Evaluating the Solvent Stark Effect from Temperature‐Dependent Solvatochromic Shifts of Anthracene

Abstract The solvent Stark effect on the spectral shifts of anthracene is studied with temperature‐dependent solvatochromic measurements. The Stark contribution Δv Stark to the absorption shift Δv p in polar solvents is measured to be Δv Stark=(53±35) cm−1, in reasonable agreement with dielectric continuum theory estimate of 28 cm−1, whereas the major shift Δv p∼300 cm−1 presumably originates from the solute quadrupole. We pay attention to the accurate correction of Δv p for the nonpolar contribution that is crucial when the shifts are modest in magnitude.


Introduction
When a nondipolar polarizable solute like anthracene, perylene or trans-stilbene is immersed in a polar solvent, it is stabilized via interaction with fluctuating solvent electric field e by energy E a ¼ À ah e 2 i=2 where α is the solute polarizability, and the averaging is taken over all solvent configurations. [1,2] This stabilization is commonly called the solvent Stark effect. [3][4][5][6][7] Karlström and Halle [1] applied a fluctuation approach for dielectrics with a spherical cavity of radius a to obtain the following exact result E a ¼ À 3 2 RT � ln½ 1 À a l c q l ð Þ= 1 À a l c l ð Þ � (1) Here multipolar susceptibilities c l, c q l are related to slow (orientational) and fast (quantum, electronic) degrees of freedom of the solvent with dielectric constant e and refractive index n; R = 8.31 J/mol/K is the gas constant and T temperature; l = 1 or 2 corresponds to dipolar or quadrupolar solute, or with l � 3 to higher multipoles. For a nondipolar solute and when a l c q l , a l c l ! 1, one gets where f p ¼ ½2ðeÀ 1Þ=ð2eþ1Þ À 2ðn 2 À 1Þ=ð2n 2 þ1Þ� is the wellknown response in polar solvents. [4] A very similar expression for E a was also derived by Scaife. [2] The Stark effect directly results in solvatochromic shifts for absorption where the polarizability difference appears because α differs in ground (g) and excited (e) electronic state of the solute; the superscript FC abbreviates "Franck-Condon" indicating that α e is taken at ground state (S 0 ) chromophore geometry. Hereinafter, we apply the following units: Δv in cm À 1 , a in Å, α in Å 3 , dipole moment μ in D, quadrupole moment Q in DÅ. Energies E and shifts Δv are in eV, kJ/mol or cm À 1 with the relations between the units Just to give an idea of the expected shift (4), one has 3f p /2 � 1, ða FC e À a g Þ~15 Å 3 , a = 5 Å, RT = 2.44 kJ/mol = 204 cm À 1 at T = 20°C, that predicts quite a small value Δv Stark~2 0 cm À 1 .
In the late 1960s Baur and Nicol [8] suggested a different expression for the Stark shift, Dv Stark � eðeÀ 1Þ=ð2eþ1Þ which gives for ɛ > 10 a much larger shift than that by Eq. (4). Furthermore, they tried to ascribe the full observed shift from nondipolar solutes in polar solvents entirely to the Stark contribution, and even obtained a support from other workers. [9] However, Ghoneim and Suppan [3] experimentally demonstrated an inconsistency in their approach, and instead proposed quadrupolar or higher multipolar nature of the aforementioned shifts.
Since then, to the best of our knowledge, there were no attempts to quantify the Stark contribution to the solvatochromic shifts experimentally.
The aim of the present paper is to determine the weak Stark contribution to temperature-dependent solvatochromic shifts of the absorption spectra of anthracene.
In addition, we propose a simple method for correcting the shifts for the nonpolar contribution, the correction being crucial when the shifts are modest in magnitude.
The paper is organized as follows. In section 2.1 we overview the theory of solvatochromic shifts and introduce the correction procedure, section 2.2 describes our calculations, followed by section 2.3 for the experimental results and discussion.

Solvatochromic Shifts
For a dipolar solute a classical theory [4,[10][11][12][13][14] expresses absorption shifts Δv as the sum of nonpolar Δv n and polar Δv p contribution Here f n , f p is already familiar nonpolar and polar solvent response, μ, α is the solute dipole moment and polarizability, and the products of the dipole moments are to be understood as scalar products. The nonpolar part Δv n is proportional to f n and consists of inductive (the first term) and dispersive contribution. [4] The semiempirical constant C is often expressed via solute (I) and solvent (I') ionization potential, C ¼ 2II 0 =ðI þ I 0 Þ and usually is in the range of 10 eV = 80000 cm À 1 . This gives an estimate of 4000 cm À 1 for the dispersive term, and with μ g = 5 D, μ e = 10 D an estimate of 1500 cm À 1 for the inductive term. Regarding the polar part Δv p , it is proportional to f p and represents the dipolar shift in polar solvents. With the above μ g = 5 D, μ e = 10 D, the dipolar shift Δv p reaches 1000 cm À 1 .
When like in our case, the solute dipoles vanish, μ g � μ e � 0, the shifts Δv are strongly dominated by the dispersive contribution, which exceeds the expected Stark shift by two orders of magnitude.
Baur and Nicol [8] and Gerhold and Miller [9] plotted experimental shifts Δv against the calculated shifts given by a sum of Δv n and their Stark term Dv Stark � eðeÀ 1Þ=ð2eþ1Þ�RT a FC e À a g À � =a 3 . It is however quite clear, that small (in percentages) errors in the huge dispersive contribution Δv n may completely mask the effect of interest.
We therefore apply below a different approach. First, we note that both Δv n and Δv p contribute to the shift Δv in polar solvents. On the other hand in nonpolar solvents, Δv p vanishes completely since ɛ = n 2 and f p = 0, and hence Δv n can be fully determined with a set of nonpolar solvents only with the slope B n being calculated from a linear fit of Δv n against f n . Having this result at hand one can get rid of the nonpolar contribution to Δv In our case Δv p does not contain the dipolar part, but presumably contains Stark Δv Stark and quadrupolar [3] Δv Q part where the traceless quadrupole tensors Q are according to Buckingham [15] and their products should be understood as the double inner products. The quadrupolar response f Q = [3(ɛÀ 1)/ (3ɛ + 2)À 3(n 2 À 1)/(3n 2 + 2)] is slightly different from f p , but in the realm of more common solvents, where 1.4 < n 2 < 2.6 (from perfluoroalkanes to CS 2 ) and n 2 � ɛ < 111 (formamide) f Q deviates from f p by at most 6 %, the highest discrepancy to be expected for extreme cases, perfluoroalkanes and compounds with high ɛ and n 2 . Therefore, one can safely substitute f p to obtain In experiment the polar shifts Δv p from nondipolar chromophores are in no case negligible and reach 300 cm À 1 for anthracene, stilbene or diphenylbutadiene. We believe, following Suppan, [3] that these shifts originate from quadrupolar [15] or higher multipolar contribution, as shall be discussed in detail in our forthcoming article.
Regarding the Stark contribution, although it is much smaller in magnitude, it can be derived from temperaturedependent shifts Δv p (T).
A further very helpful comparison is between Stark 3 and nonpolar B n = Cða FC e À a g Þ=2a 3 slope. As seen, the both depend on the solute parameters in a similar fashion. This allows one to exclude the solute radius a (not well-defined in the continuum dielectric theory) and to express B Stark through the well-measured quantity B n that provides an improved estimate for the Stark shift. With ða FC e À a g Þ = 16.5 Å 3 , [16][17][18] and taking I = 7.4 eV for anthracene, I' = 10.4 eV for n-pentane, one calculates C = 69740 cm À 1 . And with our experimental B n = 3150 cm À 1 (see Figure 2) this gives at T = 293 K

Calculations
Vacuum static polarizabilities for S 0 and S n states are computed with two approaches. The first uses the CC2 approximation of coupled-cluster theory with a aug-cc-pVTZ basis set, [19] and corresponding auxiliary basis set [20] using the ricc2 module of the Turbomole program package version 7.0.2. [21] The second approach involves the RI-XMCQDPT2 quasi-degenerate perturbation theory [22] implemented in the Firefly V8.2 software [23] which is partly based on the GAMESS(US) package. [24] The perturbation corrections are applied on top of the CASSCF (14e,14o)/aug-cc-pwCVTZ reference where the active space encompasses all the 14 π-orbitals. To suppress the intruder state effects, the intruder-state-avoidance (ISA) parameter is set 0.02 a.u. The anthracene polarizabilities have already been calculated. [16][17][18] Pavlovich [16] considered the temperature-dependent shifts of absorption in frozen glassy alcohols where the Stark effect and the dispersive contribution were added up. Mathies and Albrecht [17] performed electric field perturbation spectroscopy in a frozen medium, and Bendkowsky et al. [18] measured the quadratic Stark effect in jet-cooled molecules.
Our XMCQDPT2 and RI-CC2 computations confirm the first absorbing excited state to be indeed S 1 dominated by the HOMO!LUMO excitation. At the CASSCF level it emerges incorrectly as S 3 . The second (after S 1 ) bright absorbing state turns out to be S 6 which is almost degenerate with S 5 at the XMCQDPT2 level. It involves a mixture of several single excitations. Taking into account that the task of accurate description of the higher-lying excited states would require at least further augmentation of the basis set, the XMCQDPT2 calculations were primarily focused at the task of more accurate description of the S 0 and S 1 states. In view of that, we use the CASSCF reference averaged over the five lowest singlet roots and include 13 states in the XMCQDPT2 model space. Resulting vertical gas-phase excitation energies of 3.39 eV for S 0 !S 1 and 4.88 eV for S 0 !S 6 are in a good agreement with the experiment.
The calculated polarizabilities are given in Table 1 (x is the long axis of the anthracene molecule, y is the short one, and zthe perpendicular one). As seen Δα for S 0 !S 1 equals 15.9 Å 3 , in agreement with both the present experiment (see below) and the earlier estimates. [16][17][18] A very close result was previously obtained in the relaxed RI-CC2/aug-cc-pVTZ calculations. [25] At the same time, considering the S 0 and S 1 states separately, our values in Table 1 are lower than the previous CCSD(T) estimates for S 0 [26] (interestingly, the discrepancy is almost entirely associated with α xx ) and than the above RI-CC2 data for S 1 . [25] ) Unfortunately there is an understandable lack of reliable polarizability benchmarks even for the ground state of anthracene, as can be seen from a survey of the previous condensed-phase experimental data. [18] The XMCQDPT2 polarizability of the S 6 state is considerably underestimated. This obviously results from the coupling to S 5 which is placed by the calculation only a few meV below S 6 . Thus, an accurate computational treatment of S 6 requires very precise energies of the other states and possibly even an explicit consideration of the relevant vibronic levels. Our present RI-CC2 data are however qualitatively correct, suggesting a 40 % increase in the polarizability from S 1 to S 6 .

Experimental Shifts and Discussion
Absorption spectra of anthracene in solution are recorded at T = 10, 20, 30, 40, 50°C with 0.02 nm step both in the visible (S 0 !S 1 ) and in the UV (S 0 !S 6 ). Anthracene is chosen as the probe because its narrow sub-bands (see Figure 1) allow for high accuracy � 1 cm À 1 of the spectral shifts in the visible. The nonpolar and polar solvents used are collected in Table 2 (see  ref. 27 for the full solvent properties).
Typical S 0 !S 1 absorption spectra of anthracene are displayed in Figure 1. They consist of well resolved vibronic bands, with the 0-0 band peaked at 26650 cm À 1 in n-pentane. The spectra in nonpolar (top) and polar (bottom) solvents are shifted relative to n-pentane for best coincidence in the red part, including the 0-0 and 0-1 band. We estimate the accuracy   Table 2.
of such determined shifts to be � 1 cm À 1 . The shifts are indicated in units 1000 cm À 1 . Similarly, shifted UV spectra for the bright S 0 !S 6 electronic transition are displayed in Figure S1, Supporting Information (SI).

Figure 2 shows plots Δv n (f n ) and
Δv p (f p ) for the S 0 !S 1 and S 0 !S 6 band of anthracene (top and middle frame), and for the S 0 !S 1 band of highly polar dye C153.
Let us consider the S 0 !S 1 band of anthracene first. Nonpolar shifts Δv n (top left) show a nice linear behavior along nhydrocarbons, from n-pentane to n-hexadecane. Note that 2methylbutane (tm) and cyclohexane (ch) apparently deviate from the linear fit. The deviation is systematic and is also observed with other solvatochromic probes.
Switching to polar solvents at right of Figure 2, one sees that directly measured shifts Δv (black squares) reveal strong scatter that prevents from a satisfactory fit. As discussed above, this scatter is mainly due to the nonpolar contribution Δv n = B n f n. The subtraction (9) eliminates that contribution from Δv and results in Δv p shown by the open squares. These allow now for a good linear fit with slope B p = (272 � 13)cm À 1 .
For comparison, the bottom frame of Figure 2 shows the shifts from highly polar C153. [28] While its nonpolar slope B n = 5180 cm À 1 is comparable with that for anthracene, the polar slope B p = 2800 cm À 1 is by factor 10 larger, in which case the f n contribution to Δv can be safely neglected.
Next, the experimental ratio B p /B n~0 .1 is about 10 times larger than B Stark /B n = 0.009 estimated by Eq. (12). That is, the solvent Stark effect is expected to contribute about 10 % of the observed shift Δv p in polar solvents.   Table 1.
Consider now the shifts of the S 0 !S n band (n = 6 from our calculations) displayed in the middle frames of Figure 2. Here the nonpolar slope B n = (6560 � 190) cm À 1 is twice steeper than that for the S 0 !S 1 transition, in approximate agreement with the calculated higher polarizability in S n (compared to S 1 , see Table 1). Turning to the polar slope B p = (190 � 40) cm À 1 , we note that it is 1.5 times smaller than that for the S 0 !S 1 transition, contrary to what is expected if the slope would depend on the polarizability. Hence the polar and nonpolar shifts in anthracene are of different nature, consistent with the above assumption that Δv p originate mainly from the solute quadrupole (rather than from the solute polarizability).
To isolate the Stark shift Δv Stark we measure the S 0 !S 1 absorption spectra of anthracene at different temperatures. The results are presented in Figure 3 with nonpolar shifts Δv n (T) shown at left and polar shifts Δv p (T) at right, the corresponding slopes B n , B p being indicated as inserts. Figure 4 summarizes our results on the temperaturedependent shifts. Here the slope B p (T) is shown as function of temperature. Despite large error bars, the fit gives dB p /dT = (0.18 � 0.12) cm À 1 /K, in good agreement with dB Stark /dT =  . Despite large error bars, the fit gives dB p /dT = (0.18 � 0.12) cm À 1 /K, in agreement with dielectric continuum theory, dB Stark /dT = 0.096 cm À 1 /K (Eq. (12)). This corresponds to B Stark = (53 � 35) cm À 1 at T = 293 K. Directly measured shifts Δv(T), without subtracting the nonpolar contribution, result in dB p /dT = (À 0.07 � 0.14) cm À 1 /K ( Figure S4), implying no apparent temperature dependence. Hence the subtraction of the nonpolar contribution is crucial when the shifts in polar solvents are modest in magnitude. 0.096 cm À 1 /K (Eq. (12)) estimated by the dielectric continuum theory. We therefore ascribe the temperature dependence in Figure 4 to the Stark contribution that corresponds to B Stark = (53 � 35) cm À 1 at T = 293 K.
It is worth noting that directly measured shifts Δv(T), without subtracting the nonpolar contribution, result in dB p / dT = (À 0.07 � 0.14) cm À 1 /K (see Figure S4), that means in fact no actual temperature dependence. Thus, the subtraction of the nonpolar contribution is crucial when the polar shifts are modest in magnitude.

Conclusion
In summary, we derived the Stark shift of the S 0 !S 1 band of anthracene from temperature-dependent solvatochromic absorption shifts. The obtained derivative of the Stark slope dB Stark /dT = (0.18 � 0.12) cm À 1 /K corresponds to B Stark = (53 � 35) cm À 1 at T = 293 K, that constitutes approximately 10-20 % of the full slope B p in polar solvents observed for anthracene and many other nondipolar chromophores. The measured Stark shift is in good agreement with the estimate from dielectric continuum theory.
To calculate the true shifts Δv p in polar solvents, it is necessary to subtract from directly measured shifts Δv the nonpolar contribution which can be precisely determined by solvatochromic measurements in nonpolar solvents. The subtraction is especially necessary when the shifts Δv p are modest in magnitude that is usually the case for nondipolar or weakly polar chromophores.