,

~('. :.r,

')•

/-

A'

• 'l/,

"

/

... '

/,.

,/

, .

MODEL (OTENTIALS~D MOLECULAR RYDBERGS7~IES-

'.

. ,, .

,.

-'- , .. '.

;i......

"

t) a /'

, '

'j / . '~ .,

,A Thests

Submitted ~ ~eschool of Graduate Studies

,in Partial Fulfilment of the Requirements

for the 'DegreeffR "

Doctor of Philosoph~•

/

McMaster University

August 1973

','-.

•'D, Richard Frank .Greening 1974,

, ,

, ,

"

i

DEDICATION

•

/

"

-To·' my PareD.ts'

"

,

. ,

"

, .._,.'"An~here are trees and I know theirgn~rled

surface~ wa~erand I feel its taste. These scentsof grass and stars at night, certain evenings when,the heart relaxes--how shall 'r negat~ t~is worldwhose power and stren9th r fee:!.? Yet all,the 'knowledge on earth ~i11 give ~e nothing to ass'ure ~,

me that this loIorld is mine. You describe·it to meand you teach me,to classify it. You enum~rate itsl~ws and in my thirst for knowledge J\admi~tnatthey are 'true. You ,take apart its mechanism.,and myhope increases., At, 'the final stage you teacl), me thaI:this wondrous and multicolored universe cari.be re­duced to the ele'ctron. All this is 909d ana,}r waitfor you to continue. ,But; you tell me of an invisibleplanetary.system in which electrons gravitate arounda nucleus. . You explain this world to m~ wi th an Jinage.r realize then that you have been reduced to poetry:r shall never know. Have r the time to become indig­nant? You have, ~lready changed theories •• ;. n'

}

Albert CamusThe Myth of Sisyph~s

/

, \ii

DOCTOR OF PHILOSOPHY (1973)(Chemistry)

,co McMASTER UNIVERSITY

Hamilton, Ontario

",

f

TITLE: Model Potentials and, Molecular Rydberg Series.

AUTHOR: Fra~Richard Greening,'B.Sc. (King's College, London)"

SUPE;RVISOR:, Professor G.W. King \'

NUMBER OF PAGES:'ix, 162, \ ,

SCOPE AND CONTENTS:, ,

M0del gote~tial calculations have~been carried out on

the_ng~POUand np~~ Rydberg seri~s of CO2 ,CS 2 and

Cse 2 . T~e mOlec11ar PQ~ential was represented: by a super- ­

position <?f atomic model potentials 'which were calibrated to

atomic d,ata. The RydDez;g M.p. ";as expandedabo,ut the molecular

mid"':point in, ,

, many, membe rs'

a linear,-'combination ,of hydrogen. functions and\

of 'a Rydberg'series wer~ obtained in a single

'calculation.on,a com~uter" The results df the calculations

were used to check'pre~ipus aS5ig~ment~ofRydbergse~ies in, ,, '

The previously unreported vacuum u-v,spectrum ~f'C02 and CS 2 •.4" 0

"" CSe 2 was observed in tI:e'regiono ,

from 1200-2000 A"and analysed

using the,model calculations.

, ,

•

ii i

>,

,J

, '

-- ACKNOWLEDGEMENT.S

~.- .'

)

"L wish to thanlt Dr. G.W~ King for allowing me complete,,

fr~edom in my research while estill providing the adJice and

enc:ouragement that was often needed •. I wish also to thank •

Dr. D.P. Santry for considerabl~ assistance with the compu-

tational aspects' of ~y research~

:, I'wish to thank my'research colleagues, Dr. E.J. Finn

and Dr. P. Pichat, for·experimental assistance and helpful. .

discussions, and I ·thank all the other members of this research

group in the period 1968-73 for ~eir interest in my work:'

Dr. G. Kidd, Dr. P.R. MCLean~ "0. Gra!1g~, .R.C. Meathe,rall,,

E.R •. Farnworth, R. Lemanczyk,. A. Van Putten ~ M. Danyluk' and

oR, Judge.

I am very gratefut to Mrs. Jan Coleman fbr her

through the financial•J • ~

assistance of .the Department of .Chemistry of McMaster University.

excellent work in typing t,..his. thesis.

This research was made ~s~

& ' .Finally, I thank my wife,Suzanne, who showed me where

to look among the garbage and the flowers, and my daughter?

Hillarie who showed me sweet innocence., .. ,

"

,I

.'

..,

\

,

TABLE OF CONTENTS'

33

Page.

1

3 \

3Q

7'

12

'-=!

the CalculatiOn

Tobe Factorisation of Schrodinger'sEquation forExci~ed States'.~ .

T~e Atomic Effective Potential' "'.i.,

,Solutions of the ~ne-electron -\.Schrodinger Equation ~o~ a Coulomb.Field· 'lThe Variation'Principle and theDetermination of Atomic Model

•Potentials

,

n

1.1

- "

Atomic Model Potentials pndof Molecular Rydberg Series

1.2

1.3

­.'

CHAPTER 2

INTRODUCTION

/CHAPTER l "Model Potentials a~d the Calculation ofAtomic Rydberg Series

2.~, In~roduction 33

2.2 The United-Atom Approach' 34 v-

,

2.3

2.4

The Separated Atom Approach. . ~ ..Expansion of lhe Molecular RydbergWave function

45

51

,2 ;5 The Calculation of Rydberg Series inLinear Triatomic Molecules Using aSeparated Atom Model

o.' .•.

55

CHi\PTER 3

,2.6 'Two-cenl'le Expansions of CO2 Rydberg

wavefunctions.~,

'Rydberg Trans i tions 'in CO2

' . .--

3.l Coupling 'Schemes for the RydbergStates of CO2 @ ,,

66 '

79

79

~ ,;. ,3.2 Selection Rules for Rydberg Trans{tions 92

in CO2

3.3 Changes in Geometry in the Excited St-ate 96

v

Rydberg Series in'~02

.. -.

125

Page

103

lD3 • ..

104

124

124.

,.,14.1

141

142

the 145

Spectrum' 147

" 156

158

, ., . (

of the.

6.2 The Vacuum u-v Spectrum of CSe 2. .6~3Model Potential Calculations of

Rydberg series of. CSe 2

6.4 Analysis·of the cse 2 Vacuum~u-v

6.1 Previous Work"

4.1 Experimental Studies of the RydbergExcited States of CP2., ,

. "'.\0..

4.2 Theoretical Studies of the L~wer

EX~ited States. of 'C02. . .'Rydberg ~eries inCS 2 .

,'5.1 prev~ous Experimental and TheoreticalWork •. .

5.2 ~odel potentiai Calculations~ydbergSeri~ ofCS2

~Rydbe~~series in CSe 2CHAPTER 6

CHAPTER 5

-CHAPTER 4.

. CONCLt,lSIONS

BIBLIOGRAPHY

'.

.;

.)

•

; -

.•

• Vi..

J

.. \

\""

LIST OF FIGURES

Figure

1.I

Title

Curve of Best ',Fitting Pavameters forOxygen ns(3So) Rydberg Series

\Page,

25

,1. I!

1. III

2.I

2. II

..~.III

2.IV·

2.V

3.I

Hydrogen 3s Function R3 (I') Compared witha,Hydrogenic fupct{on' SRjs(r) Having theSame Energy as an Oxygen 3s Orbital

Comparison of Model Wavefunctions withOtherW~vefUnctions, .

Co-ordinates, for ,the United':'At()m, Calculationr

Co-ordinate Transforination for ,the: United­Atom Ca~culation

,,!odel Potenti'~l,s for, CO2 and CS2+ '

,H2 co-orflinates " , .

The' Eff~t~f I~cluding p Functions' inTwo-,c?ire o~n,s MO's . " .

Correlation of St-atesBetween Ideal (A,S)and Ideal'(OC.,w), Coupling' for a, ,(::g) 30uConfiguration" ..",

29'

37

40

50

51

78

86

91Correlatic:m of States Between Ideal. (A,S I and(nC,w) Coupling for a (::g> 3::u Configur.ation .,

The Approximate Form of the 3pdu and 3p,"u .... 108C02 Rydberg MO's

I. !

oSpectrum from 1000-1400 A Showing• n£X Assignmentu

122"-

138

144

154

,155 '

•

Showing

..A Showing

•

o

CSe2 Spectrum from 1700-2000 A" + nt, Assignments

g

csc 2 ,Spectrum from 1300-1600"9 - nnAss,i9n1r·cnts

C021::

g• 0

C~2 Spectrum from 1300-1900 ,A Showing"g • ntXAssignments

.- Effect of ~C£ and CH2CtZ Impurities onCSc 2 Spec~ru.'ll

J . •

•

-/•

,, \

LIS~ OF TABLES

Table,

l.I

, Title,

Observed and Calculated Term' Values of theOxygen n5(3so) RYdberg -Series'

23.'

Expansion Coefficients for Calculated Oxygenns Series

Atomic Model Potential Parameters used in theCalculation of CO2 , CS2 and Cse 2 Rydberg Terms'

~-purityof Hi+ Wave functions at R=4.4 a.U.z=1/2

I.II

2. I'

"2.II

2. III

Calculated apd Observedof H20

npbl Rydberg Ser.ies

24

.-

Ove+lap Integrals for Hydrogen Fuhctions at 70Ra 2.0 and R=4.0·a~u.

2.IV

2. V

~.VI

Calcul~ted Rydberg Terms for CO2

CalcUlated nso Rydberg Terms for @Cs

64

65

139

123

139\

0-

2.VII

2.VIIl

3.I

3. I II

4.1

4.11

5.1

5. II

5.III

Smallest Eigenvalues for g Hydrogen s-functions 70'at R=2.0 and -R=4.0 a.u.

.,'Calculated One and Two-Centre Rydberg Terms 74 0

'for CO2 ~

Selection Rules for Rydberg Tr~nsrtions in CO2 ~4

M.O, Cbre Precursors 'for the nsog, npou and 100np~u Rydberg Orbitals of C02

The Change in Orbital Energy, 6E, on ~nding 102CO 2 ' ... jCalcu lated RydbergTerrns for CO 2 (Averages) 110

A CO:':lparison of Some Calculated and Obse!;,ved'Rydberg Series in CO2

Calculated Terms for CS 2

Price ~ s Series I and II for CS 2 Compared toca1cula~ed-np'u Ser~es

Assiqnments of ~~e CS2 Spectr~~ in the Regio~140

1325':1825 J\

viJ..i .'

Additional Rydberg Series o~ CSe 2 in theRegion 1300-1900 A

•Table,

G.!

\

G.I.II"

j,

Title

Calculated Terms for cse2p~incipal Ry~erg seri~ of CSe 2Region 1300-1500 A

in the

Page•146

149.,

~

153

I ..

r·

•

, ..

tix

/

/

•

...,:;:...

,

INTRODUCTION

.Molecuiar ab~rption spectra in the far.ultraviolet

~ .

spectra should.exhibitseries of bands. with similar behaviour

to atomi~ line :s~~r-a,nd which fit a Rydberg fo~ula.4

,,) . +f. \!~ is the frequency. of an -absorption band in wave- ,

, cKurnber units, labelled by a running n1.1mber n, -and \J",' is the

"

•

atoms and ~ecules arise from ~h~

electron to an orbit located m~tly.. :.4_

Bohr p;red~cted that molecular absorption.

puts ide al'\ ..iOIHc core.

-'excitation of'n valence

. "> O. .J

'(. /Le: a:~ lwaVelengthS below 2000 A) were first obtained by

, .T: Lyman in the years between 1900 and 1914.

The 'first ttieol(etical unde~standing ofo thej'ie spe'ctra

was provided' at .ab~ the' same time~ by N. Bohr2 , 3, who' fiugges~ed~ .

. that optical' spect.ra in

,~ .

fr,equency of, the series ',limit,' then Rydberg's formula is:

\) III \) _

n '"R

2(n-lJ )

where R is called Rydberg's constant and lJ is approximately

\. constant within a gi"ven series. Because,the lines of ,the •

. h}'d'rogen spcctr\llll fit tne simpli fied. Rydbe~g formula,'"

•

II

Rvn

.'c \'~""_ ~ ,

, n'.where n is ,the principal quantum number,

quantum defect and rcprcscntsthc' cffect

p~rt of the ion~corc.

1

•

" .... .;

" '::

,.• , ' •

"Bohr's important'WO!K was followed by, the i~entificat~~n

numerous RYdberg~eri~s. in ~e absorption'spectra of

.~.2 (

ldiatomic molec,ules such as H2 , N2/CO and ,W.C. PrIce and

~o-worker~5 f",J.nd' ,serfes iI) the~ol~'!tOItlic molecules HiO', H;j!S,

CO2 , CSi" CH3I, C

6H

6etc. ~,'

--At the time of:wrLtirig, Rydberg series have beeh

. 'identified in over' 100 different mo~ec!lles '(see, Reference .5 ,. '-.

,analysi~ and classifica~ion of Rydberg series ii) polyatomic

"calculation of Ryciber'g terms in ,linear triatomic mo1.es;ules

~ ... . ",

,is applied to the analysis of Ry~e~g serie~ in CO2 ,

I

, mole,cules.

I'll thiftC-thesis a theoretical model is ;develo~ fdr the "

and

..:), .

, ..

,

:

--:

"

:--~

, ,

j"

, , '".'

,, '

•

•CHAP-TER 1

;' Model!

Potentialsr<

and the Calculation of Atomic Rydberg Serieso

.~.

• The FM:torisation of Schrodinger' s Equation for Excited' States'. ..,

mol~cular Schrodinger eq~tion.1 .

. . 6M. Born and R.J. Oppenheimer have shown that a good'

approxim~tion~hichg~eatlY~implifi~S the m~th~matical des-,~. . ..

. ', -, . '":"crip1:>ic;n OfI1l0.l:cul'a~ states is the ~eparation of eI~ctr6nio,

't . A~and.nuclear mO~ions in the full

~"'. c. I

---~"';.

This equa t:lon is:

( ,

where H is ,the Hami).toniarr'operator,· '.'• '. .6 ~ .'. • • .

E iil the" totalmol~curar energy, ,• I • 0" • •

, ,

~ . .-;

HfrNJR,e,"¢}'iMOl (rNiR,a,¢)I

(;,.~

= EMol 'fMol (~N'R,a,<:\l,,.1. 1

o

r~, ~epresen\~ t;he ~,elt;ctroni~" ,

an,d ·R,a,Q r,epres'ent the ,_r~~. , ., ,

co-ordi'nates of. internuclear

. separation Me-cr1:entation •

{1. 2."y 1" (rN',Rl

e ~c "

, ,.',

• 'i' .. 'l(rN,R,e,~)' •.~ ;"~o .. ....

. , ' ' ~

e,Born~ppc~eirner approximation leads 'to the factorisation, ,

~ ",... .....' • ','. .'Q • . • ~. . •

iFf 'i'MO.linto, the prQduct ~f ~ ~lectronicfunction"¢~~ec(rN,R~"a

\, :,' vib,r~tion~l fun~~iO~Xvib (Ft,) :. and' ,4 rotati~nal fuhctf6n Srot (~, {l ,. - . t(? . ,

i .<l; ':

:lchr04inger t s equa'tien 1,'1 '-can now be written in terms of ant ;'. " ~

. :'.~l~c~~onic,waVofunctibn dcsc~ibing ,the ~~ti6n of .th~ eldctrons \, . .. .',' ". . . . .: - . ~. "

" in' the !ield'~f fixdd nuciei:!'

"i.o

t·,,~.

i:

I • .,, ..

- . ~..,

,

3

, Q ; •

" ,

'.

4

- ",

,'.,

o

where E is the electronic energy."

The.electronic Schrodinger !!quation 1.3 is usuallesolved by' using the orbital, approximation, in which-the,.,N. ., ~~

.~ I ;:.\

electrons are. assigned ·.N one-electron functions WI' .w2

' '"

•called orbitals.

. .•• , ';'N (r" , R). ..

A.'mpHfi.d '0;"'" of <he o)b,~al .pprox'm.t'on " part'o''':::,approprLate for the description of molecular states in,which

~~

one electron is located ~ostly outside an ionic core •

.The Bo"rn-Oppenheimer' electronic 'Schrodinger equation for< ~

.~··molecule having'O nuclei with position co-ordinates R;(,a=1,2 ••• 0).," • I

and nuclear charges Za' Zb' ... may: be written,

'ZaZb"'"Ral)

N'O- l: l:

k. a

o+ l:, a<b

}~ a E "elec " elec

in the .... form,.-~.->~;.:...'" .

1.5

If one electron i~' positioned wellout~ide the remaining (core)

elect~ons then on~ of the r IS, say r l , is large compared to the

. I 3.'others~In this case equation 1.S may be wr tten

.fr.'......r· •

0 N 0 N N 0-,'~

. 1 2 ~ 1 1 2 Za~-1 " l: + 1: 1 l: ""It 1: 1: - ..

'1 rIa )(-2 r llt It-2 rita'!- a It-2 ' a ,

'..---N 0

,1 ZaZb

l: 1: •• E'¢elec+ + R;; . "elec - 1.6·iaz<lt rik. a<b

. ,"

)

'r5

,Since r lk is large compared to all other rik,s' r lk will be

approximately the same for all k\.and,

so that"

N 1r - =

k=2 r ik1.7

N-l·-+rl ,

1 N 2[-'" . r v..< k=2' k

Upon separating the 'variables in .equati0<!1. 1. 8. two 'equatiC?ns

are obtained, one.describing the motion of the ex~ited electron, ,

+Nr

i=z<k1-- -r ik

Qr

a<bE IJielec, L8

in the fiela of the molecular-ion represented by lJiand theex .

other defining. the state of the molecular ion lJiion ' i.e,

and

1.9

Q tl, 1 2 r ~+ N-l' ~ .;, . (r R) 1.l0a\-2 \\ - -} l;i(rl,R} .. cr . r l ex .e:l\ . ex 1·"a l,a

'r

. 1 N • 2 1'1 Q Za 1'1 1Q ZaZb1: "It 1: r r , ,. } .;.: (rN I,R)\-2 - + ~ . Rab1t",2 It .. 2 rita i .. 2<1t r ik a<b 10n -a

• 1.10b

The exactnes's. of the . product' resolution described by. /

/0/equatinns l.l0a and 1.10b d~nds on the validity of the

I

'\