Reaction dynamics on highly excited states

Reaction dynamics on highly

excited states

c

Johanna Brinne Roos, Stockholm 2009

ISBN 978-91-7155-869-5

Printed in Sweden by Universitetsservice US-AB, Stockholm 2009

Distributor: Department of Physics, Stockholm University

Abstract

In this thesis I have performed theoreti al studies on the rea tion

dy-nami s in few-atom mole ules. In parti ular, I have looked at rea tion

pro esses in whi h highly ex ited resonant states are involved. When

highlyex itedstatesareformed,the dynami sbe omes ompli atedand

approximationsnormallyusedin hemi alrea tionstudiesarenolonger

appli able.

To al ulate the potential energy urve for some of these states as a

fun tionofinternu lear distan e,a ombinationofstru ture al ulations

and s attering al ulations have to be performed, and the rea tion

dy-nami son the potentials hasbeen studied using both time-independent

andtime-dependent methods.

Thepro esses thathave been studied and whi h aredis ussed inthis

thesisareion-pairformationinele tronre ombinationwithH

+

3

,

disso ia-tivere ombinationandion-pairformationofHF

+

,mutualneutralization inH

+

+F

−

ollisionsanddisso iativere ombinationofBeH

+

.Isotope

List of Papers

Thisthesisisbasedonthe followingpapers, whi h arereferredto inthe

text bytheir Romannumerals.

I Ion-pair formation in ele tron re ombination with

H

+

3

Å.Larson,J.RoosandA. E. Orel

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

II Ele tron ollisions with

H

+

3

: ion-pair formation

J.B.Roos,Å.Larson and A.E. Orel

Phys.Rev. A,76, 042703(2007)

III Disso iative re ombination of

HF

+

J.B.Roos,Å.Larson and A.E. Orel

Phys.Rev. A,78, 022508(2008)

IV Resonant ion-pair formation in ele tron

re ombina-tion of

HF

+

J.B.Roos,A.E. OrelandÅ. Larson

(Manus ript, submitted to Phys.Rev. A(2009))

V Mutual neutralization of

H

+

_{+ F}

−

J. B. Roos, J. Zs. Mezei, K. Shilyaeva, N. Elander and Å.

Larson

(Manus ript, to be submittedto Phys.Rev. A(2009))

VI Disso iative re ombination of

BeH

+

J.B.Roos,M. Larson, Å.Larsonand A.E. Orel

(Manus ript, submitted to Phys.Rev. A(2009))

VII Ion-pair formation in ele tron re ombination with

mole ular ions

Å.Larson,J.B. Roos, M. Stenrupand A.E. Orel

J.Phys.: Conf. Ser.,88, 012065(2007)

Publications not included in the thesis

VIII A dire t, lo al model of disso iative re ombination

of

HF

with

H

+

3

Å.Larson,J.B. Roosand A.E. Orel

A epted byJ.Phys.: Conf. Ser.(2009)

X Resonan es in disso iative re ombination: trends

and patterns

A.E.Orel, Å.Larson,J.B.Roos,V.NgassamandJ.Royal

A epted byJ.Phys.: Conf. Ser.(2009)

Contributions by the author

I Iperformedthe stru ture al ulations, arriedoutthe

diaba-tizationofthe al ulatedadiabati potential energysurfa es

anddeterminedtheele troni ouplings.Ialso al ulatedthe

lassi al rea tion path on theion-pair potential energy

sur-fa e whi h is then used for the semi lassi al al ulation of

theion-pairrea tion rossse tion.

II The multi onguration time dependent Hartree method

used to propagate the wave-pa kets in this paper requires

that all potential energy surfa es and ouplings are in

produ t form. I have developed a fun tion that has the

required form and works well for all of the diabati

potentials al ulated earlier and I then tted these

potentialsto thatfun tion.Ihave alsoperformedthetting

of all ele troni ouplings and autoionization widths to

appropriateforms.

III Iperformed all al ulations and wrotethepaper.

IV Iperformed all al ulations and wrote thepaper. In

parti -ular,Idevelopedadiabatizationmethodinwhi hboth

qua-sidiabati potential energy surfa es and ele troni oupling

are al ulated simultaneously.

V Imodiedthe al ulations usedto studymutual

neutraliza-tion pro esses, performed al ulations using the potentials

and ouplings we reported in paper IV and analyzed the

stru turesinthe rossse tion.

VI I performed the diabatization of the al ulated adiabati

potential energy surfa es and al ulated the ele troni

ou-plings.I also performedthe dynami al al ulations and

al- ulated the rossse tion.

Contents

1

Introduction

. . . .

13

1.1 Dissociative recombination

. . . .

13

1.2 Resonant ion-pair formation

. . . .

17

1.3 Mutual neutralization

. . . .

19

2 Theory

. . . .

23

2.1 Potential energy curves and couplings

. . . .

23

2.1.1

Approximations and definitions

. . . .

23

2.1.2

Structure calculations

. . . .

28

2.1.3

Scattering calculations

. . . .

31

2.1.4

Diabatization

. . . .

40

2.1.5

Extrapolation

. . . .

45

2.2 Reaction dynamics

. . . .

46

2.2.1

The driven Schrödinger equation

. . . .

46

2.2.2

Wavepacket dynamics

. . . .

54

2.2.3

The log-derivative method

. . . .

61

2.2.4

Identifying resonances

. . . .

62

3 Results and Discussion

. . . .

65

3.1 Paper I

. . . .

65

3.2 Paper II

. . . .

67

3.3 Paper III

. . . .

69

3.4 Paper IV

. . . .

71

3.5 Paper V

. . . .

75

3.6 Paper VI

. . . .

78

3.7 Paper VII

. . . .

1. Introduction

In this thesis I have studied rea tions that are of importan e for the

physi s of plasmas where mole ular ions are present. There isa la k of

understanding of how these ions are formed, ex ited and destroyed in

elds su h as astrophysi s and plasma-based te hnologies, and this has

been a strongmotivationfor this resear h.

The abundan e of H

+

3

indiuse mole ular louds,for example,

trou-bled resear hers for de ades until Kokoouline et al. in 2003 in luded a

previously overlooked mole ular ee t in their theoreti al treatment of

the disso iative re ombination of H

+

3

[1℄ and their new results yielded

goodagreement between theory andexperiments.

In et hing plasmas, undesirable rest produ ts are sometimes reated

that an be harmful to the environment [2℄ and to prevent this from

happeningitisimportanttoknowmoreaboutthepro esseswithinthese

plasmas.

The disso iative re ombination (DR), resonant ion-pair formation

(RIP) and mutual neutralization (MN) rea tions are similar in many

ways and in the remaining part of this hapter I will give a short

introdu tion to thesepro esses.

1.1

Dissociative recombination

Disso iativere ombination(DR)isthepro esswhereamole ular ation

re ombines with an ele tron to form a highly ex ited neutral mole ule

thatdisso iatesinto neutral fragments.

One might ask oneselfhow it an be possible for an ele tron, whi h,

relativetothesmallestmole ularion,isthesizeofanant ompared toa

elephant,tohavesu hanimpa tonamole ule?Su hapossibilityexists,

however,wheneverthe times ale oftherea tion orrespondstothatof a

mole ular vibration of the ion. The ele tron an then atta h to theion

to formahighly ex itedneutral state thatisstrongly repulsive,a

prop-ertythatimmediatelyfor esthemole uleto disso iate.Thedisso iation

pro ess is fast ompared to ompeting pro esses su h as relaxation by

photonemissionand thereforedominates.

Theneutralintermediatestateformedinthisrea tionis alleda

reso-nantstate.Itisnotstableinthetraditionalsense,sin e therealsoexists

theneutral mole ule reemits the ele tron ba kto the ionization

ontin-uumandfallsba kintoavibrationalstateoftheion.Thispro esso urs

on the same times ale asdisso iation and also needs to be onsidered.

Theprobability forele tron apture into andautoionization froma

par-ti ular state is related to the autoionization width, whi h is inversely

proportional to the lifetime of this state. When the resonant state has

rossedtheiongroundstate,autoionizationisnolonger possibleandthe

resonant state be omesenergeti allystable towardsautoionization.

A possible disso iation path is illustrated inFig. (1.1). Here, theion

islabeledAB

+

,theresonantstateAB

∗∗

andtheresultingDRfragments

atlargeinternu lear distan esarelabeledwithA+B.Theprobabilityof

autoionization from the resonant state is indi ated by adding a width

to theresonant state potential above the ionstate potential. In theory,

however,theautoionizationwidthisrepresentedasa omplexpartofthe

state,whi hiszerowhenthestateisele troni allystable.Thehorizontal

line indi ates the ground vibrational state of the ion where the pro ess

might start, andthe arrowsindi ate thedire tion oftherea tion.

Fig. (1.1) is a very simplied pi ture of a DR event. In general the

initial ondition of the ion is more orre tly des ribed by some kind

of distribution among of vibrational and rotational states. There might

an innite number of Rydberg states whi h onverge to this state. A

Rydbergstate hasthe same ongurationasthe ionplusan ele tronin

an outerorbital. Resonant states areRydbergstates whi h onvergeto

ele troni ally ex ited states of the ion, and sothere are also an innite

numberofthese.However,belowa ertainenergy,thenumberislimited

althoughmight still be large.

In1950,SirDavidBateswastherstpersontosuggestthatthede ay

inthelevelofionization intheionosphere ouldbeexplainedbyDR[3℄.

In order to do so, he used a diabati representation. In a diabati

rep-resentation states of the same symmetry are allowed to ross, and are

oupled to ea h other by ele troni ouplings. In this diabati pi ture,

and in ea h symmetry, we an have a large number of resonant states

rossing an innite number of Rydberg states. All of these states are

oupledto ea h otherbyele troni ouplings.

TherearetwomodesofDR,adire tandanindire tone.Intheformer,

rst proposed by Bates [3℄, the ele tron is resonantly aptured by the

ionand losesits energy to ele troni ex itation of the resulting neutral

mole ule. This is the pro ess illustrated by the potential energy urves

showninFig.(1.1).Inthe latterpro ess,theindire tmodeproposedby

Bardsley[4℄, theele tron looseitsenergy to rovibroni ex itationofthe

resulting neutral mole ule.In otherwords, the ele tronis aptured into

aRydbergstate andifthisstate then rossesarepulsive state,

predisso- iationispossible.Predisso iationisenhan ed iftheele troni oupling

between the rossing states is large. This is the pro ess illustrated in

Fig.(1.2).Here, the Rydbergstate is labeledbyAB

∗

.

Thedire tandindire tmodeoftheDRpro ess anberepresentedby

AB

+

₊

e

−

→

AB

∗∗

→

A

+

B (1.1) AB

+

₊

e

−

→

AB

∗

→

AB

∗∗

→

A

+

B (1.2)

respe tively. These two pro esses o ur simultaneously and

ompeti-tion between them has to be onsidered for a orre t des ription [5℄.

InFig.(1.1)andFig.(1.2) onlyonedisso iationpathisdisplayed.

How-ever, as mentioned earlier, there exist an innite number of Rydberg

states below the ion and there an also be a large number of resonant

statesthat rosstheionpotential.Allofthestatesthatareenergeti ally

open for disso iationat a given energy and asymptoti allyformneutral

fragments areavailable DR hannels and ux from resonant states an

be redistributed among all of the states that ele troni ally oupled to

these.Byenergeti allyopenfordisso iation, Imeanthattheintera tion

energy between the ion and the ele tron, measured relative theground

vibrational state of the ionwhere the rea tion starts, is larger than the

There an alsoexist statesthat asymptoti ally formioni fragments,

andthispro essisex ludedfromDR.Whendisso iationintothis

han-nel happens, the pro ess is referred to as resonant ion-pair formation

(RIP).Thisrea tion willbe dis ussedinthe following se tion.

For simpli ity, Ihave only onsidered a diatomi mole ule ABwhere

thepotential energy urvesonly depend on one internal oordinate,the

internu lear distan e.Inthe aseofpolyatomi mole ules,thedynami s

takespla eonmultidimensionalsurfa esandthe fragmentsAandB an

be mole ular or atomi .

Weinvestigatearea tionby al ulatinga rossse tion.A rossse tion

isa measure of the probability for a rea tion to o ur and isan energy

dependent quantity. For an ele tron re ombination pro ess su h asDR,

inthesimplest ase theenergy isgiven bythe ollision energy between

theinitial ele tronandthemole ular ion.Fluxthatinitiallyis aptured

into the resonant states and that is not lost due to autoionization will

be redistributed among all states below the ion. The ross se tion for

DR is the sum of partial ross se tions of all neutral hannels. Sin e

theabsolutemajorityof statesbelowtheionarestatesdisso iatinginto

neutralfragments,theex eptionbeingtheion-pairstate,itisreasonable

toassumethata rossse tionforDR al ulatedbyex ludingele troni

nalstate distributions are to be determined, it is of great importan e

to in lude theele troni ouplingsbetween the states.

When the ele tron re ombines with the mole ular ion there is a

Coulomb intera tion between theion andtheele tron. It wasshown by

E. P. Wigner in 1948 that for a Coulomb attra tion between olliding

parti les, at low ollision energies the ross se tionfor the rea tion will

be inversely proportionalto the energy [6℄.

From this short introdu tion of DR itis not hard to understand that

thetheoreti altreatment ofthispro essisfarfromtrivial.Inthetheory

se tionofthisthesis,theapproximationsandmethodsusedfor

perform-ingthese al ulationswill be dis ussed.

1.2

Resonant ion-pair formation

Resonant ion-pair formation (RIP) is a similar pro ess to DR ex ept

that oppositely harged pairs of fragments are formed. The dire t and

indire tmode of RIP an be representedby

AB

+

₊

e

−

→

AB

∗∗

→

A

+

₊

B

−

(1.3) AB

+

₊

e

−

→

AB

∗

→

AB

∗∗

→

A

+

₊

B

−

(1.4) respe tively.

The ele tron anity of an atom or mole ules is the energy released

whenthe extraele tron isdeta hed froma singly hargednegative ion.

Assumethatthemole ularionAB

+

disso iatesintofragmentsA

+

₊

B.If

theele tron anityofthe Bfragment intheA

+

₊

B

−

hannelissmaller

thanthedisso iationenergyoftheinitialion,theasymptoti limitofthe

ion-pairstate liesabove thegroundvibrationalstate oftheionand this

hannel is then energeti ally losed for disso iation at zero intera tion

energy.ThisisillustratedinFig.(1.3). In ontrastto theneutralstates

belowtheion, the ion-pairstateis notat intheasymptoti region due

to the Coulomb attra tion between the ion pair, and this is shown in

Fig.(1.3).

Dierentexperimentalte hniqueshavebeen developed overtheyears

tostudyrea tionssu hasDRandRIP.Oneofthemostsu essful

meth-ods inuseis theionstoragering[7℄.Here, themole ular ionsarestored

underhigh-va uum onditionsathighkineti energiessothatI-Ra tive

ions have time to relax down to the ground vibrational state through

photon emission. In this way, a well dened initial ion-state is formed.

Theionsarethen ollidedina ontrolledmannedoverasmallintera tion

region with old ele trons ina o-parallel merged beams onguration.

The bending magnets in the storage ring make it easy to separate the

Figure 1.3: S hemati potentialenergy urvesfortheRIP pro ess. D

0

(AB

+

) isthedisso iationenergyfortheionandEA(B)istheele tronanityforB

alsoawelldenednalstatefortheRIPpro ess.StudyingRIPpro esses

theoreti ally and omparing results with experimental measurements is

ne essary if we are to rea h a deeper understanding of the underlying

quantum ee ts that ontribute to the stru ture whi h an be seen in

therea tion rossse tion.

Stru tures in the ross se tion ould arise from, for example,

interferen e between dierent pathways and tunneling whi h are both

purelyquantum me hani al ee ts. These phenomena areillustrated in

Fig. (1.4). It is the oupling between a Rydberg state and a resonant

state that makes it possible at the rossing point for ux whi h is

initially in the resonant state to ouple out onto the Rydberg state. If

the same urves ross again, the pro ess an o ur again and due to

the fa t that there are now several ompeting pathways to rea h the

same point, interferen e between the various ux pa kets will o ur.

Without su h rossings, tunneling through the barrier between states

anstill o ur. Stru turesinthe rossse tionthataredue to tunneling

pro esses are alled shape resonan es and stru tures with their origin

in interferen e between dierent pathways are alled Stü kelberg [8℄

se tion.

By omparing our al ulated RIP ross se tions with those

experi-mentally measured in ion storage rings, su h as CRYRING in

Sto k-holm [9,10℄, and TSR inHeidelberg [11, 12℄ we have rea hed a deeper

understandingintheimportan eofhaving welldes ribedpotentialsand

ouplingsinour al ulations.

1.3

Mutual neutralization

Mutualneutralization (MN)o urs onthesame set ofpotentialsasthe

ele tronre ombinationpro essesDRandRIP.Butinthis ase,the

rea -tionstartsoutintheion-pairlimitandgoestowardssmallerinternu lear

distan es. An ele tron is transferred between the negative and positive

ions,and after ree tion towards theinner walls of the potentials,

neu-tral fragments are formed. In a quasidiabati pi ture, where the

ele -troni ouplingsarelo alizedatthe urve rossings,anele trontransfer

eventismostlikelytoo urattheinterse tionbetweenthe ion-pairand

neutral state potentials.

TheMN pro essis illustratedinFig.(1.5)and anberepresentedby

A

+

₊

B

−

The neutral fragments formed inthe rea tion might be in formed in

their ground or ex ited states. Note that if the ion-pair state does not

rossthe groundstate, there is no a essto this state, and only one of

theneutral fragments an be formed in its ground state,see Fig. (1.5).

From thepotentials in this gure we an see that in this ase, the rst

hannel, the ground state A(1)+B(1), is not available sin e there is no

rossingand hen e no a essto theex ited states. The se ond hannel,

A(2)+B(1), isopen at zero ollision energy sin e this limit is belowthe

ion-pair limit A

+

₊

B

−

and so the ion-pair state must ross this state

twi esothereisanenhan edpossibilityofinterferen ebetweendierent

pathways for this hannel.

To rea hthethirdand forth hannel, A(3)+B(1)and A(4)+B(1),the

ollision energy must be higher than their orresponding threshold

en-ergies E

th

(1)

and E

th

(2)

. This means that the ross se tions for these hannels arezero for energiesbelowE

th

(1)

and E

th

(2)

respe tively.

In summary, the shape of the ross se tion for a MN rea tion very

mu hdependsonthepositionoftheion-pairstaterelativetotheneutral

states.However,quantumee tssu hastunnelingandinterferen es will

also ontributewith moreor lesspronoun ed stru tures.

The rea tion of H

+

+ H

−

has served as a ben hmark for MN studies

mid-fties by Bates and Lewis [13℄and sin e then more rigorous

treat-ments of the rea tion have been published [14,15, 16℄. This parti ular

rea tion ano urat zero ollisionenergy sin einthis asetheion-pair

limitlies above someof theneutral hannels.

InpaperV,we studytheMN ofH

+

₊

F

−

.Our al ulations showthat

an ex ess energy is needed for the rea tion to o ur, i.e. at non-zero

ollisionenergies. However, otherstru ture al ulations ofthe potential

energy urvesof HFshowan interse tionbetween theground state and

theion-pair state,and ifthis is the ase, no ex essenergy isneededfor

thisrea tion to o ureither.

Finally,notethattheautoionizationpro esseshavenotbeenin luded

in our theoreti al treatment of MN. In the example urves shown in

Fig.(1.5) theresonant state liesabove theionAB

+

urve and an then

be rea hed even at zero ollision energy if the ion-pair limit is above

the ground vibrational state of the ion. Autoionization may therefore

ontributeandour theoreti almodelhastobedeveloped toin lude this

possibility.

DESIREE is a double ele trostati storage ring whi h is now

un-der onstru tion in Sto kholm [17℄. It is designed for experiments with

merged beams of positive and negative ions and one of the aimsof the

apparatus is to allow mutual neutralization rea tions to be studied at

low ollision energies. Using an imaging te hnique [18℄ it will also be

possible to determine the nalstate distributions of the produ ts from

MNrea tions.Comparingour al ulatedMN rossse tionswiththe

or-responding experimental rossse tion measured with DESIREE would

provide us withmoreinformation onthevalidity ofthe approximations

2. Theory

In this hapter, I will dis uss the theoreti al ba kground to the papers

in luded in the thesis. For simpli ity and larity in the derivations of

the equations, I will des ribe the ase of a diatomi mole ule with an

internu lear distan e

R

whenthe equations inthis hapter are derived.

Coordinates of all ele trons are represented by

r

.The generalization to

polyatomi mole ules, where

R

be omesave tor,isstraightforward but

theequations aremore ompli ated. Atomi units are usedthroughout

this thesis unless otherwise stated, i.e.,

~

= k

e

= m

e

= e = 1

,where

~

is the redu ed Plan k onstant,

k

e

is the Coulomb onstant,

m

e

is the

ele tronmassand

e

istheelementary harge.

2.1

Potential energy curves and couplings

Inthisthesis, the out ome ofa hemi alrea tionis investigatedbyrst

al ulatingpotential energy urves and ouplings for themole ular

sys-temand thenstudyingthe nu lear dynami s onthese urves. Thebasis

fortheideathatnu leardynami stakespla eonpotentialenergy urves

isthefamousBorn-Oppenheimer approximation[19℄,inwhi hthe

time-independentS hrödingerequationofamole uleisseparatedintoan

ele -troni andanu lear part.Apotentialenergy urveisthen al ulated by

solvingtheele troni S hrödingerequationatdierentxedinternu lear

distan es.In order to go beyond theBorn-Oppenheimer approximation

dierent representations of the ele troni states of a mole ule an be

used.Themost ommon istheadiabati representation, wherethe

ele -troni Hamiltonianisdiagonalized.For somepurposeshowever,abetter

option an be to usea non-diagonal diabati representation of the

ele -troni states. In a diabati representation, the diagonal elements of the

ele troni Hamiltonian are the diabati potentials and the o-diagonal

elementstheele troni ouplingsbetween thesestates.

2.1.1

Approximations and definitions

Thenon-relativisti Hamiltonianof adiatomi mole ule hastheform

H = −

_2µ

1

∇

2

R

+ H

el

,

H

el

= −

1

2

n

el

X

i=1

∇

2

r

i

+

Z

A

Z

B

R

+

n

el

X

i=1

n

el

X

j>i

1

r

ij

−

n

el

X

i=1



Z

A

r

Ai

+

Z

B

r

Bi



.

(2.2)

The rst term in Eq. (2.1) is the nu lear kineti energy operator

T

N

(R, θ, ϕ)

, and this an be separated further into a rotational and a

vibrational term whi h des ribe therotation of thenu lear axisaround

the enter-of-mass of the mole ule and thevibrations along thenu lear

axis, respe tively. The internu lear separation ve tor

(R, θ, ϕ)

denotes

thespatial orientation ofthemole ule.The rstterminEq. (2.2)isthe

ele tron kineti energy operator

T

e

(r)

. The remaining terms are the

ele trostati potential energy,

V (R, r)

, onsistingof thenu lear-nu lear

and ele tron-ele tron repulsions and the ele tron-nu lear attra tion.

µ

istheredu ed massofthetwo nu lei,

A

and

B

,andisdened as

µ =

M

A

M

B

M

A

+ M

B

,

(2.3)

and

Z

A

and

Z

B

arethe nu lear hargestates.

To nd solutions

ψ(R, θ, ϕ; r)

to the time-independent S hrödinger

equation

Hψ(R, θ, ϕ; r) = Eψ(R, θ, ϕ; r),

(2.4)

thetotal wavefun tion an beapproximated bythe produ tof an

ele -troni anda nu lear wave fun tion

ψ(R, θ, ϕ; r) = χ(R, θ, ϕ)φ(R, r).

(2.5)

Thisistheso alledBorn-Oppenheimerprodu tand isinsertedintothe

time-independentS hrödingerequation, Eq.(2.4).Usingtheassumption

thatthe ele trons move mu h faster than the mu h heavier nu lei, this

equation an be separated into a nu lear and an ele troni part. The

ele troni S hrödinger equation for adiatomi mole ule hastheform

H

el

φ

a

i

(R, r) = E

i

a

φ

a

i

(R, r),

(2.6)

where

R

no longer isa variablebut a parameter.By solving this

diago-nalequationatdierentnu lear positions,anadiabati potentialenergy

urve

E

a

i

(R)

is formed.The potential energy urve is alled adiabati 

sin eitisassumedthattheele tronsrespondimmediatelytothemotion

ofthe nu lei.The nu lear time-independent S hrödinger equation

[T

N

+ E

i

a

]χ

a

i

(R, θ, ϕ) = E

i

χ

a

i

(R, θ, ϕ),

(2.7)

des ribes the nu lear motion on the state

i

. It an be the vibrational

orrotational motionofbound ele troni statesand alsodisso iation

tenagood approximationfor ele troni ground statessin e thesearein

generalwellseparatedinenergyfrom theex ited states.For theex ited

states,ontheotherhand,thepotentialenergy urvesarenotalwayswell

separatedfromea hotherandthestatesintera tmoreeasily.Inthese

re-gions,nu leardynami swillnotfollowtheele troni statesadiabati ally

andtheBorn-Oppenheimer approximationbreaksdown.

The intera tion of the nu lear and ele troni motion is responsible

both for the ouplings between the dierent neutral ele troni states

andalsofor autoionization. When des ribingpro essessu h as

disso ia-tive re ombination, ion-pairformation,and mutualneutralization,these

ouplings are ru ial and soit is absolutely ne essaryto go beyond the

Born-Oppenheimer approximationifthese rea tions areto be studied.

Inonesu hmodelwhi hgoesbeyondtheBorn-Oppenheimer

approxi-mation,theadiabati stateswillget oupledbythekineti energy

opera-tor

T

N

(R, θ, ϕ)

.Asmentionedearlier,thenu learkineti energyoperator

anbe dividedinto a vibrationaland a rotationalpart

T

N

(R, θ, ϕ) = T

vib

(R) + T

rot

(θ, ϕ),

(2.8) where

T

vib

(R) = −

1

2µR

2

∂

∂R



R

2

∂

∂R



(2.9) and

T

rot

(θ, ϕ) = −

1

2µR

2



1

sin θ

∂

∂θ



sin θ

∂

∂θ



+

1

sin

2

θ

∂

2

∂ϕ

2



.

(2.10)

Assume that Eq. (2.6) has been solved for all

R

and that a full set

of adiabati ele troni states have been obtained. The solutions an be

takenasorthogonalandnormalized andassu htheyprovidea omplete

basisforfun tionsdenedovertheele troni spa e.Thetotalwave

fun -tion an thenbeexpanded as

ψ(R, θ, ϕ; r) =

∞

X

i=1

χ

a

_i

(R, θ, ϕ)φ

a

_i

(R, r),

(2.11)

where the nu lear wave fun tions

χ

a

i

(R, θ, ϕ)

have been in luded

in order to span the whole onguration spa e. If this expansion is

inserted in Eq. (2.4), and that expression is multiplied from the left

with

φ

a

j

(R, r)

and integrated over the ele troni oordinates a oupled

time-independent S hrödinger equation for the adiabati states an be

obtained

[T

N

− E

i

a

]χ

a

i

−

X

j



1

µ

F

a

ij

· ∇

R

+

1

2µ

G

a

ij



χ

a

_j

= Eχ

a

_i

,

(2.12)

Internuclear distance

Potential energy

R

x

φ

a

1

,

φ

d

1

φ

a

2

,

φ

d

2

φ

a

1

,

φ

d

2

φ

a

2

,

φ

d

1

Figure 2.1: S hemati potentialenergy urves lose toa rossing. Solid lines are adiabati potential energy urvesand dashed lines are diabati potential energy urves.

φ

a

and

φ

d

representtheadiabati and diabati ele troni wave fun tionsrespe tivelyand

R

x

istheavoided rossingpoint.

where

F

_ij

a

_{= hφ}

a

_i

_|∇

R

|φ

a

j

i

(2.13)

G

a

_ij

_{= hφ}

a

_i

_|∇

2

_R

_|φ

a

_j

_{i .}

(2.14)

Theo-diagonalelements

F

a

ij

and

G

a

ij

a tas ouplingsbetweendierent

ele troni statesandarereferredtoasrstderivativeandse ond

deriva-tive non-adiabati ouplingsrespe tively.

F

a

ij

ispurelyo-diagonal while

G

a

_ij

alsohasdiagonalterms,whi haresometimesreferredtoasadiabati

orre tions. Theo-diagonal ontributions from

G

a

ij

arenormally mu h

smallerthanthoseof

F

a

ij

andarethereforeoftennegle ted. Itwasshown

in 1929 by Neumann and Wigner that the adiabati potential energy

urves,for adiatomi mole ule, orrespondingtoele troni statesofthe

samesymmetry annot ross[20℄.Thisisreferredto asthenon- rossing

rule.Instead,two adiabati potential energy urvesofthesame

symme-tryrepelea hotherwhenthey ome lose.Thepointof losestapproa h

is alledan avoided rossing point and inFig.(2.1) isindi ated by

R

x

.

Thestrongest ontributiontothenon-adiabati ouplingintheregion

ofanavoided rossing omesfromtheradialpartof

F

a

se ond derivative oupling elements a ton

χ

a

i

. This type of dierential

ismore di ult to solve than one ontaining ouplingswhi h have only

potentialform. Furthermore,thenumeri alevaluationofthese ouplings

analso benontrivial.

These issues an be ir umvented byusing diabati states dened in

su h a way that the rst derivative ouplings disappear. The rst

re-ported use of su h states, that later would be referred to as diabati

states, or a tually quasidiabati states, was by Zener in 1932 [21℄. The

adiabati ele troni eigenfun tions have a typi al hange of hara ter

lose to an avoided rossing. Zener assumed that the basis fun tions

used were approximative eigenfun tions of the ele troni Hamiltonian

without this hara teristi , and this is also indi ated in Fig (2.1). As

the name diabati suggests, these states do not adjust adiabati ally to

hanges intheinternu lear distan e.

ThestatesusedbyZenerarereallyquasidiabati ,sin etherst

deriva-tiveradial ouplingsdonotvanish ompletelywiththisapproa h.Ashas

been pointedout byC.A.Meadand D.G.Truhlar[22℄,every omplete

setof ele troni statesfullling

F

_ij

d

_{(R) = hφ}

d

_i

_|

∂

∂R

|φ

d

j

i = 0

(2.15)

mustbeindependentofR.However,thebasissetusedina al ulationis

both nite and in omplete and thena diabatization within thenumber

ofstates onsidered anbeperformed. Thefollowingsteps arerequired.

First,thediabati statesare expandedintheadiabati statesas

φ

d

_i

(R, r) =

M

X

j=1

φ

a

_j

(R, r)T

T

ij

(R) ,

(2.16)

where

T

ij

is an element in the orthogonal adiabati to diabati

trans-formation matrix (ADTM) and

M

isthe number of adiabati statesto

betransformed.Theexpansionistheninsertedintothestri tlydiabati

ondition,Eq. (2.15),andaftersome manipulations thefollowing

dier-ential equation for the ADTMmatrix is obtained

d

dR

T

+ F

a

_T

_{= 0 .}

(2.17)

This dierential equation an be solved with the following boundary

ondition

lim

R→∞

T(R) = 1,

(2.18)

whi hmakestheadiabati anddiabati potentialenergy urvesidenti al

furthermore, that the rst derivative ouplings between the adiabati

stateswith

i < M

andthosewith

i≥M

arezero,it anbeshownthatthe

se ondderivative ouplingsbetweenthediabati statesarealsozero[23℄.

The oupledS hrödingerequationinthestri tlydiabati representation

be omes

T

N

χ

d

i

−

X

j

V

_ij

d

χ

d

_j

= Eχ

d

_i

,

(2.19)

where thediabati potential matrix elementsaregiven by

V

_ij

d

_{= hφ}

d

_i

_|H

el

|φ

d

j

i .

(2.20)

In ontrast to theadiabati potential matrix,thediabati potential

ma-trix is not diagonal. The diagonal elements are the diabati potential

energy urves

E

d

i

(R) = V

ii

(R)

. The o-diagonal elements will ouple

the diabati ele troni states. These ouplings are alled the ele troni

oupling elements

c

ij

(R) = V

ij

(R), i 6= j

.Therelationship between the

adiabati and the diabati potential matrix isgiven by

V

d

= T

T

V

a

T

.

(2.21)

In the ele tron re ombination pro esses that have been studied in the

work presented here, there exist an innite number of Rydberg states

onverging to the ion ore. Furthermore, there are a large number of

resonant states that are embedded in the ontinuum of the ion plus a

free ele tron. In general, these resonant states will ross both the

ion-ization ontinuum and the Rydberg states and this situation makes a

stri tdiabati representationimpossibleto use.Toresolvethis problem,

a kind of quasidiabati representation, in whi h theele troni oupling

elementsarelo alized totheregion of avoided rossings,hasbeen used.

The methods that have been employed to determine the quasidiabati

potentials andthe ouplings will be des ribed inalater se tion.

2.1.2

Structure calculations

Sin e the states involved in the pro esses under investigation onsist

of both ele troni ally stable states belowthe ion and ele troni ally

un-stable, autoionizing states, above the ion, the potential energy urves

annot be al ulated ompletely with the quantum hemistry methods

normally used for this purpose. S attering al ulations need to be

per-formedonthosestatesabovetheiontodeterminethetruepositionsand

theautoionizationwidthsoftheresonantstates.Belowtheion,however,

whereallstatesareele troni allystable, thepotentialenergy urves an

be al ulated using onventional stru ture al ulations. In the

s attering al ulations withthose obtained from stru ture al ulations,

thesamebasisset,typeoforbitals andquantum hemistry methodthat

has been used for the target wave fun tion, must also be used in the

stru ture al ulations on the neutral ele troni states. To obtain

a u-rateresults from the stru ture al ulations agoodbasisset and a large

expansion of the wave fun tion are needed. Unfortunately, due to the

s attering method used,thereis arestri tionon thesize ofthebasisset

andtheexpansionthat anbeusedandthismeant thatthe al ulations

on the potentials of the stable states were less a urate. However, the

benet ofthes attering results ompensates for this disadvantage sin e

it provides both the autoionization width and a mu h more a urate

resultfor theposition oftheresonant states.

Thequantum hemistry methods [24,25℄thathave been used to

al- ulate thepotentials of the stablestates inthis thesis will bedes ribed

hereand thes attering al ulationsare des ribedinthe nextse tion.

IntheHartree-Fo k(HF)method,alsoreferredtoastheself- onsistent

eldmethod,theele troni wavefun tion isexpressedasaSlater

deter-minant

φ

SD

=

1

√

N

el

!





















ξ

1

(1)

ξ

2

(1)

. . .

ξ

N

el

(1)

ξ

1

(2)

ξ

2

(2)

. . .

ξ

N

el

(2)

. . . . . . . . . . . .

ξ

1

(N

el

) ξ

2

(N

el

) . . . ξ

N

el

(N

el

)





















,

(2.22)

where

ξ

i

(j)

isthemole ularspinorbital

i

forele tron

j

.Thespatialparts

of the mole ular spin orbital onsist of mole ular orbitals (MOs) that

are onstru ted by linear ombinations of atomi orbitals. The optimal

wave fun tionisobtainedbyminimizing thefollowingexpressionforthe

energy

E

SD

= hφ

SD

|H

el

| φ

SD

i .

(2.23)

Otherele troni stru ture al ulationsbeginwithaHartree-Fo k(HF)

al ulation.Fromthis,the mole ular orbitals(MOs)anda groundstate

solution an be obtained though ele tron orrelation ee ts are not

a - uratelyin luded.The ongurationintera tion(CI)methodisthe

sim-plestwaytoin ludethis orrelationandtothereforeimprovethe

Hartree-Fo ksolution.Thisisamethodwellsuited for al ulatingex itedstates

of small mole ules. The wave fun tion is then onstru ted as a linear

ombinationof Slater determinants

φ

CI

= φ

SD

+

X

i

c

i

φ

i

,

(2.24)

where

φ

SD

is the initial Hartree-Fo k wave fun tion and

φ

i

are Slater

thesolutionto the Hartree-Fo kequation.

For alineartrialwave fun tion,thevariational prin iplenowleadsto

solving the se ular equations for the CI oe ients. In this ase there

willbeasmanysolutionsasthereare ongurationsintheCIexpansion.

The solution with the lowest energy is the ground state and the other

solutions orrespondto ex ited states.

The eigenfun tions of the ele troni Hamiltonian are simultaneously

eigenfun tionsof the spinoperator and soanalternative approa h isto

use ongurationalstatefun tions(CSFs),whi harespinadaptedlinear

ombinationsofSlater determinantsdesignedtobeeigenfun tionsofthe

S

2

operator inthe CIexpansion.

In a full CI (FCI) al ulation a omplete set of determinants is

gen-erated by distributing the ele trons among all of the orbitals and then

in ludedintheCIexpansion,Eq.(2.24).ThenumberofSlater

determi-nantsin reases very rapidly both withthenumberofele trons and also

withthenumberof orbitals. A FCIexpansion istherefore suitable only

for thesmallest ele troni systems.

For larger ele troni systemsthe FCIexpansion has to be trun ated.

When designing smaller onguration spa es, it is important to

dis-tinguish between stati and dynami orrelation. Stati orrelation is

treated by retaining the dominant ongurations of the FCI expansion

aswellasthosethatarenearlydegeneratewiththedominant

ongura-tions.These ongurationsarereferredtoasthereferen e ongurations

of theCI wave fun tion, and they span thereferen e spa e. Dynami al

orrelation is treated by adding ongurations whi h are generated by

ex itations outof the referen espa e to the wave fun tion.

Thereferen espa einamultireferen eCI(MRCI) al ulation should

ontain all those ongurations thatmaybe ome importantto des ribe

the potential energy urves needed to model a ertain rea tion. The

MRCIwave fun tion isgenerated byin luding all ongurations inthis

referen espa easwellasthose ongurationsgeneratedbyex itationsof

thereferen eele trons intothe virtualorbitals. Itis ommonto in lude

all singleand double ex itations out of thereferen e spa e, resulting in

aMRsingles-and-doubles CI(MRSDCI)wave fun tion.

The s attering al ulations ne essary to obtain the potential energy

urvesabovetheionarefarmoretime onsumingthanthestru ture

al- ulations arriedout for those statesbelowtheion. To redu ethe

om-putational time required for the s attering al ulations, but still

main-taining good a ura y for the potential energy urves, natural orbitals

For aCIwavefun tion onstru tedfromorbitals

φ

i

,theele tron

den-sity fun tion

ρ

an bewritten as

ρ =

X

i

X

j

ρ

ij

φ

∗

i

φ

∗

j

,

(2.25)

where the oe ients

ρ

ij

are a set of numbers whi h form the density

matrix.NOsareorbitalsthatredu ethis densitymatrix

ρ

toadiagonal

form

ρ =

X

k

b

k

φ

∗

k

φ

∗

k

,

(2.26)

and the oe ients

b

k

arein this ase, the o upation numbers of ea h

orbital. A CI expansion based on su h orbitals will generally have the

fastest onvergen e. A CI al ulationis rst arriedout using theMOs

obtained froma SCF al ulation. The onstru ted densitymatrix

ρ

ij

is

thendiagonalizedandtheNOsaredetermined.InthesubsequentMRCI

al ulation, the referen e spa e is built up by using the NOs and this

provides a ompa t representation of theorbitals thatis well suited for

thefollowings attering al ulations.AMRCI al ulationusingtheNOs

instead of the arbitrary basis set will give a wave fun tion onsisting

only of those ongurations built up from natural orbitals with large

o upation number.

2.1.3

Scattering calculations

S attering events an o urbetween dierent kinds of parti les. In this

thesis,Ihave onsidereds attering betweenan ele tronandamole ular

ionin theele tron re ombinationstudies and also s attering between a

ation and an anion in the ase of mutual neutralization. The ele tron

s attering al ulations have been performed in order to determine the

parameters of the resonant states above the ion. These parameters are

theenergy positionand theautoionizationwidthat a xedinternu lear

distan e

R

for a parti ular state. In studying the s attering of an

ele -tronand mole ular ion,the omplex Kohn variational method hasbeen

used [26, 27℄ and this method is outlined later in this se tion. For the

mutualneutralization pro ess,the potential energy urveshave already

been determined and the dynami s on these urvesis investigated with

nu lear s attering al ulations. The method used in performing these

al ulationsis des ribed furtherinthe se tion onRea tion Dynami s.

General

Consider the s attering pro esses between an in oming ele tron and a

mole ulartargetshowns hemati allyinFig.(2.2).Oneofthedi ulties

ins attering al ulations [28℄is how to handlenon-spheri ally

spheri ally symmetri intera tion potential

V (r)

, where

r

now denotes

theradialdistan ebetweentheele tronandthetarget. Theentry

han-nel onsists of the ion in its ground vibrational level and an in oming

ele tronwithwavenumber

k

travelingalongthe ollisionaxis.One

exam-pleexit hannel anbetheioninitsgroundvibrationora rovibrational

ex itedlevelandtheele trontravelingwithadierentwavenumberand

inadire tion

k

′

.Atlargeinternu leardistan es,thes atteredwave

fun -tionshould onsistofan in omingplanewave intheentran e hannel

j

andoutgoing spheri alwavesintheenergeti allyavailableexit hannels

i

. If the in oming plane wave is taken as traveling along the ollision

axis(thez-axis)the boundary onditionfors attering from hannel

j

to

hannel

i

an be written as

χ

d

_i

_{(r, θ) ∼}

r→∞

δ

ij

e

ik

j

z

₊

f

ij

(θ)e

ik

i

r

r

,

(2.27)

where the azimuthal angle

ϕ

dependen e has been omitted due to the

ylindri alsymmetryaroundthe ollisionaxis.Thefun tion

f

ij

(θ)

isthe

s attering amplitude and

k

i

is the asymptoti wave number. The ross

se tion for therea tion is obtained by integrating the dierential ross

se tion, given by the absolute square of the s attering amplitude, over

allsolidangles

σ

ij

=

2πk

i

k

j

Z

π

0

|f

ij

(θ)|

2

_{sin θ}

d

θ .

(2.28)

To determine the s attering amplitudes,

f

ij

(θ)

,the asymptoti form of

thes atteredwavefun tion,Eq.(2.27),isexpandedintermsofLegendre

polynomials,

P

l

(cos θ)

,andradialwavefun tions. Theradial wave

fun -tions in the asymptoti limit an be written as a linear ombination of

anin oming wave and an outgoing wave multiplied byas attering

ma-trixelement,

S

o

ij

,whi h measures the responseofthe target.When this

expressionis inserted into the partialwave expansion,and thenal

thefollowing expression for the s attering amplitude

f

ij

(θ)

isobtained

f

ij

(θ) =

i

2(k

i

k

j

)

1/2

∞

X

l=0

(2l + 1) δ

ij

− S

ij,l

o

P

l

(cos θ) .

(2.29)

Thesupers ript

o

onthes atteringmatrixelement

S

o

ij,l

indi atesthat en-ergeti allyopen hannels aretreated.Inserting theexpressionfor

f

ij

(θ)

into thatfor the rossse tion, Eq.(2.28),yields

σ

ij

=

∞

X

l=0

σ

ij,l

,

(2.30) where

σ

ij,l

=

π

k

2

j

(2l + 1)



δ

ij

− S

ij,l

o





2

.

(2.31)

Consider a totally elasti ollision in one ofthe hannels

i

. The ross se tionfor this hannel,

σ

e

= σ

ii,l

,be omes

σ

e

=

π

k

2

∞

X

l=0

(2l + 1)|1 − S

l

|

2

,

(2.32)

where

S

l

= S

ij,l

is thes attering, or

S

,matrix,thequantity

1 − S

l

= T

l

isthetransition, or

T

,matrix.

Attheresonantenergy,

E

res

,theele tron anbetemporarily aptured

by the ioninto a nearly bound state giving rise to a sharp variation in

the rossse tion,i.e. aresonan e. The termresonan e iswidelyusedin

theliterature, butunfortunately,oftenwithdierentmeanings. Sowhat

isit? Supposethat the ele tron is aptured, or temporarily trapped by

theion,at someenergy

E

res

.Atthisenergy theelasti rossse tionwill

have a pole in the

S

(or

T

) matrix and, therefore, give rise to a sharp

variation inthe rossse tion lose to the resonant energy

E

res

.A

non-redundant poleinthe

S

(or

T

)matrix isalsoazeroof theJost fun tion

f

l

[28℄sin e

S

l

(k) =

f

l

(−k)

f

l

(k)

.

(2.33)

Furthermore,azerooftheJostfun tion,orapoleinthe

S

or

T

matrix,

also orrespondsto apoint inthe omplex

k

-plane.Three dierent kind

ofstatesexistdependingonwhere the point

¯

k

islo atedinthe omplex

k

-plane:

bound state,

k = iκ

¯

, with

κ > 0

virtual state,

¯

k = −iγ

, with

γ > 0

resonant state,

¯

Sin e

E

isproportionalto

k

2

,aresonantstatewillhave omplexenergy

with

E = E

res

− iΓ/2

. If

E

is real, the probability of the state an be

written as

|ψ(r, t)| =





φ(r)e

−iEt/~







2

= |φ(r)|

2

,

(2.34)

but,on theotherhand,if

E

is omplex

|ψ(r, t)| =





φ(R)e

−i(E

res

−iΓ/2)t/~







2

= |φ(r)|

2

e

−Γt/~

.

(2.35)

Thus,aresonant statehasalifetimeand isnotastationary stateofthe

Hamiltonianfor the system.

TheJost fun tion,

f

l

(k)

,hasa zeroat a resonan e. Nearsu h a

reso-nan etheJostfun tion anbeexpanded about

¯

k = k

res

− iγ

,a ording

to

f

l

(k) =



df

l

dk



¯

k

(k − ¯k).

(2.36)

Provided that

¯

k

is fairly lose to the real axis, Eq. (2.36) is a good

approximation. Thephaseshift for a xedvalueof

l

,

δ

l

(k)

,is given by

δ

l

(k) = arg(f

l

) = − arg



df

l

dk



¯

k − arg(k − ¯k)

(2.37)

≡ δ

bg,l

+ δ

res

(k)

(2.38)

where

δ

bg,l

is the slowly varying ba kground ontribution to the phase

shiftand

δ

res

(k)

isthe resonant partofthephaseshiftthatvaryrapidly.

δ

res

(k)

isthe angleshowninFig.(2.3).When

k

isin reasedsu hthatit movespastthepositionof

k

¯

,theresonantpartofthephaseshift,

δ

res

(k)

, will in reasefrom

0

to

π

.The loserthat

k

¯

isto the real axis, themore

qui klythis in reasehappens. Ifthephaseshiftinsteadisexpressedas

afun tion of

E

,wehave that

δ(E) ≈ δ

bg,l

+ δ

res

(E),

(2.39)

and

sin δ

res

(E) =

Γ/2

[(E − E

res

)

2

+ (Γ/2)

2

]

1/2

.

(2.40)

In the spe ial ase where the ba kground phase

δ

bg,l

is zero,

σ

l

(E) ∝

sin

2

_(δ

l

(E))

,and the expressionfor thepartial rossse tionisgiven by

σ

l

(E) ∝

Γ

2

/4

(E − E

res

)

2

+ (Γ/2)

2

.

(2.41)

k

-plane.

In an a tual ele tron-mole ular ion s attering event, the intera tion

potential,

V (r)

,isnon-spheri al andthe

S

(or

T

)matrix will have both

l

and

m

indi es. Resonant statesappearassharpvariations inthe ross

se tion, and even more learly when the eigenphase sum is studied. As

afun tion of energy,the eigenphasesum isgivenby:

δ(E) =

X

l,m

δ

l,m

(R, E),

(2.42)

where

R

is the xed internu lear distan e at whi h the al ulation is

performed.Thesuddenshiftoftheeigenphasesumwith

π

attheresonant

energy

E

res

is illustrated in Fig. (2.4), whi h shows the results of an

ele trons attering al ulationfor

e

−

₊

HF

+

in

1

_A

1

symmetryat

R = 1.2

a.u..The omplexKohn variationalmethodwasusedinthis al ulation

andthis methodis des ribed inthe nextse tion.

When tting the eigenphase sum inthe neighborhood of a resonan e

tothe following Breit-Wignerform, both theposition,

E

res

(R)

,andthe

autoionizationwidth,

Γ(R)

,oftheresonan eatinternu lear distan e,

R

, areobtained.

δ(E) = δ

res

(E) + δ

bg

(E)

= tan

−1



Γ

2(E − E

R

)



+ a + bE + cE

2

.

(2.43)

Here,

δ

res

isthe ontribution fromtheresonan eto the eigenphasesum

and

δ

bg

istheba kground ontribution.Toobtainthepotentialenergyof

theresonant state,the ioni potential hasto be added to theresonan e

energy

V

res

(R) = V

ion

(R) + E

res

(R).

(2.44)

Complex Kohn variational method

Inthe omplex Kohn variational method [26,27℄, a trial wave fun tion

0

0.05

0.1

0.15

0.2

10

2

Interaction energy (H)

Cross section (cm

2

)

0

0.05

0.1

0.15

0.2

−2

−1

0

1

2

Interaction energy (H)

Eigenphase sum (rad)

(a)

(b)

Figure 2.4: Resonan esfoundin elasti ele trons atteringwith

HF

+

in

1

_A

1

symmetry and at an internu lear distan e

R = 1.2 a

0

. (a) shows the ross

usingashort-rangespheri allysymmetri potential,

V (r)

,butthe

appli- ationto more realisti ases will alsobementioned.Ifthe partialwave

radialS hrödinger equation

Lφ

l

(r) = 0,

(2.45) where

L =



−

1

₂

_dr

d

₂

+

l(l + 1)

r

2

+ V (r) −

k

2

2



,

(2.46)

isinserted into the Kohnfun tional,

I

,dened as

I[φ

l

(r)] = hφ

l

|L|φ

l

i =

Z

∞

0

φ

l

(r)Lφ

l

(r)dr,

(2.47)

then the fun tional

I

is zero if the wave fun tion is the exa t solution

to Eq. (2.45). Conversely, if a trial wave fun tion

φ

t

l

is inserted, the

fun tional

I

diersfromzero.Assumethefollowingboundary onditions

for

φ

t

l

φ

t

_l

(0) = 0

(2.48)

φ

t

_l

_{(r → ∞) ∼ F}

l

(kr) + λG

l

(kr),

(2.49)

where

F

l

and

G

l

are two linearly independent solutions of the radial

S hrödinger equationwith

V (r) = Z/r

.Therst onditionrequiresthat

thetrialwavefun tion tobe niteattheoriginwhereasthese ond

on-dition onstrains the ele tron to behave as an outgoing spheri al wave.

The parameter

λ

is a linear oe ient. In order to examine how the

fun tional

I

hanges when the exa t solution of the radial S hrödinger

equation

φ

l

(r)

isrepla edbythetrialwave fun tion,

φ

t

l

(r)

,theresidual

between thesewave fun tions, dened as

δφ

l

(r) ≡ φ

t

l

(r) − φ

l

(r)

(2.50)

withtheboundary onditions

δφ

t

_l

(0) = 0

(2.51)

δφ

t

_l

_{(r → ∞) ∼ λG}

l

(kr)

(2.52)

isinserted into the Kohnfun tional. The resultisgiven by:

δI =

Z

∞

0

δφ

l

Lφ

l

dr +

Z

∞

0

φ

l

Lδφ

l

dr +

Z

∞

0

δφ

l

Lδφ

l

dr.

(2.53)

UsingEq.(2.45) we anseethattherst terminthis expressioniszero.

Integratingthese ondtermbypartsandmakinguseofEq.(2.48)-(2.50)

gives

δI = −

k

₂

W δλ +

Z

∞

0

W = F

l

(r)

d

dr

G

l

(r) − G

l

(r)

d

dr

F

l

(r)

(2.55) and

δλ = λ − λ

t

. Here,

λ

t

is the variational parameter whereas

λ

is

theexa t value. Eq.(2.54) isknown astheKato identity.Thisequation

providesastationaryprin ipleforapproximating

λ

sin etheexa tvalues

of

λ

an beobtained intermsoftheapproximate valueofthefun tional

I[φ

t

_l

]

and atermwhi hisse ondorderintheerrorofthewave fun tion.

Thestationaryprin iple of approximating

λ

is givenby

λ

s

= λ

t

+

2

kW

Z

∞

0

δφ

t

_l

(r)Lδφ

t

_l

(r)dr.

(2.56)

The trial wave fun tion should des ribe the ele tron well in both the

intera tion region andin theasymptoti region and,for thatreason,

φ

t

l

is hosento be

φ

t

_l

(r) = f

l

(r) + λ

t

g

l

(r) +

n

X

i=1

c

i

φ

i

,

(2.57)

where

φ

i

aresquareintegrable fun tionsand

f

l

(r → ∞) ∼ F

l

(kr)

(2.58)

g

l

(r → ∞) ∼ G

l

(kr).

(2.59)

The oe ients,

c

i

and

λ

t

aredetermined fromthefollowingvariational

onditions

∂λ

s

∂c

i

=

∂λ

s

∂λ

t

= 0.

(2.60)

Insertingthetrialwave fun tion given byEq.(2.57) intoEq. (2.56) and

takingthederivativewithrespe tto

c

i

gives

Z

∞

0

φ

i

Lφ

t

l

dr = 0, i = 1, ..., n.

(2.61) Similarly,with

λ

t

,we obtain

Z

∞

0

g

l

Lφ

t

l

dr = 0, i = 1, ..., n.

(2.62)

By olle tingthebasisfun tions

g

l

(r)

and

φ

i

(r)

intoasingleset

{φ

i

}, i =

0, ..., n

, with

φ

0

≡ g

l

,and denoting the linear parameters

{λ

t

_{, c}

1

, ..., c

n

}

bytheve tor

c

,Eq.(2.61)andEq.(2.62) anbeexpressedina ompa t form

where

M

isa matrix withelements

M

ij

=

Z

∞

0

φ

i

Lφ

j

dr, i, j = 0, ..., n

(2.64)

and

s

is ave tor withelements

s

i

=

Z

∞

0

φ

i

Lf

l

dr, i = 0, ..., n.

(2.65)

The stationary value of

λ

s

is obtained by inserting Eq. (2.63) into

Eq.(2.56)whi h gives

λ

s

=

2

kW

Z

∞

0

f

l

Lf

l

dr − sM

−1

s



.

(2.66)

Finally,if

g

l

(r)

is hosento be theoutgoing fun tion

h

+

l

(r)

,dened as

h

+

_l

(r) = i[F

l

(kr) − iG

l

(kr)]

1

√

k

,

(2.67)

thentheWronskian redu es to

W = −1/k

andEq. (2.66) redu es to an

expressionfor the

T

-matrix,

λ

s

_{= T}

s

l

= e

iδ

l

sin δ

l

[28℄

T

_l

s

_{= −2}

Z

∞

0

f

l

Lf

l

dr − sM

−1

s



.

(2.68)

The matrix

M

is now omplex symmetri and its inverse is generally

non-singular at realenergies [29℄,hen ethename omplexKohn.

To use the omplex Kohn variational prin iple for ele tron-mole ular

ions attering,forwhi htheintera tionpotentialisnon-spheri al,amore

appropriatetrialwavefun tionhastobe hosen.Thistrialwavefun tion

mustbe exibleenough todes ribethe behaviorbothintheintera tion

region andalso asymptoti ally.

For this reason, the ele troni trial wave fun tion of the (

N + 1

)-ele trons attering systemis expandedas[26℄

Ψ

=

X

Γ

A[Φ

Γ

(x

1

...x

N

)F

Γ

(x

N +1

)] +

X

µ

d

µ

Θ

µ

(x

1

...x

N +1

).

(2.69)

Therstsuminthisexpressionis alledtheP-spa e portionofthewave

fun tion.The sumrunsovertheenergeti ally opentarget states, where

Φ

Γ

are the N-ele tron target eigenstates and

x

i

the spa e-spin

oor-dinates. The operator

A

antisymmetrizes the oordinates of the target

ands atteredele trons.Inthese ondsum, the

Θ

µ

aresquare-integrable

N + 1

onguration state fun tions(CSFs). Thesefun tionsareusedto

des ribe short-range orrelations and theee ts of losed hannels and

fun tion.

Two lasses of terms are in luded in Q-spa e. The rst lass of

fun -tionsisthesetof all(

N + 1

)-ele tronCSFsthat anbeformedfromthe

a tive spa e of target orbitals. They will relax any onstraints implied

by the strong orthogonality between the s attering fun tions,

F

Γ

, and

thetarget orbitals and are alledpenetration terms.A se ond lassof

terms, alledCIrelaxationterms, analsobein luded.Inthis lasswe

have all losed hannels that an be formed from the set of target

on-gurations used to build P-spa e. These CI relaxation terms, together

with the penetration terms, onstitute the orrelation part of the trial

wave fun tion.

Whenthe trialwave fun tion is insertedinto the fun tional, and this

fun tionalisminimizedtoobtaintheunknown oe ients,the

T

matrix

and thus the eigenphase sum, given by Eq. (2.42), will ontain both

l

and

m

indi es. Thekind of behaviordisplayed inFig.(2.4) an thenbe

obtained forthe rossse tion andtheeigenphase sum.

2.1.4

Diabatization

After the adiabati potential energy urves have been obtained, below

theionbystru ture al ulationandabovetheionbyele tron-s attering

al ulations, these urvesare thendiabatized. As mentioned earlier,we

arerestri ted to dierent typesof quasidiabatizations sin e inthe

rea -tions we arestudying thereare aninnite numberof intera ting states.

Themethods thathave been usedfor thepurpose ofquasidiabatization

will be dis ussed next. However, before this dis ussion, Iwill briey

in-trodu e mole ular symmetry sin e stru tureand s attering al ulations

inmost asesare arriedoutinasymmetrythatisnotthetruesymmetry

ofthemole ule.

Restriction of molecular symmetry in the calculations

Thesymmetryofmole ulesisdes ribed intermsofelementsand

opera-tions.Elementsaregeometri entities su hasaxes,planesand pointsin

spa eusedtodenesymmetryoperations.Operationsinvolvethespatial

re-arrangement of atoms in a mole ule by rotation about an axis (

C

n

),

by ree tion through a plane (

σ

), or by inversion through a point (

i

).

A rotation, ree tion or inversion operation will be alled a symmetry

operation if, and only if, the new spatial arrangement of the atoms in

themole ule isindistinguishable fromtheoriginal arrangement.

The point group, or symmetry group, is the name of a a olle tion

ofsymmetryelementspossessed byamole ule.Ea h ommon olle tion