Simulating the Fluorescence of the Locally Excited state of DMABN in Solvents of Different Polarity

Abstract

DMABN (4-(N,N-dimethylamino) benzonitrile) is a luminescent probe that can be used for tracking changes in the surrounding solvent due to the large change in polarity between its ground and excited states. An important characteristic of DMABN is that it exhibits dual fluorescence with two different emission energies that can be monitored, allowing for better characterization of the surrounding system. The first excited state is called the locally excited (LE) state and is characterized by a movement of charge over the conjugated ring structure. In nonpolar solvents and in the gas phase, the fluorescence of DMABN is entirely attributed to the transition from the near-planar LE state. In more polar environments, emission occurs from both the LE and a second excited state corresponding to a twisted intramolecular charge-transfer (ICT) structure. For the sake of simplicity, this work only considers transitions between the ground and the LE state. Molecular mechanical force field models of DMABN in its ground and LE state have been developed to investigate the sensitivity of the LE state to the polarity of solvent. Both nonpolarizable and polarizable force fields were developed to simulate the molecule in a series of ten different solvents of different polarity. The calculated Stokes shift of DMABN increases with the increasing orientation polarizability of the surrounding solvent, which is the expected trend as seen in experimental studies.

Graphical Abstract

Introduction

The small molecule 4-(N,N-dimethylamino)benzonitrile, or DMABN, is a valuable luminescent probe for tracking changes in the surrounding solvent due to the large change in polarity between its ground and excited states. The molecule has an overall fairly rigid conjugated section with two nitrogen on either end of this active site, while the amino group has different optimized twist angles for the ground and excited states (Figure 1). Applications of DMABN as a spectroscopic probe include determining the critical micelle concentration of a surfactant,¹ detecting dissolved organic matter concentrations based on the variance of the phototransformation rate constant of DMABN,² and investigating solvation structure and dynamics of ionic liquids like imidazolium with the time-resolved infrared spectroscopy of DMABN.³ Because its fluorescence is affected by its restricted geometry in a polymer matrix, DMABN is also a useful probe of the molecular weight of polymers.^4,5

Figure 1:

The structure of DMABN (4-(N,N-dimethylamino)benzonitrile)

DMABN exhibits the rare phenomenon of dual fluorescence, showing one “normal” band that is rather insensitive to solvent polarity and is attributable to a near-planar locally excited (LE) structure of DMABN, and one “anomalous” band that is assigned to a twisted intramolecular charge transfer (ICT) structure and appears in solvents of sufficiently high polarity. The two different emission energies produce overlapping peaks of varying intensity depending on the polarity of the environment,^6–8 allowing for better characterization of the surrounding system. In nonpolar solvents, the excitation and emission are determined by the LE state. In more polar solvents, a second peak corresponding to the ICT state appears and partially overlaps with the LE emission spectra. As the polarity of the surroundings is increased, the peak of the ICT state becomes more prominent until it dominates the fluorescence spectra.

Experimentally it can be difficult to differentiate the fluorescence of the two excited states, especially in highly polar solvents, but theoretically the contribution of the LE state can be easily isolated. Many studies have been performed to investigate the specific mechanism of the fluorescence cycle, especially in regards to the interconversion between excited states.^9–11 The ICT state is commonly associated with a significant change in the angle of the amino group, referred to as the “twisted internal charge transfer” (TICT) state.¹² In addition to the amino twisting along its bond with respect to the rest of the molecule, there are also studies that investigate a wagging motion of the amino up or down out of the plane of the conjugated ring.¹³ QM/MM simulation studies of DMABN in water and acetonitrile support the notion that the TICT structure is reached only by conversion from the LE state as the amino group is twisted, which allows for a breakdown of spectral contributions between the two states based on geometry.^14,15 Furthermore, it has even been proposed that an intermediate state exists between the LE and ICT states, where an electron moves from the phenyl orbital to the nitrile group, based on TD-DFT calculations that did not show an expected energy transition.¹⁶ However, CASPT2 calculations from a QM/MM study still predict the expected absorption bands for the LE state without any additional states considered.¹⁷ These changes in geometry are certainly favored by the ICT state, but can still occur while the molecule is in the LE state, likely facilitating interconversion. The parameterization of the LE state should involve particular consideration of this geometrical feature.

Due to it primarily being the result of local excitation of the benzyl ring, the LE state has been depicted as being much more similar to the ground state. It is either perfectly planar or with a less extreme twist of the amino substituent, having a low oscillator strength especially in comparison to the ICT state. Several experimental studies support the planar LE state by IR spectroscopy¹⁸ and rotational contour analysis,¹⁹ as well as semi-empirical²⁰ and CASSCF calculations.¹³ Though there does exist contrasting evidence of the ideal configuration actually being with a twist of the amino group with respect to the conjugated structure of 22° or more.²¹ A TDDFT study yielded a pretty flat energy profile of this angle with a slight minimum at 32.5°.²² Additional steps were taken to find the optimal twist of the ground and LE state of DMABN to be reflected in the produced parameter sets, but largely the geometry optimization remained the same during a transition.

DMABN has been studied experimentally in a variety of different solvents, often displaying their results as overlapping spectra to show the shift in the maximum intensity of its absorption and emission. While there are some discrepancies between the various studies, it is possible to draw general trends. There are exceptions where the general positive trend matching increasing dielectric with increasing wavelengths is not followed by a given solvent. Neubaur et al.²³ studied DMABN dissolved in hexane, diethylether and acetonitrile (which is a dielectric range of 1.9 to 35.7) showing an increase in the wavelengths of both the absorption and emission from the LE state as the dielectric increases over this range. Atsbeha et al.²⁴ measured the absorption and emission for the LE state in cyclohexane, dioxane, dichloromethane and acetonitrile, which covers approximately the same dielectric range. The absorption maximum wavelength mostly follows the same trend, but dichloromethane has the largest wavelength despite not having the highest dielectric. The LE emission data from this study starts with the lowest wavelength matching with DMABN in the lowest dielectric but the rest of the values essentially plateau instead of steadily increase. The data reported by Haidekker et al.²⁵ includes emission wavelengths in benzene, ethylene glycol, glycerol and dimethylsulfoxide, which generally increase with dielectric, but dimethylsulfoxide was found to yield a smaller LE emission wavelength than glycerol.²⁵ In summary, the general trends appear to be increases in wavelength with increasing dielectrics for both the absorption and emission, though there is definitely some ambiguity as to exactly how a specific solvent will behave and whether or not it consistently follows that trend.

In large part, the impact of induced polarization on fluorescence is interpreted within the framework of the Lippert-Mataga theory.^26,27 The theory assumes that the fluorescing molecule, occupying a spherical cavity of radius $a$ in a continuum dielectric solvent, is making transitions between a ground state with dipole ${\mu }_{\mathrm{g}\mathrm{s}}$ and and an excited state with dipole ${\mu }_{\mathrm{e}\mathrm{s}}$, and relates the Stokes shift (the difference between the absorption and emission wavelengths) of a fluorescing molecule, $\mathrm{\Delta }\tilde{\nu }$, to properties of the surrounding solvent modeled as a continuum dielectric,

$$\mathrm{\Delta }\tilde{\nu }=\mathrm{\Delta }{\tilde{\nu }}^0+\frac{2}{hc{a}^3}\mathrm{\Delta }f{\left({\mu }_{\mathrm{e}\mathrm{s}}-{\mu }_{\mathrm{g}\mathrm{s}}\right)}^2$$ (1)

where $\mathrm{\Delta }{\tilde{\nu }}^0$ is the Stokes shift of the molecule when in vacuum, and

$$\mathrm{\Delta }f=\left(\frac{\epsilon -1}{2\epsilon +1}-\frac{{\epsilon }_{\mathrm{\infty }}-1}{2{\epsilon }_{\mathrm{\infty }}+1}\right)$$ (2)

is called the orientation polarizability. This term represents the part of the surrounding dielectric that is associated with the reorientation of the dipole of the solvent molecules. It is expressed in terms of the static dielectric constants of a given solvent $\epsilon$, and the high-frequency contribution ${\epsilon }_{\mathrm{\infty }}$ associated with the nearly instantaneous redistribution of electrons. While the QM description of the electronic states of DMABN has often been the focus of previous computational studies, the interactions with the solvent have typically been represented on the basis of simple continuum models. For instance, Eq.(1) treats the solute molecule as a spherical cavity of radius $a$ inside the continuum solvent. Even in the context of an implicit solvent representation, it is understood that more sophisticated considerations are required to account for both the the granularity of the solvent molecules and the irregular shape of the solute-solvent boundary.^28,29 Ultimately, to go beyond such approximations, it is necessary to use detailed atomic models of the solvent molecules.

The goal of the present work is to investigate the effects of the surrounding solvent on the electronic transitions of DMABN. For the sake of simplicity, the present treatment considers only transitions between the ground state and the LE state. Transitions to the ICT state are not included, implying that the scope of the work remains somewhat incomplete from the viewpoint of understanding all solvent effects on the fluorescence of DMABN. Particular attention is given to the orientation polarizability as defined by Eq.(2). The present study, aimed at characterizing a broad range of solvent of different polarity, relies primarily on molecular mechanical force fields and classical molecular dynamics simulations. In that sense, the goal and approach differs from that of previous studies of DMABN in water and acetonitrile based on sophisticated QM/MM semiclassical surface hopping simulations.^14,15 For nonpolarizable additive force fields, the specific contribution from electronic polarization is accounted for in an average way and ${\epsilon }_{\mathrm{\infty }}$ is equal to 1, causing the right-side term to disappear entirely. For a polarizable force field, the value of ${\epsilon }_{\mathrm{\infty }}$ is not equal to 1. To address this issue, molecular dynamics (MD) simulations are generated for the ground and excited state using nonpolarizable and polarizable molecular mechanical force fields, and the absorption and emission energies are calculated.

Methods

Solvent Models

Ten pure solvent systems were considered: benzene, diethylether, 1,1,1-trichloroethane, 1,1-dichloroethane, acetone, ethanol, methanol, acetonitrile, dimethylsulfoxide and water. The density, heat of vaporization and dielectric constants for both the nonpolarizable and polarizable models of the solvent molecules were determined and compared to the corresponding experimental values. Experimental data for the heat of vaporization and density of neat liquids was taken from the National Institute For Standards and Technology (NIST) Chemistry WebBook (https://webbook.nist.gov/chemistry). Experimental data for the static dielectric constant and index of refraction of neat liquids was taken from SpringerMaterials Interactive (https://materials.springer.com/). Density of a given solvent is determined from,

$$\rho =\frac{m{N}_{\mathrm{m}\mathrm{o}\mathrm{l}}}{N_A⟨V⟩}$$ (3)

where $N_{\text{mol}}$ is number of molecules, $m$ is the molecular mass, $N_A$ is Avogadro’s number, and $⟨V⟩$ is average volume. The heat of vaporization is determined from,

$$\mathrm{\Delta }{H}_{\text{vap}}=-\frac{⟨U_{\text{liq}}⟩+p⟨V_{\mathrm{l}\text{iq}}⟩}{N_{\text{mol}}}+⟨U_{\text{gas}}⟩+RT$$ (4)

where $⟨U_{\text{liq}}⟩$ is the potential energy ensemble average of a simulated box of solvent, $p$ and $⟨V_{\text{liq}}⟩$ refer to the constant pressure and ensemble averaged volume of that liquid simulation, $N_{\text{mol}}$ is the number of molecules in that liquid simulation, $⟨U_{\mathrm{g}\mathrm{a}\mathrm{s}}⟩$ is the potential energy of a single molecule in vacuum, $R$ is the gas constant and $T$ is the constant temperature that both the liquid and gas simulations were carried out under. The contribution of the $p⟨V_{\text{liq}}⟩$ term is negligible, meaning that an accurate calculation of the heat of vaporization can be found with the remaining terms after its removal. The dielectric constant is determined from,^30–32

$$\epsilon ={\epsilon }_{\mathrm{\infty }}+\frac{⟨{\mathbf{M}}^2⟩}{3{\epsilon }_0{k}_{\mathrm{B}}T⟨V⟩}$$ (5)

where $\mathbf{M}={\sum }_i{\mathit{\mu }}_i$ is the sum of all the molecular dipole moments of the solvent molecule. The optical or dynamic dielectric constant, ${\epsilon }_{\mathrm{\infty }}$, is an induced polarization term equivalent to the square of a solvent’s refractive index. The refractive index arises from electron polarizability and is thus a high-frequency term accounting for nearly instantaneous change in the induced dipoles. In a nonpolarizable simulation, there is no electronic polarization occurring when the nuclei do not move, so this term is equal to 1. In the polarizable models, the instantaneous electronic polarization ${\epsilon }_{\mathrm{\infty }}$ can be determined by again using Eq.(5) to analyze a simulation where the nuclei are held fixed and only the Drude particles are able to move.³³

Each solvent was represented as a box built with Packmol³⁴ to be 40 × 40 × 40 ų, where the number of solvent molecules was determined by its experimental density at 298 K. The solvent boxes were energy-minimized, equilibrated at constant volume for 5 ns, and equilibrated for an additional 5 ns under constant pressure with NAMD.³⁵ The average volume of the box during the constant pressure portion of the simulation was used to compute the density of the solvent using Eq.(3). The heat of vaporization was calculated from the average potential energy of the box of pure liquid and the average potential energy of a single molecule of the given solvent in vacuum using Eq.(4). The single molecule simulation was run for 1 ns at 298 K, the same temperature of the box simulation. The potential energy values were extracted directly from the NAMD output for all systems. The static dielectric properties of the pure solvents were characterized from the liquid box simulations using Eq.(5). The CHARMM program³⁶ was used to calculate the quadratic fluctuations of the average total dipole from each frame from the simulation, separated into its $xyz$ components. To determine the dynamic dielectric constant ${\epsilon }_{\mathrm{\infty }}$, ten independent simulations starting from random configurations of the initial equilibration were simulated for 10 ps each at a temperature of 10 K with fixed nuclei (only the Drude particles of the system were allowed to move freely). Finally, the dynamic dielectric constant ${\epsilon }_{\mathrm{\infty }}$ was added to the contribution from the dipole fluctuations associated with the term $⟨{\mathbf{M}}^2⟩$ following Eq.(5) to obtain the total static dielectric constant for the polarizable solvent models.

Most of the force field parameters used to characterize each solvent, both nonpolarizable and polarizable Drude, were taken from the Toppar files used in CHARMM-GUI.³⁷ However the polarizable Drude versions of acetonitrile and dimethylsulfoxide were not available in that list of parameters. The parameter optimization was accomplished by starting with an estimation from GAAMP³⁸ followed by an optimized fitting to their respective Lennard-Jones parameters, heat of vaporization and density.^39,40

Parameterization of DMABN

The parameterization of DMABN was accomplished using an expanded version of GAAMP (General Automated Atomic Model Parameterization).³⁸ This program is designed to improve upon an initial guess from CGenFF (CHARMM General Force Field)⁴¹ or GAFF (general Amber force field)⁴² for small molecules with additional calculations of the electrostatic potential of the overall molecule, and optimization of the more mobile dihedral angles and potential hydrogen bond donors and acceptors. GAAMP was used to parameterize the ground state force field for DMABN, using its default settings and fitting its initial values to the results of a CGenFF guess of its partial charges. Each step optimizing its geometry were performed with HF/6–31G*. The electrostatic potential calculations for the polarizable model used B3LYP/aug-cc-pVDZ.

The geometry of DMABN in its excited states is still somewhat contentious, almost entirely surrounding the position of the amino group (specifically to what degree it is twisted and whether or not it is wagging out of the plane of the aromatic ring). The ground state is generally accepted to be consistently flat in the plane. And while there exists evidence that supports the LE state as being flat as well,^13,18–20 it has also been reported to exhibit a slight angle of its amino group.^21,22 In contrast, the ICT state is commonly associated with a much greater twist of the amino group. Additionally, a consensus on the exact interconversion mechanism between the two excited states has not been reached. The prevailing model is based on femtosecond stimulated Raman spectroscopy studies that elucidated an initial relaxation of the LE state in about a third of a picosecond, followed by a rapid internal conversion between the LE and ICT state occurring within 2 picoseconds, and finally vibrational relaxation of the ICT state that occurs over the course of 6 picoseconds involving twisting of the amino group and other geometry changes.^43,44 However the scope of this paper is to investigate solely the LE state and observe its sensitivity to the polarity of a solvent environment.

The program ORCA^45,46 was used to perform CASSCF calculations and isolate the excited state of DMABN. For DMABN there are 12 electrons and 11 orbitals in its complete active space, however it has been shown in computational studies that a CASSCF(6,7) calculation (in which 6 electrons and 7 orbitals are considered in the active space) is sufficient.⁴⁷ This reduced active space size was used to save computation time. Additionally, the orbital set reached by the CASSCF(6,7) study reported by Xu et al.⁴⁷ was used to aid in the process of recognizing and rotating in the necessary orbitals to the active space. The final selection of orbitals in the active space for the excitation of DMABN are shown in Figure 2, achieved with a def2-SVP def2-SVP/C calculation with the convergence criteria of orbstep SuperCI and switchstep DIIS included. These seven orbitals share six electrons, where each orbital is considered to have partial occupancy rather than be fully occupied with two electrons or vacant with zero. The first three orbitals are practically full, with occupancies approaching two while the other four have occupancies close to zero. The ground state would essentially have either full or empty orbital occupancy. The transition from ground to the first excited state can be observed as a decrease of occupancy in the bonding orbitals and an increase in occupancy of the antibonding orbitals. This set of orbitals, representing the excited state, shows this shift most prominently with its 1.34 occupancy in the third orbital and the 0.68 in the fourth orbital. Once the set of active orbitals was settled upon, matching the set described by Xu et al.,⁴⁷ the electrostatic potential from the CASSCF ab initio calculations for a large number of positions surrounding the excited molecule was then determined. This was then used as the target data in the ESP charge fitting protocol utilized in GAAMP together with additional fitting with test water molecules placed near potential hydrogen bond donors and acceptors.

Figure 2:

Active orbitals of the excitation of DMABN - converged to through CASSCF calculations. The biggest change in occupancy during the transition from ground to LE state occurs in the third and fourth orbitals: the third orbital is a bonding orbital that drops from 2 to 1.34 electron occupancy while the fourth is an antibonding orbital that increases from 0 to 0.68 electron occupancy

The only structural feature of DMABN expected to significantly shift due to the change in electronic state is the twist angle of the amino group.^12,48 A dihedral scan was performed with ORCA for every ten-degree rotation of the amino substituent with respect to the conjugated ring. This revealed the optimum degree of twist for the two states of interest which was then directly compared with short simulations, performed with NAMD,³⁵ of the molecule locked in those configurations. For the ground state the optimal angle did indeed center around zero, favoring a conformation where the amino group doesn’t twist more than about 10° and strongly resisting a twist of 50° or more. The excited state on the other hand showed a local minima around zero, with wells at around 10°, as well as another minimum at closer to 50°. These dihedral scans are included in Supplemental Figure S1 for the ground state and Supplemental Figure S2 for the excited state. Some manual manipulation of the dihedral parameters in both the nonpolarizable and polarizable force fields was performed to better reflect these QM dihedral scans.

The dipole of the ground and the first excited singlet found by Xu et al.,⁴⁷ calculated with CASSCF (6, 7) at with a 6–311G* basis set, is 6.41 and 6.25 Debye respectively.⁴⁷ Another study using CASPT2 calculated dipoles of 7.36 Debye for the ground state and 7.58 Debye for the LE state.¹³ Both of these studies found the difference in dipole to only be about 0.2 Debye. The result of this study’s CASSCF step also found a difference in dipole between the two states of about 0.2 Debye: 6.815 and 6.665 Debye for the ground and the LE state. After creating the parameter files the dipole of the isolated DMABN molecule in each snapshot of the molecule simulated in solvent was determined as well. The dipole for the nonpolarizable model of the ground state is 7.53 Debye, and 8.81 Debye for the polarizable model. The excited state was built based on the CASSCF data and thus reflects the similar but slightly smaller LE state dipole. The dipole for the excited state nonpolarizable model was 6.78 Debye and 6.04 Debye for the polarizable model of the excited state. Despite the similar dipoles found via CASSCF, the work of Jamorski et al. show that the values of the dipoles heavily depend on the method used to calculate it.⁴⁹ There is also experimental uncertainty. A range of 5–7 Debye for the ground state and 6–11 Debye for the LE state is generally accepted.^47,49 The dipoles of the models used in this study are based on the CASSCF calculations, and produce ground and excited dipoles at similar values to those seen in other studies using that method. All nonpolarizable and Drude polarizable force fields for DMABN in its ground and LE excited states parameter files are given in https://github.com/RouxLab/Fluorescence-of-DMABN.

Simulations of DMABN in Solvent

The program Packmol³⁴ was used for generating the initial starting positions of DMABN in each solvent 40 Å cube. The program NAMD³⁵ was used to generate 5 ns trajectories at 298 K under constant volume of each of the ten DMABN-solvent system using the nonpolarizable and polarizable force fields optimized for the ground state of DMABN. Each of these runs was then repeated with the same settings, except now using the nonpolarizable and polarizable force fields optimized for LE state of DMABN. The absorption and emission steps are entirely to and from the LE state, removing the complication added by the possibility of transforming into the ICT state and fluorescing from that instead. In total, twenty independent nonpolarizable simulations and twenty independent polarizable simulations were generated.

According to the Franck-Condon principle, which separates electronic movements from any nuclear motion, the instantaneous electronic transition can be calculated for any set of nuclear coordinates from a snapshot of simulation. The conversion between the ground and the LE excited state are performed by switching between force field parameter files, simulating the instantaneous electronic transition, and then the subsequent relaxation is on the new potential energy surface. The force field representation of the surrounding solvent is not changed, regardless of the ground or excited state of DMABN.

The average absorption and emission energy was calculated as

$$\mathrm{\Delta }{E}_{\mathrm{a}\mathrm{b}}={⟨\left(E_{\mathrm{e}\mathrm{s}}-E_{\mathrm{g}\mathrm{s}}\right)⟩}_{\left(\mathrm{g}\mathrm{s}\right)}$$ (6)

for the absorption process, and

$$\mathrm{\Delta }{E}_{\mathrm{e}\mathrm{m}}={⟨\left(E_{\mathrm{e}\mathrm{s}}-E_{\mathrm{g}\mathrm{s}}\right)⟩}_{\left(\mathrm{e}\mathrm{s}\right)}$$ (7)

for the emission process. In Eqs.(6) and (7), $E_{\mathrm{g}\mathrm{s}}$ and $E_{\text{es}}$ represent the total potential energy of the system with DMABN in its ground or excited state, respectively, and the subscripts of the brackets indicate an equilibrium average in the corresponding state. The energy differences are always $E_{\text{es}}$ minus $E_{\mathrm{g}\mathrm{s}}$, the only difference being whether the configuration is taken from a simulation of the ground or excited state. For each process, the peak absorption or emission wavelengths are given as, $\lambda =hc/\mathrm{\Delta }E$. For the purpose of analysis, the energy contribution from the interaction between DMABN and the surrounding solvent was also determined by subtracting the energy gap of the isolated molecule for each configuration. In the case of the nonadditive polarizable model, it is important to relax the Drude particles to their energy minimum positions for each configuration in order to recovers the correct SCF limit. The energy differences of a given configuration from a trajectory based on the polarizable Drude force field are between the excited state energy after minimizing the position of the Drude particles with the force field parameters for the excited state, and the ground state energy after minimizing the position of the Drude particles with the force field parameters for the ground state.

The energies extracted in the present study are vertical transitions, based on the Franck-Condon principle^50–52 that allows for the separation of the electronic transition from any movement of the nuclei due to a difference in timescale. However, in the context of a molecular mechanical force field, the total energy of the molecule in a given electronic state is not meaningful. For instance, the bonds and angles terms are accounted for only with quadratic function with zero energy at the equilibrium values, implying that the total energy is essentially arbitrary. To match the total energy from QM, it is necessary to offset the total molecular mechanical energy of the molecule. In principle, a single offset constant would be sufficient to allow the determination of the energy difference between the ground and excited states, $\mathrm{\Delta }E$. However, the observed optical spectrum of a molecule is also affected by a number of subtle factors, including non-radiative QM relaxation associated with the coupling between the electronic transition and the nuclear motions.^53–56 Differences in the molecular geometry at the energy minimum as well as the curvature of the potential energy surface between the ground and excited states give rise to an internal reorganization energy, yielding a difference between the absorption and emission frequency that is expected to be on the order of about 2,000 cm⁻¹. For the sake of simplicity, these effects are subsumed into environment-independent empirical offset constants for the excitation and emission in the present work.^57,58 Thus, the present study is built on the assumption that there is one solvent-independent offset constant for matching all the absorption data in different solvents, and a different solvent-independent offset constant for matching all the emission data in the different solvents. Because of the QM factors affecting both the absorption and emission processes, two empirical energy offset constants expected to differ by about 5 kcal/mol are needed to model a given fluorescent molecule. Accounting for all QM effects is certainly essential for a complete and accurate simulation of optical spectra, but since the focus of this work is a characterization of the influence of the molecular environment on the excitation and emission wavelengths, the present simplification is justified.

Offset energy constants for the absorption and emission processes were estimated empirically by comparing with experimental data for the nonpolarizable and polarizable force fields. All the experimental data used this study is included in Supplemental Table S3. The exact values of these offsets are determined by fitting the energies directly with the available experimental data for the solvents it has in common with those considered in this study. The average energy gaps corrected with the added offset constant are then converted into the absorption wavelengths, emission wavelengths and Stokes Shift (the difference between the emission and absorption wavelengths). The exact value used for the offset is obtained by fitting the absorption and emission data for benzene, diethylether, ethanol, acetonitrile, dimethylsulfoxide and water from experimental studies.^{7,23–25,59–61} For the nonpolarizable model, the optimal offsets are 102.569 kcal/mol for the absorption and 85.022 kcal/mol for the emission. For the polarizable model, the optimal offsets are 84.319 kcal/mol for the absorption and 68.547 kcal/mol for the emission. the coupling between the ground and LE state of DMABN with the solvent is accurately represented.

Results and Discussion

Pure Solvents

The calculated densities for each solvent considered for the nonpolarizable and polarizable models are reported in Supplemental Table S1 where they are compared to experimental sources. Looking first at the nonpolarizable representations of the ten solvents, the predicted density averaged around 3.75% error as compared to experiment, where the closest prediction was for the density of water which had an error of less than one percent. The polarizable model was very similar in these regards, being the most accurate for its approximation of water’s density and having a smaller average percent error of around 1.7%. The heat of vaporization for each solvent, calculated for the nonpolarizable and polarizable models, are displayed in Supplemental Table S2. When considering all ten solvents, the heats of vaporization calculated for the nonpolarizable model overall has an average percent error of about 6.3%, though it is particularly accurate for benzene and water. The polarizable model was also found to be particularly close with its estimation of water’s heat of vaporization, and when considering all ten solvents an average percent error of 6.8% was found.

The static dielectric constants calculated for each solvent with both the nonpolarizable and polarizable force fields are generally in-line with experimental values, as shown in the first section of Table 1 and Figure 3.

Table 1:

Static dielectric ($\epsilon$), dynamic dielectric (${\epsilon }_{\infty }$) and orientation polarizability ($\Delta f$) of pure solvent for the nonpolarizable and polarizable force fields as compared to experimental values - note that no nonpolarizable dynamic dielectric constant was determined. The solvents are listed in order of increasing experimental dielectric constant.

	Static Dielectric			Dynamic Dielectric		Orientation Polarizability
Solvent	Exp	Nonpol	Drude	Exp	Drude	Exp	Nonpol	Drude
Benzene	2.271	1.026	2.228	2.253	2.165	0.002	0.009	0.007
Diethylether	4.240	4.719	4.870	1.822	1.958	0.165	0.356	0.166
Trichloroethane	7.083	5.221	5.491	2.064	1.400	0.194	0.369	0.270
Dichloroethane	10.125	7.568	8.565	2.007	1.694	0.229	0.407	0.259
Acetone	20.493	20.048	16.947	1.846	2.061	0.284	0.463	0.250
Ethanol	24.852	15.300	22.252	1.850	1.515	0.290	0.453	0.339
Methanol	32.613	25.001	27.788	2.097	2.068	0.266	0.471	0.266
Acetonitrile	35.688	19.815	39.683	1.807	1.799	0.304	0.463	0.308
Dimethylsulfoxide	46.826	66.236	58.683	2.188	2.657	0.263	0.489	0.226
Water	78.355	96.614	82.256	1.777	1.662	0.320	0.492	0.338

Figure 3:

Static dielectric $\left(\epsilon \right)$, dynamic dielectric $\left({\epsilon }_{\mathrm{\infty }}\right)$ and orientation polarizability $\left(\mathrm{\Delta }f\right)$ of pure solvent for the nonpolarizable (red) and polarizable (blue) force fields graphed against experimental values

In almost every case, the dielectric constants from the polarizable force field are closer to those found in experiment, especially in regards to the differentiation of some of the middling solvents like acetone, ethanol and methanol. Additionally, the polarizable model as compared to the nonpolarizable model has a much more accurate estimation of the two extremes (benzene and water). The solvents are ordered in terms of the increasing static dielectric constant reported in experiment, an order which is maintained by the polarizable Drude values. In contrast, the nonpolarizable model has many instances where the order is not the same as for the experimental values.

Greater differences between the two theoretical models appear when considering the dynamic dielectric constant, ${\epsilon }_{\mathrm{\infty }}$. For the nonpolarizable force field, ${\epsilon }_{\mathrm{\infty }}$ is constrained to 1 by the nature of the model. For the polarizable force field, the value can be calculated using Eq.(5) from the Drude dipole fluctuations with all nuclei fixed and then averaged over a number of configurations.³³ The calculated dynamic dielectric constants for the polarizable model are reported for each solvent in the second section of Table 1. Each term is pretty close to the expected experimental values, which are determined by squaring a generally accepted value of the refractive index of each solvent. The dynamic dielectric constant was also used in the calculation of the static dielectric constant calculated for the polarizable Drude solvent models that was shown in the first section of Table 1, contributing to its improved estimation of the static dielectric constant as compared to those found with the nonpolarizable force fields.

A classic Lippert plot commonly shows the Stokes Shift as a function of the solvent orientation polarizability $\mathrm{\Delta }f$, defined by Eq.(2). In a continuum dielectric model, a Lippert plot is expected to reveal the general linear trend of an increasing difference between the absorption and emission wavelengths of the molecule of interest as the dielectric of the solvent increases. The seemingly small difference of incorporating a dynamic dielectric constant whose value is closer to 1.5 or 2 (in contrast to 1) results in a noticeable difference between the estimations of the orientation polarizabilities as calculated by the nonpolarizable and polarizable models. As observed from the rightmost section of Table 1, the orientation polarizabilities for the polarizable model are much closer to experimental data. It is important to note that the nonpolarizable model of pure solvent assumes the dynamic dielectric constant (and thus also the index of refraction) to be equal to 1 in all cases. It follows that estimates of the orientation polarizability from the nonpolarizable model are noticeably shifted to higher values as compared to the experimental data. For example, the nonpolarizable model’s orientation polarizability of diethylether is found to be even larger than the experimental orientation polarizability of water. Because of this, each solvent orientation polarizability found with the polarizable force fields is much more in line with the experimentally accepted value for that given solvent than that found with the nonpolarizable force fields.

DMABN in Solvent

Absorption and Emission

The absorption, emission and Stoke shifts of DMABN in various solvents calculated from the nonpolarizable and polarizable models are reported in the corresponding columns in Table 2. It should be noted, however, that the comparison of calculated and experimental absorption and emission wavelengths depends on our ability to ascribe the transitions involving the LE and ICT states unambiguously. In the present analysis, we relied on the absorption data from Figure 4b of Neubauer et al.,²³ and from Table 1 and Figure 6 of Atsbeha et al.²⁴ For nonpolar solvent (e.g., hexane), there are distinct absorption and emission peaks, corresponding to transitions to or from the LE state. However, in the case of more polar solvents (e.g., acetonitrile), there are overlapping emission peaks though it is clear which one corresponds to the LE state or the ICT state.^23,24 In all cases, however, there is generally a single absorption peak. In that sense, it is more accurate to say that the available experimental data absorption peak corresponds to transitions from the ground state to either the LE or the ICT state.

Table 2:

Absorption, Emission and Stokes shift of DMABN in solvent using nonpolarizable and polarizable Drude force fields. Experimental wavelengths included for comparison when available.

Solvent	Absorption (nm)			Emission (nm)			Stokes Shift (nm)
	NonP	Pol	Exp	NonP	Pol	Exp	NonP	Pol	Exp
Benzene	289.2	289.5		358.3	352.0	352.7a	69.1	62.5
Diethylether	288.6	289.9	287.1b	357.7	354.2	348.6b	69.1	64.3	61.5b
Trichloroethane	290.8	289.1		358.5	354.2		67.7	65.1
Dichloroethane	287.2	287.9		356.1	354.2		68.9	66.3
Acetone	285.0	286.8		356.2	354.6		71.2	67.9
Ethanol	285.4	284.6		356.7	354.6	354c	71.2	70.0
Methanol	281.8	282.7		355.2	355.2		73.4	72.6
Acetonitrile	285.0	283.8	292.4b 291.1c	356.5	355.5	359.7b 353b	71.5	71.8	67.3b 61.9 c
Dimethylsulfoxide	281.0	283.5		354.5	354.9	362.6a	73.5	71.5
Water	277.1	274.4		352.4	365.0	360.4d 365.9d	75.2	90.6

^areference 25

^breference 23

^creference 24

^dreference 7.

Figure 4:

Absorption, emission and Stokes shifts (in nm) for DMABN. The entries are listed in order of increasing experimental dielectric constants for the ten solvents considered in this study (1=benzene, 2=diethylether, 3=1,1,1-trichloroethane, 4=1,1-dichloroethane, 5=acetone, 6=ethanol, 7=methanol, 8=acetonitrile, 9=dimethylsulfoxide, 10=water). Results from the simulations based on a nonpolarizable (red) and polarizable (blue) force fields, are plotted together with experimental data (black dots,²³ black squares,²⁴, black upward triangles²⁵ and black downward triangles⁷ ). The data is taken from Table 2.

Figure 6:

Relaxation of DMABN in three sample solvents immediately after excitation - using the nonpolarizable (red) and polarizable (blue) force fields.

From Table 2, it is observed that the absorption wavelength calculated from both the nonpolarizable and polarizable models generally decreases from benzene to water as the dielectric constant of the solvent increases. Such a trend appears to be contradicted by the experimental data for diethylether to acetonitrile.^23,24 Although the difference is fairly small for these two solvents (287.1 nm and 291.1–292.4 nm), the increase in absorption wavelength with the dielectric constant of the solvent is also displayed when taking hexane into consideration.²³ However, a different experimental study shows that, despite its smaller dielectric constant, the absorption wavelength in dichloromethane is actually greater than in acetonitrile.²⁴ Therefore, it is difficult to draw a definitive conclusion about the dependence of the absorption wavelength with respect to the polarity of the solvent due to the sparsity of experimental data.

The situation is a little better regarding the emission process in various solvents, making a more meaningful comparison with the computational results possible. The emission wavelength from experimental studies do not systematically display that with an increased dielectric constant the emission of DMABN would increase as well, but overall they follow this general trend. From hexane to acetonitrile an increase of about 20 nm of the emission of DMABN has been reported.²³ A similar comparison of solvents saw an initial increase of about 13 nm from cyclohexane to 1,4-dioxane but a plateau over the rest of the solvents ending in acetonitrile.²⁴ In the case of the nonpolarizable models, the calculated emission wavelength with respect of the solvent dielectric constants also trends in the opposite direction compared to experimental data.^{7,23–25,59–61} While the experimentally measured emission wavelength in benzene (low dielectric) and water (high dielectric) are 352.7 nm and 360.4–365.9 nm, respectively, the corresponding results calculated from the nonpolarizable models are 358.3 and 352.54 nm, in disagreement with the data. However, the results from the polarizable models are generally in much better agreement with experimental data. With the exception of diethylether and dimethylsulfoxide, the emission wavelength is indeed increasing for the low to high dielectric solvents in the sequence benzene, ethanol, acetonitrile, and water.²⁴ Notably, the emission wavelength in benzene and water calculated from the polarizable force field is 352.0 nm and 365.0 nm, which is in excellent agreement with the data. Overall, a greater difference between the polarizable and nonpolarizable models is observed with respect to the emission wavelength. This is particularly obvious when looking at benzene and diethylether. The experimental emission in benzene has been reported to be 352.7 nm²⁵ which is much closer to the calculations based on the polarizable models of 352.0 nm than the 358.3 nm from the nonpolarizable model. Similarly, the experimental emission in diethylether has been reported to be 348.6 nm²³, which is closer to the 354.2 nm calculated with the polarizable model than the 357.7 nm calculated with the nonpolarizable model. A direct comparison of the calculations with available experimental data is provided in Supplemental Figures S11 and S12.

The total range of the absorption wavelength (from benzene to water) for the nonpolarizable model is about 12 nm, and only about 4.5 nm when considering the range to be from benzene to acetonitrile. For the polarizable model the total range is closer to 15 nm, while the range ending in acetonitrile is 6 nm, in better agreement with experiments. It is clear that the nonpolarizable models understate the effect of an increased dielectric on the absorption activity of DMABN. The average absorption and emission energies, standard deviations, and statistical uncertainties for the overall system, as well as the internal energy of DMABN and its interaction with the solvent are provided in Supplemental Tables S4, S5, S6, and S7. The energy differences taken of the isolated DMABN in the different poses extracted from the simulations in each solvent system results in average values of around 97.2 kcal/mol and 96.4 kcal/mol for the nonpolarizable and polarizable model, respectively. These values are consistent with a simple equilibrium simulation of the ground state of DMABN in vacuum (97.1 kcal/mol and 96.4 kcal/mol for the nonpolarizable and polarizable models, respectively). The small fluctuations in the internal energy of DMABN, regardless of the solvent, indicate that the changes in the total energy are almost entirely due to the electrostatic coupling to the surroundings in response to the electronic transition of DMABN. The trends of the interaction energy between DMABN and the solvent mirrors the energy difference in the complete system with a general increase with increasing dielectric. This observation seems reasonable because DMABN is a small rigid molecule that does not undergo large changes in conformation. Although the ground state is generally accepted to be perfectly planar, some twisting of the amino group and other fluctuations in the other angles occur during the simulations. This can be seen directly in the tabulated standard deviations of the energy differences of the isolated DMABN molecule. These slight variations in DMABN itself and in the specific interactions of the solvent with DMABN resulted in a range of reported energy differences, necessitating the average energies for each DMABN-solvent combination being used in analysis. In the case of the emission process, the internal energy difference of the isolated DMABN (79.4 kcal/mol and 79.9 kcal/mol for the nonpolarizable and polarizable model, respectively) is slightly different than the absorption frames and the internal energy averages. This is explained by the fact that the force field for the excited state allow for slightly more twist of the amino group (Supplemental Figures S1 and S2). These values are consistent with the energy differences found for simulations of the excited state of DMABN in vacuum: 79.4 kcal/mol and 79.8 kcal/mol for the nonpolarizable and polarizable models, respectively. Regardless, the interaction energy with the solvent, as in the case of the absorption process, mirrors the changes in the total energy emission. Hence, the total energy differences for the emission is also dominated by the interaction energy between DMABN and the surrounding solvent.

Stokes Shifts

Figure 5 shows the Stokes shift of DMABN as a function of the orientation polarizability $\mathrm{\Delta }f$ of various solvents calculated from the nonpolarizable and polarizable models. Despite the disagreement with experimental data with respect to the absorption and emission processes, the Stokes shift increased with the increasing orientational polarizability of the solvent that surrounds the DMABN molecule for both the nonpolarizable and polarizable models. It is notable, however, that the results do not produce a perfectly linear progression. The deviations from linearity are most likely due to specific interactions between the fluorescent DMABN molecule and the surrounding solvent molecules. Experimental studies considering different mixtures of two solvents to reduce the impact of these effects often yield Lippert plots that tend to be more strictly linear.⁶² The closer match in terms of range and trend indicate that the polarizable models based on the Drude force field yields more accurate results than the nonpolarizable force field. According to the Lippert-Mataga Eq.(1), the Stokes shift is directly related to dielectric properties of the surrounding solvent, which are included in the orientation polarizability $\mathrm{\Delta }f$ defined in Eq.(2).^26,27 The static dielectric constants of the nonpolarizable and polarizable models are not drastically different from one another. However, the effect of the dynamic dielectric constant ${\epsilon }_{\mathrm{\infty }}$ is critical when considering its impact on the orientation polarizability $\mathrm{\Delta }f$. It may also be noted that the results among different experimental studies are sometimes inconsistent. An example of this is apparent with the Stokes shift, where one study shows that in diethylether the Stokes shift is 61.5 nm²³ while in another study the Stokes shift in acetonitrile is practically identical at 61.9 nm.²⁴ Comparisons of the simulated models to experimental results is more appropriate for experimental results performed by the same study. Looking at the two Stokes shift from Neubauer et al. (61.5 nm in diethylether and 67.3 nm in acetonitrile), the polarizable model better matches (64.3 nm in diethylether and 71.8 nm in acetonitrile) than the nonpolarizable model (69.1 nm in diethylether and 71.5 nm in acetonitrile). Considering how the polarizable model was consistently closer to the available experimental emission wavelengths and that the absorption wavelengths are extremely similar between the two models, it is not surprising that the polarizable model better approximates the Stokes shifts as well.

Figure 5:

Stokes shift of DMABN calculated as a function of $\mathrm{\Delta }f$, the orientation polarizability of various solvents. The results from nonpolarizable (red) and polarizable (blue) force fields are plotted together with experimental data (black). The experimental data is given in Supplemental Table S3. A direct one-to-one comparison of the Stokes shift data with all available experimental data is provide in Supplemental Figure S13.

The sensitivity of the polarizable model results in a greater range with respect to changes in the calculated Stokes shifts. Compared to the nonpolarizable model, the least polar solvent of benzene has a smaller Stokes shift and the most polar solvent of water has a larger Stokes shift. According to the nonpolarizable model, the energy transitions to and from the locally excited (LE) state first singlet of DMABN hardly change based on its environment. From benzene to water, the change in the Stokes shift is only 6.13 nm (from benzene to dimethylsulfoxide it is 4.44 nm) for the nonpolarizable model. In contrast the polarizable model predicts the Stokes shift to change from benzene to water by 28.07 nm (from benzene to dimethylsulfoxide it is 8.97 nm). The greater range of the polarizable Drude data is overall more in-line with experiment, making it a better representation of the spectral properties of DMABN than the results when using a nonpolarizable force field. The DMABN in water data as represented by the Drude polarizable force field seems to overestimate the Stokes shift, and that point is a bit of an outlier from the linear trend produced by the rest of the solvents (Figure 5). Removing this data point from the plot yields a linear trend for the polarizable model that results in a better match of the available experimental data than for the nonpolarizable model. Considering how closely the DMABN emission wavelength in water calculated from the polarizable model is to experimental data, this could be due to inaccuracies in the calculated absorption wavelength (which is not directly compared to any experimental data) or this Stokes shift is actually correct for the system Unfortunately, due to the lack of experimental measurements, it is unclear if the behavior of the DMABN-water system based on the polarizable force field reflects an inaccuracy of the calculation or a genuine feature of the system.

Dynamical relaxation after excitation

The absorption and emission wavelengths are calculated according to Eqs.(6) and (7), $E_{\mathrm{g}\mathrm{s}}$ and $E_{\text{es}}$, as simple averages of energy differences. While these averages are well defined, it is of interest to appreciate the magnitude of the energy fluctuations occurring within the system. Histograms of the instantaneous absorption and emission energy differences extracted from the simulations are shown as Supplemental Figures S3, S4. S5, and S6. The broad distributions provide clear indications that the absorption and emission processes are taking place within highly dynamical and fluctuating environments. Similarly, one could calculate the time-correlation function of the energy gap from the instantaneous absorption and emission energy differences. Within the context of a linear-response approximation, it is possible to exploit this type of information to predict the time-dependent electronic transition.^63,64 Alternatively, one may explicitly simulate the relaxation of the energy gap following an electronic transition.

To illustrate this point, fifty different snapshots were extracted from the ground state simulation in methanol, dimethylsulfoxide and water. Then each snapshot is treated as the starting point of a new simulation of the excited state. After each femtosecond of simulation, the energy gap between the ground and the LE state are measured and averaged over the fifty iterations in the given solvent for a total of 1000 fs. The initial energy gap at $t=0$ was subtracted to more clearly display the relaxation process over the course of the simulation. The dynamical relaxation after excitation was examined with both the nonpolarizable and polarizable force fields. Figure 6 shows the relaxation of the total energy of the system as a function of time after an excitation in each of the three solvents considered (methanol, dimethylsulfoxide and water), nonpolarizable results graphed in red and polarizable in blue. In all cases, a sharp decline occurs in the first couple hundred femtoseconds, which then levels off as the simulation continues. This trend is most noticeable in the simulations with water, where the total energy change is larger than in the other two solvents. For the nonpolarizable models, there is a decrease of about 4.5 kcal/mol during the relaxation in water while there is a decrease of about 2.5 kcal/mol in the two other solvents. The same trend is observed with the polarizable model, however the magnitude of the change is noticeably larger than was seen for the nonpolarizable force field. The decrease in energy is about 9.5 kcal/mol in water, while it is 4.5 kcal/mol in methanol and 3.5 kcal/mol in dimethylsulfoxide. As with the average instantaneous energy differences, the polarizable force field predicts a larger energy difference range between solvents.

Conclusion

Using molecular mechanical force fields optimized against QM data of the ground and excited state, we carried out MD simulations of the transition between the ground and excited state of the fluorescent molecule DMABN in solvents of different polarity. In practice, the instantaneous electronic transition is carried out by switching the ground state for the corresponding excited state force field parameters. A Lippert plot allows for a visual representation of the effect of changing the surrounding solvent on the spectra of DMABN. The Lippert plot extracted from the simulations follow the expected trend of an increase in the Stokes shift corresponding to an increase in the polarizability of the surroundings. Both nonpolarizable and polarizable Drude force fields were used to model DMABN in a series of solvents, and both resulted in Lippert plots with a positive trend. There are deviations from the main trend, but since the solvents are explicitly included in the systems, the Lippert plot is not expected to coincide perfectly with a straight line. On the other hand, the limited quantitative agreement of the computations with experimental data is somewhat disappointing. One possible limitation is the fact that the absorption wavelengths calculated simply as energy differences between the LE state and the ground state should include overlapping contributions from the LE and ICT states, which are not accounted for in the present treatment. However accounting for induced polarization improves the representation of these systems, especially in regards to the interactions between DMABN and more polar solvents. Though this series of MD simulations were performed for a relatively simple molecule that exhibits a small energy transition between its ground and first excited state, it can be clearly seen that this methodology can be applied effectively to more complex fluorescent probes and in more complicated environments.