Two-Photon Voltmeter for Measuring a Molecular Electric Field**

We present a new approach for determining the strength of the dipolar solute-induced reaction field, along with the ground- and excited-state electrostatic dipole moments and polarizability of a solvated chromophore, using exclusively one-photon and two-photon absorption measurements. We verify the approach on two benchmark chromophores N,N-dimethyl-6-propionyl-2-naphthylamine (prodan) and coumarin 153 (C153) in a series of toluene/dimethyl sulfoxide (DMSO) mixtures and find that the experimental values show good quantitative agreement with literature and our quantum-chemical calculations. Our results indicate that the reaction field varies in a surprisingly broad range, 0–107 V cm−1, and that at close proximity, on the order of the chromophore radius, the effective dielectric constant of the solute–solvent system displays a unique functional dependence on the bulk dielectric constant, offering new insight into the close-range molecular interaction.


Materials and 1-Photon Absorption Measurements
Toluene (> 99.9%), DMSO (> 99.9%), C153 (99%) and PRODAN (98%) were purchased from Sigma-Aldrich and used as received. Linear spectroscopic measurements were performed on a Perkin Elmer Lambda 950 spectrophotometer in 1 cm spectroscopic grade quartz cuvettes for all mixtures of Toluene and DMSO in ratios given in Table S1. Extinction coefficients were measured by the weight method. Table S1: Solvent mixtures used for each chromophore (e.g. 7:1 indicates 7 parts Toluene to 1 part DMSO by volume).
X indicates that a 1PA and 2PA spectroscopic measurements were performed in that solvent mixture.

2-Photon Absorption Measurements
Spectral shapes were obtained using the two-photon excited fluorescence method by employing a tunable optical parametric oscillator (OPO, Insight DeepSee, Spectra Physics) with a step size of 1 nm over the ranges of 680-900 nm (PRODAN) and 700-1025 nm (C153). Fluorescence was collected at a right angle to the incident beam with a PMT. Power dependence was measured at each wavelength with a stepper motor-controlled variable neutral density filter wheel (Thorlabs) that measured the fluorescence intensity for 20 steps over the full range of the wheel (OD 2). The intercept corresponding to the linear fit to the loglog plot of the fluorescence vs. incident power data was used to determine the relative value of the 2PA cross section at each wavelength. Only those points were used where the slope of the power dependence was verified to be in the range 2.00 ± 0.05. Correction for the beam shape was determined by imaging the beam at each wavelength with a CCD camera (Stingray, Basler). Pulse durations were measured using a home-built second harmonic generation autocorrelator using BBO as the doubling material, and were in the range 90-120 fs. The pulse repetition rate of the laser is 80 Mhz, and was chopped down to 70 Hz using a mechanical chopper wheel to alleviate thermal effects in the sample. Excitation power values were adjusted for the solvent absorption based on the linear spectroscopic measurements of each respective mixture. Absolute cross section values were measured for PRODAN (C153) at 750 nm relative to BDPAS (Fluorescein) using the reference method as described in detail earlier [1] .

2-Level Model Description
Degenerate two-photon absorption is formally described via second order perturbation theory with the sumover-states expression [2] ( where σ2 is the 2PA cross section, ν/2 is the two-photon excitation energy, νim is the transition energy between states i and m, h is Planck's constant, c is the speed of light in vacuum, n is the refractive index of the medium, e is the polarization unit vector of the optical electric field, im mj , µ µ are the dipole moments between states i and m and between states m and j, respectively (note that if i = m or if m = j, these are permanent dipole moments), ( ) j g ν is the transition line shape of the final state, f is the local field factor, taken here to be the Lorentz factor, 2 2 3 n + , and Ω represents isotropic averaging over all orientations of the absorbing molecules relative to the direction of e . For the 2-level system approximation, i = 0, j = 1, and only the indices m = 0,1 are included in the summation, so that where 10 µ ∆ is the permanent dipole moment difference between the ground (0) and first excited (1) states, and β is the angle between the 10 µ ∆ and 10 µ vectors. Note here that the averaging over Ω assumes an isotropic distribution of molecular orientations in solution. Finally, using the relationship between the transition moment and molar extinction coefficient, ( ) M ε ν [3] , where NA is Avogadro's number, and assuming that 0 β ≈ , the 2-level 2PA cross section is given as Rearranging for 10 µ ∆ and converting into the units of Eqn. (1) in the main paper, .

MATLAB Optimization Routine Description
The model line shape is calculated based on Eq. (10) from the main text, where the peak transition energy and width are extracted from the fit to Eq. (7) for approximately 250 million combinations of the 5 unknown parameters over a broad input value interval (see below). The program then subtracts these model results from the experimental data to calculate the residual, and stores the product of the absolute value of the residuals (peak and width) in an array corresponding to the parameter values used for that particular calculation. The parameter interval ranges are then iteratively reduced until convergence of the results is achieved within the expected experimental uncertainty (±10%). The coordinates of the minimum value in this final 5-D array then correspond to the parameter values that best reproduce the experimental spectra, according to our model. Note that the experimental data are sufficient to solve simultaneously for 5 parameters, given that for convergence, a global fit must be obtained for the values of the peak transition energy and its first 2 derivatives vs. ε, as well as the values for the width and its first derivative vs. ε (see Figures S4, S5). The initial parameter variation intervals for both chromophores were the same and are given as follows:

S S a
: Flow chart outlining MATLAB parameter optimization procedure.

Computational Details
Ground state conformational analysis of Prodan and Coumarin C153 compounds in a gas phase has been carried out at BP86/TZVPP level using the Turbomole [4] software package. The resulting structures were re-optimized at B3LYP/6-311G(d,p) level in Gaussian09 [5] and the absence of imaginary frequencies was confirmed. Vertical excited state properties of the lowest singlet states were obtained from Gaussian09 TDDFT calculations using the Coulomb Attenuated Method CAM-B3LYP functional and the 6-311++G(d,p) basis set for the ground state geometries of Prodan and Coumarin 153. For Prodan, adiabatic properties of the first excited singlet state were also explored by optimizing the structure of the excited state at TD/CAM-B3LYP/6-311++G(d,p) level. The effects of solvation on excited state properties of Prodan were studied in toluene and dimethyl sulfoxide (DMSO) using linear response (LR) equilibrium IEF-PCM solvation (toluene and DMSO), as well as non-equilibrium LR and state-specific IEF-PCM solvation (DMSO) combined with the TDDFT approach. Excited-state dipole moments were obtained from relaxed excited-state electron densities computed by Z-vector method as implemented in Gaussian09. Excited-state polarizability was computed as the numerical derivative of the dipole moment. The isotropic average of the polarizability in the ground and excited states is used for comparison in Table 1. Unless specified otherwise, the results correspond to the lowest energy conformers of Prodan and Coumarin 153 (ground state gasphase reference). Figure S1 shows the measured 1PA (solid line) and 2PA (black markers) spectra for PRODAN (left) and C153 (right), in all measured solvent mixtures. C153 was measured in fewer mixtures for the sake of expediency, once it was determined that this was sufficient for reasonable determination of the parameters.    Figures S4 and S5 show the dependence of the 1PA peak energy (Gaussian fit for the lowest energy transition) on the dielectric constant for PRODAN (left) and C153 (right). The symbols are experimental data and the red curves show the fits to the data.   Figure S6 shows the dependence of the 1PA lowest energy band width (Gaussian fit for the lowest energy transition) on the dielectric constant for PRODAN (left) and C153 (right). The symbols are experimental data and the red curves show the fits to the data.  Figure S7 shows the final predicted 1PA spectra (black markers) and fits to the experimental spectra (red lines). The model predictions almost exactly reproduce the experimental data. This is to be expected, as we optimized the parameters in order to reproduce the 1PA spectral data in the MATLAB routine.

Discussion of Cavity Model:
The empirical formula for the power dependence of the interior dielectric constant on that of the bulk solvent was chosen in order to account for close-range intramolecular interactions while still maintaining the conditions defining the reaction field. These conditions maintain that: 1) in ε ε ≤ for all dielectric constants so that the reaction field points in the direction of 0 µ and therefore lowers, instead of increasing, the energy of the system, and; 2) = , so that in the vacuum limit the reaction field goes to zero.
It is commonly assumed that the interior of the cavity is vacuum, but this does not account for the effect of close-range interactions which can sometimes play a dominant role. We take the approach of assuming that the "dielectric" response of the solute molecule is related to the dielectric properties of the solvent. How the value for p is related to the type and magnitude of close-range interaction remains to be determined. However, an interesting comparison can be made between our power dependence and the Lippert-Mataga orientational polarizability function ∆f, given in Eq. (S6), where n is the solvent refractive index, taken here to be the average value of 1.488 [6] . Figure S8 compares the ε dependence of f from Eq. It can be observed that the ∆f function provides a reasonable estimate to the behavior, but saturates more rapidly than our function f as a consequence of the constant (vacuum) values for the interior dielectric constant and refractive index. Changing the value of p slightly from 0.65 to 0.8 yields a lower scaling for the reaction field, perhaps indicative of a different electrostatic response to the solvent environment.
Uncertainty Estimation: 1) Experimentally determined parameters (νvac, ∆νvac, ∆µvac, ∆α): The error in νvac ( Figure S5) is expected to be primarily determined by the accuracy of the Gaussian fitting. This is because: 1) f(p) was optimized in order to minimize the standard error between the linear fits to the peak transition energy dependence (8 cm -1 and 15 cm -1 for PRODAN and C153, respectively); and 2) The lack of resolution of the peaks in the absorption band prevented locating the peaks with an accuracy less than these standard errors. The uncertainty in νvac is therefore estimated at ~5% of the width of the total 1PA band, or ±200 cm -1 .
The main contribution to the uncertainty in the accuracy of our optimized parameter values is due to that associated with the reference standards used for the determination of 2PA cross section values. This uncertainty is estimated at ±10% of σ2. This corresponds to approximately ±5% of ∆µ, and is a systematic error expected to propagate to all solvent mixtures. Therefore, the result of this error is a systematic discrepancy in ∆µvac of ±5%.