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On the Color of Bridge-Substituted Cyanines Seth Olsen J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.6b10340 • Publication Date (Web): 21 Nov 2016 Downloaded from http://pubs.acs.org on November 26, 2016

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The Journal of Physical Chemistry

On the Color of Bridge-Substituted Cyanines Seth Olsen* School of Mathematics and Physics, The University of Queensland, Brisbane QLD 4072 Australia AUTHOR INFORMATION Corresponding Author *[email protected] Present Addresses †If an author’s address is different than the one given in the affiliation line, this information may be included here.

ABSTRACT Theories of color in cyanine dyes have evolved around the idea of a “resonance” of structures with distinct bonding and charge localization.

Understanding the emergence of

“resonance” models from the underlying many-electron problem remains a central issue for these systems. Here, the issue is addressed using a maximum-entropy approach to valence-bond representations of state-averaged complete-active space self-consistent field (SA-CASSCF) models. The approach allows calculation of energies and couplings of high-energy valence-bond structures that mediate superexchange couplings and chemical bonding. A series of valencebond Hamiltonians for a series of bridge-substituted derivatives of Michler’s hydrol blue (a monomethine cyanine) is presented.

The Hamiltonians are approximated with a simple linear

model parameterized by the Brown-Okamoto σp+ parameter of the bridge substituent.

A

quantitative lower bound on σp+, beyond which a resonant cyanine-like ground state won’t exist,

ACS Paragon Plus Environment

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is presented. The large effective coupling in two-state resonance models emerges from superexchange associated with either covalent bonding or charge-carrier delocalization, with the former contribution significantly the stronger.

The results provide ab initio justification for

empirical diabatic models of methine optical response.

They are of general interest for

understanding the response in cyanines.

Introduction Color-constitution relationships in cyanine dyes have been of interest to chemistry since the beginning of modern chemical industry.1-2 Interest in monomethine cyanines such as in Scheme 1 as been rekindled their use as biophotonic fluorescence imaging markers3-7. The bisdimethylanilino group that is common to all the dyes in Scheme 1 has also appeared as an electron-donating group in organic photovoltaic sensitizers8, motivating interest in the optoelectronic properties of derived dyes. The low-energy electronic structure of symmetric monomethines such as in Scheme 1 has been modeled as a resonance between structures that are distinguishable by charge location and bond alternation.2, 9 This is shown in the Scheme. A characteristic feature of two-state resonance models is the large (1 – 2 eV) effective coupling matrix element between the structures, which is larger than expected based on orbital overlap between charge centers10-13. The origins of the large coupling are resolved in models that include additional higher-energy symmetric structures that couple the resonant pair via superexchange.14 The introduction of higher states seems at first to introduce too many parameters for a model of only the lowestenergy band – but the idea is saved by the observation that such expanded models also describe higher-order optical processes pertaining to higher excited states.15-17 A recent theoretical


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analysis suggested that superexchange mediated by at least two high-energy structures may contribute to the resonance coupling of dyes in Scheme 118. The precise origin of the large effective coupling remains an open question. An early success of molecular orbital theories of cyanine dyes was the Dewar-Knott color rule.19-21 For dyes in Scheme 1, which have an excess π-electron relative to half-filling, raising the electronic potential at the bridge should raise the S0–S1 adiabatic energy gap.19-21 One way to raise the potential at the bridge would be via inductive interactions brought on by a substituent group. The Brown-Okamoto subsituent parameter σp+ measures the ability of the substituent to stabilize hole density at the reaction center in cases where π-electron resonance can contribute22. Substituents with more negative σp+ parameters23-24 are expected to raise the potential at the bridge, and blue-shift the electronic gap. Figure 1 confirms that solution-state absorbance energies of dyes in Scheme 1 show good linear anti-correlation (R2 > 0.95) with the BrownOkamoto substituent parameter σp+.23-24 Complete active space valence-bond (CASVB) representation of complete active space selfconsistent field (CASSCF) calculations offer a route to direct ab initio valence-bond models of chemical processes25-27. The ability to represent a CASSCF solution in a valence-bond representation is guaranteed for single-state CASSCF calculations28. Multiple excited states of arbitrary symmetry can be treated when a state averaged CASSCF (SA-CASSCF) is used29, but the requirement that the variational condition be invariant to the CASVB I transformation seriously constrains the weightings that can be used in the SA-CASSCF30. Equilibration of SACASSCF weights in a canonical-ensemble scheme guarantees invariance to transformation into a CASVB representation30. The approach allows CASVB models with an arbitrary number of


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excited states within the same self-consistent field, but still maintains variational focus on an energy scale. The scale is introduced as the reciprocal of the Lagrange multiplier conjugate to the Hamiltonian, which acts as an effective “temperature”31-32. A recent CASVB study of lowenergy electronic structure in para-substituted green fluorescent protein (GFP) chromophores used a similar approach33.

Scheme 1. Dyes studied in this paper.


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Figure 1.

Solution-state absorbance (colors as shown) and SA-CASSCF S0-S1 adiabatic

excitation energies (black dots) plotted against the Brown-Okamoto substituent parameter σp+. SA-CASSCF excitation energies were uniformly shifted down by 1.46 eV to pin the excitation energy of Michler’s hydrol blue to its experimental absorbance maximum in dichloromethane. Solution-state data in the indicated solvents were taken from the literature. Experimental data was from references 34-41. The experimental excitation energies are tabulated with references in the supplement. Separate linear regressions of the experimental data (solid line) and the SACASSCF data (dotted line) are shown.


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Description and benchmark of SA-CASSCF model This paper uses a complete active space self-consistent field (CASSCF) model previously advocated for the low-energy electronic structure of monomethines.18, 42-46 The model maps the electronic structure of a monomethine cyanine onto the 3-orbital allyl anion π-electron system, as shown in Figure 1, using an appropriate set of localized frontier orbitals. When this 3-orbital system is filled with 4 electrons, this model generates the 6-dimensional many-body Hilbert space spanned by the states shown at Figure 2, bottom. Although the dyes in Scheme 1 are cations, the physical charge carrier is an electron because the π-electron system is more than half-filled. A three orbital space with four (or two) electrons is the minimal complete quantum model that can describe transport of an excess charge and a covalent bond-pair. Earlier work used this active space model, uniformly weighted over either two43-44 or three18, 45 states, for dyes in Scheme 1. This work uses a new strategy that allows resolution of all of the CASVB states that mediate chemical bonding and superexchange coupling in these systems. I will show that this provides a more transparent chemical interpretation, because the variation of the CASVB models for different dyes becomes very simple when these high-energy structures are resolvable.


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Figure 2. Illustration of CASVB model. (Top) Boys-localized active space orbitals from a SA6CAS(4,3) calculation with canonical-ensemble weighting equilibrated at a “temperature” of 2.5 eV. Calculations at 2.0 eV and 3.0 eV gave visually identical orbitals. (Bottom) CASVB states built over the orbitals. Explicit corresponding structures are shown for the case of Michler’s hydrol blue (R=H).


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The SA-CASSCF results in this paper were generated using a self-consistent maximumentropy (canonical ensemble) weighting over all six states generated in the four-electron, threeorbital CAS expansion (see Figure 2).47-48 At convergence, the state-averaged density matrix Γ is a maximum-entropy ensemble with the SA-CASSCF Hamiltonian as sufficient statistic:

(0.1) where f is the free energy (which is minimized at convergence) in units of β-1, I is the identity on the Hilbert space generated by the CAS expansion, H is the self-consistent SA-CASSCF effective Hamiltonian, and β is a Lagrange multiplier conjugate to the Hamiltonian. The reciprocal of the Lagrange multiplier β-1 defines a characteristic energy scale that acts as an effective electronic “temperature”. I use quotes to distinguish this from the physical temperature, which it is not. All calculations were performed in Molpro49. Equilibration was realized using an iterative weight-updating loop written in Molpro’s internal scripting language49. Convergence to the solutions shown in Figure 2 was robust for most dyes for “temperatures” spanning visible photon energies. All calculations were performed at the ground state of an MP2 model50. I used Dunning’s cc-pvdz basis set51. An earlier work44 using a twostate-averaged (but otherwise identical) model showed that results obtained with cc-pvdz were quantitatively identical to those obtained with the (much smaller) MIDI! basis set52, indicating the model is converged (in the operational sense) with respect basis set size for the properties studied here.


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One advantage of using a canonical-ensemble weighting scheme in SA-CASSCF is that concepts from the methods of free-energy relationships in physical organic chemistry22 can be used for comparative analysis of results. There is an isolobal analogy53-54 between the Boyslocalized55 active space orbitals for dyes in Scheme 1 – this is shown in Figure 2, top. The Boyslocalized orbitals are the orbitals that maximize the sum of distances between charge-centroids – a thorough modern account of the Boys localization has been contributed by Subonik and coauthors55. I will invoke the isolobal analogy, shown in Figure 2, to define a common abstract Hilbert space, spanned by the CASVB states that correspond via the analogy. In the common abstract Hilbert space, the ensemble state of each dye satisfies independently its own thermodynamic identity

(0.2)

,

where S=-lnΓ Γ is the surprisal (an operator whose expectation is the von Neumann entropy) and the subscript denotes the dye labeled by its substituent R. Once the notion of a common abstract Hilbert space is invoked, the concepts of addition and subtraction between states of dyes become naturally defined, as needed to deploy the tools of linear free energy relationships towards the quantitative comparison of the SA-CASSCF calculations.33 To benchmark the ability of the SA-CASSCF model to describe color trends for dyes in Scheme 1, I shifted uniformly the SA-CASSCF S0->S1 electronic gaps by 1.46 eV (to pin the gap of MHB to its value in dichloromethane35) and compared these against a basket of solution-state data reported in the literature35-37, 39, 41, 56-57. This comparison is shown in Figure 1. The SACASSCF data quantitatively captures the trend of the solution-state absorbance on the substituent parameter σp+.


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Results The main results of this paper are shown in Figures 3 and 4. Figure 3 shows diabatic state energies of all CASVB states calculated for dyes at a “temperature” of 2.5 eV. Error bars give the variation in each as the “temperature” is varied in the range 2.0-3.0 eV. The energies are fit to linear relationships against the Brown-Okamoto substituent parameter σp+. Root-mean-square deviations from the fit are shown. As the electrodonation power of the substituent increases (decreasing σp+), so that the S0–S1 excitation energy increases, Figure 3 shows that CASVB state energies vary in understandable ways. CASVB states with an electron pair on the bridge are destabilized, and CASVB states with a hole on the bridge are stabilized, relative to single occupation. The figure shows also that there are two points in the relevant parameter range where the CASVB states cross. One is where |202> and |121> cross. The other is where |202> crosses below the “resonating pair” (|211>,|112>). These state crossings indicate that reduction to a two-state or three-state resonance model with well-defined diabatic states could not work over the entire range of the substituent parameter shown. The error bars in Figure 3 represent the variation in the corresponding energy when the “temperature” is varied in the range 2.0-3.0 eV, showing that the dependence on the chosen “temperature” is weak. The CASVB couplings could be classified into three distinct groups based on magnitude. A histogram of CASVB couplings for all dyes at “temperature” 2.5 eV is in Figure 4. The distribution is clearly trimodal. The group of couplings with largest magnitude (~-2.5 eV) are the couplings in the set (, , , ) These couplings connect the covalent resonating pair to their bond-polarized ionic counterparts. They


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are associated with covalent bonding interactions. The group with second largest magnitude (~1.5 eV) are the set ( , ). They represent delocalization of the excess electron onto the bridge. Remaining couplings were small (S1 gap or with the Brown-Okamoto substituent parameter σp+. A main point of Figure 4 is that couplings with significant magnitude correspond to transfers across bonded groups. This is a result because the SA-CASSCF model is capable of describing more complex coupling scenarios. Although the only significant couplings are those that would occur in a tight-binding model, the SA-CASSCF model need not respect alternancy symmetry because the transfer elements are not strictly one-particle operators58. The intuitive physical reason for the inequality of matrix elements for transfer of an electron on (Fig 4, red, orange bars) or off (Figure 4, green, yellow bars) of the bridge is that spin constraints are different for the two processes. For transfer of an electron onto the bridge, the transferred electron is constrained to have spin anti-parallel to those on both the bridge and the donor ring. For the case of transfer off the bridge the transferred electron is only constrained to have spin anti-parallel to its partner in the singlet.


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. Figure 3. This figure displays CASVB diabatic state energies (relative to average) for dyes in Scheme 1, plotted against the Brown-Okamoto substituent parameter σp+. CASVB states are indicated using graphical labels as in Figure 2. Linear regression fits are shown with RMSD. For dyes with asymmetric substituents (R=OH, CH3), the fit was taken against the average energy of corresponding pairs of CASVB states. Error bars show the range of state energies spanned as the “temperature” of the CASVB ensemble is scanned in the range 2.0-3.0 eV.


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Figure 4. This figure shows the distribution of CASVB Hamiltonian coupling matrix elements. Couplings were calculated using a SA-CAS(4,3) ensemble equilibrated at a “temperature” of 2.5 eV. The couplings fall into three distinct groups. Strong (>0.5 eV) couplings are color-coded and labelled using graphical labels in Figure 2. The group of largest-magnitude (~–2.5 eV) couplings is associated with covalent chemical bonding interactions; these link covalent states with their bond-polarized ionic counterparts. The second-largest-magnitude group (~–1.5 eV) is associated with charge delocalization between the rings and the bridge. All other couplings are less than 0.5eV in magnitude, and are not color-coded or individually labeled.


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Discussion The results in Figures 3 and 4 suggest that CASVB Hamiltonians for bridge-substituted derivatives of Michler’s hydrol blue be coarsely approximated by the parametric family (0.3)

 6.30 − 1.16σ + p  0   + H (σ p )  0 =  eV 0   −2.5  0 

0

0

0

6.30 − 1.16σ +p

0

0

0

1.66 − 1.26σ +p

0

0

0

0 −2.5

−1.5 −1.5

1.08 + 1.54σ −2.5 −2.5

  0 −2.5   −1.5 −1.5   −2.5 −2.5  0 0   0 0  −2.5

+ p

0

,

where the linear fits in Figure 3 have been shifted so that the fit line for the covalent resonating configurations (|211>,|112>) in Figure 3 defines the zero of energy. The ordering of indexes from left to right (or top down) is (|022>, |220>,|121>,|202>,)112>,|211>). The constant couplings were determined as the mean of the distinct groups identified in Figure 4. This Hamiltonian has a simple, chemically transparent structure. It’s diagonal elements depend linearly on the substitution parameter σp+, as suggested in Figure 3, and the couplings are independent of the parameter, as in Figure 4. The only couplings considered in Equation (0.3) are the large ( and |112> are below |202>, which can only be true for for substituents with σp+ > -0.7 according to the model Equation (0.3). For substituents with σp+ ≤ -0.7, the excited state is still dominated by the out-of-phase “twin” state60 |211> – |112>, but the ground state becomes dominated by |202> so a two-state “resonance” model will no longer describe well the ground and first excited states. The model therefore anticipates that the ground state of auramine-O (R=NH2) is not normally written using the resonance notation in Scheme 1, but rather as in Scheme 2.7, 41, 61-68

Scheme 2. Ground state of auramine-O. The grouping of CASVB couplings in Figure 4 suggests an origin for the large effective coupling in two-state resonance models of cyanine dyes. In these models, the bare coupling between charge-resonant covalent structures (eg. Scheme 1) is assumed to be small, but is magnified by superexchange via symmetric excited-state diabatic strutures.14, 18 Our results suggest that there are two distinct channels that contribute to the coupling, associated with the diabatic CASVB states |202> and |121>. Either of these channels would lead to effective


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coupling at second order between the resonating pair. The former, stronger, channel is associated with covalent-bonding interactions and the latter is associated with charge-carrier delocalization. A notion of resonance via dual, distinct channels was introduced by Painelli and coworkers in the context of a empirical four-state model of polymethines.15-16 In their model, as in our results, the channels were distinguished by virtual states with either hole or electron-pair density on the bridge. Painelli’s group’s analysis suggested that the two channels give rise to distinct nonlinear properties of the dye in question (excited-state absorbance vs. two-photon absorbance).15-16 My results in Figure 3 show the gap between the ionic, asymmetric CASVB states (|022> and |220>) and other CASVB states is maintained over the parameter range in Figure 3. It is larger than the relevant couplings in Figure 4, so a perturbation-theoretic reduction of the complete sixstate CASVB model to an effective four-dimensional model may be a sane approach to the development of analytical models. Conclusion I have made some observations about the origin of color in cyanine dyes. Such dyes are traditionally described using a resonance of asymmetric structures such as in Scheme 1. The strong coupling between resonating structures arises via superexchange through structures with symmetric charge distribution. I have reported a series of CASVB Hamiltonians for a set of bridge-substituted methines using a new weighting strategy that allows resolution of high-energy diabatic states. I have shown that the solutions to the minimal quantum model that could describe the resonance in Scheme 1 is analogous for all the dyes in Scheme 1 and has the same structure as “resonance” (or essential-state15-16) models. This provides new theoretical


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justification for such models. I have shown that there is a simple and chemically transparent approximation to the CASVB Hamiltonians for the dyes in Scheme 1 where the coupling is linearly parametrized by the Brown-Okamoto σp+ parameter of the bridge substituent and the coupling is constant. The model predicts a lower bound of σp+ < –0.7 below which a two-state resonance model of the ground and first excited state will not apply. My results have clarified the origin of the large effective coupling in two-state resonance models. I have shown that the coupling arises from superexchange via two chemically distinct channels associated with either covalent bonding or charge-carrier delocalization. The covalent bonding channel is significantly the stronger of the two. The results will be of general interest to research in cyanine electronic structure and opto-electronic properties. ASSOCIATED CONTENT Details of quantum chemistry calculations, including molecular geometires (Å), SA-CASSCF energies, natural orbital graphics and associated occupation numbers. AUTHOR INFORMATION Corresponding Author *Email: [email protected] Notes Any additional relevant notes should be placed here. The authors declare no competing financial interests. ACKNOWLEDGMENT


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This work was supported by Australian Research Council (ARC) Discovery project grants DP110101580 and DP160102425. Computations were carried out at the National Computational Infrastructure facility at ANU using time allocated to Merit Allocation Scheme grants m03 and n62. Referencess 1.

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amyloid fibril detection. Biochemistry 2011, 50, 3451-3461. 4.

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Essential-state model for polymethine dyes: Symmetry breaking and optical spectra. J. Phys. Chem. Lett. 2010, 1, 1800-1804. 16. Sissa, C.; Jahani, P. M.; Soos, Z. G.; Painelli, A., Essential state model for two-photon absorption spectra of polymethine dyes. ChemPhysChem 2012, 13, 2795-2800. 17. Hu, H.; Przhonska, O. V.; Terenziani, F.; Painelli, A.; Fishman, D.; Ensley, T. R.; Reichert, M.; Webster, S.; Bricks, J. L.; Kachkovski, A. D., et al., Two-photon absorption spectra of a near-infrared 2-azaazulene polymethine dye: solvation and ground-state symmetry breaking. Phys. Chem. Chem. Phys. 2013, 15, 7666-7678. 18. Olsen, S.; McKenzie, R. H., A three-state effective Hamiltonian for symmetric cationic diarylmethanes. J. Chem. Phys. 2012, 136, 234313. 19. Dewar, M. J. S., 478. Colour and constitution. Part I. Basic dyes. J. Chem. Soc. 1950, 1950, 2329-2334. 20. Knott, E., The colour of organic compounds. Part I. A general colour rule. J. Chem. Soc. 1951, 1024-1028. 21. Förster, T., Discussion on molecular spectra. Rev. Mod. Phys. 1963, 35, 572-575. 22. Anslyn, E. V.; Doherty, D. A., Substituent Effects. In Modern Physical Organic Chemistry, University Science Books: 2006; pp 441-472. 23. Brown, H.; Okamoto, Y., Substituent constants for aromatic substitution. J. Am. Chem. Soc. 1957, 79, 1913-1917.


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TOC GRAPHIC


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Scheme 1. Dyes studied in this paper. 102x64mm (300 x 300 DPI)


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Figure 1. Solution-state absorbance (colors as shown) and SA-CASSCF S0-S1 adiabatic excitation energies (black dots) plotted against the Brown-Okamoto substituent parameter σp+. SA-CASSCF excitation energies were uniformly shifted down by 1.46 eV to pin the excitation energy of Michler’s hydrol blue to its experimental absorbance maximum in dichloromethane. Solution-state data in the indicated solvents were taken from the literature. Experimental data was from references 34-41. The experimental excitation energies are tabulated with references in the supplement. Separate linear regressions of the experimental data (solid line) and the SA-CASSCF data (dotted line) are shown. 139x132mm (300 x 300 DPI)



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Figure 2. Illustration of CASVB model. (Top) Boys-localized active space orbitals from a SA6-CAS(4,3) calculation with canonical-ensemble weighting equilibrated at a “temperature” of 2.5 eV. Calculations at 2.0 eV and 3.0 eV gave visually identical orbitals. (Bottom) CASVB states built over the orbitals. Explicit corresponding structures are shown for the case of Michler’s hydrol blue (R=H). 85x181mm (300 x 300 DPI)


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Figure 3. This figure displays CASVB diabatic state energies (relative to average) for dyes in Scheme 1, plotted against the Brown-Okamoto substituent parameter σp+. CASVB states are indicated using graphical labels as in Figure 2. Linear regression fits are shown with RMSD. For dyes with asymmetric substituents (R=OH, CH¬3), the fit was taken against the average energy of corresponding pairs of CASVB states. Error bars show the range of state energies spanned as the “temperature” of the CASVB ensemble is scanned in the range 2.0-3.0 eV. 119x125mm (300 x 300 DPI)



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Figure 4. This figure shows the distribution of CASVB Hamiltonian coupling matrix elements. Couplings were calculated using a SA-CAS(4,3) ensemble equilibrated at a “temperature” of 2.5 eV. The couplings fall into three distinct groups. Strong (>0.5 eV) couplings are color-coded and labelled using graphical labels in Figure 2. The group of largest-magnitude (~–2.5 eV) couplings is associated with covalent chemical bonding interactions; these link covalent states with their bond-polarized ionic counterparts. The second-largestmagnitude group (~–1.5 eV) is associated with charge delocalization between the rings and the bridge. All other couplings are less than 0.5eV in magnitude, and are not color-coded or individually labeled. 136x124mm (300 x 300 DPI)


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Scheme 2. Ground state of auramine-O. 46x27mm (300 x 300 DPI)


Color in Bridge-Substituted Cyanines - ACS Publications - American

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