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Valence-bond non-equilibrium solvation model for a twisting monomethine cyanineSean McConnell, Ross H. McKenzie, and Seth Olsen Citation: The Journal of Chemical Physics 142, 084502 (2015); doi: 10.1063/1.4907758 View online: http://dx.doi.org/10.1063/1.4907758 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/142/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Many-body effects on the structures and stability of Ba2+Xe n (n = 1–39, 54) clusters J. Chem. Phys. 141, 154308 (2014); 10.1063/1.4896607 Photoisomerization among ring-open merocyanines. II. A computational study J. Chem. Phys. 140, 224311 (2014); 10.1063/1.4881259 Impact of intramolecular twisting and exciton migration on emission efficiency of multifunctional fluorene-benzothiadiazole-carbazole compounds J. Chem. Phys. 134, 204508 (2011); 10.1063/1.3594047 Conical Intersections, charge localization, and photoisomerization pathway selection in a minimal model of adegenerate monomethine dye J. Chem. Phys. 131, 234306 (2009); 10.1063/1.3267862 Valence bonds in solid hydrogen under pressure J. Chem. Phys. 109, 10316 (1998); 10.1063/1.477727

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THE JOURNAL OF CHEMICAL PHYSICS 142, 084502 (2015)

Valence-bond non-equilibrium solvation model for a twistingmonomethine cyanine

Sean McConnell, Ross H. McKenzie, and Seth Olsena)School of Mathematics and Physics, The University of Queensland, Brisbane 4072, Australia

(Received 4 December 2014; accepted 28 January 2015; published online 24 February 2015)

We propose and analyze a two-state valence-bond model of non-equilibrium solvation effects onthe excited-state twisting reaction of monomethine cyanines. Suppression of this reaction is thoughtresponsible for environment-dependent fluorescence yield enhancement in these dyes. Fluorescenceis quenched because twisting is accompanied via the formation of dark twisted intramolecularcharge-transfer (TICT) states. For monomethine cyanines, where the ground state is a superpositionof structures with different bond and charge localizations, there are two possible twisting pathwayswith different charge localizations in the excited state. For parameters corresponding to symmetricmonomethines, the model predicts two low-energy twisting channels on the excited-state surface,which leads to a manifold of TICT states. For typical monomethines, twisting on the excitedstate surface will occur with a small barrier or no barrier. Changes in the solvation configurationcan differentially stabilize TICT states in channels corresponding to different bonds, and thatthe position of a conical intersection between adiabatic states moves in response to solvation tostabilize either one channel or the other. There is a conical intersection seam that grows along thebottom of the excited-state potential with increasing solvent polarity. For monomethine cyanineswith modest-sized terminal groups in moderately polar solution, the bottom of the excited-statepotential surface is completely spanned by a conical intersection seam. C 2015 AIP PublishingLLC. [http://dx.doi.org/10.1063/1.4907758]

I. INTRODUCTION

Monomethine cyanines have a distinguished role in thehistory of chemical science and industry.1,2 The description oftheir optical properties was an early target of quantum elec-tronic structure models.3,4 The optical properties of monome-thine dyes and push-pull polyenes can be described by two-diabatic-state models based on the concept of resonance be-tween configurations with opposing charge and bond locali-zation.3,6–11 The resonating diabatic states are shown for fourtypical symmetric monomethine dyes in Fig. 1.

A distinguishing feature of monomethine cyanines is theenvironmental sensitivity of their fluorescence emissionyield.13 The dyes are practically non-fluorescent in fluid solu-tions, but can become highly fluorescent when bound tobiomolecules or other constraining media.5,14–20 This under-lies the use of monomethine cyanines as fluorescent turn-onlabels in biological imaging. The fluorescence enhancementis interpreted as suppression of a bond-twisting process in theexcited state, which would otherwise lead to dark twisted in-tramolecular charge-transfer (TICT) states.21 The TICT statesin monomethine cyanines are dark, so population of thesestates is associated with ultrafast loss of transient fluorescence.The TICT are characterized by a small adiabatic gap, andnonadiabatic decay can occur by conical intersections thatoccur at twisted geometries.22

Enhancement of fluorescence due to suppression of ex-cited-state twisting motion is the defining characteristic of a

a)Electronic mail: [email protected]

“molecular rotor.”23,24 This behavior is not limited to monome-thine cyanines. An example of a non-methine molecular rotoris the amyloid fibril-sensing dye Thioflavin-T.23 This moleculeundergoes twisting in the excited state, allowing ultrafast in-ternal conversion via TICT states with small or vanishing elec-tronic gap.21 The TICT states in Thioflavin-T and Auramine-O are dark, so that the twisting reaction is accompanied by afast transient fluorescence decay and negligible steady-statequantum yield. The fluorescence is enhanced upon bindingto biomolecular environments or in viscous solution. Thenonradiative decay rates of Thioflavin-T and the monome-thine dye Auramine-O have been extensively characterized byHuppert and coworkers in a series of glass-forming alcohols,under conditions spanning 10 orders of magnitude in theviscosity.25,64 These studies supported earlier indications26

that the nonradiative decay time could be estimated from thedamped rotational Stokes drift of a spherical rotor down alinear potential through an angle of one radian. The reactioncoordinate is taken to be rotation of a double bond to form adark TICT state. A useful estimate of the rotational timescale(which describes the non-radiative decay timescale) was foundto be

τrot =2ηV

torque, (1)

where η is the medium viscosity, V is the effective volume ofthe rotor (20 Å3 was taken for Thioflavin-T or Auramine-O),and torque is the mean torque on the rotor over a rotation of oneradian. A key point was that the non-radiative decay timescalewas found to depend linearly on the medium viscosity. A linear

0021-9606/2015/142(8)/084502/14/$30.00 142, 084502-1 © 2015 AIP Publishing LLC


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084502-2 McConnell, McKenzie, and Olsen J. Chem. Phys. 142, 084502 (2015)

FIG. 1. Some symmetric monomethine cyanine dyes. The ground state ofthese cyanines is an even superposition of states |R⟩ and |L⟩ with differentcharge and bond localizations. In Michler’s hydrol blue (upper left), the phys-ical charge carrier is an electron, but a hole in other dyes shown, explainingthe different disposition of bond and formal charge.

viscosity dependence of the nonradiative decay rate was alsoobserved in experiments on the triphenylmethane dye crystalviolet, which decays via a similar mechanism.28–30

Different possible situations for a one-dimensional molec-ular rotor system with dark TICT states are illustrated sche-matically in Figure 2. For twisting that is both favorable in theexcited state and without barrier, one expects ultrafast (1 ps)decay of fluorescence as the TICT state is formed from thebright Franck-Condon (FC) state. Provided depletion of TICTexcited state is not rate-limiting, which our results support(see below), ground state recovery should track the fluores-cence decay. For twisting that is favorable but activated, oneexpects slower fluorescence decay. Again, if the depletion ofthe excited TICT states is not rate-limiting, the ground staterecovery should follow the fluorescence decay. When TICTstate formation is not favorable, we expect that they do notform on a relevant timescale and the molecule is fluorescent.If our assumption about fast depletion of the TICT excited-state was wrong, so that population accumulates on the excitedstate, then the same pattern would appear in the ultrafast fluo-rescence, but we would expect the ground state recovery tolengthen independently.

For molecular rotors with the dimensions characteristicof monomethine cyanines, the timescale for generation of acharge-transfer state by twisting and the timescale for solvationof the same charge-transfer state cannot be reliably separated.The nonradiative decay times for Thioflavin-T and Auramine-O in glass-forming solvents scale with the transverse Debyedielectric relaxation time,25 but are 1-2 orders of magnitudesmaller.25 This is consistent with nonradiative decay times thatare comparable to the Debye longitudinal relaxation time. Thelongitudinal relaxation time is the appropriate timescale todescribe dielectric relaxation following a sudden change inthe molecular charge density.31 For alcoholic solutions, the

FIG. 2. Three distinct possible outcomes for an excited-state twisting reac-tion leading to a dark TICT state. The reaction coordinate is taken to be a sin-gle bond-twisting angle θ, which connects the FC region on the excited-statepotential energy surface with the TICT state at θ = π2 . (left) If the reaction isdownhill and barrierless, fast non-radiative decay is expected, with fast (fs-ps)decay of fluorescence in solutions of low (1 cP) viscosity. (middle) If there isa barrier along the reaction coordinate, but the reaction is still favorable, thenslower non-radiative decay is expected, with the rate depending on the barrierheight. For barriers less than a few thermal energy units kBTr , only transientfluorescence is expected; for larger barriers, steady-state fluorescence maybe observed. (right) For an uphill reaction, the TICT states do not formspontaneously in significant amounts. In this case, the dye will decay byemission of fluorescence.

longitudinal relaxation time will be 10 times smaller thanthe transverse timescale,32 which is comparable to estimatedtwisting times. Although solvent-dependent effects can beobserved at sub-ps timescales for Auramine-O, these were as-signed to a distinct dynamical origin.27 Heisler and colleaguesshowed for Auramine-O in nanoconfined water droplets thatrestriction of solvation by the confinement slowed the fluores-cence upconversion decay.33 Results reported by Kondo andcoworkers for Auramine-O in bulk solutions indicated thatthe dominance of solvation dynamics vs reaction coordinatefriction in determining the fluorescence upconversion decaycould not be determined for several solvents except water,for which solvation dynamics was dominant.34 The N,N ′-dimethylaniline rotors in Auramine-O are identical to thoseof Michler’s hydrol blue in Figure 1, and so the timescales fordamped twisting are expected also to be the same.

If the timescales for solvation and for twisting are compa-rable, then a faithful description of the physics must treatthese processes at a comparable level of approximation. Asuccessful yet simple approach for the one-dimensional rotorN,N ′-dimethylbenzoaminonitrile (DMABN), which forms anemissive TICT state on a timescale much longer than com-mon for monomethine cyanines, has been to develop two-dimensional models coupled to a twisting coordinate and asingle solvation coordinate that gauges the polarization stateof the medium.57–60

The identification of monomethine cyanines as molec-ular rotors raises an interesting question, because the groundstates of these molecules are a superposition of structures withdifferent π-bond localizations (cf. Figure 1). In a moleculewhere the double bond can be in multiple places, how does onechoose the appropriate reaction coordinate to describe doublebond twisting in the excited state? Ab initio studies of theexcited state surface in monomethine cyanines indicate thatthere are not one but two low-energy twisting channels on theexcited-state potential energy surface.35

General models of TICT electronic structure suggest thatas the bond twisting proceeds, the lowest two adiabatic states


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should become polarized and form a TICT system.21,22,36,37

This picture is upheld by more detailed quantum chemicalmodels.35,38 However, twisting one or the other bond willdestabilize the diabatic states by different amounts—becausethe π-bonds are in different places. Accordingly, the polarityof the TICT adiabatic states will depend upon the bond thatis twisted.22,37 The dependence of the TICT polarity on theidentity of the twisted bond can also be explained usingsimple molecular orbital theories.36 The correlation betweenthe path taken and the TICT polarity suggests that a polarenvironment can distinguish the twisting pathways, and couldinfluence their relative yields. This has been suggested asthe mechanism that underlies reversible photochromism andfluorescence switching in fluorescent proteins.39 Fluctuationin the charge-coordination environment has been predicted toinfluence the bifurcation and availability of twisting pathwaysin photoactive yellow protein chromophores.40–42

Modern theoretical photochemistry suggests the location,topography, and energetics of conical intersection seams be-tween adiabatic electronic states are important for under-standing the mechanism and yield of photochemical reac-tions.43–48 Ab initio calculations and general models indi-cate that conical intersections can occur in monomethinedyes in configurations where one bond or more bonds istwisted, so that the π orbitals on the fragments across the bondoverlap weakly.22,35,37 The non-adiabatic coupling (to nucleardisplacement) diverges at conical intersections, so that internalconversion is expected to be dominated by configurations nearthese points. Unlike transition states that separate differentbasins on a given surface, conical intersections are multidi-mensional seams embedded in the configuration space of themolecule.45 Although conical intersections can sometimes bediscussed in terms of their minimum-energy configurations,there are cases where the relevant nonadiabatic dynamicsis thought to occur far removed from the minimum-energyregions of the intersection seam.49,50 Since the twisted conicalintersections in monomethine dyes arise from intersectionof diabatic states with different charge localization, changesin the solvent configuration could very plausibly affect thepositioning, topology, and/or energetics of the seam.

The effect of polar solvation on the structure of conicalintersection seams in a retinal protonated Schiff base model hasbeen studied in a series of papers by Hynes and coworkers.53–56

In these papers, it was shown that solvent effects determinedthe placement and energy of conical intersection seams, aswell as the coordinates that accessed them. A tendency forcharge-transfer conical intersections to be stabilized by sol-vents excited-state potential in solvated systems has beenobserved in excited-state dynamics simulations.51,52

In this paper, we seek to develop a better intuition forhow non-equilibrium solvation affects the excited-state twist-ing process in monomethine cyanines, with special attention tothe issue that there are two possible twisting pathways that canbe distinguished by the solvent.

One reason why it is important to consider simple modelsis that simple models make it easy to keep track of how theideas that go into the model relate to the predictions that comeout. That way, a challenge to the model can be framed as achallenge to the underlying ideas, rather than to some detail of

the model parameterization. In this paper, we focus on only afew ideas that are simple extensions of the old notion of cyanineelectronic structure as a resonance between structures withopposing charge- and bond-order localizations (cf. Figure 1).

Our model describes favorable excited-state twisting withno or a small barrier for typical monomethine cyanines insolution, leading to a manifold of TICT configurations withsmall or no electronic gap. The driving force for twisting inthe isolated dye is small for dyes with absorption in the red,and increases with increasing absorption energy for a given π-bond stabilization energy. The model predicts that solvationof products along one twisting pathway tends to destabilizethe alternate pathway, so that the model can describe solvent-driven selection of distinguishable TICT channels. The posi-tion and energy of a conical intersection in the TICT mani-fold are shown to change with the solvent configuration. Thefraction of TICT states that can be made to coincide withthe intersection seam grows with the solvent reorganizationenergy. For couplings corresponding to typical monomethinecyanines in polar solvents, there is no minimum or “basin” thatcan be associated with an adiabatically excited TICT system.Instead, the bottom of the excited state potential coincideswith a conical intersection for monomethine cyanines in evenmoderately polar solvents.

Here, the paper is organized as follows: in Sec. II, we intro-duce the model, describing its physical content andparameterization. Section III discusses the potential energysurfaces (PESs) given by the model. We show that the PESs ofthe model can be qualitatively categorized according to a phasediagram and describe relevant features of the phase diagram.Section IV discusses results in the context of experiments onthe nonradiative decay of monomethines. Section V concludes.

II. MODEL HAMILTONIAN

The model Hilbert space is two-dimensional and spannedby diabatic basis states distinguished by charge and bridge π-bond order as in Figure 1. In this basis, the Hamiltonian iswritten as (2),

Ĥ = ĤDye + ĤSol

= *,

Jsin2θL −β cos θL cos θR−β cos θL cos θR Jsin2 θR

+-

ĤDye

+ λ

*....,

(s + 1

2

)20

0(

s − 12

)2+////-

+

(1√

n− 1

)12r

ĤSol

. (2)

The Hamiltonian is comprised of two subcomponents, adye Hamiltonian ĤDye, which is a function of the bridge-bondtwist angles θL and θR and a solvent interaction HamiltonianĤSol, which is a function of a single solvent configurationcoordinate s, which interpolates between equilibria with |L⟩ (s= −1) and |R⟩ (s = 1). The parameters β, J, and λ are the elec-tronic coupling strength, the π-bond stabilization energy, and


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the diabatic solvent reorganization energy. The solvent inter-action term ĤSol includes a configuration-independent scalarsolvation energy that depends on the refractive index n ofthe medium and the Born cavity radius r characterizing therelevant terminal heterocycle. This term is the same for bothdiabatic states in a symmetric dye and does not vary withthe configuration of the dye or solvent; it can be removed byresetting the zero of energy.

The transformation between the diabatic and adiabaticbases can be parameterised as a rotation by an angle φ,

|0⟩|1⟩

=

cos φ sin φ− sin φ cos φ

|L⟩|R⟩

. (3)

After resetting the zero of energy to eliminate the scalar solva-tion energy, the eigenvalues of Hamiltonian (2) are

E0 = Ē −∆E2

, E1 = Ē +∆E2

,

where

Ē =12

(λ

2�s2 + 1

�+ J

�sin2 θL + sin2 θR

�),

∆E =(2β cos θL cos θR)2 + �J �sin2 θL − sin2 θR� + λs�2.

A. Dye Hamiltonian

The diabatic energies vary with the twisting of the bridgebonds through the angles θL and θR. The dependence reflectsdistinguishable π-bond localizations in the diabatic states inFigure 1. Each diabatic state has π-bonding character localizedon a distinct bridge bond. The π-bond stabilization energy J isthe energy required to twist a π -bond to θ = π2 , breaking thebond,

L ��ĤDye�� L

= Jsin2 θL, (4)

R ��ĤDye�� R

= Jsin2 θR. (5)

We are concerned with the interaction of twisting andsolvation displacements, because the corresponding timescalesin monomethine cyanines appear to be non-separable. A literalinterpretation of the valence-bond structures in Figure 1 im-plies also strong coupling to bond-stretching displacements.Such displacements are important for understanding linear andnonlinear spectra of these and similar systems.6,7,12,61–63

The off–diagonal coupling has one term representingcharge transfer between rings via sequential transfers acrossboth bridge bonds. The direct transfer element between therings is taken to be negligible, so that the lowest order cou-plings between the (many-body) diabatic states go as theproduct of transfer terms for each bond, and the modulationof transfer across each bond goes as the overlap between π-orbitals on the bonding fragments. Similar strategies have beenapplied to the nonlinear solvation of systems with activatedTICT formation along a single twisting coordinate coupled toa solvation mode,

L ��ĤDye�� R

= −β cos θL cos θR. (6)

The parameter β determines the magnitude of the elec-tronic coupling between diabatic states. It is equal to the adia-

batic gap at the (planar) ground state geometry. The form ofEqs. (4) and (6) can be justified using valence bond theory,66–69

in so far as the single-particle transfer element goes as the over-lap between atomic p orbitals. The parameterization assumesa well-defined nodal plane associated with an sp2-hybridizedmonomethine bridge, so that the twist angles (θl, θr) can beunambiguously defined.

B. Parameterisation of the dye Hamiltonian

The electronic coupling parameter β is equal to the elec-tronic gap when the dye is planar and the solvent is unpo-larized, which is true at the ground state minimum. Sincemonomethine dyes have strong, narrow absorbance lines, thisparameter can be estimated from absorbance spectroscopy asthe energy corresponding to the band maximum. Alternatively,it can be estimated using quantum chemistry calculations,

∆E(0,0) = 2β. (7)The absorbance of monomethine cyanines is usually sharp andsolvatochromism is negligible.71 For dyes such as in Figure 1,with absorbance in the visible, an appropriate value for βwould be in the range 0.8-1.7 eV.

The π-bond energy J is equal to the adiabatic gap if onebond is twisted and the other is planar (in an unpolarizedsolvent),

∆E (±π/2,0) = ∆E (0,±π/2) = J. (8)This suggests the parameter J is in-principle estimable fromexperiments, if transient fluorescence from the excited TICTstate could be collected in sufficiently non-polar solvents. Un-fortunately, the transition dipole strength also vanishes at TICTstates, making spectroscopic estimation difficult or unfeasiblein most cases. In such cases, it may be easiest to resort tocalculation of the gap using quantum chemistry computationsat a suitable model of the twisted state geometry.

General considerations suggest that the parameter Jshould be in the range 1.0-2.0 eV. This estimate comes from theidentification of J with the energy associated with twisting aπ-bond in a diabatic state with clearly defined bond alternation.To the extent that the energy of a localized double bond istransferable, we expect that J should be at most comparable tothe ground-state twisting energy for a molecule with definitesingle-double bond alternation in the adiabatic ground state.The reason for the “at most” in the previous sentence is thatthe initial state for bond-twisting in a molecule with definiteground state bond alternation is near the minimum for therelevant diabatic state, this will not be the case in a symmetricmonomethine where the ground state geometry is ambiguous.A reasonable reference system of similar size to the mole-cules in Figure 1 might be an asymmetrically substitutedstilbene. The singlet ground state barriers for stilbene and 4-styrylpyridinium have been reported as 1.78 eV and 1.69 eV,respectively.72 Another route to a similar conclusion would beto note that the π-bond stabilization energy in Hückel theoryis just the resonance integral β, which is usually found to bein the range 1-2 eV.73 In any case, we are led to the conclusionthat an appropriate value for the barrier height will be 1 eV.


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For some systems, calculations of the diabatic barrierheight are available, as calculations of the adiabatic gap at asuitable model TICT geometry. For near-resonant models ofthe green fluorescent protein (GFP) chromophore, which arenot strictly symmetric but electronically almost so, this yieldsa barrier height of 0.9-1.0 eV.22 For the case of Michler’s hydrolblue, we have an unpublished value of the twisted gap calcu-lated using multi-state and multi-reference perturbation theorythat is 1.0 eV. For a slightly truncated model of the monome-thine cyanine NK88 (cf. Figure 1), Santoro and coworkers havecalculated twisted gaps in the range 1.3-1.5 eV using differentelectronic structure approaches.35 These values for the diabaticbarrier height J are consistent with the arguments above.

C. Dye–solvent interaction

The interaction with the solvation field is described usinga model of charge transfer between conducting spheres ina linear dielectric, with cavity radii and intercavity distancechosen to reflect the geometry of a typical monomethine withmodest-sized rings. This model is well-established in the elect-ron transfer literature.75 It has also been successfully appliedto model the anomalous solvatochromism of charged push-pull polyenes in solution.74 The diabatic solvent reorganizationenergy λ is calculated as the reorganization energy of chargetransfer between the spheres. Charged push-pull polyenes arechemically similar to monomethine dyes (cf. Figure 1) and are,essentially, strongly asymmetric polymethine cyanines.

The configuration of the solvent is parameterised by asingle solvent coordinate s. Those configurations which are inequilibrium with the diabatic states |L⟩ and |R⟩ (cf. Figure 1),we denote by s = 1 and s = −1, respectively. The free energiesof the diabatic states are shown in Figure 3.

The solvation interaction ĤS is diagonal in the diabaticbasis. The relevant matrix elements are

L ��ĤS�� L

= G0 + λ

(s + 1

2

)2, (9)

R ��ĤS

�� R= G0 + λ

(s − 1

2

)2, (10)

where λ is the dielectric reorganization energy and G is theBorn solvation energy of a charged conducting sphere of radiusr embedded in a linear dielectric (in CGS units)

G0 =(

1ϵ− 1

)e2

2r. (11)

For a symmetric monomethine, the cavity radii will be the sameand the solvation energy terms contribute only to the trace ofthe Hamiltonian. They can be removed by resetting the zero ofenergy to G0.

The physical interpretation of the solvent reorganisationenergy λ is the energy released by the non–adiabatic relaxationof the system to the minimum energy configuration of |R⟩ ats = 1 following excitation from |L⟩ at s = −1, the expressionfor this is74,75

λ ≡ λ(r,R) =(

1ϵ∞− 1ϵ

)e2

(1r− 1

R

), (12)

FIG. 3. A schematic diagram of the solvation interaction in our model.Shown are the (quadratic) potential energy surfaces of the diabatic andadiabatic states along a single solvation coordinate s, which gauges thepolarization of the solvent. The solvent is in equilibrium with diabatic state|R⟩ (green line) at s = 1 and in equilibrium with state |L⟩ (red line) ats =−1. The diabatic gap is modulated by the torsion angles of the dye,θL and θR (cf. Figure 6) through the torsion coupling constant J , whichis the energy of twisting the π−bond to the bridge. The splitting of theadiabatic ground (blue) and excited (orange) states has contributions from thediabatic gap and the diabatic coupling matrix element. The diabatic couplingis modulated by the torsion angles through the electronic coupling constantβ, and is equal to half of the electronic excitation energy at the ground state(planar) minimum. At the solvent configuration where the diabatic statescross, the adiabatic states are split by an amount equal to twice the diabaticcoupling. The diabatic solvent reorganization energy λ is the energy requiredto displace one diabatic state from its own equilibrium solvent configurationto the equilibrium configuration of the other. It is the same for both diabaticstates.

where r is the cavity radius, R is the distance between cavitycenters, and ϵ and ϵ∞ are the static and high-frequency dielec-tric constants of the medium. We have assumed the cavity radiiare the same and that the transferred charge is equivalent to thefundamental charge e.

For symmetric monomethine dyes such as shown inFigure 1, we expect R ' 2r and cavity radii r ∼ 2 − 4 Å.Table I shows the reorganisation energies at these values ofr and R. Values of r → 2 Å are representative of smallerendgroups, such as in dyes used as models of barrierlessreaction kinetics.76,77

Solvent reorganisation energies for a conducting-spheremodel of monomethines with reasonably small rings (likeFigure 1), as outlined in Table I, are in excess of 1 eV, close to orlarger than typical values of J, according to Sec. II B. Huppertand coworkers have used an approximate volume of 20 Å3

to calculate the nonradiative decay rate of auramine-O andThioflavin-T, corresponding to a sphere of radius 2 Å. Thishappens to be just under half of the experimentally measuredtransition dipole moment for Michler’s hydrol blue.78 Theagreement in estimates of the rotor size is satisfying, since oneis estimated from a hydrodynamic model of the non-radiativedecay time,79 while the other is obtained from an absorbancespectroscopy experiment.78

We will now summarize the ideas that have gone intomodel Hamiltonian equation (2).

• The model is built on a Hilbert space spanned by struc-tures with distinct charge and π-bond localizations.


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TABLE I. Reorganisation energies (in eV) representative of monomethine cyanines in some typical solvents. Thevalues of the cavity size and endgroup separation (r,R) are shown in brackets. The ranges of r and R are chosento span realistic ranges for monomethine cyanines with endgroups containing 1-2 rings with small substituents.

Compound ϵ ϵ∞ (4,8) (4,9) (3,6) (3,7) (2,4) (2,5)Water 78.4 1.9 1.0 1.1 1.3 1.5 2.0 2.4Acetonitrile 36.6 1.8 1.0 1.1 1.3 1.5 1.9 2.3Methanol 33.0 1.8 1.0 1.1 1.3 1.5 1.9 2.3Dichloromethane 8.9 2.0 0.7 0.8 0.9 1.1 1.4 1.7Chlorobenzene 5.7 2.3 0.5 0.5 0.6 0.7 0.9 1.1

This notion of the low-energy electronic structure ofcyanines has a history in chemical theory that goes backto the early part of the last century.2,71

• We have introduced the idea that π-bonds have barriersto rotation that are not present in single (σ) bonds.This concept is not new, and is one of the most recog-nized basic facts of structural organic chemistry. Thefunctional form of the barrier to rotation that we haveused can be derived from valence-bond theoretical argu-ments.69 The π-bond stabilization energy can beestimated to be in the range 1-2 eV based on textbookarguments.73

• The inter-ring coupling is a product of transfers ontoand off of the bridge. This reflects the idea that the direct(i.e., single-particle) inter-ring charge transfer couplingis negligible, which goes back as far as Pauling.80 Sincethe direct coupling vanishes, the coupling goes as aproduct of transfers across the bridge bonds in lowestorder. The dependence on the variables θL and θR acrosseach bond goes as the overlap of local π fragment or-bitals across the respective bond. A lucid early descrip-tion of the many-particle origin of the large (β 1 − 2 eV)coupling in cyanines was given by Moffitt.81 Essentialaspects of this analysis have recently been confirmedagain using a quasi-diabatic analysis of the electronicstructure of series of monomethine cyanines.22

• We have introduced the notion that the solvent hasdifferent equilibrium positions for the different diabaticstates. The solvation interaction is described using amodel of charge transfer between conducting spheresin a linear dielectric, with sphere radius and inter-sphere distances characteristic of the monomethinecyanine. This model, which is quite well-establishedin the chemical physics literature related to electron-transfer reactions,75 has also been used in a descriptionof anomalous solvatochromism in charged push-pullpolyenes (asymmetric polymethine cyanines).65 Wehave neglected the contribution of solvation effects tothe coupling element, as has also been done in otherstudies of nonequilibrium solvation during photoiso-merization reactions.56

III. RESULTS

A. General remarks and classification of PES

The PESs given by the model can be qualitatively classi-fied according to the “phase diagram” in Figure 4. The figure

classifies potential energy surfaces according to three yes/noquestions. These are

• Will TICT states form spontaneously after electronicexcitation?

• Is there a barrier to twisting on the excited state surface?• Do gapped TICT states exist? A TICT state is gapped

if it cannot be brought into coincidence with a conicalintersection seam for any configuration of the solvent.

For parameters corresponding to typical monomethine cy-anines in solvents of weak to moderate polarity, our modeldescribes favorable (spontaneous) formation of TICT states onthe excited-state PES. Twisting should be barrierless for dyeswith electronic excitation energy above the far-red, but a barriercan exist for monomethines with weak electronic coupling orwith weak π-bonding interactions. Numerical estimates thatwe have performed for limiting cases suggest the barrier, if itexists, is probably small (a few kBT , thermal energy at roomtemperature).

Our model can describe the differential stabilization oftwisting channels by the solvation field in a monomethine and

FIG. 4. This “phase diagram” summarizes the qualitative features of PESsprovided by the model in distinct regimes of the parameters J, λ, β. Theparameter regimes characteristics of typical monomethine cyanines are insidethe dotted ellipse. PES given by the model is classified according to the dis-position with respect to three lines drawn. Above the blue line, spontaneousformation of dark TICT states is expected. Below the line, the dye will decayby fluorescence. To the left of the green line, the (planar) FC configurationis stable (left) on the exited-state PES. If TICT states are still favorable, thenthere is a barrier to reach them. To the right of the line, the planar (FC) stateis unstable on the excited state surface, and twisting occurs without barrier.Below the red line, there are TICT states that are adiabatically gapped for allsolvent configurations. Above the red line, the TICT states are ungapped, inthe sense that the CI can be made to coincide with any TICT geometry forsome configuration of the solvent.


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can also describe movement of the conical intersection betweenthe twisting channels in response to changes in solvent configu-ration. This is an important aspect of the model, which is worthhighlighting now. This behavior is summarized in Figure 5.This shows that our model, despite its simplicity, can describethe environmental selection of different twisting channels by anenvironmental field, such as has drawn interest in the literatureon fluorescent proteins39 and photoactive yellow proteins.41

Our model suggests that, for typical monomethine cya-nines with modest-sized endgroups (a single substitutedheterocycle) in typical solvents (for which we expect J / λ),there is no minimum on the excited state potential energysurface. The bottom of the excited state potential correspondsto a conical intersection seam. This seam grows along the“valley floor” of the excited state PES. For monomethine

cyanines with modest-sized rings (containing one substitutedring, r ∼ 2 Å), there are no gapped TICT states in solventsof even moderate polarity. For monomethine cyanines in weaksolvents and/or with large rings, our model predicts that gappedTICT states may exist. If diffusion within the channels isslow enough, then these gapped TICT states may transientlyaccumulate excited-state population.

B. When are TICT states favorable? (Blue linein Fig. 4)

The twisted internal charge-transfer configurations/statesin our model occur where the coupling matrix element vanishes−2β cos θL cos θR = 0. This occurs on the lines θL = π2 andθR =

π2 , limiting focus to the range indicated in Figure 6. We

FIG. 5. Changes in the potential surfaces of the dye in response to changes in the solvent configuration. For an unpolarized solvent, such as at the ground stateminimum, the excited state PES is ambiguous with respect to the L and R TICT channels (cf. 6), and the CI between adiabatic states is positioned ambiguously,where both bonds are equally twisted. As the solvent polarizes in one or the other direction, one channel is stabilized over another and the CI moves into thestabilized channel. This corresponds to positioning the CI at dye configurations where one bond is twisted (θ = π2 ) and the other is less twisted (θ <

π2 ). For

moderate polarization of the solvent, the CI is located in one of the twisting channels. At a critical polarization s =± Jλ , the CI reaches the “end” of the TICTchannel, where one bond is twisted and the other planar. For polarizations beyond this, the adiabatic gap reappears and the excited state is destabilized. Thesolvent coordinate is bound in the interval [−1,1], so that not all regimes are accessible if λ < J . The parameters used here were chosen for Michler’s hydrolblue (β = 1.0 eV, J = 1.0 eV, r = 2.3 Å,R = 4.6 Å) in acetonitrile (λ = 1.6 eV).


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FIG. 6. A map of the “asymmetric unit” in dye configuration space, withlabelling of landmarks referred to in the text. The space, spanned by thetwisting angles shown at top, is periodic and infinite. In this paper, we focussolely on the patch shown, which is spanned by intervals of [0, π2 ] in both thetwisting angles θL and θR. The corners of the patch are landmarks frequentlymentioned by acronym in the text: the (FC) configuration, the left-twistedconfiguration (LT), the right-twisted configuration (RT), and a “hula-twist”(HT) configuration70 where both bonds are twisted. Initial photoexcitedpopulation would be prepared at FC because the transition is bright thereand it coincides with the ground state minimum. The boundaries of the patchopposing FC and connecting the twisted configurations are where the TICTconfigurations are found. The TICT configurations are divided into the “L”TICT channel, consisting of configurations where θL = π2 and the “R” TICTchannel (θR = π2 ). Conical intersections between the adiabatic states in ourmodel only occur in the TICT channels. Our model neither does distinguishconcerted (even) and “hula” (odd) combinations of the torsions nor does itdistinguish between different cis/trans isomers of a given dye.

will divide the TICT states by dividing them into two “chan-nels”: the R channel (θR = π2 ) and the L channel (θL =

π2 ). The

points corresponding to these channels are also indicated inFigure 6.

TICT states will only form spontaneously on a rele-vant timescale if they are energetically favorable. For a sin-gle coordinate, this corresponds to either of the two left panelsof Figure 2. The excited-state energy (relative to FC) ∆E1(θL, θR, s) = E1(θL, θR, s) − E1(0,0,0) is defined in the R andL channels as

∆RE1 (θL, s) = J2�sin2 θL + 1

�+ λ

( s2

)2+

12|λs − Jcos2 θL | − β, (13)

∆LE1 (θR, s) = J2�sin2 θR + 1

�+ λ

( s2

)2+

12|λs + Jcos2 θR| − β. (14)

The absolute value function introduces a derivative discon-tinuity where its argument is zero. In this case, the argumentis the adiabatic gap and the derivative discontinuity gives theposition of a conical intersection within the TICT channels.

The excited-state energy in the L channel can be writtenanalytically on either side of the conical intersection as

∆LE1 (θR, s) = J + λ(( s

2

)2+

s2

)− β,

0 ≤ Jcos2 θR + λs,(15)

∆LE1 (θR, s) = Jsin2 θR + λ(( s

2

)2− s

2

)− β,

Jcos2 θR + λs ≤ 0,(16)

and in the R channel as

∆RE1 (θL, s) = Jsin2 θL + λ(( s

2

)2+

s2

)− β,

0 ≤ λs − Jcos2θR,(17)

∆RE1 (θL, s) = J + λ(( s

2

)2− s

2

)− β,

−Jcos2 θL + λs ≤ 0.(18)

The situation is described visually in Figure 7, whichshows the energies in the R channel for some solvent config-uration that puts the conical intersection (CI) in that channel.Since the coupling vanishes in the TICT channels, the diabaticcrossing point is also a conical intersection between adiabaticstates. We can see immediately that diabatic state |R⟩ favorsunwinding the angle θL, while |L⟩ is invariant to θL. The coupl-ing vanishes in the channel, so the lowest adiabatic excited stateenergy in the channel can never be lower than the adiabaticexcited state energy at the geometry RT, corresponding toθL = 0.

It follows that the lowest excited-state energy (relativeto FC) in the R channel equals the excited-state energy at

FIG. 7. Movement of the conical intersection in the TICT channels can beunderstood by examining the diabatic states within a single channel (here, theR channel, for which θR = π2 , for a solvent configuration favoring |R⟩). Thediabatic coupling vanishes everywhere in the figure, so the adiabatic statesare given by taking the lowest and highest values at every point, and thecrossing point is a conical intersection between adiabatic states. The potentialfor diabatic state |R⟩ in the channel is flat, because the π-bond in this stateis already twisted. The potential for |L⟩ rises as θL→ π2 . For an unpolarizedsolvent (such as at the ground state minimum), the CI is located at HT, aconfiguration where both bonds are twisted. As the solvent moves to stabilize|L⟩, the excited state energy at LT decreases, and the CI moves from HTtowards RT (to the left of the figure). When the solvent stabilization λs = J ,the CI has reached RT. For larger polarizations of the solvent, the π-bondstabilization energy can no longer compete with the energy of the solventinteraction. Then, the adiabatic gap reopens and the CI disappears.


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RT. Analogous reasoning says that the lowest energy in theL channel is the energy at LT. The energies at these points arewritten on either side of the conical intersection as

∆LTE1 (s) = J + λ(( s

2

)2+

s2

)− β, 0 ≤ λs − J, (19)

∆LTE1 (s) = λ(( s

2

)2− s

2

)− β, λs − J ≤ 0, (20)

while the excited-state energy at RT, which similarly boundsthe energy in the R channel from below, we have

∆RTE1 (s) = λ(( s

2

)2+

s2

)− β, 0 ≤ λs − J, (21)

∆RTE1 (s) = J + λ(( s

2

)2− s

2

)− β, λs − J ≤ 0. (22)

We can see that the lowest energy at RT is achieved forλ < J at s = 1, whereas for J ≥ λ, it is achieved at s = J

λ,

which is at the conical intersection.It follows that the lowest energy that is achievable in the

TICT channels is the energy at RT (or LT) at the solventconfiguration which brings the conical intersection to thatpoint. This energy is given (relative to FC) by

∆Emin1 = J − β −λ

4. (23)

This equation gives the lowest (most favorable) excited-state energy that can be obtained for any TICT state in ourmodel (relative to FC). If it is negative, then there exist TICTstates that are energetically accessible from FC. If it is positive,then the TICT states are not accessible from FC. In the formercase, the TICT states are expected to form spontaneously in theexcited state on a timescale that will depend on the interveningpotential (cf. Figure 2, difference between two left-hand panelsand Sec. III C). In the latter case, the molecule will remain onthe excited state in a region of strong radiative coupling andwould be expected to decay by fluorescence. If the energy atRT/3 vanishes, then there is no driving force for twisting. Inthe case of a flat, barrierless surface, Eq. (1) is still useful ifthe average torque over a radian is taken as thermal energy atroom temperature, kBT . In that case, for a rotor with r ∼ 2 Åin a medium with viscosity 1 cP, the resulting timescale is stillin the picosecond range. Non-radiative decay should still winover fluorescence, even if the potential is completely flat (fortypical solutions at room temperature).

C. Is there a barrier? (Green line in Fig. 4)

The excited-state twisting process in monomethine cya-nines has been invoked as an example of a barrierless viscosity-controlled process.82 Quantum chemistry calculations onmonomethine cyanines and similar systems usually indicate nobarrier or small barrier.35,83,84 The accepted limits of quantumchemistry estimates of excited-state energy differences are ofthe order of a few kBTr (even where the Born-Oppenheimerapproximation is appropriate), so predictions of a barriersmaller than this are not conclusive.85

Here, we examine what can be expected, with respect tothe presence or absence of a barrier to twisting, on the basis ofthe few enumerable assumptions included in our model.

The question of whether or not there is a barrier to twistingboils down to the sign of the eigenvalues of the second-derivative matrix (Hessian) with respect to the twistingdisplacements at FC. The solvent coordinate is always boundin the adiabatic excited-state of our model (as can be verifiedby consideration of Figure 3 and remembering that λ ≥ 0). TheHessian is diagonal at FC, and the force constants associatedwith the two twisting displacements are equal by symme-try. The excited-state force constants with respect to thesedisplacements have the particularly simple form

∂2E1(0,0,0)∂θ2R

=∂2E1(0,0,0)

∂θ2L= J − β. (24)

Negativity of the twisting force constants at FC impliesaccessibility of TICT states, as can be seen by comparingEqs. (24) and (23). If J − β ≤ 0, then the model predicts bar-rierless descent to the TICT manifold. This would correspondto the case of the far left panel in Figure 2. If 0 < J − β ≤ λ4 ,then FC is bound but the TICT states are still favourable, andtwisting proceeds with a barrier (middle panel of Figure 2). Ifthe TICT states are inaccessible ( λ4 < J − β), then the mole-cule will decay by fluorescence (right panel, Figure 2).

Since the adiabatic transition is dipole allowed at FC,but forbidden in the TICT channels, the transient fluorescencedecay measures exit from the region of FC. We expect thepresence of the barrier to lengthen the fluorescence decay.However, the time to internal conversion on the far side ofthe barrier should be independent of the barrier height, sothat in the activated regime, we expect other contributions tothe ground-state recovery time to be unaffected. The ground-state recovery time would be expected to be dominated by thefluorescence decay time in this case.

Comparison of Eqs. (23) and (24) shows that J ≥ β im-plies that twisting is unfavorable in sufficiently non-polar sol-vents. However, Table I shows also that even for relatively non-polar solvents, we still have λ ≥ 0.5 eV. For Michler’s hydrolblue (cf. Figure 1), we have the interesting case of J ∼ β, sothat the driving torque for twisting in gas phase is expectedto vanish. We have tested this assertion for Michler’s hydrolblue using multi-reference perturbation theory estimates of theexcited-state energy at the ground state minimum (a modelof FC) and at a twisted excited-state minimum (a model ofRT or LT) and found that this is true to within the expectedaccuracy of the calculations. In this case, solvation in weaklypolar solvents such as, e.g., chlorobenzene will induce a bias of∼ 0.2 eV, which should be sufficient to quench fluorescence.

The reddest dyes with absorbance in the visible wouldhave β ∼ 0.8 eV. We have numerically calculated the barrierheight for a dye with β = 0.8 eV, J = 1.0 eV with dimen-sions comparable to Michler’s hydrol blue (r = 2 Å, R = 4 Å)in chlorobenzene (λ = 0.8 eV) and found it to be less than3kBT . Assuming an attempt frequency of ∼ 1 ps for a torsionalbarrier, this would not be sufficient to engender steady-statefluorescence in the dye for moderately polar solvents λ ∼ 1 eV.

The observation of a barrier to twisting in our modelmust be approached with awareness that we are leaving outpotentially important intramolecular degrees of freedom. Forexample, our model does not consider bond vibration displace-ments, to which the states in these systems are known to


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be coupled. Laage and Hynes have suggested a form for thediabatic coupling that is formally identical to the solvationmodel we use here.65 It has been pointed out that the couplingof solvation and stretching displacements through the sameelectronic degree of freedom (the diabatic gap) can amplifythe effects of either.9 If a solvation coordinate was includedin this way, with a reorganization energy of order ∼ 0.1 eV, itwould act to reduce the barrier height and increase the drivingforce for reaction. This may lower the barrier to the point thatit becomes irrelevant for typical monomethine cyanines.

We conclude that our model predicts barrierless or near-barrierless excited state twisting for typical solvated monome-thine cyanines, but we are unable to rule out the possibilitythat a monomethine cyanine may exist, if the dye has near-IRabsorbance and large π-bonding interactions that would showsteady-state fluorescence in sufficiently non-polar solvents.

D. Do gapped TICT states exist? (Red line in Fig. 4)

We call at particular TICT state (geometry) gapped if itcannot be made to coincide with the conical intersection seamfor any configuration of the solvent.

It has been suggested that the timescales observed in ultra-fast spectroscopic experiments on monomethine cyanines maybe interpreted on the basis of a partitioning of the dynamicsbetween (fast) descent into a basin corresponding to a twistedconfiguration, followed by diffusion to the sink or conicalintersection.86 This notion presupposes the existence of anadiabatic twisted minimum on the excited state surface, inwhich diffusion could occur. In this section, we address thequestion of whether a true (i.e., gapped) excited state minimumcan exist in the TICT manifold.

To answer the question, we analyze the extent and locationof conical intersections in the model. The conditions for inter-section of adiabatic states is that the diabatic gap and couplingvanish simultaneously. This gives

− 2β cos θL cos θR = 0 (25)

for the coupling and

J�sin2 θL − sin2 θR

�− λs = 0. (26)

for the gap.Equation (25) is only (and always) satisfied in the TICT

channels, so that conical intersections will only ever occur inthe channels.

Equation (26) can be used to derive the solvent config-uration that brings the conical intersection into coincidencewith any point in the TICT channels. This is, for the L and Rchannels,

sCI(θL, θR) = Jλ

cos2 θR, |θL | = π2 ,−cos2 θL, |θR| = π2 .

(27)

This shows that the CI will be in the R channel for s ≥ 0and the L channel for s ≤ 0. Again, we have concentratedsolely on the “patch” shown in Figure 6; a more general state-ment would include periodic images of the CI.

It is clear that (26) can only be solved for s ≤ Jλ

if J≤ λ, since the range of s is limited to the interval [−1,1]. Forsolvent configurations s ≥ J

λ, no conical intersection exists (cf.

Figure 5, bottom row).For any set of model parameters J, β, λ such that λ ≤ J,

the range of motion of the CI into the channels is limited toangles θCImin ≤ θL,R ≤

π2 . Accordingly, there is an adiabatic gap

remaining at angles θL,R less than some minimum angle θCIminfor which the CI may exist. If TICT states exist that are gappedfor all solvent configurations, then these may accumulate pop-ulation on the excited state surface, which will decay followingdiffusion to the vicinity of the CI seam.

If the adiabatic excited state PES does possess a minimum,and if this basin is sufficiently removed in energy and config-uration space from the intersection seam such that transientsteady-state population accumulates there, we would expectthe ground state recovery time to increase without consummatelengthening of the fluorescence decay time. This is because theTICT states are dark, so accumulated population in the TICTchannels will not contribute to the fluorescence.

Table I says that, for a dye with dimensions similar toMichler’s hydrol blue (r ∼ 2 − 3 Å , R ∼ 4 − 7 Å), the reor-ganization energy should be of order 1 eV even for weaklypolar solvents. Accordingly, we expect that θCImin ∼ 1 for thesesolvents. This implies that the conical intersection can accessmost of the TICT configurations in the channels. However,even for an angular interval of ∼ 0.1 rad, we expect that thediffusion time in a medium of viscosity 1 cP might be 10− 100 ps, and so this should be detectable as a delay in theground state recovery time that is not apparent in the fluores-cence decay time.

For dye-solvent systems with λ ≥ 1 eV, the CI can bebrought into coincidence with any point in the TICT chan-nels. Since the TICT channels represent the bottom of theexcited-state potential (cf. discussion relevant to blue linein Figure 4, above), this implies that a CI spans the bottomof the excited state potential of the model in this regime.This regime is the one that we expect to be relevant to mostspectroscopic experiments using monomethine cyanines inmoderate- to strongly polar solvents (such as, e.g., small-chainalcohols).

The growth of access to the TICT channels by the CIcan be dramatically visualized by taking the fast-solvent limit,where the solvent is allowed to equilibrate to the dye at everyfixed dye configuration. This limit yields two-dimensionalpotential energy surface such as shown in Figure 8 for Mich-ler’s hydrol blue. In the fast-solvent limit, the coincidence ofthe CI with the bottom of the excited-state PES is apparentas a “zipping up” of the TICT states as the seam extends intothe channels. The results shown in Figure 8 show that evenfor weakly polar solvents, the intersection can already extendover most of the TICT configurations. The PES given foracetonitrile in the figure is the same for all solvents with λ ≥ J.This is because solvent configurations that are sufficientlypolarized to re-introduce an excited-state gap also raise theexcited-state energy, so that the force on the solvent coordinateat these configurations will drive the system back towardsthe intersection. Accordingly, these solvent configurations arenever sampled in the fast-solvent limit.


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FIG. 8. Potential energy surfaces for Michler’s hydrol blue (β = 2.0 eV, J = 1.0 eV) in the fast-solvent limit, in gas phase (left), chlorobenzene (middle), andacetonitrile (right). Reorganization energies and static dielectric constants for each solvent are shown. In the fast solvent limit, growth of the CI seam alongthe bottom of the excited state PES appears as a “zipping up” of the CI seam along the TICT states. This is because, for any TICT configuration, the solventconfiguration that yields the lowest excited-state energy is the one that brings the CI to that point. The fast-solvent limit is taken at any dye configuration byletting the solvent equilibrate to the excited state of the dye at that configuration. Shown are fast-solvent-limit PES for Michler’s hydrol blue (J = 1.0 eV,β = 1.0 eV, r = 2.3 Å, R = 4.6 Å) in gas phase (left), a weakly polar solvent (chlorobenzene, middle), and a strongly polar solvent (acetonitrile, right). Evenfor chlorobenzene, the CI seam spans most of the TICT configurations. The PES shown for acetonitrile is identical in any solvent for which λ ≥ 1, since at thatpoint, the TICT states are already “all zipped up” (J = 1 eV for this case). The fast-solvent limit is not physically realistic for these systems, but provides goodillustration of the growth of the CI seam in solvents with increasing susceptibility.

The phenomenon of “zipping up” of intersection seamsin the fast-solvent limit has also been described by Burghardtand co-workers for a model of the photoisomerization of aprotonated Schiff base.53 This phenomenon can be traced inboth cases to the diagonal form of the solvation coupling usedin Hamiltonian equation (2).

Erez et al. have showed for the molecular rotorsThioflavin-T and Auramine-O that the nonradiative decaytimescale is 1-2 orders of magnitude slower than the transversedielectric relaxation timescale in a variety of glass-formingsolvents.25 This is consistent with a twisting timescale that isof the same order as the longitudinal Debye relaxation time.The longitudinal time will be an order of magnitude lowerthan the transverse time in lower alcohols. The longitudinalrelaxation time is the relevant time for solvation following achange in the charge distribution.31 These points emphasizethat only qualitative insights can be obtained from consideringthe fast-solvent limit, which is not believably achieved in realexperiments on symmetric monomethines.

IV. DISCUSSION

We have proposed a valence-bond model of non-equilib-rium solvation effects on the excited-state twisting reaction of ageneral monomethine cyanine dye. The model has been madeintentionally simple, so as to explore the consequences of a fewspecific ideas.

The solvation model is similar to a previously publishedmodel of anomalous solvatochromism charged push-pull poly-

enes.65 It differs by the introduction of potentials for twistingdistinct π-bonds in the diabatic states, and the introduction ofa twist-dependent coupling that goes as product of π orbitaloverlaps across the bridge. The functional forms used can bejustified using valence-bond models.69

Our results suggest that the concept resonance in dyes,traditionally used to describe optical properties,2–4,6,7,9,63,81

may also explain non-radiative decay. Since the resonancenotation of Figure 1 describes a transfer between states withdifferent charge- and bond-order localizations, the locationof the barrier is different in the diabatic states. The correla-tion of charge and π-bond order localizations has importantconsequences for the structure of the excited state PES in themodel, and the response to non-equilibrium solvation. Theassumption of a coupling that varies as a product of overlapsacross the bridge bonds is a consequence of the negligibilityof the direct (single particle) inter-ring charge-transfer matrixelement, which is traditionally assumed.80

The model broadly reproduces features of the ground andexcited state PES that are observed in more complicated elec-tronic structure calculations of monomethine cyanines35,84 andrelated systems.22,39,41,42,87 These studies indicate that there aretwo energetically accessible twisting channels on the excitedstate surface, and that these channels are distinguished by thepolarity of the adiabatic TICT transitions within the chan-nels,35,84 such that control of the pathway yield by environ-mental interactions is possible.41,42,87

For parameter regions characteristic of monomethine dyes(cf. Figure 1), there is a single charge-transfer conical


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intersection in the model that moves in response to changesin solvent configuration. For an unsolvated dye or a dye in anunpolarized solvent, the intersection occurs at a geometry ofthe dye where both bonds are twisted. This is an ambiguousposition with respect to the two TICT channels. It correspondsto either a concerted- or “hula-” twisted geometry.70 As thesolvent configuration changes, so as to solvate better one orthe other diabatic state, the intersection moves into one of theTICT channels. This means that the intersection will occur ata geometry where one bond is twisted more than the other.

In general, ab initio calculations on methine dyes showthat minimal-energy conical intersections in symmetricmonomethine cyanines occur at geometries where both bondsare twisted, but where the twist distribution is not symmetric.35

This is also true in non-symmetric methines with very smalldiabatic gap.37 The reason why our model predicts a sym-metric twist distribution can be traced to the neglect of otherintramolecular displacements that could break symmetry inthe any real specific case. In any case, our results suggest thatintersections in methines can move in response to solvation,and will move to the bottom of the excited-state PES forsolvents of moderate polarity. Decay via different twistingchannels is favoured for distinct configurations of the solvent.Migration of conical intersections towards the bottom of theexcited state potential in solvated organics was observed inearly quantum mechanical/molecular mechanical (QM/MM)wavepacket simulations of solvated photoisomerization in me-thines.88 Our model gives a simple and transparent descriptionof this effect for the case of monomethine cyanines.

Our model predicts that configurations with mixed twistdistributions (for example, the “hula-twist” configuration HT)have relatively low energy on the excited-state PES. In partic-ular, these geometries are accessible from FC in gas phase.This is contraindicated by ab initio calculations on monome-thine cyanines,35 as well as non-symmetric monomethineswith small diabatic gap.37 The physical origin of the discrep-ancy is that, in our model, there is no additional penalty forbreaking both π bonds at once. This could be added to ourmodel easily, but would not qualitatively alter our main results.

We have provided a simple expression for the lowest en-ergy possible in the TICT channels, relative to FC, that canoccur in our model. This is Eq. (23). This can be used inconjunction with timescale formula Eq. (1) to yield a timescalefor rotation of one of the rotors in a monomethine. An estimateof the relevant torque in this case is the lowest energy inthe channel (Eq. (23)) divided by π2 radians. This yields themodified timescale formula

τTICT =πηV∆Emin1

. (28)

From this timescale, assuming no other non-radiativedecay channels and also accounting for the existence of tworotors in monomethine cyanines, we can obtain an estimate ofthe fluorescence quantum yield Φ f as

Φ f =1

2τradτTICT

+ 1. (29)

For Michler’s hydrol blue (β = 1 eV, J = 1 eV, r = 2 Å,R = 4 Å, V = 34 Å3) in methanol (η = 0.6 cP), dichloro-

methane (η = 0.45 cP), and chlorobenzene (η = 0.8 cP), thisapproach yields quantum yield estimates of 2 × 10−4, 2 × 10−4,and 7 × 10−4, respectively, assuming a radiative timescale ofτrad = 2 ns. These estimates are in reasonable agreement withmeasured values of 2 × 10−4, 3 × 10−4, and 5 × 10−4, respec-tively.91

Equations (29), (28), and (23) together suggest that thefluorescence quantum yield of a monomethine cyanine islimited by the relative value of the driving torque in the excitedstate and the hydrodynamic friction. For a given dye chromo-phore (i.e., given β, J), the driving torque will be modulatedby the reorganization energy as Eq. (23), so that decreasing thereorganization energy of the environment of the dye will, in theabsence of other considerations, give greater fluorescence, aswill increasing the viscosity. Alternatively, for a given medium,the driving force will be increased by reducing the electroniccoupling or by increasing the hydrodynamic volume.

Assuming a π-bond stabilization energy of J = 1 eV, aswe have here, implies that only monomethines with near-IRabsorbance will have inaccessible TICT states. Polar solva-tion can only accelerate TICT formation (in one channel oranother). In a structured environment, such as provided by aprotein, it is possible that the elastic properties of the proteinmay provide an additional potential barrier, but there seemslittle scope for engendering fluorescence in fluid solution bythe manipulation of the driving torque.

On the other hand, if the rotation timescale is linear in theviscosity as suggested by Eq. (1), and given that the timescalefor diffusion on a flat potential of ∼1 ps for, e.g., Michler’shydrol blue, one expects that a 1000-fold increase in the vis-cosity should be sufficient to achieve significant fluorescence.This is consistent with changes in the fluorescence of Michler’shydrol blue recorded in a series of water-glycerol mixturessampling this viscosity range.91 It is also broadly consistentwith the radiative lifetime of Auramine-O and Thioflavin-Tover a broader range.25

Both the driving torque and the hydrodynamic frictiondepend on the spatial extent of the endgroups. The drivingtorque varies inversely with r (cf. Eq. (12)), while the hydro-dynamic friction scales with the volume (i.e., with r3). Ac-cordingly, varying the spatial extent of the endgroups withbulky substituents may be an effective way to tune the solutionfluorescence quantum yield of monomethine cyanines.

Symmetric monomethines have been used as experimentalmodels for the study of barrierless viscosity-controlled reac-tions.26,82,86,89,90 For the cyanine 1122C, Yartsev and coworkershave cited evidence of a thermalized excited state population.86

Specifically, a fast (1 ps) component in the dynamics has beenattributed to twisting motion towards an excited state basin,followed by depletion from the basin through the sink on thelonger (10 ps) timescale.86 For a dye such as 1122C, with ringsabout double that of Michler’s hydrol blue, the reorganizationenergy in alcohol solution may be sufficiently low as to allowthe existence of gapped TICT states, which could supportadiabatic excited-state population diffusion. For a dye of thissize in methanol, we have estimated the timescale for TICTformation (as above, but with r = 4 Å, R = 8 Å), and found itto be several picoseconds. However, the timescale for diffusionon a flat potential (such as would be expected for population


130.102.158.13 On: Tue, 09 Jun 2015 00:29:50


diffusing on the adiabatic excited state in a gapped region of theTICT channels) is several tens of ps. For dyes in moderatelypolar solution, for which the driving torque will be ' 10kBT ,the timescale for TICT formation may be distinctly faster thanthe timescale for flat potential diffusion of an adiabaticallyexcited TICT population. This suggests that our model maydescribe the transient accumulation of adiabatically excitedTICT population for dyes with rings comparable to those in1122C. Further elaboration of this possibility would seem torequire time-dependent simulations, and is beyond the scopeof this paper.

V. CONCLUSION

We have proposed and analysed a valence-bond modelof non-equilibrium solvation in a monomethine cyanine dye.The lowest adiabatic transition in these systems is tradition-ally described using a concept of resonance between quan-tum states with distinct charge and bond localizations. Thisresults in strong coupling to the twist degrees of freedom anda twist-dependent charge-transfer character of the transition,all of which our model captures. It also describes several fea-tures of the potential energy surfaces that have been observedwith more complicated electronic structure models,35 includ-ing the occurrence of multiple low-energy TICT channels onthe excited state surface, correlation between twisting andcharge-transfer polarizations of the transition in the channelsand the existence of low-energy conical intersections in theTICT channels. We believe that our model may be the simplestthat captures these three effects.

Our modelling suggests that solvent effects are signifi-cant on the potential energy surfaces that govern photoexcitedtwisting in monomethine cyanines. Polar solvation generallystabilizes TICT states, and can increase the driving forces forTICT formation. We have provided a simple expression for thelowest possible energy (relative to FC) that can exist in theTICT channels of our model (Eq. (23)).

Our modelling suggests that TICT formation will be favor-able for typical monomethine cyanines in solvents sufficientlypolar to allow good solvation. TICT formation should be bar-rierless for dyes with absorbance above 600 nm (for whichβ ≥ J ≈ 1.0 eV). For dyes with absorbance at lower photonenergy, our modelling predicts a small barrier that may existon the excited-state surface. This barrier should be less than afew kBT for dyes with visible absorbance. If a barrier exists,then our model also predicts that excited-state twisting will beunfavorable for the same dye in vacuum.

A conical intersection grows along the bottom of theexcited-state surface in solvents with increasing relative dielec-tric. For monomethine cyanines with modest-sized rings inmoderately polar solvents, our modelling predicts that noadiabatically excited TICT state can exist, as the bottom ofthe excited-state PES is completely spanned by a conicalintersection seam.

ACKNOWLEDGMENTS

This work was supported by Australian Research CouncilDiscovery Project No. DP110101580. We thank A. Painelli, J.T. Hynes, and T. J. Martínez for helpful discussions.

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