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Theoretical studies of the spectra and two-photon absorption cross sections for porphyrin and carbaporphyrins
Chemical Physics Letters 373 (2003) 197–206 www.elsevier.com/locate/cplett
Theoretical studies of the spectra and two-photon absorption cross sections for porphyrin and carbaporphyrins Xiao-Juan Liu a, Ji-Kang Feng a
a,b,*
, Ai-Min Ren a, Xin Zhou
a
State Key Laboratory of Theoretical and Computational Chemistry, Institute of Theoretical Chemistry, Jilin University, Changchun 130023, PR China b The College of Chemistry, Jilin University, Changchun 130023, PR China Received 27 January 2003; in final form 26 March 2003
Abstract The geometric and electronic structures of porphyrin and a series of carbaporphyrins have been theoretically studied using the time-dependent density functional theory (TDDFT). The two-photon absorption cross sections (TPACS) of these molecules are computed using the ZINDO-SOS formula. The calculated results indicate that when the N atom is substituted by the C atom, the molecular center is enlarged and the absorptions are red-shifted and that porphyrin, carbaporphyrin, opp-dicarbaporphyrin and adj-dicarbaporphyrin show TPACS in small absorption area, while tetracarbaporphyrin has fairly large two-photon absorptions in comparatively larger area, which may lead it to many practical applications. Ó 2003 Elsevier Science B.V. All rights reserved.
1. Introduction Porphyrins and their analogous participate in a wide variety of important biological processes, including oxygen and electron transport (the hemes) and photosynthesis (the chlorophylls). Consequently, considerable attention has been attracted throughout the 20th century [1]. The robust characteristic associated with the porphyrins are due in character [2]. One view about the origin of aromaticity in the porphyrin is attributed
*
Corresponding author. Fax: +86-431-8945942. E-mail address: [email protected] (J.-K. Feng).
to the presence of an 18p-electron delocalization pathway [3], the other view suggests that the Ôlone pairÕ electrons of the two pyrrole-type nitrogens are also crucial to the aromatic character of porphyrins and that the system should be considered a 22p-electron aromatic system [4]. Obviously, the nitrogen electrons must have an important influence on the electronic structure of the porphyrins and are in relation to the macrocycleÕs chemical, physical, and spectroscopic properties. Two-photon absorption (TPA) is a process wherein two photons are absorbed simultaneously. Characteristic features are adherent to even-parity selection rules and quadratic intensity dependence. Compounds with large two-photon cross sections are of interest in applications as diverse as 3D
0009-2614/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0009-2614(03)00577-3
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optical data storage [5], optical limiting [6], and fluorescence microscopy [7]. These applications use two key features of two-photon absorption, namely, the ability to create excited states with photons of half the nominal excitation energy, which can provide improved penetration in absorbing or scattering media, and the I2 dependence of the process, which allows for excitation of chromophores with a high degree of spatial selectivity in three dimensions through the use of a tightly focused laser beam. The carbaporphyrins were first discovered accidentally as by-products of the acid-catalyzed pyrrole–aldehyde cyclocondensation rout to porphyrins in 1994 [8]. Afterwards, several routes have been made to synthesize the carbaporphyrins [9–11]. These experiments have concentrated on studies of their one-photon absorptions (OPAs) and ligand properties. The two-photon absorptions have been noted in porphyrin, [12] furthermore, two-photon spectroscopy has proved to be of great utility in studying the electronic excited states of the porphyrins. However, systematic theoretical studies on the geometries and absorption properties, especially on the two-photon absorptions of these series of carbaporphyrins have not been done. In this Letter, we studied five molecules in which the N atoms are substituted by C atoms gradually through the calculation of their geometries, spectroscopies and two-photon absorption properties, aiming at investigating the relationship of structure–properties for this system.
energy and oscillator strength calculations were carried out using time-dependent density functional response theory, as implemented in the GA U S S I A N 98 program. The TDDFT calculations, carried out at the B3LYP structures, were done using the same basis set and effective core potentials as those used in the ground-state DFT calculations. The DFT has been quite successful in predicting the structures and spectra of porphyrin [22–27]. The effect of basis set on the excitation energies and oscillator strengths has been examined and was found to be rather small [28]. So in this Letter, we take 6-31g basis set. TPA cross section is related to the imaginary part of the third-order polarizability by the expression dðxÞ ¼
8p2 hx2 4 L Im cðx; x; x; xÞ; n2 c 2
ð1Þ
where h is PlankÕs constant divided by 2p; x is photon frequency; c is the speed of light; n is the refractive index of the medium and L is a local field factor (equal to 1 for vacuum). The sum-over-states (SOS) expression to evaluate the components of the third-order polarizability cijkl can be induced out using perturbation theory and density matrix method. By considering a power expansion of the energy with respect to the applied field, the cijkl Cartesian components are given by 4p3 cijkl ð xr ; x1 ; x2 ; x3 Þ ¼ 3 P ði; j; k; l; xr ; x1 ; x2 ; x3 Þ 3h " XXX m6¼0 n6¼0 p6¼0
2. Theoretical method The structures of all the studied molecules have been predicated using the density functional theory (DFT) with the 6-31g basis set. DFT calculations were carried out using BeckeÕs three-parameter hybrid functional, [13,14] referred to as B3LYP. Time-dependent density functional theory (TDDFT) [19–21], which combines the advantages of density functional theory and time-dependent formalism allowing the accurate determination of excited state properties, has been implemented in several quantum chemistry packages [15–18]. The excitation
hojli jmihmjlj jnihnjlk j pih pjll joi ðxmo xr iCm0 Þðxno x2 x3 iCn0 Þ xpo x3 iCp0
XX m6¼0 n6¼0
# hojli jmihmjlj joihojlk jnihnjll joi ; ðxmo xr iCmo Þðxno x3 iCno Þðxno þ x2 iCno Þ
ð2Þ where P ði; j; k; l; xr ; x1 ; x2 ; x3 Þ is a permutation operator defined in such a way that for any permutation of ði; j; k; lÞ, an equivalent permutation of ðxr ; x1 ; x2 ; x3 Þ is made simultaneously. The xr ¼ x1 þ x2 þ x3 is the polarization response frequency; x1 ; x2 ; x3 indicate the frequencies of the perturbing radiation fields (considering the degenerate TPA, x1 ¼ x2 ¼ x and x3 ¼ x).
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The i; j; k and l correspond to the molecular axes x, y and z; m, n and p denote excited states and o, the ground state. The lj is the jð¼ x; y; zÞth component of the dipole operator; ðhmjlj jni ¼ hmjlj jni hojlj joiÞ; ðh=2pÞ xmo is the transition energy between the m and o states. The Cmo is the damping factor of excited state m. We consider that the higher the excited state, the shorter its lifetime will be, and express the damping in electronvolts as expression Cmo ¼ 0:08
xmo : xlo
ð3Þ
We proceed in the following steps: (1) optimization of the molecular structure with the B3LYP/6-31g method, (2) calculations of the transition energies, oscillator strength using both the TDDFT and INDO/CI methods and (3) evaluation of the SOS expression for the thirdorder nonlinear susceptibility c, the summation is performed over 256 states, which is sufficient to obtain converged values of the molecular thirdorder nonlinear susceptibilities in the case of the compounds investigated here. On this basis, we gained the TPA cross sections of the studied molecules.
199
3. Results and discussion 3.1. The design of the compounds and the geometry optimization The investigated molecules are shown in Fig. 1, including free base porphyrin and four carbaporphyrins in which the N atoms are substituted by C atom from one (carbaporphyrin), two (existing two isomers – opp-dicarbaporphyrin and adj-dicarbaporphyrin) to four (tetracarbaporphyrin). The geometries were optimized under D2h symmetry restrictions for porphyrin and opp-dicarbaporphyrin, C2v symmetry for carbaporphyrin and adj-dicarbaporphyrin, and D4h symmetry for tetracarbaporphyrin. The bond lengths and the distances between the opposite atoms in the center are shown in Fig. 2. Through comparing the multiplication of the distances between the two atoms (N, N or N, C or C, C), it is found that when C replace the N atoms, the hole of the center is enlarged: for example, the distance between N and C, N and N in carbaporphyrin is 0.4056 and 0.4380 nm, (0:4056 nm 0:4380 nm ¼ 0:1777 nm2 ) while in porphyrin it is 0.4069 and 0.4230 nm, (0:4069 nm 0:4230 nm ¼ 0:1721 nm2 ) and for
Fig. 1. The investigated molecules and their corresponding chemical names.
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Fig. 2. The bond length and the distance between the opposite atoms in the center. Unit: 1010 m.
the case of tetracarbaporphyrin in which the four N atoms are all substituted by C atoms, the C–C distances are 0.4380 nm ð0:4380 nm 0:4380 nm ¼ 0:1918 nm2 Þ. We attribute this phenomenon in part to the H atom which is added to the C atom – the H atom has some interactions with the opposite and two adjacent H atoms, which lead to the enlargement of the hole center. The bond lengths also have some changes, which can be seen from Fig. 2. 3.2. One- and two-photon absorptions We calculated the one-photon absorption using two methods: first by means of TDDFT, and then by semiempirical method INDO/CI. Only porphyrin has the available experimental values, so for porphyrin we compared the calculated values by the two methods with the experimental values, while for the other molecules we only compared the values obtained by the two methods. All these results have been listed in Table 1. For porphyrin, the calculated absorptions by the two methods are
consistent with the experimental values, and in the INDO/CI calculation, we take the standard parameters for all the molecules. Generally, the more the N atoms are substituted by C atoms, the more red-shifted the one-absorptions are. 3.2.1. Analysis and discussion of the calculated results by TDDFT We first analyze the one-photon absorptions calculated by TDDFT. Electronic structures are fundamental to the interpretation and understanding of the absorption spectra. In Table 2, we listed the symmetries and energies of the frontier orbitals for all the molecules and in Fig. 3, in which the orbitals are plotted in order to compare clearly. Furthermore, the highest occupied molecular orbitals (HOMOs) and the lowest unoccupied orbitals (LUMOs) calculated by the TDDFT method have also plotted in Fig. 4. The carbaporphyrins in which N atoms are replaced by C atoms gradually from one to two, their HOMOs (b1 ) coefficients are largely localized on the center C atom, meso carbon atoms and only a little on the
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Table 1 The calculated one-photon absorption values by TDDFT and INDO/CI methods for all the molecules and the experimental values for porphyrin Porphyrin
Carbaporphyrin
opp-Dicarbaporphyrin
adj-Dicarbaporphyrin
Tetracarbaporphyrin
INDO/CI k (nm)
TDDFT k (nm)
INDO/CI k (nm)
TDDFT k (nm)
631
696 406 362
684 554 460 412 410 362 333
815
419 385 337 327 271 210
624 540 499 412 390
TDDFT INDO/CI Experiment TDDFT INDO/CI TDDFT k (nm) k (nm) k (nm) k (nm) k (nm) k (nm) 545 373 352 327 321 284 279 241 197
616 517 354 340
614–626a 512–532a 372–396b 340c
287 281 237 196
596 433 396 366 348 326 318 259 207
658 432 395 345 325 320 264 206
292 269 211
334 326 301 223 192
414 357 302 276
299 277 216 198
205 199
INDO/CI k (nm) 553 413 345 315 287 266 204
a
Data from optical spectra in vapor and different solvents [29–31]. Band maxima from optical spectra in vapor and different solvents [29,31,32]. c Gas-phase absorption spectrum [29]. b
three N atoms (shown in Fig. 4), while its LUMO (a2 ) receives a major contribution from the meso, Ca and Cb atoms (the atom label is shown in Fig. 1 taking porphyrin as an example), it can be con-
cluded that the substitutions should make effect on the HOMOs. But it is not the case for tetracarbaporphyrin: the LUMO coefficients are localized on the four center and meso C atoms while the
Table 2 The calculated orbital energies by TDDFT (eV) Porphyrin Symmetry
Energy
Carbaporphyrin
adj-Dicarbaporphyrin
opp-Dicarbaporphyrin
Tetracarbaporphyrin
Symmetry
Symmetry
Symmetry
Symmetry
Energy
Energy
Energy
Energy
Virtual orbitals b1g 1.78 b2g 1.71 b3u 1.23 )0.68 au b1g )2.27 b2g )2.30
b1 a2 b1 a2 b1 a2
1.81 1.69 1.24 )0.70 )2.23 )2.30
b1 a2 a1 b1 b1 a2
1.71 0.90 1.24 )0.68 )2.24 )2.31
b2g ag b3u au b1g b2g
1.67 1.35 1.28 )0.3 )2.21 )2.31
eg eg a1g b1u eg eg a2u
1.96 1.96 1.92 )0.76 )2.03 )2.03 )3.84
Occupied orbitals )5.16 b3u au )5.41 b1g )6.36 b3u )6.51 )7.02 b2u ag )7.03 b2g )7.18 b3u )7.23 )7.59 b2g b1g )7.79 au )8.89
b1 a2 b1 b1 a2 b1 a1 a2 b1 a2 b2
)4.94 )5.40 )5.95 )6.51 )7.23 )7.27 )7.27 )7.55 )7.78 )8.87 )9.52
b1 a2 a2 b1 a2 b1 b1 a2 b1 a1 b2
)4.86 )5.27 )5.61 )6.16 )7.42 )7.49 )7.55 )7.72 )8.93 )9.25 )9.31
b3u au b1g b3u b2g b3u b2g b1g ag b3g b1g
)4.83 )5.40 )5.73 )6.18 )7.29 )7.32 )7.50 )7.77 )8.84 )9.54 )9.62
a1u b2u eg eg a2u eg eg b1u b1g b2g
)5.23 )5.73 )5.76 )5.76 )6.52 )7.55 )7.55 )8.79 )9.38 )9.47
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Fig. 3. Molecular orbitals indicating the TDDFT predicted values.
HOMO is contributed largely from the Ca and Cb atoms, so its LUMO is significantly influenced by the substitution, in Fig. 3 it is shown that the LUMO is lowered about 1.54 eV compared with porphyrin. The energy gap is largely reduced and the absorption of tetracarbaporphyrin is red-shifted. Based on the analysis above we find that there is a qualitative change in the electronic structure for tetracarbaporphyrin, and it will show quite different absorption properties from the other molecules. Opp-dicarbaporphyrin and adj-dicarbaporphyrin in which C atoms substitute two N atoms, their LUMOs are almost at the same energy levels (in Table 2) and the HOMO in oppdicarbaporphyrin ()4.825 eV) is a little higher than in adj-dicarbaporphyrin ()4.858 eV). Fig. 3 indicates that the first few frontier orbitals of the two isomers are similar to each other, but the other higher orbitals of the two isomers are different from each other, the two isomers should show different one-photon absorption. On the other hand, the difference of their symmetries adj-dicarbaporphyrin is attributed to the point symmetry of C2v which has no symmetry center while oppdicarbaporphyrin is D2h that has symmetry center, in addition to their different electronic structure, their two-absorption properties are also different from each other. We notice that the tetracarbaporphyrin with D4h point symmetry in which the four N atoms are all replaced by C atoms, its frontier orbitals are completely different from the
former molecules, as can be seen from Fig. 3, its LUMO is significantly lowered compared with LUMOs of the other molecules, and its absorption properties will be red-shifted and show very different situation. In addition, the famous four-orbital model [33] which have been succeed in predicting the origins of the Q and B bands of metalloporphyrins. Obviously, we can see from Fig. 3, this four-orbital model is fit to porphyrin, because the other frontier orbitals are separated by a comparatively large energy gap from the four orbitals; but for carbaporphyrin, opp-dicarbaporphyrin and adj-dicarbaporphyrin, the fourorbital model is not applicable because the HOMO, HOMO-1, HOMO-2 and HOMO-3 are contiguous; as for the tetracarbaporphyrin, the four-orbital model obviously is not applicable as well. 3.2.2. One- and two-photon absorptions by INDO/ CI Now, we analyze the calculated one- and twophoton absorptions obtained by INDO/CI method. The INDO/CI calculations employed here includes single (S) excitations between 14 occupied orbitals and 14 unoccupied orbitals, and double (D) excitations among the two highest occupied and two lowest unoccupied molecular orbitals (SDCI), in all the summation is performed over 256 excited states. It is well known that the one- and two-photon absorptions adhere to difference selection rules.
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Fig. 4. The HOMO and LUMO orbitals for the studied molecules.
For centro-symmetric molecules, the one- and two-photon absorptions mutually exclusive: onephoton absorptions are allowed between the opposite parity, while the two-photon absorptions are allowed between the similar parity. As for molecules without center-symmetry, there is no selection limitation. For our investigated molecules: porphyrin, opp-dicarbaporphyrin and tetracarbaporphyrin have center-symmetry while adj-
dicarbaporphyrin and carbaporphyrin are not center-symmetric. The calculated one-photon absorptions are listed in Table 1. The calculated results by INDO/CI method are in agreement with the ones obtained by the TDDFT method. We should notice that from the view point of the calculated results, INDO/CI results are more consistent with the experimental values than TDDFT results. Table 1
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indicate that when C atoms replace the N atoms one by one, the absorptions are red-shifted from 616 nm (for porphyrin), to 658 nm (for carbaporphyrin), to 696 nm (for opp-dicarbaporphyrin) and to 684 nm (for adj-dicarbaporphyrin). We should notice that the tetracarbaporphyrin shows a very different absorption property, which is acscribed to the qualitative changes in its geometric and electronic structures when the four N atoms are replaced by the four C atoms. The two-photon absorptions and the absorption cross sections are listed in Table 3. The absorption of light by matter is a consequence of the interaction of an electromagnetic molecule. For the molecule that has central symmetry, a change in the parity between the initial and final states (wave functions) is required for every photon involved in the transition for electric dipole transitions. Thus the selection rule for TPA is different from that of OPA. One change of parity is required for a one-photon transition, while twophoton transitions must have initial and final states with the same parity. According to the rules above, for every molecule, we selected several excited states that two-photon absorption allowed as the two-photon absorption final states and calculated the two-photon absorption cross sections dðxÞ , then we obtain the maximum dðxÞ for every molecule (shown in Fig. 5). We would like to discuss the two-photon absorptions, compared with the one-photon absorptions, for the center symmetrical molecules (porphyrin and opp-dicarbaporphyrin) and noncenter symmetrical molecules (carbaporphyrin and adj-dicarbaporph-
yrin). For the porphyrin, the two-photon absorptions are positioned at 598 nm, while one-photon absorptions next to the two-photon absorption are the absorptions of 616 and 517 nm, these two onephoton absorptions are positioned far from the two-photon absorption. Similarly, for the oppdicarbaporphyrin with the same point symmetry (D2h ), the two-photon absorptions are positioned at 595 and 400 nm, which are also far away from one-photon absorptions (see Table 1). Contrary to the two molecules above, for the carbaporphyirn and adj-dicarbaporphyrin (both molecules are attributed to the C2v symmetry and there is no symmetry center), when comparing the one- and two-photon absorptions, we found that the oneand two-photon absorption peaks are close to each other; for the carbaporphyrin, the two-photon absorptions are positioned at 602 and 431 nm, and the latter is close to the one-photon absorptions of 432 nm; for the adj-dicarbaporphyrin three positions exist strong two-photon absorptions – 609, 457 and 412 nm, and the 457 and 412 nm absorptions are close to the one-photon absorptions 460 and 412 nm, respectively (shown in Table 1). At last, we should notice the two-photon absorption of tetracarbaporphyrin, its two-photon absorption is significantly different from the other molecules. If we compared with the ones of the porphyrin, we found that the two-photon absorption cross sections are largely enhanced in tetracarbaporphyrin, and the absorption is blueshifted. So for tetracarbaporphyrin and its derivatives, we can expect some new properties, which may lead it to many other applications in practice.
Table 3 The calculated two-photon absorptions by INDO/SDCI–SOS Compounds
Transition states
kTPA (nm)
d 1050 cm4 s photon1
Porphyrin (D2h ) Carbaporphyrin (C2v )
1 ! 13 1 ! 13 1 ! 32 1 ! 13 1 ! 27 1 ! 40 1 ! 13 1 ! 41 1 ! 52
598 602 431 609 457 412 595 400 404
59.04 8.32 568.40 8.14 601.68 2207.47 2.14 3146.36 1185.69
adj-Dicarbaporphyrin (C2v ) opp-Dicarbaporphyrin (D2h ) Tetracarbaporphyrin (D4h )
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205
Fig. 5. The relationship of the calculated two-photon absorption cross section dTPA ð1050 cm4 s photon1 Þ with the energy (eV).
It is obvious seen from Fig. 5 that two-photon absorptions of the porphyrin, carbaporphyrin, opp-dicarbaporphyrin and adj-dicarbaporphyrin exist only in a very limited area, while for tetracarbaporphyrin with only one peak value, its twophoton absorption exist in a large area, which is useful in many practical applications, such as in optical limiting.
4. Conclusions In this Letter, we have theoretically studied the geometries, one- and two-photon absorptions of porphyrin and a series of carbaporphyrins in which the N atoms are substituted by C atoms gradually, using both TDDFT and semi-empirical methods INDO/SDCI. The TDDFT method is proven to be
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an excellent alternative for the calculation of excitation energies, and the INDO/SDCI method is verified to be applicable for the investigated molecules. The calculated results indicate that the substitution of N atoms with C atoms not only lead to an enlargement of the molecular center in geometry but also to a red-shifting in the one-photon absorptions (except for the tetracarbaporphyrin), and there is a significant difference in two-photon absorption properties between the molecules with center-symmetry and the ones without centersymmetry. Especially, for the tetracarbaporphyrin, there is a qualitative change in its electronic structures and spectral properties. The calculated results indicate that although the tetracarbaporphyrin show a little smaller two-photon absorption cross section than other carbaporphyrin, it show a fairly large two-photon absorption cross section at the high-energy level in a wider area (shown in Fig. 5 for tetracarbaporphyrin) than porphyrin, this property may lead tetracarbaporphyrin derivatives to more practical applications, for example, in optical limiting.
Acknowledgements This work is supported by the National Nature Science Foundation of China (Nos. 20273023 and 90101026) and the Key Laboratory for Supramolecular Structure and Material of Jilin University.
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Theoretical studies of the spectra and two-photon absorption cross sections for porphyrin and carbaporphyrins Xiao-Juan Liu a, Ji-Kang Feng a
a,b,*
, Ai-Min Ren a, Xin Zhou
a
State Key Laboratory of Theoretical and Computational Chemistry, Institute of Theoretical Chemistry, Jilin University, Changchun 130023, PR China b The College of Chemistry, Jilin University, Changchun 130023, PR China Received 27 January 2003; in final form 26 March 2003
Abstract The geometric and electronic structures of porphyrin and a series of carbaporphyrins have been theoretically studied using the time-dependent density functional theory (TDDFT). The two-photon absorption cross sections (TPACS) of these molecules are computed using the ZINDO-SOS formula. The calculated results indicate that when the N atom is substituted by the C atom, the molecular center is enlarged and the absorptions are red-shifted and that porphyrin, carbaporphyrin, opp-dicarbaporphyrin and adj-dicarbaporphyrin show TPACS in small absorption area, while tetracarbaporphyrin has fairly large two-photon absorptions in comparatively larger area, which may lead it to many practical applications. Ó 2003 Elsevier Science B.V. All rights reserved.
1. Introduction Porphyrins and their analogous participate in a wide variety of important biological processes, including oxygen and electron transport (the hemes) and photosynthesis (the chlorophylls). Consequently, considerable attention has been attracted throughout the 20th century [1]. The robust characteristic associated with the porphyrins are due in character [2]. One view about the origin of aromaticity in the porphyrin is attributed
*
Corresponding author. Fax: +86-431-8945942. E-mail address: [email protected] (J.-K. Feng).
to the presence of an 18p-electron delocalization pathway [3], the other view suggests that the Ôlone pairÕ electrons of the two pyrrole-type nitrogens are also crucial to the aromatic character of porphyrins and that the system should be considered a 22p-electron aromatic system [4]. Obviously, the nitrogen electrons must have an important influence on the electronic structure of the porphyrins and are in relation to the macrocycleÕs chemical, physical, and spectroscopic properties. Two-photon absorption (TPA) is a process wherein two photons are absorbed simultaneously. Characteristic features are adherent to even-parity selection rules and quadratic intensity dependence. Compounds with large two-photon cross sections are of interest in applications as diverse as 3D
0009-2614/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0009-2614(03)00577-3
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optical data storage [5], optical limiting [6], and fluorescence microscopy [7]. These applications use two key features of two-photon absorption, namely, the ability to create excited states with photons of half the nominal excitation energy, which can provide improved penetration in absorbing or scattering media, and the I2 dependence of the process, which allows for excitation of chromophores with a high degree of spatial selectivity in three dimensions through the use of a tightly focused laser beam. The carbaporphyrins were first discovered accidentally as by-products of the acid-catalyzed pyrrole–aldehyde cyclocondensation rout to porphyrins in 1994 [8]. Afterwards, several routes have been made to synthesize the carbaporphyrins [9–11]. These experiments have concentrated on studies of their one-photon absorptions (OPAs) and ligand properties. The two-photon absorptions have been noted in porphyrin, [12] furthermore, two-photon spectroscopy has proved to be of great utility in studying the electronic excited states of the porphyrins. However, systematic theoretical studies on the geometries and absorption properties, especially on the two-photon absorptions of these series of carbaporphyrins have not been done. In this Letter, we studied five molecules in which the N atoms are substituted by C atoms gradually through the calculation of their geometries, spectroscopies and two-photon absorption properties, aiming at investigating the relationship of structure–properties for this system.
energy and oscillator strength calculations were carried out using time-dependent density functional response theory, as implemented in the GA U S S I A N 98 program. The TDDFT calculations, carried out at the B3LYP structures, were done using the same basis set and effective core potentials as those used in the ground-state DFT calculations. The DFT has been quite successful in predicting the structures and spectra of porphyrin [22–27]. The effect of basis set on the excitation energies and oscillator strengths has been examined and was found to be rather small [28]. So in this Letter, we take 6-31g basis set. TPA cross section is related to the imaginary part of the third-order polarizability by the expression dðxÞ ¼
8p2 hx2 4 L Im cðx; x; x; xÞ; n2 c 2
ð1Þ
where h is PlankÕs constant divided by 2p; x is photon frequency; c is the speed of light; n is the refractive index of the medium and L is a local field factor (equal to 1 for vacuum). The sum-over-states (SOS) expression to evaluate the components of the third-order polarizability cijkl can be induced out using perturbation theory and density matrix method. By considering a power expansion of the energy with respect to the applied field, the cijkl Cartesian components are given by 4p3 cijkl ð xr ; x1 ; x2 ; x3 Þ ¼ 3 P ði; j; k; l; xr ; x1 ; x2 ; x3 Þ 3h " XXX m6¼0 n6¼0 p6¼0
2. Theoretical method The structures of all the studied molecules have been predicated using the density functional theory (DFT) with the 6-31g basis set. DFT calculations were carried out using BeckeÕs three-parameter hybrid functional, [13,14] referred to as B3LYP. Time-dependent density functional theory (TDDFT) [19–21], which combines the advantages of density functional theory and time-dependent formalism allowing the accurate determination of excited state properties, has been implemented in several quantum chemistry packages [15–18]. The excitation
hojli jmihmjlj jnihnjlk j pih pjll joi ðxmo xr iCm0 Þðxno x2 x3 iCn0 Þ xpo x3 iCp0
XX m6¼0 n6¼0
# hojli jmihmjlj joihojlk jnihnjll joi ; ðxmo xr iCmo Þðxno x3 iCno Þðxno þ x2 iCno Þ
ð2Þ where P ði; j; k; l; xr ; x1 ; x2 ; x3 Þ is a permutation operator defined in such a way that for any permutation of ði; j; k; lÞ, an equivalent permutation of ðxr ; x1 ; x2 ; x3 Þ is made simultaneously. The xr ¼ x1 þ x2 þ x3 is the polarization response frequency; x1 ; x2 ; x3 indicate the frequencies of the perturbing radiation fields (considering the degenerate TPA, x1 ¼ x2 ¼ x and x3 ¼ x).
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The i; j; k and l correspond to the molecular axes x, y and z; m, n and p denote excited states and o, the ground state. The lj is the jð¼ x; y; zÞth component of the dipole operator; ðhmjlj jni ¼ hmjlj jni hojlj joiÞ; ðh=2pÞ xmo is the transition energy between the m and o states. The Cmo is the damping factor of excited state m. We consider that the higher the excited state, the shorter its lifetime will be, and express the damping in electronvolts as expression Cmo ¼ 0:08
xmo : xlo
ð3Þ
We proceed in the following steps: (1) optimization of the molecular structure with the B3LYP/6-31g method, (2) calculations of the transition energies, oscillator strength using both the TDDFT and INDO/CI methods and (3) evaluation of the SOS expression for the thirdorder nonlinear susceptibility c, the summation is performed over 256 states, which is sufficient to obtain converged values of the molecular thirdorder nonlinear susceptibilities in the case of the compounds investigated here. On this basis, we gained the TPA cross sections of the studied molecules.
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3. Results and discussion 3.1. The design of the compounds and the geometry optimization The investigated molecules are shown in Fig. 1, including free base porphyrin and four carbaporphyrins in which the N atoms are substituted by C atom from one (carbaporphyrin), two (existing two isomers – opp-dicarbaporphyrin and adj-dicarbaporphyrin) to four (tetracarbaporphyrin). The geometries were optimized under D2h symmetry restrictions for porphyrin and opp-dicarbaporphyrin, C2v symmetry for carbaporphyrin and adj-dicarbaporphyrin, and D4h symmetry for tetracarbaporphyrin. The bond lengths and the distances between the opposite atoms in the center are shown in Fig. 2. Through comparing the multiplication of the distances between the two atoms (N, N or N, C or C, C), it is found that when C replace the N atoms, the hole of the center is enlarged: for example, the distance between N and C, N and N in carbaporphyrin is 0.4056 and 0.4380 nm, (0:4056 nm 0:4380 nm ¼ 0:1777 nm2 ) while in porphyrin it is 0.4069 and 0.4230 nm, (0:4069 nm 0:4230 nm ¼ 0:1721 nm2 ) and for
Fig. 1. The investigated molecules and their corresponding chemical names.
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Fig. 2. The bond length and the distance between the opposite atoms in the center. Unit: 1010 m.
the case of tetracarbaporphyrin in which the four N atoms are all substituted by C atoms, the C–C distances are 0.4380 nm ð0:4380 nm 0:4380 nm ¼ 0:1918 nm2 Þ. We attribute this phenomenon in part to the H atom which is added to the C atom – the H atom has some interactions with the opposite and two adjacent H atoms, which lead to the enlargement of the hole center. The bond lengths also have some changes, which can be seen from Fig. 2. 3.2. One- and two-photon absorptions We calculated the one-photon absorption using two methods: first by means of TDDFT, and then by semiempirical method INDO/CI. Only porphyrin has the available experimental values, so for porphyrin we compared the calculated values by the two methods with the experimental values, while for the other molecules we only compared the values obtained by the two methods. All these results have been listed in Table 1. For porphyrin, the calculated absorptions by the two methods are
consistent with the experimental values, and in the INDO/CI calculation, we take the standard parameters for all the molecules. Generally, the more the N atoms are substituted by C atoms, the more red-shifted the one-absorptions are. 3.2.1. Analysis and discussion of the calculated results by TDDFT We first analyze the one-photon absorptions calculated by TDDFT. Electronic structures are fundamental to the interpretation and understanding of the absorption spectra. In Table 2, we listed the symmetries and energies of the frontier orbitals for all the molecules and in Fig. 3, in which the orbitals are plotted in order to compare clearly. Furthermore, the highest occupied molecular orbitals (HOMOs) and the lowest unoccupied orbitals (LUMOs) calculated by the TDDFT method have also plotted in Fig. 4. The carbaporphyrins in which N atoms are replaced by C atoms gradually from one to two, their HOMOs (b1 ) coefficients are largely localized on the center C atom, meso carbon atoms and only a little on the
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Table 1 The calculated one-photon absorption values by TDDFT and INDO/CI methods for all the molecules and the experimental values for porphyrin Porphyrin
Carbaporphyrin
opp-Dicarbaporphyrin
adj-Dicarbaporphyrin
Tetracarbaporphyrin
INDO/CI k (nm)
TDDFT k (nm)
INDO/CI k (nm)
TDDFT k (nm)
631
696 406 362
684 554 460 412 410 362 333
815
419 385 337 327 271 210
624 540 499 412 390
TDDFT INDO/CI Experiment TDDFT INDO/CI TDDFT k (nm) k (nm) k (nm) k (nm) k (nm) k (nm) 545 373 352 327 321 284 279 241 197
616 517 354 340
614–626a 512–532a 372–396b 340c
287 281 237 196
596 433 396 366 348 326 318 259 207
658 432 395 345 325 320 264 206
292 269 211
334 326 301 223 192
414 357 302 276
299 277 216 198
205 199
INDO/CI k (nm) 553 413 345 315 287 266 204
a
Data from optical spectra in vapor and different solvents [29–31]. Band maxima from optical spectra in vapor and different solvents [29,31,32]. c Gas-phase absorption spectrum [29]. b
three N atoms (shown in Fig. 4), while its LUMO (a2 ) receives a major contribution from the meso, Ca and Cb atoms (the atom label is shown in Fig. 1 taking porphyrin as an example), it can be con-
cluded that the substitutions should make effect on the HOMOs. But it is not the case for tetracarbaporphyrin: the LUMO coefficients are localized on the four center and meso C atoms while the
Table 2 The calculated orbital energies by TDDFT (eV) Porphyrin Symmetry
Energy
Carbaporphyrin
adj-Dicarbaporphyrin
opp-Dicarbaporphyrin
Tetracarbaporphyrin
Symmetry
Symmetry
Symmetry
Symmetry
Energy
Energy
Energy
Energy
Virtual orbitals b1g 1.78 b2g 1.71 b3u 1.23 )0.68 au b1g )2.27 b2g )2.30
b1 a2 b1 a2 b1 a2
1.81 1.69 1.24 )0.70 )2.23 )2.30
b1 a2 a1 b1 b1 a2
1.71 0.90 1.24 )0.68 )2.24 )2.31
b2g ag b3u au b1g b2g
1.67 1.35 1.28 )0.3 )2.21 )2.31
eg eg a1g b1u eg eg a2u
1.96 1.96 1.92 )0.76 )2.03 )2.03 )3.84
Occupied orbitals )5.16 b3u au )5.41 b1g )6.36 b3u )6.51 )7.02 b2u ag )7.03 b2g )7.18 b3u )7.23 )7.59 b2g b1g )7.79 au )8.89
b1 a2 b1 b1 a2 b1 a1 a2 b1 a2 b2
)4.94 )5.40 )5.95 )6.51 )7.23 )7.27 )7.27 )7.55 )7.78 )8.87 )9.52
b1 a2 a2 b1 a2 b1 b1 a2 b1 a1 b2
)4.86 )5.27 )5.61 )6.16 )7.42 )7.49 )7.55 )7.72 )8.93 )9.25 )9.31
b3u au b1g b3u b2g b3u b2g b1g ag b3g b1g
)4.83 )5.40 )5.73 )6.18 )7.29 )7.32 )7.50 )7.77 )8.84 )9.54 )9.62
a1u b2u eg eg a2u eg eg b1u b1g b2g
)5.23 )5.73 )5.76 )5.76 )6.52 )7.55 )7.55 )8.79 )9.38 )9.47
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Fig. 3. Molecular orbitals indicating the TDDFT predicted values.
HOMO is contributed largely from the Ca and Cb atoms, so its LUMO is significantly influenced by the substitution, in Fig. 3 it is shown that the LUMO is lowered about 1.54 eV compared with porphyrin. The energy gap is largely reduced and the absorption of tetracarbaporphyrin is red-shifted. Based on the analysis above we find that there is a qualitative change in the electronic structure for tetracarbaporphyrin, and it will show quite different absorption properties from the other molecules. Opp-dicarbaporphyrin and adj-dicarbaporphyrin in which C atoms substitute two N atoms, their LUMOs are almost at the same energy levels (in Table 2) and the HOMO in oppdicarbaporphyrin ()4.825 eV) is a little higher than in adj-dicarbaporphyrin ()4.858 eV). Fig. 3 indicates that the first few frontier orbitals of the two isomers are similar to each other, but the other higher orbitals of the two isomers are different from each other, the two isomers should show different one-photon absorption. On the other hand, the difference of their symmetries adj-dicarbaporphyrin is attributed to the point symmetry of C2v which has no symmetry center while oppdicarbaporphyrin is D2h that has symmetry center, in addition to their different electronic structure, their two-absorption properties are also different from each other. We notice that the tetracarbaporphyrin with D4h point symmetry in which the four N atoms are all replaced by C atoms, its frontier orbitals are completely different from the
former molecules, as can be seen from Fig. 3, its LUMO is significantly lowered compared with LUMOs of the other molecules, and its absorption properties will be red-shifted and show very different situation. In addition, the famous four-orbital model [33] which have been succeed in predicting the origins of the Q and B bands of metalloporphyrins. Obviously, we can see from Fig. 3, this four-orbital model is fit to porphyrin, because the other frontier orbitals are separated by a comparatively large energy gap from the four orbitals; but for carbaporphyrin, opp-dicarbaporphyrin and adj-dicarbaporphyrin, the fourorbital model is not applicable because the HOMO, HOMO-1, HOMO-2 and HOMO-3 are contiguous; as for the tetracarbaporphyrin, the four-orbital model obviously is not applicable as well. 3.2.2. One- and two-photon absorptions by INDO/ CI Now, we analyze the calculated one- and twophoton absorptions obtained by INDO/CI method. The INDO/CI calculations employed here includes single (S) excitations between 14 occupied orbitals and 14 unoccupied orbitals, and double (D) excitations among the two highest occupied and two lowest unoccupied molecular orbitals (SDCI), in all the summation is performed over 256 excited states. It is well known that the one- and two-photon absorptions adhere to difference selection rules.
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Fig. 4. The HOMO and LUMO orbitals for the studied molecules.
For centro-symmetric molecules, the one- and two-photon absorptions mutually exclusive: onephoton absorptions are allowed between the opposite parity, while the two-photon absorptions are allowed between the similar parity. As for molecules without center-symmetry, there is no selection limitation. For our investigated molecules: porphyrin, opp-dicarbaporphyrin and tetracarbaporphyrin have center-symmetry while adj-
dicarbaporphyrin and carbaporphyrin are not center-symmetric. The calculated one-photon absorptions are listed in Table 1. The calculated results by INDO/CI method are in agreement with the ones obtained by the TDDFT method. We should notice that from the view point of the calculated results, INDO/CI results are more consistent with the experimental values than TDDFT results. Table 1
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indicate that when C atoms replace the N atoms one by one, the absorptions are red-shifted from 616 nm (for porphyrin), to 658 nm (for carbaporphyrin), to 696 nm (for opp-dicarbaporphyrin) and to 684 nm (for adj-dicarbaporphyrin). We should notice that the tetracarbaporphyrin shows a very different absorption property, which is acscribed to the qualitative changes in its geometric and electronic structures when the four N atoms are replaced by the four C atoms. The two-photon absorptions and the absorption cross sections are listed in Table 3. The absorption of light by matter is a consequence of the interaction of an electromagnetic molecule. For the molecule that has central symmetry, a change in the parity between the initial and final states (wave functions) is required for every photon involved in the transition for electric dipole transitions. Thus the selection rule for TPA is different from that of OPA. One change of parity is required for a one-photon transition, while twophoton transitions must have initial and final states with the same parity. According to the rules above, for every molecule, we selected several excited states that two-photon absorption allowed as the two-photon absorption final states and calculated the two-photon absorption cross sections dðxÞ , then we obtain the maximum dðxÞ for every molecule (shown in Fig. 5). We would like to discuss the two-photon absorptions, compared with the one-photon absorptions, for the center symmetrical molecules (porphyrin and opp-dicarbaporphyrin) and noncenter symmetrical molecules (carbaporphyrin and adj-dicarbaporph-
yrin). For the porphyrin, the two-photon absorptions are positioned at 598 nm, while one-photon absorptions next to the two-photon absorption are the absorptions of 616 and 517 nm, these two onephoton absorptions are positioned far from the two-photon absorption. Similarly, for the oppdicarbaporphyrin with the same point symmetry (D2h ), the two-photon absorptions are positioned at 595 and 400 nm, which are also far away from one-photon absorptions (see Table 1). Contrary to the two molecules above, for the carbaporphyirn and adj-dicarbaporphyrin (both molecules are attributed to the C2v symmetry and there is no symmetry center), when comparing the one- and two-photon absorptions, we found that the oneand two-photon absorption peaks are close to each other; for the carbaporphyrin, the two-photon absorptions are positioned at 602 and 431 nm, and the latter is close to the one-photon absorptions of 432 nm; for the adj-dicarbaporphyrin three positions exist strong two-photon absorptions – 609, 457 and 412 nm, and the 457 and 412 nm absorptions are close to the one-photon absorptions 460 and 412 nm, respectively (shown in Table 1). At last, we should notice the two-photon absorption of tetracarbaporphyrin, its two-photon absorption is significantly different from the other molecules. If we compared with the ones of the porphyrin, we found that the two-photon absorption cross sections are largely enhanced in tetracarbaporphyrin, and the absorption is blueshifted. So for tetracarbaporphyrin and its derivatives, we can expect some new properties, which may lead it to many other applications in practice.
Table 3 The calculated two-photon absorptions by INDO/SDCI–SOS Compounds
Transition states
kTPA (nm)
d 1050 cm4 s photon1
Porphyrin (D2h ) Carbaporphyrin (C2v )
1 ! 13 1 ! 13 1 ! 32 1 ! 13 1 ! 27 1 ! 40 1 ! 13 1 ! 41 1 ! 52
598 602 431 609 457 412 595 400 404
59.04 8.32 568.40 8.14 601.68 2207.47 2.14 3146.36 1185.69
adj-Dicarbaporphyrin (C2v ) opp-Dicarbaporphyrin (D2h ) Tetracarbaporphyrin (D4h )
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Fig. 5. The relationship of the calculated two-photon absorption cross section dTPA ð1050 cm4 s photon1 Þ with the energy (eV).
It is obvious seen from Fig. 5 that two-photon absorptions of the porphyrin, carbaporphyrin, opp-dicarbaporphyrin and adj-dicarbaporphyrin exist only in a very limited area, while for tetracarbaporphyrin with only one peak value, its twophoton absorption exist in a large area, which is useful in many practical applications, such as in optical limiting.
4. Conclusions In this Letter, we have theoretically studied the geometries, one- and two-photon absorptions of porphyrin and a series of carbaporphyrins in which the N atoms are substituted by C atoms gradually, using both TDDFT and semi-empirical methods INDO/SDCI. The TDDFT method is proven to be
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an excellent alternative for the calculation of excitation energies, and the INDO/SDCI method is verified to be applicable for the investigated molecules. The calculated results indicate that the substitution of N atoms with C atoms not only lead to an enlargement of the molecular center in geometry but also to a red-shifting in the one-photon absorptions (except for the tetracarbaporphyrin), and there is a significant difference in two-photon absorption properties between the molecules with center-symmetry and the ones without centersymmetry. Especially, for the tetracarbaporphyrin, there is a qualitative change in its electronic structures and spectral properties. The calculated results indicate that although the tetracarbaporphyrin show a little smaller two-photon absorption cross section than other carbaporphyrin, it show a fairly large two-photon absorption cross section at the high-energy level in a wider area (shown in Fig. 5 for tetracarbaporphyrin) than porphyrin, this property may lead tetracarbaporphyrin derivatives to more practical applications, for example, in optical limiting.
Acknowledgements This work is supported by the National Nature Science Foundation of China (Nos. 20273023 and 90101026) and the Key Laboratory for Supramolecular Structure and Material of Jilin University.
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