Letter pubs.acs.org/JPCL
Excited-State Absorption from Real-Time Time-Dependent Density Functional Theory: Optical Limiting in Zinc Phthalocyanine Sean A. Fischer,† Christopher J. Cramer,‡ and Niranjan Govind*,† †
William R. Wiley Environmental Molecular Sciences Laboratory, Pacific Northwest National Laboratory, P.O. Box 999, Richland, Washington 99352, United States ‡ Department of Chemistry, Supercomputing Institute and Chemical Theory Center, University of Minnesota, Minneapolis, Minnesota 55455, United States ABSTRACT: Optical-limiting materials are capable of attenuating light to protect delicate equipment from high-intensity light sources. Phthalocyanines have attracted a lot of attention for optical-limiting applications due to their versatility and large nonlinear absorption. With excited-state absorption (ESA) being the primary mechanism for optical limiting behavior in phthalocyanines, the ability to tune the optical absorption of ground and excited states in phthalocyanines would allow for the development of advanced optical limiters. We recently developed a method for the calculation of ESA based on realtime time-dependent density functional theory propagation of an excited-state density. In this work, we apply the approach to zinc phthalocyanine, demonstrating the ability of our method to efficiently identify the optical limiting potential of a molecular complex.
T
he proliferation of intense light sources has led to a commensurate increase in the need to protect lightsensitive equipment. Optical-limiting materials show a strong attenuation of light transmission at high input intensities.1,2 Porphyrin- and phthalocyanine-based materials have attracted a great deal of attention for optical-limiting applications due to their large nonlinear optical properties, fast response times, and ability to tune their optical properties through chemical modifications.3−10 A previous study examined the optical-limiting potential of free-base phthalocyanine through a series of real-time timedependent density functional theory (RT-TDDFT) simulations by applying increasingly intense electric fields.11 This approach allowed the authors to predict the fluence dependence of the material’s absorption spectrum and enabled them to estimate the threshold intensity of the material; however, the influence of the triplet manifold was not examined, and many simulations, at different field strengths, were required to determine the fluence dependence. With excited-state absorption (ESA) being the suggested mechanism of optical-limiting behavior in phthalocyanines,4,11 our recently introduced method for the calculation of ESA with RT-TDDFT12 is well suited to study their optical-limiting potential. The optical-limiting behavior in these molecules is often interpreted by means of a five-state model (Figure 1). An intense light source creates a substantial population of the excited state. Molecules in the excited state can then absorb an additional photon to be further excited. This population and further excitation of excited states is the main reason for nonlinear absorption under intense light for phthalocyanines. The interplay of the duration of the incident pulse, the excited© XXXX American Chemical Society
Figure 1. Left: Geometry of ZnPc optimized at the B3LYP/6-31G* level on the singlet ground state. Right: Schematic energy level diagram for ZnPc, consisting of the ground singlet state (S0), first excited singlet state (S1), first excited (with respect to S0) triplet state (T1), and the higher excited singlet (Sn) and triplet (Tn) states.
state lifetime, and the intersystem crossing rate determines whether singlet or triplet ESA is relevant. For zinc phthalocyanine (ZnPc), triplet quantum yields between 0.58 and 0.98 and intersystem crossing lifetimes between 4.5 and 5.6 ns have been reported,13 indicating that for nanosecond or longer pulses triplet−triplet ESA would be observed, while for shorter pulses singlet ESA would be observed. A necessary, albeit insufficient, condition for effective optical limiting is for the ESA cross-section to be larger than the ground-state absorption (GSA) cross section in the relevant region of the spectrum. Additional conditions, which must also Received: February 8, 2016 Accepted: March 23, 2016
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where c is the speed of light. The absorption cross section can then be obtained from the dipole strength function via the relation
be met for a material to be an effective optical limiter, include a high linear transmission over a broad spectral range, a lowintensity threshold, and a high saturation fluence.4 The first additional criterion is trivially included in our calculations. The ability to calculate the intensity threshold was demonstrated in the work previously mentioned,11 and so we feel our current contribution is complementary to that work. A high saturation fluence is typically accomplished through a high concentration of absorbers in the beam path and therefore is related to the solubility of the material or whether the material is a pure liquid or has the ability to produce a high-quality film. Even though our calculations do not address the latter two conditions, these conditions are rendered irrelevant if the first two conditions (ESA larger than GSA and high linear transmission) are not met. In this work we examine the ability of our approach for the calculation of ESA12 to characterize the optical-limiting potential of materials, using ZnPc (Figure 1) as an application. We provide a brief overview of our RT-TDDFT-based method here and direct the reader to our previous works for additional details.12,14,15 The basic premise of our approach is the propagation of an excited-state density with RT-TDDFT. Linear-response TDDFT (LR-TDDFT) is used to obtain the excited-state density, including the effects of orbital relaxation, and we represent that density through the one-particle reduced density matrix in an atomic orbital basis. We use the density matrix form of the TDDFT equations of motion to propagate the excited-state density in time ∂ P′(t ) = [F′(t ), P′(t )] ∂t
σ(ω) =
2π 2ℏ2 S(ω) mec
(5)
where me is the electron mass. The GSA spectrum of phthalocyanines consists of two prominent absorption bands, the Q band in the visible and the B (or Soret) band in the ultraviolet.16 Originally these bands were thought to be composed of π−π* transitions.16 More recent work has shown that while this is the case for the Q band, the B band also has contributions from n−π* transitions.17 The yellow to violet parts of the spectrum between these two bands lack any significant absorption. This feature of phthalocyanines is one of the reasons that they are attractive for optical-limiting applications: While there is transmittance at these wavelengths under low-intensity light, a nonlinear absorption appears under high-intensity light.3−8 As can be seen from Figure 2 and Table 1, all of the tested functionals predict very similar Q bands for the GSA (solid red
(1)
The prime indicates quantities in an orthogonal basis, with P′ being the density matrix and F′ being the Fock matrix. Absorption spectra are calculated from the fluctuations of the time-dependent dipole moments δμ(t ) = Tr[P(t )D] − μref
(2)
where P is the density matrix in the atomic orbital basis, D is the dipole moment matrix in the atomic orbital basis, and μref is a reference dipole moment. For a calculation on the ground state, the reference dipole is the equilibrium dipole moment of the molecule. For a calculation on the excited state, the combination of the change in reference state from the ground state to the excited state coupled to approximate exchangecorrelation functionals results in the reference dipole being time-dependent.12 It is computed from a simulation starting from the excited-state density but without an applied electric field. The ratio of the Fourier transforms of the dipole moment fluctuations and applied electric field gives the dipole polarizability tensor αjk(ω) =
Figure 2. Simulated absorption spectra for the singlet ground state (S0), the first singlet excited state (S1), and the lowest energy triplet state (T1). The calculations were performed with the exchangecorrelation functional indicated on the plot. The spectra have been normalized to the Q band of each ground-state spectrum.
Table 1. Positions of Prominent Features in the Groundand Excited-State Absorption Spectraa exp. gasb exp. DMSOc B3LYP M06 CAM-B3LYP BHLYP
δμj (ω) Ek (ω)
(3)
4πω Tr[Im[α(ω)]] 3c
S0→S2 (B)
S1→Sn
T1→Tn
1.88 1.85 2.09 2.02 2.04 2.06
3.80 3.59 3.67 3.84 4.20 4.32
NA 1.97, 2.56 2.31 2.43 2.28, 2.78 2.24, 2.84
NA 2.58 2.37, 2.57 2.54 2.18, 3.01, 3.24 2.31, 3.34
a All values are given in electron volts (eV). bFrom ref 16. cFrom ref 13. (ESA spectra were measured only up to 2.76 eV.)
where j and k index the Cartesian component of the dipole moment and applied electric field polarization and E(ω) is the Fourier transform of the applied electric field. For a delta function field, the Fourier transform is simply the strength of the applied field. The dipole strength function is then calculated as S(ω) =
S0→S1 (Q)
line in the plots); however, the location and relative intensity of the B bands are much more sensitive, with the percentage of Hartree−Fock (HF) exchange correlating with the energy and intensity of the B band. This was previously noted in studies of the ground-state absorption spectrum of ZnPc and Znporphyrin, with the cause being related to a stabilization of
(4) 1388
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The Journal of Physical Chemistry Letters the nitrogen n-orbitals relative to the phthalocyanine πorbitals.17,18 The gap between the Q and B bands predicted by the M06 functional is closest to that observed in the gasphase measurement.16 B3LYP underestimates this gap somewhat, while CAM-B3LYP and BHLYP both overestimate the gap between the bands. Turning to the ESA presented in Figure 2, we again see significant sensitivity to the choice of functional used in the calculation. Functionals with a higher percentage of HF exchange predict excited-state absorption that occurs higher in energy. Collected in Table 1 are the current results along with those determined from the transient absorption measurements of Savolainen et al.13 Experimentally, two peaks were observed in the singlet ESA spectrum and a single peak was observed in the triplet ESA spectrum. It should be noted that the transient absorption results of Savolainen et al. were only measured up to 2.76 eV. For the singlet ESA, B3LYP and M06 both predict a single peak that falls in between the experimentally measured ones. BHLYP and CAM-B3LYP both predict two features in the singlet ESA; although, each occurs 0.2 to 0.3 eV higher in energy than the peaks in the experimental spectrum. For the triplet ESA, experimentally a single feature was observed, although this feature was extremely broad. B3LYP predicts a dual-peak feature at approximately the correct energy, while M06 predicts a single peak (albeit with a shoulder) also at the correct energy. BHLYP and CAM-B3LYP both predict lowenergy peaks that were not observed experimentally. The other peaks observed in the triplet ESA spectra from BHLYP and CAM-B3LYP are beyond the energy range of the experimental measurement. From these results, the M06 functional appears to give the best agreement with experiment for all three absorption spectra. While there are now a number of “rules of thumb” with respect to expected accuracies of various density functionals19,20 when predicting ground-state absorption spectra, more data will be required to assess whether the sensitivity of different excitedstate absorptions to such features in functionals as percentage HF exchange, range separation, and so on is precisely analogous. We note that the triplet ESA can be considered a ground-state calculation in the triplet spin manifold. Our RTTDDFT results for the triplet are equivalent to those that would be obtained from LR-TDDFT calculations, as is the case for the S0 spectra. To focus in on the optical-limiting potential of ZnPc, we have computed the ratio of unnormalized ESA to unnormalized GSA and displayed the results in Figure 3. These results represent a best-case scenario as they assume complete population of the relevant excited state. A result above the red line corresponds to increased absorption in the excited state, whereas a result below the red line corresponds to increased transmission in the excited state. While the details depend on the functional used in the calculation, it is clear from the calculations that ZnPc is capable of significant optical limiting behavior in either the singlet or triplet state across much of the spectral range of the optical window in the GSA spectrum, in agreement with experimental results.8,13 Concentrating on the M06 results, which we previously suggested as the most accurate, we can see that we would expect increased transmission of red and orange light in the singlet excited state, with a corresponding increase in absorption of yellow, green, and blue light. For the triplet excited state, we expect an increase in the transmission of red,
Figure 3. Ratio of the singlet and triplet ESA to the ground state. The calculations were performed with the exchange-correlation functional indicated on the plot, and the results are plotted on a logarithmic scale. Above the red line corresponds to higher absorption in the excited state, while below the red line corresponds to higher absorption in the ground state.
orange, and yellow light with a decrease in the transmission of green, blue, and violet light. These results show that ZnPc has the intrinsic ability to be a limiting material for light in the yellow to violet range, depending on the duration of the incident light pulse. In the present work we have demonstrated how our recently introduced method for the calculation of ESA spectra can be used to determine the optical-limiting potential of a molecular complex. Our method allows the absorption spectrum of the ground singlet state, excited singlet state, and excited triplet state to be calculated within a single framework, RT-TDDFT. With these data, molecules and modifications to molecules can be screened according to the relative absorbance in the ground and excited states, thereby potentially guiding the design and creation of new optical limiting materials. We have observed sensitivity of the calculated ESA spectra to the functional, and in future work we plan on taking a closer look at the performance of our method in general as well as the performance of different functionals. Additionally, our approach opens up a new space for the development of exchangecorrelation functionals specifically designed to accurately capture the excited-state response of molecular complexes.
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COMPUTATIONAL METHODS All calculations were carried out with the development version of NWChem.21 For the current study, eq 1 was integrated using a second-order Magnus propagator with a time step of 0.2 atomic units (au) and a simulation time of 1000 au (∼24.2 fs). Delta function electric fields with a field strength of 1 × 10−4 au were used to excite the molecule. The geometry of ZnPc was optimized at the B3LYP22/6-31G(d)23−25 level in the singlet ground state (see Figure 1). The same basis set was used for the RT-TDDFT simulations, along with the B3LYP, BHLYP,26 CAM-B3LYP,27 and M0628 exchange-correlation functionals. The B3LYP and BHLYP functionals are global hybrids with a fixed percentage of Hartree−Fock (HF) exchange (20 and 50%, respectively). The CAM-B3LYP functional is a range-separated hybrid functional with 19% HF exchange at short range and increasing to 65% HF exchange at long range. The M06 functional is a hybrid meta functional with a fixed percentage of 1389
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HF exchange (27%) and a dependence on the electronic kinetic energy density. The time-dependent dipole moments were damped by an exponential function with a time constant of 250 au before taking the Fourier transform, resulting in peaks in the absorption spectra with a full width at half-maximum of ∼0.22 eV. The initial ground singlet state (S0) and lowest energy triplet state (T1) densities were obtained via solution of the selfconsistent field equations for the relevant spin symmetries, while the lowest energy excited singlet state (S1) density was obtained from a LR-TDDFT calculation on the S0 state. All absorption spectra were calculated via RT-TDDFT simulations starting from the relevant density. For the M06 functional, second and higher derivatives of the functional have not been implemented in NWChem, making calculation of the singlet excited-state density not possible. For the calculation of the singlet ESA with the M06 functional, we started with the singlet excited-state density calculated with the B3LYP functional. As discussed in our previous work,12 the absorption spectrum calculated from RT-TDDFT is robust with respect to variations in the initial density. We now show that this idea easily extends to the case of calculating the initial density with one functional and propagating with another functional. In particular, we have calculated the singlet excited-state absorption for ZnPc with the B3LYP functional starting from the B3LYP, PBE0, or LDA excited-state density. The results are displayed in Figure 4,
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Division of Chemical Sciences, Geosciences, and Biosciences and the Office of Advanced Scientific Computing Research through the Scientific Discovery through Advanced Computing (SciDAC) program under Award Numbers KC-030106062653 (S.A.F., N.G.) and DE-SC0008666 (C.J.C.). The research was performed using EMSL, a DOE Office of Science User Facility sponsored by the Office of Biological and Environmental Research and located at the Pacific Northwest National Laboratory (PNNL). PNNL is operated by Battelle Memorial Institute for the United States Department of Energy under DOE contract number DE-AC05-76RL1830. The research also benefited from resources provided by the National Energy Research Scientific Computing Center (NERSC), a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No.DE-AC02-05CH11231, and resources provided by PNNL Institutional Computing (PIC).
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Figure 4. Comparison of the singlet excited-state absorption spectrum calculated from various combinations of functionals. The first functional listed indicates the functional used in the calculation of the excited-state density, while the second functional listed indicates the functional used in the propagation of the excited-state density. The results indicate that it is the functional used for propagation that determines the spectrum: almost the identical spectrum results when the B3LYP functional is used for propagation no matter which functional is used to calculate the starting density.
where the first functional given in the key is the one used for calculating the initial density and the second functional is the one used for propagation. We have also displayed the results for the PBE0 and LDA functionals. It is clear from the figure that it is the functional used for propagation that determines the computed spectrum (thereby further validating our approach for S1 with the M06 functional).
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REFERENCES
(1) Tutt, L. W.; Boggess, T. F. A Review of Optical Limiting Mechanisms and Devices Using Organics, Fullerenes, Semiconductors and Other Materials. Prog. Quantum Electron. 1993, 17, 299−338. (2) Sun, Y.-P.; Riggs, J. E. Organic and Inorganic Optical Limiting Materials. From Fullerenes to Nanoparticles. Int. Rev. Phys. Chem. 1999, 18, 43−90. (3) Calvete, M.; Yang, G. Y.; Hanack, M. Porphyrins and Phthalocyanines as Materials for Optical Limiting. Synth. Met. 2004, 141, 231−243. (4) Hanack, M.; Schneider, T.; Barthel, M.; Shirk, J. S.; Flom, S. R.; Pong, R. G. S. Indium Phthalocyanines and Naphthalocyanines for Optical Limiting. Coord. Chem. Rev. 2001, 219−221, 235−258. (5) Perry, J. W.; Mansour, K.; Marder, S. R.; Perry, K. J.; Alvarez, D.; Choong, I. Enhanced Reverse Saturable Absorption and Optical Limiting in Heavy-Atom-Substituted Phthalocyanines. Opt. Lett. 1994, 19, 625−627. (6) Perry, J. W.; Mansour, K.; Lee, I.-Y. S.; Wu, X.-L.; Bedworth, P. V.; Chen, C.-T.; Ng, D.; Marder, S. R.; Miles, P.; Wada, T.; et al. Organic Optical Limiter with a Strong Nonlinear Absorptive Response. Science 1996, 273, 1533−1536. (7) Wei, T. H.; Hagan, D. J.; Sence, M. J.; Van Stryland, E. W.; Perry, J. W.; Coulter, D. R. Direct Measurements of Nonlinear Absorption and Refraction in Solutions of Phthalocyanines. Appl. Phys. B: Lasers Opt. 1992, 54, 46−51. (8) Santhi, A.; Namboodiri, V. V.; Radhakrishnan, P.; Nampoori, V. P. N. Spectral Dependence of Third Order Nonlinear Optical Susceptibility of Zinc Phthalcyanine. J. Appl. Phys. 2006, 100, 053109. (9) de la Torre, G.; Vázquez, P.; Agulló-López, F.; Torres, T. Role of Structural Factors in the Nonlinear Optical Properties of Phthalocyanines and Related Compounds. Chem. Rev. 2004, 104, 3723−3750. (10) O’Flaherty, S. M.; Hold, S. V.; Cook, M. J.; Torres, T.; Chen, Y.; Hanack, M.; Blau, W. J. Molecular engineering of peripherally and axially modified phthalocyanines for optical limiting and nonlinear optics. Adv. Mater. 2003, 15, 19−32. (11) Cocchi, C.; Prezzi, D.; Ruini, A.; Molinari, E.; Rozzi, C. A. Ab Initio Simulation of Optical Limiting: The Case of Metal-Free Phthalocyanine. Phys. Rev. Lett. 2014, 112, 198303. (12) Fischer, S. A.; Cramer, C. J.; Govind, N. Excited State Absorption from Real-Time Time-Dependent Density Functional Theory. J. Chem. Theory Comput. 2015, 11, 4294−4303.
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. 1390
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