Tuning Range-Separated Density Functional Theory for Photocatalytic Water Splitting Systems

1.2 MB·English

by  Olga S. Bokareva

Tuning Range-Separated Density Functional Theory
for Photocatalytic Water Splitting Systems
∗
Olga S. Bokareva, Gilbert Grell, Sergey I. Bokarev, and Oliver Kühn
InstitutfürPhysik,UniversitätRostock,Universitätsplatz3,D-18055Rostock,Germany
5
1 E-mail: [email protected]
0
2

n
Abstract
a long-living triplet states are then available for fur-
J
therreactions. Besideslight-harvestingproperties,
7
2 We discuss the system-specific optimization of TM complexes are also used as direct catalysts
long-range separated density functional theory for water splitting; for recent reviews on homoge-
]
h (DFT) for the prediction of electronic properties neousphotocatalysissee,e.g. refs4,5.
p relevant for a photocatalytic cycle based on an To develop new efficient and stable photocat-
-
m Ir(III) photosensitizer (IrPS). Special attention is alytic systems, understanding of their primary
e paidtothecharge-transferproperties,whichareof photoreaction steps and especially excited state
h
c key importance for the photoexcitation dynamics, propertiesisrequired.6 Atthemoment,mostcom-
s. butandcannotbecorrectlydescribedbymeansof putationally demanding studies of sizeable TM
c conventional DFT. The optimization of the range- complexesappliedincatalysisareperformedwith
i
s separation parameter using the ∆SCF method is DFT in combination with the B3LYP functional
y
h discussed for IrPS including its derivatives and and its extension in the time-domain in the lin-
p
complexes with electron donors and acceptors ear response formulation (TDDFT).7–15 Besides
[
used in photocatalytic hydrogen production. Par- efficiency it is the absence of system-dependent
2
v ticular attentionis paid tothe problems arising for parameters that need to be determined first (such
3 a descriptionof mediumeffects bymeans of apo- as the active space in multi-reference methods),
6
larizablecontinuummodel. which makes (TD)DFT attractive. The power of
8
5 the DFT method to reproduce different electronic
0
ground state properties even for rather large sys-
. 1 Introduction
1
tems is well documented.16 Concerning excited
0
5 state properties of TM complexes, however, there
1 Transition metal (TM) organometallic complexes
appearstobenounequivocalopinion;forareview
: havefoundawiderangeofcatalytic,medical,and
v see ref17. For an Iridium (III) heteroleptic com-
i biologicalapplications. Inviewofthegrowingde-
X plex, for instance, we have shown that the excited
mandinenergy,oneofthemostperspectiveappli-
r state energies are strongly dependent on the em-
a cations of TM complexes is their role in systems
ployed exchange-correlation functional.18 Never-
for conversion and storage of solar light energy
theless, the TDDFT approach remains to be the
into chemical form. Here, a variety of homoge-
mostattractiveone.
neous and heterogeneous schemes have been sug-
Many of the practical problems are related to
gested.1–3 TM complexes are attractive because
the erroneous description of charge transfer (CT)
of their notable spin-orbit coupling. This facili-
states.19,20 To overcome this drawback, different
tatestheabsorptionofsunlight,whoseenergycan
schemes have been proposed, such as, e.g., scaled
be stored in charge-separated triplet states. These
hybrids21,22 and range-separated hybrid function-
∗Towhomcorrespondenceshouldbeaddressed als.23,24
1
2 OPTIMALLYTUNEDRANGE-SEPARATEDFUNCTIONALS 2
2 Optimally Tuned Range-
Benchmarking DFT results for TM complexes
in solution is hampered by the facts that there
Separated Functionals
are no high-resolution experimental data avail-
able and only a limited number of theoretical ap-
DFT has its formal roots in the theorems of Ho-
proaches might be applicable to generate a pre-
henbergandKohn30,31 anditcanbeconsideredas
sumably more accurate reference. At the mo-
an in principle exact approach. In practice, it is
ment, multi-reference perturbation theory is the
hampered by the lack of the exact exchange cor-
best choice for reference calculations of excited
relation(XC)functional,E [ρ((cid:126)r)],withρ((cid:126)r)be-
XC
state properties. However, in multi-reference ap-
ingtheelectrondensity. Thereseemstobenofirst
proaches the problem of a sufficiently large and
principleroutetoE [ρ((cid:126)r)]andthereforeitisusu-
XC
balanced active space has to be addressed for ev-
ally constructed on the basis of model systems or
erysystem.18
fittedtoexperimentalreferencedata. Excitedstate
In this paper, we focus on the tuning of the
calculationsperformedbyTDDFTinthelinearre-
rangeseparationparameterforthelong-rangecor-
sponse approximation employ XC functionals ob-
rected LC-BLYP functional as applied for a par-
tainedfortheelectronicgroundstate.
ticular photocatalytic system developed by Beller
All caveats and the sometimes spurious be-
et al.25 It consists of a heteroleptic Ir(III) photo-
haviour of the DFT approach originate from the
sensitizer, triethylamine (TEA) as a sacrificial re-
approximation of the unknown XC kernel. There
ductant and a series of iron carbonyls (Fe-cat) as
are three main problems: First, in the r →∞ limit
water reduction catalyst (Figure 1). In addition,
the potential and hence the density itself possess
hybrid systems of IrPS and silver nanoparticles
an incorrect behaviour. In principle, at large dis-
are considered.26–28 Finally, the scope is broad-
tances the potential should be dominated by the
ened by including results on modified IrPSs and
exchange term decaying as −1/r, while the cor-
copper based PS which could replace noble-metal
relation term decays as ≈ −1/r4.32,33 However,
containing PS, see ref.29 and references therein.
for the local density approximation and to a lesser
Although the choice of the system is arbitrary, its
extent for the generalized gradient approximation
general properties as well as the basic reaction
(GGA),theexchangetermdecaysexponentially.34
steps are common for TM photocatalysis. There-
Second, the approximate nature of the exchange
fore, the present study provides some general in-
term does not cancel the Coulomb interaction of
sight into the role of tuning the range separation
the electron with itself at large r as it is the case
parameterinthesesystems.
fortheHartree-Fock(HF)method(self-interaction
In the following section, we start with a brief
error). Third, a well-known deficiency of ap-
introduction in the theory of long-range separated
proximate Kohn-Sham approach is that the funda-
functionals. We then present our results on tuning
mental gap, defined as a difference between ion-
the range-separation functional for IrPS and joint
ization potential (IP) and electron affinity (EA),
IrPS-X systems, where X is a reactant responsible
differs notably from the orbital energy difference
for a particular process, i.e. light absorption, elec-
ε −ε . 35 This can be explained by the
HOMO LUMO
trontransferandrecoveryofthegroundstate. The
finitejumpoftheKohn-Shamcorrelationpotential
influence of the reference geometry, environmen-
for a statistic ensemble with variable number of
talandbasisseteffectsarediscussed. Inparticular,
electrons while passing through integer (N) num-
the applicability of the ω-tuning together with the
ber of electrons, the so-called derivative disconti-
Polarizable Continuum Model (PCM) is critically
nuity. 35–39 According to Koopmans’ theorem for
analyzed.
theDFTcase,33,36,40,41 theHOMOcorrespondsto
the IP, but the LUMO is generally more strongly
bound than in HF theory and cannot be related to
theEA.
Considering TDDFT calculations of electronic
excitation energies, the error correlates with the
2 OPTIMALLYTUNEDRANGE-SEPARATEDFUNCTIONALS 3
TEA+
IrPS- Fe-cat ½H
ox 2
+
TEA
N
Ir
IrPS*
N N
2
IrPS
IrPS Fe-cat H+ [Ir(ppy)(bpy)]+
2
h red
Ag
-
n
+ OC CO
+ OC Fe CO
+ N N H
H Ir Ir
N2 Fe-cat(1)
Ir O N N O N N
N N
H2 OC H CO -
2 OC Fe Fe CO
2 OC C CO
2 O
[Ir(ppy)(en)]+ [Ir(pox)(bpy)]+ [Ir(pbox)(bpy)]+ Fe-cat(2)
2 2 2
+
HC Ph
+ 3 -
PPh OC CO
2 N
N N OC Fe CO
Ir Ir O Cu
N OC H CO
N N N N PPh OC Fe Fe CO
2 OC C CO
2 2 H3C Ph O
[Ir(ppy)(bpym)Ir(ppy)]2+ [Cu(phen*)(xant)]+ Fe-cat(3)
2 2
Figure 1: Scheme of photocatalytical water splitting introduced in ref25 and structural formulas of pho-
tosensitizersandwaterreductioncatalystsstudiedinthiswork.
overlap of the donor and acceptor orbitals, being andnon-localparts
the largest for CT and Rydberg states. 19,20 For
1 1−Γ(ωr ) Γ(ωr )
vanishingoverlapbetweentheseorbitals(e.g.,due 12 12
= + , (1)
tolong-rangeCT),theexcitationenergyisreduced r12 r12 r12
to the orbital energy difference, being a poor esti-
whereΓ(ωr )isasmoothrange-separationfunc-
12
mate in case of DFT, which is in contrast to HF
tion,whichdampstheexchangecontributionfrom
theory.
the density functional and complements it with
The remedy for the erroneous exchange poten-
exact exchange. Examples are the Yukawa ker-
tial could be the substitution of the approximate
nel43,44 e−ωr/r or the error function24,45 kernel
density-dependentexchangeenergywiththeexact
erf(ωr)/r. Suchanapproacheliminatesthespuri-
orbital-dependentone42 withingeneralized Kohn-
ous behaviour because the exact exchange has the
Shamtheory.34,35,42Theexactexchangecanbein-
correct asymptotic character and cancels the self-
cludedinafixedmannerlikeinhybridfunctionals
interactionexactly.46–51
(e.g. B3LYP)orweightedwithafunctiondepend-
Certain standard values for the range-separation
ing on the inter-electron distance r . The latter
12 parameter ω have been established in refs52,53
case is implemented in range-separated function-
and they are used like universal constants in pop-
als via splitting the Coulomb operator into local
ular quantum chemical programs.54,55 The value
0.33bohr−1 was determined by a least square fit
2 OPTIMALLYTUNEDRANGE-SEPARATEDFUNCTIONALS 4
to empirical data for first- to third-row atoms52 erroraccordingto61,62
and later refined to 0.47bohr−1 for larger sets
(cid:113)
of small molecules.53 (for other test sets, see J∗(ω)= J2(ω)+J2(ω). (7)
0 1
also refs.49,50,56,57) However, it is clear that for
molecules,whichhavenotbeenpartofthetraining Alternatively, for long-range CT systems where
sets, a more accurate description will be provided donor and acceptor units can be clearly distin-
bychoosingasystem-dependentω. Anoptimalω guished, ground states of neutral donor and nega-
for a particular system can be determined by en- tivelyionizedacceptorcanbeconsideredinatun-
suringthattheenergyoftheHOMOorbitalequals ingapproach59,60 basedonMulliken’srule.63
the negative of the IP, a relation that would be ful- It is important to stress that the optimal ω is
filledfortheexactfunctional.58 density and thus system dependent. Thereby,
However,inthefollowingwewillusealternative the dependence of ω on the density for range-
the so-called ∆SCF method,49,59,60 where the IP separated functionals is much more pronounced
and EA are calculated as the differences between than that of the scaling factor for hybrid function-
ground state (gs) energies of systems with N and als.34 Since the range-separated functionals elim-
N±1electrons,i.e. inate self-interaction46–51,64 and have the correct
asymptotic form of non-local exact exchange, the
IPω(N)=Eω(N−1)−Eω(N), (2) correct Coulomb-like behaviour of the energy is
gs gs
assured and thus CT and Rydberg excitation ener-
gies are improved. Additionally, since the tuned
IPω(N+1)=EAω(N)=Eω(N)−Eω(N+1). range-separation parameter improves the funda-
gs gs
mental gap for the N electron system, one can ex-
(3)
pect also a better description of electronic excita-
Thisyieldstheseparatetuningconditions
tion energies in TDDFT, where the leading term
J (ω)=|εω (N)+IPω(N)|, (4) readsasεHOMO−εLUMO.
0 HOMO
In the past it was shown in numerous publi-
cations the the optimally tuned functionals in-
deed improve the description of different molec-
J (ω)=|εω (N+1)+EAω(N)|. (5)
1 HOMO ular properties such as IPs, fundamental and op-
tical gaps, and CT and Rydberg transition en-
Inordertoobtainaproperdescriptionofthefun-
ergies.20,35,40,42,50,52,53,59–61,65–82 The accuracy of
damental gap, the functions J (ω) and J (ω) for
0 1
optimally tuned range-separated functionals was
IP and EA should be minimized simultaneously.
critically tested in ref61 for basis set variation
For the present one-parameter formulation this re-
as well as prediction of relative energies of spin
quirestominimizethegeneralfunction
states, binding energies, and the form of potential
J(ω)=J (ω)+J (ω). (6) energysurfaces.
0 1
Finally, we note that a generalized form of LC-
NotethatingeneralJ(ω)canbenon-zeroevenfor partitioning of Coulomb interaction proposed by
exactfunctionals,becauseitisdefinedforsystems Yanai for CAM-B3LYP56 is often applied for
withdifferentnumbersofelectrons. non-empirical optimal tuning as well (see, e.g.
Although minimizing J(ω) was applied exten- refs76,79–83). Additionally, the simultaneous two-
sively for tuning the range-separation parameter, parameter optimization of short- and long-range
in ref61 it was stressed that the ω-values are very separation parameters was proposed in order to
close to minima of J (ω), eq 4, or J (ω), eq 5, minimize the delocalization error. Successfully
0 1
dependingontheparticularformofthedependen- applications provided an improved description of
ciesofε andthetotalenergyonω. Toavoid molecularproperties.79,80,82
HOMO
suchabiasedbehaviour,itwassuggestedtoapply
a least-square approach to minimize the resulting
4 RESULTSANDDISCUSSION 5
I rPS Water reduction catalysts IrPS
ar d
e
e r
l [Ir(ppy)(en)]+ i
c 2 v
u a
t
n i
v
bi Fe-cat(2) e
Fe-cat(1)
Fe-cat(3) s
[Ir(ppy)(bpym)Ir(ppy)]2+
2 2
[Cu(phen*)(xant)]+
[Ir(ppy)(bpy)]+
2
Ag
20
Ag [Ir(pbox)2(bpy)]+ [Ir(pox)2(bpy)]+
10
Silver clusters
Figure2: Overviewofalloptimizedstructuresofthosemoleculesthatwereusedforω-tuning.
3 Computational Details
The impact of reference geometry used for tun-
ing of ω was studied on the example of IrPS.
The tuning of the range-separation parameter ω The tuning for the slightly different geometries,
according to Eqs 6 and 7 has been done for the which have been obtained within LC-BLYP with
LC-BLYP functional.45,52,84 It includes the Becke thetwostandard(0.33and0.47)andtheoptimized
exchange85andacorrelationpartofGGA-typees- (0.18) ω, led to the same value of 0.18bohr−1. In
tablishedby Lee, Yangand Parr.86,87 This scheme the following, the structures optimized with LC-
uses eq 1 with an error function kernel as sug- BLYP with optimal ω have been utilised for tun-
gested by Hirao et. al..45 For the case of the par- ing (see Figure 2). For the complexes IrPS−X
ent IrPS compound, the results with a tuned ω (X=TEA or Fe-cat) the geometries of the consti-
are compared to those obtained with the standard tuing parts have been first optimized separately
system-independent values of 0.33 bohr−1 and and then placed at fixed positions without further
0.47bohr−1 as well as with BLYP and B3LYP88 optimization (for further details see ref89). For
functionals. systems containing silver clusters, the geometries
All calculations have been performed using the wereoptimizedinrefs26–28. ThestandardTDDFT
LANL2DZ ECP basis set for Ir, Fe, Cu, and Ag formalism was applied for excited state calcula-
and the 6-31G(d) basis set for all other atoms. In tions. Solvent effects have been included within
ref61 it was pointed to a strong dependence of the the PCM approach.90 All calculations were done
optimal ω on the inclusion of diffuse function in withtheGaussian09setofprograms.55
case of simple atomic and diatomic systems. To
test the influence of diffuse basis functions for the
4 Results and Discussion
present example of IrPS, calculations have been
performed with 6-31+G(d) and 6-31++G(d) basis
sets. The inclusion of one or two diffuse function 4.1 Optimization of the Range-
has led to decrease of ω by 0.01bohr−1. Since Separation Parameter
this effect appears to be of minor importance, be-
low only results for the 6-31G(d) basis set will be Theoptimalvaluesoftherange-separationparam-
presented. eter ω for molecules and complexes related to the
4 RESULTSANDDISCUSSION 6
Figure 3: Left panel: The functions defined in eqs 4–7 used for optimization of the range-separation
parameterω forIrPS−Ag . Rightpanel: SelectedexamplesofJ(ω)forIrPSindifferentenvironments.
2
studied photocatalytic systemfor hydrogen gener- blet electronic state, the unpaired electron was
ation (Figure 1) as well relevant literature data for previously shown to be localized on π(bpy)∗ or-
similar substances are collected in Table 1. In the bital.91 FortheIrPS−Ag ,theinternaloxidation-
10
following,wewilldiscussparticularaspectsofthe reduction occurs for ω > 0.25bohr−1: the un-
optimizationforthechosentargetsystems. pairedelectronmovesfromπ(bpy)∗ toσ∗(Ag).
Exemplary, in the left panel of Figure 3 details As IrPS forms no stable complexes, IrPS−X,
ofthetuningprocedurefortheIrPS−Ag caseare neither with TEA nor with the iron-catalysts89,
2
presented. Since J (ω) and J (ω) describe sys- the relative position of constituents might be flex-
0 1
tems with different numbers of electrons, in gen- ibly varied. Similar to the procedure in ref89
eral, their minima do not coincide. The shown we have varied the position of X=TEA or Fe-cat
example is the only case among the studied sys- on a sphere around IrPS. Thereby we found that
tems, where the minimum of J(ω) is not well- choosing different locations does not have an im-
defined due to an almost flat region from 0.18− pact on resulting ω. However, changing the dis-
0.25bohr−1, where J (ω) and J (ω) upon sum- tance between IrPS and X led to some variation
0 1
mation compensate each other. If the least-square of optimal ω, see Table 1. Interestingly, the op-
function J∗(ω) is applied instead, a minimum at timal ω value was found to increase in the range
0.19bohr−1canbeeasilylocated. Forallothertest 0.23−0.32bohr−1withincreasingthedistancebe-
systems collected in Table 1, the minima of J and tween IrPS and TEA from 7 to 12 Å. While the
J∗ were very close to each other; J∗ led in some minimal value of J (ω) remains at 0.18bohr−1,
0
cases to a decrease of ω by 0.01−0.02bohr−1. theminimumofJ (ω)occursatlargerω. Theus-
1
SincetheuseofJ∗ wasnotcrucialinourcase,we age of J∗(ω) instead of J(ω) provides a narrower
focusedontheresultsobtainedbyapplyingJ,asit range of 0.23−0.28bohr−1. In passing we note
wasthemainapproachofpreviousinvestigations. that in refs49,61 it was shown that an optimization
In the right panel of Figure 3 selected exam- ofω foreachgeometryandsubsystemcanleadto
plesofJ(ω)forIrPSindifferentenvironmentsare size-inconsistency or unphysical potential energy
presented. The curve for IrPS(vac) represents a curvesinexcitedstates.
case with regular behaviour, i.e. there is a dis- For comparison, the standard52,53 system-
tinct minimum. For IrPS−Ag the region af- independent ω = 0.33bohr−1 and ω =
10 1 2
ter 0.25bohr−1 corresponds to a changed order of 0.47bohr−1 are shown in both panels of Fig-
HOMO and HOMO-1 orbitals for N+1-electron ure 3 by vertical lines. As can be seen from
system. For the isolated reduced IrPS in the dou- that figure and Table 1, the optimally tuned ω
4 RESULTSANDDISCUSSION 7
values for compounds relevant in photocataly-
sis are substantially lower than those determined
for diatomics and small molecules.52,53 Since
ω−1 reflects a characteristic distance for switch-
ing between short- and long-range parts or, in
Table 1: Optimized range-separation parame-
other words, an effective electron screening (de-
ters, ω, defined by the minumum of J(ω), eq
localization) length, previously optimal ω val-
6, for different molecules depicted in Figure
ues were found to decrease with increasing sys-
1. Forcomparison,literaturedataonmolecules
tem size and conjugation length.60,69,70,72,73,77,92
relevantforphotovoltaicsareprovided.
However, in some cases the dependence was not
Compound Optimalω [bohr−1] monotonous and strongly varied for systems with
IrPS 0.18 different electronic structure.70 In the present
IrPS(THF,PCM) 0.05 study, the Ir(III) complexes have various sizes,
IrPS+28THF 0.15 being smallest for [Ir(ppy) (en)]+ and largest for
2
[Ir(ppy) (en)]+ 0.18 [Ir(ppy) (bpym)Ir(ppy) ]+. However, only a mi-
2 2 2
[Ir(pox) (bpy)]+ 0.18 nordependenceofω onthesizeofligandsaround
2
[Ir(pbox) (bpy)]+ 0.18 the central Ir atom and the size of molecules (sil-
2
[Ir(ppy) (bpym)Ir(ppy) ]+ 0.14 ver clusters, iron carbonyls) bound to the PS was
2 2
[Cu(phen∗)(xant)]+ 0.16 found(cf. Table1).
Ag 0.42
2
Ag 0.22
10 4.2 Effect of Solvation
Ag ≈0.20a
20
IrPS−Ag 0.18–0.25b By requiring Koopmans’ theorem to be satisfied
2
IrPS−Ag 0.16 simultaneouslyforsystemswithN andN+1elec-
10
IrPS−Ag 0.16 trons where HOMO and LUMO are on differ-
20
TEA 0.31 ent fragments, we automatically improve the de-
IrPS−TEA 0.23–0.32c scription of electron transfer. This implies that
Fe-cat(1) 0.24 thelong-rangecorrectedfunctionalwithoptimized
Fe-cat(2) 0.19 range-separation parameter should have a predic-
Fe-cat(3) 0.17 tive power to study also redox properties. To de-
IrPS-Fe-cat(1) 0.18c scribe redox (electron transfer) reactions in pho-
IrPS-Fe-cat(2) 0.17–0.18c tocatalysis, solvents effects need to be included.
IrPS-Fe-cat(3) 0.16–0.17c However, to the best of our knowledge, so far no
C 70 0.21 ω-tuninginthepresenceofasolventwasreported.
60
phthalocyanine70 0.16 The green curve with square symbols in Figure 3
Si-nanocristals(5-15Å)69 0.10–0.24 showsJ(ω)forIrPS,solvatedviathePCMmodel
coumarinedyes60 0.17–0.21 (THF solvent). This yields a very small opti-
pentacene-C 81 0.21 mal ω =0.05bohr−1. Similar computations have
60
nucleobases78 0.27-0.31 beendoneforTEAandFe-cat(3),resultinginval-
ues 0.10 and 0.05 bohr−1, respectively. Note that
aBecause of convergence problems, only
such a small value of ω implies that there is al-
tentative a value is given; busing J(ω), eq
most no long-range correction from exact HF ex-
6, a flat dependence is obtained, see text and
change. Interestingly, the inclusion of 28 explicit
Figure 3;
THFmoleculesaroundIrPS,whichcorrespondsto
cdifferentgeometriesareused,seetext. onesolvationshell,ledtoω =0.15bohr−1 similar
tovacuumcase.
According to Janak’s theorem93 the redox en-
ergy upon changing the number of electrons from
4 RESULTSANDDISCUSSION 8
Figure4: ∆SCFredoxenergies∆E(dashedlines)andorbitaleigenvalues−ε (solidlines)for
HOMO/LUMO
different charge states (+2,+1, and 0) of IrPS computed with pure BLYP and LC-BLYP with ω optimiza-
tion in vacuum and PCM solvent. The eigenvalue of the β spin-orbital was used for LUMO(+2) all other
valuescorrespondtotheα spin-orbital.
N−1toN canbeobtainedinintegralformas deficiencies.
Thatiswhyoneneedstoincludemuchlessexact
N
(cid:90) exchange into the LC-BLYP functional (smaller
∆E = ε(n)dn. (8)
ω) to satisfy Koopmans’ theorem and derivative
N−1 discontinuity condition as is seen from Figure 3
and Table 1. The orbital eigenvalues in the right
In the absence of self-interaction and delocaliza-
panel of Figure 4 for the solvent case are almost
tion errors, the orbital energies ε stay constant
constant, the deviations being probably due to a
between integer numbers of electrons, thus satis-
residual delocalization error. Importantly, the ω
fying the condition of ω optimization, eq 2. In
value optimized for the vacuum does not fulfil the
Figure 4 redox and HOMO/LUMO energies are
condition of constant orbital energies in solvent
shown for different functionals and optimization
and vice versa. Further examples of Fe-cat(3) and
conditions. Forthetunedω,∆E =−ε (N)=
HOMO TEA leading to the same conclusion can be found
−ε (N−1) are given in central panel of Fig-
LUMO intheSupportingInformation.
ure4forIrPSinvacuum. Whenusingtheconven-
tional BLYP functional this condition is not satis-
4.3 Triplet Stability
fied (left panel). However, if we include implicit
PCMsolvationthejumpinenergiesofHOMO(N)
The nature and energetic position of the lowest
and LUMO(N −1) is mitigated by the response
triplet state of PSs need to be calculated very ac-
of the polarizable continuum. This leads to much
curately since this state plays a key role in photo-
better estimates of redox energies obtained with
processes as a dominant long-living and emitting
conventional functionals with PCM as compared
excited state. However, the calculation of triplet
with the vacuum.94 In other words, in terms of
states represents a challenge for the TDDFT ap-
the derivative discontinuity the PCM solvent ef-
proachwithlong-rangecorrectedhybridfunction-
fectively leads to a partial mitigation of the E
XC
4 RESULTSANDDISCUSSION 9
the exact exchange part, which is a common be-
haviourforTDDFT.18 Fortripletstates,thisrising
is compensated by the decrease of energies upon
increase of instability resulting in maxima around
0.25−0.40bohr−1.
4.4 Electronic Absorption Spectra
A comparative analysis of excitation and differen-
tial absorption spectra of IrPS in THF calculated
by means of local (BLYP), hybrid (B3LYP), and
long-range corrected LC-BLYP with standard and
both optimal (for vacuum and THF) ω values is
provided in Figure 6. In the TDDFT calculations,
Figure 5: Energies of the lowest five singlet and the 80 lowest singlet-singlet states have been in-
triplet excited states and the lowest eigenvalue of cluded and assigned. A phenomenological broad-
thegroundstatestabilitymatrix(λ)independence ening by a Gaussian lineshape with a width of 0.1
on the range separation parameter ω for IrPS in eVwasassumed.
vacuum. Overall, the features of the absorption spec-
trum are well reproduced by B3LYP and LC-
BLYP with optimal ω = 0.18bohr−1. However,
als because of symmetry breaking instabilities of
absorption and difference spectra could only be
thegroundstatesolution.95,96 Typically,thelarger
properlyassignedsimultaneouslyusingLC-BLYP
amount of exact exchange leads to more pro-
(ω = 0.18bohr−1).91 It should be noted, that be-
nounced problems with instabilities of the ground
sides the overall shape of spectra, which can be
state wave function. This results in a divergence
described by the conventional hybrid functional,
of TDDFT excitation energies and an imaginary
the quality of description of CT states and band
energy of the lowest triplet state.97 As shown in
gap characteristics is only assured for the long-
ref72, the non-empirical tuning of ω could avoid
rangefunctionalwiththeproperlytunedω param-
the instabilities through the stabilizing the ground
eter. The excessive amount of HF exchange (as in
statesolution,butthisissueshouldbecarefullyan-
LC-BLYP with both standard ω values) as well as
alyzedfortheparticularsystemunderstudy. Since
complete absence of exact exchange (as in BLYP)
the instability problems are generally more pro-
led to shapes of absorption and difference spec-
nounced for the HF method than for DFT,72,95 the
tra of IrPS deviating from experiments. Naturally,
application of a smaller value for ω should result
the results of LC-BLYP with optimal ω tuned for
in a more stable ground-state solution. In Figure
PCM are very close to those of pure BLYP be-
5, the dependency of the smallest eigenvalue of
cause of only small amount of exact exchange is
triplet stability matrix95 on the range-separation
included. Hence,LC-BLYPwithω =0.05bohr−1
parameter, λ(ω), for the case of IrPS is given; the
optimized in a PCM solvent model does not agree
energies of lowest triplet and singlet energies are
withexperiment,similartotheparentBLYPfunc-
providedaswell.
tional which is known for wrong prediction of en-
It can be clearly seen that for the present sys-
ergiesofCTstates.
tem the maximum of stability λ corresponds
max Thelatterobservationdeservessomemorecom-
to the optimal ω value. The λ value is notably
ments. Inviewofthederivativediscontinuity,ω =
higher for the standard values ω . 52,53 This im-
1,2 0.05bohr−1 obtained for the PCM model gives an
plies that LC-BLYP with properly tuned range-
improveddescriptionascomparedtothegasphase
separation parameter ensures a more reliable es-
value (see Figure 4, right panel). In contrast, us-
timate of triplet state energies. The singlet state
ingthegasphaseω =0.18bohr−1 togetherwitha
energies systematically rise with the increase of
PCMmodelyieldsan“overscreening”(seeFigure
5 CONCLUSIONS 10
Figure 6: Left panel: Experimental (optical density, OD) and theoretical (TDDFT, oscillator strength)
electronic excitation spectra. Right panel: Difference absorption spectra (reduced species - oxidized
species) obtained theoretically and by transient absorption experiments of IrPS in THF-solution. Experi-
mental spectra were measured by Lochbrunner et al. and had been published together with the calculated
BLYPandB3LYPspectrainrefs18,91. AllcalculationsincludedTHFwithinthePCM.
4, middle panel). At this point, one should note CT states. The actual optimized value, ω =
that under the influence of a polar solvent the en- 0.18bohr−1, was found to be nearly the same
ergies of CT and local states shift differently from for various Ir and Cu-based PS as well as for
their vacuum values because of substantially dif- IrPS complexes with silver clusters and iron car-
ferent dipole moments. That is why, to correctly bonyls. Using the ω optimization it was shown
predicttheelectronicabsorptionspectruminsolu- that the tuning for supermolecular compound sys-
tion,oneneedstouseω tunedinvacuumandtoin- tems,whereanelectronrelaybetweenconstituents
clude solvent effects via PCM. In other words, we isofinterest,givesanimprovementoftheenergies
observe that, when using the PCM model, for ob- ofHOMOandLUMOorbitalslocalizedondiffer-
tainingareasonableagreementbetweenmeasured ent fragments. Thus, this approach is in principle
andcalculatedabsorptionspectraonehastosacri- suitableforaccuratemodellingofredoxmolecular
ficethederivativediscontinuitycondition. properties.
Theoptimizationoftherange-separationparam-
eter is usually performed for molecules in vac-
5 Conclusions
uum. Here, for the first time, an optimization in
the presence of a solvent described by the PCM
The present investigation of molecules and com-
modelhasbeendone. Fortheused∆SCFapproach
plexes relevant to photochemical water splitting
this yielded some unexpected results, namely that
points to the need for a system-specific optimiza-
the optimal ω tends to zero. As a consequence
tion of the range-separation parameter within the
there is almost no long-range correction and the
LC-BLYP approach. It provides an improved
quality, e.g. of electronic excitation spectra, cor-
band gap characteristics, increased ground state
responds to that of the BLYP functional. Since
stability, and a correct asymptotic behaviour of