First-Principles Simulations of One- and Two-Photon Absorption Band

Article pubs.acs.org/JPCB

First-Principles Simulations of One- and Two-Photon Absorption Band Shapes of the Bis(BF2) Core Complex Robert Zaleśny,*,† N. Arul Murugan,‡ Guangjun Tian,‡,⊥ Miroslav Medved’,¶ and Hans Ågren‡ †

Department of Physical and Quantum Chemistry, Faculty of Chemistry, Wrocław University of Technology, Wyb. Wyspiańskiego 27, PL-50370 Wrocław, Poland ‡ Division of Theoretical Chemistry and Biology, School of Biotechnology, Royal Institute of Technology, SE-10691 Stockholm, Sweden ¶ Department of Chemistry, Faculty of Natural Sciences, Matej Bel University, Tajovského 40, 974 01 Banská Bystrica, Slovak Republic ABSTRACT: Motivated by the outstanding properties of bis(BF2) core complexes as fluorophore probes, we present a systematic computational study of their vibrationally resolved one- and two-photon absorption spectra in vacuum and in solution. Electronic and vibrational structure calculations were performed using the coupled cluster CC2 method and the Kohn−Sham formulation of density functional theory (DFT). A nonempirical estimation of the inhomogeneous broadening, accomplished using the polarizable embedding (PE) approaches combined with time-dependent DFT and CC2 methods, is used as a key ingredient of the computational protocol employed for simulations of the spectral features in solution. The inhomogeneous broadening is also determined based on the Marcus theory employing linear response and state-specific polarizable continuum model (PCM) methods. It is found that the polarizable embedding CC2 and the state-specific PCM are the most successful approaches for description of environmental broadening. For the 11Ag → 11Bu transition, the non-Condon effects can be safely neglected and a fair agreement between the simulated and experimental band shapes is found. In contrast, the shape of the vibrationally resolved band corresponding to the two-photon allowed 11Ag → 21Ag transition is largely dominated by non-Condon effects. A generalized few-level model was also employed to analyze the mechanism of the electronic two-photon 11Ag → 21Ag excitation. It was found that the most important optical channel involves the 11Bu excited state. Ramifications of the findings for general band shape modeling are briefly discussed.

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INTRODUCTION The 4,4-difluoro-4-bora-3a,4a-diaza-s-indacene, or boron-dipyrromethene (BODIPY), complex is recognized as a versatile fluorophore with many rewarding applications owing to its facile synthesis and excellent spectroscopic properties referring to narrow absorption and emission bands, good photostability, and high molar extinction coefficients or quantum yields. Synthetic routes encompass not only incorporation of heavy atoms (iodine, bromine, selenium, sulfur, or lanthanides) but also heavy atomfree BODIPY photosensitizers which are of recent interest because of their use in environment-friendly applications. It has also been demonstrated that BODIPY derivatives can be employed as effective fluorescent chemosensors and fluorescent chemodosimeters for ions (e.g., Hg2+, Cu2+).1,2 Some of the synthesized compounds exhibit a highly selective chelation enhanced fluorescence effect.1 BODIPYs can also be functionalized to work as fluorescent pH probes by substitution with pHdependent moieties.3 There have also been attempts to use the materials based on dipyrromethene and azadipyrromethenes for organic photovoltaic applications.4 Some studies were performed with a consideration that these dyes can be applied as potential photosensitizers in dye-sensitized solar cells.5 To further extend © XXXX American Chemical Society

the broad range of applications, there are also attempts to synthesize other compounds with BF2 moiety.6 It should moreover be underscored that BODIPYs have huge potential for bioimaging7−9 and that many efforts have been reported in the synthesis of BODIPY-based photosensitizers for photodynamic therapy.10,11 The applications of BF2-carrying compounds in bioimaging exploit the two-photon induced emission process. Recently, Li et al.12 have synthesized a bis(BF2) core complex containing 1,8naphthyridin derivative (cf. molecule 2, Scheme 1). As demonstrated by these authors, the complex in question exhibits yellow-green emission with high quantum yield, and the corresponding two-photon absorption (TPA) cross section at 730 nm roughly equals 100 GM. A high two-photon action cross section (the product of fluorescence quantum yield and the twophoton absorption cross section) indicates the potential utility of this bis(BF2) core complex, and presumably other BF2-carrying compounds, for highly efficient two-photon excited fluorescence, Received: October 6, 2015 Revised: December 7, 2015

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DOI: 10.1021/acs.jpcb.5b09726 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry B

and Brown is that for the functionals which deliver the smallest errors in excitation energy (BLYP and PBE) the values of R2 are quite small (roughly equal to 0.8). On the other hand, much larger values of R2 are found for range-separated functionals and hybrid GGA functionals. This is line with the results presented earlier by Chibani et al.29 It seems that the consensus on the choice of exchange-correlation functional for the studies of optical properties of BF2-carrying compounds has not yet been reached. Thus, in the present study we employ several exchangecorrelation functionals to predict the band shapes in the onephoton spectrum and compare them with the results of CC2 calculations and experimental data. On the basis of the thus selected functional, we then simulate the two-photon absorption spectrum and perform the analysis of the multiphoton absorption mechanism. This paper is organized as follows: First, we present the results of calculations of the lowest-energy one-photon absorption band corresponding to the 11Ag → 11Bu transition for bis(BF2) core complex (2) (and its unmethylated form, 1), including vibrational fine structure. The inhomogeneous broadening for 11Ag → 11Bu and 11Ag → 21Ag transitions is estimated based on the Marcus theory as well as using polarizable embedding (PE) model combined with TD-DFT and CC2 methods. In the following section, on the basis of the one-photon absorption results, we simulate the vibrational fine structure of the absorption band corresponding to two-photon 11Ag → 21Ag transition. Before concluding, we analyze the two-photon absorption mechanism with the aid of the generalized few-level model of Alam et al.34,35

Scheme 1. Molecules Investigated in the Present Study

allowing for applications of these systems in fluorescence microscopy. To fully exploit properties for this important class of fluorophores by maximization of two-photon absorption cross section, it becomes imperative to understand the nonlinear spectrum and a sometimes quite complicated multiphoton absorption mechanism. Quantum-chemistry studies of twophoton absorption in BF2-carrying compounds are scarce;13 hence, the reliability of electronic structure methods in predicting two-photon absorption strength and band shapes is yet to be addressed. Likewise, not much is known about optical channels involved in TPA for these compounds. The primary goal of the present study is to perform computer simulations to determine the one- and two-photon absorption spectrum of bis(BF2) core complexes in vacuum and in solution (dichloromethane (DCM) and acetonitrile (ACN) solvents) and to analyze the multiphoton absorption mechanism for the longwavelength part of the spectrum. Moreover, the role of nonCondon effects in the TPA spectra of BF2-carrying compounds has not yet been examined, although there are reports that they are important for two-photon absorption of other π-conjugated molecules.14−18 To elucidate this subject, we will simulate the absorption band corresponding to the two-photon 11Ag → 21Ag transition including the Franck−Condon (FC) and Herzberg− Teller (HT) contributions (including a linear term in the HT expansion19). For a presentation of recent developments in modeling optical band shapes for molecules in solution, we refer to refs 20−25. In contrast to the scarce availability of studies concerning TPA, there is much more data regarding the performance of quantum-chemistry methods in predicting onephoton spectra of BF2-carrying compounds, including the vibrational fine structure of absorption bands.26−32 Recently, Momeni and Brown presented the results of calculations of excitation energies for a large set of BODIPYs, performed using time-dependent density functional theory (TDDFT) as well as post-Hartree−Fock wave functions.33 These authors employed nine exchange-correlation functionals, including BLYP, PBE, B3LYP, PBE0, LC-BLYP, LC-PBE, CAMB3LYP, ωB97X-D, and LC-ωPBE, as well as ab initio wave function methods: CIS, CIS(D), EOM-CCSD, SAC-CI, LCC2*, CASSCF, CASPT2, CCSDR(T), and CCSDR(3). Momeni and Brown reported that mean absolute errors for TD-DFT excitation energies span from 0.364 eV (BLYP) to 0.572 eV (LC-ωPBE). Much better accuracy was obtained for some of the post-Hartree−Fock methods, i.e., mean absolute error (employing the cc-pVDZ basis set) was found to be 0.148 eV (LCC2*), 0.187 eV (SAC−CI), and 0.161 eV (CASPT2). This parallels the conclusions reported by Chibani et al., who reported that CIS(D) and SOS-CIS(D) methods clearly outperform TDDFT.32 Another important conclusion presented by Momeni

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COMPUTATIONAL DETAILS Simulations of One-Photon Absorption Spectra. In the present study we determined vertical excitation energies using TD-DFT and the RI-CC2 method. For this, we used Gaussian 0936 (TD-DFT) and Turbomole37 (RI-CC2) programs. Several exchange-correlation functionals were employed, including B3LYP, CAM-B3LYP, PBE0, BHandHLYP, and M06-HF. Vibrational fine structure of the band corresponding to the 11Ag → 11Bu transition was determined including the Franck− Condon (FC) as well as the Herzberg−Teller (HT) contributions. To determine the FC spectra, we employed a linear coupling model (LCM)38,39 also referred to as the independent mode displaced harmonic oscillator model (IMDHO).40,41 LCM assumes no changes to the curvature of the corresponding potential energy surfaces (PES) in the two electronic states and no mixing of vibrational modes. The vibrational structure is determined by the displacements (Δk) between the two sets of PESs which can be evaluated as Δk =

Fk ωk

2

,

Fk =

∂E , ∂Q k

k = 1, ···, N (1)

Here, ωk and Qk are the vibrational frequency and the normal coordinate of mode k, respectively; E is the total energy of the excited state, and N is the number of normal modes. For vibrational structure calculations, the geometries of two studied molecules (in the ground and electronic excited state 11Bu) were optimized with tight convergence criteria so that the root-meansquare (RMS) gradient was less than 10−7 Hartree/Bohr. To determine the non-Condon contributions to the one-photon absorption band, the transition moment nuclear derivatives were evaluated numerically by computations of the property at a set of displaced geometries. The stability of numerical differentiation B


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The Journal of Physical Chemistry B was controlled with the aid of the Rutishauser−Romberg algorithm (employing the displacements ±2nh, where h = 0.004 bohr and n = 0, 1, 2). Calculations based on LCM were performed with the orca_asa module which is a part of the Orca program,42 while other simulations of one-photon absorption spectra were performed using the FCclasses program.43,44 Simulations of Two-Photon Absorption Spectra. Twophoton absorption is a third-order process related to the imaginary part of the second hyperpolarizability.45−47 The second-order transition moment, Sab, can be identified from the single residue of the quadratic response function.48 At a twophoton resonance, probed by two monochromatic photons (ω= ωf 0/2), the two-photon absorption transition matrix from the ground state to an excited state |f⟩ is given by Sab =

⎡ ⟨0|μ ̂ |n⟩⟨n|μ ̂ |f ⟩ a b

∑⎢ n>0

⎢⎣ ωn0 − ωf 0 /2

+

⟨0|μb̂ |n⟩⟨n|μâ |f ⟩ ⎤ ⎥ ωn0 − ωf 0 /2 ⎥⎦

Two sets of systems were prepared in this study, namely, 2 in ACN and 2 in DCM. In each case, the molecular structure for 2 is based on the geometry optimized in the respective solvents. Furthermore, charges were obtained taking into account the specific solvent environment. In particular, the molecular structure and charges for 2 were obtained by employing density functional theory with BHandHLYP exchange-correlation functional and the 6‑311G(d,p) basis set, with solvent description using polarizable continuum model55 as implemented in Gaussian 09 software.36 It is notable that the charges for 2 were obtained through fitting to the molecular electrostatic potential as implemented in the CHELPG procedure56 in Gaussian 09. A similar protocol has been used to obtain the molecular structure and charges for the two solvents. The molecular structure and the charges obtained for 2 in solvents as described above have been used in the molecular dynamics simulations. The description of nonbonded interactions of these molecules is based on general amber force-field (GAFF).57 The input structure for the 2−ACN solute−solvent system was composed of a single molecule of 2 and 2364 molecules of the solvent. The simulation box dimensions were approximately 65.3 × 54.0 × 56.6 Å3. Likewise, the 2−DCM system was composed of a single molecule of 2 and 2080 solvent molecules in an orthorhombic box with approximate dimensions 67.4 × 71.3 × 60.0 Å3. Minimization runs were initially performed to remove any hot spots in the prepared input structure. Then simulations in the isothermal−isobaric ensemble were carried out keeping the temperature at 300 K and the pressure at 1 atmospheric pressure. The time step for the integration of equation of motion was set to 2 fs. A cutoff of 12 Å was used for calculating nonbonded interactions for 2 in both solvents. The electrostatic interactions beyond this cutoff distance have been taken into account using the Ewald summation method. The convergence of the total system density and energies has been used to decide the time scale for equilibration run. For example, followed by an equilibration run that lasted for 2 ns, a production run of total time scale 20 ns has been carried out for 2 in both solvents. During the whole simulation, only the solvents are treated as flexible molecules and molecule 2 has been treated as a rigid molecule retaining its geometry as in the initial structure during the entire simulations. As discussed in a previous paper,58 this has been done to avoid double counting the contributions from vibrational modes to the broadening of the absorption spectra of the molecule in solution. The final trajectory that corresponds to 10 ns has been used for preparing input files for the subsequent TD-DFT/MM and PERI-CC2 computations, performed uisng the Dalton and Turbomole programs, respectively. We have employed a polarizable electronic embedding scheme to describe the interactions between solute−solvent subsystems in the calculation of optical properties. It requires, in addition to charges, the polarizabilities distributed on the atomic sites for the solvent molecules. The isotropic polarizabilities distributed on the atomic sites for both solvents have been computed using the LoProp approach as implemented in MOLCAS software.59 The TD-DFT/MM and PERI-CC2 calculations have been carried out for 100 configurations selected at a regular interval, and the one- and two-photon properties were computed from linear and quadratic response theory.

(2)

In the case of linearly polarized light, the two-photon absorption strength for an isotropic sample can then be expressed as49 ⟨δ TPA ⟩ =

1 15

∑ (SaaSbb* + 2SabSba* ) ab

(3)

The calculations of electronic ⟨δTPA⟩ were performed at the DFT and RI-CC2 level of theory using Dalton50,51 and Turbomole37 programs, respectively. In the present study, we also simulated the two-photon absorption spectrum of molecule 2 in solution corresponding to the 11Ag → 21Ag transition, including vibrational fine structure. Similarly to the one-photon absorption spectra calculations, we employed LCM38,39 for this purpose. Simulations of the nonlinear spectrum were performed using the DynaVib program.52 Nuclear derivatives of the second-order transition moment were evaluated numerically by computations of Sab at a set of displaced geometries followed by the transformation to the normal coordinate basis. The stability of numerical differentiation was controlled with the aid of the Rutishauser− Romberg algorithm (employing the displacements ±2nh, where h = 0.004 bohr and n = 0,1,2). Numerical derivatives of the Sab tensor as well as normal mode displacements were computed including the solvent effects by means of the linear response polarizable continuum model (PCM). The stick vibronic twophoton absorption spectrum was then convoluted with a Gaussian line shape to include inhomogeneous broadening (see below). Solvent Effects. To compute the one- and two-photon absorption spetra of BF2 complexes in solvents (ACN and DCM), we employ in this study an integrated approach involving molecular dynamics (to account for the sampling effect) and electronic structure calculations. We carried out molecular dynamics (MD) simulations for 2 in the two solvents and selected configurations at a regular interval to be used in the hybrid TD-DFT/MM calculations to compute their contributions to the optical properties. The MD simulations account for the finite temperature, pressure effects, and sampling over all solute−solvents configurational degrees of freedom. The importance of sampling over various configurations and incorporating the mutual polarization effect between the solute and solvent systems in the calculation of optical properties of molecules in solution has already been demonstrated (e.g., see ref 53 and references therein). The molecular dynamics simulations for 2 in solvents have been carried out using Amber14 software.54

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RESULTS AND DISCUSSION We will start the discussion with an assessment of singledeterminant methods in the analysis of electronic structure of the studied compounds. To justify the reliability of the TD-DFT and C


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The Journal of Physical Chemistry B Table 1. Vertical and 0−0 Excitation Energies (in eV) for 1 in the Gas Phasea CC2 1 Ag → 1 Bu 11Ag → 21Ag 1

1

BHandHLYP

B3LYP

CAM-B3LYP

PBE0

M06-HF

Ev

E0−0

Ev

E0−0

Ev

E0−0

Ev

E0−0

Ev

E0−0

Ev

E0−0

2.90 3.58

2.54 −

3.65 4.24

3.37 −

2.91 3.35

2.69 −

3.47 4.08

3.20 −

3.02 3.50

2.80 −

3.90 4.55

3.60 −

a

Note that the same level of theory was used to determine geometry in the ground state and electronic structure. All results of calculations were obtained employing the cc-pVTZ basis set.

this work it has been evaluated based on the dimensionless normal mode displacements. The difference between the vertical and 0−0 transition for 1 varies from 0.22 eV (B3LYP and PBE0) to 0.36 eV (CC2). BHandHLYP, CAM-B3LYP, and M06-HF functionals yield intermediate values that lie in the range of 0.27− 0.30 eV. There is a rather large spread of vibrational reorganization energy values across the studied set of approaches. This might suggest differences in vibrational fine structure of the absorption bands predicted by these methods. Indeed, the simulated band for molecule 1 in the gas phase, corresponding to the 11Ag → 11Bu transition, shows a different shape for B3LYP/ PBE0 and the remaining employed methods (see Figure 1). The intensity of the 0−0 peak decreases with respect to the intensity of neighboring peaks on passing from B3LYP/PBE0 through BHandHLYP/CAM-B3LYP/M06-HF to the CC2 model. Although the simulated spectra shown in Figure 1 correspond to the molecule being in the gas phase, the stick spectra were

CC2 methods for investigating the OPA and TPA processes, the two lowest singlet excited states (11Bu and 21Ag) for molecule 1 in vacuum have been checked against their possible doubleexcitation character. First, we have used a double excitation diagnostic %;2 based on the biorthogonal norm calculated using the excitation amplitudes in the spin−orbital basis as proposed by Hättig et al.60 It was suggested61 that the second-order methods such as CC2, ADC(2), and CIS(D) can be trusted, provided that the weight of the double-excitation contributions to an excited state is smaller than 15%. At the CC2/cc-pVTZ level, %;2 values for both singlet states of the molecule 1 are roughly 13−14%, which is just below the proposed limit. To support these findings, we have performed for this system CASSCF and CASPT2 calculations using the MOLCAS program59 and employing the cc-pVDZ basis set considering 14 active electrons in 16 (2-6-5-3) orbitals that were selected on the orbital energy basis, i.e., all orbitals with energy −0.45 eV < ϵ < 0.15 eV were included in the complete active space (CAS). Seven (1-2-2-2) orbitals with energies −0.55 eV < ϵ < −0.45 eV were considered as inactive. The CASSCF results confirm that both 11Bu and 21Ag excited states can reliably be described as single-electron excitations, namely, the respective weights of dominant single excited configurations are 85% and 82%. One-Photon Absorption Spectra. The summary of gasphase calculations for molecule 1 is presented in Table 1. We will first focus on the analysis of the 11Ag → 11Bu transition. There is rather significant spread of vertical excitation energies (Ev), i.e., from 2.90 eV (CC2) up to 3.90 eV (M06-HF) . B3LYP and PBE0 exchange-correlation functionals yield vertical excitation energies close to the value predicted by the CC2 method, and the corresponding difference is only 0.01 eV (B3LYP) and 0.12 eV (PBE0). Much larger deviations (with respect to the CC2 method) are found for other functionals, i.e., 0.57 eV (CAMB3LYP), 0.75 eV (BHandHLYP), and 1.0 eV (M06-HF). These results are in line with recent findings presented by Momeni and Brown, who studied excitation energies for a large set of BODIPYs and aza-BODIPYs.33 The vertical excitation energy was also computed for the 11Ag → 21Ag transition (cf. Table 1). Contrary to the 11Ag → 11Bu excitation, the B3LYP and PBE0 functionals yield excitation energies (3.35 and 3.50 eV, respectively) smaller than that predicted by the CC2 method (3.58 eV). Likewise, the BHandHLYP, CAM-B3LYP, and M06HF overshoot the value of Ev with respect to the CC2 result. We have also estimated the 0−0 excitation energy for the 11Ag → 11Bu transition assuming shifted harmonic potentials in both electronic states: E0 − 0 = E v −

∑ k

ωk 2 Δk 2

(4)

where k runs over all vibrational normal modes and Δk is the normal mode displacement for mode k between the two electronic states. The second term on the right-hand side of eq 4 is referred to as the vibrational reorganization energy,62 and in

Figure 1. Absorption band corresponding to 11Ag → 11Bu transition for molecule 1 in the gas phase. Simulations were based on LCM using the cc-pVTZ basis set. All spectra are shifted so that the 0−0 transition is located at 0 cm−1. See text for details regarding broadening. D


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The Journal of Physical Chemistry B convoluted with a Gaussian profile [with standard deviation (σ) equal to 212 cm−1; half-width at half-maximum (HWHM) = 250 cm−1] to mimic the spectral broadening in solution. In what follows, we will demonstrate that the assumed broadening is realistic and that it is within the range of values predicted for 2 by PERI-CC2 and Marcus theory. Because M06-HF, CAM-B3LYP, and BHandHLYP yield similar band shapes (close to that predicted by the CC2 method) for the lowest-energy transition for molecule 1 in the gas phase, we selected the latter functional for further analyses. Figure 2 contains more results of simulations for molecule 1 in the gas phase performed to validate the reliability of LCM and

Figure 3. Absorption band corresponding to 11Ag → 11Bu transition for molecules 1 and 2. All spectra were computed at the BHandHLYP/6311G(d,p) level of theory based on LCM and are shifted so that the 0−0 transition is located at 0 cm−1. See text for details regarding broadening.

convoluted the stick spectra (broadened by Lorentzian, where HWHM = 10 cm−1) with a Gaussian spectral profile, where standard deviation was in the range of 200−500 cm−1. For each value of σ, we computed a parameter quantifying the differences in experimental and simulated band shapes: ⎡ ∫ [I exptl(ν) − I simul(ν)]2 dν ⎤1/2 FC ⎥ β=⎢ ⎢⎣ ⎥⎦ ∫ [I exptl(ν)]2 dν

Figure 2. Absorption band corresponding to 11Ag → 11Bu transition for molecule 1 in the gas phase simulated using BHandHLYP functional. All spectra are shifted so that the 0−0 transition is located at 0 cm−1. See text for details regarding broadening.

(5)

To compute β, by definition equal to zero for identical spectral profiles, the experimental and simulated bands were imposed in such a way that the features corresponding to the 0−0 transition were located at the same energy. The dependence of β on the standard deviation is shown in Figure 4. It follows from this figure

the importance of non-Condon effects for the 11Ag → 11Bu transition. Likewise, the stick spectra were convoluted with a Gaussian profile (σ = 212 cm−1; HWHM = 250 cm−1). First, it should be highlighted that LCM predicts correctly the spectral features in the Franck−Condon spectrum, although there are some variations in the band shoulder ratio. Considering the rigid molecular skeleton of 1, this result does not come as surprise. Second, the non-Condon (i.e., Herzberg−Teller) contributions have negligible effect on the band shape. We have also checked that the use of the 6-311G(d,p) basis set, although roughly 40% smaller in size than the cc-pVTZ basis set, does not lead to any significant deterioration of results. Henceforth, the smaller basis set will be used in the electronic and vibrational structure calculations. Because the one-photon absorption spectrum of 2 was measured in DCM and ACN, we checked if the assessment of our theoretical protocol performed for 1 in the gas phase is reliable. For this purpose, we compared the band shape for the 11Ag → 11Bu transition determined using the LCM and the PCM approach for 1 in the gas phase with spectra for 2 in the gas phase and ACN/DCM solutions (see Figure 3). As seen, there are only insignificant variations in the band shape due to both solvent as well as the structural changes (i.e., the presence of methyl groups in 2). Again, for the sake of consistency, the results shown in Figure 3 correspond to the broadening represented by a Gaussian profile (σ = 212 cm−1; HWHM = 250 cm−1). To compare the results of simulations with experimental data, we made an attempt to extract the total broadening from experimental spectra. For this purpose, we have computed the Franck−Condon spectra of 2 in ACN and DCM solutions and

Figure 4. Dependence of β parameter (see eq 5) for molecule 2 on the standard deviation corresponding to the Gaussian-like broadening. See text for more details.

that the best match between the simulated and experimental spectra in DCM and ACN solutions is obtained for σ equal to 370 and 385 cm−1, respectively. It should be underscored that these values not only correspond to the inhomogeneous broadening but also include the effect of unknown instrumental broadening and possibly other broadening mechanisms. The spectra simulated based on thus estimated broadening are shown E


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broadening estimated based on the Marcus theory, the half-width at half-maximum corresponding to Gaussian line shape, Γ, is21,22

in Figure 5. The shape of the simulated spectra is close to the measured one, and the insignificant differences, as highlighted

Γ = 2 λkBT ln 2

(6)

where kB is the Boltzmann constant and T is temperature. λ is the solvent reorganization energy, and it was computed using the PCM method as the difference of the free-energy in the nonequilibrium and equilibrium regimes.21,22 This approach was employed in combination with the linear response (LR) formulation as well as three state-specific (SS) variants of the continuum solvation model: the corrected linear response PCM (cLR-PCM),64 the vertical excitation method (VEM),65 and the state-specific approach proposed by Improta, Barone, Scalmani, and Frisch (hereafter denoted as SS-PCM(IBSF)).66 Let us note that the cLR approach can be viewed as the first-order approximation step of the self-consistent iterative VEM scheme applied here in its diagonal version (VEM(d)) as implemented in the VEMGAUSS program.67 The SS-PCM(IBSF) scheme is a self-consistent alternative to VEM, but in IBSF the excited-state wave function is no more orthogonal to the ground state, which can bring undesirable effects, e.g., in the case of solvatochromic shifts for charge-transfer transitions.68 We also underline that the SS-PCM schemes mainly provide an electrostatic correction to the LR approach neglecting the exchange repulsion and dispersion interaction terms. The comparison of broadening parameters is presented in Table 2. When discussing these data,

Figure 5. Experimental and simulated absorption band corresponding to 11Ag → 11Bu transition for molecule 2 in dichloromethane (DCM) and acetonitrile (ACN) solution. Experimental spectra were taken from ref 12 and normalized70 (ϵ(ν)/ν). Simulated spectra were determined at the BHandHLYP/6-311G(d,p) level of theory based on LCM. All spectra are shifted so that the 0−0 transition is located at 0 cm−1. See text for details regarding broadening.

previously, might be largely attributed to the approximations introduced in LCM and, to a lesser extent, to the neglect of nonCondon effects. It should not be overlooked that there is also an inherent source of error due to inaccuracy of TD-DFT combined with the BHandHLYP functional. The experimental shift of 0−0 transition energy is insignificant on passing from DCM to ACN, and it is only 0.04 eV. Moreover, the BHandHLYP/6-311G(d,p) 0−0 energies for 1 and 2 (gas phase) are close to each other. Hence, one may safely compare the experimental 0−0 transition energy for 2 in DCM solvent (2.62 eV) with the estimates for 1 in vacuo (see Table 1). We note that the 0−0 energy predicted by the CC2 model is only 0.08 eV smaller than the experimental value, thus indicating the reliability of this method. Finally, it is fair to mention that 0−0 energies were determined for molecule 2 in DCM solution by Chibani et al., who performed a statistical study for a set comprising 25 molecules.63 Following our previous study,58 we also made an attempt to estimate the inhomogeneous broadening based on the nonempirical approach. For this purpose, we have employed several methods, including (i) density functional self-consistent quantum mechanics/molecular mechanics theory (TD-DFT/ MM), (ii) polarizable-embedding CC2 method (PERI-CC2), and (iii) Marcus theory combined with the polarizable continuum model. In the case of the TD-DFT/MM and the PERI-CC2 approaches, we determined the standard deviation corresponding to the distribution of vertical excitation energy for 100 snapshots extracted from rigid-body MD simulations. Moreover, it was also checked for 10 solute (2)−solvent (ACN) configurations that the average error in 11Ag → 11Bu excitation energy computed at the PERI-CC2/cc-pVDZ level is 0.07 eV with respect to the PERI-CC2/cc-pVTZ results. The standard deviation corresponding to error in determining excitation energy for these 10 snapshots using the cc-pVDZ basis set is only 0.001 eV (8 cm−1), thus indicating that the smaller basis set is reliable for determining the inhomogeneous broadening at the CC2 level. In the case of the inhomogeneous

Table 2. Summary of PE/TD-DFT and PCM Calculations at the BHandHLYP/6-311G(d,p) Level of Theory for 2 in Dichloromethane (DCM) and Acetonitrile (ACN) Solutiona DCM Ev [eV]

ACN −1

σ [cm ]

PE-TDDFT PERI-CC2 LR-PCM cLR-PCM SS-PCM(IBSF) VEM(d)

3.62 3.08 3.61 3.69 3.79 3.71

PE-TDDFT PERI-CC2 LR-PCM cLR-PCM SS-PCM(IBSF) VEM(d)

4.27 3.82 4.26 4.30 4.41 4.33

Ev [eV]

1 A g → 1 Bu 118 3.65 139 3.12 559 3.64 161 3.71 264 3.82 290 3.73 11Ag → 21Ag 131 4.29 179 3.85 471 4.29 161 4.32 307 4.45 308 4.36 1

σ [cm−1]

1

153 162 662 194 311 340 179 218 570 190 358 359

a

Ev is the vertical excitation energy; in the case of PE calculations, the numbers correspond to average values for 100 snapshots. σ is the standard deviation corresponding to the Gaussian distribution.

two aspects should not be overlooked. First, the centrosymmetric molecule 2 is a challenge for the methods employed to estimate inhomogeneous broadening because the effect of polar environment on the bright 11Ag → 11Bu transition is rather insignificant (cf. Figure 5 comparing experimental spectra of 2 in DCM and ACN solutions). Hence, the inhomogeneous broadening is rather small. Second, as highlighted above, the values of σ estimated based on fitting simulated spectra to experimental results contain, to some (unknown) extent, also the instrumental broadening. We may thus conclude that σ equal to 370 cm−1 (DCM) and 385 cm−1 (ACN) is the upper bound for the inhomogeneous broadening. On the basis of the data presented F


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The Journal of Physical Chemistry B in Table 2, one finds for the 11Ag → 11Bu transition that the LRPCM substantially overestimates the value of σ while cLR-PCM underestimates it. Both SS-PCM(IBSF) and VEM(d) deliver the numbers roughly in the middle between cLR-PCM and LRPCM. The values of σ estimated based on the cLR-PCM approach are close to the TD-DFT/MM. It is worth highlighting that PERI-CC2 improves upon TD-DFT/MM. A similar pattern is found for the 11Ag → 21Ag transition. It should be noted that for this transition the broadening is slightly larger than that for the 11Ag → 11Bu transition. We also note that among the PCM variants only the self-consistent SS approaches (SS-PCM(IBSF) and VEM(d)) predict the increase in broadening on passing from 11Ag → 11Bu to 11Ag → 21Ag transition. Hence, there is a qualitative agreement between these methods and the polarizable embedding methods, which take into account also the solvent−solute interactions other than electrostatics. This indicates possible compensation of the nonelectrostatic terms which can be expected to be relevant in the case of a large quadrupolar molecule such as 2. The value of inhomogeneous broadening for 11Ag → 21Ag transition estimated using the SSPCM(IBSF) model will be used in simulations of vibronic twophoton absorption spectrum presented in the next section. Two-Photon Absorption Spectra. Before we analyze the band shape and optical channels for the 11Ag → 21Ag transition, we first assess the performance of density functionals for computing TPA strengths. For this purpose, we employed the CC2/cc-pVTZ level of theory to determine the reference values for molecule 1 in the gas phase. The results of calculations, performed using the B3LYP, BHandHLYP, and CAM-B3LYP functionals at the CC2/cc-pVTZ geometry are shown in Table 3.

Figure 6. Absorption band corresponding to the two-photon 11Ag → 21Ag transition for molecule 2 in ACN. The calculations were performed at the BHandHLYP/6-311G(d,p) level of theory. The intensity of the most prominent feature in the total spectrum is normalized to 1. The stick vibronic two-photon absorption spectrum, determined based on the linear response PCM, was convoluted with a Gaussian line shape to include inhomogeneous broadening. See text for details regarding broadening.

Table 3. Averaged Two-Photon Absorption Probability for Molecules 1 and 2a

Figure 7. Schematic respresentation of the three-level model for 11Ag → 21Ag transition for molecules 1 and 2.

⟨δTPA⟩ [au] CC2/cc-pVTZ B3LYP/cc-pVTZ CAM-B3LYP/cc-pVTZ BHandHLYP/cc-pVTZ BHandHLYP/6-311G(d,p) BHandHLYP/6-311G(d,p) (gas) BHandHLYP/6-311G(d,p) (DCM) BHandHLYP/6-311G(d,p) (ACN)

1 3.75 × 104 2.96 × 104 1.08 × 104 1.12 × 104 1.16 × 104 2 1.03 × 104 1.02 × 104 0.84 × 104

the DFT level, and the use of the 6-311G(d,p) basis set is wellsupported. This is an important finding because the application of a large basis set could be prohibitive in computations of geometric derivatives of second-order transition moment. We will now focus on the band shape corresponding to the two-photon allowed 11Ag → 21Ag transition. In particular, we elucidate the importance of non-Condon effects. As already demontrated by Figure 2, these contributions are rather negligible in the case of one-photon 11Ag → 11Bu transition. Experimental two-photon absorption spectrum reported by Li et al.12 for molecule 2 in ACN solution was determined with unsatisfactory resolution; thus, any vibrational fine structure is not revealed. The results of simulations of the vibrational fine structure of the two-photon absorption band corresponding to the 11Ag → 21Ag transition for molecule 2 in ACN solution are shown in Figure 6. The stick vibronic TPA spectra was convoluted with the Gaussian profile representing inhomogeneous broadening. The corresponding standard deviation (σ = 358 cm−1, HWHM = 422 cm−1) was estimated for the 11Ag → 21Ag transition based on the SS-PCM approach (see Table 2). Note that the indicated value of σ was used to convolute the Gaussian line shape with the stick spectra represented as a function of transition energy (and not photon energy). The breakdown of the total intensity into FC, HT, and the mixed FC/ HT terms shows that all of them contribute substantially to the total TPA spectrum. Similarly to other π-conjugated molecules,17 there are significant differences in the FC and HT profiles.

a

Results for 1 correspond to the gas phase and were obtained at the CC2/cc-pVTZ geometry.

The latter two functionals predict the ⟨δTPA⟩ values roughly 3.5 times smaller than that computed using the CC2 method. This result falls into expectations; it has been recently demonstrated by Beerepoot et al. for fluorescent protein chromophores that the CAM-B3LYP yields TPA strengths 1.5−3.0 times smaller than the coupled cluster reference values.69 On the other hand, the CC2 method to a much lesser degree overestimates the CCSD TPA strengths (only up to 1.4). Although BHandHLYP underestimates the TPA strengths, it yields correct band shapes for 11Ag → 11Bu transition (contrary to B3LYP). Moreover, our recent study suggests, at least for 4-nitroaniline, that the BHandHLYP and CAM-B3LYP yield TPA vibronic profiles much closer to the CC2 results in comparison with the B3LYP functional.18 Finally, as demonstrated in Table 3, the effect of basis set on the two-photon absorption strength is insignificant at G


Article


absorption strength may rely on increasing μ01 and μ12 values, 1 simultaneously decreasing the detuning factor (ω1 − 2 ω2 ).

To unravel the optical channels involved in two-photon 11Ag → 21Ag transition for molecule 2, a generalized few-state model proposed by Alam et al.34,35 was employed. In this study, we include three electronic states: ground (11Ag denoted as |0⟩), intermediate (11Bu denoted as |1⟩), and final (21Ag denoted as |2⟩) (see Figure 7). Note that the excited state |2⟩ is also included as an intermediate state in this model. The two-photon absorption probability corresponding to the 11Ag → 21Ag transition is given by 2←0 δ TLM = δ11 + δ 22 + 2δ12

where 11

(8)

2

δ 22 =

2 4 ⎛ |μ02 μ22 | ⎞ 02 ) + 1) ⎟ (2cos2(θ22 ⎜ 15 ⎝ ω2 ⎠

(9)

⎛ ⎞ 8 ⎜ |μ01||μ02 ||μ12 ||μ22 | ⎟ 22 12 (cos θ02 cos θ01 δ = 15 ⎜ ω2 ω1 − 1 ω2 ⎟ ⎝ ⎠ 2 12

(

)

01 12 12 22 + cos θ02 + cos θ02 cos θ22 cos θ01 )

(10)

ℏω i stands for the excitation energy from the ground state to the excited state |i⟩, μij = ⟨i|μ⃗ |j⟩, and θklij is the angle between (transition) dipole moments μij and μkl. δ2←0 TLM was determined for molecule 2 in the gas phase and using the polarizable embedding TD-DFT in ACN and DCM solutions. For the solvent calculations we selected solute− solvent configurations corresponding to absorption band maximum, i.e., Ev = Ev. The results of the calculations of δ2←0 TLM and its breakdown into various contributions are shown in Table 4. Since |0⟩ and |2⟩ have even parity then, δ12 and δ22 vanish for Table 4. Breakdown of the Two-Photon Absorption Probability of Molecule 2 Using Generalized Three-State Modela δ11 gas DCM ACN a

1.74 × 10 1.62 × 104 1.32 × 104 4

δ22

δ12

δ2←0 TLM

0.00 0.06 0.13

0.00 7.80 × 101 8.57 × 101

1.74 × 104 1.64 × 104 1.34 × 104

SUMMARY AND CONCLUSIONS

■

AUTHOR INFORMATION

This paper is motivated by the versatility and applicability of boron-dipyrromethene (BODIPY) complexes and other BF2carrying molecules as commonly used fluorophores. To unravel what is required to capture their optical spectra, we carried out a systematic computational study of vibrationally resolved oneand two-photon absorption spectra of the recently synthesized12 bis(BF2) core complex and its unmethylated derivative in vacuum and in solution (in ACN and DCM solvents). The onephoton absorption spectra were simulated using the CC2 method as well as by means of the density functional theory. In the latter case, several exchange-correlation functionals were employed, including B3LYP, PBE0, CAM-B3LYP, BHandHLYP, and M06-HF. It is demonstrated that the latter three functionals are much more successful than B3LYP or PBE0 in predicting the band shape corresponding to the intense onephoton allowed 11Ag → 11Bu transition. In general, a fair agreement between the simulated and experimental band shapes for 11Ag → 11Bu transition has been found, while the vertical excitation energies predicted by CAM-B3LYP, BHandHLYP, and M06-HF largely deviate from the CC2 result with a difference span of 0.57−1.0 eV. We have also shown that Herzberg−Teller contributions have only minor influence on the vibrational fine structure of the 11Ag → 11Bu band. On the other hand, non-Condon effects play a major role in the case of twophoton allowed 11Ag → 21Ag transition for the bis(BF2) core complex. A generalized few-level model has also been employed to analyze the mechanism of the electronic two-photon 11Ag → 21Ag excitation. It has been found that the most important optical channel involves the 11Bu excited state. This is a practical hint for studies aiming at maximizing the two-photon absorption cross section for derivatives of bis(BF2) core complex. The nonempirical estimation of inhomogeneous broadening is a key ingredient of computational protocols employed for simulations of one- and two-photon absorption spectra in solution. For this purpose, and following our earlier study,58 we have employed a polarizable embedding approach combined with TD-DFT and CC2 methods. Moreover, the inhomogeneous broadening has also been determined based on the Marcus theory employing linear response-PCM, corrected linear response PCM, as well as state-specific PCM methods. A comparison with experimental data reveals that the polarizable embedding CC2 and SS-PCM are the most successful approaches for the description of environmental effects, thus becoming a solid foundation for purely ab initio simulations of multiphoton vibronic spectra of molecules in solution. We believe that the current study, applied to an important chromophore system, illustrates what can be accomplished, as well as what is required, by current state-of-the-art simulations of optical spectra, both with respect to excitation energies and band shapes.

(7)

⎛ ⎞2 4 ⎜ |μ01 μ12 | ⎟ (2cos2(θ1201) + 1) δ = 15 ⎜⎝ ω1 − 1 ω2 ⎟⎠

■

All values are given in au.

molecule 2 in the gas phase. It turns out that the dominant contribution is due to the 11Bu state, i.e., the value of δii for other intermediate states is a few orders of magnitude smaller. Moreover, there is a fair agreement between the value of ⟨δTPA⟩ estimated based on the three-state model (1.74 × 104 a.u.) and the exact response theory value (1.03 × 104 a.u.). The PE calculations, which rely on discrete representation of solvent, lead to noncentrosymmetric perturbations of electronic density of the solute. Thus, δ12 and δ22 do not vanish, although they are negligible in comparison with the dominant δ11 contribution. A solvent-induced decrease of averaged two-photon absorption strength predicted by the response theory (cf. Table 3) is correctly reproduced based on the three-level model, i.e., the TPA ⟨δTPA gas ⟩/⟨δACN⟩ ratio is 1.23 and 1.30, respectively. As already highlighted, the optical channel involving the 11Bu state is the crucial one. Hence, possible routes to increase the two-photon

Corresponding Author

*E-mail: [email protected]. Tel.: +48 713204472. Present Address ⊥

(G.T.) School of Sciences, Yanshan University, Qinhuangdao, Hebei 066004, China

H


Article

The Journal of Physical Chemistry B Notes

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The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS Financial support from the National Science Centre (Grant 2013/09/B/ST5/03550) is acknowledged. The calculations were performed in part at the Wroclaw Center for Networking and Supercomputing. This work was also supported by the Swedish Infrastructure Committee (SNIC) for the project “Multiphysics Modeling of Molecular Materials”, SNIC2015-1610.

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First-Principles Simulations of One- and Two-Photon Absorption Band

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