Time-Dependent Density Functional Theory Investigation of Electric

Guillermo Nieto-Malagón , Julio M. Hernández-Pérez , Rubicelia Vargas , Jorge Garza. International Journal of Quantum Chemistry 2012 112 (21), 3552...
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J. Phys. Chem. B 2005, 109, 13921-13927

13921

Time-Dependent Density Functional Theory Investigation of Electric Field Effects on Absorption Spectra of Meso-Meso-Linked Zinc Porphyrin Arrays: Role of Charge-Transfer States Katsunori Nakai, Riadh Sahnoun, Tsuyoshi Kato, Hirohiko Kono,* and Yuichi Fujimura Department of Chemistry, Graduate School of Science, Tohoku UniVersity, Sendai 980-8578, Japan ReceiVed: February 11, 2005; In Final Form: April 24, 2005

By using time-dependent density functional theory, we calculated the transition energies of a zinc porphyrin monomer and its meso-meso-linked arrays. In line with the prediction of the molecular exciton model, the calculated splitting energy of the Soret band increased as the number of linked porphyrins increased. We then examined how the transition energies of the dimer array were shifted by an applied electric field. For reproduction of an electroabsorption spectrum (EA), i.e., the field-induced change in absorption intensity, a model Hamiltonian constructed from five states is proposed. It is concluded for the dimer that the fieldinduced coupling between the lower-energy Soret band Se and the lower-lying ionic character (charge-transfer) states is responsible for the experimentally observed blue shift of Se as well as the second-derivative profile in the EA spectrum.

1. Introduction Artificial porphyrin supramolecules with high degrees of thermal stability and chemical stability have been used to mimic the photosynthetic system for possible applications such as optical switches, sensors, and solar cells.1-3 Furthermore, through the use of different types of linkers to connect the meso positions of porphyrin molecules, various types of covalently linked porphyrin arrays have been synthesized with the aim of developing molecular conductive wires and optoelectronic devices.4-6 Intriguing electronic structures and spectroscopic properties of such types of porphyrin have opened up an active research field to experimentalists7-12 and theoreticians.13-21 Directly meso-meso-linked porphyrin arrays recently synthesized by Osuka et al. have attracted much attention because the arrays exhibit strong exciton splitting in their Soret bands.7 From spectroscopic measurements, it has been found that the Soret band of an array is split into two bands.8 The split component located in the lower-energy region is further shifted downward with increases in the number of linked porphyrins, while the other split component located in the higher-energy region remains almost unchanged. These spectroscopic features, which are in rather good agreement with the prediction of the molecular exciton model,10,22 have inspired researchers to study energy and charge-transfer processes in excited states of mesomeso-linked porphyrin arrays.8,23 The properties of excited states of porphyrin arrays can be investigated by electric field modulation spectroscopy.24,25 The absorption intensity changes according to the strength of an applied electric field. The field-induced change in absorption intensity as a function of transition energy is called an electroabsorption (EA) spectrum. For an isolated absorption band in a nonoriented sample, the field-induced change in absorption intensity can be expressed in terms of changes in the electric permanent dipole moment and polarizability upon photoexcitation, denoted by ∆µ and ∆R, respectively.24 ∆µ and ∆R can be estimated by fitting a simulated EA spectrum to the observed one. Ohta et al. have found that the EA spectrum around the split lower-energy Soret band of a meso-mesolinked zinc porphyrin array doped in a poly(methyl methacryl-

ate) (PMMA) polymer film has a second-derivative profile of the absorption spectrum.8 In the framework of the isolated absorption band model, the existence of a second-derivative profile means that ∆µ upon excitation to the lower-energy Soret band is nonzero.24 Since the permanent dipole moment of a nondegenerate electronic state is zero for meso-meso-linked porphyrin arrays essentially possessing D2d or D2h symmetry, the origin of nonzero ∆µ should be clarified. Ohta et al. have also found that the lower-energy Soret band is blue-shifted by an electric field, while the higher-energy Soret band is redshifted. In a theoretical study, Kim and Osuka calculated the energies of excited states of the dimer by the semiempirical method INDO/S-SCI.23 They found that eight charge-transfer states, known as ionic character states,26 are located just above the lower-energy Soret band. By calculating the second hyperpolarizability using the sum-over-states approach at the INDO/SSCI level, Matsuzaki et al. have recently shown that the secondderivative profiles of EA spectra of porphyrin arrays can be reproduced by taking into account the coupling of the lowerenergy Soret band with nearby ionic character states.21 These semiempirical treatments have prompted us to investigate the field-induced level shifts of the Soret bands and to elucidate the role of ionic character states at different theoretical levels. In this study, employing time-dependent density functional theory (TDDFT), we calculated the transition energies of a zinc porphyrin monomer and its meso-meso-linked arrays in a zero field. The applicability of TDDFT to porphyrins was assessed by comparing the calculated transition energies with experimental results. We then investigated how the energies of individual excited states of the monomer and its dimer array were shifted by an electric field. The obtained ∆R values were compared with the experimental values. From analysis of the results of TDDFT calculations, we then constructed a five-state effective Hamiltonian for a dimer in terms of dipole transitionallowed states in the Soret band and two ionic character states. The effective Hamiltonian contains field-induced couplings among states and locations of ionic character states as parameters. Systematic comparisons of TDDFT calculations using different exchange-correlation functionals have led to appropri-

10.1021/jp050720y CCC: $30.25 © 2005 American Chemical Society Published on Web 07/01/2005

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Figure 2. Transition energies of Soret bands in a zero field: (a) experimental values for Z1, Z2, and Z3 porphyrin compounds taken from ref 8 and (b) values calculated by using TDDFT(B3LYP/6-31G(d)) for Z1′, Z2′, and Z3′ model compounds. The results of TDDFT calculations capture the experimentally observed feature that the energy of the lower Soret band Se decreases, and the energy of the higher band L hardly shifts from the Soret band of Z1 as the number of linked porphyrin units increases.

Figure 1. Structures of the zinc porphyrin monomer (Z1), mesomeso-linked zinc porphyrin dimer (Z2), and trimer (Z3). Ar denotes 3,5-di-tert-butylphenyl. The three molecule-fixed axes (x,y,z) for Z1 are shown in part a, and those for Z2 and Z3 are shown in part b. In the present theoretical study, the Ar groups are all substituted by H atoms. The corresponding model compounds are designated in the text by Z1′, Z2′, and Z3′.

ate choices of those parameters. Using the constructed Hamiltonian, we discuss in the present paper the origins of blue shift of the lower-energy Soret band and the second-derivative profile of the EA spectrum.

by the dihedral angle formed between two adjacent porphyrin rings in an array,23 we choose a perpendicular arrangement of planar rings. The optimized geometries were used for TDDFT calculations of excited-state energies. For comparison, we used the pure local type of exchange-correlation (xc) functional and the so-called hybrid type of xc functional. Two gradient-corrected xc functionals, BP8629,32 and PBEPBE,33,34 were used for the former, and B3LYP and PBE1PBE33,34 were used for the latter. A hybrid xc functional contains a term that is proportional to the HartreeFock type of exchange energy for Kohn-Sham orbitals. To investigate the effects of an electric field on the absorption spectra of the monomer and dimer model compounds, we calculated the energies of the ground and excited states in the presence of an electric field. The molecular polarizabilities and permanent dipole moments of the ground and excited states were evaluated from the field-induced energy shifts.

2. Computational Details The structures of a zinc porphyrin monomer (Z1) and its meso-meso-linked porphyrin dimer (Z2) and trimer (Z3) arrays are shown in Figure 1. The molecule-fixed coordinate systems (x, y, and z) for the monomer and the arrays are shown in Figure 1. On the basis of the fact that the effects of substituents on the absorption spectra of arrays are negligible,27 the 3,5-di-tertbutylphenyl groups, abbreviated in Figure 1 by Ar, are all substituted by H atoms in the present theoretical study. The model compounds corresponding to Z1, Z2, and Z3 are denoted by Z1′, Z2′, and Z3′, respectively. All of the electronic structure calculations were performed using the Gaussian 98 suite of programs.28 For porphyrins, it has been found that the basis set has only a little effect on the features of the absorption spectra obtained by TDDFT calculations;15 a valence double-ζ with a polarization function added to the heavy atoms is sufficient. We confirmed by comparison of the results obtained from 6-31G(d) and 6-31G+(d) that the changes in transition energies are negligible. This indicates that the diffuse function has no significant effects on the energies of the states. Therefore, the 6-31G(d) basis set was used in all calculations. The geometries in the ground electronic states of the model compounds were optimized at the B3LYP29-31 level of density functional theory (DFT). Since the lower-energy Soret band, which we are mainly interested in, is known to be unaffected

3. Results and Discussion 3.1. Absorption Spectra in a Zero Field. The Soret band of a porphyrin array splits into two bands. The transition energies of the two experimentally observed dominant peaks in the Soret band regions of Z2 and Z3 are plotted in Figure 2a together with the Soret band peak of Z1. As the number of linked porphyrin units increases, the energy of the lower band Se decreases, while the energy of the higher band L hardly shifts from the Soret band of Z1. This tendency is explained in terms of the molecular exciton model in which band Se is attributed to a dipole-allowed exciton band and band L is attributed to excitation localized in one of the porphyrin units.12 The transition energies of the model compounds calculated by TDDFT/B3LYP are also shown in Figure 2b to check the accuracy of the TDDFT methodology for the calculation of excited-state energies of porphyrin arrays. Although all of the calculated transition energies are shifted upward by ∼0.5 eV, Figure 2b features the experimentally observed increase in the splitting energy between L and Se upon polymerization. The calculated splitting energy for Z3′ is larger than that for Z2′ by a factor of x2, as predicted by the molecular exciton model. Band L of Z2′ is 2-fold degenerate due to the perpendicular arrangement of the two porphyrin rings.10,22,35 The transition energies to bands L and Se of Z2′ calculated by various TDDFT functionals are tabulated in Table 1 together

TDDFT of Zinc Porphyrin Arrays

J. Phys. Chem. B, Vol. 109, No. 29, 2005 13923

TABLE 1: Transition Energies of the Soret Bands of the Model Meso-Meso-Linked Porphyrin Dimer, Z2′ method

Soret band L

Soret band Se

splitting energy

B3LYP PBE1PBE

Hybrid xc Functionals 3.51a (352.7)b 3.34 (371.1) 3.57 (346.7) 3.40 (364.7)

0.17 0.17

BP86 PBEPBE

Pure Local xc Functionals 3.35 (369.8) 3.18 (389.8) 3.36 (368.9) 3.18 (389.6)

0.17 0.18

SAC-CIc experimentald

3.46 (358.3) 3.00 (413.6)

0.19 0.26

3.27 (379.1) 2.74 (452.4)

a

Energies are given in units of electronvolts. b Shown in parentheses are the transition energies in nanometers. c From ref 13. d From ref 8.

with the results of the SAC-CI calculation performed by Miyahara et al.13 and experimental results.8 All of the calculated energies fall within a difference of ∼0.2 eV; all of the methods give nearly the same splitting energy of the Soret band with an average deviation of 0.08 eV from the experimental value. Not surprisingly, all of the calculated transition energies are overestimated. Yet, the results presented in Table 1 indicate that the TDDFT functionals employed in this study can capture the characteristic features of the Soret bands of porphyrin arrays in a zero field.13,16,18,36-38 3.2. Effects of an Electric Field on Soret Band Transition Energies. The field-induced shift (Stark shift) of the ground state was calculated by DFT, and those of the Soret excited states were calculated by TDDFT. If a state under consideration is not an ionic character state, the Stark shift is given by the equation ∆E(ε) ) -1/2Rε2, where ε is the strength of an applied electric field and R is the molecular polarizability of the state. 3.2.1. Monomer. We first calculated the Stark shifts of states of Z1′ for a case where an electric field is applied parallel to the porphyrin plane (Figure 1a). The x-axis components of polarizability calculated using the B3LYP functional were 439(4πε0a03) for the ground state and 511(4πε0a03) for the Soret excited state, where ε0 is the permittivity of vacuum and a0 is the Bohr radius. The difference between the two values is 72(4πε0a03). Since Z1′ can be approximately treated as an oblate symmetric top molecule, the difference averaged over spatial random orientation is estimated to be 72(4πε0a03) × 2/3 ) 48(4πε0a03),39 which is close to the experimentally observed change in molecular polarizability upon photoexcitation to the Soret band, 56(4πε0a03).8 This suggests that TDDFT is useful for quantitative reproduction of the Stark shift of the Soret excited state. 3.2.2. Dimer. As expected, our calculations showed that the main distortion of the electronic cloud of Z2′ was induced by the electric field component parallel to the z-axis (the long axis of the dimer) defined in Figure 1b. Therefore, we calculated the state energies of Z2′ in the presence of an electric field applied parallel to the z-axis. The transition energies of the Soret bands L and Se calculated at the TDDFT/B3LYP level are plotted in Figure 3 as a function of the electric field strength ε. We varied ε from 0 to 8 × 10-4Eh/ea0 (4.1 × 108 V/m), where Eh and e are the hartree and elementary charge, respectively. A z-polarized electric field splits the 2-fold degenerate band L into two levels, L1 and L2. This indicates that a field-induced dipole moment extending over two porphyrin units is created even in the case of a perpendicular arrangement of the two units. As the field strength increases, the gap in energy between the two states increases, as shown in Figure 3. It should be mentioned that the polarizability averaged over L1 and L2 calculated by TDDFT/B3LYP, which is 1400(4πε0a03), is significantly larger than twice that of the Soret excited state of

Figure 3. Electric field dependence of the transition energies of L1, L2, and Se in the Soret band for the dimer model compound Z2′. The transition energies were calculated by TDDFT(B3LYP/6-31G(d)). The 2-fold degenerate band L is split into two levels, L1 and L2, when the electric field is applied parallel to the z-axis defined in Figure 1b. As in the experiment, Se is blue-shifted in the presence of a z-polarized electric field, while L (L1 and L2) is red-shifted.

Z1′. The difference is 380(4πε0a03) ≈ 1400(4πε0a03) - 2 × 511(4πε0a03). Similarly, the polarizability of the ground state of Z2′, which is estimated by DFT/B3LYP to be about 1150(4πε0a03), is larger than twice that of the ground state of Z1′ by 270(4πε0a03). If the field-induced interactions between porphyrin units are small, then the polarizability is expected to be proportional to the number of units in an array. Therefore, these excesses in polarizability upon dimerization also indicate field-induced charge migration between porphyrin units. The calculated polarizability of Z2′ is larger in the Soret excited state L than in the ground state by ∼250(4πε0a03), which is as large as the experimentally observed change upon photoexcitation to L, 250(4πε0a03).8 It is noteworthy in Figure 3 that the transition energy of Se increases, while the transition energies of L decrease on average. This tendency is in agreement with the experimental fact that Se is blue-shifted in the presence of an electric field, while L is red-shifted.8 The blue shift of Se and the red shift of L obtained by TDDFT/B3LYP have also been reproduced by using another hybrid type of xc functional, PBE1PBE. In contrast, the transition energy of Se calculated by TDDFT with a pure local xc functional such as BP86 or PBEPBE decreases with increases in the electric field strength. It is therefore expected that in the TDDFT calculation the hybrid and pure local xc functionals provide different effective interactions between the Soret band state Se and other states. In subsection 3.4, we show that the dependence of the transition energy of Se on the electric field strength is governed by the locations of ionic character states explained in the following subsection. 3.3. Ionic Character States of the Dimer. We next examine another type of excited state of Z2′, i.e., ionic character states. Both TDDFT/B3LYP and TDDFT/PBE1PBE gave nearly the same results for the energies and Stark shifts; therefore, only the results of TDDFT/B3LYP are presented in this subsection. The TDDFT/B3LYP energies of states located in the Soret and Q-bands are plotted in Figure 4 as a function of the electric field strength. From the results of TDDFT/B3LYP calculation for Z2′, we could also locate eight states between the Soret band and Q-band. For clarity, the TDDFT/B3LYP energies of only four states chosen from the eight states are plotted in Figure 4. It is clear that the changes in the energies of the four states are linearly proportional to the electric field strength, while the energies of the Soret band and Q-band decrease quadratically

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Figure 4. Electric field dependence of state energies of Z2′ obtained by TDDFT(B3LYP/6-31G(d)). Eight ionic character (or charge-transfer) states are located between the Soret band and the Q-band. The change in the energy of an ionic character state is linearly proportional to the electric field strength, while the energies of the Soret band and Q-band decrease quadratically with respect to the field strength. The eight ionic character states are classified into four pairs of states assigned to symmetry species A1, A2, B1, and B2 in the C2V point group. The TDDFT/B3LYP energies of only A1 and A2 ionic character pairs are plotted in the figure. The solid, long-dashed, dashed, and dotted lines denote the energies of states belonging to A1, A2, B1, and B2 symmetry species, respectively. The energies are offset by the energy of the ground state in a zero electric field.

with respect to the field strength. The energies of the other four states, which are not displayed in Figure 4, change in proportion to the electric field strength in a similar way. Hence, these eight states are regarded as ionic character states. The eight states are classified into four pairs of ionic character states assigned to symmetry species A1, A2, B1, and B2 in the C2V point group; such classification is applicable even to porphyrin arrays in an electric field applied parallel to the z-axis. For a given pair of two ionic character states, the energy of one state decreases and that of the counter state increases as the field strength increases, as shown in Figure 4. The four pairs are not coupled to each other by a z-polarized electric field because the electronic coordinates parallel to the z-axis are assigned to A1 in C2V. In Figure 5, we present two diagrams of Kohn-Sham molecular orbital (MO) energies of Z2′ to show the origin of the eight ionic character states. The eight orbitals from HOMO-3 to LUMO+3 of Z2′ play a major role in the electronic structures of low-lying excited states. The energies of the eight orbitals of Z2′ in a zero field are shown in Figure 5a; those in a nonzero field (ε * 0) are shown in Figure 5b. Each MO can be labeled as a symmetry species of C2V such as a2, b1, or b2. In the case where ε is not equal to 0, as in Figure 5b, each MO is characterized by a MO localized in either the left or the right porphyrin unit. A localized MO is represented approximately by one of Gouterman’s four orbitals of Z1′.40 The four orbitals, i.e., HOMO-1, HOMO, and the 2-fold degenerate LUMO of Z1′, are assigned to a1u, a2u, and (egx,egy) of the D4h point group, respectively. The four orbitals are schematically represented in Figure 6. The label in parentheses in Figure 5b represents the symmetry species of the correspond-

Nakai et al.

Figure 5. Energies of the eight Kohn-Sham MOs from HOMO-3 to LUMO+3 of Z2′ (a) in a zero electric field and (b) in a nonzero field of strength ε ) 4 × 10-4Eh/ea0. Each MO is labeled as a symmetry species of C2V such as a2, b1, or b2. In part b, each MO is characterized by an MO localized in either the left or the right porphyrin unit. A localized MO is represented approximately by one of Gouterman’s four orbitals of Z1′,40 which are HOMO-1, HOMO, and the 2-fold degenerate LUMO of Z1′, assigned to a1u, a2u, and (egx,egy) of the D4h point group, respectively. The label in parentheses in part b represents the symmetry species of the corresponding Gouterman’s orbital. The superscript L (or R) is used to denote that the MO is localized in the left porphyrin unit (or right one). In the presence of a nonzero field, the 16 electron configurations constructed from the eight localized MOs are classified into two categories: eight electron configurations of localized excitation represented by gray dotted lines with an arrow and eight configurations of interunit electron transfer represented by gray solid lines with an arrow.

Figure 6. Schematic representation of Gouterman’s four orbitals of the monomer Z1′. The MOs are sketched as seen from above. The signs of the atomic orbitals represented by open circles are opposite to those represented by filled circles.

ing Gouterman’s orbital. The superscript L (or R) is used to denote that the MO is localized in the left porphyrin unit (or right one). In the presence of a nonzero field, the 16 electron configurations constructed from the eight localized MOs are classified into two categories: eight electron configurations of localized excitation represented by gray dotted lines with an arrow (associated with Soret and Q-bands) and eight configurations of interunit electron transfer represented by gray solid lines with an arrow in Figure 5b. As expected, the eight configurations of interunit electron transfer are classified into four pairs designated by A1, A2, B1, and B2 of the C2V point group. It is found from the calculated transition density that these eight configurations are the main components of the eight ionic character or charge-transfer states under consideration. For instance, two electron configurations of interunit electron transfer, Φ(aR1u feLgy) and Φ(aL1u f eRgy), belong to A1; the main component of an A1 ionic character or charge-transfer state is a linear combination of these two configurations. To confirm the nature of the eight ionic character states, we estimated the dipole moment of each state from the gradient of the state energy with respect to the field strength. In the presence

TDDFT of Zinc Porphyrin Arrays of an electric field, for instance, the dipole moments of the A1 paired states and those of the A2 paired states (Figure 4) are (6.38ea0 (16.2 D) and (9.9 ea0 (24.8 D), respectively. The sign of the dipole moment of a charge-transfer state is determined by the amount of interunit electron-transfer configurations. At large ε > 4 × 10-4 Eh/ea0, one electron-transfer configuration Φ(aR1u f eLgy) is the dominant configuration in the lowest A1 ionic character state. The dipole moment and charge distribution of the lowest state of the A2 symmetry species were also calculated using DFT/ B3LYP. In the presence of an electric field, as expected, the calculated dipole moment of the lowest A2 state is nonzero (∼5.3ea0 (13.5 D)). By natural population analysis, we found that in this state a charge of 0.6e migrates from unit to unit. Since the field-induced energy shift of the second lowest A2 state is opposite to that of the lowest A2 state, the same amount of charge is transferred in the second lowest A2 state from unit to unit inversely in comparison with the charge transfer in the lowest A2 state. Since the other three pairs of ionic character states (A1, B1, and B2) show almost the same field-induced shifts as those in the case of the A2 paired states, we conclude that in these eight ionic character states a fraction of charge is transferred from unit to unit if the z-axis component of the electric field is nonzero. 3.4. Field-Induced Blue Shift of the Lower Soret Band. In the hybrid xc functional case, the Soret band Se of Z2′ is blue-shifted by an electric field, as was shown in subsection 3.2. In this subsection, we reveal the origin of the field-induced blue shift of Se. Blue shift of Se means that the Stark shift |∆E(ε)| of Se is smaller than that of the ground state. This is an exceptional case in which the polarizability of Se is smaller than that of the ground state. The polarizability of Se calculated by TDDFT/B3LYP for Z2′, 566(4πε0a03), is only slightly larger than that of the Soret excited state of Z1′, 511(4πε0a03). The small polarizability of Se in Z2′ suggests that the field-induced couplings between the A1 symmetry state Se and the lowerlying A1 ionic character states are strong enough to raise the transition energy of Se. The TDDFT calculations with pure local xc functionals (BP86 or PBEPBE) also showed that there exist four pairs of ionic character states characterized by the electron-transfer configurations discussed in subsection 3.3. However, the energies of the eight ionic character states are shifted downward by about 0.5 eV compared to the values obtained by using a hybrid xc functional (B3LYP or PBE1PBE). In line with this finding, several research groups have pointed out that the energies of charge-transfer states calculated by TDDFT with a pure local xc functional are significantly underestimated.14,16,41,42 Since the energy gap between the Soret band Se and the A1 ionic character states is smaller in the hybrid xc functional case, the mutual mixing induced by an electric field is relatively strong. As a consequence, the transition energy of Se is blue-shifted in the hybrid xc functional case and red-shifted in the pure local xc functional case. In the pure local xc functional case, ionic character states other than the lowest eight ionic character states that we have discussed appear near Se. However, the main electron configurations of these higher ionic character states are different from those of Se. Consequently, the higher ionic character states are only weakly coupled to Se. 3.5. Reproduction of the EA Spectrum of the Dimer by a Five-State Model. We now present a five-state model for Z2′ that reproduces the features of the experimentally observed electroabsorption (EA) spectra of Z2 as well as the field-induced

J. Phys. Chem. B, Vol. 109, No. 29, 2005 13925 blue shift of the Soret band Se. To this end, it is necessary to investigate how the states are coupled to each other by an electric field. In a zero field, each electronic state can be assigned to a symmetry species of the D2d point group. In this case, the symmetry species of the Soret excited state Se is B2; those of the upper and lower A1 ionic character states in Figure 4 are B2 and A1, respectively. We denote these upper and lower ionic character states in a zero field by u and l, respectively. The main ionic characters of u and l, Ψu,l, adapted to the D2d point group, are represented by

Ψl,u )

1 [Φ(aR1u f eLgy) ( Φ(aL1u f eRgy)] x2

(1)

D2d symmetry-adapted forms of the eight ionic character states have also been presented by Matsuzaki et al.21 Since the electric field is of B2 symmetry species in D2d, Se is most strongly coupled to l, while it is not directly coupled to u. It should be noted that Se is not coupled to the other three ionic character pairs. Following the above coupling scheme, we constructed a fivestate effective Hamiltonian H of states L1, L2, Se, u, and l as

(

EL1(ε) 0 H(ε) ) 0 0 0

0 EL2(ε) 0 0 0

0 0 ESe - 1/2R0ε2 0 εµSe,l

0 0 0 Eu - ∆E εµu,l

0 0 εµSe,l εµSe,l El - ∆E

)

(2)

where EL1(ε) and EL2(ε), which are functions of the field strength ε, are the energies of L1 and L2, respectively; ESe is the energy of Se in a zero field; Eu and El are the energies of the states u and l, respectively; µSe,l is the field-induced coupling between Se and l; µu,l is the field-induced coupling between u and l; ∆E is an adjustable parameter that shifts the energies of u and l; R0 is polarizability of Se. The TDDFT/B3LYP values were used for Eu and El, 2.78 and 2.72 eV, respectively. The field-induced coupling µu,l was determined so that the value of µu,l reproduces the field-induced changes in the energies of the two A1 ionic character states in Figure 4. Fitting was performed in a region of field strength ε > 4 × 10-4 Eh/ea0 in Figure 4, where the energies of the two A1 ionic character states change linearly with ε. The optimized value of µu,l is 6.38ea0 (16.2 D). Among all of the TDDFT calculations, with a pure local or hybrid xc functional, the largest energy gap between Se and l was obtained by using pure local xc functionals. In the case of pure local xc functionals, the coupling between Se and l is insignificant. Since the molecular polarizability R0 of Se in eq 2 can be regarded as the polarizability of a “noninteracting” state Se, we chose the value of R0 ) 1257(4πε0a03) obtained by using a pure local xc functional BP86. We finally determined µSe,l so that the field-induced shift of Se in the case of TDDFT/ B3LYP is reproduced. The optimized value of µSe,l is 1.59ea0 (4.1 D). As in the results obtained by the CI-singles method,43 we found that the oscillator strength of the transition to Se calculated by using TDDFT/B3LYP is overestimated. Therefore, the calculated oscillator strength of Se is modified in such a way that the relative peak heights of the experimentally observed bands L and Se are reproduced for the zero-field case (while the oscillator strengths of u and l are negligible). Using the

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modified oscillator strength of Se, we estimated the oscillator strengths of the eigenstates obtained by diagonalization of the matrix H(ε). To calculate EA spectra, we then inserted the energies hνj for the transitions to the jth excited state obtained by diagonalization of H(ε), offset by the ground-state energy Eg(ε), and the oscillator strengths Fj into the absorption profile A(ν) as a function of absorption frequency ν

A(ν) )

∑j Fjg(ν - νj)

(3)

where g(ν) is a line-shape function. We used a Gaussian function for g as

g(ν - νj) )

1

x2πσ

[

]

(ν - νj)2

exp -

2σ2

(4)

where the bandwidth σ was estimated from the observed absorption spectra to be 450 cm-1 (0.06 eV).8 Since the precise locations of u and l are still difficult to determine by modern quantum chemical calculation or TDDFT calculation, the value of ∆E was varied so as to reproduce the features of the observed EA spectra. In the measurement of EA spectra, the porphyrin monomer and arrays were doped in a PMMA polymer film.8 Since the dielectric constant of PMMA is as small as 3.6, we assume that the electrostatic interaction of a solute molecule with PMMA is negligible and that the solvation effects on the locations of excited states are also negligible. This assumption is justified by the fact that the absorption spectrum in a PMMA polymer film is nearly identical to that in a nonpolar solvent such as a benzene solution (of which the dielectric constant is 2.3).44 We present the results of two typical cases of ∆E, ∆E ) 0 and ∆E ) 0.0216Eh (0.59 eV), where the electric field strength ε is fixed at ε ) 2.0 × 10-4Eh/ea0 (1.0 × 108 V/m). In the case of ∆E ) 0, where the energies of the states obtained by TDDFT/ B3LYP are used, the band shape of the EA spectrum around the peak of band Se is a first-derivative form of the absorption spectrum, as shown in Figure 7. This is due to the relatively weak interaction between Se and l states. However, if ∆E is increased to 0.0216Eh, then the band shape of the EA spectrum around the peak of Se is not a first-derivative form but nearly a second-derivative form of the absorption spectrum as was experimentally observed. This approach enables us to estimate the amount of contamination of u and l in the lower-energy Soret band. Although the contamination in the case of ∆E ) 0.0216Eh is as small as 5%, it significantly affects the profile of the EA, as can be clearly seen in Figure 7. The choice of ∆E ) 0.0216Eh means that the energy of u is lower than that of ESe by only ∼0.05 eV and the u and l states overlap with the band profile of Se. The ascent of the energies of u and l is in accordance with the conclusions reached by other groups regarding the locations of long-range chargetransfer states or ionic character states.14,45,46 It has been shown that the current TDDFT cannot capture the effect of the longrange electrostatic interaction between separated charges unless a nonlocal exchange functional is introduced;14,45,46 that is, the values of the transition energies of charge-transfer states are greatly underestimated by TDDFT using any local xc functional and are improved by TDDFT/(a hybrid xc functional) due to its nonlocal part in the exchange functional. Recently, Tsuneda et al. have proposed a new scheme to include the effects of long-range electrostatic interaction in the exchange functional of DFT45 or TDDFT.46 The results obtained by the long-range

Figure 7. Calculated electroabsorption (EA) spectra of Z2′ at an electric field strength of 2.0 × 10-4Eh/ea0 (1.0 × 108 V/m). The thin broken line stands for the calculated absorption spectrum A (in arbitrary units) as a function of transition energy. The EA spectrum ∆A is the fieldinduced change in absorption intensity. The bold solid line denotes the EA spectrum for ∆E ) 0.0216Eh (0.59 eV), and the bold dotted line denotes the EA spectrum for ∆E ) 0.0Eh, where ∆E is an adjustable parameter that shifts the energies of the upper and lower A1 ionic character states (u and l) in a zero field. In the case of ∆E ) 0, where the energies of the states obtained by TDDFT/B3LYP are used, the band shape of the EA spectrum around the peak of band Se is a first-derivative form of the absorption spectrum. However, if ∆E is chosen to be 0.0216Eh so that the energy of u is lower than the energy of Se by only ∼0.05 eV, then the band shape of the EA spectrum around the peak of Se is not a first-derivative form but nearly a secondderivative form of the absorption spectrum as was experimentally observed. The vertical thin line plotted at the peak of the Se band is inserted as a visual guide.

correction scheme for relatively small systems such as C2H4C2F4 have shown that the transition energies of charge-transfer states should be larger than the values obtained by TDDFT/ B3LYP. In both cases of ∆E, the transition energy of Se is blue-shifted, as observed experimentally. However, the transition energy of Se is red-shifted for the case of pure local xc functionals, as mentioned in subsection 3.2. In the case where the energies of the states obtained from TDDFT/(a pure local xc functional BP86) calculation (∆E ) - 0.0216Eh) are used, the five-state model predicts the red shift of the transition energy of Se as in the results of the full TDDFT/BP86 calculation. For a wide range of values of ∆E, the results of the five-state model are therefore consistent with the field-induced shifts of the Se transition energy obtained in full TDDFT calculations. The five-state model can be applied to the case where the energies of u and l are higher than that of Se. In this case, the transition energy of Se calculated from the five-state model is found to be redshifted. The results of the present simulation hence indicate that the following two conditions need to be fulfilled for the locations of the lowest two A1 ionic character states. First, the A1 ionic character states u and l should be in the vicinity of Se so that the EA spectrum near the peak of Se takes a second-derivative form of the absorption spectrum. Second, unlike the results obtained by INDO/S-SCI,21,23 the positions of the A1 ionic

TDDFT of Zinc Porphyrin Arrays character states should be below Se to reproduce the fieldinduced blue shift of Se. 4. Conclusions By using DFT and TDDFT, we investigated the excited states of a zinc porphyrin monomer Z1 and its meso-meso-linked porphyrin dimer Z2 and trimer Z3. In the dimer and trimer compounds Z2′ and Z3′ modeled after Z2 and Z3, the Soret band is split into the lower-energy band Se and the higher-energy band L. The splitting energy between L and Se in Z3′ is larger than that in Z2′ by a factor of x2, which is in accordance with the prediction of the molecular exciton model. We next calculated the energies of states in the presence of an electric field. The long-axis component (z-component) of the polarizability of Z2′ calculated by TDDFT/B3LYP is larger in the Soret excited state L than that in the ground state by ∼250(4πε0a03), which is consistent with the experimentally observed change upon photoexcitation to L. Overall, the Stark shifts of the states characterized mainly by local excitation in one porphyrin unit, such as the state L, are satisfactory reproduced by TDDFT/(a hybrid xc functional B3LYP or PBE1PBE). In the case of Z2′, from the TDDFT/(hybrid xc functionals) calculation, four pairs of ionic character states assigned to symmetry species A1, A2, B1, and B2 in C2V were located between the Soret and Q-bands. The two states in a given pair are coupled to each other by a z-polarized electric field. From a comparison with the results obtained by TDDFT/(a pure local xc functional BP86 or PBEPBE), we attributed the experimentally observed blue shift of the transition energy of the A1 symmetry state Se to its field-induced coupling with the lowerlying A1 paired ionic character states. Using the five-state effective Hamiltonian constructed for the dimer Z2′ case, we demonstrated that the experimentally observed second-derivative form of the electroabsorption (EA) spectrum of the dimer array is reproduced only if the A1 ionic character states are in the vicinity of Se. Although the degree of field-induced mixing of the coupled states is estimated to be as small as 5%, the EA spectrum takes a second-derivative form of the lower-energy Soret band profile. The positions of the A1 ionic character states are also required to be below Se to reproduce the field-induced blue shift of Se. Acknowledgment. We gratefully acknowledge helpful discussions with Professor T. Tsuneda about his and his collaborator’s treatment of the long-range interaction in DFT. Thanks are also due to Professor N. Ohta for his valuable discussion. This work was supported in part by a grant-in-aid for scientific research (no. 16350001) and a grant-in-aid for scientific research on priority areas, “Control of Molecules in Intense Laser Fields” (area no. 419) from the Ministry of Education, Culture, Sports, Science and Technology, Japan. R.S. acknowledges the receipt of a JSPS grant (P02353). References and Notes (1) Wagner, R. W.; Lindsey, J. S.; Seth, J.; Palaniappan, V.; Bocian, D. F. J. Am. Chem. Soc. 1996, 118, 3996. (2) Wagner, R. W.; Lindsey, J. S. J. Am. Chem. Soc. 1994, 116, 9759. (3) Martin, R. E.; Diederich, F. Angew. Chem., Int. Ed. 1999, 38, 1350. (4) Seth, J.; Palaniappan, V.; Johnson, T. E.; Prathapan, S.; Lindsey, J. S.; Bocian, D. F. J. Am. Chem. Soc. 1994, 116, 10578. (5) Crossley, M. J.; Burn, P. L. J. Chem. Soc., Chem. Commun. 1991, 1569. (6) Anderson, H. L. Chem. Commun. 1999, 2323. (7) Osuka, A.; Shimidzu, H. Angew. Chem., Int. Ed. Engl. 1997, 36, 135.

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