Sensitivity of the photophysical properties of organometallic complexes to smallchemical changesA. C. Jacko, B. J. Powell, and Ross H. McKenzie Citation: The Journal of Chemical Physics 133, 124314 (2010); doi: 10.1063/1.3480981 View online: http://dx.doi.org/10.1063/1.3480981 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/133/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Magnetic properties of phthalocyanine-based organometallic nanowire Appl. Phys. Lett. 101, 062405 (2012); 10.1063/1.4744437 Disposition of the axial ligand in the physical vapor deposition of organometallic complexes J. Vac. Sci. Technol. A 28, 795 (2010); 10.1116/1.3377140 Color tunable organic light-emitting diodes by using europium organometallic complex Appl. Phys. Lett. 89, 251108 (2006); 10.1063/1.2405420 Noncovalent interactions between organometallic metallocene complexes and single-walled carbon nanotubes J. Chem. Phys. 125, 154704 (2006); 10.1063/1.2349478 Effect of platinum on the photophysical properties of a series of phenyl-ethynyl oligomers J. Chem. Phys. 122, 214708 (2005); 10.1063/1.1924450

Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 130.102.82.69 On: Fri, 07 Oct 2016

04:31:18

Sensitivity of the photophysical properties of organometallic complexes tosmall chemical changes

A. C. Jacko,a� B. J. Powell, and Ross H. McKenzieCentre for Organic Photonics and Electronics, School of Mathematics and Physics,The University of Queensland, Brisbane QLD 4072, Australia

�Received 23 May 2010; accepted 30 July 2010; published online 28 September 2010�

We investigate an effective model Hamiltonian for organometallic complexes that are widely usedin optoelectronic devices. The two most important parameters in the model are J, the effectiveexchange interaction between the � and �� orbitals of the ligands, and ��, the renormalized energygap between the highest occupied orbitals on the metal and on the ligand. We find that the degreeof metal-to-ligand charge transfer character of the lowest triplet state is strongly dependent on theratio �� /J. �� is purely a property of the complex and can be changed significantly by even smallvariations in the complex’s chemistry, such as replacing substituents on the ligands. We find thatsmall changes in �� /J can cause large changes in the properties of the complex, including thelifetime of the triplet state and the probability of injected charges �electrons and holes� formingtriplet excitations. These results give some insight into the observed large changes in thephotophysical properties of organometallic complexes caused by small changes in the ligands.© 2010 American Institute of Physics. �doi:10.1063/1.3480981�

I. INTRODUCTION

Organometallic complexes are being developed as opti-cally active materials for devices such as organic light emit-ting diodes �OLEDs� �Refs. 1–4� and organic photovoltaics�OPVs�.5–8 The functionality of such complexes depends ontheir excited state properties. Of particular interest is theemission process, key to the function of OLEDs,

electron + hole → C� → C + photon,

where C� �C� denotes the complex in its excited �ground�state. The reverse process, converting a photon into electronand hole excitations, is the key process in OPV cells such asdye-sensitized solar cells �Gratzel cells�.5 To understand andcontrol these processes �in the design of new complexes� oneneeds to understand the relevant excited states of the com-plex.

Two types of excited state typically dominate the behav-ior of organometallic materials �such as those illustrated inFig. 1�: ligand centered �LC� states and metal-to-ligandcharge transfer �MLCT� states. Isolated ligands have a“bright” high energy �UV� singlet transition �associated witha �→�� excitation, equivalent to a LC transition�. This tran-sition has a molar absorptivity of order 104 mol−1 cm−1.11

The corresponding �→�� triplet transition has a muchlower energy and is “dark.”11 When ligands are bonded to atransition metal to form a complex new singlet transitions inthe visible region are observed �with molar absorptivity oforder 103 mol−1 cm−1�.11 These new features are typicallyattributed to MLCT transitions. The significant oscillatorstrength associated with MLCT singlet states arises becauseof the hybridization of metal orbitals with �� �unoccupiedligand� orbitals, known as back-bonding.12–15

The presence of the heavy transition metal ion producesa spin-orbit interaction which mixes singlets and triplets withseveral important effects.16 First, this interaction allowsstates with dominant triplet character to decay radiatively.Second, it also allows a rapid ��50 fs� intersystem crossingbetween singlet and triplet dominated states.17 In organiccomplexes with no transition metal ion the spin-orbit cou-pling is much weaker, so triplet excitations tend to decay vianonradiative decay paths.18–20

The character of the emitting state in organometalliccomplexes has attracted considerable interest and debate. Itis generally agreed to be predominantly triplet �due to itslong radiative lifetime, which is associated with phosphores-cence�. However for a variety of complexes it has been la-beled variously as LC, MLCT, or as a LC-MLCT hybrid.21–23

For example, Li et al.3 considered a family of OLED com-plexes and claimed that the emitting state is a superpositionof a singlet MLCT state and a triplet LC state. Such spinhybridization is only possible due to spin-orbit coupling. Inthe case of very strong spin-orbit coupling, the predomi-nantly triplet state can have a comparable intensity to singletemission.23 Yersin et al.24,25 claimed that the amount of zero-field splitting �ZFS� in the emitting triplet reflects the amountof MLCT character in the state, and that there is “an empiri-cal correlation between the amount of ZFS and the com-pound’s potential for its use as emitter material in anOLED.”25 It has been suggested that the key effect of in-creasing MLCT character is to enhance the effect of spin-orbit coupling, which in turn increases the radiative rate oftriplet excited states.26

A key aspect to understanding the photophysical proper-ties of complex molecular materials is identifying the rel-evant frontier molecular orbitals and their interactions withone another. This allows one to define an effective Hamil-a�Electronic mail: jacko@physics.uq.edu.au.

THE JOURNAL OF CHEMICAL PHYSICS 133, 124314 �2010�

0021-9606/2010/133�12�/124314/13/$30.00 © 2010 American Institute of Physics133, 124314-1

Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 130.102.82.69 On: Fri, 07 Oct 2016

04:31:18

tonian which involves just a few parameters. Well knownexamples of this approach involve the Hückel, Hubbard,Heisenberg, and Pariser–Parr–Pople models for conjugatedpolyenes.27 With regard to organometallic complexes this ap-proach has been applied to mixed valence binuclear systemsincluding magnetic atoms in proteins �Hubbard and doubleexchange models�,28 molecular magnets,29 and Anderson im-purity models for cobalt based valence tautomers.30

Such semiempirical approaches have significant advan-tages which mean that they nicely complement first prin-ciples approaches such as DFT and high level ab initio quan-tum chemistry.31 First, effective Hamiltonians can helpreveal features that are common to a diverse range of mate-rials. Second, since the number of degrees of freedom in themodel is significantly less than in the actual material, onedoes not necessarily have to make the approximations onewould be forced to make if one did a complete quantumsimulation of the actual material. For example, for large mo-lecular systems, one can also describe the nuclear dynamicsquantum mechanically32 and include the effect of the envi-ronment such as the solvent.33,34 Such models can captureuniversal behavior that is not sensitive to microscopic de-tails. For example, the single impurity Anderson model candescribe the Kondo effect in a diverse range of systems in-cluding magnetic impurities in metals, quantum dots in semi-conductor heterostructures, carbon nanotubes, and singlemolecule transistors.35,36

Here we develop and examine a model Hamiltonianwhich reproduces the key photophysical properties of orga-nometallic complexes. We apply the model to complexes inwhich the low energy physics is dominated by one ligand�see for example Refs. 15, 26, and 37�. We find that there aretwo key parameters in describing the low energy photophys-ics of these complexes - J, the spin-exchange interaction be-tween ligand � and �� orbitals, and ��, the renormalizedenergy gap between the ligands highest occupied molecular

orbital �HOMO� and highest occupied metal orbital. Weshow that, through small changes to these parameters, singlechemical substitutions can cause significant changes in thetriplet excited state lifetime and the probability of triplet for-mation, causing large variations in efficiency. This paper isorganized as follows. In Sec. II we introduce our modelHamiltonian, discussing its parameters and an appropriatebasis in which to investigate its properties. In Sec. III weanalyze some approximate and exact solutions of the Hamil-tonian, building an understanding of the key features of themodel. In Sec. IV we determine the effect of MLCT charac-ter on the radiative properties of the excited states. In Sec. Vwe study the probability of finding an excitation in the tripletstate after charge injection. Section VI presents some con-cluding remarks.

II. MODEL HAMILTONIAN

In principle, one should determine an effective Hamil-tonian by integrating out high energy states. However, ex-plicitly carrying out this procedure is prohibitively expensivefor all but the smallest molecules.38 Therefore, in order toinvestigate correlation effects in these organometallic com-plexes, we reduce the size of the basis set in a way motivatedby our understanding of the important physical processes.

To this end we study the following model Hamiltonian:

H = − JS�H · S�L + ��=↑,↓

�M�† M� + �L�

†L� − tH�H�† M� + M�

†H��

− tL�L�† M� + M�

†L�� + UHnH↑nH↓ + ULnL↑nL↓

+ UMnM↑nM↓ + VHLnHnL + VHMnHnM + VMLnMnL, �1�

where nH���H�†H�, nL���L�

†L�, nM ���M�† M�,

S�H=�����H�†�� ���H�� , S�L=�����L�

†�� ���L�� , �� ���= ���,��

x , ��,��y , ��,��

z �, where ��,��s is the �� ,��� element of

the s Pauli spin matrix, �s. In what follows it will be usefulto have intuitive descriptions of the creation operators H�

† ,L�

† , and M�† . Thus, we will refer to the state H�

† vac �wherevac is the state with �nH= �nM= �nL=0� as the “HOMOlevel of the ligand,” call L�

† vac as the “LUMO level of theligand,” and call M�

† vac the “metal orbital.” It is importantto stress, however, that the level H�

† vac �L�† vac� may well

be very different from the highest occupied �lowest unoccu-pied� molecular orbital that one would find in a Hartree–Fock calculation. Similarly, M�

† vac is likely to be very dif-ferent from the atomic d orbital in the isolated transitionmetal.27 As such one cannot describe the coefficients of theHamiltonian in terms of one electron orbitals like UH

=��dr1dr2�H↑� �1��H↑�1��e2 / r1−r2��H↓

� �2��H↓�2�.Indeed, it is not possible, in general, to associate any

static “orbital distribution” with any of these states as orbitalrelaxation will occur when the physical state of the complexchanges. Rather, orbital relaxation �and higher order pro-cesses� are captured in the renormalization of the effectiveparameters of the model.27,39 However, one expects that themodel states correspond rather more closely to ones intuitionof how a HOMO, LUMO, and metal orbital behave than theactual Hartree–Fock orbitals do. Thus, this may provide newinsights to the problems at hand.31

FIG. 1. Examples of organometallic complexes for optoelectronic applica-tions. Both Pd�thpy�2 and Ir�ppy�3 are the basis of many complexes used inorganic light emitting diodes �Refs. 3 and 9�, while the benchmark N3 dye�Ref. 10� is used in organic dye-sensitized photovoltaic cells.

124314-2 Jacko, Powell, and McKenzie J. Chem. Phys. 133, 124314 �2010�

Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 130.102.82.69 On: Fri, 07 Oct 2016

04:31:18

One might intuitively interpret the parameters as fol-lows: J is the effective exchange interaction between theligand HOMO and the ligand LUMO, � is the effective dif-ference in the energies of the metal orbital and the ligandHOMO, � is the effective HOMO-LUMO gap, tH �tL� is theeffective hopping amplitude between the metal orbital andthe ligand HOMO �LUMO�, Ui is the effective Coulombrepulsion between two electrons in orbital i �wherei� H ,L ,M��, and Vij is the effective Coulomb repulsion be-tween an electron in orbital i and another in orbital j. Thisinterpretation would seem to be equivalent to making anintermediate neglect of differential overlap �INDO�approximation40 after the physically motivated basis set re-duction. However, we do not now make the Hartree–Fockapproximation made in INDO calculations. Note that theseparameters are renormalized from the values that would befound from direct computation. It is known that the param-eter values in small organic molecules, like the ligands con-sidered here �cf. Fig. 1�, can be significantly changed due tothis renormalization.39,41–44 Indeed, we will not attempt tocalculate the values of the parameters here. Instead we con-struct a semiempirical theory by comparing our model toexperimental data in the Appendix.

In general, several ligands and metal orbitals could beinvolved in the low energy physics of the organometalliccomplexes we are investigating. For simplicity, here we fo-cus on complexes in which one ligand dominates the lowenergy physics �for example Ru�NH3�2Cl2�bqdi� �Ref. 15�,Ru�dcbpy��bpy�2 �Ref. 37�, or Pt�cnpmic� �Ref. 26��, inter-acting with only one fully occupied metal orbital, and leavethe discussion of models with more metal and ligand sites fora later publication. In the noninteracting ground state �nH= �nM=2 and �nL=0. Thus we consider a three site modelwith four electrons.

A. Basis states

One expects that the LUMO will be much higher in en-ergy than the metal state ��−�� tL, see Appendix�. There-fore, the effect of tL on the eigenstates will be small com-pared to the other interactions. As such nL is almost aquantum number so it is convenient to work in a basis ofstates which are eigenstates of nL.

We define a “reference state” for this model,

0 � H↑†H↓

†M↑†M↓

†vac ,

the nL=0 state, with energy E0= �0H0=2�+UH+UM

+4VHM, and where vac is the vacuum state.We will also choose our basis states to be eigenstates of

the exchange interaction, J, that is, singlets and triplets. Wedefine the nL=1 MLCT triplet states as

3MLCT1 � �L↑†M↓0, L↓

†M↑0,1�2

�L↑†M↑ − L↓

†M↓�0� ,

and the ligand centered �LC� triplet states as

3LC1 � �L↑†H↓0, L↓

†H↑0,1�2

�L↑†H↑ − L↓

†H↓�0� ,

where the prefix “3” indicates the spin degeneracy and thesuffix “1” is the value of nL in these states �since these aretriplets between an electron and a hole in the reference state,the Sz=0 states appear to have the opposite phase relation tousual�. Similarly, the MLCT singlet is

1MLCT1 �1�2

�L↑†M↑ + L↓

†M↓�0 ,

and the LC singlet is

1LC1 �1�2

�L↑†H↑ + L↓

†H↓�0 .

When we go to the nL=2 states we have three singlets, theMLCT

1MLCT2 � L↑†L↓

†M↓M↑0 ,

the LC

1LC2 � L↑†L↓

†H↓H↑0 ,

and the metal-HOMO �MH� singlet

1MH2 �1�2

L↑†L↓

†�H↑M↓ − H↓M↑�0 ,

and only one kind of triplet,

3MH2 � �L↑†L↓

†H↓M↓0, L↑†L↓

†H↑M↑0,

1�2

L↑†L↓

†�H↓M↑ + H↑M↓�0� .

When examining the results of our model, it will be helpfulto consider numerical results for a particular set of parametervalues. In the Appendix we estimate parameter values rel-evant to complexes of interest for optoelectronic applications�such as those in Fig. 1�. We use the following parameter setas a typical example of these values: J=1 eV, �=3 eV, tH

=0.1 eV, tL=0.1 eV, UM =UH=UL=U=3 eV, VHL=VHM

=VLM =V=3 eV �note that the approximation here that UH

=VHL is physically reasonable as the H†vac and L†vacstates are delocalized and have a large spatial overlap; a de-tailed derivation of this result is given in the Appendix�. It isnot possible to estimate � �the energy gap between theHOMO and the metal orbital� from what is known experi-mentally about the ligand as it is, intrinsically, a property ofthe complex. Indeed, we will show below that many of theimportant properties of the complex depend sensitively on �.For consistency and concreteness we take �=0.5 eV whenwe wish to investigate how the properties of the complexvary with another parameter. In the supplementary informa-tion we show that our main conclusions are robust to varia-

124314-3 Photophysical properties of organometallics J. Chem. Phys. 133, 124314 �2010�

Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 130.102.82.69 On: Fri, 07 Oct 2016

04:31:18

tions of these parameters in physically reasonable ranges andpresent calculations for many alternative parameter sets.

Figure 2 shows that for this typical set of parameters thatnL is indeed almost a quantum number for the singlet states,particularly in the low energy singlets. It is useful to writethe Hamiltonian matrix in the basis defined above. We canwrite the singlet sector of the Hamiltonian in terms of blocks

of constant nL, HSnL, and the matrices that couple them, TS

nL,nL�,

HS �� H0 TS0,1S TS

0,2

TS1,0 HS

1 TS1,2

TS2,0 TS

2,1 HS2� , �2�

and similarly for the triplet sector of the Hamiltonian,

HT �� HT1 TT

1,2

TT2,1 HT

2� , �3�

where TSnL,nL��= TS

nL�,nL.The only nL=0 basis state is the reference state,

H0= �2�+UH+UM +4VHM�. The singlet nL=1 block is

HS1 =�

1MLCT1 1LC1� + � + UH + 2VHL + 2VHM + VLM tH

tH � + 2� +3J

4+ UM + VHL + 2VHM + 2VLM

� , �4�

where the states above the matrix indicate the basis in which we write the matrix. The triplet nL=1 block is

HT1 =�

3MLCT1 3LC1� + � + UH + 2VHL + 2VHM + VLM tH

tH � + 2� −J

4+ UM + VHL + 2VHM + 2VLM

� , �5�

the singlet nL=2 block is

HS2 = �

1MLCT2 1MH2 1LC2

2� + UH + UL + 4VHL �2tH 0

�2tH 2� + � + UL + 2VHL + VHM + 2VLM �2tH

0 �2tH 2� + 2� + UL + UM + 4VLM ,� , �6�

FIG. 2. The eigenstates of the singlet part of the model Hamiltonian, Eq.�1�, from lowest energy �1� to highest �6�, in terms of singlet basis stateswhich are eigenstates of nL, using the example parameter set J=1 eV, �=3 eV, �=0.25 eV, tH=0.1 eV, tL=0.1 eV, UM =UH=UL=U=3 eV, andVHL=VHM =VLM =V=3 eV �see Appendix for more details�. It is clear thatfor this set of parameters the eigenstates are predominantly one basis stateand that the coupling between blocks of different nL is a small effect. Theseresults hold for physically reasonable parameter values. See supplementarymaterial for the equivalent figure at different parameter values �Ref. 45�.Figures S1–S6 in the supplementary material show that this picture does notbreak down except near �avoided� level crossings and that the first excitedstate has a large overlap with the MLCT1 basis state for a wide range ofparameters around our typical set.

124314-4 Jacko, Powell, and McKenzie J. Chem. Phys. 133, 124314 �2010�

Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 130.102.82.69 On: Fri, 07 Oct 2016

04:31:18

and there is only one state in the triplet nL=2 block, the

3MH2 state, HT2 = �2�+�+UL+2VHL+VHM +2VLM�. The

coupling matrices are

TS0,1 � �− �2tL,0� ,

TS1,2 � �− �2tL 0 0

0 tL 0� ,

TS0,2 = �0,0,0� ,

among the singlet states, and for the triplets

TT2,1 � �0,tL� .

III. APPROXIMATE AND EXACT SOLUTIONS OF THEMODEL HAMILTONIAN

We will now investigate both the analytical solutions ofEq. �1� in the limit of tL=0, tH�0 and the exact solutionsnumerically, i.e., we find the full configuration interactionsolution, to gain some insight into the behavior of the model.A typical value of � is around 3 eV, much larger than thetypical tL values of 0.1 eV �see Appendix�. This means thatthe hybridization between states of different nL will be smallcompared to the hybridization between states of the same nL,as is clear from Fig. 2. As such, we expect the approximationthat tL=0 is close to the exact solution. In the limit tL=0 all

of the TnL,nL� matrices are null, and hence nL is a good quan-tum number.

The lowest excited states will come from the nL=1 sub-space, provided � is large, as we expect it to be in complexeswith OLED and OPV applications �see the Appendix�. Theeigenvalues of this subspace are

ES,T�1�� =

1

2�2� + 3�� + �3 − 4S�

J

4− 2UM + 4UH + 6VHL

+ 2VHM� �1

2���� + �3 − 4S�

J

4�2

+ 4�tH�2�1/2,

�7�

where

�� � � + UM − UH + VLM − VHL �8�

is the effective HOMO-metal energy gap relevant for thelowest excited states with S=0 �S=1� for the eigenvalues ofthe singlet states �triplet states�. ��, J, and tH are the keyenergy scales that determine the properties of the nL=1states. For states with purely nL=1 character, varying �� isequivalent to varying any one of �, UM, UH, VLM, or VHL asgiven by Eq. �8�. If we relax the approximation tL=0 weneed to deal with states with nL�1. If tL�0, then in prin-ciple ��, J, and tH are no longer the only energy scales—onealso needs to consider the individual effects of each of thedirect Coulomb terms. For our typical parameter set, the ef-fect of tL on the lowest excited states is quite small. Thismeans that �� is still the key parameter, as it is in the tL=0case, so long as �� does not vary too much �cf. supplemen-tary information, Figs. S7–S17�.45

Figure 3 illustrates the character of the nL=1 singlet andtriplet states. If ���−J /4� / tH is small then the LC and MLCTtriplets will be strongly mixed �note that this does not requirethat tH is large compared to any of the parameters�. On theother hand, in the absence of interactions, the level crossingwould occur at �=0, the point at which the one electronH†vac and M†vac states are degenerate. Figures 4 and 5show that the nL=1 states are well separated from the otherstates for our typical parameter set �see Appendix for moredetails�. It is important to stress that a large hybridization ofthe metal d and ligand � orbitals is neither necessary norsufficient to have a triplet excited state with mixed MLCTand LC character. This is different to what is discussed inRefs. 4 and 46. It is well known47,48 that density functionaltheory �DFT� tends to overestimate the delocalization ofelectron density in excited states. As such the conclusionsdrawn from DFT with regards to changes in electron density,for example identifying transitions as MLCT or LC, may notbe reliable.

In the exact solution, neither nL nor nH are good quan-tum numbers. There is not much insight to be gained fromthe analytic solutions to this Hamiltonian, so we proceed byusing some typical parameter values �discussed further in theAppendix� and investigating around these values numeri-cally. All level crossings are avoided, as is clear from Figs. 4and 5. These figures also show that the eigenstates are almostpure basis states everywhere except very near the avoidedcrossings �listed in Table I�, since tH and tL are small com-pared to all the other Hamiltonian parameters. Hence, �nL�0,1 or 2 for each of the singlet eigenstates, except near theavoided level crossings.

Table I lists all the �avoided� level crossings in ourmodel, as well as the equivalent conditions for a noninteract-ing model. J plays a key role in determining the level cross-ing for most of the states, so the singlet and triplet spectra arevery different. By comparison, the level crossing points forthe noninteracting states �one electron states� only depend on

FIG. 3. Schematic of the energy and character of the nL=1 excited states inthe regimes ��J /4, ���J /4, and ��J /4, where ����+UM −UH+VLM

−VHL. As �� increases, the two triplet states approach the energy of thesinglet MLCT state. At ��=J /4 there is an avoided crossing and the tripletstates change character, with the lowest energy triplet state going from pre-dominantly LC when ��J /4, to predominantly MLCT when ��J /4.This can be seen for some explicit parameter values in Fig. 5.

124314-5 Photophysical properties of organometallics J. Chem. Phys. 133, 124314 �2010�

Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 130.102.82.69 On: Fri, 07 Oct 2016

04:31:18

the one electron site energies. Thus, singlets and triplets aredegenerate in the noninteracting model. The differences be-tween the mixing of the singlet states and that of the tripletstates are apparent from Figs. 4 and 5.

Figure 4 shows for our typical set of parameters how theeigenvalues change as a function of � and how the singleteigenstates are dominated by one basis state except nearavoided crossings. Note that for the typical parameter set nosinglet avoided crossings occur for �2 eV, i.e., in therange of � values found in the ligands with applications inOLED and OPV devices �see Appendix�. Figure 5 shows thecharacter of the triplet states as a function of � and how thetriplet states can be strongly hybridized even away fromavoided crossings. These results indicate the importance ofthe exchange interaction J in determining the character andenergy of the excited states. As is apparent from Figs. 4 and5, for a wide range of reasonable parameter values the S1 andS2 states of our model are almost pure 1MLCT1 and 1LC1basis states, respectively. As such, one can calculate the gapbetween these states to be ��+3J /4. With an independentestimate of J based on isolated ligand data �see Appendix�,one can estimate �� from the energies of the S1 and S2 statesfound in absorption spectra, and compare it to the valuefound from the singlet-triplet gap �Eq. �17��. This allows fora self-consistency check on the model and an estimate of itsaccuracy.

IV. EFFECT OF THE MLCT CHARACTER OF THETRIPLET EXCITED STATE ON ITS RADIATIVEPROPERTIES

Experimental data indicate that the emitting state is pre-dominantly triplet. However, there has been considerable de-bate over the exact character of the state, i.e., whether it isLC, MLCT, or a hybrid.21–23 The degree of MLCT characterin the triplet excited state has been discussed by Yersin et al.as a key indicator of a compounds potential as an OLEDemitter.11,24,25 Here we examine how the MLCT characterchanges in the lowest singlet and triplet states and how thesechanges in character might affect the radiative properties ofthe triplet states.

In an OLED device one wants to maximize the radiativedecay rate compared to the nonradiative decay rate. It is notpossible to predict how the nonradiative decay rate will varyfrom the current model, but we can examine how to increasethe radiative rate via the increase of the transition dipolemoment. For a triplet �phosphorescent� emitter, increasingthe triplets transition dipole moment requires significantsinglet-triplet mixing via spin-orbit coupling, which can onlyoccur in the presence of the heavy transition metal core.16

The spin-orbit interaction couples electrons in atomic orbit-als of different angular momentum, allowing spin flips be-tween these orbitals and hence coupling singlets and triplets.

FIG. 4. Singlet excitation energies as a function of the ligand HOMO-LUMO gap �. The triplet excitation energies �dashed blue lines� are alsoshown for comparison. The color of each curve shows how the character ofthe eigenstates changes with �. Around the example value of �=3 eV thereis no mixing of the lowest excited singlets, meaning they are nearly purebasis states. There are four pairwise avoided crossings and one avoidedcrossing involving three levels. These occur at the maximal hybridizationconditions listed in Table I. Away from the avoided crossings, the eigen-states are predominantly the basis states for these values of the parameters.Note that all level crossings are avoided. This plot is for our example pa-rameter values �cf. Fig. 2 and the Appendix�. Figures S7, S9, S11, S12, andS15–S17 in the supplementary information show that for the lowest excitedstates, this picture holds for a wide range of parameters around our typicalparameter set �Ref. 45�.

FIG. 5. The triplet excitation energies as a function of the ligand HOMO-metal orbital gap ��. The singlet excitation energies �dashed blue lines� arealso shown for comparison. The color of each curve shows how the charac-ter of the eigenstates changes with �. One can see the 3MLCT1-3LC1 hy-bridization point occurs at ��=J /4=0.25 eV as expected, while the3MH2-3LC1 hybridization occurs at very large values of �� for this param-eter set. The location of the 3MLCT1-3LC1 maximum hybridization point at0.25 eV puts it in the likely range of � values, meaning that in realisticcircumstances the lowest triplet state will probably be a hybrid state. Incontrast, avoided crossings in the lowest excited singlet state occur at ��1 eV, outside the reasonable range in these complexes. This plot is forour example parameter values �cf. Fig. 2 and the Appendix�, varying �between 0 and 1 eV. Figures S8, S10, S13, and S14 in the supplementaryinformation show that this picture holds for a wide range of parametersaround our typical parameter set and reinforce the relevance of �� as the keyparameter for the nL=1 excited states �Ref. 45�.

124314-6 Jacko, Powell, and McKenzie J. Chem. Phys. 133, 124314 �2010�

Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 130.102.82.69 On: Fri, 07 Oct 2016

04:31:18

In our model we only have one metal orbital �which is reallysome renormalized effective orbital we have identified as“metal”�, so we insert a singlet-triplet coupling explicitly be-tween states with an unpaired spin in the metal orbital tomimic the true effect of a spin-orbit interaction.

The triplet state Tm has a transition dipole moment MTm

to the ground state that is given �to first order in the spin-

orbit coupling Hamiltonian HSO� by

MTm = �n

�TmHSOSnETm − ESn

MSn, �9�

where ETm is the energy of the triplet state, ESn is the energyof the singlet state Sn with transition dipole moment to theground state MSn, and the sum runs over all singlet states �seep. 271 of Ref. 49 for more details�. This expression is only

valid if �TmHSOSn / �ETm −ESn�1. If the singlet and tripletare nearly degenerate, then one needs to exactly solve theentire singlet-triplet Hamiltonian with explicit spin-orbitcoupling.

Figure 5 and Figs. S2–S6 of the supplementary informa-tion show that the energy of the MLCT singlet lies betweenthe two nL=1 triplet states over a wide range of possibleparameter values, while the LC singlet is higher in energy.These figures also show that when the lowest triplet state ispredominantly MLCT �i.e., ��J /4�, and is nearly degener-ate with the singlet MLCT state. In this regime the perturba-tion expression above is no longer valid. For reasonable pa-rameter values the MLCT singlet is the closest singlet �inenergy� to the lowest triplet state �see Figs. 4 and 5 and Figs.S7–S17 of the supplementary information�.45 If we assumethat the only contribution to the transition dipole moment ofthe lowest triplet state comes from the singlet which is nearlypure MLCT we can rewrite Eq. �9� as

MT1 = M1MLCT1�T13MLCT1ET1 − E1MLCT1

�3MLCT1HSO1MLCT1 .

�10�

Figure 6 shows the amount of MLCT character in the lowestenergy triplet state �3MLCT1 T1 as a function of �� /J.

We write our eigenstates as

Si � cSi

0 0 + cSi

1MLCT11MLCT1 + ¯ , �11�

so the ith singlet states transition dipole moment to theground state S0 is

TABLE I. Level crossing points for many-body states and one electron states in the one metal one ligand model and the Hamiltonian element coupling them.Some of the hybridizations occur indirectly �i.e., the states are not directly connected by a Hamiltonian element but can be at higher orders in perturbationtheory�. The first block of conditions is that between states with a Hamiltonian element directly coupling them �coupling at first order in perturbation theory�.The second block is that between states which have no direct coupling in the Hamiltonian �higher orders in perturbation theory�. The final block is of thehybridization conditions for the noninteracting model �UH=UL=UM =VHM =VHL=VLM =J=0�.

States Degeneracy conditions Coupling

3MLCT1, 3LC1 ��=J /4 tH

1MLCT1, 1LC1 ��=−3J /4 tH

3LC1, �3MH2 �=��−J /4+UH−UL+VHM −VLM tL

1LC1, 1MH2 �=��+3J /4+UH−UL+VHM −VLM tL

0, 1MLCT1 �=��+UH+2VHM −VHL−2VLM�2tL

0, 1MLCT2 2�=2��+2UH−UL−UM +4VHM −2VHL−2VLM�2tL

1MLCT1, 1MLCT2 �=��+UH−UL−UM +2VHM −VHL�2tL

1LC1, 1MLCT2 �=2��+3J /4+UH−UM −UL−VHL+2VHM Indirect1LC1, 1LC2 �=3J /4−UL+2VHM +VHL−2VLM Indirect0, 1MH2 2�=��+2UH−UL+3VHM −VHL−3VLM Indirect

H†vac and M†vac �=0 tH

L†vac and M†vac �=� tL

FIG. 6. Degree of MLCT character in the lowest energy triplet state�T1 3MLCT1 as a function of �� /J. As �� /J increases, the MLCT statebecomes favorable and begins to dominate the lowest triplet state. At thepoint �� /J=1 /4 the state is half MLCT, half LC ��3MLCT1 T1= �3LC1 T1�1 /�2�. Note that around our typical choice of parameters,varying �� is equivalent to varying �. To give the variation in �� in thisfigure we use our typical parameter values �cf. Fig. 2 and the Appendix�, andvary � between 0 and 1 eV.

124314-7 Photophysical properties of organometallics J. Chem. Phys. 133, 124314 �2010�

Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 130.102.82.69 On: Fri, 07 Oct 2016

04:31:18

MSi � e ��,��

cSi

�cS0

����r�� , �12�

where � and �� are any of the basis states. We expect thatdue to the spatial separation between the metal and ligandorbitals, we can set �0r1MLCT1=0 and similarly for otherterms involving separated excitations. The only terms weexpect to remain are diagonal terms �=�� and intraligandexcitations, i.e., �0r1LC1. In the range of our typical pa-rameter values the lowest excited singlet state has almostpure 1MLCT1 character, whereas the ground state is almostpure 0, so the dominant eigenstate coefficients will be cS0

0

and cS1

1MLCT1. In the range of our typical parameter values, we

find that cS1

0 cS0

0 �−cS1

1MLCT1cS0

1MLCT1, so we can approximate

the lowest singlet’s transition dipole moment as

MS1 = ecS1

0 cS0

0 �r� , �13�

where �r���1MLCT1r1MLCT1− �0r0. Inserting Eq. �13�into Eq. �10� we find

MT1 = ecS1

0 cS0

0 �T13MLCT1ET1 − E1MLCT1

�r��3MLCT1HSO1MLCT1 .

�14�

ecS1

0 cS0

0 �T1 3MLCT1 / �ET1 −E1MLCT1� will vary with our pa-

rameter values, while �r��3MLCT1HSO1MLCT1 is a func-tion only of our basis states and the strength of the spin-orbitcoupling on the metal atom.

The radiative decay rate of the triplet, found via the Ein-stein A coefficient, is

1

�T�

2 3

3�0hc3 MT12, �15�

where � is the triplet-ground state energy gap and �0 is thepermittivity of free space.50 This quadratic dependence onMT1 further amplifies the effects of small changes in ��.

Apart from the Hamiltonian parameters, there is onlyone free parameter in the expression for the singlet lifetime�found from Eq. �13� via the Einstein A coefficient�, which isthe distance of the transition, �r�. We solve our model withthe typical parameter values and find that to reproduce asinglet radiative lifetime of �10 ns �see, for example, Ref.51� we must have �r�=20 Å. This distance is about twice thesize of a Pd�thpy�2 or Ir�ppy�3 molecule. In other similarcharge transfer excitations, one often finds that the geometri-cal distance between the assumed “donor” and “acceptor”fragments is not well correlated with the dipole length.52 Inthis case, the large value of �r� may be due to our neglect ofthe �0r1LC1 term in calculation of the singlets transitiondipole moment. The approximation that there is a large spa-tial separation between the metal and ligand orbitals couldalso be the source of the discrepancy, as the effective orbitalsof our model may not be as localized as our labels suggest.

We choose a reasonable value for the strength of the

spin-orbit coupling, �3MLCT1HSO1MLCT1�100 cm−1

=0.012 eV �cf. Ref. 53� use the value �r�=20 Å discussedabove, and plot the calculated the lowest exited singlet and

triplet states lifetimes as a function of �� /J in Fig. 7. Figure7 shows that the triplet lifetime �T changes rapidly as a func-tion of �� /J as long as �� /J1 /4, i.e., the triplet state is not

within �3MLCT1HSO1MLCT1 of the singlet energy �thecondition for perturbation theory to be valid�. Changing �� /Jfrom 0.1 to 0.25 decreases the triplet’s radiative lifetimeby an order of magnitude. For example with

�3MLCT1HSO1MLCT1=100 cm−1, the triplets radiativelifetime changes from 17 to 1.5 �s. These values of �T arecomparable to those found experimentally in variousorganometallic complexes, for example �1 �s for Ir�ppy�3

�Ref. 54�, or �100 �s for octaethyl-porphyrin platinum�II��Ref. 55�.

One can understand the apparently exponential changein �T as follows. We are explicitly in the regime where T1 isdominated by 3LC1. If we treat the MLCT component ofT1, �T1 3MLCT1, perturbatively we find it contains a fac-tor �ET1 −E3MLCT1�−1= �ET1 −E1MLCT1�−1. Thus the transitiondipole moment of the triplet contains a factor �ET1

−E1MLCT1�−2, so the lifetime varies with �ET1 −E1MLCT1�4.Over the small regime we vary ��, �ET1 −E1MLCT1� decreaseslinearly with ��, as seen in Fig. 5. In this small range, a largepower will be approximately linear on a semilogarithmicplot.

These results show that the excited state character andlifetime can be sensitively dependent on changes in Hamil-tonian parameters. This implies that small changes to thechemistry of a complex, for example replacing a single hy-drogen atom on a ligand molecule with a fluorine atom,could result in large changes in the molecules photophysicalproperties and, hence, the efficiency of optoelectronic de-vices made from these molecules. Observations of preciselythis effect have been made in several systems, for example in

FIG. 7. Triplet radiative lifetimes �T �upper three curves, blue� and singletlifetimes �lowest curve, red� as a function of �� /J for various values of HSO

solved to first order in perturbation theory. We have chosen �r�=20 Štoreproduce a singlet lifetime of order 10 ns. The triplet lifetime decreasesrapidly as �� /J increases, up until the point where the lowest singlet andtriplet are nearly degenerate �i.e., both MLCT� at which point the perturba-tive solution becomes invalid. As the strength of the spin-orbit couplingincreases, the triplet lifetime rapidly decreases. We use our typical parametervalues �cf. Fig. 2 and the Appendix�, varying � between 0 and 0.25 eV.

124314-8 Jacko, Powell, and McKenzie J. Chem. Phys. 133, 124314 �2010�

Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 130.102.82.69 On: Fri, 07 Oct 2016

04:31:18

a series of blue phosphorescent iridium complexes in Ref. 4,and in a series of complexes based around N-heterocycliccyanosubstituted carbenes in Ref. 26. Reference 26 foundthat changing the ligand altered the singlet-triplet gap, andcorrelated this change in the gap with the changes in theradiative lifetime of the triplet state, finding the �T� �ET1

−E1MLCT1�2 relationship we predict in the regime where thespin-orbit coupling can be included perturbatively, i.e., for�� /J1 /4.

In the regime �� /J1 /4, where the lowest excited sin-glet and triplet are both of MLCT character, the dominantsplitting between them comes from an exchange interactionbetween the metal orbital and the ligand orbitals, an interac-tion we have neglected in this model. The relative size of thisintersite exchange and the spin-orbital coupling element willdetermine the excited state properties in this regime. Spin-orbit coupling may also have important effects on the nonra-diative lifetime of the excited state, but these effects cannotbe studied within the current model. One can expect thatsince nonradiative processes are thermally activated, smallchanges in energy barriers can cause exponentially largerchanges in the rate.

V. CHARGE INJECTION

An important problem to understand in organometalliccomplexes is the energetics of charge injection and extrac-tion, and the influence of the character of the statesinvolved.31 When electrons �holes� are injected into a bulksample of our organometallic complex at an anode �cathode�,a five �three� electron state is formed on a complex. Whenthese oppositely ionized complexes are near each other theycan return to charge neutral states �charge recombination�with one complex in its neutral ground state, and the other ina neutral excited state. It is the photoemission from this ex-cited state which is desired in an OLED device. We wouldlike to predict the relative population of each emitting ex-cited state.

The typical lifetime of a singlet state is of the order of 10ns and for a triplet it is much longer �around 10 �s�. It isbelieved that in complexes such as these a singlet-triplet in-tersystem crossing can occur in 50 fs.17 This is more thanfive orders of magnitude greater than either excited state life-time. As such, the population of the excited states has plentyof time to become a thermal population. This means we canpredict the probability of finding either triplet or singlet ex-cited states in the thermal population based on the energydifference between those states.

The probability of the excited state being a triplet state ata temperature T given by a Boltzmann distribution is

PT =1

1 + 13e−�ES−ET�/�kBT�

, �16�

while the probability of finding it in a singlet state is PS=1− PT.

Increasing �� rapidly suppresses the triplet population,cf. Fig. 8. As �� increases, the probability of finding theexcitation in the triplet state decreases, toward the limit of75%. The triplet probability depends on precisely what

ligand and metal one uses to construct the complex due tothe probabilities sensitivity to ��. At ��=0.5 eV on the curvewith J=1.17 eV the probability of finding the excitation inthe triplet state is above 90%. The often quoted proportion of75% triplets, 25% singlets �see, for example, Ref. 56� is onlyreached in the limits �� /J→� or T→�.

In the limit tL=0 we can find the lowest singlet �triplet�excited state by solving Eq. �4� �Eq. �5��. Thus we have theenergy of the state with recombined charges forming an ex-cited singlet

ES =1

2�2� + 3�� +

3J

4− 2UM + 4UH + 6VHL + 2VHM�

−1

2���� +

3J

4�2

+ 4�tH�2,

or an excited triplet

ET =1

2�2� + 3�� −

J

4− 2UM + 4UH + 6VHL + 2VHM�

−1

2���� −

J

4�2

+ 4�tH�2.

The energy gap is

ES − ET =1

2�J +���� −

J

4�2

+ 4�tH�2

−���� +3J

4�2

+ 4�tH�2� . �17�

Thus we see again that small variations in parameter valuescan have large effects, in this case shifting the triplet prob-ability exponentially as shown in Fig. 8. As �� increases,there are two competing effects on the photoluminescent ef-ficiency: one is the suppression of triplet production down to75% seen in Fig. 8 and the other is the rapid decrease in

FIG. 8. Probability of injected charges forming a triplet excited state vs ��

for various values of J. This probability is highly sensitive to �� /J. It is clearthat increasing �� suppresses triplet formation. The point at which the rapiddecline from probability 1 begins is �� /J=1 /4, the point at which the tripletstate gains large MLCT character, bringing its energy close to that of thelowest excited singlet state. The values of J correspond to the ligands ppy�J=2.12 eV�, thpy �J=1.54 eV�, fluorene �J=1.17 eV�, and bzq�J=0.88 eV�, all at 300 K. This plot was made with the typical parametervalues �cf. Fig. 2 and the Appendix�, varying � from 0 to 2 eV.

124314-9 Photophysical properties of organometallics J. Chem. Phys. 133, 124314 �2010�

Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 130.102.82.69 On: Fri, 07 Oct 2016

04:31:18

triplet lifetime seen in Fig. 7. The condition ����J� neces-sary for a large triplet transition dipole moment also sup-presses the probability of the formation of a triplet excitedstate.

VI. CONCLUSIONS

We have investigated an effective model Hamiltonian fororganometallic complexes in electronic devices. We haveseen that while the lowest singlet state is typically a nearlypure MLCT state, the lowest triplet states character varies,changing from LC to MLCT with a highly hybridized regionbetween. This variation in triplet character is strongly depen-dent on the ratio �� /J. Importantly, �� is purely a property ofthe complex and will depend sensitively on the ligand chem-istry. The strong LC-MLCT mixing in the lowest triplet statemeans that a small shift in parameter values can cause largechanges in the properties of the state �changing �� by a factorof 2 changes the triplet lifetime by almost an order of mag-nitude�. This sensitive dependence provides an explanationfor the large observed changes in the photophysical proper-ties of organometallic complexes caused by small changes inthe ligands �such as changing a single substituent atom onthe ligand�. As well as having a direct effect on the lifetime,the change in excited state energy which accompanies thechange in hybridization causes a shift in the probability offinding the excitation in the triplet state. As �� increases, thetriplet decay rate increases by orders of magnitude while thetriplet probability decreases by at most 33%.

ACKNOWLEDGMENTS

We are grateful to Arthur Smith for critically reading thismanuscript and to Paul Burn, Arthur Smith, Paul Shaw,Lawrence Lo, and Seth Olsen for helpful discussions. B.J.P.was the recipient of an Australian Research Council �ARC�Queen Elizabeth II Fellowship �Project No. DP0878523�.R.H.M. was the recipient of an ARC Australian ProfessorialFellowship �Project No. DP0877875�.

APPENDIX: ESTIMATED PARAMETER VALUES

We would like to understand the values of and relation-ships between the Hamiltonian parameters �J, �, tH, tL, UH,UL, UM, VHL, VHM, and VLM�. We begin with a two-site ex-tended Hubbard model, modeling the ligand as two sites �forexample the two phenyl rings in a biphenyl ligand� occupiedby two electrons

H = − t���

�c1�† c2� + c2�

† c1�� + U��n1↑n1↓ + n2↑n2↓�

+ V� ��,��

n1�n2��, �A1�

with ni��ci�† ci�. By neglecting an exchange interaction be-

tween these localized orbitals, we are making the completeneglect of differential overlap approximation in this smallbasis set.57 By transforming to a basis of delocalized statesH�

† ��c1�† +c2�

† � /�2 and L�† ��c1�

† −c2�† � /�2, one finds that58

H = − t��nH − nL� − JS�H · S�L −J

2�H↑

†H↓†L↑L↓ + L↑

†L↓†H↑H↓�

+ UH�nH↑nH↓ + nL↑nL↓� + VHL�nH↑nL↓ + nH↓nL↑

+ nH↑nL↑ + nH↓nL↓� , �A2�

where

J = U� − V�, �A3�

UH = UL =U� + V�

2, �A4�

VHL =U� + 3V�

4. �A5�

1. Intraligand exchange: J¶1 eV

The first excited singlet state energy of the two-site Hub-bard model for the ligand is ES=VHL+ �3J /4�, and the firstexcited triplet state energy is ET=VHL− �J /4�. Thus we caneasily estimate J as

J = ES − ET. �A6�

Table II shows the value of J estimated in this way for sev-eral common ligands. This data shows that using a value ofJ�1 eV is realistic for investigating our model. While thepossible values of J vary widely, the effects of this variationin the lowest excited states are identical to variations in ��.Rusanova et al.15 found exchange interactions of the samemagnitude �around 1 eV�, having performed a semiempiricalINDO/S analysis on a series of ruthenium complexes. Theyalso assert that the degree of singlet-triplet splitting is a mea-sure of � backbonding. This feature is naturally reproducedin our model as the only way to have singlet-triplet splittingin our model is via the LC1 states, the only states directly

TABLE II. Data for several common ligands. Experimental energies for the lowest visible singlet and tripletstates, the calculated spin-exchange J using Eq. �A6�, and HOMO-LUMO gap � using Eq. �A7�.

Ligand ReferenceS1

�eV�T1

�eV�J

�eV��

�eV�

thpy 59 4.08 2.54 1.54 3.31ppy 59 4.99 2.87 2.12 3.93bzq 59 3.57 2.69 0.88 3.13Biphenyl p. 108, 60 4.33 2.84 1.49 3.59Carbazole p. 111, 60 3.60 3.05 0.55 3.33Fluorene p. 118, 60 4.11 2.94 1.17 3.53

124314-10 Jacko, Powell, and McKenzie J. Chem. Phys. 133, 124314 �2010�

Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 130.102.82.69 On: Fri, 07 Oct 2016

04:31:18

split by J. Figures S9 and S10 of the supplementary infor-mation show that our specific choice of J has no effect on thequalitative conclusions drawn in this paper.45

2. H-L splitting: �¶3 eV

The energy difference between H↑†H↓

†0 and L↑†L↓

†0 is2t�, and they are coupled by J /2. Thus the eigenstates will beseparated by an energy �4t�

2+J2 /4. Table II shows that J /2 isaround 0.5 eV, while the excited singlet to ground state gapis around 4 eV. We know the gap �4t�

2+J2 /4 will be greaterthan the singlet ground state gap of �4 eV. Since we knowJ /2 is an order of magnitude smaller than this energy, wemake the approximation that the states H↑

†H↓†0 and L↑

†L↓†0

are eigenstates �implying that �= t��. We find the groundstate is H↑

†H↓†0 with energy −�+UH. The ground state-

singlet gap is �S= �J /2�+� and the ground state-triplet gapis �T=−�J /2�+� �using UH−VHL=J /4�. Thus we find

� =�S + �T

2. �A7�

We apply this to spectral data from isolated ligands to esti-mate the value of � in Table II, finding that ��3 eV is arealistic value to use in the investigations of the properties ofthe model Eq. �1�. Figures S7, S9, S11, S12, and S15–S17 ofthe supplementary information show that our specific choiceof � within the range of possible values in Table II has noeffect on the lowest excited states and therefore no effect onthe conclusions is drawn here.45

3. Direct Coulomb interactions on the ligand: UH¶VHL¶3 eV

There is an empirical relationship between the U� on thelocalized sites and the V� between the localized sites, interms of the intersite spacing R,

V�−1 = R + U�

−1, �A8�

in atomic units �p. 20 of Ref. 40�. If we substitute Eq. �A3�we find that

U� =J

2+�J2

4+

J

R. �A9�

Figure 9 shows that U� should be somewhere between 2.5and 3.75 eV, given the possible range of R and a typicalvalue of J=1 eV. It is worth noting here that a factor of 3change in bond length is only a factor of 1.5 in the magni-tude of U.

For J=1 eV we have

UH = U� − 12 eV, VHL = U� − 3

4 eV, �A10�

hence

UH = VHL + 14 eV. �A11�

We expect UH to be in the range 3.25–2 eV �based on theabove range for U��. UH will be much larger than 0.25 eV,hence we can approximate UH�VHL. This analysis makes itseem reasonable to choose UH=VHL=3 eV for our typical

parameter set used to investigate the model. Reference 15evaluates direct Coulomb integrals for a series of Ru com-plexes, finding values �4.5 eV �calculated via semiempir-ical INDO/S computations�, similar to the values found fromthe above discussion. As discussed in Ref. 39, this kind ofcalculation is at best a reasonable upper bound for the valueof the parameters in an effective low energy Hamiltonian.Figures S7 and S8 of the supplementary information showthat relaxing the assumption that UH=VHL does not causeany qualitative changes in the solutions of the model.45

4. Direct Coulomb interactions involving the metalsite: UM¶UH ,VHL¶VHM

Since the HOMO and LUMO are in the same locationand have the same on-site Coulomb repulsion, we find thatthe intersite Coulomb repulsion between the ligand orbitalsand the metal will be equal,

VHM−1 = VLM

−1 = RHM +2

UH + UM. �A12�

We expect that R12�RHM �where R12 is the distance betweenthe two sites of our model of the ligand and RHM is thedistance between the ligand and the metal�. As long as UM

�UH, we will have VHM =VLM �VHL=UH. This reduces thesix parameters for the direct Coulomb integrals �UH, UL, UM,VHL, VHM, and VLM� to just two �UH and UM�.

If we were to assume that UH�UM then each four elec-tron basis state gains an energy 6UH relative to the case withno Coulomb interactions. Thus it is clear that in this approxi-mation the direct Coulomb interactions have no qualitativeeffect on the solutions to this Hamiltonian in the four elec-tron subspace.

For our typical parameter values, the nL=1 states arewell separated from the nL�1 states. This means that whilewe are investigating the lowest excited states �the nL=1states� varying �� captures all the same physics as varying �,UM, UH, VLM, and VHL individually. As such, it is convenientto choose UM =UH and VHL=VHM and then investigate theeffects of changing ��.

FIG. 9. Predicted U� as a function of R with the constraint that we matchour typical value of J=U�−V��1 eV. The dotted vertical red lines are atthe carbon-carbon bond length in benzene 1.4 Å=2.65aB and the distancebetween benzene ring centers in biphenyl, 3.2 Å.

124314-11 Photophysical properties of organometallics J. Chem. Phys. 133, 124314 �2010�

Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 130.102.82.69 On: Fri, 07 Oct 2016

04:31:18

Figures S1–S4, S6–S8, and S11–S17 of the supplemen-tary information show that varying � and UM �equivalent tovarying VLM� cause no qualitative changes to the solutions ofthe model in reasonable parameter ranges.45 We must in-crease UM more than 3 eV above the typical value beforethere are any qualitative changes to the lowest excited stateswhich would alter the conclusions drawn here �see Figs. S3and S4�.45

5. Hopping integrals: tH¶ tL¶0.1 eV

Using the standard semiempirical parametrization �forexample, p. 551 of Ref. 61�, along with Hückel HOMO andLUMO orbitals of an isolated ligand and experimentalcarbon-metal and nitrogen-metal bond lengths, we estimatethe values of tH and tL for various ligands given in Table III.Note that the variation in the t values is almost completelydue to the differences between the Hückel orbitals �the varia-tion due to the different experimental bond lengths is �1%�.

6. HOMO-metal splitting, ε, is a property of thecomplex

� is a property only of the whole complex which is dif-ficult to predict a priori. In Fig. 4 we choose a value of �=0.25 eV for the sake of concreteness. Figures S6 and S15–S17 in the supplementary information show that this choicehas no effect on our conclusions regarding the lowest excitedstates.45 Figures S1–S4, S6–S8, and S11–S17 of the supple-mentary information show that varying � and UM and hence�� causes no qualitative differences in the solutions of themodel, as discussed above in the section on direct Coulombintegrals involving the metal site.45

1 R. H. Friend, R. W. Gymer, A. B. Holmes, J. H. Burroughes, R. N.Marks, C. Taliani, D. D. C. Bradley, D. A. Dos Santos, J. L. Brédas, M.Lögdlund, and W. R. Salaneck, Nature �London� 397, 121 �1999�.

2 S. R. Forrest, Nature �London� 428, 911 �2004�.3 J. Li, P. I. Djurovich, B. D. Alleyne, M. Yousufuddin, N. N. Ho, J. C.Thomas, J. C. Peters, R. Bau, and M. E. Thompson, Inorg. Chem. 44,1713 �2005�.

4 S.-C. Lo, C. P. Shipley, R. N. Bera, R. E. Harding, A. R. Cowley, P. L.Burn, and I. D. W. Samuel, Chem. Mater. 18, 5119 �2006�.

5 B. O’Regan and M. Grätzel, Nature �London� 353, 737 �1991�.6 A. Hagfeldt and M. Grätzel, Acc. Chem. Res. 33, 269 �2000�.7 C. Dimitrakopoulos and P. Malenfant, Adv. Mater. �Weinheim, Ger.� 14,99 �2002�.

8 G. Fernández, L. Sánchez, D. Veldman, M. M. Wienk, C. Atienza, D. M.Guldi, R. A. J. Janssen, and N. Martín, J. Org. Chem. 73, 3189 �2008�.

9 H. Yersin, D. Donges, W. Humbs, J. Strasser, R. Sitters, and M. Glas-beek, Inorg. Chem. 41, 4915 �2002�.

10 M. Nazeeruddin, A. Kay, I. Rodicio, R. Humphry-Baker, E. Mueller, P.Liska, N. Vlachopoulos, and M. Gratzel, J. Am. Chem. Soc. 115, 6382�1993�.

11 H. Yersin and W. J. Finkenzeller, Highly Efficient OLEDs with Phospho-rescent Materials �Wiley, New York, 2008�, Chap. 1.

12 F. R. Hartley, Chem. Soc. Rev. 2, 163 �1973�.13 C. Creutz and H. Taube, J. Am. Chem. Soc. 91, 3988 �1969�.14 J. R. Reimers and N. S. Hush, J. Phys. Chem. 95, 9773 �1991�.15 J. Rusanova, E. Rusanov, S. I. Gorelsky, D. Christendat, R. Popescu, A.

A. Farah, R. Beaulac, C. Reber, and A. B. P. Lever, Inorg. Chem. 45,6246 �2006�.

16 E. M. Kober and T. J. Meyer, Inorg. Chem. 21, 3967 �1982�.17 S. Yoon, P. Kukura, C. M. Stuart, and R. A. Mathies, Mol. Phys. 104,

1275 �2006�.18 M. Segal, M. A. Baldo, R. J. Holmes, S. R. Forrest, and Z. G. Soos, Phys.

Rev. B 68, 075211 �2003�.19 A. L. Burin and M. A. Ratner, J. Chem. Phys. 109, 6092 �1998�.20 M. Reufer, M. Walter, P. G. Lagoudakis, A. B. Hummel, J. S. Kolb, H. G.

Roskos, U. Scherf, and J. M. Lupton, Nature Mater. 4, 340 �2005�.21 R. Schwarz, G. Gliemann, L. Chassot, P. Jolliet, and A. von Zelewsky,

Helv. Chim. Acta 72, 224 �1989�.22 M. G. Colombo, A. Hauser, and H. U. Guedel, Inorg. Chem. 32, 3088

�1993�.23 S. Lamansky, P. Djurovich, D. Murphy, F. Abdel-Razzaq, H.-E. Lee, C.

Adachi, P. E. Burrows, S. R. Forrest, and M. E. Thompson, J. Am. Chem.Soc. 123, 4304 �2001�.

24 H. Yersin and D. Donges, Top. Curr. Chem. 214, 81 �2001�.25 W. J. Finkenzeller, T. Hofbeck, M. E. Thompson, and H. Yersin, Inorg.

Chem. 46, 5076 �2007�.26 S. Haneder, E. D. Como, J. Feldmann, J. M. Lupton, C. Lennartz, P. Erk,

E. Fuchs, O. Molt, I. Münster, C. Schildknecht, and G. Wagenblast, Adv.Mater. �Weinheim, Ger.� 20, 3325 �2008�.

27 B. J. Powell, e-print arXiv:0906.1640.28 G. Blondin and J. J. Girerd, Chem. Rev. �Washington, D.C.� 90, 1359

�1990�.29 H. Nagao, M. Nishino, Y. Shigeta, T. Soda, Y. Kitagawa, T. Onishi, Y.

Yoshioka, and K. Yamaguchi, Coord. Chem. Rev. 198, 265 �2000�.30 M. X. LaBute, R. V. Kulkarni, R. G. Endres, and D. L. Cox, J. Chem.

Phys. 116, 3681 �2002�.31 A. C. Jacko, R. H. McKenzie, and B. J. Powell, “Models of organome-

tallic complexes for optoelectronic applications,” J. Mater. Chem. �2010�.32 G. A. Worth and L. S. Cederbaum, Annu. Rev. Phys. Chem. 55, 127

�2004�.33 F. Terenziani, A. Painelli, C. Katan, M. Charlot, and M. Blanchard-

Desce, J. Am. Chem. Soc. 128, 15742 �2006�.34 J. Gilmore and R. H. McKenzie, J. Phys.: Condens. Matter 17, 1735

�2005�.35 A. C. Hewson, The Kondo Problem to Heavy Fermions �Cambridge Uni-

versity Press, Cambridge, 1997�.36 L. Kouwenhoven and L. Glazman, Phys. World 14, 33 �2001�.37 P. G. Bomben, K. C. D. Robson, P. A. Sedach, and C. P. Berlinguette,

Inorg. Chem. 48, 9631 �2009�.38 K. F. Freed, Acc. Chem. Res. 16, 137 �1983�.39 E. Scriven and B. J. Powell, J. Chem. Phys. 130, 104508 �2009�.40 P. Fulde, Electron Correlations in Molecules and Solids, 3rd ed.

�Springer, Berlin, 1995�.41 O. Gunnarsson, Alkali-Doped Fullerides: Narrow-Band Solids with Un-

usual Properties �World Scientific, Singapore, 2004�.42 G. Brocks, J. van den Brink, and A. F. Morpurgo, Phys. Rev. Lett. 93,

146405 �2004�.43 L. Cano-Cortés, A. Dolfen, J. Merino, J. Behler, B. Delley, K. Reuter,

and E. Koch, Eur. Phys. J. B 56, 173 �2007�.44 E. Scriven and B. J. Powell, Phys. Rev. B 80, 205107 �2009�.45 See supplementary material at http://dx.doi.org/10.1063/1.3480981 for

further figures illustrating the robustness of our conclusions to the choiceof parameter values.

46 P. J. Hay, J. Phys. Chem. A 106, 1634 �2002�.47 A. Ghosh, JBIC, J. Biol. Inorg. Chem. 11, 712 �2006�.48 A. J. Cohen, P. Mori-Sanchez, and W. Yang, Science 321, 792 �2008�.49 R. M. Hochstrasser, Molecular Aspects of Symmetry �Benjamin, New

York, 1966�.50 R. C. Hilborn, e-print arXiv:physics/0202029v1.51 W. Gawelda, A. Cannizzo, V.-T. Pham, F. van Mourik, C. Bressler, and

M. Chergui, J. Am. Chem. Soc. 129, 8199 �2007�.

TABLE III. Values of tH for various platinum complexes, estimated with asemiempirical parameterization and using HOMOs and LUMOs fromHückel model calculations of isolated ligands using the parameterization onpage 284 of Ref. 62. The ligand—metal bond lengths for the complexescome from crystallographic data contained in the references in the table.

Pt thpya ppyb bzqc

tH �eV� 0.09 0.08 0.08tL �eV� 0.11 0.11 0.06

aReference 63.bReference 64.cReference 65.

124314-12 Jacko, Powell, and McKenzie J. Chem. Phys. 133, 124314 �2010�

Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 130.102.82.69 On: Fri, 07 Oct 2016

04:31:18

52 L. Grisanti, G. D’Avino, A. Painelli, J. Guasch, I. Ratera, and J. Veciana,J. Phys. Chem. B 113, 4718 �2009�.

53 I. B. Bersuker, Electronic Structure and Properties of Transition MetalCompounds: Introduction to the Theory �Wiley, New York, 1996�.

54 M. A. Baldo, S. Lamansky, P. E. Burrows, M. E. Thompson, and S. R.Forrest, Appl. Phys. Lett. 75, 4 �1999�.

55 M. A. Baldo, D. F. O’Brien, Y. You, A. Shoustikov, S. Sibley, M. E.Thompson, and S. R. Forrest, Nature �London� 395, 151 �1998�.

56 Y. Sun, N. C. Giebink, H. Kanno, B. Ma, M. E. Thompson, and S. R.Forrest, Nature �London� 440, 908 �2006�.

57 J. A. Pople, D. P. Santry, and G. A. Segal, J. Chem. Phys. 43, S129�1965�.

58 V. Bonačić-Koutecký, J. Koutecký, and J. Michl, Angew. Chem., Int. Ed.Engl. 26, 170 �1987�.

59 M. Maestri, D. Sandrini, V. Balzani, L. Chassot, P. Jolliet, and A. VonZelewsky, Chem. Phys. Lett. 122, 375 �1985�.

60 M. Montalti, A. Credi, L. Prodi, and M. T. Gandolfi, Handbook of Pho-tochemistry �Taylor & Francis, London, 2006�.

61 W. A. Harrison, Electronic Structure and the Properties of Solids: ThePhysics of the Chemical Bond �Dover, New York, 1989�.

62 J. P. Lowe and K. Peterson, Quantum Chemistry �Academic Press, NewYork, 2005�.

63 J. Breu, K.-J. Range, A. von Zelewsky, and H. Yersin, Acta Crystallogr.,Sect. C: Cryst. Struct. Commun. 53, 562 �1997�.

64 L. Chassot, E. Müller, and A. von Zelewsky, Inorg. Chem. 23, 4249�1984�.

65 P. Jolliet, M. Gianini, A. von Zelewsky, G. Bernardinelli, and H.Stoeckli-Evans, Inorg. Chem. 35, 4883 �1996�.

124314-13 Photophysical properties of organometallics J. Chem. Phys. 133, 124314 �2010�

Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 130.102.82.69 On: Fri, 07 Oct 2016

04:31:18