Ultrafast Formation of the Charge Transfer State of Prodan Reveals Unique Aspects of the Chromophore Environment

Abstract

Lipophilic dyes such as laurdan and prodan are widely used in membrane biology due to a strong bathochromic shift in emission that reports the structural parameters of the membrane such as area per molecule. Disentangling of the factors which control the spectral shift is complicated by the stabilization of a charge-transfer-like excitation of the dye in polar environments. Predicting the emission therefore requires modeling both the relaxation of the environment and the corresponding evolution of the excited state. Here, an approach is presented in which (i) the local environment is sampled by a classical molecular dynamics (MD) simulation of the dye and solvent, (ii) the electronically excited state of prodan upon light absorption is predicted by numerical quantum mechanics (QM), (iii) the iterative relaxation of the environment around the excited dye by MD coupled with the evolution of the excited state is performed, and (iv) the emission properties are predicted by QM. The QM steps are computed using the many-body Green’s function in the GW approximation and the Bethe–Salpeter equation with the environment modeled as fixed point charges, sampled in the MD simulation steps. The comparison to ultrafast time-resolved transient absorption measurements demonstrates that the iterative molecular mechanics (MM)/QM approach agrees quantitatively with both the polarity-dependent shift in emission and the time scale over which the charge transfer state is stabilized. Together the simulations and experimental measurements suggest that the evolution into the charge transfer state is slower in amphiphilic solvents.

Graphical Abstract

INTRODUCTION

Polarity sensitive dyes, 4-(2-(6-(dibutylamino)-2-naphthalenyl)ethenyl)-1-(3-sulfopropyl)-hydroxide (di-4-ANEPPS), and the naphthalene derivatives 2-dimethylamino-6-propionyl-naphthalene (prodan), 2-dimethylamino-6-lauroylnaphthalene (laurdan), and C-laurdan are widely used as reporters of membrane structure in both model and cell membranes.^1,2 All are lipophilic, so that they partition to the membrane and have spectral properties that depend on membrane structure, including lipid packing, membrane thickness, and chain ordering. The sensitivity of the Stokes shift of the prodan family is especially dramatic with a difference of 50 nm when comparing emission in a highly ordered and tightly packed gel-phase membrane to a less ordered fluid membrane.¹ This strong signal, together with tolerable toxicity, underlies an enormous literature using these dyes to study membrane structure over the past 40 years.^1–3

The pronounced Stokes shift of the prodan-derived dyes is due to a transition from an excitonic state to a charge transfer (CT) state that is extremely sensitive to the electronic environment.⁴ Applications to biomembranes have traditionally relied on a comparison of the relative fluorescence intensity at two different wavelengths (termed “generalized polarization” or GP), because the implementation is straightforward and is easily combined with confocal microscopy.^1,5 More recently, methods have been introduced that make use of the full fluorescence spectrum⁶ and of the fluorescence lifetime (phasor methods).⁷ There have been some attempts to elucidate the excited state dynamics employing ultrafast spectroscopic methods.⁸ The stabilization of the charge transfer state may include changes in both hydrogen bonding⁹ and local polarization¹⁰ due to solvent reorientation, complicating an unambiguous interpretation of the experimental results.

Prior numerical work on the spectral properties of prodan (Figure 1) and related dyes has in a few cases used wave function based methods,^11,12 but most studies have been performed using time-dependent density functional theory (TD-DFT).^11,13–18 While some attempts have been made to include the effects of one¹⁴ or many^16,17 solvent molecules explicitly, a majority of studies rely on implicit approaches such as polarizable continuum models (PCM).^{11–13,15,18} Although most of these studies give results that are in qualitative agreement with polarity-dependent spectroscopies, they fall short in two important respects given the goal of simulating spectra in heterogeneous membrane environments. First, conventional DFT functionals are not well-suited to CT states due to the lack of proper long-range interactions.¹⁹ This can be overcome when special system-tuned corrections, e.g., CAMB3LYP as in ref 18, are taken into account,²⁰ but particularly for the emission spectra, the tuning may also depend on the specifics of the environment before, during, and after the relaxation of the excited state. Second, even though a continuum treatment of the environment can be tuned to represent the effects of a molecular embedding, it will not capture the relaxation dynamics and coupled stabilization of the CT excited state in a complex (e.g., fluid membrane vs gel phase), dynamic, and atomistic environment. For a reliable prediction of the absorption and time-dependent emission spectra, it is therefore desirable to choose a simulation approach that combines an accurate quantum-mechanical method for the calculation of the excited states and an explicit, atomistic environment model without the necessity of tuning either of those.

Figure 1.

Chemical structure of 2-dimethylamino-6-propionylnaph-thalene (prodan).

In the following, ultrafast spectroscopy of prodan is combined with state-of-the-art molecular dynamics/quantum mechanics simulations in order to resolve the time evolution of the excited state, overcoming the limitations of previous work. Femtosecond transient absorption of prodan in different solvents is applied to measure the solvent-dependent subpicosecond formation of the CT state. The optical excitations of the dye are treated using many-body Green’s functions²¹ within the GW approximation²¹ and the Bethe—Salpeter equation²² (GW-BSE) which yields accurate descriptions of localized excitonic and CT excitations on an equal footing, without the need for system- and environment-specific adjustments.^23–25

An iterative molecular dynamics/GW-BSE scheme is used to alternately relax the solvent environment and evolve the excited state. Excellent agreement between the simulated and measured spectral dynamics provides confidence in the integrated experimental/numerical approach and promises a reliable interpretation of spectra in more complex samples. The femtosecond absorption measurements reveal a new feature of the excited state dynamics: There are two relaxation processes (picoseconds and tens of picoseconds) in hydrocarbon alcohol solvents like octanol and methanol, but there is only a single fast relaxation in solvents like hexane, acetone, and water. This feature appears to be polarity-independent.

METHODS

Theory.

Electronic excitations of a closed-shell system with a net spin of zero are calculated using GW-BSE. This approach builds upon a DFT³⁴ ground state calculation in which Kohn–Sham (KS) energy levels ${\epsilon }_i^{\text{KS}}$ and wave functions $|{\varphi }_i^{\text{KS}}\rangle$ are obtained as solutions to

$$[{\widehat{h}}_0+{\widehat{V}}_{\text{XC}}]|{\varphi }_i^{\text{KS}}\rangle ={\epsilon }_i^{\text{KS}}|{\varphi }_{\text{i}}^{\text{KS}}\rangle$$ (1)

where ${\widehat{h}}_0={\widehat{T}}_0+{\widehat{V}}_{\text{ext }}+{\widehat{V}}_{\text{H}}.{\widehat{T}}_0$ is the kinetic energy operator. ${\widehat{V}}_{\text{ext}}$ is the external potential. ${\widehat{V}}_{\text{H}}$ is the Hartree potential. ${\widehat{V}}_{\text{XC}}$ is the exchange-correlation potential.³⁵

Quasiparticle Excitations.

First, quasiparticle (QP) states representing independent single-particle (electron or hole) excitations are calculated within the GW approximation²¹ from

$$[{\widehat{h}}_0+\widehat{\mathrm{\Sigma }}({\mathrm{Є}}_i^{\text{QP}})]|{\varphi }_i^{\text{QP}}\rangle ={\epsilon }_i^{\text{QP}}|{\varphi }_i^{\text{QP}}\rangle$$ (2)

where the $|{\varphi }_i^{\text{QP}}\rangle$ are QP wave functions and the ${\epsilon }_i^{\text{QP}}$ are the excitation energies of quasielectron and quasihole states. In eq 2, the approximate exchange-correlation potential of KS-DFT as in eq 1 has been replaced by the self-energy $\widehat{\mathrm{\Sigma }}(E)$ which is a nonlocal, energy-dependent operator. Within the GW approximation it is calculated as

$$\widehat{\mathrm{\Sigma }}=iGW$$ (3)

which is a convolution of the single-particle Green’s function G and the screened Coulomb interaction $W={Є}^{-1}{v}_{\text{C}}$, where v_C = |r − r′|⁻¹ is the bare Coulomb interaction and Є(r, r′, ω) is the microscopic dielectric function calculated within the randomphase approximation. These quasielectron and quasihole states are typically expanded in terms of KS states as

$$|{\varphi }_i^{\text{QP}}\rangle =\sum _j{C}_{ij}|{\varphi }_j^{\text{KS}}\rangle$$ (4)

Assuming that the Kohn–Sham states approximate the GW quasiparticle states, i.e., $|{\varphi }_i^{\text{QP}}\rangle \approx |{\varphi }_i^{\text{KS}}\rangle$, the quasiparticle energies can be obtained perturbatively²⁵ according to

$$\begin{gathered} {\epsilon }_i^{\text{QP}}={\epsilon }_i^{\text{KS}}+\mathrm{\Delta }{\epsilon }_i^{\text{QP}} \\ ={\epsilon }_i^{\text{KS}}+\langle {\varphi }_1^{\text{KS}}|\widehat{\mathrm{\Sigma }}({\epsilon }_i^{\text{QP}})-{\widehat{V}}_{\text{XC}}|{\varphi }_i^{\text{KS}}\rangle \end{gathered}$$ (5)

In eq 5, both the correction terms $\mathrm{\Delta }{\epsilon }_i^{\text{QP}}$ and $\widehat{\mathrm{\Sigma }}$ (via G and W) depend on ${\epsilon }_i^{\text{QP}}$, requiring an iterative procedure until self-consistency in the eigenvalue ${\epsilon }_i^{\text{QP}}$ is reached.

Coupled Electron–Hole Excitations.

Quasiparticle excitations, as above, correspond to charged excitations of the system. Neutral excitations which involve a coupled electron–hole pair, e.g., after the absorption of a photon, are not included in this framework and need to be described using a two-particle wave function.^25,36 The S-th excitation, χ_S, can be written as the linear combination of quasiparticle product states as

$${\chi }_S\left({\mathbf{r}}_{\text{h}},{\mathbf{r}}_{\text{e}}\right)=A_{\mathrm{vc}}^S{\varphi }_v\left({\mathbf{r}}_{\text{h}}\right){\varphi }_c^{*}\left({\mathbf{r}}_{\text{e}}\right)+B_{\mathrm{vc}}^S{\varphi }_c\left({\mathbf{r}}_{\text{h}}\right){\varphi }_v^{*}\left({\mathbf{r}}_{\text{e}}\right)$$ (6)

where r_h and r_e are the hole and electron coordinates, respectively; v and c run over occupied and unoccupied single-particle states, respectively; and $A_{vc}^S$ and $B_{\mathrm{vc}}^S$ are the resonant and antiresonant electron–hole amplitudes, respectively. They and the associated excitation energy Ω_S are obtained by solving the Bethe–Salpeter equation, which can be cast in the form of a non-Hermitian eigenvalue problem

$${\widehat{H}}^{\text{BSE}}|{\chi }_S\rangle ={\mathrm{\Omega }}_S|{\chi }_S\rangle$$ (7)

or in matrix form

$$\left(\begin{array}{ll}{H}^{\text{res}}& K\\ -K& -H^{\text{res}}\end{array}\right)\left(\begin{array}{l}{A}^S\hfill \\ B^S\hfill \end{array}\right)={\mathrm{\Omega }}_S\left(\begin{array}{l}{A}^S\hfill \\ B^S\hfill \end{array}\right)$$ (8)

For spin-singlet excitations, as considered in this work, the diagonal and off-diagonal blocks of the Hamiltonian in eq 8 are determined in the noninteracting quasiparticle basis according to

$$\begin{gathered} H_{vc,v^{\prime }{c}^{\prime }}^{\text{res}}=D_{vc,v^{\prime }{c}^{\prime }}+2H_{\mathrm{vc},v^{\prime }{c}^{\prime }}^{\text{x}}+H_{vc,v^{\prime }{c}^{\prime }}^{\text{d}} \\ {K}_{cv,v^{\prime }{c}^{\prime }}=2H_{cv,v^{\prime }{c}^{\prime }}^{\text{x}}+H_{cv,v^{\prime }{c}^{\prime }}^{\text{d}} \end{gathered}$$ (9)

with

$$\begin{gathered} D_{\mathrm{vc},v^{\prime }{c}^{\prime }}=\left({\epsilon }_c-{\epsilon }_v\right){\delta }_{v,v^{\prime }}{\delta }_{c,c^{\prime }} \\ {H}_{\mathrm{vc},v^{\prime }{c}^{\prime }}^{\text{x}}=\int d^3{\mathbf{r}}_{\text{h}}{d}^3{\mathbf{r}}_{\text{e}}{\varphi }_c^{*}\left({\mathbf{r}}_{\text{e}}\right)\varphi \left({\mathbf{r}}_{\text{e}}\right)v_{\text{C}}\left({\mathbf{r}}_{\text{e}},{\mathbf{r}}_{\text{h}}\right){\varphi }_{c^{\prime }}\left({\mathbf{r}}_{\text{h}}\right){\varphi }_{v^{\prime }}^{*}\left({\mathbf{r}}_{\text{h}}\right) \\ {H}_{\mathrm{vc},v^{\prime }{c}^{\prime }}^d=-\int d^3{\mathbf{r}}_{\text{e}}{d}^3{\mathbf{r}}_{\text{h}}{\varphi }_c^{*}\left({\mathbf{r}}_{\text{e}}\right){\varphi }_{c^{\prime }}\left({\mathbf{r}}_{\text{e}}\right)W\left({\mathbf{r}}_{\text{e}},{\mathbf{r}}_{\text{h}},\omega =0\right){\varphi }_v\left({\mathbf{\text{r}}}_{\text{h}}\right){\varphi }_{v^{\prime }}^{*}\left({\mathbf{\text{r}}}_{\text{h}}\right) \end{gathered}$$

The contribution H^x is the repulsive exchange interaction originating from the unscreened interaction v_c, while the direct interaction H^d contains the attractive, but screened (W), interaction between electron and hole and causes the binding of the electron–hole pair. Furthermore, it is assumed here that the dynamic properties of W(ω) are negligible and the computationally less demanding static approximation ω = 0 is sufficient.

Electrostatic Embedding.

As discussed in the Introduction, the optical excitations of prodan are influenced by the structural details of the extended solvation shell. Emission involves coupled electronic and structural relaxation, which occurs on a time scale of picoseconds and involves more nuclei than can be treated quantum mechanically. For this reason, a mixed approach is used, in which the excited state of the dye is calculated within GW-BSE embedded in the electrostatic background of the solvent sampled by classical molecular dynamics. The coupling of the solvent and the solute in the evaluation of the excitations within GW-BSE is performed using a molecular mechanics (MM) representation of the electrostatic potential. Specifically, the solvent molecules in the MM region are represented by the static atomic point charge Q^a at position R_a where a indicates the atoms in the MM region. The classical electrostatic potential

$$V_{\text{MM}}(\mathbf{r})=\sum _{a\in \text{MM}}\frac{Q^a}{\mid \mathbf{r}-{\mathbf{R}}_a\mid }$$ (10)

from the MM region is added to ${\widehat{V}}_{\text{ext }}$ in ${\widehat{h}}_0$ in eqs 1 and 2, respectively.

Relaxation of the environment and the corresponding evolution of the excited state are handled by an iterative process described in the next section.

COMPUTATIONAL DETAILS

Classical MD.

Partial charges for the prodan ground state have been obtained following the standard procedure for the CHARMM family of force fields.^37,38 Initial guesses for partial charges were obtained from the CHARMM general force field (CGenFF)³⁷ via the Paramchem web server.^39,40 The initial charges did not yield a ground state dipole moment in agreement with available data, so partial charge optimization was performed. The geometry was optimized at the MP2/6–31G* level.⁴¹ Single-point energies for interactions with water were computed for TIP3P water molecules at several locations at the HF/6–31G* level. Water interaction energies are scaled by a factor of 1.16 to be relevant for the bulk phase.³⁷ The level of theory and the scaling of energy are done to maintain the compatibility with CHARMM’s additive force fields. Scaled HF/6–31G* model compound–water interaction energies and the computed dipole moment are used as target data, and the partial charges are adjusted until a best-fit among the QM and MM computed interaction energies and dipole moment is obtained. See the Supporting Information for more details, including the final values of the partial charges following optimization.

Six different systems were built with Packmol,⁴² each of which included a single prodan molecule in six different bulk solvents: 7200 TIP3P water, 2225 ethanol, 1770 acetone, 826 octanol, 3212 methanol, or 990 hexane. All MD simulations were run with GROMACS 5.1.1⁴³ using the CHARMM36 force field.^37,38 Steepest descent energy minimization was followed by 20 ns NVT (where N is the constant particle number, V is the volume, and T is the temperature; Nose–Hoover⁴⁴ for temperature coupling at 298 K), 30 ns NPT (where N is the costant particle number, P is the pressure, and T is the temperature) equilibration (Berendsen⁴⁵ for temperature and isotropic Parrinello–Rahman⁴⁶ pressure coupling for 298 K and 1 atm pressure, respectively), and then 300 ns production simulation (Nose–Hoover temperature coupling⁴⁴ and Parrinello–Rahman pressure coupling⁴⁶). Lennard-Jones⁴⁷ interactions were cut off using a switching function between 10 and 12 Å, and the particle mesh Ewald⁴⁸ method with a 12 Å cutoff radius was used for long-range electrostatics.

Excited State Calculation and Iterative Solvent Relaxation.

GW-BSE calculations for singlet excited states were run using VOTCA-XTP²⁵ which allows the interface to run Gaussian09 for the preceding ground state DFT calculation. All of the following results have been obtained based on DFT calculations using the PBE0 hybrid functional⁴⁹ and the cc-pVTZ basis set with its optimized auxiliary basis⁵⁰ to express two-electron integrals within the resolution-of-identity technique. KS orbitals (748) are used within the RPA, and product functions for the BSE are formed using 61 occupied and 687 unoccupied orbitals. The frequency dependence of the self-energy is evaluated using a generalized plasmon-pole model.⁵¹ Quasiparticle energies are determined self-consistently, as discussed above, until changes are smaller than 10⁻⁵ Hartree.

Special care needs to be taken to avoid spurious effects arising from the inconsistencies of the classical and quantum molecular potential energy surface when GW-BSE calculations are performed on configurations sampled by classical MD. Deviations in equilibrium bond lengths can change excitation energies by several tenths of an electronvolt. VOTCA-XTP, therefore, postprocesses the structures read from the MD trajectory and replaces the rigid parts of prodan, i.e., its naphthalene core, the carbonyl groups, and the methylene groups, with copies optimized on the PBE0/cc-pVTZ level for the DFT S₀ ground state (for absorption) and the GW-BSE S₁ excited state (for emission). For information about ground state and excited state geometries, see Table S1 in the Supporting Information.

The MM/GW-BSE absorption and emission calculations are performed as shown in the flowchart in Figure 2. First, 25 configurations are randomly sampled from the 300 ns classical NPT MD simulation of prodan in bulk solvents. Absorption energy from the S₀ → S₁ state is computed. Then, the initial calculation of the S₁ → S₀ emission energy is performed for the same 25 configurations. For each, the partial charges of the S₁ state are determined with the CHELPG method by constraining the partial charges of terminal hydrogens to 0.09e (where e is the elementary charge). Then, the partial charges of the S₁ state are used to continue classical MD for another 1 ps to relax the solvent environment in the presence of the changed dipole moment of the dye. The GW-BSE emission calculation is performed again in the new solvent background, and the procedure is repeated for 50 iterations.

Figure 2.

Prodan excitation and solvent relaxation workflow.

EXPERIMENTAL SECTION

Prodan was obtained from Invitrogen, and all solutions for the transient absorption (TA) measurements were saturated. Spectroscopic grade solvents were obtained from Fisher Chemicals and were dried over sodium sulfate before use.

Solutions of prodan (1.25 μM) were prepared from a stock solution in acetone. After allocation, the acetone was allowed to dry off in open air (about 30 s, aliquot was 50 μL), leaving prodan in the vial. The desired solvent was added to this sample and shaken well. This was done immediately before measurement for each solution. A scanning fluorometer (Fluoromax-4, Horiba) with excitation at 350 and 410 nm and a resolution of 5 nm was used. The spectra were recorded with a resolution of 1 nm. To ensure that the system was working properly and consistently, a water blank was measured before and after experimental measurements, and the emission peak at 397 nm was observed with consistent wavelength and intensity.

Femtosecond transient spectra were recorded using a setup previously discussed.⁵² Briefly, the system is based on a 10 kHz regenerative Ti:sapphire amplifier. White light was generated in a 1 mm sapphire window and compressed to a group velocity mismatch of less than 2 fs across the spectrum using chirped-mirrors (Laser Quantum) and fused silica wedges. The pump pulse at 340 nm (3.65 eV) was generated by the second harmonic generation from the output of a noncollinear parametric amplifier. The instrument response function of the setup was 25 fs.

Transient maps (Figure 3 and Figures S5 and S6) were analyzed by fitting spectra at a range of time steps to a Gaussian function. The peak maximum was plotted as a function of time. The shift of the maximum with time was fitted by one or two exponential functions to extract time-constants.

Figure 3.

Prodan TA-map in ethanol. The black line indicates the maximum of the stimulated emission peak.

RESULTS

Simulation Results: Prodan Emission and Dipolar Evolution in Six Different Solvents.

We begin with a short discussion about the composition of the excited states in vacuum calculations. As shown in Figure 4a, the S₁ absorption excitation as obtained from a GW-BSE calculation in the ground state geometry is formed mainly by a HOMO → LUMO transition (66%) with a smaller contribution (27%) coming from a HOMO → LUMO + 1 transition. After relaxation of the geometry following S₁ excitation for emission, as shown in Figure 4b, the quasiparticle HOMO (LUMO) level result is 0.16 eV (0.25 eV) higher (lower) in energy, and the emitting state is determined to be 84% by a LUMO → HOMO transition and is red-shifted by 0.4 eV. Note that even though the quasiparticle excitation energies are changed, the character of the involved orbitals remains practically the same compared to the calculation in the ground state geometry.

Figure 4.

Quasiparticle energy levels $({\epsilon }_i^{\text{QP}})$ and contributions of interlevel transitions to the electron–hole wave functions for (a) absorption and (b) emission in vacuum on optimized geometries, as well as on sample MD/GW-BSE relaxed geometries in (c) hexane and (d) water solvent. Isosurfaces of selected quasiparticle orbitals (isovalue ±10⁻², red/blue) and difference densities of the excited state (isovalue ±5 × 10⁻⁴ e, orange/green) are shown as insets.

Turning now to the combined MM/GW-BSE results, Figure 5 shows the evolution of the S₁ state dipole moment of prodan (a) and S₁ to S₀ emission energy (b) for the 50 steps of the iterative relaxation procedure in six different solvents of differing polarity. In every case, both the excited state dipole and the emission energy plateau after (at most) 20 ps, indicating convergence of the iterative MM/GW-BSE relaxation protocol. During the 1 ps MD simulation steps, the solvent nuclear degrees of freedom relax in response to the excited state partial charges. In the least polar solvent (hexane), the dipole moment of the dye in the S₁ state changes from 5.5 to 10.98 Debye. In more polar solvents, the dipole moment evolves to a larger value (24 Debye in the most polar environment): The excited state dipole polarizes the environment, which stabilizes the charge transfer state, polarizing the environment more. This process continues until the environment can no longer polarize. The predicted dipole moments for the CT state are within the range reported in the literature for the excited state (see Table 1), although the most polarizable environments yield dipole moments at the very upper end of this range. The dipole moments included in the table are obtained via different methods (solvatochromic shift, solvent perturbation, thermochromic shift, and microwave absorption) with the Onsager cavity radius approximated between 4.2 and 6.3 Å.

Figure 5.

(a) Prodan S₁ state dipole moment and (b) S₁ → S₀ emission energy in hexane (red), octanol (black), acetone (cyan), ethanol (magenta), methanol (green), and water (blue). Error bars and background shading indicate the standard deviation among 25 different solvent configurations at each time point.

Table 1.

Prodan Dipole Moments^a Reported in the Literature^26–33

reference	μG (D)	μE (D)
Balter et al.26	2.9	10.9
Catalan et al.27	4.7	11.7
Bunker et al.28	2.85	9.8
Kawski29	2.1/2.5	6.4/7.4
Kawski33	2.14/2.46	6.46/7.37
Kawski et al.32	2.45/2.80	6.65/7.6
Samanta and Fessenden30	5.2	10.2
Cintia et al.31	5.5	20.0

^aThe ground state and excited state dipole moment of prodan are μ_G and μ_E, respectively. They are measured in Debye (D).

The observable effect of the stabilization of the CT state is the shift to lower emission energies with increasing polarity of the solvent, shown in Figure 5b. After the solvent has fully relaxed and the data have plateaued, the S₁ to S₀ emission energy predicts the steady state emission spectra that are typically reported in the literature as a probe of the solvent environment of the chromophore. A comparison to reported values for the Stokes shift will be revisited in the Discussion.

In Figure 4c,d, we show sample quasiparticle spectra and the composition of the emitting S₁ state for two sample configurations taken from the final step of the MM/GW-BSE relaxation in hexane (least polar) and water (most polar), respectively. The presence of hexane in this particular geometry minimally lowers all quasiparticle energies compared to those of the vacuum emission case, and the energy and character of the S₁ state are hardly affected. In water, one can clearly discern more changes of the quasiparticle spectrum. In general, the level distances are increased, and most importantly, the energies of the empty states are reduced; e.g., the LUMO is lowered by 1.12 eV compared to that of the vacuum case.

Steady state absorption and emission spectra are predicted by broadening the S₀ to S₁ and S₁ to S₀ transitions assuming a Gaussian band shape with σ = 0.4 eV and then averaging the obtained energies for 25 different configurations. Figure 6 shows the absorption spectra, the emission spectrum immediately after absorption, and the emission spectrum following the 49 ps of iterative relaxation shown in Figure 5. As expected, the absorption spectra show almost no dependence on the solvent. Similarly, the emission immediately following excitation (and before the polar solvents can relax and stabilize the CT state) shows little dependence on the solvent environment. However, after the solvent is relaxed via the iterative series of MM/GW-BSE steps, the emission in polar solvents displays a strong bathochromic shift as the CT state is progressively stabilized.

Figure 6.

Normalized absorption (dashed lines), emission immediately after absorption (dashed–dotted lines), and emission after 49 ps of MM/GW-BSE relaxation (solid lines) spectra of prodan in different solvents.

Solvent Relaxation Dynamics: Comparison of Ultrafast Spectroscopy to MM/GW-BSE Approach.

Absorption and emission spectra of prodan were measured in two solvents of similar polarity (acetone and methanol) that differ in their ability to donate a hydrogen bond (as mentioned in the Introduction, prodan is a hydrogen bond acceptor (at the carbonyl), and this has an influence on spectral properties which is not captured in the present MM/GW-BSE protocol^53,54). Figure 7 compares the absorption and emission spectra in acetone (aprotic) and methanol (H-bond donor) to the experimentally measured steady state spectra. No shift or fit parameters are applied to the predicted spectra. Although the absolute positions of the lines are at a systematically higher energy in the MM/GW-BSE approach, the Stokes shifts are in nearly quantitative agreement, although the acetone calculation agrees slightly better [94.0 nm (simulation) vs 100.8 nm (experiment)] than the methanol calculation (117.6 nm calculated vs 133.9 nm measured). Note that hydrogen bonding is not explicitly accounted for in the ab initio calculation.

Figure 7.

Prodan absorption (left) and emission (right) spectra in acetone, ethanol, and methanol from simulation (solid lines) and experimental measurements (dashed lines).

Figure 8 compares the time-dependent relaxation of the environment obtained by the iterative MM/GW-BSE scheme to ultrafast measurements of the same. In both the experimental and simulated data, the relaxation dynamics in methanol is slower than in acetone. An exponential fit to the acetone data yields a single time scale of 2.63 ps (experiment) and 1.36 ps (simulation). The methanol data are better fit by a fast and a slow decay, with a time scale of 1.03 and 15.16 ps (experiment) and 1.74 and 45.31 ps (simulation). Fast and slow time scales are also observed in ethanol, with time scales of 0.18 and 13.44 ps (experiment) and 1.25 and 13.61 ps (simulation).

Figure 8.

Prodan time-resolved emission spectra (black) compared with simulation spectra in acetone (cyan), ethanol (magenta), and methanol (green).

The simulations confirm the observation of fast and slow relaxations in the alcohols but not in acetone. Fitting the relaxation of the excited state for the rest of the simulation data reveals an unexpected trend: Three very different solvents (hexane, acetone, and water) all display a single, fast time scale. In contrast, all of the hydrocarbon alcohols (octanol, ethanol, and methanol) show a fast (ps) and slow (tens of ps) relaxation. This suggests that the slow relaxation is not due to the polarization of the environment nor to the formation of hydrogen bonds. Instead, the only commonality among the solvents with a second, slower time scale is that they are all weakly amphiphilic. We speculate that the slower time scale is due to local rearrangement of the solvent to form a slightly more hydrophilic cavity for the excited chromophore. It will be interesting to see how these observations carry over to ultrafast measurements in lipid environments, which embed the chromophore in a more chemically and dielectrically diverse interface.

DISCUSSION

Prediction of the spectral properties of polarity sensitive dyes by a combined MM/GW-BSE approach yields excellent agreement with both steady state and time-resolved, ultrafast measurements. The iterative approach based on alternating MD relaxation of the solvent environment and GW-BSE prediction of the excited state leads to the progressive stabilization of a charge transfer excited state in more polar environments, resulting in a lower energy emission as polarity increases. Comparison of the Stokes shift obtained for the six solvents in Figures 5 and 6 to results presented here and in the published literature⁵³ yields an excellent, linear correlation with a slope of 1 (Figure 9), without any adjustment of the simulated data (Table 2).

Figure 9.

Comparison of Stokes shift obtained in simulation with experimental measurements from the literature⁵³ (●) and Figure 7 (■).

Table 2.

Stokes Shift of Prodan Spectra from MM/GW-BSE Calculated Absorption and Emission Energies Compared to Experimental Measurements from the Literature⁵³ and Figure 7

solvent	$E_{\text{MM}/GW-\text{BSE}}^{\text{absorption}}(\text{eV})$	$E_{\text{MM}/GW-\text{BSE}}^{\text{emission}}(\text{eV})$	ΔEMM/GW-BSE (eV)	ΔEexp (eV)
hexane	3.67	3.26	0.41	0.44
acetone	3.76	2.88	0.88	0.74
octanol	3.79	3.03	0.76	0.79
ethanol	3.75	2.77	0.98	0.91
methanol	3.71	2.75	0.96	0.93
water	3.75	2.57	1.18	1.10

Ultrafast spectroscopy directly reveals the evolution of the CT state in polar environments (both protic and aprotic), providing evidence for a fast (ca. 1 ps) and slow (ca. 10–20 ps) process. Note that both of these processes are too fast to be observed by previous applications of other time-resolved spectroscopies to prodan and related dyes.^55–58 Remarkably, the iterative MM/GW-BSE approach also agrees quantitatively with the time scale of relaxation in acetone, ethanol, and methanol when the duration of the MD steps is appropriately chosen (Table 3). A longer solvent relaxation step arbitrarily increases the simulated relaxation time scale, and a shorter step does not change the observed time scales (see Figures S4 and S5).

Table 3.

Prodan Excited State Relaxation Times Obtained from Single and Double Exponential Fits^a

solvent	τMM/GW-BSE (ps)	τexp (ps)
hexane	0.41 ± 0.19	N/A
acetone	1.36 ± 0.11	2.63 ± 1.13
water	1.74 ± 0.11	N/A
octanol	0.89 ± 0.1	N/A
	29.13 ± 15.44
ethanol	1.25 ± 0.28	0.18 ± 0.04
	13.61 ± 3.96	13.44 ± 1.66
methanol	1.74 ± 0.27	1.03 ± 0.11
	45.31 ± 54.33	15.16 ± 0.6

^aFor details about the fitting functions, see Table S2. τ_MM/GW-BSE and τ_exp are relaxation times from simulation and from experiment, respectively.

The iterative MM/GW-BSE approach presented here is designed specifically for the quantitative prediction of the spectral properties of polarity sensitive dyes that are commonly used to study biomembrane structure and dynamics. The MD portion of the calculation (if the model is well-parametrized) accounts for the heterogeneity of the environment, overcoming the limitations of continuum treatments that are more commonly used. Together, these two components yield a very highly accurate, quantitative method with the potential to reveal the mechanism of other more recently developed, less understood chromophores. Integrating the calculations with the ultrafast measurements of the excited state dynamics will provide valuable information on the nature of the emitting state for these other dyes.