Inclusion of nonadiabiatic effects in calculations on...
13
Inclusion of nonadiabiatic effects in calculations on vibrational excitation of molecular hydrogen by low-energy electron impact S. Mazevet, 1 Michael A. Morrison, 1, * Olen Boydstun, 1 and R. K. Nesbet 2 1 Department of Physics and Astronomy, The University of Oklahoma, 440 West Brooks Street, Norman, Oklahoma 73019-0225 2 IBM Almaden Research Center, 650 Harry Road, San Jose, California 95120-6099 ~Received 2 September 1998! The nonadiabatic phase matrix method offers a unified, systematic treatment of vibrational dynamics in calculations of low-energy inelastic electron-molecule cross sections. This formalism uses fixed-nuclei R matrices to describe the region of configuration space near the target but—unlike its fully adiabatic counterpart, the energy-modified adiabatic method—includes nonadiabatic effects, which are important for resonant scat- tering and near a vibrational threshold. A most stringent test of this method is e -H 2 scattering below 10 eV, where elastic and inelastic cross sections exhibit an enhancement around 3 eV which at the fixed-nuclei level involves a range of physical effects, from nonresonant to resonant scattering, as the internuclear separation varies from the smallest to largest relevant values. Here we describe an implementation of this method appropriate to such systems, an assessment of its accuracy for e -H 2 scattering, and an appraisal of the impor- tance of nonadiabatacity for the 0 →1 and 0 →2 vibrational excitations. @S1050-2947~99!08901-5# PACS number~s!: 34.80.Bm, 34.80.Gs I. INTRODUCTION Electron scattering by the hydrogen molecule has been studied extensively by both theorists and experimentalists ~for reviews and references, see Refs. @1–3#!. Although H 2 is the simplest of all neutral molecular targets, the presence of a fixed-nuclei shape resonance around 3 eV whose width varies drastically with internuclear separation presents a con- siderable challenge to the theory of vibrational excitation from the ground state @4#. Direct calculation of the 2 S u com- plex potential-energy function, using quantum chemistry techniques such as the complex self-consistent-field method @5,6#, indicate that this resonant state interacts strongly with the electron-scattering continuum for small values of inter- nuclear separation, but becomes bound for internuclear sepa- rations around 3 a 0 @7#. This 2 S u resonance enhances inelastic cross sections for excitation to low-lying vibrational states, induces a strong resonant structure in the cross sections for excitation to higher vibrational states @8#, and is primarily responsible for dissociative attachment into H and H 2 @3#. Different theoret- ical formalisms and computational methods have been used to treat various consequences of this resonance, each adapted to a particular situation. For excitation to low-lying vibra- tional states, modified adiabatic methods such as the first- order nonadiabatic method @9,10# and the energy-modified adiabatic phase matrix method ~EMAP!@11–13#, which al- low for the transfer of kinetic energy to nuclear motion dur- ing an inelastic process but neglect explicitly nonadiabatic effects, are appropriate. However, for deep inelastic scatter- ing or for dissociative attachment, it is expected that a fully nonadiabatic theory such as vibrational close coupling is necessary, in order to describe the relatively strong coupling between electronic and vibrational motion. Due to the diffi- culty of including continuum channels in a close-coupling formalism, even for e -H 2 scattering, alternative methods such as the projection operator method @3,14,15# have been used. This paper concerns the extension to extremely broad resonances such as the one in the e -H 2 system of an alterna- tive treatment, the nonadiabatic phase matrix ~NADP! method @16–18,13#. The NADP method has previously been applied to the narrower resonance structures observed in low-energy e -N 2 scattering @18#. Its advantage for electron-impact vibrational excitation of e -H 2 in the 2 S u symmetry @13# is that this method includes nonadiabatic physics within a formulation which consistently treats the entire range of physical effects that influence e -H 2 cross sections as the internuclear separa- tion R varies, from strongly resonant scattering at large R to purely background scattering at small R. The NADP method derives this advantage from the use of R matrices to treat the region near the target. ~For an alter- native R-matrix-based formalism for the inclusion of nona- diabatic effects, see Refs. @19,20#.! In a NADP calculation, body-frame fixed-nuclei R matrices are used to define phase matrices at each R. Each phase matrix is decomposed addi- tively into foreground part and background parts. The latter is then converted to a vibronic background phase matrix us- ing the EMAP approximation @21#. Following this method- ology, the foreground fixed-nuclei phase matrix is also con- verted to a foreground vibronic phase matrix by replacing functional forms determined by the electron continuum en- ergy by matrix expressions defined by an operator which explicitly contains the vibrational kinetic-energy operator. The NADP procedure per se need be applied only to the resonant symmetry; nonresonant symmetries can be treated with the EMAP method. The case of e -H 2 scattering is par- ticularly appropriate to calibrate this methodology because of the diversity of the effects in 2 S u scattering below 10 eV. In Sec. II we outline the theoretical foundation of the EMAP and NADP methods and the role of the R matrix in *Electronic address: [email protected] PHYSICAL REVIEW A JANUARY 1999 VOLUME 59, NUMBER 1 PRA 59 1050-2947/99/59~1!/477~13!/$15.00 477 ©1999 The American Physical Society
Transcript of Inclusion of nonadiabiatic effects in calculations on...
225
PHYSICAL REVIEW A JANUARY 1999VOLUME 59, NUMBER 1
Inclusion of nonadiabiatic effects in calculations on vibrational excitation of molecular hydrogenby low-energy electron impact
S. Mazevet,1 Michael A. Morrison,1,* Olen Boydstun,1 and R. K. Nesbet2
1Department of Physics and Astronomy, The University of Oklahoma, 440 West Brooks Street, Norman, Oklahoma 73019-02IBM Almaden Research Center, 650 Harry Road, San Jose, California 95120-6099
~Received 2 September 1998!
The nonadiabatic phase matrix method offers a unified, systematic treatment of vibrational dynamics incalculations of low-energy inelastic electron-molecule cross sections. This formalism uses fixed-nucleiRmatrices to describe the region of configuration space near the target but—unlike its fully adiabatic counterpart,the energy-modified adiabatic method—includes nonadiabatic effects, which are important for resonant scat-tering and near a vibrational threshold. A most stringent test of this method ise-H2 scattering below 10 eV,where elastic and inelastic cross sections exhibit an enhancement around 3 eV which at the fixed-nuclei levelinvolves a range of physical effects, from nonresonant to resonant scattering, as the internuclear separationvaries from the smallest to largest relevant values. Here we describe an implementation of this methodappropriate to such systems, an assessment of its accuracy fore-H2 scattering, and an appraisal of the impor-tance of nonadiabatacity for the 0→1 and 0→2 vibrational excitations.@S1050-2947~99!08901-5#
PACS number~s!: 34.80.Bm, 34.80.Gs
eelis
eido
ion
trthit
erep
fng
r
sap-rs
r-ti
ttell
linffi
ngs
road-
the
al
ionctsra-
of
a-
sedi-terus-
n-ingn-ichr.
ated
of
e
I. INTRODUCTION
Electron scattering by the hydrogen molecule has bstudied extensively by both theorists and experimenta~for reviews and references, see Refs.@1–3#!. Although H2 isthe simplest of all neutral molecular targets, the presenca fixed-nuclei shape resonance around 3 eV whose wvaries drastically with internuclear separation presents a csiderable challenge to the theory of vibrational excitatfrom the ground state@4#. Direct calculation of the2Su com-plex potential-energy function, using quantum chemistechniques such as the complex self-consistent-field me@5,6#, indicate that this resonant state interacts strongly wthe electron-scattering continuum for small values of intnuclear separation, but becomes bound for internuclear srations around 3a0 @7#.
This 2Su resonance enhances inelastic cross sectionsexcitation to low-lying vibrational states, induces a stroresonant structure in the cross sections for excitationhigher vibrational states@8#, and is primarily responsible fodissociative attachment into H and H2 @3#. Different theoret-ical formalisms and computational methods have been uto treat various consequences of this resonance, each adto a particular situation. For excitation to low-lying vibrational states, modified adiabatic methods such as the fiorder nonadiabatic method@9,10# and the energy-modifiedadiabatic phase matrix method~EMAP! @11–13#, which al-low for the transfer of kinetic energy to nuclear motion duing an inelastic process but neglect explicitly nonadiabaeffects, are appropriate. However, for deep inelastic scaing or for dissociative attachment, it is expected that a funonadiabatic theory such as vibrational close couplingnecessary, in order to describe the relatively strong coupbetween electronic and vibrational motion. Due to the di
*Electronic address: [email protected]
PRA 591050-2947/99/59~1!/477~13!/$15.00
nts
ofthn-
yodh-a-
or
to
edted
t-
cr-
yisg
-
culty of including continuum channels in a close-coupliformalism, even fore-H2 scattering, alternative methodsuch as the projection operator method@3,14,15# have beenused. This paper concerns the extension to extremely bresonances such as the one in thee-H2 system of an alternative treatment, the nonadiabatic phase matrix~NADP!method@16–18,13#.
The NADP method has previously been applied tonarrower resonance structures observed in low-energye-N2
scattering@18#. Its advantage for electron-impact vibrationexcitation of e-H2 in the 2Su symmetry @13# is that thismethod includes nonadiabatic physics within a formulatwhich consistently treats the entire range of physical effethat influencee-H2 cross sections as the internuclear sepation R varies, from strongly resonant scattering at largeR topurely background scattering at smallR.
The NADP method derives this advantage from the useR matrices to treat the region near the target.~For an alter-native R-matrix-based formalism for the inclusion of nondiabatic effects, see Refs.@19,20#.! In a NADP calculation,body-frame fixed-nucleiR matrices are used to define phamatrices at eachR. Each phase matrix is decomposed adtively into foreground part and background parts. The latis then converted to a vibronic background phase matrixing the EMAP approximation@21#. Following this method-ology, the foreground fixed-nuclei phase matrix is also coverted to a foreground vibronic phase matrix by replacfunctional forms determined by the electron continuum eergy by matrix expressions defined by an operator whexplicitly contains the vibrational kinetic-energy operatoThe NADP procedureper seneed be applied only to theresonant symmetry; nonresonant symmetries can be trewith the EMAP method. The case ofe-H2 scattering is par-ticularly appropriate to calibrate this methodology becausethe diversity of the effects in2Su scattering below 10 eV.
In Sec. II we outline the theoretical foundation of thEMAP and NADP methods and the role of theR matrix in
477 ©1999 The American Physical Society
e-h-
Pdy
u
a
othody
ip
ar
ri
centoal
-
e
ooet
-
y
c-ves
in-y isotic
t
,e
ns
enalof
aseon
otic
ule
of
chrreso-
erix
nt
478 PRA 59MAZEVET, MORRISON, BOYDSTUN, AND NESBET
these approximations. Section III is concerned with thimplementation fore-H2 scattering. In Sec. IV we concentrate on tests of the validity of the NADP and EMAP metods for excitation of low-lying vibrational states of H2 . Herewe report a systematic comparison of EMAP and NADtotal and differential cross sections with benchmark boframe vibrational close-coupling calculations@22#. To ensuremeaningful comparisons among these methods, all calctions use the same Hartree-Fock ground-state H2 wave func-tion, the same representation of the static, exchange,correlation polarization terms in thee-H2 interaction poten-tial, and comparable numerical precision.
II. THEORY
In the continuum Born-Oppenheimer approximation@23#,where the vibrational and rotational kinetic energy operatare neglected in equations for the scattering function,coupled radial integrodifferential equations for an electrscattered by a diatomic molecule, written in the usual bofixed reference frame with thez axis coincident with theinternuclear axis, are@1#
F d2
dr22
l ~ l 11!
r 21kb
2Gul ,l 0~r ;R!
52(l 8
@Vl ,l 8~r ;R!1Vl ,l 8~r ;R!#ul 8,l 0~r ;R!, ~1!
where the semicolon denotes the parametric status ofR, thevalue of which is fixed in this approximation. The subscrl 0 denotes the entrance channel, andkb
2/25e is the energyof the projectile in the body frame~in Hartree units!. Thequantum numberl corresponds to the electronic angulmomentum of the scattering electron, andL to its projectionalong the internuclear axis. The coupling potential matelements are Vl ,l 8(r ;R) for the static plus ~local!correlation-polarization potential andVl ,l 8(r ;R) for thenonlocal exchange operator. We suppress the dependenall matrix elements, radial functions, and scattering quaties onL and on the parity of the system, it being understothat all fixed-nuclei quantities are referred to particular vues of these quantum numbers.
The fixed-nuclei R matrix for a particular electronmolecule symmetry, internuclear separationR, and electroniccontinuum energye, is defined by@24,25#
ul ,l 0~r 0 ;R!5(
l 8Rl ,l 8~e;R,r 0!F d
drul 8,l 0
~r ;R!Gr 5r 0
.
~2!
The sum over l 8 includes values consistent with thelectron-molecule symmetry under consideration~e.g., forthe Su symmetry, l 851,3, . . . ). In the EMAP methodfixed-nuclei matrices become operators in the nuclear cdinates @18#. Matrix elements of these operators are thevaluated between vibrational wave functions. In orderavoid integrating over poles, the fixed-nucleiR matrix is re-placed by thedimensionlessfixed-nuclei phase matrix defined by the matrix relation
ir
-
la-
nd
rse
n-
t
x
ofi-d-
r-no
tanF~e;R,r 0!5kb1/2R~e;R,r 0!kb
1/2, ~3!
wherekb is a diagonal matrix of~body-frame! wave numberskb5A2e. This definition differs from that given previousl@17# by the energy-dependent factors ofkb
1/2, which renderF dimensionless.
In conventional scattering theory for a spherical interation potential, phase shifts are defined relative to free wavia the usual asymptotic boundary conditions@26#. In con-ventional electron-molecule scattering theory, where theteraction potential is nonspherical, the analogous quantitthe eigenphase sum, which is calculated from the asymptK matrix ~see Ref.@27#, and references therein!. The fixed-nuclei phase matrixF(e;R,r 0), however, is absolute in thait contains the variation withR ande due to the free-electronphase matrix. Relation~2! indicates that, for a free electronthe R matrix is diagonal with elements proportional to thinverse logarithmic derivative of Ricatti-Bessel functiowith argumentl ,
Rl ,l 8FE
~e;r 0!5j l ~kbr 0!
j l 8~kbr 0!d l ,l 8 . ~4!
As a consequence of definition~3!, the free-electron phasmatrix is a diagonal matrix whose elements are proportioto the inverse tangent of the inverse logarithmic derivativeRicatti-Bessel functions with argumentl ,
F l ,l 8FE
~e;r 0
co
a
s
acc
tsnm
ised
innt
d
rix,een
i-oteingun-Pt to
an-ua-the
a-n-
icearens,
en-asisndr-
s isis-e a
tion
po-
ator
ed
nts
y-
ns
t toesset.ted
PRA 59 479INCLUSION OF NONADIABIATIC EFFECTS IN . . .
This precursor resonance analysis can be used to depose the phase matrixF(e;R,r 0) into a rapidly varying part,the foreground phase matrixF1(e;R,r 0), and a slowly vary-ing background part. The foreground phase matrix thensumes the form
F1~e;R,r 0!5y~R,r 0!tan21F g~R,r 0!
2@e res~R,r 0!2e#Gy†~R,r 0!.
~8!
The background phase matrix is defined by subtraction a
F0~e;R,r 0
iaso’tic
h
is
th
la
a
t
ry
feis
t
a
o
ctrr
-tiod
uthyu
eromn
-cle
inale
in
outcat-ex-l
ent-ofanese
-s
ns,
wereeethe
--
heeto
tly
clei
sum
ed-
480 PRA 59MAZEVET, MORRISON, BOYDSTUN, AND NESBET
by thedipole term in the moment expansion of this potent@39#; hence we have adopted the form christened by Giband Morrison @34,39# the ‘‘better-than-adiabatic dipole’~BTAD! potential. This potential is parameter free. Analyforms for the equilibrium BTAD potentials fore-H2 aregiven in Eqs.~6! and ~7! of Ref. @41#. From the asymptoticform of this potential at eachR we extract the polarizabilitiesa0(R) and a2(R). The averages of these values over tground vibrational state are 5.4367a0
3 and 1.4575a03 , respec-
tively. These polarizabilities can be assessed by comparto the experimental values (5.426360.02)a0
3 for the spheri-cal polarizability, as determined from measurements ofrefractive index at 290 K@42#, and (1.356760.0023)a0
3 forthe nonspherical polarizability, as determined in molecubeam resonance experiments@43#.
We calculate the vibrational wave functions in the vibrtional close-coupling~VCC! and adiabatic~EMAP, NADP!calculations reported here from the Morse potential
V~R!5De~e22ax22e2ax!, ~16!
where x[(R2Re)/Re . For H2 , the dissociation energy aequilibrium is De50.1819Eh , and the parametera, deter-mined from measured spectroscopic constants@44#, is a51.4110.
Fixed-nuclei R matrices are constructed at a radiusr 0510.0a0 for a grid of 45 electronic energies which vafrom 0.005 to 10.0 eV. For H2 , this R-matrix radius ensuresthat the molecular bound-state wave functions have eftively vanished atr 0 , and consequently that exchangefully included in the inner region. For scattering in thePuandSg symmetries, fixed-nucleiR matrices are calculated ainternuclear separations 0.5a0 , 0.8a0 , 1.0a0 , 1.2a0 , 1.4a0 ,1.6a0 , 1.8a0 , 2.0a0 , 2.2a0 , 2.4a0 , and 2.5a0 . To accountfor the rapid variation of the fixed-nuclei phase matrix asfunction of internuclear separation in theSu symmetry, ad-ditional fixed nuclei R matrices are included for internucleseparations 0.7a0 , 0.9a0 ,1.1a0 , 1.3a0 , 1.5a0 ,1.7a0 , 1.9a0 ,2.1a0 , and 2.3a0 .
At each value of the fixed-nuclei electronic energy andthe internuclear separation, the fixed-nucleiR matrices, mul-tiplied by an appropriate energy factor as in Eq.~3!, arediagonalized, and fixed-nuclei phase matrices are construfrom the inverse tangents of the eigenvalues using the cosponding eigenvectors. Multiples ofp are added to the resulting eigenvalues in order to ensure a smooth variawith both energy and internuclear separation. If the fixenuclei R matrices have degenerate eigenvalues at particvalues of electronic energy and internuclear separation,additional multiples ofp are added over the whole energrange and for all internuclear separations, in order to ensa smooth variation of the phase matrix in regions whR-matrix eigenvalues exhibit avoided crossings. Far frsuch regions, the matrix of eigenvectors is close to a umatrix. In this situation, multiples ofp can be added independently to each of the eigenvalues, i.e., to the fixed-nuphase matrix. In contrast, when the eigenvalues of the fixnucleiR matrix exhibit avoided crossings, the correspondmatrix of eigenvectors deviates significantly from a unit mtrix. In such situations, the independent addition of multip
ln
e
on
e
r-
-
c-
a
r
f
ede-
n-
laren
ree
it
eid-g-s
of p to each inverse tangent of the eigenvalues resultsadding nonintegral multiples ofp to the phase matrix.
Outside theR-matrix region (r .r 0), powerful computa-tional methods facilitate solving the scattering equationsto a matching radius in the asymptotic region where the stering matrix can be extracted. In this outer region, weplicitly allow for vibrational coupling, including vibrationastates in the coupled equations forv<3. Here the interactionpotential is local, a combination of terms due to permanmultipole moments of the target~predominantly the quadrupole interaction! and terms due to polarization distortionsthe target induced by the projectile, which in this region cbe considered to move adiabatically with respect to thdistortions @45#. For low-energye-H2 scattering, one needinclude only the induced polarizability interaction@34#. Thevibronic R matrix @Eq. ~15!# at r 15100a0 is calculated fromits counterpart at the boundary of theR-matrix region r 0using standardR-matrix propagation methods@46–49#. Forlarge enough radial values (r .r 1.r 0), these adiabatic polarization potentials reduce to their analytic multipolar form@39#. In this outermost region, beyondr 1 , the coupled scat-tering equations can be solved by asymptotic expansiowhich are evaluated analytically by converting the~diver-gent! asymptotic series to continued fractions@50–52#.
To complete the description of the scattering matrix,require elements for symmetries other than the lowest th(Sg , Su , andPu), and for partial waves within these thresymmetries of order higher than are required to convergefixed-nucleiR matrix at r 0 . ~In the present fixed-nuclei calculations forr<r 0 we include four partial waves per symmetry.! We calculate these additionalK-matrix elements us-ing the first Born approximation which, because of tcentrifugal barrier term in the effective Hamiltonian for thscattering electron, gives accurate approximationsK-matrix elements at any scattering energy for sufficienhigh l ~for further discussion, see Refs.@53# and @54#!.
Figure 1 shows the variation in theSu symmetry of theprecursor eigenphase sum as a function of the fixed-nuelectronic energye and internuclear separationR. We em-phasize that this quantity is not the physical eigenphaseone would calculate from the asymptoticK matrix @28#;rather it is the analogous quantity calculated from the fixnuclei phase matrix at theR-matrix boundaryr 0510a0 . This
FIG. 1. The precursor eigenphase sum fore-H2 scattering in theSu symmetry evaluated from fixed-nuclei phase matrices atr 0
510a0 .
-e
rroo
seo-
thrn-T
l t
ts
-omon
ctor
aseatde-en-twoum,ced
isd is
tic
he
henap-
on-atrix
thehendhe.sorob-
rgyallig.
t thesheits
ceal
and
n-d
onrac-
ofto
e
ato
PRA 59 481INCLUSION OF NONADIABIATIC EFFECTS IN . . .
quantity is a smooth function of bothe andR. For each valueof the internuclear separation, theSu eigenphase sum increases rapidly with increasing energy. This behavior ispecially pronounced at large values ofR, where the precursoeigenphase sum shows a rapid variation for energies fthreshold to around 1.0 eV. This characteristic behaviorthe eigenphase sum above threshold and at internuclearrations near 2.5a0 corresponds to a true fixed-nuclei resnance whenr 0 is extended into the asymptotic region.
As the internuclear separation decreases, however,precursor resonance structure vanishes for values of inteclear separation less than 2.0a0 ; there the precursor eigenphase sum tends to a smooth function of the energy.understand the behavior of this quantity better, it is usefuconsider its counterpart for a free electron.
In Fig. 2~a!, we show the variation of diagonal elemenof the free-electron phase matrix calculated atr 510a0 forpartial wave ordersl 51, 3, and 5, the dominant contributions to theSu precursor eigenphase sum. As expected frEq. ~5!, the diagonal elements of the phase matrix are n
FIG. 2. ~a! Free-electron eigenphase shifts and their sum~solidline! for the dominant partial waves of scattering in theSu symme-try, l51, 3, and 5.~b! Fixed-nucleiSu precursor eigenphase sumsthe extreme values of internuclear separation that are relevant t0→1 and 0→2 excitations,R50.5a0 andR52.5a0 . Also shownis the sum of the free-electron eigenphases~open circles! from ~a!~solid line!.
s-
mfpa-
isu-
oo
-
linear functions of the energye that exhibit characteristicbehavior near threshold due to the energy-dependent fak5A2e. In Fig. 2~b!, we compare theSu eigenphase sumfor our smallest and largest values ofR, 0.5a0 and 2.5a0 , tothe sum of the diagonal elements of the free-electron phmatrix to theSu symmetry. This comparison indicates ththe variation of the precursor eigenphase sum is largelytermined by that of the corresponding free-electron eigphase sum. By considering the difference between thesequantities, we see that, like the asymptotic eigenphase sthe precursor eigenphase sum of Fig. 2 shows a pronounresonance structure near threshold for large values ofR. Bycontrast, for small values ofR the precursor resonancesuperimposed on a strongly decreasing background andifficult to characterize.
This point is reinforced in Fig. 3~a!, which shows thevariation of themodifiedphase matrix~the fixed-nuclei phasematrix minus the free-electron matrix! as a function ofe andR. This variation closely resembles that of the asymptoeigenphase sums calculated from theSu K matrix @13#. It isimportant to recall that the quantity shown in Fig. 3~a! de-pends on theR-matrix radius, and as such only includes teffect of the potential in the inner region,r<r 0 . Figures 3~b!and 3~c!, respectively, show the variation withR ande of thebackground and foreground phase matrices obtained wthe precursor resonance analysis described in Sec. II isplied to the modified phase matrix. These graphs demstrate that this analysis leads to a background phase mwhich varies smoothly withR ande, while the rapidly vary-ing part of the modified phase matrix is described byforeground phase matrix. This point further illustrates treason for using two different methods—the EMAP aNADP approximations—to treat vibrational dynamics in tbackground and foreground phase matrices, respectively
The variation with internuclear separation of the precurresonance energy curve and the corresponding widthtained by the resonance analysis are shown in Figs. 4~a! and4~b!. Figure 4~a! shows that the precursor resonance enecurve differs from the potential of the target molecule forinternuclear separations. The corresponding width in F4~b! increases rapidly with decreasingR, and the resonancefades smoothly into the background and ceases to affecvibrational dynamics significantly.Yet at these small valueof the internuclear separation the decomposition of tphase matrix continues to be well defined, even thoughphysical effects are negligible.Electron scattering from H2 isprototypical of situations in which a fixed-nuclei resonanhas a minimal influence on excitation to the first vibrationstate. A comparison betweenv050 to v51 cross sectionsfrom NADP and EMAP calculations in theSu symmetry istherefore particularly useful as a test of this assumptionwill be the subject of the next section.
Finally, we emphasize that the computational implemetation of relation~12! requires special care due to avoidecrossing in the eigenvalues of the vibronic foregroundR ma-trix. It should be noted that this inconvenience dependsthe system as well as on assumptions inherent in the intetion potential. It arises in the present application becausethe large range of energies under consideration, from 010.0 eV. To overcome this difficulty, irregularities in thforeground vibronic phase matrix~12! are identified by com-parison to the equivalent transformation
the
gy
to
vi-ant
se
ialter-C
and
g5
o-
c-al-
482 PRA 59MAZEVET, MORRISON, BOYDSTUN, AND NESBET
Fv,v81
~E!5(s,s8
tan21^fvuy~R,r 0!uxs&
3^xsuF g~R,r 0!
~Es2E!1~Es82E!G uxs8&
3^xs8uy†~R,r 0!ufv8&. ~17!
When found, irregular elements of the phase matrix are replaced by values from this relation. Equation~17! is obtainedby twice introducing the complete set of eigenfunctions
FIG. 3. ~a! Modified precursor eigenphase sums fore-H2 scat-tering in theSu symmetry, evaluated from fixed-nuclei phase ma-trices at r 0510a0 by subtracting the free-electron phase matrixfrom the fixed-nuclei precursor eigenphase matrix. This phase mtrix is decomposed into~b! background and~c! foreground parts, asdescribed in the text.
-
$xs(R)% in Eq. ~8!, and by choosing as the continuum enerthe arithmetic mean of the eigenvalues of the operatore res
2 e. Despite this approximation, Eq.~17! allows a directconstruction of the vibronic phase matrix without the needdiagonalize the correspondingR matrix. Where valid, thisalternative procedure thus allows a construction of thebronic foreground phase matrix even if the vibronic resonR-matrix eigenvalues manifest avoided crossings.
IV. RESULTS
In this section we investigate the validity of the phamatrix decomposition~9! and of the NADP method via asystematic comparison of NADP integral and differentcross sections against results from its fully adiabatic counpart, the EMAP method, and from fully nonadiabatic VCcalculations. We consider excitations to thev51 and 2states. The thresholds for these states are 0.51561.001 127 eV, respectively@44#.
A. Integral cross sections
Partial integral cross sections in the dominante-H2 sym-metries (Sg , Su , andPu) and their sum, as calculated usinthe NADP, EMAP, and VCC methods are shown in Fig.
a-
FIG. 4. ~a! The effective potential energy for the precursor resnance~solid line!: the sum ofe res(R,r 0) and the ground electronicstate potential energy of H2 , represented by a Morse potential~longdashed line! ~b!. Variation of the precursor resonance width funtion g(R,r 0) with internuclear separation. Both quantities were cculated at theR-matrix radiusr 0510a0 .
ults
PRA 59 483INCLUSION OF NONADIABIATIC EFFECTS IN . . .
FIG. 5. Partial integral cross sections in the~a! Sg , ~b! Su , and~c! Pu symmetries for the 0→1 vibrational excitation of H2 and~d! theirsum. Results for the EMAP~short dashed line! and NADP~long dashed line! approximations are compared with benchmark VCC res~solid line!.
ahea
omn
leat
reA
rgicoil-tio
lyur-ia-tri-atic
-ionen-sent
heed,od
ionll
for v51 and in Fig. 6 forv52. These data are presentedselected energies in Table I; a full list is available from tauthors upon request. Comparison between the NADPEMAP cross sections in Figs. 5 and 6 validates the decposition of the fixed-nuclei phase matrix into foreground abackground parts even for small values of the internucseparation, where no physical resonance appears in~asymptotic! fixed-nuclei eigenphase sum. Furthermocomparison to the VCC results shows that both the EMand NADP methods provide a reliable description ofe-H2partial cross sections even in the acutely sensitive eneregime near threshold. Cross sections in this energy reare especially sensitive to conservation of energy in thelision; violation of this requirement is the reason for the faure near threshold of the usual adiabatic-nuclear-vibraapproximation@2,55,56#.
t
nd-
dr
he,P
gyonl-
n
The EMAP approximation does not account rigorousfor the loss of kinetic energy of the continuum electron ding excitation; this energy loss is properly included in adbatic theories that use off-shell fixed-nuclei scattering maces, such as the first-order nondegenerate adiabapproximation @9,10#. However, the EMAP method improves on the conventional adiabatic-nuclei approximatby calculating elements of the vibronic phase matrix atergies appropriate to each vibrational channel. In the preimplementation, the geometric mean~14! of the eigenvaluesof the operatore are used, so precise agreement of tEMAP and VCC results is not expected. It should be nothowever, that here, as in applications of the EMAP methdirectly to the asymptoticK matrix @13#, this procedure doesprovide a reliable description of the inelastic cross sectwithout imposing the computational burden of off-she
484 PRA 59MAZEVET, MORRISON, BOYDSTUN, AND NESBET
FIG. 6. Same as 5 for the 0→2 excitation.
nsmsm
C-m
iono
0thifrexete
forccu-
art,enal
undsti-his
setForu-
fixed-nuclei scattering calculations. For both excitatioVCC, NADP, and EMAP cross sections tend to the savalues for scattering energies large enough that the loskinetic energy of the continuum electron is negligible copared to the scattering energy, i.e., above about 7.0 eV.
For the 0→1 cross section in Fig. 5, comparison of VCand NADP results confirms the ability of the NADP to incorporate nonadiabatic effects for this excitation. The coparison between EMAP and VCC results for this excitatshows that nonadiabatic effects exert minimal influencethis cross section at energies near the enhancement at 3.
More interesting is the corresponding comparison of0→2 cross sections in Fig. 6. For this excitation larger dferences are evident between NADP, EMAP, and VCCsults, especially in the magnitude and position of the mamum in theSu and total cross sections. These differencclearly indicate that the fixed-nuclei resonance at large in
,eof-
-
neV.e--i-sr-
nuclear separations influences the 0→2 cross section morethan the 0→1 cross section—hence the need to accountnonadiabatic effects to describe the former process arately.
Although the maximum value of the 0→2 NADP crosssection appears in good agreement with its VCC counterpa mild dip is evident in the NADP result at energies betwe2.0 and 3.0 eV. Within the limits of the interaction potentiused to calculate the fixed-nucleiR matrices, which is basedon a Hartree-Fock representation of the electronic grostate wave function, the NADP method appears to overemate the influence of the nonadiabatic effects slightly in tenergy region.
The accuracy of these results is limited by the discreteof internuclear separations used in the calculations.NADP and VCC calculations, the largest values of internclear separation are 2.5a0 and 2.6a0 , respectively. Further-
trf
-o
int
-r-ghor
ns,f thes.e 0sar
ore
hodd
nglergyboththatad,
est-DP
tly
a
of
tly
t
nceef-andn-
th-lo-tic-ve
fllymst,l-of
’s
pri-
n
lei
r-
a
’’he
PRA 59 485INCLUSION OF NONADIABIATIC EFFECTS IN . . .
more, the NADP and EMAP methods are based on marelations and, for theSu symmetry, require a calculation ovibrational submatrices (v,v8) of the phase matrix for vibrational quantum numbers larger than that of the final statethe excitation of interest. Thus, for the 0→2 cross sectionsin Figs. 6, vibronic phase matrices were constructed usblocks forv andv8 from 0 to 3. This may introduce a sligherror, because, fore-H2 scattering, thev53 vibrational stateis appreciable atR52.5a0 . The errors due to renormalization of the vibrational wave function on the interval of intenuclear separations considered here, and to neglect of hivalues ofR where the fixed-nuclei resonance is even mpronounced, are difficult to assess.
TABLE I. Partial integral cross sections~in a02) and their sum
~the total 0→1 cross section! for e-H2 scattering at selected enegies from the following theories: vibrational close coupling~VCC:top line for each energy!, the nonadiabatic phase matrix~NADP!method~second line!, and the energy-modified adiabatic phase mtrix ~EMAP! approximation~third line!. The NADP method is re-quired only for theSu symmetry; the ‘‘NADP total cross sectionsin the last column therefore include EMAP contributions from tSg andPu symmetries.
E Method Sg Su Pu total
0.60 VCC 0.0126 0.0211 0.0013 0.0349NADP — 0.0316 — 0.0405EMAP 0.0081 0.0149 0.0008 0.0238
0.80 VCC 0.0222 0.1331 0.0063 0.1616NADP — 0.1522 — 0.1754EMAP 0.0187 0.1124 0.0045 0.1356
1.20 VCC 0.0327 0.5449 0.0181 0.5957NADP — 0.5578 — 0.6033EMAP 0.0316 0.5152 0.0138 0.5607
1.40 VCC 0.0361 0.8241 0.0239 0.8841NADP — 0.8451 — 0.8996EMAP 0.0359 0.7960 0.0186 0.8505
1.60 VCC 0.0387 1.1187 0.0292 1.1866NADP — 1.1500 — 1.2120EMAP 0.0391 1.0860 0.0232 1.1480
2.00 VCC 0.0425 1.5974 0.0382 1.6781NADP — 1.6000 — 1.6750EMAP 0.0438 1.5420 0.0314 1.6170
3.00 VCC 0.0471 1.8415 0.0502 1.9388NADP — 1.8740 — 1.9660EMAP 0.0495 1.8520 0.0429 1.9440
4.00 VCC 0.0486 1.5967 0.0505 1.6957NADP — 1.5980 — 1.6930EMAP 0.0514 1.5950 0.0439 1.6900
5.00 VCC 0.0488 1.2505 0.0449 1.3442NADP — 1.2530 — 0.1344EMAP 0.0518 1.2550 0.0393 1.3460
7.00 VCC 0.0477 0.7464 0.0301 0.8242NADP — 0.7532 — 0.8304EMAP 0.0506 0.7541 0.0266 0.8313
8.00 VCC 0.0469 0.5885 0.0238 0.6592NADP — 0.5940 — 0.6648EMAP 0.0497 0.5946 0.0211 0.6654
ix
f
g
ere
B. Differential cross sections
Because differential cross sections~DCS’s! are more sen-sitive to the scattering dynamics than integral cross sectiothey more clearly reveal the strengths and weaknesses oapproximations inherent in the NADP and EMAP methodFigures 7 and 8 show DCS’s at selected energies for th→1 and 0→2 excitations. The most significant differencein the 0→1 DCS’s of Fig. 7 appear at 0.8 eV, an energy nethe threshold of this excitation at 0.516 eV. Figure 7~a!shows that at 0.8 eV nonadiabatic effects are somewhat mimportant for scattering anglesu>90 ° than at smallerangles. Throughout the angular range, the NADP metslightly overestimate the DCS’s, while the EMAP methounderestimates them, most noticeably foru,90 °. These dif-ferences are washed out by integration over scattering ain the integral cross sections shown in Fig. 5. As the eneincreases and nonadiabatic effects become negligible,methods reproduce the VCC cross sections well. Notethis statement holds even at 3.0 eV, the peak of the broweak resonance in this cross section.
More pronounced differences are evident at the lowenergy in Fig. 8, the 0→2 DCS’s at 1.6 eV. Both approximations underestimate these cross sections, with the NAresult closer to the VCC values at all angles. At a slighhigher energy, 2.0 eV in Fig. 8~b!, the NADP approximationproves excellent while the EMAP approximation still givesresult below the VCC cross section. By 3.0 eV, in Fig. 8~c!,all three methods give identical DCS’s. But the levelagreement at this energy is slightly misleading. As Fig. 8~d!shows, the EMAP and NADP cross sections move slighabove the VCC result except near the (p-wave! maximum at90 °, where all three agree.
With still further increases in energy~not shown!, theEMAP and NADP 0→2 DCS’s rapidly come into agreemenwith the VCC results, as happens in Fig. 7 for 0→1 DCS’sat and above 1.6 eV. For both excitations, this concurresimply reflects the relative unimportance of nonadiabaticfects for these excitations at energies above a few eV,the increasing validity of the approximate treatment of eergy conservation inherent in the NADP and EMAP meods. This concurrence is not surprising in light of the anagous phenomenon in DCS’s from conventional adiabanuclear-vibrational calculations at energies well abothreshold@56#.
At all energies, 0→2 DCS’s show the influence op-wave Su scattering in their shapes, which are essentiasymmetric about 90 °, although minor contributions froother partial waves are evident below 3.0 eV. By contrathe 0→1 DCS’s in Fig. 7 exhibit more substantial partiawave mixing at all energies. Even at 3.0 eV, the energythe maximum in the 0→1 integral cross section, the DCSreveal that the scattering is not purelyp wave, indicatingminimal influence of the large-R fixed-nuclei resonance onthis cross section, and suggesting that it may be inapproate to think of it as ‘‘resonant scattering.’’
Becausef2(R), the final-state vibrational wave functiofor the 0→2 excitation, is appreciable at larger values ofRthan isf1(R), the DCS’s for the 0→2 excitation are moreinfluenced by thep-wave shape resonance in the fixed-nucSu S matrix for largeR than are those for the 0→1 excita-
-
486 PRA 59MAZEVET, MORRISON, BOYDSTUN, AND NESBET
FIG. 7. Differential cross sections for the 0→1 vibrational excitation of H2 at ~a! 0.8, ~b! 1.6, ~c! 2.0, and~d! 3.0 eV. Results for theNADP ~long-dashed line! and EMAP~short-dashed line! approximations are compared with benchmark VCC results~solid line!.
na
th
s
ter-hys-
in
tion. Consequently, the shape of the 0→2 DCS’s moreclearly manifest the symmetric shape of ap-wave resonance~especially near 3.0 eV! than do the 0→1 DCS. This obser-vation is consistent with the increased importance of nodiabatic effects in this excitation evident in Fig. 6.
Such effects become significantly more important asfinal-state vibrational quantum number increases@8#. To il-
-
e
lustrate this effect, we conclude with Fig. 9, which showintegral cross sections for the 0→3 excitation in H2 . TheNADP cross sections show clear evidence of the characistic near-resonant structures found when nonadiabatic pics is important@15,14#. This structure is wholly absent fromthe EMAP cross sections, further verifying their originnonadiabaticity.
PRA 59 487INCLUSION OF NONADIABIATIC EFFECTS IN . . .
FIG. 8. Differential cross sections for the 0→2 vibrational excitation of H2 at ~a! 1.6, ~b! 2.0, ~c! 3.0, and~d! 3.4 eV. Results for theNADP ~long-dashed line! and EMAP~short-dashed line! approximations are compared with benchmark VCC results~solid line!.
in
ap
n ofef-fhere
i-
V. CONCLUSIONS
The NADP method has previously proven its mettlecalculations on resonante-N2 scattering @18#, where thefixed-nuclei eigenphase sum exhibits an unambiguous shresonance at all relevant internuclear separations. The
peur-
pose of the present paper is to calibrate a modified versiothe method which systematically includes nonadiabaticfects for systems in which the range of relevant values oRencompasses quite different scattering mechanisms—ranging from nonresonant scattering at smallR to resonantscattering at largeR. The key equations of the present mod
f t-
n
ixefnrethcto
to-e
u
e ofd.lo
in
he
a
ser-gyualu-dureP, isap-po-atenal
ral
m-lts,ionstona-
ng-eschDPlu-
-I.
nce
488 PRA 59MAZEVET, MORRISON, BOYDSTUN, AND NESBET
fied NADP method are the redefinition@Eq. ~3!# of the fixed-nuclei phase matrix as dimensionless, and the removal ofree-electron phase matrix~5! prior to the precursor resonance analysis of the foreground phase matrix~8!.
The key step in the NADP formalism is the replacemeof the fixed-nuclei electronic energye by the operatore[E2H(v) in the fixed-nuclei foreground phase matrF1(e;R,r 0). This replacement introduces nonadiabaticfects into a procedure that, throughout the inner regior<r 0 , is based solely on fixed-nuclei quantities; it therefoallows a transfer of energy between the kinetic energy ofvibrational motion and the continuum energy of the projetile. In e-H2 scattering, such an energy transfer is vitalhigh-lying vibrational excitations (v050→v>3) and to dis-sociative attachment.
What enables construction from the resulting operafunction F1( e;R,r 0) of the vibronic foreground phase matrix at r 0 is the introduction via closure of the complete s
$xs(R)% of eigenfunctions ofH(v)1e res(R,r 0). The effectivepotential energy for these vibrational eigenfunctions, the s
FIG. 9. Partial cross section in theSu symmetry for the 0→3excitation of H2 as calculated in the NADP~solid curve! and EMAP~dashed curve! approximations.
y
he
t
-
e-
r
t
m
of the ground-state electronic energy of H2 and the fixed-nuclei precursor resonance energy, is the potential curvFig. 4, the key computational device of the NADP methoThe expansion basis$xs(R)% corresponding to this potentiatherefore represents the effective response of the nuclei tallthe electrons, bound and continuum, in the system@57#.
Two limitations of the NADP method should be keptmind. First, although the method exploits anR-matrix for-malism to connect to formally exact VCC theory outside tR-matrix boundaryr 0 , it invokes simplifying approxima-tions insider 0 . Whereas the VCC method is limited bynecessarily incomplete vibrational basis~which omits the vi-brational continuum!, the NADP method replaces functionof operators by simplified algebraic forms that neglect ctain commutators involving the rovibrational kinetic-eneroperator. Thus it is difficult to assess the formal residerrors in a NADP result except by the type of detailed nmerical comparisons presented here. Second, the procefor evaluating off-diagonal matrix elements in the EMAmethod, which we use for the background phase matrixrather arbitrary, and the resulting formulas may not bepropriate for scattering near threshold, where long-rangetentials play an important role. This point requires separstudy in particular cases, as has been done for rotatioeffects in dipolar scattering@18#.
The thrust of the comparisons of NADP and VCC integand differential cross sections for the 0→1 and 0→2 exci-tations in Sec. IV are, first, to validate the method by deonstrating concurrence with fully nonadiabatic VCC resuand, second, to probe the breakdown of these approximatelsewhere. Introducing the fully adiabatic EMAP resultsthis picture reveals, especially in DCS comparisons, theture and importance of nonadiabatic effects for low-lyivibrational excitations of H2 . These effects, of course, become far more important for excitation to higher-lying statand for dissociative attachment. It is in the study of suprocesses, the next stage of this research, that the NAmethod as formulated and calibrated here will be most vaable.
ACKNOWLEDGMENTS
We would like to thank Dr. Wayne K. Trail for his assistance with the VCC calculations reported here, Dr. BarrySchneider for useful discussions, and the National ScieFoundation for support under Grant No. PHY–9722055.
v.
D.-
s.
@1# N. F. Lane, Rev. Mod. Phys.52, 29 ~1980!.@2# M. A. Morrison, Adv. At. Mol. Phys.24, 51 ~1988!.@3# W. Domcke, Phys. Rep.208, 97 ~1991!.@4# R. K. Nesbet, Comments At. Mol. Phys.11, 25 ~1981!.@5# C. W. McCurdy and R. C. Mowrey, Phys. Rev. A25, 2529
~1982!.@6# E. DeRose, E. A. Gislason, and N. H. Sabelli, J. Chem. Ph
82, 4577~1985!.@7# T. Gorczyca and D. W. Norcross, Phys. Rev. A42, 5132
~1990!.@8# M. Allan, J. Phys. B18, L451 ~1985!.
s.
@9# M. A. Morrison, J. Phys. B19, L707 ~1986!.@10# M. A. Morrison, M. Abdolsalami, and B. K. Elza, Phys. Re
A 43, 3440~1991!.@11# H. T. Thummel, T. Grimm-Bosbach, R. K. Nesbet, and S.
Peyerimhoff, in Computational Methods for ElectronMolecule Collisions, edited by W. M. Huo and F. A. Gianturco~Plenum, New York, 1995!, Chap. 12, pp. 265–292.
@12# H. T. Thummel, R. K. Nesbet, and S. D. Peyerimhoff, J. PhyB 25, 4533~1992!.
@13# S. Mazevet, M. A. Morrison, and R. K. Nesbet, J. Phys. B31,4437 ~1998!.
n
.
v.
f.
r-
.
,
-
s.
PRA 59 489INCLUSION OF NONADIABIATIC EFFECTS IN . . .
@14# M. Berman, C. Mu¨ndel, and W. Domcke, Phys. Rev. A31,641 ~1985!.
@15# C. Mundel, M. Berman, and W. Domcke, Phys. Rev. A32,181 ~1985!.
@16# R. K. Nesbet, inCollision Processes of Ion, Positron, Electroand Photon Beams with Matter, Latin American School ofPhysics, edited by A. C. de Azevedo e Souzaet al. ~WorldScientific, New Jersey, 1991!, pp. 94–103.
@17# R. K. Nesbet, Phys. Rev. A54, 2899~1996!.@18# T. Grimm-Bosbach, H. T. Thu¨mmel, R. K. Nesbet, and S. D
Peyerimhoff, J. Phys. B29, L105 ~1996!.@19# M. L. D. Barry, I. Schneider, and P. G. Burke, J. Phys. B12,
L365 ~1979!.@20# B. I. Schneider, M. L. Dourneuf, and V. K. Lan, Phys. Re
Lett. 43, 1926~1979!.@21# R. K. Nesbet, Phys. Rev. A19, 551 ~1979!.@22# W. Sunet al., Phys. Rev. A52, 1229~1995!.@23# M. Shugard and A. U. Hazi, Phys. Rev. A12, 1895~1975!.@24# Atomic and Molecular Processes: An R-Matrix Approach, ed-
ited by P. G. Burke and K. A. Berrington~Institute of Physics,Bristol, 1993!.
@25# B. I. Schneider, in An R-Matrix Approach to Electron-Molecule Collisions, edited by W. M. Huo and F. A. Gianturco~Plenum, New York, 1995!, Chap. 8, pp. 213–226. See Re58.
@26# J. R. Taylor,Scattering Theory~Wiley, New York, 1972!.@27# M. A. Morrison and W. Sun, inComputational Methods for
Electron-Molecule Collisions~Ref. @11#!, Chap. 6, pp. 131–190.
@28# A. Hazi, Phys. Rev. A19, 920 ~1979!.@29# R. K. Nesbet,Variational Methods in Electron-Atom Scatte
ing Theory~Plenum, New York, 1980!.@30# F. T. Smith, Phys. Rev.118, 349 ~1960!.@31# R. K. Nesbet and T. Grimm-Bosbach, J. Phys. B26, L423
~1993!.@32# J. W. Moskowitz and L. C. Snyder, inMethods of Electronic
Structure Theory, edited by I. H. F. Schaeffer~Plenum, NewYork, 1977!, p. 387.
@33# T. Dunning, J. Chem. Phys.55, 3958~1971!.
@34# T. L. Gibson and M. A. Morrison, Phys. Rev. A29, 2479~1984!.
@35# N. F. Ramsey, Phys. Rev.87, 1075~1952!.@36# A. D. Buckingham, R. L. Disch, and D. A. Dunmur, J. Am
Chem. Soc.90, 3104~1968!.@37# W. K. Trail, M. A. Morrison, W. A. Isaacs, and B. C. Saha
Phys. Rev. A41, 4868~1990!.@38# W. K. Trail, Ph.D. thesis, University of Oklahoma, 1991~un-
published!.@39# T. L. Gibson, Ph.D. thesis, University of Oklahoma, 1982~un-
published!.@40# A. Temkin, Phys. Rev.107, 1004~1957!.@41# T. L. Gibson and M. A. Morrison, J. Phys. B14, 727 ~1981!.@42# A. C. Newell and R. C. Baird, J. Appl. Phys.36, 3751~1965!.@43# K. B. MacAdam and N. F. Ramsey, Phys. Rev. A66, 898
~1972!.@44# K. P. Huber and G. Herzberg,Molecular Spectra and Molecu
lar Structure IV: Constants of Diatomic Molecules~Van Nos-trand Reinhold, New York, 1979!.
@45# M. A. Morrison and P. J. Hay, Phys. Rev. A20, 740 ~1979!.@46# J. C. Light and R. B. Walker, J. Chem. Phys.65, 4272~1976!.@47# B. I. Schneider and R. B. Walker, J. Chem. Phys.70, 2466
~1979!.@48# K. L. Baluja, P. G. Burke, and L. A. Morgan, Comput. Phy
Commun.27, 299 ~1982!.@49# L. A. Morgan, Comput. Phys. Commun.31, 419 ~1984!.@50# M. Gailitis, J. Phys. B9, 843 ~1976!.@51# C. J. Noble and R. K. Nesbet, Comput. Phys. Commun.33,
399 ~1984!.@52# R. K. Nesbet, Comput. Phys. Commun.32, 341 ~1984!.@53# M. A. Morrison, A. N. Feldt, and D. Austin, Phys. Rev. A29,
2518 ~1984!.@54# R. K. Nesbet, C. J. Noble, and L. A. Morgan, Phys. Rev. A34,
2798 ~1986!.@55# F. H. M. Faisal and A. Temkin, Phys. Rev. Lett.28, 203
~1972!.@56# M. A. Morrison, A. N. Feldt, and B. C. Saha, Phys. Rev. A30,
2811 ~1984!.@57# B. I. Schneider, Phys. Rev. A14, 1923~1976!.
