SCI with COSMO Implicit Solvation and

Nov 29, 2016 - Here, we implement the COSMO solvent model with a perturbative state-specific correction to the excited-state energies with the INDO/SC...
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Implementation of INDO/SCI with COSMO Implicit Solvation and Benchmarking for Solvatochromic Shifts Rebecca L. Gieseking, Mark A. Ratner, and George C. Schatz* Department of Chemistry, Northwestern University 2145 Sheridan Road, Evanston, Illinois 60208, United States S Supporting Information *

ABSTRACT: Accurate and rapid quantum mechanical prediction of solvatochromic shifts, particularly in systems where charge transfer plays a significant role, is important for many aspects of molecular and material design. Although the semiempirical INDO/SCI approach is computationally efficient and performs well for charge-transfer states, the availability of implicit solvent approaches has been limited. Here, we implement the COSMO solvent model with a perturbative state-specific correction to the excited-state energies with the INDO/SCI method. We show that for a series of prototypical π-conjugated molecules, our newly implemented INDO/SCI/COSMO model yields more accurate absorption energies and comparably accurate solvatochromic shifts to those computed using TDωB97XD and CIS with COSMO solvation at a substantially lower computational cost.

1. INTRODUCTION Environmental effects are well-known to greatly influence the electronic and optical properties of many molecular systems. Accurate quantum mechanical prediction of these environmental effects, in particular solvatochromic shifts, is critical for many aspects of molecular and materials design. Implicit solvent models, where the solvent is treated as a dielectric continuum that interacts with the solute via electrostatic interactions, have been widely developed and used over the past several decades.1−5 The ground-state solvent correction is typically added to the Hamiltonian as part of the self-consistent procedure. However, corrections to the excited states are more complex and often involve perturbative corrections to each excited state because the solvent is only able to partially relax on the time scale of absorption. These corrections include statespecific (SS) approaches,6−13 which correct the excited-state energies based on the excited-state electron density, and linear response (LR) approaches,9,12,14−17 which are based on the transition densities between the ground and excited states. For reasons of computational ease, SS approaches are more common for wave-function-based methods and LR approaches are more common for density functional theory (DFT) methods. Solvatochromic shifts are typically large in molecules with substantial charge redistribution upon excitation, leading to large differences between the dipole moments of the ground and excited states in dipolar molecules or large changes in the corresponding higher-order multipole moments upon excitation in quadrupolar or octupolar systems. Thus, proper treatment of states with significant charge-transfer character is important to compute accurate solvatochromic shifts. However, much of the work benchmarking these methods has focused on © XXXX American Chemical Society

DFT approaches. Although DFT is in principle an exact theory, the commonly used generalized gradient approximation (GGA) and hybrid functionals are well-known to underestimate the charge-transfer energies due to self-interaction error.18−20 Modern functionals such as range-separated hybrids can reduce the self-interaction error and improve performance for chargetransfer states if an appropriate range-separation parameter is chosen;21,22 however, empirical tuning of the range-separation parameter does not adequately reproduce the evolution of properties such as bond length alternation (BLA) in πconjugated chromophores.23 Because of this limitation, in systems with significant intramolecular24−26 or intermolecular27,28 charge transfer, these DFT-based approaches with implicit solvent often underestimate solvatochromic shifts. Thus, there is a need to evaluate solvent effects using a computationally inexpensive method that can accurately predict the energy of charge-transfer states. Although the semiempirical INDO method has been shown to accurately predict chargetransfer state energies in several cases,29,30 the availability of implicit solvent approaches is limited.31,32 Here, we implement the COSMO implicit solvent model3 with the semiempirical INDO/SCI (INDO with singles configuration interaction) method33 including a simple statespecific perturbative correction to the excited-state energies.6 We then benchmark the INDO/SCI/COSMO approach relative to experimental results for a set of 27 prototypical organic π-conjugated molecules with varying degrees of intermolecular charge transfer along the molecular backbone Received: October 17, 2016 Revised: November 23, 2016 Published: November 29, 2016 A

DOI: 10.1021/acs.jpca.6b10487 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry A (Figure 1). Our newly implemented INDO/SCI/COSMO model yields more accurate absorption energies and com-

Aμν =

1 || rμ − rν ||

The total screening energy ΔE for a molecular charge distribution Q is given by 1 1 ΔE = − f (ε)Q BA−1BQ = − f (ε)Q DQ 2 2

For the molecular ground state, the screening energy is computed as part of the SCF process and the surface charges do not need to be explicitly computed. Because the INDO Hamiltonian neglects all one-electron integrals involving two different basis functions on one atom and an operator on a different atom, solvent corrections to the one-electron integrals are added only to the terms involving the same basis function on the same atom.31 We have implemented perturbative corrections to the excited-state energies using the approach derived by Klamt.6 The dielectric constant of solvents is typically reported near the static limit, where electronic, vibrational, and orientational relaxation of the solvent can all contribute. In contrast, on the time scale of optical excitations, only the electronic component n2 of the dielectric constant is fast enough to contribute to the solvent relaxation, where n is the solvent refractive index. Thus, the dynamic relaxation of the solvent is scaled by f(n2) = (n2 − 1)/(n2 + 0.5). In the CI procedure, the solvation-corrected energy ET,solv of a transition T is6

Figure 1. Chemical structures of the donor−acceptor-substituted polyenes studied in this paper.

parably accurate solvatochromic shifts to those computed using TD-ωB97XD and CIS with COSMO solvation at a substantially lower computational cost, whereas TD-B3LYP gives a qualitatively incorrect description of the solvatochromic shifts in these systems.

2. COMPUTATIONAL METHODOLOGY 2.1. Implementation of INDO/SCI with COSMO. The COSMO solvent model3,6 for INDO/SCI was implemented into a code incorporating portions of MOPAC 7.134 and the INDO/CI code from Jeffrey Reimers35 as has been previously described for semiempirical CI approaches.3,6 MOPAC 7.1 includes ground-state COSMO corrections for semiempirical models, and the extensions of this code to the INDO Hamiltonian and the excited-state corrections are newly implemented here. Since this solvent model has been described in detail previously,3,6 we review only the details relevant for our implementation and key differences from the previous ZINDO/PCM implementation.31 In the COSMO solvent model, the solvent is treated as a continuum with a dielectric constant ε surrounding a molecular cavity. The cavity is constructed from the union of atomcentered spheres at a distance of 1.2 times the Bondi radii36 of each solute atom and segmented into discrete triangles.3 Unlike in the previous ZINDO/PCM implementation,31 the hydrogen atoms are explicitly included in the cavity construction and are not summed into the neighboring heavy atoms. The screening energy is scaled by a factor of f(ε) = (ε − 1)/(ε + 0.5) relative to a perfect conductor,3 which differs from the factor of (ε − 1)/ε typically used in PCM models.31 The interactions of the solvent surface charges with the solute are described as electrostatic interactions between the molecular electronic and nuclear charges i and the surface charges μ. The screening energy involves a matrix B that includes all such interactions, where each element Biμ is defined as 1 Biμ = || rμ − ri ||

ET ,solv = ET ,0 + f (ε)Δex DP0 −

1 2 f (n )Δex DΔex 2

where Δex is the change in the electron distribution upon excitation. The first correction to the gas-phase transition energy ET,0, proportional to f(ε), includes the change in the excited-state energies due to changes in the ground-state charge distribution; this term is automatically incorporated via the changes in the orbital energies due to ground-state solvation. The second term, proportional to f(n2), involves changes in the excitation energy related to solvent polarization upon excitation of the molecule and is computed individually for each excited state. In contrast with the previous ZINDO/PCM implementation,31 corrections to the CI matrix are only made to the diagonal elements of the CI matrix. To compute the COSMO energetic correction to the INDO/SCI excitation energies, the solvated ground-state electronic structure is first calculated self-consistently. While constructing the CI matrix, the solvent corrections to each pure excitation are computed based on the change in electron density upon excitation. Off-diagonal corrections to the CI matrix are not considered. Diagonalization of the solventcorrected CI matrix yields the solvent-corrected CI states. However, since the excited states in general involve linear combinations of excitations and thus the change in the excitedstate electron density is a linear combination of the density changes of the pure excitations, this procedure generally overestimates the solvent stabilization of the excited states. To obtain more accurate solvent-corrected state energies, the energy of each excited state is computed using the uncorrected CI matrix, and the appropriate solvent correction is computed using the change in electron density associated with the excited state. In the original implementation of this procedure,6 this two-step correction procedure yielded similar results to a onestep procedure where the solvent correction was applied only after diagonalizing the uncorrected CI matrix. However, since

where rμ and ri indicate the positions of surface charge μ and molecular charge i, respectively. Similarly, a matrix A contains all electrostatic interactions between pairs of surface charges μ and ν, defined as B

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average lengths of the double and single bonds along the carbon backbone) increases from −0.1 to +0.1 Å. The bondorder alternation (BOA) by convention has the opposite sign as BLA and decreases from +0.6 to −0.6 between the same limits. The excited-state energies are strongly dependent on BOA, and in particular the first excited-state energy Ege is lowest near the cyanine limit where BOA is close to zero.51,55 The vibronic structure within the absorption peaks also depends on BOA. In particular, the 0−0 peak dominates for molecules near the cyanine limit, whereas the absorption is distributed over several vibronic states for molecules with larger BLA.55 For most of the molecules considered here, the experimental absorption spectra have similar enough vibronic structure between the two solvents that the shift between the absorption maxima accurately represents the change in the excited-state energy;37−39 the exceptions will be discussed below. The solvatochromic shifts in donor−acceptor-substituted polyenes are strongly dependent on the molecular structure. If the donor and acceptor are relatively weak and BLA is negative, BLA will become less negative and Ege will decrease with increasing solvent dielectric constant, defined as positive solvatochromism. In contrast, when the donor and acceptor are strong enough to produce a positive BLA, an increase in the solvent dielectric constant will cause both BLA and Ege to become larger, defined as negative solvatochromism. Accurate treatment of the electronic structure and solvent effects in both the ground state and the excited state is required to properly reproduce these trends. We first examine the absorption energies of the molecules in both toluene and ethanol. In both solvents, the computational methods used are generally fairly reliable at predicting changes in the absorption energies, but all of the methods considered considerably overestimate the absorption energies (Figure 2). Notably, the INDO/SCI absorption energies are on average the closest to the experimental values, overestimating the experimental values by an average of 0.42 eV in toluene and 0.36 eV in ethanol; considering the computational simplicity of this approach, this accuracy is quite good. CIS overestimates the experimental absorption energies by an average of 1.28 eV in both toluene and ethanol, which is unsurprising as CIS is well-known to overestimate excited-state energies.56,57 The two DFT functionals ωB97XD and B3LYP overestimate the absorption energies by 0.74 and 0.64 eV, respectively, in toluene and by comparable amounts in ethanol. This discrepancy likely has several origins. The results presented here use the TDA approach; test calculations indicate that use of the full TD calculations stabilize the excited states by 0.1− 0.15 eV at the ωB97XD level and 0.15−0.2 eV at the B3LYP level but do not otherwise significantly affect the results presented in this paper. As will be discussed later, many of these molecules have structures relatively close to the cyanine limit, and typical TD-DFT approaches are well-known to overestimate absorption energies at the cyanine limit.58−60 The comparisons here are also limited to the vertical absorption energies and neglect vibrational contributions and geometric relaxation effects, which often lower the energy of the 0−0 vibronic transition.61−65 As our primary focus is on efficient computation of the solvatochromic shifts, these corrections are beyond the scope of this paper. The solvatochromic shifts are computed as the difference between the excited state energies in toluene and ethanol. Overall, the solvatochromic shifts at the ωB97XD, CIS, and INDO/SCI levels are in quite good agreement with experiment

we are focusing in particular on molecules where the solvent corrections may be quite large due to significant intramolecular charge transfer upon excitation, we have chosen to focus on the two-step procedure. 2.2. Computational Details. For this study, 27 donor− acceptor-substituted polyenes were selected, with the chemical structures shown in Figure 1. The experimental absorption maxima, and in some cases the full absorption spectra, are available for 24 of the 27 molecules in both toluene and ethanol;37−39 to our knowledge experimental data are not available for the D1-n-A2 series. We note that the experimental results have a variant of D1 with a benzyl substituent instead of methyl on the nitrogen. Since this substituent is not part of the π-conjugated system, this small structural change is not expected to significantly affect the results. The experimental solvatochromic shifts in these molecules range from −0.30 to +0.29 eV. The geometries of these structures were optimized at the ωB97XD/6-31G* level40−42 using the COSMO solvent model in both toluene (ε = 2.38; n2 = 1.99) and ethanol (ε = 24.3; n2 = 1.85). Although experimental bond lengths in solution cannot be measured, this functional has been shown to yield bond lengths of polyenes in good agreement with CCSD(T).23 The first five excited states were computed at the TD-DFT level using the ωB97XD and B3LYP43,44 functionals and the 631G* basis set within the COSMO solvent model. The first strongly absorbing excited state (typically the first excited state) was selected for comparison with the experimental absorption energy. With the use of both functionals, excited states were computed using the Tamm−Dancoff approximation (TDA)45 and the COSMO solvent model with the recently implemented perturbative linear response (ptLR) excited-state correction,9,46 which allows us to obtain the excited-state dipole moments. The excited-state energies were also computed using the COSMO solvent model with no solvent correction to the excited-state energies, the perturbative state-specific (ptSS) correction, and the linear response (LR) correction; these results are presented in the Supporting Information. The excited-state energies at the CIS/6-31G* level were also computed using the COSMO solvent model. All DFT, TDDFT, and CIS calculations were performed using Q-Chem 4.3.47 Although the implementation of COSMO in Q-Chem is not identical to our semiempirical implementation, the radii and the f(ε) terms are consistent between the two. The Wiberg bond orders48 were also computed using the ωB97XD, B3LYP, and HF approaches with the 6-31G* basis set within the NBO 5.0 program.49 The INDO/SCI excited-state energies were computed using our modified code as detailed in the previous section. All possible single excitations were generated, and the lowest 1000 excitations were included in the CI matrix. This matrix was diagonalized to obtain the first 10 excited states, and the first strongly absorbing state was considered. The Wiberg bond orders were also computed using the INDO wave function within our modified code.

3. RESULTS AND DISCUSSION Although solvatochromic shifts occur in many systems, a prototypical example is donor−acceptor-substituted polyenes (Figure 1).50−55 In these systems, as the degree of charge transfer between the two molecular ends increases from the polyene limit to the zwitterionic limit, the degree of bondlength alternation (BLA; defined as the difference between the C

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poor solvatochromic shifts relative to the experimental values, with most molecules having computed solvatochromic shifts close to zero; the origins of this error will be discussed later. We note that B3LYP is known to perform poorly for chargetransfer excitations,18,20,66 and so the large errors are perhaps unsurprising; however, we provide this comparison because this functional is still widely used. A comparison of the solvatochromic shifts at the ωB97XD and B3LYP levels with different excited-state corrections is provided in Figures S3 and S4; the choice of correction approach does not qualitatively affect the results. For the molecules with experimental solvatochromic shifts larger in magnitude than ±0.2 eV, CIS, INDO/SCI, and ωB97XD all underestimate the solvatochromic shifts; however, examination of the experimental absorption spectra37−39 suggests that large magnitudes of the experimental shifts are associated with changes in the vibronic structure and thus should not be fully reproduced by computational approaches that consider only the electronic structure. To understand the magnitudes of the solvatochromic shifts in these systems, we examine the ground-state and excited-state electronic structures. As described earlier, the solvatochromic shifts are associated with an increase in the ground-state dipole moment μg and a corresponding increase in BOA. Within this series, the molecules with large positive BOA generally have large positive solvatochromic shifts, and the computed solvatochromic shift becomes more negative with decreasing BOA as shown in Figure 4. Because the magnitudes of the solvatochromic shifts also depend on factors that are not accounted for in this simple comparison such as the polarizability of each molecule, there is significant variation in the solvatochromic shift of molecules with similar BOA. The differences in BOA between the four methodologies can be used to understand the differences in the solvatochromic shifts. We note that while the ωB97XD/6-31G* optimized geometries are used to calculate all of these bond orders and

Figure 2. Comparison between the experimental and computed absorption energies of donor−acceptor-substituted polyenes in (top) toluene and (bottom) ethanol.

as shown in Figure 3. Although INDO/SCI yields solvatochromic shifts that are too positive by an average of 0.06 eV, this deviation is fairly consistent across the series of molecules studied here. We note that ωB97XD tends to underestimate the magnitudes of the negative solvatochromic shifts, corresponding to molecules with donor D3, which is not the case for CIS or INDO/SCI. In contrast, B3LYP yields quite

Figure 3. Comparison between the experimental and computed solvatochromic shifts of donor−acceptor-substituted polyenes. D

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tendency of INDO to overestimate BOA may explain some of the overestimation of the solvatochromic shifts, since a more positive BOA in general corresponds to a more positive solvatochromic shift. For consistency with the definition of the solvatochromic shifts, we define the change in BOA (ΔBOA) for each molecule as BOA in toluene minus BOA in ethanol, which always yields a positive change (Figure 5, bottom panel); this gives a measure of how much change in polarization of the molecules occurs between the two solvents. The change in BOA is smallest for B3LYP, which is consistent with the overdelocalization of the wave function discussed earlier. HF shows a slightly larger change in BOA than ωB97XD, and INDO has the largest change in BOA, implying a larger polarizability. The solvatochromic shifts also depend on the ground-state and excited-state dipole moments μg and μe. As shown in Figure 6 (top), μg is fairly consistent across all four levels of

Figure 4. Correlation between the computed bond-order alternation in toluene and the solvatochromic shift at the ωB97XD/6-31G* level.

thus BLA does not vary between the four levels of theory, changes in the electronic structure will still result in significant changes in BOA.53 Because there is no experimental standard for the bond orders, we compare the BOA at each level of theory to ωB97XD in Figure 5 (top panel). B3LYP consistently

Figure 6. Computed (top) ground-state and (bottom) excited-state dipole moments in toluene (closed symbols) and ethanol (open symbols). Figure 5. (top) Computed bond order alternation (BOA) in toluene (closed symbols) and ethanol (open symbols). (bottom) Change in BOA between toluene and ethanol.

theory. B3LYP overestimates μg by up to 1.5 D relative to ωB97XD for the molecules with smaller dipole moments, which is consistent with the underestimation of the magnitude of BOA in these molecules. In contrast, INDO in many cases underestimates μg, particularly for the molecules with acceptor A2, which is unsurprising given the underestimation of BOA for these molecules. Much more significant differences between the four levels of theory are seen for μe. Both INDO/SCI and CIS yield μe values in reasonable agreement with those at the ωB97XD level. Since the ground-state and excited-state wave functions are largely consistent among these three levels of theory, it is unsurprising that all three yield similarly accurate solvatochromic shifts. However, B3LYP yields dramatically inconsistent μe values in both toluene and ethanol, in many

provides slightly smaller BOA values than ωB97XD, which is unsurprising given that standard hybrid functionals are known to overdelocalize the wave function in π-conjugated chromophores.23,67−69 Similarly, HF slightly overestimates BOA relative to ωB97XD, which is consistent with the tendency of HF to overlocalize the wave function.23,68 INDO gives somewhat more variation in BOA and in particular tends to overestimate the magnitude of BOA of molecules with acceptor A2, suggesting that the standard INDO parameters somewhat underestimate the acceptor strength of this end group. The E

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The Journal of Physical Chemistry A cases overestimating μe by 10 or more Debye. The inaccurate treatment of the excited state wave functions is likely the origin of the errors in the solvatochromic shifts at the B3LYP level.

(2) Kirkwood, J. G. The Dielectric Polarization of Polar Liquids. J. Chem. Phys. 1939, 7, 911−919. (3) Klamt, A.; Schuurmann, G. COSMO: A New Approach to Dielectric Screening in Solvents with Explicit Expressions for the Screening Energy and Its Gradient. J. Chem. Soc., Perkin Trans. 2 1993, 2, 799−805. (4) Cossi, M.; Rega, N.; Scalmani, G.; Barone, V. Energies, Structures, and Electronic Properties of Molecules in Solution with the C-PCM Solvation Model. J. Comput. Chem. 2003, 24, 669−681. (5) Marenich, A. V.; Olson, R. M.; Kelly, C. P.; Cramer, C. J.; Truhlar, D. G. Self-Consistent Reaction Field Model for Aqueous and Nonaqueous Solutions Based on Accurate Polarized Partial Charges. J. Chem. Theory Comput. 2007, 3, 2011−2033. (6) Klamt, A. Calculation of UV/Vis Spectra in Solution. J. Phys. Chem. 1996, 100, 3349−3353. (7) Caricato, M.; Mennucci, B.; Tomasi, J.; Ingrosso, F.; Cammi, R.; Corni, S.; Scalmani, G. Formation and Relaxation of Excited States in Solution: A New Time Dependent Polarizable Continuum Model Based on Time Dependent Density Functional Theory. J. Chem. Phys. 2006, 124, 124520. (8) Chibani, S.; Budzak, S.; Medved, M.; Mennucci, B.; Jacquemin, D. Full cLR-PCM Calculations of the Solvatochromic Effects on Emission Energies. Phys. Chem. Chem. Phys. 2014, 16, 26024−26029. (9) Mewes, J. M.; You, Z. Q.; Wormit, M.; Kriesche, T.; Herbert, J. M.; Dreuw, A. Experimental Benchmark Data and Systematic Evaluation of Two a Posteriori, Polarizable-Continuum Corrections for Vertical Excitation Energies in Solution. J. Phys. Chem. A 2015, 119, 5446−5464. (10) Cammi, R.; Fukuda, R.; Ehara, M.; Nakatsuji, H. SymmetryAdapted Cluster and Symmetry-Adapted Cluster-Configuration Interaction Method in the Polarizable Continuum Model: Theory of the Solvent Effect on the Electronic Excitation of Molecules in Solution. J. Chem. Phys. 2010, 133, 024104. (11) Cammi, R. Coupled-Cluster Theories for the Polarizable Continuum Model. II. Analytical Gradients for Excited States of Molecular Solutes by the Equation of Motion Coupled-Cluster Method. Int. J. Quantum Chem. 2010, 110, 3040−3052. (12) Caricato, M. A Comparison between State-Specific and LinearResponse Formalisms for the Calculation of Vertical Electronic Transition Energy in Solution with the CCSD-PCM Method. J. Chem. Phys. 2013, 139, 044116. (13) Caricato, M. A Corrected-Linear Response Formalism for the Calculation of Electronic Excitation Energies of Solvated Molecules with the CCSD-PCM Method. Comput. Theor. Chem. 2014, 1040− 1041, 99−105. (14) Cammi, R.; Mennucci, B. Linear Response Theory for the Polarizable Continuum Model. J. Chem. Phys. 1999, 110, 9877−9886. (15) Cossi, M.; Barone, V. Separation between Fast and Slow Polarizations in Continuum Solvation Models. J. Phys. Chem. A 2000, 104, 10614−10622. (16) Cossi, M.; Barone, V. Time-Dependent Density Functional Theory for Molecules in Liquid Solutions. J. Chem. Phys. 2001, 115, 4708−4717. (17) Cammi, R. Coupled-Cluster Theory for the Polarizable Continuum Model. III. A Response Theory for Molecules in Solution. Int. J. Quantum Chem. 2012, 112, 2547−2560. (18) Dreuw, A.; Weisman, J. L.; Head-Gordon, M. Long-Range Charge-Transfer Excited States in Time-Dependent Density Functional Theory Require Non-Local Exchange. J. Chem. Phys. 2003, 119, 2943−2946. (19) Tozer, D. J. Relationship between Long-Range Charge-Transfer Excitation Energy Error and Integer Discontinuity in Kohn-Sham Theory. J. Chem. Phys. 2003, 119, 12697−12699. (20) Dreuw, A.; Head-Gordon, M. Failure of Time-Dependent Density Functional Theory for Long-Range Charge-Transfer Excited States: The Zincbacteriochlorin-Bacteriochlorin and Bacteriochlorophyll-Spheroidene Complexes. J. Am. Chem. Soc. 2004, 126, 4007− 4016.

4. CONCLUSIONS Modeling solvent effects in quantum mechanical systems is critical to understand a variety of chemical effects. Here, we have implemented the COSMO solvent model with the INDO/SCI method. For a series of prototypical donor− acceptor-substituted polyenes, we have shown that INDO/SCI yields quite reasonable trends in the solvatochromic shifts and predicts absorption energies in good agreement with the experimental values. The INDO/SCI solvatochromic shifts are slightly more positive than the experimental values or those computed at the ωB97XD or CIS levels, which is likely due to a tendency to underestimate the ground-state charge-transfer character. In contrast, B3LYP fails to even qualitatively describe the solvatochromic shifts in these molecules due to a failure to adequately describe the charge-transfer character of the excited states. The reasonable performance of our newly implemented INDO/SCI/COSMO approach at a substantially lower computational cost than CIS or TD-DFT suggests that this method can be used for rapid screening of new systems or to evaluate solvent effects in systems large enough that higher level methods are not computationally feasible. In particular, since INDO does not suffer from the well-known limitations of typical DFT functionals in describing charge-transfer states, this method may prove a valuable tool to model solvent effects in a variety of systems with significant intramolecular or intermolecular charge-transfer character in the ground state or in the excited states of interest. Efforts are in progress to apply this methodology to charge-transfer excitations in solution.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpca.6b10487. List of experimental and computed excited-state energies and solvatochromic shifts and comparison of excitedstate energies and solvatochromic shifts with various COSMO excited-state corrections using the ωB97XD and B3LYP functionals (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected]; Tel (847)-491-5657 (G.C.S.). ORCID

Rebecca L. Gieseking: 0000-0002-7343-1253 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank Jeffrey Reimers for providing his INDO/CI code. This research was supported by DOE Grant DE-FG0210ER16153.

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REFERENCES

(1) Onsager, L. Electric Moments of Molecules in Liquids. J. Am. Chem. Soc. 1936, 58, 1486−1493. F

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The Journal of Physical Chemistry A (21) Stein, T.; Kronik, L.; Baer, R. Prediction of Charge-Transfer Excitations in Coumarin-Based Dyes Using a Range-Separated Functional Tuned from First Principles. J. Chem. Phys. 2009, 131, 244119. (22) Kuritz, N.; Stein, T.; Baer, R.; Kronik, L. Charge-Transfer-Like π-π * Excitations in Time-Dependent Density Functional Theory: A Conundrum and Its Solution. J. Chem. Theory Comput. 2011, 7, 2408− 2415. (23) Körzdörfer, T.; Parrish, R. M.; Sears, J. S.; Sherrill, C. D.; Brédas, J. L. On the Relationship between Bond-Length Alternation and Many-Electron Self-Interaction Error. J. Chem. Phys. 2012, 137, 124305. (24) Kityk, A. V. Absorption and Fluorescence Spectra of Heterocyclic Isomers from Long-Range-Corrected Density Functional Theory in Polarizable Continuum Approach. J. Phys. Chem. A 2012, 116, 3048−3055. (25) Reish, M. E.; Kay, A. J.; Teshome, A.; Asselberghs, I.; Clays, K.; Gordon, K. C. Testing Computational Models of Hyperpolarizability in a Merocyanine Dye Using Spectroscopic and DFT Methods. J. Phys. Chem. A 2012, 116, 5453−5463. (26) Rezende, M. C.; Dominguez, M. The Solvatochromism of Phenolate Betaines: Comparing Different Cavities of a Polarized Continuum Model. J. Mol. Model. 2015, 21, 212. (27) Mach, P.; Budzák, Š.; Medved, M.; Kysel, O. Theoretical Analysis of Charge-Transfer Electronic Spectra of Methylated Benzenes-TCNE Complexes Including Solvent Effects: Approaching Experiment. Theor. Chem. Acc. 2012, 131, 1−14. (28) Budzák, Š.; Mach, P.; Medved, M.; Kysel, O. Critical Analysis of Spectral Solvent Shifts Calculated by the Contemporary PCM Approaches of a Representative Series of Charge-Transfer Complexes between Tetracyanoethylene and Methylated Benzenes. Phys. Chem. Chem. Phys. 2015, 17, 17618−17627. (29) Fox, T.; Kotzian, M.; Rosch, N. Design of Rigid Donor-Acceptor Systems with a Low-Lying Charge-Transfer State. An INDO Model Study of Barrelene-Based Compounds. J. Phys. Chem. 1993, 97, 11420−11426. (30) Nazeeruddin, M. K.; Wang, Q.; Cevey, L.; Aranyos, V.; Liska, P.; Figgemeier, E.; Klein, C.; Hirata, N.; Koops, S.; Haque, S. A.; et al. DFT-INDO/S Modeling of New High Molar Extinction Coefficient Charge-Transfer Sensitizers for Solar Cell Applications. Inorg. Chem. 2006, 45, 787−797. (31) Caricato, M.; Mennucci, B.; Tomasi, J. Solvent Effects on the Electronic Spectra: An Extension of the Polarizable Continuum Model to the ZINDO Method. J. Phys. Chem. A 2004, 108, 6248−6256. (32) Caricato, M.; Mennucci, B.; Tomasi, J. Solvent Polarity Scales Revisited: A ZINDO-PCM Study of the Solvatochromism of Betaine30. Mol. Phys. 2007, 104, 875−887. (33) Ridley, J.; Zerner, M. An Intermediate Neglect of Differential Overlap Technique for Spectroscopy: Pyrrole and the Azines. Theor. Chim. Acta 1973, 32, 111−134. (34) Stewart, J. J. P. MOPAC: A Semiempirical Molecular Orbital Program. J. Comput.-Aided Mol. Des. 1990, 4, 1−105. (35) Shapley, W. A.; Reimers, J. R.; Hush, N. S. INDO/S Parameters for Gold. Int. J. Quantum Chem. 2002, 90, 424−438. (36) Bondi, A. Van Der Waals Volumes and Radii. J. Phys. Chem. 1964, 68, 441−451. (37) Kulinich, A. V.; Derevyanko, N. A.; Ishchenko, A. A. Synthesis, Structure, and Solvatochromism of Merocyanine Dyes Based on Barbituric Acid. Russ. J. Gen. Chem. 2006, 76, 1441−1457. (38) Ishchenko, A. A.; Kulinich, A. V.; Bondarev, S. L.; Knyukshto, V. N. Electronic Structure and Fluorescent Properties of MalononitrileBased Merocyanines with Positive and Negative Solvatochromism. Opt. Spectrosc. 2008, 104, 57−68. (39) Kulinich, A. V.; Derevyanko, N. A.; Mikitenko, E. K.; Ishchenko, A. A. Merocyanines Based on 1,3-Indanedione: Electronic Structure and Solvatochromism. J. Phys. Org. Chem. 2011, 24, 732−742. (40) Chai, J.-D.; Head-Gordon, M. Systematic Optimization of LongRange Corrected Hybrid Density Functionals. J. Chem. Phys. 2008, 128, 084106.

(41) Chai, J.-D.; Head-Gordon, M. Long-Range Corrected Hybrid Density Functionals with Damped Atom−atom Dispersion Corrections. Phys. Chem. Chem. Phys. 2008, 10, 6615−6620. (42) Hariharan, P. C.; Pople, J. A. The Influence of Polarization Functions on Molecular Orbital Hydrogenation Energies. Theor. Chim. Acta 1973, 28, 213−222. (43) Becke, A. D. Density-Functional Exchange-Energy Approximation with Correct Asymptotic Behavior. Phys. Rev. A: At., Mol., Opt. Phys. 1988, 38, 3098−3100. (44) Lee, C.; Yang, W.; Parr, R. G. Development of the Colle-Salvetti Correlation Energy Formula into a Functional of the Electron Density. Phys. Rev. B: Condens. Matter Mater. Phys. 1988, 37, 785−789. (45) Hirata, S.; Head-Gordon, M. Time-Dependent Density Functional Theory within the Tamm-Dancoff Approximation. Chem. Phys. Lett. 1999, 314, 291−299. (46) You, Z. Q.; Mewes, J. M.; Dreuw, A.; Herbert, J. M. Comparison of the Marcus and Pekar Partitions in the Context of Non-Equilibrium, Polarizable-Continuum Solvation Models. J. Chem. Phys. 2015, 143, 204104. (47) Shao, Y.; Gan, Z.; Epifanovsky, E.; Gilbert, A. T. B.; Wormit, M.; Kussmann, J.; Lange, A. W.; Behn, A.; Deng, J.; Feng, X.; et al. Advances in Molecular Quantum Chemistry Contained in the QChem 4 Program Package. Mol. Phys. 2015, 113, 184−215. (48) Wiberg, K. B. Application of the Pople-Santry-Segal CNDO Method to the Cyclopropylcarbinyl and Cyclobutyl Cation and to Bicyclobutane. Tetrahedron 1968, 24, 1083−1096. (49) Glendening, E. D.; Badenhoop, J. K.; Reed, A. E.; Carpenter, J. E.; Bohmann, J. A.; Morales, C. M.; Weinhold, F. NBO 5.0; p http:// www.chem.wisc.edu. (50) Marder, S. R.; Gorman, C. B.; Meyers, F.; Perry, J. W.; Bourhill, G.; Bredas, J.-L.; Pierce, B. M. A Unified Description of Linear and Nonlinear Polarization in Organic Polymethine Dyes. Science 1994, 265, 632−635. (51) Meyers, F.; Marder, S. R.; Pierce, B. M.; Bredas, J. L. Electric Field Modulated Nonlinear Optical Properties of Donor-Acceptor Polyenes: Sum-over-States Investigation of the Relationship between Molecular Polarizabilities (Alpha, Beta, and Gamma) and Bond Length Alternation. J. Am. Chem. Soc. 1994, 116, 10703−10714. (52) Gorman, C. B.; Marder, S. R. Effect of Molecular-Polarization on Bond-Length Alternation, Linear Polarizability, First and Second Hyperpolarizability in Donor-Acceptor Polyenes As a Function of Chain-Length. Chem. Mater. 1995, 7, 215−220. (53) Gieseking, R. L.; Risko, C.; Brédas, J. L. Distinguishing the Effects of Bond-Length Alternation versus Bond-Order Alternation on the Nonlinear Optical Properties of Conjugated Chromophores. J. Phys. Chem. Lett. 2015, 6, 2158−2162. (54) Gieseking, R. L.; Mukhopadhyay, S.; Risko, C.; Marder, S. R.; Brédas, J. L. 25th Anniversary Article: Design of Polymethine Dyes for All-Optical Switching Applications: Guidance from Theoretical and Computational Studies. Adv. Mater. 2014, 26, 68−84. (55) Gieseking, R. L.; Risko, C.; Marder, S. R.; Bredas, J.-L. Understanding the Relationships Among Molecular Structure, ExcitedState Properties, and Polarizabillities of Pi-Conjugated Chromophores. In WSPC Reference on Organic Electronics; 2016; Vol. 1, pp 393−419; doi 10.1142/9789813148598_0011. (56) Foresman, J. B.; Head-Gordon, M.; Pople, J. A.; Frisch, M. J. Toward a Systematic Molecular Orbital Theory for Excited States. J. Phys. Chem. 1992, 96, 135−149. (57) Stanton, J. F.; Gauss, J.; Ishikawa, N.; Head-Gordon, M. A Comparison of Single Reference Methods for Characterizing Stationary Points of Excited State Potential Energy Surfaces. J. Chem. Phys. 1995, 103, 4160−4174. (58) Send, R.; Valsson, O.; Filippi, C. Electronic Excitations of Simple Cyanine Dyes: Reconciling Density Functional and Wave Function Methods. J. Chem. Theory Comput. 2011, 7, 444−455. (59) Moore, B., II; Autschbach, J. Longest-Wavelength Electronic Excitations of Linear Cyanines: The Role of Electron Delocalization and of Approximations in Time- Dependent Density Functional Theory. J. Chem. Theory Comput. 2013, 9, 4991−5003. G

DOI: 10.1021/acs.jpca.6b10487 J. Phys. Chem. A XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry A (60) Le Guennic, B.; Jacquemin, D. Taking Up the Cyanine Challenge with Quantum Tools. Acc. Chem. Res. 2015, 48, 530−537. (61) Jacquemin, D.; Duchemin, I.; Blase, X. 0−0 Energies Using Hybrid Schemes: Benchmarks of TD-DFT, CIS(D), ADC(2), CC2, and BSE/GW Formalisms for 80 Real-Life Compounds. J. Chem. Theory Comput. 2015, 11, 5340−5359. (62) Muniz-Miranda, F.; Pedone, A.; Battistelli, G.; Montalti, M.; Bloino, J.; Barone, V. Benchmarking TD-DFT against Vibrationally Resolved Absorption Spectra at Room Temperature: 7-Aminocoumarins as Test Cases. J. Chem. Theory Comput. 2015, 11, 5371− 5384. (63) Laurent, A. D.; Le Guennic, B.; Jacquemin, D. Theoretical Spectroscopy of BASHY Dyes. Theor. Chem. Acc. 2016, 135, 173. (64) Barone, V.; Benassi, E.; Carnimeo, I. Computational Spectroscopy in Solution: Methods and Models for Investigating Complex Systems. In Quantum Modeling of Complex Molecular Systems; Springer International Publishing: Cham, 2015; pp 447−517. (65) Cerezo, J.; Santoro, F. Revisiting Vertical Models To Simulate the Line Shape of Electronic Spectra Adopting Cartesian and Internal Coordinates. J. Chem. Theory Comput. 2016, 12, 4970−4985. (66) Fabian, J. Electronic Excitation of Sulfur-Organic Compounds Performance of Time-Dependent Density Functional Theory. Theor. Chem. Acc. 2001, 106, 199−217. (67) Cohen, A. J.; Mori-Sanchez, P.; Yang, W. Insights into Current Limitations of Density Functional Theory. Science 2008, 321, 792− 794. (68) Mori-Sánchez, P.; Cohen, A. J.; Yang, W. Localization and Delocalization Errors in Density Functional Theory and Implications for Band-Gap Prediction. Phys. Rev. Lett. 2008, 100, 1−4. (69) Gieseking, R. L.; Ravva, M. K.; Coropceanu, V.; Brédas, J.-L. Benchmarking Density Functional Theory Approaches for the Description of Symmetry Breaking in Long Polymethine Dyes. J. Phys. Chem. C 2016, 120, 9975−9984.

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DOI: 10.1021/acs.jpca.6b10487 J. Phys. Chem. A XXXX, XXX, XXX−XXX