Linear and Nonlinear Optical Properties of Expanded Porphyrins: A

Article pubs.acs.org/JPCA

Linear and Nonlinear Optical Properties of Expanded Porphyrins: A DMRG Study Simil Thomas, Y. A. Pati, and S. Ramasesha* Solid State and Structural Chemistry Unit, Indian Institute of Science, Bangalore 560 012, India S Supporting Information *

ABSTRACT: We study absorption spectra and two photon absorption coefficient of expanded porphyrins (EPs) by the density matrix renormalization group (DMRG) technique. We employ the Pariser−Parr−Pople (PPP) Hamiltonian which includes longrange electron−electron interactions. We find that, in the 4n+2 EPs, there are two prominent low-lying one-photon excitations, while in 4n EPs, there is only one such excitation. We also find that 4n+2 EPs have large two-photon absorption cross sections compared to 4n EPs. The charge density rearrangement in the one-photon excited state is mostly at the pyrrole nitrogen site and at the meso carbon sites. In the two-photon states, the charge density rearrangement occurs mostly at the aza-ring sites. In the one-photon state, the C−C bond length in aza rings shows a tendency to become uniform. In the twophoton state, the bond distortions are on C−N bonds of the pyrrole ring and the adjoining C−C bonds which connect the pyrrole ring to the aza or meso carbon sites.

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show structural diversity.10−12 Effect of N confusion (inverted porphyrins) on the electronic structure has been studied by Lim et al.13 Larger EPs and heteroatom substituted porphyrins, which replace pyrrole by thiophenes or selenophene, have been synthesized and their NLO properties have also been studied.14 It is found that the two-photon absorption cross-section, σ, increases with conjugation length and also when thiophene is replaced by selenophene. Very large two-photon absorption (TPA) coefficients were observed for the heterocyclic EPs.5,6,15,16 Chandrashekar et al. have noticed that the octaporphyrins with S and Se as hetero atoms show very large TPA coefficient. There have been efforts to correlate the TPA efficiencies of large 4n and (4n+2) π-electron porphyrins.15,16 EPs are also studied to understand the difference between 4n and (4n + 2) π-electron systems and their aromaticity or antiaromaticity. Cho et al. found that the expanded porphyrins with (4n + 2) π-electrons show two distinct spectral lines, namely, Q and B bands, while those with 4n π-electrons show a single broad band.17 Ha et al. studied the photophysical properties of thiophene analogues of porphyrins using timeresolved spectroscopic methods. Expanded porphyrins are also used as photosensitizers in photodynamic therapy. These photosensitizers have specific binding capacity (to tumor cells) and can absorb light in the long wavelength region, as light of longer wavelength can penetrate deeper into the tissue. These photosensitizers after excitation undergo intersystem crossing to triplet state. From the triplet state they relax to ground state, through phosphorescence or spin exchange, generating singlet

INTRODUCTION Optical properties of porphyrin and metalloporphyrins have been a subject of active research because of the biological importance of metalloporphyrins which are present in hemoglobin and chlorophyll. Also, large π-conjugation length in porphyrin and metalloporphyrin gives rise to large nonlinear optics responses like third harmonic generation (THG),1 and two-photon absorption (TPA) cross sections.2 In recent times porphyrins and the related phthalocyanine systems have attracted attention in the context of molecular electronic conduction. Properties of these systems can be easily tuned by functionalization of the substituents on their periphery.3 In this context another class of these systems which are gaining prominence are the expanded porphyrins (EPs). The EPs have more than four pyrroles or meso-links, or both, and have been known for several years.4 They have longer π-conjugation length compared to porphyrin. Optical absorption spectra of EPs covers the entire visible range depending on the number of meso and/or number of pyrroles in the system. EPs can adopt three different geometries: regular porphyrin, where all pyrroles point inside; inverted porphyrin where one or more pyrroles point outside and the figure-8, twisted geometry which is adopted mainly by larger EPs.5,6 Generally these molecules adopt twisted geometry and hence become Hückel nonaromatic systems. Sessler et al. synthesized rubyrin, an aromatic hexaporphyrin [1.1.0.1.1.0], in 1991.7 (The nomenclature in square brackets denotes the number of carbon atoms (known as meso carbons) bridging the pyrrole units). They also synthesized nonaromatic porphyrins, amethyrin (hexaporphyrin [1.0.0.1.0.0.]) and rosarin (hexaporphyrin [1.0.1.0.1.0]).8 Stable hexaporphyrins were obtained by substituting the expanded porphyrins at meso-position by pentafluoro benzene.9 Meso-aryl substituted porphyrins on core-modification © 2013 American Chemical Society

Received: May 28, 2013 Revised: July 30, 2013 Published: July 30, 2013 7804

dx.doi.org/10.1021/jp4052352 | J. Phys. Chem. A 2013, 117, 7804−7809

The Journal of Physical Chemistry A

Article

oxygen which is highly reactive.18 Sessler et al. have developed a number of photosensitizers, which are derivatives of expanded porphyrins, known as texaphyrins.19,20 Besides their linear and nonlinear optical properties, EPs are studied for their application as ion receptors. Because of their large cavity size, they are used as anion receptors.21 Molecules can be diprotonated, and in this form, they can bind to the anions through electrostatic interaction.22−24 We report here a systematic study of linear and nonlinear optical properties of regular and inverted expanded porphyrins, by varying the number of meso bonds as well as number of pyrrole rings. This variation leads to systems having 4n or (4n + 2) π-electrons. We have employed density matrix renormalization group (DMRG) method in all our studies.25−27 We will give a brief introduction to the method in the next section, followed by results and discussion.

having one orbital which participates in conjugation. From an exact solution of the small system, we obtain the density matrix of one-half of the system by treating the other half as the environment. A small number “m” of the eigenvectors of the density matrix (DMEVs) with large densities are chosen to span the Fock space of the system. The same is repeated by interchanging the environment and the system block. All the relevant operator matrices such as site creation and annihilation operator matrices and the Hamiltonian matrix of the half-block are renormalized using the matrix with retained DMEVs as its columns. The system size is augmented by adding two additional sites between the two half-blocks and the Hamiltonian of the augmented system is constructed on the basis of the direct product of the DMEVs and the Fock states of the new sites. The dimensinality of the space is 16m2 as each new site spans four Fock states. From the desired eigen state of the augmented system, the 4m × 4m density matrix of the halfblock is constructed and diagonalized. Again, m DMEV’s of the new density matrix are retained to span the augmented half block, based on their densities. The renormalization procedure is once again effected and the whole procedure is repeated to obtain a twice augmented system. The accuracy of the DMRG method can be enhanced by resorting to a finite system DMRG algorithm. In this procedure, after reaching a final system size, the system block is progressively increased in size while simultaneously shrinking the environment block. This allows the construction of intermediate density matrices from the desired final size system rather than from a system of intermediate size. The DMRG method is predominantly a method for quasi-one-dimensional systems. It works well for the PPP model, even though it has long-range interactions, because in the PPP model, the long-range interaction terms are diagonal in real space and do not lead to hopping of electrons.32 In Figure 1 we have shown a few steps, for one of the molecules, how the expanded porphyrin molecule can be built up, starting from a 4 site system. We have employed finite DMRG method by keeping 500 density matrix eigenvectors, in all our calculations.25,26

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METHODOLOGY We have taken into consideration the σ−π separability and assumed that a single pz orbital per site is taking part in conjugation, in our Hamiltonian. We use Pariser−Parr−Pople (PPP) Model28 where the Hamiltonian is given by H=

∑ tij(ai†σ ajσ + a†jσ aiσ ) + ∑ εini i,j σ

+

1 2

i

∑ Uni i(ni − 1) + ∑ Vij(ni − zi)(nj − zj) i

i,j

(1)

In the Hamiltonian, tijs are the resonance or hopping/transfer integrals between bonded carbon sites i and j, εi is the orbital energy of the pz orbital on the ith carbon atom, and Uis are the Hubbard repulsion parameters at site i. a†iσ (aiσ) creates (annihilates) an electron of spin σ at the ith carbon atom, n̂i is the corresponding number operator, and zi is the local chemical potential at site i. Vijs are intersite interactions between charges at sites i and j, and given by the Ohno29 interpolation formula 2. Vij =

14.397 ⎛ 28.794 ⎞2 2 ⎜ ⎟ + r ij ⎝ Ui + Uj ⎠

(2)

which interpolates between Ui for rij = 0 and e2/rij for rij → ∞. In our calculations we set tij to −2.4 eV for bonded C−C pair and −2.5 eV for the C−N bond. This accounts for the slightly more diffused nitrogen 2p orbitals compared to the carbon 2p orbitals. Site energy ε is taken as zero for C atoms. There are two different types of N-atom: the pyrrole N atom which contributes the lone pair to conjugation and the aza N which contributes a single π-electron. The site energy of the pyrrole nitrogen is fixed at −12.0 eV, and for aza nitrogen, it is fixed at −2.0 eV. Hubbard interaction U is taken to be 11.26 eV for carbon and 15.0 eV for nitrogen atoms irrespective of whether they contribute one or two electrons to conjugation. The local chemical potential, zi is 1 for C and N in the aza ring, while it is 2 for N in the pyrrole ring. We have optimized the geometry using Gaussian 09 suite of programs30 using 6-31g* basis set within density functional theory (DFT) with B3LYP exchangecorrelation functional.31 The optimized structure is used for the calculation of intersite electron−electron interaction potential. DMRG method25,26 has been used to solve the long-range interacting Pariser−Parr−Pople (PPP) model. In this method, we typically start with a system of four atoms with each atom

Figure 1. Schematic diagram showing the construction of hexaporphyrin(1.1.1.1.1.1), starting from a four site ring. Positivenumber sites belong to one block and the negative-number sites to another block. Black dots represent new sites added at the DMRG iteration. Symmetry corresponding to the interchange of +ve and −ve sites is used in the calculations. 7805



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Figure 2. Structures of EPs used in the calculation.

We have used the Davidson algorithm to find a few low-lying eigen values and eigen vectors of a large sparse matrix H. These eigen vectors are used to calculate ground and excited state bond-orders and charge densities. TPA coefficients are calculated using correction vector method (CV).28,33 The sum over state expression for ijth component of the resonant TPA transition matrix element is given by ⎡ ⟨g |μ ̃ |k⟩⟨k|μ ̃ |f ⟩

Sij =

∑⎢ k

⎢⎣

i

j

ωkg − ω

+

⟨g |μj̃ |k⟩⟨k|μĩ |f ⟩ ⎤ ⎥ ωkg − ω ⎥⎦

structures of inverted EPs are given in the Supporting Information and their numbering is suffixed with “A”.) We have taken one π-electron on each carbon, nitrogen atom of the aza ring, and two π-electrons on N of the pyrrole ring which contributes a lone-pair to π-conjugation. So, overall, a pyrrole ring contributes 6 π-electrons and an aza ring contributes 5 πelectrons to conjugation. Hence, depending upon the number of pyrrole and aza rings in the molecule and number of mesobonds, we have 4n or (4n + 2) π-electronic system. In what follows, we give results of optical gap followed by ground state properties and last the TPA coefficients. To have confidence in our technique, we have carried out calculations for Hückel model using the DMRG method and compared results with the exact Hückel calculations, for hexaporphyrin molecule with two meso bonds. We find that the ground state energy from DMRG calculation is in good agreement with the exact calculation (−157.17 eV against −156.92 eV with an error of 0.16%). To check the accuracy of the eigenfunction, we have also obtained the bond orders from both methods and these are shown in Figure 3. We see that bond orders from DMRG calculations are in good agreement with the exact Hückel bond orders. Spectral Gaps in EPs. We have obtained the spectral gaps of EPs in the MS = 0 sector. Within the MS = 0 sector we have computed the one- and two-photon transition matrix elements of the excited states with the ground state. We can unambiguously label the states for which the transition dipole with the ground state is nonzero as singlet B subspace states because of spin and spatial symmetry. Similarly for states for which the TPA cross section is nonzero, excitations from the ground state can be labeled singlets in the A subspace. We present the optical gap (OPA) and two-photon gaps for 4n+2 π-systems in Table 1 and for 4n π-systems in Table 2. We first focus our attention on the one-photon gaps. What is striking is that the 4n+2 π-electron EPs have two prominent one-photon states, one less than 1.0 eV and another around

(3)

where Sij is the transition matrix element. |g⟩ is the ground state, |k⟩ is the excited state of the Hamiltonian, and |f⟩ is the state to which TPA is to be calculated. ωkg is the energy gap from the ground state to the excited state |k⟩ and ω = ωfg/2. i and j are the Cartesian coordinates; μ̃ j is the dipole dispalcement operator. Sij is calculated using the CV method. In the CV method, the correction vector |ϕ(1) i (ω)⟩ is computed by solving the set of linear algebraic equations defined by 34

(H0 − E0 − ℏω)|ϕi(1)( −ω)⟩ = μi⃗ |g ⟩

(4)

Using |ϕ(1) i (−ω)⟩ from above, it can be shown that Sij(ω) is exactly given by Sij(ω) = (⟨ϕi(1)( −ω)|μj̃ |f ⟩ + ⟨ϕj(1)( −ω)|μĩ |f ⟩)

(5)

Orientationally averaged TPA cross-section for a planepolarized light is given by the isotropic tensor δgf =

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1 15

∑ i,j=x ,y,z

(SiiS*jj + 2SijSij*) (6)

RESULTS AND DISCUSSION We have studied 4 different expanded porphyrins (regular EPs) and their corresponding inverted structures. The schematic structures of the regular molecules are shown in Figure 2. (The 7806



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Table 2. Excitation Energies, (ΔOPA and ΔTPA) (in eV), Transition Dipole Moment, |μ| (in Debye), and Two-Photon Absorption Cross Sections, δgf (in GM) (of Expanded Porphyrins with 4n π-Electronsa regular

Figure 3. Comparison of Bond orders, for molecule II, from noninteracting Hückel model using DMRG (upper half of the figure, shown in violet) and Hückel MO methods (lower half of the figure, shown in blue). The molecule has an inversion symmetry.

inverted

system

ΔOPA (|μ⃗ |)

ΔTPA (δgf)

ΔOPA (|μ⃗ |)

ΔTPA (δgf)

V (2, 4, 36)

2.26 (0.30)

2.26 (189.77)

2.17 (0.41)

2.16 (169.47)

[2.54] 2.34 (0.25)

[3100 at ℏω = 3.16 eV] -

2.30 (0.28)

2.31 (0.52)

1.79 (2.29)

1.33 (152.14)

1.75 (2.80)

1.33 (159.84)

VI (4, 2, 36) VII (6, 4, 40)

[1.23] [2.09]

∼2.0 eV, with ∼1 eV apart, except for molecule I. However, the 4n π-electron EPs have only one prominent one-photon state at ∼2.0 ± 0.3 eV. This difference is seen even in the Hückel picture. In the 4n+2 π-electron systems, the HOMO−LUMO excitation is dipole allowed, while in the 4n π-electron systems, the excitation is dipole forbidden. This result should hold good for both inverted and regular EPs as long as the structure is planar, since the Hückel Hamiltonian only depends upon bond connectivity, which is the same in regular and inverted systems. This implies that the symmetry of the excited states is retained when the electron−electron interactions are turned on. Experimentally, the electronic absorption spectra of molecules I, IIIA, IVA (4n + 2 π systems), V, and VIIA (4n π systems) are known in the literature. The one-photon gap in the 4n+2 systems indeed shows both a low energy gap in the range 1.0 eV and another one-photon gap at twice this value. On the other hand, the 4n system V shows experimental one-photon gap above 2 eV, and the system VIIA shows a low energy gap at ∼1.23 eV and a higher energy gap at ∼2.09 eV. This may be because VIIA, an inverted EP, is nonplanar and hence could lead to broken symmetry. Our studies of the 4n + 2 systems show that all these systems have two photon gap less than 2 eV except system I. The cross section for two-photon absorption is significant to the low-lying state except for system I. In experimentally synthesized systems,

[2600 at ℏω = 2.16 eV]

a The numbers in the bracket of the first column correspond to number of meso-bonds, pyrrole rings, and π-electrons, respectively. The numbers in square bracket correspond to experimental gaps.

IIA, IIIA, and IVA, the TPA cross section is very large. In the 4n system, except VII all the two-photon gaps are at ∼2.0 eV and the cross sections are much smaller than in the 4n+2 system. We also observe that the TPA coefficient is quite large if the one-photon state is lower than the two-photon state (see Tables 1 and 2). The experimentally observed TPA coefficients are higher than our theoretical values, which could be due to the fact that we could not obtain the higher two-photon states which were accessible in experiments. We also note that systems which have been synthesized and studied have higher TPA values than our theoretical values. Charge Density and Bond Orders. In order to explore the nature of the dominant one and two-photon excited states, we have analyzed the charge density and bond orders in the ground state as well as excited states. The charge density information of excited states provides the difference in charge distribution between the ground and the chosen excited state. This is important in tailoring the molecule through substitution

Table 1. Excitation Energies, (ΔOPA and ΔTPA) (in eV), Transition Dipole Moment, |μ| (in Debye), and Two-Photon Absorption Cross Sections, δgf (in GM) of Expanded Porphyrins with 4n+2 π-Electronsa regular system I (0, 4, 34)

II (2, 2, 34) III (4, 4, 38)

IV (6, 2, 38)

ΔOPA (|μ⃗|) 1.14 (0.66) 3.41 (1.49) [1.56] [1.75] [3.12] 0.61 (0.71) 1.61 (0.96) 0.99 (2.36) 1.98 (0.97)

0.89 (3.34) 2.14 (0.33)

inverted ΔTPA (δgf)

3.47 (230.33)

ΔOPA (|μ⃗|)

ΔTPA (δgf)

-

-

0.55 (1.44) 1.51 (1.33) 1.00 (1.75) 1.86 (0.59) [1.34] [2.29] 0.90 (3.88) 2.19 (0.54) [1.22] [2.19]

0.60 (1130.80)

[2100 at ℏω = 1.6 eV]

1.71 (105.80) 1.15 (6050.85)

1.98 (25091.36)

1.08 (3181.17) [9080 at ℏω = 2.36 eV] 1.08 (3766.69) [9890 at ℏω = 2.06 eV]

The numbers in brackets of the first column correspond to number of meso-bonds, pyrrole rings, and π-electrons, respectively. The numbers in square bracket correspond to experimental gaps. a

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Figure 4. Ground state charge density and bond order for molecules IV (4n+2 π-electrons) and VII (4n π-electrons). Upper panel shows the bond order and the lower panel shows the charge densities.

Figure 5. Difference in charge density (a) and bond order (b) between excited state and ground state for molecule IV (4n+2 π-electrons). Upper panel shows the difference for OPA state and lower panel that for TPA state.

bonds are strongly alternating, in the case of an aza ring. Meso bonds connected to the aza-ring are much stronger than that connected to the pyrrole ring. Meso bonds between a pyrrole and an aza ring are strongly alternating (with bond order ≈0.70 and 0.40, respectively) and those between two pyrroles or two aza rings are either weakly alternating or equal. Among the Cα− Cα bond connecting two rings, the one between two aza rings is the strongest (bond order is 0.50) followed by that between an aza ring and a pyrrole ring (bond order is ≈0.40) and that between two pyrroles is the weakest (bond order is ≈0.30). In Figure 5a and in Supporting Information, we present the charge density difference between the one-photon state with largest transition dipole and the ground state on the top part of the structures. On the bottom part of the structure we have given the charge density difference between the two-photon state, to which the TPA cross section is maximum, and the ground state. Similarly the bond order difference is shown in Figure 5b. The salient features from these figures is that in all the molecules charge density on N of aza ring is depleted while on N of the pyrrole ring it accumulates. Except for the first molecule, the charge density redistribution is greater on aza rings (especially on Ns), compared to the pyrrole ring, upon excitation. We also observe that the aza rings of 4n systems have an overall larger negative charge compared to the aza rings of 4n+2 systems. There is very little charge delocalization on pyrrole rings. Meso carbons have charge accumulation (except in the case of molecule IV), in general. A similar qualitative trend is observed for the two-photon state. However, the magnitude of charge delocalization is greater in the two-photon state.

to yield system with large transition dipoles and to tune the excitation gaps. The bond order difference will yield information about the Stark shifts in the excited states. If these are large, the excited state equilibrium will be significantly different from the ground state geometry and could result in weak fluorescence and large Stark shifts. In Figure 4 we present the ground state charge density and bond orders for one of the structures we have studied. (See Supporting Information for the other structures.) In the ground state, charge density on N is ≈1.36 ± 0.05 in pyrrole rings and in aza ring it is ≈1.14 ± 0.08. Charge density on all nitrogen atoms decreases by ≈0.07 when two of the four pyrrole rings are replaced by aza rings. In most of the molecules, charge density on all β-carbon atoms is 1.04 ± 0.02. While charge density on α-carbon atoms in aza ring is ≈1.0; it is ≈1.2 for αcarbon atoms in pyrrole rings. The total charge density on an aza ring is ≈5.2 and that on pyrrole ring is ≈5.8. Charge density on meso carbon atom between two pyrroles is greater than 1 and it is less than 1 for other types of meso carbon atoms such as those between two aza rings or an aza and a pyrrole ring. We observe a similar trend in the case of inverted porphyrins. In the ground state, in general, the Cβ−Cβ bond is the strongest bond both in aza and pyrrole rings. It is more like a double bond in the case of aza ring (bond order is ≈0.80), compared to pyrrole ring where the bond order is ≈0.7. Moreover, the bond Cβ−Cα is weaker in the aza ring (more like a single bond) than in pyrrole. The salient feature is that the Cβ−Cβ is the strongest bond and the other two C−C bonds are much weaker, in the case of aza rings; while all three C−C bonds are of similar strength in pyrrole rings, whereas C−N 7808



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(6) Gokulnath, S.; Chandrashekar, T. K. J. Chem. Sci. 2008, 120, 137. (7) Sessler, J. L.; Morishima, T.; Lynch, V. Angew. Chem., Int. Ed. Engl. 1991, 30, 977. (8) Sessler, J. L.; Weghorn, S. J.; Morishima, T.; Rosingana, M.; Lynch, V.; Lee, V. J. Am. Chem. Soc. 1992, 114, 8306. (9) G. P. M. S. Neves, M.; M. Martins, R.; C. Tome, A.; J. D. Silvestre, A.; M. S. Silva, A.; Felix, V.; A. S. Cavaleiro, J.; G. B. Drew, M. Chem. Commun. 1999, 385. (10) Srinivasan, A.; Reddy, V. M.; Narayanan, S. J.; Sridevi, B.; Pushpan, S. K.; Ravikumar, M.; Chandrashekar, T. K. Angew. Chem., Int. Ed. Engl. 1997, 36, 2598. (11) Narayanan, S. J.; Sridevi, B.; Chandrashekar, T. K.; Vij, A.; Roy, R. J. Am. Chem. Soc. 1999, 121, 9053. (12) Narayanan, S.; Srinivasan, A.; Sridevi, B.; Chandrashekar, T.; Senge, M.; Sugiura, K.-i.; Sakata, Y. Eur. J. Org. Chem. 2000, 2000, 2357. (13) Lim, J. M.; Lee, J. S.; Chung, H. W.; Bahng, H. W.; Yamaguchi, K.; Toganoh, M.; Furuta, H.; Kim, D. Chem. Commun. 2010, 46, 4357. (14) Kumar, R.; Misra, R.; Chandrashekar, T. K.; Nag, A.; Goswami, D.; Suresh, E.; Suresh, C. H. Eur. J. Org. Chem. 2007, 2007, 4552. (15) Ahn, T. K.; Kwon, J. H.; Kim, D. Y.; Cho, D. W.; Jeong, D. H.; Kim, S. K.; Suzuki, M.; Shimizu, S.; Osuka, A.; Kim, D. J. Am. Chem. Soc. 2005, 127, 12856. (16) Yoon, Z. S.; Kwon, J. H.; Yoon, M.-C.; Koh, M. K.; Noh, S. B.; Sessler, J. L.; Lee, J. T.; Seidel, D.; Aguilar, A.; Shimizu, S.; Suzuki, M.; Osuka, A.; Kim, D. J. Am. Chem. Soc. 2006, 128, 14128. (17) Cho, S.; Yoon, Z. S.; Kim, K. S.; Yoon, M.-C.; Cho, D.-G.; Sessler, J. L.; Kim, D. J. Phys. Chem. Lett. 2010, 1, 895. (18) Pushpan, S. K.; Venkatraman, S.; Anand, V. G.; Sankar, J.; Parmeswaran, D.; Ganesan, S.; Chandrashekar, T. K. Curr. Med. Chem. Anti-Cancer Agents 2002, 2, 187. (19) Sessler, J. L.; Tvermoes, N. A.; Davis, J.; P. Anzenbacher, J.; Jursikova, K.; Sato, W.; Seidel, D.; Lynch, V.; Black, C. B.; Try, A.; Andrioletti, B.; Hemmi, G.; Mody, T. D.; J., M. D.; Král, V. Pure Appl. Chem. 1999, 71, 2009. (20) Sessler, J. L.; Hemmi, G.; Mody, T. D.; Murai, T.; Burrell, A.; Young, S. W. Acc. Chem. Res. 1994, 27, 43. (21) Ravi, M.; Chandrashekar, T. K. J. Inclus. Phenom. Macrocycl. Chem. 1999, 35, 553. (22) Sessler, J. L.; Davis, J. M. Acc. Chem. Res. 2001, 34, 989. (23) Narayanan, S. J.; Sridevi, B.; Chandrashekar, T. K.; Vij, A.; Roy, R. J. Am. Chem. Soc. 1999, 121, 9053. (24) Misra, R.; Anand, V. G.; Rath, H.; Chandrashekar, T. K. J. Chem. Sci. 2005, 117, 99. (25) White, S. R. Phys. Rev. Lett. 1992, 69, 2863. (26) White, S. R. Phys. Rev. B 1993, 48, 10345. (27) Pati, S. K.; Ramasesha, S.; Shuai, Z.; Brédas, J. L. Phys. Rev. B 1999, 59, 14827. (28) Soos, Z. G.; Ramasesha, S. J. Chem. Phys. 1989, 90, 1067. (29) Ohno, K. Theor. Chim. Acta 1964, 2, 219. (30) Frisch, M. J. et al. Gaussian 03; Revision B.03, 2003. (31) Stephens, P.; Devlin, F.; Chabalowski, C.; Frisch, M. J. J. Phys. Chem. 1994, 98, 11623−11627. (32) Shaon, S.; Durga Prasad, G. V. M. L.; Ramasesha, S.; Diptiman, S. J. Phys.: Condens. Matter 2012, 24, 115601. (33) Ramasesha, S.; Soos, Z. Chem. Phys. Lett. 1988, 153, 171. (34) Wang, C.-K.; Macak, P.; Luo, Y.; Ågren, H. J. Chem. Phys. 2001, 114, 9813.

Similar to charge density variation, bond order differences are maximum in the aza ring compared to pyrrole rings, except in the first molecule. There is hardly any variation in the bond order from the ground state to excited state in the pyrrole molecules. In the case of aza rings, there is switch over of bond orders, i.e., single bonds in the ground state become double bonds and vice versa. Qualitatively, a similar trend is observed for the two-photon state as well.

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CONCLUSIONS We have carried out correlated electronic structure calculations of EPs using a PPP model to describe the interacting πelectrons. We have used the DMRG method to solve the PPP model Hamiltonian. We have employed the correction vector method which is by far the most accurate method to calculate the TPA cross sections. We find that in the 4n+2 EPs, there are two prominent low-energy one-photon excitations, while in 4n systems there is only one such excitation. The two-photon gaps in both these types of systems are at energies close to the onephoton excitations. The charge density rearrangement in the one-photon excited state is most at the aza nitrogen site and at the meso carbon sites. In the two-photon states also the charge density rearrangement occurs mostly at the aza-ring sites. The bond order changes in these states is much more striking. In the one-photon state, the C−C bond length in the aza rings shows a tendency to become uniform. A similar qualitative trend is also observed for the two-photon state.

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ASSOCIATED CONTENT

S Supporting Information *

(i) Structures of Expanded Porphyrins (EPs) used in the calculation. (ii) Simulated spectra of EPs. (iii) Charge density and bond order of Expanded Porphyrins in ground state. (iv) Difference in charge densities of one-photon and two-photon states with respect to ground state. (v) Difference in bond orders of one-photon and two-photon states with respect to ground state. This material is available free of charge via the Internet at http://pubs.acs.org/.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Department of Science and Technology, India, for financial support. Y.A.P. thanks Disha Programme for women in Science, Department of Science and Technology, India, for financial support.

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REFERENCES

(1) Nalwa, H. S.; Saito, T.; Kakuta, A.; Iwayanagi, T. J. Phys. Chem. 1993, 97, 10515. (2) Drobizhev, M.; Karotki, A.; Kruk, M.; Rebane, A. Chem. Phys. Lett. 2002, 355, 175. (3) Chou, J.-H.; Kosal, M. E.; Nalwa, H. S.; Rakow, N. A.; Suslick, K. S. Applications of Porphyrins and Metalloporphyrins to Materials Chemistry. In The Porphyrin Handbook; Kadish, K. M., Smith, K. M., Guilard, R., Eds.; Academic Press: New York, 2000; Vol. 6, p 44. (4) Jasat, A.; Dolphin, D. Chem. Rev. 1997, 97, 2267. (5) Rath, H.; Sankar, J.; Prabhu, V., Raja; Chandrashekar, T. K.; Nag, A.; Goswami, D. J. Am. Chem. Soc. 2005, 127, 11608. 7809


Linear and Nonlinear Optical Properties of Expanded Porphyrins: A

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