High-accuracy configuration-space faddeev calculations

299~

Nuclear Physics A508 (1990) 299c-304~ North-Holland

High-Accuracy

Configuration-Space N.W.

Institute

for Theoretical

Schellingerhout

Faddeev

Calculations

and L.P. Kok

Physics, University of Groningen,

Groningen,

The Netherlands

A new method for solving the bound-state configuration-space Faddeev equations is presented. This method is largely based on the (Los Alamos) spline method, but exploits the tensor structure of the resulting matrix problem. With this method accurate results can be obtained using small computers, or very accurate results using large computers. We present results obtained with our method, among which are accurate results for the bound-state Coulomb problem. 1

Introduction

The

Faddeev

equations

many

groups

to solve

group

used finite-difference

Alamos

(/Iowa)

developed

group

in configuration

method,

solution

spline

by the

exploited, 2

The

The

These total

and show some results Faddeev-Noyes

Faddeev

plitudes

coupling

that

[3].

equations

very much better

we explain

were obtained

Initially, More

the spline

than

the Faddeev-Noyes this,

[l], have been

success.

recently,

method

[2,4].

the Los We have

for the bound-state

case,

the ones used until now.

equations how the

used by

the Grenoble

and their

tensor

numerical

structure

can be

with our method.

Equations

scheme

consists

of decomposing

the wave function

9 into three

am-

$i as follows:

amplitudes wave function

equations

must satisfy

the Faddeev

can be retrieved

for the bound-state

Here we have defined from

After

by Noyes

equations

by using

the Faddeev-Noyes

describe

method.

great

to solve these

which performs

we will briefly

with

very successful

for solving

In the following

as derived

problem,

methods

has been

a new method

based on the spline

space

the trinucleon

a mass-dependent

three

problem

equations,

by using the relation can in configuration

sets of mass-weighted

factor

equal

which have a unique

to the vector

03759474 I 90 / $3.50 0 Elsevier Science Publishers B.V. (North-Holland)

Jacobi

solution

@ = $1 + $2 + $3. space

be written

coordinates

connecting

particles

The

Faddeev

as:

(x;, y;). j

[5]. The

(xi is apart

and k, and yi is

3ooc

N.W. Schellingerhout,

L.P. Kok I Configuration-space

apart from another factor equal to the vector connecting particles j and k.) Kinematical I$ describes

the interaction

assumed the potentials The usual method analysis.

particle

i to the center of mass of

details are thus removed from the equations.

between

particles

i and k.

The potential

In the above equations,

we have

to be central, but this is by no means a requirement. employed

The Faddeev

Faddeev calculations

for solving these equations begins with angular-momentum

amplitudes

are projected

+i(Xj,yi)

o:

FCY(k;,?j),

C

=

onto a set of bipolar harmonics

a

where we have used

We will not go into the details concerning harmonic ticles. one.)

particles with (iso)spin,

basis must satisfy (anti)symmetry

(Also,

requirements

in that case, the number of independent

but note that the bipolar

when dealing with identical

Faddeev

equations

The bipolar harmonic functions are often referred to as channel functions,

lation that truncates

the expansion

after n channel functions,

par-

reduces to two or and a calcu-

is referred to as an n-channel

calculation. After

the introduction

of polar coordinates

{ and some standard

manipulations,

(Ai,,

corresponding

yi

=

Psin6;,

(5)

we can write the Faddeev-Noyes

equations

as:

on channel o, v,p is the matrix

element

to channels (Y and p, and pji a set of constants representing

This elliptic partial (integro-) conditions

PCOSBi,

of the free Hamiltonian

the mass ratios of the particles.

boundary

=

- K”) &i(p, @i) - C ubp(pcos @i)#b(pl ei> P

where A;,a is the projection of the potential

Xi

Details can be found in Refs. [4] and [6]. differential

eigenvalue equation has a unique solution when

are given on a closed surface:

for all i and CL 3

Solution

The boundary

of the Faddeev-Noyes conditions

Equations

(7) are usually approximated

zero [3] or equal to the asymptotic

by assuming the Faddeev amplitudes

form [4] outside a certain finite interval.

We use a different

301c

N.W. Schellingerhout, LP. Kok I Configuration-space Faddeev calculations

approach,

which we find very robust:

by using the following

the infinite

transformation

=

r where

X is a parameter

formation

that

has the effect

function

interval

of changing

1-e+,

(8)

the asymptotic

the reduced

Faddeev

amplitudes

is satisfied

conditions

described

onto a bicubic

Note

that

this

of an exponentially

trans-

decreasing

reduces

by Eq. (8))

by choosing

Eq. (6) (or, rather, to the following

A, B, C, and D represent

the left-hand

spline basis and demanding

The

solution

to store central

the matrices memory,

but

matrix

(especially

and background

slow and has low access structure

Tensor

matrices

constructed

The boundary

these

conditions.

from the change

of variable

Da,

(9)

associated

with

n2, the potential

terms

in

a very costly

of the very large

must

procedure

amount

it is impossible

because

of memory

to store

the matrices

be used, which on small computers

We will now present

a new method,

of the needed

based

in

is often

on the tensor

(f(z,y)

Method

can

be stored

from tensor

expands

in a more

products.

the reduced

= fJz)f,(y)).

The tensor

We put asterisks

S contains represents

the values the unit

amount

calculated.

corner

of storage

(M, 8 :I

c3

to denote

consequence

onto

a basis

that

they

can be

of the solution of decomposed

method functions

channel,

a representation

the tensor where

=

points.

the tensor

AZPZ (S, 63 :I

matrices

matrix

S, thus

requires

a much

products problem

a parameter c3

The

The

representation

to solve the matrix

side and introducing

work (from the upper-

T, and 13 space).

in the collocation

Clearly,

of flops required

:Q*)a

(10)

in which spaces the matrices

functions

Ca to the right-hand

S* + N, 8

4:

by recognizing

of B is:

we have amplitude,

than

manner,

‘1 @I *n @ s* @ S’ .

in r space.

Also, the number moving

representation

of the spline

operator

amplitudes

=

on the corners

left to the upper-right

efficient

This is an immediate

Faddeev

:B:

After

satisfies

side, respectively.

In practice,

memory

speed.

that the differen-

[7,4].

of (9).

The

smaller

=

because

D).

that

of expanding

problem:

is computationally

mostly

basis

the form that results

A, the identity

of this equation involved,

(four per interval)

a spline

side, and those in the right-hand

high flop count

which consists

collocation),

points

(A-n2B-C)a

which

behavior

(or orthogonal

in a set of collocation

can be incorporated

This procedure

The

the accuracy.

eefiP to (1 - r)lrlx.

tial equation

4

into a finite interval,

p:

can be used to optimize

Now we apply the spline method

where

is transformed

of the hyperradius

are explicitly can be reduced.

A, we have:

S* + Se 8 zJ*)a.

(11)

302~

N.W. ~ch~~~inger~ut,L.P. Kok I ~on~~urat~~n-sp~e Faddeev ~al&~Iati~~

The potential

matrix

P is not decomposed,

but diagonal

two of the three spaces, so that J (containing

by the tensor method.

is the largest

matrix.

for D and 3, to calculate the storage-space

now compare the storage requirements obtained

in three spaces, Q is diagonal

the integrals)

in

We will reduction

For three identical particles, the amount of storage required

for D is N

=

32N*{N~N=)2

words,

(12)

where N,., No, and N, stand for the number of intervals in the r and B directions, number of channels, respectively. N Therefore, original

However, =

4(N~~~)2

by using the tensor method,

required

magnitude.

(The

to overhead.)

improvement

The performance

words.

(13)

D can be stored using (8Np)-*

the matrix

space, which in practice

is an improvement

of the tensor method

times the

of more than two orders of a

for the total storage requirements

puters can achieve results that previously

and the

to store J we only need

will be a little smaller, due

is therefore

very high: personal com-

could only be obtained

with very large mainframe

comuters. The nature of the spectrum of this equation (the set of potential bound state at the given energy) inverting

J (which would be very costly),

the Lanczos (M,

@ :1 @ S’ f

Although obtained

5

algorithm

the explicit

strengths that support a

enables us to solve the matrix equation without by solving the following,

equivalent

(implicitly)

problem

with

[4,8]:

N, @ :&“)-l

=P:(s*~~n~s*+s*~:J*)a

inverse of A4 @

by simultaneous

=

4%

(14)

1 @S + N ~3Q is not strictly needed, it can easily be

diagonalization

of both terms.

Results

In Table 1 we show some of the results obtained the numbers shown here are reliable,

muonic molecules we have used the parameters the t&

molecule

the University

have probably

of Florida

they use slightly different precision

(14 digits),

with our method.

from [ll].

been performed

by the Quantum

masses. Our results were obtained

the longest calculation

Theory

E-E,,

=

with a Cyber

took approximately

We define the order of convergence

For the

The most accurate calculations

[12], who obtain 12 accurate digits, but comparison

To obtain these results, we used an extrapolation convergence.

All but the last digits of

except for the numbers taken from Ref. [lo].

Project

for

group of

is hard because 205, using single

one hour.

technique, which increases the order of

5 as:

O(hc),

(15)

where h is the grid size, E the exact energy, and Eh the energy resulting from a calculation with grid size h.

N.W. Schellingerhout,

Table 1: Comparison the literature,

L.P. Kok I Configuration-space

of results obtained

with the tensor method

Best literature

System

value

Best Faddeev

8.251

e-e+e-

(au.)

0.2620050702325

(111

0.263029

Hem

(a.u.)

2.90372437705

[ll]

2.90516

WI VI

-

PI

PPCL

(eV)

253.152616808

tdp

(eV)

319.1401188

To obtain

the correct

in the matrix

energy,

However,

IcA

The helium ground-state

order of convergence effective

which must be

that are not too far from the For the Coulomb

problem,

the

(16) was obtained

using 21 channels and three

technique uses these three results

with eighth, instead of fourth order, which is the typical

for the spline method.

Note

that higher-order

when fine grids can be used. Due to the tensor method,

finer grids than usual, making the extrapolation The results from the present calculation more accurate

energy

AK, .

=

56 x 56). Th e extrapolation

a result that converges

0.26202 2.9037247

since for this case we have:

energy, for example,

different grid sizes (maximum: to obtain

the estimate

for estimates

alltogether,

8.25273

PI PO1 PO1

253.0

will give an accurate answer.

process can be abandoned

Tensor Method

319.0

we have to iterate

equation.

real value, only two iterations

value

8.251

(MeV)

iterative

to the best value found in

and the best Faddeev results.

MT-V

substituted

303c

Faaiieev calculations

results can be obtained

are only

we were able to use much

pay off significantly.

are already of considerable

without

techniques

accuracy, but even

having to resort to extreme

measures.

The

accuracy of the present helium result, for example, is limited by the real number system used by us, which has only 14 significant

digits.

In calculations

of this size, it is to be expected

that some 7 digits may be lost. The rather modest form three-body required.

With

memory

calculations

requirements

of the tensor method

on small computers,

a personal computer

make it possible to per-

when a not high degree of precision

one can obtain results to four significant

is

digits in a few

hours.

6

Conclusion

The tensor representation

of the matrices that occur in the numerical solution of the Faddeev-

Noyes equations reduces the amount of storage required by two orders of a magnitude the number of intervals in the r direction). for the applicability calculations

of configuration-space

to a considerable

use of extrapolation

This improvement Faddeev

has considerable

calculations.

First,

techniques and supercomputers

makes this method

consequences

anybody

degree of precision on his/her personal computer.

(8 times

can do

Second, the

competitive

in fields

304c

N.W. Schellingerhout.

L.P. Kok I Configuration-space

that have until now been the exclusive of non-Coulomb three-body

potentials

problem

can be totally

We wish to thank the Stichting gebruik

Supercomputers

realm of variational

is trivial.)

Finally,

Faddeev calculations

methods.

the question

(Also,

of convergence

the incorporation for the nuclear

resolved. SURF

that enabled

for granting

us to perform

the funds from the Nationaal

the calculations

Fonds

shown here.

References [l] H.P. Noyes and H. Fiedeldey, in Three-Particle Scattering in Quantum Mechanics. Proceedings of the Tezas A&M Conference, edited by J. Gillespie and J. Nutall (W.A. Benjamin, New York, 1986), pp. 195-294. [2] G.L. Payne, J.L. Friar, B.F. Gibson, and I.R. Afnan, Phys. Rev. C 22, 823 (1980). [3] A. Laverne and C. Gignoux, Nucl. Phys. &Q& 597 (1973). [4] G.L. Payne, in Lecture Notes in Physics 273, Models and Methods in Few-Body Physics, Proceedings, Lisbaa, Portugal 1986, edited by L.S. Ferreira, A.C. Fonseca, and L. Streit (Springer-Verlag, Berlin, 1987), pp. 64-99. [5] L.D. Faddeev, Zh. Eksp. Teor. Fiz. a,

1459 (1960). [Sov. Phys. JETP 12, 1014 (1961).]

[6] G.D. Bosveld and N.W. Schellingerhout, Configuration-Space Faddeeu Calculations, Report 231, Institute for Theoretical Physics, University of Groningen, Groningen, The Netherlands, unpublished. (Available upon request from the authors.) [7] C. de Boor and B. Swartz, SIAM J. Num. Anal. [8] Y. Saad, SIAM J. Num. Anal.

19,485

l,Q, 582 (1973).

(1982).

[9] J.L. Friar, B.F. Gibson, and G.L. Payne, Phys. Rev. C 24, 2279 (1981). [lo] E. Cravo and A.C. Fonseca, Few-Body Systems [ll]

8, 117 (1988).

A. Yeremin, A.M. Frolov, and E.B. Kutukova, Few-Body Systems 4, 111 (1988).

[12] S.E. Haywood, H.J. Monkhorst, and K. SzaIewicz, Phys. Rev. A 37, 3393 (1988).