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Modeling electrostatic interactions in complex systems

Anzola, Mattia · 2020

Abstract

The study of novel molecular materials provides great opportunities to solve a large amounts of needs. A careful arrangement of molecules at the nanoscale can lead to the formation of systems with unique optical, electrical and magnetic properties. Purpose of this thesis is the study and integration of novel methodologies to get a deep understanding of intermolecular electrostatic interactions in complex nanosized systems. In Chapter 1 we investigate resonant energy transfer for a pair of dyes linked to a calixarene scaffold. We make extensive use of MD simulations in two different solvents to describe the effect of solvation and conformational motions on the rate of energy transfer. Moreover, we develop a fully dynamical model, based on Monte Carlo method, to analyze the characteristic timescales of such processes and compare them with the experimental picture. In Chapter 2 we examine spectroscopic properties of molecular aggregates, testing new approaches and approximation schemes for polar and non-polar supramolecular assemblies. The first part is focused on the discussion of aggregates of polar and polarizable dyes, improving already existent models to account for vibrational coupling and hence for spectral band-shapes. We then turn attention to aggregates of non-polar chromophores, addressing the reliability of the Heitler-London approximation and presenting a model for two dimensional aggregates. Chapters 3 and 4 are focused on chiral aggregates. In Chapter 3 we investigate aggregates formed by dicyanostilbenes decorated with chiral pendants. Through the use of an hybrid approach, involving MD simulations and exciton modeling, we are able to get a deep understanding on both aggregation and spectroscopic features of these system, questioning the effectiveness of widely adopted rules to assess the system chirality from Circular Dichroism (CD) spectra. In Chapter 4 we focus attention on non-symmetric squaraine aggregates. An extensive theoretical work is discussed, devoted to the study of spectroscopic features of squaraine assemblies in solution. We present a new model for the calculation of absorption and CD spectra of squaraine complexes using a delocalized electrons approach, taking into account for both intra- and intermolecular charge transfer mechanisms.

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Modeling electrostatic interactions in complex systems

Author: Anzola, Mattia
Publisher: Università degli Studi di Parma. Dipartimento di Scienze chimiche della vita e della sostenibilità ambientale
Year: 2020
Source: https://www.repository.unipr.it/bitstream/1889/4288/3/Anzola_Mattia_Thesis.pdf
DEPARTMENT OF CHEMISTRY, LIFE SCIENCES
AND ENVIRONMENTAL SUSTAINABILITY
Doc o al P og amme in Ma e ial Science and Technology
XXXIII Cycle
MODELING SUPRAMOLECULAR
ELECTROSTATIC INTERACTIONS IN
COMPLEX SYSTEMS
Coo dina o :
P o . En ico Dalcanale
Tu o :
P o . Anna Painelli
PhD s uden :
Ma ia Anzola
2017-2020
Con en s
In oduc ion 1
1 Resonan Ene gy T ans e : a dynamical app oach 5
1.1 In oduc ion and plan o he wo k . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Fo ce Field alida ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.1 NBD................................... 11
1.2.2 NR.................................... 13
1.2.3 Pa ame iza ion o he exci ed s a e FF: QM calcula ions . . . . . 15
1.3 Molecula Dynamics simula ions . . . . . . . . . . . . . . . . . . . . . . . 18
1.3.1 The ee DA pai in chlo o o m . . . . . . . . . . . . . . . . . . . . 18
1.3.2 The bound DA pai in chlo o o m . . . . . . . . . . . . . . . . . . 21
1.3.3 The bound DA pai in DMSO . . . . . . . . . . . . . . . . . . . . . 21
1.3.4 Con o ma ional Analysis . . . . . . . . . . . . . . . . . . . . . . . . 24
1.3.5 Cha a e is ics imescales . . . . . . . . . . . . . . . . . . . . . . . . 25
1.4 A dynamical model o exci ed s a e decay . . . . . . . . . . . . . . . . . . 28
1.4.1 DecayTimes .............................. 33
1.4.2 Simula ions in DMSO . . . . . . . . . . . . . . . . . . . . . . . . . 38
1.4.3 Decay a es in s a ic con igu a ions . . . . . . . . . . . . . . . . . . 39
1.5 Conclusions................................... 43
2 Molecula Agg ega es 45
2.1 In oduc ion................................... 45
2.2 Agg ega es o pola and pola izable dyes . . . . . . . . . . . . . . . . . . 47
2.2.1 DA ch omopho es as pola and pola izable dyes . . . . . . . . . . . 47
I
2.2.2 The model o he agg ega e . . . . . . . . . . . . . . . . . . . . . 50
2.2.3 Elec onic Hamil onian: he o a ion on adiaba ic s a es . . . . . . 51
2.2.4 Accoun ing o ib a ions: he Lang-Fi so ans o ma ion . . . . 54
2.2.5 Compu a ional S a egy . . . . . . . . . . . . . . . . . . . . . . . . 55
2.2.6 Resul s ................................. 57
2.2.7 Discussion................................ 63
2.3 Agg ega es o non-pola molecules . . . . . . . . . . . . . . . . . . . . . . 65
2.3.1 The model Hamil onian . . . . . . . . . . . . . . . . . . . . . . . . 66
2.3.2 Exac diagonaliza ion app oach: calcula ing abso p ion and luo-
escencespec a ............................ 68
2.3.3 Exac esul s on ini e size sys ems: alida ing he Hei le -London
app oxima ion ............................. 71
2.3.4 Tes ing app oxima ion schemes . . . . . . . . . . . . . . . . . . . . 74
2.3.5 2Dagg ega es.............................. 89
2.4 Conclusions................................... 96
3 Chi al agg ega es o αand β-dicyanos ylbenes: chi op ical p ope ies 99
3.1 In oduc ion................................... 99
3.2 DCSB: expe imen al esul s . . . . . . . . . . . . . . . . . . . . . . . . . . 101
3.3 Molecula Dynamics ..............................103
3.3.1 De ini ion o he o ce ield . . . . . . . . . . . . . . . . . . . . . . 103
3.3.2 MD simula ions on agg ega es . . . . . . . . . . . . . . . . . . . . 105
3.4 Abso p ion and CD spec a o DCSB agg ega es . . . . . . . . . . . . . . 107
3.4.1 Themodel ...............................107
3.4.2 Calcula ed Spec a . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
3.5 Conclusions...................................111
4 Chi al agg ega es o squ aine dyes 113
4.1 In oduc ion...................................113
4.2 The h ee s a e model o squa aine dyes . . . . . . . . . . . . . . . . . . . 115
4.3 Agg ega e model: elec os a ic in e ac ions . . . . . . . . . . . . . . . . . 118
4.4 The ole o in e molecula cha ge ans e in e ac ions . . . . . . . . . . . 123
II
4.5 Calcula ion o CD spec a o agg ega es wi h delocalized elec ons . . . . 129
4.6 Molecula Dynamics modeling o chi al squa aine agg ega es . . . . . . . . 133
4.6.1 De ini ion o a o ce ield . . . . . . . . . . . . . . . . . . . . . . . . 133
4.6.2 Enhanced sampling MD on e ame s . . . . . . . . . . . . . . . . 133
4.6.3 Selec ion o s a is ically ep esen a i e s uc u es . . . . . . . . . . 134
4.7 Chi al agg ega es o squa aine dyes: abso p ion and CD spec a . . . . . . 135
4.8 Conclusions...................................136
Gene al conclusions 143
A Molecula Agg ega es 145
A.1 Oscilla o S eng h: sum ule . . . . . . . . . . . . . . . . . . . . . . . . . 145
A.2 Agg ega es o pola dyes: addi ional esul s . . . . . . . . . . . . . . . . . 147
B Molecula dynamics 151
B.1 Foundamen al pa ame e s . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
B.2 Fo ceFields...................................153
B.3 Ensembles....................................153
B.4 Enhanced Sampling Techniques . . . . . . . . . . . . . . . . . . . . . . . . 155
B.4.1 Umb ella Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
B.4.2 Hamil onian Replica Exchange . . . . . . . . . . . . . . . . . . . . 156
B.5 Simula ion condi ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
B.5.1 Chap e 1................................157
B.5.2 Chap e 3................................158
B.5.3 Chap e 4................................158
Acknowledgemen s 161
Bibliog aphy 163
Lis o Publica ions 177
III

In oduc ion
E e y known in e ac ion be ween objec s o pa icles can be asc ibed o one o he
so called ou undamen al o ces: s ong, weak, g a i a ional, and elec omagne ic.
They a e sepa a ed acco ding o he a e age s eng h o he o ce, he kind o pa icles
in ol ed in he in e ac ions, and he ange o e ec i eness. The ele omagne ic o ce, and
speci ically elec os a ic o ce in chemis y, ep esen s undoub edly he mos impo an .
Almos he o ali y o ans o ma ions in ol ing a oms and molecules, om he eezing
o wa e o complex ene gy ans e mechanisms, obey i s ules. Elec os a ic in e ac ions
depend on a ac i e o epulsi e o ces be ween a oms o molecules gene a ed by hei
elec ical cha ges. This in e ac ion is also known as Coulomb in e ac ion, named a e
physicis Cha les-Augus in de Coulomb, who i s cha ac e ized i in 1785.[1]. E en
hough he majo i y o non-bonded o ces a e elec os a ic, chemis s usually disc imina e
be ween hem on he basis o he ypes o cha ges in ol ed. Cha ged pa icles a e
esponsible o he s onges non-bonded o ces, and include elec on-nucleus, ion-ion
and ion-dipole in e ac ions. Con e sely, weake elec os a ic o ces mani es be ween
non-cha ged objec s, such as pola o pola izable molecules, and comp ehend, among
many o he s, dipole-dipole in e ac ions, Van de Waals and London dispe sion o ces.
In his wo k we will ocus on in e molecula in e ac ions ha play a majo ole in he
de ini ion o p ope ies in molecula ma e ials. The s udy o no el molecula ma e ials
p o ides g ea oppo uni ies o sol e a la ge amoun o needs. A ca e ul a angemen
o molecules a he nanoscale can lead o he o ma ion o sys ems wi h unique op ical,
elec ical and magne ic p ope ies. Molecules a anged in sup amolecula agg ega es
in e ac h ough weak non-co alen o ces, gi ing ise o collec i e elec onic s a es ha
adically change spec oscopic ea u es o hese sys ems. A la ge a ie y o ields exploi
1
INTRODUCTION
his high e sa ili y, om o ganic sola cell [2, 3, 4] o OLED echnology.[5, 6, 7] Ano he
no iceable example ha le e ages in e molecula elec os a ic in e ac ions is made up o
he so called ligh ha es ing complexes, ac ing in he e y i s s s ep o pho osyn hesis, in
which sola ligh is abso bed and e icien ly ca ied as exci a ion ene gy h ough esonan
ene gy ans e .[8, 9] These sys ems ha e been in es iga ed o a long ime since hei
a i icial eplica ions ind nume ous applica ions as powe sou ces, senso sys ems and
nano ab ica ed de ices.[10, 11, 12, 13]
O c ucial impo ance in esea ching new ma e ials, as well as in de eloping new ea-
u es in exis ing one, is he heo e ical modeling o hese sys ems and he a ionaliza ion
o hei spec oscopic p ope ies. Pu pose o his hesis is he s udy and in eg a ion o
no el me hodologies o ge a deep unde s anding o in e molecula elec os a ic in e -
ac ions in complex nanosized sys ems. In Chap e 1 we in es iga e F¨o s e RET o
a pai o dyes linked o a calixa ene sca old. In sys ems whe e he geome y o he
ch omopho e pai is no known a p io i, he a e o ene gy ans e is es ima ed con-
side ing wo limi ing egimes, s a ic and dynamic, o e ing an o e simpli ied iew o he
mechanism. Fo his eason, we make ex ensi e use o MD simula ions in wo di e en
sol en s o desc ibe he e ec o sol a ion and con o ma ional mo ions on he a e o
ene gy ans e . Mo eo e , we de elop a ully dynamical model, based on Mon e Ca lo
me hod, o analyze he cha ac e is ic imescales o such p ocesses and compa e hem
wi h he expe imen al pic u e. Subsequen chap e s a e all ocused on molecula ag-
g ega es. In Chap e 2, we examine spec oscopic p ope ies o molecula agg ega es,
es ing new app oaches and app oxima ion schemes o pola and non-pola sup amolec-
ula assemblies. The i s pa is ocused on he discussion o agg ega es o pola and
pola izable dyes, imp o ing al eady exis en models o accoun o ib a ional coupling
and hence o spec al band-shapes. We hen u n a en ion o agg ega es o non-pola
ch omopho es, add essing he eliabili y o he Hei le -London app oxima ion and p e-
sen ing a model o wo dimensional agg ega es. Chap e s 3 and 4 a e ocused on chi al
agg ega es. In Chap e 3 we in es iga e agg ega es o med by dicyanos ilbenes deco a ed
wi h chi al pendan s. Th ough he use o an hyb id app oach, in ol ing MD simula ions
and exci on modeling, we a e able o ge a deep unde s anding on bo h agg ega ion and
spec oscopic ea u es o hese sys em, ques ioning he e ec i eness o widely adop ed
2
INTRODUCTION
ules o assess he sys em chi ali y om Ci cula Dich oism (CD) spec a. In Chap e
4 we ocus a en ion on non-symme ic squa aine agg ega es. An ex ensi e heo e ical
wo k is discussed, de o ed o he s udy o spec oscopic ea u es o squa aine assemblies
in solu ion. While he possibili y o hese sys em o o m s able Cha ge T ans e (CT)
s a es has al eady been ex ensi ely discussed,[14, 15, 16] li e a u e lacks o a me hodol-
ogy able o desc ibe o a o y powe induced by CT ansi ions. We he e p esen a new
model o he calcula ion o abso p ion and CD spec a o squa aine complexes using a
delocalized elec ons app oach, aking in o accoun o bo h in a- and in e molecula
cha ge ans e mechanisms.
Du ing my hesis pe iod I spen 3 mon hs a he In e na ional Cen e o Theo e ical
Physics in T ies e unde he guidance o D . Ali Hassanali and 3 mon hs a he Rude
Boˇsko i´c Ins i u e in Zag eb unde he supe ision o D . Luca G isan i whe e I lea n
and mas e ed all he Molecula Dynamics echniques implemen ed in his essay.
3
CHAPTER 1: RESONANT ENERGY TRANSFER: A DYNAMICAL APPROACH
o he sys em we e unsuccess ul, wi h heo e ically es ima ed imescales anging om a
ew en hs o a ew hund eds s.[34] He e, while adop ing he same app oach o he VDA
es ima e, we a e able o p ope ly simula e RET imescales hanks o ou ully dynamical
app oach o RET. We will demons a e ha he RET p ocess is go e ned by a complex
in e play be ween di e en compe ing dynamical p ocesses ha include no jus he D
adia i e and non- adia i e elaxa ion, bu also he con o ma ional and sol a ion deg ees
o eedom o he sys em ha , modula ing VDA on simila imescales as RET, canno be
neglec ed in he desc ip ion o his dynamical phenomenon.
We can summa ize ou oadmap as ollows:
•build a compu a ionally eliable dynamical sys em o he D,A molecules in he
p esence o he calixa ene (clx) sca old
•de elop a ully dynamical model o he ene gy ans e p ocess and how his
compa e wi h expe imen al pic u e
•analyze he cha ac e is ics imescales o such p ocesses o unde s and he ole o
di e en physical componen s
1.2 Fo ce Field alida ion
Because o he complexi y o he sys em, we decided o s udy i s he wo ee dyes in
wa e (a e y well s udied sol en o MD). Then we p oceeded acco ding o he ollowing
scheme:
(DA-c) Dand Aunbound pai in chlo o o m
(clx-DA-c) Dand Aconnec ed o calix-[4]-a ene in chlo o o m
(clx-DA-d) Dand Aconnec ed o calix-[4]-a ene in DMSO
A eliable compu a ional model o he abo e sys ems can be buil by aking ad an age
o MD. In a MD simula ions, a oms mu ually in e ac , accoun ing o a ac i e and
epulsi e o ces as well as o chemical bonds cons ain s, gene a ing a po en ial ield.
10

1.2 Fo ce Field alida ion
The nume ical solu ion o he New on’s equa ions o mo ion gene a es a ajec o y,
showing he dynamic e olu ion o he sys em. See Appendix B o mo e de ails.
The i s s ep in any MD in es iga ion is he de ini ion o an op imal Fo ce Field (FF)
o he sys em a hand. Typically, non-biological o ganic molecules o mode a e size
a e modelled using o ce ields ha includes all a oms and desc ibe hei in e ac ions,
s a ing om ab-ini io op imized geome ies. Commonly adop ed FF a e GROMOS,
CHARMM and GAFF. GAFF will p o e he mos app op ia e choice o ou sys em.
A speci ic equi emen o ou wo k is a good ep esen a ion o he dono dye bo h in
he g ound s a e, Dand in he elaxed exci ed s a e D∗. Di e en cha ge dis ibu ions
o he same molecule in he wo di e en s a es can be eadily con e ed in o GAFF
opologies, enabling a ine uning o pa ame e s.
1.2.1 NBD
Figu e 1.3: 4-amino-1-ni obenzoxadiazole, NBD
The dono molecule o ou sys em, NBD, is a small molecule wi h a ai ly igid
s uc u e. Howe e , some geome ical pa ame e s mus be moni o ed o ensu e a ealis ic
11
CHAPTER 1: RESONANT ENERGY TRANSFER: A DYNAMICAL APPROACH
beha iou du ing simula ions. We speci ically in es iga e he geome y o amino and
ni o g oups. A e opology iles we e c ea ed o each FF, h ee di e en simula ions
a cons an numbe o molecules, olu e and empe a u e (NVT ensamble, see Appendix
B) simula ions we e pe o med in a 5 ×5×5nm3wa e box. To es he goodness o
he opology he Radial Dis ibu ion Func ion (RDF) o wa e oxygens a ound he ni o
and amine ni ogen a oms is calcula ed (Fig. 1.4). RDF desc ibes how he densi y o
su ounding a oms a ies as a unc ion o dis ance om a poin . I is usually de e mined
as he numbe o a oms A (in ou case wa e oxygens) a a dis ance om a gi en a om
B (ni o and amino ni ogen), no malized o he densi y o A a oms in he sys em:
gAB( ) = 1
ρ(A)hX
A
δ(~ A−~ B)i(1.6)
RDF esul s o GROMOS di e om hose ob ained by o he FFs, gi ing an inco ec
Figu e 1.4: Le : adial dis ibu ion unc ion o wa e oxygens a ound amino N; igh :
adial dis ibu ion unc ion o wa e oxygens a ound ni o N.
dis ibu ion o wa e oxygens a ound he amino ni ogen, while CHARMM and GAFF
esul s a e compa able (Fig. 1.4).
We modi ied some pa ame e s in GAFF opology ile o he NBD molecule in o de
o imp o e he consis ency o esul s ob ained wi h di e en o ce ields. We expec he
dyhed al angle dis ibu ion o bo h he ni o and amino g oups o be peaked a 0◦, since
12
1.2 Fo ce Field alida ion
Figu e 1.5: H-N-C-C dihed al angle dis ibu ion o di e en FFs.
he s ong dono /accep o cha ac e o he NH2/NO2g oups ensu es conjuga ion and
hence plana i y. While he H-N-C-C dihed al (shown in Fig. 1.5) al eady sa is ies ou
p edic ions, he O-N-C-C dihed al dis ibu ion p esen s a minimum a 0◦(Fig. 1.6 le ).
Fo his eason we inc eased he ele an o ce cons an , om 2.51 o 4.4 Kcal·mol−1·˚
A−2
(i.e. se ing i o he same alue as o he H-N-C-C o ce cons an ). The esul ing
dis ibu ion ob ained p ope ly peaks a 0◦, as shown in he igh side o Fig. 1.6.
1.2.2 NR
Fo NR we un analogous simula ions as desc ibed o NBD. As shown in he le panel
o Fig. 1.8, he magni ude o he dipole momen shows di e en ime p o iles depending
on he adop ed FF. A co ela ion is obse ed be ween he oscilla ions o he dipole
magni ude and he o ien a ion o he me hoxy g oup (see Fig. 1.8). In GROMOS, whe e
he me hoxy g oup is con ined in he [−45◦,45◦] egion, dipole module is ixed a ound
13
CHAPTER 1: RESONANT ENERGY TRANSFER: A DYNAMICAL APPROACH
Figu e 1.6: Dis ibu ion o dihed al angle O-N-C-C in di e en Fo ce Fields. Le : be o e
co ec ion; igh : a e co ec ion.
Figu e 1.7: me hoxy-Nile Red, NR
6.7 D, while i s ongly oscilla es when he me hoxy g oup can in e change be ween
“open” and “closed” con o ma ions (de ined espec i ely as con o ma ions whe e he
O−CH3g oup is in “cis” o “ ans” con igu a ion wi h espec o he ca bonyl g oup).
We pe o med wo di e en Quan um Mechanical (QM) calcula ion (b3lyp/6-31g(d,p),
ozen geome y) on he “closed” and “open” con o ma ion o he me hoxy g oup o
14
1.2 Fo ce Field alida ion
Figu e 1.8: Le : NR dipole momen magni ude as a unc ion o ime. Righ : me hoxy
g oup dihed al dis ibu ion.
assess he magni ude o he elec ic dipole in he wo con o me s. We ob ain 6.2899 D
in he closed con i ma ion and 7.8773 Din he open con o ma ion, in good ag eemen
wi h MD esul s. The good ag eemen wi h DFT suppo ed ou choice o GAFF as he
e e ence o ce ield, wi h he mino co ec ion o he o ce cons an ele an o he OMe
g oup ha we se o 9.17 kJmol−1nm−2, o be compa ed wi h he o ginal alue 3.77.
Wi h his new cons ain he me hoxy g oup is s ill able o o a e bu he dis ibu ion
is mo e p onounced in he 0◦and 180◦ egion (Fig. 1.9), consis en ly wi h chemical
in ui ion.
1.2.3 Pa ame iza ion o he exci ed s a e FF: QM calcula ions
Fo he D molecule we need also a FF o he exci ed D* s a e. Wo king wi h exci ed
s a es in classical molecula dynamics would in p inciple equi e a ull epa ame iza ion
o he FF. This is a non- i ial ask and de ini ely beyond he scope o ou wo k. The e-
o e we adop ed an empi ical app oach using he same GAFF pa ame iza ion de ined
o he g ound s a e, bu eplacing he g ound s a e equilib ium geome y and cha ge
dis ibu ion wi h hose ela i e o he Kasha s a e ( he lowes ib a ionally elaxed ex-
ci ed s a e). Indeed e y mino con o ma ional changes, a e obse ed, as expec ed o
15

CHAPTER 1: RESONANT ENERGY TRANSFER: A DYNAMICAL APPROACH
Figu e 1.9: Le : compa ison be ween me hoxy dihed al dis ibu ion in modi ied-GAFF
wi h FFs in Fig. 1.8; igh : dipole magni ude oscilla ion o he modi ied GAFF o ce
ield compa ed wi h “closed” (o ange) and “open” ( ed) con o ma ion dipole magni ude
as ob ained om DFT calcula ions.
NBD, a plana and igid molecule.
Fo exci ed s a e calcula ions we es ed ew le els o heo y. We used s anda d HF as
well as TD-DFT wi h wo di e en unc ionals, wi h 6-31G(d,p) basis se . Calcula ions
we e un bo h in acuum ( ac) and in chlo o o m (cl , PCM). Table 1.1 show ele an
esul s.
The diag am in Fig.1.10 schema izes he p ocess o ex ac he p ope opology om
he esul o he ab-ini io calcula ion. Molecula dynamics simula ions add ess slow
Figu e 1.10: Calcula ion pe o med o ob ain exci ed s a e cha ges on isola ed ch o-
mopho es
16
1.2 Fo ce Field alida ion
NBD CIS-HF B3LYP CAM-B3LYP Exp.
cl ( ac) cl ( ac) cl ( ac)
T ans. En. (Abs)a4.11 (4.35) 3.14 (3.32) 3.39 (3.60) 2.73
µ (Abs) 5.978 (4.272) 3.964(2.222) 4.894 (2.885)
T ans. En. (Emi) 3.70 (3.96) 2.56 (2.75) 2.91 (3.12) 2.38
µ (Emi) 5.972 (4.186) 2.682 (1.501) 4.289 (2.401)
Table 1.1: Calcula ed abso p ion and emission ene gies (eV) and ansi ion dipole mo-
men s (a omic uni s, a.u.) ele an o he lowes exci ed s a e NBD. Las column shows
expe imen al ansi ion ene gies om Re . [34].
deg ees o eedom, such as con o ma ional changes and di usion wi hin he sol en ,
much slowe han molecula elec onic and ib a ional deg ees o eedom. The e o e
he exci a ion is simula ed by simply swi ching he molecula opology ile om he one
ele an o he g ound s a e o ha ele an o he elaxed exci ed s a e. As shown in
he diag am abo e, we s a om he op imized g ound s a e, ocus on he i s exci ed
s a e, elax i s geome y, calcula e es ained elec os a ic po en ials (RESP cha ges)
and inally ans e his in o ma ion o GAFF opology.
Since bes esul s o g ound s a e calcula ions we e ob ained wi h B3LYP unc ional
in acuum, he same unc ional is adop ed in s eps 2 and 3. Fo s ep 4 ins ead, we
adop ed he HF-CIS unc ional, he one mos sui able wi h RESP cha ges. The basis se
is main ained as 6-31 G(d,p) in all calcula ions. Table 1.2 epo he di e ence be ween
g ound and exci ed s a e elec ic dipole (bo h in magni ude ∆|µ|and o ien a ion θµ),
calcula ed o di e en le els o heo y. Along wi h hem, we lis ed he a ia ion in
cha ge upon exci a ion o a ew a oms.
17
CHAPTER 1: RESONANT ENERGY TRANSFER: A DYNAMICAL APPROACH
NBD
∆|µ|D 1.15
θµdeg. 12.65
O−NO ∆q0.031
NO2∆q-0.074
NH2∆q0.053
Table 1.2: G ound and exci ed s a e RESP cha ge dis ibu ion. Magni ude and o ien a-
ion changes in elec ic dipole momen a e shown along wi h cha ge di e ences o mos
ele an a oms.
1.3 Molecula Dynamics simula ions
1.3.1 The ee DA pai in chlo o o m
As a i s s ep, we add ess MD simula ions o he unbound DA pai in chlo o o m. We
ocus on wo di e en sys ems:
-D−A
-D∗−A
We i s gene a ed a 5 ×5×5nm3chlo o o m box wi h he wo dyes andomly placed
in i . We hen p oceeded wi h a 1 ns simula ion a ixed molecule numbe , p essu e
and empe a u e (NPT simula ion), ollowed by a p elimina y NVT un o 10 ns and
a inal NVT un o 100 ns. The Po en ial o Mean Fo ce (PMF) cu e is de ined as
he ee ene gy su ace along a chosen coo dina e. Speci ically, PMF P, along a gene ic
coo dina e ξ, can be de ined as[35]:
P(ξ) = P(ξ0)−RTln D(ξ)
D(ξ0)(1.7)
whe e Ris he uni e sal gas cons an , T he empe a u e ξ0is an a bi a y e e ence
poin . The unc ion D(ξ), he dis ibu ion unc ion along he coo dina e ξ, is ob ained
di ec ly om he MD ajec o y.
18
1.3 Molecula Dynamics simula ions
Figu e 1.11: PMF ex ac ed om a 100 ns NVT simula ion in chlo o o m o DA, D*A
and DA*.
PMFs shown in Fig. 1.11 (and all he o he s) we e elabo a ed om he dis ibu ions
D(d) o he dis ance (d) be ween he D and A cen e o mass (COM), calcula ed o e
long dynamical uns (a leas 100 ns, see Appendix B o u he de ails). The PMF
p o iles in Fig. 1.11 show a e y b oad minimum a 2.58 nm, which is howe e an a i ac
due o pe iodic bounda y condi ions ( he box dimension is 5.16 nm). The peak a 0.39
nm co esponds ins ead o a π−πcon igu a ion (Fig. 1.12 le ). A second peak a 1
nm aco esponds (Fig. 1.12 igh ) o a con igu a ion cha ac e ized by a s ong H-bond
be ween he amino hyd ogen o NBD wi h he quinonoid oxygen o NR.
Umb ella Sampling
Fo ou pu pose, since RET is s ongly a ec ed by he D-A dis ance, we mus ensu e
ha all possible dis ances a e p ope ly sampled. The e o e he PMF cu e ob ained
19
CHAPTER 1: RESONANT ENERGY TRANSFER: A DYNAMICAL APPROACH
Figu e 1.19: Bidimensional his og ams o θµ(le ) and θπ( igh ) o e ch omopho e
cen e o mass dis ance in chlo o o m. Top: esul s o clx-DA. Bo om: esul s o
clx-D*A.
whe e h...iξindica es he in eg al o e ime on he en i e dynamics. Since in MD simu-
la ions we only ha e access o disc e e da a poin s sepa a ed by ime in e als ∆ , he
in eg a ion is subs i u ed by a sum as ollows:
AC (j∆ ) = 1
N−j
N−1−j
X
i=0
(i∆ ) ((i+j)∆ )
whe e iand j un o e all N ames o he simula ions. The a iables o in e es a e
he dis ance, he angle bew een he ec o s no mal o he molecula planes and he
angle be ween pe manen dipoles (d,θπand θµ), as de ined p e iously. Fo each deg ee
o eedom, an au oco ela ion cu e is calcula ed and hen each AC( ) is i ed wi h a
26

1.3 Molecula Dynamics simula ions
Figu e 1.20: Bidimensional his og ams o θµ(le ) and θπ( igh ) o e ch omopho e cen e
o mass dis ance in DMSO. Top: esul s o clx-DA. Bo om: esul s o clx-D*A.
double exponen ial (Table 1.3) o ex ac he ele an ime-scale ha will be compa ed
o he exci ed s a e decay in he nex sec ion.
CXX( ) = a0exp(b0 ) + (1 −a0) exp(b1 )
wi h X=d, θπ, θµ.
The esul s in Fig. 1.22 con i m a s ong simila i y o beha iou in chlo o o m o
he in e molecula dis ance and he o ien a ional mo ions, wi h all a iables dynamically
ac i e on a ime scale o ∼10 ns. In con as , in DMSO he o ien a ional mo ion occu s
on a as e ime scale (∼1ns) han he a ia ion o in e molecula dis ances (∼5ns).
27
CHAPTER 1: RESONANT ENERGY TRANSFER: A DYNAMICAL APPROACH
Figu e 1.21: MD simula ion snapsho o wo o he mos isi ed con o ma ions in chlo-
o o m: le , π−πs acking; igh , dye-linke H-bond
1.4 A dynamical model o exci ed s a e decay
Simula ing he elaxa ion o D∗is qui e icky. In ac , once he dono has been exci ed,
i can decay along di e en pa hs:
•Non- adia i e decay, wi h kine ic cons an kn
•Radia i e decay, luo escence, wi h kine ic cons an k ad
•Ene gy T ans e , wi h kine ic cons an kRET
The i s wo p ocesses a e ma ginally a ec ed by he sys em dynamics, while he RET
a e, being s ongly dependen on he dis ance and mu ual o ien a ion o he dyes, is
highly a iable. We a e pa icula ly in e es ed o unde s and and model he concu -
en dynamics o RET and con o ma ional mo ion, well beyond s anda d ea men s
28
1.4 A dynamical model o exci ed s a e decay
clx −DA Chlo o o m clx −DA DMSO
d i ange :100000 :100000
a00.722 0.664
b−1
011.63 5.47
b−1
11.95 0.883
θπ i ange :100000 :100000
a00.678 0.748
b−1
010.87 0.747
b−1
10.37 0.052
θµ i ange :100000 :100000
a00.550 0.877
b−1
011.76 1.02
b−1
10.68 0.067
Sampling equency: 10 ps.
Table 1.3: Pa ame e s ex ac ed om exponen ial i ing o au oco ela ion unc ions o
d,θπand θµ. All ime cons an s a e in ns.
ha ypically add ess wo limi ing cases. Speci ically, i he molecula mo ion is much
slowe han he ans e p ocess, dyes o ien a ion is app oxima ed as ozen and he
o ien a ional ac o κ2is conside ed cons an and se o a alue anging om 0 o 4,
depending on he mu ual o ien a ion o he dyes. In he opposi e case, he dono and
accep o ha e enough ime o explo e all hei a iable-space be o e exchanging ene gy,
and κ2=2
3. In ealis ic si ua ions, and especially o bound DA pai s, he si ua ion is
ac ually in e media e and calls o mo e de ailed model, as made possible by MD.
In ou app oach we con e he a e o each decay pa h o D∗( adia i e/non-
adia i e decay, RET) in o a p obabili y. The cons an a es o he adia i e and
non- adia i e decay a e se o he he expe imen al alues. Ins ead, he p obabili y de-
cay along he he RET channel is con olled by he ins an aneous alue o VDA. Plo ing
29
CHAPTER 1: RESONANT ENERGY TRANSFER: A DYNAMICAL APPROACH
Figu e 1.22: . Au oco ela ion unc ions o dis ance d,θµand θφ, be ween D and A o
clx-DA in chlo o o m (le ) and DMSO ( igh )
he alue o in e ac ion, along wi h D-A sepa a ion, as a unc ion o ime (Fig. 1.23), i
is clea how long in e -dyes dis ance is always ela ed o a negligible magni ude o VDA
while, as ch omopho es ge close , he in e ac ion can assume la ge alues, depending
on ela i e o ien a ion o ansi ion dipole momen s. Knowing, o a gene ic p ocess, he
cha ac e is ic decay ime τ, de ined as:
τ( ) = 1
k ad +kn +kRET ( ),(1.9)
we can assess an in ini esimal p obabili y Pd o e e y “ins an o ime”:
Pd =1
τd →Zτ
0
Pd = 1 (1.10)
Since MD simula ion uses nume ical in eg a ion, we only ha e access o ini e ime-s eps
∆ , and we need o ansla e Eqn. 1.10 in o a disc e e equa ion:
P∆ =1
τ∆ →
τ
X
0
P∆ = 1 (1.11)
In p ac ice, each imes ep ca ies a ac ion o he p obabili y o each o he possible
pa hs.
30
1.4 A dynamical model o exci ed s a e decay
Figu e 1.23: D-A in e ac ion VDA (black lines) and Dono Accep o dis ance d( ed lines)
plo ed as a unc ion o ime o h ee sample exci ed s a e ajeco ies. Top panels: ull
dynamics; bo om panels: magni ied sec ions o he a o emen ioned ajec o ies.
Ene gy ans e a e kRET is easily calcula ed om Eqn. 1.2, while k ad and kn
a e ex ac ed om li e ime (τD) and quan um yeld (Φ) expe imen ally measu ed on he
isola ed ch omopho e:
k ad =1
τD·Φkn =1
τD−k ad (1.12)
Once ∆ and τa e de e mined elaxa ion ime is calcula ed as ollows:
1. Ini ializa ion
A ini e g oup o ch omopho es is selec ed; numbe o simula ion s eps is se o
Ncycle =−1 and ini ial ime is de ined as = ∆ ·Ncycle
2. Time s ep
Time is inc eased by ∆ (Ncycle =Ncycle + 1)
31

CHAPTER 1: RESONANT ENERGY TRANSFER: A DYNAMICAL APPROACH
3. Decays
Fo each molecule, a andom numbe 0 ≤Rn<1 is gene a ed:
- i Rn≤1
τ, a elaxa ion occu
- else, no hing happens
4. Con inua ion
Relaxed molecules a e emo ed om he ini ial g oup, and hei decay ime (Ncycle·
∆ ) is sa ed; emaining ch omopho es a e sen back o S ep 2
5. End
When all molecules a e elaxed, he simula ion s ops
A g aphical ep esen a ion is shown in Fig. 1.24. In his example a gene ic s ep N
o an ideal sys em is illus a ed. Remembe ing he ela ion be ween li e ime and a e
cons an s (Eq.1.9), P∆ can be w i en as:
P∆ = (k ad +kn +kRET )·∆ = 3.5ns−1·0.1ns = 0.35 (1.13)
Each ime s ep is ep esen ed by a box wi h Nb= 200 balls, o di e en colou s.
The colou o each ball ep esen s he ”ac ion” pu sued by he exci ed moie y o each
imes ep, as depic ed in Fig. 1.24. Red, g een and cyan balls all ep esen a de-exci a ion
e en , espec i ely RET, adia i e decay and non- adia i e decay, while he black balls
accoun s o imes eps in which he dono s ays in he exci ed s a e. The po ion o balls
o each colou a e di ec ly ela ed o he p obabili y he exci ed dono has o ake ha
pa h. The simula ion consis s in consecu i ely ex ac ing a ball om each box, s a ing
om box numbe 0, un il a non-black ball is picked. The ou pu o he simula ion is a
dis ibu ion o elaxa ion imes, one o each molecule. Plo ing he ac ion o molecules
in he exci ed s a e pe ime, an exponen ial p o ile is ob ained. Mo eo e , being able
o disc imina e be ween he h ee di e en pa hs o elaxa ion, ou model allows o
de e mine he ene gy ans e e iciency o each sys em as he a io be ween molecules
elaxing h ough RET channel o e all he decayed molecules:
ΦRET =NRET
N o
(1.14)
32
1.4 A dynamical model o exci ed s a e decay
Figu e 1.24: A simple g aphic ep esen ing a gene ic s ep in he simula ion.
1.4.1 Decay Times
To s a wi h, we conside he isola ed D∗, se ing kRET = 0. The expe imen al li e ime
o he ee dono is τ=1
kn +k ad = 7.0ns. We selec ed a popula ion o 105molecules
and se he ime s ep o he simula ion o ∆ = 10ps. In absence o RET he decay
p obabili y is P∆ = 1.429 ·10−3and we ob ain qui e na u ally an exponen ial decay
p o ile. The calcula ed dis ibu ion o decay- imes is shown in Fig. 1.25. The cumula i e
di e ence o e he his og am, gene a ed sub ac ing o he o al exci ed s a e popula ion
he numbe o molecule elaxing a each imes ep, gi es he cha ac e is ic exponen ial
decay shown in he igh panel. We now u n on he dono -accep o in e ac ion, and
hence RET. Since he in e ac ion be ween he ch omopho es depends on he sepa a ion
and ela i e o ien a ion be ween ch omopho es, ha a y du ing dynamics, he new
p obabili y will be i sel unc ion o ime (P∆ ( )). We un a long clx-DA simula ion
(1µs) o ha e a s a is ically ele an numbe o con igu a ions, and om his se we
andomly selec 1000 con igu a ions, making su e ha he subse gi es a good eplica
o he o iginal dis ibu ion (Fig. 1.26). Fo each con igu a ion a 10 ns NVT dynamics
33
CHAPTER 1: RESONANT ENERGY TRANSFER: A DYNAMICAL APPROACH
Figu e 1.25: Le : dis ibu ion o elaxa ion imes o τ= 7.0ns; igh : no malized
exci ed popula ion as unc ion o ime o he same sys em.
Figu e 1.26: PMF cu es calcula ed om he 1000 ames used o RET analysis ( iole )
and dis ibu ion calcula ed o all ames in he ajec o y (g een).
34
1.4 A dynamical model o exci ed s a e decay
F agmen µ
xµ
yµ
z|µ |a om 1 a om 2
F ee NBD -1.1920 0.2826 0.0000 1.23 N1 N4
F ee NR -3.0375 0.5667 0.0783 3.09 C51 C62
Table 1.4: T ansi ion dipole momen componen s o he wo ch omopho es. In MD
simula ions, he di ec ion is de e mined by he e so connec ing highligh ed a oms in
Fig. 1.27.
is calcula ed, simula ing he dono exci a ion by swi ching he cha ge dis ibu ion wi h
he one o clx-D*A.
The decay imes ollowing he ins an aneous exci a ion we e es ima ed calcula ing
VDA( ), using ansi ion dipole momen s om he elaxed exci ed s a e o he g ound
s a e and om he g ound s a e o he b igh exci ed s a e, o dono and accep o espec-
i ely. We assume ha molecula p ope ies, including he ansi ion dipole momen s,
a e ma ginally a ec ed by he dynamics, so hei magni udes and o ien a ions ela i e o
he co esponding ch omopho ic uni s a e cons an . Fo each ch omopho e, wo a oms
a e selec ed whose dis ance is pa allel o he ansi ion dipole momen o ien a ion e al-
ua ed wi h ab-ini io calcula ions (Table 1.4 and Fig. 1.27). Fo each ins an aneous
exci a ion dynamics, we we e able o calcula e kRET ( ) as ollows:
kRET =V2
DA
~2cJDA (1.15)
whe e cis he speed o ligh (in m s−1) and JDA is he spec a o e lap be ween NBD
emission spec um and NIR abso p ion spec um, each no malized o uni a ea, consid-
e ed cons an in ou model[34] (1.95 ·10−6m). Following he s a egy ou lined abo e,
we inally ob ain a decay ime o each de i ed ajec o y. To u he imp o e s a is ical
accu acy we epea ed he calcula ion 100 imes o each dynamics, ending up wi h 105
decays.
Fig. 1.28 shows he calcula ed decay o he exci ed dono , and compa es he esul s
o he dynamical calcula ion o he RET pai wi h hose ob ained o he isola ed D
dye. To i he exponen ial decay o D∗we used a mul iple exponen ial i ing, wi h
35
CHAPTER 1: RESONANT ENERGY TRANSFER: A DYNAMICAL APPROACH
channels. On he con a y, in a dynamical model, he popula ion is con inuously ans-
e ed o eplenish hese ho spo s, so ha as channels s ay ac i e all along he p ocess.
Qui e in e es ingly, he di e en esul s ob ained in he s a ic and dynamic calcula ions
canno be asc ibed o di e en dis ibu ions o kRET . Indeed he his og ams calcula ed
along he s a ic and dynamic ajec o ies shown in Figu e 1.31 a e e y simila . As he
s a ic scena io only e e s o a sampling h ough g aound-s a e MD, while he dynamic
one is he esul s o he exci ed s a e (non-equilib ium) dynamics, he simila i y o he
wo dis ibu ions sugges s ha he sou ces o non-equilib ium e ec s a e ac ually a he
modes .
Figu e 1.31: Le panels: decay o he D∗popula ion calcula ed in chlo o o m ( op)
and DMSO (bo om). Dashed lines e e o he exponen ial decay o he isola ed D
species; ed lines show s a ic esul s (ob ained neglec ing he con o ma ional mo ion o
he exci ed RET pai a e exci a ion), and he black lines show he ull dynamical esul .
Righ panels show he (a ea no malized) dis ibu ions o he RET a es, esul ing om
s a ic and dynamic calcula ions o bo h sol en s
42

1.5 Conclusions
1.5 Conclusions
We p oposed an o iginal compu a ional p o ocol o he calcula ion o he RET dynamics
and quan um yields o a dono -accep o pai in wo di e en sol en s, highligh ing he
impo ance o con o ma ional luc ua ions. The p oposed app oach is alida ed agains
an ex ensi e se o expe imen al esul s a ailable o a RET-pai bound ia lexible links
o a calixa ene sca old. We ob ained a e y good compa ison wi h he expe imen and
sol ed a heo e ical p oblem associa ed wi h his sys em, whe e es ima es o VDA om
TD-DFT calcula ions on ew p eselec ed ep esen a i e con igu a ions o he D-clx-A
sys em unde es ima ed he RET li e imes by se e al o de s o magni ude.[34]
Ou app oach combines equilib ium and non-equilib ium MD calcula ions wi h TD-
DFT esul s, leading o a de ailed desc ip ion o he concu en p ocesses: D∗decay,
ene gy ans e and con o ma ional dynamics. The con o ma ional mo ion modula es
VDA, he in e molecula in e ac ion esponsible o RET. In a ully dynamical pic u e
he sys em a e pho oexci a ion is allowed o explo e con o ma ional egions whe e
VDA is la ge, hen opening as RET channels and leading o as e RET han in a s a ic
pic u e, whe e he con o ma ional mo ion is ozen.
RET in diso de ed sys ems is a delica e issue: he s anda d app oach elying on he
use o an a e age κ2 alue has been ques ioned in se e al ways. In he i s place he
ac o iza ion o he VDA in e ac ion in a e m κ2 ha only depends on he in e molecula
o ien a ion and in a e m ha only depends on he in e molecula dis ance, is inco ec
- pa icula ly i he sys em can explo e egions whe e in e molecula dis ances a e com-
pa a i ely sho . Mo e gene ally hose app oaches ep esen a oo c ude app oxima ion
o desc ibe DA in e ac ions in sys ems, like he one in es iga ed he e, whe e a complex
sup amolecula s uc u e poses se ious cons ain s o he mu ual a angemen s o he
Dand Amoie ies. We show ha hese p oblems can be easily add essed by g ound
s a e MD calcula ions, ha o e eliable in o ma ion on he con o ma ional he e ogene-
i y o he RET pai . Howe e , we also demons a ed ha his is no su icien in he
case in es iga ed he e. In such lexible sys ems, he con o ma ional mo ion modula es
in e molecula in e ac ions on a imescale ele an o RET, leading o impo an e ec s
ha canno be accoun ed o h ough an o ien a ional a e age in ei he in he s a ic o
ul a as egime.
43
Chap e 2
Molecula Agg ega es
2.1 In oduc ion
Molecula ma e ials a e cha ac e ized by in e molecula o ces much weake han he
chemical bonds inside each indi idual molecula uni . In spi e o ha , in molecula
ma e ials in e molecula in e ac ions deeply a ec op ical spec a, ha he e o e can-
no be calcula ed as he sum o molecula spec a. In e molecula cha ge ans e (CT)
was ea ly ecognized as a sou ce o imp essi e spec oscopic phenomena in abso p ion
spec a o molecula ma e ials, bo h in he isible and nea -IR spec al egions, whe e
so-called CT abso p ion bands appea [36, 37]. Vib a ional spec a a e also a ec ed by
CT, wi h he appea ance o s ong ea u es due o la ge cha ge luxes d i en by molec-
ula ib a ions o la ice modes (phonons).[38, 39, 40, 41] In molecula ma e ials wi h
in e molecula dis ances la ge han he sum o Van de Waals adii, elec ons a e lo-
calized wi hin each molecula uni and CT in e ac ions a e negligible. E en in hese
condi ion, elec os a ic in e molecula in e ac ions may ha e p ominen e ec s, d i ing
esonance ene gy ans e among di e en molecula species[17, 42, 43, 44] and ene gy
delocaliza ion among equi alen (o nea ly so) molecules in molecula c ys als and ag-
g ega es.[45, 46, 47] The physics o exci ons and o op ical spec a in molecula c ys als
was i s add essed in he seminal wo ks o C aig,[48] Da ido [49] and Ag ano ich[50].
The same physics also applies o molecula agg ega es: he g ound-b eaking disco e y
45
CHAPTER 2: MOLECULAR AGGREGATES
o he anomalous spec a o cyanine dyes in poo sol en s[51] opened he esea ch ield
o molecula agg ega es, wi h he seminal heo e ical wo k o Kasha.[52] An eno mous
body o expe imen al and heo e ical wo k can be ound in he li e a u e, as ecen ly
e iewed.[46, 47] Cu en unde s anding o op ical spec a o molecula agg ega es and
c ys als is mainly based on he so-called exci on model ha , only accoun ing o elec o-
s a ic in e ac ions among degene a e s a es, leads o a la ge educ ion o he basis se .
Analy ical solu ions ha e been ob ained o he elec onic p oblem ha o e a eliable
basis o unde s anding spec al p ope ies o molecula agg ega es and c ys als. The
app oxima ions o he exci on model we e discussed in he o iginal pape s[49, 50] and
ha e been ecen ly add essed in ela ion wi h agg ega es o pola izable molecules wi h
pola [53, 54, 55] o quad upola cha ac e .[56, 57, 58, 59]
Molecula ib a ions add ano he laye o complexi y o he physics o molecula
agg ega es and c ys als: he de o ma ion o he molecula s uc u e upon exci a ion
is esponsible o he F anck-Condon s uc u e o abso p ion and luo escence spec a
o isola ed molecules in solu ion, bu he shape o abso p ion and luo escence spec a
o molecula agg ega es o en la gely de ia es om he F anck-Condon beha io ,[60,
46] as i s ecognized in he na ow and s uc u less abso p ion and emission spec a
o agg ega es o cyanine dyes.[51] Delocaliza ion ene gies and ib a ional ene gies a e
o en compa able in molecula agg ega es and he adiaba ic app oxima ion mus be
abandoned. T ea ing he elec onic and ib a ional deg ees o eedom on he same
oo is a o midable ask. Indeed analy ical solu ions o he coupled elec onic and
ib a ional p oblem a e a ailable o an in ini e one-dimensional a ay o molecules in
he exci on app oxima ion and accoun ing o a single coupled ib a ion.[60, 61] Mos
o en, app oxima ions schemes ha e been p oposed o ea he p oblem, whose alidi y
and applicabili y need a ca e ul discussion.
In his chap e we will add ess spec al p ope ies o molecula agg ega es, discussing
i s agg ega es o pola and pola izable dyes, ex ending a p e ious wo k[53, 62] o
accoun o ib a ional coupling and hence o spec al band-shapes. We will hen u n
a en ion o agg ega es o non-pola , ye pola izable, dyes, add essing he eliabili y
o he Hei le -London app oxima ion. Finally we will b ie ly add ess he beha io o
wo dimensional agg ega es. This chap e is based on a published wo k on agg ega es
46
2.2 Agg ega es o pola and pola izable dyes
o non pola dyes (Anzola M, Di Maiolo F, Painelli A. Op ical spec a o molecula
agg ega es and c ys als: es ing app oxima ion schems, PCCP 2019), and a second
pape (in p epa a ion) on agg ega es o pola and pola izable dyes.
2.2 Agg ega es o pola and pola izable dyes
2.2.1 DA ch omopho es as pola and pola izable dyes
Conjuga ed dyes wi h an elec on-dono (D) and accep o (A) g oup ep esen a la ge
amily o dyes o in e es o se e al applica ions, anging om non-linea op ics[63],
molecula elec onics[64, 65], OLED[5] e c. In e amolecula in e ac ions in agg ega es
o hese dyes conside ably al e hei spec al p ope ies[54, 55, 66] One o he mos
widely used app oxima ion o desc ibe DA dyes is he so called Mulliken model[36, 39,
67], in which a dye is desc ibed in e ms o wo elec onic s a es, a neu al (|DAi) and
a zwi e ionic (|D+A−i) s a e. The unde lying hypo hesis is ha highe exci ed s a es
a e loca ed a oo la ge ene gies o be ele an . Indeed he dyes o in e es ypically
ha e he lowes exci ed s a e in he isible egion o he spec um, while highe exci ed
s a es a e in he ul a iole egion. De ining 2z0as he ene gy di e ence be ween he
wo s a es and −τ he mixing ma ix elemen , he Hamil onian eads
ˆ
H= 0−τ
−τ2z0!= 2z0ˆρ−τˆσx(2.1)
In he ollowing we will se τ= 1 as he ene gy uni .
The Hamil onian can be eadily diagonalized, gi ing he eigen alues Eg=z0−
pz2
0+τ2and Ee=z0+pz2
0+τ2and eigens a es
|gi=p1−ρ|DAi+√ρD+A−
|ei=−√ρ|DAi+p1−ρD+A−(2.2)
whe e ρ=hg|ˆρ|gimeasu es he weigh o he zwi e ionic s a e in he g ound s a e, and
hence he molecula pola i y. I s dependence on z0is (see also Fig 2.1a):
ρ=1
2−z0
2pz2
0+ 1 (2.3)
47

CHAPTER 2: MOLECULAR AGGREGATES
Figu e 2.1: The isola ed (gas-phase) dye. Top: he wo esona ing s uc u es. (a) The ρ
dependency om z0. (b) he ansi ion ene gy Ω and he ansi ion dipole momen as a
unc ion o ρ. (c) The po en ial ene gy su aces o a sys em wi h ε = 0.4 and z0= 0.7.
(d) The adiaba ic PES calcula ed o he same sys em as in panel (c).
48
2.2 Agg ega es o pola and pola izable dyes
To add ess op ical spec a we need a de ini ion o he dipole momen ope a o (ˆµ).
Following Mulliken [36], we neglec all ma ix elemen s o he dipole momen ope a o in
he chosen basis bu µ0, he la ge dipole momen associa ed wi h he zwi e ionic s a e.
Acco dingly, he dipole momen ope a o eads:
ˆµ=µ0ˆρ= 0 0
0µ0!(2.4)
All spec al p ope ies can be exp essed as unc ion o ρ. Speci ically, he ansi ion
ene gy and he ansi ion dipole momen s ead:
~Ω = 1
ρ(1 −ρ)=1
β(2.5)
hg|ˆµ|ei=µ0β(2.6)
and he mesome ic dipole momen , he di e ence be ween he pe manen dipole momen s
in he exci ed and g ound s a es eads
∆µ=µ0(1 −2ρ) = µ0α(2.7)
The dependence o ansi ion ene gy and dipole momen on ρis shown in Fig. 2.1b. The
∆µ(ρ) dependence, a s aigh line, is no shown.
To p ope ly add ess he shape o op ical spec a o pola agg ega es, molecula i-
b a ion coupled o he elec onic sys em mus be aken in o accoun . As ske ched in Fig
2.1(c,d), we in oduce a single e ec i e ib a ional coo dina e Q, and assign o he wo
basis s a es wo ha monic po en ial ene gy cu es wi h he same cu a u e and displaced
minima. The ele an Hamil onian eads:
ˆ
H= 2z0ˆρ−τˆσx+~ω (ˆa†ˆa+1
2)−g(ˆa†+ ˆa)ˆρ. (2.8)
whe e ω and ga e he equency and he elec on- ib a ion coupling o he e ec i e
ib a ional mode, espec i ely (see Fig. 2.1c). The coo dina e and i s conjuga ed mo-
men um a e exp essed in second quan iza ion as ollows
Q= ~
2ω
(ˆa†
i+ ˆai)
P=i ~ω
2(ˆa†
i−ˆai) (2.9)
49
CHAPTER 2: MOLECULAR AGGREGATES
The ib a ional elaxa ion ene gy (see Fig. 2.1c) is ε =g2/ω . In he adiaba ic ap-
p oxima ion he wo-dimensional elec onic Hamil onian is diagonalized o each Q o
ge he Q-dependen adiaba ic eigens a es whose ene gy as a unc ion o Qis shown in
Fig. 2.1d.
I we a e only in e es ed in adiaba ic esul s a equilib ium Q
¯
Q= 2ω
~
g
ω2ρ(2.10)
we may sol e he equilib ium adiaba ic Hamil onian ha educes o he wo-s a e elec-
onic Hamil onian in Eq. 2.1 bu wi h he ene gy gap be ween he N and Z s a es ha
sel -consis en ly depends on he ρ: 2z= 2z0−2ε ρ[68]. This sel -consis en p oblem is
easily sol ed o ge he ρ(z0) o ixed  (see Fig. 2.1a).
2.2.2 The model o he agg ega e
To model an agg ega e we conside a linea o de ed a ay o NDA dyes, imposing pe-
iodic bounda y condi ions o minimize ini e size e ec s, while p ese ing ansla ional
symme y. In line wi h he s anda d model o desc ibe agg ega es, we assume ha in e -
molecula dis ances a e la ge enough o neglec he o e lap be ween o bi als on nea by
molecules, so ha elec ons a e ully localized on he molecula uni s, and in e molecula
in e ac ions a e only due o elec os a ic o ces. The agg ega e Hamil onian is he sum
o he molecula Hamil onians plus a e m ha accoun s o elec os a ic in e molecula
in e ac ions:
ˆ
H=X
i
(2z0ˆρi−τˆσx,i) + 1
2X
i,j
Vij ˆρiˆρj+~ω X
i
(ˆa†
iˆai+1
2)−gX
i
(a†
i+ ˆai)ˆρi(2.11)
whe e Vij is he elec os a ic in e ac ion be ween wo molecules in hei zwi e ionic
s a e, and i,j un o e all agg ega e uni s.
The adop ed model o dipola agg ega es, despi e i s semplici y, implies a p oblem
ela ed o i s dimension. Indeed, he dimension o he elec onic basis g ows as 2N,
allowing o ea ai ly la ge agg ega es (up o mo e han 10 molecules). Bu ib a ions
mus be accoun ed o in a non-adiaba ic app oach o p ope ly deal wi h agg ega es,
and i jus 5 ib a ional quan a a e conside ed pe molecule, one ends up wi h a basis
50
2.2 Agg ega es o pola and pola izable dyes
inc easing as 10N, making he p oblem in ac able al eady a N∼3−4. In he ollowing
we will discuss s a egies o deal wi h his p oblem, be o e showing selec ed esul s.
2.2.3 Elec onic Hamil onian: he o a ion on adiaba ic s a es
We s a ou discussion ocusing on he elec onic Hamil onian, co esponding o Eq.
2.8 wi h g=0:
ˆ
Hel =X
i
(2z0ˆρi−τˆσx,i) + 1
2X
i,j
Vij ˆρiˆρj(2.12)
This Hamil onian, w i en on he diaba ic DA e D+A−basis, is ai ly simple, bu we
need o accoun o all 2Ns a es al eady o desc ibe he g ound s a e. In ac each
molecula uni is desc ibed in he g ound s a e by a linea combina ion o he DA and
D+A−s a e (see equa ion 2.2). The e o e i is con enien o o a e he elec onic basis
and ew i e he Hamil onian on he adiaba ic basis, gand e. O cou se he o a ion does
no a ec he basis dimension. Howe e , as i will be discussed below, a p ope choice
o he adiaba ic basis will help us o be e unde s and he physics o he sys em, also
allowing o a con olled unca ion o he basis, while main aining he quali y o he
esul s.
Following a p e ious wo k[69] we de ine wo new ope a o s, linea combina ions o
Pauli ope a o s σx,z, whose p ojec ions a e de ined by he pa ame e ρ:
ˆ
Sx,i =−2βˆσz,i +αˆσx,i
ˆ
Sz,i =αˆσz,i + 2βˆσx,i (2.13)
whe e i uns on he molecula si es and α= 1 −2ρand β=pρ(1 −ρ) a e he same
a iables in oduced p e iously. The abo e ope a o s a e hen exp essed in e ms o
c ea ion and annihila ion ope a o s:
ˆ
Sx,i = 1 −2ˆ
b†
iˆ
bi
ˆ
Sz,i = (ˆ
b†
i+ˆ
bi) (2.14)
The ope a o ˆ
b†
ic ea es an exci a ion on si e i, by u ning he molecule om s a e |gi
o |ei, while ˆ
bides oys he exci a ion. As discussed by Ag ano ich,[70] hese ope a o s
51
CHAPTER 2: MOLECULAR AGGREGATES
wi h low and in e media e pola i y (le and cen al panels), bu ed-shi s in he case o
a la ge ly pola dye ( igh panel). These appa en ly c azy esul s, possibly sugges ing
he ailu e o he exci on pic u e, a e indeed ela ed o a bad choice o he e e ence s a e.
A la ge pa o he shi in ac is no exci onic in o igin, bu is ela ed o he e ec s
ha su ounding cha ges ha e on he ene gy o he s a es. This is easily calcula ed
in he m app oxima ion, sol ing he p oblem o an isola ed dye eeling he po en ial
om he su ounding dyes. Repulsi e in e molecula in e ac ions leads o a educ ion
o he pola i y o each dye in he agg ega e (see ig. 2e) and hence o a a ia ion o he
equency o he abso p ion band. The p ope e e ence o he exci on model is indeed
ep esen ed by he m equency. Speci ically, o he dye in he le panels o Fig. 2.4,
he ionici y dec eases om 0.19 in he gas phase o a m alue o 0.17. Acco dingly,
he maximum o he abso p ion blueshi s, sligh ly educing he exci on shi . Simila
conside a ions apply o he dye in he middle panels, whose ionici y is educed om
0.64 in he gas phase o 0.5 in he m app oach. Fo ρ= 0.5 ( he so-called cianine
limi ) he Condon ib a ional coupling (p opo ional o he squa ed mesome ic dipole
momen ) anishes leading o he disappea ance o he ib onic s uc u e o abso p ion
and luo escence bands. Mo e inspi ing is he case o he zwi e ionic dye in igh column
o Fig. 2.4. He e he dec ease o he ionici y om 0.76 in he gas phase o 0.61 in he
m app oxima ion is esponsible o a la ge ed-shi o he abso p ion band. Taking as
p ope e e ence he m equency, a blue-shi o he abso p ion band is obse ed o he
agg ega e, ully in line wi h i s H cha ac e , as due o epulsi e (V > 0) in e molecula
in e ac ions.
Fluo escence in H-agg ega es comes om elec onic s a es a he bo de o he B il-
louin zone and a e only allowed due o he coupling o ib a ional modes. As a esul ,
e y weak and la gely ed-shi ed bands a e obse ed, bu wha we no ice he e is ha ,
since he domina ing (Condon) e m accoun ing o elec on- ib a ion coupling anishes
in he cyanine limi , he luo escence in ensi y is anishingly small in his limi .
A simila analysis can be done o he agg ega es in Fig. 2.5, co esponding o he
case o weak a ac i e in e molecula in e ac ions (V=−1). In ense emission bands
and anishing S okes shi s in he agg ega e a e ully in line wi h J-agg ega e beha io .
The edshi o abso p ion (and emission) bands obse ed o he dyes in he le and
58

2.2 Agg ega es o pola and pola izable dyes
Figu e 2.4: H agg ega e, V= 1, ε = 0.4, ω = 0.17: op and bo om panel show
calcula ed abso p ion and luo escence spec a. In ensi ies pe molecules a e epo ed
in a bi a y uni s. The weak luo escence spec a o he agg ega e a e mul iplied by he
ac o shown in he igu e. Le panels e e o a sys em wi h z0= 0.8, co esponding
o an ionici y o he isola ed dye ρ= 0.19 ha deac eses in he m app oxima ion o
ρ= 0.17. Middle panels: z0=−0.3, gas phase ρ= 0.64, m ρ= 0.50. Righ panels:
z0=−0.6, gas phase ρ= 0.76, m ρ= 0.61.
59
CHAPTER 2: MOLECULAR AGGREGATES
Figu e 2.5: J agg ega e, V=−1, ε = 0.4, ω = 0.17: op and bo om panel show
calcula ed abso p ion and luo escence spec a. In ensi ies pe molecules a e epo ed in
a bi a y uni s. Le panels e e o a sys em wi h z0= 1.0, co esponding o an ionici y
o he isola ed dye ρ= 0.15 ha inc ease in he m app oxima ion o ρ= 0.21. Middle
panels: z0= 0.7, gas phase ρ= 0.21, m ρ= 0.50. Righ panels: z0= 0.3, gas phase
ρ= 0.36, m ρ= 0.82.
60
2.2 Agg ega es o pola and pola izable dyes
middle panels o Fig. 2.4 a e again in line wi h a J-agg ega e beha io . The mos
s iking esul s is howe e ecognized again o he mos pola molecule (ρ= 0.36 in
he gas phase) in he igh panels o Fig. 2.4: he e in ac he exci on band mo es o
he blue wi h espec o he gas-phase molecule. Bu again his anomalous beha io is
simply ela ed o he choice o a w ong s a ing poin . In he agg ega e, he m solu ion
o he p oblem d i es he molecule deep in he ionic egime wi h ρ= 0.82. This implies
a la ge blue shi o he abso p ion and luo escence bands, so ha , when aking as
e e ence he gas phase molecule, an appa en blue-shi o he exci on band is obse ed,
ha ac ually co esponds o a ed-shi when he p ope e e ence is conside ed, in line
wi h he a ac i e na u e o he in e ac ions. We also no ice ha o he zwi e ionic
sys em, when he w ong e e ence s a e is conside ed, he in ensi y o he ansi ions
(bo h abso p ion and luo escence) dec eases and he ib onic s uc u e becomes mo e
p ominen , in s iking con as wi h he J-na u e o he agg ega e. This inconsis ency
is howe e qui e na u ally sol ed i he p ope m e e ence is conside ed: in all cases
he spec al in ensi y inc eases when going om he m dye o he agg ega e, while he
ib onic s uc u e becomes less and less p ominen . Qui e in e es ingly, esul s in he
cen al panel o Fig. 2.4 e e o a dye wi h ionici y ρ= 0.16 in he gas phase ha is
d i en o he cyanine limi , ρ= 0.50 when embedded in he agg ega e. Once again, in
he cianine limi he ib onic s uc u e o abso p ion and luo escence bands disappea s.
Medium and S ong Coupling
We will now add ess he cases o medium and s ong coupling. Fig. 2.4 show abso p ion
spec a calcula ed o H-agg ega es in he medium (V= 1.6) and s ong-coupling (V=
2.0) egimes. I u ns ou ha Ne= 4 is he minimum numbe o exci on s a es needed
o con e gence, Ne= 3 esul s a e o ally un enable, wi h he only excep ion o he
sys ems ha in he m app oaxima ion ha e ρ= 0.5. In hese condi ions in ac all e ms
in Eq.2.18 p opo ional o 1−2ρ anish. Acco dingly, he ib onic s uc u e disappea s,
as discussed abo e, as well as all e ms ela ed o he mesome ic dipole momen . Fo
he elec onic pa , he Hamil onian in he ρ= 0.5 limi educes o ha ele an o non-
pola agg ega es and mos o he anomalous e ec s associa ed wi h agg ega es o pola
and pola izable dyes a e washed ou . Once con e gence is eached, ini e size e ec s a e
61
CHAPTER 2: MOLECULAR AGGREGATES
ma ginal o la gely neu al dyes, as well as o dyes in he cyanine limi , bu become
ele an o zwi e ionic dyes.
Figu e 2.6: H agg ega e abso p ion spec a. All esul s e e o a sys em wi h  = 0.4
and ω = 0.17. Top panels show esul s o V=−1.6, om le o igh : z0= 0.8, gas
phase ρ= 0.21, m ρ= 0.15; z0=−0.6, gas phase ρ= 0.84, m ρ= 0.5; z0=−1.0,
gas phase ρ= 0.9, m ρ= 0.62. Bo om panels show esul s o V=−2.0, om le o
igh : z0= 0.8, gas phase ρ= 0.21, m ρ= 0.14; z0=−0.8, gas phase ρ= 0.88, m
ρ= 0.5; z0=−1.2, gas phase ρ= 0.92, m ρ= 0.61. Fo N= 4 he ull elec onic basis
is conside ed, Ne= 4.
Mo e in e es ing is he case o J-agg ega es, whe e elec os a ic in e molecula in e -
ac ions lead o in iguing phenomena.[53, 62] Fig. 2.7 shows abso p ion and luo escence
spec a calcula ed o a sys em wi h V=−1.6, co esponding o he cu e in Fig. 2.2
ha ma ks he bounda y be ween he no mal (weak coupling) and he bis able (s ong
coupling) egime. Much as in he weak case, he appa en ly anomalous beha io ob-
se ed when compa ing agg ega e spec a wi h spec a calcula ed o he isola ed dye
62
2.2 Agg ega es o pola and pola izable dyes
a e elie ed i he p ope e e ence sys em is conside ed, co esponding o he m so-
lu ion. In all cases in ac he agg ega e spec um is ed-shi ed wi h espec o he
ele an m spec um. The mos impo an di e ence wi h espec o he weak coupling
is he appea ance o ini e size e ec s, wi h N= 6 esul s di e ing om N= 4, poin ing
o exci ons wi h la ge delocaliza ion. Mo eo e , o ge con e gence o N= 6 a leas
Ne= 4 is needed (see Appendix A) in sha p con as wi h he weak coupling case. Qui e
in e es ingly, ini e size e ec s a e ma ginal o he sys em desc ibed in middle column
o ig. 2.7 whe e he m ionici y is 0.5. As discussed abo e, in his limi , he anishing
o e ms p opo ional o 1 −2ρno only kills he main ib onic coupling e m, bu also
educes he elec onic pa o he Hamil onian o ha o agg ega es o non-pola dyes.
This is e en mo e e iden in he s ong coupling limi in ig. 2.8, showing spec a
calcula ed o V=−2.0. Simila conside a ions apply as in he medium-coupling egime,
bu in his case N= 6 esul s do no con e ge un il he maximum numbe o exci ons
Ne= 6 is accoun ed o in he calcula ion, o in o he e ms, he comple e elec onic
basis is conside ed (see Appendix A). This immedia ely ells us ha he exci on-exci on
in e ac ion e m ( he i s e m in he las line o Eq.2.18, lowe s he ene gy o mul iex-
ci on s a es ha ge mixed wi h he lowes exci ed s a es gi ing a sizable mul iexci onic
cha ac e o he s a e, as ex ensi ely discussed in e s. [53, 62] Again, his e m anishes
o a sys em wi h a m ionici y ρ= 0.5, so ha o his sys em (middle panel o ig. 2.8)
he N= 6 esul s al eady con e ge a Ne= 3.
2.2.7 Discussion
Ex ending a p e ious wo k[53] o accoun o molecula ib a ions, as needed o p ope ly
add ess spec al bandshapes, a wo-s ep app oach is in oduced o he desc ip ion o
op ical spec a o agg ega es o pola and pola izable molecules. The i s s ep is he
de ini ion o he p ope e e ence s a e as he m solu ion o he p oblem. Basically,
he g ound s a e pola i y o each dye is sel -consis en ly de ined by he pola i y o he
su ounding dyes, leading o inc eased pola i y o a ac i e in e molecula in e ac ions
and educed pola i y o epulsi e in e ac ions. O cou se all molecula p ope ies (in-
cluding ansi ion equencies and dipole momen s) a e a ec ed by his a ia ion. These
m s a es de ine he p ope e e ence s a es o he exci on model. The molecula ge-
63

CHAPTER 2: MOLECULAR AGGREGATES
ome y is also a ec ed by he molecula pola i y and he co ec e e ence s a e o
he ib a ional p oblem is de ined ia a Lang-Fi so ans o ma ion ha simply ans-
la es he o igin o he ib a ional coo dina es o he equilib ium posi ion ele an o he
cha ge dis ibu ion o he molecule inside he agg ega e. Since molecula ib a ions in
u n a ec he molecula pola i y, his leads o ano he sel -consis en in e ac ion. While
his may look as a di icul p oblem, i boils down o a simple sel -consis en diagonal-
iza ion o a wo by wo Hamil onian i he molecules a e desc ibed in an essen ial s a e
pic u e.[77, 55]
The essen ial s a e model adop ed he e has been ex ensi ely alida ed agains expe i-
men and desc ibes in a e y e ec i e way he low-ene gy spec al p ope ies o push-pull
dyes accoun ing o en i onmen al e ec s in solu ion[68, 78, 79] and agg ega es[80, 55],
ilms[81] and c ys als.[72, 77] In he con ex o his wo k, howe e , we unde line ha he
model elies on simila app oxima ion as he s anda d exci on model, accoun ing o a
single elec onic exci a ion and a single ib a ional mode pe molecule. A a iance wi h
he s anda d exci on model, howe e , he p oposed model ully accoun s o he molec-
ula pola izabili y and o he dependence o he g ound and exci ed s a e molecula
geome y on he molecula pola i y.
Once he p ope e e ence s a e is de ined, se e al in e ac ion e ms a e ecognized
in he Hamil onian ha can be classi ied as exci onic, when conse ing he exci on
numbe , and ul aexci onic when mixing s a es wi h a di e en numbe o exci ons.[53]
The ib a ional coupling lead o an exci onic e m ha co esponds o he Condon
coupling in he exci on model, bu u ns ou p opo ional o 1−2ρ, and he e o e anishes
in sys em whose m ionici y is close o 0.5: in hese sys em he ib onic bandshape is
washed ou . The ul aexci onic e m exchanges ib a ional quan a and exci ons and has
ma ginal spec oscopic e ec s in he weak coupling limi as shown in Fig. 2.9 and 2.10,
ha compa e exac esul s ob ained in he weak and s ong coupling egimes o H and
J agg ega es wi h hose ob ained supp essing he non-Condon ib onic coupling in he
Hamil onian in Eq. 2.18. Non-Condon co ec ions gi e ise o sizable e ec s only in he
s ong coupling egime.
As o exci onic e ms o igina ing om elec os a ic in e ac ions, we ecognize e ms
∝ρ(1−ρ), i.e. p opo ional o he squa ed ansi ion dipole momen o he m molecules:
64
2.3 Agg ega es o non-pola molecules
hese e ms a e esponsible o he exci on hopping. O he e ms appea ∝(1 −2ρ),
i.e. p opo ional o he mesome ic dipole momen , ha accoun o exci on-exci on
in e ac ions. These las e ms anish when he m molecula ionici y is close o 0.5, and
he sys em educes o an agg ega e o non-pola dyes. The exci on app oxima ion wo ks
easonably well o weak coupling bu becomes clea ly un enable in he s ong coupling
egimes (see ig. 2.9 and 2.10).
Indeed, when inc easing he coupling, ul aexci onic e ms en e in o play wi h pa -
icula ly imp essi e e ec s in J-agg ega es, whe e bis abili y egions a e obse ed in he
m solu ion.[53, 62] Fini e size e ec s become impo an in hese condi ions and he
exci on basis canno be educed o accoun o jus he i s ew exci on s a es (up o
3 exci ons a e enough o ge con e ged esul s in he weak coupling limi ). Indeed, he
lowes exci ed s a e in hese condi ions canno be desc ibed, no e en app oxima ely,
as a s a e wi h a single exci on, a he i co esponds o a s a e whe e se e al exci ed
molecules clus e oge he in a mul iexci on s a e.[53, 62]
2.3 Agg ega es o non-pola molecules
Agg ega es o non-pola dyes a e ypically desc ibed in e ms o he exci on model.
Speci ically, he model assumes ha each molecule in he agg ega e can be ei he on
he g ound |gio exci ed |eis a e, bo h s a es ha ing a negligible pe manen dipole
momen . Elec os a ic in e molecula in e ac ion hen only imply ansi ion dipole mo-
men s, hg|ˆµ|ei=µ . Rele an in e ac ions en e he agg ega e Hamil onian in wo di -
e en e ms: an Hei le -London (HL) e m ha is esponsible o he exci on hopping,
and a non-HL e m ha mixes s a es whose exci on numbe di e s by wo uni s. When
in e molecula in e ac ions a e much smalle han he exci on ene gy, one can neglec all
e ms in he Hamil onian ha mix s a es wi h di e en ene gy. The non-HL e ms a e
hen neglec ed, leading o he s anda d exci on model. The HL app oxima ion is e y
use ul and widely adop ed as i leads o an eno mous educ ion o he elec onic basis.
Indeed, while o he comple e model one should accoun o 2Ns a es (Nis he numbe
o molecules in he agg ega e) in he HL app oxima ion, as long as one is in e es ed o
linea spec al p ope ies, only he subspace wi h a single exci on is o ele ance and he
65
CHAPTER 2: MOLECULAR AGGREGATES
basis has dimension N.
In physical e ms, imposing he HL app oxima ion amoun s o ully neglec he
molecula pola izabili y, imposing ha he na u e o he molecula g ound and exci ed
s a es is no a ec ed by in e molecula in e ac ions. As i will be discussed below, his
app oxima ion, while usually leading o accep able spec a, leads o some undamen al
p oblem. Speci ically he sum ule o he oscilla o s eng h and o he o a ional
s eng h in chi al agg ega es[82] a e b oken. In he ollowing, we will see how hese
issues can be sol ed elaxing he HL app oxima ion.
2.3.1 The model Hamil onian
We conside a one dimensional a ay o Nnon-pola molecules assuming pe iodic bound-
a y condi ions. Each molecule is ei he in he g ound, |gio exci ed |eis a e, he wo
s a es being sepa a ed by an ene gy ~ω0. An in e nal ib a ional coo dina e ˆqiis in-
oduced pe molecule and he wo elec onic s a es a e assigned ha monic po en ial
ene gy su aces wi h he same equency, ω , bu displaced minima. The s eng h o he
elec on- ib a ion coupling is measu ed by he ib a ional elaxa ion ene gy, λ, ha ,
being ela ed o he Huang-Rhys ac o , S=λ/~ω , can be ex ac ed om he analysis
o he abso p ion o luo escence bandshape o he isola ed dye in solu ion.[25] The
elec os a ic in e ac ion be ween wo molecules a si es iand j eads:[50]
Ji,j =µ2
4πεd3
ij
Dij,(2.21)
whe e Dij is a geome ical ac o ha only depends on he ela i e o ien a ion o he
ansi ion dipole momen s on si es iand j, while dij is he dis ance be ween he wo
molecula si es. Dij can assume posi i e alues ( epulsi e in e ac ions) o nega i e alues
(a ac i e in e ac ions). In he ollowing, we will add ess one-dimensional molecula
agg ega es wi h one molecule pe uni cell. We will impose pe iodic bounda y condi ions
as o main ain ansla ional symme y. In hese condi ions, in e molecula in e ac ions
only depend on he ela i e dis ance be ween molecules and we de ine Jm=Ji,i±m. Two
ex eme cases will be conside ed: (a) nea es -neighbo in e ac ions wi h J1=Jand
Jm= 0 o m > 1; (b) unsc eened long- ange Coulomb in e ac ions wi h J1=Jand
Jm=J[sin(π/N)/sin(mπ/N)]3 o m > 1, wi h Nbeing he numbe o molecula uni s.
66
2.3 Agg ega es o non-pola molecules
In ei he case a single pa ame e , J, measu ing he s eng h o he nea es -neighbo
in e ac ion, ully de ines he model.
The magni ude o µ is expe imen ally accessible om he oscilla o s eng h o he
g→e ansi ion measu ed o he isola ed dye in solu ion:
ge =2
3
me
~e2ω0µ2
,(2.22)
whe e meis he elec on mass, e he elec on cha ge and ω0 he equency o he g→e
ansi ion. Al e na i ely, he ansi ion dipole momen can be ob ained om quan um
chemical calcula ions, wi h he added alue o ge ing in o ma ion abou he dipole
momen o ien a ion wi h espec o he molecula ame.
Wi h hese de ini ions, he Hamil onian o a linea a ay o Nmolecules eads:
ˆ
H=X
ihE−λ(ˆa†
i+ ˆai)iˆni+~ω X
iˆa†
iˆai+1
2
+X
m
Jm(ˆ
b†
iˆ
bi+m+h.c.) + X
m
Jm(ˆ
b†
iˆ
b†
i+m+h.c.),(2.23)
whe e iand j un on he Nmolecula si es. The ope a o s ˆa†
i, ˆaia e he boson ope a o s
associa ed wi h he ha monic oscilla o on si e i. The ope a o ˆ
b†
ic ea es an exci a ion
on si e i, by u ning he molecule om s a e |gi o |ei, while ˆ
bides oys he exci a ion
( hese ope a o s obey Paulion algeb a as discussed in Sec ion 2.2).
The i s e m in he abo e Hamil onian desc ibes he molecula p oblem, whe e E
is he e ical exci a ion ene gy o he molecule in he agg ega e. I may di e om
~ω0, he ansi ion ene gy in he isola ed molecule, due o local ield e ec s, bu we will
neglec hese co ec ions in he ollowing, se ing E=~ω0. The las wo e ms accoun
o in e molecula in e ac ions: he same in e ac ion Jmis esponsible o he hopping
o he exci a ion om si e i o i+m(and ice e sa) and o he simul aneous c ea ion
(des uc ion) o wo exci a ions on si es iand i+m. As men ioned abo e, he hopping
e m mixes s a es wi h he same numbe o exci ons, i.e. s a es ha ing he same diagonal
ene gy, while he las e m in he abo e Hamil onian desc ibes he in e ac ion among
s a es whose ene gy di e s by 2Eand, in he HL app oxima ion, i is neglec ed. O
cou se, he HL app oxima ion wo ks well o J2E.
Be o e closing his Sec ion, i is ins uc i e o compa e he Hamil onian o non -
pola dyes in Eq. 2.23 wi h he Hamil onian desc ibing pola and pola izable dyes in
67
CHAPTER 2: MOLECULAR AGGREGATES
2.3.4 Tes ing app oxima ion schemes
Exac diagonaliza ion app oaches a e limi ed o small agg ega es, up o 7 si es o he
comple e model and up o 10 si es in he HL app oxima ion. The elec onic basis is
compa a i ely small, g owing wi h Nin he HL app oxima ion and as 2Nin he comple e
model. Indeed he basis blows up because o he ib a ional s a es: accoun ing o jus
h ee ib a ional quan a pe si e would mul iply by a 3N ac o he basis dimension. I
is he e o e e y impo an o discuss app oxima ion schemes o cu he basis dimension
and pa icula ly so o he ib a ional s a es. Indeed he HL elec onic basis is al eady
e y small, while we al eady discussed how he elec onic basis in he comple e model
can be educed by ixing a maximum numbe o exci ons (Me= 3 seems o wo k p e y
well in mos cases o in e es in his s udy, e en i his app oxima ion is un enable o
clus e s o pola and pola izable dyes in medium o s ong coupling egimes.
Recen ly,[55] discussing J-agg ega es o pola dyes, we ealized ha o la gely delo-
calized exci ons only he ib a ional modes in he close p oximi y o he cen e o he
B illouin zone a e e ec i ely coupled o he elec onic deg ees o eedom so ha , ins ead
o accoun ing o Nlocal ha monic oscilla o s, easonable esul s a e ob ained accoun -
ing o he single oscilla o wi h q= 0. This o cou se leads o an eno mous educ ion
o he basis dimension. The Hamil onian in Eq. 2.26 shows ha accoun ing o jus he
q= 0 mode one ob ains a simila coupling Hamil onian as o he isola ed molecule, bu
wi h he s eng h o he coupling educed o λ/√N. As a esul , in his app oxima ion
he same bandshape is calcula ed o J and H agg ega es, as shown in he op panels o
Fig. 2.13, whe e we show abso p ion spec a calcula ed in he HL app oxima ion o he
same model pa ame e s as in Fig. 2.11.
A ull decoupling o he q= 0 ib a ional mode is expec ed in he in ini e chain limi
and a washing ou o he ib onic s uc u e in ei he J o H agg ega es o in ini e size.
Adding he wo nea es modes o he q= 0 mode in he B illouin zone (middle panels
o Fig. 2.13) imp o es he ag eemen and adding 2 mo e modes ( o a g and o al o 5
delocalized ib a ions, bo om panels) gi es e y good esul s o J agg ega es and an
accep able ag eemen o H-agg ega es. Cu ing ib a ional modes in he Fou ie space
wo ks in p inciple o he comple e as well as o he s anda d exci on model, bu , apa
om he simples case whe e only he q= 0 mode is accoun ed o , he app oach is
74

2.3 Agg ega es o non-pola molecules
di icul o implemen in he comple e model. Mo eo e his app oxima ion is expec ed
o wo k well o la gely delocalized exci ons. O cou se o localized exci ons o in he
p esence o diso de he app oach could only wo k i many modes (possibly all) in he
ecip ocal space a e in oduced, making he app oxima ion useless.
A use ul and widely adop ed app oach o educe he ib a ional space is he so-
called ew-pa icle app oxima ion,[83, 84] ha has been ex ensi ely applied by Spano[46,
47] in he 2-pa icle app oxima ion (2PA) o 3-pa icle app oxima ion (3PA) la o s.
I wo ks in he eal space, so ha i does no equi e a symme ic o o de ed sys em,
bu only applies in he HL app oxima ion whe e all ele an basis s a es ha e a single
exci on. In he 2PA app oxima ion, he basis is cu imposing a maximum numbe neo
ib onic exci a ions on he elec onically exci ed s a es (no ice ha hese ib onic s a es
e e o he displaced ha monic oscilla o as ele an o he elec onically exci ed s a e).
Mo eo e a maximum numbe o ib a ional quan a n can be p esen in jus a single
addi ional si e, di e en om he si e bea ing he exci on.
In he h ee pa icle app oxima ion (3PA), one accoun s o ib a ional exci a ions
occu ing on up o wo si es. Two di e en app oxima ion schemes a e possible o bo h
2PA and 3PA, a small- ange (s ) scheme, whe e ib a ional exci a ions a e only allowed
in he nea es si es o he si e bea ing he exci on (Fig. 2.15), o a long- ange (l ) scheme,
whe e ib a ional si es can be sp ead all o e he agg ega e. O cou se he 2PA-s o
he 3PA-s only apply when he exci on model accoun s o nea es neighbo in e ac-
ions, while one mus eso o l -schemes when accoun ing o long- ange elec os a ic
in e ac ions.
To be speci ic, he 2PA basis se is:
|ψ2PAi=|n, ˜ν, νli, l 6=n, (2.32)
whe e nma ks he si e whe e he exci on esides and ˜νcoun s he numbe o ib a ional
quan a in he displaced oscilla o associa ed wi h he same si e. The numbe s νl6=ncoun
he ib a ional quan a in he undisplaced ha monic oscilla o on si e l. In he s la o
o 2PA, l=n±1, while in he l la o , lcan assume any alue di e en om n. The
diagonal ene gy o he 2PA s a es is easily calcula ed as ~ω0+ (˜ν+νl)~ω . O diagonal
75
CHAPTER 2: MOLECULAR AGGREGATES
ma ix elemen s, accoun ing o he in e ac ion be ween di e en si es, a e:
hn, ˜ν, νl|ˆ
H|m, ˜µ, µki=J ˜ν,µn ˜µ,νmY
i6=n,m
δνi,µi,(2.33)
whe e ˜
i,j is he F anck-Condon ac o measu ing he o e lap be ween he ib a ional
le el o exci ed s a e ˜
iand ib a ional le el o g ound s a e j. Finally, he ansi ion
dipole momen is calcula ed as ollows:
µ ans
i=hG|ˆµ|ψii=X
n,˜ν
cn,˜νhG|ˆµ|n, ˜ν, 0i=X
n,˜ν
cn,˜νµ0 ˜ν,0.(2.34)
Mo ing o he 3PA, he ele an basis se eads |ψ3P Ai=|n, ˜ν, νl, νl0i,l, l06=n.
Acco dingly, he diagonal ene gy is ~ω0+ (˜ν+νl+νl0)~ων, while he Hamil onian o -
diagonal ma ix elemen s a e:
hn, ˜ν, νl, νl0|ˆ
H|m, ˜µ, µk, µk0i=J ˜ν,µn ˜µ,νmY
i6=n,m
δνi,µi.(2.35)
Fig. 2.16 compa es abso p ion spec a calcula ed ia exac diagonaliza ion and wi h
he 2PA-s and 3PA-s o an agg ega e o 10 molecules, desc ibed by he s anda d exci-
on model, wi h inc easing s eng h o nea es -neighbo in e ac ions, J. Fo J-agg ega es
he 2PA and 3PA app oxima ions wo k p e y well up o medium-la ge in e ac ions, bu
o H-agg ega es, he app oxima ion is poo al eady o in e ac ions o medium s eng h.
Simila esul s hold ue o emission spec a in Fig. 2.17. Mo ing o 2PA-l o 3PA-l
does no change he pic u e, as expec ed.
The l ex ensions o he 2PA and 3PA app oaches has o be in oked o sys ems whe e
long- ange Coulomb in e ac ions a e accoun ed o . Howe e he 3PA-l basis is e y
la ge, making i impossible o deal wi h agg ega es wi h mo e han 6 si es. The e o e
Fig. 2.18 and Fig. 2.19 compa e exac and 2PA-l esul s o abso p ion and emission
spec a, escpec i ely, o 10 si e agg ega es wi h long- ange in e molecula in e ac ions.
76
2.3 Agg ega es o non-pola molecules
Figu e 2.7: J agg ega e, V=−1.6,  = 0.4, ω = 0.17: op and bo om panel show
calcula ed abso p ion and luo escence spec a. In ensi ies pe molecules a e epo ed in
a bi a y uni s. Le panels e e o a sys em wi h z0= 1.5, co esponding o an ionici y
o he isola ed dye ρ= 0.09 ha inc eases in he m app oxima ion o ρ= 0.10. Middle
panels: z0= 1.0, gas phase ρ= 0.16, m ρ= 0.50. Righ panels: z0= 0.5, gas phase
ρ= 0.33, m ρ= 0.90. Fo N= 4 he ull elec onic basis is conside ed, Ne= 4. Fo
N= 6 only con e ged esul s a e shown wi h Ne= 4
77
CHAPTER 2: MOLECULAR AGGREGATES
Figu e 2.8: J agg ega e, V=−2.0,  = 0.4, ω = 0.17: op and bo om panel show
calcula ed abso p ion and luo escence spec a. In ensi ies pe molecules a e epo ed in
a bi a y uni s. Le panels e e o a sys em wi h z0= 1.5, co esponding o an ionici y
o he isola ed dye ρ= 0.09 ha inc eases in he m app oxima ion o ρ= 0.11. Middle
panels: z0= 1.2, gas phase ρ= 0.12, m ρ= 0.50. Righ panels: z0= 1.0, gas phase
ρ= 0.16, m ρ= 0.87.Fo N= 4 he ull elec onic basis is conside ed, Ne= 4. Fo
N= 6 only con e ged esul s a e shown wi h Ne= 4
78
2.3 Agg ega es o non-pola molecules
Figu e 2.9: H agg ega es wi h N= 6. Top panel show weak-coupling esul s, V= 1
o he same alues o model pa ame e s as in Fig. 2.4; bo om panels show esul s
o s ong coupling, V= 2, o he same pa ame e s as in he bo om panels o Fig.
2.6. In all panels blue lines show con e ged esul s o he o al Hamil onian, dashed
black cu es show esul s ob ained neglec ing he non-Condon elec on- ib a ion cou-
pling e m, con inuous black lines show esul s o he exci on model, i.e. supp essing
all ul aexci onic e ms in he Hamil onian.
79

CHAPTER 2: MOLECULAR AGGREGATES
Figu e 2.10: J agg ega es wi h N= 6. Top panel show weak-coupling esul s, V=−1
o he same alues o model pa ame e s as in Fig. 2.5; bo om panels show esul s o
s ong coupling, V=−2, o he same pa ame e s as in Fig. 2.8. In all panels blue
lines show con e ged esul s o he o al Hamil onian, dashed black cu es show esul s
ob ained neglec ing he non-Condon elec on- ib a ion coupling e m, con inuous black
lines show esul s o he exci on model, i.e. supp essing all ul aexci onic e ms in he
Hamil onian.
80
2.3 Agg ega es o non-pola molecules
Figu e 2.11: Abso p ion spec a calcula ed o he comple e Haml onian in Eq.2.23 only
accoun ing o nea es neighbo in e ac ions. Resul s a e ob ained o N=6, ~ω0= 2.0 eV,
λ=0.17 eV, ~ω =0.17 eV and |J|=0.255 eV. The ib a ional basis is unca ed se ing he
maximum numbe o o al ib a ional quan a M =6. The elec onic basis is unca ed
ixing he maximum numbe o exci ons Me o a alue anging om 1 o 4. Resul s o
Me=1 coincide wi h hose ob ained in he HL app oxima ion. Top and bo om panels
e e o H-agg ega es (J > 0) and J-agg ega es (J < 0), espec i ely. In bo h panels
he dash-do ed line shows he spec um ele an o non-in e ac ing molecules (i.e., he
monome limi ).
81
CHAPTER 2: MOLECULAR AGGREGATES
Figu e 2.12: Emission spec a calcula ed o he he same sys em as in Fig. 2.11
82
2.3 Agg ega es o non-pola molecules
Figu e 2.13: Calcula ed abso p ion spec a o J and H agg ega es (le and igh panels,
espec i ely) calcula ed in he HL app oxima ion o he same model pa ame e s as in
Fig. 2.11. Resul s e e o agg ega es o 10 molecules, black lines show nume ically exac
esul s, ob ained wi h M = 4, magen a lines show esul s calcula ed only accoun ing
o he q= 0 mode ( op panels), q∈ {−π
5,0,π
5}(middle panel), q∈ {−2
5π, −π
5,0,π
5,2
5π}
(bo om panels).
83
CHAPTER 2: MOLECULAR AGGREGATES
Figu e 2.21: C ys al s uc u e o BF38: (a) laye ed s uc u e; (b) highligh s o he
wo di ec ions o in e ac ions inside a single laye . S uc u es o BF31 and GK08 a e
comple ely analogous.
(kx, ky), whe e kxand kya e de ined as:
kx=2π
Nx
sxky=2π
Ny
sy(2.36)
Nx(y) ep esen s he numbe o molecules in he x(y) dimension, and sx(y)is anologous
o i s monodimensional coun e pa . The Hamil onian is w i en in he ecip ocal space,
as in Eq.2.26. The Paulion c ea ion and annihila ion ope a o s a e:
ˆ
bK=ˆ
bkxky=1
pNxNyX
m,n
eimkxeinkyˆ
bmn (2.37)
ˆ
b†
K=ˆ
b†
kxky=1
pNxNyX
m,n
eimkxeinkyˆ
b†
mn (2.38)
whe e i(j) uns on he Nx(Ny) molecules. The exci on Hamil onian in he ecip ocal
space eads:
H2D=X
K
ˆ
b†
KbKE+ 2cos(kx)Jx+ 2cos(ky)Jy+~ω X
Qˆa†
QˆaQ+1
2
−λ
√NX
K,Q
(a†
Qb†
KbK+Q+h.c.) (2.39)
90

2.3 Agg ega es o non-pola molecules
whe e b†
Kc ea es an exci on wi h wa e ec o ~
K, whose ene gy is ep esen ed by he
quan i y in squa e pa en hesis, N ep esen he o al numbe o molecules o ming he
agg ega e (N=Nx·Ny) and ~
Q= (qx, qy) ep esen s he wo dimensional coun e pa o
he ecip ocal space ib a ions desc ibed in Eq. 2.26. Selec ion ules o abso p ion and
emission a e he same as in one dimension. Abso p ion is only possible om he g ound
s a e o a o al-symme ic s a e ( ~
K= (0,0)), while he symme y o he luo escen s a e
( he lowes exci ed s a e) depends on he sign o he in e ac ions. Speci ically, he e a e
ou possibili ies o he wa e ec o o he luo escen s a e, as shown in Table 2.1. whe e
JxJy~
K
+ + (0,0)
+ - (0, kmax
y)
- + (kmax
x,0)
- - (kmax
x, kmax
y)
Table 2.1: Possible alues o ~
K o he lowes exci ed s a e in bidimensional agg ega es.
kmax
i, he highes alue o he wa e ec o , depends on he numbe o molecules in he
idimension (Ni):
kmax
i=


π, i Nie en
Ni−1
Niπ, i Niodd.
(2.40)
Reliable alues o he Jxand Jyin e ac ions a e ex ac ed in wo s eps. Fi s we
use TD-DFT (B3LYP unc ional, 6-31g(d,p) basis se ) o calcula e he magni ude and
o ien a ion o he ansi ion dipole momen o he isola ed molecules (see Table 2.2).
Then, c ys allog aphic da a a e exploi ed o calcula e he in e ac ions be ween ansi ion
dipole momen s on di e en molecules, in he dipola app oxima ion:
J12 =1
4πε0η2 3
12
[(~µ1·~µ2)−3
2
12
(~µ1·~ 12)(~µ2·~ 12)]|2(2.41)
Speci ically, we calcula ed in e ac ions Jbe ween all nea es neighbo couples in he
c ys al. Fo all he h ee sys ems, wo majo in e ac ion a e ound as epo ed in Table
2.3 o η2= 2.
91
CHAPTER 2: MOLECULAR AGGREGATES
Molecule B igh Exc. S a e Ene gy (eV) T ans. Dipole Momen (D)
BF31 3.27 5.056
BF38 2.69 5.259
GK08 3.42 4.578
Table 2.2: Exci ed s a e ene gy and ela i e ansi ion dipole momen magni ude o he
3 molecules unde in es iga ion.
Molecule Jx(eV) Jy(eV)
BF31 0.0059 -0.030
BF38 0.053 0.028
GK08 0.061 0.021
Table 2.3: In e ac ion ene gies calcula ed o he h ee molecula c ys als
Figu e 2.22: No malized emission spec a calcula ed o he a ge molecules in THF
solu ion.
The single laye s a e hen modeled using a 5 ×3 bidimensional agg ega e o a
g and o al o 15 molecules. Vib a ional ene gies ω and elec on-phonon couplings λ
a e ex ac ed om expe imen al monome emission spec a (shown in Fig. 2.22). The
di e ence in ene gy be ween he 0 −0 and 0 −1 ib onic peaks gi es he ib a ional
ene gy, while he a io be ween 0 −0 and 0 −1 peak in ensi ies is p opo ional o he
92
2.3 Agg ega es o non-pola molecules
coupling λ. In ac , ecalling he exp ession o he F anck-Condon coe icien s:
h0|ni2=Sne−S
n!wi h S=ε
ω
,(2.42)
and knowing ε =λ2/ω , he elec on-pho on coupling is ob ained as:
λ=√Sω wi h S=h0|1i2
h0|0i2.(2.43)
In Fig. 2.23, 2.24 and 2.25, expe imen al abso p ion and emission spec a a e compa ed
o calcula ed spec a. The monome abso p ion ene gies (Ein Eq. 2.39) a e selec ed
Figu e 2.23: Compa ison be ween expe imen al measu ed spec a, in g een, and cal-
cula ed spec a om Hamil onian in Eq. 2.39, in iole , o he b 31 c ys al. Le :
abso p ion spec a; igh : emission spec a.
in o de o ma ch he posi ion o expe imen al c ys al abso p ion peaks. Fo bo h BF38
and GK08 he compa ison be ween calcula ed and expe imen al spec a show a ema k-
ably good ma ch, bo h in abso b ion and emission. The bandshapes o he ansi ions,
as well as he S okes shi is sa is ac o ily ep oduced. The esul s acqui e addi ional
alue emembe ing ha he pa ame iza ion o he Hamil onian is ob ained, excep o
he exci on ene gies, using elec on-phonon couplings di ec ly ex ac ed om monome
spec a in solu ion and e alua ing he Jxand Jyin e ac ions om ab-ini io calcula ions
and expe imen al c ys allog aphic da a. On he con a y he modeliza ion o BF31,
93
CHAPTER 2: MOLECULAR AGGREGATES
Figu e 2.24: Compa ison be ween expe imen al measu ed spec a, in g een, and cal-
cula ed spec a om Hamil onian in Eq. 2.39, in iole , o he BF38 c ys al. Le :
abso p ion spec a; igh : emission spec a.
Figu e 2.25: Compa ison be ween expe imen al measu ed spec a, in g een, and cal-
cula ed spec a om Hamil onian in Eq. 2.39, in iole , o he GK08 c ys al. Le :
abso p ion spec a; igh : emission spec a.
while leading o an accep able ma ch in abso p ion spec um, ail in he desc ip ion o
luo escence, ega ding bo h posi ion and shape o he spec um. The p oblem wi h his
sys em can be due o he p esence, in he wo di ec ion, o a posi i e and a nega i e
94
2.3 Agg ega es o non-pola molecules
in e ac ion, which leads o a mo e cumbe some landscape ha equi e bigge dimensions
o he simula ed agg ega e.
95

CHAPTER 2: MOLECULAR AGGREGATES
2.4 Conclusions
In his chap e we ha e discussed e ec i e models o molecula agg ega es. Fi s we
ha e add essed linea agg ega es o pola and pola izable molecules, ex ending a p e ious
wo k[69, 62] o accoun o elec on- ib a ion coupling. A cle e choice o he basis,
de ined on he adiaba ic elec onic s a es in he m app oxima ion and accoun ing o
he ib a ional displacemen ia a Lang-Fi so ans o ma ion, allows o la gely educe
he basis dimension, so ha , also exploi ing ansla ional symme y, we a e able o ea
ai ly la ge sys ems, wi h up o 6 molecules. Apa om his impo an echnical esul ,
he p ope choice o he basis amoun s o a p ope choice o he e e ence s a e. The
appa en ly e a ic beha io obse ed o agg ega es o pola and pola izable molecules,
whe e ed-shi s a e some imes obse ed o H-agg ega es and blue-shi s a e some imes
obse ed o J-agg ega es,[66] ac ually esul s om a poo choice o he e e ence s a e.
Exci onic (and ul aexci onic) e ec s, including shi s, mus be e alua ed agains he
ansi ion equencies ob ained in he m app oxima ion, i.e. aking in o accoun he
la ge a ia ion o he na u e o he pola izable dye when inse ed in a la ice o pola
dyes. Once he p ope e e ence is selec ed, J and H agg ega es always gi e ise o ed and
blue-shi ed abso p ion band. Simila ly, anomalous e ec s on bandshapes and in ensi ies
a e elie ed when he p ope e e ence s a e is conside ed. Ou analysis demons a es
ha , o agg ega es o pola and pola izable dyes, when he p ope e e ence is aken,
he exci onic app oxima ion wo ks easonably a leas in he weak coupling egime.
Ul aexci onic e ec s a e impo an in he s ong couplig egime and pa icula ly so in
J-agg ega es, whe e mul iexci onic s a es become p ominen .[69, 62]
Fo agg ega es o non-pola dyes we we e able, exploi ing symme y and a cle e
compu a ional implemen a ion, o ea ai ly la ge sys ems, as needed o alida e se -
e al app oxima ion schemes. Mo e impo an ly we we e able o add ess a undamen al
p oblem o he exci on model ha gene a es spec a ha do no obey he sum ule o
he oscilla o s eng h. This is sa ely asc ibed o he HL app oxima ion. We demon-
s a ed ha he HL app oxima ion, he main app oxima ion adop ed in he exci on
model o agg ega es o non-pola dyes, leads o good es ima es o ansi ion ene gies
because o a cancella ion o e o s, while al eady in he weak in e ac ion egime, he
HL app oxima ion leads o o e es ima ed abso p ion (and luo escence) in ensi ies o
96
2.4 Conclusions
H-agg ega es and unde es ima ed in ensi ies o J agg ega es. O cou se o la ge in e -
ac ions, ul aexci onic e ec s a e also ecognized in he equencies o he bands.
Ha ing demons a ed ha he exci on model leads o easonable esul s o weak
in e ac ions, we inally applied he model o desc ibe op ical spec a o h ee c ys als o
o ganic dyes, wi h good esul s, pa icula ly in iew o he minimal numbe o adjus able
model pa ame e s.
97
Chap e 3
Chi al agg ega es o αand
β-dicyanos ylbenes: chi op ical
p ope ies
3.1 In oduc ion
Objec s ha canno be supe imposed o hei mi o image, echnically objec s whose
symme y g oup does no con ain any imp ope axis (including S1 he mi o plane
and S2 he in e sion cen e ), a e called chi al ( om he g eek wo d o hand). Chi-
al sys ems show special p ope ies when in e ac ing wi h ligh . A he linea o de
(weak elec omagne ic ields) he Op ical Ro a o y Dispe sion (ORD)[86, 87] measu es
he ( equency-dependen ) di e ence be ween he e ac i e index o he le and igh
ci cula ly pola ized ligh and Ci cula Dich oism (CD)[88, 89] measu es he di e ence
in he co esponding ex inc ion coe icien s.[90, 91, 48] O cou se ORD and CD spec-
a a e ela ed h ough in eg al exp essions simila o he K ame s K ¨onig ela ions.[92]
Chi ali y and chi op ical ac i i y a e obse ed in in insically chi al molecules, including
molecules wi h chi al cen e s as well as chi al s uc u es like helicenes. Howe e , non-
chi al molecules may gi e ise o chi al esponses is some condi ions as (a) unde he
99
CHAPTER 3: CHIRAL AGGREGATES OF αAND β-DICYANOSTYLBENES
and β-DCSB molecules wi h non-chi al subs i uen s[99, 100] and s a ou MD simula-
ion om a con igu a ion ob ained a anging he eigh molecules o he αand β-DCSB
de i a i es mimiking s acks ex ac ed om he ele an c ys al s uc u es, as shown in
Fig. 3.6. We no ice ha , since he c ys al s uc u es a e ob ained o non-chi al sys ems,
he s a ing con igu a ion is non-chi al.
Figu e 3.6: Ini ial con igu a ions o α- and β-DCSB.
Molecula dynamics simula ions a e hen un wi h a i s minimiza ion s ep, ollowed
by a 30 ns NPT (see Appendix B) equilib a ion (needed o allow molecules o a ange
in he he modynamically s able s uc u e) and a inal 200 ns NPT main un wi h a
1 ps imes ep. In he poo sol en (wa e ) he molecules eo ganize o minimize he
in e ac ion wi h he sol en . The cen al ch omopho ic uni s in e ac ia π-πs acking
while he side chi al chains, wi h asymme ic in e ac ions, impa an o e all o sion
o he whole sup amolecula agg ega es. The high lexibili y o he molecules oge he
wi h he la ge numbe o uni s, esul in a somewha messy con o ma ional landscape.
Ne e heless, a p e e en ial packing is clea ly de ec ed o he 4 in es iga ed sys ems. In
Fig. 3.7 he mos ep esen a i e s uc u es a ising o each o he long ajec o ies a e
p esen ed.
All sys ems a e cha ac e ized by a well-de ined helical s uc u e, shaped by in e ac-
ions o side pendan s. The RR enan iome s (ei he αo β-DCSB) show a le -handed
106

3.4 Abso p ion and CD spec a o DCSB agg ega es
helix o ganiza ion, while SS enan iome s o m a igh handed helix. The MD esul s,
Figu e 3.7: Mos ele an s uc u es o he 4 sys ems unde in es iga ion.
showing ha αand β-DCSB subs i u ed wi h he same chi al pendan agg ega e in
sup amolecula s uc u es wi h he same handedness bu showing opposi e CD signal
is a i s con i ma ion o he alidi y o he new chi ali y ule, s a ing ha he sign o
CD spec a depends no only on he chi ali y o he sys em, bu also on he sign o
in e molecula in e ac ions. Howe e , mo e eliable esul s and a be e unde s anding
o he phenomenon can be ob ained ia a de ailed calcula ion o abso p ion and CD
spec a o he agg ega es.
3.4 Abso p ion and CD spec a o DCSB agg ega es
3.4.1 The model
To model ou agg ega e we adop he s anda d exci on model o non-pola dyes (i.e.
we impose he HL app oxima ion, see Chap e 2) and neglec he coupling o molecula
ib a ion. As discussed p e iously, he exci on model gi es accep able esul s o op ical
spec a o non-pola dyes and, o la gely diso de ed sys em, he in o ma ion abou he
ib onic s uc u e is smea ed ou by inhomogeneous b oadening e ec s.
107
CHAPTER 3: CHIRAL AGGREGATES OF αAND β-DICYANOSTYLBENES
Wi h hese app oxima ions, he exci onic Hamil onian o a gene ic agg ega e o N
iden ical molecules eads:
ˆ
H=X
n
E0|nihn|+X
nm
Vnm|nihm|(3.1)
whe e |niiden i ies he basis s a e in which he n h molecule is in i s exci ed s a e
(|g1, g2. . . en. . . gNi), wi h ene gy E0, and Vnm is he dipole-dipole in e ac ion e m:
Vnm =1
4πε0n2| n,m|2~µ n
0·~µ m
0−3(~µ n
0·~ nm)(~µ m
0·~ nm)
5
n,m
(3.2)
whe e ~
µn
0is he ansi ion dipole momen o he exci a ion on he n- h molecule and
~ nm is he ec o dis ance be ween si es nand m. In he ollowing we will use he molec-
ula ansi ion dipole momen s ob ained by TD-DFT (B3LYP/6-31G**) as discussed in
Sec ion 3.3, ancho ed o he molecula si e.
The diagonaliza ion o he exci onic Hamil onian on he Nbasis gi es he exci onic
eigens a es accoun ing o he linea combina ion o singly exci ed s a es. T ansi ion
dipole momen , om he g ound s a e o he keigens a e, ~µ k
, is ob ained as linea
combina ions o molecula ansi ion dipole momen s:
~µ k
=X
n
~µ kn
=X
nhk|ni~µ n
0(3.3)
whe e k uns on he eigens a es and ~µ n
0is he dipole momen associa ed o he |gi →
|ei ansi ion in he n h molecule. The abso p ion spec um is calcula ed assigning a
Gaussian lineshape o each ansi ion:
A(ω) = ~ωX
k~µ k

2e
(~ω−Ek)2
2σ2(3.4)
whe e Ekis he ene gy o he ansi ion om he g ound o he keigens a e.
In he p oposed model, elec ons a e localized on each molecula uni . In his case, as
i s demons a ed in a classical wo k by Condon[91], CD spec a can be calcula ed om
he knowledge o ansi ion dipole momen s. In ac , in his limi , ansi ion magne ic
dipole momen s, needed o calcula e CD spec a (see Chap e 4 o u he de ails), can
be exp essed in e ms o ansi ion dipole momen s. Following Condon we de ine he
108
3.4 Abso p ion and CD spec a o DCSB agg ega es
o a ional s eng h o each ansi ion om he g ound o he k-eigens a e as
Rk=−EkX
nm
~ nm ·(~µ kn
×~µ km
) (3.5)
whe e ~µ kn
is he dipole momen o he |0i → |ki ansi ion ela i e o molecule n.
Replacing Eq. 3.3 in Eq. 3.5, we ob ain he explici exp ession o Rk:
Rk=−EkX
nm
~ nmhn|kihm|ki·(~µ n
0×~µ m
0) (3.6)
CD spec a, measu ing he di e ence be ween exc inc ion coe icien o le and igh
ci cula ly pola ized ligh , is simply ob ained assining a Gaussian lineshape wi h wid h
σ o each ansi ion:
∆(ω)∝X
k
Rke
(~ω−Ek)2
2σ2(3.7)
3.4.2 Calcula ed Spec a
The ou pu o MD simula ions is a collec ion o con igu a ions. Assuming ha he
ansi ion dipole ec o s calcula ed o he isola ed molecules s ay cons an du ing he
dynamics wi h espec o he molecula ame o each monome , we can calcula e, o each
con igu a ion o he agg ega e, he elec os a ic in e ac ions en e ing he Hamil onain
(see Eq. 3.2). The alues o E0, co esponding o he ansi ion ene gy o he isola ed
monome , a e ex ac ed om expe imen al alues in Fig. 3.2 and a e se o 3.37 eV
o αde ia i es and 3.10 eV o βde i a i es. The Hamil onian is hen diagonalized
and ansi ion ene gies, ansi ion dipole momen s and o a ional s eng hs a e inally
ob ained.
Wi h hese in o ma ion, abso p ion and CD spec a a e calcula ed (Eq. 3.7, se ing
σ=0.07 eV). The esul ing spec a a e hen a e aged o e a la ge po ion o dynamics (60
ns). Resul s o he abso p ion and and CD spec a o he 4 sys ems unde in es iga ion
a e shown in Fig. 3.8 and 3.9.
Calcula ed spec a and expe imen al da a p esen ed in Fig. 3.2 a e gene ally in
good ag eemen . Ro a ional s engh s ex ac ed om ajec o ies mimic accu a ely ex-
pe imen al CD spec a. Speci ically, we ob ain he co ec signs o CD spec a o he
4 simula ions. This esul , oge he wi h he mos ele an s uc u es p esen ed in Fig.
109
CHAPTER 3: CHIRAL AGGREGATES OF αAND β-DICYANOSTYLBENES
Figu e 3.8: Top panels: abso p ion spec a o bo h enan iome s o α-DCSB compa ed
o he monome (black cu e). Bo om panels: co esponding CD spec a.
3.7, demons a e he eliabil y o ou hyb id me hod in he modeliza ion o hese complex
sys ems. Resul s o β-DCSB a e e y good: abso p ion spec a o bo h enan iome s
a e e y simila , com i ming ha he equilib ium geome y is ob ained o bo h species,
and show a e y clea blue-shi wi h espec o he monome . Resul s o α-DCSB a e
less sa is ac o y. While he spec um o he R ena iome has i s main peak sligh ly
ed shi ed wi h espec o he monome , bo h R and S sys ems p esen a no negligible
band a highe ene gy, deno ing he o ma ions o H ype clus e s. Mo e c i ically, he
abso p ion spec a o he wo α-enan iome s di e conside ably, sugges ing ha he MD
simula ion has no ye ully con e ged o he equilib ium. Fo his eason, while esul s
o all sys ems a e p omising, calcula ions on αde i a i es a e s ill on going.
110
3.5 Conclusions
Figu e 3.9: Top panels: abso p ion spec a o bo h enan iome s o α-DCSB compa ed
o he monome (black cu e). Bo om panels: co esponding CD spec a.
3.5 Conclusions
In his Chap e we sugges ed a no el app oach o he spec oscopic cha ac e iza ion o
complex agg ega es in solu ion. We p oposed a hyb id me hod, based on he combina ion
o MD simula ions and exci on models o calcula e abso p ion and ci cula dich oism
spec a o a amily o dicyanos ilbenes (DCSB) de i a i es. In pa icula , we ocused
ou in es iga ion on 4 sys ems: α-DCSB (R and S enan iome s) and β-DCSB (R and
S enan iome s) ha show di e en spec oscopic beha io depending on he na u e o
he ch omopho ic uni and he chi ali y o he side chains. Wi h he use o MD we
s udied he agg ega ion o hese sys ems in solu ion. S a ing om comple ely achi al
a angemen s o monome s, ex ac ed om c ys allog aphic da a on ela ed sys ems, well
de ined clus e s a e ob ained, wi h an o e all helici y ha only depends on he chi ali y
o side chains. Geome ic in o ma ions we e ex ac ed om classic ajec o ies and hen
111

CHAPTER 3: CHIRAL AGGREGATES OF αAND β-DICYANOSTYLBENES
p ocessed in o de o pa ame ize an exci on model. Abso p ion and CD spec a a e
hen calcula ed o he 4 sys ems o in e es , a e aging o e housands o con o ma ions.
We ob ain CD spec a in good ag eemen wi h expe imen al da a. The hyb id echnique
p oposed in his Chap e ep esen s a eliable me hod o he eplica ion o chi op ical
p ope ies in unusual chi al assemblies, whe e he ine in e play be ween ch omopho es
na u e and pe iphe al ligand chi ali y con ibu es o complex ou comes.
112
Chap e 4
Chi al agg ega es o squ aine dyes
4.1 In oduc ion
Squa aines a e a widely in es iga ed amily o o ganic dyes, cha ac e ized by a eso-
nance s abilized s uc u e whe e he cen al squa yl ing, a s ong elec on-accep o
uni , is conjuga ed o wo elec on-dona ing g oups, as in he examples shown in Fig.
4.1. Squa aines ha e in e es ing spec oscopic p ope ies, wi h in ense and sha p abso p-
ion and emission bands in he ed and nea -IR spec al egions and la ge wo-pho on
c oss-sec ions. They a e in es iga ed o applica ions in dye-sensi ized sola cells[3, 4],
in colo ime ic senso s [109] and non-linea op ics.[110] Mo eo e , hei igid conjuga ed
s uc u e allows he o ma ion o s able agg ega es, [111, 112] whose unsusual spec o-
scopic p ope ies u he widen he possible ange o applica ions. The s udy o sel -
o ganiza ion and agg ega ion beha iou o squa aine dyes is a ai ly ho opic in ecen
yea s. [113, 16, 114, 14]
Chi al agg ega es o squa aine dyes we e ob ained ∼15 yea s ago ia he sup amolec-
ula a angmen o squa aine dyes deco a ed a he wo sides by chi al g oups[115, 112].
The chi al sup amolecula a angmen is demons a ed by he obse a ion o cha ac e -
is ic CD spec a o he agg ega e s uc u e, in he egion o he squa aine abso p ion,
CD-silen o he non-agg ega ed dye. New a en ion on chi al squa aine agg ega es
was called by a ecen pape om he g oup o Manuela Schieck [114], showing how
113
CHAPTER 4: CHIRAL AGGREGATES OF SQURAINE DYES
Figu e 4.1: a): example o wo squa aine dyes; b): schema ic ep esen a ion o he
squa aine D-A-D s uc u e.
agg ega es deposi ed on ilms gi e ise o e y la ge CD signals.
In his chap ed we desc ibe an ex ensi e heo e ical wo k ha , de o ed o he analysis
o abso p ion and CD spec a o chi al squa aine agg ega es in solu ion, ep esen s a i s
undamen al s ep o unde s and he beha io o agg ega es deposi ed on ilms. This wo k
is done in collabo a ion wi h expe imen alis s and speci ically wi h esea che s in he
g oup o P o esso s Manuela Schiek (Johannes Keple Uni e si y o Linz, Aus ia) and
A ne L¨u zen (Uni e si y o Bonn, Bonn). Fig. 4.2 (cou esy o PhD s uden Jenni e
Zablocki) shows ep esen a i e abso p ion and CD spec a o agg ega es in solu ion.
Indeed a e y la ge numbe o esul s a e a ailable on agg ega es o squa aine dyes wi h
di e en subs i uen s (Fig. 4.3). Fo all agg ega es abso p ion spec a show agg ega ions
ea u es bo h o he blue and o he ed o he monome band, loca ed a 650 nm. We
will dub he wo ea u es as H and J bands, espec i ely, e en i hei na u e (as i will
be demons a ed below) is di e en .
Ini ially, he wo bands we e asc ibed o wo ea u es associa ed wi h he exci on
spli ing o he monome band. This howe e would equi e a ai ly la ge in e molecula
in e ac ions. Mo eo e , his in e p e a ion con as s sha ply wi h he obse ed CD spec-
a. Indeed i he H and J bands we e he wo ea u es due o he exci on spli ing, one
would ha e obse ed in CD spec a a single bisigna ed signal wi h opposi e sign a he
114
4.2 The h ee s a e model o squa aine dyes
Figu e 4.2: Abso p ion (le ) and CD ( igh ) spec a measu ed o he P oSQ-SS-C10
molecule in me hanol upon addi ion o wa e .
H an J band loca ions. Ins ead, he CD spec um shows wo dis inc bisigna ed signals,
one in he egion o he H and and one in he egion o he J band. The possibili y ha
he H and J bands a e due o he simul aneous p esence o di e en kinds o agg ega es
in he same sample is no e y likely, due o he consis en obse a ion o he wo ea u es
in samples ob ained in di e en expe imen al condi ions and using dyes wi h di e en
subs i uen s. Abso p ion and CD spec a o chi al squa aine agg ega es hen call o a
ca e ul modeliza ion ha will be he opic o his chap e .
4.2 The h ee s a e model o squa aine dyes
The spec oscopic beha iou o quad upola DAD dyes, and speci ically o squa aines,
can be a ionalized adop ing an essen ial s a e model, based on 3 elec onic basis s a es
[116]. The h ee o hogonal s a es a e a neu al s a e |Ni, and wo degene a e s a es,
|Z1iand |Z2i, co esponding o he wo zwi e ionic s uc u es D+A−D and DA−D+.
On his basis, he elec onic Hamil onian is:
ˆ
Hel = 2z0ˆρ−τˆσ(4.1)
115
CHAPTER 4: CHIRAL AGGREGATES OF SQURAINE DYES
Figu e 4.6: Abso p ion ( op-le ) and CD (bo om-le ) spec a o he same sys em in
Fig. 4.5, wi h a ini e il angle α= 25◦.
122

4.4 The ole o in e molecula cha ge ans e in e ac ions
Figu e 4.7: The model dime s udied by Collison and Spano wi h highligh ed in amolec-
ula and in e molecula cha ge ans e s.
4.4 The ole o in e molecula cha ge ans e in e ac ions
In a ecen pape Spano and Collison s udied non-chi al agg ega es o squa aine dyes
cha ac e ized by an abso p ion spec um showing p ominen abso p ion ea u es bo h
o he ed and o he blue o he monome abso p ion spec um.[14]. They asc ibed
his obse a ion o he p esence o a cha ge ans e (CT) in e ac ion be ween adjacen
molecules. This wo k inspi ed us o in es iga e in e molecula CT as a sou ce o he
anomalous spec al ea u es obse ed in ou chi al agg ega es. Speci ically, Spano and
Collison, conside ed a dime s uc u e as in Fig. 4.7, and accoun ed o a CT in e ac ion
be ween he adjacen D and A si es loca ed in he wo di e en molecules. Indeed o he
s uc u es can also be conside ed, including e.g. hose d awn in Fig. 4.5a.
O cou se, when accoun ing o in e molecula CT in e ac ion he basis mus be
enla ged o accoun o s a es wi h cha ge sepa a ion be ween he wo molecules, so ha
in addi ion o he 9 localized s a es in Eq. 4.9, o he s a es en e in o play. Speci ically,
Spano and Collison p oposed o add he s a es ob ained conside ing ha each dye could
123
CHAPTER 4: CHIRAL AGGREGATES OF SQURAINE DYES
be in any o he ollowing s a es:
D+AD |C1i
DAD+|C2i
D+A−D+|C3i(4.10)
DA−D|Ai
wi h Cns a es bea ing a posi i e cha ge (ca ions) and Abea ing a nega i e cha ge
(anion). O cou se dime s a es mus be o e all neu al so ha one adds 6 s a es o he
9 localized s a es in Eq. 4.9, o a g and o al o 15 s a es.
While Spano and Collison de ini ely had he good in ui ion abou he possible ole
o in e molecula CT in squa aine agg ega es, he adop ed basis is a om comple e.
Indeed hey ully dis ega d he elec onic spin deg ees o eedom, basically building a
model ha implici ely ea s elec ons as spinless e mions. This is easily unde s ood
adop ing he alence-bond basis whe e elec ons a e pai ed in single s a es.[123, 124] Fig.
4.8 shows a e y simple example e e ing o a s a e wi h wo zwi e ionic molecules,
indeed a s a e ha is al eady p esen in he localized basis as s a e |Z2Z1i. I CT
in e ac ions a e dis ega ded, elec ons can only be pai ed wi hin each single molecula
uni ( he basis s a es de ined as di ec p oduc o he h ee basis s a e o each dye
cons i u e a comple e basis se ), bu , i CT in e ac ions a e accoun ed o , a second
s a e mus be conside ed o accoun o he wo di e en possibili ies o pai elec ons
in single s a es.[123, 124]
The alence bond basis, cons uc ed accoun ing o he subspace wi h o al spin ze o,
would co espond o he smalles basis o ou sys em. Howe e a p oblem a ises since he
alence bond basis is non-o hono mal, making he p ocedu e o calcula e obse ables
and ansi ion dipole ai ly cumbe sone. We he e o e adop a simple app oach, using
he so-called eal-space basis, whe e elec ons a e accomoda ed in he di e en D/A si e,
selec ing only s a es wi h Sz= 0, whe e Szmeasu es he o al spin componen along he
z di ec ion. The eal space basis is la ge han he alence bond-basis, bu , in iew o
he ai ly complex calcula ions equi ed o CD spec a (see Sec ion 4.7) we s ick on i
o ou calcula ions. Speci ically, he eal space basis o a squa aine dime comp ises 53
basis s a es, ep esen ed in he bi - ep esen a ion by in ege numbe s, as shown in Fig.
124
4.4 The ole o in e molecula cha ge ans e in e ac ions
Figu e 4.8: a) Example o bi-zwi e ionic s a e as p oposed by Spano and cowo ke s; b)
When in e molecula cha ge delocaliza ion is aken in o accoun , wo single s a es a e
needed o ep esen he same cha ge dis ibu ion.
4.9
To p oceed, we mus de ine he Hamil onian accoun ing o in a and in e molec-
ula CT and o elec os a ic in e molecula in e ac ions. We p opose a Hubba d-like
Hamil onian as ollows:
ˆ
H=X
mX
i
εiˆnm
i+U
2X
mX
i
ˆnm
i(ˆnm
i−1) + 1
2X
mn X
ij
Vmn
ij ˆqm
iˆqn
j(1 −δij)
− X
mX
i6=jˆ
bm
ij +ˆ
b†m
ij −βX
m,m+1 X
ij ˆ
bm
ij +ˆ
b†m
ij (4.11)
whe e ˆnm
icoun s he elec ons on si e i(= 1,2,3) o molecule m, ˆqm
imeasu es he on-
si e cha ge wi h ˆqm
i= 2 −ˆnm
i o si es D (i=1 o 3) and ˆqm
i=−ˆnm
i o si es A
(i=2). Mo eo e , nm
i=Pσˆc†m
iσ ˆcm
iσ coun s he elec ons on si e io molecule mand
ˆc†m
iσ and ˆcm
iσ c ea e and annihila e and elec on wi h spin σon si e i, m. In amolecula
and in e molecula CT a e de ined by he hopping in eg als and βwi h he hopping
ope a o (an an i-He mi ian ope a o ) de ined as ˆ
bmn
ij =Pσˆc†m
iσ ˆcn
jσ. A a iance wi h he
Hamil onian in Eq. 4.6, in his Hamil onian bo h in e and in amolecula elec os a ic
125
CHAPTER 4: CHIRAL AGGREGATES OF SQURAINE DYES
Figu e 4.9: The 53 diag ams composing he eal s a e basis o a squa aine dime .
The i s column numbe s he diag ams, he second and hi d columns a e he in ege
numbe s ep esen ing each basis s a e in decimal and bina y ep esen a ion, espec i ely.
Each o he 6 si es is ep esen ed by wo digi s in he bina y numbe , 00 s ands o a
accum si e, 01 and 10 o a spin up and a spin down elec on, espec i ely, 11 o a
doubly occupied si e. In o he e ms, o each si e, he i s digi coun s he numbe o
spin down, he second digi coun s he numbe o spin up.
126
4.4 The ole o in e molecula cha ge ans e in e ac ions
in e ac ions a e accoun ed o , wi h Vmn
ij designed as in Eq. 4.8. To ensu e ha he
Hubba d Hamil onian in he abo e equa ion educes o he Hamil onian in Eq. 4.6
o β= 0 we se =τ/√2 and ix on si e ene gies so ha 2z0= 2εi−U−Vwhe e
V=Vmm
12 =Vmm
23 is he modulus o he elec os a ic in e ac ion be ween cha ges on D
and A si es in he same molecule (o cou se ully de ined by he D-A dis ance). Finally,
we se U, he epulsion be ween wo elec ons esiding on he same si e, o a e y la ge
alue as o make i s alue i ele an , and o ensu e ha s a es wi h doubly cha ged si e
D2+ and A2−ha e e y la ge ene gies and can be sa ely dis ega ded.
The abo e Hamil onain is w i en and diagonalized on he eal-space basis o ge
ele an eigens a es and eigen alues. The calcula ion o abso p ion spec a only equi es
he de ini ion o he dipole momen ope a o , ha eads:
ˆ
~µ =−X
mX
i
~ m
iˆnm
i(4.12)
whe e ~ m
iis he posi ion ec o o si e io molecule m. On he chosen basis he dipole
momen ope a o is diagonal. To calcula e abso p ion spec a we calcula e ansi ion
dipole momen s om he g ound o he exci ed s a es and calcula e abso p ion spec a
assigning a Gaussian lineshape wi h wid h σ= 0.08 o each ansi ion:
A(ω)∝~ωX
E|hE|ˆµ|Gi|2e
−(~ω−~ωEG)2
2σ2(4.13)
Fig. 4.10 shows abso p ion spec a calcula ed o wo di e en dime s wi h aligned
molecules wi h exac ly he same model pa ame e s as in Fig4.5 bu in oducing a ini e
β= 0.6. I is clea ha o bo h dime s, accoun ing o in e molecula CT in e ac ions
leads o a spli ing o he band wi h he appea ance o wo ansi ions, one o he blue
and one o he ed o he monome abso p ion band. We obse e ha he abso p ion
spec um a ising om he con igu a ion B is e y simila o expe imen al da a p esen ed
in Fig. 4.2.
To be e unde s and he na u e o he double picked abso p ion spec um we gi e a
close look o he ansion dipole momen s. Speci ically, we calcula e abso p ion spec a
o he dime in Fig. 4.10B accoun ing o pola ized adia ion. We o ien ed he dime
in o de o ha e he molecules aligned along he x-axis and packed along z. The dipole
momen o ansi ions lead by in amolecula CT ( e m τ) will be pa allel o x-axis
127

CHAPTER 4: CHIRAL AGGREGATES OF SQURAINE DYES
Figu e 4.10: Abso p ion spec a (le ) o a model squa aine dime in he same con ig-
u a ions in oduced in Fig. 4.5, compa ed o he monome . The CT in e ac ions aken
in o accoun o each con igu a ion a e depic ed wi h g een dashed lines.
128
4.5 Calcula ion o CD spec a o agg ega es wi h delocalized elec ons
(same di ec ion o he D-A a ms), while in e molecula CT ( e m β) will be conco dan
wi h he packing di ec ion, he z-axis. In Fig. 4.11 he o al abso p ion spe um o
he dime ( op panel) is plo ed oge he wi h abso p ion spec a ob ained wi h x- and
z-pola ized ligh (middle and bo om panels). I is e iden how he in ensi y o bo h
ansi ions is go e ned by in amolecula , a he han in e molecula , cha ge mig a ion.
The wo cha ac e is ic ansi ions seen in CT dime s a ise om he mixing o CT and
exci on s a es. Howe e he CT weigh in he g ound s a e is negligible so ha CT s a es
do no con ibu e o he in ensi y o ansi ions om he g ound s a e. In o he e ms
he in ensi y o he localized ansi ion edis ibu es in wo s a es, he ed and he blue
band.
4.5 Calcula ion o CD spec a o agg ega es wi h delocal-
ized elec ons
Resul s in he p e ious sec ion sugges s ha accoun ing o in e molecula CT in e -
ac ion is p obably he key o unde s and he s ange spec oscopic beha io o chi al
squa aine agg ega es. Howe e , o p oceed we mus a ack a non- i ial p oblem, i.e.
he calcula ion o CD spec a in a molecula agg ega e whe e, due o in e molecula CT
in e ac ions, elec ons a e no localized on he molecula uni s. This makes i impos-
sible o use he app oach desc ibed in Sec ion 4.3 ha , based on he classical Condon
wo k,[91] elies on he de ini ion o he agg ega e dipole momen as he sum o molecula
dipole momen s. This app oach o cou se is inadequa e o delocalized elec ons.
Acco ding o Condon[91], CD spec a can be calcula ed om he o a ional s eng h
associa ed o each ansi ion Ri. The o a ional s eng h plays o CD spec a he
same ole as he oscilla o s eng h in abso p ion spec a, so ha CD spec a can be
calcula ed, once he Ria e known o each ansi ion assigning a spec al bandshape
( ypically Gaussian o Lo en zian) o each ansi ion. I |giis he g ound s a e he
o a ional s eng h o he |gi→| i ansi ion is
R = Imnhg|ˆµ| i·h |ˆ
M|gio(4.14)
whe e ˆµand ˆ
Ma e he elec ic and magne ic dipole ope a o s, espec i ely. While
129
CHAPTER 4: CHIRAL AGGREGATES OF SQURAINE DYES
Figu e 4.11: Top panel: abso p ion spec um o dime in Fig. 4.10B. Middle panel:
abso p ion calcula ed o he same sys em wi h adia ion pola ized along he x-axis.
Bo om panel: abso p ion calcula ed o he same sys em wi h adia ion pola ized along
he z-axis.
he de ini ion o elec ic dipole momen ope a o is s aigh o wa d (see Eq, 4.12), he
de ini ion o he magne ic dipole momen ope a o is mo e delica e. The magne ic dipole
130
4.5 Calcula ion o CD spec a o agg ega es wi h delocalized elec ons
is p opo ional o he o al angula momen um:
ˆ
~
M∝ −ˆ
~
L=−X
k
ˆ
~
lk(4.15)
whe e k uns on all elec ons. The angula momen um o k-elec on is ela ed o i s
linea momen um ˆ
~pkby
ˆ
~
lk=ˆ
~ k׈
~pk(4.16)
Finally he linea momen um can be ob ained as
ˆ
~pk=−i
~hˆ
~µk, Hi(4.17)
The p oblem is ha in a eal-space desc ip ion we canno add ess he p ope ies o a
single elec on. To o e come he p oblem we s a e alua ing he o al linea momen-
um. Fo he sake o cla i y we explici ly w i e he exp ession o only one o he h ee
componen s:
ˆ
Px=−i
~[ˆµx, H] (4.18)
whe e ˆµxis he xcomponen o he o al dipole momen in Eq. 4.12. The dipole momen
ope a o commu es wi h he i s h ee e ms o he Hamil onian in Eq. 4.11, while i
does no commu e wi h he hopping e ms. O e all
ˆ
Px=i
~X
mn X
ij
δi,j±1(xn
j−xm
i)ˆ mn
ij τδmn +β(1 −δmn) (4.19)
whe e he bond eloci y is an an i-He mi ian ope a o
ˆ mn
ij =ˆ
bmn
ij −ˆ
b†mn
ij =ˆ
bmn
ij −ˆ
bnm
ji (4.20)
I we calcula e ˆ
~
R׈
~
P, whe e ˆ
~
R, he global posi ion ope a o , is iden ical o he o al
dipole momen ope a o in Eq. 4.12, we do no ge he o al angula momen um in Eq.
4.15, as ~
R×~
Pdoes con ain bo h one-elec on and wo-elec on e ms. We hen de ine
ˆ
~
Lselec ing ou o he ˆ
~
R׈
~
Pope a o only he one-elec on e ms. Acco dingly:
ˆ
Lx=−i
~X
mn X
ij
δi,j±1(zn
j−zm
i)(ym
iˆ
bmn
ij −yn
jˆ
bnm
ji )
−(yn
j−ym
i)(zm
iˆ
bmn
ij −zn
jˆ
bnm
ji )τδmn +β(1 −δmn) (4.21)
131
CHAPTER 4: CHIRAL AGGREGATES OF SQURAINE DYES
Figu e 4.14: Modeliza ion o P osq-SS in o a schema ic ep esen a ion o DAD quad upo-
la dye (same modelizazion applies o P oSQ-RR).
Figu e 4.15: The h ee pa ame e s used o de ine ela i e o ien a ion o squa aine pai s.
F om le o o igh : plane dis ance d, shi angle θSand il angle θT
138

4.8 Conclusions
Figu e 4.16: His og ams ep esen ing he dis ibu ion o he h ee geome ic pa ame e s.
F om op o bo om: d,θSand θT. In each panels he dis ibuions ela i e o he 3
nea es neighbo s pai s o he e ame a e shown.
139
CHAPTER 4: CHIRAL AGGREGATES OF SQURAINE DYES
Figu e 4.17: Dime abso p ion (le ) and CD spec a ( igh ) calcula ed o di e en alue
o in e molecula Cha ge T ans e in eg al β.
140
4.8 Conclusions
Figu e 4.18: T ime abso p ion (le ) and CD spec a ( igh ) calcula ed o di e en
alue o in e molecula Cha ge T ans e in eg al β.
141
CHAPTER 4: CHIRAL AGGREGATES OF SQURAINE DYES
Figu e 4.19: Te ame abso p ion (le ) and CD spec a ( igh ) calcula ed o di e en
alue o in e molecula Cha ge T ans e in eg al β.
142
Conclusions
In his Thesis we ex ensi ely discussed in e molecula elec os a ic in e ac ions wi h
special emphasis on molecula agg ega es and FRET. The p esen ed heo e ical wo k
p o ides a se ies o ools wi h he pu pose o imp o ing he desc ip ion o complex
sup amolecula sys ems. Speci ically, we demons a ed how he combined use o MD,
exci on and essen ial s a e models can p o ide in e es ing esul s and a eliable desc ip-
ion o op ical p ope ies o a a ie y o sys ems.
In he i s pa o he disse a ion we in es iga ed, o a selec ed pai o ch o-
mopho es, he ela ions be ween ene gy ans e e iciencies and slow deg ees o eedom
connec ed o con o ma ional mo ion and sol a ion. To his aim we p oposed a new com-
pu a ional me hod o he calcula ion o he dynamics and RET quan um yields o a
RET pai in wo di e en sol en s, highligh ing he impo ance o con o ma ional luc u-
a ions. Ou app oach, combining equilib ium and non-equilib ium MD simula ions wi h
ab-ini io esul s, o e s a de ailed desc ip ion o all he p ocesses ha simul aneously a -
ec he decay a es o ou sys em. We alida ed he eliabili y o he p o ocol agains an
ex ensi e se o a ailable expe imen al esul s, wi h e y good esul s. This wo k sol ed
a majo issue ela ed o he cha ac e iza ion o RET a es in sys ems wi h high lexibil-
i y, emphasizing he impo ance o a dynamical app oach o he desc ip ion o complex
decay mechanisms and opening new oppo uni ies o hei heo e ical ea men .
The second pa o he Thesis, ocused on molecula agg ega es, aims o ex end he
po olio o ele an models and compu a ional echniques. We conduc ed a de ailed
s udy on agg ega es o pola and pola izable molecules, es ing di e en app oxima-
ion app oaches and hei limi a ions. Mo eo e , we p oposed a simple ye accoun able
me hod o he calcula ion o op ical spec a o bidimensional agg ega es, whose desc ip-
143

CONCLUSIONS
ion is usually e y complex. Wi h espec o chi al molecula agg ega es, we ex ensi ely
in es iga ed spec oscopic and sup amolecula ea u es o wo sys ems, whe e he sub-
le in e play be ween in e molecula elec os a ic in e ac ions and chi al sup amolecula
a angemen s esul s in non-ob ious ou comes. Speci ically, we ocused a en ion on ag-
g ega es o non-pola and quad upola ch omopho es. We sugges ed a no el app oach
o he cha ac e iza ion o a amily o dicyanos ilbenes de i a i es (non-pola ). The hy-
b id me hod p oposed, based on he combina ion o MD simula ions and exci on models,
simula es well he chi op ical p ope ies obse ed expe imen ally. As o agg ega es o
quad upola dyes, a g oup o squa aines chi al de i a i es, showing unique abso p ion
and ci cula dich oism spec a upon agg ega ion, ha e been in es iga ed. Th ough ac-
cu a e modeliza ion, we we e able o mimic he sup amolecula a angemen s o he
molecula uni s by he use o molecula dynamics simula ions, We hen p esen ed a new
model, able o accoun o CT in e molecula in e ac ions, ha o e s an accu a e ool
o he desc ip ion o spec oscopic p ope ies o squa aine agg ega es in solu ion.
To conclude, his Thesis o e s new heo e ical app oaches o he desc ip ion o
in e molecula elec os a ic in e ac ions ha a e o ou s anding impo ance in he ield o
inno a i e molecula ma e ials. Complex mechanisms, such as esonan ene gy ans e
and cha ge ans e in e ac ions, ha e been in es iga ed in a comple ely di e en ligh ,
leading o a mo e p o ound unde s anding o hei p ope ies and oppo uni ies.
144
Appendix A
Molecula Agg ega es
A.1 Oscilla o S eng h: sum ule
To ela e he o al oscilla o s eng h o a sys em o a g ound-s a e expec a ion alue,
we de ine he eloci y dipole ope a o , ˆ :
i~ˆ = [ˆµ, ˆ
H],(A.1)
whe e, o simpli y he no a ion, we ha e supp essed he ec o no a ion on bo h ˆ and
ˆµope a o s. The oscilla o s eng h associa ed wi h he G→E ansi ion is:
EG =2
3
me
~e2ωEGhG|ˆµ|EihE|ˆµ|Gi.(A.2)
To elimina e he ansi ion equency om he abo e exp ession we use:
ihG|ˆ |Ei=ωEGhG|ˆµ|Ei(A.3)
and i s complex conjuga e, hus ge ing:
F=X
E
EG =−ime
3~e2hG|[ˆµ, ˆ ]|Gi,(A.4)
hus p o ing Eq.2.28 in he main ex . Fu he mo e, using Eqs.2.26 and 2.27 (main ex )
and emembe ing he Paulions algeb a (see Eq. 2.15 in he main ex ), he commu a o
145
APPENDIX A
[ˆµ, ˆ ] can be easily calcula ed. In pa icula , we ha e:
[ˆµ, ˆ ] = 1
i~hˆµ, [ˆµ, ˆ
H]i,
=1
i~µ2
0X
i
[~ω0−λ(ˆa†
i+ ˆai)](4ˆni−2),(A.5)
ha is Eq.2.30 in he main ex .
146
APPENDIX A
A.2 Agg ega es o pola dyes: addi ional esul s
Figu e A.1: The same esul s as in Fig. 2.7, main ex (N= 6), accoun ing o di e en
alue o Ne.
147