# Ab initio electronic and optical spectra of free-base porphyrins: The role of electronic correlation.

**ABSTRACT** We present a theoretical investigation of electronic and optical properties of free-base porphyrins based on density functional theory and many-body perturbation theory. The electronic levels of free-base porphine (H(2)P) and its phenyl derivative, free-base tetraphenylporphyrin (H(2)TPP) are calculated using the ab initio GW approximation for the self-energy. The approach is found to yield results that compare favorably with the available photoemission spectra. The excitonic nature of the optical peaks is revealed by solving the Bethe-Salpeter equation, which provides an accurate description of the experimental absorption spectra. The lowest triplet transition energies are in good agreement with the measured values.

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**ABSTRACT:**Soft X-ray spectroscopy (SXS) techniques such as photoelectron spectroscopy, soft X-ray absorption spectroscopy and X-ray emission spectroscopy are efficient and direct tools to probe electronic structures of materials. Traditionally, these surface sensitive soft X-ray techniques that detect electrons or photons require high vacuum to operate. Many recent in situ instrument developments of these techniques have overcome this vacuum barrier. One can now study many materials and model devices under near ambient, semi-realistic, and operando conditions. Further developments of integrating the realistic sample environments with efficient and high resolution detection methods, particularly at the high brightness synchrotron light sources, are making SXS an important tool for the energy research community. In this progress report, we briefly describe the basic concept of several SXS techniques and discuss recent development of SXS instruments. We then present several recent studies, mostly in situ SXS experiments, on energy materials and devices. Using these studies, we would like to highlight that the integration of SXS and in situ environments can provide in-depth insight of material's functionality and help researchers in new energy material developments. The remaining challenges and critical research directions are discussed at the end.Advanced Materials 05/2014; · 14.83 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**Many-body Green's function perturbation theories, such as the GW and Bethe-Salpeter formalisms, are starting to be routinely applied to study charged and neutral electronic excitations in molecular organic systems relevant to applications in photovoltaics, photochemistry or biology. In parallel, density functional theory and its time-dependent extensions significantly progressed along the line of range-separated hybrid functionals within the generalized Kohn-Sham formalism designed to provide correct excitation energies. We give an overview and compare these approaches with examples drawn from the study of gas phase organic systems such as fullerenes, porphyrins, bacteriochlorophylls or nucleobases molecules. The perspectives and challenges that many-body perturbation theory is facing, such as the role of self-consistency, the calculation of forces and potential energy surfaces in the excited states, or the development of embedding techniques specific to the GW and Bethe-Salpeter equation formalisms, are outlined.Philosophical Transactions of The Royal Society A Mathematical Physical and Engineering Sciences 01/2014; 372(2011):20130271. · 2.89 Impact Factor - SourceAvailable from: pubs.acs.orgDavid A Egger, Shira Weissman, Sivan Refaely-Abramson, Sahar Sharifzadeh, Matthias Dauth, Roi Baer, Stephan Kümmel, Jeffrey B Neaton, Egbert Zojer, Leeor Kronik[Show abstract] [Hide abstract]

**ABSTRACT:**Density functional theory with optimally tuned range-separated hybrid (OT-RSH) functionals has been recently suggested [Refaely-Abramson et al. Phys. Rev. Lett. 2012, 109, 226405] as a nonempirical approach to predict the outer-valence electronic structure of molecules with the same accuracy as many-body perturbation theory. Here, we provide a quantitative evaluation of the OT-RSH approach by examining its performance in predicting the outer-valence electron spectra of several prototypical gas-phase molecules, from aromatic rings (benzene, pyridine, and pyrimidine) to more complex organic systems (terpyrimidinethiol and copper phthalocyanine). For a range up to several electronvolts away from the frontier orbital energies, we find that the outer-valence electronic structure obtained from the OT-RSH method agrees very well (typically within ∼0.1-0.2 eV) with both experimental photoemission and theoretical many-body perturbation theory data in the GW approximation. In particular, we find that with new strategies for an optimal choice of the short-range fraction of Fock exchange, the OT-RSH approach offers a balanced description of localized and delocalized states. We discuss in detail the sole exception found-a high-symmetry orbital, particular to small aromatic rings, which is relatively deep inside the valence state manifold. Overall, the OT-RSH method is an accurate DFT-based method for outer-valence electronic structure prediction for such systems and is of essentially the same level of accuracy as contemporary GW approaches, at a reduced computational cost.Journal of Chemical Theory and Computation 05/2014; 10(5):1934-1952. · 5.39 Impact Factor

Page 1

Ab initio electronic and optical spectra of free-base porphyrins:

The role of electronic correlation

Maurizia Palummo,1,2,a?Conor Hogan,1,2,3Francesco Sottile,1,4Paolo Bagalá,2and

Angel Rubio5

1European Theoretical Spectroscopy Facility (ETSF)

2Department of Physics, University of Rome “Tor Vergata”, Via della Ricerca Scientifica 1,

00133 Roma, Italy

3CNR-INFM-SMC, University of Rome “Tor Vergata”, Via della Ricerca Scientifica 1, 00133 Roma, Italy

4Laboratoire des Solides Irradies, Ecole Polytechnique, CEA/DSM, CNRS,

91128 Palaiseau, France

5Nano-Bio Spectroscopy Group and ETSF Scientific Development Centre, Dpto. Física de Materiales,

Universidad del País Vasco, Centro de Física de Materiales CSIC-UPV/EHU-MPC, Spain; DIPC,

Av. Tolosa 72, E-20018 San Sebastián, Spain; and Fritz-Haber-Institut der Max-Planck-Gesellschaft,

D-14195 Berlin-Dahlem, Germany

?Received 10 April 2009; accepted 24 July 2009; published online 24 August 2009?

We present a theoretical investigation of electronic and optical properties of free-base porphyrins

based on density functional theory and many-body perturbation theory. The electronic levels of

free-base porphine ?H2P? and its phenyl derivative, free-base tetraphenylporphyrin ?H2TPP? are

calculated using the ab initio GW approximation for the self-energy. The approach is found to yield

results that compare favorably with the available photoemission spectra. The excitonic nature of the

optical peaks is revealed by solving the Bethe–Salpeter equation, which provides an accurate

description of the experimental absorption spectra. The lowest triplet transition energies are in good

agreement with the measured values. © 2009 American Institute of Physics.

?DOI: 10.1063/1.3204938?

I. INTRODUCTION

Porphyrins constitute an important class of ? conjugated

organic chromophores that play a fundamental role in numer-

ous biological and chemical processes1–3and have recently

found wide application in developing technologies. Promis-

ing memory devices have recently been demonstrated in

which porphyrins were used to functionalize nanowires.4–6

Their oligomers and solid aggregates are of growing interest

for optoelectronic devices, solar cells, and light-harvesting

devices, as well as having applications in nonlinear

optics.7–12It is not surprising therefore that, in addition to the

numerous experimental studies appearing in the literature,

several semiempirical and ab initio theoretical studies, based

on time-dependent density functional theory ?TDDFT? and

quantum chemistry techniques, have been carried out in or-

der to characterize the fundamental electronic and optical

properties of these molecules.13–17

The UV/optical spectra of all porphryins are generally

quite similar, being characterized by a number of weak bands

or peaks in the optical range ?the Q bands?, and a relatively

strong band in the UV region ?the Soret or B band?.18The

simplest interpretation of porphyrin spectra is given by the

Gouterman four-orbital model, a semiempirical configuration

interaction scheme involving excitations from the two high-

est occupied molecular orbitals ?HOMOs? to the two lowest

unoccupied orbitals ?LUMOs?.19In spite of its success, how-

ever, not all spectral features can be explained by the model.

In fact, ab initio quantum-mechanical approaches are re-

quired to gain a thorough knowledge of the excited state and

photophysical properties of these molecules, which, despite

the numerous technological applications, are not completely

understood.

Previous quantum-chemical studies of the porphyrin

class of molecules illustrated the important role played by

electronic correlation in describing their excited state prop-

erties. Excitation energies of free-base porphine have been

reported using a variety of techniques, including multicon-

figurational second-order perturbation theory ?CASPT2?,15

multireference second-order perturbation theory,20symmetry

adapted cluster-configuration interaction ?SAC-CI?,21–24and

similarity transformed equation-of-motion coupled-cluster

?STEOM-CC? approaches,25generally obtaining a precision

of the order of 0.1–0.3 eV.

In the present work, we use an analogous approach

based on many-body perturbation theory ?MBPT? ?Ref. 26?

?namely the so-called GW method and the Bethe–Salpeter

equation ?BSE??, which achieved much success over recent

years within the domain of solid-state physics, frequently

yielding excitation energies within 0.1–0.3 eV of the experi-

mental values when applied to systems ranging from bulk to

zero dimensional.27–33The GW/BSE method has not been

widely applied, however, to the study of ?-conjugated low

dimensional and molecular systems. Work carried out in this

direction34–39furthermore illustrated that some of the usual

assumptions made in the application of the method, such as

the use of LDA/GGA wave functions as a starting point37or

the Tamm–Dancoff approximation,40are not always valid for

a?Electronic mail: maurizia.palummo@roma2.infn.it.

THE JOURNAL OF CHEMICAL PHYSICS 131, 084102 ?2009?

0021-9606/2009/131?8?/084102/7/$25.00© 2009 American Institute of Physics

131, 084102-1

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Page 2

molecular systems.41The applicability to the porphyrin class

of molecules is therefore far from obvious and the present

study aims to further knowledge in this direction.

Besides being of general academic interest, the tech-

nique should prove to be important for investigating the ex-

citonic character in extended porphyrin oligomers or solid

aggregates, for which the use of the less computationally

demanding TDDFT approach has been questioned,42but

which is naturally accessible within GW/BSE. In this work

we concentrate on the electronic and optical properties of

two such porphryins, namely, free-base porphine ?H2P?, the

basic building block of all porphryins, and a phenyl deriva-

tive, tetraphenylporphyrin ?H2TPP?. This study aims, there-

fore, to be a first step toward a complete ab initio analysis of

the exciton character in porphyrin systems and in particular

forunderstandinghow this

moving from isolated porphyrins to their oligomers or solid

aggregates.

character changes when

II. METHODOLOGY

In the present MBPT scheme, the DFT Kohn–Sham

?KS? eigenvalues and eigenfunctions are used as a starting

point for constructing the one-particle and two-particle

Green’s functions including all relevant aspects of electronic

interaction and correlation. The key quantity is the electron

self-energy operator, which can be evaluated very accurately

for many materials in the GW approximation.27The one-

particle Green’s function describes quasiparticle ?QP? excita-

tions ?i.e., the individual excitation of electrons and holes?

while the two-particle Green’s function describes coupled

electron-hole excitations and is thus required for describing

the optical spectrum.28

Results presented in this work are based on the follow-

ing three-stage approach. As a first step, the geometrical

structures of the two isolated molecules ?H2P and H2TPP?

are relaxed using DFT ?Ref. 43? within the general gradient

approximation ?DFT-GGA? in the Perdew–Burke–Ernzerhof

PBE parametrization functional.44We use a plane-wave ap-

proach, as implemented in the

package,45with norm-conserving pseudopotentials and a ki-

netic energy cutoff of 70 Ry. Fictitious molecule-molecule

interactions occurring in the repeated cell approach are elimi-

nated by using, after convergence tests, a vacuum thickness

of more than 10 Å. The relaxed geometries compare well

with other similar calculations found in the literature and

with the experimental data.46,47In particular, the external

phenyl groups of H2TPP cause an in-plane distortion of the

porphyrin ring, without any appreciable out-of-plane distor-

tion, in agreement with Ref. 48. We then calculate, at the

optimized geometries, all the KS eigenvalues and eigenvec-

tors up to 15 eV above the HOMO energy using LDA,49in

order to reach a good convergence in the excited-state calcu-

lations. The GGA and LDA eigenvalues, for the same fixed

geometry, were found to be very similar, within the order of

0.02–0.03 eV.

In a second step, we perform GW calculations using the

YAMBO code50in order to obtain the real QP energies Ei

QUANTUM-

ESPRESSO

QPas

corrections to the KS eigenvalues Ei

well known expression:27

KSusing the following

Ei

QP= Ei

KS+

1

1 − ?i

??i

KS???Ei

KS? − Vxc??i

KS?,

?1?

where the index i runs over the occupied ?holes h? and un-

occupied ?electrons e? states. ??i

tions, ?iis given by

KS? are the KS eigenfunc-

?i= ??i

KS?d?/dE?Ei

KS??i

KS?,

?2?

i.e., the linear coefficient in the energy expansion of the self-

energy ?, which is itself the product of the KS Green’s func-

tion G times the screened Coulomb interaction W obtained

within the random phase approximation ?RPA?.51Vxcis the

usual DFT exchange-correlation potential. A boxlike cutoff

in the long-range Coulomb potential is used at this stage in

order to simulate truly isolated molecular excited states. This

technique is essential52for reaching good convergence ?be-

low 0.1 eV? in the self-energy calculations.

In the final step of our approach, we calculate the optical

spectra including excitonic effects and self-energy correc-

tions by means of solving the BSE. By expanding the states

over the KS basis, the solution of the BSE can be mapped

onto aneigenvalue problem

Hamiltonian:28,53

Hexc=?

− ?Hcoupl??

for theexcitonic

Hres

Hcoupl

− ?Hres???,

?3?

where the resonant part,

Hres= ?Ee

QP− Eh

QP??e,e??h,h?+ ?eh?K?e?h??,

?4?

is Hermitian. The part in the lower right is denoted antireso-

nant. K=W−2v is the excitonic kernel, with W and v being

the screened and bare Coulomb interaction, where the factor

2 comes from the spin degeneracy.53The coupling part

Hcoupl= ?eh?K?e?h???5?

is symmetric and describes the interaction between the reso-

nant and antiresonant parts, or in other words, between the

e-h pairs at positive and negative ?antipairs? energies ?see

Ref. 41 for a more detailed description of the notation?. Here

electron-hole antipairs are denoted by e?h?, while Eh

and Ee

occupied and unoccupied states, respectively. As is often

done in this framework, we replace the QP eigenfunctions

?e?,?h? with the KS ones ??h

have shown that this approximation may not work well in

highlyanisotropicsystems37

materials,54the results presented below demonstrate that it is

reasonable for describing the low lying excitations of sys-

tems such as the studied porphyrins, as was also illustrated

elsewhere for azobenzene.41

Once the eigenvectors and eigenvalues E?of the exci-

tonic Hamiltonian Hexcare obtained, the photoabsorption

cross section is obtained from

QP,?h?

QP,?e? refer to the QP energies and eigenstates of the

KS?,??e

KS?. While several works

orstrongly correlated

084102-2Palummo et al. J. Chem. Phys. 131, 084102 ?2009?

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Page 3

???? ??

?

?D??2??? − E??,

?6?

where D?=???i? ? ·r ??0? are the electron-hole optical strengths,

? is the light polarization direction, ?0? is the ground state,

and ??? is the generic excited state expanded in term of

electron-hole pairs and antipairs. If the electron-hole interac-

tion is neglected, the excitations are given by vertical transi-

tions between independent electron and hole states with D?

=?h?i? ? ·r ??e?, so that Eq. ?6? reduces to the well known

independent-particle Fermi golden rule expression.

Although the above approach is based on a local or

semilocal DFT ground state calculation, it is important to

realize that it does not inherit all the associated pathologies.

In particular, the short-range exchange-correlation potential

appearing in LDA and GGA is not present in the self-energy

or, more importantly, the BSE. In these methods, the

screened Coulomb interaction is used instead, yielding a cor-

rect description of the long-range 1/r behavior. Thus, the

BSE can in principle describe charge transfer as well as

Rydberg states.55–57

III. RESULTS

According to the Gouterman model, the HOMO ?b1u?,

HOMO−1?au?, LUMO?b2g?, and LUMO+1?b3g? states are

the most important ones involved in the Q and B optical

peaks. In Fig. 1 we report probability distribution isosurfaces

for each of these four states as they occur for the H2TPP

molecule. The inclusion of phenyl rings does not change

their character with respect to free-base porphine. We con-

firmed that the orbital character and ordering are consistent

with other DFT-LDA calculations,46an observation that does

not change if local or semilocal exchange-correlation poten-

tials are used. All these findings attest to the robustness of

the Gouterman model in describing the order and character

of these energy levels based on the symmetry of the porphy-

rin molecular orbitals. Nevertheless, we will show below

how a proper description of the optical response and photo-

emission spectra requires a more sophisticated theoretical

treatment beyond this simple empirical model.

In Fig. 2 we show the optical spectrum of the H2TPP

molecule obtained at the independent-particle level ?or

equivalently, within the RPA?, where a sum over the KS

transitions, according to a Fermi golden rule description, is

considered. Two strong peaks are visible at 1.75 and 2.15 eV.

The former peak derives from b1u↔b2gand b1u↔b3gtran-

sitions, while the latter derives from a1u↔b2gand au↔b3g

transitions. The RPA optical spectrum of H2P, not reported

here, appears very similar with a small blueshift of the two

peaks ?of about 0.2 eV?. As expected, the optical spectra

obtained at this level of approximation are in complete dis-

agreement with the experimental data, which feature almost

forbidden Q bands in the visible region and very intense B

bands in the near-UV region.

These results stress the need to overcome the single-

particle scheme and mix the single-particle transitions. Such

mixing can be achieved by means of configuration interac-

tion techniques, the TDDFT approach, or the present GW

+BSE approach. As we will see below, the four-level mixing

scheme proposed in the Gouterman model appears valid for

accounting for the Q bands and, to a lesser extent, the B

bands, in agreement with published results based on

quantum-chemical or TDDFT schemes.16,58

The computed QP energies for the isolated H2P and

H2TPP molecules are compared with the KS ?DFT-LDA?

energies in Fig. 3. The typical linear relation that is often

found between the two sets of eigenvalues in many semicon-

ductor and insulating materials ?both in bulk and in low di-

mensional systems? is only partially reproduced in the

present molecular systems. The GW calculation opens the

electronic HOMO-LUMO gap in H2P to 5 eV ?1.97 eV is the

corresponding DFT-LDA gap?, while a QP gap of 4.39 eV is

obtained in H2TPP ?the DFT-LDA gap being 1.75 eV in this

case?. In Fig. 4 we compare the levels of the H2P, obtained

using the GW method, with experimental UV photoemission

spectroscopy ?UPS? data found in the literature.59

For comparison, we also report KS eigenvalues as ob-

tained using a local ?LDA?, a semilocal ?GGA?, and a hybrid

exchange-correlation functional ?B3LYP ?Ref. 60??. As these

methods do not satisfy Koopman’s theorem, their eigenval-

ues should not be directly interpreted as electron removal/

addition energies. Nevertheless, a comparison with the GW

FIG. 1. Plots of ???2of the four levels that mainly par-

ticipate in the optical response of H2TPP, as obtained at

the DFT-LDA level of approximation. From left to right

are reported the two highest occupied and the two low-

est unoccupied states. The x axis coincides with the

direction of the central N–H bonds.

1

23

4

5

Energy (eV)

Absorption (arbitrary units)

HOMO

HOMO-1

LUMO, LUMO+1

HOMO-1

LUMO, LUMO+1

LUMO, LUMO+1

x polarization

y polarization

-1

0

1

2

3

4

5

LDA states (ev)

HOMO

FIG. 2. Absorption spectrum of the H2TPP molecule as obtained at the

independent-particle level ?RPA?. Spectra according to x and y polarizations

are almost identical. An artificial Lorenztian broadening of 10 meV has been

used. Inset: DFT-LDA energy levels.

084102-3 Electronic and optical spectra of porphyrinsJ. Chem. Phys. 131, 084102 ?2009?

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Page 4

eigenvalues illustrates the importance and magnitude of the

self-energy corrections to the starting DFT eigenvalues ?that

could then be considered as an order 0 approximation to the

QP energies?. The theoretical description of the experimental

UPS peaks is clearly much improved when QP energies are

considered. It is interesting that the hybrid functional scheme

also gives a very good description of the occupied states.

This finding is in agreement with recent studies of

porphryins61

andofother

molecules.62,63

Optical spectra of the H2P and H2TPP molecules, com-

puted at the GW+BSE level of approximation, are presented

in the top panels of Figs. 5?a? and 5?b?, respectively. The

theoretical spectra are compared with the corresponding ex-

perimental data from Ref. 18 reproduced in the bottom pan-

els. Since the computed absorption for light polarized per-

pendicular to the central ring is found to be negligible, we

report only the in-plane ?x and y? components. Note that

vibrational coupling effects are not included in the present

calculation and hence the Qx?0,1? and Qy?0,1? replicas,

present in the experimental curves, are always absent in the

theoretical spectra.

Inspection of Fig. 5?a? shows a reasonably good agree-

ment between theory and experiment for the H2P molecule.

extended carbon-based

The Qx?0,0? and Qy?0,0? peaks appear at 1.98 and 2.3 eV,

respectively, in good agreement with the experimental exci-

tation energies reported at 1.98–2.02 and 2.33–2.42 eV.18

Very intense optical peaks are obtained in the UV range

around 3.3 eV. Their position and shape are in reasonable

agreement with the experimentally observed Soret bands at

3.13–3.33 eV.18The level of accuracy reached for these low

energy transitions is similar to that obtained in other ab initio

approaches. Some previously computed excitation energies

are reported in Table I for comparison with the GW+BSE

and experimental data. Other very intense transitions are

found near 4.0 eV, which may correspond to the experimen-

tally observed Nxand L bands at 3.65 and 4.25 eV. However,

the assignment is not clear in this case, as their intensities are

overestimated with respect to the lower energy peaks. This

may be due to a lack of convergence in the present GW

calculations for states lying close to the continuum of mo-

lecular states: By underestimating their lifetime, the calcula-

tions yield sharper resonances that those seen experimentally.

Large oscillator strengths have also been reported for the N

band in Ref. 21, although the accuracy of that work has been

questioned.25

The comparison with experiment further improves for

the H2TPP case, where the experimental absorption spectrum

of H2TPP, shown in Fig. 5?b?, is very well reproduced by the

theoretical Bethe–Salpeter calculation. The Qx?0,0? and

Qy?0,0? peaks appear at 1.88 and 2.15 eV, nicely reproduc-

ing the experimental transitions reported at 1.86 and 2.27

eV.18The experimentally observed Soret band is located at

-4-2

KS energy (eV)

024

-6

-4

-2

0

2

4

6

8

Energy (eV)

H2P

-4 -2

KS energy (eV)

024

-6

-4

-2

0

2

4

6

8

H2TPP

KS

QP

FIG. 3. Quasiparticle levels ?open red squares? plotted as a function of the

KS levels of the H2P ?left panel? and the H2TPP molecules ?right panel?.

The KS eigenvalues ?filled black dots? are also reported for comparison.

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

Expt. UPSGW B3LYPGGALDA

Energy (eV)

FIG. 4. Electronic states ?in eV? obtained within DFT and quasiparticle

schemes, compared with experimental UPS experimental data from Ref. 59.

The highest occupied orbital has been aligned to zero eV in each case.

Qx

B

Qy

(a) H2P

GW+BSE

x

y

x

y

Qx

Qy

B

(b) H2TPP

GW+BSE

2345

Energy (eV)

Absorption (arbitrary units)

Qx

Qy

B

Nx

L

Expt.

2345

Energy (eV)

Qx

Qy

B

Expt.

Nx

FIG. 5. Absorption spectra of the H2P ?left? and H2TPP ?right? molecules.

The top panels show theoretical spectra obtained at the GW+BSE level of

approximation for x ?blue? and y ?red, shaded? light polarizations. Experi-

mental gas phase spectra, as reported in Ref. 18, appear in the bottom

panels. Left of the vertical dashed line, the intensity of each spectrum has

been multiplied by a factor of 10 for clarity. An artificial Lorenztian broad-

ening of 10 meV has been used.

084102-4Palummo et al.J. Chem. Phys. 131, 084102 ?2009?

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Page 5

3.08 eV, slightly redshifted with respect to the free-base por-

phine, and is in very good agreement with theoretical predic-

tion. Finally, we note that the BSE absorption curves show

an optical anisotropy splitting in the B band of about 0.3 and

0.15 eV for the H2P and H2TPP molecules, respectively.

From the calculated values of the electronic gaps ?5.0

and 4.3 eV, for the H2P and H2TPP molecules, respectively?

it is clear that the optical spectra feature strong excitonic

effects, with estimated exciton binding energies of about 3.0

and 2.5 eV. This is further illustrated in Fig. 6, where we

compare a GW+RPA calculation of the H2TPP absorption

spectrum with the result of the full GW+BSE calculation,

for the x-polarization. It is interesting, therefore, that the ex-

citonic transitions associated with the Q bands are found to

derive from a mixing of the single particle transitions from

the HOMO?1 to LUMO+1 and from the HOMO?1 to

LUMO states, in agreement with the Gouterman model. For

the B and N bands, however, single-particle transitions from

the HOMO?2 to LUMO+1 also contribute, and hence

involve states beyond the standard four involved in the

Gouterman picture.

Often ?and especially in extended systems? it is found

that the resonant part of the excitonic Hamiltonian, Hres, is

adequate for describing the optical excitations correctly. This

corresponds to the so-called Tamm–Dancoff approximation40

and is equivalent to neglecting the interaction between the

e-h pairs and antipairs. In Fig. 6 we illustrate the influence of

the coupling term Hcouplon the final optical spectrum of

H2TPP. The effect is found to be quite large, both in the

energetic peak positions and in the spectral lineshape. These

findings are consistent with the conclusions of Grüning

et al.41for other carbon-based molecules and similar obser-

vations in other molecular systems.64For this reason we have

chosen to include the coupling term in all GW+BSE spectra

appearing in this work.

Due to their importance in emission processes and in

photobiology or medical applications such as photodynamic

therapy, it is also interesting to investigate how the present

approach is able to reproduce the characteristics of the triplet

excitons. Within the BSE approach, triplet excitons can be

calculated by simply considering Keh=W. In Table II we

compare our results for the lowest triplet excitons in both

molecules with the experimental values and some of the the-

oretical ones as taken from the literature.

We find that the energetic position of the lowest triplet

excitons appears in very good agreement with the available

experimental data and with other theoretical approaches for

the H2P and H2TPP molecules. Spatial analysis of the exci-

tonic wave function reveals that the lowest singlet and triplet

excitons have different character: This is illustrated in Fig. 7

for the H2P molecule. While the lowest singlet exciton re-

sults from a mixing of the four Gouterman states ?HOMO,

HOMO?1, LUMO, LUMO+1?, the triplet exciton is instead

a pure mixing of the HOMO and LUMO states. From our

previous discussion and the results shown in Tables I and II,

we can conclude that the present solid-state based scheme

?GW+BSE? provides singlet and triplet excitations of por-

phyrins to within 0.2 eV of experiment, therefore having the

very same range of accuracy as the best QC approaches dis-

cussed in the introduction.

TABLE I. Excitation energies ?in eV? of the H2P molecule obtained using different theoretical approaches and

compared with the experimental peak positions.

MethodReference

Qx

Qy

Bx

By

Expt.

GW+BSE

TDDFT ?LDA?

CASPT2

NEVPT2

SAC-CI

STEOM-CC

TDDFT ?B3LYP?

18 1.98–2.02

1.98

1.97

1.63

2.04

1.81

1.70

2.27

2.33–2.42

2.3

2.1

2.11

2.51

2.10

2.59

2.44

3.13–3.33

3.3

3.0

3.08

3.22

3.47

3.63

3.33

3.13–3.33

3.5

3.0

3.12

3.30

3.69

3.74

3.41

This work

This work

15

24

22

25

17

23

4

5

Energy (eV)

Absorption (arbitrary units)

GW+RPA

GW+BSE resonant only

(Tamm Dancoff)

GW+BSE full

(x-polarization

only)

FIG. 6. Absorption spectra of the H2TPP molecule ?x-polarization only?

computed at various levels of approximation: GW+RPA, resonant only

?Tamm–Dancoff approximation?, and full GW+BSE as reported previously.

An artificial Lorenztian broadening of 10 meV has been used.

TABLE II. Theoretical energetic positions ?in eV? of the first triplet ?T?

exciton for the H2P and H2TPP molecules. The experimental data are also

reported for comparison.

MethodReferenceH2PH2TPP

Expt.

GW+BSE

TDDFT ?B3LYP?

CASPT2

STEOM-CC

QMC

65–67

This work

17

15

25

68

1.56–1.58

1.6

1.46

1.52

1.19

1.6

1.45

1.5

084102-5Electronic and optical spectra of porphyrinsJ. Chem. Phys. 131, 084102 ?2009?

Downloaded 19 Feb 2010 to 161.111.180.191. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp

Page 6

It is useful to compare the GW+BSE results with a

complementaryapproach. We

calculation69for the H2TPP molecule, within the adiabatic

local density approximation. The results, shown in Fig. 8

?see also Tables I and II?, illustrate a reasonable agreement

between the two methods for both Q and Soret bands, in

particular with regard to the peak positions. The relative in-

tensities of the x and y polarizations are also fairly consistent

between the two approaches, although we note that the TD-

DFT calculations appear to overestimate the relative inten-

sity of the Q and Soret bands in comparison with the experi-

mental data.

performedaTDDFT

IV. CONCLUDING REMARKS

In conclusion, we calculated by means of first-principles

MBPT the charged and neutral electronic excitations of the

isolated H2P and H2TPP molecules. The available photo-

emission and optical absorption measurements are well de-

scribed by this approach, which also naturally provides a

complete picture of the e-h coupling, the singlet and triplet

excitations and the real space extension of the excitonic

wave functions. The character and energy of the singlet and

triplet lowest energy excitons turn out to be in good agree-

ment with experiment and other ab initio calculations, with

accuracy comparable to quantum chemical methods. As a

byproduct, we performed TDDFT calculations. The results

show that a simple approximation such as ALDA can already

give important insights about the optical spectrum of such

molecules. The present study should represent the first step

toward a complete ab initio analysis of the change in exciton

character when moving from isolated porphyrins to their oli-

gomers or to solid aggregates.

Note added in proof.

In a recently published work,71the

GW method was also used to calculate the ionization poten-

tial of the H2TPP molecule. The work provides further evi-

dence that the MBPT scheme can be applied with success to

the study of these molecular systems.

ACKNOWLEDGMENTS

The authors are grateful to Dr. Simona Silaghi and Dr.

Norbert Esser for useful discussions, as part of the ETSF

pilot user project n. 12. We acknowledge funding by the

European Community through e-I3 ETSF project ?INFRA-

2007-1.2.2: Grant Agreement No. 211956? and by the

Spanish MEC ?Grant No. FIS2007-65702-C02-01?, “Grupos

Consolidados UPV/EHU del Gobierno Vasco” ?Grant No.

IT-319-07?, CSIC and by MIUR-PRIN2007. Supercomputing

support is acknowledged from The Barcelona Supercomput-

ing Center, “Red Espanola de Supercomputacion” and

SGIker ARINA ?UPV/EHU?, and the CINECA Supercom-

puting Center through CNR-INFM projects ?accounts

cne0fm2h and cne0fm2v?.

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084102-7Electronic and optical spectra of porphyrinsJ. Chem. Phys. 131, 084102 ?2009?

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