Interpretation of the electronic absorption spectrum of free-baseporphin using time-dependent density-functional theory

Dage Sundholm

Department of Chemistry, P.O. Box 55 (A.I. V irtasen aukio 1) FIN-00014 Helsinki, Finland

Received 9th March 2000, Accepted 17th March 2000Published on the Web 17th April 2000

The electronic absorption spectrum of free-base porphin has been studied at density-functional theory (DFT)level using the time-dependent perturbation theory approach. The optimization of the molecular structure wascarried out using the BeckeÈPerdew functional and split-valence quality basis sets augmented by polarizationfunctions. In the calculation of the electronic excitation energies, the same functional was employed while thebasis set was further augmented by di†use s, p, and d functions. The calculated absorption spectrum agreeswith recent ab initio and DFT calculations and with experiment. The B-band must be assigned to the strong

and the transitions. Two additional weak transitions and are found below the B3 1B2u 3 1B3u (2 1B2u 2 1B3u)band. However, these states have not been obtained at CASSCF (CASPT2) and DFT/MRCI levels and mightbe an artifact of the functional used. The lowest symmetry-allowed tripletÈtriplet transition is calculated to be1.58 eV as compared to the experimental value of 1.56È1.58 eV.

I. IntroductionThe electronic excitation spectra of porphyrins and chlorinsare of central importance for many biological processes ofwhich perhaps the most prominent is the capture of light bychlorophyll. The ultra-violet/visible (UV/VIS) spectra ofporphyrins and chlorins have been the subject for numerousexperimental and computational studies.1h4 The basic build-ing unit for porphyrins is free-base porphin (porphyrin) whoseUV/VIS spectrum in the visible region consists of four weakbands usually called the Q bands. In the UV region fourbands have also been detected. The very strong band in thenear UV region is called the Soret or B band, and the threeother UV bands are known as the N, L and M bands. Theinterpretation of the porphyrin and chlorin UV/VIS spectrarelies on GoutermanÏs four orbital model5h9 which considersonly the two highest occupied and the two lowest unoccupiedmolecular orbitals. More sophisticated models to interpret theUV/VIS spectra also include vibronic coupling e†ects.9

Computationally, the electronic absorption spectrum offree-base porphin has more recently been studied at semi-empirical,10,11 density-functional theory,12h15 and ab initiolevels of theory. Baker and Zerner10 and Baraldi et al.11 havemade extensive studies of the excitation spectrum of free-baseporphin using semi-empirical conÐguration interaction andrandom phase approximation methods (CS INDO CI,INDO/S CIS, and INDO/S RPA). and co-Mercha� nworkers16h18 and Kitao and co-workers19 have studied theexcitation spectrum at multiconÐguration second-order per-turbation theory (CASPT2) level which usually provideshighly accurate excitation energies for conjugated organicmolecules such as benzene, furan and pyrrole.17 Free-baseporphin with its 24 p valence molecular orbitals and 26 p elec-trons is a real challenge for correlated ab initio methods.Other ab initio studies of the excitation spectrum of free-baseporphin include symmetry-adapted-cluster conÐgurationinteraction (SAC-CI),20h22 conÐguration interaction singles(CIS),23h25 and single- and multireference conÐguration inter-action (SCF-CI and MRCI) calculations.26,27 Bartlett and co-workers have calculated the excitation energies at the

similarity-transformed equation-of-motion coupled-clusterand perturbation levels (STEOM-CC and STEOM-PT).25,28

Bauernschmitt and Ahlrichs24 calculated the Ðrst four exci-tation energies for free-base porphin using the time-dependentperturbation theory approach based on the HartreeÈFockapproximation (random phase approximation, RPA, orHartreeÈFock linear response, HF-LR) and its density-functional generalization (TD-DFT) employing several func-tionals such as local density approximation (S-VWN),generalized gradient approximations (B-P), and hybrid func-tionals (B3LYP). Their results showed that the excitationenergies obtained at the TD-DFT level are in much betteragreement with the experimental ones than those obtained atthe RPA (HF-LR) and the CIS levels.

Recently, we found at the TD-DFT level a one-to-one cor-respondence between the calculated excitation energies andthe measured absorption maxima for chlorophyll a.29 Thesame TD-DFT method24,30,31 has been employed here to cal-culate all the electronic excitation energies for free-baseporphin below 5 eV which is the upper limit for reliable exci-tation energies when using the present functional. The aim ofthis work is to calculate the excitation energies below 5 eV offree-base porphin and to use them to understand the UV/VISspectra of porphyrins.

II. Computational methods

The molecular structure of free-base porphin (porphyrin) hasbeen optimized at density functional level employing theresolution of the identity approach.32 In the RI approach, theelectron density is approximated by an expansion in atom-centered auxiliary basis sets which avoids the evaluation offour-center two-electron integrals. Depending on the size ofthe molecule, the computational costs are reduced by a factorof four to ten by using the RI-DFT approach. In both thegeometry optimization and the subsequent calculations of theelectronic excitation spectrum, the BeckeÈPerdew (B-P)functional33h35 as implemented in TURBOMOLE36 has beenemployed.

DOI : 10.1039/b001923m Phys. Chem. Chem. Phys., 2000, 2, 2275È2281 2275

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The excitation energies have been obtained at densityfunctional level using the time-dependent perturbation theoryapproach (TD-DFT)24,30,31,37h44 which is based on theRungeÈGross theorem45 and related to the HartreeÈFocklinear response (HF-LR) or random phase approximation(RPA) method.46 In the present implementation of theTD-DFT method,24,30,31 the RI approach is used for thetreatment of Coulomb type matrix elements and thereby oneachieves a signiÐcant reduction in the computational costs.For comparison, some of the lowest excitation energies werealso calculated using the three-parameter LeeÈYangÈParrfunctional47 (B3LYP) proposed by Becke.48

III. Basis setsIn the geometry optimization, the Karlsruhe split-valencebasis sets49 augmented with polarization functions50 wereemployed (SVP). In the calculation of the excitation energies,the basis sets were augmented by di†use s, p, and d functions(SVP] di†). The exponents of the di†use functions are : 0.04(s), 0.3 (p) for H, 0.06 (s), 0.07 (p), 0.3 (d) for N, and 0.04 (s),0.05 (p), 0.3 (d) for C. The optimized auxiliary basis sets usedin the RI-DFT calculations consist of 4s2p/2s1p for H and of8s3p3d1f/6s3p3d1f for C, N, and O.32 All calculations havebeen performed in symmetry with the molecule placed inD2hthe xy plane and the pyrrole hydrogens in the x direction. Tocheck the basis-set convergence also triple zeta valence basissets augmented with polarization functions (TZVP) wereemployed.51

IV. ResultsThe molecular structure and the Ðrst few singlet and tripletexcitation energies of free-base porphin have been calculatedat the DFT level as described in the previous sections. The

optimized molecular structure of free-base porphin can bedownloaded from our Internet page.52

A. Singlet states

In Table 1, the four Ðrst transition energies for free-baseporphin are compared to previously calculated and experi-mental excitation energies. As seen in Table 1, the addition ofdi†use basis functions hardly a†ects the excitation energies.For the four lowest states, the largest di†erence between theresults obtained with the three basis sets is only 0.02 eV. Thesecond state of symmetry is the only state whose energyB1ushows a signiÐcant basis-set dependence. Its excitation energyis 0.8 eV smaller when the SVP] di† basis sets are employed.

The excitation energies of the two Ðrst transitions of 2.13and 2.25 eV for the and states, respectively, are both1B3u 1B2uin relatively good agreement with the reported experimentalexcitation energies of 1.98È2.02 and 2.33È2.42 eV, respectively.For comparison, the excitation energy obtained at CASPT2level for the Ðrst excited state is about 0.35 eV too low, whilethe second state was in a somewhat better agreement withexperiment.16 The agreement between the present calculatedand measured excitation energies is surprisingly good for suchan inexpensive approach as TD-DFT. The two Ðrst excitationenergies obtained at the TD-DFT level di†er 0.1È0.2 eV fromreported experimental excitation energies.

The two subsequent excited states also belong to the 1B3uand irreducible representations. The corresponding exci-1B2utation energies are 2.94 and 2.96 eV, respectively. The twostrongest transitions appear at 3.36 and 3.42 eV. Judging fromthe values of the excitation energies and the oscillatorstrengths these must be assigned to the very strong B band.This means that the excitation energies of the transitions at2.94 and 2.96 eV are either too small as compared to experi-ment and cannot be detected due to the strong B-band or,alternatively, they are hidden under the Q-bands. Very

Table 1 Theoretical and experimental singlet excitation energies (eV) for free-base porphin

Methoda 1 1B3u 1 1B2u 2 1B3u 2 1B2u Refs.

TD-DFT B-P, SVP] di† 2.13 2.25 2.94 2.96 This workTD-DFT B-P, SVP 2.13 2.25 2.94 2.96 This workTD-DFT B-P, TZVP 2.13 2.26 2.95 2.97 This workTD-DFT B3LYP, SVP 2.24 2.39 3.27 3.45 This workDFT/MRCI BHLYP,VDZP 1.94 2.38 3.07 3.17 [15]DFT/SCI B3LYP, VTZP 2.03 2.28 3.56 3.89 [14]DFT/MRCI BHLYP 1.94 2.41 3.12 3.23 [13]DFT B-P, TZ2P STO 2.16 2.29 2.98 3.01 [12]SAC-CI 1.81 2.10 3.47 3.69 [19]CASPT2 1.63 2.11 3.08 3.12 [16]SAC-CI 1.77 2.01 3.73 3.47 [21]STEOM-CC 1.75 2.40 3.62 3.47 [28]SAC-CI 1.75 2.23 3.59 3.79 [22]CIS 2.48 2.60 4.67 4.46 [25]STEOM-PT 2.20 3.01 3.93 4.03 [25]STEOM-CC 1.72 2.61 3.66 3.77 [25]SAC-CI 1.75 2.23 3.55 3.56 [20]TD-DFT S-VWN, SVP 2.18 2.32 2.99 3.00 [24]TD-DFT B-P, SVP 2.17 2.30 2.98 3.01 [24]TD-DFT B3LYP, SVP 2.27 2.43 3.32 3.49 [24]CIS, SVP 2.41 2.53 4.48 4.73 [24]RPA, SVP 1.72 1.89 3.95 3.95 [24]CASPT2 1.20 2.74 3.74 3.98 [17]CS INDO CI 2.14 2.49 3.59 4.00 [11]CASPT2 1.70 2.26 2.91 3.04 [18]MRCI 1.97 2.40 3.41 3.24 [27]CIS 2.56 2.71 4.66 4.91 [23]INDO/S CIS 1.70 2.06 2.37 3.52 [10]INDO/S RPA 1.46 1.97 2.99 3.03 [10]SCF-CI 2.27 3.42 4.72 5.29 [26]Exp. 1.98È2.02 2.33È2.42 3.13È3.33 3.13È3.33 [54, 56, 57]

a DFT/SCI : A combination of DFT and CI methods including only singles excitations. DFT/MRCI: A generalization of DFT/SCI to a multi-reference singles and doubles CI treatment. BHLYP\ BeckeÏs ““half-and-half ÏÏ functional.53 See also text.

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recently Parusel and Grimme15 performed DFT with multi-reference conÐguration interaction calculations (DFT/MRCI)and found that there is a signiÐcant double excitation charac-ter in the B-band transitions. They did not obtain any addi-tional states between the Q- and B-bands. Therefore, it islikely that the two extra states, obtained at 2.95 eV, are part ofthe strong B-band rather than hidden under the Q-band. Onealternative that should not be ruled out is that the two statesare an artifact of the functional used. A comparison of theexcitation weights shows that the present and2 1B3u 2 1B2ustates are superÑuous as compared to CASSCF16 and DFT/MRCI15 calculations.

In Table 2, the TD-DFT calculations using the B3LYPfunctional are summarized. The excitation spectra obtainedusing the B3LYP and B-P functional, respectively, are in qual-itative agreement. However, there are a few signiÐcant di†er-ences in the obtained spectra. The two additional and1B3ustates obtained using the B-P functional do not appear1B2uat the B3LYP TD-DFT level. At the B3LYP TD-DFT level,the excitation energy of the Ðrst n ] p transition is 0.75 eVlarger than the B-P TD-DFT value. The transition isL

yobtained at 4.29 eV as compared to the B-P TD-DFT value of4.94 eV, while the experimental value is 4.68 eV. As seen inTable 1, the excitation energies obtained at the B3LYPTD-DFT level agree much better with the DFT/MRCI resultsthan with the excitation energies calculated at the B3LYPDFT single CI level. This indicates that some higher-orderexcitation e†ects are considered at the TD-DFT level. The

excitation spectrum calculated using the B3LYP functionalalso agrees somewhat better with experiment than the excita-tion spectrum obtained at the B-P TD-DFT level. However,the RI-DFT approximation cannot be used in combinationwith hybrid functionals such as B3LYP. This prevents theapplicability of the B3LYP TD-DFT method to large systemssuch as chlorophylls and molecular complexes containingseveral chlorophyll molecules.

In the gas phase, the Ðrst four absorption maxima are mea-sured at 627, 575, 511.5, and 483.5 nm54 which correspondingto transition energies of 1.98, 2.16, 2.42 and 2.56 eV. The tran-sitions are assigned as and where the sub-Q

x0, Q

x1, Q

y0 Q

y1

scripts denote the excitation direction and the superscript thevibrational band. For comparison, the calculated excitationenergies of the two transitions are 2.13 and 2.25 eV respec-tively, which should be compared to the experimental valuesof 1.98 and 2.42 eV.

The Ðfth excitation energy can be interpreted as a veryweak n] p* transition. The sixth and seventh excitation ener-gies are two strong near-degenerated transitions at 3.36 and3.42 eV, respectively. These transitions must be interpreted asthe B band which experimentally appear at 3.13È3.33 eV. Thenext two bands are found at 3.72 eV and can be assigned tothe and transitions. The tenth transition is probably theN

xN

yband and the eleventh is again a weak n] p* transition,Lxwhile the twelfth could be the transition. Note that theL

yL

ytransition is much higher in energy than the corresponding Lxband which is also seen experimentally. At the B3LYP

Table 2 The nine lowest singlet excitation energies (eV) of free-base porphin, the corresponding wavelengths (nm), and oscillator strengths of thetransitions from the ground state calculated at the TD-DFT level using the B3LYP functional ; the SVP basis sets were employed

Assignment Calculation Experiment

State Energy Wavelength Oscillator strength Energy Wavelength Oscillator strength

1 1B3u Qx

2.24 554 0.000 18 1.98 627 0.02a1 1B2u Q

y2.39 517 0.000 04 2.42 511.5 0.07a

2 1B3u Bx

3.27 379 0.40 3.33 3.72 1.15a2 1B2u B

y3.45 359 0.61 3.33 372

3 1B2u Ny

3.70 335 0.55 3.65 340 \0.1a3 1B3u N

x3.79 327 0.82 3.65 340

1 1B1u n ] p 4.03 308 0.0012 È È4 1B2u L

y4.29 289 0.12 4.68 265 0.1a

4 1B3u Lx

4.40 282 0.11 4.25 292

a For both the transitions of the band.54

Table 3 The twelve lowest singlet excitation energies (eV) of free-base porphin, the corresponding wavelengths (nm), and oscillator strengths ofthe transitions from the ground state calculated at the TD-DFT level are compared to experimental results ; the SVP ] di† basis sets wereemployed

Assignment Calculation Experiment

State Energy Wavelength Oscillator strength Energy Wavelength Oscillator strength

1 1B3u Qx

2.13 582 0.0009 1.98 627 0.02a1 1B2u Q

y2.25 550 0.0006 2.42 511.5 0.07a

2 1B3u B3x

2.94 422 0.12 È È2 1B2u B3

y2.96 419 0.04 È È

1 1B1u n ] p 3.28 378 0.0004 È È3 1B2u B

y3.36 369 0.90 3.33 372 1.15a

3 1B3u Bx

3.42 363 0.78 3.33 3724 1B3u N

x3.72 334 0.13 3.65 340 \0.1a

4 1B2u Ny

3.72 334 0.15 3.65 3405 1B3u L

x4.28 290 0.21 4.25 292 0.1a

2 1B1u n ] p 4.80 258 0.004 È È5 1B2u L

y4.94 251 0.085 4.68 265

a For both the transitions of the band.54

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TD-DFT level, the band is below As seen in Table 3,Ly

Lx.

the present interpretation results in an easy assignment of thehigh-energy part of the spectrum.

The comparison of the present B-P TD-DFT transitionenergies and oscillator strengths with the measured spectrumwould suggest that the bands in the experimental free-baseporphin spectrum correspond to the following transitions :

n ] p*, and and n ] p*Qx, Q

y, B3

x, B3

y, B

xB

y, N

xN

y, L

x,

and Magnetic circular dichroism (MCD) spectroscopicLy.

measurements on free-base porphin55 gave negative intensitiesfor the two Ðrst peaks and positive intensities for the two sub-sequent ones. This shows that the two Ðrst excited states areprobably of the same symmetry i.e. they are transitions.Q

xFor the same reason the third and the fourth peaks are Qytransitions. This is supported by Kim and Bohandy56 who

spectroscopically found that the Ðrst Q band is the andQx0

third band must be a transition though its identiÐcationQywas less straightforward. The two weak transitions andB3

xB3

y,

can not be seen experimentally due to the presence of thestrong B-band in the vicinity or alternatively they are an arti-fact of the functional used.

Even though the intensity of the and bands areB3x

B3yseveral order of magnitude smaller than the experimental

oscillator strength, Gisbergen et al.12 assigned them as the Bband. This assignment is probably incorrect. This assignmentalso a†ects their interpretation of the rest of the excitationspectrum.

The energy di†erences between and as well asBx

Bybetween and are too small for a deÐnitive mutual ord-N

xN

yering of them. Experimentally, is below The n] p*Bx

By.55

transitions are weak and difficult to detect due to the vicinityof the strong B band. The n ] p* transitions have not yetbeen found experimentally. In Table 3, the obtained excitationenergies and the corresponding oscillator strengths are com-pared to experiment. In Fig. 1, the experimental and com-puted spectra are compared. Note that the calculatedintensities of the and transitions have been multipliedQ

xQ

yby 100. The experimental intensities for the Q band have beenmultiplied by 5. Since the experimental oscillator strengths arereported as the sum of both transitions in the andQ

xQ

ybands, the experimental intensities of the Q bands are splitroughly in the same ratio as they appear in the experimentalspectrum. As seen in Fig. 1, the high-energy part of the calcu-lated spectrum is somewhat shifted to higher energies as com-pared to the experiment. The transition energies to thesymmetry forbidden singlet transitions below 5 eV are givenin Table 4.

In Table 5, the most important excitation weights are given.The weights are obtained as the square of the excitation coeffi-cients. Since the deexcitation coefficients are much smaller andcontribute at most 2.5%, their weight contributions have notbeen taken into account. All conÐguration weights smallerthan 5% have been neglected. For the Ðrst excited andB3ustates, the TD-DFT, DFT/MRCI and CASSCF excitationB2u

Table 4 The lowest forbidden singlet transitions (eV) for free-base porphin and the corresponding wavelengths (nm) calculated at the TD-DFTlevel ; the SVP] di† basis sets have been employed

State Energy Wavelength State Energy Wavelength

1 1B1g 2.90 428 4 1B1g 4.08 3041 1Ag 2.94 422 4 1Ag 4.26 2911 1Au 3.25 382 2 1B3g 4.29 2891 1B2g 3.25 382 2 1Au 4.54 2731 1B3g 3.27 379 2 1B2g 4.54 2732 1B1g 3.39 366 5 1Ag 4.57 2723 1B1g 3.61 343 3 1B3g 4.67 2652 1Ag 3.63 341 3 1B2g 4.94 2513 1Ag 3.68 337

Table 5 The excitation energies (E in eV) and the excitation weights (W in %) of the most important excitations for the excited states offree-base porphin as compared to CASSCF calculations

State E ConÐguration W (TD-DFT) W (CASSCF) W (DFT/MRCI)

1 1B3u 2.13 5b1u] 4b2g 61.9 37 482au] 4b3g 37.3 34 33

2 1B3u 2.94 4b1u] 4b2g 68.3 27 302au] 4b3g 22.0 28 32

5b1u] 4b2g 9.1 13 123 1B3u 3.42 2au] 4b3g 36.7 9 14

4b1u] 4b2g 27.5 29 355b1u] 4b2g 27.9 22 18

4 1B3u 3.72 3b1u] 4b2g 92.4 55 615 1B3u 4.28 3b3g] 3au 90.1 È 371 1B2u 2.25 5b1u] 4b3g 58.2 41 49

2au] 4b2g 41.1 39 362 1B2u 2.96 4b1u] 4b3g 83.5 17 26

2au] 4b2g 8.0 30 285b1u] 4b3g 7.8 21 18

3 1B2u 3.36 2au] 4b2g 48.3 7 175b1u] 4b3g 31.8 È 124b1u] 4b3g 15.1 32 403b1u] 4b3g 6.5 15 È

4 1B2u 3.72 3b1u] 4b3g 90.5 6a 655 1B2u 4.94 3b2g] 3au 87.8 È È

1au] 4b2g 5.4 È È1 1B1u 3.28 17b2u] 4b3g 99.9 È È2 1B1u 4.80 5b1u] 21ag 99.9 È È

a The largest CASSCF contribution of 51% comes from the conÐguration.4b1u] 4b3g

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Fig. 1 A schematic comparison of the calculated and measuredUV/VIS spectra of free-base porphin. The intensities of the calculated

and transitions have been multiplied by 100, while a scalingQx

Qyfactor of 5 has been used for the experimental intensities of the Q

band.

weights are relatively similar. For the higher excited states theTD-DFT and CASSCF weights di†er.16 For example, at theTD-DFT level the is completely dominated (90.5%) by4 1B2uone single excitation while at CASSCF level its(3b1u] 4b3g),contribution is only 6% and the largest contribution comesfrom the excitation.4b1u] 4b3gAs seen from the excitation weights in Table 5, the second

and states do not appear in the CASSCF calcu-1B3u 1B2ulation. The excitation weights for the third states of these sym-

metries agree well with the excitation weights of the secondstates obtained at CASSCF level. This supports the presentinterpretation that the and states with large3 1B2u 3 1B3uoscillator strengths are assigned to the B band. The DFT/MRCI excitation weights obtained by Parusel and Grimme15agree well with excitation weights obtained at CASSCFlevel.16 The present and states are not obtained2 1B2u 2 1B3uat the DFT/MRCI level either. The largest di†erence betweenthe excitation weights obtained at the DFT/MRCI andCASSCF levels appears for the state.4 1B2uThe Ðrst states of and symmetry consist mainly ofB3u B2uexcitations from HOMO and HOMO[ 1 to LUMO andLUMO] 1 which agrees with the results of the CASSCFcalculations16 and with GoutermanÏs four orbital model.5h9However, the second state of these symmetries are dominatedby excitations from HOMO [ 3 to LUMO and LUMO] 1.The third and states which are assigned to the B bandB3u B2uinvolve excitations from the three highest occupied orbitals tothe two lowest virtual ones. The two highest andB3u B3ustates considered in this work involve excitations from lower-lying occupied orbitals to LUMO and LUMO] 1. Then ] p* transitions are each completely dominated by onesingle excitation.

There are four occupied orbitals with orbital energiesbetween 5.16 and 6.12 eV. This means that more than fourlow-lying excited states can be expected to appear. TheLUMO part of the orbital-energy spectrum is more remi-niscent of the spectra obtained at other levels of theory. Twoof the empty orbitals are slightly lower than the otherLUMOs which is also reÑected in the excitation character ofthe excited states. The three lowest states of and sym-B2u B3umetries consist mainly of excitations from HOMO,HOMO[ 1, and HOMO [ 3 to LUMO and LUMO] 1.The excitation weights show that the two lowest states can beinterpreted using the four-orbital model while the four-orbitalmodel does not seem to be the correct model for the interpre-tation of the rest of the UV spectrum of the free-base porphy-rin.

B. Triplet states

In Table 6 the lowest triplet transition energies are given. Theparity allowed transitions from the lowest triplet state arefound in the rightmost columns. The lowest triplet state is cal-culated to be 1.58 eV above the ground state which is in excel-lent agreement with the measured values of 1.56È1.58 eV.58h60

Table 6 Triplet excitation energies (eV) of free-base porphin below 5 eV and the corresponding wavelengths (nm) obtained at the TD-DFT levelare compared to experimental values (eV). The SVP] di† basis set was employed. The parity allowed tripletÈtriplet dipole transitions from the

state are given in the rightmost columns1 3B2uState Energy Wavelength State Energy Wavelength Exp.a

1 3B2u 1.58 783 1 3B1g 2.56 484 1.581 3B3u 1.74 713 1 3Ag 2.63 4712 3B2u 1.97 631 2 3B1g 2.97 4172 3B3u 1.99 622 1 3B3g 3.14 3953 3B3u 2.63 471 1 3B2g 3.19 3883 3B2u 2.78 446 3 3B1g 3.25 382 3.231 3B1u 3.14 395 2 3Ag 3.28 3781 3Au 3.19 388 3 3Ag 3.34 3714 3B2u 3.29 377 4 3Ag 3.66 3394 3B3u 3.44 361 4 3B1g 3.70 3355 3B3u 3.88 319 5 3B1g 3.82 3242 3Au 4.50 275 5 3Ag 3.92 3175 3B2u 4.56 272 6 3B1g 4.05 3066 3B3u 4.69 264 2 3B2g 4.50 275 4.546 3B2u 4.72 263 7 3B1g 4.61 2692 3B1u 4.79 259 6 3Ag 4.77 260 4.81

8 3B1g 4.90 2533 3B2g 4.94 251

a Refs. 58 and 61.

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Table 7 The Ðrst IPs (eV) obtained using KoopmansÏ theorem and the negative orbital energies, [e, of the LUMOs (eV) of each irreduciblerepresentation for free-base porphin calculated at DFT level using the SVP] di† basis sets

MO IP Label MO [e Label

20ag 6.46 HOMO[4 21ag 0.34 LUMO]314b1g 9.02 HOMO[12 15b1g [0.14 LUMO]93b2g 6.71 HOMO[6 4b2g 3.23 LUMO]13b3g 6.00 HOMO[2 4b3g 3.24 LUMO2au 5.38 HOMO[1 3au 1.93 LUMO]25b1u 5.16 HOMO 6b1u 0.24 LUMO]417b2u 6.46 HOMO[5 18b2u 0.13 LUMO]617b3u 8.99 HOMO[11 18b3u 0.20 LUMO]5

Other high-lying occupied orbitals Other low-lying unoccupied orbitals

4b1u 6.12 HOMO[3 5b2g [0.06 LUMO]73b1u 6.77 HOMO[7 5b3g [0.06 LUMO]82b2g 7.12 HOMO[82b3g 7.32 HOMO[91au 8.22 HOMO[10

The tripletÈtriplet excitation spectrum shows absorptions at752, 419, 384, and 310 nm60,61 which corresponds to tran-sition energies of 3.23, 4.54, 4.81, and 5.58 eV, respectively,relative to the singlet ground state. Since the present methoddoes not provide oscillator strengths for tripletÈtriplet tran-sitions the triplet spectrum is more difficult to assign than thesinglet spectrum. Since the lowest triplet state belongs to B2u ,all transitions from to ungerade states are, for parity1 3B2ureasons, forbidden. The Ðrst allowed tripletÈtriplet transitionsare found at 2.56 eV and 2.63 eV The next(3B1g) (3Ag).allowed tripletÈtriplet transitions from the state are1 3B2uobtained at 2.97 eV and 3.14 eV as compared to(3B1g) (3B3g)the experimental result of 3.23 eV.60,61 Between 3.19 and 3.34eV four symmetry allowed triplet states are obtained and twoat 3.66 and 3.70 eV, respectively. It is not possible to decidewhich of the allowed transitions are experimentally seen sinceeven though the transitions are formally allowed their oscil-lator strengths can di†er by several orders of magnitude.

C. KoopmansÏ theorem

Most density functionals are not able to describe correctly theasymptotic behavior at long distances which is reÑected onthe orbital energy of the highest occupied molecular orbital

The orbital energy of the highest occupied molecular(eHOMO).orbital, calculated at density functional level is ineHOMO ,general too small which means that KoopmansÏ theorem isnot fulÐlled. For the exact density functional, KoopmansÏtheorem must be fulÐlled, at least for the highest occupiedmolecular orbital (HOMO),62h64 but also for the highestoccupied molecular orbital of each irreducible representa-tion.63 This deÐciency of commonly used density functionalsimplies that it is not possible to obtain reliable values for exci-tation energies of highly excited states, the excitation energiesof which are larger than For free-base porphin, theeHOMO .obtained is [5.16 eV. The highest excited state con-eHOMOsidered in this work is the band whose energy is 4.94 eVL

yabove the ground state. The Ðrst ionization potentials of allthe irreducible representations and the negative orbital energyof the lowest unoccupied molecular orbitals (LUMO) of eachirreducible representation are given in Table 7.

Tozer and Handy65 as well as van Leeuwen et al.66 havepresented new functionals that fulÐll KoopmansÏ theoremapproximately and possibly can provide reliable excitationenergies also for states close to the ionization threshold. Whenthe excitation energy of an excited state is larger than eHOMOany value for the excitation energy can be obtained by choos-ing the basis set. For example by adding di†use basis func-tions in order to obtain a good representation of the Rydbergorbitals as described in ref. 17, the excitation spectrum for

pyrrole, obtained using the BeckeÈPerdew33h35 functional,will contain 18 nearly degenerated states of symmetry1A2between 5 and 6 eV.67 Pyrrole has one or at most two 1A2states in this energy region.17

V. ConclusionsThe low-lying bands in the absorption spectrum of free-baseporphin have been studied at TD-DFT level using three di†er-ent basis sets. The obtained excitation energies are almostindependent of the basis-set size. The obtained results arecompared with available ab initio, DFT, and experimentaldata for singlet and triplet states. Judged from the excitationweights and the oscillator strengths, the B-band must beassigned to the and the transitions, while the3 1B2u 3 1B3uand the states seems to be superÑuous as com-2 1B2u 2 1B3upared to CASSCF (CASPT2) and DFT/MRCI calculations.The lowest symmetry allowed tripletÈtriplet transition,obtained at 1.58 eV, is in excellent agreement with the experi-mental result of 1.56È1.58 eV.

AcknowledgementWe are indebted to Prof. R. Ahlrichs for a recent version ofTURBOMOLE.

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