^{1,a)}and Ross H. McKenzie

^{1}

### Abstract

We propose a minimal model Hamiltonian for the electronic structure of a monomethine dye, in order to describe the photoisomerization of such dyes. The model describes interactions between three diabatic electronic states, each of which can be associated with a valence bond structure. Monomethine dyes are characterized by a charge-transfer resonance; the indeterminacy of the single-double bonding structure dictated by the resonance is reflected in a duality of photoisomerization pathways corresponding to the different methine bonds. The possible multiplicity of decay channels complicates mechanistic models of the effect of the environment on fluorescent quantum yields, as well as coherent control strategies. We examine the extent and topology of intersection seams between the electronic states of the dye and how they relate to charge localization and selection between different decay pathways. We find that intersections between the and surfaces only occur for large twist angles. In contrast, intersections can occur near the Franck–Condon region. When the molecule has left-right symmetry, all intersections are associated with con- or disrotations and never with single bond twists. For asymmetric molecules (i.e., where the bridge couples more strongly to one end) the and surfaces bias torsion about different bonds. Charge localization and torsion pathway biasing are correlated. We relate our observations with several recent experimental and theoretical results, which have been obtained for dyes with similar structure.

This work was partially supported with funds from the Australian Research Council Discovery Project No. DP0877875. We thank Anthony Jacko and Michael Smith for readings of the manuscript. We additionally thank Anthony Jacko for bringing to our attention an error in the eigenvalues formulas. We acknowledge helpful discussions with Steven Boxer, Paul Brumer, Irene Burghardt, Dan Cox, Philippe Hiberty, Noel Hush, Todd Martínez, Stephen Meech, Ben Powell, Jeff Reimers, Fritz Schaefer, Shason Shaik, and Tom Stace. Some of the graphics were generated with VMD.^{92}

I. INTRODUCTION

II. ELECTRONIC STRUCTURE OF MONOMETHINE DYES

III. MODEL HAMILTONIAN

IV. INTERPRETATION OF ANGLES AND PARAMETERS IN TERMS OF A MOLECULAR “TOY” MODEL

V. ANALYTIC FORMULAS FOR EIGENENERGIES AND EIGENVECTORS

VI. FULL AND PARTIAL LOCATION OF THE INTERSECTION SEAMS

VII. INTERSECTIONS CAN ONLY OCCUR AT TWISTED GEOMETRIES

VIII. WHEN THE COUPLING TO THE BRIDGE IS THE SAME FOR BOTH ENDS, ALL INTERSECTIONS LIE ALONG CON- OR DISROTATORY COORDINATES

IX. INTERSECTIONS OCCUR NEAR THE FRANCK-CONDON POINT WHEN THE COUPLING BETWEEN THE ENDS IS THE SAME AS THEIR COUPLING TO THE BRIDGE.

X. WHEN THE COUPLING TO THE BRIDGE IS DIFFERENT FOR DIFFERENT ENDS, THE SURFACE BIASES TWISTING ONE BOND MORE THAN THE OTHER

XI. INTERSECTIONS SEPARATE REGIONS OF DISTINCT CHARGE LOCALIZATION

XII. CHARGE LOCALIZATION AND BOND-SELECTIVE TWIST BIASING OCCUR TOGETHER

XIII. DISCUSSION

XIV. CONCLUSION

### Key Topics

- Chemical bonds
- 25.0
- Fluorescence
- 12.0
- Proteins
- 12.0
- Solvents
- 9.0
- Surface dynamics
- 7.0

## Figures

Examples of monomethine dyes. The molecules resonate between Lewis structures which invert bond alternation and redistribute the formal charge. Examples include, from top to bottom, the symmetric monomethine cyanine dye 1122C, the symmetric monomethine cyanine dye NK88, the asymmetric monomethine cyanine dye thiazole orange, and the chromophore of the GFP, an asymmetric diarylmethine oxonol dye.

Examples of monomethine dyes. The molecules resonate between Lewis structures which invert bond alternation and redistribute the formal charge. Examples include, from top to bottom, the symmetric monomethine cyanine dye 1122C, the symmetric monomethine cyanine dye NK88, the asymmetric monomethine cyanine dye thiazole orange, and the chromophore of the GFP, an asymmetric diarylmethine oxonol dye.

The four isomers of the asymmetrical dye thiazole orange, which differ by isomerism of the bridge. Thiazole orange is the least symmetrical of the three example molecules in Fig. 1, and so all isomers are distinguishable. They are labeled according to usual organic chemistry nomenclature.

The four isomers of the asymmetrical dye thiazole orange, which differ by isomerism of the bridge. Thiazole orange is the least symmetrical of the three example molecules in Fig. 1, and so all isomers are distinguishable. They are labeled according to usual organic chemistry nomenclature.

We illustrate three conceivable situations, where the shape of potential energy surfaces, and their interactions, could influence photochemistry. On the left, two states are biased toward the same pathway and ensuing dynamics may converge to a common evolution. In the middle, the electronic states are close in energy at the FC region, and are not biased strongly, so there is ambiguity in the pathways and no clear connection between the dynamics and the initially excited state. On the right, the states are biased differently, so that the dynamics on the different states diverge. Other situations are also conceivable.

We illustrate three conceivable situations, where the shape of potential energy surfaces, and their interactions, could influence photochemistry. On the left, two states are biased toward the same pathway and ensuing dynamics may converge to a common evolution. In the middle, the electronic states are close in energy at the FC region, and are not biased strongly, so there is ambiguity in the pathways and no clear connection between the dynamics and the initially excited state. On the right, the states are biased differently, so that the dynamics on the different states diverge. Other situations are also conceivable.

Localized-orbital active space representations for monomethine dye systems. For every monomethine dye, there is a “methine adapted” three-orbital solution to the state-averaged complete active space self consistent field problem. Monomethine cyanine dyes (left) have a two-electron solution, and diarylmethine dyes (right) possess a four-electron solution. In either case, the many-electron state space is six-dimensional and has a natural valence-bond structure in the localized representation (bottom). The energetic ordering of the localized orbitals is inverted in the two dye classes.

Localized-orbital active space representations for monomethine dye systems. For every monomethine dye, there is a “methine adapted” three-orbital solution to the state-averaged complete active space self consistent field problem. Monomethine cyanine dyes (left) have a two-electron solution, and diarylmethine dyes (right) possess a four-electron solution. In either case, the many-electron state space is six-dimensional and has a natural valence-bond structure in the localized representation (bottom). The energetic ordering of the localized orbitals is inverted in the two dye classes.

A toy model using simple orbitals, where orbital overlaps have a similar functional dependence on and to the matrix elements in our model Hamiltonian, which are shown.

A toy model using simple orbitals, where orbital overlaps have a similar functional dependence on and to the matrix elements in our model Hamiltonian, which are shown.

Representation of relevant coordinates in terms of the geometrical model in Fig. 4 (top) and as vectors in the plane (bottom). Single bond twists change one angle while leaving the other constant. The conrotatory and disrotatory twist coordinates are antisymmetric and symmetric combinations of the single bond twists, respectively. The parity of the combination coordinates depends on the “handedness” convention used to define the torsion angles, which here uses left and right hand rules for left and right torsion angles. Changing the handedness of the definition for one of the bonds is equivalent to interchanging the parameters and . In the context of an untwisted molecular frame with symmetry, conrotatory twisting preserves symmetry and breaks symmetry, while the disrotatory twist breaks symmetry and preserves symmetry.

Representation of relevant coordinates in terms of the geometrical model in Fig. 4 (top) and as vectors in the plane (bottom). Single bond twists change one angle while leaving the other constant. The conrotatory and disrotatory twist coordinates are antisymmetric and symmetric combinations of the single bond twists, respectively. The parity of the combination coordinates depends on the “handedness” convention used to define the torsion angles, which here uses left and right hand rules for left and right torsion angles. Changing the handedness of the definition for one of the bonds is equivalent to interchanging the parameters and . In the context of an untwisted molecular frame with symmetry, conrotatory twisting preserves symmetry and breaks symmetry, while the disrotatory twist breaks symmetry and preserves symmetry.

The analytic eigenvalues of a symmetric matrix can be expressed by two parameters, and , which are polynomials of first and higher traces of the matrix. The space of all degeneracies between the eigenvalues can be expressed as a relationship between these two parameters. (Top) Eigenvalues plotted over a plane spanned by the parameters and . The conditions and (region shown at bottom) are sufficient to guarantee real eigenvalues of a matrix and are equivalent to symmetry and positive definiteness of the matrix. When the inequality is strong, three nondegenerate eigenvalues exist. On the boundary (highlighted in black) at least two of the eigenvalues are degenerate. degeneracies occur on the part of the boundary. degeneracies occur on the region of the boundary, and a three-state intersection occurs at .

The analytic eigenvalues of a symmetric matrix can be expressed by two parameters, and , which are polynomials of first and higher traces of the matrix. The space of all degeneracies between the eigenvalues can be expressed as a relationship between these two parameters. (Top) Eigenvalues plotted over a plane spanned by the parameters and . The conditions and (region shown at bottom) are sufficient to guarantee real eigenvalues of a matrix and are equivalent to symmetry and positive definiteness of the matrix. When the inequality is strong, three nondegenerate eigenvalues exist. On the boundary (highlighted in black) at least two of the eigenvalues are degenerate. degeneracies occur on the part of the boundary. degeneracies occur on the region of the boundary, and a three-state intersection occurs at .

Conical intersections in the space spanned by the angles , and the affine parameter [Eq. (36)]. (Top) (red) and (yellow) intersection seams at constant , 1.0 (center), and 1.25 (right). When , there are intersections at the corners of the unit cell for *all values of * . This is accompanied by a change in the curvature of the seam. (Bottom) The location of the seam at different values of . When , the and intersections lie along perpendicular lines in the plane. When , this behavior changes, and both and intersections lie along a single line. Black arrows indicate the direction of motion of the seams with increasing Intersections were visualized using Eq. (30), with different branches colored according to the sign of [Eqs. (32) and (33)].

Conical intersections in the space spanned by the angles , and the affine parameter [Eq. (36)]. (Top) (red) and (yellow) intersection seams at constant , 1.0 (center), and 1.25 (right). When , there are intersections at the corners of the unit cell for *all values of * . This is accompanied by a change in the curvature of the seam. (Bottom) The location of the seam at different values of . When , the and intersections lie along perpendicular lines in the plane. When , this behavior changes, and both and intersections lie along a single line. Black arrows indicate the direction of motion of the seams with increasing Intersections were visualized using Eq. (30), with different branches colored according to the sign of [Eqs. (32) and (33)].

Relationships between conical intersection seams (top) and charge-localization in the adiabatic states of the model. (Left) Conical intersection seams are shown for varying at constant , . Cross sections are shown at , and 1.0. (Right) Population (absolutely squared amplitude) of the fragment diabatic sites are plotted over the torsion plane. Diabatic state populations were used as a measure of the charge distribution, and were used as convex coordinates in a Red-Blue-Green (RGB) color map. Areas where the cross sections intersect the neighborhood of the seam [Eq. (30), ] are shown by filled red and yellow regions. Charge localization is generally twist dependent, and regions of different localizations on the state are separated by intersections impinging on that state.

Relationships between conical intersection seams (top) and charge-localization in the adiabatic states of the model. (Left) Conical intersection seams are shown for varying at constant , . Cross sections are shown at , and 1.0. (Right) Population (absolutely squared amplitude) of the fragment diabatic sites are plotted over the torsion plane. Diabatic state populations were used as a measure of the charge distribution, and were used as convex coordinates in a Red-Blue-Green (RGB) color map. Areas where the cross sections intersect the neighborhood of the seam [Eq. (30), ] are shown by filled red and yellow regions. Charge localization is generally twist dependent, and regions of different localizations on the state are separated by intersections impinging on that state.

Synergistic relationship between energetic biasing of different bond torsions (top) and twist dependence of the charge localization (bottom). (Top) Biasing in the potential energy surface is introduced by asymmetric bridge-end couplings. When the coupling of the bridge to both ends is symmetric (top center), the curvature at the FC point is equivocal with respect to both single-bond twists. Introducing asymmetric coupling alters the curvature to favor one bond over the other (top left and right). (Bottom) Introducing asymmetric bridge-end coupling also exerts changes on the charge distribution over the fragments and its dependence on the twist. When the coupling is symmetric , charge localization follows the twist distribution, for con- and disrotatory twists the charge is spread over the left and right fragments, but for single-bond twists, it localizes on one side or the other in response to small displacements (bottom center)*. *When the dye is not symmetric , the charge is localized at FC. The polarity at FC depends on the sign of . The polarity is insensitive to small displacements from FC—twisting one of the bonds maintains the polarity at FC throughout the available range; twisting about the other induces polarity reversal, but only at large angles. In the context of a solvent reaction field, this means that progress along one coordinate may require a dielectric reorganization event, while the other may not. Note that the domain of the coordinates is out of phase relative to Figs. 8 and 9, so that the Frank-Condon point is in the middle of the square.

Synergistic relationship between energetic biasing of different bond torsions (top) and twist dependence of the charge localization (bottom). (Top) Biasing in the potential energy surface is introduced by asymmetric bridge-end couplings. When the coupling of the bridge to both ends is symmetric (top center), the curvature at the FC point is equivocal with respect to both single-bond twists. Introducing asymmetric coupling alters the curvature to favor one bond over the other (top left and right). (Bottom) Introducing asymmetric bridge-end coupling also exerts changes on the charge distribution over the fragments and its dependence on the twist. When the coupling is symmetric , charge localization follows the twist distribution, for con- and disrotatory twists the charge is spread over the left and right fragments, but for single-bond twists, it localizes on one side or the other in response to small displacements (bottom center)*. *When the dye is not symmetric , the charge is localized at FC. The polarity at FC depends on the sign of . The polarity is insensitive to small displacements from FC—twisting one of the bonds maintains the polarity at FC throughout the available range; twisting about the other induces polarity reversal, but only at large angles. In the context of a solvent reaction field, this means that progress along one coordinate may require a dielectric reorganization event, while the other may not. Note that the domain of the coordinates is out of phase relative to Figs. 8 and 9, so that the Frank-Condon point is in the middle of the square.

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