Computational study of the anion photoelectron spectra of fexn (x = 0, s and n = 3, 4) clusters

Van Tan Tran

FACULTY OF SCIENCE
DEPARTMENT OF CHEMISTRY
DIVISION OF QUANTUM CHEMISTRY AND PHYSICAL CHEMISTRY
CELESTIJNENLAAN 200F BOX 2404
B-3001 HEVERLEE, BELGIUM

ARENBERG DOCTORAL SCHOOL
FACULTY OF SCIENCE

COMPUTATIONAL STUDY OF THE ANION PHOTOELECTRON SPECTRA OF FeXn (X = O, S AND n = 3, 4) CLUSTERS

Computational Study of the
Anion Photoelectron Spectra of FeXn
(X = O, S and n = 3, 4) Clusters

Van Tan Tran

December 2013

Promoter:
Prof. Dr. Marc Hendrickx

Dissertation presented in partial
fulfilment of the requirements for the
degree of Doctor in Chemistry
December 2013


Computational Study of the

Anion Photoelectron Spectra of FeXn
(X = O, S and n = 3, 4) Clusters
Van Tan Tran

Jury:

Dissertation presented in partial

Prof. Dr. Arnout Ceulemans, chair

fulfilment of the requirements for

Prof. Dr. Marc Hendrickx, promotor

the degree of Doctor in Chemistry

Prof. Dr. Luc Van Meervelt
Prof. Dr. Minh Tho Nguyen
Prof. Dr. Ewald Janssens
Prof. Dr. Paul Geerlings
(Vrije Universiteit Brussel)

December 2013


© Katholieke Universiteit Leuven – Faculty of Science
Celestijnenlaan 200F box 2404, B-3001 Heverlee (Belgium)

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the publisher.
D/2013/10.705/83
ISBN 978-90-8649-668-6


Preface

“The most beautiful experience we can have is the
mysterious. It is the fundamental emotion which stands
at the cradle of true art and true science.”
A LBERT E INSTEIN
This work would not have been possible without the help from many wonderful people
who gave their support in different ways. To them I would like to express my deepest
gratitude and sincere appreciation.
First and foremost, I would like to express my gratitude to my supervisor, Prof. Marc
Hendrickx, for his patient guidance and enthusiastic encouragement during my PhD.
I am especially grateful to Prof. Tran Thanh Hue at the Hanoi National University of
Education in Vietnam for introducing me to the world of computational chemistry
and for his advice and assistance in keeping my progress on schedule.
I would like to send special thanks to Prof. Minh Tho Nguyen and Prof. Thierry
Verbiest for the courses during my first year of my doctoral studies. I’m really benefited
from the material offered.
I would like to thank the jury members for taking time reading my thesis. Their
suggestions and corrections really improve my the thesis.
Many thanks I would like to send to my colleagues at Dong Thap University who
provided all the convenient conditions for me to study abroad. Especially, I would like

to thank Dr. Tran Quoc Tri who does a lot of teaching work when I disappear from the
university.
i


II

P REFACE

I am deeply grateful to financial supports from the 322 Scholarship Foundation of
Vietnamese Government and from the KU Leuven. Without these financial supports, I
could not have a chance to finish the thesis.
I would like to thanks Rita Jungbluth for all her kind help for the administration of
my study in Leuven. Also, I would wish to thank Hans Vansweevelt for his support
concerning all possible computing difficulties I encountered during my PhD.
I am indebted to all my friends here in Leuven, thank you for your understanding
and encouragement in my many moments of crisis. Your friendship makes my life a
wonderful experience.
Lastly, I dedicate this thesis to my parents, my wife and my son who supported and
encouraged me to keep going to finalize this thesis.
Van Tan Tran

Leuven, December 2013


Samenvatting

In dit proefschrift worden de structurele en elektronische eigenschappen van
FeXn −/0 ( X = O, S en n = 3–4 ) clusters onderzocht door gebruik te maken
van verschillende computationele kwantumchemische methoden. Deze clusters

zijn relevant om een beter begrip op te bouwen in allerlei sterk uiteenlopende
domeinen zoals onder andere diverse industriële katalytische processen en tal van
biochemische processen. Vanwege de ingewikkelde elektronische structuur van
dit soort van clusters, wat meestal het geval is voor transitiemetaalverbindingen, is
gebleken dat enkel een combinatie van verschillende elektroncorrelatiemethoden
zoals DFT, CASPT2 en RCCSD(T) toelaat een afdoende beschrijving te geven van de
bestudeerde verbindingen. Hierbij wordt elke computationele methode gebruikt
ter berekening van die eigenschappen waarvoor ze het best geschikt is. Op deze
wijze was het mogelijk om de onderliggende ionisaties zoals die optreden in de
beschikbare experimentele anionfoto-elektronspectra, te identificeren. Omgekeerd
was het eveneens mogelijk deze experimentele gegevens te gebruiken om de
kwaliteit van de rekenmethoden in te schatten.
Hoofdstuk 1 geeft een overzicht van de technieken die in de experimentele
studies aangewend worden. Hieruit blijkt dat anionfoto-elektronspectroscopie
onmiskenbaar één van de belangrijkste methoden is voor het bestuderen van de
structurele en elektronische eigenschappen van kleine clusters die een transitiemetaalcentrum bevatten. Inderdaad, in de literatuur kan een groot aantal spectra voor
deze soort clusters teruggevonden worden, die weliswaar waardevolle informatie
bevatten betreffende verschillende spectroscopische parameters maar niet steeds
een eenduidige conclusie toelaten aangaande de onderliggende geometrische en
elektronische structuur. Tot heden is de interpretatie van de foto-elektronspectra
vooral uitgevoerd op DFT-niveau, zodat heel wat vragen onbeantwoord bleven.
iii


S AMENVATTING

IV

Het volgende hoofdstuk beschrijft in detail de basisprincipes van foto-elektronspectroscopie en de aangewende kwantumchemische technieken. De elektronische
selectieregels die nodig zijn voor de interpretatie van de spectra worden in detail

afgeleid. Ook het Franck–Condonprincipe dat in dit werk wordt toegepast om de
waargenomen vibrationele progressies te simuleren, wordt eveneens geïntroduceerd.
Alle aangewende computationele kwantumchemische methoden, zoals DFT,
CASPT2, RASPT2 en RCCSD(T) worden op een kwalitatieve wijze omschreven.
Het derde hoofdstuk toont aan hoe deze computationele technieken worden
aangewend om de elektronische structuur van de FeO3 en FeO3 − clusters te
onderzoeken. Meer specifiek, geometrieën van alle relevante spinmultipliciteiten
werden zonder enige symmetriebeperkingen geoptimaliseerd op het BP86/QZVPniveau en verder verfijnd met de CASPT2- en RCCSD(T)-methoden.

Beide

bevestigen dat alle laaggelegen elektronische toestanden die relevant zijn voor de
beschrijving van het foto-elektronspectrum overeenkomen met of sterk gelijken
op een vlakke D 3h -structuur zonder bindingen tussen de drie zuurstofionen.
Afhankelijk van de gebruikte rekenmethode, kan de grondtoestand van het
FeO3 − -anion ofwel 2 E of 4 A2 zijn.

CASPT2 berekent het 4 A2 als de laagste

energietoestand, terwijl RCCSD(T) het 2 E als grondtoestand voorspelt.

De

twee laagste bindingsenergiebanden van de foto-elektronspectrum van FeO3 −
kunnen zonder twijfel alleen worden toegeschreven aan één-elektron ionisaties
vanuit de 2 E -toestand. De eerste band is het resultaat van een overgang naar
de 1 A1 -grondtoestand van FeO3 , terwijl de tweede band afkomstig is van de
eerste aangeslagen 3 E -toestand.

Uit een harmonische vibrationele analyse

van de symmetrische stretching mode bleek dat de waargenomen vibrationele
progressies van deze twee banden in het experimentele foto-elektronspectrum
ook in overeenstemming zijn met de RCCSD(T)-assignatie. Een moleculaire
orbitaalanalyse leidde overduidelijk tot de conclusie dat de elektronische structuur
van de grondtoestanden van de anionische en neutrale clusters respectievelijk
overeenkomen met een oxidatietoestand +5 en +6 voor ijzer.
De relatieve stabiliteit van alle laaggelegen isomeren van de FeO4 −/0 -clusters werden
bestudeerd in hoofdstuk 4. Voor zowel de anionische en neutrale clusters, bleek
het bepalen van de meest stabiele structuur een veeleisende taak. Zowel DFT
als CASPT2 plaatsen de doublettoestand van het tetraëdrische O4 Fe-isomeer dat
opgebouwd is uit vier onafhankelijke O2− atomaire liganden, aanzienlijk lager, tot


S AMENVATTING

V

0,81 eV, dan de doublettoestand van het η2 -(O2 )FeO2 − . Dit laatste isomeer bezit
slechts twee atomaire O2− -liganden en één moleculair O2 2− -ligand dat zijdelings aan
het ijzerkation is gebonden. De RCCSD(T)-methode reduceert dit energieverschil
tot minder dan 0,01 eV. Enkel deze gelijke stabiliteit van de grondtoestanden van
O4 Fe− en η2 -(O2 )FeO2 − leidt tot een volledige assignatie van de experimentele
foto-elektronspectra van FeO4 − . De laagste bindingsenergieband (X-band) wordt
toegeschreven aan de ionisatie 2 A1 naar 1 A1 van het η2 -(O2 )FeO2 − , terwijl de
eerstvolgende hogere energieband (A-band) het gevolg is van de overgang van 2 E
naar 1 A1 tussen de O4 Fe−/0 -conformaties. Voor een specifiek isomeer, berekent
CASPT2 de beste ionisatie-energieën. De hoogste piek in de A-band met de zwakste
intensiteit, kan eventueel worden toegeschreven aan de overgang van 2 A2 naar 3 A2
van η2 -(O2 )FeO2 . Beide progressies in het experimentele spectrum zijn het resultaat

van ionisaties vanuit de antibindende orbitalen met overheersend ijzer-3d-karakter.
Een Franck–Condonsimulatie van de waargenomen vibrationele progressies zoals
deze werd uitgevoerd met BPW91, bevestigde de voorgestelde assignaties.
Geometrische structuren van FeS3 en FeS3 − met spinmultipliciteiten variërend
van singlet tot octet werden in hoofdstuk 5 geoptimaliseerd op het B3LYPniveau, waardoor twee laaggelegen isomeren voor deze clusters konden worden
geïdentificeerd. Het planaire isomeer bezit een D 3h -symmetrie en bevat drie
S2− -atomaire liganden (S3 Fe−/0 ), terwijl de C 2v structuur, naast een atomair
S2− -ligand een S2 2− -ligand bevat dat zijdelings gebonden is aan het ijzerkation:
een η2 -(S2 )FeS isomeer.

Vervolgens werden de energieverschillen tussen de

verschillende toestanden van deze twee isomeren geschat door het uitvoeren van
geometrie-optimalisaties met de multireferentie CASPT2-methode. Verschillende
concurrerende structuren voor de grondtoestand van de anionische cluster werden
herkend op dit niveau. De relatieve stabiliteiten werden ook geschat door singlepoint RCSSD(T)-berekeningen uitvoeren op de B3LYP-geometrieën. Het 5 B2 werd
ondubbelzinnig aangeduid als de grondtoestand van het neutrale complex. De aard
van de grondtoestand van het anion daarentegen is aanzienlijk minder zeker. De
14 B2 -, 24 B2 -, 4 B1 - en 6 A1 -toestanden werden allemaal gevonden als laaggelegen
η2 -(S2 )FeS− -toestanden. Ook het 4 B2 van S3 Fe− heeft een vergelijkbare CASPT2energie. Hiermee in tegenstelling, plaatsen B3LYP en RCCSD(T) gezamenlijk deze
S3 Fe− toestand op een veel hogere energie. Energetisch, kunnen op het CASPT2niveau alle banden van de foto-elektronspectra van FeS3 − gereproduceerd worden
als ionisaties vanuit ofwel de 4 B2 - of de 6 A1 -toestand van het η2 -(S2 )FeS− . Echter,


S AMENVATTING

VI

uit de Franck–Condonsimulaties, die verkregen werden door een harmonische
vibrationele analyse uit te voeren op het B3LYP niveau, blijkt dat alleen de

ionisatie van 4 B2 naar 5 B2 , waarbij de structuur η2 -(S2 )FeS behouden blijft, de
beste overeenkomst-qua vibrationele progressie bezit met de X-band van het
experimentele foto-elektronspectrum.
De B3LYP, CASPT2 en RCCSD(T) computationele methoden werden eveneens in
hoofdstuk 6 succesvol aangewend voor de interpretatie van de foto-elektronspectra
van de FeS4 − -stoichiometrie door het berekenen van de geometrische structuren
van alle mogelijke laaggelegen FeS4 −/0 -isomeren.
2

−

(η -(S2 ))2 Fe -isomeer met twee S2

2−

De 4 B1g -toestand van het

-moleculaire liganden zijdelings gebonden op

een D 2h -wijze aan het centrale ijzer(III)kation, wordt eenduidig als grondtoestand
van de anionische cluster voorspeld en de experimentele foto-elektronspectra
werden met CASPT2 toegewezen als afkomstig van dit isomeer. De complexe
vibrationele structuur van de laagste energie X-band is het resultaat van ionisatietransities naar de 3 B3g -, 5 B1u - en 5 B1g -toestanden van de neutrale cluster, die
energetisch erg dicht bij elkaar gelegen zijn. Een analyse van de CASSCF-orbitalen
geeft een quasi ontaarding aan van de niet-bindende 3d-orbitalen van het ijzerkation
en de π∗ -valentie-orbitalen van het moleculaire S2 2− -ligand. Alle experimenteel
waargenomen hogere ionisatie-energiebanden kunnen theoretisch toegekend
worden als zijnde afkomstig van de voorgestelde anionische grondtoestand door
onthechting van een elektron uit één van deze ijzer(III)- of ligandorbitalen.

Abbreviations

ADE

adiabatic detachment energy

ANO

atomic natural orbital

CASPT2

complete active space second order perturbation theory

CASSCF

complete active space self-consistent field

CC

coupled-cluster

CCSD

coupled-cluster with single and double excitations

CCSD(T)

coupled-cluster with single and double and

perturbative triple excitations

CCSDT

coupled-cluster with single, double, and triple excitations

CISD

configuration interaction including single and double excitations

CISDT

configuration interaction including single, double,
and triple excitations

CSF

configuration state function

DFT

density functional theory

FCF

Franck–Condon factor

GGA

generalized gradient approximation

GTF

Gaussian type function

GTO

Gaussian type orbital

HF

Hartree–Fock

LDA

local-density approximation

LSDA

local-spin density approximation

MCSCF

multi-configuration self-consistent field

meta-GGA

meta-generalized gradient approximation

MPn

Møller–Plesset perturbation theory of order n

MRCI

multi-reference configuration interaction

vii


S AMENVATTING

VIII

PES

photoelectron spectroscopy

RASPT2

restricted active space second order perturbation theory

RASSCF

restricted active space self-consistent field

RE

relative energy

RHF

restricted Hartree–Fock

ROHF

restricted open-shell Hartree–Fock

RCCSD(T)

restricted coupled-cluster with single and double and
perturbative triple excitations

UHF

unrestricted Hartree–Fock

VDE

vertical detachment energy

ZPE

zero-point energy


Contents

Contents

ix

List of Figures

xi

List of Tables

xv

1

Introduction

1

2

Photoelectron spectroscopy and computational approach

15

2.1

Basic principles of photoelectron spectroscopy

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

15

2.2

The photoelectron spectrometer . . . . . . . . . . . . . . . . . . . . .

16

2.3

Interpretation of photoelectron spectra . . . . . . . . . . . . . . . . .

17

2.4

Selection rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

2.5

Electronic selection rules . . . . . . . . . . . . . . . . . . . . . . . . . .

20

2.6

Vibrational selection rules . . . . . . . . . . . . . . . . . . . . . . . . .

22

2.7

Interpretation of the spectra: electron binding energy . . . . . . . . .

25

2.8

Interpretation of the spectra: Franck–Condon simulations . . . . . .

26

2.9

The Schrödinger equation . . . . . . . . . . . . . . . . . . . . . . . . .

27

2.10 Wave functions for many-electron systems . . . . . . . . . . . . . . .

28

2.11 The Hartree–Fock method . . . . . . . . . . . . . . . . . . . . . . . . .

30

2.12 Electron correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

2.13 Coupled-cluster theory . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

2.14 CASSCF, RASCSF, CASPT2, and RASPT2 methods . . . . . . . . . . . .

33

2.15 Density functional theory . . . . . . . . . . . . . . . . . . . . . . . . .

36

ix


CONTENTS

X

3

4

5

6

7

2.16 Basis sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

In search for the ground state of FeO3 −

43

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

3.2

Computational methods . . . . . . . . . . . . . . . . . . . . . . . . . .

45

3.3

Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

3.4

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

Which is the most stable isomer of FeO4 − ?

67

4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

4.2

Computational methods . . . . . . . . . . . . . . . . . . . . . . . . . .

70

4.3

Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

4.4

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87

Nearly degenerate low-lying electronic states of FeS3 −

91

5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

5.2

Computational methods . . . . . . . . . . . . . . . . . . . . . . . . . .

93

5.3

Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

5.4

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

Unraveling the complex X band of the photoelectron spectra of FeS4 −

115

6.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.2

Computational methods . . . . . . . . . . . . . . . . . . . . . . . . . . 116

6.3

Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

6.4

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

General conclusions and perspectives

137

A Supplements to chapter 5

149

B Supplements to chapter 6

151

List of publications

153

List of Figures

2.1

The principal components of a photoelectron spectrometer. . . . . .

2.2

Photoelectron spectrum of FeO3 − as recorded with 4.66 eV photon
energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3

2.4
3.1

18

Potential energy curves of diatomic molecules illustrating the vibrational structure of photoelectron spectra. . . . . . . . . . . . . . . . .

23

Illustration of the CASSCF and RASSCF wave function. . . . . . . . .

35

(a) Choice of the coordination system for the CASPT2 calculations.
(b) Qualitative orbital energy scheme for the valence d orbitals. . . .

3.2

17

45

Structures (bond distances in Ångstroms and bond angles in degrees)
and relative energies (eV) of FeO3 and FeO3 − as obtained by
BP86/QZVP calculations. M is the spin multiplicity with even values
for the anions and odd values for neutral clusters. . . . . . . . . . . .

3.3

47

Structures (bond distances in Ångstroms and bond angles in degrees)
and relative energies (eV) for the low lying states of FeO3 and FeO3 −
as obtained by CASPT2 geometry optimizations. . . . . . . . . . . . .

3.4

CASPT2 (using small ANO-RCC basis sets) potential energy curves of
the symmetric Fe–O bond stretch (D 3h symmetry). . . . . . . . . . .

3.5

49

50

Pseudonatural molecular orbital plots and their occupation numbers
for the 4 B2 state (FeO3 − ) as calculated by CASSCF (small ANO-RCC
basis sets). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

53


LIST OF FIGURES

XII

3.6

Photoelectron spectrum of FeO3 − taken from ref 10.

Abscissa:

binding energies in electronvolts. Ordinate: relative electron intensities. 57
3.7

Simulated vibrational progression on the basis of harmonic Franck–
Condon factors for the symmetric stretch of the three lowest energy
electron detachment processes. Peak positions derived from CASPT2
energies, BP86/QZVP zero-point energies, and CASPT2 symmetric
stretch frequencies. Abscissa: binding energy in eV. Ordinate: relative
transition probabilities. . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.1

Coordinate systems for FeO4 −/0 as employed during the CASPT2 and
RCCSD(T) calculations. . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2

61

71

Structures, symmetries, and relative energies of FeO4 −/0 as calculated
at the BPW91/aug-cc-pVTZ level. M is the spin multiplicity with even
values for the anions and odd values for neutral clusters. . . . . . . .

4.3

74

Potential energy curves for the ground states of O4 Fe−/0 calculated
at the CASPT2 level for the symmetric stretching mode of the Fe–O
bond in Td symmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4

CASSCF pseudo-natural orbitals of the active space for the 2 A1 state
of O4 Fe− and their occupation numbers in parentheses. . . . . . . .

4.5

79

CASSCF pseudo-natural orbitals of the active space for the 2 A2 state
of η2 -(O2 )FeO2 − and their occupation numbers in parentheses. . . .

4.6

76

80

Franck–Condon simulations by using the harmonic vibrational
frequency analyses at the BPW91 level. (a) Simulation of the X band
starting at 3.30 eV, (b) simulation of the intense peaks of the A band
starting at 3.84 eV, and (c) single peak, that is, tentative assigned as the
highest energy peak of the A band. Abscissa: vibrational frequency in
cm−1 . Ordinate: relative transition probabilities. . . . . . . . . . . . .

4.7

83

Photoelectron spectrum of FeO4 − as recorded with 4.66 eV photon
energy (taken from Ref. 7). On the abscissa, the binding energies are
given in eV and arbitrary units for the intensity in the ordinate. . . .

84


LIST OF FIGURES

5.1

XIII

Choice of coordinate systems for the CASPT2 and RCCSD(T) calculations and qualitative orbital energy schemes for the valence 3d
orbitals as derived from the CASPT2 results for (a) S3 Fe−/0 and (b)
η2 -(S2 )FeS−/0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.2

94

Calculated structures for the relevant low-energy spin multiplicities
of FeS3 and FeS3 − at the B3LYP/QZVP level and their relative energies
in electronvolts. M is the spin multiplicity with even values for the
anions and odd values for neutral clusters. . . . . . . . . . . . . . . .

5.3

96

Photoelectron spectrum of FeS3 − taken from ref 11 as recorded using
laser detachment photons of 193 nm. Arrows indicate the calculated
CASPT2 vertical detachment energies expressed in electronvolts. . . 102

5.4

CASSCF natural orbitals for the 4 B2 state of the S3 Fe− isomer classified
according to their symmetry and natural occupation numbers in

parentheses.
indicated.

5.5

The type of predominant iron 3d orbitals is also

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

CASSCF natural orbitals for the 4 B2 initial state of the η2 -(S2 )FeS−
isomer classified according to their symmetry and natural occupation
numbers in parentheses. The type of predominant iron 3d orbitals is
also indicated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.6

Calculated Franck–Condon factors at the B3LYP level for the η2 -(S2 )FeS
isomer. Relative transition probabilities (in arbitrary units) for the (a)
4

B2 A 5 B2 and (b) 6 A1 A 5 B2 ionizations. The inset in panel a depicts

the experimental vibrational progression of the X band as obtained
with 355-nm detachment photons in ref 11.
6.1

. . . . . . . . . . . . . . 109

Coordinate systems of D 2h (a) and D 2d (b) structures of (η2 -(S2 ))2 Fe−/0 ,
and C 2v structure of η2 -(S3 )FeS−/0 (c) as used in the DFT, CASPT2 and

RCCSD(T) calculations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6.2

Qualitative orbital diagram showing the state average CASSCF
pseudo-natural orbitals of the 4 B1g ground state of the (η2 -(S2 ))2 Fe−
cluster. The occupation of the orbitals refers to the leading configuration of this state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123


LIST OF FIGURES

XIV

6.3

Photoelectron spectra of FeS4 − as recorded at 193 nm (a) and 355 nm
(b) photon energies.[14] Abscissa: binding energies in electronvolts.
Ordinate: relative electron intensities. Arrows represent the vertical
detachment energies (VDE) calculated at the CASPT2 level. . . . . . 126

6.4

Qualitative state diagram explaining the origin of the X, A and F bands.
Because of the small exchange energy between the metal 3d orbitals
and the π∗ orbitals, there is a small splitting between the 3 B1u and
5

B1u states. The original larger splitting between the 5 B1g on the

one hand and the 3 B1g and 3 B3g (3d exchange energy) is reduced by

configuration interaction. . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.5

B3LYP Franck–Condon factor simulations for the 4 B3g A 3 B3g (a),
4

B3g A 5 B1u (b), 4 B3g A 5 B1g (c) ionizations starting at 3.20, 2.99,

and 3.19 eV, respectively. Abscissa: vibrational frequency in cm−1 .
Ordinate: relative transition probabilities. Displacement vectors of
the vibrational modes responsible for the corresponding calculated
vibrational progressions are depicted as insets. . . . . . . . . . . . . . 131
A.1

CASPT2 optimized structures and relative energies in eV of the lowlying states of FeS3 −/0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

B.1

State-average CASSCF pseudo-natural orbitals and the electron
occupation numbers in the active space of the 6 A1 state of D 2d
(η2 -(S2 ))2 Fe− cluster with two S2 ligands perpendicular together. . . 151

B.2

State-average CASSCF pseudo-natural orbitals and the electron
occupation numbers in the active space of the 6 A1 state of η2 -(S3 )FeS−
cluster. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152


List of Tables

3.1

Relative energies (RE), harmonic vibrational frequencies (cm−1 ), and
intensities (km/mol) for the two studied isomers of FeO3 and FeO3 −
at the BP86/QZVP level. . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2

Vertical detachment energies (VDE) from the 4 B2 state as calculated
by CASPT2 with small ANO-RCC basis sets. . . . . . . . . . . . . . . .

3.3

48

50

Vertical detachment energies (VDE) from the 2 A2 state as calculated
by (a) CASPT2 with small ANO-RCC basis sets and (b) CASPT2 with
large ANO-RCC basis sets. . . . . . . . . . . . . . . . . . . . . . . . . .

3.4

51

Relative CASPT2 energies for the iron trioxides FeO3 and FeO3 − . (a)
CASPT2 single point calculations with small ANO-RCC basis sets at
BP86/QZVP geometries. (b) CASPT2 geometry optimizations with
small ANO-RCC basis sets. (c) CASPT2 single point calculation with

large ANO-RCC basis sets at geometries (b). . . . . . . . . . . . . . . .

3.5

Mulliken population analysis charges for low-lying states of the FeO3
and FeO3 − clusters as obtained from the CASPT2 wave functions. . .

3.6

55

Relative energies (eV) of the low-lying states of O3 Fe−/0 as calculated
with RCCSD(T). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.1

52

59

Relative energies (RE) and frequencies of O4 Fe−/0 calculated at
BPW91/aug-cc-pVTZ level. (a) Calculated from the doublet of O4 Fe− ,
(b) calculated from the doublet of η2 -(O2 )FeO2 − , and (c) calculated
from the doublet of η1 -(O2 )FeO2 − . . . . . . . . . . . . . . . . . . . . .
xv

73


LIST OF TABLES

XVI

4.2

Estimated CASPT2 adiabatic and vertical detachment energies. (a)
Relative energies with respect to the 2 E (2 A1 , 2 A2 ) state of O4 Fe− as
obtained by using the BPW91 optimized structures. For a specific
spin multiplicity of a particular conformation the lowest energy
difference represents an estimate for the adiabatic detachment energy.
(b) Ditto (a) but calculated from the 2 A2 state of η2 -(O2 )FeO2 − . (c)
Vertical detachment energies from the 2 E (2 A1 , 2 A2 ) ground state of
O4 Fe− . (d) Vertical detachment energies from the 2 A2 ground state of
η2 -(O2 )FeO2 − . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.3

78

Basis set influence on the calculated relative energies (eV) for various
states. RCCSD(T) results obtained for the triple (3-ζ), quadruple (4-ζ),
and quintuple (5-ζ) zeta basis sets by using the corresponding BPW91
equilibrium structures. . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.1

85

Relative energies (REs) and harmonic vibrational frequencies for the
considered spin multiplicities of FeS3 and FeS3 − as calculated at the

B3LYP/QZVP level. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.2

97

Relative energies (REs) and harmonic frequencies for the totally
symmetric vibrational modes of the low-lying states of FeS3 −/0 as
obtained by CASPT2 geometry optimizations. . . . . . . . . . . . . .

5.3

99

RCCSD(T) relative energies (eV) with respect to the 4 B2 state of
η2 -(S2 )FeS− for relevant low-lying states of FeS3 −/0 . (a) For the
smallest basis set (aug-cc-pVTZ) only the valence electrons of iron
(3d and 4s) and sulfur (3s and 3p) are correlated, whereas for the two
larger basis sets, aug-cc-pwCVTZ and aug-cc-pwCVQZ, the outer core
of iron (3s and 3p) is also correlated. (b) Calculation did not converge
because of persistent oscillations.

5.4

. . . . . . . . . . . . . . . . . . . . 100

Vertical relative energies (VREs) for the lowest one-electron ionizations as calculated for the 4 B2 state (upper part) and 2 B1 state
(lower part) of the S3 Fe− isomer by multistate (two-root) CASPT2. (a)
Denotes the molecular orbital that is ionized for the specific electrondetachment process. (b) Relative energy with respect to 2 A2 at the
geometry of 2 B1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

LIST OF TABLES

5.5

XVII

Vertical detachment energies (VDEs) for the lowest one-electron
ionizations as calculated for the 6 A1 state (upper part) and the 4 B2
state (lower part) of the η2 -(S2 )FeS− isomer by multistate (four-state)
CASPT2. (a) Denotes the molecular orbital that is ionized for the
specific electron-detachment process with its predominant character
in parentheses. (b) Calculated as an average of two states. (c)
Calculated as an average of six states. (d) Calculated as an average of
eight states. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6.1

B3LYP relative energies (RE) and structural parameters of low-lying
states for different isomers of the FeS4 −/0 stoichiometries. . . . . . . 120

6.2

Relative energies (RE) and structural parameters of low-lying states of
(η2 -(S2 ))2 Fe−/0 and η2 -(S3 )FeS−/0 as calculated by CASPT2 geometry
optimizations. (a) single-point calculations employing the B3LYP
geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

6.3

Relative energies (RE) and structural parameters for low-lying states
of (η2 -(S2 ))2 Fe−/0 and η2 -(S3 )FeS−/0 as calculated with RCCSD(T). (a)
Geometry optimization with the triple-ζ basis sets without iron outercore correlation, (b) single-point calculations with the quadruple-ζ
basis sets including iron outer-core correlation. . . . . . . . . . . . . 124

6.4

Vertical detachment energies (VDEs) of (η2 -(S2 ))2 Fe− as calculated
with CASPT2. (a) Orbitals from which an electron is removed during
the ionization process from the 4 B1g ground state. . . . . . . . . . . . 128



Chapter 1

Introduction

Transition metal-containing compounds are very important for our modern society.
The presence of these compounds is found in industrial catalysis, medicine, and
biological storage and transport.[1–3] According to the periodic table, transition
metals are all the elements of the d-block, including groups 3 to 12. In compounds,
these metals have an ability to lose multiple electrons from their valence nd
and (n + 1)s orbitals to create cations which usually have open-shell electronic
configurations. Because of their rather exceptional electronic structures, including
electronic near-degeneracy effects, transition metal compounds have a wide range
of physical and chemical applications. However, a detailed investigation of the
electronic structures of these rather large molecular systems represents an extremely
important challenge for understanding their properties. On the other hand, the
smaller sized transition metal-containing compounds in the gas phase such as

MCn −/0 , MOn −/0 and MSn −/0 , which contain one transition metal center, have been
extensively investigated by theoretical and experimental techniques.[4–18] Since
these compounds are synthesized by clusterification processes, they are frequently
denoted in the literature as clusters. Further, because they are stable chemical
identities in the gas phase, it is also acceptable to describe them as molecules or
more specifically as transition metal complexes. In spite of their small sizes, these
clusters are of importance because they represent fundamental building blocks for
larger clusters. Therefore, getting an insight into the electronic structures of small

1


2

I NTRODUCTION

clusters could be beneficial for understanding the properties of larger transition
metal compounds.
Experimentally, these small clusters are generated in plasma reactions of laser
ablated pure transition metals that are mixed with a helium carrier gas containing
a suitable gas, or are produced by laser vaporization of solid targets of various
composition.[5, 6, 9, 12–18] In particular, in order to create iron oxide clusters [6, 13,
14], an intense pulsed laser beam is concentrated onto a pure iron target to produce
a plasma containing iron atoms and ions. A helium carrier gas with 0.5% O2 is mixed
with the plasma to produce Fex O y −/0 clusters. Otherwise, iron sulfide clusters[16]
are produced by laser vaporization of a mixed Fe/S target (10/1 ratio) in the presence
of a helium carrier gas. Transition metal carbides Fex C y −/0 clusters[9, 17, 18] are
formed either by plasma reactions between laser-ablated metal atoms or ions and a
helium carrier gas containing 5% CH4 or by laser vaporization of a transition metal
carbide target. Generally, as a result of these uncontrollable clusterification reactions,

different uncharged and charged clusters containing a different number of atoms are
observed. Additionally, each specific stoichiometric cluster is able to have several
isomers of almost the same stability. Therefore, in subsequent steps a particular
stoichiometric cluster is selected and its structural and electronic properties can be
probed by an appropriate experimental technique.
In the experiments that we will consider in this work, only singly charged anionic
clusters containing a single transition metal center are selected, which can be
achieved by using a time-of-flight mass spectrometer.[15, 19–21] During this mass
selection, all equally charged anionic clusters are accelerated by an electrostatic field
to have equally kinetic energies. These clusters are then allowed to move through the
free-flight drift tube towards the detector. Depending on the mass-to-charge (m/z)
ratio, different clusters strike on the detector at different times. Clusters with smaller
mass-to-charge ratio will fly faster and reach the detector first, while clusters with
larger mass-to-charge ratio will arrive later. Measuring the time needed for a cluster
to fly to the detector in a very accurate way, usually in the rage of nanoseconds (ns),
allows to separate clusters with different mass-to-charge ratios.
Alternatively, following the mass selection, the electronic structures of the selected
anionic clusters and of the corresponding neutral clusters can be investigated
by using photoelectron spectroscopy.[15, 21–23] In this step, a specific anionic
cluster is irradiated by a laser beam which induces ionization. By collecting the


I NTRODUCTION

3

detached electrons and measuring their kinetic energies, a photoelectron spectrum
is recorded which typically contains a series of peaks and bands. In the most
simple interpretation, each band in a photoelectron spectrum corresponds to a
removal of one electron from a particular orbital. The starting point position of

each band is usually interpreted as the adiabatic electron detachment energy of
the corresponding band. High resolution photoelectron spectra are observed by
lowering the energy of the photons of the detachment laser beam, while higher
ionization energies are accessible by increasing the photon energy.
More specifically, anion photoelectron spectroscopy is widely used to investigate the
electronic structure of neutral transition metal compounds. Indeed, an electron can
be detached from any molecular orbital of the anionic ground state, provided that
the photon energy of the beam is high enough. Therefore, some low-lying electronic
states of the neutral cluster are seen in the photoelectron spectrum. Moreover,
when the structural difference between the initial anionic and final electronic
state is large enough, vibrational progressions are additionally observed, which
are usually the result of transitions between the vibrational ground state of the
anionic electronic ground state and several vibrational levels of the neutral final
state. In these circumstances the vibrational frequencies of the final neutral state
are measurable. In some spectra hot bands at the low energy side of a band are seen.
Since these arise from a detachment out of an excited vibrational level, they afford
vibrational frequencies for the initial state of the anionic cluster.
As a conclusion it is possible to state that anion photoelectron spectra provide
information about the electron affinity, the relative energies of low-lying states
and vibrational frequencies of the anionic and neutral cluster. However, such
information is not sufficient for a clear understanding of the geometric and
electronic structures of the compounds studied. For instance, it is not possible
to determine with absolute certainty which isomers of the cluster contribute to
the spectra. Therefore, quantum chemical calculations are needed to obtain the
structural, electronic, and vibrational properties of the considered clusters.
In addition to photoelectron spectroscopy, the vibrational properties of small
clusters have also been investigated by infrared matrix isolation spectroscopy.[4, 5,
8, 11] All products formed in the reactions between laser-ablated metal atoms and a
noble gas carrier containing a second suitable gas, are trapped within a solid matrix
of the noble gas at very low temperatures. In this way vibrational absorption bands

4

I NTRODUCTION

of different clusters have been recorded by infrared spectroscopy. This technique is
very useful for studying clusters because it isolates different isomers into the frozen
matrix, prevents them from further chemical interaction, and therefore renders
evidence for the existence of different isomers as they are produced during the
formation process. Also, frequencies of asymmetric vibrational modes, which cannot
be recorded by photoelectron spectroscopy, can be observed. By employing this
infrared matrix isolation spectroscopy to investigate the reaction between iron
atoms and oxygen molecules, the existence of the side-on and the end-on bound
dioxygen-iron dioxide complexes, which are respectively denoted as η1 -(O2 )FeO2
and η2 -(O2 )FeO2 , was proposed.[5, 7, 8] Otherwise, the reactions between small
sized sulfur molecules with laser-ablated iron atoms were also studied with the
same technique, and allowed to identify the η2 -(S2 )2 Fe cluster, which contains two
S2 ligands side-on bound to iron.[11] For this identification the comparison of the
observed vibrational frequencies with density functional theory results proved to be
very helpful.[7, 8, 11]
As already mentioned, computational chemistry, which uses quantum mechanics to
directly calculate many different molecular properties, represents a very valuable
tool to investigate the electronic structures of small transition metal-containing
clusters.[7, 11, 18, 24–27] Geometric and electronic structures, potential energy
surfaces, harmonic vibrational frequencies, and ionization energies are a few
examples of what computational chemistry can predict. The calculated results
have the potential to assign the recorded photoelectron spectra. It is very wellknown that clusters containing transition metals still remain as a difficult challenge
for quantum chemists, because of the complex electron correlation effects in this
kind of systems. Qualitatively, the correlation energy is divided into the dynamical

correlation which arises from the correlated motion of each electron with every
other electron, and a non-dynamical (static) part which originates from a certain
number of frontier molecular orbitals that are quasi-degenerate. The latter effect
occurs frequently in transition metal complexes.
In order to recover a sufficient amount of dynamical correlation energy, electron
correlation methods such as perturbation theory (MP2, MP3,. . . ), configuration interaction (CISD, CISDT,. . . ), or coupled-cluster theory (CCSD, CCSD(T), CCSDT,. . . )
must be used. These methods are usually called post-Hartree–Fock methods because
they use a single reference Hartree–Fock wave function as a starting point. In spite