Time-dependent density functional study of the electronic spectra of oligoacenes in the charge states −1, 0, +1, and +2

Abstract

We present a systematic theoretical study of the five smallest oligoacenes (naphthalene, anthracene, tetracene, pentacene, and hexacene) in their anionic, neutral, cationic, and dicationic charge states. We used density functional theory (DFT) to obtain the ground-state optimised geometries, and time-dependent DFT (TD-DFT) to evaluate the electronic absorption spectra. Total-energy differences enabled us to evaluate the electron affinities and first and second ionisation energies, the quasiparticle correction to the HOMO–LUMO energy gap and an estimate of the excitonic effects in the neutral molecules. Electronic absorption spectra have been computed by combining two different implementations of TD-DFT: the frequency–space method to study general trends as a function of charge-state and molecular size for the lowest-lying in-plane long-polarised and short-polarised π → π∗ electronic transitions, and the real-time propagation scheme to obtain the whole photo-absorption cross-section up to the far-UV. Doubly ionised PAHs are found to display strong electronic transitions of π → π∗ character in the near-IR, visible, and near-UV spectral ranges, like their singly charged counterparts. While, as expected, the broad plasmon-like structure with its maximum at about 17–18 eV is relatively insensitive to the charge-state of the molecule, a systematic decrease with increasing positive charge of the absorption cross-section between ∼6 and ∼12 eV is observed for each member of the class.

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Time-dependent density functional study of the electronic spectra
of oligoacenes in the charge states ?1, 0, +1, and +2
G. Mallocia,*, G. Mulasa, G. Cappellinib,a, C. Joblinc
aINAF – Osservatorio Astronomico di Cagliari–Astrochemistry Group, Strada 54, Localita ` Poggio dei Pini, I-09012 Capoterra (CA), Italy
bCNR – Sardinian Laboratory for Computational Materials Science, CNISM and Dipartimento di Fisica, Universita ` degli Studi di Cagliari,
Cittadella Universitaria, Strada Prov. le Monserrato-Sestu Km 0.700, I-09042 Monserrato (CA), Italy
cCentre d’Etude Spatiale des Rayonnements, Universite ´ Toulouse 3, CNRS, Observatoire Midi-Pyre ´ne ´es, 9 Avenue du Colonel Roche,
31028 Toulouse Cedex 4, France
Received 23 February 2007; accepted 20 July 2007
Available online 9 August 2007
Abstract
We present a systematic theoretical study of the five smallest oligoacenes (naphthalene, anthracene, tetracene, pentacene, and hexa-
cene) in their anionic, neutral, cationic, and dicationic charge states. We used density functional theory (DFT) to obtain the ground-state
optimised geometries, and time-dependent DFT (TD-DFT) to evaluate the electronic absorption spectra. Total-energy differences
enabled us to evaluate the electron affinities and first and second ionisation energies, the quasiparticle correction to the HOMO–LUMO
energy gap and an estimate of the excitonic effects in the neutral molecules. Electronic absorption spectra have been computed by com-
bining two different implementations of TD-DFT: the frequency–space method to study general trends as a function of charge-state and
molecular size for the lowest-lying in-plane long-polarised and short-polarised p ! p*electronic transitions, and the real-time propaga-
tion scheme to obtain the whole photo-absorption cross-section up to the far-UV. Doubly ionised PAHs are found to display strong
electronic transitions of p ! p*character in the near-IR, visible, and near-UV spectral ranges, like their singly charged counterparts.
While, as expected, the broad plasmon-like structure with its maximum at about 17–18 eV is relatively insensitive to the charge-state
of the molecule, a systematic decrease with increasing positive charge of the absorption cross-section between ?6 and ?12 eV is observed
for each member of the class.
? 2007 Elsevier B.V. All rights reserved.
Keywords: Acenes; Electronic absorption; Density functional theory; Time-dependent density functional theory
1. Introduction
Polycyclic aromatic hydrocarbons [1a,1b] (PAHs) are a
large class of conjugated p-electron systems of fundamental
importance in many research areas of chemistry as well as
in astrophysics and materials science. The carbon skeletons
of PAHs may be considered as small pieces of graphite
planes and, as such, they have been proposed as precursors
to extended carbon networks such as fullerenes and carbon
nanotubes [2a,2b]. PAHs are of high interest in environ-
mental chemistry due to their carcinogenicity and their
ubiquity as air pollutants produced by the combustion of
organic matter [3a,3b]. In the astrophysical context, PAHs
are found in carbonaceous meteorites [4a] and in interplan-
etary dust particles [4b]. Based on the astronomical obser-
vation of IR, visible and UV spectroscopic features, neutral
and charged PAHs are thought to be the most abundant
molecules in space after molecular hydrogen and carbon
monoxide [4c].
Oligoacenes, or simply acenes, are a subclass of catacon-
densed PAHs (with all carbon atoms on the periphery of
the ring system) consisting of fused benzene rings joined
in a linear arrangement. In their crystalline state these
0301-0104/$ - see front matter ? 2007 Elsevier B.V. All rights reserved.
doi:10.1016/j.chemphys.2007.07.046
*Corresponding author. Tel.: +39 070 675 4915; fax: +39 070 510171.
E-mail address: gmalloci@ca.astro.it (G. Malloci).
www.elsevier.com/locate/chemphys
Available online at www.sciencedirect.com
Chemical Physics 340 (2007) 43–58

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organic semiconducting materials have received particular
attention in the field of electronics and photonics
[5a,5b,5c,5d]. Acenes and their derivatives are being
increasingly used as active elements in a variety of opto-
electronic devices such as organic thin-film field-effect tran-
sistors [6a,6b], light-emitting diodes [7a,7b], photovoltaic
cells [8a,8b], and liquid crystals [9]. Organic electronics
based on functionalised acenes and heteroacenes is pres-
ently a very active field of research [10a,10b].
Since the early work by Clar [1a,1b] and Platt [11a,11b],
there has been a wide-ranging interest in the electronic
properties of PAHs using different spectroscopic tech-
niques, such as absorption [12a,12b,12c,12d,12e,12f,12g,
12h,12i,12j], electron-energy-loss [13a,13b,13c,13d], fluo-
rescence [14a,14b,14c,14d,14e,14f], polarisation [15a,15b,
15c], photoelectron [16a,16b,16c,16d,16e,16f], and photo-
ion mass spectrometry [17a,17b,17c,17d]. Thanks to these
studies, the electronic spectra of neutral PAHs are known
to be composed of two main regions: (i) the broad plas-
mon-like excitation peaking at ?17–18 eV, which involves
p ! r*, r ! p*, r ! r*, and Rydberg spectral transitions,
and (ii) the single-particle excitation part below a few eV,
where the lowest energy singlet–singlet p ! p*transitions
occur. The four lowest transitions of neutral PAHs are usu-
ally described by the Clars’s notation p, a, b, b0[1a], or1La,
1Lb,
[11a]. In catacondensed PAHs these transitions are charac-
terised by the following intensities and oscillator strengths:
p, weak, f ? 0.01 ? 0.1; a, very weak, f ? 0.001; b, very
strong, f ? 1.0; b0, medium strong, f ? 0.1 ? 1.0 (e.g.,
[17d]). In the case of oligoacenes, in particular, the transi-
tion dipole moment for the p-band lies along the short-axis
of the molecule, while for the a and b-bands it lies along the
long-axis. In the following we will refer to these transitions
as ‘‘short-polarised’’ and ‘‘long-polarised’’, respectively.
Charged PAHs have been the subject of extensive spec-
troscopic studies in frozen glassy organic solids [18a,18b].
These experiments showed that PAH radical cations and
anions: (i) display intense optical transitions at lower ener-
gies than their parent molecule, and (ii) have very similar
electronic spectra, in qualitative agreement with the parti-
cle–hole equivalence in the pairing theorem of Hu ¨ckel’s the-
ory [18a]. Currently, the interest on charged PAHs comes
mainly from basic research in astrophysics because PAHs
are expected to exist in space in different charge states
depending on the physical conditions (e.g., UV flux, elec-
tron density, etc.) of the host environment [4c]. This has
motivated a large amount of laboratory work based first
onmatrixisolationspectroscopy
19e,19f,19g,19h,19i,19j,19k] and, more recently, laser mass
spectroscopy [20a,20b,20c], spectroscopic studies of the
molecules trapped in helium droplets [21a,21b,21c,21d],
and a high sensitivity photo-absorption technique in free
jets called cavity ring-down spectroscopy [22a,22b,22c,
22d,22e,22f, 22g].
From an astrophysical point of view the knowledge of
the electronic absorption spectra of PAHs in all their rele-
1Bb,
1Baaccording to the empirical model of Platt
[19a,19b,19c,19d,
vant charge states is of fundamental importance for our
understanding of their photophysics in space. While this
concerns the whole energetic range excitable in a typical
interstellar environment, i.e., from the visible to the far-
UV, very few experimental data are available for charged
PAHs in this spectral range due to the limitations which
are intrinsic to the laboratory techniques more widely used.
As a part of a more extensive study [23a,23b,23c,23d,23e]
towards the knowledge of the spectral properties of a large
sample of PAHs to be modelled in astrophysical environ-
ments [24a,24b,24c], we report in this paper a detailed
study of the electronic absorption spectra of the five small-
est oligoacenes naphthalene, anthracene, tetracene, penta-
cene, and hexacene in the charge states most relevant for
astrophysical applications, i.e., ?1, 0, +1, and +2. The
geometries of the molecules considered are sketched in
Fig. 1. The theoretical methods we used for both ground-
state and excited-state calculations have been validated
for the basic aromatic unit benzene in its neutral form,
for which a large amount of spectroscopic data are
available.
There have been many systematic studies of benzenoid
hydrocarbons, including acenes, using quantum-chemical
calculations to obtain, e.g., heats of formation [25a], infra-
red spectra [25b], and C–H bond dissociation energies
[25c]. While the electronic properties of neutral acenes in
the near-IR, visible, and near-UV spectral ranges are thor-
oughly characterised up to the sizes of hexacene [26a,
26b,26c,26d,26e,26f,26g,26h,26i,26j,26k,27a,27b] and some
studies have been done for larger acenes [28a,28b,28c,28d],
a comparatively smaller amount of work exists for their
correspondingmono-cations
29g,29h] and mono-anions [29e,29f,30a,30b]. To the best
of our knowledge, however, the electronic spectra of
charged PAHs in the far-UV spectral domain have not
[29a,29b,29c,29d,29e,29f,
Fig. 1. Oriented geometries of the molecules considered. From top to
bottom: naphthalene, anthracene, tetracene, pentacene, and hexacene. The
molecules are supposed lying in the x–y plane, the y-axis being the longer
one.
44
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been measured to date. In addition, a detailed study of the
electronic excitation properties of PAHs in their doubly
ionised state has been missing until now. First proposed
20 years ago [31a,31b,31c], the possible presence of PAH
dications in the interstellar medium has recently received
further support based on the proposition that these mole-
cules could contribute to the luminescence observed in
the red part of the visible spectrum in many interstellar
sources [32]. While laboratory measurements of the yield
of optical fluorescence and phosphorescence by these spe-
cies are needed to test this hypothesis, the determination
of their electronic absorption spectra is motivated.
We used density functional theory (DFT) [33a,33b,33c,
33d,33e,33f,33g,33h,33i,33j,33k] and its time-dependent
extension (TD-DFT) [34a,34b,34c,34d,34e,34f, 34g,34h,
34i,34j,34k], which are methods of choice for this type of
investigations for large molecules. Since excitation energies
and oscillator strengths within TD-DFT can be computed
following two different strategies (see Section 2.2 for
details), we used both of them to obtain: (i) the whole abso-
lute photo-absorption cross-sections up to the soft X-ray
region near 30 eV, and (ii) the positions, the oscillator
strengths, and the leading configurations of the lowest-
lying valence p ! p*short-polarised and long-polarised
electronic transitions, falling in the visible/near-UV region,
as a function of molecular size and charge state. In case (i),
due to its numerical stability, we used the adiabatic local-
density approximation in the parametrisation of Perdew
and Zunger [33c], which has proven to yield reliable results
for the dynamical polarisability of conjugated molecules
[23e,34g]. In case (ii) we used the hybrid B3LYP functional
[33i] and the gradient-corrected BLYP functional [33d,33e],
which are widely used in the study of PAHs to obtain a
large number of molecular parameters, such as structures
and energetics [26a,35a], vibrational spectra [35a,35b,35c],
ionisation spectra [26c,26e], ionisation energies [23c,26a,
35f,26k], electron affinities [23b,30a,35g,35k], electronic
excitations[29d,26d,26f,26g,29f,30b,26k],
structure of absorption spectra [26h,26i], hydrogen dissoci-
ation [29g,29h], and Jahn–Teller effects [35d,35e,35h,
35i,35j].
Although the approximate exchange-correlation func-
tionals we used have been developed for the electronic
ground-state, they are also routinely employed in TD-
DFT calculations, and their application usually yields
accurate results for low-lying valence–excited states of both
closed-shell [34c,34d] and open-shell [34e,34f] species. It is
known, however, that these functionals show the wrong
asymptotic behaviour, decaying faster than 1/R (i.e., expo-
nentially) for large distances R from the nuclei. Among the
well known and documented limitations of these methods
[34j,34k] are: (i) the correct description of Rydberg
[36a,36b,36c], doubly excited [37a,37b], and charge-trans-
fer [38a,38b,38c] excited-states, (ii) the failure for large,
extended p-systems such as polyacetylene fragments and
oligoporphyrins [39], and (iii) the system-size-dependent
errors found for the lowest short-polarised excited-states
thevibronic
(p-bands in Clar’s notation) of neutral oligoacenes [26g].
Furthermore, the oscillator strengths computed by TD-
DFT are considered to be only in qualitative agreement
with experiments. In a rigorous assessment of the quality
of TD-DFT molecular oscillator strengths, for example,
they were found to be reasonable but not in quantitative
agreement with reliable experimental and theoretical values
for small molecules such as CO, N2, and CH2O [34h].
In spite of the above-mentioned failures of the level of
theory we used, it is known to be sufficient to identify the
most intense lowest-lying electronic excitations of neutral
PAHs [26b,26f] as well as radical ions [29d,29e,29f,30a,30-
b,40a,40b,40c], which are found to match closely with the
available experimental data both in terms of positions
and intensity ratios. Despite the non-physical exponential
fall-off of the exchange-correlation functional used, in fact,
these states all involve excitations to and from delocalised,
valence p-orbitals, which are not significantly affected by
the shape of the exchange-correlation potentials in the
asymptotic region [29f]. In particular, it has been shown
that interesting trends exist in the vertical excitation ener-
gies and the oscillator strengths for homologous series of
PAHs [40a,40b,40c]. For the transitions with the largest
oscillator strength in the oligorylenes perylene, terrylene,
and quaterrylene, for example, a net increase of the oscilla-
tor strength per unit mass of carbon along the series has
been found [40a]. This result might have important impli-
cations in astrophysics with respect to the long-standing
unsolved problem of the diffuse interstellar bands, about
300 unidentified absorption features observed in the near-
UV, visible, and near-IR spectra of stars obscured by inter-
stellar dust [41].
Moreover, we used the so-called delta-self-consistent-
field (DSCF) approach [33g], evaluating total-energy differ-
encesbetween theself-consistent
performed for the neutral and charged systems to obtain:
(i) the vertical and adiabatic electron affinities and the first
and second ionisation energies; (ii) the quasiparticle correc-
tion (quasiparticle energies are associated with the addition
or removal of an electron) to the highest occupied molecu-
lar orbital (HOMO) – lowest unoccupied molecular orbital
(LUMO) gap. This quantity is related to molecular hard-
ness, the analogue of the band gap of solids, defined as half
the difference between the ionisation potential and the elec-
tron affinity, which is a key property characterising the
chemical behaviour and reactivity of a molecule. The com-
parison between the optical gap, i.e., the lowest singlet–
singlet excitation energy obtained with TD-DFT, and the
quasiparticle corrected HOMO–LUMO gap enabled us
to estimate the excitonic effects (due to the electron–hole
interaction) in the neutral molecules.
The paper is organised as follows. Section 2 contains the
technical details for the calculation of the ground-state
properties (Section 2.1) and the electronic absorption spec-
tra (Section 2.2). The results we obtained are presented in
Section 3 and discussed in Section 4. Our concluding
remarks are reported in Section 5.
field calculations
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2. Computational details
2.1. DFT calculations
The calculation of the excitation energies and the elec-
tronic absorption spectra required the previous knowledge
of the ground-state optimised geometries. For this part of
the work we used the Gaussian-based DFT module of
the NWCHEM package [42]. Geometry optimisations were
performed using a basis-set with the smallest addition of
diffuse functions, namely the 6 ? 31 + G*basis, a valence
double zeta set augmented with d polarisation functions
and s and p diffuse functions for each carbon atom.
We used the hybrid B3LYP functional, a combination
of the Becke’s three parameter exchange functional [33i]
and the Lee–Yang–Parr gradient-corrected correlation
functional [33e]. Although hybrid DFT functionals are
computationally more expensive than other exchange-cor-
relation functionals in the local density or generalised gra-
dient approximations, B3LYP results for ground-state
properties are known to be markedly more accurate com-
pared with experiment for a large number of systems
including PAHs in general [35a,35b] and oligoacenes in
particular [26a,26c,26e,26k]. This is confirmed for neutral
benzene, whose optimised bond-lengths we obtain at the
B3LYP/6 ? 31 + G*level are rCC= 1.399 A˚, and rCH=
1.087 A˚, to be compared with the empirical equilibrium
(re) recommendedvalues
0.0010 A˚and 1.0802 ± 0.0020 A˚[43a]. Analogously, the
ground-state rotational constant we found is ?5700
MHz, in agreement with the empirical equilibrium (Be)
determination of 5731.73 MHz [43a].
From the structural relaxations performed for both neu-
tral and charged systems, we computed via total-energy dif-
ferences the adiabatic electron affinities and the adiabatic
single and double ionisation energies. At the optimised
geometry of the neutral molecule we evaluated also the ver-
tical electron affinity (EAv) and the vertical first ionisation
energy (IEv). This enabled us to obtain the quasiparticle
(QP) corrected HOMO–LUMO gap of the neutral systems
considered, which is rigorously defined within the DSCF
scheme [33g] as
of,respectively,1.3914 ±
QP1
gap¼ IEv? EAv¼ ENþ1þ EN?1? 2EN;
ð1Þ
ENbeing the total energy of the N-electron system. We
used also the following approximate expression [33f]:
QP2
gap¼ ?Nþ1
where ?j
results obtained using the above Eqs. (1) and (2) tend to
coincide as the system gets larger and the orbitals more
delocalised. The B3LYP/6 ? 31 + G*level of theory shows
good agreement with experiments for the electron affinities
of PAHs [23b,30a,35k], but it is known to be unable to pre-
dict their absolute ionisation energies with chemical accu-
Nþ1? ?N
N;
ð2Þ
iis the ith eigenvalue of the j-electron system. The
racy (±0.1 eV) [23c,26a,26k,35f]. This has been discussed
indetailbyKadantsevetal.[26k],whoconcluded thatabet-
ter description of the electron correlation is needed to repro-
ducetheexperimentalIEs.
computationalschemeonecouldemploymany-bodypertur-
bation theory in the so-called Hedin’s GW approximation
[44a]. This method, in which the QP energies are calculated
from the self-energy operator of the system (given as the
product of the Green’s function G and the screened Cou-
lomb interaction W), gives results in excellent agreement
with the available experiments for many materials (see e.g.,
[44b]). To assess the reliability of our DSCF QP-corrected
HOMO–LUMO gaps, we compared them with the GW re-
sults obtained for the oligoacenes [28d]. The QP-corrected
HOMO–LUMO gap of our benchmark benzene molecule
is10.59 eV(IEv= 9.20 eV,EAv= ?1.39 eV),thatcompares
favorablywiththeGWresults[28d]of10.59 eV(first-princi-
plescalculations using Gaussian-type orbitals) and 10.46 eV
(DFT-based tight-binding
and with the experimental value of ?10.36 eV (IEv=
9.24384 ± 0.00006 [43b], EAv= ?1.12 ± 0.03 eV [43c]).
Switching to another
calculations), respectively,
2.2. TD-DFT calculations
Thanks to the good compromise between accuracy and
computational costs, compared to many-electron wave-
function-based ab initio methods, TD-DFT is the most
widely used approach to compute the excitation energies
of such complex molecules as PAHs [26b,26d,26f,26g,26h,
26i,28b,29d,29e,29f,30b,40a,40b,40c]. In this study we used
two different implementations of TD-DFT in the linear
response regime, in conjunction with different representa-
tions of the wavefunctions:
(1) the real-time propagation scheme using a grid in real
space [34g], as implemented in the OCTOPUS computer
program [45a,45b,45c,45d].
(2) the frequency–space implementation [34e] based on
the linear combination of localised orbitals, as given
in the NWCHEM package [42].
In the first scheme (1) the time-dependent Kohn–Sham
equations are directly solved in real-time and the wavefunc-
tions are represented by their discretised values on a uni-
form spatial grid. The static Kohn–Sham wavefunctions
are perturbed by an impulsive electric field and propagated
for a given finite time interval. In this way, all of the fre-
quencies of the system are excited. The whole absolute
photo-absorption cross-section r(E) then follows from
the dynamical polarisability a(E), which is related to the
Fourier transform of the time-dependent dipole moment
of the molecule. The relation is
rðEÞ ¼8p2E
where h is Planck’s constant, IfaðEÞg is the imaginary part
of the dynamical polarisability, and c the velocity of light in
hc
IfaðEÞg;
ð3Þ
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vacuum. The dipole strength-function S(E) is related to
r(E) by the equation:
SðEÞ ¼mec
phe2rðEÞ;
ð4Þ
meand e being respectively the mass and charge of the elec-
tron. S(E) has units of oscillator strength per unit energy
and satisfies the Thomas–Reiche–Kuhn dipole sum-rule
Ne¼RdESðEÞ, where Neis the total number of electrons.
once is particularly useful for astrophysical applications,
for which the whole absorption spectrum is needed.
In the most widely used frequency–space TD-DFT
implementation (2), based on the linear response of the
density–matrix, the poles of the linear response function
correspond to vertical excitation energies and the pole
strengths to the corresponding oscillator strengths [34b].
With this method computational costs scale steeply with
the number of required transitions and electronic excita-
tions are thus usually limited to the low-energy part of
the spectrum. From a computational point of view the
advantages of the real-time propagation method are dis-
cussed, e.g., in Ref. [45d]. On the other hand, the main
drawbacks of the real-time approach are that: (i) no infor-
mation is given on dipole-forbidden singlet–singlet and sin-
glet–triplet transitions, and (ii) one does not obtain
independent information for each excited state, such as
its irreducible representation of the point group of the
given molecular system, and the description of the excita-
tions in terms of promotion of electrons in an orbital
picture.
We performed the OCTOPUS calculations in the local-den-
sity approximation, with the exchange-correlation energy
density of the homogeneous electron gas [46a] parametr-
ised by Perdew and Zunger [33c]. The ionic potentials are
replaced by norm-conserving pseudo-potentials [46b]. We
used a grid spacing of 0.3 A˚and determined the box size
by requiring each atom to be at least 4 A˚away from its
edges. We furthermore added a 1 A˚
boundary, which quenches spurious resonances due to
standing waves in the finite simulation box used to confine
the molecules [34g,45c]. We used a time integration length
T = 20⁄ eV, corresponding to an energy resolution of ⁄/
T = 0.05 eV. For the numerical integration of the time evo-
lution we used a time step of 0.002⁄ eV, which ensured
energy conservation with good numerical accuracy.
The TD-DFT calculations with NWCHEM were performed
at the same level B3LYP/6 ? 31 + G*used to obtain the
ground-state geometries. Although basis set convergence
is not yet expected at the level we used, our results for
the neutral systems are almost coincident with the ones
obtained in Ref. [26k] using the larger 6 ? 311++G**basis,
which is supplemented with a third layer of valence func-
tions and includes polarisation and diffuse functions on
both carbon and hydrogen atoms. We thus believe our the-
oretical predictions to be sufficiently accurate for the pur-
poses of this work. In the case of neutral benzene, we
The great advantage of obtaining the whole response at
thick absorbing
predict the strong p ! p*transition
6.96 eV with an oscillator strength of 1.22, in good agree-
ment with the measured band position in vapour-phase
of 6.94 eV with an f-value of 1.2 [43d].
In order to assess the choice of this specific exchange-
correlation functional, at the B3LYP optimised geometries,
we used also the gradient-corrected BLYP functional
[33d,33e], the same approach used in previous studies of
PAH ions [29d,40a,29f,30b]. With both methods we
restricted ourselves to the first 20 singlet–singlet roots.
Since we are interested in the behaviour of the lowest-lying
permitted in-plane long-polarised and short-polarised elec-
tronic transitions as a function of charge-state and molec-
ular size, in the following we report only the first five
electronic transitions. The complete set of electronic excita-
tion energies and oscillator strengths computed at both
B3LYP and BLYP levels, including also the optically inac-
tive ones, are available in our online database of the com-
puted spectral properties of PAHs [23d].
1A1g!1E1u at
3. Results
3.1. Static properties
Geometry optimisations with NWCHEM were performed
using tight convergence criteria, that are specified by max-
imum and root mean square gradient thresholds of
1.5 · 10?5and 1.0 · 10?5atomic units, respectively, and
maximum and root mean square thresholds of the Carte-
sian step respectively of 6.0 · 10?5and 4.0 · 10?5atomic
units. According to previous studies [29b,29d] the lifting
of the molecular symmetry D2hof the neutral molecules
is not expected to lead to optimised geometries with lower
symmetry for the corresponding charged species. We
indeed confirmed that the structural parameters obtained
for naphthalene, anthracene, and tetracene in the charge-
states ?1,+1, and +2, ignoring any apparent symmetry
and adopting the D2h symmetry, are coincident within
numerical errors. We therefore assumed the above D2h
constraint for all subsequent calculations in the paper.
Our B3LYP/6 ? 31 + G*geometry optimisations for the
neutral molecules give structural parameters in good agree-
ment with those previously published [26a,26e,26k]. In par-
ticular, although a larger basis was used in Ref. [26k], the
two sets of results are almost coincident and compare fairly
well with the available X-ray data. We do not discuss here
the changes of the single bond lengths and bond angles
occurring in the charged species, compared to the corre-
sponding neutral ones. Depictions of the structures of each
molecule considered, in which internal coordinates are
shown and compared, are given as supplementary material
to the paper. Instead, Fig. 2 presents in a collective way the
structural variations relative to the neutral molecules,
expressed in terms of the percentage variations of the rota-
tional constants A and B. These latter quantities are pro-
portional to the inverse of the principal momenta of
inertia corresponding to the in-plane short and long-axis
G. Malloci et al. / Chemical Physics 340 (2007) 43–58
47

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of the molecules, respectively: A = (h/8p2c)Ishort, B = (h/
8p2c)Ilong. Note that in Fig. 2 we omitted the results corre-
sponding to naphthalene anion, since gas-phase naphtha-
lene is unable to bind an additional electron in its
LUMO state [43c]. The ground-state optimised geometries
of all of the molecules considered, both Cartesian and
internal coordinates, are freely available in our online data-
base of the computed spectral properties of PAHs [23d].
All neutral and singly charged species were computed as
singlet and doublet, respectively, while for dications we
computed both their singlet and triplet ground-states.
The adiabatic and vertical values of electron affinities and
single and double ionisation energies as obtained via
total-energy differences are given in Table 1 and compared
with the available experimental data [47,31c]. As shown in
Table 1, for all of the five molecules considered in their
doubly ionised state, our calculations predict the total
energy of the singlet state to be lower than that of the trip-
let state by ?0.5–1.0 eV. Fig. 3 displays the computed elec-
tron affinities and first and double ionisation energies as a
function of size, and compares them with the available lab-
oratory data.
3.2. Excitation energies and electronic absorption spectra
Since its first applications, the real-time TD-DFT
method in real space was proven to give good results for
neutral benzene [34g,45c], compared with the experimental
spectrum recorded in the energy range 6–35 eV [12e].
Applying this same approach to a large sample of PAHs,
we already showed [23a,24b] our results to be in good
agreement up to photon energies of about 30 eV with the
experimental data obtained for a few neutral PAHs with
the synchrotronradiation
[12h,12i]. The comparison between the theoretical spectra
obtained with the
OCTOPUS code and the experimental
photo-absorption cross-sections of neutral anthracene
(C14H10) and benzene in the gas-phase are shown in Figs.
4 and 5. The latter experimental spectra are in good agree-
ment with the ones obtained with the synchrotron radia-
facility ofSUPERACO
Fig. 2. Percentage variation of the rotational constants A (top panel) and
B (bottom panel) relative to the neutral counterparts for anions (asterisks),
cations (triangles), and dications (diamonds), as a function of molecular
size. We omit the entries for naphthalene anion, which is known not to
form a stable anion in the gas-phase.
Table 1
Adiabatic and vertical values (in parentheses), all data in eV, of electron affinities and single and double ionisation energies of the oligoacenes considered in
this work (C4n+2H2n+4, n = 2, 3, 4, 5, 6) as obtained through total-energy differences at the B3LYP/6 ? 31 + G*level
Number of cycles Electron affinity First ionisation energyDication state Double ionisation energy
Ad. (Vert.) Exp.Ad. (Vert.) Exp.Ad. (Vert.)Exp.
2
?0.26(?0.38)
?0.20 ± 0.05a
7.80(7.89)8.144 ± 0.001b
Singlet
Triplet
20.99(21.35)
21.45(21.57)
21.5 ± 0.2
30.53(0.43)0.530 ± 0.005c
7.02(7.09)7.439 ± 0.006d
Singlet
Triplet
18.70(18.95)
19.70(19.80)
–
41.08(1.00)1.067 ± 0.043e
6.49(6.55)6.97 ± 0.05f
Singlet
Triplet
17.15(17.34)
17.96(18.11)
18.6 ± 0.2
51.48(1.41)1.392 ± 0.043e
6.12(6.16)6.589 ± 0.001g
Singlet
Triplet
16.03(16.18)
16.67(16.80)
17.4 ± 0.2
61.78(1.72)–5.83(5.87)6.36 ± 0.02f
Singlet
Triplet
15.18(15.30)
15.68(15.78)
–
For comparison we list also the experimental adiabatic electron affinities and adiabatic single ionisation energies taken from the NIST Chemistry
WebBook [47], as well as the adiabatic second ionisation energies photon-impact measurements [31c].
aExtrapolated from the EAs of naphthalene–water clusters determined via photoelectron spectroscopy [16e].
bFrom laser threshold photoelectron spectroscopy [16c].
cFrom photodetachment photoelectron spectroscopy [16d].
dFrom two-laser photoionisation supersonic jet mass spectrometry [17a].
eEstimated from gas-phase electron attachment free energies with the electron-transfer equilibria technique [48].
fFrom gas-phase photoelectron spectroscopy [16b].
gFrom high-resolution gas-phase photoelectron spectroscopy [16f].
48
G. Malloci et al. / Chemical Physics 340 (2007) 43–58

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tion from the electron accelerator DESY for anthracene
[12g], and benzene [12e], respectively. Since the theoretical
spectrum is averaged over the three x, y, and z polarisa-
tions, each single contribution is also shown in Figs. 4
and 5. As examples, the spectra computed for tetracene
(C18H12) and pentacene (C22H14) in the four charge states
considered are given in Figs. 6 and 7. The spectra for all
of the other molecules under study can be found in our
online database [23d]. Fig. 8 shows the comparison
between the integrated values in the range 6–12 eV of the
Fig. 3. Computed adiabatic ionisation energies and electron affinities of
the studied oligoacenes as a function of size. The corresponding
experimental values are represented by the filled symbols (see Table 1).
Fig. 4. Comparison between the computed (solid black line) photo-
absorption cross-section r(E) of neutral anthracene (C14H10) and the
corresponding gas-phase absorption spectrum (dotted line, taken from
Refs. [12h,12i]). The contributions corresponding to polarisations along
the x-axis (in-plane short), y-axis (in-plane long), and z-axis (out-of-
plane), are marked in gray, dark gray, and light gray, respectively. Units
are megabarns, 1 Mb = 10?18cm2.
Fig. 5. Same as Fig. 4 for neutral benzene. Note that, due to the symmetry
of the molecule, the curves corresponding to x and y polarisations are
obviously coincident.
Fig. 6. Computed photo-absorption cross-section r(E) of tetracene
(C18H12) in neutral (black), anionic (dark gray), cationic (gray) and
dicationic (light gray) charge-state, as obtained with the real-time real-
space implementation of TD-DFT.
Fig. 7. Same as Fig. 6 for pentacene (C22H14).
G. Malloci et al. / Chemical Physics 340 (2007) 43–58
49

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individual dipole strength-functions S(E) (see Eq. (4))
divided by the number of carbon atoms in each molecule
as a function of molecular size.
The first few permitted electronic transitions of each
molecule, as obtained at the B3LYP/6 ? 31 + G*and
BLYP//B3LYP/6 ? 31 + G*levels with the TD-DFT fre-
quency–space implementation of NWCHEM are reported in
Tables 2–6, and compared with the available experimental
data that we could find in the literature. We deliberately
omitted the large amount of photoelectron data available
for neutral species [16a,16b,16c,16d,16e,16f]. The use of
such data for spectral assignments of the so-called Koop-
mans transitions of radical cations has been already dis-
cussed many times [29c,29d,29e,29f]. Electronic excited
states are classified under the point-group D2h and the
ground-state symmetry is specified for each charge-state.
As to the character of the excited electronic states, we ana-
lyse the nature of the corresponding transitions in terms of
the occupied and virtual molecular orbitals that have been
interchanged between the ground and the excited electronic
states [34j]. The above description is computationally con-
venient and straightforward for states which are well
described with only one or two significantly contributing
‘‘excited’’ Slater determinants [34j]. This is the case for
the electronic excited states reported in Tables 2–6. The
Fig. 8. Comparison between the integrated values in the range 6–12 eV of
the individual dipole strength-functions S(E) (see Eq. (4)) divided by the
number of carbon atoms for the oligoacenes anions (asterisks), neutrals
(crosses), cations (triangles), and dications (diamonds) considered, as a
function of molecular size.
Table 2
Singlet–singlet lowest-lying permitted excitation energies (in eV) of naphthalene anion, neutral, cation, and dication as obtained via frequency–space TD-
DFT
State (pol.) ExcitationBLYPB3LYPExp.
Anion (ground-state2B3g)
12B2u(z)
12B1u(y)
22B2u(z)
22B1u(y)
12Au(x)
p0! p?
p0! p?
p0! p?
p0! r?
p?1! p?
2
0.66(0.001)
1.48(0.022)
1.85(0.005)
2.30(0.071)
2.73(0.001)
0.99(0.001)
1.58(0.034)
2.15(0.006)
2.49(0.069)
2.89(<0.001)
–
?1.5a
–
?2.5a
–
7
8
2
1
Neutral (ground-state1Ag)
11B3u(x,p)
11B2u(y,a)
21B2u(y,b)
21B3u(x)
11B1u(z)
p?1! p?
p?2! p?
p?2! p?
p?2! p?
p?2! p?
1
1, p?1! p?
1, p?1! p?
2
4.06(0.045)
4.20(<0.001)
5.62(1.155)
5.80(0.146)
5.77(0.008)
4.36(0.061)
4.44(<0.001)
5.85(1.260)
6.08(0.199)
6.24(0.016)
4.45(0.102)b, 4.45(0.109)c, 4.44d
3.97(0.002)b, 4.0c, 3.98d
5.89(1.3)b, 5.89(1.3)c, 5.86d
6.14(0.3)b, 6.0c, 6.09d
2
2
3
–
Cation (ground-state2Au)
12B2g(y)
12B3g(x)
22B3g(x)
22B2g(y)
32B2g(y)
p?2! p?
p?3! p?
p0! p?
p?1! p?
p0! p?
0
2.15(0.042)
2.77(0.006)
3.51(0.051)
3.74(0.012)
4.30(0.008)
2.14(0.053)
2.98(0.006)
3.59(0.064)
3.91(0.032)
4.61(0.006)
1.84d, 1.85(0.052)e, 1.84(0.011)f
2.72d, 2.72(0.010)e, 2.69(0.001)f
3.29d, 3.25(0.016)f
4.02d, 4.03(0.024)f
4.49d, 4.52(0.060)f
0
1
1
2
Dication (ground-state1Ag)
11B2u(y)
11B1u(z)
11B3u(x)
21B2u(y)
21B1u(z)
p?2! p?
p?4! p?
p?3! p?
p?1! p?
r?3! p?
1
2.85(0.061)
2.71(<0.001)
3.32(0.062)
5.06(0.833)
4.95(0.001)
3.00(0.090)
3.25(<0.001)
3.67(0.090)
5.28(0.865)
5.58(0.002)
–
–
–
–
–
1
1
2
1
The corresponding oscillator strengths are given in parentheses. Polarisation of the bands are denoted according to Fig. 1 as x, y, and z for in-plane short,
in-plane long, and out-of-plane polarised bands, respectively. The description of each excitation is based on the B3LYP results and is given in terms of the
occupied and virtual molecular orbitals contributing significantly to it [29d,29f]. We report also the available experimental data for comparison.
aAbsorption in glassy organic solid [18a].
bGas-phase absorption [12c].
cElectron-energy-loss spectroscopy [13b].
dAbsorption in neon-matrix [19d].
eGas-phase absorption [20b].
fAbsorption in argon-matrix [19b].
50
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use of more sophisticated and physically more appealing
ways to obtain information about an electronic transition,
such as difference density or attachment/detachment den-
sity plots [34j], is outside the scope of the present work.
We used the same notation as in Refs. [29d,29f] where
the p orbitals are numbered in the order of increasing ener-
gies and p?1, p0ðp?
occupied p orbital, the singly occupied (unoccupied) p orbi-
tal, andthe lowestdoubly
respectively.
Figs. 9 and 10 display, respectively, the B3LYP/
6 ? 31 + G*positions and the corresponding oscillator
strengths of the lowest in-plane short-polarised (p-bands
in the neutral molecules) and long-polarised (a-bands in
the neutral molecules) electronic transitions as a function
of the number of benzene units and the charge-state of
the molecule.
0Þ, and p?
1denote the highest doubly
unoccupied
p
orbital,
3.3. Quasiparticle-corrected HOMO–LUMO gap of the
neutral species
For each of the neutral molecules considered at the
B3LYP/6 ? 31 + G*level, we report in Table 7 the com-
parison between the HOMO–LUMO gap EKS
gapobtained
as difference of Kohn–Sham eigenvalues, the excitation
energy of the HOMO–LUMO transition ETD-DFT
by TD-DFT, and the corresponding experimental value
Eexp
the results of Eqs. (1) and (2) with the DFT-based tight-
binding GW data of Ref. [28d], QPDFT-GW
the corresponding experimental value obtained as the dif-
ference between the experimental EAs and first IEs given
in Table 1, QPexp
binding energy Ebindis estimated through the difference
QP1
gap
, which is compared with its corresponding
experimental value QPexp
displayed in Fig. 11 as a function of molecular size.
gap
as given
gap(p-bands in Tables 2–6). In the same table we compare
gap
, as well as with
gap¼ IEexp? EAexp. The theoretical exciton
gap? ETD-DFT
gap? Eexp
gap. All of these quantities are
4. Discussion
Fig. 2 shows that structural variations between charged
species and their respective neutral counterparts display the
same well-defined trend as a function of molecular size: a
general decrease for Arel(with the possible exception of
anthracene anion), and an increase of Brelis observed for
all charge-states considered. More specifically, the largest
structural changes are observed for the anions, a conse-
quence of the strongly antibonding character of the LUMO
Table 3
Same as Table 2 for anthracene in its ?1, 0, +1, and +2 charge-states
State (pol.)ExcitationBLYPB3LYP Exp.
Anion (ground-state2B1u)
12Ag(z)
12B3g(y)
22Ag(z)
12B2g(x)
22B2g(x)
p0! p?
p0! p?
p0! p?
p ? 1 ! p?
p0! p?
1
1.23(0.001)
1.83(0.110)
1.88(0.001)
2.12(0.008)
2.78(0.008)
1.56(0.001)
1.90(0.137)
2.17(0.002)
2.20(0.012)
3.01(0.010)
–
?1.7a
–
?2.4a
–
5
7
0
11
Neutral (ground-state1Ag)
11B3u(x,p)
11B2u(y,a)
21B3u(x)
21B2u(y,b)
11B1u(z)
p?1! p?
p?2! p?
p?1! p?
p?2! p?
p?1! p?
1
1, p?1! p?
6
1, p?1! p?
7
2.92(0.039)
3.60(<0.001)
4.59(<0.001)
4.88(1.782)
4.83(0.001)
3.21(0.058)
3.85(<0.001)
5.06(<0.001)
5.14(1.992)
5.24(0.002)
3.27(0.1)b, 3.45c, 3.43d
3.47–3.60–3.45e, 3.84f
2
–
2
4.84(2.28)b, 5.24c
–
Cation (ground-state2B2g)
12Au(y)
12B1u(x)
22B1u(x)
22Au(y)
32Au(y)
p?2! p?
p0! p?
p?4! p?
p?1! p?
p0! p?
0
1.87(0.085)
2.21(0.004)
2.91(0.044)
3.22(0.017)
3.69(0.016)
1.93(0.108)
2.32(0.007)
3.17(0.058)
3.39(0.054)
4.02(0.010)
1.71a, 1.73(0.076)g, 1.75h
2.02(0.018)g
2.83a, 2.90(0.026)g
3.52(0.104)g
3.95(0.187)g
1
0
1
2
Dication (ground-state1Ag)
11B2u(y)
11B3u(x)
21B2u(y)
11B1u(z)
21B3u(x)
p?2! p?
p?3! p?
p?1! p?
p?8! p?
p?4! p?
1
2.31(0.139)
3.02(0.052)
4.38(1.368)
3.92(<0.001)
4.96(0.002)
2.45(0.206)
3.39(0.078)
4.63(1.444)
4.66(<0.001)
5.59(0.001)
–
–
–
–
–
1
2
1
2
aAbsorption in glassy organic solid [18a].
bAbsorption in n-heptane solution [11b].
cGas-phase absorption [12g].
dFluorescence jet spectroscopy [14b].
eMagnetic circular dichroism measurements in solvents [15a,15b,15c].
fTwo-photon absorption in solution [12d].
gAbsorption in argon-matrix [19e].
hGas-phase absorption [22d].
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of the neutral counterpart. The variations in both cations
and dications are nearly equal and remain small. A short-
ening along the short-axis (Arel> 0) compared to the neu-
tral molecules is always observed. The anions appear to
be primarily distorted along the long-axis showing a
lengthening along it (Brel< 0) for all sizes. These findings
agree with a Hartree–Fock study of naphthalene and
anthracene cations [25b], and a DFT study of anthracene
anion performed with both hybrid and gradient-corrected
exchange-correlation functionals [30a].
Table 1 and Fig. 3 confirm for the oligoacenes the good
agreement found for the whole class of PAHs between the
B3LYP/6 ? 31 + G*electron affinities and the available
experimental data. The differences between computed and
observed first ionisation energies are of the same order of
magnitude as in previous analyses [26a,26k], and seem to
increase slightly at increasing molecular size (from ?4%
for naphthalene to ?10% for hexacene). Double ionisation
energies are found to change more rapidly along the series
than single ionisation energies, and the relative errors of
our calculations, in comparison with the only three exper-
imental data available, increase from ?2% for naphthalene
to ?8% for anthracene and tetracene.
As shown in Figs. 4 and 5, the real-time real-space TD–
DFT method provides results in very good agreement with
the available experimental data for neutral species up to
about 30 eV. The broad plasmon-like excitation peaking
atabout17.5 eV,which
r ! r*, and Rydberg spectral transitions, is well repro-
duced both in position and width. However, as previously
discussed in the case of neutral benzene [34g], the use of the
finite simulation box and the absorbing boundary at its
edges does not give a satisfactory treatment of continuum
effects producing spurious structures. The contribution of
the three possible polarisations to the total absorption
cross-section varies considerably. While the low-energy
part is due to in-plane-polarised electronic transitions (x
and y axes), the contribution of the z-axis perpendicular
to the plane of the molecule is significant only above a
few eV. These features are found to be common to all
PAHs [23a,23b,23c,23d,23e]. In the case of oligoacenes,
due to their special symmetry, the strongest absorption (b
band in the neutral molecules) corresponds to long-axis
polarisation, which can be simply understood as the classi-
cal resonance in a conducting rod [11a]. From Figs. 6 and 7
it is seen that the broad plasmon-like structure with its
maximum at 17–18 eV is relatively insensitive to the
charge-state of the molecule. On the other hand, the
charge-state of the molecule shows up in the low-energy
range and, interestingly, the onset of this broad absorption
moves blue-ward and becomes steeper with increasing posi-
tive charge. As shown in Fig. 8 this translates into a sys-
involves
p ! r*,
r ! p*,
Table 4
Same as Table 2 for tetracene in its ?1, 0, +1, and +2 charge-states
State (pol.)Excitation BLYPB3LYPExp.
Anion (ground-state2B3g)
12Au(x)
12B1u(y)
12B2u(z)
22B1u(y)
22B2u(z)
p?1! p?
p0! p?
p0! p?
p0! p?
p0! p?
0
1.53(0.009)
1.68(0.164)
1.68(0.001)
2.40(<0.01)
2.58(0.001)
1.60(0.014)
1.77(0.209)
2.06(0.001)
2.62(<0.001)
2.91(0.001)
1.69a
1.50a
1.91a
–
–
1
3
10
11
Neutral (ground-state1Ag)
11B3u(x,p)
11B2u(y,a)
21B3u(x)
21B2u(y,b)
31B3u(x)
p?1! p?
p?3! p?
p?1! p?
p?3! p?
p?4! p?
1
1, p?1! p?
5
1, p?1! p?
1
2.16(0.031)
3.21(0.001)
3.98(<0.001)
4.34(2.317)
4.35(0.010)
2.45(0.049)
3.47(0.002)
4.59(<0.001)
4.62(2.691)
4.92(<0.001)
2.62(0.08)b, 2.60(0.11)c, 2.72d
3.12d
–
4.55(1.85)b, 4.50(1.75)c
–
3
3
Cation (ground-state2Au)
12B2g(y)
12B3g(x)
22B3g(x)
22B2g(y)
32B2g(y)
p?1! p?
p0! p?
p ? 4 ! p?
p ? 2 ! p?
p ? 5 ! p?
0
1.59(0.129)
1.58(0.008)
2.67(0.029)
2.81(0.025)
3.17(0.005)
1.70(0.169)
1.70(0.012)
3.03(0.041)
3.05(0.084)
3.52(0.005)
1.43a, 1.43e
1.65a, 1.66e
3.14a, 3.16e
–
–
1
0
1
0
Dication (ground-state1Ag)
11B2u(y)
11B3u(x)
21B2u(y)
11B1u(z)
31B2u(y)
p?1! p?
p?4! p?
p?5! p?
p?8! p?
p?2! p?
1
1.97(0.226)
2.80(0.042)
3.65(0.093)
3.46(<0.001)
3.86(1.709)
2.11(0.335)
3.16(0.065)
4.05(0.626)
4.11(<0.001)
4.16(1.353)
–
–
–
–
–
1
1
1
2
aAbsorption in glassy organic solid [18a].
bAbsorption in n-heptane solution [11b].
cAbsorption in benzene solution [12a].
dAbsorption in gas-phase [14a].
eAbsorption in argon-matrix [19c,19g].
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Table 5
Same as Table 2 for pentacene in its ?1, 0, +1, and +2 charge-states
State (pol.) ExcitationBLYP B3LYPExp.
Anion (ground-state2B1u)
12B2g(x)
12B3g(y)
12Ag(z)
22Ag(z)
22B3g(y)
p?1! p?
p0! p?
p0! p?
p0! p?
p ? 1 ! p?
0
1.10(0.008)
1.50(0.224)
1.96(0.001)
2.53(<0.001)
2.65(0.016)
1.16(0.013)
1.60(0.286)
2.35(0.001)
2.81(0.001)
2.86(0.118)
1.06a, 1.37c
1.40a, 1.42c
–
–
2.82a
1
3
8
4
Neutral (ground-state1Ag)
11B3u(x,p)
11B2u(y,a)
21B3u(x)
21B2u(y,b)
31B3u(x)
p?1! p?
p?3! p?
p?4! p?
p?3! p?
p?1! p?
1
1, p?1! p?
1
1, p?1! p?
4
1.63(0.023)
2.94(0.004)
3.33(<0.001)
3.93(2.706)
3.66(0.021)
1.91(0.041)
3.21(0.005)
3.95(<0.001)
4.24(3.346)
4.26(0.003)
2.12(0.08)b, 2.28c, 2.31d
3.73c
–
4.00(2.20)b, 4.40c
–
3
3
Cation (ground-state2B2g)
12B1u(x)
12Au(y)
22Au(y)
22B1u(x)
22B1u(x)
p0! p?
p?1! p?
p?2! p?
p?4! p?
p?4! p?
1
1.17(0.007)
1.42(0.182)
2.61(0.034)
2.60(0.019)
2.85(0.001)
1.25(0.012)
1.50(0.238)
2.80(0.122)
2.85(0.013)
3.10(0.022)
1.27c
1.31c
2.92c
–
–
0
1
0
0
Dication (ground-state1Ag)
11B2u(y)
11B3u(x)
21B2u(y)
21B3u(x)
31B2u(y)
p?1! p?
p?4! p?
p?5! p?
p?3! p?
p?2! p?
1
1.73(0.324)
2.61(0.029)
3.17(0.080)
3.20(0.011)
3.44(2.022)
1.86(0.483)
2.98(0.050)
3.57(0.240)
3.66(0.001)
3.75(2.186)
–
–
–
–
–
1
1
2
2
aAbsorption in glassy organic solid [18a].
bAbsorption in n-heptane solution [11b].
cAbsorption in neon-matrix [29e].
dAbsorption in gas-phase [12j].
Table 6
Same as Table 2 for hexacene in its ?1, 0, +1, and +2 charge-states
State (pol.)ExcitationBLYPB3LYPExp.
Anion (ground-state2B3g)
12Au(x)
12B1u(y)
22Au(x)
12B2u(z)
22B1u(y)
p?1! p?
p0! p?
p?2! p?
p0! p?
p?1! p?
0
0.78(0.006)
1.34(0.284)
2.41(<0.001)
2.14(<0.001)
2.51(0.043)
0.83(0.011)
1.43(0.370)
2.47(<0.001)
2.56(0.001)
2.71(0.149)
–
–
–
–
–
1
1
3
2
Neutral (ground-state1Ag)
11B3u(x,p)
11B2u(y,a)
21B3u(x)
31B3u(x)
41B3u(x)
p?1! p?
p?3! p?
p?4! p?
p?1! p?
p ? 2 ! p?
1
1, p?1! p?
1
1.24(0.017)
2.75(0.009)
2.78(0.001)
3.07(0.028)
3.35(0.017)
1.51(0.034)
3.02(0.010)
3.39(<0.001)
3.69(0.010)
3.82(0.061)
1.90a
2.80a
–
–
–
3
4
2
Cation (ground-state2Au)
12B3g(x)
12B2g(y)
22B3g(x)
22B2g(y)
32B3g(x)
p0! p?
p?1! p?
p?3! p?
p?2! p?
p?4! p?
1
0.84(0.006)
1.27(0.246)
2.30(<0.001)
2.43(0.046)
2.52(0.007)
0.90(0.010)
1.34(0.317)
2.40(<0.001)
2.63(0.167)
2.92(0.021)
–
–
–
–
–
0
1
1
0
Dication (ground-state1Ag)
11B2u(y)
11B3u(x)
21B3u(x)
21B2u(y)
31B2u(y)
p?1! p?
p?4! p?
p?3! p?
p?5! p?
p?2! p?
1
1.55(0.440)
2.43(0.009)
2.66(0.018)
2.82(0.062)
3.10(2.203)
1.67(0.652)
2.83(0.031)
3.05(0.001)
3.21(0.130)
3.44(2.642)
–
–
–
–
–
1
2
1
2
aExtrapolated from solution spectra to the gas-phase (see compilation in Ref. [26k]).
G. Malloci et al. / Chemical Physics 340 (2007) 43–58
53

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tematic decrease with increasing positive charge of the
absorption cross-section in the energy gap between the p
and r plasmon-like structures (between ?6 and ?12 eV).
This same behaviour has been shown for a larger sample
of PAHs [23c]. In addition, as expected, the observed scat-
ter between the different charge-states decreases with
increasing molecular size.
Concerning the visible–UV part of the spectrum, as
reported in Tables 2–6, we checked both the sensitivity of
ourTD-DFTcalculations
exchange-correlation functionals, and their reliability in
comparison with the available experimental data. General
agreement is found between BLYP and B3LYP which yield
the same ordering of the strongest dipole-allowed excited
states. On the average, the BLYP energies are found to
be systematically smaller by 0.1–0.6 eV compared to the
corresponding B3LYP results, our data confirm the spec-
tral assignment done in previous studies for neutral [26k],
and singly charged species [29d,29f]. From the more accu-
rate experimental data reported for the anions, neutrals,
and cations we found that the mean relative deviation of
the B3LYP functional is of the order of 6%, compared to
the mean relative deviation of about 7% given by BLYP.
We therefore consider only the former set of results in
the following analysis.
Thefrequency–space implementation
enabled us to gain some insight into the nature of the first
few electronic excitations. Focusing only on dications,
reported here for the first time, we find that the lowest
in-plane long (y) and short-axis (x) polarised bands corre-
spond, respectively, to the HOMO-1 ! LUMO (p?2! p?
and HOMO-2 ! LUMO (p?3! p?
thalene andanthracene,
(p?1! p?
cene, pentacene, and hexacene. Figs. 9 and 10 show inter-
esting trends as to the behaviour of the lowest-lying
electronic transitions as a function of the size of the mole-
cule and its charge-state. The positions of both in-plane
short and long-polarised excitations are found to shift to
lower energies with molecular size for all charge-states con-
sidered (Fig. 9). We observe the well-known similarities in
the electronic absorption spectra between anionic and cat-
ionic PAHs, as well as the systematic shifts in band posi-
tion when going from the cation to the anion [18a,29f].
The sign and the magnitude of these shifts, attributed to
the different effect of the r-electrons in both ions [18a],
are reproduced by TD-DFT for the most intense bands
[29f], i.e., the lowest-lying y bands in oligoacenes. For
example, the measured blue-shifts of 0.07 eV (from
1.43 eV of the 12B2gstate of the cation to 1.50 eV of the
12B1ustate of the anion, data in organic solid [18a], see
Table 4) and 0.11 eV (from 1.31 eV of the 12Austate of
the cation to 1.42 eV 12B3g state of the anion, data in
Ne-matrix [29e], see Table 5), when going from the cation
to the anion of tetracene and pentacene, respectively, com-
pare favorably with our computed blue-shifts of 0.07, from
1.70 to 1.77 eV, and 0.10 eV, from 1.50 to 1.60 eV (0.05 and
to theuseofdifferent
of TD-DFT
1)
1) transitions for naph-
and HOMO ! LUMO
1) and HOMO-3 ! LUMO (p?4! p?
1) for tetra-
Fig. 9. Calculated positions (B3LYP/6 ? 31 + G*) of the lowest short-
polarised (crosses, p-bands in the neutral species) and lowest long-
polarised (diamonds, a-bands in the neutrals) electronic transitions in the
five oligoacenes considered as a function of the size and charge-state of the
molecule.
Fig. 10. Same as Fig. 9 for the corresponding oscillator strengths.
54
G. Malloci et al. / Chemical Physics 340 (2007) 43–58

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Author's personal copy
0.08 in Ref. [29f]). However, this is not the case for the low-
est-lying x bands, for which our theoretical predictions
seem to present a mismatch. In the case of tetracene and
pentacene, e.g., the experimental blue-shifts of 0.04 eV
(from 1.66 eV of the 12B3gstate to 1.7 eV of the 12Austate,
Table 4), and 0.10 eV (from 1.27 eV of the 12B1ustate to
1.37 eV of the 12B2gstate, Table 5), are predicted in this
study to be red-shifted by ?0.10, from 1.70 to 1.60 eV,
and ?0.09 eV, from 1.25 to 1.16 eV, respectively (?0.06
and ?0.05 in Ref. [29f]).
We confirm the system-size-dependent errors found for
the short-polarised transitions (p bands) in the neutral sys-
tems [26g,26k]. More specifically, from a comparison with
the available experimental data reported in Tables 2–6 we
find that while the relative error in the position of the a
band is always of the order of 10%, in the case of the p
band this error increases from 2% for naphthalene to
20% for hexacene (see the triangles on the top panel of
Fig. 11). On the other hand, we find the position of the b
band to be reproduced with a good precision with relative
errors of, at most, 4%. In the case of the radical anions and
cations, we could not find clear trends as to the perfor-
mances of our TD-DFT/B3LYP results. On the average,
the relative errors are larger for the long-polarised bands
compared to the short-polarised ones for both anions
(12% vs. 9%) and cations (16% vs. 7%).
The oscillator strengths f we obtain for the oligoacenes
are also found to display systematic changes as a function
of molecular size (Fig. 10). Unlike the neutral molecules, in
all charged species the lowest parallel (y) transitions have
larger f-values compared to the corresponding lowest per-
pendicular (x) ones. The oscillator strengths of the x-polar-
ised transitions display small changes with the number of
benzene units being always of the order of 0.01 for cations
and anions, and decreasing from 0.06 to 0.03 and from 0.09
to 0.03 for neutrals and dications, respectively. The oscilla-
tor strengths obtained for the y-polarised transitions are
found to increase with the molecular size for all the
charge-states considered. In particular, while the increase
for the neutral molecules is from 1 · 10?5to 1 · 10?2when
going from naphthalene to hexacene, for the charge-states
?1, +1, and +2 the corresponding values go from 0.03 to
0.37, 0.05 to 0.32, and from 0.09 to 0.65. Therefore, analo-
gously to singly charged species, doubly ionised PAHs have
strong absorption features in the near-IR, visible, and near-
UV spectral ranges, a result that might be relevant in the
astrophysical context.
Orbital energy differences are well-defined zeroth-order
approximations to electronic excitation energies [33j]. As
shown in Table 7, the HOMO–LUMO energy gap
obtained directly as difference of Kohn–Sham eigenvalues
gives a better description of the optical gap, i.e., the posi-
tion of the p-band, with relative errors in the range 1–
6%, compared to TD-DFT. As already discussed, in this
latter case a system-size-dependent error is known to exist
[26g], with an increase from 2% to 20% in the relative error
as found in this study when going from naphthalene to
Table 7
Comparison between the B3LYP/6 ? 31 + G*results (all values in eV) for the HOMO–LUMO energy gap EKS
eigenvalues, the excitation energy of the HOMO–LUMO transition as given by TD-DFT, ETD-DFT
in Tables 2–6)
gapobtained as difference of Kohn–Sham
, and its corresponding experimental value Eexp
gapgap(p-bands
n
EKS
gap
ETD-DFT
gap
Eexp
gap
QP1
gap
QP2
gap
QPDFT-GW
gap
QPexp
gap
Ebind
Eexp
bind
3.89
3.46
3.18
2.89
–
2
3
4
5
6
4.74(4.75)
3.54(3.55)
2.74(2.75)
2.19(2.19)
1.78(1.78)
4.36(4.35)
3.21(3.21)
2.45(2.44)
1.91(1.90)
1.51(1.50)
4.45
3.45
2.72
2.31
1.90
8.27(8.29)
6.66(6.60)
5.55(5.56)
4.75(4.76)
4.15(4.16)
8.12
6.58
5.50
4.72
4.13
8.0
6.6
5.5
4.8
4.3
8.34
6.91
5.90
5.20
–
3.91
3.45
3.10
2.84
2.64
The QP–gap evaluated using Eqs. (1) and (2) is compared with the DFT-based tight-binding GW data of Ref. [28d], QPDFT?GW
experimental value QPexp
gapobtained as the difference between the experimental EAs and first IEs given in Table 1. The theoretical and experimental exciton
binding energy Ebindare estimated as QP1
gap
and QPexp
parentheses.
gap
, as well as with the
gap? ETD?DFT
gap? Eexp
gap, respectively. For comparison, the results of Ref. [26k] are reported within
Fig. 11. Top panel: optical gap ETD-DFT
B3LYP/6 ? 31 + G*level (open triangles) and DSCF QP-corrected
HOMO–LUMO gap QP1
function of molecular size. Bottom panel: exciton binding energy Ebind
estimated as QP1
gap
(open squares). In both figures, the
corresponding experimental values are represented by the filled symbols
(see Table 7).
gap
as obtained via TD-DFT at the
gapcomputed via Eq. (1) (open diamonds), as a
gap? ETD-DFT
G. Malloci et al. / Chemical Physics 340 (2007) 43–58
55

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Author's personal copy
hexacene. The DSCF QP-corrected HOMO–LUMO gaps
of neutral oligoacenes as computed at the B3LYP/
6 ? 31 + G*level, compare very well with the DFT-based
tight-binding GW calculated ones independently from the
size of the system, with discrepancies no larger than 3%.
These values, however, when compared to experiments
appear to be affected by the same systematic error found
for the TD-DFT results (see the diamonds on the top panel
of Fig. 11). On the other hand, since these errors cancel
each other in the evaluation of the exciton binding energy,
we obtain an accurate estimate for Ebind. Appreciable exci-
tonic effects due to both quantum confinement and reduc-
tion of screening are found, with values ranging from
3.9 eV for naphthalene to 2.8 eV for hexacene.
5. Concluding remarks
We presented a systematic theoretical study of the five
smallest oligoacenes, i.e., naphthalene, anthracene, tetra-
cene, pentacene, and hexacene in the charge-states most
relevant for astrophysical applications, namely ?1, 0, +1,
and +2. From the ground-state structural relaxations per-
formed at the B3LYP/6 ? 31 + G*level we computed the
electron affinities, the first and second ionisation energies,
the quasiparticle correction to the HOMO–LUMO gap
of the neutral systems, and an estimate of the excitonic
effects in this class of compounds. Good agreement is
found with the available experimental data as well as with
previous theoretical results. To study the electronic absorp-
tion spectra we used a compendium of the TD-DFT theo-
retical framework in both real-time, to obtain the whole
photo-absorption cross-sections in a single step, and fre-
quency space, to study general trends as a function of
charge-state and molecular size for the lowest-lying valence
p ! p*in-plane long-polarised and short-polarised elec-
tronic transitions. The main step forward achieved in this
work with respect to previous theoretical analyses lies (i)
in the spectral range covered, that extends up to the far-
UV for both neutral and charged PAHs, and (ii) in the first
detailed study of doubly ionised species, largely unexplored
so far. The interest on PAH dications by the astrophysical
community has been recently renewed by the proposal that
they could be plausible candidates to explain a red fluores-
cence observed in many interstellar sources [32]. We find
that dications, like their singly charged counterparts, dis-
play strong electronic transitions of p ! p*character in
the near-IR, visible, and near-UV spectral ranges. As
expected, the broad plasmon-like structure peaking at
about 17.5 eV is found to be relatively insensitive to the
charge-state of the molecule, but we interestingly find a sys-
tematic decrease with increasing positive charge of the
absorption cross-section between about 6 eV and about
12 eV. Since the latter spectral signature is a general prop-
erty of all PAHs [23c], a comparison with astronomical
extinction curves could provide an additional observational
handle for estimating the average charge-state of interstel-
lar PAHs.
Acknowledgements
G. Malloci acknowledges financial support by Regione
Autonoma della Sardegna. G.C., G.M., and G.M.
acknowledge financial support by MIUR under project
PON-CyberSar. We thank the authors of OCTOPUS for mak-
ing their code available under a free license. We acknowl-
edge the High Performance Computational Chemistry
Group for the use of NWCHEM, A Computational Chemistry
Package for Parallel Computers, Version 4.7 (2005),
PNNL, Richland, Washington, USA. Part of the simula-
tions were carried out at CINECA (Bologna).
Appendix A. Supplementary data
Supplementary data associated with this article can be
found, intheonline
j.chemphys.2007.07.046.
version,atdoi:10.1016/
References
[1] (a) E. Clar, Aromatische Kohlenwasserstoffe, Springer, Berlin, 1941;
(b) E. Clar Polycyclic Hydrocarbons, Vols. 1 and 2, Academic Press,
London, 1964.
[2] (a) R.F. Curl, Nature 363 (1993) 14;
(b) C.J. Pope, J.A. Marr, J.B. Howard, J. Phys. Chem. 97 (1993)
11001.
[3] (a) R.G. Harvey, Polycyclic Aromatic Hydrocarbons: Chemistry and
Carcinogenesis, Cambridge University Press, Cambridge, 1991;
(b) M. Rehwagen, A. Mu ¨ller, L. Massolo, O. Herbarth, A. Ronco,
Sci. Total Environ. 348 (2005) 199.
[4] (a) J.H. Hahn, R. Zenobi, J.L. Bada, R.N. Zare, Science 239 (1988)
1523;
(b) S.J. Clemett, C.R. Maechling, R.N. Zare, P.D. Swan, R.M.
Walker, Science 262 (1993) 721;
(c) A.G.G.M. Tielens, The Physics and Chemistry of the Interstellar
Medium, Cambridge University Press, Cambridge, 2005.
[5] (a) M. Bendikov, F. Wudl, D.F. Perepichka, Chem. Rev. 104 (2004)
4891;
(b) K. Hummer, P. Puschnig, C. Ambrosch-Draxl, Phys. Rev. Lett. 92
(2004) 147402-1;
(c) K. Hummer, C. Ambrosch-Draxl, Phys. Rev. B 71 (2005) 081202-
1;
(d) K. Hummer, C. Ambrosch-Draxl, Phys. Rev. B 72 (2005) 205205-
1.
[6] (a) S.F. Nelson, Y.Y. Lin, D.J. Gundlach, T.N. Jackson, Appl. Phys.
Lett. 72 (1998) 1854;
(b) H. Klauk, M. Halik, U. Zschieschang, G. Schmid, W. Radlik, J.
Appl. Phys. 92 (2002) 5259.
[7] (a) H. Aziz, Z.D. Popovic, Appl. Phys. Lett. 80 (2002) 2180;
(b) Y. Kan, L. Wang, H. Yunchuan, W. Guoshi, Q. Yong, Appl.
Phys. Lett. 84 (2004) 1513.
[8] (a) G.K.R. Senadeera, P.V.V. Jayaweera, V.P.S. Perera, K. Tennak-
one, Sol. Energy Mater. Sol. Cells 73 (2002) 103;
(b) S. Yoo, B. Domercq, B. Kippelen, Appl. Phys. Lett. 85 (2004)
5427.
[9] I. Shiyanovkaya, K.D. Singer, V. Percec, T.K. Bera, Y. Miura, M.
Glodde, Phys. Rev. B 67 (2003) 035204.
[10] (a) J.E. Anthony, Chem. Rev. 106 (2006) 5028;
(b) M. Winkler, K.N. Houk, J. Am. Chem. Soc. 129 (2007) 1805.
[11] (a) J.R. Platt, J. Chem. Phys. 17 (1949) 484;
(b) H.B. Klevens, J.R. Platt, J. Chem. Phys. 17 (1949) 470.
[12] (a) A. Bree, L.E. Lyons, J. Chem. Soc. (1960) 5206;
(b) L.E. Lyons, G.C. Morris, J. Mol. Spectrosc. 4 (1960) 480;
56
G. Malloci et al. / Chemical Physics 340 (2007) 43–58