Environmental effects on a conical intersection: A model study

Excited-state processes at conical intersections (CIs) involving charge transfer phenomena can depend sensitively on the influence of a polar and polarizable environment.

We propose here a formulation to describe the chromophore–environment interaction for such situations.

In a model study, we focus on an extension of the two-electron two-orbital model by V. Bonačić-Koutecký, J. Koutecký, and J. Michl [Angew. Chem., Int. Ed. Engl., 1987, 26, 170], which yields a diabatic model for the S1–S0 CI in protonated Schiff bases and related systems, and describes the charge properties and charge translocation phenomena associated with this CI.

The electrostatic effects of the environment, which are expected to strongly affect the CI topology, are accounted for by a dielectric continuum model.

This translates to the image of free energy surfaces for the coupled chromophore–environment system represented by molecular coordinates plus a solvent coordinate.

The environment's impact on the location and character of the CI is investigated.

The limiting situations of “frozen” and equilibrium solvation effects are examined.


Conical intersections (CIs), i.e., multidimensional, cone-like topological structures at degeneracies between Born–Oppenheimer surfaces, are known to play a key role in the ultrafast dynamics of photoexcited molecular systems.1–4

Their great relevance for biochemical and biological systems has been confirmed over the past decades.5,6

By their particular topology, CIs act as “photochemical funnels” which effectively and selectively induce transitions between electronic states.

While such processes have been extensively studied in gas phase species, the influence of an environment on conical intersections has not yet been systematically investigated.

For example, the recent study involving two of us in refs. 7–9 is the first explicit investigation of solvent effects on the ground state reaction path near a CI, and the QM/MM type calculational approaches of refs. 10, 11 provide an avenue to the microscopic description of both ground and excited states in an environment.

In the present contribution, we discuss a new approach to modeling the environment's influence on the excited-state processes at a conical intersection.

The importance of understanding environmental effects on the processes at and near CIs is highlighted by a number of recent studies of biologically relevant systems, notably the chromophores of the rhodopsin family12–14 and other photoactive proteins,15–20 porphyrin-quinone complexes,21 as well as the building blocks of DNA.22,23

These studies reveal the crucial role of CIs and suggest that a solvent or protein environment has a pronounced impact on the electronic structure and CI topology.

Due to the environment, CIs can indeed be substantially displaced, or may even disappear.

Given the complexity of these systems, a key step in the theoretical analysis is to select the relevant features which determine the energetics and dynamics of the chromophore–environment supermolecular system.

In the present work, we focus on situations where the CI involves a charge transfer process and therefore couples strongly to the environment's electrostatic properties.

This is of relevance, in particular, for the S1–S0 CI in protonated Schiff bases (PSBs) like retinal, which plays a key role in the chromophore's cistrans isomerization and can be associated with the translocation of a positive charge.24

Charge transfer has also been shown to characterize the earliest events in the photoactive yellow protein, PYP, photocycle.10

The theoretical model developed within is designed to capture the electrostatic aspects of the chromophore–environment interaction and is thus specifically adapted to such situations.

At the level of description adopted here (which allows for considerable generalization), the model involves (i) a description of the environment as a dielectric continuum (Section 2) which translates to a solvent coordinate picture, previously exploited in models for charge transfer reactions strongly coupled to large amplitude nuclear motion, see e.g.refs. 7, 25–29, in conjunction with (ii) a representation of the chromophore's electronic structure in a diabatic basis of states which are constructed such as to preserve their charge character in the nonadiabatic coupling region.

This type of basis is best adapted to formulate the interaction with a polar and polarizable environment.

(The term diabatic is here meant in a loose sense, to describe a basis which has “smooth” properties and approximately diagonalizes the nuclear kinetic energy operator.1,2)

In this first effort, we focus on a simple realization of such a basis in terms of valence bond (VB) type states suitable for the description of PSBs, as an extension of the two-electron two-orbital model by Bonačić-Koutecký, Koutecký, and Michl5,6,24,30 (Section 3).

The description of the environment as a dielectric continuum leads to an augmented electronic structure problem yielding free energies; the latter carry a parametric dependence on the chromophore's internal degrees of freedom plus a solvent coordinate representing the environmental polarization field.

The solvent coordinate thus appears as an additional degree of freedom which, for the coupled free energy surfaces at the CI, plays a substantial role in determining the CI topology and the concerted chromophore–environment dynamical path.

In the present context, we examine the two extreme cases of a “frozen” solvent vs. equilibrium solvation situation (Section 4).

While the present work focusses on the case of a small chromophore in solution phase, the theoretical framework we are developing is generalizable to take into account the specific nature of the environment, by referring to microscopic solvent coordinate concepts,31–34 with emphasis on the environment's heterogeneous character.

This opens up an avenue to treat very general types of system–environment situations, in particular (i) complex molecular systems, even gas phase species, characterized by a local site which interacts with an intramolecular “bath” involving charged and polarizable groups, and (ii) large molecules in a solvent environment, where the local site of interest interacts with an effective environment comprising intramolecular and solvent components.

For a chromophore embedded in a protein via electrostatic, hydrogen bonding and covalent interactions, the local environment bears features of both cases (i) and (ii).

The present work maps out the very first steps towards such complex environments.

The paper is organized as follows.

Section 2 addresses the general theoretical formulation leading to a free energy surface description of the combined chromophore–environment system.

Section 3 introduces the extended two-electron two-orbital model we use to describe the coupled S1–S0 states in protonated Schiff bases.

In Section 4, we address the charge properties, free energy surfaces, and solvent effects for the CI model.

Section 5 summarizes some future perspectives and concludes.

Free energy formulation for the combined chromophore–environment system

We adopt a description of the environment via a dielectric continuum model, i.e., the environment is described in terms of a polarization field interacting with the electric field created by the solute.

The polarization field can be represented as the sum of two components, P(x) = Pel(x) + Por(x) where x denotes a field point in the solvent, and the sum refers to an electronic (“fast”) component Pel associated with the electronic solvent degrees of freedom and an orientational (“slow”) component Por associated with the solvent dipoles' reorientation in response to the solute field.

This latter component determines the non-equilibrium effects in the solvation dynamics: There is no requirement in general that the orientational polarization be in equilibrium with the molecular solute's charge distribution in its dynamical evolution.

As detailed in refs. 35 and 36, the interacting solute–solvent system can be characterized via a free-energy functional G[ψel, Pel, Por].

Here, the interaction with the environmental polarization fields is added to the molecular electronic Hamiltonian, thus defining an augmented electronic structure problem.

Minimization of G yields solutions which depend parametrically on two types of “slow” variables: the internal nuclear coordinates of the chromophore R, and the slow polarization component Por.

These solutions G(R,Por) are adiabatic free energies, by analogy with the adiabatic energies of the conventional wavefunction description.

Importantly, a nonequilibrium process can be described in terms of the dynamics on such free energy surfaces G(R,Por).

This necessitates the introduction of a kinetic energy for Por.25,26,37–41

Time scales of nonequilibrium solvation are generally similar to the internal molecular degrees of freedom dynamics: picoseconds to femtoseconds.

Free energies in the Born–Oppenheimer limit for the electronic polarization

While the solution to the combined solute–solvent electronic structure problem is non-trivial, and in general necessitates a quantum treatment of the electronic polarization field35, we focus here on a particular limiting situation: The solvent electronic polarization adjusts rapidly to the solute's electronic configuration and can therefore be adiabatically eliminated.

In electron transfer language, the electronic polarization is fast compared to the transferring electron.

This limit is characterized by the condition 2β/ℏωel ≪ 1 in terms of the electronic coupling β, which sets the electron transfer time scale, and the characteristic solvent electronic polarization energy ℏωel.

This condition is applicable to an important class of charge transfer reactions in various solvents.

This Born–Oppenheimer (BO) limit35,42 leads to the following, simplified, form of the free energy functional35GBO[ψel, Por] = Hmol[ψel,Por] + Gint[ψel,Por] + Gel[ψel] + Gor[Por]with the following contributions:

(i) the expectation value of the electronic Hamiltonian H0 of the isolated molecular system, Hmol[ψel,Por] = 〈ψel|H0|ψel〉for the solution phase solute electronic wavefunction ψel(xel;R,Por) which has a parametric dependence on the nuclear degrees of freedom R and on the orientational polarization field Por;

(ii) the solute–solvent interaction Gint, i.e., interaction between the polarization Por and the solute electric field ψ, Gint[ψel,Por] = –(1 – 4πC–1el)∫ dx ψ·Porwhere Cel = 4πε/(ε – 1), with ε the optical dielectric constant;

(iii) the electronic polarization contribution Gel, for an electronic polarization Pel field which instantaneously equilibrates to the solute electric field ψ,

(iv) the orientational self-free energy contribution Gor, with Cor = 4π(1 + ε0 – ε)/(ε0 – ε) where ε0 is the static dielectric constant.

We note that the free energies resulting from minimizing the functional eqn. (2) can be formulated as the eigenvalues of an effective Hamiltonian.35

In the above expressions, the electric field ψ generated by the solute is35,36ψ(x; R) = –∇φ(x; R) with the potential produced at point x in the solvent by the solute electronic density ρ(x′;R).

The latter is the single-particle density derived from the electronic wavefunction, ρ(x1; R) = N∫ dx2…dxNψel(x1,…,xN;R)ψel*(x1,…,xN; R)

We will focus within on the case where the wavefunction is expressed in a basis of states {ϕj(xel;R)}, each of which corresponds to a fixed charge distribution.

These states are what we term valence bond (VB) states and constitute an appropriate basis for including the interaction with the external polarization fields.

The solute charge distribution can thus be written28,35,36 with component densities ρij(x1,R) = N∫ dx2…dxNϕi(x1,…xN;R) × ϕ*j(x1,…,xN;R).

This partitioning carries over to the potential φ = ∑ijφij and to the electric field, ij = –∇φij, such that the solute–solvent interaction takes the form.

Thus, components of the interaction free energy can be associated with the field generated by the component solute charge distributions ρij.

The orientational polarization also couples to the off-diagonal components i ≠ j, corresponding to the “exchange” field components,35 giving rise to off-diagonal terms in the diabatic representation for the free energy and leading to a diabatic coupling renormalization35,36,43,44.

Solvent coordinate formulation

The interaction free energy partitioning eqn. (10) is taken as a basis to define solvent coordinates associated with each component.35,36

To calculate the individual contributions, knowledge of the nonequilibrium polarization field Por is required, and necessitates approximations in the absence of a complete representation of the field.

One useful approach is the representation of Por in terms of linear combinations of equilibrium contributions.

In particular, the nonequilibrium polarization can be expressed, in a chemical reaction context, in terms of a single solvent coordinate z37,41 which weights the equilibrium contributions in the reactant (R) and product (P) states, Por(x;R) = (1 – z)Peqor,R(x;R) + zPeqor,P(x;R) Here, the off-diagonal components i ≠ j of eqn. (10) have been neglected.35

(We will later identify R and P in the CI context).

With eqn. (11) for the polarization field, the “Marcus-like” free energy parabolas are obtained, displaced along the solvent coordinate z.

The free energies of eqn. (12), denoted by a subscript s, comprise components in the interaction free energy Gint of eqn. (4), the electronic solvent free energy Gel of eqn. (5), and the orientational self free energy Gor of eqn. (6).

The GR,Peq(R) are the equilibrium free energies for the R and P states, i.e., the free energies with Por equilibrated to the charge distributions in R and P, respectively.

The “solvent force constant” is ks(R) = k0sdx[Peqor(x, R) – Peqor, P(x, R)]2, with k0s = 4π/(1 – ε/ε), and is directly related to the reorganization energy λ(R) = ks(R)/.237,41

Note that all relevant quantities – in particular, the equilibrium polarization fields, the solvent force constant, and the reorganization energy – carry a dependence on the internal molecular coordinates R, due to the solute electric field ψ dependence on R.

A model for the S1–S0 conical intersection in protonated Schiff bases

Conical intersections of charge transfer type, i.e., involving a transition between an ionic and a covalent state, constitute a generic class of conical intersections in polar double bond systems.45

Among these, as anticipated in the Introduction, protonated Schiff bases (PSBs) may be considered a paradigm example, for which S1–S0 isomerization can proceed via a CI, for both the CC and CN bonds.5,6,24,30,46

While the chromophores in question in fact exhibit a complicated electronic structure, with various CIs of different types, notably of one-bond flip (OBF) and hula-twist (HT) type,45 we focus here exclusively on an OBF mechanism which only involves a twist around the CC double bond, rather than a concerted twist/flip process for several neighboring bonds.

The CIs in question occur at a twisted geometry of the double bond – often close to 90°30,47–50 or, according to our model, at exactly 90°24,30 – and thus correspond to a biradicaloid structure.

The two-electron two-orbital model developed by Bonačić-Koutecký and Michl24,30 in the late 80s explains the occurrence of such CIs by a degeneracy, at a biradicaloid geometry, of a covalent (“dot–dot”) and ionic (“hole-pair”) type configuration.

A key observation of the model is that the electronegativity difference across the double bond, measured by a parameter δ,24,30 determines whether a CI can actually be found: With increasing values of δ, the S1 state is stabilized until an S1/S0 degeneracy is met at a threshold value.

The model thus explains that, while no CI is found for twisted ethylene (δ = 0), in the absence of asymmetric perturbations as induced, e.g., by pyramidalization, CIs do occur for PSBs.

Two-state model including solvent

The relevant PSB electronic configurations can be represented in an approximate fashion in a basis of localized p orbitals, which are best adapted to the biradicaloid structure at the twisted geometry.

A minimal model for the S1–S0 conical intersection can be formulated in terms of the two singlet states24,30 with the notation A ≡ pA and B ≡ pB for two p orbitals on neighboring carbon centers, and |Σ〉 = (1/√2)(α1β2 – β1α2) the singlet spin configuration.

The |AB〉 state corresponds to the dot–dot configuration mentioned above while the |B2〉 state is one of two hole-pair configurations.

(The complementary |A2〉 configuration is assumed to be shifted to high energies, due to the effect of the asymmetry parameter δ.24,30)

A detailed description of the model is given in .ref. 51

The two configurations eqn. (13) have different locations of the positive excess charge carried by the PSBs.

If one thinks of the following partitioning in terms of the left and right hand PSB moieties (see also refs. 49, 47), where the vertical bar indicates a cut through the CC double bond, the charge characteristics of the basis states may be described in a short-hand notation,i.e., the positive charge is localized on the nitrogen end for the |AB〉 state and on the hydrocarbon end for the |B2〉 state.

A charge translocation24 therefore takes place upon the transition between these two states.

The {|AB〉,|B2〉} states represent a charge-localized, valence bond (VB) type basis, and provide a simple realization of a more general type of diabatic representation which preserves the basis states' charge character in the nonadiabatic coupling region.

At the 90° twisted geometry, the |AB〉 and |B2〉 states are uncoupled due to symmetry reasons51 and describe biradicaloid configurations.

Twisting away from θ = 90° induces a non-zero overlap and, hence, π-binding effects between the p orbitals on neighboring centers, such that the |AB〉 and |B2〉 states are mixed.

The twisting coordinate θ therefore acts as a coupling, or symmetry-breaking, coordinate.24,30,51

The solution phase diabatic free energy model we adopt takes the following form, where the coordinate r (associated with CC stretch motion) plays the role of a tuning coordinate,1 which only contributes to the diagonal terms and thus modulates the energy gap between the two diabatic states.

The solvent coordinate z also acts as a tuning coordinate via the diagonal terms in eqn. (16).

Generally, the electronic coupling γ(θ) is not identical to the coupling in the isolated chromophore; it also includes the interaction with the orientational polarization field eqn. (10), γ(θ) = γ0(θ) – (1 – 4πC–1el)∫dxex·Porvia the “exchange” field component ex related to the off-diagonal component density ρij with i = |AB〉 and j = |B2〉.

(We have suppressed the explicit dependence of γ on Por(x;r,θ) in the notation).

The solvent-induced component thus leads to a renormalization35 of the coupling.

Further, at a CI, the solvent-induced coupling can have a symmetry-breaking effect: i.e., a coupling can be induced by the solvent for θ = 90° while γ0(θ = 90°) = 0 by symmetry for the isolated chromophore.

In this case, the CI is shifted away from θ = 90°.

We consider this type of effect no further here.

The model eqn. (16) should be taken to correspond to the simplest scenario which gives a qualitatively correct picture for the S1–S0 intersection in PSBs.

Different interpretations are currently given as to whether an effective two-mode model is in fact viable to describe the dynamics in various retinal analogs and other PSBs.

While Robb, Olivucci and co-workers indeed supported a two-state-two-mode model in their recent work,49 Martínez and co-workers47 provided evidence that, besides the torsion and the CC stretching mode (or a concerted stretching motion across the PSB carbon skeleton), the C–N stretch plays an important role in the branching space of the S1–S0 conical intersection.

Our model could be extended in view of these findings.

Parameterization of the model

The model eqn. (16) is parameterized in an approximate fashion for the internal modes (r, θ) and solvent coordinate z; Table 1 lists the relevant parameter values.

While we do not explicitly make contact with the electronic structure side in the present study, we suggest as a future strategy to fit ab initio data to the model under consideration.

The diagonal contributions VAB(r,θ) and VB2(r,θ) in eqn. (16) include (i) a Morse form in the stretch coordinate r, (ii) a θ-dependent shift term which displaces the minimum of the Morse potential for increasing θ, and (iii) a diagonal angle-dependent term chosen according to the study by Tennyson and Murrell for ethylene.52

Hence, VAB(r,θ) = VABmorse(r) + VABshift(r,θ) + VABθ(θ)with the individual termsVABmorse(r) = DAB(1 – exp[–αAB (r – r0,AB)])2where r0,AB is the CC equilibrium distance at the planar geometry, and an angle dependent shift potentialVABshift(r,θ) = ξAB(r)sin2θwhich is adjusted to shift the minimum of the Morse curve to its appropriate value at θ = 90°.

Here, ξAB(r) = cABshift (r – r0,AB)exp(–|r – r0,AB|/σr) with cABshift = –2DABα2ABΔrAB, for a shift ΔrAB in the minimum geometry.

Note the exponential which is added to attenuate the effect of the linear term for increasing distance from the minimum geometry.

Further, the diagonal contribution in the torsion is given asVABθ(θ) = cABθcos2θwith cABθ < 0 such that the diabatic ground-state potential is bonding for θ = 0°, see .ref. 52

Similarly, the diagonal potential for the hole-pair state |B2〉 reads , with definitions analogous to eqns. (19), (20) and (21), in addition to an energy shift ΔEB2 with respect to the |AB〉 state.

The coupling between the |AB〉 and |B2〉 states is given as γ(θ) = c(1)γcos θ + c(3)γcos3θin accordance with the models developed for ethylene by Domcke, Manthe and co-workers53 and, for the contribution in cos θ, by Tennyson and Murrell.52

The physical origin of the term in cos θ is a resonance integral, proportional to the overlap between the p orbitals on the neighboring centers;24,54 the third-order term should be understood as a higher-order correction conforming to the required symmetry.

We restrict the present analysis to the term in cos θ.

Finally, the solvent free energies GABs and correspond to two displaced parabolas in accordance with eqn. (12), with the same equilibrium free energies and force constants ks, The free energies eqn. (23) are analogous to, e.g., the model developed in refs. 55, 56 for charged polyenes.

An estimate for the solvent force constant ks = 2λs (here assumed to be independent of the internal coordinates), with λs the solvent reorganization energy, can be obtained from the Marcus–Born model37,41 where the solute charge distribution is surrounded by a cavity in the dielectric continuum.

The reorganization energy λs is then related to the work required to transfer charges from the reactant to the product cavity.

Here, a value of ks = 2.5 eV is used, which has been derived for acetonitrile8 but is also applicable in an approximate fashion to other solvents.

Two-state model for the coupled S0/S1 states in PSBs

The following analysis is based on the two-state, three-mode model of eqn. (16), with the twisting coordinate θ as a coupling mode, the CC stretch coordinate r as a tuning mode, and the solvent coordinate z as an additional tuning mode.

We will successively address the resulting potential surfaces and their topology for the isolated chromophore (Section 4.1), the associated charge distributions resulting from the mixing of the {|AB〉,|B2〉} states and their decoupling at the 90° twisted geometry (Section 4.2), and the inclusion of the solvent (Section 4.3).


Fig. 1 shows the coupled S1 and S0 surfaces for the isolated chromophore, obtained by diagonalizing the matrix eqn. (16) excluding the solvent contribution.

The surfaces feature a CI at θ = 90°, which coincides with the minimum of the upper adiabatic surface.

Fig. 2a shows a cut of the potential surface along the torsional coordinate, at a value of the CC stretch coordinate corresponding to the intersection point, r = r0CI.

The cut illustrates that while the coupling vanishes for θ = 90°, the diabatic vs. adiabatic surfaces increasingly diverge when moving towards the planar geometry.

Fig. 2b shows a complementary cut for the CC stretch potential at the 90° twisted geometry.

We pause to note that while our analysis focuses on the conical intersection region, the work by Robb, Olivucci and co-workers48,49 shows that the earliest dynamical events in the isomerization of short-chain retinal analogs (subsequent to the Franck–Condon (FC) excitation to S1 at planar geometry) involve a pronounced displacement in the skeletal stretching modes.

Our perspective relates to the second phase of the dynamical process which predominantly involves a displacement in the torsional coordinate, towards the conical intersection point.

In this context, we note that the upper adiabatic surface of our model does not give an accurate representation of the Franck–Condon region: for the retinal analogs studied in refs. 48 and 49, the S1 surface is bound in the torsional angle for the initial planar geometry.

For a complete analysis of the reaction path, crucial for the understanding of the femtosecond to picosecond events, the model Hamiltonian has to be modified to account for this feature.

Evolution of charge distributions at the CI

Since the basis functions |AB〉 and |B2〉 are associated with fixed charge distributions, the charge properties of the adiabatic states S0 and S1 as a function of geometry can be derived from an eigenvector decomposition in terms of the VB type components.

While the {|AB〉,|B2〉} states mix substantially for geometries away from θ = 90°, they decouple at the conical intersection.

Hence, a pronounced polarization effect is expected when moving towards the conical intersection.

This effect is illustrated in Fig. 3, for the isolated chromophore adiabatic surfaces as depicted in Fig. 1.

The figure shows the decomposition of the eigenvector corresponding to the S1 state in terms of its {|AB〉,|B2〉} components, as a function of the twist angle θ and the tuning coordinate r.

A step function-like feature is observed for θ = 90° along r, reflecting that the adiabatic states correspond to different diabatic branches to the left and right of the conical intersection point (see also Fig. 2b).

Thus, while the S1 surface is predominantly of |B2〉 character close to the planar geometry, and is entirely |B2〉 along the line θ = 90°, for r < r0CI, the same surface is of |AB〉 character at and near θ = 90°, for r > r0CI.

As can be inferred from Fig. 3, the discontinuity is smoothed out away from θ = 90°.

An entirely complementary picture holds for S0, with the roles of the diabatic |AB〉 and |B2〉 states exchanged.

The S0 surface is thus predominantly of |AB〉 character, apart from the charge reversal near θ = 90° for r > r0CI.

The prominent features of the charge distributions of the adiabatic states S0 and S1 can be summarized as follows: (i) a “polarization” effect when moving towards θ = 90° at a given r, as a signature of the |AB〉 and |B2〉 state decoupling, and (ii) a sudden charge character change when traversing the CI region along r, for values of θ near θ = 90°.

This change is step function-like at θ = 90°.

The above observations in part agree with the picture described by Robb, Olivucci and co-workers48,49,57,58 and the results obtained by Martínez and co-workers47 for different retinal analogs.

Solvent effects

Properties and topology of the free energy surfaces

The S0 and S1 free energy surfaces derived by diagonalizing the matrix potential eqn. (16) depend on the two tuning modes r and z, in addition to the coupling mode θ.

Hence, a seam of conical intersections results, i.e., a line in the (r,z) space for θ = 90°, depicted in Fig. 4.

The minimum energy CI (MECI) is located at (r = 1.54 Å,z = 0.55); the slight departure from the midpoint z = 0.5 can be shown to follow from an analytic analysis.51

(Note also the discussion in ref. 1 regarding the general tendency towards lowering of the MECI energy upon adding further tuning modes).

In the present system, the MECI point in fact represents the global minimum of the S1 surface.51

As can further be inferred from Fig. 4, no CI exists for values of z below a certain limiting value, here z < 0.21.

As z increases, the intersection point is shifted to smaller r values.

The locations (r,z) along the CI seam are determined by the condition ΔV(r, θ = 90°) = –ΔGs(z) for a degeneracy of the two adiabatic states, given a vanishing coupling (θ = 90°), with ΔV(r, θ) = VB2(r,θ) – VAB(r,θ) and the solvent-induced free-energy gap The required degeneracy at the CI seam is thus achieved by matching the respective energy gaps induced by the two tuning coordinates r and z.

According to eqn. (25), the solvent-induced free energy gap ΔGs(z) is a linear function of z and vanishes for z = 1/2.

The maximum value of ΔGs(z) which allows for a CI to occur corresponds to z = 1, for a solvent equilibrated to the |B2〉 state.

This point, at (r = 1.33 Å,z = 1,θ = 90°), coincides with the unique equilibrium point (EQCI) along the CI seam, see the discussion below.

Importantly, the equilibrium CI point does not minimize the S1 free energy, i.e., the EQCI and MECI points do not coincide.

This reflects the particular character of the CI topology, as further detailed below (Section 4.3.3). in connection with an equilibrium solvation perspective.

The CI seam plays a key role for the S1 topology and, hence, the excited-state dynamics.

The combined dynamical evolution in the internal coordinates and solvent coordinate z will ultimately tend towards the CI seam and the MECI point.

In a manner depending on the nonadiabatic dynamics details, this will be followed by a path involving the solvent coordinate on the adiabatic ground state.

In general, motion in the solvent coordinate z will be of a non-equilibrium character, i.e., the orientational polarization will not be that appropriate to the actual charge distribution in the chromophore as the latter evolves in r and θ.

This necessitates consideration of the full three-dimensional surfaces G(r,θ,z).

An approximate dynamical scenario for the nonequilibrium case is addressed in .ref. 51

In the remainder of this section, we focus on the solvent's effect on the chromophore for two limiting situations: (i) a “frozen” solvent polarization (Section 4.3.2). where the solvent coordinate is fixed throughout, imposing a pronounced nonequilibrium solvation situation, and (ii) the case of equilibrium solvation (Section 4.3.3). which implies extremely rapid solvent motion, adiabatically adjusting to the solute charge distribution.

Both cases allow for a reduction from the three-dimensional (r,θ,z) space to effective free energy surfaces Geff(r, θ) in two dimensions.

Frozen solvent perspective

For a frozen solvent polarization, determined by a particular value z, a single point on the CI seam is selected.

Due to the correlation between the two tuning coordinates r and z along the seam, the value of z determines the location of the CI in r.

Fig. 5 illustrates the adiabatic S0 and S1 surfaces as a function of r and θ, for two cases which represent equilibrium situations for the uncoupled |AB〉 and |B2〉 states at θ = 90°: (i) a solvent equilibrated to the |AB〉 charge distribution, zABeq = 0 and (ii) a solvent equilibrated to the |B2〉 charge distribution.

For each fixed z value, a nonequilibrium solvation situation applies for all geometries away from θ = 90°, since the solvent polarization is not allowed to adjust to the actual solute charge distribution.

With either condition eqn. (26) or eqn. (27), the solvent-induced free energy gap eqn. (25) takes a fixed value, of ΔGs(zABeq) = ks/2 and , respectively.

The former does not allow for a degeneracy according to eqn. (24), such that the CI is actually lost for z = zABeq = 0.

For , a CI does exist, at a value of r = 1.33 Å determined from eqn. (24).

This point is in fact the EQCI point referred to above.

For z = 1, the shift in location of the CI point leads to a change in CI topology from “peaked” for the isolated chromophore to “sloped”,59 further illustrated by the one-dimensional cut in Fig. 6.

In this case, the minimum of the upper adiabatic (S1) surface along r no longer coincides with the CI; see also the discussion on the role of CI topology in refs. 60, 61.

Equilibrium solvation perspective

The condition of equilibrium solvation identifies the particular points on the free energy surfaces which correspond to a solvent orientational polarization field which is always in equilibrium with the chromophore's charge distribution at each given geometry (r, θ).

These equilibrium points z = zeq(r, θ) are derived from the free energy condition ∂Gn(r, θ, z)/∂z = 0 where the index n corresponds to the adiabatic state in question.

We focus in the following on the situation where the solvent is equilibrated to the upper adiabatic surface S1.

For the matrix potential eqn. (16), analytical diagonalization yields an explicit form for the adiabatic surfaces GS0(r, θ, z) and GS1(r, θ, z) and the corresponding eigenvectors.

From this follows for the free energy derivative with respect to z, for S1, with c|B2(r, θ, z) the |B2〉 component of the S1 eigenvector.

From the minimum free energy condition and eqn. (28), the geometry dependent equilibrium values are thus found as a solution to

The case θ = 90°, with a vanishing diabatic coupling, represents a particular instance of eqns. (28) and (29).

For this angle, eqn. (29) is the equilibrium condition for the diabatic states, (i) for the S1 branch corresponding to |B2〉, for r ≤ rEQCI, and (ii) zABeq = 0 for the S1 branch corresponding to |AB〉, for r > rEQCI.

Note that the EQCI point at (rEQCI = 1.33 Å, zEQCI = 1) is the unique equilibrium point lying on the CI seam.

(In general, all points fulfilling the degeneracy condition eqn. (24) for or z = zABeq represent EQCI points; the latter case is excluded in the present system as discussed in Section 4.3.2)..

Fig. 7a shows the (r, θ) dependent solutions of eqn. (29) and illustrates the step function-like behavior versusr, at θ = 90°, at the EQCI point.

The picture is the analog, for S1 equilibrium solvation, of Fig. 3a for the isolated chromophore.

Detailed comparison of the two figures reveals that the equilibrium solvation case further exhibits a discontinuity in θ, at θ = 90°, along the whole |AB〉 branch of S1(r > rEQCI).

This requires further explanation, now given.

To understand the origin of the discontinuity, we consider in Fig. 8 a projection of the three-dimensional G(r, θ, z) surfaces along z, at fixed r = r0CI = 1.59 Å and θ = 90° (panel a) vs.θ = 90° + ε (panel b).

While the free energy minimum for the uncoupled case (a) corresponds to the diabatic minimum at zABeq = 0, the adiabatic S1 minimum in (b) is located at .

The limiting value as θ approaches 90° is defined by the crossing of the diabatic states, and hence, the point (r = r0CI, z = 0.5) along the CI seam, i.e., .

Since this limiting value does not coincide with the equilibrium point zABeq = 0 at θ = 90°, a discontinuity arises.

A similar observation holds for all r values (within the range of existence of a conical intersection) such that the equilibrium points on the upper adiabatic surface tend towards the z values along the CI seam, Only at the EQCI point does the zseam value coincide with the true equilibrium value, zEQCI = zseam(r = 1.33 Å) = 1.

Fig. 7b illustrates the limiting behavior eqn. (30).

While the overall seam does not conform to an equilibrium condition, its role as a limiting value according to eqn. (30) nonetheless turns out to be crucial for the topology of the free energy surfaces.

This is illustrated in Fig. 9 for the equilibrium solvation free energies .

At θ = 90°, the free energies of both states approach the degeneracy at the CI seam, as a result of converging to zseam.

Since the S1 and S0 surfaces thus tend towards a line of degeneracies, the topology is markedly different from the surfaces of Fig. 1 and Fig. 5 which exhibit a single CI point.

Conditions of equilibrium solvation with respect to S1 are therefore expected to induce enhanced nonadiabatic transition probabilities to the S0 ground state as compared with either the isolated chromophore or frozen solvent conditions.

Conclusions and outlook

We have introduced a theoretical formulation to describe the environment's impact on a conical intersection (CI), which in this first effort has focussed on the environment's influence on the existence, location and character of the CI.

This formulation describes the interacting chromophore–environment system in situations where electrostatic interactions are dominant such that the environment's orientational polarization field interacts strongly with the chromophore.

The combined chromophore–environment system is characterized by free energy surfaces in the space of internal molecular coordinates plus one or several solvent coordinate(s).

A first sketch is given of applying this formulation to the coupled S1/S0 states in protonated Schiff bases.

The perspective we have developed is based on a free energy model including the solvent as an additional “tuning” mode which modulates the energy gap and displaces the CI.

The impact of a solvent can be substantial: For example, if the solvent is equilibrated to a solute state characterized by the charge distribution pertaining to one or the other diabatic state and then frozen, the conical intersection can be entirely lost, or its topology may be altered (here, from “peaked” to “sloped”).

For the opposite limit of extremely rapid solvent motion, equilibrium solvation free energies can be defined whose topology is determined by the CI seam: i.e., a line of intersections, rather than a single point in (r,θ) space.

Pronounced implications for the dynamics are thus expected for the different solvation limits.

In addition to these effects, the solvent can also have off-diagonal contributions, solvent-induced couplings, which have a symmetry-breaking effect at the geometries where the coupling intrinsic to the chromophore vanishes.

A detailed discussion of the PSBs and related systems in the context of the present model will require an ab initio based parameterization including all relevant modes.

In addition, the model should be further developed in view of the following aspects: (i) Extension of the free energy picture to molecular level interactions, beyond the dielectric continuum description (This leads to the definition of generalized, microscopic solvent coordinates31–34 and paves the way for the application of the present approach to complex environments), (ii) the development of general strategies to define an appropriate diabatic basis representation, chosen such as to exhibit a well-defined charge character, and (iii) a dynamical description including the solvent, leading to the image of a combined dynamical path for the chromophore–environment supermolecular system.

This necessitates the combination of the nonequilibrium dynamics in the solvent coordinate with the quantum dynamics of the chromophore.

Given the above elements, the present approach should contribute to clarifying various aspects of the environment's impact on the subpicosecond and picosecond dynamics in the PSBs.

This relates, in particular, to the pronounced differences in the dynamics observed in solution phase as compared with the native protein environment: For the retinal chromophores of rhodopsin and bacteriorhodopsin, and similarly for the chromophore of the photoactive yellow protein, the decay observed in solution is much slower (of the order of 10 ps)18,63,62 than the decay observed for the native protein (of the order of 200–500 fs).12–15,17

For both the solution phase and protein environments, an interesting comparison will arise from the comparison with QM/MM type approaches as developed, e.g., in ref. 10.