampling variance is

observed to have little dependence on the path integral

method or the value of K. The sampling variance of the

perturbative corrections does vary strongly with method as

well as K and K.

The C-FPI, MT-FPI, and ZOP-FPI schemes are observed

to yield poor accuracy and slow convergence as compared to

the other methods. The FOP-FPI method performs reason-

ably well as a function of the number, K, of terms in the

Fourier series, but is extremely expensive and thus is always

less efficient than either the PA-FPI or TT-FPI schemes. TheTT-FPI scheme is observed to be both efficient and accurate.

The new RF-FPI method performs similarly to the TT-FPI

scheme. Both methods converge well without requiring gra-

dients or Laplacians, and the absolute variance of their per-

turbative corrections is relatively small compared to the

other methods. The PA-FPI and TTI-FPI methods yield good

results but at considerable expense due to the need to calcu-

late Laplacians and gradients respectively; thus these two

methods are only likely to be efficient choices for low-

dimensional systems. On balance we expect the TT-FPI and

RF-FPI schemes to be the most widely useful of the methods

considered here.

ACKNOWLEDGMENT

Partial support for this work was provided by the Na-

tional Science Foundation under Grant No. CHE97-25965.

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Documents

Steven L. Mielke and Donald G. Truhlar- A new Fourier path integral method, a more general scheme for extrapolation, and comparison of eight path integral methods for the quantum mechanical