Electron Recombination with Small Molecular Ions

Small Molecular Ions

Johanna Brinne Roos

TheoreticalChemistry

School of Biotechnology

Royal Institute ofTechnology

SmallMolecularIons

Licentiatethesis

c

JohannaBrinneRoos,2007 ISBN:978-91-7178-681-4

PrintedbyUniversitetsserviceUS AB,

Stockholm, Sweden, 2007 Typesetin L A T E Xbytheauthor.

InthisthesisIhavetheoreticallystudiedelectronrecombinationprocesseswith

smallmolecularions. Inthesekindofprocessesresonantstatesareinvolved. To

calculatethepotentialenergyforthese statesasafunction ofinternuclear

dis-tance, structurecalculationsandscatteringcalculationshavetobeperformed.

So far I have been studying the ion-pair formation with in electron

recombi-nation with

H

+

3

. The cross section for this process has been calculated using dierent kind of models, both a time dependent quantum mechanical and a

semiclassical. I have also studied the direct process of dissociative

recombi-nation of

HF

+

. To calculate the total cross section for this process, wehave

performedwavepacket propagation onthirtyresonant statesandsummed up

theindividual crosssectionsfor thesestates. Thecrosssectionsforboththese

processes have a similar appearance to those measured experimentallyin the

Paper I:

Ion-pair formation in electron recombinationwith

H

+

3

Å. Larson,J.RoosandA. E.Orel

Phil. Trans. R.Soc. A, 364,2999(2006)

Paper II:

Electroncollisionswith

H

+

3

: ion-pair formation

J.B.Roos,Å.LarsonandA.E.Orel

(Manuscript,tobesubmittedto Phys. Rev. A)

Paper III:

Dissociativerecombination of

HF

+

J.B.Roos,Å.LarsonandA.E.Orel

I wouldrstliketothankHansÅgrenforgivenmetheopportunityto

accom-panyÅsaLarson,mysupervisor,whenshemovedtotheTheoreticalChemistry

departmentin 2006.

ÅsaLarsonandAnnOrel,Ireallyenjoyworkingwithyou. Beingamother

oftwoitisextrainspiringtohaveyouasrolemodels.

The thesis is based on three theoretical papers, one published and two in

manuscriptform.

InpaperI,Iwasinvolvedinthestructurecalculations. Iperformeda

diaba-tizationof thecalculatedadiabaticpotentialenergysurfaces and extrapolated

these surfaces to their asymptotic limits. I was also involved in the

calcula-tion of the electronic couplings between these surfaces and the extrapolation

of these. I alsocalculatedthe classicalreactionpath ontheion-pair potential

energysurface. ThispathIthenextractedfromallcalculatedpotentialsurfaces

andtheachievedcurveswereusedforasemiclassicalcalculationoftheion-pair

reactioncrosssection.

InpaperII,wewereusingMCTDHmethodtopropagatethewavepackets.

Thismethodrequiresthatthepotentialsurfacesandcouplingstobeinproduct

form. Ihavedevelopedafunctionthathastherequiredformandworkswellfor

allthediabatic potentialscalculatedearlierandttedthese tothatfunction. I

wasalsoperformingthettingofallelectroniccouplingstoappropriateforms.

Inpaper III,I have been performing allthestructure calculationsand the

scatteringcalculationsforthestatesofsingletsymmetry. I haveperformedthe

diabatizationofthepotentials. Ihavedeterminedtheasymptoticlimitsforthe

systemand extrapolatedallpotentialsandautoionizationwidths using

appro-priate methods. Finally Iperformedwavepacket propagationontheresonant

states of the system and calculated thecross section for the reaction. I have

1 Introduction 1

1.1 Molecularionsin spaceand industry . . . 1

1.2 Electronrecombinationprocesses . . . 2

2 Theoreticaltreatment 5 2.1 TheBorn-Oppenheimerapproximation. . . 5

2.2 Potentialsurfacesandcouplings. . . 6

2.2.1 Structurecalculations . . . 10

2.2.2 Scatteringcalculations . . . 15

2.2.3 Extrapolation. . . 18

2.3 DissociationDynamics . . . 21

2.3.1 Wavepackettreatment . . . 21

2.3.2 TheMCTDHmethod . . . 26

3 Resultsand Discussion 28 3.1 PaperI . . . 28

3.2 PaperII . . . 29

Introduction

1.1 Molecular ions in space and industry

Dissociative recombination and ion-pair formation are key processes in the

physicsofplasmaswheremolecularionsarepresent. Inionstoragerings,plasma

environmentcanberesembled,and these ringshavebeenused successfullyto

study the above mentioned reactions experimentally [1]. We are comparing

ourtheoreticalcalculatedcrosssection forthesereactionswithmeasuredcross

sectionsusingtheion-storagerings.

Lowtemperatureplasmashavemadeasignicantimpactonsocietyduring

thelast half centuryand improvedthe quality oflife. Fluorescentlightisone

example found in many homes today and plasma ion sources are used to

im-plantionsintomaterialssuchassemiconductorchipsforthecomputerindustry.

Manylow-temperatureplasmaapplications involvecomplexreactionsbetween

electronsandahostofatomic,molecular, andionicspecies. Thesespeciesare

found in highly excited states not encountered in non plasma environments.

Studyingthedynamics onthesestatescangiveusefulinformationtoscientists

modelingthese plasmas[2].

Oneofthemostimportantprocessesindeterminingthespeciccomposition

ofanyionosphereisthedissociativerecombinationofmolecularions.

Dissocia-tiverecombinationiseectivelythe onlyelectronlossmechanismin planetary

ionospheres and thus it playsakey role in the daily variationsin ionospheric

electrondensities. Itisalsotheprimarylossmechanismformolecularions,and

isresponsibleforsignicantalterationsinthepopulationsofthevariousatomic

andmolecularspeciespresent. Sir DavidBateswasin 1950thersttosuggest

that the decayof ionization from the ionosphere couldbe explainedby

disso-ciativerecombination. In order to dosohewasusinga set ofdiabatic states,

1.2 Electron recombination processes

Forrelativelowinteractionenergies,thereareingeneralonlyafewdissociation

processesthatoccur.

Indissociativerecombination (DR),the electronis resonantlycaptured by

theion,loosingitsenergyeitherto electronic-ortorovibronicexcitation

exci-tationoftheresultingneutralmolecule. Theformeroftheseprocessesisknown

asthedirect modeofDRandthelatterastheindirectmodeofDR.

In the direct mode of DR, rst proposed by Bardsley [4], the electron is

captured into a double excited state of the molecule, which has a repulsive

potentialcurve,see gure1.1. Inthisstatethemoleculecaneitherautoionize,

re-emitting the electronto the autoionization continuum, orit can dissociate

intofragments. Theprocesscanberepresentedby:

AB

+

_{+ e}

−

→ AB

∗∗

→ A + B.

(1.1)

Intheindirectmodeof DR, theelectroniscaptured intooneofthe highly

excited Rydberg state converging to the the initial ion, see gure 1.2. The

excessenergyistransferredintovibrationalorrotationalexcitationoftheionic

core. If there is a second state of the same type asin the direct mode that

crosstheRydbergstate,predissociationis possible. TheRydbergstatecouple

to the resonant state by electronic coupling and the process is completed by

dissociationalongthisstate. Thisprocesscanberepresentedby:

AB

+

+ e

−

→ AB

∗

→ AB

∗∗

→ A + B.

(1.2)

InDR, bydenition,neutralfragmentsareformed. Theion-pairformation

process, sometimes referred to as RIP (Resonant Ion-Pair formation), is the

samekindofprocessasDRexceptthationicfragmentsareformed. Therelevant

that, if the electron anity for the

B

fragment in the

A

+

_{+ B}

−

channel of

RIP is larger than the dissociation energy of the initial ion, this channel is

openfordissociationatzerointeractionenergy. Thedirect RIPprocesscanbe

representedby:

AB

+

+ e

−

→ AB

∗∗

→ A

+

+ B

−

,

(1.3)

andtheindirectprocessofRIPcanberepresentedby:

AB

+

+ e

−

→ AB

∗

→ AB

∗∗

→ A

+

+ B

−

.

(1.4)

Inelectronrecombinationwiththemolecular ion

AB

+

, thereis aCoulomb

attractionbetweentheinitialionandtheelectronandthereforethecrosssection

will behigh at low interactionenergies. Itwasshown in 1948 by E.P Wigner

that foraCoulombattractionbetweencollidingparticlesthecrosssection will

decreasewith

1/E

atsmall interactionenergies[5].

With increasing interaction energy additional processes can occursuch as

dissociativeexcitation (DE). Thisis aninelastic scatteringprocess wherepart

ofthekineticenergyoftheelectronistransferredtoelectronicexcitationofthe

ion. If thisstateis repulsiveneutralandionicfragmentswillbeformed. This

reactioncanberepresentedby:

AB

+

+ e

−

→ (AB

+

)

∗

+ e

−

→ A + B

+

+ e

−

Figure 1.3: Schematic potentialcurvesfor theRIP process.

D

0

(AB

+

₎

is the

Theoretical treatment

Bycalculatingthepotentialenergysurfaceandstudythenucleardynamicson

thatsurface,theoutcomeofchemicalreactionscanbestudied. Ifthepotential

energy surfaces are coupled, dierent outcomes(products) of the reactionare

possible. With wavepacketswecanstudythemotiononcoupledpotential

en-ergysurfacesandwecaninpracticestopthereactionatanytimeandanalyze

how theinitial wavepacket haspropagated, spread outand been distributed

among theelectronic states. Wavepacketsare thus adoortowardsbetter

un-derstanding of quantum mechanics and towards quantum control of reaction

dynamics.

2.1 The Born-Oppenheimer approximation

Ifallnonelectrostaticinteractionsareignored,theHamiltonianforanymolecule

canbewrittenas

H

en

= T

n

+ T

e

+ V

en

+ V

ee

+ V

nn

,

(2.1)

where

T

n

is the nuclear kinetic energy,

T

e

the electronic kinetic energy,

V

en

theelectron-nuclearelectrostaticinteraction,

V

ee

theelectron-electronCoulomb repulsionand

V

nn

thebarenucleus-nucleusCoulombrepulsion.

ThefamousBorn-Oppenheimerapproximation[6],basedontheassumption

that the electronsin general move much faster than themuch heavier nuclei,

separatesthetimeindependentSchrödingerequationintoonenuclear-andone

electronic part.

Theelectronic Schrödingerequationforadiatomicmoleculehastheform

[H

e

− E

i

e

(R)]Φ

i

(R, r) = 0,

(2.2)

where

H

e

= T

e

+ V

en

+ V

ee

+ V

nn

.

(2.3)

Here, R is the internuclear distance. It appear no longer asavariable in the

equation,insteaditisaparameter. BysolvingtheelectronicSchrödinger

equa-tionatdierentnuclearpositions,aBorn-Oppenheimerpotentialcurve,

E

e

i

(R)

statessincetheseareingeneralwellseparatedfromtheexcitedstates. Forthe

excited states, on the other hand, the potential surfaces are not always well

separatedfromeachotherandthestatesinteractmoreeasily. Inthoseregions,

nucleardynamicswillnotfollowtheBorn-Oppenheimerstatesadiabaticallyand

theBorn-Oppenheimerapproximationbreaksdown.

Theinteractionof thenuclearand electronicmotionsisresponsibleforthe

couplingsbetweendierentneutralelectronicstatesandalsoforautoionization.

Whendescribingprocessesasdissociativerecombinationandion-pairformation,

these couplingsarecrucial.

Thenucleartime-independentScrödingerequation

[T

n

+ E

e

j

(R)]χ

j

(R) = E

j

χ

j

(R)

(2.4)

describesthenuclearmotiononstate

j

. Itcanbethevibrationalorrotational motionofboundelectronicstatesandalsodissociationdynamicsondissociative

states. Inthisthesis,thedissociationdynamicshasbeentreatedwithdierent

methods, seesection2.3.

2.2 Potential surfaces and couplings

If we add a small diagonal correction, like a perturbation, to the electronic

energy calculated with the Born-Oppenheimer approximation, that take into

accounttheweakcouplingbetweentheelectronicandnuclearmotions,weend

upwiththeadiabaticpotentialsurface:

E

ad

i

(R) ≡ E

i

e

(R) + hΦ

ad

i

|T

n

|Φ

ad

i

i.

(2.5) The electronic part of the Hamiltonian in this approximation is diagonal, a

propertythat makesthesepotentialsfairlyeasytocalculate

hΦ

ad

i

|H

e

|Φ

ad

j

i = E

i

e

(R)δ

ij

.

(2.6)

Theodiagonalelementsofthenuclearkineticenergyoperatorinthis

approx-imationdierfromzeroandcausecouplingsbetweentheadiabaticstates

hΦ

ad

i

|T

n

|Φ

ad

j

i 6= 0.

(2.7) Thesearecalledthenon-adiabaticcouplingelements. Itwasshownin1929,by

NeumannandWignerthattheadiabaticpotentialenergycurves,foradiatomic

molecule,correspondingtoelectronicstatesofthesamesymmetrycannotcross

[7]. Thisisreferredtoasthenon-crossingrule. Instead,twoadiabaticpotential

energycurvesofthesamesymmetryrepeleachotherwhentheycomeclose. The

pointofclosestapproachiscalledanavoidedcrossingpoint,

R

x

in gure2.1. Thestrongestcontributiontothenon-adiabaticcoupling,givenbyequation

(2.7), in the regionof an avoided crossingcomes from arst derivativeradial

couplingterm[8], thatforeach internuclearcoordinatehastheappearance

−

1

µ

hΦ

ad

i

|

d

dR

|Φ

ad

j

i

r

d

dR

.

(2.8)

Byincludingnon-adiabaticcouplingtermswewillgetanadiabaticSchrödinger

adiabatic potentialcurvesand dotted lines fordiabatic potential curves.

Φ

ad

and

Φ

d

representtheadiabaticanddiabatic electronicwavefunctions.

dierentialequation ofthat kindismoredicultto solvethanonecontaining

only couplings of potential form. In addition, the numerical evolution of the

couplingscanalsobenontrivial.

Byinsteadusediabaticstatesdenedinsuchawaythattherstderivative

couplingsdisappearwecangetaroundthisproblem. Therstonetousestates

that later would be referredto asdiabatic states wasZener in 1932 [9]. The

adiabaticelectroniceigenfunctionshaveatypicalchangeofcharacterclosetoan

avoidedcrossing. Zenerassumedthatthebasisfunctionsusedwere

approxima-tiveeigenfunctionsoftheelectronicHamiltonianwithoutthischaracteristic,as

showningure2.1. Asthenamediabaticindicates,thesestatesdonotadjust

adiabaticallytochangesoftheinternucleardistance.

However,ashas beenpointedout byC. A. Mead and D. G. Truhlar [10],

everycompletesetofelectronicstatesfullling

hΦ

di

i

|

d

dR

|Φ

di

j

i

r

= 0

(2.9)

mustbeindependentofR,butinacalculation,thebasissetusedisbothnite

andincompleteandthenadiabatizationwithinthenumberofstatesconsidered

ispossible.

Thediabaticstateswillbecoupledbytheo-diagonalelementsofthe

elec-tronicHamiltonian:

c

ij

= hΦ

di

i

|H

e

|Φ

di

j

i

r

6= 0.

(2.10)

Ifwestartbyassumingthat theadiabaticand diabaticstatesareequalfar

matrixVtothediabaticpotentialmatrixUcanbedoneasfollows[11]

U

_{= MVM}

−

1

,

(2.11) where

M

₌



cos[γ(R)]

sin[γ(R)]

− sin[γ(R)] cos[γ(R)]



.

(2.12)

TheadiabaticpotentialmatrixVforthetwostateproblem is

V

₌



E

ad

1

0

0

E

ad

2



,

(2.13)

sotheelementsofthediabaticpotentialmatrixUwillbecome:

U

11

=

E

1

di

(R) = E

1

ad

(R) cos

2

γ(R) + E

2

ad

(R) sin

2

γ(R)

U

22

=

E

1

di

(R) = E

1

ad

(R) sin

2

γ(R) + E

ad

2

(R) cos

2

γ(R)

U

12

=

U

21

= c

12

= c

21

=

1

2

(E

ad

2

(R) − E

1

ad

(R)) sin(2γ(R))

(2.14)

Thus, thediagonal elements of thediabatic potential matrix are the diabatic

potentials and the o-diagonal elements are the electronic couplings. From

the equationabovewecansee that, at the curvecrossing point

R

x

, the angle

γ = π/2

andtheelectronic couplingwillbegivenbyhalf thedistancebetween theadiabaticstates,arelationthatisusedintheLandau-Zenermodel,applied

in paperI.

ForallR,theangle

γ(R)

canbeobtainedfromthecongurationinteraction (CI)coecients. Ifweonlyhavetwointeractingadiabaticstatesdominatedby

two congurations, we havethe following relation between the adiabatic and

diabaticstates:



Φ

ad

1

Φ

ad

2



=



C

11

C

12

C

21

C

22

 

Φ

di

1

Φ

di

2



.

(2.15)

Sincethe adiabaticanddiabaticwavefunctions areorthonormaltheabove

re-lationcanalsobewritten as



Φ

ad

1

Φ

ad

2



=



cos[γ(R)] − sin[γ(R)]

sin[γ(R)]

cos[γ(R)]

 

Φ

di

1

Φ

di

2



,

(2.16)

andwecanseethattheangle

γ(R)

isgivenby

tan[γ(R)] =

C

21

(R)

C

11

(R)

.

(2.17)

The above expressions are developed for diatomic molecules and canbe used

if the potential curves are well separated. In paper I and II, we have also

usedtheaboverelationstoestimatethecouplingbetweentheion-pairpotential

energysurfaceinelectronrecombinationwith

H

+

3

andthelowerRydbergstates convergingto

H

+

3

. Inthis paperweused the radialJacobi coordinates

r

and

Figure 2.2: Eectivequantum numbersforRydbergstatesof

1

_Σ

+

symmetry

convergingtothegroundstateof

HF

+

one,

r

, xed. The electronic coupling between the states,

c

12

, wascalculated withequation(2.14).

In dissociative recombination, the electron is captured by the ion into a

dissociativemetastable statethat sometimescrossthewholemanifoldof

Ry-dbergstatesconvergingtotheinitialion. Itismetastable inasensethatitis

associatedwith alifetime andhenceawidth

Γ

. ForthehigherRydbergstates thatareverycloseinenergy,itisoftennotpossibletoassumethatthecrossing

statesonlycoupletwo-by-two.Wecanhoweverestimatetheelectroniccoupling

between theresonant andRydberg states by scaling the autoionizationwidth

Γ(R)

[12]

hΦ

ryd

(R, r)|H

e

|Φ

d

(R, r)i

r

=

r

Γ(R)

2π

1

(n

∗

_(R))

3/2

.

(2.18) Here,

n

∗

(R)

istheeectivequantumnumberoftheRydbergstatethat, accord-ingtoMulliken'sformula[13]isgivenby

n

∗

(R) = n − µ(R) =

1

p2(E

ion

(R) − E

ryd

(R))

.

(2.19)

In this expression,

n

and

µ

are the principal quantum number and quantum defect of the Rydberg state respectively,

E

ion

(R)

is the initial ion potential energyand

E

ryd

(R)

istheRydbergstatepotential. Ingure(2.2),weshowthe eectivequantum numbersofthediabatiziedRydbergstatesofHF.

Ifthekineticenergy ishigh at acurvecrossingand wehavealarge

adiabaticrepresentationwouldbetterimitatethetrueevolvementofthesystem.

Inelectronrecombinationprocesses,wenormallyhavetherssetofconditions,

andwethereforechose adiabaticrepresentation.

2.2.1 Structure calculations

Conguration interaction Calculation

The electronic structure calculation starts with a Hartree-Fock (HF)

calcula-tion,fromwhichmolecularorbitals(MOs)andagroundstatesolutionthatnot

accuratelyinclude correlationisobtained.

The conguration interaction method (CI) is the simplest way to include

electroncorrelationandtoimprovetheHartree-Focksolution. Thisisamethod

wellsuited forcalculatingexcitedstatesofsmallmolecules.

ThewavefunctionisconstructedasalinearcombinationofSlater

determi-nantsorcongurationstatefunctions(CSF)

Ψ

CI

=

N

SCF

X

m=1

C

m

Ψ

m

.

(2.20)

CSFs are createdby distributingthe electrons in the MOs obtained from the

Hartree-Focksolution

Ψ

CI

= C

o

|HF i +

occ.

X

i

virt.

X

r

C

i

r

Ψ

r

i

+

occ.

X

i<j

virt.

X

r<s

C

ij

rs

Ψ

rs

ij

+ . . . .

(2.21)

where

occ.

standsforoccupiedorbitalsand

virt.

forvirtualorbitals. The vari-ationalprincipleis thenusedforsolvingtheelectronicSchrödingerequation

E

var

=

R Ψ

∗

H

e

Ψdτ

R Ψ

∗

_Ψdτ

≥ E

0

true

(2.22)

Foralineartrialwavefunction,thevariationalprincipleleadstosolvingthe

secularequationfortheCIcoecientsordiagonalizingtheCImatrix.

ForafullCI(FCI),thecompletesetofdeterminantsgeneratedby

distribut-ing the electrons among all orbitals is included in the expansion(2.21). The

numberof Slater determinants increaseveryrapidly with thenumber of

elec-trons and with the numberof orbitals. A FCI expansionis therefore suitable

onlyforthesmallestelectronicsystems. InpaperIandII,afullCIcalculation

wasused to determinethe adiabaticpotentialsof

H

3

situated belowthe ionic groundstateof

H

+

3

.

Forlargerelectronicsystem,theFCIexpansionhastobetruncated. When

designing smaller congurationspaces, it is important to distinguish between

staticand dynamicalcorrelation. Static correlationis treatedbyretaining the

dominant congurationsof theFCI expansionaswell asthose that arenearly

degeneratewiththedominantcongurations. Thesecongurationsarereferred

to as referencecongurations of theCI wave function, andthey span a

refer-ence space. Dynamical correlation is treated by adding to the wavefunction

allcongurationsthatmaybecomeimportantintheregionofaenergysurface

neededtodescribeacertainreaction. TheMRCIwavefunctionisgeneratedby

addingtoallcongurationsinthisreferencespaceallexcitationsuptoagiven

level from each reference conguration. In paper III, the MRCI method was

usedyodeterminetheHFpotentialsbelowthegroundstateof

HF

+

.

Validation

Below the ion state potential, structure calculationscan be used to calculate

theessentialpotentialenergysurfaces ofthesystemunderstudy.

Thisisnotpossibleforthestatesabovetheion,sincetheseresonantstates

are associatedwith awidth, related to thelifetime of these states. They are

referredto asresonantstates sincetheyshowupasa sharpvariation, a

reso-nance, in thecross sectionfor elasticscattering betweenthe electronand ion.

Withastructurecalculation,wecannotbesurethatwehavereceivedtheexact

positionoftheresonantstate. Thepositionwillbewithinthewidth,andifthe

widthisnarrow,astructurecalculationwillnotgiveasignicanterror. Onthe

otherhand,weneedtocalculate thewidthtondoutifwemakeasignicant

error,andforthiswehavetorunscatteringcalculations.

Scatteringcalculationsarefarmoretimeconsumingthanstructure

calcula-tions. Tond theresonance,weneedto perform anenergyscan andsincethe

resonancescanbeverynarrowweneedtouseaverynegridtomakesurewe

donotmissaresonance. Theresonancealsohastobewellresolved,andforthis

at least ten energy points aroundthe resonanceare needed. In paper III, for

HF,eachenergypointtookapproximatelytwohourstocalculate. Tominimize

the time put into the calculation, it is therefore a good idea to start with a

structurecalculationtogetanapproximate position oftheresonance. Forthis

purpose,structure calculationstodeterminetheresonantstatesarejustied.

Choice ofbasis set and wave functionin DR and RIP

Indissociativerecombinationandion-pairformation,therearesometimesmany

dissociativestatesandRydbergstatesofdierentsymmetriesthatareimportant

for the process. In paper III, where dissociative recombination with

HF

+

is

studied, for example,allof the30calculatedresonantstatescontributeto the

totalcrosssection. Itcanalsobethecasethattheresonantstatecoupletothe

whole manifold of Rydberg states of the same symmetry. To represent all of

these states correctly, largebasis sets haveto be used. ForHF, upto 15000

congurations had to be included. For somesystems morethan 20 adiabatic

roots have to be calculated for each symmetry and internuclear distance to

obtainallneutralstatesneededbothbelowand abovethe groundstateofthe

ion.

Continuumstates

Abovethegroundstateoftheion,thecalculatedstatescanbeeitherthe

reso-nantstatesweare interestedin orit canbecontinuum states,i.e. stateswith

the same character asthe ground state of the ion but with anextra electron

diabatizationoftheadiabaticstatesareperformed,thesestatesareremoved.

The choice ofsymmetry

The symmetry of molecules is described in terms of elements and operations.

Elements aregeometric entities such asaxes, planes and points in space used

to dene symmetryoperations. Operationsinvolvethe spatial re-arrangement

of atoms in amoleculebyrotationabout anaxis,

C

n

, by reectionthrougha plane,

σ

or by inversionthroughapoint,

i

. A rotation,reectionorinversion operation will becalled a symmetryoperationif, and onlyif, the new spatial

arrangementoftheatomsinthemoleculeisindistinguishablefromtheoriginal

arrangement.

Thepointgroup,orsymmetrygroup,isthename ofaacollectionof

sym-metryelementspossessedby amolecule. Eachcommon collectionof elements

isrepresentedbyasimplesymbol,calledtheSchöniesnotation,indicatingthe

typeofreectionsymmetryand theorderoftheprincipalrotationalaxis.

The

C

2v

notation is indicating that it exist one

C

2

axis and two vertical planescontainingthisaxis. Byexcludingbendingof themolecule,wecouldin

paperIII carryoutourcalculationsonthe

H

3

moleculein

C

2v

symmetry[14]. ThesymmetrygroupofHFis

C

∞

v

, butin paperIIIwecarryoutcalculations onthesystemin

C

2v

symmetrysincewefreelycanchosethenumberofvertical reectionplanescontainingtheprincipalaxis.

In

C

2v

symmetry,wecalculatestatesof

A

1

,

A

2

,

B

1

and

B

2

symmetries. The stateswegetin

B

1

and

B

2

symmetriesaredegenerateforHF.Ifwewouldlike to go from

C

_∞

v

symmetry to

C

2v

symmetry, that is transform into states of

Σ

+

,

Σ

−

∆

and

Π

symmetries,thiscanbedoneasfollows:

1. Removestatesthat occurinboth

A

1

and

A

2

from

A

1

. -Thestatesthataretakenawayare

∆

states.

-Theremainingstatesare

Σ

+

states.

2. Removethe

∆

statesfrom the

A

2

states. -Theremainingstatesareof

Σ

−

symmetry.

3. Degeneratestates

B

1

and

B

2

givedegeneratestatesof

Π

x

and

Π

y

symme-tries.

Diabatization

Thenon-crossing ruletellsus thattwoadiabaticstatesof thesamesymmetry

cannot cross [7]. Ingure 2.3 the resultingHF states of acalculation in

1

_A

1

symmetry are shown. Before adiabatizationis performedon these states, we

have to separate the states of

1

_Σ

+

and

1

_∆

symmetry and remove continuum

states. That yieldsadiabaticstatesof

1

_Σ

+

symmetryand

1

_∆

symmetry.

A trivial diabatization can be done by at each internuclear distance,

R

i

, identify which root to connect with a root calculated at the greatest lower

internucleardistance

R

i−1

. If wehaveclearavoidedcrossingsin theadiabatic states,thereceiveddiabaticstateswillhavesomeofthischaracteristicin them

Figure 2.3: HFpotentialsof

1

_A

1

symmetry. Resultsfrom MRCIcalculations

havean avoided crossing, equations(2.17) and (2.15)canbe usedif itcanbe

assumedthat thepotentialscoupletwo-by-two.

Adiabatizationcanbedonebytrackingthedominantcongurationswhen

theinternucleardistance

R

isvaried. TheRydbergstatesallhavethesame con-gurationasthegroundstateoftheionplusanextraelectroninaouterorbital.

Theresonantstateshavecongurationsthat diersfrom thoseoftheRydberg

states. InthiswaywecanseparatetheresonantstatesfromtheRydbergstates.

If we haveindications of avoided crossings betweenthe resonant statesat

someinternucleardistance,

R

,wecanatapointwherethestatesarewell sepa-ratedpickthemostprobablecongurationfor thelowest stateand diabatizise

thestatesbytrackingthat congurationtowardssmallerorlargerinternuclear

distancesusingthegeometrydependence oftheCIcoecients. Itcanhowever

bethe case that other stateshave thesame congurationin them too, which

canmakethisproceduredicult. InpaperIII,wefoundthattheconguration

associatedwith the ion-pair in HF was the

(1σ)

2

_(2σ)

2

_(3σ)

1

_(4σ)

1

_(1π)

4

cong-uration. This conguration was howeveralso present in other diabatic states

of the same symmetry, see gure (2.4). This gure shows in what states the

(1σ)

2

_(2σ)

2

_(3σ)

1

_(4σ)

1

_(1π)

4

congurationappear. The square of the CI

coe-cientsforthiscongurationintheion-pairstatearealsoshown.

Ifwewouldonlyhaveonediabatic statecontainingacertainconguration,

wecouldinsteadofusing equation(2.17)makethediabatic statesmootherby

coecientsforthecongurationasaweight

V (R) =

P

i

C

i

2

(R)V

i

(R)

P

i

c

2

i

(R)

.

(2.23)

This method wasused tocreate thediabatic potentialsurfacefor theion-pair

statein

H

3

belowtheionsurfaceinpaperI.

For HF(paperIII),weonlyused thetrivialmethodto diabatizatethe

adi-abaticstates. Theresultfor

1

_Σ

symmetryisshownin gure2.5. Here wecan

seethatsomecharacteristicforanavoidedcrossingstillremainsinsomeofthe

diabaticstates.

Inafuture study, we areplanningperformamorecareful diabatizationof

theHF potentialsandcalculate the electroniccouplings betweentheresonant

Figure 2.5: Diabatic potentials of

1

_Σ

+

symmetry. Diabatization of results

fromMRCIcalculations.

2.2.2 Scattering calculations

Theresonantstatesabovethegroundstateoftheionisassociatedwithawidth,

related to the lifetime of the state. Due to this, these statescannot be

stud-ied usingconventionalstructurecalculations. Weare calculatingthe potential

energy,

V (R)

, andtheautoionizationwidth,

Γ(R)

, withscatteringcalculation, using aComplex-Kohnvariational method [15, 16]. Inthese calculations, the

elastic scattering process between an incoming electron and the ion target is

studied, see gure2.6. At the resonant energy

E

res

(R)

, the ion canbe tem-porarilycapturedintoanearlyboundstate,which willcauseasharpvariation

ofthecrosssection,aresonance.

A boundary condition of the of the scattered electronic wave function is

givenby

lim

r→∞

Ψ

k

(r) = e

ik·r

₊

e

ikr

r

f (Θ, Φ),

(2.24)

where

Θ

and

Φ

arethepolarandazimuthalanglesofscatteringrelativetothe directionof incident,

ˆ

k

and

f (Θ, Φ)

isthescatteringamplitude. Thedierentialcrosssectionforthereactionisgivenby

dσ

dΩ

= |f(Θ, Φ)|

2

_,

(2.25)

inte-gratingthedierentialcrosssectionoversolidangles:

σ

e

=

Z

dΩ|f(Θ, Φ)|

2

_.

(2.26)

Fora sphericalsymmetricpotential, therewill be no

Φ

dependence in the scatteringamplitudeanditispossibleto expandit inthecompletesetof

Leg-endrepolynomials,

P

l

(cos Θ)

:

f (Θ) =

i

2k

∞

X

l=0

(2l + 1)(1 − S

l

)P

l

(cos Θ),

(2.27)

where

S

l

is thescattering,or

S

, matrix,which inthecaseofcentral eld scat-teringonlyisaone-by-onematrix,orasinglefunctionof

k

,and

l

istheangular momentum [17].

Thetotalelasticcrosssectionwill become

σ

e

=

π

k

2

∞

X

l=0

(2l + 1)|1 − S

l

|

2

,

(2.28)

wherethequantity

1 − S

l

= T

l

isthetransition, or

T

,matrix.

Thetermresonanceiswidelyusedintheliterature,butunfortunatelywith

dierentmeanings. Itisconvenienttothinkofaresonanceasapoleinthe

S

(or

T

)matrix. Apole inthe

S

(or

T

)matrixcorrespondtoapointinthecomplex

p

-plane(

p = ¯

hk

). Apointinthe

p

-planecancorrespondtothreedierentkind ofstates:

boundstate,

p = iκ

, with

κ > 0

virtualstate,

p = −iγ

, with

γ > 0

resonancestate,

p = ±β − iγ

, with

β, γ > 0

When the electronis scattered towardsthe molecular ion, the interaction

potential is non-spherical and the

S

(or

T

) matrix will have both

l

and

m

indicies.

Aresonantstatewillthusshowupasasharpvariationofthecrosssectionat

k

res

accordingtoequation(2.28). Theywillhowevershowupmoreclearlywhen theeigenphasesumisstudied. Theeigenphasesumisrelatedtotheeigenvalues

Figure 2.7: Resonancesin DRwith

HF

+

:

1

_A

1

symmetryat internuclear dis-tance

R = 1.2 a

0

and

R = 1.3 a

0

(

e

iδ

lm

) of the

S

matrix. As a funct function of energy,

E = ¯

h

2

_k

2

_/2m

, the

eigenphasesumisgivenby:

δ(E) =

X

l,m

δ

l,m

(R, E),

(2.29)

where

R

isthexedinternucleardistanceatwhichthecalculationisperformed. At everyenergy where there is a resonance,

E

res

, this eigenphase sumjumps suddenlywith

π

. Thiscanbeseeningure2.7. Thisgurealsocleariesan ad-ditionaladvantageofthescatteringcalculations,bytheshapeoftheresonances

it is possible to determine if there exist a crossing between diabatic resonant

statesornot. For

R = 1.2 a

0

wehaveaverynarrowresonanceat

E = 0.1375 H

, thatisnotresolvedat

R = 1.3 a

0

. Thisresonanceisof

1

_∆

symmetry,whilethe

otherare of

1

_Σ

symmetry. Inthecalculationcarried outin

1

_A

2

symmetrythe gridwas neenoughto resolvethisresonance.

The

N +1

electronwavefunctionforthesystemshouldsatisfythetime inde-pendentSchrödingerequation. ThetimeindependentSchrödingercanhowever

notbesolvedforsuchasystemsoinsteadweset upatrialwavefunctionwith

unknowncoecients. ThesecoecientsaredeterminedwiththecomplexKohn

variationalmethodand a

T

matrixisobtained. Fromthis

T

matrix,thecross sectionandeigenphasesumcanbecalculated.

Tobeableto mergetheresultsof thescattering calculationsand structure

calculationsweare performingboth calculationswith thesame program. We

Wignerformwegetboththeposition

E

res

(R)

andtheautoionizationwidth

Γ

(R)

oftheresonanceatinternucleardistance

R

δ(E)

= δ

res

(E) + δ

bg

(E)

= tan

−

1



Γ

2(E − E

R

)



+ a + bE + cE

2

.

(2.30)

Here,

δ

res

isthecontributionfromtheresonancetotheeigenphasesumand

δ

bg

isthebackgroundcontribution.

Toobtainthepotentialenergyoftheresonantstate,theionicpotentialhas

tobeaddedtotheresonanceenergy

V

res

(R) = V

ion

(R) + E

res

(R).

(2.31)

2.2.3 Extrapolation

Scattering calculation are as mention earlier far more time consuming than

structure calculations. First the resonance must be localized, which can be

tricky if it is narrow, then the resonancemust bewell resolved, so the

Breit-Wignerttingwillnotyieldanincorrectresult. Tottheresonance,atleastten

energy pointsare neededto becalculatedclose aroundtheresonance,and for

HF,eachveenergypointstookapproximatelythesamecomputationaleortas

calculating25adiabaticroots. Inaddition,theBreit-Wignerttingtakestime.

Structurecalculationsarehowevertimeconsumingtoo,anddependingonwhich

basissetandnumberofcongurationsthathavebeenusedinthecalculationthe

potentialsurfacecalculatedwillbemoreaccurateatsomeinternucleardistances

than at another. Careful extrapolation is thereforeneeded, bothof potential

surfaces,electronic couplingsandautoionizationwidths.

Towardssmaller internuclear distances

Manypotentialcurvesofdiatomicmolecules canbewelldescribedbyaMorse

potential:

V (R) = D

e

[1 − e

−

α(R−R

_e

)

]

2

+ T,

(2.32)

where

R

istheinternucleardistance,

D

e

isthedissociationenergy,

R

e

the equi-libriumbondlength,

α

isrelatedtothebondforceconstantand

T

thepotential minimumrelativetoareferenceenergylevel,suchasthepotentialminimumof

thegroundstateof thesystem. AMorsepotentialisoftenagood

approxima-tion aroundthe potentialminimum and sometimes also at largerinternuclear

distancesbut doesnotbehaveproperlyatsmallinternucleardistances.

AMorse-typepotentialthat behavesbetter atsmall internuclear distances

is

V (R) = D

e



1 −

e

αR

e

_{− 1}

e

αR

_{− 1}



2

+ T.

(2.33)

This potentialwas usedwhen the potentaial curvesfor HFwere extrapolated

towards smaller internuclear distances in paper III. This function was

how-everusedonly atinternuclear distanceswhere theFranck-Condonoverlapwas

insignicant. That is, inside the region where the electron capture and the

AtomicStates MolecularStates

S

g

+ S

g

or

S

u

+ S

u

Σ

+

S

g

+ S

u

Σ

−

S

g

+ P

g

or

S

u

+ P

u

Σ

−

, Π

S

g

+ P

u

or

S

u

+ P

g

Σ

+

_{, Π}

S

g

+ D

g

or

S

u

+ D

u

Σ

+

_{, Π, ∆}

S

g

+ D

u

or

S

u

+ D

g

Σ

−

, Π, ∆

S

g

+ F

g

or

S

u

+ F

u

Σ

−

, Π, ∆, Φ

S

g

+ F

u

or

S

u

+ F

g

Σ

+

_{, Π, ∆, Φ}

P

g

+ P

g

or

P

u

+ P

u

Σ

+

_{(2), Σ}

−

, Φ(2), ∆

P

g

+ P

u

Σ

+

, Σ

−

(2), Φ(2), ∆

P

g

+ D

g

or

P

u

+ D

u

Σ

+

_{, Σ}

−

(2), Φ(3), ∆(2), Φ

P

g

+ D

u

or

P

u

+ D

g

Σ

+

_{(2), Σ}

−

, Φ(3), ∆(2), Φ

P

g

+ F

g

or

P

u

+ F

u

Σ

+

_{(2), Σ}

−

, Φ(3), ∆(3), Φ(2), Γ

P

g

+ F

u

or

P

u

+ F

g

Σ

+

_{, Σ}

−

(2), Φ(3), ∆(3), Φ(2), Γ

D

g

+ D

g

or

D

u

+ D

u

Σ

+

_{(3), Σ}

−

(2), Φ(4), ∆(3), Φ(2), Γ

D

g

+ D

u

Σ

+

(2), Σ

−

(3), Φ(4), ∆(3), Φ(2), Γ

D

g

+ F

g

or

D

u

+ F

u

Σ

+

_{(2), Σ}

−

(3), Φ(5), ∆(4), Φ(3), Γ(2), H

D

g

+ F

u

or

D

u

+ F

g

Σ

+

_{(3), Σ}

−

(2), Φ(5), ∆(4), Φ(3), Γ(2), H

Towardsasymptoticlimits

Theasymptoticformofanion-pairstateisnotatinin theasymptoticregion

dueto theCoulombattractionbetweentheion-pair. Instead,thepotentialfor

suchastatehasthefollowingform asymptotically:

V

res

(R) = V

f inal

−

1

R

−

α

2R

4

,

(2.34)

where

V

f inal

is the asymptotic energy limit and

α

is the polarizability. For HF,weassumedinpaperIIIthat theion-pairpotentialobtainthisasymptotic

form at

R ≥ 20 a

0

. We have structure calculationsout to

R = 9 a

0

. Spline interpolationisusedtoconnectthetworegions.

Ifwehaveadiatomicmolecule,theasymptoticenergylimitsforthe

poten-tial energyare obtainedfrom spectroscopicdata forthe separatedatoms. By

examining thecalculatedpotentialcurvesand using theWigner-Witmer rules

(seetable2.1)[18],wecandeterminewhichasymptoticleveleachresonantstate

goesto. TheWigner-Witmerrulesgiveusthenumberofelectronicstates,ofa

givensymmetry,thatisassociatedwithaspecicasymptoticlimit.

Inaddition, themultiplicityofthemolecularstateshavetobedetermined.

Therelationbetweenthemultiplicitiesoftheatomicstatesandthemultiplicities

forthemolecularstatesaregivenin table2.2.

Finally,theasymptoticenergylevelshavetobeshiftedrelativetosome

refer-encepointinthecalculatedpotentialsandinterpolationbetweenthecalculated

AtomicStates MolecularStates

Singlet+singlet Singlet

Singlet+doublet Doublet

Singlet+triplet Triplet

Doublet+doublet Singlet,triplet

Doublet+triplet Doublet,quartet

Doublet+quartet Triplet,quintet

Triplet+triplet Singlet,triplet,quintet

Triplet+quartet Doublet,quartet,sextet

Quartet+quartet Singlet,triplet,quintet,septet

method. A suitablemethod isnotalwaysaspline,sincethedistance between

thecalculatedpotentialsandtheasymptoticlimitcanbelargeandasplinecan

giveunphysicaloscillatingbehaviourintheintermediateregion.

The ion-pairsurface in

H

3

Fortheion-pairsurfacein

H

3

,calculatedinpaperI,wehavealongthereaction coordinate,

z

,theasymptoticlimit

H

+

2

+ H

−

. Forlargez,equation(2.34)give

ustheasymptoticformoftheion-pairsurface

V (r, z) −→ E(r) −

1

_z

−

_2z

α

4

.

(2.35)

Here

E(r)

is thepotentialcurveof

H

+

2

and

α

thepolarizibility of

H

−

. When

the other coordinate,

r

, is increased, we will have the three body breakup,

H

+

_{+ H}

−

+ H

, asymptotically. Inthis case,wehaveaCoulombattraction be-tween

H

+

and

H

−

andweassumethattheion-pairpotentialhasthefollowing

form:

V (r, z) −→ E

0

−

1

√

r

2

_{+ z}

2

−

α

2(r

2

_{+ z}

2

₎

2

,

(2.36) where

E

0

is thedissociationenergy of

H

+

2

. Ifweinvestigatethe limitwhere

r

goesto zero,

H

+

_{+ H}

−

goesto theunited atom limit

2

_He

+

, and wewill have

as a function of

z

, the potential curve for

2

_HeH

with the asymptotic limit

2

_He

+

_{+ H}

−

. With calculatedpotential energypointsaround the minima and

withtheabovelimitsitispossibletoconstructthepotentialion-pairsurfaceof

H

+

₃

withasuitableextrapolationmethod.

Wecould not usea twodimensional spline to connectthe calculated data

withtheasymptoticlimitsforthesamereasonsasdiscussedearlier. Insteadwe

tted thecalculatedpointsandlimits to atwodimensional function of Morse

character. Theion-pairpotentialcurvein HeHhad been calculatedearlierby

Larsonand Orel [19]. And the potential curvefor

H

+

V (R) =

n

X

j=0

V

j

e

−

jα(R−R

_e

)

,

(2.37)

where

n = 2

, withthecoecients

V

0

= T + D

e

,

V

1

= −2D

e

and

V

2

= D

e

. By including terms up to

n = 5

it was possible to t the above sum to the

H

+

2

potential. If termsup to

n = 9

wasincludedthe function gotexible enough forttingtheion-pairpotentialcurveofHeH.Aproductofthesetwofunctions

V (r)

and

V (z)

canbewrittenas

V (r, z) =

n

i

X

i=0

n

j

X

j=0

V

ij

e

−

iα

r

(r−r

e

)−jα

z

(z−z

e

)

,

(2.38)

where

n

i

= 5

and

n

j

= 9

. The above function has 49 coecients that need to be determined by tting. In paper I,weused this function to extrapolate

the abinitio calculated ion-pairto its asymptoticlimits. Figure 2.8 showthe

ion-pairsurfacein twodimensions. Also,theclassicalpathwithstartingpoint

at theequilateralgeometryof

H

3

isdisplayed. This pathwascalculatedusing Hamilton'sequationsofmotion.

InpaperII,thesurfacewasused directly,sinceithastherequiredproduct

form fortheMCTDHmethod usedforpropagatingwavepacketsinthis paper

[see section2.3.2and equation(2.61)]. Theother, forion-pairformation,

rele-vantpotentialsurfaces in paperII werealsotted successfullyto thefunction

given by equation (2.38). For the same reason, the electronic couplings, and

autoionizationwidthswerettedtoasumofGaussianfunction products:

c(r, z) =

m

i

X

i=0

m

j

X

j=0

c

ij

e

−

α

_r

(r−r

_i

)

2

−

α

_z

(z−z

_i

)

2

,

(2.39)

where3to6Gaussianfunctionsofeachdimensionwereincludeddependingon

theshapeofthecouplings.

2.3 Dissociation Dynamics

Whenthepotentialsurfacesandcouplingshavebeencalculated,thenuclear

dy-namicscanbestudied. Thisisdonebysolvingthetime-dependentSchrödinger

equationforthesystem. Thefollowingchapterwillcontaininformationonhow

the electron capture, propagation of wave packets and autoionization from a

resonantstatecanbedescribed. Information onhowthe crosssectionfor the

reactioncan beobtainedwillalsobegiven.

2.3.1 Wave packet treatment

The wave packet

The solutionto the Schrödinger equation for afree particle is given by plane

waves. Accordingto theprincipleof linearsuperposition,alinearsuperosition

Figure 2.8: Theion-pairsurfacein electronrecombinationwith

H

+

3

. Theline starting at the equilateral geometry of

H

+

3

indicates the classicalpath on the potentialsurface.

asuperpositionin which thewavefunction hasanappreciablemagnitudeonly

overarelativelysmallinterval(areafortwodimensions)andisfallingorapidly

outsidethisinterval[21]. Inonedimension,awavepacket isoftheform

Ψ(R, t) =

√

1

2π

Z

Φ(k)e

i(kR−ω(k)t)

dk,

(2.40)

where

k

is the wave number, and

ω(k)

the angular frequency. If the angular frequencyisexpandasaTaylorseriesaboutthecentralfrequency

k

0

,therst termsintheserieswillbe

ω(k) = ω

0

+

 dω

dk



k

0

(k − k

0

) +

1

2

 d

2

_ω

dk

2



k

0

(k − k

0

)

2

+ ... .

(2.41)

The rstterm is just thecentral frequency, the coecientin thesecond term

is the group velocity, the third term will give us the dispersion of the group

velocity,thespreadingofthewavepacket,andhighertermswillgiveushigher

orderdispersions.

With wavepacket methods,it ispossibletostudy thestatedistributionat

anytimeduring thepropagation andthereforetheygiveenormousinsightinto

the mechanismof the reaction. Thewavepackets, thepotentials,the

autoion-ization widthsand thecouplingsarein these methodsdened ona

R

-grid. In onedimension,thegridisgivenby

Whenthe electronrecombines withthe ion, wavepackets areinitiated onthe

resonantstates. If theelectronic capture is fastcompared with themotion of

the nuclei, an assumptionthat normallyholds, theinitial wave packet on the

resonantstate

i

isgivenby

Ψ

i

(t = 0, R) =

r

Γ

i

(R)

2π

χ

v=0

(R),

(2.43)

where

χ

v=0

(R)

isthegroundvibrationalwavefunctionfortheion,and

Γ

i

(R)

is theautoionizationwidth ofresonantstate

i

[22]. AccordingtoFermi'sGolden rule,the

pΓ

i

/2π

factoristheelectroniccouplingelementbetweentheresonant stateandtheautoionizationcontinuum.

Thevibrational wave fuction forthe ion canbedetermined using a

nite-dierencemethod[23],wherethesecondorderderivativeisapproximatedby

d

2

dR

2

χ(R

i

) ≈

1

(∆R)

2

[χ(R

i+1

− 2χ(R

i

) + χ(R

i−1

)],

(2.44)

whichwilltransformthetimeindependentSchrödingerequationintoasolvable

tridiagonalmatrixeigenvalueequation.

Autoionization

Whenthemoleculebeginits fragmentation, theelectroncanbere-emitted,or

autoionize. Thiscanhappenaslongas

R

issmallerthanthecrossingpoint,

R

x

, betweentheresonantstateandtheion. Atsucientlyhightotalenergywecan

assumethatwehaveclosurefortheenergeticallyopenvibrationallevels. That

is,themoleculecanautoionizeintoallenergeticallyopenvibrationallevels. In

this case wecan apply the boomerang model [24, 25], where autoionization

from theresonantstates isincluded byletting thepotentialsbecomecomplex

abovetheionpotential

V

i

(R) = E

i

(R) − i

Γ

i

(R)

2

.

(2.45)

If the total energy is not suciently high, which is typically the case if

the resonantstatecross theion closeto its minimum, thecontribution to the

Hamiltonian ofthe autoionizationhaveto becalculatedfor eachenergy point

usinganon-localoperator[4,26]

OΨ = −iπ

r

Γ(R)

2π

X

v

j

hχ

v

j

(R)|

r

Γ(R)

2π

|Ψ(R)i

R

χ

v

j

(R),

(2.46)

wherethesumisoverallenergeticallyopenvibrationallevels.

Propagation

The wave packets are propagated by solving the time-dependent Schrödinger

equation. Inonedimensionwehave

i

∂

Ψ(R, t) =







Ψ

1

(R, t)

Ψ

2

(R, t)

. . .







(2.48)

andtheHamiltonianforcoupledpotentialsis givenby

H =







T

n

+ V

1

(R)

c

12

(R)

· · ·

c

21

(R)

T

n

+ V

2

(R) · · ·

. . . . . . . . .







.

(2.49)

If the potentials

V

i

(R)

in theHamiltonian are diabatic,

c

ij

are theelectronic couplings,givenbyequation(2.11)and

c

ij

= c

ji

. Ifthepotentialsareadiabatic, thecouplingelementsaredominatedbytherstderivativeradialcouplingterm,

equation(2.7),and

c

ij

= −c

ji

.

Thepotentials,theautoionization,thecouplingsandtheinitialwavepacket

arerepresentedbytheirvaluesonthechosengrid. Thekineticenergyoperator

in the Hamiltonian isapproximatedusing thenite dierence method, likein

equation (2.44). The solution to the time dependent Schrödinger equation is

givenby

Ψ(R, t) = e

−

i(t−t

0

)H

_{Ψ(R, t}

0

)

(2.50)

where

exp(−i(t − t

0

)H) = exp(−i∆tH)

istheevolutionoperator. Propagation methodsand crosssection

The propagation of thewavepacket can beperformed with dierent

numeri-cal methods, suchastheCrank-Nicholsonmethod, the SplitOperator Fourier

Transform(SOFT)method, theChebychevpropagationmethodandtheMulti

CongurationTimeDependentHartree(MCTDH)method. IntheSOFTmethod

theevolutiontimeisdividedintoanumberofslices. Foreachslice,thealgoritm

transforms back and forth between the momentum space and position space.

The potentialoperatoris treatedin the position space whilethe kinetic

oper-ator is treatedin themomentum space. If thenumber of slices is suciently

high, convergencecanbereached.

IntheChebychevmethod,theevolutionoperatorisexpandedin a

polyno-mialseriesintheoperator

exp[−i∆tH]

. Thebestapproximationisachievedby an expansionbased on complexChebychev polynomials [27]. The Chebychev

method was used succesfully in paper I when the propogation wasperformed

on the ion-pair surface alone, but when couplings to the other states where

includedweencounterednumericalproblems and themethod wasabandoned.

Insteadweestimatedthelossofuxto(andregainingfrom)theothersurfaces

by calculatinga diabatic survivalprobability alongthe classicaltrajectory on

theion-pairpotentialusingtheLandau-Zenermodel[9]. WhentheChebychev

method was used in paper I, thecross section was obtainedby analyzing the

uxthrougaplaneintheasymptoticregion

(z = z

stop

)

verticaltothereaction coordinate

z

.

In paper II,we are using the MCTDH method instead of the Chebychew

method topropagatewavepacketsonthecoupledpotentialsurfaces,thistime

Figure 2.9: Wavepacketevolutiononthe

1

1

_Π

resonantstateof

HF

+

IntheCranck-Nicholsonmethod[28]theevolutionoperatorisapproximated

withtheCayleyform

e

−

i∆tH

≈

1 − i

1

2

∆tH

1 + i

1

₂

∆tH

(2.51)

Thisapproximationisbothstableandunitary,incontrarytoexplicitmethods,

whereonlytermsuptorstorderisincludedin theexpansionoftheevolution

operator in a Taylorseries. This method was used in paperIII to propagate

wavepacketsontheresonantpotentialcurves. Figure2.9showsthetime

evolu-tionof theinitial wavepacket onthelowest

1

_Π

resonantstateof

HF

using the Crank-Nicholsonmethod.

When the wave packets have been propagated out in the asymptotic

re-giontheyareprojectedontoenergynormalizedwavefunctionsfortheseparated

atoms. Thisgivesus thetransitionmatrixelement. Fortheresonantstate

i

,it isgivenby

T

i

(E) = lim

t→∞

hΦ

E

(R)|Ψ

i

(R, t)i

R

,

(2.52)

andthecrosssectionforstate

i

isgivenby[29,22]

σ

i

(E) =

2π

E

g|T

i

(E)|

2

_,

(2.53)

σ

tot

=

X

i

σ

i

.

(2.54)

ForthestatesofHFthatdissociateintoneutralfragmentsanddonothave

abarrierinthepotentialcurve,weuseplanewavesasenergynormalized

wave-functions:

Φ

E

(R) =

r

µ

2πk

e

ikR

_.

(2.55)

Here

µ

is thereducedmassand

k

isthewavenumberrelatedto theenergyof thedissociatingfragments.

Ifapotentialhasabarrierwefoundthatpartofthewavepacketgottrapped

beforethisbarrierand didnotreachtheasymptoticregion. Ifwethenproject

ontoplanewavesunrealinteferensepatternsshowupinthecrosssection. Also,

thecrosssectionwillbetimedependent. Instead,energynormalized

eigenfunc-tionsareusedintheprojectionoftheasymptoticwavepackets. Thesefunctions

werealsousedfortheion-pairstate,thathasalongrangeCoulombpotential.

The bound vibrational wave functions and the free energy eigenfunctions

˜

Φ

v

(R)

are calculated with the same method that was used to calculate the groundvibrationalwavefunctionfor

HF

+

. Thismethodalsogiveustheenergy

eigenvalues

E(v)

˜

. The energy normalized wave functions for these states are givenby:

Φ

E

(R) =

d ˜

E(v)

dv

!

−

1/2

˜

Φ

v

(R).

(2.56) 2.3.2 The MCTDH method

The MulticongurationTimeDependentHartree (MCTDH) programpackage

[30] is developed to study wavepacket dynamics in many dimensions. It has

succesfullybeenusedtostudyupto24degreesoffreedom[31]. InpaperIIthis

methodisusedtostudytheion-pairformationprocessinelectronrecombination

with

H

+

3

includingtwodimensions. ThebasicideaofMCTDHisthatthenuclear wavepacketcanbewrittenasasumofseperableterms. Intwodimensionthis

becomes

Ψ(t, r, z) =

n

r

X

i=1

n

z

X

j=1

A

ij

(t)ϕ

(r)

i

(r, t)ϕ

(z)

j

(z, t)

(2.57)

where

A

ij

denotetheMCTDHexpansioncoecientsand

ϕ

(q)

i

arethe single-particlefunctions. IftheHamiltonianofthesystemcanbewrittenasasumof

onedimensionaloperators,onlyoneterminequation(2.57)hastobeincluded

Ψ(t, r, z) = A(t)ϕ

(r)

(r, t)ϕ

(z)

(z, t)

(2.58)

and the Time Dependent Hartree (TDH) method is obtained. In the TDH

method, the correlationbetween thetwonucleardegreesof freedomis not

and

n

r

, n

z

= 10

single particle functions are taken into account in equation (2.57).

Eachsingleparticlefunction,

φ

(q)

i

isrepresentedintermsofaprimitivebasis:

φ

(q)

i

(q, t) =

N

q

X

k=i

c

(q)

_ik

χ

(q)

_k

(q).

(2.59)

Thetimeindependentbasisfunctionsareforcomputationaleciencychosenas

thebasisfunctionsofdiscretevariablerepresentation(DVR) [31].

Byinsertingtheansatz(2.57)intothetimedependentSchrödingerequation,

timedependentcoupleddierentialequationsfor thecoecients

A

ij

(t)

aswell as the single particle functions

φ

(q)

i

(q, t)

are obtained. The solution of these equations are not unique. Uniqueness is achieved by imposing constraints on

thesingleparticlefunctionssuch askeepingthemorthonormalforalltime.

In order for the MCTDH method to be eective the Hamiltonian of the

system must be written as a sum of products of single coordinate operators.

This is typicallythe casefor the kineticenergy operator. Forexample, using

the same set of coordinates (Jacobi coordinates) as in paper II, the kinetic

energyoperatorhastheform

ˆ

T = −

_2µr

1

∂

2

∂r

2

−

1

2µz

∂

2

∂z

2

.

(2.60)

Alsothe potential energy surfaces,electronic couplings and autoionization

widthsmust bettedintothedesiredproductform

V (r, z) =

m

r

X

i=1

m

z

X

j=1

V

ij

f

i

(r)g

j

(z).

(2.61)

Insection2.2.3,itwasdescribedhowthiswasperformedfortheion-pairsurface

relevantforpaperII.

Thewavepackets are propagated outin the asymptoticregion. Thecross

section for ion-pair formation and dissociative recombination is calculated by

integratingtheux absorbedbyacomplexabsorbingpotentialsituatedin the

Results and Discussion

3.1 Paper I

Inthis paper,the processof resonantion-pairformation(RIP) in electron

re-combination with

H

+

3

is studied. Weare performing structure calculationsin

C

2v

symmetry since it had earlier beenfound that the resonantstates of the systemwerestronglyrepulsivewithrespecttobothJacobicoordinatesbut

rel-ativelyatwhentheanglewasvaried.

Theion-pair state is identied by following the

(1a

1

)

2

_(2a

1

)

2

conguration and atransformation from adiabaticto diabatic statesis performedusing the

geometry dependence of theCI coecients. The electronic couplingsbetween

theresonantstatesandbetweentheion-pairstateandthelowerRydbergstates

are obtainedusing atwo-by-twotransformationusingthe CIcoecients. The

electronic couplings betweenthe higherRydberg statesand the ion-pairstate

at small internuclear distances are estimated using a scaling of theelectronic

coupling between the ionization continuum and the ion-pair state. At large

internuclear distances the two-by-two transformation and the CI coecients

wereused to obtainthe electronic coupling betweentheion-pair stateand all

Rydbergstatesincluded.

The electroncapture induce wave packets on the tworesonant states and

autoionization isincludedby lettingthe potentialsbecomecomplexabovethe

ionpotentialsurface. Therealpartofthepotentialsurfaceistheenergyandthe

imaginarypart is negative with amagnitude given by half the autoionization

width. Thepositionsandwidths oftheresonantstateshadbeencalculatedin

anearlierstudy.

Thedissociationdynamicsontheion-pairsurfaceandonthetwocoupled

res-onantstatesaredescribedbysolvingthetimedependentSchrödingerequation

usingtheChebyshevpropagatorandthecrosssectionisobtainedbyanalyzing

theux throughaplaneat thereactioncoordinate

z = z

stop

. Toestimatethe luss of ux to the Rydberg states, the survivalprobability along theclassical

pathwascalculatedusingtheLandau-Zenermodel.

In an earlier one dimensional study of the reaction, a double peak

struc-turein thecrosssectionwasobtained. This structurewasnotobservedin the

measured cross section at CRYRING and wasdescribed asan inteference

yields across sectionwithamagnitudethat is approximatelysix timeshigher

thenthemesuredcrosssectionandwithapeakshiftedtowardssomewhatlower

interaction energiesrelative to the measured cross section. When the second

resonantstateisincluded, theamplitudeof thecrosssectionis increasedbya

smallamountwhentheChebyshevmethodisused. Whenthelossofuxtothe

Rydbergstatesis estimatedbytheLandau-Zener model, theamplitudeofthe

crosssectionbecomessmallerthenthemeasuredcrosssection. Whenincluding

ux recoveryat the second crossing,the cross sectionincreasesslightly but is

still smaller than the measured cross section and peaked at lower interaction

energies.

Theconclusionsarethat inorderto describethedynamicsofthisreaction,

at leasttwodimensionsneedtobeincluded. Theseconddimensionwillsmear

outthe interferenceeects. Toobtainacomplete pictureof thedynamics, all

calculatedstateshavetobeincludedin thewavepacketpropagation.

3.2 Paper II

Inthispaper,weusetheMultiCongurationTimeDependentHartree(MCTDH)

methodtostudythedynamicsoftheion-pairprocessinelectronrecombination

with

H

+

3

.

Inorder tousethismethod thepotentialsurfaces,electronic couplingsand

autoionization widths need to be tted. The potential surfaces calculated in

paperI arettedtoatwodimensionalfunction ofMorse-characterandthe45

unknowncoecientsofthisfunctionwereoptimized. Theautoionizationwidths

aswellas theelectronic couplingsto theRydberg statescalculatedin paperI

are tted with products of gaussian functions. For theautoionization widths

andtheelectroniccouplingsbetweentheresonantstates,totally15coecients

are included. Since theelectronic couplings betweenthe ion-pair surface and

theRydbergstatesaremorespreadout,36coecientswereoptimized.

Thedissociateduxontheion-pairstateisanalyzedthroughacomplex

ab-sorbingpotential. Suchabsorbingpotentialsarealsousedfortheothersurfaces

to avoidreections towardsthe edge ofthe grid. Thecross sectionweobtain

usingtheMCTDHmethod,whenwepropagatethewavepacketontheion-pair

surfacealone,isshiftedtowardssmallerenergiesrelativetothecrosssectionwe

obtainwiththeChebychevmethod. ThecrosssectioncalculatedusingMCTDH

alsohasaslightlysmalleramplitude. TheMCTDHmethodisseveralordersof

magnitude faster and convergence is easierachieved. A draw backis that the

potentialsandwidths needto betted.

The cross section obtainedwhen awavepacket is propagated on the

ion-pairstatealoneusingtheMCTDHprogramisaboutafactorofsixlargerthan

the measured cross section. When the second resonantstate is included, the

crosssectiondropswith afactorof0.27. Whenthediabaticion-pairstatehas

crossedthe groundstateof theion, it will alsocross themanifoldof Rydberg

states below. We are hereincludingelectronic couplingsbetween theion-pair

stateandthefourlowestRydbergstates. Asexpected,uxwillbelostthrough

this interaction and cause the magnitude of the cross section to drop. When

the electronic couplingsto the Rydberg statesare included, themagnitude of