Electronic Properties of a New Two-Photon Absorbing Fluorene

J. Phys. Chem. C 2009, 113, 20719–20724

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Electronic Properties of a New Two-Photon Absorbing Fluorene Derivative: The Role of Hartree–Fock Exchange in the Density Functional Theory Design of Improved Nonlinear Chromophores Ivan A. Mikhailov,† Mykhailo V. Bondar,‡ Kevin D. Belfield,§ and Arte¨m E. Masunov*,†,| NanoScience Technology Center, UniVersity of Central Florida, 12424 Research Parkway, Suite 400, Orlando, Florida 32826, Institute of Physics, National Academy of Sciences of Ukraine, Prospect Nauki 46, KieV-28, 03028, KieV, Ukraine, Department of Chemistry and CREOL, College of Optics and Photonics, UniVersity of Central Florida, P.O. Box 162366, Orlando, Florida 32816-2366, and Department of Chemistry and Department of Physics, UniVersity of Central Florida, P.O. Box 162366, Orlando, Florida 32816-2366 ReceiVed: July 20, 2009; ReVised Manuscript ReceiVed: September 18, 2009

One- and two-photon absorption (2PA) properties of a new fluorene derivative with diphenylamino and 2-(2′hydroxyphenyl)benzothiazole substituents were investigated theoretically using the density functional theory approach with different types of functionals and experimentally by two-photon induced fluorescence methodology. The effect of the exchange-correlation functional choice on the description of 2PA properties of fluorenes was analyzed. The best agreement of the experimental and theoretical 2PA spectra was obtained with a custom modification of the hybrid M05 functional that included 35% of Hartree-Fock exchange. This functional is recommended for reliable prediction of nonlinear optical properties of conjugated molecules. A chemical modification of the studied compound is suggested to increase the 2PA cross section. 1. Introduction Fluorene-type molecules with high fluorescence quantum yield and efficient two-photon absorption (2PA) cross sections have the potential for a variety of emerging applications, such as multiphoton fluorescence microscopy,1,2 three-dimensional optical data storage and microfabrication,3,4 optical power limiting,5,6 two-photon molecular fluorescence sensing and switching,7,8 and so forth. Therefore, it is important to investigate the electronic properties of this class of molecules. In this paper we report one-photon absorption (1PA) and 2PA spectra for 5-(2-(9,9-bis(2-(2-methoxyethoxy)ethyl)-2-(diphenylamino)-fluoren-7-yl)vinyl)-2-(benzothiazol-2-yl)phenol, called hereinafter compound 1 and shown in Figure 1. The spectral purity and chemical structure was confirmed by nuclear magnetic resonance spectroscopy. Details of the synthesis and comprehensive experimental analysis are published elsewhere.9 The spectra were measured in a weak-polar solvent, hexane. Calculations for 1 and some of its rotamers were carried out in vacuum. To improve nonlinear properties, some chemical modification of compound 1 is suggested. 2. Theory Density functional theory (DFT) in the well-known KohnSham (KS) formalism became the method of choice in the study of the electronic structure of organic molecules.10 Although DFT is exact in principle, its accuracy is limited by the approximate form of the energy functional. The exchange-correlation part * Corresponding author. E-mail: [email protected]. † NanoScience Technology Center, University of Central Florida. ‡ National Academy of Sciences of Ukraine. § Department of Chemistry and CREOL, College of Optics and Photonics, University of Central Florida. | Department of Chemistry and Department of Physics, University of Central Florida.

Figure 1. Chemical structure of 1 for which the measurments were done.

Figure 2. The molecules studied: (a) complete structure 1a; (b,c,d) substituents that replace the boxed one in 1a and make rotamers 1b, 1c, and 1d. The x-direction is defined from the left to the right-hand side of this figure.

of this functional is usually designed on the basis of satisfying a number of physical constraints and/or fitting to an empirical data set. Although systematic improvements of the exchangecorrelation functional is not possible, there has been steady progress in the development of approximate functionals, which were designed in order to meet additional physical constraints. The local (local density approximation or LDA), semilocal or gradient-dependent (generalized gradient approximation or

10.1021/jp906875b CCC: $40.75  2009 American Chemical Society Published on Web 11/04/2009

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Figure 3. 1PA spectrum for 1a obtained using TD-DFT with various exchange-correlation functionals. The experimental data for 1 in hexane is also plotted.

Figure 4. 2PA spectrum for 1a obtained using TD-DFT with various exchange-correlation functionals. The experimental 2PA spectrum for 1 in hexane is also plotted. The dotted lines connect absorption maxima (denoted by triagles) to guide the eye. They show trends in intensities and positions of the peaks with changes in the fraction of HFX in the DFT functionals. The roman numerals label the 2PA maxima discussed in the paper.

GGA), and kinetic energy dependent (meta-GGA) functionals were developed as rungs of the Jackob ladder.11 An important breakthrough was accomplished by Becke,12 who proposed to also include a fraction of the nonlocal orbital-dependent exchange energy, calculated with Hartree-Fock (HF) theory. The original (B3LYP)12 as well as the more recent (BMK, M05, and M05-2X)13,14 hybrid functionals allow for calculations of many molecular properties with nearly chemical accuracy at a relatively low computational cost.15-17 One of the widely used extensions of DFT for description of the excited states is time-dependent density functional theory (TD-DFT). It is based on approximate solution of the timedependent KS equations to first order (linear response) in the external electric field. The dependence of the functional on the frequency of the external field is usually ignored. This approximation is called an adiabatic TD-DFT and used in this paper. An added advantage of the KS formalism is an op-

Mikhailov et al.

Figure 5. 1PA and 2PA spectra of 1a, calculated at the TD-M051.25X/pc-1 level of theory. Experimental results for 1 are presented by circles. TD-DFT predictions according to the SOS expression with 28 states are plotted with solid lines. The vertical strokes show the wavelengths corresponding to the excited states, labled accordingly. The dotted line is plotted for the 2PA spectra obtained with four essential states: 1, 2, 3, and 10.

portunity to use a familiar concept of orbitals (single-electron wave functions) to analyze the electronic structure. The transition densities obtained from TD-DFT calculations can be expressed in the basis of KS orbitals. This allows one to analyze the electronic structure of the excited states. We refer to a transition energy as the energy difference between an excited state and the ground state. The electric dipole moment associated with the transition between two states is called the transition dipole moment. Using TD-DFT, one obtains transition dipole moments and transition energies from the ground state to a manifold of excited states. These values are sufficient to calculate a 1PA spectrum. The following expression can be used for the molar absorption coefficient ε(ω) at arbitrary angular frequency ω (in s-1) of the incident light simulating the 1PA spectrum:

ε(ω) )

10-3 4π2ω N ln 10 A 3pc

∑ ∑ |〈f|µR|0〉|2gf(ω) x,y,z

f

(1)

R

Here the summation on f runs over all the excited states, c is the speed of light, NA is Avogadro’s number, g is the line shape function, and 〈f|µR|0〉 is the transition dipole moment between the ground |0〉 and fth excited state |f〉. The normalized shape function

gf(ω) )

Γf 1 π (ω - ω)2 + Γ2 f0 f

(2)

corresponds to the fth state with the line width Γ usually taken as an empirical constant equal to 0.1 eV for all the excited states. The predictions of 2PA are more involved. For a single linear polarized laser beam, the 2PA cross section, averaged over isotropic molecular orientation can be expressed in centimetergram-second (cgs) units as follows:18

Electronic Properties of a New 2PA Fluorene Derivative

σ(2)(ω) )

16π3ω2 15c2

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x,y,z x,y,z

∑ ∑ ∑ (M fRRM ββf* + 2M Rβf M Rβf* )gf(2ω) f

R

β

(3) Here, R,β ) x,y,z, and M is a so-called 2PA transition moment, expressed as a sum over states (SOS) through transition dipoles between the states:

f MRβ )

1 2p

∑ k

[

〈f|µβ |k〉〈k|µR |0〉 〈f|µR |k〉〈k|µβ |0〉 + ωk0 - ω - iΓk ωk0 - ω - iΓk

]

(4)

In this expression, the summation runs over all the excited states; and the transition dipole moments between the fth and kth excited states (permanent dipoles, when f ) k) are

〈f|µR |k〉 ) 〈f|µR |k〉 - 〈0|µR |0〉δfk

(5)

To predict a 2PA spectrum, one needs transition dipole moments between excited states and permanent dipole moments for the complete manifold of excited states. These values can be obtained only if one retains the quadratic terms in the external field while solving the time-dependent equations. This quadratic approximation uses the states obtained in the linear response

approximation as the basis, and involves summation over infinite number of these states. This summation can also be done implicitly by solving the linear equations, as it is done in quadratic-response DFT (QR-DFT).19 Alternatively, one can carry out an explicit summation over a sufficient number of the lowest excited states.20 The state-to-state transition dipoles and permanent excited state dipoles can also be approximated using the a posteriori Tamm-Dancoff approximation (ATDA), and substituted in eqs 3-5 to predict 2PA. The ATDA was proposed recently and validated numerically against ab initio results for π-π* excited states of some linear polyenes.21 Transition energies and ground-to-excited transition dipoles in ADTA remain identical to the ones obtained in full linear response TD-DFT calculations. At the same time, ATDA allows calculations of the state-to-state transition dipoles without solving the equations of full QR-DFT. Thus, it can be used for a computationally inexpensive prediction of linear and nonlinear properties of organic molecules. 3. Computational Details All quantum-chemical calculations were performed using the Gaussian 03 rev. E1 suite of programs.22 To save computer time, the side chains in the 9-position of the fluorene ring in 1 were replaced with methyl groups, resulting in the simplified structure 1a. This simplification is justified since aliphatic substituents are not conjugated with the aromatic system and, therefore, are not expected to exhibit any noticeable effect on the electron

TABLE 1: Transition Energies (Eex in eV) of Excited States, Their Leading Configurations, and x-Projection (Defined in Figure 2) of Transition Dipole Moments [in D, See Eq 5] Obtained at the ATDA-M05-1.25X/pc-1 Level of Theorya max

state

Eex

leading configurations

S0

S1

S2

S3

S10

I II III IV

S1 S2 S3 S10

2.90 3.53 3.83 4.48

77(1-1′)-8(2-1′)+4(1-2′) 70(2-1′)+9(1-1′)-9(3-1′) 73(1-2′)-7(1-1′)-4(1-3′) 58(2-2′)+7(1-6′)-6(1-5′)

13.68 -4.84 3.71 0.46

10.70 9.36 -9.84 1.97

9.36 8.48 5.41 -9.31

-9.84 -5.41 12.67 4.60

1.97 -9.31 4.60 3.10

a Corresponding maxima (column Max) of the 2PA spectra are labeled in Figure 4 with roman numerals. The state notations are taken from Figure 5, the zeroth state corresponds to the ground state. The configurations are listed in the form w(i-k′), where (i-k′) represents a singly excited determinant obtained from the KS ground state by the electron transition from the ith occupied to the k′th vacant orbital, and w is the configuration weight in %. The ground state dipole is 0.92 D; its x-projection (equal to 0.40 D) is subtracted from the x-component of the permanent dipoles according to eq 5.

Figure 6. KS orbitals for 1a.

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structure or photophysical properties of the π-electron system.23,24 The ground-state geometries of this and some other molecules (shown in Figure 2) were optimized using the B3LYP exchangecorrelation functional and pc-1 basis set.25,26 This level of theory was recently shown27 to produce a good bond length alternation parameter, when compared with X-ray crystal structures. The 1PA and 2PA spectra were calculated at the TD-DFT/ pc-1 level of theory with various exchange-correlation functionals. The permanent and state to state transition diploes were obtained using ATDA, implemented in the locally modified version of the Gaussian 03 code. The SOS expressions 1 and 3 were used with the total number of states equal to 28. The highest excitation in this set corresponded to a wavelength of 226 nm, which was by 1.2 eV higher than the highest excitation energy measured in the 2PA experiment. The range of the experimental 2PA spectrum from the first maximum to the shortest wavelength edge is about 1.3 eV. The fraction of HF exchange (HFX) in the M05 hybrid functional was increased to 35% with nonstandard Gaussian options IOp(3/76 ) 0650003500, 3/78 ) 0930009300), corresponding to 35% HFX, and 65% meta-GGA exchange. This HFX fraction is 125% of that of the standard M05, and is considerably smaller than the respective fraction in the standard M05-2X (200% of the standard M05). The resulting exchangecorrelation functional is referred to as M05-1.25X in the following. 4. Experimental 1PA and 2PA Measurements The steady-state absorption spectrum of 1 was obtained in spectroscopic-grade hexane at room temperature with an Agilent 8453 UV-visible spectrophotometer in 10 mm path length quartz cuvettes with dye concentrations of C ∼ 2 × 10-5 M. The value of fluorescence quantum yields of the new fluorene derivative was measured relative to 9,10-diphenylanthracene in cyclohexane.28 The 2PA spectrum of 1 was determined over a broad spectral region by a typical two-photon-induced fluorescence (2PF) method relative to Rhodamine B in methanol as a standard.29 A PTI QuantaMaster spectrofluorimeter and femtosecond Clark-MXR CPA-2010 laser pump with an optical parametric generator/amplifiers (TOPAS, Light Conversion), with pulse duration ≈ 140 fs (fwhm), tuning range 580-940 nm, pulse energies e0.15 µJ, and 1 kHz repetition, rate were used. 2PF measurements were performed in 10 mm fluorometric quartz cuvettes with dye concentrations of ∼3 × 10-5 M in hexane. The experimental fluorescence excitation and detection conditions occurred with negligible reabsorption processes, which could effect 2PA measurements. For purity of the experiment, quadratic dependence of 2PF intensity on the excitation power was verified for each excitation wavelength. We report 2PA cross sections for photon wavelengths longer than 580 nm. At shorter wavelengths, the fluorescence could be depleted by one-photon stimulated emission from the first singlet excited state. As a result, quadratic dependence due to 2PA is difficult to observe, and therefore results at shorter wavelength are not reported. 5. Results and Discussions Molecule 1 has two single bonds that give rise to four planar conformer structures, shown in Figure 2 after geometry optimization at the B3LYP/pc-1 level of theory. The total energy (including nuclei repulsion) of the most stable isomer (a) is obtained to be -2202.4327 hartree, while the isomers (b, c, and d) are 12.1, 13.6, and 9.2 kcal/mol higher in energy, respectively. The lowest energy isomer is stabilized by the

Mikhailov et al. intramolecular H-bond OH · · · N. Its electronic structure was further studied in this paper. We compare the TD-DFT predicted 1PA spectrum for 1a with the experimental one for 1 in Figure 3. One can see that a high HFX fraction in the functionals (such as M05-2X with 56% of HFX) leads to a blue shift, whereas the low fraction (M05 with 28% of HFX) results in a red shift relative to the experimental absorption maximum. At the same time, the higher the HFX fraction in the functional, the smaller is the intensity ratio of the first-to-second absorption maxima counting from the longest wavelength edge of the spectrum. For instance, the B3LYP functional (which has 20% of HFX and is not shown in Figure 3) produces two maxima at 490 and 390 nm with almost equal intensities. The similar trend was described previously.30 The first excited state calculated for rotamers 1b, 1c, and 1d is blue-shifted by 14, 4, and 7 nm with respect to the one for 1a. The best transition energy for the first excited state is predicted by the BMK functional with 40% of HFX, while the M05 functional with 27% of HFX better reproduces the relative intensities of the first two maxima in 1PA spectra. The 2PA spectrum of 1a obtained with different DFT functionals is plotted in Figure 4 along with the experimental spectrum for 1 in hexane. The spectra calculated with BMK and M05 functionals both have a maximum around 600 nm, but differ in the number of maxima found in 650-750 nm range. Two maxima are predicted by the M05 functional, while the BMK functional produces only one. The functional M05-1.25X, with an intermediate fraction of HFX, reveals the reason for this difference. It helps to connect the peaks, thus demonstrating that an increase in the fraction of HFX leads to a uniform blue shift of the absorption maxima; meanwhile, their intensities vary in different ways. The corresponding trends are marked by dotted lines in Figure 4. The maximum at 690 nm (III) for the M05 functional is shifted to 650 nm for M05-1.25X, and to 610 nm for BMK, simultaneously increasing in intensity. The peak at 590 nm (IV) for M05 moves to a shorter wavelength and out of the 600-900 nm window reported in the experiment. The origin of this disagreement between the theory and experiment may be the adiabatic approximation or/and the selfinteraction error of the exchange-correlation functionals used here. The best agreement of the theory and experiment was obtained with a new M05-1.25X functional, defined in the Computational Details section. It is used in the remaining calculations. Our analysis of the permanent and state-to-state transition dipoles identified four essential excited states that contribute to the 2PA spectrum. The 2PA spectrum obtained in the four-state model is compared with the full SOS model including 28 states in Figure 5. The observed quantitative agreement confirms that the four chosen states are sufficient. For those states, labeled 1, 2, 3, and 10 in Figure 5, we list the transition energies, configurations, and dipole moments in Table 1. In order to describe configurations of excited states, we use shorthand notations 1, 2, ..., m for the occupied and 1′, 2′, ..., m′ for the vacant orbitals, with highest occupied molecular orbital (HOMO) being 1, and lowest unoccupied molecular orbital (LUMO) being 1′. The KS orbitals 1, 2, 3, 1′, 2′, 3′, 5′, 6′ that are active in the selected states are plotted in Figure 6. According to our analysis of eq 4, a main contribution to the 2PA transition moment Mfxx for all the final states f ) S1, S2, S3, S10 comes from large dipole moments S0fS1 and S1ff. This corresponds to the term of the SOS expression with k ) S1. The other two large contributions to the 2PA transition moment come from the states k ) S2, S3;

Electronic Properties of a New 2PA Fluorene Derivative

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TABLE 2: Positions and Values of Four Maxima in 2PA Spectra for All the Structures Drawn in Figure 2 and Figure 7 Simulated at the ATDA-M05-1.25X/pc-1 //B3LYP/pc-1 Level of Theory 2PA maximum one photon wavelength, nm

2PA cross section, GM

molecule

I

II

III

IV

I

II

III

IV

1a 1b 1c 1d 1am

852 825 844 838 911

700 677 687 689 717

646 620 641 627 671

552 542 549 546 562

179 114 172 113 353

446 451 499 568 577

844 887 882 948 715

1031 696 957 804 816

however, for the first three 2PA maxima (f ) S1, S2, S3) those contributions have an opposite sign to the first term. These negative contributions from states S2 and S3 reduce the 2PA cross section at the I, II, and III maxima. Opposite to that, at maximum IV (f ) S10) all the states contribute to eq 4 together with an equal sign, making state S10 the most active 2PA state. The first 2PA maximum, at 425 nm, is due to combination of the large transition dipole moment S0fS1 and large permanent dipole of the first excited state. This state is mostly the (HOMO-LUMO) transition. The HOMO is localized on the fluorene moiety and phenyl rings, whereas the LUMO is shifted to the phenol and benzothiazole (Figure 6). In order to improve 2PA absorption into S1, one has to increase the difference in permanent dipoles between the first excited and the ground states. To that end, we suggest the following molecular modifications: an amino group attached to the fluorene and nitro group attached to the benzothiazole should enforce the charge transfer character of the lowest excitation. The resulting chemical structure 1am is shown in Figure 7. According to the M05-1.25X/pc-1//B3LYP/pc-1 level of theory, this modification leads to an increase in the x-projection of the ground state dipole moment from 0.40 D for compound 1 to 6.25 D for compound 1am. However, the S1 permanent dipole for 1am increases even more, making the difference between x-projections of S1 and S0 permanent dipoles equal to 15.04 D. 2PA maxima for the modified molecule 1am and four different 1 conformers, defined in Figure 2, are listed in Table 2. From the data in Table 2, one can see that the first maximum of the 2PA spectrum in the modified molecule 1am has indeed increased in comparison with the 1a spectrum; the second maximum has slightly increased, while the other two maxima decreased. All the maxima have acquired a red shift. The obtained red shift and increase in 2PA cross section can be considered as an advantage for practical applications. All considered rotamers 1a-1d have similar 2PA spectra. The excitation wavelengths differ by less than 30 nm. Among them, compound 1a has largest cross sections at I and IV maxima. The closest to it is isomer 1c, which is, similar to 1a, stabilized by the intramolecular H-bond OH · · · S. The other isomers, 1b and 1d, have comparable 2PA cross sections. 6. Conclusions We presented a detailed analysis of the electronic structure for the ground and excited states in a push-pull substituted

Figure 7. Chemical structure of 1am, nitro-amino-substituted 1a.

fluorene derivative. Both linear and 2PA spectra were predicted with the recently developed ATDA to the second-order TDDFT in combination with the SOS approach. The variations in the predicted spectra due to the fraction of HFX in different hybrid exchange-correlation functionals were systematically analyzed, and the new functional that fits the experimental data the best was proposed. The 1PA and 2PA spectra for the investigated rotamers were predicted to be quite close to the ones for the most stable isomer. This allows one to conclude that the hydrogen bond between the N or S atom and the H atom from the OH group of compound 1 or from the solvent does not exert much influence on the spectra. Four essential electronically excited states were identified and shown to be sufficient to describe 2PA spectra in the entire range of the wavelengths. The analysis of these states in terms of KS orbitals, permanent and state-to-state transition dipoles allowed us to propose a molecular modification that is predicted to improve the 2PA absorption cross section by the factor of 2. The experimental test of the prediction is currently under way. Acknowledgment. This work is supported in part by the National Science Foundation (CCF-0740344 and CHE-0832622), the Civilian Research and Development Foundation (UKB22923-KV-07), and the Ministry of Education and Science of Ukraine (Grant M/49-2008). Research was performed using (1) the Stokes HPCC facility at the UCF Institute for Simulation and Training (IST), (2) the Bethe SMP server at the UCF NanoScience Technology Center (NSTC), and (3) the Bassi supercomputer at the National Energy Research Scientific Computing Center (NERSC), a DOE Office of Science user facility at Lawrence Berkely National Laboratory. References and Notes (1) Bhawalkar, J. D.; Swiatkiewicz, J.; Pan, S. J.; Samarabandu, J. K.; Liou, W. S.; He, G. S.; Berezney, R.; Cheng, P. C.; Prasad, P. N. Scanning 1996, 18, 562. (2) Zipfel, W. R.; Williams, R. M.; Christie, R.; Nikitin, A. Y.; Hyman, B. T.; Webb, W. W. Proc. Natl. Acad. Sci. U.S.A. 2003, 100, 7075. (3) Belfield, K. D.; Ren, X. B.; Van Stryland, E. W.; Hagan, D. J.; Dubikovsky, V.; Miesak, E. J. J. Am. Chem. Soc. 2000, 122, 1217. (4) Corredor, C. C.; Huang, Z. L.; Belfield, K. D.; Morales, A. R.; Bondar, M. V. Chem. Mater. 2007, 19, 5165. (5) Charlot, M.; Izard, N.; Mongin, O.; Riehl, D.; Blanchard-Desce, M. Chem. Phys. Lett. 2006, 417, 297. (6) Lin, T. C.; He, G. S.; Zheng, Q. D.; Prasad, P. N. J. Mater. Chem. 2006, 16, 2490. (7) Akiba, M.; Dvornikov, A. S.; Rentzepis, P. M. J. Photochem. Photobiol. A: Chem. 2007, 190, 69. (8) Maeda, H.; Tierney, D. L.; Mariano, P. S.; Banerjee, M.; Cho, D. W.; Yoon, U. C. Tetrahedron 2008, 64, 5268. (9) Morales, A. R.; Schafer-Hales, K. J.; Yanez, C. O.; Bondar, M. V.; Przhonska, O. V.; Marcus, A. I.; Belfield, K. D. ChemPhysChem, in press. (10) Toro, C.; Thibert, A.; De Boni, L.; Masunov, A. E.; Hernandez, F. E. J. Phys. Chem. B 2008, 112, 929. De Boni, L.; Toro, C.; Masunov, A. E.; Hernandez, F. E. J. Phys. Chem. A 2008, 112, 3886. Patel, P. D.; Masunov, A. E. J. Phys. Chem. A 2009, 113, 8409. Patel, P. D.; Mikhailov, I. A.; Belfield, K. D.; Masunov, A. E. Int. J. Quantum Chem. 2009, 109, 3711. Mikhailov, I. A.; Belfield, K. D.; Masunov, A. E. J. Phys. Chem. A 2009, 113, 7080. Toro, C.; De Boni, L.; Yao, S.; Ritchie, J. P.; Masunov,

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