The J-Matrix Method.Abdulaziz D. Alhaidari docx

Similar to the R-matrix method, but unlike the L2 Fredholm method, this approach made it possible to take full account of the reference Hamiltonian H0bygiving a full solution of the refe

Abdulaziz D Alhaidari · Eric J Heller ·

Editors

The J-Matrix Method

Developments and Applications

Foreword by Hashim A Yamani and Eric J Heller

IndustryRiyadh 11127Saudi Arabiahaydara@sbm.net.saMohamed S AbdelmonemKing Fahd University

of Petroleum & MineralsDept of PhysicsDhahran 31261Saudi Arabiamsmonem@kfupm.edu.sa

ISBN: 978-1-4020-6072-4 e-ISBN: 978-1-4020-6073-1

Library of Congress Control Number: 2008921100

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2008 Springer Science+Business Media B.V.

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Although introduced 30 years ago, the J -matrix method has witnessed a resurgence

of interest in the last few years In fact, the interest never ceased, as some authorshave found in this method an effective way of handling the continuous spectrum

of scattering operators, in addition to other operators The motivation behind the

introduction of the J -matrix method will be presented in brief.

The introduction of fast computing machines enabled theorists to perform lations, although approximate, in a conveniently short period of time This made itpossible to study varied scenarios and models, and the effects that different possibleparameters have on the final results of such calculations The first area of researchthat benefited from this opportunity was the structural calculation of atomic andnuclear systems The Hamiltonian element of the system was set up as a matrix in aconvenient, finite, bound-state-like basis A matrix of larger size resulted in a betterconfiguration interaction matrix that was subsequently diagonalized The discreteenergy eigenvalues thus obtained approximated the spectrum of the system, whilethe eigenfunctions approximated the wave function of the resulting discrete state.Structural theorists were delighted because they were able to obtain very accuratevalues for the lowest energy states of interest

calcu-Of course, the result of diagonalization also gives information on the ing discrete states, including those that lie in the energy continuum The fact thatthe approximation yields ‘discrete’ scattering states could not be helped, since theHamiltonian is represented by a finite matrix The situation is worsened by the factthat the eigenfunctions of these “discrete” continuum states are bound-state-like, asall members of the basis set used in constructing the Hamiltonian have this property.This was deemed unnatural, and led theorists to believe that this data did not consti-tute information that was useful for the calculation of scattering This belief almostput a stop to the application of basis-set techniques for the solving of scatteringproblems However, a major turnaround occurred as a result of the work by Haziand Taylor [1]

remain-Hazi and Taylor took the natural step of asking whether a use could be made ofthe bound-state-like basis to describe resonances, which resemble bound states eventhough they are actually scattering states This led to the “Stabilization method”:real discrete energy eigenvalues closest to the resonance energy become stable asthe parameters of the calculation are varied This development rekindled confidence

v

among theorists for the belief that the discrete energy eigenvalues and tions may indeed contain useful information about the continuous spectrum of theHamiltonian.

eigenfunc-The Reinhardt group at Harvard was in the meantime developing a computationalscheme to extract scattering information from the Fredholm determinant [2] For a

given short range potential V , the Fredholm determinant D(z) is defined as:

with the phase ofD(z) being the negative of the phase shift δ(E) in the limit z →

E + iε It also satisfies the dispersion relation

whereD(E + iε) = d(E) − iπ A(E), and the functions d(E) and A(E) are both

real and analytic Encouraged by the stabilization results, the group worked with the

matrix representation of H and H0in a finite L2-basis set,{φ n}N−1

n=0 Therefore, theFredholm determinant has the approximate value

results

This development added some analytical tools to the predominant numerical

tools available for use in the L2-Fredholm method For instance, the discrete

eigen-function of the finite N × N Hamiltonian H0may now be written as a finite sum of

L2-basis, with the orthogonal polynomials as Fourier-like expansion coefficients:

Heller of the Reinhardt group, who was also studying ways of enhancing the

R-matrix method of scattering, proposed a way to improve the accuracy of the L2Fredholm calculation He first proposed that the exact sine-like solution of the ref-

-erence Hamiltonian H0be expanded in the complete L2-basis This could formally

be done since the Fourier-like expansion coefficients were obtainable, even

analyt-ically Similar to the R-matrix method, but unlike the L2 Fredholm method, this

approach made it possible to take full account of the reference Hamiltonian H0bygiving a full solution of the reference problem in the complete basis Furthermore,

in the spirit of the R-matrix method, Heller and Yamani proposed that the potential

representation be limited to a finite subset of the complete basis This is the basic

idea behind the J -matrix method [4], so called because the matrix representation

of the operator J = H0 − E in certain Laguerre or oscillator basis functions is

tridiagonal (Jacobi)

The analogy with the matrix method [5] is strong The language of the matrix in configuration space is translated for the J -matrix into the language of

R-function space The full model Hamiltonian and waveR-function are written for the

R-matrix method and J -matrix method as follows, in their respective languages:

where s n and c n are the expansion coefficients of the sine-like and cosine-like

solu-tions of the reference Hamiltonian in the complete L2 basis In both methods, the

resulting scattering matrix S(E) is exact for the approximate model Hamiltonian ˜ H

This contrasts with the use of variational schemes to solve the scattering problem,

which basically seek an approximate solution to the exact Hamiltonian H The latter

methods sometimes lead to the existence of anomalous pseudo-resonance behavior

of the calculated cross sections This phenomenon is not present in J -matrix

Here g N −1,N−1 (E) is the (N −1,N−1) element of the inverse of the matrix

represen-tation of (H − E) in the finite basis, and J nm (E) ≡ (H0− E) nm Results show that

S(E) is a highly accurate approximation of the S-matrix of the exact Hamiltonian.

Most notably, S(E) has the following two desirable properties:

(a) S(E) is a smooth function of E, even as it assumes the values of the eigenvalues

{E i}N−1

i=0 or



E i0N−1

i=0 In fact, at the points{E i}N−1

i=0 , S(E) has the special value

This is a noteworthy result, which states that the S-matrix (cross section, or

phase shift) for the approximate Hamiltonian ˜H could be calculated exactly, at

positive discrete energies, by knowing only the coefficients of the

wavefunc-tion of the reference Hamiltonian H0 in the basis Yamani and Abdelmonemtook the set{S(E i)}N−1

i=0 , representing the value of S(E) on the real (scattering)energy axis, and analytically continued it to the lower complex energy plane

to search for shallow resonances [7] These are characterized by poles of S(z)

in the second sheet of the complex energy plane The same authors showedthat a similar, and slightly more complicated result, holds in the multi-channelcase [7]

(b) The diagonalization of H in the finite L2basis needs to be done only once to

enable the calculation of S(z) over the entire continuum range of energy.

The first application of the J -matrix method was in the Tempkin-Poet model

of an S-wave e-H scattering analysis [8] The target and incoming projectile aredescribed on the same basis, resulting in a finite number of target channels A six-state calculation was able to reproduce previous results using different theoreticalmethods, and the calculation improved when the finite basis was enlarged Implicit

in this model is the approximation of the target by a finite number of channels;some, if any, represent the bound states, while the rest represent the continuum

of scattering states This naturally leads to anomalous behavior of the scatteringcross section when the scattering energy is increased, so that more discrete chan-nels become “open.” Of course, this should not be the case The correction of this

anomalous behavior is still an outstanding problem for the J -matrix method and

for similar methods that approximate the target ionization region by a finite ber of channels Bray and Stelbovics showed that the anomalous behavior tends todisappear as the number of target channels is increased [9]

num-Another early application of the J -matrix method of scattering was carried out by

Broad and Reinhardt, who calculated e-H−scattering cross-sections and accuratelyreproduced resonance positions [10]

Further developments in the J -matrix method ensued:

(1) The matrix elements of Green’s function associated with the reference

Hamil-tonian H0 were evaluated by Heller, showing the analogy to the configurationspace result [6]:

finite Green’s function g N −1,N−1 (E) and only considered quantities associated with G0(E) [11] In so doing, they showed that the Lippman-Schwinger equa- tion T = V – VGV could be solved over a continuous range of energies, without

re-diagonalization of matrices every time the energy is varied

(3) Horodecki achieved a relativistic generalization of the J -matrix method [12].

This generalization was refined by Alhaidari et al [13]

(4) Yamani et al made an important generalization of the J -matrix method to any convenient L2-basis, without any significant loss of the advantages provided bythe method [14].

(5) Vanroose et al [15] enhanced the method, especially in the treatment of longrange potentials, by introducing additional terms in the three-term recursionrelation that takes into account the asymptotic behavior of the potential.(6) Alhaidari et al [16] present an alternative, but equivalent, approach to the regu-

larization of the reference problem in the J -matrix method It is an integration

approach, which was found to be more direct and transparent than the classical

differential approach They also developed the relativistic J -matrix method of

scattering for spin 12Dirac projectile with position-dependent mass [17].The contributions in the present volume are aimed at giving a brief account ofrecent developments, and some selected applications, of the method in atomic andnuclear physics:

In Part II, convergence issues are revisited by Igashove who makes a hensive study of the convergence of the Fourier expansion of the wavefunction inthe oscillator basis After investigating the effects of the regularization procedure onconvergence, Broeckhove et al propose an alternative regularization approach in the

compre-J -matrix method and demonstrate the resulting improvements On the other hand,

a method for the accurate evaluation of the S-matrix for multi-channel analytic and non-analytic potentials in complex L2 bases is presented in the same Section by

Yamani and Abdelmonem Using the tools of the relativistic J -matrix, Alhaidari

obtains analytic expressions for the scattering phase shift of separable potentialswith Laguerre-type form factors Shirokov and Zaytsev study an interesting problem

that is best addressed in the language of the J -matrix They show that non-local

in-teraction models could result in “isolated bound states” embedded in the continuumwith positive as well as negative energy

In Part III, Knyr et al use the J -matrix method as a universal approach for the

description of the process involving the ionization of atoms They succeed in dressing the difficult problem of correctly describing the continuous spectrum eigen-

ad-functions in the scattering of three charged particles using the J -matrix method Papp exploits the J -matrix structure of the Coulomb Green’s matrix to solve the

Faddeev-type integral equations of the three-body Coulomb problem In the cle by Yamani and Abdelmonem, three approximation methods are proposed thatendow the complex scaling method with the ability to compute resonance partialwidths for multi-channel problems

arti-In Part IV, Lurie and Shirokov give a review of their recent work on the

three-body loosely-bound nuclear systems within the J -matrix approach with an extended

oscillator basis They apply their investigation to11Li and6He nuclei Furthermore,

Shirokov et al construct the nucleon–nucleon interaction by means of the J -matrix

version pertaining to inverse scattering theory, where they eliminate the problem ofambiguity of the interaction by postulating tridiagonal and quasi-tridiagonal forms

of the potential matrix In the same Section, Arickx et al use their proposal of

modifying the J -matrix method (to account for coupling to long range interactions)

to cluster descriptions of light nuclei The method is applied to6He and6Be nucleifor scattering and reaction problems Finally, in Part V, Johnson and Holder present

a generalization of the density functional theory, which is widely used in chemicalphysics applications, using a theoretical framework whose structure parallels that of

the J -matrix method.

It is the hope of the editors that this volume will provide the interested reader

with enough material for him or her to appreciate the advantages of the J -matrix

method, and to encourage its use and development into a viable method for ical calculation of nuclear and atomic systems

References

1 A U Hazi and H S Taylor, Phys Rev A 2, 1109 (1970)

2 W P Reinhardt, D W Oxtoby, and T N Rescigno, Phys Rev Lett 28, 401 (1972);

T S Murtaugh and W P Reinhardt, J Chem Phys 57, 2129 (1972)

3 C Schwartz, Ann Phys (NY) 16, 36 (1960)

4 E J Heller and H A Yamani, Phys Rev A 9, 1201 (1974); H A Yamani and L Fishman,

J Math Phys 16, 410 (1975)

5 A M Lane and R G Thomas, Rev Mod Phys 30, 257 (1958); A M Lane and D Robson, Phys Rev 178, 1715 (1969)

6 E J Heller, Phys Rev A 12, 1222 (1975)

7 H A Yamani and M S Abdelmonem, J Phys A 26, L1183 (1993); 27, 5345 (1994); 28,

2709 (1996); 29, 6991 (1996)

8 E J Heller and H A Yamani, Phys Rev A 9, 1209 (1974)

9 I Bray and A T Stelbovics, Phys Rev Lett 69, 53 (1992)

10 J T Broad and W P Reinhardt, J Phys B 9, 1491 (1976)

11 H A Yamani and M S Abdelmonem, J Phys B, 30, 1633 (1997); 30, 3743 (1997)

12 P Horodecki, Phys Rev A 62, 052716 (2000)

13 A D Alhaidari, H A Yamani, and M S Abdelmonem, Phys Rev A 63, 062708 (2001)

14 H A Yamani, A D Alhaidari, and M S Abdelmonem, Phys Rev A 64, 042703 (2001)

15 W Vanroose, J Broeckhove, and F Arickx, Phys Rev Lett 88, 010404 (2002)

16 A D Alhaidari, H Bahlouli, M S Abdelmonem, F Al-Ameen, and T Al-Abdulaal, Phys.

Lett A 364, 372 (2007)

17 A D Alhaidari, H Bahlouli, A Al-Hasan, and M S Abdelmonem, Phys Rev A 75, 062711

(2007)

Part I Two of the Original Papers

New L2 Approach to Quantum Scattering: Theory 3Eric J Heller and Hashim A Yamani

J-Matrix Method: Extensions to Arbitrary Angular Momentum

and to Coulomb Scattering 19

Hashim A Yamani and Louis Fishman

Part II Theoretical and Mathematical Considerations

Oscillator Basis in the J-Matrix Method: Convergence of Expansions,

Asymptotics of Expansion Coefficients and Boundary Conditions 49

S.Yu Igashov

Scattering Phase Shift for Relativistic Separable Potentials

with Laguerre-Type Form Factors 67

A.D Alhaidari

Accurate Evaluation of the S-Matrix for Multi-Channel Analytic

and Non-Analytic Potentials in Complex L2 Bases 83

H.A Yamani and M.S Abdelmonem

J-Matrix and Isolated States 103

A.M Shirokov and S.A Zaytsev

On the Regularization in J-Matrix Methods 117

J Broeckhove, V.S Vasilevsky, F Arickx and A.M Sytcheva

xiii

Part III Applications in Atomic Physics

The J-Matrix Method: A Universal Approach to Description

of Ionization of Atoms 137

V.A Knyr, S.A Zaytsev, Yu.V Popov and A Lahmam-Bennani

J-Matrix Green’s Operators and Solving Faddeev Integral Equations

for Coulombic Systems 145

Z Papp

The Use of a Complex Scaling Method to Calculate Resonance

Partial Widths 173

H.A Yamani and M.S Abdelmonem

Part IV Applications in Nuclear Physics

J-Matrix Approach to Loosely-Bound Three-Body Nuclear Systems 183

Yu.A Lurie and A.M Shirokov

Nucleon–Nucleon Interaction in the J-Matrix Inverse Scattering

Approach and Few-Nucleon Systems 219

A.M Shirokov, A.I Mazur, S.A Zaytsev, J.P Vary and T.A Weber

The Modified J-Matrix Approach for Cluster Descriptions

of Light Nuclei 269

F Arickx, J Broeckhove, A Nesterov, V Vasilevsky and W Vanroose

Part V Other Related Methods: Chemical Physics Application

A Generalized Formulation of Density Functional Theory with Auxiliary Basis Sets 311

Benny G Johnson and Dale A Holder

Index 353

Two of the Original Papers

New L Approach to Quantum

Scattering: Theory

Eric J Heller and Hashim A Yamani∗

Abstract By exploiting the soluble infinite tridiagonal (Jacobi)-matrix problem

generated by evaluating a zero-order scattering Hamiltonian H0 in a certain L2basis set, we obtain phase shift, wave functions, etc., which are exact for a full

Hamiltonian H in which only the potential V is approximated Only bound–bound (L2) matrix elements of the Hamiltonian and finite matrix manipulations are needed

The method is worked out here for s-wave scattering using Laguerre basis functions.

Kato improvement of the results and necessary generalizations to many channels aretreated

1 Introduction

In atomic and nuclear scattering, it is often desirable to use Slater (Laguerre) oroscillator (Hermite) basis functions This chapter is the first of several in which wepresent a new method for performing scattering calculations entirely with square-

integrable (L2) functions We develop techniques in which we attempt to take full

advantage of the analytic properties of a given Hamiltonian and also of the L2

basis which is used to describe the wave function Specifically, in what follows,

we develop the basic theory using Laguerre-type basis functions appropriate for

s-wave scattering In the following chapter [1], we will apply the method to

electron-hydrogen elastic s-wave scattering below the n = 2 threshold, and to inelastic

radial-limit scattering calculations above and below the ionization threshold

Our basic approach is to treat an uncoupled Hamiltonian H0exactly in the space

spanned by the complete L2basis The remaining part of the Hamiltonian (i.e., the

E.J Heller

Department of Chemistry and Physics, Harvard Univeristy, Cambridge, MA 02138, USA e-mail: heller@physics.harvard.edu

∗Supported by a fellowship from the College of Petroleum and Minerals, Dhahran, Saudi Arabia.

Reprinted with permission from:

E.J Heller and H.A Yamani, New L2 Approach to Quantum Scattering: Theory, Phys Rev A 9, 1201–1208, (1974) Copyright (1974) by the American Physical Society.

potential) is approximated to some desired degree of accuracy, Vapprox, such that the

resulting Hamiltonian H0+ Vapproxis also exactly soluble in the complete L2space.Phase shifts and cross sections can then be extracted from the resulting wave func-tionψ E This wave function has the desirable property of being an exact solution

to a well-defined scattering Hamiltonian If Vapprox is a good representation of theexact potential, and if second-order accuracy is desired, thenψ E may be considered

as a trial wave function in the standard variational formulas

By an exact solutionχ E to the Hamiltonian H0+ V , we mean of course,

(H0+ V − E) |χ E  = 0. (1)

In a space of complete L2functions{φ n }, where χ Eis expanded asχ E =∞0 b n φ n,

Eq (1) is equivalent to

m | (H0+ V − E) |χ E = 0 (2)

for all m = 0, 1, 2, ∞ For most potentials considered in scattering theory, it will

not be possible to satisfy Eq (2)

However, consider the basis set{φ m}∞

m=0such that

φ m (r ) = (λr)e −λr/2L1m(λr), (3)whereλ is a scaling parameter In Section 2.1, we show that by writing χ0

properties of the Jacobi matrix representation of H0in the L2basis play a central

role in our method For this reason we call our approach the Jacobi (or J -) matrix

method

In Section 2.2, we construct a solutionψ E for

m|H0+ V N − E|ψ E  = 0, m = 0, 1, , ∞ (5)

where V N is an N × N matrix representation of V in the set {φ n}, thus achieving

the goal of obtaining an exact solution to the Hamiltonian with an approximatingpotential In Section 3, we employ this wave function as a trial function in theKato variational formula [2] In Section 4, the necessary extension to multi-channelscattering is developed In Section 5, a brief discussion is presented

2 Potential Scattering

Our task in this section is to determine the coefficients b0

nof the expansion ofχ0

E in

Eq (4) in terms of our basis set{φ n } Substituting the expansion for χ0

E in Eq (4)results in an infinite matrix problem for the set{b0

2x s0− s1= 0, for n = 0. (8b)Equation (8a), being a second-order difference equation, naturally has two lin-early independent solutions However, Eq (8b) provides a boundary condition and

thereby completely determines the s n’s Equation (8a) is the recursion relation isfied by the Chebyschev polynomials [3], and Eq (8b) gives us those polynomials

sat-of the second kind Therefore, we may write



s n



in x A similar analysis of H0in the basis{φ n} has been provided by Schwartz [4]

The expression forχ0

Since we have now solved Eq (4) exactly in the basis set, it is not surprising that

the s n ’s are simply the expansion coefficients of sinkr (with E = 1

2k2) in terms oftheφ n ’s [5] Note that although we have used a discrete (L2) basis, H0nonethelesshas a continuous spectrum This stems from the fact that the set{φ n} is infinite and

behaves as coskr when r → ∞ Since the c n’s form an independent solution to

Eq (8a), they satisfy the following equation:

2x c n − c n−1− c n+1 = 0, n ≥ 1 (8c)with the boundary condition

The requirement that ˜C (r ) → cos(kr) as r → ∞ means that β = −1 With this

value forβ, it is easily verified that c n = − cos(n + 1)θ satisfies Eqs (8c) and (8d).

Therefore, ˜C(r ) now reads

2.2 Adding an Approximating Potential

One way to introduce an approximation to V is to truncate the representation of V

in the basis{φ n } to an N × N matrix; we call this new potential V N:

•

•

0

—00

approach the problem from a different viewpoint, noting that V N couples only the

first N functions φ m , m = 0, 1, , N − 1, to each other Thus outside the space

spanned by these N basis functions, we expect the sine-like and the cosine-like

solutions, derived in Section 2.1, to be valid Therefore we write our solution as

ψ = ˜⌽ + ˜S + t ˜C, (17)

where ˜⌽ =N−1

n=0 ˜a n φ n , ˜S is the sine-like expansion χ0

E of Eq (10), and ˜C is the

cosine-like solution of Eq (13) The unknown coefficient t, then, corresponds to the tangent of the phase shift caused by V N Since the ˜a n’s are yet to be determined, we

can absorb the first N terms in the expansion of ˜S and ˜ C into ˜a n’s, writing

ψ E = ⌽ + S + tC, (18)where

four cases: first, the N − 1 conditions arising from m = 0, 1, , N − 2; second,

the case for m = N − 1; third, the condition for m = N; and last, the remaining set

of conditions arising from m = N + 1,N + 2, , ∞.

The first case leads to the N− 1 equations

So far we have (N + 1) equations in (N + 1) unknowns It would seem that

we are left with an infinite number of equations arising from case four with nocorresponding unknowns Therefore, if we claim thatψ E = ⌽ + S + tC is an exact

solution for H0+ V N , then the remaining case-four equations, for m = N + 1, N +

m|H0+ V N − E m | (H0− E) |S + tC (21)

if m ≥ N + 1 Equation (21) follows from the fact that V N is defined to be zero in

this region of Hilbert space, and because (H0− E) is tridiagonal in the basis {φ n},

and therefore does not connect the N terms in the expansion of ˜ ⌽ or the first N

terms in the expansion of ˜S and ˜ C with φ m for m ≥ N + 1 Furthermore, for each

m ≥ N + 1 the right-hand side of Eq (21) leads to the three-term recursion relation

(8a) and (8c) for the coefficients c n and s n Therefore the right-hand side of Eq (21)

vanishes identically Thus, we now have exactly (N+ 1) equations to determine the

(N + 1) unknowns {a n , t} Hence the form (18) for ψ E is indeed capable of giving

an exact solution to Eq (15)

Equations (20) can be written in matrix form as

column shown and with the right-hand side driving term Equation (22) can be

im-mediately solved for t by standard techniques An illuminating formula for tan δ = t can be obtained by pre-diagonalizing the inner N × N matrixH0+ V N − E

with the energy-independent transformation⌫, where

˜

⌫H0+ V N − E⌫nm = (E n − E) δ nm (23)Augmenting⌫ to be the (N + 1) × (N + 1) matrix

(E m − E) In arriving at Eq (25), we have used the

fact that s n = sin(n + 1)θ and c n = − cos(n + 1)θ Note that the entire energy

dependence of the phase shift is given analytically by Eq (25) It is interesting that

at the N Harris eigenvalues [7] E m, tanδ becomes simply

tanδ(E m)= tan(N + 1)θ(E m) (26a)

Also at the N − 1 points E μwhereν(E μ)= 0, we have

tanδ(E μ)= tan Nθ(E μ). (26b)

Equation (28) is just the distorted-wave Born formula, where V N is the distorting

potential and V Ris the perturbation which has been excluded from the calculation oftanδ t In order to perform the integral in Eq (28), bound–free and free–free matrixelements of the potential are required An approximation to the Kato correction (28)

which involves only bound-bound matrix elements of V is considered in the next

chapter [1]

4 Many-Channel Scattering

In this section, we extend the previous potential scattering formulas to allow sion with targets possessing internal states Basically we will be treating the close-

colli-coupling equations, employing an s-wave Laguerre set to describe the projectile

wave function in each channel As in the close-coupling formalism, we can treatexchange by the addition of a nonlocal potential

Assuming the target possesses coordinates which we collectively call ρ, the

Schr¨odinger equation for the many-particle wave function⌰ reads

If the target has a dense or continuous spectrum, the method of pseudo-target statesmay be employed [8] This is done in the following chapter for the case of electron-hydrogen scattering

As in the one-particle case, we will not be able to solve Eq (29) for⌰ exactly in

the Hilbert space which is spanned by the set

forα, α ≤ N c and n ≤ N α − 1, n ≤ N α − 1 We define ˜V αα

nn ≡ 0 otherwise The

number N α is the truncation limit in the channelα and N c is the total number ofchannels which are allowed to couple

To determine an S matrix, we will need N0independent solutions⌰αof Eq (29),

where N0is the number of open channels In the same spirit as in the single-channelcase, we expand⌰αas

The quantities k α are the channel momenta k α = √2|E α − E| where E α is the

channel energy appearing in Eq (30) Analogous to the one-channel case, the S α’s

and C α’s for open channels (see below for closed channels) are

α =1N α The remaining unknowns R αα are of course the elements

of a reactance matrix which may be used to determine the S matrix as

S = (1 + i[R]) (1 − i[R])−1, (37)

where [R] is the N0× N0open channel part of R.

The sum in Eq (34) should formally be extended to infinity, but since ˜V ≡ 0

for any channelα > N c , R αα vanishes forα or α > N c Actually the sum needonly be over open channels But if we do things this way, we place the burden ofdescribing the exponentially decaying closed-channel asymptotic behavior on theinternal function⌽α Near the threshold of channelα = N0+1, however, the decay

will be so slow that⌽α will be incapable of properly describing the asymptotic

behavior Fortunately, we can allow the sum to include the closed channels up to N c because we can find functions C α (r ) that can describe the asymptotic behavior in

the closed channels in the same way that C α (r ) and S α (r ) do for open channels The

proper asymptotic form in a closed channelα is e −|k α |r A function which has thisasymptotic form can be obtained by combining the expressions for ˜S α and ˜C α for

The form (38b) is really not needed, since for k α > λ α

2 the decay is rapid enough

to be described by⌽α

We wish to show that⌰ of Eq (34) is capable of describing the exact solution

for the problem

We will not consider the four cases for m as is done in the single-channel case, but show instead that for m > N β the left-hand side of Eq (41) also vanishes

identically First, the potential term vanishes by the definition of ˜V for m > N β.

m and the first N βfunctionsφ(β)

n Then Eq (41) becomes

But this is automatically true for m > N β in direct analogy with the single-channel

case, because of the three-term recursion relation satisfied by the coefficients of S β and C β The remainder of the equations; i.e., Eq (41) for m ≤ N β, lead to the samenumber of conditions as there are unknowns The totality of these equations can be

organized in a similar fashion as done in the form (22), with the L2matrix elements

of ˜V appearing in a large inner block One extra row and column are added to this

block for each channelα ≤ N c The right-hand side driving term and the solution

“vector” containing the a βn α ’s and R αα ’s have as many columns as open channels

The R matrix can then be obtained by solving the resulting linear equations As

before, the calculation may be facilitated by a pre-diagonalization of the inner blockusing an energy-independent transformation

ization of R-matrix theory [6], we have divided the Hilbert space into two

func-tion spaces We have an inner space consisting of those funcfunc-tions coupled by

is already solved and consists of the remaining functions (φ n , n = N, , ∞) It is interesting to compare the Wigner R matrix which has the form

eigen-H0is tridiagonal and is also treated exactly Other types of basis sets can also be

used in the R-matrix method In general, however, the ability to account for H0,exactly is lost The same is true in the present method, in the analogous situationwhen other basis functions not belonging to the tridiagonal set{φ n} are used

As in the R-matrix approach, we expect to find no Kohn-type pseudo-resonances [11] appearing in computed cross sections Because in some sense V N uniformly

approximates V , we expect ψ N to uniformly approximateψ as N → ∞ A series

of ever denser, but narrower pseudo-resonances as found by Schwartz in this limitseems rather unlikely This conjecture has been borne out by extensive calculations,some of which appear in the next chapter [1]

It is interesting to note that our truncated potential V N leads to a separable kernel

[12] in the T -matrix equation

T = V + V G0T

It is easy to see that T possesses non-vanishing matrix elements only in the same

L2subspace as that of V N This means that we can solve the finite matrix problem

a less convenient algorithm The kernel of Eq (43a) is energy dependent and must

be regenerated, and the linear equations must be resolved, at each new value of thetotal energy

Finally, we compare the Jacobi-matrix approach to the recently developed L2Fredholm techniques [13–15] Both approaches enjoy the advantage of requiring

only L2 matrix elements of the potential The L2 Fredholm method employs thedevices of analytic continuation [13], dispersion correction [14], and contour rota-tion [15] These techniques can be viewed as supplying, in an approximate fashion,

information about the continuous spectrum of H0which is not explicitly contained

in a finite L2 matrix representation Unfortunately, the amount of information

concerning H0that can be extracted decreases with the number of basis functionsper channel, and this can cause difficulties when basis size is a restriction On the

other hand, approximate treatment of H0 can be advantageous when for examplechannel threshold details are of no interest or are unwanted artifacts of particular

models In the J -matrix method, H0is accounted for exactly independent of basissize Thus we start with a large part of the problem “diagonalized” and the full

analytic structure of the S matrix is built into the problem, raising the hope that

quite small basis sets will be sufficient for many problems The analytic nature ofthe solutions allows variational corrections to be made and provides a solid footingfor further theoretical work

We now summarize the steps necessary to perform a calculation with the J -matrix method First, the potential V N (or ˜V ) is evaluated in the Laguerre basis set; and is

then added to the N × N tridiagonal representation of H0− E To this inner matrix

we add one extra row and column, for each asymptotic channel, containing matrix

elements of H0and the cos(n + 1)θ terms The right-hand side “driving” terms are

similarly constructed with the sin(n + 1)θ terms The resulting linear equations can

be solved efficiently if a pre-diagonalizing transformation⌫ is applied to the inner

matrix as in Section 2 If desired, the matrix elements of H0+ V N − E can be

eval-uated in the Slater set (λr) n e −λr/2, n = 1, 2, , N, since these are just transformed

Laguerres Then a different transformation⌫ will be necessary to pre-diagonalizethe inner matrix

In the following chapter we apply the method presented here to s-wave

electron-hydrogen scattering model The generalization of the method to all partial waves forboth Laguerre and Hermite basis sets has been derived and will be the subject of

a future publication The case where H0, contains the termα/r (i.e., the Coulomb

case) is also worked out for Laguerre sets

Acknowledgments We are grateful for helpful discussions with Professor William P Reinhardt

and his support of this work We have also benefited greatly from conversations with L Fishman,

A Hazi, T Murtaugh, and T Rescigno This work was supported by a grant from the National Science Foundation.

References

1 E.J Heller and H.A Yamani, following paper, Phys Rev A 9, 1209 (1974).

2 T Kato, Progr Theor Phys 6, 394 (1951).

3 M Abramowitz and I.A Stegun, Handbook of Mathematical Functions (Dover, New York,

1965).

4 C Schwartz, Ann Phys (N.Y.) 16, 36 (1961).

5 E.J Heller, W.P Reinhardt, and Hashim A Yamani, J Comp Phys 13, 536 (1973).

6 H Feshbach, Ann Phys (N.Y.) 19, 287 (1962).

7 F.E Harris, Phys Rev Lett 19, 173 (1967).

8 P.G Burke, D.F Gallaher, and S Geltman, J Phys B 2, 1142 (1962).

9 See for example, A.M Lane and R.G Thomas, Rev Mod Phys 30, 257 (1958); A M Lane and D Robson, Phys Rev 178, 1715 (1969).

10 P.J.A Buttle, Phys Rev 160, 719 (1967); P.G Burke and W.D Robb, J Phys B 5, 44 (1972); E.J Heller, Chem Phys Lett 23, 102 (1973).

11 C Schwartz, Phys Rev 124, 1468 (1961) For a comprehensive review of the algebraic ods for scattering, including a discussion of pseudo-resonances, see D.G Truhlar, J Abdallah,

meth-and R.L Smith, Advances in Chemical Physics (to be published).

12 R.G Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966),

p 274.

13 W.P Reinhardt, D.W Oxtoby, and T.N Rescigno, Phys Rev Lett 28, 401 (1972); T.S Murtaugh and W.P Reinhardt, J Chem Phys 57, 2129 (1972).

14 E.J Heller, W.P Reinhardt, and H.A Yamani, J Comp Phys 13, 536 (1973); E.J Heller, T.N.

Rescigno, and W.P Reinhardt, Phys Rev A 8, 2946 (1973); H.A Yamani and W.P Reinhardt

Phys Rev A11, 1144 (1975).

15 T.N Rescigno and W.P Reinhardt, Phys Rev A 8, 2828 (1973); J Nuttall and H L Cohen, Phys Rev 188, 1542 (1969).

Angular Momentum and to Coulomb ScatteringHashim A Yamani∗and Louis Fishman

Abstract The J -matrix method introduced previously for s-wave scattering is

extended to treat theth partial wave kinetic energy and Coulomb Hamiltonians within the context of square integrable (L2), Laguerre (Slater), and oscillator(Gaussian) basis sets The determination of the expansion coefficients of the con-

tinuum eigenfunctions in terms of the L2 basis set is shown to be equivalent to thesolution of a linear second order differential equation with appropriate boundaryconditions, and complete solutions are presented Physical scattering problems areapproximated by a well-defined model which is then solved exactly In this manner,the generalization presented here treats the scattering of particles by neutral andcharged systems The appropriate formalism for treating many channel problemswhere target states of differing angular momentum are coupled is spelled out in

detail The method involves the evaluation of only L2 matrix elements and finitematrix operations, yielding elastic and inelastic scattering information over a con-tinuous range of energies

1 Introduction

In two previous publications [1, 2] (referred to as I and II) the J -matrix (Jacobi

ma-trix) method was introduced as a new approach for solution of quantum scatteringproblems As discussed in I, the principal characteristics of the method are its use

H.A Yamani

Ministry of Commerce & Industry, P.O Box 5729, Riyadh 11127, Saudi Arabia

e-mail: haydara@sbm.net.sa

∗Supported by a fellowship from the College of Petroleum and Minerals, Dhahran, Saudi Arabia.

Reprinted with permission from: J-matrix Method: Extension to Arbitrary Angular Momentum

and to Coulomb Scattering, H.A Yamani and L Fishman, Journal of Mathematical Physics 16, 410–420 (1975) Copyright 1975, American Institute of Physics.

A.D Alhaidari et al (eds.), The J-Matrix Method, 19–46. 19

C

Springer Science+Business Media B.V 2008

of only square integrable (L2) basis functions and its ability to yield an exact tion to a model scattering Hamiltonian, which, in a well-defined and systematicallyimprovable manner, approximates the actual scattering Hamiltonian The method isnumerically highly efficient as scattering information is obtained over a continuousrange of energies from a single matrix diagonalization.

solu-The development of the J -matrix method as presented in I is based primarily upon the observation that the s wave kinetic energy,

in the above basis is tridiagonal (i.e., J or Jacobi

matrix) and that the resulting three-term recursion scheme can be analytically solved

yielding the expansion coefficients of both a “sine-like” ˜S(r ) and a “cosine like”

˜

C(r ) function The J -matrix solutions ˜S(r ) and ˜ C(r ) are used to obtain the exact

solution of the model scattering problem defined by approximating the potential V

by its projection V N onto the finite subspace spanned by the first N basis functions.

That is, the exact solution⌿ of the scattering problem,



H0+ V N − k2

2

is obtained by determining its expansion coefficients in terms of the basis set{φ n}

subject to the asymptotic boundary condition,

⌿ → ˜S(r) + tan δ ˜C(r), (4)whereδ is the phase shift due to the potential V N

.This chapter is intended to generalize the formalism developed in I in three areas.First, the results of I are extended to all partial waves, in which case the uncoupledHamiltonian becomes theth partial wave kinetic energy operator,

in the Laguerre function space of Eq (6), which again yields a Jacobi form and

is subsequently analytically soluble It is noted that the analysis of the CoulombHamiltonian in the oscillator set of Eq (7) does not lead to a Jacobi form

For the solution of these problems, a general technique is developed which duces the solution of the infinite recurrence problem for the asymptotically “sine-

re-like” J -matrix eigenfunction to the solution of a linear second order differential

equation with appropriate boundary conditions An asymptotically “cosine-like” lution which obeys the same differential equation with different boundary conditions

so-is then constructed The fact that both J -matrix solutions obey the same recurrence

scheme is essential to the success of the method as an efficient technique for solvingscattering problems [1]

The program of the chapter is as follows: In Section 2.1, the generalized H0

problem is considered and a general procedure for obtaining the expansion cients of the sine-like and cosine-like functions in terms of the basis sets is outlined

coeffi-In Section 2.2, the general method is illustrated in detail for the case of the radialkinetic energy in a Laguerre basis The analogous results for the oscillator basisand for the Coulomb problem are outlined in Sections 2.3 and 2.4, respectively Thedetails of the Coulomb derivation are given in the Appendix Section 3 contains theapplication of the results thus obtained to potential scattering problems This sec-tion presents a formula which allows for the computation of phase shifts Section 4

presents the natural generalization of the J -matrix method to multi-channel

scatter-ing Finally, Section 5 contains a brief discussion of the over all results and tions for applications and areas of further theoretical interest

within the framework of the L2function space{⌽n} in such a manner as to obtain

both an asymptotically sine-like and asymptotically cosine-like function The two

J -matrix solutions, ˜S(r ) and ˜ C(r ), form the basis for the asymptotic

representa-tion of the scattering wavefuncrepresenta-tion associated with the full problem It will also

be required that the expansion coefficients of both ˜S(r ) and ˜ C(r ) satisfy the same

three-term recursion scheme,

whereξ = kr −π2 in the free particle case andξ = kr +tln(2kr)−π2+σ in

the Coulomb case In the above, the definitions t = −zk and σ  = arg ⌫(+1−it)

have been used [3]

Since the basis set{φ n} is complete for functions regular at the origin, ⌿0

reg can

be expanded as

⌿0 reg≡ ˜S(r) =∞

, the{s n} satisfy a three-term

recur-sion relation of the form,

whereη is the energy variable defined by η = kλ and g(η) is a function dependent

upon the particular choice of{φ n } and H0 Differentiating Eq (14) with respect to

x , where x is a function of the energy variable η appropriate to the particular case,

leads to a differential difference equation of the form

For the case of the Laguerre function space, b 2n = 0, while for the oscillator function

space the general form of Eq (16) is appropriate Combining Eqs (15) and (16)yields a linear second order differential equation of the form

A(x ) d

2s n

d x2 + B(x) ds n

d x + D(x)s n = 0 (17)with two linearly independent solutionsχ1andχ2and a general solution of the form

s n = α1χ1+ α2χ2. (18)

Equation (15b) determines s n to within a normalization constant The advantage

of the differential equation approach is that a cosine-like solution ˜C (r ), whose

ex-pansion coefficients will also satisfy the differential equation (17), can be readilyconstructed

The cosine-like J-matrix solution,

irreg, and (3) to have its sion coefficients{c n } satisfy Eq (15b) This immediately means that ˜C(r) cannot

expan-satisfy the homogeneous differential equation (10) By choosing ˜C(r ) to satisfy the

inhomogeneous differential equation



(21)may be used to obtain the solution [5]

reg(r )

 ∞

r

dr ⌿0 irreg(r ) ¯φ0(λr )$

,

(22)

irregand is independent of r and β is a free parameter [6] ˜C(r) as given by Eq (22)

is regular at the origin and with the choice

where l( η) is a function which depends upon the form of β and upon any terms that

were divided out in the derivation of the homogeneous recursion relation, Eq (15).Equation (24) is to be contrasted with the homogeneous initial condition

which occurs in the sine-like J -matrix solution It may be shown from Eqs (22)

and (23) that the set{c n} satisfies a differential difference equation analogous to

Eq (16), which when combined with Eqs (15a) and (24) leads to the tial equation (17) The application of the inhomogeneous initial condition given by

differen-Eq (24), and an additional boundary condition specific to the case being considered,determine the two integration constantsγ1andγ2in the solution

c n = γ1χ1+ γ2χ2. (25)

2.2 Radial Kinetic Energy: Laguerre Basis

For the case of the radial kinetic energy and a Laguerre basis, the detailed

construc-tion of the J -matrix soluconstruc-tions is given following the general technique out lined in

Section 2.1 The Hamiltonian is

m|H 0− k2

2  ˜C(r )

= 0, m = 1, 2, , ∞, (28b)are solved where

r→∞sin(kr − π2)+ tan δ cos(kr − π2), (30)

δ being the scattering phase shift.

From the boundary conditions of Eq (29), ˜S(r ) is designated the “sine-like”

J -matrix solution, and ˜ C(r ) the “cosine-like” J -matrix solution The sine-like

solu-tion is discussed in Secsolu-tion 2.2.1, where the recurrence relasolu-tion for the coefficients

{s n } is solved explicitly, giving closed form expressions The discussion of ˜C(r)

is somewhat more complex: In Section 2.2.2, a function ˜C(r ) with the appropriate

cosine-like behavior is constructed such that, for n > 0, the expansion coefficients

{c n } obey the same recursion scheme as the set {s n}; a fact that is an essential

in-gredient of the J -matrix method as will be seen in Sections 3 and 4 and has been

discussed in I and II

may be taken to be regular at r = 0 and sine-like asymptotically, that is,

functions L2n +1(λr) [4], be reduced to the Jacobi (J-matrix) form

The expansion coefficients{s n } which satisfy the matrix equation J · s = 0 may be

determined by the solution of the three-term recursion relation

2x (n +  + 1)u n (x ) − (n + 2 + 1)u n−1(x ) − (n + 1)u n+1(x )= 0 (35a)with the initial condition

where

s (x )=⌫(n + 1)⌫(n + 2 + 2)u (x ) (36)

coeffi-In Section 2.2, the general method is illustrated in detail for the case of the radialkinetic energy in a Laguerre basis The analogous results for the. .. basisand for the Coulomb problem are outlined in Sections 2.3 and 2.4, respectively Thedetails of the Coulomb derivation are given in the Appendix Section contains theapplication of the results... reduced to the Jacobi (J-matrix) form

The expansion coefficients{s n } which satisfy the matrix equation J · s = may be

determined by the solution of the three-term