976 J. Chem. Theory Comput.

2007, 3, 976-987

Dependence of Spurious Charge-Transfer Excited States

on Orbital Exchange in TDDFT: Large Molecules and
Clusters

R. J. Magyar

NIST Center for Theoretical and Computational Nanosciences (NCTCN),

Gaithersburg, Maryland 20899, and Theoretical DiVision, Los Alamos National
Laboratory, Los Alamos, New Mexico 87545

S. Tretiak*

Theoretical DiVision and Center for Nonlinear Studies, Los Alamos National
Laboratory, Los Alamos, New Mexico 87545

Received September 9, 2006

Abstract: Time-dependent density functional theory (TDDFT) is a powerful tool allowing for
accurate description of excited states in many nanoscale molecular systems; however, its
application to large molecules may be plagued with difficulties that are not immediately obvious
from previous experiences of applying TDDFT to small molecules. In TDDFT, the appearance
of spurious charge-transfer states below the first optical excited state is shown to have significant
effects on the predicted absorption and emission spectra of several donor-acceptor substituted
molecules. The same problem affects the predictions of electronic spectra of molecular
aggregates formed from weakly interacting chromophores. For selected benchmark cases, we
show that today’s popular density functionals, such as purely local (Local Density Approximation,
LDA) and semilocal (Generalized Gradient Approximation, GGA) models, are qualitatively wrong.
Nonlocal hybrid approximations including both semiempirical (B3LYP) and ab initio (PBE1PBE)
containing a small fraction (20-25%) of Fock-like orbital exchange are also susceptible to such
problems. Functionals that contain a larger fraction (50%) of orbital exchange like the early
hybrid (BHandHLYP) are shown to exhibit far fewer spurious charge-transfer (CT) states at the
expense of accuracy. Based on the trends observed in this study and our previous experience
we formulate several practical approaches to overcome these difficulties providing a reliable
description of electronic excitations in nanosystems.

I. Introduction the optical properties of large molecular systems containing

hundreds of atoms.1-3 Combined the two have been suc-
Density functional theory (DFT) is one of the most promising cessful in determining ground-state geometries, vibrational
tools to provide an accurate yet computationally feasible spectra, and photoexcitations of many small molecules.4,5
electronic-structure theory for large nanoscale systems, and These days, accurate measurements are now readily available
the time-dependent extension of DFT (TDDFT) makes
through various absorption and emission time- and frequency-
possible perhaps the only first principles approach to access
resolved spectroscopies even for large molecular systems and
their assemblies. The understanding and controlled manipu-
* Corresponding author e-mail: serg@lanl.gov. lation of optical properties of nanoscale molecular clusters
10.1021/ct600282k CCC: $37.00 © 2007 American Chemical Society
Published on Web 04/20/2007
Spurious Charge-Transfer Excited States in TDDFT J. Chem. Theory Comput., Vol. 3, No. 3, 2007 977

may lead to many technological applications such as light- TDDFT kernel constitutes a specific case of nonadiabatic
harvesting and photovoltaic devices6,7 and precision chemo- approximation.23
and biosensors8,9 to name just a few. Several comprehensive One of the greatest challenges to the universal applicability
reviews of quantum-chemical methodology for UV-visible of practical TDDFT is the inability of popular density
spectra of large molecules has been published recently (e.g. functionals (and their kernels) to routinely and accurately
refs 10 and 11). So there is a clear need to apply TDDFT describe certain excitations with a long-range spatial
here. extent.24-27 For example, charge-transfer (CT) states cor-
Density functional theory formally permits the expression respond to excited states in which the photoexcited hole and
of the total ground-state energy and other properties of a electron do not greatly overlap due to incorrect long-range
quantum many-body system as functionals of the electron functional asymptotics and the missing discontinuity of the
density alone and provides a formally exact scheme for approximate xc-potentials with respect to particle number.
solving the many-body problem.12 TDDFT is an extension Subsequently, TDDFT often predicts charge-transfer states
of density functional theory in which many-body excitations of substantially lower energy and below optical states. A
are associated with the poles of the exact density re- heuristic way to understand this is to consider the KS orbital
sponse.3,13,14 Practical density functionals are approximate excitations as zeroth-order approximations to the true excita-
and typically make use of an auxiliary orbital-based Kohn- tions of the system. The TDDFT linear response formalism
Sham (KS) scheme in which the kinetic energy contribution can be thought of as the inclusion of an effective mixing
of the total energy is expressed as an orbital functional. The determined by the exchange-correlation kernel between the
only contribution to the total energy that must be approxi- pair of occupied and unoccupied KS orbitals that constitutes
mated is the exchange-correlation contribution. This contri- the excited state. With the most commonly used approximate
bution is often approximated by functionals which depend kernels, such as LDA and GGA models, many bound exciton
on the electronic density locally (Local Density Approxi- states are not described at all.27,28 This is because the kernal
mation, LDA) or semilocally (Generalized Gradient Ap- is local and the orbital overlap is negligible. This problem
proximation, GGA) in a way convenient for calculations. is further complicated by the neglect of an important
However, more accurate functionals require further long- derivative discontinuity in most functionals. This results in
range information about the density which is contained in a mismatch of ionization potentials between the donor and
the Kohn-Sham orbitals. A step toward a general orbital acceptor portions of molecule.29 Thus, the energies of CT
exchange-correlation functional is the exact-exchange (EXX) states are frequently so significantly lower than what they
which has recently become popular in the solid state-physics should be that these states often become the lowest energy
community. Results within EXX should exhibit both the states in calculated electronic spectra.
proper asymptotic behavior for the effective potential, ∼1/r These TDDFT shortcomings have been well identified and
for finite systems, and a derivative discontinuity as the explored in the case of small molecular systems and local
number of particles changes through integer values. Properly approximations. In this work, we explore how DFT’s
describing these could mean more accurate band-gaps and predictive power changes when we simulate the ground and
optical spectra. We note too that long-range effects will excited states of larger molecules and aggregates. In par-
become important when treating the excited states of the ticular, we investigate the appearance of low-lying unphysical
jellium model. These problems are discussed in detail in refs CT states which depends on the choice of density functional.
14-16. We first focus on ground- and excited-state geometries in
Orbital exchange in the DFT formalism is the exact orbital the substituted molecules. Such calculations constitute a
contribution of exchange to the exchange-correlation func- common approach to investigate the coupling of electronic
tional and can be written as a Fock-like integral over the and vibrational degrees of freedom (frequently quantified as
KS orbitals.17,18 The rewards for using orbital exchange are Frank-Condon overlaps) and to calculate spectroscopically
tempting and include more realistic potentials, better decay- observed vibrational structure in absorption and fluorescence
ing KS orbitals, and more accurate excited states. However, spectral line shapes.30,31 We next study electronic interactions
extracting the local potential and its derivative kernel from mediated by aggregation in molecular assemblies. Computa-
this exact-exchange functional is difficult and in practice, is tion of small molecular aggregates is a typical technique to
often done approximately by replacing the local exact- determine interchromophore interactions in molecular
exchange potential by a nonlocal Hartree-Fock exchange. assemblies.32-34 This allows determination of aggregation
All current routine applications of TDDFT are based on the type (H or J) and electronic coupling energies. The calculated
adiabatic approximation, where the memory effects in the information is useful for building reduced Frenkel exciton
time-dependent density evolution are neglected.19,20 The models, for example, one for Light-Harvesting Complex II
adiabatic approximation, for example, makes it impossible (LHCII).35,36 Such tasks extend well beyond a standard simple
to describe highly correlated electronic states such as those case of ground-state geometry optimizations followed by
having predominantly double character.21 It has been argued excited-state calculations and constitute the area where the
that omission of nonadiabatic effects is also solely respon- performance of TDDFT has not been consistently tested yet.
sible for the charge-transfer problems.22 In this paper, we We chose representative molecules with properties that
show that long-range orbital-based exchange improves upon could be useful for several important technological applica-
the semilocal approximations when describing charge-transfer tions: donor-donor and donor-acceptor substituted mol-
states. This can be rationalized by noting that the hybrid ecules and aggregates of thiophene oligomers and bacteri-
978 J. Chem. Theory Comput., Vol. 3, No. 3, 2007 Magyar and Tretiak

performed in the gas phase using the SV basis set and one
of a set of density functionals; see below. The SV and similar
6-31G basis sets are known to be an efficient blend of
accuracy and manageable size for large conjugated mol-
ecules.48 For the thiophenes and chlorophyll calculations we
used geometries extracted from the experimental X-ray
crystallographic data.49,50 In addition to the single molecule
limit, we consider thiophene dimers (2Th4) generated from
the coordinates of two nearest neighbor pairs in the unit cell.49
For the single molecules, two Th4 geometries exist in the
crystal structure. One structure is slightly more elongated.
The calculated excited-state spectra are essentially identical
for both geometries. Similarly, the coordinates of Râ Bchls-a
pair (2Bchl-a) have been extracted from Rs. molischianum
crystal data.50 At the obtained geometries we calculate
Figure 1. Chemical diagrams of the molecules studied:
corresponding excited-state structures up to 20 lowest excited
donor-acceptor (DA) and donor-donor (DD) substituted
states using the Gaussian 03 package51 with a 6-31G basis
chromophores, thiophene oligomer (Th4), and bacteriochlo-
rophyll a (Bchl-a). set and density functionals corresponding to the methods used
for geometry optimizations (unless specified otherwise).
ochlorophylls. Substituted organic chromophores are promising We use several common density functionals with vari-
candidates in several areas, such as nonlinear optical response ous fractions of orbital exchange, namely HF (100%),
applications.37,38 The thiophenes have been exploited for BHandHLYP (50%), PBE1PBE (25%), B3LYP (20%),
applications in organic electronics related to novel display, TPSS-H (10%), TPSS (0%), BP86 (0%), and SVWN (0%).
photovoltaic, and lighting technologies.39,40 Finally, the The set represents a gradual decreasing fraction of exchange.
chlorophylls constitute the main pigment in the biological The treatments of correlation in each functional are not
photosynthetic light-harvesting complexes.35,36 Previous theo- comparable and do not scale with the given fractions of
retical work within DFT on some of these systems has been exchange; however, the effects of correlation are assumed
carried out within the DFT formalism for donor-acceptor to be smaller than the relative error in the exchange amount.
molecules,41,42 thiophenes,43 and bacteriochlorophylls.44,45 General trends seen as a function of the fraction of orbital
Here our calculations examine the qualitative and quantitative exchange should persist regardless of the details of the
aspects of TDDFT for excited-state molecular electronic correlation functional. We expect that functionals, including
structures. The results are compared for several commonly exchange, are less dramatically affected by CT problems.
used functionals with varying fractions of Hartree-Fock However, when the faction of exchange becomes large, these
(HF) exchange as implemented in standard quantum chemical functionals typically provide less accurate energetics because
packages. By tuning the fraction of HF exchange we hope the cancellation of errors between exchange and correlation
to gain insight into how the orbital dependence of the is reduced.52 For example, the BHandHLYP functional53
exchange function affects the description of electronic excited combines semilocal exchange-correlation with orbital ex-
states. change in a 50-50 ratio. By construction, this functional
Details of our numerical modeling are presented in section handles a large fraction of long-range exchange exactly but
II. In section III we analyze computational results obtained fails to describe correlation in a compatible way.
via different DFT approximations. Finally, we discuss the To interpret computational trends we use a transition
trends that emerge and summarize our findings in section orbital analysis, which allows precise identification and
IV. visualization of the electronic excitations in question.54 These
transition orbitals provide a graphical real-space representa-
II. Computational Methodology tion of the transition densities associated with the molecular
Figure 1 shows the chemical structures of the molecules electronic excitations computed with TDDFT. This analysis
considered, including donor-acceptor DA ((E)-4-(4-(meth- offers the most compact description of a given transition
ylsulfonyl)styryl)-N,N-diphenylbenzenamine) and donor- density in terms of its expansion in single KS transitions.
donor DD (4,4′-[(1E,1′E)-2,2′-(biphenyl-4,4′-diyl) bis(ethene- The Gaussian 03 code was locally modified to be able to
1,1′-diyl)]bis(N,N-dimethylbenzenamine)) substituted mole- perform the transition orbital analysis.
cules, thiophene oligomer (Th4), and bacteriochlorophyll-a
(Bchl-a).
Ground-state geometry optimizations of the charge-neutral, III. Results and Discussion
singlet states of the substituted molecules have been done Through a series of benchmark calculations, we highlight
with the Turbomole program suite.46 The lowest excited- the difficulties appearing when we apply TDDFT to molec-
state geometries were obtained next using Turbomole as well. ular systems of sizes important to nanoscale applications.
This code is able to search for a minimum over the TDDFT An important issue with using TDDFT for larger molecular
excitation energy surface with respect to nuclear coordinates systems is the introduction of spurious low-energy CT states.
using analytic gradient techniques.47 All optimizations were In the following subsections, we investigate several fre-
Spurious Charge-Transfer Excited States in TDDFT J. Chem. Theory Comput., Vol. 3, No. 3, 2007 979

Figure 2. The calculated optical excitation energy as a function of the percentage of orbital exchange for the ground-state (GS)
and the first excited-state (ES) optimized geometries of the donor-donor (DD) and donor-acceptor (DA) molecules just the GS
for the 2Th4 and 2Bchl-a dimers.

quently unexpected manifestations of this problem and A. Donor-Acceptor Substituted Chromophores. We

explore their implications. begin this investigation by examining a relatively large
We arbitrarily label three types of singlet excited states: (approximately 70 atoms) molecular donor-donor (DD)
‘optical’, ‘dark’, and ‘charge transfer’. The optical excitation system. This molecule has a long π-conjugated backbone
has a nonvanishing oscillator-strength, which means that it and donors (etholanimes) at both ends of the backbone. This
is possible to detect this excited state spectroscopically. The is perhaps the easiest test case for TDDFT and offers the
dark state refers to a low-energy electronic excitation with best opportunity for the theory to provide accurate results.
vanishing oscillator strength, that will not appear in the The molecule has no strong electron attractors, and we do
optical absorption. A CT state is a specific case of ‘dark’ not expect large-scale charge transfer with electrons and holes
state with a highly ionic nature where the photoexcited migrating from/to opposite ends of the molecule. The charge
electron and hole have significant spatial separation. The transfer should be of a mild nature from the donors to the
oscillator strengths of CT states are expected to be weak conjugated backbone. In the upper left-hand corner of Figure
because the overlap between the charge-transfer state and 2, we plot the energy of the lowest excited state versus
the ground state is extremely small and so are the transition functional for the DD molecule at both ground- and excited-
dipole moments between these states. For example, in organic state optimal geometries. The respective calculated excitation
molecular crystals, CT states are usually higher in energy energies and their oscillator strengths are listed in Table 1.
than the relevant Frenkel-exciton band of optical and dark The experimental values are from refs 42 and 56. Overall,
states.55 However, DFT methods drastically underestimate functionals with a moderate fraction of orbital exchange such
the energies of these states. For this reason, all three types as B3LYP provide the best agreement with the experimental
of states in dimers might be mixed, and their clear distinction absorption and fluorescence maxima, neglecting the proper
is not possible. The CT states become particularly trouble- treatment of vibronic, temperature, and nonpolar solvent
some in the optimization of the excited-state geometries when effects.
the target excited state is not clearly constrained by sym- The lowest-energy (band gap) transition in such conjugated
metry, and level crossings may occur. molecules typically has a remarkable oscillator strength and
980 J. Chem. Theory Comput., Vol. 3, No. 3, 2007 Magyar and Tretiak

Table 1. Calculated Excitation Energies (eV) of the the short-range description embodied in the BP86 functional.
Donor-Donor (DD) and Donor-Acceptor (DA) However, there is a quantitative difference in the energetics.
Compoundsa This is consistent with the well-known fact that local density
molecule method % exchange GS ES functionals are known to perform poorly for certain properties
DD HF 100 3.88 (2.56) 3.24 (2.58) in the limit of longer conjugated chains.28 The conjugation
DD BHandHLYP 50 3.38 (2.72) 2.90 (2.80) length in this donor-donor (DD) molecule is perhaps too
DD PBE1PBE 25 2.89 (2.43) 2.58 (2.63) short for these semilocal functional shortcomings to become
DD B3LYP 20 2.76 (2.29) 2.48 (2.53) visually dramatic, but they remain energetically significant.
DD BP86 0 2.20 (1.61) 2.1 (1.2) A more difficult test case is the donor-acceptor (DA)
DA HF 100 4.27 (1.57) 3.39 (1.64) molecule in which the charge transfer between the donor
DA BHandHLYP 50 3.61 (1.40) 3.15 (1.50) and acceptor regions is energetically favorable and pro-
DA PBE1PBE 25 3.02 (1.06) 2.33 (0.13) nounced. It is hard to properly describe the underlying long-
DA B3LYP 20 2.89 (0.99) 2.13 (0.08) range interactions behind CT using a semilocal functional.
DA BP86 0 2.34 (0.72) 1.29 (0.02)
The upper right-hand corner of Figure 2 shows scaling of
a The corresponding oscillator strengths f are given in parentheses.
the lowest singlet-state energy for the DA molecule, and the
GS and ES refer to the ground-state and the first excited-state
optimized molecular geometries, respectively. The SV basis set was calculated data are presented in Table 1. As for the DD
used for all calculations. Experimental absorption and fluorescence molecule, the excitation energy calculated at the ground-
maxima are ωabs ) 3.0 eV and ωfl ) 2.7 eV for DD and ωabs ) 3.2 state geometry dramatically increases with the increasing
eV and ωfl ) 2.7 eV for DA. fraction of exchange. This state is optically allowed having
characteristically appears as a main peak in the linear substantial oscillator strength. The HF transition orbitals
absorption spectra.41,42 The calculated excited-state energy shown in Table 2 reveal this excitation of π-π nature
shifts to the blue with an increase of the fraction of orbital delocalized between donor and acceptor. Steric distortion
exchange. Likewise, the oscillator strength of the transition prevents participation of terminal phenyls, and the state has
follows the fraction of exchange. These trends are expected only weak CT character. In contrast, BP86 transition orbitals
since exact exchange cancels Coulomb self-interaction, and, display strong CT character in this excited state (see Table
thus, a larger fraction of orbital exchange corresponds to a 2). Since the transition is delocalized, the overlap between
contracted core. The valence states are also more tightly electron and hole wave-functions is small, which results in
bound by the orbital exchange potential, but the effect is a reduced oscillator strength compared to the other methods.
less extreme compared to the core contraction, since the HF BP86 shows to the incorrect physical behavior of semi-
virtual orbitals are calculated for the N+1 electron system local functionals. The accurate numbers should lie some-
and their energies correspond to electron affinities rather than where in between HF and BP86 extremes, and this is where
excitation energies. Consequently, the KS orbitals become the hybrid TDDFT excels.42 The situation with semilocal
widely separated. This is consistent with results seen from functionals becomes even more problematic after excited-
noble-gas solids and atoms treated by exact-exchange DFT.57 state optimization. While the functionals with a large
Since HF has even more weakly bound virtual orbitals than percentage of orbital exchange (HF and BHandHLYP)
EXX, we expect the effect to be somewhat overemphasized. behave normally, other methods converge to low-lying
The energy of the first singlet state calculated at a relaxed spurious CT states, where spatial overlap between an electron
excited-state geometry exhibits similar trends as a function and a hole orbital is negligible (see Table 2), making the
of the fraction of orbital exchange. During excited-state excitation optically forbidden (Table 1). A significant rear-
geometry optimization the bonds along the conjugated rangement of the molecular geometry is observed during
backbone stretch, so that the bond-length alternation caused excited-state geometry optimization, and this facilitates the
by uneven distribution of π-electronic density reduces.28 This crossover to the ionic CT state with a huge 1 eV Stokes
allows the electrons to become more delocalized, and the shift.
excited-state energy is reduced. The magnitude of the To rationalize that such ionic states are artifacts of the
observed Stokes shift and the respective geometry changes method, we recall that the ground- and lowest-energy excited
intensify with an increase of the fraction of exchange. electronic states of substituted push-pull molecules are often
To analyze the nature of the discussed excited state, Table described as a combination of neutral and zwitterionic basis
2 displays the transition orbitals of the DD molecule obtained states represented by the corresponding molecular resonance
at the HF and BP86 limits, respectively. The two represent forms.58,59 The zwitterionic state assumes full separation of
extremes in the locality of the functional. BP86 has only positive and negative charges, and, consequently, it is
semilocal density-based exchange, while HF relies on the optically forbidden. The excited state usually possesses
full orbital exchange. For the DD molecule, we see no greater zwitterionic basis state character than the ground state.
qualitative difference between the HF and BP86 transition The molecular structures also become more zwitterionic in
orbitals: they represent delocalized electronic state with character as solvent polarity increases.59 The semilocal
electronic density slightly shifting toward the middle of the functionals and functionals with a small fraction of orbital
molecule upon photoexcitation. Transition orbitals describing exchange do predict the ground state to be the mixture of
emitting states are qualitatively similar to that of the such states; however, the excited state is described to be of
absorbing state. We note that the BP86 orbitals are slightly solely zwitterionic character (see Table 2). In contrast, the
more delocalized compared to the HF ones. This is due to DA molecule exhibits pronounced fluorescence properties,42
Spurious Charge-Transfer Excited States in TDDFT J. Chem. Theory Comput., Vol. 3, No. 3, 2007 981

Table 2. Hartree-Fock and BP86 Transition Orbitals of the Lowest Excited State for the Ground-State (GS) Geometries of
the Donor-Donor (DD) and Donor-Acceptor (DA) Moleculesa

a These orbitals represent the full orbital exchange and semilocal functional limits, respectively. GS refers to the ground-state optimized

molecular geometry.

pointing to the strong component of the neutral basis in the is used. In practice, these molecules are often studied in
excited state. The observed emission stems from the lowest solution, so we estimate solvent effects using the Polarizable
vibronic state, and Kasha’s rule is not violated. Thus all Continuum Model (PCM) based on the Integral Equation
available experimental data point that the DA compound does Formalism61-64 for toluene. Adding PCM stabilized the
not exhibit dual-fluorescence properties as observed on other vertical absorption maximum by 0.15 eV to the red with the
molecular systems such as 4-(dimethyl)aminobenzonitrile most dramatic effects for the BP86 GGA kernel (see Table
(DMABN).60 Moreover, computational excited-state geom- 3). Subsequently, using extended basis sets and solvent
etry optimizations of the bright and the CT state for selected models leads to the overall 0.1-0.25 eV red-shifts of the
GGA functionals (not shown) do not change the relative state excitation energies compared to plain 6-31G calculations in
ordering. An opposite extreme is observed at the HF level vacuo without changing the essential photophysics. This is
when both ground and excited states have a dominant neutral a typical picture for extended molecules with delocalized
component. Subsequently, the final result has a strong π-orbitals, in contrast to small molecules of a few-atoms,
dependence on the fraction of orbital exchange used in the where large basis sets are necessary.
functional, which lessens the predictive power of TDDFT B. Molecular Aggregates. In section IIIA, we have shown
in applications to the polar substituted molecules. that TDDFT may incorrectly predict excited-state properties
Finally, in order to assess the basis set dependence, we for large molecules with donor-acceptor character. Another
performed several single point calculations on the DA interesting case, where such problems persist, is molecular
molecule with split-valence basis sets of increasing complex- aggregates. The separation distances are even larger for
ity. The simplest is the 6-31G basis, then the same with aggregates, and the nonlocality should play an even more
polarization functions (6-31G*), then with diffuse functions important role. Modeling of molecular aggregates and
(6-31+G), and with both (6-31+G*). Table 3 shows the assemblies can frequently be reduced to computations on
lowest optical excitation energy for both the original Tur- characteristic dimers extracted from the underlying aggregate
bomole SV-optimized geometry obtained for a given func- structure. This allows for the understanding of electronic
tional in vacuo and then the native geometry. The latter is couplings,33,65 excited state, and charge dynamics34 seen in
the optimized structure consistent with the level of theory ultrafast optical probes. This information in important for
and basis set used for excited-state calculations. Adding light-harvesting and photovoltaic applications of the materi-
diffuse functions alone shifted the vertical absorption spec- als. According to molecular orbital and Hückel theory, every
trum by a maximum of 0.13 eV to the red, and adding excited state of an isolated molecule should split into a nearly
polarization functions shifted the spectrum by 0.1 eV to the degenerate pair of states for the dimer (the Davydov’s pair).
red as well with the most dramatic basis set effect for pure The magnitude of splitting characterizes intermolecular
HF. These changes are less dramatic when native geometry coupling. Depending on the orientation of the molecules,
982 J. Chem. Theory Comput., Vol. 3, No. 3, 2007 Magyar and Tretiak

Table 3. Calculated Vertical Excitation Energies (eV) of Table 4. Calculated Excitation Energies (eV) of
the Donor-Acceptor (DA) Compound within Several Levels Thiophene Oligomer (1Th4) and Its Dimer (2Th4)a
of Theorya % optical dark no. of CT
method/basis set SV geometry native geometry molecule method exchange state state states

HF/6-31G 4.31 (1.59) 4.45 (1.56) 1Th4 HF 100 3.59 (1.47)

HF/6-31G* 4.21 (1.57) 4.43 (1.48) 1Th4 BHandHLYP 50 3.36 (1.28)
HF/6-31+G 4.18 (1.45) 4.33 (1.41) 1Th4 PBE1PBE 25 3.14 (1.26)
HF/6-31+G* 4.07 (1.43) 4.22 (1.40) 1Th4 B3LYP 20 3.04 (1.22)
HF/6-31G PCM 4.19 (1.70) 4.35 (1.60) 1Th4 TPSS 10% 10 2.98 (1.21)
HF/6-31+G* PCM 3.96 (1.57) 4.11 (1.44) 1Th4 TPSS 0 2.82 (1.13)
BHandHLYP/6-31G 3.71 (1.40) 3.66 (1.42) 1Th4 BP86 0 2.77 (1.09)
BHandHLYP/6-31G* 3.66 (1.44) 3.66 (1.42) 1Th4 SVWN 0 2.77 (1.08)
BHandHLYP/6-31+G 3.61 (1.33) 3.61 (1.33) 2Th4 HF 100 3.67 (2.21) 3.47 0
BHandHLYP/6-31+G* 3.51 (1.33) 3.49 (1.36) 2Th4 BHandHLYP 50 3.45 (2.19) 3.21 0
BHandHLYP/6-31G PCM 3.59 (1.54) 3.51 (1.56) 2Th4 PBE1PBE 25 3.24 (2.19) 2.97 1
BHandHLYP/6-31+G* PCM 3.42 (1.50) 3.45 (1.44) 2Th4 B3LYP 20 3.15 (1.5) 2.88 2
PBE1PBE/6-31G 3.12 (0.99) 3.04 (1.06) 2Th4 TPSS 10% 10 3.08 (1.35) 3
PBE1PBE/6-31G* 3.05 (0.97) 3.05 (1.08) 2Th4 TPSS 0 2.92 (1.94) 4
PBE1PBE/6-31+G 3.01 (0.99) 2.96 (1.03) 2Th4 BP86 0 2.86 (1.86) 4
PBE1PBE/6-31+G* 2.99 (1.02) 2.97 (1.05) 2Th4 SVWN 0 2.86 (1.83) 4
a The corresponding oscillator strengths f are given in parentheses.
PBE1PBE/6-31G PCM 2.99 (1.11) 3.03 (1.08)
PBE1PBE/6-31+G* PCM 2.87 (1.13) 2.91 (1.11) Experimental absorption maximum is ωabs ) 2.7 eV for Th4 in
solution.
B3LYP/6-31G 2.98 (0.90) 2.91 (0.99)
B3LYP/6-31G* 2.95 (0.93) 2.93 (1.01) of charge-transfer states below the first optical singlet. For
B3LYP/6-31+G 2.90 (0.89) 2.90 (0.98) example, LDA gives at least 4. For the BP86 GGA, the first
B3LYP/6-31+G* 2.87 (0.91) 2.92 (0.99) optical state is given as state 6 instead of 2. In this case,
B3LYP/6-31G PCM 2.85 (1.02) 2.77 (1.14)
Figure 2 shows these states as stars. They are heavily mixed
B3LYP/6-31+G* PCM 2.73 (1.04) 2.75 (1.18)
with dark and optically allowed excitonic states (e.g., Table
BP86/6-31G 2.39 (0.62) 2.34 (0.72)
5). We further notice that these spurious charge-transfer states
BP86/6-31G* 2.37 (0.62) 2.33 (0.71)
disappear (shift upward in energy) with increasing separation
BP86/6-31+G 2.34 (0.62) 2.27 (0.69)
BP86/6-31+G* 2.31 (0.63) 2.31 (0.72)
between monomers for hybrid functionals with small amount
BP86/6-31G PCM 2.25 (0.72) 2.20 (0.86)
of orbital exchange (B3LYP and PBE1PBE) but carry on
BP86/6-31+G* PCM 2.16 (0.74) 2.17 (0.91) for all LDA and GGA models (not shown). Subsequently, a
a The corresponding oscillator strengths f are given in parentheses.
small fraction of the HF exchange would help to predict the
correct monomer limit of excitation energies in the aggregate
either the lower or higher energy state would acquire a larger at large intermolecular separations (>5 Å) when the inter-
oscillator strength. These two classes are classified as J or chromophore orbital overlap vanishes but fails at smaller
H aggregates, respectively.55 In this section, we investigate distances. In contrast, the CT problems persist for GGA
both cases. semilocal functionals at all intermolecular separations, which
First, we examine the thiophene dimers (2Thio4) exem- agrees with the results obtained for the ethylene dimer (off-
plifying the H-aggregate case. In the lower left-hand corner site excitations).67 The failure of semilocal functionals to
of Figure 2, we plot the calculated excitation energies for properly describe and characterize these low-lying states
thiophene dimers compared to the single molecule limit. means that TDDFT would provide a highly unreliable
Table 4 shows the respective data. The experimental result description of electronic coupling and excitonic effects in
is from ref 66. Thiophenes form H-aggregates in crystals. such molecular aggregates.
Experimentally, this is exemplified by a blue-shifted absorp- In a second example of molecular assemblies, we consider
tion and small fluorescence yield compared to the corre- a form of chlorophyll (Bchl-a) and its dimer (2Bchl-a) from
sponding properties of the monomer. Parallel orientation of the light-harvesting system of purple bacteria. The compu-
the molecules in a dimer results in a positive intermolecular tational results are shown in the lower right-hand corner of
coupling term. This shows up as a higher-lying optical state Figure 2 and Table 6, and the experimental numbers are from
(in phase combination of monomer’s wave functions) than ref 68. The Râ Bchls-a pair forms a near J-aggregate complex
the dark state (out of phase combination of monomer’s wave as evidenced by the red-shifted first optical states in the
functions). This picture holds relatively well for functionals dimer. In this case, the dark states lie above the optical ones.
with a large fraction of orbital exchange (HF and BHandH- The full circular LHCII complex has, however, different
LYP): the Davydov’s states split equally up and down from properties suitable for efficient light-harvesting.35,36,68 We see
the isolated molecule limit (see Figure 2) and appear as a an interesting trend for the chlorophyll monomers. As more
linear combination of the monomer states in the transition orbital exchange is included, the predicted singlet energy is
orbitals (see Table 5). With less than 50% exchange, several reduced. This is probably because of a more localized nature
spurious charge-transfer states appear. A smaller fraction of of the excited state and compact 2D size of the molecule.
exchange generally means a larger number This trend may be rationalized by noting that exchange
Spurious Charge-Transfer Excited States in TDDFT J. Chem. Theory Comput., Vol. 3, No. 3, 2007 983

Table 5. Transition Orbitals of Thiophene Dimer (2Th4)a

a The top two and bottom two rows represent Hartree-Fock and BP86 limits. States 1 and 2 in the HF limit refer to the dark and optical

transitions in the Davydov’s pair, respectively. States 1 and 6 in the BP86 limit show charge-transfer character in the lowest dark and optically
allowed transitions, respectively.

Table 6. Calculated Excitation Energies (eV) of for orbital based functionals and a larger gap for semilocal
Bacteriochlorophyll a (1Bchl-a) and Râ Bchls-a Dimer ones. In the dimer, we observe the expected superposition
(2Bchl-a)a of the monomer’s states in the transition orbitals for HF and
% optical dark CT BHandHLYP methods. As an example, Table 7 shows the
molecule method exchange state state states first dominant pair of transition orbitals localized on one
1Bchl-a HF 100 1.21 (0.42) chromophore for the first and second excited state in the
1Bchl-a BHandHLYP 50 1.81 (0.40) HF limit. The second pair (not shown) involved the second
1Bchl-a PBE1PBE 25 1.90 (0.36) molecule, respectively. Below 50% exchange, however,
1Bchl-a B3LYP 20 1.89 (0.35) charge-transfer states begin to contaminate the optical spectra
1Bchl-a TPSS 10% 10 1.91 (0.33)
(see Table 7) making it impossible to reliably identify the
1Bchl-a TPSS 0 1.88 (0.30)
states involved. Formation of such ghost states have been
1Bchl-a BP86 0 1.85 (0.27)
observed before in chlorophylls treated with TDDFT based
1Bchl-a SVWN 0 1.84 (0.27)
2Bchl-a HF 100 1.12 (0.90) 1.32 (0.08) 0
on GGA models.69,70 Finally we note that recently developed
2Bchl-a BHandHLYP 50 1.71 (0.89) 1.85 (0.04) 0 functionals based on meta-GGA extensions (TPSS and TPSS-
2Bchl-a PBE1PBE 25 1.90 (0.54) 2 hybrid) are subject to the same charge-transfer problems as
2Bchl-a B3LYP 20 1.88 (0.61) 2 the other approaches for both thiophene and chlorophyll
2Bchl-a TPSS 10% 10 1.88 (0.67) 2 dimers (Tables 4 and 6). The large fraction of the orbital
2Bchl-a TPSS 0 1.84 (0.64) 4 exchange turns out to be the only factor affecting the quality
2Bchl-a BP86 0 1.81 (0.62) 4 of the results.
2Bchl-a SWVM 0 1.85 (0.64) 4
Recent TDDFT study of a much smaller system, the
a The corresponding oscillator strengths f are given in parentheses.
ethylene dimer, shows a similar problem with CT states in
The experimental absorption maximum of Qx band of Bchls-a is at
about 1.6 eV for a light-harvesting antenna. the symmetric dimers.67,71,72 Notably, in our examples of
2Thio4 and 2Bchl-a, the monomers are not identical.
creates a steeper optical potential comparable to the core Consequently the observed CT states have nonzero net charge
contraction. The net effect would be a smaller optical gap transferred between chromophores (see Tables 5 and 7),
984 J. Chem. Theory Comput., Vol. 3, No. 3, 2007 Magyar and Tretiak

Table 7. Transition Orbitals of Râ Bacteriochlorophyll a Dimer (2Bchl-a)a

a The top two and bottom two rows represent Hartree-Fock and BP86 limits, respectively. States 1 and 2 in the HF limit refer to the dark and

optical transitions in the Davydov’s pair, respectively. States 1 and 5 in the BP86 limit show charge-transfer character in the lowest dark and
optically allowed transitions, respectively.

opposite to what has been observed in a perfectly symmetric some chemical defect such as an sp3 kink. These conclusions
case due to symmetry reasons.67 In any case, it is an fully agree with the general analysis in refs 67, 71, and 72.
inaccurate xc-potential that leads to the appearance of
unphysical low-lying states, generally dubbed as ‘CT states’,
IV. Conclusions
In this study, we have examined the performance of the
irrespectively if the actual charge is transferred or not upon
TDDFT approach for calculations of excited states. The
excitation. We emphasize that physical analysis of coupling
benchmark systems include several polar donor-acceptor
between various molecular regions is very important when
substituted compounds and molecular aggregates. These are
deciding whether TD-DFT is applicable to ‘problematic’ typical examples of nanosized molecules currently in the
cases or not: Small overlap between individual subsystems focus of applied quantum chemistry. For all of the systems
of a large system is particularly susceptible to the CT studied, we have employed a wide range of modern func-
problems. Either vanishing overlap (e.g., large separations tionals including state-of-the-art meta-GGAs. Overall, today’s
between monomers) or ‘strong’ communication (e.g., fully available density functionals often do not handle charge-
π-conjugated bridge) will help to offset the CT failures (i.e., transfer states properly. We see that no amount of sophistica-
here hybrid functionals with small amount of the orbital tion in the currently available semilocal functionals describes
exchange may work). For example, the CT problems in the CT states successfully. These problems, however, can
donor-acceptor substituted compounds would be even more be overcome by including a larger fraction of orbital
severe if the donor/acceptor subsystems are separated by exchange, but this is at the expense of accuracy. We point
Spurious Charge-Transfer Excited States in TDDFT J. Chem. Theory Comput., Vol. 3, No. 3, 2007 985

out that exact-exchange in density functional theory and Acknowledgment. This work was carried out under
Hartree-Fock exchange are not identical, and, in particular, the auspices of the National Nuclear Security Administration
they differ in their representation of virtual orbitals. EXX of the U.S. Department of Energy at Los Alamos National
DFT virtual orbitals are more strongly bound than their HF Laboratory. The research at LANL is supported by the Center
analogies. This difference means that EXX-based virtual for Nonlinear Studies (CNLS) and the LANL LDRD
orbitals for the LUMO should be more localized than the program. This support is gratefully acknowledged.
HF analog. Consequently, our results will overemphasize the
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