convergent close-coupling calculations of electron

PHYSICAL REVIEW A VOLUME 46, NUMBER 11 1 DECEMBER 1992

Convergent close-coupling calculations of electron-hydrogen scattering

Igor Bray*Electronic Structure of Materials Centre, School of Physical Sciences, The Flinders University of South Australia,

G.P.O. Box 8200, Adelaide 5002, AustraLia

Andris T. StelbovicsiCentre for Atomic, Molecular and Surface Physics, School of Mathematical and Physical Sciences,

Murdoch University, Perth 6150, Australia(Received 6 April 1992)

The convergence of the close-coupling formalism is studied by expanding the target states in anorthogonal L Laguerre basis. The theory is without approximation, and convergence is establishedby simply increasing the basis size. We present convergent elastic, 2s, and 2p differential crosssections, spin asymmetries, and angular-correlation parameters for the 2p excitation at 35, 54.4, and100 eV. Integrated and total cross sections as well as T-matrix elements for the first five partialwaves are also given.

PACS number(s): 34.80.Bm, 34.80.Dp, 34.80.Nz

I. INTRODUCTION

The electron-hydrogen scattering problem is an idealtesting ground for any scattering theory as this is theonly electron-atom scattering problem where the targetwave functions are known exactly. This relatively sim-ple problem continues to attract considerable interest asthere are still significant discrepancies between theoryand experiment.

The close-coupling equations for electron-hydrogenscattering, in principle, provide a complete description ofthe scattering process if they are solved without approx-imation and if correct thr""-body boundary conditionsare applied to the channels with three free particles inthe asymptotic region. They are derived by taking an ex-pansion of the electron-hydrogen atom wave function overthe complete set of target states. Because there are aninfinite number of discrete and continuum target states,methods must be devised in order to render the equationsnumerically soluble. One method which suggests itself isto replace the integration over the continuum states ofthe close-coupling equations by a numerical quadrature.The convergence of such a method can be determined byincreasing the order of the quadrature until scatteringamplitudes are stable to a specified accuracy, for exam-ple I'%%uo. This approach has two drawbacks Firstly, th. isquadrature still leaves an infinite sum over the discrete in-elastic states of the target which needs further treatment.Secondly, a quadrature formula applied in the most obvi-ous way will use regular Coulomb functions in the calcu-lation of the potentials. Numerical difficulties in calcu-lating integrals involving transitions between continuumchannels have made this an unattractive proposition todate.

There is, however, another approach to evaluating thesum over discrete and integral over continuum targetstates which treats them on an equal footing and is re-ferred to as the pseudostate method. A set of Lz func-

tions is chosen in which to diagonalize the target Hamil-tonian so that the wave functions for the channels overwhich we calculate T-matrix amplitudes are accuratelydetermined. The remaining states from such a diagonal-ization have negative and positive eigenenergies. The as-sumption implicit in this method is that, when the set offunctions is carefully selected, as their number increaseswithout limit, they form a basis for the Hilbert space ofthe target. Thus the expectation is that answers one ob-tains by solving the coupled-channel equations, when thesum and integral over the true target states are replacedby a discrete sum over the pseudostates, will convergewith sufficiently large basis sets.

In terms of testing the basic assumptions of the pseu-dostate method and understanding its theoretical jus-tification the work of several groups deserves mention.Early numerical calculations for the electron-hydrogenproblem utilizing pseudostates were carried out by Burkeand Webb [I]. They demonstrated that the inclusion ofa few pseudostates significantly reduced the cross sec-tions for scattering, bringing them into better agreementoverall with experiment. It was soon realized that withpseudostates one often has the problem that spuriousresonance features are introduced into the model crosssections. A study of the convergence properties of pseu-dostate sets was undertaken by Burke and Mitchell [2].They considered the model of electron-hydrogen scatter-ing that only treats states of zero orbital angular mo-mentum. Progressing systematically by including morestates, they concluded from their study of the singletL = 0 partial wave amplitude that the model calcu-lations were converging except in the neighborhood ofthe pseudoresonance features. This work while exploringthe numerical advantages of pseudostates did not pur-sue the deeper question of the nature of the quadraturerules that pseudostate target expansions induced, nor didthey attempt to establish the relationship between pseu-dostates and true target continuum states. The pioneer-

46 6995 1992 The American Physical Society

6996 IGOR BRAY AND ANDRIS T. STELBOVICS 46

ing work in this connection was carried out in a series ofpapers proposing a method commonly referred to as theJ-matrix method of atomic scattering [3]. Through theuse of a I aguerre function basis the nature of the quadra-ture could be explored. The quadrature rules were of aGaussian type and the underlying orthogonal polynomi-als were shown by Yamani and Reinhardt [4] to be thoseof the Pollaczeck class. They were also able to demon-strate the manner in which each pseudostate derived froma finite subset of the Laguerre basis is related to the con-tinuum functions; apart from an overall normalizationconstant, the pseudostates were the Fourier expansionsof the continuum functions truncated to the N-functionbasis set. The nature of the convergence was studied byStelbovics and Winata [5], who showed the convergencerate was very slow with the pointwise convergence beingof conditional type only.

An important advance in the theory of this model ofelectron-hydrogen scattering was provided by Poet [6],who solved it using a method which relied on the separa-ble nature of the potential of the three-body Schrodingerequation. The solutions of the differential equation were,as a result, obtainable analytically and the physicalthree-body scattering wave function could be constructedby matching linear combinations of the solutions to thephysical boundary conditions. The resulting equationswere simpler than those of the close-coupling formulationand could be solved to a high degree of accuracy. A latercomparison by Poet [7] confirmed the earlier conclusionsby Burke and co-workers concerning the convergence ofthe pseudostate method. Further studies involving basissets comprising Slater functions were undertaken by Ozaand Callaway [8, 9] for this model. They considered basissets of up to nine states and found that pseudoresonancebehavior was still present although the size and extentof the pseudothreshold behavior appeared diminished forthe largest basis. This conclusion was in the same veinas that deduced earlier by Heller and Yamani [10] in a10-state J-matrix calculation.

The work described so far left an important questionunanswered. Were pseudoresonances a necessary featureof the close-coupling method, which could only be elimi-

nated by some averaging technique as used in [8]? Brayand Stelbovics [ll] demonstrated this was not the case.By extending the Laguerre basis size to as many as 30states, they showed that close-coupling equations yieldconvergent results at all energies, for scattering and ion-ization channels, in complete agreement with Poet [6,7]. This is a most important result as it means thatclose-coupling equations are valid at all energies, and thatpseudoresonances are simply an indication of the lack ofconvergence in amplitudes as a function of the numberof basis states.

With this information we apply the close-couplingformalism using the Laguerre basis states to the full

electron-hydrogen problem, i.e., we treat the full set ofpartial waves. This leads to very-large-scale calcula-tions that require an expansion of the momentum-spacemethod of solution of the close-coupling equations used

by McCarthy and Stelbovics [12], upon which this workis based.

In Sec. II we present the close-coupling formalism andaddress the issue of nonunique solutions. In Sec. IIIwe show how to solve the Lippmann-Schwinger equationminimizing the amount of computational resources nec-essary for a particular basis set. In Sec. IV we present ourresults and compare with experiment and other theories,followed by conclusions in Sec. V.

II. THEORY

In this section we give a derivation of the close-couplingequations in operator form, with sufficient boundary con-ditions to ensure the uniqueness of the T-matrix coupledequations. The derived form carries over directly to theform applicable to finite-basis expansions of the targetstates.

The Hamiltonian for the nonrelativistic electron-atomic hydrogen scattering problem is given by

H = K&+v&+K2+v2+v3,

where Kq and K2 are, respectively, the projectile- andtarget-electron kinetic energy operators, and vq, v2,and vs are the projectile-proton, target-proton, andprojectile-target potentials.

The solution of the full Schrodinger equation

(E —H)l@s) = 0 (2)

requires the determination of a wave function possessingthe symmetry

(rq, rq) = (—1) 4 (rz, rq)= ( 1) P„@ (r—y, r2)) (3)

A. Exact target states

In the close-coupling method the wave function (3) isexpanded over the complete set, discrete and continuous,of target states P, (r) as

(4)

Here the states P, (r) are the exact eigenstates of the tar-get Hamiltonian with

(6)(~.+")l~,) =~, l~,).The functions fs(r), by consideration of (Pzgkl4' ) in

(4) and (5), must satisfy

(&~ lfl') = (—1)'(W~lf,'). (7)

In standard close-coupling formulations, see Ref. [13]for example, the symmetry condition (3) is imposed bymaking the expansion (4) manifestly symmetric by writ-

where S is the total spin and P„ denotes the space ex-change operator.

46 CONUERGENT CLOSE-COUPLING CALCULATIONS OF. . .

ing

+(-1) 4'(ri)F; (»)]. (8)

However, such an expansion does not lead to a uniquesolution for the Fs(r); if one takes a particular solutionof (8) [for example, the fs(r) is one such solution] thenso are

Fs(r) = fs(r) + ) c,,P, (r)

Using (13) we express (2) as

0 = (E —H)I&l@s)=(E—H)-,'[1+(—1) P„]IzlC ), (16)

where we introduced the operator [1+i—1)eP„]/2, which

commutes with H, to symmetrize l@ ) numerically. Ifl@s) already had the required symmetry then this oper-ator would be redundant. Without the condition (15) itamounts to the explicit symmetrization (8) which yieldsnonunique solutions.

Using Iz as a projection operator we write (16) as

0 = I2(E —H) [1 + (—1) P„]Izl4' )

for discrete i, and which we split to get

I,(E K, --K, -~,)I,le ) =I,V'r, ]e ), (18)

for continuous i, where the arbitrary constants cs satisfy

cs + (-1)'c,', = 0. (»)Note that the cs. are only defined for either i and j bothdiscrete or both continuous.

To avoid this problem of nonunique solutions for Fs(r)we impose the condition (7) to the manifold of solutions.It is seen this, in conjunction with (11) selects the so-lution Fs = fs We ar.e now in a position to deriveour close-coupling equations. We avoid summation overchannel indices throughout by using projections over thecomplete set of target states labeled in an obvious man-ner for coordinates of electrons 1 and 2. Defining theidentity operators Iq and I2 by

w ere

V = vs+vs+ (—1) (H —E)P„ (»)We impose condition (15) with the aid of (14) in the Eterm above,

( 1) EI2P—„I2l@ ) = ( 1) EIzI—qIqP„I3]@ )

= EI~Izl~')= (—1) E(1 —8)IzP„Iz]@ )

+E8I~I3]@s), (20)

where 8 is an arbitrary constant and incorporates condi-tion (15) for 8 g 0. So instead of using Vs in (18) weuse Vs(8) where

V'(8) = v, + ~, E8I, + (--1)s[H - E(1 8)]P„-

(21)

)/. I 4')(0

we write the multichannel expansion (4) as

(12) Rather than solving explicitly for the functions fs(r)we form the Lippmann-Schwinger equation for the T ma-trix

l~') = I21~')

and hence

(13) Ts (kglI3VS(8)I2]@s&yl)

where lkP) are the asymptotic states satisfying

(22)

P„l@s)= I&P„I2]4's). (14)

Writing Itl@ ) instead of l@s) explicitly indicates thatthe multichannel expansion for the complete wave func-tion is being used. The symmetry condition (7) can thenbe expressed more usefully as

Iz]@' ) = (—1) IqP„Iz]@ ). (»)

Iz(E —Kq —K2 —v2)I3]kp) = 0, (23)

and where the notation (+) indicates outgoing sphericalwave boundary conditions. Combining (18), (22), and(23) the T matrix for the transition of the target in stateP;, to state P; on impact of projectile ko is then givenby the Lippmann-Schwinger equation

(ky lTs]y k ) (ky lVs(8)]y k ) )I d3k ( O''I ( )14" )( O''I I4'o c)(24)

Any nonzero 8 will have implemented the symmetry con-dition (15), and leads to a unique answer independent of8. So even though the V-matrix elements have an ar-bitrary constant the solution of the integral equation isindependent of this constant. This is confirmed by ournumerical investigations.

Previous work done by McCarthy and Stelbovics [12]using momentum-space coupled channel formalism forelectron-atom scattering did not employ the symmetrycondition (15). This is equivalent to taking 8 = 0 above.The nonuniqueness manifested itself as instability in theoff-the-energy-shell T matrix only, and so did not cause

6998 IGOR BRAY AND ANDRIS T. STELBOVICS

great concern. Stelbovics [14] has shown analytically thatthis is indeed the case by studying the nature of the ho-mogeneous solutions of the 8 = 0 Lippmann-Schwingerequation (24). He then used this analysis to obtain newV-matrix elements, similar to (21), which yield unique so-lutions off the energy shell as well. Here we have derivedthe new form in a simpler way. Even though on-shell am-plitudes are unique even if e = 0, we find that numericalinstability occurs when a large number of channels is in-volved. This problem disappears whenever we introducea nonzero 8. It has been utilized in all our calculations,where we have taken 8 = 1/2.

B. Square-integrable states

In practice no numerical method for solving the cou-pled T-matrix equations in the form (24) has yet beenimplemented. The difficulty is the fact that in order tosolve this integral equation it must be closed by allow-ing the index i to run over the same complete range asi', which leads to singular V-matrix elements wheneverboth i and i' are in the continuum. In principle this isnot an insurmountable problem as there is an integralover these elements, but numerically this has proved tobe an unattractive option to date.

A number of approximate methods can be applied.The simplest is to truncate the range of i' to some fi-

nite set of bound states, thus ignoring the effect of otherbound states and the continuum. Unfortunately theireffect is important and their omission leads to noncon-vergent amplitudes. Alternatively one can use Feschbachformalism to include the effect of the truncated states byadding a complex nonlocal polarization potential which iscalculated from the truncated states. The latter has been

I

)/ P, (rz)f, (ri) = lim ) P; (rz)f; (ri).J i=1

(26)

With these definitions, the sum and integral in (12) andthe Lippmann-Schwinger equation (24) become a singlesum over N, with the target states and energies beingreplaced by IP,. ) and s, , respectively. So instead of Iiand Iz, we define

I"=).I 4"")(4,

N

I,"=) ly," )( &

and have

(27)

the major approach of Bray, Konovalov, and McCarthy[15] who have attempted to calculate the polarization po-tential ab initio with fewer and fewer approximations.

The approach that is taken in this work is to diago-nalize the hydrogen target Hamiltonian in a set of Ifunctions which when extended to completeness form abasis for the target Hilbert space. The use of I~ functionseliminates the problem of singular continuum-continuumV-matrix elements. Also most importantly, with a knownbasis the convergence of the expansions can be studiedin a systematic manner with increasing number of basisfunctions.

We introduce a finite set of N square-integrable statesIPN) which satisfy

(25)

and have the property

(kyNITSNIyNk ) (kyNIVSN(g)IyNk ) + ) dskj i I ( )I i' ( i' I I ip )E —e —k'~+ z0i/=1

for the physical T-matrix elements. The projection op-erator Ii is replaced by IiN in calculating the matrix el-ements of V (8) in (21), and Iz is replaced by I~ in

(23) which yields the sN energies in the Green's functionof (28).

The states IPN) are obtained by diagonalizing the tar-get Hamiltonian (25), for each l of the target electron, inthe Laguerre basis (A,.i(r) which has the form

A(k —1)!( (2l + 1 + A:)!

x exp( —Ar/2)I q'+, (Ar),

(A.)'+'

(3O)

where the I&+i (Ar) are the associated Laguerre poly-nomials, and I!- ranges from 1 to the basis size N~. The

where for the physical T-matrix elements of interest wemust have lg, ) = IPN) and lg, , ) = IPN) to sufficientlyhigh precision. With these definitions we have

(ky, lTsly, .k, ) = h (kyNITsNIy, ".k, )

I

constant A is arbitrary and is chosen so that the lowest

energy states are essentially the exact hydrogen eigen-states. For convenience take A = 2 for all l. This givesthe exact 1s state from the diagonalization for No ) 1.The rate of convergence to other exact hydrogen boundstates for this A as a function of Ni has been given byBray, Konovalov, and McCarthy [15]. We have estab-lished that convergence is independent of the value of A

in [11], though the rate of convergence would certainlybe affected by choosing an inappropriate value.

This choice of basis we consider to be very important.Unlike the Slater basis, the Laguerre basis is orthogonal,and so does not suffer from any linear dependence prob-lems as the basis size is increased. With this basis we areable to obtain as many as 100 orthonormal states upondiagonalization of the Hamiltonian for each /. Thus, thisbasis is ideal for convergence studies. Furthermore, thenature of the quadrature in (26) has been studied in de-

tail by a number of authors, see Refs. [4, 16, 17] for exam-

ple. It is a Gaussian-type quadrature and the underlying

orthogonal polynomials are of the Pollaczeck class. It

46 CONVERGENT CLOSE-COUPLING CALCULATIONS OF. . . 6999

can be shown that the weights of the negative energyL2 states converge to unity in Eq. (26) in the limit oflarge N and that limN ~P, = P;, limN ~ f, = f,This ensures that the limiting procedure (29) gives thecorrect T-matrix amplitudes (24) for the transitions tothe 1s, 2s, and 2p levels. We also mention for complete-ness (although it has no bearing on the calculations re-ported here) that the positive energy L2 states cannotconverge to the non-L2 regular Coulomb functions. Itcan be shown [4, 5] that the Gaussian weight functiontogether with the positive-energy weights in the quadra-ture (26) define a renormalization of the Lz states so thatthey approximate the true continuum functions for thefirst few oscillations in coordinate space at each positivepseudostate energy. A proper consideration of such nor-

I

malizations would be required if one was to use the L'~

approximated T-matrix equations to model difFerentialionization cross sections.

III. SOLUTION OF THELIPPMANN-SCHWINGER EQUATION

We shall omit the bars as well as explicit N~ depen-dence in the V-matrix elements to simplify the notation.Subsequent discussion refers only to the Lz states, andthere should be no confusion with the exact target eigen-states.

The partial wave Lippmann-Schwinger equation corre-sponding to (28) for the reduced T-matrix elements is

(Lk~&tn II T II notpkpLp) = (Lk~&~n II V (6) II notokoLo)

) dk'k' " (L'k't'n'll

Ts~II nplpk L ) (31)

A ) j

where the projectile is denoted by linear momentum k and orbital momentum L, while the target state is denoted byprincipal quantum number n and orbital angular momentum t. The total orbital angular momentum is denoted byJ, and

E = s„,~, + kp/2 = s„~ + k„&/2 (32)

is the on-shell energy. The V-matrix elements are evaluated using (21).It is our aim to perform very large multichannel calculations so it is important to reduce the amount of computer

resources necessary to solve (31). Instead of solving (31) directly, which involves complex T-matrix elements, werewrite it as

(L4i&n II T II notokoLo) = (L4«n II V (8) II nptpkpLp)

NIi V" n' ' 'L'+) ) P dk'k"( "'"ll () II" )(L'k't'n'IIT" lln & k L)&00 0 0l' L' n'=1 &n'l'—

NL I

—i~ ) .k~ i (Lk~(tn II V"(e) IIn't'k~ i L')(L'k t

t'n'llT"

II notokoLo),

(33)

where the symbol P indicates that the integral is of the principal value type, and k„ i is defined for 1 & n' & NP ( N~for which

k„i = /2(E —s„i ) (34)

is real. In this case we say that the channel n't, ' is open (there are ¹&,of these), and if E ( s„ t we say that thischannel is closed. We can use purely real arithmetic if we introduce the K-matrix formulation by letting

N

(Lk„tin II K II nplpkpLp) = ) ) (Lk„)in II T IIn't'k„ t L')

x (6i i,6r, r.„6~~, + ivrk~ l'(L'k&'t'f n'll K II nolokoLp)) .

(35)With this definition (33) becomes

(Lk i« II K II no&okoLo) = (Lk t« II V (8) II noto"oLo)

+) ) I dk'k" ( "' (L'k't' 'IIK" II t k L ) (36)t', L' n'= j E —e ~)~ —k'~ 2n

IGOR BRAY AND ANDRIS T. STEI.BOVICS 46

dk'(4~ i Ik')(k'~4~ ~ ) (38)

is satisfied for each n' = 1 to Laguerre basis size Np tothree significant figures.

Denoting the combinations of Lk„~ln by f, Lokolsnoby i, and L'k'l'n' by n we write (36) as

KsJ Vs J(8) + ) ut VsJ(8)KsJ (39)

where the single sum over n contains the sum over N„ t

quadrature points kI". ' in (37), and the single index onI I

m„ indicates the complete range of m". ' . To solve thisequation we form a closed set of linear equations by let-ting f run over the same range as n. We replace f by n'to indicate this and have

V„; (8) = ) b„„—ur„V„„(8) K„,

= ) 6„„/m„—V„„(8) m„K„, . (40)

As both b„„/m„and VsJ(8) are symmetric on inter-change of n and n' we need to solve the linear system ofthe form AX = B, where A is a real symmetric matrix.

which is solved for the K matrix using real arithmetic,and the T matrix is obtained by solving the much smallerset of equations (35).

There are a number of methods for calculating princi-pal value type integrals. McCarthy and Stelbovics [12)used an on-shell subtraction to remove the singularity,and so they used a single quadrature rule to representthe integral over k' in (36). This method does not workvery well when there is a large number of channels. Thisis due to the fact that as the singularity is difFerent foreach n'l' one of the quadrature points often falls near thesingularity resulting in a very large subtraction whichloses precision. We get around this difficulty by choosinga different quadrature rule for each n'l' Fu. rthermore,rather than using a subtraction, we simply ensure thatfor each open channel n'l' the singularity is exactly inthe middle of an interval which contains an even numberof Gaussian points. Thus, for some arbitrary function I'we write

F k" Ndk'k' = ) m" ' F(k" '

)E —s„( —k'z/2 j=1

(37)I ~lwhere the weights m" ' contain Gaussian-type weights as

well as the Green's function. In practice we take N„ t

to be the same for each channel n'l' In our. calculationsour largest Laguerre basis size has N~ = 13 (see next sec-tion), in which case we find that we require the number ofquadrature points N„Ill to be as big as 55. It is generallythe case that the larger the basis size the more quadraturepoints are necessary. This is unfortunate as it means thatthe size of the matrices grows more rapidly than wouldbe the case if only the basis size was increased. To de-termine approximately how many quadrature points arenecessary we check that the identity

We do this using LAPACK routines [18] that store A incompact form. This way we use the minimal amount ofstorage space. The usage of real arithmetic and symmet-ric matrices reduces the storage by a factor of 4. This isinvaluable as for our 70-channel calculations (see below),the matrix A occupies —30M of storage [(55x 70)2 x 4/2],where 4 is the number of bytes required to store a realnumber, and we divide by two since the symmetric matrixis stored in compact form. This allows the linear systemto be solved in the core memory of our IBM RS6000/540computer which has 64M of core memory. Storage of thedirect (no S dependence) and exchange (have S depen-dence) V-matrix elements (21), also real and symmetric,adds a further 60M of required storage space.

The required K&s~ are calculated by direct substitutionof the solution of (40) for the KS~ into (39). Havingcalculated the K&s~ we recover the Tg,

~ by solving (35)which in the simplified notation we write as

Kq,' ——) Tq~q', (6'l, + i~kg Kq~,', ) . (41)

flNote that the sum in (39) contains all channels, open andclosed, as well as the quadrature points, whereas the sumin (41) contains only the open channels.

This procedure is carried out for each value of S (S = 0for singlet, S = 1 for triplet) of partial wave J, with as

many partial waves taken as necessary for convergenceto better than 1%. Typically, we take J = 0 to 80 withhigher partial waves taken care of by either an extrap-olation or by analytic Born subtraction as discussed byMcCarthy and Stelbovics [12].

The relation between the reduced T-matrix elementsand various physical observables may be found in Refs.[12, 19]. For completeness, we present the definitions ofthe calculated observables here. Writing the amplitudefor the excitation of the 2p mt = n (n = —1,0, 1) substateas a„, the A, R, and I parameters are given by

A = (apao)/o's» R—:Re(apaq)/os» (42)I =—Im(aoa&)/o2„= Lz/~8, —

~he~~ rr2„ is the difFerential cross section, the notation() indicates a spin weighted sum, and L~ is the angu-lar momentum transferred perpendicular to the scatter-ing plane (see review of Anderson, Gallagher, and Hertel[20]). The spin asymmetry A„~ for the nl channel is re-lated to the ratio of triplet to singlet scattering for thatchannel r„t by

(43)+ Tnl

The relationship between the I and L~ parameters isparticularly worth noting given the remarkable quanti-tative agreement between experiment and the coupled-channel optical (CCO) theory of Bray and McCarthy [21]in electron-sodium scattering for this and other param-eters This agreem. ent is found for singlet, triplet, andaveraged spin states. As the CCO theory [21] is an ap-proximation to the CC theory presented here we also ex-pect excellent quantitative agreement with such param-eters in the simpler case of electron-hydrogen scattering.Unfortunately, this does not prove to be the case.


IV. RESULTS

The major discrepancy between theory and experimentfor electron-hydrogen scattering is for angular correlationparameters A and R of the 2p excitation. So we apply ourtheory to those energies where there are most measure-ments of these parameters namely at 35 and 54.4 eV. Wealso look at the higher energy of 100 eV to get a broadenergy spectrum.

Our aim is to present a series of large basis close-coupling calculations which demonstrate convergence ofthe close-coupling approach. In the partial wave solutionof the close-coupling equations discussed above, conver-gence with target states, denoted by nl, must be demon-strated separately for n and I.

In Figs. 1—9 the three left-hand pictures have targetstates with /m~ = 2, and examine the convergence as Ntis increased for each l (n = l + 1 to N1). The largest N1,which we take to be the same for each l, that our localcomputers can handle for this l ~ is 13, i.e., 13s, 12p,and lid states. This leads to a 70-channel calculation asfor J & l an s state leads to a single channel, a p stategenerates 2 channels, and a d state generates 3 channels.These calculations are denoted by 70CC. In the threecenter pictures convergence is examined for fixed Nt, butwith increasing l up to and including f states, i.e. , l~s„=3. The largest calculation that we could perform whichhad f states has Nt = 10 for each l, i.e., 10s, 9p, 8d, and7f states. This leads to an 80-channel (80CC) calculationfor J & 3. Note that even though the 80CC calculationhas 10 more channels than the former, it has a total of34 states, while the 70CC treats 36 states. The threeright-hand pictures compare our results with some other

theories and if available, experiment. We do not presentour quantitative results as they are extremely extensivebut they may be readily obtained by corresponding withthe authors.

In Fig. 1 we look at the differential cross sections forthe ls, 2s, and 2p channels at 35 eV. We find excel-lent convergence as a function of both N~ and I „ forall channels. The convergence is excellent at all anglesperhaps with the exception of the very backward angles.There is good agreement with other nonperturbative the-ories, the coupled-channel optical model (6CCO) of Bray,Konovalov, and McCarthy [19] and the intermediate en-

ergy R-matrix method (IERM) of Scholz et aL [22]. Theexact second-order theory (DWB2) of Madison, Bray,and McCarthy [23] is a little above the other theories,but as it is a perturbative theory we do not expect it tohave converged at this relatively low energy.

Figure 2 looks at the convergence in the spin asymme-tries (43) at 35 eV. We see that for these parameters theconvergence is again very good, but is not quite as rapidas for the differential cross sections. Convergence at thevery backward angles for the 2p channel hss not beenachieved which is probably due to the difficulty associ-ated with getting a convergent ratio of difFerential crosssections where they are very small. Comparison withother theories shows quite a bit of variation. There isconsiderable agreement with the IERM theory and to alesser extent with the 6CCO theory. The DWB2 resultshave quite a difFerent quality at the intermediate anglesfor the 2s and 2p channels.

In Fig. 3 we look at the angular correlation parame-ters, for which there is an extensive set of measurements.Once again we see that convergence as a function of N1 is

10

to 10

10

10

0

10CO

CV

OQ)tQ

10

10

Q)0

cu 10

4-1Q

U10

10

OCC

2p, lid

2CC

p, 8d

2CC

~ 3d

C

8d, 1f

C

8d

OCC

2p, lid

FIG. 1. DifFerential cross sections at 35eV for the 1s, 28, and 2p channels calculatedwith the indicated number of Laguerre basis states. The left-hand column has calcula-tions which include d states, so convergenceis examined as the basis size is increased foreach l. The center column fixes the basissize for each l, and demonstrates convergenceas l is increased from s and p states to dsnd f states. The right-hand column com-pares the 70CC calculation with other theo-ries. The coupled-channel optical calculationof Bray, Konovslov, sud McCsrthy [19], in-termediate energy R-matrix method of Scholzet al. [22], snd the distorted-wave second-order Born calculation of Madison, Bray, andMcCsrthy [23] sre denoted by 6CCO, IERM,sud DWB2, respectively. Quantitative re-sults may be obtained by corresponding withthe authors.

40 80 40 80 120 160 0

scattering angle (deg)120 160

IGOR BRAY AND AND RIS T. STELBOVICS 46

0.0

—7GCC

. 3s, ' 2p, . x d

—80CC

1 Cs, 9p, 8d, 7f)GCC

13s, 12p, !ld

-0.2

(U O. B

0.6

—7CCC —80CC 7Grr

0 4

s

60.2 .

E

0.0 "

-0.2

FIG. 2. S in aspin asymmetries at 35 eV fe or

with the ip channels calculated

wi t e indicated number of La estates. See Fi . 1

o aguerre basise ig. or more detail and defin'-

tion of other theories.e 1-

0.4

7A//444 —8GCC —7CCC

C4

0.2

0.0 ~

40 BO 120 160 0 120 160 0

scattering angle E,'deg)

40 BO 120 '160

very good. The convergence as a function

p'' g

the forward angles for thy e ween all theories and experiment at

gence of the close-cou l'es or t e R parameter.r. Here conver-

coup mg results is clearly established )

and furthermorore, they are in corn lete ath oth th o }M. Aangles is much b tt

ries. greement with exxpersment at other

h th o t te er, but wh this s

n ra e is much smaller is at th th

sory ss very good at forward angles, but gets

1.0

O. B

0.6

0. 4

0. 4

QE

C4

0.2O

u ~

o. o,

FIG.. 3. Angular correlationf ~ ~e p excitation of hydro encalculated 'th the '

wi t e indicated number ofguerre basis states. See Fi .ee ig. 1 for more detail

e nition of other theories.surements are du

cones. The mea-re ue to Slevin et at [26]. .

0. 2

0. 0

40

/Gs, 9o

120 160 0 80 120 160 0

scattering angle (deg)

40 BO 120 160


to be very poor at intermediate to backward angles .Having looked at pictorial results it is instructive to

examine the actual T-matrix elements for the most sig-nificant partial waves, as well as the integrated and to-tal cross sections. In Table I we present the 35-eV T-matrix elements for the 52CC, 70CC, and 80CC calcu-lations, and compare them with the IERM results. Wealso give the integrated 1s, 2s, 2p, and total cross sec-tions, which show convergence and good agreement withIERM. The IERM TR-matrix elements [22] are related to

our T-matrix elements by

TR = 2—vriy kfk, T, (44)

where kf and k; are the final and incident linear momentaof the projectile. The difference in the 70CC and 80CCresults gives a good indication of the convergence, sincethe two calculations not only treat each partial wave witha different number of states, but also treat a difFerentnumber of partial waves. If the 70CC and 80CC calcu-

TABLE I. T-matrix elements (real and imaginary parts) for the first five partial waves J, and integrated and total crosssections (in units of ao) at 35 eV. The numbers in brackets indicate powers of 10.

Channel 52CC 70CC 80CC IERM1$-1$ls-2sls-2pls-ls1$-2$1$-2p

-2][-2][-2]-2]-2]-3)

-6.74[-1.191.31

-8.63[-1.09[-6.18[

-2]-2]-2]-1]-3]-2)

-8.37[1.83[2.73[

-1.41[2.46[1.71[

-6.86[-1.24[1.16[

-8.61[-1.03[-6.11[

-2]-8.26 [-2]-2] 1.78[-2]-2] 2.79[-2]-2]-1.41[-1]-2] 1.95[-3)-3) 1.72[-2]

-6.75[-1.28[1.21[

-8.63[-1.09[-6.40[

-2]-2]-2]-2]-2]-3]

-8.39[1.87[2.73[

-1.41[2.34[1.71[

-2]-2]-2]-1]-3]-2]

-1.36[-2] 1.65[-2]1.32[-2] 2.94[-2]

-1.03[-2] 1.19[-3]-6.79[-31 1.73[-2]

ls-ls1$-2s1s-2p1s-2@+1$-ls1$-2$1$-2p1$-2@+

-2]-2]-2]-2]-2]-3]-4)-3]

-2.24[1.51 [1.07[1.14[

-6.83[3.90[

-4.64[6.48[

-2)-2]-2]-4]-2]-2]-3)-3)

-2.01[1.73[1.24[

-1.54[-3.14[1.40[3.50[6.06[

-2)-2]-2]-2]-2]-3]-4)-3)

-2.24[1.51[1.06[1.15[

-6.87[3.69[

-S.67[6.68[

-2)-2]-2]-4]-2]-2]-3]-3]

-2.00[1.78[1.28[2.72[

-3.17[1.39[3.66[6.30[

-2.25[1.52[1.07[1.14[

-2)-2]-2]-2]

-6.84[-2]3.81[-3]

-5.01[-4)6.61[-3]

-2.01[-2)1.78[-2]1.24[-2]

-4.96[-4]-3.14[-2]1.41[-2]3.55[-3]6.06[-3]

1.45[-2] 1.73[-2]1.17[-2] 1,32[-2]1.26[-2] 5.12[-4]

3.46[-3] 1.36[-2]1.08[-4] 4.02[-3]7.31[-3] 6.93[-3]

ls-lsls-2s1s-2p1$-2p+ls-lsls-2$1$-2p1$-2p+

-1.43[-2]6.51[-3]2.38[-2]

-1.o2[-s]-2.70[-2]9.31[-3]6.96[-3)4.21[-3]

-1.79[3.93[1.07[

-2.76[-7.31[7.37[1.07[

-1.22[

-2]-3]-2]-3]-3)-3)-2]-3]

-2]-3]-2]-4]-2]-3]-3]-3)

-1.46[6.39[2.36[

-4.69[-2.72[9.19[6.96[4.27[

-2]-3]-2]-3][-3]-3]-2]-3]

-1.76[4.05[1.14[

-2.53[-7.367.37[1.08[

-1.06[

-2]-3]-2]-4]-2]-3]-3]-3]

-1.44[6.54[2.38[

-4.79[-2.71[9.31[6.92[4.24[

-2]-3]-2]-3]-3)-3]-2]-3]

-1.79[4.03[1.12[

-2.80[-7.32[7.44[1.08[

-1.30[

5.38[-3] 3.36[-3]2.53[-2] 1.17[-2]2.56[-4]-3.40[-3]

8.64[-3) 6.97[-3]7.53[-3] 1.11[-2)5.60[-3]-1.03[-3]

1$-lsls-2s1s-2p1$-2@+ls-ls1$-2$1$-2p1$-2p+

-1.01[4.67[2.42[2.14[

-4.90[2.75[1.00[

-2.14[

-2]-3]-3]-5]-3]-3]-2]-3]

-1.10[-2]1.oo[-s]2.45[-2]

-3.07[-3]-1.20[-2]5.44[-3]1.57[-2]2.08[-4]

-1.12[1.11[2.41[

-3.00[-1.215.32[1.58[3.09[

-2]-3]-2]-3][-2]-3)-2]-4]

-9.92[-3]4.76[-3]2.43[-3]

-1.57[-4]-4.94[-3]2.72[-3]1.01[-2]

-2.21[-3]

-1.10[1.18[2.45[

-3.03-1.215.38[1.56[1.56[

-2]-3]-3]-4]-3]-3]-2)-3]

[-sl[-2)-3]-2)

4]

-1.58[-4.88[2.81[1.02[

-2.12[

-2]-1.01[-3] 4.58[-2] 2.71[

1.56[-4] 4.60[-3]2.43[-2] 2.71[-3]

-2.92[-3] 8.86[-4]

4.35[-3] 2.50[-3]1.62[-2) 1.03[-2)3.92[-4]-2.15[-3]

ls-1$1$-2s1$-2p1s-2p+1$-1$1s-2$1$-2p1$-2@+

~2s0'2p

0'g

[-sl[-4][-2][-sl[-3][-sl-2]-3]

-7.452.301.98

-2.86-7.011.721.82[

-1.64[

-5.12[4.96[

-5.79[7.97[

-3.70[2.03[5.43[

-1.20[

-3]-3]-4]4]

-3]-3]-3]-3]

0]-1]0]1]

5.33[2.48[2.26[1.07[

-3][-5]-21

-3]-3]-3]-2]-3]

-7.50[-1.891.97[

-2.83[-7.03[1.62[1.81 [

-1.64[

-5.16[5.08[

-4.50[1.14[

-3.70[1.96[5.54[

-1.18[

-3]-3]-4]-3]-3]-3]-3]-3]

5.37[ 0]2.50[-1]2.27[ 0]1.08[ 1]

-7.37[-3]3.27[-4]1.98[-2]

-2.53[-3]-7.06[-3)1.60[-3]1.81[-2]

-1.63[-3]

-3]-3]-4]-4)-3]-3)-3]-3]

0]-1]0]1]

5.33[2.52[2.23[1.07[

-4.98[4.76[2.92[7.55[

-3.68[2.04[5.63[

-1.16[

-3.30[-4] 5.39[-3]1.97[-2]-4.84[-4]

-2.77[-3) 1.73[-3]

1.16[-3) 2.02[-3]1.85[-2] 5.56[-3]

-1.29[-3]-1.02[-3]

2.58[-1]2.06[ 0]

IGOR BRAY AND ANDRIS T. STELBOVICS 46

nV

- n

O

C/3

cD(f)

tnO

21C

I. 2

0

, 8d, 2!

FIG. 4. Differential cross sections at 54.4eV for the 1s, 2s, and 2p channels calculatedwith the indicated number of Laguerre basisstates. The calculation of van Wyngaardenand Walters [24] is denoted by vWW40CC.See Fig. 1 for more detail and definition ofother theories. The measurements for theelastic channel have been interpolated fromWilliams [27], and for the inelastic channelsare due to Williams [28].

(D

(D

Q. -

Q

n

40 80 120 160 0 40 80 120 160 0 40 80 120 160


0. 0

I I',

235, !2p—80C"!". -, 9p, 8 "', l;

I OLC

13s, 22p, 1!d

-0. 2 .

0.8

I

0.6I

f

CQ 0(D

0.2EE

o. o

-0.2

00.2

1 CI nl'V '4

!35, .'2.;, l'!c

, n/-rV L. L

3s, !2p, '!d/ /

', Cs, 9p, 8c

—80CC

!Os, 9p, 8c, t!

—80CC

!0s,9p, 8d, '7 f

——52CC

Os, 9p, 8d

—70CC

!3s, '. 2p, !!d

—&OCC

.'3s/!2p/c, //VLL v

vWWI!OCC

FIG. 5. Spin asymmetries at 54.4 eVfor the 1s, 2s, and 2p channels calculatedwith the indicated number of Laguerre ba-sis states. The calculation of van Wyn-gaarden and Walters [29, 30] is denoted byvWW40CC. See Fig. 1 for more detail anddefinition of other theories.

0. 0

40 80 120 160 0 40 80 120 160 0 40 80 120 160



lations are significantly difFerent then comparison withthe 52CC elements gives an indication as to whether thelack of convergence is due to insufficient N~ or t ~. Wesee that, in general, agreement between the theories isvery good; however, on occasion the IERM differs signif-icantly from the others, which are in good agreement witheach other. The IERM method involves matching outsidethe interaction region, uses T-matrix averaging, and in-cludes corrections based on the plane-wave second-orderBorn approximation to make allowance for configurationsnot included in the IERM expansion of the three-bodyscattering wave function. As our method has no such ap-proximations we suggest this is a possible source for thediscrepancies.

The differential cross sections for the ls, 2s, and 2pchannels for electron hydrogen scattering at 54.4 eV aregiven in Fig. 4. Convergence is clearly demonstrated.Agreement with experiment and other theories is quitegood, though our results are systematically a little lowerthan the measurements. At this energy as well as at100 eV there is a large pseudostate calculation of vanWyngaarden and Walters [24]. This is a 40CC calcula-tion with N~ = 8 and l ~ = 2, where they also triedto take care of higher t by employing a distorted-wavesecond-order Born approximation. They used Slater-typeorbitals for their basis which was chosen to get goodsecond-order Born terms. We denote their results byvWW40CC.

The corresponding ssymmetries are given in Fig. 5.Here we see that while convergence for the ls and 2schannels is evident at all angles, it is not so clear forthe 2p channel. Clearly larger calculations are necessary

to establish convergence at all angles for the asymme-

try of the 2p channel. Agreement with other theories israther mixed with the DWB2 calculation standing outmost from the others.

The 54.4-eV angular correlation parameters are givenin Fig. 6. Very good convergence is evident at all anglesfor each parameter. Agreement with experiment, two in-

dependent sets of measurements, is very good at forwardangles, but poor at intermediate angles for A, and back-ward angles for the R parameter. The difFerent theoriestend to support each other, as do the two sets of measure-ments. The agreement with the measurements of the Iparameter is also particularly disappointing since there is

very good agreement between the various theories. Hav-

ing established convergent close-coupling results we areforced to claim that the close-coupling theory is unableto explain the existing measurements of the angular cor-relation parameters.

Again it is instructive to look at the T-matrix elementsgiven in Table II together with the integrated and to-tal cross sections. We see that agreement between the70CC, 80CC, and IERM theories is generally very good,though there are some significant differences on occasion.Given the nature of the agreement of the T matrices wewere puzzled by the oscillations at backward angles of theIERM A and R parameters. Since IERM is a hybrid the-ory which uses partial-wave T-matrix elements for J & 5from an entirely separate pseudostate calculation, we cal-culated the angular correlation parameters for the IERMmethod using the T-matrix elements for J & 5 from our70CC calculation. The resulting A and R were free of theoscillations at the backward angles, but otherwise very

(U

0.4

0.20.4

S-I(D

E0.2

re0

O

0.0G)

OO

0.2

—70CC —80CC

70CC

13s, 12p, lid——6CCO

o~ --. vWW40CC

--- DWB2

—&OCC

FIG. 6. Angular correlation parametersfor the 2p excitation of hydrogen at 54.4 eVcalculated with the indicated number of La-guerre basis states. See Figs. 1 and 4 fordefinition of other theories and more detail.The measurements denoted by o are due toWilliaxns [28,31], those denoted by C3 are dueto Weigold, Frost, and Nygaard [32].

0.0

5s, lp, 3d 10s, 9p

40 80 40 80 120 160 0



similar to those presented. This confirmed our suspicionthat the oscillations in the IERM results are likely to bedue to numerical problems. The integrated cross sectionsshow remarkable convergence, and are in excellent agree-ment with IERM, and good agreement with experiment.

In Fig. 7 we present the differential cross sections at100 eV. Clearly our results have converged, and are invery good agreement with other theories. Agreementwith the elastic measurements is a little disappointing

as all of the models lie a little lower than the experiment,but are in complete agreement with each other.

The corresponding asymmetries are given in Fig. 8.As at 54.4 eV, the covergence for the 1s and 2s channelsis obtained at all angles, whereas it hss not been estab-lished for some angles of the 2y channel. Agreement withthe 6CCO and vWW40CC results is quite good, but it issurprising to see a large discrepancy with the DWB2 cal-culations at this relatively high energy. Whenever there

TABLE II. T-matrix elements (real and imaginary parts) for the first five partial waves J, and integrated and total crosssections (ao) at 54.4 eV. The numbers in brackets indicate powers of 10.

Channel 52CC 70CC 80CC IERM1$-1$1$-2$ls-2pls-ls1$-2$ls-2p

-2]4]

-3]-2]-3]-4]

-5.56[-5.88[9.50[

-7.54[-5.79[3.91[

-2]-2]-2]-2]-3]-2]

-6.11[1.62[1.58[

-8.93[7.85[1.32[

-5.65[-1.33[1.02[

-7.54[-5.73[3.46[

-2]-3]-2]-2]-3]4]

-6.16[1.55[1.64[

-8.94[7.68[1.32[

-2]-2]-2]-2]-3]-2]

[-2]-6.12[-5.59-1.69[9.35[

-7.54[-5.96[2.98[

1.69[1.47[

-8.95[7.85[1.30[

-3]-3]-2]-3]-4]

-2]-2]-2]-2]-3]-2]

-2.00[-3] 1.54[-2]1.07[-2] 1.72[-2]

-6.02[-3] 6.78[-3]-8.89[-5] 1.41[-2]

1$-1$ls-2s1s-2p1s-2p+ls-lsls-2sls"2p1$-2p+

-2.33[1.SO[6.73[6.71[

-5.08[5.31[8.85[4.89[

-2]-1.72[-2] 1.20[-3] 5.80[-3] 1.01[-2]-2.19[-3] 1.04[-4] 3.35[-3] 3.86[

-2]-2]-3]-3]-2]-2]-3]-3]

-2]-2]-3]-3]-2]-2]-3]-3]

-2.421.286.666.93[

-5.10[5.24[8.43[4.92[

1.32[-2.21[1.O5[3.38[3.93[

-3]-2]-3]-4]-3]

[-2]-1.82[[-2] 1.16[[-3] 5.73[

-2.39[-2]1.3O[-2]6.49[-3]6.53[-3]

-5.09[-2]5.27[-S]7.47[-4]4.78[-3]

-2]-2]-3]-3]

-2.19[1.04[3.36[3.60[

-1.73 [-2]1.21[-2]5.73[-3]5.25[-4]

1.24[-2] 1.16[-2]7.40[-3] 6.46[-3]7.39[-3] 1.33[-3]

4.80[-3] 1.01[-2]9.98[-4] 3.76[-3]5.27[-3] 4.48[-3]

ls-ls1$-2$1$-2p—1$-2@+ls-lsls-2$1$-2p1$-2p+

-1.25[-2]-1.28[-2]8.90[-3] 3.97[-3]1.39[-2] 5.89[-3]5.00 [-4]-1.34[-3]

-2.32 [-2]-6.73[-3]8.13[-3] 5.81[-3]6.30[-3] 6.70[-3]2.60 [-3]-5.03[-4]

-2]-3]-2]-4]-2]-3]-3]-3]

-1.21[9.12[1.42[5.14[

-2.33[8.20[6.25[2.68[

-1.29[3.98[5.90[

-1.42[-6.77[5.87[6.67[

-4.96[

-2]-3]-3]-3]-3]-3]-3]-4]

-1.27[8.99[1.38[4.84[

-2.32[8.10[6.17[2.52[

-2]-3]-3]-3]-3]-3]-3]-4]

-2]-1.28[-3] 4.04[-2] 5.91[-4]-1.54[-2]-6.79[-S] 5.76[-S] 6.73[-S]-6.28[

8.38[-3] 3.50[-3]1.52[-2] 6.O2[-3]1.73[-3]-1.48[-3]

7.79[-3] 5.37[-3]6.71[-3] 6.96[-3]3.77[-3]-1.06[-4]

ls-ls1$-2s1$-2pls-2@+ls-lsls-2s1$-2p1$-2p+

-8.42 [-3]-8.92 [-3]4.5O[-3] 2.62[-S]1.64[-2] 2.05[-3]

-1.98[-3]-5.88[-4]-1.1O[-2]-4.37[-3]6.17[-3] 2.58[-3]1.14[-2] 6.08[-3]7.75 [-5]-1.39[-3]

-8.57[2.73[2.44[

-6.49[

-3]-3]-2]-3]-2]-3]-2]-5]

-8.26[4.61[1.65 [

-1.90[-1.10[6.10[1.14[8.46[

-3]-3]-3]-4]

-4.29[-S]2.66[-3]6.08[-3]

-1.S7[-3]

-8.87[2.58[2.10[

-7.06[-4.32[2.60[6.05[

-1.45[

-3]-3]-3]-4]-3]-3]-3]-3]

-8.35[-3]4.68[-3]1.63[-2]

-2.00[-3]-1.10[-2]6.19[-3]1.12[-2]2.31[-5]

4.03[-3] 2.51[-3]1.68[-2] 2.32[-3]

-1.49[-3]-4.53 [-4]

5.43[-3] 2.35 [-3]1.18[-2] 6.25[-3]5.85 [-4]-1.19[-3]

ls-lsls-2s1$-2p1$-2@+ls-lsls-281$-2p1$-2@+

0 ls&2s

&2p

Og

1.76(12)[-1]2.26( 9)[0]

-3]-3]-4]-4]-3]-3]-3]-3]

-5.73[2.23[1.56[

-2.83[-6.12[3.64[1.S5[

-1.42[-3]-1.05[

3.01[ 0]2.08[-1]2.32[-1]

8.58 [ 0]

-3]-5.69[-3] 2.62[-2]-2.17[-3] 1.17[-3]-3.53[-3] 1.52[-2] 3.73[

-3]-3]-2]-3]-3]-3]-2]-3]

-5.82[2.26[1.54[

-2.76[-6.14[3.63[1.34[

-1.42[

-3]-3]-5]-4]-3]-3]-3]-3]

0]-1]0]0]

3.03[2.08[2.32[8.56[

-5.57[2.69[

-9.73[1.50[

-3.51 [1.55 [3.71[

-1.06[

-5.72[-3]2.45[-3]1.56[-2]

-2.52[-3]-6.17[-3]3.57[-3]1.35[-2]

-1.41[-3]

3]-3]-4]4]

-3]-3]-3]-3]

0]-1]0]0]

3.02[2.12[2.26[8.52[

-5.49[-2.68[3.07[1.50[

-3.48[1.58[3.85[

-1.02[

1.39[-3] 2.73 [-3]1.57[-2]-2.66[-4]

-2.58[-3] 6.42[-4]

3.09[-3] 1.52[-3]1.39[-2] 4.02[-3]

-1.13[-3]-9.22 [-4]

2.O7[-1]2.27[ 0]

The estimates of the 2s and 2p integrated cross sections are due to van Wyngaarden and Walters [24].


10 —70CC

CO

CD

10

1O'tg

0cg 10

10

CQ

N

OCDCO

10

OS-IO 10

10

10',CD

4-Ig 10

U

10

lid 8d, 7f

Bd

C

, 11d

40CC FIG. 7. Differential cross sections at 100eV for the 1s, 2s, and 2y channels calculatedwith the indicated number of Laguerre basisstates. See Figs. 1 and 4 for definition of othertheories and more detail.

10

10 I

40 80 120 160 0 40 80 120 160 0


40 120 160

0.0

—7OCC

1 3s, 1 2p, lid—BOCC

los, 9p, 8d, 7f—70CC

13s, 12p, lid

-0. 1

0.&

CDCD

0.2

CD

rn

0. 1

EE

0.0

0.2—70CC —8OCC —70CC

FIG. 8. Spcn asymmetnes at 100 eVfor the 1s, 2s, and 2y channels calculatedwith the indicated number of Laguerre ba-sis states. See Figs. 1 and 5 for definition ofother theories and more detail.

0. 1

Q

0.0

-0. 1

40 80 120 160 0 40 80 120 160 0


40 80 120 160


is a problem with convergence it is instructive to look atthe T-matrix elements which are given together with theintegrated and total cross sections, which demonstrateexcellent convergence, in Table III. Comparing 52CCand 70CC elements for the Is-2p channel we see excellentagreement from which we would have expected almost

indistinguishable asymmetries. To 6nd the cause of theproblem we replaced just the J = 0 T-matrix elementsof the 52CC calculation with those of the 70CC calcu-lation, i.e., a change of at most 1%. We found that theresultant asymmetry varied from the 52CC by as muchas 10% at the backward angles. As the differential cross

TABLE III. T-matrix elements (real and imaginary parts) for the first five partial waves J, andintegrated and total cross sections (a&&) at 100 eV. The numbers in brackets indicate powers of 10.

J 80 0

Channel

1$-1$18-2818-2p18-1818-2$18-2p

52CC

-2]-3]-3]-2]-3]-3]

-3.89[9.97[7.30[

-4.74[7.47[7.16[

-4.40[-2]2.75[-3]5.21[-3]

-5.47[-2]-6.74[-4]2.50[-3]

70CC

-2]-3]-3]-2]-4]-3]

[-2)-3]-3][-2)-3]-3]

-3.859.92[7.33[

-4.727.34[7.09[

-4.43[2.69[5.22[

-5.48[-5.83[2.51[

80CC

-2]-3]-3]-2]-4]-3]

-4.40[2.44[5.34[

-5.47[-7.45[2.61[

-3.91[-2]1.00[-2]6.69[-3]

-4.74[-2]7.49[-3]7.01[-3]

1 0 ls-1818-2$1s-2p—1$-2p+ls-ls1$-2$1s-2p1$-2p+

-1.22[-2]6.55[-3]2.43[-3]1.09[-3]

-1.32[-2]

-2]-3]-3]-3]-2]

-2.23[8.27[2.78[3.06[

-3.38[4.86[1.09[2.68[

-3] 6.24[-3]-3] 2.16[-3]-3] 1.86[-3]

-2.25[8.21[2.79[3.03[

-3.39[

-2]-3]-3]-3]-2]

4.85[-3]1.10[-3]2.64[-3]

-2]-3]3]3]

-2]-3]-3]3]

-1.25[6.53[2.44[-1.07[-

-1.34[6.29[2.19[1.84[-

-2]-3]-3]-3)-2]-3]-3]-3]

-2.25[8.22[2.83[3.12[

-3.38[4.85[1.11[2.70[

-2]-3]-3]-4]

-1.23[6.60[2.32[8.57[

-1.33[-2]6.28[-3]2.10[-3]1.75[-3]

2 0 ls-18ls-281s-2p-1$-2p+18-181$-28ls-2pls-2p+

-3]-3]-3]-4]-3]-3]-3]-5]

-7.87[2.84[2.73[

-3.41[-5.16[3.72[3.27[

-3.01[

-1.15[-2]7.25[-3]6.40[-3]6.51[-4]

-1.78[-2]5.94[-3]4.04[-3]1.31[-3]

-2]-3]-3]-4]-2]-3]-3]-3]

-1.18[7.20[6.32[5.75[

-1.80[5.92[3.98[1.26[

-3]3]3]

-4]-3]-3]-3]-5]

-7.83[2.95[-2.67[-

-3.98[-5.21[3.75[3.23[

-4.79[

-2]-3]-3]-4]-2]-3]-3]-3]

-1.17[7.26[6.30[5.44[

-1.78[5.97[4.03[1.22[

-7.84[2.97[2.60[

-4.78[-5.17[3.77[3.20[

-1.30[

-3]-3]-3]4]

-3]-3]-3]-4]

3 0 1$-1$ls-281s-2p-ls-2p+1$-1$ls-28ls-2p1$-2p+

3]3]

-3]4]

-3]-3]-3]-6]

-6.63[-5.30[-8.48[

-8.29[--9.50[5.21[6.43[6.09[

-5.75[-1.53[1.70[

-4.28[--3.19[2.04[2.96[

-5.78[

3]-3]-3]4]

-3]-3]-3]4]

-3]3]3]

-4]-3]-3]-3]4)

-3][-3)-3][-4)-3)[-31

[-31-6]

-6.60[5.308.43[

-8.33-9.51[5.216.395.87[

-5.73[1.54[-1.76[-

-3.85[-3.17[2.04[2.97[

-5.69[

3]-3]-3]4]

-3]-31-3]4]

-6.73[5.35[8.30[

-8.83[--9.52[5.23[6.34[

-6.20[

-3]-5.71[--3] 1.62[-3] 1.59[4]-4.66[-

-3)-3.17[-3] 2.08[-3] 2.87[-5]-6.22[

4 0 1$-181$-2$ls-2p1$-2p+1$-181$-2s1$-2p18-2p+

&1s

&2s

1.85[ 0]1.22(13)[-1]"1.94( 9)[ 0]6.84[ 0]

-4.28[-1.09[7.37[

-2.49[--2.58[1.16[2.16[

-6.01[

3]-3]-4]4]

-3]-3]-3]-4]

0]-1]0]0]

1.40[1.39[1.98[6.11[

-4.13[-3]3.58[-3]9.28[-3]

-1.?2[-3]-5.36[-3]3.95[-3]7.86[-3]

-9.40[-4]

-3]-3]-3]-3]-3]-3)[-3]-4]

-4.18[3.56[9.27[

-1.68[-5.38[3.93[7.87

-9.42[

-3]-3]-4]-4]-3]-3]-3]-4]

0]-1]0]0]

1.41[1.39[1.98[6.11[

-4.28[1.15[6.97[

-2.44[-2.58[1.18[2.14[

-5.95[

-4.28[-3.69[9.00[

-1.63[--5.41[3.98[7.73[

-9.54[

3]-3]-3)3]

-3]-3]-3]4]

-2.56[1.22[2.08 [

-5.95[

-3]-3]-3]-4]

1.41 [1.40[1.96[6.15[

0]-1]0]0]

-4.21 [-3]1.22[-3]7.24[-4]

-1.68[-4]

The semiempirical estimates of the integrated elastic and total cross section are due to de Heer,

McDowell, and Wagenaan [25].The estimates of the 2$ and 2p integrated cross sections are due to van Wyngaarden and Walters

[24].

46 SE-COUPLINGG CALCULATIONS OF. . .CONVERGENT CLO

1.0

O. B

0.6

o. 4

O. 2

o. 4Q)

(DE

G4

O~ 0.2

CD

OU

o.o

—70CC —BOCC —7OCC

13s, l2p, lid

n ular correlat&on parametersFIG. 9. g""""y-'citation of hy rogen

1 d4f d6ith the indica e

gu tates. See Figs. an

H d W 'old d

er theories an moree due to oo,measurements ar

Dixon [33].

0.2

0.0

-O. 2 -- 22CC10s, 9ps, 4p, 3d

160 0120 160 0 40 80 120

le (deg)scattering angle

120 160

all for the largerk gn les is so sm11 variation in the lowes pa

d 1 ofh1 i tio b teter. This explains t e argeparameter.

these angles.o c w

theories at t etal cross sections wi

ih 1

ooce. Agreemengsee exce

0 lower

erimenth xperimen .

elas icby a ou

t' M thi 1 tKdeterminatson u

'th the e pen e tal esti-cro

t d o tio ireement wi

vermenmates of the 2s an d 2p integrate c

lar correlation pa-ook at the 100-eV angu aas beenh t onege h een

ement with other theories a

surements are only aLJ )

~

mb the high par ia wy

tia y giv0(I

lit d b. t. th. . ; ...,...l. d fI,

in this section.

of convergence we oe obtainedindicative o ef the general tren o coin these studies.

V. CONCLUSIONS

a frameworker has been to develop aM .1 ti . Mi

ure close-coup mg

hi hi i, h mbl

b li i dol bsys

lculations have e'lities available.

and our ca c

tdiffe ntial, i tegrat i d 1

efor elastic sca

r iesscattering to the 2s and 2p }eve}s

alculations at all energies ortween oursections, convergence is o

th o Wh t t expan

'

C 1 1

h tth o tioeec ise si ni6cant y c a

closeP D

hdth oit htensive basis sesets. e

fs rovided a more stringlete.

ent test o1't d to

h b' o d

AB dIbt' d7

tt lt'n its size we o ain

u toa10 0

g

1e change in their va ues a


So if higher precision was required at these angles thenperhaps even higher l states would be required. All ourcalculations are unable to reproduce the deep minima atbackward angles in A for the 35- and 54.4-eV experimentsand there is a similar divergence in the R parameter withthe 54.4-eV experiments. While we are unable to claimconvergence to a 1% level, it does appear that our corre-lations will not be altered substantially by more extendedcalculations, certainly not sufBciently to explain the dis-crepancy with experiment.

Spin asymmetries proved to be the most sensitive pa-rameters to target basis sets. For basis sets restrictedto s, p, and d states, convergence with basis is achievedto within a few percent over most of the angular range.The elastic channel asymmetry has a single minimumand rises at backward angles. Discrepancies of up to 10%were obtained in the 100-eV result for angles greater than120'. The 2s spin asymmetries have a node at forwardangles which is followed by a steady rise at backward an-gles. Convergence rates are similar to the 1s channel.The 2p channel shows most structure. The calculationsover s, p, and d basis states indicate a well-defined for-ward peak and subsequent nodal structure that is similarfor the 52CC and 70CC models. The main differences areagain at extreme backward angles where the cross sec-tion is very small. When f states are added to the 52CCmodel, their efFect is significant in altering the magni-tude of the forward peak of the 2p channel but not itsposition, and in addition varies the shape and structureof the remaining peaks. The convergence was reasonablydemonstrated for the 35-eV results but becomes progres-sively worse at 54.4 and 100 eV. We conclude that statesof higher partial waves are needed at 54.4 and 100 eVin order to obtain better convergence for the 2p channelspin asymmetry.

At this stage, where our results and experiment dis-agree we have good agreement with other theories, thenonperturbative IERM of Scholz et aL, vWW40CC ofvan Wyngaarden and Walters, CCO of Bray and co-workers and the perturbative (exact to second order)

DWB2 of Madison, Bray, and McCarthy. Where there issignificant discrepancy between the theories, particularlythe asymmetries, there is no experiment. It would bemost helpful if new measurements of the angular correla-tion parameters as well as measurements of the hydrogenasymmetries were available. Given the excellent quanti-tative agreement in electron-sodium scattering measure-ments of the I~ = 2~—2I parameter and the CCO the-ory of Bray and McCarthy [21], which is an approxima-tion to the convergent CC theory pesented here, we arenot too perturbed by the present discrepancies betweentheory and experiment in electron-hydrogen scattering.

Our overall conclusion is that the Laguerre basis ex-pansions applied here are adequate to obtain fully con-vergent scattering amplitudes in the close-coupling equa-tions. In the calculations reported, we have demon-strated convergence of individual partial-wave T-matrixamplitudes for channels of interest to better than 5% levelover all the energy range considered with few exceptions.For parameters which are made from the summed par-tial waves we achieve convergence in some cases to bet-ter than 2%. We have demonstrated this for difFerentialcross sections and to a lesser extent the angular corre-lations. Larger basis expansions are needed to obtainconvergent spin asymmetries at all angles and energiesto better than 10%. Achieving significantly better con-vergence for the amplitudes, say to a 1% level, is likelyto require a large scaling up of our present calculationsbased on our experience [11]with the Poet model whichrequired 30 l = 0 states. Extrapolating to our currentmodels, it is reasonable to suggest that a 200CC calcula-tion could be required, which is well within the capacityof modern supercomputers.

ACKNOWLEDGMENTS

We thank Professor Ian McCarthy for many useful dis-

cussions. We would also like to acknowledge supportfrom the Australian Research Council.

' Electronic address: igoresm. cc.flinders. edu. aut Electronic address: stelboviatsibm. csu. murdoch. edu. au

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J. Du Croz, A. Greenbaum, S. Hammarling, A. McKen-ney, S. Ostroucnov, and D. Sorensen, LAPACK User'sGuide (Society for Industrial snd Applied Mathematics,Philadelphia, PA, 1992).

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convergent close-coupling calculations of electron

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