ELSEVIER Computer Physics Communications 105 (1997) 108-126

Computer Physics Communications

Application of the Feynman-Kac path integral method in finding the ground state of quantum systems

J .M. R e j c e k a, S. D a t t a a, N.G. F az l eev a,b, J .L. F r y a, A. K o r z e n i o w s k i a a The University of Texas at Arlington, Arlington, TX 76019, USA

b Department of Physics, Kazan State University, Kazan 420008, Russia

Received 10 March 1997

Abstract

Numerical methods of applying the Feynman-Kac path integral approach to quantum mechanics are presented. The methods are demonstrated on simple quantum mechanical systems, including the hydrogen atom, the simple harmonic oscillator and infinite square wells. New analytic results for the Wiener integrals are obtained and compared with numerical results. A measure of the statistical uncertainty is introduced and rates of convergence are investigated. Implementation of the method on both serial and parallel computers is discussed. @ 1997 Elsevier Science B.V.

1. Introduction

The motivation for the Feynman-Kac path integral formulation comes from the difficulty of defining a measure for the real-time Feynman path integral. Feynman extended the principal of least action of Lagrange from classical to quantum mechanics. He made two basic physical assertions. First, for any quantum process in the absence of measurements, it is just the transition amplitude and not directly the probability which is expressed as the sum of contributions from the partial processes of transition. Second, the weight with which the contributions from particular paths are counted is related to the action of the particular path. Feynman's path integral approach can be formally described by the expression

(e - - iHt /h~)(Ri ) = / eia(~~ do), (1)

B(R2;RI)

where H is the Hamiltonian operator, ~ ( R t ) is the wave function at the initial time t = 0 and position o)(0) = Rt, A(o)) is the classical action of the path to, / ' /(R2;Rt) = {o)(s) I 0 < s < t, o)(0) = Rt, o)(t) = R2} denotes the set of all paths starting at Rt and ending at point Rz and do) is a measure on /2(R2;Rt). The notation reads that the time evolution operator e -iHt/h operating on an initial function ~p evaluated at some point Rt is equal to the sum over all paths to in .O(R2;RI), weighted by a function of the action per path and employing some measure do). Unfortunately, the path spaces of interest are infinite-dimensional and no Lebesque-type measure do) exists for real A(o)). Exner [ 1] states a theorem asserting that the Feynman-type measures cannot exist because the exponential term in Eq. (1) is wildly oscillating unless A(o)) is purely

0010-4655/97/$17.00 (~) 1997 Elsevier Science B.V. All rights reserved. PHS0010-4655(97)00061-1

s Rejcek et aL/Cvmputer Physics Communications 105 (1997) 108-126 109

imaginary. Therefore, the path integrals under consideration denoted by f dto are not integrals in the Lebesque or Riemann sense. Several approaches [ 1 ] have been attempted in order to define meaning to Eq. (1) . Most approaches try to bypass this difficulty by defining a suitable class of functions ~,(Rl) whose behavior would be smooth enough in some sense to cancel the influence of the oscillations. Hopefully this class would include physically meaningful functions.

One approach at obtaining finite Feynman path integrals involves analytical continuation in time. Consider the Hamiltonian operator H = -�89 + V and assume that tile potential V is continuous. Instead of e -itH, o n e

may try to get a path integral expression for e -m. Eq. (1) becomes

!

(e-mf)(Rl)= f exP(�89 (2) #1( R2 ;RI ) 0

where A(to) = �89 fo([&(s)[2 - V(to(s)))dto is the action for the Brownian motion path to and Dto is a probability measure on the space of all Brownian motion trajectories starting at to(0) = Rt and ending at ~o(t) = R2. Expressions like the right-hand side of Eq. (2) were introduced by Wiener in order to describe Brownian motion and related processes previously formulated by Einstein and Smoluchowski [2]. The difference between Eq. (1) and Eq. (2) is that the Wiener integrals have a mathematically well-defined measure. Wiener measure and Wiener integrals comprise a large portion of modern probability theory and stochastic processes and have been intensively studied. Kac applied Wiener measure to Feynman path integrals to obtain what is now known as the Feynman-Kac formula,

t

(e-tlttp)(R) = f exp (- f V(to(,))dr)O(to(t)) dl~(~o) , (3a)

a(R) o

where V is the potential in the Hamiitonian operator H = - � 8 9 2 + V, .O(R) is the set of all continuous paths to on the interval [0, t] in N-dimensional i'eal space ~R N such that to(0) = R , / z ( t o ) denotes Wiener measure (i.e., a probability measure on the set of all possible trajectories of Brownian motion in ~rr initially started at position R), and ~ is any function which is Lebesque square integrable denoted ~, E L2(~N). The notation Lt ' (~ N) stands for the set of all functions f (r) which are Lebesque measurable on some family of Borel subsets B of ~N and have the property that fB ]flP dNr < oo for 1 _< p _< oo and B E B. A rigorous proof of the Feynman-Kac formula and limitations of its application are given by Simon [3].

The Feynman-Kac formula is an expected value of a path integral with respect to Wiener measure. If every random path to(t) initially begins at some position r0 = to(0), the function ff is defined to be a delta function, ~p(to(t)) = ~(to(t) - t o ) . Substituting this into Eq. (3a) yields

I

S(t, ro) ---- f exp ( - f V(to(r) ) dr)6(to(t) - ro) dtz(to) o(~) 0

o r

-S(t, ro) =-- Ero [exp ( - / V(to(r) ) dr)] , (3b)

0

where Ero(') is the expected value of the Wiener integral with ro = to(0) the initial position of a Wiener process (Brownian motion).

The Feynman-Kac path integral method was first applied in finding the ground state energy of a physical system by Donsker and Kac [4]. For some physical systems which can be described by a time independent

l I0 J.M. Rejcek et at../Computer Physics Communications 105 (1997) 108-126

Hamiltonian H = [ - �89 2 + V(r)] , with V(r) > 0 and using a Green's function method, Kac [5] showed a relationship between the Wiener integral and a linear expansion of the eigenfunctions and eigenvalues of the related Hamiltonian,

-S(t, ro) = ~ [g,,(ro) / C,n(r) dUr]e -a"' , (4) ~N

where the sets {~,,,(r)} and {~/,,} are the eigenfunctions and eigenvalues of the related Hamiltonian. The same result [6] is obtained by considering the N-dimensional diffusion equation

Ou(t, ro, r) = �89 ' ro, r) - V(r)u( t , to, r ) , Ot

where V 2 = aZ/ar~ + aZ/ar~ + . . . + a2/a?u. With the initial condition u(0, to, r) = 6 ( r 0 - r) , it can be shown that for all solutions where it(t, to, r) is Lebesque integrable, denoted it(t, to, r) 6 L I, then f~tu u( t, to, r) dtCr = S(t , to). Using the eigenfunctions and eigenvalues for the time independent Hamiltonian H = [ - � 8 9 V(r)] , the Wiener integral can again be written by Eq. (4). The ground state eigenvalue can be found from Eq. (4) by considering the limit as t goes to infinity,

" ~ ' = - I i m ( l l n [ ' S ( t ' r ~ " t - - . o o (5)

In his original work Kac [4] used two finite times and dropped higher-order terms to get a numerical approxi- mation given by

1 In [S( t l , r0 ) ] (6) ,tl ,~ t 2 - t l [S( t2,ro)J

There are two severe limitations with the derivations shown above. First, note that the Green functions analysis is limited to positive potentials, V(r) > 0. And second, for the Wiener integral -S(t, ro) to be represented as an eigenfunction expansion, it is necessary for u(t, ro, r) 6 L I. Every complete orthogonal basis set on L 2 can represent any L 2 function, but not every L I function: u(t, ro, r) E L 2 ~;. u(t , ro, r) 6 L l, unless the region of integration is bounded, in which case L 2 C L I by the Cauchy-Schwartz inequality.

The Feynman-Kac path integral method first considered by Kac includes a broader range of application than has been defined by the potentials described above. Even if S( t , ro) is not representable as an eigenfunetion expansion it is still true that the lowest eigenvalue is given by Eq. (5). Donsker and Varadhan [6] showed that i t determined by Eq. (5) for dimension N = I will be the lowest value of all possible Lebesque square integrable functions. Generalization to dimensions N > 1 was given rigorous mathematical justification by Korzeniowski [8] and is written more generally as

( l ) {/ / } At = l i m - t ln[S(t , r0) ] = inf V(r)q~2(r) dOr + �89 (Vq~(r), Vq~(r)) dNr , (7)

where ~7 = (a/ar~, a/arz . . . . . O/OrN) is the gradient and ( ) is the scalar product in ~U. Generalization of Eq. (7) to the class of potential functions for which the Feynman-Kae path integral

method is valid has been given by Simon [9]. These include most physically interesting potentials, both positive and negative, and in particular those potentials with Ir[ - t singularities (which include the Coulomb potential). Simon's analysis deals with the properties of Schr/Sdinger semigroups defined by e -m, where H is the Hamiltonian for a system. This is an evolution mapping of solutions of initial value problems for Eq. (3a) above. An initial value function f ( R ) is defined at t = 0 and the semigroup operator e -tit determines the value of each point for increasing t. Simon has shown [10] that the Feynman-Kae path integral method is valid

J.M. Rejcek et al./Computer Physics Communications 105 (1997) 108-126 l i t

for any potential that is an element of Kato Space,/C~v [ 11 ]. Note that the harmonic and Coulomb potentials belong to Kato space. In addition, Simon [ 12] proved many general properties for functions in Kato space including a theorem that for every function in Kato space the semigroup operator e -tn bounded in U' is also bounded in L q for 1 < p < q < co.

The Feynman-Kac path integral method was used in recent literature by Orr et al. [7] to calculate the ground state energies of the first five atoms in the periodic table. The purpose of this paper is to study the numerical utility of the more general result, Eq. (5), for exactly solvable quantum systems and to provide the computational details of the method used in [7]. In Section 2 the numerical approximation of Wiener integrals is introduced and illustrated with a one-dimensional example. Section 3 identifies the sources of error using the Feynman-Kac method in the calculation of the ground state of a quantum system and introduces a measure of the statistical uncertainty in the calculation. Section 4 considers simple one-dimensional quantum systems using the harmonic and infinite square well potentials as a benchmark in order to test convergence rates and measure the uncertainty in numerical calculation accurately. Section 5 extends the calculations to three-dimensional examples. Section 6 discusses how the computer algorithms were implemented, sources of computer error and discusses observations and conclusions about the method.

2. Numerical approximation of the Wiener integrals

Let the integral of a continuous function A(x) in the Wiener measure be defined as

!

(/ ) 3( t , xo) =- Exo A ( w ( r ) ) dr , (8)

0

where Exo(') is the expected value of the Wiener integral with xo the initial position of a random walk to(t) in one dimension. One can approximate the continuous distribution in time with discrete time steps,

l 1 nl t .

P(n , t, xo) = n ~ V(xi + xo) .~ J V(~o(~') ) dr, (9) i=l 0

where xi are discrete positions in space determined by the ith step of a random walk xi = si/x/~, with si = ql + q2 + q3"'" qi and {qj} is a set of independent identically distributed random variables, each having a mean of zero and standard deviation of one. The parameter n determines the step size Ax from

qi unit length (lO) A xi = V ~

One trial path P(n , t, xo) is walked if we take nt = N discrete steps and evaluate Eq. (9) for the set of resulting positions {xi I i = 0, 1,2 . . . . }. The path is viewed as a continuous piecewise linear function starting at x0 and is obtained by joining line segments from the positions {xi}. Let Pj(n, t , xo) be the results evaluated from Eq. (9) for the jth trial path. For .Af trials define the average over all paths as

At"

j = l "

Eq. (8) can be approximated by -S(t, xo) .~ -S,(t, xo), with the approximation becoming exact in the limit as n goes to infinity: S ( t , x0) = lim,,--,oo [S,( t , x0) ].

112 J.M. Rejcek et al./Computer Physics Communications 105 (1997) 108-126

The statistical error for finite sample sizes .iV" can be estimated using the central limit theorem which states

lim ~ ~}" lPj - - S.(t, xo) = N(O, 1) , (12) Ac--~ o'.(t, x o ) l ~ J

where ~r,(t, xo)/v/-~ = B'S,(t, xo) is the standard deviation ofSn( t , xo), N(0, 1) is a random variable with the standard normal distribution and o'n(t, xo) is the standard deviation of the underlying distribution of P (n , t, xo). Eq. (12) is true regardless of the underlying distribution function for P(n , t, xo). Therefore, if the sample size is large enough, the average value approaches the normal distribution about the true mean value.

An unbiased measure of the standard deviation o-,(t, xo) is defined by

Ac At" cr2(t, xo).~ 1 l ~-~[pi2(n,t, xo)_~n(t , xo)] ~--~ ( P i ( n , t, xo) - S . ( t , xo ) )2 - .AT - 1

i=1 i=1

If the sample size is large enough, the distribution of -Sn(t, xo) is approximately normal and a confidence interval can be estimated from

-Sn(t, xo) - z6-Sn(t, xo) <_ S( t , xo) < -S,(t, xo) + z6-Sn(t, xo),

with P0 = P{IS,,(t, x0)I < z~-S,(t, xo)} the probability of lying between the intervals [-zS-Sn(t, xo), +z6-Sn(t, xo)] on the normal distribution. For this paper, one standard deviation confidence interval is re- ported and therefore, z = 1.0 and po = 0.68.

Consider as an example of a Wiener integral the continuous function A(x) = cos(x) and its Wiener integral,

1

- (I ) $ ( 1 , 0 ) = Eo cos(co(z)) d r , (13)

0

where co(r) is Brownian motion starting at xo = 0. This integral can be calculated exactly, as shown by Korzeniowski [ 13], $ ( 1 , 0 ) = 2 ( I - e -1/2) -~. 0.786938. Compare this result to the Lebesque measure of a

single path fo I cos(r ) dr = s in( l ) ~ 0.84147, which can be written

I I

--l~( l'O) = EPt ( f c~ ) dT") ( ic~ ) dr) dpt(co)

where p# (~o) is a probability measure on .O(0) = {co(T)10 < 7- < 1, co(0) = 0} concentrated with probability one on the identity function I ( . ) , i.e. co(t) = 1(.) -- t with pl ( l ( . ) ) = 1. The comparison demonstrates the distinctly different numerical results that can be obtained using different integral measures.

The example can be approximated with the numerical simulation described above. Using a computer-based random number generator to define Bernoulli trials, random walks for t = 1 can be calculated. The step size Ax for this system is determined from Eq. (10) as

I unit length A X = '

where n is an integer. Table 1 shows the results for a random walk calculation using five different step sizes. Fig. 1 shows a plot of the numerical calculation versus the step size including one standard deviation 6-S,(t, xo) error bars. Note how the number of correct significant figures increases as the step size is decreased. Ideally several step sizes should be used in a calculation in order to determine the convergence at a given time t.

J.M. Rejcek et al./Computer Physics Communications 105 (1997) 108-126

Table I Numerical calculation for the Wiener integral equation (13)

113

n .IV" ~,([.o) ~,(t.o) ~,(l.O)/x/-~

4 18.0 0.7065283 0.247 0.000058 25 9.0 0.7739805 0.214 0.000071

100 4.0 0.7835828 0.210 0.000100 400 7.0 0.7861916 0.208 0.000079 900 9.0 0.7866659 0.208 0.000069 oo oo 0.7869386 0.208 0

.A/" is the sample size in millions of paths, S n ( l , 0 ) is the numerical result for the Wiener integral at t = 1 and x0 = O, ~ 'n ( l ,0 ) is the standard deviation of the paths P(n , t, xo) and o'n( 1 , 0 ) / ~ is the standard deviation (the uncertainty) of the calculated value of the Wiener integral.

WIENER INTEGRAL

0.790

Q

t~ 0.785

0.780

0

ExAcr

' ' I : : ; I : : : : : 1 ~ l 1 1 : : : : :

100 200 300 400 500 600 700 800 000 1 CO0 1100 1200 1300 1400 1500

Fig. 1. Wiener integral equation (13) as a function of n, with an enlarged scale to show the uncertainty. Open circles are computed values with +1o" error bars. The dashed line is the exact limit and the solid line is a least square fit.

However, a step size of n = 900 was exclusively used for all the examples in this paper. This was done because n = 900 was a practical limit in scale that would produce results in a reasonable time yet large enough that calculations of the Wiener integral are accurate to several significant figures.

From Eq. (3b) the Wiener integral for the Feynman-Kac formula is denoted,

-S(t, Xo) = E{e - Jo' v(w(r))a~}. (14)

114 J.M. Rejcek et al./Computer Physics Communications 105 (1997) 108-126

The continuous distribution in time of the integral can be approximated with discrete time steps,

I nt

(n, t, xo) = t / V ( w ( r ) ) dr ~ 1 ~ V(xi) , (15) A J n 0 i=1

where xi is again the ith step of a random walk as defined above. Let

Pi(n, t, x0) = exp[ -A i (n , t, xo) ]

denote the ith trial path result. For .A/" trials Eq. (14) is approximated by

.,v"

-- 1 ~-~ Pi(n,t , xo) S ( t , xo) ~ Sn(t, xo) ,~ ~ i=1

Using Eq. (5) and the approximate Wiener integral S,,(t, x0), an estimate of the ground state eigenvalue of the system can be written

'~' ~ lira ( I - ) ,-oo - t l n [ S " ( t ' x ~ . (16)

3. Uncertainty and error propagation

Any quantity calculated as a function of several variables ,~j (Si, $2 . . . . ), each measured with some uncertainty 8Si will also possess an uncertainty dependent on the accuracy of the measurements of Si. This is known as the propagation of error. Let t3,~ be the uncertainty in calculation of At. Then an estimate of the uncertainty resulting from the error propagation can be calculated from Gauss's Error Propagation Law [ 14],

( aa I,]

where cov(SiSj) is the covariance between any two independent variables and the partial derivatives and variances are the long run values available from existing data.

The uncertainty in Ai is a function of the uncertainty in the calculation ofS,,(t , xo). For a fixed value of the time t, the uncertainty 8,~t is given by

= ( aA, ~-S(t, Xo)- 1 ~-snCt, xo) t5,~1 \ a3n-~, x0 ) ./ t Sn(t, x o ) " (17)

If a plot of ,~j versus t is made, the uncertainty of ,~l can be represented with error bars. In the plot, one observes the trend as t gets large. A linear fit of the data can give an extrapolated value for the ground state as t gets large. Note that there are only three sources of error:

(i) uncertainty in the calculation of Sn(ti, xo) (finite sample size); (ii) the approximation of-S(t i, XO) with -Sn(ti, xo) (finite step size);

(iii) the approximation of l imt -c~( -~ In[Sn(t, x0)]) with an extrapolation using finite t. Exactly solvable examples which may be used to study these errors are given in Sections 4 and 5 below.

ZM. Rejcek et aLIComputer Physics Communications 105 (1997) 108-126

4. Application of the Feynrnan-Kac method to one-dimensional quan tum systems

115

This section considers simple quantum systems, the harmonic and infinite square well potentials, as a benchmark in order to test convergence rates, determines appropriate parameters for convergence and measures the uncertainty in numerical calculation more accurately.

4.1. The infinite square well

Consider a single quantum particle with elementary charge and mass M moving in one dimension and confined to a well of finite dimension L with an infinite potential at the boundary of the well. The Schr(Jdinger equation for the system can be written as

[ ' : ] - ~Ox--- 5 + V ( x ) G ( x ) = a , , G ( x ) ,

where

- L L V(x) = O, 2--7 <- x <_ 2--s' V(x) = co , otherwise.

Here s and U are scaling parameters such that x t = sx and E = ,~U. Then U = h2/Ms 2 becomes the natural unit of energy. The eigenfunctions are known exactly and are in the form of sines and cosines,

~ n ( x ) = cos , n = 1 , 3 , 5 . . . . (18)

~p,,(x) = sin , n = 2 , 4 , 6 . . . .

The eigenvalues are given by

"/T2 h2n 2 7r2112U(~) 2 E, = 8M(L/2)2 - ~ (19)

From Eq. (4) an explicit eigenfunction expansion can be found for S( t , Xo) in the form

oo

- g ( t , xo ) x - , ~ -~. , = 2_., t.ne , (20) n=l

where Cn = d/n(xo)f~ ~n(x)dx . Substituting Eqs. (18) and (19) into Eq. (20) gives the Wiener integral as

S(t,x.0) = g ~ (2-mn--ii cos - . (21) m= l

This represents the probability of a path remaining inside the well for a one-dimensional Brownian motion walk as a function of t and starting at some initial position x0 in the box. This is a generalization of the result given by Kac [ 15] for the probability of a one-dimensional Brownian motion walk remaining interior to a region of length L and beginning in the center, x0 = 0. To the authors' knowledge this is a new result and is related to the two absorber problem [ 16] in mathematical statistics. Previously in Kac's original paper the probability distribution function F(8) for the two absorber problem was also addressed. This function can be written

F ( 6 ) = Prob{ max Iw(~-)l ~ ,~} = P(~-, > 1 ) , o<r<l

116 J.M. Rejcek et aL/Computer Physics Communications 105 (19971 108-126

where ~s = inf{t I Iw(t)l = 6} is the first time the Brownian motion w(t) exits from the interval [ - -6 ,+6 ] starting at the origin and P(rs > 1) is the probability that the time required to escape from the interval [ - 6 , +6] exceeds one. The distribution function is given by

F(6) = _4 ~ ( - 1 ) m+l e_(~/s~,-)(2m_t)2 (22) "n" ,,=l ( 2 m - 1)

Eq. (22) can be generalized [ 17] for any time t > 0 and region 6 -- L/2 by substituting 6 = L/2x/7 into Eq. (22), which yields

l,m , = 7r ,,=l (-2-mm----~ e-(~r'/2L')C2m-I)"" (231

Fixing the energy scale to U = 1, the length scale to s = 1 and beginning at xo = 0 in Eq. (21) also gives Eq. (23).

For an explicit numerical example, let L = 10 atomic units, the particle mass be equal to an electron mass M = 9.11 x 10 TM kg and consider a random walk beginning at x0 = 0, the middle of the well. Let the scale be in atomic units s = 0.529166 x 10 - I~ meters. This implies the unit of energy is one Hartree. The eigenvalues are

77.2112 A n - 200 ' n = 1 , 2 , 3 . . . . .

with the ground state AI = "z2/200 ~ 0.049348U. Using Eq. (21) the exact solution to the Wiener integral is

S ( t , 0 ) = 4 ~ ( - 1 ) "+l e_~/20o)t2m_l)2 t (24) ,,,=l ( 2 m - i-)

For a numerical calculation, the step size Ax for this system is determined from Eq. ( I0 ) ,

l s 0.529166 x 10 - t ~ Ax = . ~ = v/B meters,

where n is an integer. A random walk calculation using n = 900 was performed. Substituting Eq. (20) into Eq. (16) yields

_ _ - - 1 tl ln[S(t, ro)] = At - t l n [ C i + C2e (a~-a')t + C3e (az-a3)t + . . . ] .

When t becomes large enough higher-order terms can be neglected, yielding

In [S ( t , x0) ] ln[Cl ] = ,~l ( 2 5 )

t t

Fig. 2 shows a plot of the calculation of - In[,S(t, xo) ] / t versus t including lo- error bars for the uncertainty. Note that the error bars are so small for this calculation that they do not appear. Four million p__aths were needed for this convergence. Because the exact Wiener integral is given by Eq. (24), a plot of - l n [ , S ( t , xo)]/t versus t is also shown. In addition, a least squares fit of the data to Eq. (25) starting at t = 28 is shown along with the exact eigenvalue. The least square fit yields the values In(Ci) = 0.2386(41) and At = 0.04855(12), where the parentheses denote the uncertainty in the final two figures. In the limit as t gets large the plot will approach Am = 0.04855(12)U. This compares with the exact value of A1 = 0.049348U. The results show that good convergence can be made using the Feynman-Kac method. Note that there is a bias in the plotted data due

J.M. Rejcek et aL /Computer Physics Communications 105 (1997) 108-126 ! 17

ONE PARTICAL INFINITE SQUARE WELL GROUND STATE IN ONE DIMENSION

0.055

0.050

0.045

0.040 ..... o..._o . - - - ~ 1 7 6 ............. o ............. o

0.035

~., 0.030

"~ 0.025

o Data

" : / I ~, 0.020 .............. Linear Fit

I 0.015 / ~ . . . . . . . . Ground S ~

O.OlO

0.005

0 . 0 0 0 ' ' ' ' ' ' . . . . . . . ' . . . . . . . . . .

0 2 4 6 8 l0 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62

t (Hartree "t)

Hg. 2. Plot of - I n [ - S , ( t , xo)l / t versus t for the one particle infinite square well ground state in one dimension. The solid curve is the

analytic value of the Wiener integral, open circles are the numerical ly calculated values with - t - l t r error bars, The dotted curve is a fit to

Eq. (25) ( l inear in I/t) and the horizontal dashed line is the exact ground state eigenvalue.

to the finite step size used which has lowered the calculated energy value. The actual error in the extrapolated eigenvalue is an order of magnitude larger than the statistical error. This is a consequence o f a systematic error associated with the finite step size. This error can be reduced to the size of the statistical error by extrapolating on step size as shown in Table 1.

4.2. The simple harmonic oscillator

Consider a particle with elementary mass M moving in a harmonic potential defined as V(x I) i - J z = ~K.~ . The Schr(Sdinger equation rescaled to the standard form for use in the Feynman-Kac integral becomes

20X2 q-fl2X2 ~t(X) = A~p(X) ,

where f12 = kMs4/2h2 is a scaling parameter which implies the unit o f length is s = ~/2h2fl2/kM and

U = v ~ h / ~ f l 2 becomes the unit o f energy. The eigenfunctions for this Hamiltonian are given by

~ , ( x ) = ~ : x / ~ H n ( v / - ~ x ) e - a X 2 / 2 n = 0, 1 ,2 ,3 , . (26)

where ce 2 = 2fl 2 and Hn(V~X) are the Hermite polynomials. The eigenvalues are given by

118 J.AL Rejcek et aL /Computer Physics Communications 105 (1997) 108-126

E, = v '~/3U(n + �89 (27)

Substituting Eqs. (26) and (27) into Eq. (4) yields

H2m( V/-~xo ) e-aXe/2 -S ( t , xo ) = v ~ e - V/-i put /2 ~"~ e - v'~ pUt2m (28)

22ram [ m--O

Eq. (28) is the general solution to the Wiener integral for the harmonic oscillator in one dimension. For x0 = 0 Eq. (28) yields

c~ (_ l )m ( 2m ) e_V~/3Ut2m= ~/sech ( v/~ /3Ut). (29) ~(t'O) =V~e-V~Pu'12~-~ 22" \ m m=0

Therefore, the Wiener integral can be represented by a simple closed form solution. Kac [ 18] derived the same result by a different approach.

Consider an explicit numerical example. Define the harmonic potential by V = x '2, where k = 2Jim 2, the particle mass M = 9.11 x 10 -31 kg of one electron and consider a random walk beginning at xo = 0. Let 13 = 1, then the unit of length s becomes ls = 3.324 x 10 - I~ meters and a = v'2. The unit of energy becomes 1U = 1.1053 x 10 -19 J. The eigenvalues become

,~,, = v/'2(n + �89 n =0 , 1,2,3 . . . . .

with the ground state energy ,~0 = �89 ,-~ 0.707106U. For a numerical calculation, the step size for this system becomes

3.324 • 10 - m Ax = meters,

where n = 900. Fig. 3 shows a plot of -ln['S(t, xo)]/t versus t including lo" error bars, a plot of the exact result -In[-S(t, xo)]/t versus t using Eq. (29) and the least square fit, Eq. (25), starting at t = 4 with In[Ci ] = 0.34627(58) and ~o = 0.70774(10). This compares with the exact value of h0 = 0.707106U. Thirty- four million paths were needed to reduce the uncertainty for larger times. Note that the error bars grow as t gets large and there is a l / t trend for the convergence to the ground state. In fact, in the limit that the time approaches infinity, the variance also grows without bound for finite sample sizes. This problem in Monte Carlo based methods can be overcome using importance sampling techniques, which can actually reduce the variance as a function of t. Note, however, that it is possible to achieve finite precision with finite effort as demonstrated in the examples above.

5. Application of the Feynman-Kac method to three-dimensional quantum systems

The~s physical problems are natural extensions from one to three dimensions. Like the one- dimensional examples, the exact solutions for the eigenfunctions, eigenvalues and Wiener integrals are also known. These are important because they illustrate how the method works in multiple dimensions and provide a benchmark for assessing accuracy with more complicated potentials.

5.1. One particle hz an infinite square well

Consider a single quantum particle with elementary charge and mass M moving in three dimensions confined to a well of finite dimensions Lx, L.~., Lz, with an infinite potential at the boundary of the well. The Schr6dinger equation for the system can be written

J.M. Rejcek et al./Conlputer Physics Communications 105 (1997) 108-126 119

t~

I

0.72

0.71

0.70

0.69

0.68

0.67

0.66

0.65

0.64

0.63

0.62

0.61

0.60

0.59

0.58

0.57

0.56

0.55

0.54

0.53

0.52

ONE PARTICLE IIARMONIC WELL GROUND STATE IN ONE DIMENSION

/ o Data

Exact

............... Linear Fit

. . . . . . . . G round State

0 2 4 6 8 l0 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62

t (U-') Fig. 3. Plot of - In{S(t, xo)]/t versus t for the one particle harmonic well ground state in one dimension with U = I. 1053 x I 0-19 Joules. The notation is the same as Fig. 2.

[ - � 8 9 + V ( r ) ] g ' ( r ) = A , : / ' ( r ) ,

where r = x~ + y.~ + zk is the three-dimensional position of the particle, V2(r) = a2/Ox 2 + a2/ay 2 + oa /& 2 is the Laplacian operator in Cartesian coordinates and V(r) is the infinite square well potential given by

- L x < x < Lx - L v Lv - L z < Z < Lz v = o , 2--7- -2- - ; ' 2-7 <- y <- 2--7- - V = oo, otherwise,

where s is the Unit of length and U = h2/Ms 2 is the unit of energy. The eigenfunctions are known exactly and can be ~v.ritten in terms of three independent one-dimensional infinite square well solutions

~tt,m,n( r) = ~t(x)~Om(y)~n( Z ) , (30)

where ,t0t(x), ~. , (y) and ~n(z) are the eigenfunctions for the one-dimensional well given by Eq. (18). The eigenvalues are given by

,:r~h2(l 2 m 2 n2"~ 7r2Us2(l 2 m 2 s e"~ 2 9

Substituting Eq. (30) into Eq. (4), it can be shown that the Wiener integral is just the product of three one-dimensional infinite square well solutions

120 J.M. Rejcek et al./Computer Physics Communications 105 (1997) 108-126

-8( t, ro ) = -8( t, xo)-8( t, yo)'8( t, Zo) , (31 )

where 8 ( t , xo), 8 ( t , yo) and 8 ( t , zo) are each defined by Eq. (21). This represents the probability of a Brownian motion walk remaining inside a three-dimensional box with sides (Lx, Ly, Lz) as a function of t

and starting at some initial position ro = x07 + Y0.~ + zoO: in the box. If the walk is started at the center of the box (ro = 07 + 0) + 0k') and the dimensions of the box are equal (Lx = Ly = L z), then the equation reduces to

8 ( t , 0 ) = ( 2 m - 1) e-t=~U~DL2)t2m-I)2r " (32) nl=]

For a numerical example let the dimensions of the well be Lx = Ly = L z = 20 atomic units and consider a random walk beginning in the center of the well. Let the length scale be in atomic units, s -- 0.529166 x I0 - I~ meters. This implies the unit of energy is one Hartree. The eigenvalues are given by

2 �9 It,,,,,,, = 8 0 o ( / + m 2 + n 2) , l , m , n = 1,2,3 . . . .

with the ground state Al,l,i = 3"rr2/800 ~ 0.037011U. Using Eq. (32) the exact solution to the Wiener integral is

( - 1 ) e_t=2/8oo)t2,,,_ O~t] 8 ( t , 0 ) = _ (-2"mn---ii- J " (33)

For a numerical calculation, the step size Ar = ( A x , Ay, A z ) for this system is determined to be

0.529166 x 10 - I~ Ax = Ay = Az = V6 i meters,

where n = 900. Fig. 4 shows a plot of the calculated result including one sigma error bars, a plot of the exact result using Eq. (33) and a plot of the least square fit to Eq. (25) starting at t = 30 gives In [ C! ] = 0.62295 (18) and AI = 0.035110(40). This compares with the exact value of Ai = 0.037011U. Two million paths were needed in order to reduce the uncertainty in the calculation for large t. The difference between the calculated and exact eigenvalues can be reduced by calculating higher times and by decreasing the step size of the walk.

5.2. The hydrogenic atom

Consider a single particle with elementary charge and mass M moving in a three-dimensional Coulomb potential defined as

Zke 2 V ( r ) = - - - - ,

[rl

where Z. is an integer, k is the Coulomb constant and e is the electron charge. The Schr/Sdinger equation for the system can be written

[ - - � 8 9

with s = h2/Mke 2 the unit of length and U = h2/Ms 2 the unit of energy. Solutions to this equation are known exactly. The eigenfunctions g',am are the hydrogenic solutions given by

t~n 3/2 /(n-l-I)[ t - a , r [ 2 21+1 m q-"ntm(r,O, fb) = x/zV-ff-~ U n(-~Sa,~i~i (ctnr) e Ln_t_,(ctnr)Y t ( 0 , ~ ) , (34)

J.M. Rejcek et al./Computer Physics Communications 105 (1997) 108-126 121

ONE PARTICLE INFINITE SQUARE WELL GROUND STATE IN THREE DIMENSIONS

0.040

0.035

0.030

0.025

~- 0.020 / / ' " ' " " ' " " ............. .... ....... ......... ~o

L~, 0.015 !

0.010

0.005

0.000 ; I : : ; 0 2 4 6 8 IO 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62

t (Hartree "l)

Fig. 4. Hot of - l n lS ( t , xo)]/t versus t for the one particle infinite square well ground state in three dimensions. The notation is the same

as Fig. 2.

where a,, = 2Z/ns and /'2/+1 (X) are the associated Laguerre polynomials and Ytm(0,$) are the normalized ~n--I--I spherical harmonics. The indices range from - l < m < l, 0 < l < n - 1 with n = 1,2, 3 . . . . . The eigenvalues are given by

Z2Mk2e 4 Z2U En = 2h2n 2 = - 2n--- T . (35)

An eigenfunction expansion o f -S( t , ro) in L 2 is sought from Eq. (4). Unfortunately, for this problem it can be shown [ 19] that no such expansion exists. (See the discussion following Eq. (6) in Section 2.) While the expansion coefficients C, in Eq. (5) can be computed, the series diverges. Nevertheless, even though no convergent solution of S ( t , to) as an eigenfunction expansion exists, the limiting value of - ln [S( t , ro) ]/t as t gets large must converge to the exact value of AI, as noted in Eq. (7) above.

For a numerical example, let Z = 1, the potential for the hydrogen atom. Choose the origin to be at the center of the well and let the scale be in atomic units s = 0.529166 • 10 - m meters which makes the unit of energy I U = 1 Hartree. The eigenvalues are given by

1 An= 2n 2 , n = 1 , 2 , 3 . . . . .

I with the ground state AI = - 3 -

122 J.Af. Rejcek et aL/Computer Physics Communications 105 (1997) 108-126

ItYDROGEN GROUND STATE

-0.45

.0.50

.0.55

!

.0.60

-0.65

-0.70

.-0.75

.0 .80

-0.85

-0.90

.... . . . ~ " .... ....

.....-"

.,.~ O"

..:"

...."

/

: o Data o :

: ............... Linear Fit ?

../ ........ Ground State

- - , [ I J [ , I I I , I t I , I t I ~ I , I , I , I ~ I ' I

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

t (Hartree "t)

F i g . 5. P l o t o f - - ln['S(t, xo)]/t v e r s u s t f o r h y d r o g e n g r o u n d s ta te . T h e n o t a t i o n is t h e s a m e a s F i g . 2 .

Consider a random walk beginning at an initial position r0 = ~ (~ - . ~ + Ic). The step size Ar = (Ax, Ay, Az) for this system is determined to be

0.529166 x 10 -z~ Ax = Ay = AZ = x/n meters ,

where n = 900. The results are shown in Fig. 5. The least square fit using Eq. (25) started at t = 24, and results in In[Ci ] = 1.327(26) and ,~l = - 0 . 4 9 8 1 ( 2 9 ) . This compares with the exact value of Al = -0 .50000U. Eight million paths were needed in order to reduce the uncertainty in the calculation.

6. P r o g r a m imp lemen ta t ion

In computing the results obtained in this paper, several computer and arithmetic techniques were implemented for efficient production of the simulations.

6.1. Sample size

In order to overcome some of the difficulties associated with computing a large number of paths, such as computer failure or input parameter errors, the sample sizes were deliberately left small and accumulated over real time to produce a large sample size. The sampling has been implemented in the following way. Let

J.M. Rejcek et aL/Computer Physics Communications 105 (1997) 108-126 123

I00

~(~r lOO)= ~ ~ P, i=1

denote the Wiener integral for .A/" = 100 trial paths Pi. Note that the Wiener integral for ./V" = 1000 trials can be calculated by computing 10 sets of 100 trial paths,

1000 [ 100 200 t t0000 ]

~(~= 1ooo) = - - ~C P, = "L ~ ~C P, + ~ ~C P , " ' ~ ~ P'l" i=1 i=1 i=101 i=901 J

"S(.Af = 1000) = ~o[Sl(Af= 100) + S2(.A/'= 100) +--.SL0(./V" = 100)1.

In general, for M sets of N trial paths, .Af = MN and

1 M -S(Af = M N ) = --~ ~ - S , ( A f = N ) . (36)

i=1

Using Eq. (36), a sample size of.N" = 106 paths can be calculated from M = I0 sets of N = 105 trial paths or M = 100 sets of N = 104 trial paths.

The uncertainty in the Wiener integral is given by the variance

= O'0 4-~'

where o-o is the standard deviation of the distribution and is given by

H H I ~ - ~ ( P i - - S ( ' I V ' = N ) ) 2 .N "! -1 ~ - '~ [P2- 'S2( 'N '=N)] (37) o - ~ v _ ' ~ . - T = - �9

i=1 /=1

Let the sum of the square of paths for the ith sample size of A/" = N be denoted by

N

8,~(~ = ~) = ~ P?. (38) j=l

Substituting Eq. (38) into Eq. (37) for sample sizes of.N" = M N yields

1 8~(.~" = N ) - M , V - S 2 ( H = M N ) . ( 3 9 ) o'~ ~ M N----~ ~ i=!

Thus, rather than trying to accumulate one large set of .fir = M N trials, we have accumulated M sets of

82(A r ~.:N) and S2(At" = N) for small sets of trials N. The average over the entire set of samples and the corresponding uncertainty in the Wiener integral have been obtained from Eqs. (36) and (39).

6.2. Overflow and underflow errors

The singularity from the Coulomb potential at zero can cause overflow and underflow errors. However, it can be shown [20] that a random walk that does not begin at the origin in three or more dimensions never passes through the origin. Therefore, steps should be taken before the calculation to insure that the distance between a random walker and the origin never gets too small. One way to implement this is to imbed a conditional "IF" statement to throw out any walk that gets too close to the origin. However, this can significantly effect

124 J.AL Rejcek et al./Computer Physics Communications 105 (1997) 108-126

the efficiency of calculations, since there would be wasted effort. In this paper, since Bernoulli trials are used to approximate the random walks, each step in a random walk lands on a point from an evenly spaced grid in the N-dimensional space defined by the physical system studied. For the hydrogen atom, the random walkers were placed on a three-dimensional grid that has been offset from zero to assure no simulated path would pass through the origin.

Due to the presence of exponential terms like e -vCx) in the calculations, over- and underflows can also occur. This can be a real problem for potentials such as the one used for the simple harmonic oscillator discussed in Section 4.2, where the quadratic term in the potential assures large contributions to the exponential term as the walker drifts away from the origin. In addition, a subtle shift in the distribution will occur if the over- and underflows are discarded. For the harmonic oscillator problem, as t gets large the machine will truncate the answer of a step contribution to zero. If the statistical parameters are close in magnitude to the underflow limit of the machine, then the resulting statistical analysis will be biased.

Where this becomes a problem depends on the computer's under- and overflow capabilities. An extensive study has been made to find how large t has to be in order for this underflow problem to occur for the one-dimensional harmonic oscillator problem in Section 4.2 using the machines listed in Table 2. It has been found that underflow truncations begin to be seen for t > 50. In this paper statistically significant data has been obtained for t < 40 and therefore this underflow problem has not been encountered. However, if larger times, higher dimensions or more particles are considered, then underflow truncation may become a problem. There are effective ways to deal with this problem, including rescaling the Schr6dinger equation into more agreeable units and taking advantage of the property of the exponential, e -~ = eAe ~'-a, where A is a constant in order to translate large or small numbers back into the computational domain of the computer.

6.3. The random number generators

Initially the imbedded random number generator in the VAX cluster was used to calculate paths. A concern began to arise whether the period of this generator was being overrun due to the large number of random numbers needed in each calculation. To increase the period of the generator two additional generators [21,22] were evaluated for implementation. There was no measurable statistical difference in answers found using the various generators. For this paper the random number generator in [22] has been used because it has the largest cycle time and according to the reference passes all the stringent tests of randomness.

6.4. CPU thne for the simulations

The amount of time needed to simulate a physical system varies according to the number of particles, the number of space dimensions and the potential. Table 2 shows the time per megapath for the various machines used. Fig. 6 shows a plot of the CPU time needed to simulate the infinite square well problem in several dimensions with one and two particles. The real time can be reduced considerably using massively parallel machines, where each dimension and particle can be calculated simultaneously.

7. Conclusions

This paper has presented computational details of applying the Feynman-Kac path integral method of quantum mechanics. Simple examples were employed to make the numerical approximations encountered subject to rigorous tests. Results for exactly soluble quantum systems without nodes in the ground state have been presented. In this case it has been demonstrated that various approximations in the numerical applications can be accurately accessed and controlled, so that arbitrary accuracy may be obtained with sufficient effort. It has been stressed that the method can be applied to a wide range of problems with arbitrarily strong correlations and

J.M. Rejcek et aL/Computer Physics Communications 105 (1997) 108-126

Table 2 Comparison of CPU time needed to compute one million paths per particle dimension per unit o f time t and step size n = 900.

125

Machine Millions of instruction per second CPU (minute)

VAX 4000 model 90 27 82 CONVEX C-220 40 38 DEC 3000/500 alpha chip 150 12 Connection Machine CM-2 a 8

a Due to the parallel architecture of the Connection Machine, performance ratings in instructions per second do not accurately measure machine performance. In addition, a massively parallel machine can be configured to simultaneously compute several particle dimensions, therefore, as the number of particles and dimension increase for a physical system, CPU time increases less rapidly for parallel machines than for serial machines.

CPU TIME

I000.00

100.00

= 10.00 0

E

1.O0

O.lO

0.01

x . /

../

~f~'" ]

f ...X"

/-" /"

X /-"

..-"

1.1 o~$\o~+ " .......... ~ ~ I~-'~"

?.,s~ ...u . # ~ -su .z, ......... c ~; ........ x~ ..........

,1.s b~'-'~'....... . 0 , i .? / ' , -~ '~0 A ............. " X" .+,' .r'l-"" ......,~t .......

I I I i I I I I t++ I t , , , , , ~ . . . . . . . . . , , , , , . . . . . . . . . . . . . , . . . . . , . . . . . . . , , ,

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60

t ( U - t )

Fig. 6. CPU time for the infinite square well calculations.

interactions, and it is easily implemented. Convergence to several significant figures may require large amounts of computer time, so importance sampling is recommended for systems containing more than one particle to improve the efficiency of the method. Many particle systems become more complicated when nodal structures are not understood or when nodes of importance sampling trial functions are approximate. These considerations will be addressed in another paper.

While illustrating the applications of the method this paper has introduced standard statistical methods for measuring sampling errors. New analytic results for the Wiener integrals have been obtained and compared with

126 J.3L Rejcek et al./Computer Physics Communications 105 (1997) 108-126

numerical approximation of the path integral in order to determine convergence rates and measure the statistical uncertainty. The only approximations in the numerical calculations are the step size in the Wiener path, finite number of paths sampled, and finite time t for each path. These errors can be accurately accessed and may be controlled so that given enough computer time any desired accuracy may be obtained.

The difference between the usual Lebesque and Wiener measure has been emphasized with a specific example. In many cases the Wiener measure of a function may exist where the Lebesque measure does not, which lead Kae to modify the original Feynman path integral method. It has been stressed that numerical application of the Feynman-Kac method is independent of the existence of eigenfunction expansions of the path integral. This has been illustrated in the case of atomic hydrogen, where the eigenfunction expansion of the Wiener integral diverges, but the Feynman-Kac solution converges to the correct ground state energy.

The computer code which ran the simulations presented in this paper consist of about 150 lines of FORTRAN 77 code. It is easily implemented and highly transparent. Readers interested in learning how to apply the Feynman-Kac method to quantum systems may obtain a copy of the basic code for the method from the authors.

Acknowledgements

This work was supported in part by TheRobert A. Welch Foundation and the Texas Advanced Research Program. The authors would also like to thank Wai Kwok Kwong, Pratap Pattnaik and David Orr for their helpful discussions and Lee Sawyer for his assistance with the DEC 3000/500 alpha chip computer.

References

[ ! ] P. Exner, Open Quantum Systems and Feynman Integrals (Reidel, Boston, 1985) ch. 5, p. 217. 121 A, Einstein, Annalen tier Physik, Ser. 4, XVII (1905) pp. 549-560;

M. Smoluchowski, Bulletin de I'Academie des Sciences de Cracovie, No. 3 (March 1906) pp. 202-213. [3] M. Reed and B. Simon, Methods of Modem Mathematical Physics, Vol. II (Academic Press, New York, 1975) pp. 279-281. [41 M.D. Donsker and M. Kac, J. Res. Natl. Bur. Stand. 44 (1950) 581. 151 M. Kae, On some connections between probability theory and differential and integral equations, in: Proe. Second Berkeley Symposium

(Berkeley Press, 1951 ). [6] M.D. Donsker and S.R.S. Varadhan, Asymptotic evaluation of certain Wiener integrals for large time, Proe. Int. Conference on

Function Space Integration (Oxford Univ. Press, Oxford, 1975) pp. 15-33. 17] A. Korzeniowski, J.L. Fry, D. On" and N.G. Fazleev, Phys. Rev. Lett. 69 (1992) 893. [8] A. Korzeniowski, On diffusions that cannot escape from a convex set, Stat. & Prob. Lett. 8 (1989) 229-234. 191 B. Simon, SchrOdinger Semigroups, Bull. Am. Math. Soc. 7 (1982) 447-526.

[ 10] B. Simon, Schrtidinger Semigroups, Bull. Am. Math. Soc. 7 (1982) p. 459. [ 11 ] B. Simon, SchrOdinger Semigroups, Bull. Am. Math. Soc. 7 (1982) p. 453. [ 121 B. Simon, Schr6dinger Semigroups, Bull. Am. Math. Soc. 7 (1982) pp. 456, 460. [ 13] A. Korzeniowski and D.L. Hawkins, Probal. Eng. & Inform. Sciences 5 (1991) 10l. [ 14] Propagation of Error, Vol. 2, Encyclopedia Of Statistical Sciences (Wiley, New York, 1982) p. 549. [ 15] P. Erdos and M. Kac, On certain limit theorems of the theory of probability, Bull. Am. Math. Soc. 52 (1946) 292-302. [ 16] R.V. Mises, Wahrscheinlichkeitrechnung (Deuticke, Leipzig, Vienna, 1931) pp. 499-506. [17] J.M. Rejcek, Applications of the Feynman-Kac Path Integral Method to Find the Excited States of Quantum Systems, Ph.D.

dissertation (University of Texas, Arlington, 1995) p. 40. [ 181 M. Kac, On distributions of certain Wiener functionals, Trans. Am. Math. Soc. 65 (1949) 1-13. [ 19] J.M. Rejcek, Applications of the Feynman-Kac Path Integral Method to Find the Excited States of Quantum Systems, Ph.D.

dissertation (University of Texas, Arlington, 1995)pp. 60-62. [20] J.M. Rejcek, Applications of the Feynman-Kac Path Integral Method to Find the Excited States of Quantum Systems, Ph.D.

dissertation (University of Texas, Arlington, 1995) p. 66. [21] W.H. Press and S.A. Teukolsky, Portable random number generators, Comput. Phys. 6 (5) (1992) 522-524. [22] G. Marsaglia and A. Zaman, Toward a Universal Random Number Generator (Supercomputer Computations Research Institute,

Florida State University, Tallahassee, FL, Sept. 1987). [23] G. Fishman, Multiplieative congruential random number generators with modulus 2/~: an exhaustive analysis for ,B = 32 and a partial

analysis for/~ = 48, Math. Comput. 54 (189) (1990) 331-344.