Ab initio study of the resonant electron attachment to the F[sub 2] molecule

JOURNAL OF CHEMICAL PHYSICS VOLUME 117, NUMBER 23 15 DECEMBER 2002

Ab initio study of the resonant electron attachment to the F 2 moleculeV. Brems,a) T. Beyer, and B. M. NestmannTheoretical Chemistry Department, University of Bonn, Wegelerstraße 12, D-53115 Bonn, Germany

H.-D. Meyer and L. S. CederbaumTheoretical Chemistry Department, University of Heidelberg, Im Neuenheimer Feld 229,D-69120 Heidelberg, Germany

~Received 1 July 2002; accepted 23 September 2002!

Dissociative attachment to and vibrational excitation of diatomic molecules by electron impact isdiscussed within the projection operator approach. The present method lifts the assumption ofseparability of the discrete-continuum coupling termVde(R), i.e., it is no longer required to writeit as a product of a function depending on coordinateR and energye separately. The method isapplied to the2Su resonant dissociative electron attachment to and vibrational electron excitation ofthe F2 molecule. The requiredab initio data have been computed using a recently developedalgorithm. This algorithm is based on the Feshbach–Fano partitioning technique and theR-matrixmethod~FFR!. The FFR method is discussed in the context of this particular application. ©2002American Institute of Physics.@DOI: 10.1063/1.1521127#

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I. INTRODUCTION

Experimental and theoretical investigations of resonelectron scattering processes off polyatomic molecules ingas phase are of great interest for modeling ionized gaseprecise knowledge of these processes is required for plaphysics,1 modeling of planetary atmospheres and the intstellar medium,2 and laser physics.3 More about moderntechnological, chemical and biological applications of renant electron attachment may be found in Ref. 4.

Much progress has been made in experimental studieelectron-molecule collisions in the gas phase.5–8 In particu-lar, electron beams are produced with a precision of ameV and enable a detailed analysis of the electron-molecollision process.9–11

Apart from the well-known stabilization12–14 andStieltjes imaging15,16 methods, several theoretical tools aavailable in order to model electron resonant processes fab initio data: ~1! the Feshbach–Fano partitioninmethod17,18 which has been applied to the treatment of tresonant Auger decay of core-excited molecules19 and ofPenning ionization,18,20 ~2! the complex absorbing potentiamethod21–23 which has been recently applied to (HF4

cluster24 and several long-lived dianions,25 ~3! the Schwingermultichannel variational method26 which has been recentlapplied to fluoromethanes and tetrafluoroethene,27,28 ~4! thecomplex Kohn method recently applied to CO2,29,30 and~5!the R-matrix theory31 which has been lately applied to Cl2O~Ref. 32! and H2.33 Moreover, new methods have been dveloped in order to determine the energy-dependent widttemporary anions fromL2 ab initio methods.34 Numerousreferences to recent publications in the field may be foun

a!Author to whom correspondence should be addressed. On leavethe Theoretical Chemistry Department, University of Heidelberg,Neuenheimer Feld 229, D-69120 Heidelberg, Germany. Electronic [email protected]

10630021-9606/2002/117(23)/10635/13/$19.00

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the literature listed above. A larger review is availableRefs. 35 and 36.

In particular, a method has been developed to applyFeshbach–Fano partitioning scheme to theR-matrixmethod.37 We will refer to this method as the FeshbachFano-R-matrix ~FFR! method. This method has been applito several molecules at the equilibrium geometry of tground state (N2 , C3H6 , and N2O)37 and to the CF3Cl1e2 system.38–40The application of the FFR method to thcomputation of the dissociative electron attachment~DA!and vibrational excitation~VE! cross sections of F2 , which isthe topic of this article, is the second application of the Fmethod, taking the nuclear dynamics into account.

The FFR algorithm makes possible to extract from tR-matrix results the resonant potential curve and the cosponding coupling terms with the electronic continuuwhich are required in order to perform the subsequnuclear dynamics calculations. For each energye of the col-liding electron and each internuclear distanceR, thisab initiomethod determines the coupling termsVde(R) ~8! whichcouple the discrete stateuwd& ~which is a square-integrablrepresentation of the resonance! to the continuum states~which are orthogonal to the discrete state and corresponthe background scattering processes!.

In case Vde(R) is small and smoothly dependent oe,41–43 the width of the resonance in the fixed-nuclei~FN!approach, is approximatively given by44 @Eq. ~4.33!#

G resFN~R!'2puVded~R!~R!u2, ~1!

with ed(R) ~58! the vertical target electronic excitation energy to the discrete stateuwd&. The definition of the complexSiegert FN resonance pole within the Feshbach–Fano frawork is discussed in Sec. IV A.

The computation of DA and VE cross sections requia precise investigation of the nuclear dynamics subseqto the electron collision. A very efficient algorithm, th

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10636 J. Chem. Phys., Vol. 117, No. 23, 15 December 2002 Brems et al.

Schwinger–Lanczos~SL! method, has been applied bHoracek and co-workers45–47to this problem. This algorithmhas been optimized on molecular systems48,49~and referencestherein! using theR-e-separability ansatz ofVde(R), i.e., thatVde(R)5 f (e)g(R)44 @Eq. ~4.61!#. In the case the dipole moment of the target depends onR,50 it is possible to extend thisapproach by using the slightly different ansatzVde(R)5 f (e;R)g(R). Both functionsf (e) and f (e;R) are analyti-cal expressions obeying the Wigner threshold rules51 and arechosen such that their Hilbert transform is knowanalytically.44

The results presented in this article are the applicatiof a code extending Hora`cek’s method beyond the two limitations mentioned above@separability ofVde(R) and analyti-cal expression off (e)]. This has been made possible bimplementing a numerical Hilbert transform algorithm pulished elsewhere.41 This progress is important since the FFmethod providesab initio coupling terms which may be noeasily fitted to aR-e-separable analytical function. A way tbypass this difficulty has been published elsewhere.40,52 Theapproach which is proposed in this work is more genesince it does not make any hypothesis on the separabilitVde(R) with respect to the variablesR ande.

The fluorine molecule, as well as most of the dihalogeis of particular technological interest~see, e.g., Ref. 53!.Nevertheless, very few experiments have been done withspect to the DA process. It has been observed that thecross sections of F2 ,54–57Cl2 ,58 and I2 ~Ref. 59! behave likee21/2 at the thresholde50. This result could not yet be successfully interpreted by theoreticians.44,60–69Possible expla-nation schemes have been listed in Ref. 6. Very recenexperimental results70 have demonstrated that the DA crosections of Cl2 are proportional toe1/2 for e,50 meV. Thisobservation sheds a new light on the previous resultsmotivates new investigations of the electron-dihalogen cosion reactions.

The F2 molecule~like Cl2 , Br2 , or I2) exhibits a curvecrossing between the neutral ground state potential andanionic ground state potential which is located very closethe equilibrium geometry of the neutral ground state pottial. Hence, the electron attachment process to the nemolecule in the vibrational ground state can be expectebe sensitive with respect toVde(R) for smalle andR close tothe crossing point. The study of the DA and VE of F2 con-sequently requires a careful investigation taking thresheffects into account on the basis ofab initio data.

The projection-operator approach is briefly presentedSec. II A. The particular application of the FFR formalismthe electron scattering off F2 is presented in Sec. II B. The Smethod is briefly summarized in Sec. II C. Section III detasome technical computational aspects of the electronic stture and dynamics of the system. We discuss in Sec. IV Aposition and the lifetime of the resonance as a function ofinternuclear distance. The cross sections at different levethe theory are presented in Sec. IV B and are comparedexperimental data.


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II. THEORY

A. The projection-operator approach

After the resonant electron-molecule collision, the intmediate metastable anion F2

2 decays by fragmentation into Fand F2 ~DA! or by electron detachment into the vibrationaexcited F2 molecule~VE!. We consider a resonant process2Su symmetry and therefore restrict our presentation to tsymmetry. Moreover, we assume thep-wave scattering com-ponent (l 51,m50) dominates the resonant process.

The DA and VE processes may be summarized byfollowing scheme:

F2~X 1Sg ,v i !1e2⇒F22~X 2Sm!

⇒H ⇒F~2p5 2P° !1F2~2p6 1S! ~DA!

⇒F2~X 1Sg ,v f !1e2 ~VE!,~2!

wherev i and v f label, respectively, the initial and final vibrational state of F2 .

Due to conservation of the total energyE, we have theidentity

Ev i1e i5E5H K2

2m1Dd ~DA!

Ev f1e f ~VE!

, ~3!

wheree i and e f are the energy of the incident electron athe emitted electron, respectively,Ev is the energy of thetarget molecule in its vibrational stateuxv&, K2/2m is thekinetic energy of the fragments, andDd is their electronicenergy. The difference betweenDd and the dissociation asymptoteD0 of the potentialV0 corresponding to the neutramolecule in its electronic ground stateuw0& gives the elec-tron affinity ~EA! of F:

eEA5D02Dd . ~4!

Under the assumption that the resonant electron scaing process is dominated by one partial wave,44 the integralDA and VE cross sections are given, respectively, by44 @Eqs.~4.25! and ~4.26!#:

sDA52p3

e inu^CK

~2 !uVde iuxv i

&u2, ~5!

sVE52p3

e inu^xv i

uVde iG~2 !~E!Vde f

* uxv f&u2. ~6!

The parametern is the degeneracy of the resonant electrostate and the Green’s functionG(2)(E) is given by

G~2 !~E!5~E2TN2Vd2F ~2 !!21, ~7!

where F (2) denotes the level shift operator defined belo@see Eq.~16!#. If m is the reduced mass of F2 , the nuclearkinetic energy termTN is given by2(1/2m)(d2/dR2). Thenuclear wave functionuCK

(2)& is the solution with ingoingboundary conditions for the optical potentialVd1F (2) cor-responding to the DA channel.

The interaction responsible for electron attachmentwell as for electron detachment is expressed b44

@Eq. ~2.32!#:


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10637J. Chem. Phys., Vol. 117, No. 23, 15 December 2002 Electron attachment to F2

Vde5^wduHelubgwe~1 !& ~8!

whereHel is the electronic part of the Hamiltonian assoated with the colliding system. The stateubgwe

(1)& ~Ref. 44! isa scattering eigenstate of the electronic background Hatonian withVe5V01e as eigenvalue andu°we& as incomingasymptote. The stateu°we& represents F2 in the stateuw0&plus a free electron of kinetic energye in a partial wave withl 51, m50. u°we& is an eigenstate of °Hel obtained fromHel

by switching off the interaction of the target molecule withe scattered electron. The background Hamiltonian isprojection ofHel onto the scattering subspace orthogonauwd&. The square-integrable quantum stateuwd& representsthe electronic wave function of the metastable anionic stIt is usually referred to as the discrete component ofresonance and can be considered as an approximation tassociated Siegert resonance state.71 Our particular choicefor uwd& is based on the FFR method which will be detailin Sec. II B. The expectation value ofHel with respect touwd&, given by

Vd5^wduHeluwd&, ~9!

behaves like a diabatic potential and smoothly dependsR.72 This is in contrast with the associated Siegert pole71

which may strongly depend onR.44 In the course of thefragmentation, the incident electron becomes attached toof the fragments. Hence, forR→`, uwd& becomes an eigenstate ofHel with eigenvalueDd andVde ~8! vanishes asymptotically.

In general, the partitioning formalism is based on tdefinition of two complementary projectorsP and Q suchthat

P1Q51. ~10!

The subspaces onto whichP andQ project are sets of stateobeying the continuum and the bound state boundary cotions respectively.P andQ are interpreted as the projectoon the background and the resonant subspaces, respectThis formalism is referred to as the Feshbach–Famethod.73–77 In the isolated resonance case~the theory ofoverlapping resonances has been recently reviewed78!, theQprojector is defined as a projector onto the single discstateuwd&:

Q5uwd&^wdu. ~11!

The P projector is defined by Eq.~10!. In the one chan-nel case~one partial wave!, it may be written

P5E0

`

deubgwe~1 !&^bgwe

~1 !u. ~12!

This partitioning, defined by the choice ofuwd&, is mean-ingful if Vde ~8! is small and smoothly depends one ~Ref.44!. Detailed analyzes of these hypotheses have beensented elsewhere.37,41–43 A time-dependent approach habeen published recently.79,80 A precise definition ofuwd& isgiven is Sec. II B in the FFR framework.

The nuclear wave functionuCK(2)& appearing in Eq.~5!

is the scattering eigenvector of the complex enerdependent nonlocal effective Hamiltonian


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Heff~2 !5TN1Veff

~2 ! ~13!

with ingoing-wave boundary condition and eigenvalueE.The potentialVeff

(2) models the dissociation following thelectron-molecule collision in competition with electron dtachment. It is often referred to as the optical potential.77,81Itcan be written as44 @Eq. ~4.13!#

Veff~2 !5Vd1F ~2 !. ~14!

The operatorF (2) is a complex energy-dependent nonlocoptical potential. It accounts for the electronic decay of tanionic molecular system. In a first approximation, timaginary part ofF (2) plays the role of a complex absorbinpotential located in the interaction region. Its real part mbe interpreted as a level shift operator taking into accountneglect of the background continuum wave functionsthe construction ofuwd&. F (2) is defined by44 @Eqs. ~4.11!and ~4.30!#

F ~2 !~R,R8;E!

5 limh→01

^wduQHP1

E2 ih2PHPPHQuwd& ~15!

5D~R,R8;E!1 iG~R,R8;E!/2. ~16!

The real width and shift functions are respectively defined

G~R,R8;E!

52p(v

Vd,E2Ev~R!xv~R!xv* ~R8!Vd,E2Ev

* ~R8!, ~17!

D~R,R8;E!51

pPE

2`

`

dE8@E2E8#21G~R,R8;E8!/2 ~18!

5HE8→EG~R,R8;E8!/2. ~19!

The symbolsP and H stand for Cauchy’s principal part othe integral and the Hilbert transform respectively. Properof the Hilbert transform are detailed in Ref. 82. The algrithm applied in this work in order to compute numericalthe Hilbert transform has been published elsewhere.41

The computation ofsDA ~5! andsVE ~6! may thereforebe divided into two main steps:

Electronic structure: The definition of the discrete statuwd& and the computation of the complementary continustatesubgwe

(1)& as a function of the interatomic distance. Thcorresponding integralsVd ~9! and Vde ~8! have then to becomputed. This will be discussed in Sec. II B.Nuclear dynamics:The computation of the stateuCK

(2)& de-termines the molecular dynamics of the dissociation reactIt requires the solution of a nonlocal Schro¨dinger equationdefined by the effective Hamiltonian~13!. This will be dis-cussed in Sec. II C.

B. The Feshbach–Fano- R-matrix „FFR… method

The purpose of this section is to show how the discrstateuwd& may be defined within the framework ofR-matrixtheory. The relationship between the Feshbach–Fanoproach and theR-matrix method has been discussed in Re


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37 and 72. Somewhat different approaches have beenlished in Refs. 67, 83, and 84. These approaches are basa resonant parameterization of theR-matrix. Another view-point on the relationship between both formalisms is psented in Ref. 77. Since the FFR method has been detelsewhere,37 only a simplified presentation adapted to the2molecule is presented.

The R-matrix theory85–87 is a well-known method inelectron-atom and electron-molecule scattering~see, e.g.,Refs. 31, 88–92!. The particular implementation of thR-matrix method used in our calculations has been descrin Refs. 93–95 and applied to several molecusystems.37–40,96–100The main idea ofR-matrix theory is todivide the space in two parts by a sphereV centered on themolecule. The radiusr V of V has to be large enough tcontain the target system entirely such that all exchangeteraction terms between target and projectile electrons caneglected outsideV. Inside V, the modified electronicHamiltonianHel

V including the Bloch operator86 provides adiscretization of the scattering continuum

HelVuwk

V&5VkVuwk

V&. ~20!

The wave functionsuwkV& are set to zero if at least one ele

tron coordinate is larger thanr V . Let M be the number ofcomputed eigenstates andHV the linear space spanned buwk51,...,M

V &. In the following, uwk51,...,MV & is supposed to be a

complete basis set.The obtained electronic statesuwk

V& are restrictions toVof the scattering eigenstatesuwe

kV

(1)& of the electronic Hamil-

tonian Hel at the discretized electron energiesekV5Vk

V

2V0 . They are normalized to unity on the sphereV,^wk8

V uwkV&5dk8k . InsideV we have the identity

HeluwkV&5Vk

VuwkV&. ~21!

Since the discrete componentuwd& is assumed to vanishoutsideV, we representuwd& as a linear combination of thuwk

V&

uwd&5(k

ckVuwk

V& ~22!

where theckV areM unknowns.

In order to determine theckV by a well-defined set of

equations, we select a spectral domainS res where the dis-crete state is expected to interact with the backgroundrestrict the expansion~22! to the N wave functions whichenergiesVk

V are lying inS res. If we define the projector

PresV 5 (

VkVPSres

uwkV&^wk

Vu, ~23!

this ansa¨tz can be written

PresV Q5Q. ~24!

In the present case where the resonance is located atelectron energy, we set

S res5@V0 ,VNV# with N,M ~25!

and therefore


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ckV50 for all k.N. ~26!

In order to obtain the required set of equations, we hato introduce some criterium defining a resonance. A renance can be defined, at least in the isolated case, as anounced variation of the cross sections contrasting withslowly varying shape of the background component. The crectness of the partitioning of the Hilbert space is judgedthe flatness of the generated background cross sectionS res ~25!. For a givenuwd&, the background cross sectionare determined from the projector

PV51V2Q ~27!

@whereQ has been defined in Eq.~11! and 1V is the identityon HV] and the spectrum of theV-restricted backgroundHamiltonian

PVHelVPVubgwk

V&5bgVkVubgwk

V&. ~28!

Due to Eq.~27!, the rank ofPVHelVPV is equal toM21. It

has thereforeM21 eigenvalues. Moreover, from Eq.~24!,the spectrum ofPVHel

VPV is identical, outside ofS res ~25!,to the spectrum ofHel

V ~21!, i.e., (1V2PresV ) PVHel

VPV (1V

2PresV )5(1V2Pres

V ) HelV (1V2Pres

V ).In order to define its flatness, the background cross s

tions have to be compared with the cross sections of a simphysical system known to have no resonance inS res ~25!.The simplest system which exhibits this property is the fscattering problem, i.e., where the Coulomb interactiontween the molecule and the scattered electron is switchedThe corresponding Schro¨dinger equation is

°HelVu°wk

V&5°VkVu°wk

V&, ~29!

where °HelV is the restriction toV of °Hel . The statesu°wk

V&are the products of a free one-electron state~quantized ac-cording to theR-matrix boundary conditions! times uw0&.Their energies °Vk

V are given by the sum ofV0 and theenergies °ek

V of the free one-electron state quantized by trestriction toV. TheV-restricted free projector is defined b

°PV5(k

u°wkV&^°wk

Vu. ~30!

In order to enable the comparison, it is required to solve~29! for °Vk

VPS res ~25!. In the isolated resonance case, it hbeen shown on some examples37 that it is possible to chooseS res such that the number of eigenvalues °Vk

V lying in S res isequal toN21. In general, the number of resonances inS res

is defined in the FFR method by the number ofR-matrixpolesVk

V of surplus compared to the number of poles °VkV in

S res.We define the free scattering to besimilar to the back-

ground scattering inS res ~25! if

PVPresV 5°PVPres

V . ~31!

The relation~31! is fulfilled in particular if there existsa unitary matrix which maps$ubgwk51,...,N21

V &% onto$u°wk51,...,N21

V &%. This is the case ifPVHelVPV is close to

°HelV in the sense of the perturbation theory inS res. Since the

basis setu°wkV& is complete, the ansatz~31! makes sense only

if N is not too large, i.e., ifN is not such thatuwd&


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V&^°wkVuwd&. Such a largeN would imply °PVQ

5Q and the identities~24! and~31! would not be consistenwith the definition~27!.

From the definition ofPV ~27! and the ansa¨tze ~24! and~31!, we obtain the set of equations we were looking for

°PVQ50, ~32!

⇔(k51

N

^°w lVuwk

V&ckV50 for l 51,...,N21. ~33!

The (N21)N coefficients^°w lVuwk

V& may be computedor estimated from the results of the diagonalizations~20! and~29!.37 Solving Eq.~33! definesuwd& in a unique way. In thecase of interacting resonances, the number of equations~33!is less thanN21 and the solutions of Eqs.~33! span a linearmanifold which dimension is equal to the number of inteacting resonances withinS res.

In order to compute the discrete-continuum coupliterm ~8!, the first step is to evaluatewduHelubgwk

V& in theuwk

V& basis set. The extension to the continuous energy strum is readily done. This is a standard result ofR-matrixtheory37,72,101that the eigenstates ofPHelP insideV can beobtained from the eigenstates ofPVHel

VPV by linear combi-nation,

ubgwe~1 !&5(

kbek

V ubgwkV&, ~34!

where the coefficientsbekV are defined in terms of theM poles

bgVkV of theR-matrix and the amplitudes of the statesubgwk

V&at the boundary ofV. We therefore have

Vde5 (k51

N21

bekV ^wduHelubgwk

V&. ~35!

C. The Schwinger–Lanczos „SL… method

The stateuCK(2)& appearing in Eq.~5! is the solution of

the nonlocal Schro¨dinger equation defined by the effectivpotentialVeff

(2) ~13! with in-going boundary condition,

S 2d2

dR2 12mVeff~2 !2K2DCK

~2 !~R!50. ~36!

The corresponding Green’s function appearing in Eq.~6! isthe solution of

S 2d2

dR2 12mVeff~2 !2K2D ^R8uG~2 !~E!uR&52md~R2R8!.

~37!

The purpose of this section is to present the applicationthe SL method in order to solve Eqs.~36! and ~37!. Themethod has been largely discussed in Refs. 45 and 47order to simplify the notation we setDd50.

In a first step, local approximations to Eqs.~36! and~37!are solved. A local approximation to the nonlocal effectipotentialVeff

(2) ~14! is given by

Veffloc~2 !~R;E!5E

0

`

Veff~2 !~R,R8;E!dR8. ~38!


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The local versions of Eqs.~36! and ~37! are integrated asVolterra equations:102

CKloc~R!5

sinK~R2R0!

K1E

R0

R sinK~R2R8!

K

3CKloc~R8!2mVeff

loc~2 !~R8!dR8, ~39!

^RuGlocu&5^Ru°G~2 !u&1ER0

R sinK~R2R8!

R

3^R8uGlocu&2mVeffloc~2 !~R8!dR8. ~40!

°G(2) is the free Green’s function for in-going boundaconditions. The kernel sin@K(R2R8)#/K is referred to as thefree Green’s function for regular boundary conditions.81 Theinteratomic distanceR0 should rigorously be equal to zeroNevertheless, it is important to choose a nonvanishingR0

value in order to avoid numerical divergences in integr~39!, ~40!, and~42!. In practice, one choosesR0 such that the

semiclassical penetration probability103 @Eqs.~5! and~50!# ofthe wave function through the repulsive potentialVeff

loc(2) isvanishing forR,R0 .

For largeR, we have

limR→`

CKloc~R!5 1

2iK@~11A2!eiKR2~11A1!e2 iKR#,

~41!

with the integralsA6

A65ER0

`

CKloc~R8!e6 iKR82mVeff

loc~2 !~R8!dR8. ~42!

The energy normalized wave function with in-going bounary conditions is given by

CKloc~2 !~R!5A2Km

p

1

11A2 CKloc~R!. ~43!

This normalization implies a normalized in-going flux ofragments. Since, the skew-Hermitian part ofVeff

(2) is definedto be positive in Eq.~16!, uCK

loc(2)& corresponds to an absorption of flux during the collision process. For largeR, Gloc

defined by Eq.~40! is a linear combination of exp(iKR) andexp(2iKR). The Green’s functionGloc(2) with in-goingboundary condition is obtained by

^RuGloc~2 !u&5^RuGlocu&2CCKloc~R!, ~44!

where the scalarC is defined by the boundary conditiolimR→`^RuGloc(2)u&}exp(2iKR).

Once the local approximations to Eqs.~36! and~37! aresolved. The nonlocal results are obtained by solving thelowing Lippmann–Schwinger equations:

uCK~2 !&5uCK

loc~2 !&1Gloc~2 !UuCK~2 !&, ~45!

G~2 !~E!5Gloc~2 !~E!1Gloc~2 !UG~2 !~E!, ~46!

with the operatorU, the nonlocal part ofVeff(2) , defined as

^RuUu&5F E0

`

Veff~2 !~R,R8;E!^R8u&dR8G

2Veffloc~2 !~R;E!^Ru&. ~47!


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TABLE I. Exponents and coefficients of the compact basis set.

Contracteds Uncontr.s Contractedp Uncontr.d

23 340 0.000 757 2.692 65.66 0.037 012 3.413431 0.006 081 1.009 15.22 0.243 943 0.85757.7 0.032 636 0.3312 4.788 0.808 302 Uncontr

209.2 0.131 704 0.264 162 133 3 Uncontr. p 1.8

66.73 0.396 24 1.732123.37 0.543 627 0.6206

23.37 0.264 893 0.2078.624 0.767 925

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The SL method consists in expanding Eqs.~45! and~46!in the Lanczos basis set$ugi&%. This basis set is definethrough the following recursive scheme:

ug1&5uf°&^f°* uUuf°&21/2, ~48!

b i ugi 11&5Gloc~2 !Uugi&2a i ugi&2b i 21ugi 21&. ~49!

The starting functionuf°& is uCKloc(2)& or Gloc(2)u&. The

parametersa i andb i are defined so that the$ugi&% diagonal-izesU and tridiagonalizesUGloc(2)U. Their explicit valueshave been published elsewhere.45,47The stateu f * & defined byits wave function^Ru f * &5^Ru f &* has been introduced inEq. ~48!. This is required by the complex symmetric sparepresentation ofU.

After I iterations of the SL algorithm, if we setuf& equalto uCK

(2)& or G(2)u& and uf& I to their I th estimations, weobtain from Eqs.~45! and ~46! expanded in$ugi 51,...,I&%:

uF& I5(i 51

I

f i ugi& ~50!

with the definitions

f 15Q1^CKloc~2 !* uUuCK

loc~2 !&1/2, ~51!

f i5b i 21f i 21Qi , ~52!

Qi512a i2b i24¯412a I2b I

2. ~53!

The Schwinger–Lanczos expansion~50! may be rewrit-ten as

uf& I5(i 51

I

hi~Gloc~2 !U ! i uCKloc~2 !& →

I→`

uf& ~54!

with hi a set of scalars. The convergence properties ofSchwinger–Lanczos algorithm have been discuselsewhere.45 This expansion is stable in a large numbercases and does not require the potentialU to be a smallperturbative term.

III. COMPUTATIONAL DETAILS

We use the one particle Gauss function basis setsented in Tables I and II. The compact basis set~Table I! isidentical to the basis set used and discussed in Refs. 69104. The diffuse functions listed in Table II have been takfrom Ref. 99. This choice has been done in order to enathe comparison with published complex absorbing poten

ec 2002 to 129.206.85.195. Redistribution subject to A

ed

f

e-

ndnlel

~CAP! results.69 The selected diffuse functions have beoptimized in order to model the scattering wave functionsthe sphereV with r V510 Bohr.105

The electronic structure of F2 in its ground state at SCFlevel is summarized in Table III forR52.67 Bohr. The mo-lecular orbitals~MO! are defined in two steps. The HartreeFock equations are solved in the compact basis set~Table I!.The obtained MO’s define the compact MO’s. The cotinuum MO’s are expanded in the complete basis set~com-pact and diffuse functions! and are orthogonalized with respect to the compact MO’s.

The static exchange~SE! level of computation for the2Su state ofF2

2 consists in expanding this state in a setconfigurations defined by adding one electron to the grostate configuration in each unoccupiedsu orbitals, i.e.,nsu

orbitals forn53,...,21. The SE plus polarization~SEP! levelof computation consists in adding to the SE set of confirations all configurations defined by exciting one electrfrom the occupied orbitals~in the SE configurations! to oneunoccupied compact orbital.

The first four eigenstatesuwk51,...,4V & of Hel

V ~20! are ob-tained at the SEP level of approximation. The eigenvalVk51,...,4

V lie in the spectral domain 0–14 eV aboveV0 . Weidentify this spectral domain toS res and setN54 in Eq.~25!. The next onesuwk5N11,...,M

V & (M523) are linear com-binations of the eigenstates estimated at the SE level oftheory orthogonalized touwk51,...,N

V &.The coefficients °w l

VuwkV& required to solve Eq.~33!

have been estimated using an approximation schemelished in Ref. 37.

The potential energy curvesbgVkV ~28! and Vd ~9! are

plotted in Fig. 1 together withVkV ~20! andV0 . One observes

that the potentialVd intersects the potentialV0 at RthresholdFFR

TABLE II. Exponents of the diffuse basis set centered on the center of m

s p d

0.130 137 0.104 354 0.111 2520.101 981 0.082 488 0.089 4120.080 524 0.065 610 0.072 3610.063 367 0.052 121 0.058 4960.049 648 0.041 110 0.047 0050.038 298 0.032 122 0.037 3970.029 151 0.024 635 0.029 2250.021 901


veoe

s

thi

ne

titn

otn

ti

e-coe-

-42.ng

re-ande-ro-b-

thecalathisucer

by

s

te

are

ete-


52.41 Bohr. For largerR, the lowest lying state is turninginto a bound state which is identified withuwd&. For Rsmaller thanRthreshold

FFR , the eigenvaluesVkV exhibit in Fig. 1

two avoided curve crossing. The potential energy curbgVk

V ~28! andVd ~9! can be considered as a diabatizationthe Vk

V curves~20!. In the present SE-SEP framework, thelectron affinityeEA ~4! is equal to 13.5 eV. This value imuch larger than the experimental value, 3.4012 eV.106 Thisdiscrepancy is due to the wrong asymptotic behavior ofSE-SEP wave functions. The quality of the wave functionsmuch better in the neighborhood of the electronic groustate equilibrium position~see Fig. 3 for comparison with thCI results!.

The fixed-nuclei width functionGFN(e)52puVdeu2 ispresented in Fig. 2 as a function of the electronic kineenergye and the interatomic distanceR. One observes thathe functionVde(R) is not separable, i.e., may not be writteas a product of one function depending ofe and one functionof R. Nevertheless the hypothesis of separability of bvariables44 is reasonable in model studies and may be seea realistic first approach to fullab initio treatments.40

Once the discrete-continuum coupling termVde ~8! isestimated on a grid of energye and interatomic distanceR,the nonlocal complex energy-dependent optical potenF (2) is generated by applying Eq.~16!. In order to calculatethe width operator defined by Eq.~17!, the vibrational wavefunctions xv(R) are computed by expansion in a sinFourier discrete variable representation basis set using adue to Hora`cek.47 In order to compute the shift operator d

FIG. 1. EigenvaluesVkV of Hel

V , bgVkV of PVHel

VPV, and expectation valueVd of Hel with respect touwd& are plotted together with the neutral stapotentialV0 .

TABLE III. Compact molecular orbitals from SCF. Koopmans’ energiesin Hartree.

MO Occupation atomic orbitals Energy

1sg 2 1s11s 226.42811su 2 1s21s 226.42792sg 2 2s12s 21.76402su 2 2s22s 21.49741pu 4 2pp12pp 20.81063sg 2 2ps22ps 20.75121pg @p* # 4 2pp22pp 20.66673su @s* # 0 2ps12ps 0.0855


sf

esd

c

has

al

de

fined by Eq.~19!, we apply a particular numerical implementation of the Hilbert transform discussed in Refs. 41 andIt requires the two-dimensional interpolation of the coupliVde on thee-R grid. Outside of thee-R grid defined by theabinitio calculations, a rough estimate of the coupling isquired. In this domain, the coupling has to be continuousto vanish asymptotically. In order to fulfill these requirments the discrete-continuum coupling function is set pportional to the functions listed in Table IV. We have oserved that the cross sections weakly depend on the wayextrapolation is performed. The computation of the optipotentialF (2)(R,R8;E) ~16! on a three-dimensional grid isprohibitive task. Nevertheless one can easily see thatoperator is in practice quasilocal, i.e., that, in order to redthe computational effort, it needs only to be estimated foRandR8 such that

uR2R8u<dD for D~R,R8;E!, ~55!

uR2R8u<dG for G~R,R8;E!/2. ~56!

The series defining the shift~18! and width ~17! operatorshave to be truncated. This truncation may be smoothedintroducing sigma factors.47 The series~18! and~17! are notuniformly converging for any pairR and R8. Convergenceis possible only if bothR and R8 are in the domain@Rvmax

2 ,Rvmax

1 # defined by the classical turning points ofV0

estimated at the energyevmaxof the last vibrational state

taken into account in Eqs.~18! and~17!. The rate of conver-gence is larger the closerR and R8 are to the equilibriumposition of V0 . Therefore, in order to eliminate spuriou

FIG. 2. The fixed-nuclei width function as a function ofe andR in Bohr.

TABLE IV. Functions used in order to extrapolate the square of the discrcoupling functionuVdeu2 ~8! outside of thee-R domain defined by theabinitio calculation.

Domain Function Parameters

R<2 Bohrexp2S R2Rlow

s lowD 2 Rlow52.05 Bohr,

s low50.2 Bohr

R>2.769 Bohrexp2

R

l large

l large50.2 Bohr

e>14.4 eVexp2S e2ehigh

shighD 2 ehigh511.4 eV,

shigh510 eV


ite

th

exi-

it

ithrio

tfao

°

azeabth

eut

-ce

. 4.valoutbe

on

to.theofent

ajoris

es-

f

oten-

d


nonphysical oscillations, it is important to choose a finvalue for the parametersdD anddG . The parameterdD is ingeneral smaller thandG . The values of the parametersvmax,dD anddG are adjusted until convergence is reached. Incase of F2 , we have obtainedvmax530, dD50.05 Bohr anddG50.1 Bohr.

IV. RESULTS

A. The fixed-nuclei complex potential

The Siegert resonance pole at fixed nucleizresFN is ob-

tained by solving the following equation44 @Eq. ~3.1!#:

zresFN2ed2DFN~zres

FN!1 iGFN~zresFN!/250 ~57!

with the definitions

ed5Vd2V0 , ~58!

GFN~e!52puVdeu2, ~59!

DFN~e!5He8→eGFN~e8!/2. ~60!

If u]eGFNu!1, and if we defineeD

FN as the solution of

eDFN2ed2DFN~eD

FN!50, ~61!

i.e., as the pole of theK-matrix44 @Eq. ~3.4!#, and

zresFN5e res

FN2 iG resFN/2, ~62!

we have42,107

e resFN'eD

FN2GFN~eD

FN!]eGFN~eD

FN!/4

12]eDFN~eD

FN!1O~G res

FN3!, ~63!

G resFN'

GFN~eDFN!

12]eDFN~eD

FN!1O~G res

FN2!. ~64!

At the energyeDFN corresponding to the resonance w

have]eDFN(eD

FN)&0.01. Hence, we have, to a good appromation,

e resFN'eD

FN'ed , ~65!

G resFN'GFN~ed!. ~66!

We compare the FFR results with results obtained wthe complex absorbing potential~CAP! method. Both FFRand CAP methods~as well as most methods concerned wthe problem of computing resonances like, e.g., extecomplex scaling,108 stabilization method,109 see Ref. 34 for areview! are based on the introduction of some perturbationthe original scattering problem in the outside region, i.e.,away of the interaction domain, in order to turn the resnance state into a bound state. In the FFR method thePV

~30! components are projected out ofuwkV& in order to define

the discrete stateuwd& ~32!. In the CAP method, one addsCAP at the boundary of the interaction region and optimithe CAP parameters until some complex eigenvalues stlize in the complex plane. The stabilization points areresonance poles.

The potentialsVd andV0 are compared in Fig. 3 to thcorresponding ones taken from Ref. 69. The CAP comption has been performed forR>2.28 Bohr. The CAP method


e

h

r

or-

si-

e

a-

does not provide theV0 potential but the crossing point betweenV0 andVd is defined as the point where the resonanwidth vanishes.

The obtained resonance widths are compared in FigBoth methods provide quite different results. On the interfrom 2.28 to 2.35 Bohr the FFR resonance widths are aba factor of 2 larger than the CAP ones. However, it shouldnoticed that in the two calculations the potential curvesV0

andVd are treated on different levels with respect to electrcorrelation. This causes different values fored . If one de-fines ed using the FFR results anded from Ref. 69, bothcurves are parallel to each other on the interval from 2.282.4 Bohr. Earlier results62,65,66have also been plotted in Fig4. The results published in Ref. 66 are obtained withinR-matrix framework at the SEP level using a small setSlater basis functions~STO!. One observes in Fig. 4 that thwidths obtained in Ref. 66 are very similar to the preseresults. Both curves are parallel to each other. The mdifference is the estimation of the crossing point whichshifted of 0.15 Bohr.

In Fig. 5, G resFN is shown as a function ofe obtained by

defining Rd(e) as the inverse function ofed(R). One ob-serves, for energies lower than 1 eV, thatG res

FN(E) estimatedfrom FFR method is one order of magnitude larger than

FIG. 3. The ground state potentialV0SCF, which is computed at SCF level o

the theory, and the discrete state potentialVdFFR ~9!, which is computed in the

FFR framework at the SEP level, are compared to the corresponding ptials V0

CI andVdCAP obtained in Refs. 104 and 69, respectively.

FIG. 4. G resFN(R) estimated from Eq.~64! using distinct definitions ofed ~58!

is compared to earlier results: CAP~Ref. 69!, complex SCF~Ref. 65!, Stielt-jes moment~Ref. 62!, R-Matrix ~Ref. 66!. The potentials used are plottein Fig. 3.


rg

ve

eFid

u-ab

lel ob

an

re

d

r thetoThe

be-unt

n.g

is

lhes

ults

sh

nuu

om


timated from CAP method. One interesting point is that, ifed

is taken from Ref. 69 to evaluateed(R), both curves areparallel to each other on more than 1 eV. In this enedomain, the Wigner threshold rules@which predictsG res

FN(E)}E3/2] are valid. This is quite surprising since we haGFN(R,E)}E3/2 ~59! at a givenR on the interval from 0 to0.1 eV only~see Fig. 2!.

Finally, the discrepancy between the results obtainwithin both methods should be commented on. The Fmethod is able to yield the nonlocal, energy-dependent wwhereas the CAP method can compute only the local~Sieg-ert! width G res

FN(R). Thus the FFR-width is used when calclating the cross sections with the projection operator formism. However, the neutral and ionic potential curves maytaken from either calculation. In the FFR approach the etronic correlation is taken into account up to the SEP levethe theory. In the CAP method the correlation is treatedmultireference configuration interaction~MRDCI!. The CAPpotential curves are thus likely to be of higher quality ththe FFR potential curves.

FIG. 5. GFN@Rd(e)# defined as in Fig. 4.Rd(e) is the inverse function ofed(R) ~58!. The threshold behavior which is expected from Wigner’s threold rules is shown by two straight lines.

FIG. 6. DA cross sections at different levels of the theory. The correspoing ionic and neutral potential curves as well as the discrete-contincoupling terms are obtained within the FFR framework~see Sec. III!. Ex-perimental data from Ref. 54.


y

dRth

l-ec-fy

B. The DA and VE cross sections

The DA cross sections at different level of the theory acompared with experiment54 in Fig. 6. In order to analyze theresults a ‘‘Franck–Condon’’-like approximation is includein Fig. 6. This crude approximationsDA

FC consists in approxi-matingVeff

(2) ~14! by Vd ~9! in Eq. ~36!. This approximationneglects the electron-detachment process occurring afteelectron-molecule collision. It provides an upper boundthe cross sections calculated with electron detachment.local approximation is given by

sDAloc 5

2p3

e inu^CK

loc~2 !uVde iuxv i

&u2, ~67!

with uCKloc~2!& defined by Eq.~43!. It is observed that, in the

local approximation, the cross sections are systematicallylow the exact result. Taking the nonlocal effects into accoleads to the wave functionCK

(2)(R) which is ‘‘less ab-sorbed’’ in the interaction region thanCK

loc~2!(R) ~43!. Thisshows that the optical potentialF (2) ~16! does not simplyplay the role of a CAP localized in the interaction regioThe separable approximationsDA

sep has been obtained usinthe algorithm described in Refs. 40 and 52. This methodexact if Vde ~8! is a separable function ofR ande but doesnot require an explicit fit ofVde to a separable analyticaexpression. This approximation is very good and matcperfectly the exact results for energies larger than 1 eV.

One observes in Figs. 7 and 8 that the obtained res

-

d-m

FIG. 7. DA cross sections for different definitions of theV0 and Vd ~9!potentials. The potentials are plotted in Fig. 3. Experimental data frRef. 54.

FIG. 8. Logarithmic plot of Fig. 7.


umlts

lebu

sen

e

ra-

pe

tes

entaveres

tionsinto

ef-by

ons

r-vi-gs.

ery4.

e ofingaleakcon-A

nuu


are very sensitive on the definition of the potential curvesV0

andVd ~9!. This is due to the fact thatVd is crossingV0 veryclose to the inner classical turning point ofV0 ~Fig. 3!. Theset of curves (V0

SCF,VdFFR) and (V0

CI ,VdCAP) reproduce quite

well the experimental results54 on the spectral domaine i

[email protected],2 eV#. It appears clearly that theX 2Su resonancescattering has a dominating contribution to the DA spectrin this domain. This qualitative result confirms resuobtained from ab initio methods62 and modelcalculations.64,67,68 Model studies are quite difficult in thecase of F2. This is due to the rapid variation ofVde ~8! withrespect toR in the interaction region~Fig. 2! which makes itdifficult to guess an analytical form forVde from some par-tial information only like, e.g., the values of the Siegert pozres

FN ~62! in the interaction region. The cross sections pulished in Ref. 62 have larger values than the present resand are larger than the experimental values54 for e i

.0.1 eV. This may be due to a lower quality of the basisused in Ref. 62 to describe the continuum wave functioand, therefore, to an over estimation ofVde ~8!.

The features observed experimentally fore i,0.5 eV54,56

cannot be modeled within thep-wave hypothesis. Somother electronic state thanX 2Su is likely to be taken intoaccount in order to model such process. Nevertheless bnew experimental results70 with respect to the DA cross sections of Cl2 suggest that, if the F2– Cl2 analogy is correct, thepeak observed at the threshold may be due to some ex

FIG. 9. VE cross sections at different levels of the theory. The correspoing ionic and neutral potential curves as well as the discrete-contincoupling terms are obtained within the FFR framework~see Sec. III!.

FIG. 10. VE spectra for different definitions of theV0 and Vd potentials.The potentials are plotted in Fig. 3.


s-lts

ts

nd

ri-

mental artifact. The structure observed fore i.2.5 eV mustbe interpreted within a model taking excited electronic staof F2 into account.110,111

The VE cross sections are presented in Fig. 9 at differlevels of the theory. The superscripts used in Fig. 9 hbeen introduced in the discussion of Fig. 6. If one compaFigs. 9 and 6, it appears that the estimated VE cross secless depend on the way the non local effects are takenaccount than the DA cross sections do. Forv f53 the threeestimations of the VE cross sections taking the nonlocalfects into account are, in Fig. 9, hardly distinguishableeye.

Figure 10 shows that the estimated VE cross sectistrongly depend on the potential curves used.

Since theV0SCF potential is not expected to model co

rectly the excited vibrational states, computations, takingbrational excitation into account, are presented in Fi11–14 with the set of curves (V0

CI ,VdCAP). The shape and

intensities of the VE spectra presented in Fig. 11 are vdifferent from the model study results published in Ref. 6

The DA and VE spectra for vibrationaly excited F2 mol-ecule are presented in Figs. 12 and 13. The temperaturthe target has been taken into account in Fig. 14 by applythe Boltzmann distribution law to the first 20 first vibrationstates of the target. The obtained results show a very wdependence of the spectra on the temperature. This is intradistinction with the results obtained recently for the Dcross sections of Cl2 within a semiempirical approach.112

d-m

FIG. 11. VE spectra.

FIG. 12. DA spectra for various target vibrational states.


et.oif-

peaciant

seesecP

he

the

ec-re-

derion

hes

ntns

entA

ssre-VEhis

hisss.-S.-rk

-ip,

s’’ofrchofc-e:sel-hey

li-

. S.n.,

s.,ol-

s.

er


V. CONCLUDING REMARKS

Recently developed methods~FFR and SL! have beenextended in order to study the F2 X 2Su resonant electronscattering on anab initio basis. The DA and VE spectra havbeen computed for various vibrational states of the targe

The results have been obtained using methods beythe standard local ‘‘boomerang model’’ approximation. Dferent approximation schemes have been tested~Figs. 6 and9!. We have shown that the estimated cross sections destrongly on the way the nonlocal effects are taken intocount. This is particularly true for the DA cross sectionsthe target is initialy in the vibrational ground state. This feture is typical of molecules which ionic ground state potetial surface intersects the ground state potential surface invicinity of the ground state equilibrium geometry.

The cross sections have been computed for differentof potential curves~computed at SCF and CI level for thneutral state, and SEP and CAP-CI for the ionic state,Fig. 3! and for a unique discrete-continuum coupling funtion Vde ~8! computed within the FFR framework at the SElevel ~Fig. 2!. A strong dependence of the results on t

FIG. 13. VE spectra for various target initial and end vibrational state

FIG. 14. DA and VE spectra for various end vibrational states and tempture.


nd

nd-

f--he

ts

e-

details of the potential curves has been observed~Figs. 7, 8,and 10!. This dependence is clearly more important thannon local effects. We have assumed thatVde does not dra-matically depend on electronic correlation. Taking the eltronic correlation interaction in a more consistent wayquires the computation ofVde at a higherab initio level.Such a computation should be made very carefully in orto handle in a well balanced way the electronic correlatwithin the n-electron neutral target and the (n11)-electronionic scattering states.

The DA spectrum of F2 for electron kinetic energye i

around 1 eV is dominated by the contribution due to tX 2Su resonant scattering. This qualitative trend confirmprevious studies.62 The quantitative results are quite differefrom previous estimations and experimental investigatiowould be welcomed in order to test the validity of the prescomputations. In particular an experimental study of the Dcross sections for small electron kinetic energye would beimportant in order to check whether thee1/2 proportionalityrecently observed at the threshold for Cl2 ~Ref. 70! is alsovalid in the case of F2 .

To our knowledge, no experimental fluorine VE crosections are available in the current literature. It was thefore not possible to compare our results concerning theprocess to experiment. Experimental investigations of tprocess would be of course very welcomed.

ACKNOWLEDGMENTS

The authors are thankful to Professor J. Hora`cek ~Prag!for useful discussions and providing the source code ofprograms for computing the dynamics of the DA proceOne of the authors~V.B.! is indebted to Professor W. Domcke ~Munich! for illuminating discussions and ProfessorD. Peyerimhoff~Bonn! for her hospitality. We are also thankful to J. Breidbach, A. Thiel, S. Sukiasyan, and Dr. R. Scho~Heidelberg! for technical support. V.B. gratefully acknowledges the financial support of the NATO science fellowshthe Graduate Program~Graduiertenkolleg! ‘‘Modeling andScientific Computing in Mathematics and Natural Scienceof the Interdisciplinary Center for Scientific Computingthe University of Heidelberg, and the German ReseaCouncil ~DFG! for research grants at the UniversitiesHeidelberg and Bonn. H.-D.M. and L.S.C. gratefully aknowledge support through the ‘‘DFG ForschergruppSchwellenverhalten, Resonanzen und nichtlokale Wechwirkungen bei niederenergetischen Streuprozessen.’’ Talso acknowledge fruitful discussions with H. Hotop~Kaiser-slautern!, who made Ref. 70 available to us, prior to pubcation.

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Documents

Ab initio study of the resonant electron attachment to the F[sub 2] molecule