Table Of ContentConservation laws, radiative decay rates, and excited state localization in
organometallic complexes with strong spin-orbit coupling
B. J. Powell
Centre for Organic Photonics and Electronics, School of Mathematics and Physics,
The University of Queensland, Brisbane, Queensland, 4072, Australia
There is longstanding fundamental interest in 6-fold coordinated d6 (t6 ) transition metal com-
2g
plexes such as [Ru(bpy) ]2+ and Ir(ppy) , particularly their phosphorescence. This interest has
3 3
increasedwiththegrowingrealisationthatmanyofthesecomplexeshavepotentialusesinapplica-
tions including photovoltaics, imaging, sensing, and light-emitting diodes. In order to design new
complexes with properties tailored for specific applications a detailed understanding of the low-
5 energy excited states, particularly the lowest energy triplet state, T , is required. Here we describe
1
1 a model of pseudo-octahedral complexes based on a pseudo-angular momentum representation and
0 showthatthepredictionsofthismodelareinexcellentagreementwithexperiment-evenwhenthe
2
deviationsfromoctahedralsymmetryarelarge. Thismodelgivesanaturalexplanationofzero-field
n splittingofT1andoftherelativeradiativeratesofthethreesublevelsintermsoftheconservationof
a time-reversal parity andtotal angular momentummodulo two. We show that the broad parameter
J regime consistent with the experimental data implies significant localization of the excited state.
5
] I. INTRODUCTION Alongstandingquestioninthesecomplexesiswhether
h
the excited state is localized to a single ligand or delo-
p
calized [2]. The main semi-empricial approach to under-
- Six-fold coordinated d6 (t6 ) transition metal com-
m 2g standing organometallic complexes is ligand field theory.
plexes, such as those shown in Fig. 1a,b, share many
Onceallofthespatialsymmetriesarebrokenthereislig-
e common properties. These include their marked similar-
h and field theory is limited to a perturbative regime near
ities in their low-energy spectra [1], cf. Table I, and the
c approximatesymmetries,thismakesanaccuratedescrip-
. competition between localization and delocalizsation in tion of localised excited states challenging.
s
their excited states [2]. Beyond their intrinsic scientific
c In this paper we describe a semi-empirical approach,
i interest,understandingandcontrollingthisphenomenol-
s based on the pseudo-angular momentum approach that
y ogy is further motivated by the potential for the use has found widespread use in, e.g., interpreting electron
h of such complexes in diverse applications including dye- paramagnetic resonance experiments. We derive con-
p sensitized solar cells, non-linear optics, photocatalysis,
servation laws based on the total angular momentum
[ biological imaging, chemical and biological sensing, pho-
(pseudo plus spin) that apply even when the pseudo-
todynamic therapy, light-emitting electro-chemical cells
1 octahedral and trigonal symmetries are strongly broken.
v and organic light emitting diodes [1, 3–7]. As many of These conservation laws imply selection rules for radia-
7 these applications make use of the excited state prop-
tive emission. We show that this model reproduces the
3 erties of these complexes a deep understanding of the
experimentally measured trends in the radiative decay
7 low-energy excited states, particularly the lowest energy
rates and excitation energies for all of the complexes for
0
triplet state, T , is required to enable the rational design
0 1 which we have data to compare with in the literature.
of new complexes.
. These trends are insensitive to the parameters of the
1
Coordination complexes where there is strong spin- model studied. Finally, we show that for the wide pa-
0
5 orbit coupling (SOC) present a particular challenge to rameter range compatible with experiment the pseudo-
1 theory because of the need to describe both the ligand angular momentum model predicts significant localiza-
: field and the relativistic effects correctly. There has tion of the excited state.
v
been significant progress in applying relativistic time-
i
X dependent density functional theory (TDDFT) to such
r complexes;butsignificantchallengesremain,forexample II. THE PSEUDO-ANGULAR MOMENTUM
a correctly describing the zero-field splitting [8–11]. There MODEL.
hasbeenlessrecentfocusontheuseofsemi-empiricalap-
proaches,suchasligandfieldtheory[8,12–14]. However, It has long been understood [16] that the three-fold
semi-empiricalapproacheshaveanimportantroletoplay degeneratestatescanberepresentedbyanl=1pseudo-
[15]. Firstly, they provide a general framework to un- angularmomentum. Perhapsthebestknownexampleof
derstand experimental and computational results across this are the t states of a transition metal in an octa-
2g
whole classes of complexes. Secondly, when properly pa- hedral ligand field. In the d6 complexes considered here
rameterised they can provide accuracy that is compet- thet orbitalsarefilled,whereasthee -orbitalsarehigh
2g g
itive with first principles methods. Thirdly, they can lyingvirtualstates. Thereforeweonlyincludethet or-
2g
provide general design rules that allow one to effectively bitals in the model described below.
target new complexes for specific applications. The complexes listed in Table I have 6-fold coordi-
2
a) b) can label such states by two l = 1 pseudo-angular mo-
menta,whichwedenoteL andL respectively. Wewill
H L
only discuss this assignment for three real space HOMO
spin orbitals, hm – it is trivial to extend the following
t2g
analysis to the LUMOs. By referring to these states as
‘HOMOs’ and ‘LUMOs’ we are adopting the language of
molecular orbital theory. However, we note that so-long
as the hm are three local states related by rotations of
t2g
2π/3 the discussion below goes through regardless of the
degree of correlations in the states. It is therefore conve-
nient to work in second quantised notation, so we define
(cid:104)r|a† |0(cid:105)=hm , where |0(cid:105) is the ground state, |r(cid:105) is the
m t2g
state with a hole at position r; spin labels are supressed.
We introduce three ‘Bloch’ operators defined by
1 (cid:88)
b+ =sgnk(−k)√ a† ei2πkm/3, (2)
k 3 m
m
c) d)
where k ∈ {−1,0,1}. Finally we identify the states cre-
ated by the Bloch operators with the eigenstates of Lz ,
H
i.e., (cid:104)0|b Lz b†|0(cid:105)=k. The phase pre-factors [sgnk(−k)]
k H k
inthedefinitionoftheBlochoperatorsarerequiredtoal-
lowthisassignmentandmaintaintherequiredbehaviour
FIG. 1: The structures of two important pseudo-octahedral under time reversal symmetry.
transitionmetalcomplexes: a)[Ru(bpy)3]2+ andb)Ir(ppy)3, As the LUMOs are pure ligand orbitals the exchange
wherebpyisbipyridineandppyis2-phenylpyridyl. Sketches
interactionwillbedominatedbytheexchangeinteraction
ofthec)π andd)π∗ orbitalsofabpyligandwiththereflec-
betweentheligandπ andπ∗-orbitals,J . Incontrastthe
tion plane marked by the dashed line. It is clear that these π
SOC on the metal, λ , is much stronger than the SOC
correspond to the bonding and antibonding combinations of d
on the ligands. Therefore states with one hole in the
singly occupied molecular orbitals of a pyridine radical.
HOMO and one electron in the LUMO are described by
the Hamiltonian
nated metal atoms, but the ligands break the octahe- H =JS ·S +λL ·S , (3)
o H L H H
dral symmetry. In complexes with D symmetry, e.g.,
3
[Ru(bpy) ]2+, theligandshaveareflectionsymmetry, cf. whereS isthe(net)spinoftheelectronsintheHOMO,
3 H
Fig. 1c,d. For a single bpy ligand the highest energy S ,isthespinoftheelectronintheLUMO,J (cid:39)J sin2θ
L π
ligand π-orbitals are even under this reflection whereas and λ(cid:39)λ cos2θ. Thus we expect positive J and λ.
d
the lowest energy π∗-orbitals are odd under the same If the excited state is sufficiently long lived for the
reflection, as one would expect from simple symmetry geometry to relax it will be unstable to a Jahn-Teller
arguments [13]. Therefore, linear combinations the π- distortion, which lifts the degeneracy. In terms of the
orbitals transform corresponding to the t representa- pseudo-angular momenta this can be represented via the
2g
tionofO ,whereastheπ∗-orbitalsformarepresentation terms
h
of t . Therefore, π-orbitals mix effectively with the oc-
cup1ieudmetald(t2g)orbitalsbutπ∗-orbitalsdonot. Thus HJT = δQ(cid:2)(LxH)2−(LyH)2(cid:3)+γQ(cid:2)(LxL)2−(LyL)2(cid:3)
the highest occupied molecular orbitals (HOMOs), hm , +kQ2, (4)
t2g
of the complex will have a significant contribution from
both the ligand π-orbitals, πm , and the metal-t or- where Q is the coordinate of the rhombic distortion per-
bitals, dm . Neglecting smalletr2gcontributions from2ogther pendicular to the C -axis of the complex, δ (γ) is the
t2g 3
ligand or metal orbitals, we have coupling constant to the HOMOs (LUMOs) and k is the
spring constant of the Jahn-Teller mode.
hm (cid:39)dm cosθ+πm sinθ, (1)
t2g t2g t2g It is helpful to briefly discuss the Jahn-Teller effect in
where θ parameterises the degree of mixing, and m ∈ the pseudo-angular momentum language, as this is not
{1,2,3} labels the ligands and symmetry equivalent lin- entirely intuitive. Consider the term
ear combinations of d-orbitals. In contrast, the lowest
1
unoccupied molecular orbitals (LUMOs) of the complex [(Lx )2−(Ly )2]= [(L+)2+(L−)2] (5)
will be almost pure ligand π∗-orbitals. H H 2 H H
Low energy excited states can be well approximated
In terms of the Bloch operators L† = 2(b†b +b†b )
byasingleholeintheHOMOmanifoldandasingleelec- H 1 0 0 −1
tron in the LUMO manifold [17]. As both the HOMOs and L−H =2(b†−1b0+b†0b1). Hence,
andLUMOsofthecomplexarethree-folddegenerateone
3
E [cm−1] E [cm−1] τ (1/kI) [µs] τ (1/kII) [µs] τ (1/kIII) [µs]
I,II II,III I R II R III R
Ir(biqa) 14 64 107 (114) 5.6 (5.7) 0.36 (0.38)
3
Ir(ppy) (in PMMA) 12.2 113 154 (175) 15 (17) 0.33 (0.34)
3
Ir(ppy) (in CH Cl ) 19 151 116 6.4 0.2
3 2 2
Ir(dm-2-piq) (acac) 9.5-10 140-150 80-124 6.5-8.6 0.33-0.44
2
[Os(phen)] (dppm)]2+ 16 106 95 13 0.6
2
[Os(phen) (dpae)]2+ 21 92 100 10 0.7
2
Ir(piq)(ppy) 16 91 64 10.5 0.3
2
Ir(4,6-dFppy) (acac) 16 93 44 9 0.4
2
Ir(pbt) (acac) 6 97 82 25 0.4
2
Ir(piq) (acac) 9 87 47 8 0.3
2
[Os(dpphen) (dpae)]2+ 19 75 92 9 0.7
2
[Os(phen) (DPEphos)]2+ 16 68 104 14 0.9
2
[Os(phen) (dppe)]2+ 19 55 107 12 0.9
2
Ir(piq) (ppy) 9 56 60 6.4 0.44
2
[Os(phen) (dppene)]2+ 18 46 108 15 1.1
2
[Ru(ppy) ]2+ 8.7 52 230 8 0.9
3
Ir(piq) 11 53 57 5.3 0.42
3
Ir(4,6-dFppy) (pic) 9 67 47 21 0.3
2
Ir(thpy) (acac) 3.5 31 113 35 1.5
2
Ir(ppy) (ppy-NPH ) 6 21 188 19 1.8
2 2
Ir(ppy-NPH ) 6 20 177 15 1.4
2 3
Ir(ppy)(ppy-NPH ) 6 17 163 20 2
2 2
Ir(btp) (acac) 2.9 22 150 58 2
2
Ir(btp) (acac) 2.9 11.9 62 19 3
2
Ir(s1-thpy) (acac) 3 13 128 62 3
2
Ir(ppy) (CO)(Cl) <1 <1 300 85 9
2
[Rh(ppy) ]3+ - - 4.5×103 1.35×103 650
3
TABLE I: Key spectroscopic data for pseudo-octahedral d6-complexes. E is the energy gap between the two lowest energy
I,II
substates of T , E is the energy gap between the two highest energy substates of T and the total lifetime of substate
1 II,III 1
m τ = (km +km )−1, where km and km and the radiative and non-radiative lifetimes of substate m. For Ir(ppy) and
m R NR R NR 3
Ir(biqa) wealsolist1/km (inbold)which,unsurprisinglygiventhehighphotoluminescentquantumyieldsinthesecomplexes,
3 R
showsthesametrendasτ . Wearenotawareofmeasurementsofkm inotherrelevantcomplexes. Notethatinallcomplexes
m R
E < E and τ > τ > τ , which suggests that kI < kII < kIII. To avoid selection bias we have included all and
I,II II,III I II III R R R
only those pseudo-octahedral d6-complexes included in Table 2 of the recent review by Yersin et al. [1]. The two rows for
Ir(btp) (acac) correspond to different sites.
2
(Lx )2−(Ly )2 = b†b +b† b
H H 1 −1 −1 −1
1(cid:104) (cid:105)
= − 2a†a −a†a −a†a +2(a†a +a†a )−a†a −a†a −a†a −a†a . (6)
3 1 1 2 2 3 3 2 3 3 2 1 2 2 1 1 3 3 1
It is therefore clear that this physics of H is that of In general the Jahn-Teller distortion could also induce
JT
the T ×t Jahn-Teller problem [or, once trigonal terms atrigonalcomponentofthedistortion[whichwouldcou-
are included, below, the (A+E)×e pseudo-Jahn-Teller ple to (Lz)2, where ν = H or L], however this does not
ν
problem] and that this distortion corresponds to the so- produce any qualitatively new features and so, for sim-
called E distortion in the notation of, e.g., section 5 of plicity, we neglect it below. Thence, the form of H is
θ JT
Ref. [18]. The E distortion corresponds to terms pro- constrainedtotheformgivenabovebysymmetryas: (1)
ε
portional to (1/2)[(L+)2 −(L−)2] = i{Lx ,Ly }, where terms that are proportional to odd powers of Lβ, where
H H H H ν
curly brackets indicate anticommutation. β =x, y or z break time reversal symmetry and so may
4
not appear in the Hamiltonian for scalar Q and (2) for
l=1 any even power of Lβ is proportional to (Lβ)2.
ν ν
However, the complexes in Table I are not octahedral,
but trigonal. In terms of the pseudo-angular momenta,
this introduces the additional terms
H =∆(Lz )2+Γ(Lz)2, (7)
t H L
where∆(Γ)istheenergydifferencesbetweentheHOMO
and HOMO-1 (LUMO and LUMO+1) in the trigonal
ground state, S , geometry. Indeed, it immediately fol-
0
lowsfromtimereversalsymmetrythatthetrigonalterms
in the Hamiltonian are constrained to take this form.
t2g → a1 + e and t1u → a2 + e on lowering the sym- FIG. 2: Energy eigenvalues of Ho for states with Lz = 0.
metry from O to D . Therefore, the two pairs of e At λ = 0 the singlets have E = 3J/4 and the triplets have
h 3
statesareallowedtoweaklymix,stabilisingtheLz =±1 E =−J/4. Forλ>0thelabels“singlet”and“triplet”areno
H longer strictly defined (in their usual sense) nevertheless the
states and destabilising the Lz = ±1 states. Thus one
L relatively small energy shifts suggest that these labels retain
expects that both ∆ and Γ will be positive [13]. The
some meaning, this claim is supported by directly examining
approximate D symmetry of the complexes with lower
3 thecharacteroftheeigenstates. Itisinterestingtonotethat,
symmetry, e.g. C3, complexes considered here means already in the octahedral problem, the lowest energy (non-
that we expect both parameters to remain positive for degenerate)statehasnosingletcontributiontoitswavefunc-
all of the complexes considered here [19]. Thus the ef- tion for any value of λ, thus radiative transitions from this
fective pseudo-angular momentum Hamiltonian for the state are forbidden.
low-energy excitations is
H =H +H +H . (8)
o t JT of H . (i) L does not couple to any of the other vari-
o L
ables. Therefore, L2 and Lz are good quantum num-
By definition Q = 0 in the S geometry and, by suit- L L
0 bers. (ii) We can define a ‘total’ angular momentum,
ably rescaling the parameters, one may define Q = 1 in
I = L + S, where S = S + S . I2 and Iz com-
the T geometry. Similarly any trigonal component to H H L
1 mute with H therefore I and Iz are also good quantum
the Jahn-Teller distortion can be taken simply to shift o
numbers.
the value of ∆ (Γ). Therefore, up to constants, in the T
1 We plot the energies of the exact solutions of H in
geometry the effective electronic Hamiltonian is o
Fig. 2 (Table II gives the basis used for all calculations
H = JS ·S +λL ·S +∆(Lz )2+Γ(Lz)2 in the paper). For simplicity Fig. 2 shows only the so-
H L H H H L lutions with Lz =0 – because L is decoupled from the
+δ[(Lx )2−(Ly )2]+γ[(Lx)2−(Ly)2]. (9) L L
H H L L other angular momenta it can be immediately seen that
the other solutions simply triple the degeneracies of all
As well as describing systems displaying a Jahn-Teller
states. Note that, firstly, the spectrum of H is not very
distortion, this model is also appropriate for heterolep- o
similar to those of the pseudo-octahedral complexes we
tic complexes. Indeed for appropriate choices of ∆, δ, Γ
are seeking to model. However, this model is an impor-
and γ one can parameterise arbitrary energy differences
tant stepping stone to understanding the full Hamilto-
of the frontier orbitals. We discuss the values of these
nian. Secondly, the eigenstates can be classified by their
parameters in the Appendix. On the basis of this dis-
total angular momentum quantum number, I, and, as
cussion, for Ir(ppy) , we take λ/J =0.2 and ∆/J =0.5,
3 H is SU(2) symmetric, have the expected 2I+1 degen-
with J ∼ 1 eV; δ (cid:46) ∆ and γ (cid:46) Γ below. Clearly, for o
eracy. Thirdly, all of the singlets have I =1; as L =1
example,λisstronglydependentonthetransitionmetal H
and,bydefinition,singletshaveS =0. Thismeansthat,
in question. However, our main qualitative results are
regardlessofhowstrongtheSOCis,thesingletscanonly
insensitivetothevaluesoftheseparameters–toempha-
mix with the I = 1 triplets. Therefore radiative decay
size this we explore a wide range of other parameters in
from the I = 0 and I = 2 triplets is forbidden by the
the sup. info.
conservation of I.
III. RESULTS B. Trigonal model
A. Octahedral model In Fig. 3 we plot the spectrum of the trigonal model
with no Jahn-Teller distortion, H +H . Again, for sim-
o t
Before considering the full pseudo-angular momentum plicity, we only show the solutions with Lz = 0. In this
L
model, H, it is important to understand the symmetries case each state has partners with Lz =±1 that have en-
L
5
Relationship to eigenstate Singlets
Name of H when λ=0, T Iz |SzSzLz (cid:105) mixed with
o L H H
|I,Iz,S(cid:105) in full model
|Sz(cid:105) |1,0,0(cid:105) -1 1 √12(|↑↓⇒(cid:105)−|↓↑⇒(cid:105)) -
|Sx(cid:105) √12(|1,1,0(cid:105)−|1,−1,0(cid:105)) 1 -1 12(|↑↓⇑(cid:105)−|↓↑⇑(cid:105)−|↑↓⇓(cid:105)+|↓↑⇓(cid:105)) -
|Sy(cid:105) √12(|1,1,0(cid:105)+|1,−1,0(cid:105)) -1 -1 12(|↑↓⇑(cid:105)−|↓↑⇑(cid:105)+|↑↓⇓(cid:105)−|↓↑⇓(cid:105)) -
|T1(cid:105) |0,0,1(cid:105) 1 1 √1 (|↑↑⇓(cid:105)+|↓↓⇑(cid:105))− √1 (|↑↓⇒(cid:105)+|↓↑⇒(cid:105)) None
3 6
|Tz(cid:105) |1,0,1(cid:105) -1 1 (cid:104) √12(|↑↑⇓(cid:105)−|↓↓⇑(cid:105)) (cid:105) |Sz(cid:105)
|Tx(cid:105) √12(|1,1,1(cid:105)−|1,−1,1(cid:105)) -1 -1 12 |↑↑⇒(cid:105)−|↓↓⇒(cid:105)− √12(|↑↓⇑(cid:105)+|↓↑⇑(cid:105)−|↑↓⇓(cid:105)−|↓↑⇓(cid:105)) |Sy(cid:105)
(cid:104) (cid:105)
|Ty(cid:105) √12(|1,1,1(cid:105)+|1,−1,1(cid:105)) 1 -1 12 |↑↑⇒(cid:105)+|↓↓⇒(cid:105)− √12(|↑↓⇑(cid:105)+|↓↑⇑(cid:105)+|↑↓⇓(cid:105)+|↓↑⇓(cid:105)) |Sx(cid:105)
|Tz2(cid:105) |2,0,1(cid:105) 1 1 (cid:104) √16(|↑↑⇓(cid:105)+|↓↓⇑(cid:105))+ √13(|↑↓⇒(cid:105)+|↓↑⇒(cid:105)) (cid:105) None
|Txz(cid:105) √12(|2,1,1(cid:105)−|2,−1,1(cid:105)) -1 -1 12 |↑↑⇒(cid:105)−|↓↓⇒(cid:105)+ √12(|↑↓⇑(cid:105)+|↓↑⇑(cid:105)−|↑↓⇓(cid:105)−|↓↑⇓(cid:105)) |Sy(cid:105)
(cid:104) (cid:105)
|Tyz(cid:105) √12(|2,1,1(cid:105)+|2,−1,1(cid:105)) 1 -1 12 |↑↑⇒(cid:105)+|↓↓⇒(cid:105)+ √12(|↑↓⇑(cid:105)+|↓↑⇑(cid:105)+|↑↓⇓(cid:105)+|↓↑⇓(cid:105)) |Sx(cid:105)
|Txy(cid:105) √12(|2,2,1(cid:105)−|2,−2,1(cid:105)) -1 1 √12(|↑↑⇑(cid:105)−|↓↓⇓(cid:105)) |Sz(cid:105)
|Tx2−y2(cid:105) √12(|2,2,1(cid:105)+|2,−2,1(cid:105)) 1 1 √12(|↑↑⇑(cid:105)+|↓↓⇓(cid:105)) None
TABLE II: The basis set used in this paper. The wavefunctions are given in the form |SzSzLz (cid:105) with ↑ (↓) indicating
L H H
Sz = +1/2 (−1/2) and ⇑, ⇒ and ⇓ indicating Lz = 1,0 and −1 respectively. Iz = (−1)Iz. In this table we list only the
ν H
LzL =0 [Lz =(−1)LzL =1] states. Each state has two partners with LzL =1 and hence Lz =−1. The latter two, but not the
former, mix under the action of the full Hamiltonian.
ergies that are higher by Γ and display twice the degen- numbers contribute to substates II and III, but all of
eracy of the Lz =0 state. The trigonal terms break the the singlets are forbidden from mixing with substate I
L
SU(2) symmetry of the octahedral model and therefore by the combination of time reversal symmetry and the
lift the three- and five-fold degeneracies. The calculated conservation of Iz. Hence, the I→0 transition remains
spectraarenowlikethosecalculatedfromfirst-principles forbiddeninthetrigonalmodel. Bothexperiments[1,20]
for relevant complexes. For example, if trigonal symme- and relativistic TDDFT calculations [8–11] find that the
try is enforced for, e.g., [Os(bpy) ]2+, Ir(ppy) , Ir(ptz) radiative rates for the transitions II→0 and III→0 are
3 3 3
relativistic TDDFT calculations predict that SOC splits more than an order of magnitude faster than that for
T into a non-degenerate state (I) and, at slightly higher I→0,cf. TableII.Thesmallnon-zerodecayrateforI→0
1
energies,apairofdegeneratestates(IIandIII)[8,9,11]. may arise from either Herzberg-Teller coupling [1, 20] or
mixingofstateIwithhigherenergysingletstates, which
We saw above that in the octahedral model radiative
arenotincludedinthepseudo-angularmomentummodel
decay from the lowest energy excited state (I→0) is for-
[8–11].
bidden by the conservation of I. Because H breaks the
t
SU(2) symmetry of the octahedral model I2 no longer
commutes with H, nevertheless Iz and Lz remains a
L
good quantum numbers for the trigonal model. Fur- C. Full model
thermore, the Hamiltonian is time reversal symmetric,
therefore the parity of an eigenstate under time reversal, Finally, weturntothefullpseudo-angularmomentum
T =±1, is also a good quantum number. Note however, model, H [Eq. (9)]. Iz does not commute with H .
JT
thatIz doesnotcommutewithtimereversalsoitisnot, However, (Lx )2−(Ly )2 = 1[(L+)2+(L−)2], where the
H H 2 H H
ingeneral,possibletoformstatesthataresimultaneously ladder operators are given by L± = Lx ±iLy , there-
H H H
eigenstatesofboth. However, onemaydefinestatesthat fore Iz is conserved modulo two. Thus, Iz is conserved
are simultaneous eigenstates of the T and Iz = (−1)Iz. even for a trigonal system that has undergone a Jahn-
Therefore, we take these as our quantum numbers, cf. Teller distortion, cf. Table II. Similarly Lz is conserved
L
Table II. modulo two, which gives rise to the quantum number
For all parameters studied substate I is composed of Lz =(−1)LzL.
thebasisstate|T (cid:105)admixedwith|T (cid:105)andhasquantum We plot the spectrum of Lz = 1 states in Fig. 4a. In
1 z2
numbers Iz = T = +1, Lz = 0 whereas states II and Fig4b,weplotthesameresults,butonlyshowthethree
L
III are a degenerate pair with Iz = −1, T = ±1, Lz = lowestenergysubstates, I-III,whichareofprimarytech-
L
0, cf. Fig. 3, whose largest contributions come from nological interest. One sees that the although states II
|T (cid:105)and|T (cid:105). Thesingletstateswiththesamequantum and III are degenerate at δ =0, a Jahn-Teller distortion
x y
6
FIG. 3: Solution of the pseudo-angular momentum model FIG. 4: Solution of the pseudo-angular momentum model
of a trigonal complex. a) spectra for λ = J/5 and varying of a complex with broken trigonal symmetry - due either to
∆/J;b)spectrafor∆=J/2andvaryingλ/J. Thequantum chemicalmodificationorexcitedstatelocalization. Panel(a)
numbers of the states are also indicated. In both panels the shows the full spectrum for states with Lz = 1. Panel (b)
stateswithquantumnumberslabelledasT =±1aretwo-fold shows only the T1 substates, which are our primary concern.
degenerate. The eigenstates with L = ±1 (not shown for Here we take ∆=J/2 and λ=J/5.
z
clarity) have the same properties except that their energies
are increased by Γ and all of the degeneracies are doubled
corresponding to the two values of L =±1.
z
Hamiltonian, |ψ (cid:105), is given by
m
km = 4mee4αEm3 (cid:88) (cid:104)S |µ |S (cid:105)(cid:104)S |ψ (cid:105) (10)
R 3(4πε )2(cid:126)3 0 β n n m
0
β∈{x,y,z}
rapidlyliftsthisdegeneracyandforreasonablevaluesofδ n∈{x,y,z}
onefindsthatthereisamuchsmallerenergygapbetween
where E is the excitation energy of the mth state, and
substates I and II than between II and III. This is what m
α is the fine structure constant. Because of the underly-
is observed experimentally [1, 20, 21] in a huge range of
ing octahedral symmetry we take (cid:104)S |µ |S (cid:105) to be inde-
complexes(TableI).Wewillseebelowthatthissplitting 0 β n
pendent of n and further we assume that the zero field
is the signature of the localization of the excitation to a
splitting is small compared to the S → T excitation
single ligand. 0 1
energy, i.e., that E (cid:39) E (cid:39) E . It is also convenient
I II III
to define
Note, in particular, that substate I remains an admix-
tnuurmeboefrs|TT1(cid:105)=wiIthz |=Tz2L(cid:105)za=nd+|T1.x2−Ays2(cid:105)noanned ohfasthqeusainntgulemt kRS = 49m(4eπeε4α)2E(cid:126)m33 (cid:88) (cid:104)S0|µβ|Sn(cid:105) (11)
states have these quantum numbers, cf. Table II, this 0
β∈{x,y,z}
state is forbidden from mixing with any of the singlet
n∈{x,y,z}
states in the model by conservation of T, Iz and Lz.
ThereforesubstateIremainsapuretripletandisforbid- this corresponds to the radiative decay rate for a pure
denfromdecayingradiatively,irrespectiveofthestrength singlet with an excitation energy equal to that of the T
1
of the SOC. manifold.
We plot the radiative decay rate in Fig. 5. State I is
The radiative rate of the mth eigenstates of the full dark – as expected from the conservation laws derived
7
0.25 III,(cid:84)(cid:61)1,(cid:73)z(cid:61)(cid:45)1
0.20 II, (cid:61)(cid:45)1, z(cid:61)(cid:45)1
(cid:84) (cid:73)
J 0.15 I, (cid:61)1, z(cid:61)1
(cid:144) (cid:84) (cid:73)
Ν
(cid:88) 0.10
0.05
0.00
0.0 0.2 0.4 0.6 0.8 1.0
∆ (cid:68)
FIG. 5: The radiative decay rates of the three substates of
FIG. 6: The degree of localization in the three
T . The conservation of T, Iz, and Lz leads to the absence
1 substates of T , ν ∈ {I(cid:144), II, III}, where Ξ =
of radiative decay for state I. It can be seen that once the (cid:68) (cid:12) 1 (cid:16) (cid:17)(cid:12) (cid:69) ν
Jahn-Tellerdistortionbecomessignificanttheradiativedecay (cid:80)σ ν(cid:12)(cid:12)a†0σa0σ− 21 a†1σa1σ+a†2σa2σ (cid:12)(cid:12)ν . Here, as above,
ratefromstateIIissignificantlysmallerthanthatfromstate we take ∆=J/2 and λ=J/5.
III, in good agreement with experiment (cf. Table I). Here,
as above, we take ∆=J/2 and λ=J/5.
pation of site 0.) Nevertheless, for δ > 0, one observes
a rapid increase in Ξ whereas Ξ grows only rather
II III
above. Furthermore, once the Jahn-Teller distortion be-
slowly.
comes significant one finds that the radiative decay from
It is therefore clear that the pseudo-angular momen-
state II is significantly slower than the radiative decay
tummodelpredictssignificantlocalizationforvaluesofδ
fromstateIII.Thisisinpreciselywhatisobservedinex-
compatible with the observed experimental results that
periments [1, 20, 21] on pseudo-octahedral d6 complexes
kI <kII <kIII and E <E , cf. Table I. We there-
(cf. Table I). R R R I,II II,III
fore conclude that all of the complexes in Table I show
It is straightforward to understand both the changes
significant localization in their excited states.
in energy and the radiative rates of states II and III. For
δ > 0 (δ < 0 simple reverses these effects) the trigonal
perturbation lowers the energy of (stabilises) states that
IV. CONCLUSIONS
are antibonding between the Iz =±1 orbitals, e.g., |S (cid:105)
x
and|T (cid:105),andraisestheenergyof(destabilises)thosethat
x
arebondingbetweentheIz =±1orbitals, e.g., |S (cid:105)and The pseudo-angular momentum model gives a natu-
y
ral explanation of the zero-field splitting observed in a
|T (cid:105). It is clear from Table II that whereas |S (cid:105) and |T (cid:105)
y x y wide range of pseudo-octahedral d6 organometallic com-
areevenundertimereversal|T (cid:105)and|S (cid:105)areodd. Thus,
x y
plexes. Furthermore, the conservation laws, and hence
SOC mixes |S (cid:105) with |T (cid:105) and |S (cid:105) with |T (cid:105). Hence
x y y x
selection rules, inherent in the model give a natural ex-
thetrigonaldistortionincreasestheenergydifferencebe-
planationoftherelativeradiativedecayratesofthethree
tween the triplet and singlet basis states that contribute
sublevelsofT . Westressthatnoneoftheresultsderived
to state II (i.e., |S (cid:105) and |T (cid:105) for δ > 0); whereas trigo- 1
x y
here rely on perturbation theory – therefore these con-
nal symmetry reduces the energy difference between the
clusions hold even when the departures from octahedral
triplet and singlet basis states that contribute to state
or trigonal symmetry are large. This immediately ex-
III (|S (cid:105) and |T (cid:105) for δ > 0). Thus the symmetry of
y x
the model dictates that kI < kII < kIII, as is observed plains why the properties of the T1 states are so similar
R R R inbothhomolepticandheterolepticcomplexes. Further-
experimentally [1, 20, 21], see Table I.
more,forparameterscompatiblewiththeexperimentally
Finally, we turn to the question of localization in the
measured energies and radiative rates of the substates of
excited state. To measure this we define
T , the pseudo-angular momentum model predicts that
Ξψ =(cid:88)(cid:28)ψ(cid:12)(cid:12)(cid:12)(cid:12)a†0σa0σ− 12(cid:16)a†1σa1σ+a†2σa2σ(cid:17)(cid:12)(cid:12)(cid:12)(cid:12)ψ(cid:29).(12) erex1mciiaatniosnwseIlladnedloIcIaalirseeds.troTnhgulys lwoecacliosnecdlu–dealtthhoautgahllIoIIf
σ
the complexes in Table I show significant localization in
We plot Ξ for the three substates of T in Fig 6. The their two lowest energy excited (sub)states.
ψ 1
lowest energy excitation, I, is completley delocalized for It is interesting to note that when the radiative rates
δ = 0 but rapidly localizes for δ > 0. It is interesting of individual sublevels, km, have been measured, rather
R
to note that both Ξ and Ξ are non-zero for δ = 0. than excited state lifetimes, τ , it is found that the rel-
II III m
However, for δ = 0 states II and III are degenerate and ative rates are in good accord [1, 20], cf. Table I. This is
Ξ =−Ξ , consistent with trigonal symmetry. (A neg- consistentwiththehighphotoluminscentquantumyields
II III
ative value of Ξ indicates that the state disfavors occu- observedinthesecomplexes. Thissuggeststhatthenon-
ψ
8
radiativedecayratesoftheindividualsublevelsaredeter- with. θ ∼ π/4 as the HOMOs are found to have about
mined by similar conservation laws. Therefore, it would 50 % metallic weight [8, 10, 11, 22]. Nozaki [8] found
be interesting to investigate non-radiative decay rates in that for Ir λ = 550 meV and sin2θ = 0.4, yielding
m
a suitable extension of the pseudo-angular momentum λ = 220 meV. Smith et al. [11] considered the C S
3 0
model. geometryandfoundfromgroundstatecalculationsshow
Wenotethatthepseudo-angularmomentummodelde- that the gap between the HOMO and HOMO-1 is 140
scribed above can be naturally extended to understand meV (LUMO and LUMO+1 is 90 meV), which may be
the properties other molecules and complexes where the taken as an estimate of ∆ (Γ). However several authors
low-energy excited states correspond to transition be- Nozaki [8, 17, 23] have noted that the values of ∆ and
tween degenerate or approximately degenerate states. Γ are difficult to calculate from first principles – there-
fore these numbers should be treated with some caution
and are likely to be underestimates as interactions sig-
Acknowledgments nificantly increase the effective values of ∆ and Γ. For
ppy it has been estimated [23] based on the absorption
IthinkSamGreer,AnthonyJackoandRossMcKenzie spectra,emissionspectra,andemissionlifetimes[24]that
forhelpfulconversations. Theworkwassupportedbythe Jπ ∼ 2 eV; and for an isolated Ir ion λd ∼ 0.43 eV [25].
Australian Research Council under grant FT130100161. Taking θ ∼π/4 yields J ∼1.4 eV and λ∼300 meV. For
concreteness we take λ = J/5 and ∆ = J/2 in the main
text. However, our results are insensitive to the values
V. APPENDIX: ESTIMATION OF of these parameters – to demonstrate this we explore a
PARAMETERS range of other parameters in the sup. info. δ and γ are
not straightforward to estimate from previous work and
While we will not make a detailed parameterization of will be left as free parameters, however as the C3 sym-
this model – an idea of the relevant parameter ranges metry remains evident even in the T1 geometry of the
can be obtained from previous experiments and density excited this suggest that δ (γ) is not significantly larger
functional calculations. Ir(ppy) has been particularly than ∆ (Γ).
3
widely studied and so is an ideal material to compare
[1] H.Yersin,A.F.Rausch,R.Czerwieniec,T.Hofbeck,and [14] E. M. Kober, and T. J. Meyer, Inorg. Chem. 21, 3967
T. Fischer, Coord. Chem. Rev. 255, 2622 (2011). (1982).
[2] K.D.Demadis,C.M.Hartshorn,andT.J.Meyer,Chem. [15] J. R. Reimers, ‘Computational methods for large sys-
Rev. 101, 2655 (2001). tems: electronic structure approaches for biotechnology
[3] S.Reineke,M.Thomschke,B.Lu¨ssem,andK.Leo,Rev. and nanotechnology’ (Wiley, 2011).
Mod. Phys. 85, 1245 (2013). [16] A.Abragam,andB.Bleaney,‘Electronparamagneticres-
[4] Q.Zhao,F.Li,andC.Huang,Chem.Soc.Rev.39,3007 onance of transition ions’ (Clarendon Press, 1970).
(2010). [17] A. C. Jacko, and B. J. Powell, Chem. Phys. Lett. 508,
[5] Y. You and S. Y. Park, Dalton Trans. 1267 (2009). 22 (2011).
[6] A. Kapturkiewicz, in ‘Electrochemistry of Functional [18] I. B. Bersuker, and S. A. Borshch, in Adv. Chem. Phys.
SupramolecularSystems’477-522(JohnWileyandSons, Vol.81(Eds.I.PrigogineandS.A.Rice)703-782(1992).
Inc., 2010). [19] Notethat,incontrast,KoberandMeyer[14]takeΓ<0,
[7] P. L. Burn, S. C. Lo, and I. D. W. Samuel, Adv. Mater. whichmeansthattheirresults,fromconventionalligand
19, 1675 (2007). field theory, are very different from ours.
[8] K. Nozaki, J. Chin. Chem. Soc. 53, 101 (2006). [20] T.Hofbeck,andH.Yersin,Inorg.Chem.49,9290(2010).
[9] K.Nozaki,K.Takamori,Y.Nakatsugawa,andT.Ohno, [21] H. Yersin, W. Humbs, and J. Strasser, Coord. Chem.
Inorg. Chem. 45, 6161 (2006). Rev. 159, 325 (1997).
[10] E.Jansson,B.Minaev,S.Schrader,andH.˚Agren,Chem. [22] P. J. Hay, J. Phys. Chem. A 106, 1634 (2002).
Phys. 333 157 (2007). [23] A. C. Jacko, B. J. Powell, and R. H. McKenzie, Chem.
[11] A. R. G. Smith, P. L. Burn, and B. J. Powell, Phys. Lett. 508, 22 (2011).
ChemPhysChem 12, 2429 (2011). [24] M. Maestri, et al. Chem. Phys. Lett. 122, 375 (1985).
[12] A.C.Jacko,R.H.McKenzie,andB.J.Powell,J.Mater. [25] N. V. Vugman and S. R. Nogueira, Bras. Fisica, 606-
Chem. 20, 10301 (2010). 611(1989);A.C.Jacko,(PhDThesis,TheUniversityof
[13] A. Ceulemans, and L. G. Vanquickenborne, J. Am. Queensland, 2011).
Chem. Soc. 103, 2238 (1981).
