Four-Electron, Three-Orbital Model for the Low-Energy Electronic

Article pubs.acs.org/JPCA

Four-Electron, Three-Orbital Model for the Low-Energy Electronic Structure of Cationic Diarylmethanes: Notes on a “Pauling Point” Seth Olsen* School of Mathematics and Physics, The University of Queensland, Brisbane QLD 4072, Australia S Supporting Information *

ABSTRACT: We examine a four-electron, three-orbital complete active space self-consistent field (SA-CASSCF) and multistate multireference perturbation theory (MS-MRPT2) model of the electronic structure associated with the two lowest-lying electronic excitations of a series of cationic diarylmethanes related to Michler’s Hydrol Blue. These dyes are of interest because of the sensitivity of their excited-state dynamics to environmental influence in biological and other condensed phases. We show that the model corresponds to an easily understandable physical approximation where the dye electronic structure is mapped onto the π-electron system of an allyl anion. We show that reported trends in solutionstate absorbance bands and transition dipole moments associated with the first two electronic excitations can be described within reasonable accuracy by the model. We also show, for Michler’s Hydrol Blue, that the four-electron, three-orbital model provides a more balanced description of the electronic difference densities associated with electronic excitation calculated with the full πelectron space than can be achieved with active space models intermediate between these limits. The valence excitation energies predicted by the model are not sensitive to the underlying basis set, so that considerable computational savings may be possible by using split-valence basis sets with a limited number of polarization functions. We conclude that the model meets the criteria for a “Pauling Point”: a point where the cancellation of large errors leads to physically balanced model, and where further elaboration degrades, rather than improves, the quality of description. We advocate that this Pauling Point be exploited in condensed-phase dynamical models where the computational overhead associated with the electronic structure must kept to a minimum (for example, nonadiabatic dynamics simulations coupled to QM/MM environmental models). (Φfl ∼ 10−5) in solution but increases by up to 6 orders of magnitude when the dyes are bound to biomolecular hosts.3,5,7 Fluorescence increases are also observed in viscotic environments, through a somewhat smaller dynamic range.4,8 The binding-dependent fluorescence increase can be used for fluorescent labeling of biological samples. These dyes have been used as microenvironment-sensitive reporters in a variety of nonbiological chemical and materials science studies.4,8−10 The bis-N,N′-dimethylanilino motif also appears as the electron-donating moiety in promising metal-free organic dyes that have been used in dye-sensitized solar cells (cf. dyes 1−3 in Table 1 of ref 11). The usefulness of dyes in Scheme 1 hinges in a crucial way on the interaction between the dye excited state and its environment, as manifested through the dynamics. Accordingly, there is a standing need for simplified models of the low-energy electronic structure models that can be used in the context of condensed-phase dynamical models (e.g., on-the-fly nonadiabatic dynamics simulations coupled to hybrid QM/MM

1. INTRODUCTION Cationic tri- and diarylmethane dyes based on an bis-(N,N′dimethyl)-aniline skeleton (Michler’s Hydrol Blue, cf. Scheme Scheme 1

1) have a long history of human use.1,2 Recent interest has been driven by the sensitivity of their excited-state dynamical response to environmental influence in biological and other condensed phases.3−7 The fluorescence quantum yield of triand diarylmethanes and monomethine cyanines is very small © 2012 American Chemical Society

Received: December 21, 2011 Revised: January 5, 2012 Published: January 5, 2012 1486

dx.doi.org/10.1021/jp212295s | J. Phys. Chem. A 2012, 116, 1486−1492

The Journal of Physical Chemistry A

Article

−C6H5).23 For all other dyes, this paper is the first report of calculations of this type. The low-lying electronic spectra of Michler’s Hydrol Blue, Malachite Green, and Auramine-O (cf. Scheme 1, X = −H, −C6H5, NH2) have been previously studied with time-dependent functional theory techniques3,8,17,24 and with semiempirical configuration interaction models.16,17,25 Several groups have proposed three-state and/or threeorbital models for the phenomenological description of the electronic spectra of dyes related to those in Scheme 1. Moffitt has highlighted the analogy between the electronic structure of such systems with the resonance-theoretic and molecularorbital descriptions of the π-electron system of formamidinium ion.26 An analogy between the low-energy excitations of the green fluorescent protein chromophore (closely related to dyes in Scheme 1) and allyl anion species has been discussed by Krylov and co-workers27 and also by the author.28−31 A general three-state model of solvent effects on the spectra of polymethine dyes has been described by Painelli and coworkers.32

environmental models12). In such models, there is a significant incentive to minimize the overhead associated with the calculation of the subsystem electronic structure. At the same time, the need for extraordinarily accurate electronic structure is somewhat relaxed, because there is little point in having an electronic structure model that is more accurate than the environmental model or the dynamical propagation model to which it is coupled. In this paper, we assess the performance of a four-electron, three-orbital approximate model for the low-energy electronic structure of the dyes in Scheme 1. We argue that the model strikes a reasonable balance between physical realism and computational cost, and we advocate its use in, e.g., QM/MM on-the-fly dynamical models of condensed-phase behavior. We will do this by showing that this model corresponds to a solution to a particular quantum chemical problem that can be obtained for all of the dyes in Scheme 1, and by showing that the results obtained with this solution can describe, with reasonable accuracy, the known excited-state behavior of the dyes. We will also show that the reduced level of description we advocate provides a balanced approximation to much more detailed computational treatments of the electronic structure, at much lower cost. We anticipate that the models we highlight will find productive use in the implementation of models of the photodynamical response in condensed-phase environments.

3. COMPUTATIONAL QUANTUM CHEMISTRY PROTOCOL For each dye in Scheme 1, we obtained ground-state minimum energy geometries by optimization on the potential surface produced by Møller−Plesset second-order perturbation theory33 and a cc-pvdz basis set34 (MP2//cc-pvdz). The Cartesian coordinates of these geometries are available in an online supplement. For each dye at its ground-state minimum geometry, we obtained a solution of the four electron, three orbital, threestate-averaged complete active space self-consistent field35 problem, using the same basis set (SA3-CAS(4,3)//cc-pvdz). Equal weights were applied to the three states in the average. The SA3-CAS(4,3) solutions for the dye family are analogous (see Supporting Information). The initial orbitals for the SACASSCF procedure were the UHF charge-natural orbitals36 obtained for the doubly cationic (i.e., ionized) radical species. This is exactly the same protocol that we have used in a previous study of dyes related to the green fluorescent protein chromophore system.31 The SA-CASSCF states were used as reference states for a second-order multistate,37 multireference perturbation theory (MS-MRPT2) calculation38 of the excitation energies and (first-order) dipole observables. Only 32 orbitals could be correlated in the MS-MRPT2, for which a level shift39 0.1 atomic units was also used. Remixing of the reference space by the perturbation theory was not found to be significant; our choice of this method is guided by the need for a benchmark for later work, where such effects are likely to be significant.12 To assess the possibility of further computational cost savings, we repeated the entire suite of calculations outlined above (i.e., MP2 optimizations, SA-CASSCF, and MS-MRPT2 calculations) with the (much smaller) MIDI! basis set.40 The accuracy of SA-CASSCF and MS-MRPT2 techniques has been the assessed in several works,41−43 but even the most favorable assessments42 indicate it is unreasonable to expect accuracy better than ±0.1 eV. To assess the accuracy of the active space model, we performed a set of SA-CASSCF calculations on Michler’s Hydrol Blue, using much larger active spaces (up to and including the full π-electron space). These calculations were performed with the cc-pvdz basis set and were performed at the MP2 ground-state geometry obtained with this basis set. Three

2. REVIEW OF PREVIOUS WORK The dyes in Scheme 1 have been investigated before, using both experimental and theoretical chemical techniques. When the substituent at the methine position is hydrogen (i.e., X = H in Scheme 1), we obtain Michler’s Hydrol Blue. The lowenergy solution-state absorbance spectrum of Michler’s Hydrol Blue has been reported by Barker, Hallas and co-workers13,14 (98% acetic acid solvent) and by Looney and Simpson15 (glacial acetic acid). Calculations of the low-lying spectrum of Michler’s Hydrol Blue have been reported by Baraldi and coworkers16 (using a semiempirical configuration interaction singles model), by Fabian17 (using semiempirical configuration interaction and time-dependent density functional theory models), by Griffiths18 (using a semiempirical configuration interaction model), and by Kitts and co-workers3 (using timedependent density functional theory). The (X = −C6H5) derivative of Michler’s Hydrol Blue is called Malachite Green. The low-energy solution-state absorbance spectrum of Malachite Green has been reported by Barker and co-workers13 (98% acetic acid) and by Looney and Simpson15 (glacial acetic acid). The (X = −NH2) derivative of Michler’s Hydrol Blue is called Auramine-O. The low-energy solution-state absorbance spectrum of Auramine-O has been reported by Adam19 (glacial acetic acid) and by Singh and co-workers20 (acetonitrile and ethanol). The low-energy solution-state absorbance of the (X = −CH3) derivative of Michler’s Hydrol Blue has been reported by Barker and Hallas14 (98% acetic acid). The (X = −OH) derivative of Michler’s Hydrol Blue is the conjugate base of Michler’s Ketone. The low-lying solution-state absorbance of this species has been reported by Castelino and Hallas21 (10−12 M HCl in ethanol). The low-lying solution-state absorbance spectra of the (X = −CN) and (X = −F) derivatives of Michler’s Hydrol Blue have been studied by Il’chenko and coworkers22 (nitromethane). To our knowledge, correlated ab initio wave function-based electronic structure calculations have only ever been reported for the excited states of Malachite Green (cf. Scheme 1, X = 1487



Article

additional calculations with successively larger active spaces (SA3-CAS(8,7), SA3-CAS(12,11), and SA3-CAS(16,15)) were performed starting with the same initial orbital set as used above (i.e., the UHF charge natural orbitals of the radical dication). At each increment of the active space, two additional degenerate pairs of orbitals (each containing one symmetric and one antisymmetric orbital) were appended to the active spaceone pair in the occupied space, and one pair in the virtual space. This is equivalent to adding one occupied and one virtual local ring orbital per ring per increment. Our basis set does not contain diffuse functions and cannot be expected to reliably describe the electronic structure of, e.g., Rydberg excited states. This does not really concern us because (1) we are advocating an approximate electronic structure model for use in simulations in condensed phases, and (2) the molecules are cations. If a reliable prediction of high-resolution gas-phase electronic spectra were our goal, then the first of these two considerations would no longer apply. All computations were performed using the MOLPRO software.44 We emphasize that we are exploring a set of analogous particular solutions to the SA3-CAS(4,3) quantum chemical problem for these dyes. There are generally multiple solutions to SA-CASSCF problems with a given active space size and state-averaging protocol, which can be reached from different initial guesses. The initial guess we used is described above. In the interest of facilitating reproducibility, we include graphics of state-averaged natural orbitals and corresponding occupation numbers in the Supporting Information. These should be sufficient to identify the target solutions.

Figure 1. Structure of the four-electron, three-orbital active space model showing that the model maps the electronic structure of cationic diarylmethanes onto the π-electron system of an allylic anion species. (Top) Boys localized state-averaged orbital representation of the active space is approximately localized as an allyl, with π-electron orbitals localized to either the ring or the bridge. (Bottom) pseudocanonical state-averaged orbital representation is analogous to the highest bonding, nonbonding and lowest antibonding orbitals of an allyl system. Only the active space for Michler’s Hydrol Blue is shown; active spaces for other dyes have a qualitatively identical structure.

4. RESULTS 4a. Structure of the Self-Consistent Field Solutions. One of the points that we seek to emphasize here is that the SA-CASSCF solutions we have obtained for the dyes have a straightforward structure corresponding to an easily understandable (and chemically motivated) approximation. Specifically, the four-electron, three orbital active space model maps the electronic structure of the diarylmethane cations in Scheme 1 onto the π-electron system of an allyl anion. This mapping can be explored from either a localized orbital viewpoint (where the local orbitals of the dye are mapped onto carbon p orbitals of the allyl) or a delocalized viewpoint (where the pseudocanonical orbitals map onto the highest bonding, nonbonding and lowest antibonding orbitals of the allyl). This analogy for symmetric cyanines has been introduced by Moffitt.26 It has been applied to the closely related green fluorescent protein chromophore system by Krylov and coworkers27 and by the author.28−31 Although Figure 1 explicitly shows these results only for the parent Michler’s Hydrol Blue dye, it is also true of all other dyes in Scheme 1; orbitals for other dyes are shown in the Supplement. 4b. Comparison with Experimental Spectral Data. Figure 2 features a comparison of the excitation energies of the two lowest-lying electronic excitations computed with the SA3CAS(4,3)*MS-MRPT2 model with experimental solution-state absorbance data available from the literature.13−16,19−22 The data used to generate the figure are tabulated in numerical form in the Supporting Information. The first thing that we point out is that the absolute agreement of the predicted S0−S1 excitation is, for all dyes except Malachite Green (X = −C6H5), within 0.2 eV of the solution-state absorbance maxima. This is about as good as

should be systematically expected by these methods.43 For many dyes, the computed excitation energy is within 0.1 eV of the solution-state absorbance band maximum. This error should be considered in light of the fact that we are comparing purely electronic (i.e., Born−Oppenheimer) excitation energies for isolated molecules with data for dyes in a collection of solvents at ambient thermodynamic conditions. The agreement with experimental S0−S2 absorption band maxima is not as good as for the S0−S1 excitation. Here, the difference between the model and experiment is in the range 0.2−0.4 eV. The relatively poorer accuracy of the S0−S2 electronic excitation energy should be considered in light of the fact that experimental data are only available for four of the dyes in the set, and also that the corresponding absorbance band is characterized by its broad shape and much lower spectral intensity relative to the (very bright) S0−S1 band.45 Figure 3 shows a comparison of the S0−SN transition dipole moment norms with solution-state data. Solution-state transition moment norms in glacial acetic acid solution have been measured for three of the dyes in Scheme 1.15,19 These are Michler’s Hydrol Blue (X = −H), Auramine-O (X = −NH2), and Malachite Green (X = −C6H5). Though the model provides a slight overestimation, it is generally quite reasonable; the model is clearly able to reproduce the trend of transition dipoles over the series. The quantum chemical model readily reproduces the experimentally determined polarizations of the two low-lying transitions of the diarylmethanes.2,15,19,45 Specifically, the S0−S1 transition is polarized along a vector pointing from one ring to another (for this reason, this band is sometimes referred to as the x band in the spectroscopic literature45). The S0−S1 1488



Article

Figure 2. Comparison between calculated and experimental low-lying (S0−S1,S0−S2) excitation energies for dyes in the series. Experimental solution-state first (x band; squares) and second (y band; diamonds) absorbance band maxima were taken from the literature (cf. refs 13−16 and 19−22) and are color-coded by solvent in which they were recorded (as shown). The dotted line represents perfect agreement, and the deviation of calculated and experimental values is expressed as vertical distance. Error bars are set uniformly at ±0.1 eV and represent most favorable assessment of the accuracy of the underlying quantum chemistry (cf. refs 41−43). There are multiple values for solution-state experiments, but only one for computational predications, so all points on a horizontal line correspond to a single dye in multiple solvents.

Figure 3. Comparison between calculated S0−SN transition dipole norms (e Å) with their solution-state values measured in glacial acetic acid (cf. refs 15 and 19). S0−S1 transitions (x band) are indicated with squares, and S0−S2 transitions (y band) with diamonds. Error bars represent reported uncertainty in the experimental value. The dotted line represents perfect agreement. In all cases, the calculated polarizations are consistent with the experimental ones (see text).

smallest molecule in the set, we can apply the most expensive methods to this case. Figure 4 summarizes key results of the expanded active-space computations. We highlight two aspects. The first is that the excitation energies for both excitations decrease monotonically as the active space is expanded. The excitation energy lowering for each state is approximately linear in the log number of configurations generated by the CAS expansion. There is no particular reason to expect that this linear behavior is general, but it does indicate for this particular system that the correlation contributions to the excitation energies are distributed more-or-less evenly over orbitals of the π-electron system. The second point is more surprising. This point is that the electron density difference isosurfaces calculated with successively larger active spaces do not converge monotonically as the active space is increased. Indeed, the agreement with the complete π-space calculation gets progressively worse for the intermediate approximations between the SA3-CAS(4,3) model and the complete SA3-CAS(16,15) model. The degradation in quality of the approximation is more apparent for the S0−S2 excitation than for the S0−S1 excitation. First, the SA3-CAS(8,7) S0−S2 difference density shows a distortion with respect to the pseudosymmetry axes of each aniline ring, with more density being drawn from the side of each ring that is on the “inside” of the triangle outlined by the methine bridge. This feature is qualitatively different from the corresponding difference density from the complete π-electron calculation, wherein the difference density is much more symmetric with respect to the pseudosymmetry axes of the N,N′-dimethylaniline moieties.

transition is polarized along a vector drawn between the methine bridge and a point midway between the rings (y band, respectively45). For dyes whose substituent supports the C2 symmetry of the common bis-N,N′-dimethylaniline frame, the excited states belong to different irreps, and the orthogonality is exact. For dyes with asymmetric substituents (i.e., X = −CH3, −OH in Scheme 1), it is approximate. We point out that in no case was the molecular symmetry enforced in any of the calculations that we describe here. It is worth noting that the experimental data collected in Figure 3 (for solvents with bulk dielectric ranging from 6.5 to 39.4 at room temperature) indicate a general insensitivity of the relevant absorption wavelengths to the effects of dielectric solvation. This is particularly apparent for the data on Auramine-O (X = −NH2), wherein the experimental band maximum recorded in glacial acetic acid19 (dielectric constant ε = 6.2) overlaps with the band maximum recorded in acetonitrile20 (ε = 37.5). 4c. Comparison with More Elaborate SA-CASSCF Models. In this section, we argue that the SA3-CAS(4,3) model is a good approximation to SA-CASSCF models with larger active spaces. We will focus on Michler’s Hydrol Blue molecule, because this represents the common electronic space that all the dyes share. The argument we present should be applicable to all of the dyes in the set, but in addition there may be specific details that arise from the electronic structure of the individual substituents. Our choice is also motivated in part by computational considerations; Michler’s Hydrol Blue is the 1489



Article

Figure 5. Excitation energy predictions afforded by the four-electron, three-orbital model that are robust with respect to reduction of the basis set, offering the potential for even lower computational overhead. Shown are S0−S1 (red) and S0−S2 (blue) excitation energies evaluated by MS-MRPT2 (filled circles) calcluations, as well as by the underlying SA3-CAS(4,3) reference (unfilled circles) using either a cc-pvdz (horizontal axis) or MIDI! (vertical axis) basis set. The cc-pvdz basis is a double-ζ basis set with polarization functions on all atoms. The MIDI! basis is a split-valence basis set with polarization functions on nitrogen atoms. Error bars are set at ±0.1 eV for all data points in both directions and represent the most favorable estimate of the accuracy of the underlying quantum chemistry protocol (cf. refs 41−43).

Figure 4. Changes in the SA-CASSCF excitation energies and associated difference densities of Michler’s Hydrol Blue as the active space is expanded. The S0−S1 and S0−S2 excitation energies predicted using (from left) SA3-CAS(4,3), SA3-CAS(8,7), SA3-CAS(12,11), and SA3-CAS(16,15) calculations are graphed against the log of the number of generated configurations, and associated electron difference isodensity surfaces are overlaid. The excitation energies decrease monotonically but the difference densities do not, and the fourelectron, three-orbital SA-CASSCF provides the best qualitative approximation to the difference densities of the full π-electron active space calculation.

For the SA3-CAS(12,11) case, the distortion of the S0−S2 difference density represents an unphysical breaking of the symmetry of the molecular frame. This effect is most obvious in the S0−S2 difference density, but it is also present in the S0−S1 difference density and can be seen by close inspection of the figure. The behavior shown in Figure 4 suggests that, for the purposes of modeling interaction with a condensed-phase environment, the four-electron, three-orbital model presents a more robust level of approximation than any other active space short of the full π-electron Hilbert space. This makes it an attractive alternative for situations where the computational expense of the full π-electron calculation needs to be avoided. 4d. Robustness with Respect to Basis Set Reduction. The results presented above suggest that the four-electron, three-orbital model is a reasonable approximate level of description for the low-energy electronic structure of cationic diarylmethanes that balances physical realism and computational expense. Here, we show that even further reductions in the computational overhead can be achieved through the reduction of the underlying basis set. Figure 5 shows a quantitative comparison between the electronic excitation energies calculated using the four-electron, three-orbital model with two basis sets of very different size: ccpvdz and MIDI!. The cc-pvdz (which has been used for all results above) is a double-ζ basis set with polarization functions on all atoms. For the dyes in Scheme 1, this leads to between 371 (Michler’s Hydrol Blue) and 475 (Malachite Green)

contracted basis functions (660 and 850 primitives, respectively). The MIDI! basis is a split-valence basis set with polarization functions on nitrogen and oxygen atoms. This set leads to between 230 (Michler’s Hydrol Blue) and 285 contracted functions (360 and 462 primitives, respectively). Figure 5 shows that there is no significant systematic difference in the excitation energies calculated using either set within the context of an otherwise identical quantum chemical protocol.

5. DISCUSSION We have described a series of analogous low-rank SA-CASSCF models for the low-energy electronic structure of a series of cationic diarylmethane dyes derived from Michler’s Hydrol Blue. The utility of these dyes arises from the sensitivity of their excited-state properties and dynamics to environmental influence in condensed phase. We have shown that this model strikes a reasonable balance between physical realism and computational cost, and we advocate in condensed-phase dynamical simulations (e.g., hybrid QM/MM on-the-fly nonadiabatic dynamics models12) where the computational overhead associated with the dye subsystem must be kept to a minimum, and where errors in the environmental model and/ or dynamical propagation model limit the benefits to be accrued from more detailed electronic structure ansätze. We have shown that the model corresponds to a straightforward physical approximation where the electronic structure of the dye is mapped onto the π-electron system of an allyl anion (or, alternatively, a formamidinium cation) species. 1490



Article

structure be represented for both rings without breaking active space symmetry. If the progressive distortion of the electronic difference densities is understood in this way, then it is clear that the relatively balanced description provided by the lowest SA3CAS(4,3) level is due to a cancellation of errors. This is not an ideal situation, but one must weigh this against the implication that a truly faithful description of the electronic structure requires invocation of the entire π-space of Michler’s Hydrol Blue (leading to a configuration expansion that is 6 orders of magnitude larger)! Lastly, we have shown that the results obtained within the SA3-CAS(4,3) model are not sensitive to reduction of the basis set. This observation is of immediate interest with respect to, e.g., the use of the model in hybrid QM/MM on-the-fly nonadiabatic simulations, because it means that even greater reduction of the computational overhead is achievable. It also yields physical insights, because it suggests a limited role differential for core polarization effects (there is only one core function in MIDI!, and two in cc-pvdz) and left−right correlation effects (there are polarization effects only on nitrogen and oxygen in MIDI!, but cc-pvdz has these on all atoms). The robustness of the electronic structure with respect to this reduction in the basis set shows that the electronic structure is “converged” in the operational sense of the word. In a series of upcoming publications, we will be applying the SA3-CAS(4,3) model studied here to gain physical insights into structure−property relationships in the cationic diarylmethanes, to the potential energy surfaces describing the photoisomerization reaction, and to the dynamics on these surfaces.

This suggests that there is considerable physical insight (in addition to computational savings) that can be gained from the study of the model. We have already begun to exploit this opportunity in our studies of, e.g., photoisomerization on the excited-state surface.46 The photoisomerization reaction plays a central role in the sensitivity of the excited-state properties to environmental influence in these3,4,6,7,10 and related47−49 systems. We have shown that the electronic energies and transition dipole moments that are obtained using this particular family of SA-CASSCF models are in reasonable agreement with known solution-state spectroscopic results. Comparison against gasphase results would be beneficial as well (and would provide a better gauge of the absolute accuracy of the model), but as far as we know, there are no such data available. The agreement is best for transition into the S0−S1 electronic state, which is the state of interest for most applications. Broadly speaking, the data in Figure 2 indicate that the model we describe here does provide a useful level of approximation that is consistent for a range of dyes whose absorbance spans the visible spectrum. For this reason, we thought it important that the model be brought to the attention of the community. In addition to an assessment of the model against experimental spectra, we have also been able to assess its quality against more elaborate SA-CASSCF models for the particular case of Michler’s Hydrol Blue. This has led to the counterintuitive result that the SA3-CAS(4,3) approximation provides a better approximation to excitations of the full πelectron system than is provided by intermediate levels of approximation! This result is particularly worthy of note, because it highlights the failure of a (usually, implicit) assumption that larger active spaces are a priori better approximations than smaller ones. It seems that, in the SA3CAS(4,3) model, we have uncovered what Löwdin called a “Pauling Point”: a point where competing errors balance, and where further expansion of the model leads to degradation rather than improvement (that is, until the next such point is achieved).50 In the context of suggesting an approximation for use in condensed-phase models, the behavior highlighted in Figure 4 is particularly interesting. Electrostatic interactions are an important component of solvation. It is very important that a quantum chemical model is ablequalitatively, at the very leastto provide a balanced approximation to the differences in electronic density that accompany electronic excitation. Figure 4 shows that the next best thing to a four-electron, threeorbital model is the full π-electron space. The latter calculation is very intensive, though, and impractical for use as part of a larger environmental and/or dynamical model. There is significant physical insight to be gained from the results of Figure 4. We propose that the pseudosymmetry and real symmetry breaking that is observed there can be understood in terms of a competition between the variational benefits offered by orbital polarization and from the ability to describe aromatic electronic structure. At the SA3-CAS(8,7) level, the active space is still not sufficient to describe aromatic electronic structure on the aniline rings, but there is some variational benefit to be gained by polarization of the ring. At the SA2-CAS(12,11) level, there is enough flexibility to describe aromatic stabilization on one ring, but this requires breaking the C2 symmetry of the dye skeleton. Only at the SA3CAS(16,15) level can the aromaticity of the π-electronic

6. CONCLUSION We have provided an argument in favor of the use of a fourelectron, three-orbital SA-CASSCF model for the description of the low-energy electronic structure of cationic diarylmethanes derived from Michler’s Hydrol Blue. We anticipate that this model will find use in models of condensed-phase photophysical processes in these dyes.

■

ASSOCIATED CONTENT

S Supporting Information *

Cartesian coordinates (Å), SA-CASSCF state energies and dipole moment operators, MS-MRPT2 state energies, and state-averaged natural orbitals with occupation numbers, comparisons of canonical and localized active orbitals for different dyes, and tabulated data to accompany Figures 2−5. This information is available free of charge via the Internet at http://pubs.acs.org.

■ ■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected].

ACKNOWLEDGMENTS This work was supported by Australian Research Council Discovery Grants DP110101580 and DP0877875. Computations were carried out at the National Computational Infrastructure Facility under the auspices of a Merit Allocation Scheme Grant (m03). We thank Prof. Anna Painelli, Dr. Christian Evenhuis, and Prof. Ross H. McKenzie for illuminating discussions. Molecular graphics were rendered using VMD.51 1491



■

Article

(43) Schreiber, M.; Silva-Junior, M. R.; Sauer, S. P. A; Thiel, W. J. Chem. Phys. 2008, 128, 134110. (44) MOLPRO, version 2010.1, a package of ab initio programs, H.-J. Werner., P. J. Knowles., G. Knizia., F. R. Manby., M. Schütz., and others, see http://www.molpro.net. (45) Lewis, G. N.; Bigeleisen, J. J. Am. Chem. Soc. 1943, 65, 2102. (46) Olsen, S.; McKenzie, R. H. J. Chem. Phys. 2009, 131, 234306. (47) Clark, F. T.; Drickamer, H. G. J. Phys. Chem. 1986, 90, 589. (48) Conyard, J.; Kondo, M.; Heisler, I. A.; Jones, G.; Baldridge, A.; Tolbert, L. M.; Solntsev, K. M.; Meech, S. R. J. Phys. Chem. B 2011, 115, 1571. (49) Tolbert, L. M.; Baldridge, A.; Kowalik, J.; Solntsev, K. M. Acc. Chem. Res. 2011, DOI: 10.1021/ar2000925. (50) Löwdin, P. O. Pure Appl. Chem. 1989, 61, 2185. (51) Dalke, H. W.; Schulten, K. J. Mol. Graph. 1996, 14, 33.

REFERENCES

(1) Rose, R. J. Chem. Educ. 1926, 3, 973. (2) Griffiths, J. Colour and constitution of organic molecules; Academic Press: London, 1976; pp 240−270. (3) Kitts, C. C.; Beke-Somfai, T.; Nordén, B. Biochemistry 2011, 50, 3451. (4) Brey, L. A.; Schuster, G. B.; Drickamer, H. G. J. Chem. Phys. 1977, 67, 2648. (5) Conrad, R. H.; Heitz, J. R.; Brand, L. Biochemistry 1970, 9, 1540. (6) Kondo, M.; Heisler, I. A.; Meech, S. R. J. Phys. Chem. B 2010, 114, 12859. (7) Babendure, J. R.; Adams, S. R.; Tsien, R. Y. J. Am. Chem. Soc. 2003, 125, 14716. (8) Rafiq, S.; Yadav, R.; Sen, P. J. Phys. Chem. B 2010, 114, 13988. (9) Kondo, M.; Heisler, I. A.; Meech, S. R. Faraday Discuss. 2010, 145 IS, 185. (10) Vogel, M.; Rettig, W. Ber. Bunsen-Ges. Phys. Chem. 1985, 89, 962. (11) Mishra, A.; Fischer, M. K. R.; Bauerle, P. Angew. Chem., Int. Ed. 2009, 48, 2474. (12) Virshup, A. M.; Punwong, C.; Pogorelov, T. V.; Lindquist, B. A.; Ko, C.; Martínez, T. J. J. Phys. Chem. B 2009, 113, 3280. (13) Barker, C. V.; Bride, M. H.; Hallas, G.; Stamp, A. J. Chem. Soc. 1961, 1285. (14) Barker, C. C.; Hallas, G. J. Chem. Soc. 1961, 1395. (15) Looney, C. W.; Simpson, W. T. J. Am. Chem. Soc. 1954, 76, 6293. (16) Baraldi, I.; Carnevali, A.; Momicchioli, F. Chem. Phys. 1992, 160, 85. (17) Fabian, J. Dyes Pigments 2010, 84, 36. (18) Griffiths, J.; Pender, K. Dyes Pigments 1981, 2, 37. (19) Adam, F. C. J. Mol. Spectrosc. 1960, 4, 359. (20) Singh, C.; Modak, B.; Mondal, J. A.; Palit, D. K. J. Phys. Chem. A 2011, 115, 8183. (21) Castellino, R.; Hallas, G. J. Chem. Soc. B 1971, 1468. (22) Il’chenko, A. Y.; Koval’chuk, R. E.; Yagupol’skii, L. M. Zh. Org. Khim. 1973, 9, 1744. (23) Nakayama, A.; Taketsugu, T. J. Phys. Chem. A 2011, 115, 8808. (24) Preat, J.; Jacquemin, D.; Wathelet, V.; André, J. M.; Perpète, E. A. Chem. Phys. 2007, 335, 177. (25) Matsuura, A.; Sato, H.; Sotoyama, W.; Takahashi, A.; Sakurai, M. J. Mol. Struct. (THEOCHEM) 2008, 860, 119. (26) Moffitt, W. E. Proc. Phys. Soc. A 1950, 63, 700. (27) Bravaya, K. B.; Grigorenko, B. L.; Nemukhin, A. V.; Krylov, A. I. Acc. Chem. Res. 2011, DOI: 10.1021/ar2001556. (28) Olsen, S. J. Chem. Theory Comput. 2010, 6, 1089. (29) Olsen, S.; McKenzie, R. H. J. Chem. Phys. 2011, 134, 114520. (30) Olsen, S.; McKenzie, R. H. J. Chem. Phys. 2009, 130, 184302. (31) Olsen, S.; McKenzie, R. H. Chem. Phys. Lett. 2010, 492, 150. (32) Terenziani, F.; Przhonska, O. V.; Webster, S.; Padilha, L. A.; Slominsky, Y. L.; Davydenko, I. G.; Gerasov, A. O.; Kovtun, Y. P.; Shandura, M. P.; Kachkovski, A. D.; Hagan, D. J.; Van Stryland, E. W.; Painelli, A. J. Phys. Chem. Lett. 2010, 1, 1800. (33) Azhary, A. E.; Rauhut, G.; Pulay, P.; Werner, H.-J. J. Chem. Phys. 1998, 108, 5185. (34) Dunning, T. H. J. Chem. Phys. 1989, 90, 1007. (35) Werner, H.-J.; Meyer, W. J. Chem. Phys. 1981, 74, 5794. (36) Pulay, P.; Hamilton, T. P. J. Chem. Phys. 1988, 88, 4926. (37) Finley, J.; Malmqvist, P. Å.; Roos, B. O.; Serrano-Andrés, L. Chem. Phys. Lett. 1998, 288, 299. (38) Celani, P.; Werner, H. J. Chem. Phys. 2000, 112, 5546. (39) Roos, B. O.; Andersson, K. Chem. Phys. Lett. 1995, 245, 215. (40) Easton, R. E.; Giesen, D. J.; Welch, A.; Cramer, C. J.; Truhlar, D. G. Theor. Chem. Acc. 1996, 93, 281. (41) Roos, B. O.; Andersson, K.; Fülscher, M. P.; Serrano-Andrés, L.; Pierloot, K.; Merchán, M.; Molina, V. J. Mol. Struct. (THEOCHEM) 1996, 388 IS, 257. (42) Azizi, Z.; Roos, B. O.; Veryazov, V. Phys. Chem. Chem. Phys. 2006, 8, 2727. 1492


Four-Electron, Three-Orbital Model for the Low-Energy Electronic

Recommend Documents