Non-Condon Effects on the Doubly Resonant Sum Frequency

Dec 30, 2013 - However, a double-resonant effect can occur if the IR laser is resonant with ... They also developed a theory for DR-SFG and used this ...
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Non-Condon Effects on the Doubly Resonant Sum Frequency Generation of Rhodamine 6G Philip A. Weiss, Daniel W. Silverstein, and Lasse Jensen* Department of Chemistry, The Pennsylvania State University, 104 Chemistry Building, University Park, Pennsylvania 16802, United States S Supporting Information *

ABSTRACT: We report first-principles simulations of the doubly resonance sumfrequency generation (DR-SFG) spectrum for rhodamine 6G (R6G). The simulations are done using a time-dependent formalism that includes both Franck−Condon (FC) and Herzberg−Teller (HT) terms in combination with time-dependent density functional theory (TDDFT) calculations. The simulated spectrum matches experiments, allowing a detailed assignment of the DR-SFG spectrum. Our work also shows that nonCondon effects are important and the DR-SFG spectrum of R6G is highly dependent on both FC and HT modes. This is surprising as R6G is known to be a strong FC resonant Raman scatterer. The simulations predict an orientation where the xanthene plane of R6G is perpendicular to the surface with binding through one of the ethyl amine groups. Our results show the importance of first-principles simulations for providing a detailed assignment of DR-SFG experiments, especially for large molecules where such an assignment is complicated due closelying vibrational modes. SECTION: Spectroscopy, Photochemistry, and Excited States ingly, they found that only the modes at 1573 and 1657 cm−1 were strongly resonance Raman active. This is rather surprising because the selection rules of SFG typically state that the modes have to be both IR and Raman active. Furthermore, because experimental and theoretical studies have shown that R6G exhibits very strong RRS,16−18 one would expect that only modes that are Raman active would be observed. To understand this, we report first-principles simulations of DR-SFG for R6G based on a time-dependent formalism in combination with time-dependent density functional (TDDFT) calculations. This approach includes both FC and Herzberg−Teller (HT) effects and has previously been used to successfully describe RRS and resonance hyper-Raman scattering of R6G.19,20 Using this approach, we will show that the observation of IR active modes is due to non-Condon effects in the DR-SFG. While these non-Condon effects are unimportant in RRS of R6G, they become detectable in DRSFG due to a specific orientation of the R6G relative to the surface along with the relatively strong IR activity for those modes. In contrast to the simulations of traditional vibrational SFG, which can be calculated directly from the polarizability and dipole derivatives, not much has been reported on simulations of DR-SFG.21−23 Most recently, Zheng et al.23 have presented theory and simulations of R-1,1′-bi-2-naphthol at the water interface. They used a sum-over-states approximation based on TDDFT calculations and included both FC and HT

V

ibrational sum-frequency generation (SFG) has become an important spectroscopic technique for characterizing the orientation of molecules at surfaces and interfaces due to its selectivity for noncentrosymmetric systems.1−5 SFG was first reported in 1987 by Shen6,7 and relies on overlapping a visible (ωVIS) and tunable infrared (ωIR) laser at the surface of the sample while monitoring the response of the sum-frequency (ωSFG = ωVIS + ωIR). This is a second-order nonlinear optical process, which is forbidden in centrosymmetric bulk media, making the technique interface-specific. In traditional vibrational SFG, the IR laser is tuned in resonance with molecular vibrations, providing information about the molecular geometry. However, a double-resonant effect can occur if the IR laser is resonant with a molecular vibration and the visible laser with an electronic transition.8,9 In doubly resonant (DR)-SFG, a coupling of the vibrational and electronic transitions occurs similarly to that in resonance Raman scattering (RRS). Thus, DR-SFG offers the same advantages as RRS but with the additional benefit of being surface-specific, making it highly suitable for characterization of low-coverage surface species.10−14 Previously, experimental DR-SFG was performed on rhodamine 6G (R6G) on fused quartz by tuning the visible laser through the first electronic transition.8,9 Raschke et al.8,9 observed the DR-SFG of R6G for four bands at 1514, 1573, 1614, and 1657 cm−1. The mode at 1614 cm−1 was assigned to the phenyl group, whereas the other three modes were assigned to the xanthene ring. They also developed a theory for DR-SFG and used this to interpret the DR-SFG signal by fitting the experimental data based on a Franck−Condon (FC) description and available relative IR intensities.8,9,15 Interest© 2013 American Chemical Society

Received: November 22, 2013 Accepted: December 30, 2013 Published: December 30, 2013 329

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between the ground state |0⟩ and the excited state |k⟩, E is the corresponding energy for that state, ω is the energy of either the SFG or IR laser, Γelk is the electronic lifetime, and Γvib L is the vibrational lifetime. The first term in the parentheses describes the transition polarizability and the second term the IR transition. The first-hyperpolarizability can be split into multiple terms through a Taylor series expansion of the coordination dependence of the electronic transition dipole moment as β = A + B + ..., where the A term describes the FC contribution and the B term is the first HT contribution. β can be converted to a time-dependent formalism so that the sum over all intermediate vibrational states K is avoided.24,26,27 Full details of the derivation are given in the Supporting Information. The FC term is then given by

contributions, although, the HT terms were found to be insignificant. Here, we present simulations that are based on the time-dependent formalism pioneered by Heller and coworkers.24−28 The time-dependent formalism offers the advantage of recasting the explicit summation over the intermediate vibrational states into a time integral. This makes the method computationally attractive for dealing with large systems. We have previously implemented a timedependent wave packet formalism for one-photon absorption, two-photon absorption, resonance Raman, and hyper-Raman spectroscopies, and here, we extend this formalism to describe DR-SFG.29,30 When nonresonant effects are insignificant, the signal of the SFG is related to the square of the macroscopic second-order nonlinear susceptibility (2) 2 ISFG ∝ |χIJK |

(1)

where χ(2) IJK is (2) χIJK =

∑ Ns⟨RIαRJβRKγ⟩βαβγ αβγ

ℏ 2ωL

AL = −

e

(2)

R is Euler angle rotational elements, Ns is the molecular surface density, and βαβγ is the microscopic first-hyperpolarizability. In Figure 1, the definition of the Euler angles is given. The tilt

∑ (μα0k )eq (μβk0 )eq i ∫

0

k

i(ϵ I0 + ωSFG)t − gk (t )

dt ×

∞

⟨L0|Ik(t )⟩

(μγ00 )L ′ iΓvib L

(4)

Similarly, the HT term for the hyperpolarizability is given by B1L = −∑ ∑ k

×i

∫0

1 (μ0k )eq (μβk 0 )b ′ ωbωL α

( Lb + 1 ⟨(Lb + 1)0 |Ik(t )⟩

Lb ⟨(Lb − 1)0 |Ik(t )⟩)e i(ϵ I0 + ωSFG)t − gk(t ) dt

+ ×

b ∞

ℏ 2

(μγ00 )L ′ iΓLvib

(5)

and B2L = −∑ ∑ k

×i

Figure 1. Definition of the Euler angles. θ is the tilt angle between the z axis of the reference geometry and the new geometry, while the ψ is the rotation around that axis. The image of the molecule was generated using VMD.31

×

∫0

a ∞

ℏ 2

1 (μ k 0 )eq (μαk 0 )a ′ ωaωL β

Ia + 1 ⟨(Ia + 1)0 |Lk (t )⟩e i(ϵ I0 + ωSFG)t − gk(t ) dt

(μγ00 )L ′ iΓvib L

(6)

eq where (μ0k α ) is the transition dipole moment evaluated at the m′ ground-state equilibrium position, (μ0k is the derivative of α) the transition dipole moment with respect to the normal mode, and Qm is the normal mode coordinate. For clarity, the HT term has been split into two separate terms. The form of the wave packet integrals are well-known in the literature and may be solved analytically assuming the independent mode harmonic oscillator model.27,30,32 Transforming the resulting hyperpolarizbility tensor, the nonlinear susceptibility is obtained with respect to the Euler angles. Depending on the experimental polarization of light, different orientational information of the normal modes is observed and may be obtained by rotationally averaging the hyperpolarizability tensor with respect to the axis normal to the surface.33 Most commonly, SFG experiments are probed with SSP polarized light where the SFG and visible light electric field vectors are perpendicular to the surface and the IR electric field vector lies parallel to the surface.

angle θ corresponds to the angle between the z axis in the reference geometry and the z axis for the molecular frame of reference z axis, while the twist angle ψ represents the z rotation axis with respect to the molecular coordinates. The first-hyperpolarizability may be determined from the molecular quadratic response function. Under the Born− Oppenheimer approximation and assuming that both electronic and vibrational states are on resonance, the first-hyperpolarizability for DR-SFG may be written as L βαβγ (− ω VIS − ωIR ; ω VIS , ωIR )

⎛ ⟨I0|μα0k |Kk⟩⟨Kk|μβk 0 |L0⟩ ⎞ ⟨L0|μγ00 |I0⟩ ⎟ = ⎜⎜∑ el ⎟ vib ⎝ K , k Ek − E0 − ωSFG − iΓ k ⎠ E L0 − E I0 − ωIR − iΓL (3)

where I0 and L0 are the ground- and excited-state vibrational states respectively, μ0k α is the electronic transition dipole integral 330

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Figure 2. (a) Comparison between the simulated DR-SFG spectrum convoluted with a Lorentzian at 10 cm−1 fwhm, an incident excitation wavelength of 577 nm with a θ = 72°, ψ = 63° orientation, and experimental data taken from ref 8. Simulated spectra including only FC or HT terms are also shown. The spectra are offset for clarity. (b) The comparison between the simulated DR-SFG spectrum and the normal SFG spectrum of R6G. (c) The reference orientation of R6G and (d) the predicted orientation (θ = 72°, ψ = 63°) relative to an interface in the xy plane. Images of the molecules were generated using VMD.31

1579 cm−1 on the ethyl amine groups, and the mode at 1507 cm−1 on the methyl groups with contributions from the xanthene ring and the ethyl amine group. See the Supporting Information for depictions of these normal mode vibrations. In Figure 2, we also compare the simulated off-resonance SFG spectrum with the DR-SFG spectrum for R6G in the wavelength region between 1200 and 1750 cm−1. Excitation into the S1 transition leads to an enhancement of the SFG signal by about a factor of 104. Intriguingly, this is much less than the enhancement of about 106 found in RRS,16−18 even though one would expect similar enhancements. Furthermore, the DR-SFG spectrum looks drastically different with several new modes appearing. Thus, DR-SFG of R6G provides, in addition to a resonance enhancement, complementary information to the off-resonance SFG spectrum. Also depicted in Figure 2 is the breakdown of the simulated spectrum into contributions arising from FC terms or HT terms. Surprisingly, we see that the modes at 1579 and 1657 cm−1 arise from FC scattering, whereas the modes at 1507 and 1621 cm−1 are due to HT scattering. While the strongest band at 1657 cm−1 is a FC active mode, the two modes caused by HT scattering are almost as intense. This is rather unexpected as R6G is a very strong FC scatterer in RRS. To understand the unusual DR-SFG spectrum of R6G, we have collected in Table 1 the IR, RRS, and DR-SFG intensities.

In Figure 2, we plot the simulated DR-SFG spectrum of R6G in the wavelength region between 1500 and 1750 cm−1. The simulations are done using a visible laser excitation at 577 nm, which corresponds to resonance with the lowest excited state S1 for R6G in IR−vis DR-SFG (see the Computational Details section). For comparison, we also included the experimental spectrum obtained on a quartz substrate using a 590 nm excitation.8 We have not included any effects arising from interactions with the quartz substrate in the simulations. The simulated spectrum is obtained using an orientation characterized by θ = 72° and ψ = 63°, which was found to give the best match between theory and experiments. The θ angle corresponds to the tilt of the xanthene plane relative to the surface and the ψ angle to the rotation of the xanthene plane in the surface plane. See Figure 2d for the illustration of the orientation relative to the surface. We find generally good agreement between the simulated spectrum and the experiments. The asymmetric line shape observed in the experimental spectrum can be attributed to interference with the nonresonant background, χ(2) NR, which is not accounted for in the simulations. The DR-SFG spectrum of R6G in the region between 1500 and 1750 cm−1 is characterized by four major peaks at 1507, 1579, 1621, and 1657 cm−1 and is in good agreement with the experimental spectrum. The modes at 1621 and 1657 cm−1 are localized on the xanthene ring, the mode at 331

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Table 1. IR, RRS, and DR-SFG Intensities for R6G Obtained at the Predicted Orientationa mode

IR

RRS

DR-SFG

b′ (μ10 x )

b′ (μ10 y )

b′ (μ10 z )

Δ

assignment

1503 1507 1514 1535 1561 1573 1579 1581 1607 1621 1657 1729

0.01 0.58 0.03 0.34 0.02 0.45 0.25 0.02 0.01 1.11 0.33 0.27

0.94 278.10 13528.70 57.31 12487.49 1.73 8459.70 1290.92 1462.50 181.40 21791.12 257.21

0.00 23020.41 1625.19 8836.85 474.83 61.11 13444.61 1726.63 724.58 24656.62 37880.80 292.96

0.00 −0.28 0.00 −0.13 0.00 −0.02 0.00 0.00 0.00 0.25 0.00 0.00

0.01 0.00 0.25 0.00 −0.05 0.00 0.02 −0.01 0.00 0.00 −0.23 −0.01

0.00 −0.06 0.00 0.02 0.00 −0.01 0.00 0.00 0.00 0.05 0.00 0.00

0.00 0.00 −0.14 0.00 0.14 0.00 0.12 0.05 0.05 0.00 −0.20 0.02

Met, Eth Met,Xan,Eth Xan,Eth Xan,Eth Xan Eth Eth Phe Phe Xan Xan Est

a

b′ Physical parameters of R6G used for simulations at the reference orientation where (μ10 α ) refer to derivatives of the transition dipole moment with respect to normal mode displacement, and Δ is the dimensionless displacement between the harmonic ground- and excited-state energy wells. The IR intensity is in atomic units, RRS in 10−30 cm2/sr, DR-SFG in 10−80 m8/V2, and the transition dipole moment derivatives in atomic units. Normal mode assignments are xanthene ring (Xan), methyl groups (Met), ethylamine (Eth), phenyl group (Phe), and ester (Est). The S1 excitation energy was calculated to be be 2.89 eV (shifted to 2.34 eV to match the experimental absorption in solutions.) with a transition dipole moment of (0.01632, −3.24101, 0.00303) au.

Figure 3. The orientational dependence of DR-SFG intensity for two of the major R6G vibrational modes represented by the Euler angles θ (tilt) and ψ (twist). The peaks are normalized to the maximum at 1657 cm−1. The predicted orientation is marked with an “x”.

intensity. Raschke et al.8,9 previously explained the DR-SFG spectrum assuming a FC description for all modes and available relative IR intensities. However, here, we show that the two modes at 1507 and 1621 cm−1 derive their DR-SFG intensity from a non-Condon RRS mechanism. The results presented here show that first-principles simulations of the DR-SFG spectrum can provide detailed assignment of the experimental spectrum. This is particularly important for large molecules such as R6G, which have many close-lying vibrational modes. Because R6G is almost Cs symmetric with a single mirror plane perpendicular to the xanthene ring, the FC active modes are symmetric with respect to the mirror plane, while the HT active modes are antisymmetric with respect to the mirror plane. On the basis of the symmetry, we would expect the IR and Raman modes to be almost mutually exclusive. This is found to be true for most of the modes collected in Table 1 except for the modes at 1579 and 1657 cm−1. These two modes are both IR and Raman active and thus appear strong in the DR-SFG spectrum. For the FC modes, the IR intensities result from changes in the ground-state dipole moment perpendicular to the xanthene plane (x and z axes; see Figure 2c for the definition of the axes), whereas the RRS intensities arise from coupling with the S1 transition dipole moment, which is oriented along the xanthene plane (y axis). In the case of the

The dimensionless displacements and transition dipole moment derivatives used to calculate the RRS and DR-SFG intensities are also listed. The selection rules of SFG dictate that the modes have to be both IR active and Raman active to appear in the spectrum. We see that this is the case for the modes at 1579 and 1657 cm−1, which have strong IR and RRS intensities. Both modes are symmetric vibrations in the xanthene plane that couple strongly to the S1 transition and should show a significant RRS intensity. However, this is not the case for the modes at 1507 and 1621 cm−1, which are antisymmetric vibrations relative to the xanthene plane and do not couple strongly with S1. Therefore, these two modes have low RRS intensities. This is also reflected in Table 1 where the electron−phonon coupling (Δ) parameters for the different vibrations are listed. This shows that the modes at 1579 and 1657 have significant electron−photon coupling with the S1 transitions whereas the two other modes do not couple to this electronic state. The mode that appears in the experimental spectrum at 1614 cm−1 was previously assigned to the phenyl group.8 However, our results indicate that this is not the case as the phenyl mode (found in the simulations at 1607 cm−1) has strong RRS intensity but is not IR active and thus not responsible for the DR-SFG band. The two modes at 1507 and 1621 cm−1 have significant IR intensity but only weak RRS 332

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excitation wavelengths. We see that the four strong modes still dominate the spectra. However, in contrast to the experiments, we see a second intensity maximum at about 530 nm due to the 0−1 transition. This second maximum is well-establish in the RRS spectrum of R6G from both theory and experiments.17,18 In the original simulations, a much larger damping factor for the 0−1 transition was used to avoid this extra maximum. Because the data used in the simulations correctly predict both the RRS and RHRS spectrum,19,20 we believe that the reason that the 0−1 transition is not observed is due to interactions between R6G molecules on the surface. It is well-known that aggregates of R6G lead to dimer absorption that coincides with the vibronic band in the absorption spectrum.35 Support for this comes from the absorption spectrum presented in ref 8, which does not show the characteristic ratio between the shoulder and main absorption band found for the free R6G molecule. Furthermore, the nonresonance background was also found to be wavelength-dependent with a maximum at the 0−0 transition. Neither of these effects are included in the simulations and thus could explain the discrepancy. Further work is needed to determine the effects of interactions with the substrate and other R6G molecules. In summary, we have reported first-principles simulations of DR-SFG for R6G based on a time-dependent formalism in combination with TDDFT calculations. In order to obtain good agreement between simulations and experimental results, both FC and HT terms were needed. Contrary to previous assignment, our work showed that non-Condon effects should be considered for DR-SFG, and remarkably, both FC and HT modes for R6G have comparable intensities in the spectrum. This is surprising because R6G is known as a strong RRS scatterer, and one would expect the FC modes to dominate the spectrum. The generally weaker HT terms were shown to be important in the DR-SFG because the predicted orientation of R6G has the xanthene ring perpendicular to the surface through one of the ethyl amine groups. This particular orientation biases HT modes while disfavoring the generally stronger FC modes. The results presented here show that first-principles simulations of the DR-SFG spectrum can provide detailed assignment of the experimental spectrum for large molecules such as R6G with many close-lying vibrational modes.

HT modes, the changes in the dipole moment are parallel with the xanthene plane (y axis), and the RRS intensity arises from a HT scattering mechanism through the transition dipole moment derivatives perpendicular to the xanthene plane (x and z axis). The experimental spectrum of R6G is obtained using SSP polarization, which probes vibrations with components of the IR dipole moment perpendicular to the surface and components of the Raman transition moment parallel to the surface. Thus, observations of FC modes would indicate that the xanthene ring is oriented parallel to the surface, and observations of HT modes indicate that the xanthene ring is perpendicular to the surface. Therefore, observation of both sets of modes in the experimental spectrum indicates that the orientation of R6G relative to the surface should be something in between. However, it is still surprising that both set of modes are observed because the IR intensities are similar among all observed modes, whereas the RRS intensity is drastically different, indicating that the FC modes should dominate the spectrum. This unusual behavior can be understood by considering the angular dependence of the FC and HT modes. In Figure 3, the DR-SFG intensity for the modes at 1621 and 1657 cm−1 is plotted varying the tilt (θ) and twist (ψ) angles relative to the reference geometry in Figure 2c. For comparison, the intensities have been normalized to the maximum of the 1657 cm−1 mode. As expected, the maximum intensity of the FC modes is about 100 times more intense than the maximum intensity for the HT modes. Also, the angular dependence is such that the maximum of the FC modes coincides with the minimum of the HT modes and vice versa. The HT modes are found to be more sensitive to the tilt angle, θ, than the FC modes, which most likely is a result of the fact that the HT modes require an orientation where the xanthene plane is perpendicular to the surface. An “optimum” orientation was obtained by matching the relative intensity of the four main peaks to that found in the experiments. This was done by selecting orientations where the intensity of the three modes at 1579, 1621, and 1657 cm−1 relative to the mode at 1507 cm−1 found in the simulations matched the experimental relative intensity within a tolerance of ∼50%. The relative intensities were found to be very sensitive to the orientation, and thus, this comparison leads to a small set of suitable orientations characterized by θ = 72 ± 3°, ψ = 63 ± 3°, as shown in Figure 2c. Because the effect of the substrate is not included in the simulations, there are other sets of angles related by symmetry that also provide an equally good match. These values are comparable to multiple polarized second-harmonic generation (SHG) experiments, which found an orientation of θ = 60° ± 2° and 55−60°, ψ = 50.2 ± 3°.34,35 Therefore, both the DR-SFG and SHG experiments indicate that R6G binds to the surface with the xanthene plane in an upright position through one of the ethyl amine groups. From Figure 3, we see that this orientation is close to a maximum intensity for the HT modes and a minimum intensity for the FC modes. Consequently, the specific orientation of R6G explains the unusual DR-SFG spectrum, which allows for the simultaneous detection of both FC and HT modes with comparable intensities. Although the results presented in this work are specific to R6G, we expect that non-Condon effects will be of importance in the DR-SFG for other molecules where the IR and Raman modes are distinct from each other. In the Supporting Information, we present the DR-SFG excitation profile of R6G by scanning over different visible

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COMPUTATIONAL DETAILS The electronic structure parameters needed for simulating the DR-SFG spectrum were taken from our previous work on describing the resonance hyper-Raman spectrum of R6G.19,20 This set of parameters has previously been shown to correctly predict both the RRS and RHRS spectra of R6G and thus provide a reliable description of the electronic structure of R6G in the ground and first-excited state probed in this work. The computational details are briefly summarized here. The groundstate equilibrium structure and normal mode frequencies were calculated using the B3LYP functional and 6-311G* basis set using the NWChem program package.36 The vibrational modes were scaled by 0.98 to better match the experimental results. The excitation energy gradients needed for evaluating the dimensionless displacements were calculated using three-point numerical differentiation of the vertical excitation energies for the three lowest states using NWChem. The excited-state lifetime parameters were obtained to match the experimental one-photon and two-photon absorption line shapes. For the DR-SFG simulations, the lowest three excited states were included, although the intensity arose predominately from the 333

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first excited state. The polarizability derivatives needed to simulate the normal SFG spectrum were obtained from threepoint numerical differentiation using the Dalton program.37 The SFG signal is expressed in terms of the macroscopic nonlinear second-order susceptibility, χ(2). The χ(2) is obtained by averaging the molecular hyperpolarizability in terms of the IR, visible, and SFG light polarization as detailed in ref 33. In this work, all spectra were generated assuming SSP light polarization; however, the full first-hyperpolarizability tensor is constructed so that other polarization combination can easily be studied if experimental data is available. Also, experimental results show that R6G is resonant at 590 nm for DR-SFG, while R6G typically has an maximum absorbance at around 530 nm.8 The red shift indicates that the molecule is first vibrationally excited followed by an electronic excitation and therefore corresponds to IR−vis DR-SFG. In all of the simulations, the effect of the substrate is neglected.

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(9) Hayashi, M.; Lin, S. H.; Raschke, M. B.; Shen, Y. R. A Molecular Theory for Doubly Resonant IR−UV−Vis Sum-Frequency Generation. J. Phys. Chem. A 2002, 106, 2271−2282. (10) Miyamae, T.; Ito, E.; Noguchi, Y.; Ishii, H. Characterization of the Interactions Between Alq3 Thin Films and Al Probed by TwoColor Sum-Frequency Generation Spectroscopy. J. Phys. Chem. C 2011, 115, 9551−9560. (11) Miyamae, T.; Takada, N.; Tsutsui, T. Probing Buried Organic Layers in Organic Light-Emitting Diodes under Operation by ElectricField-Induced Doubly Resonant Sum-Frequency Generation Spectroscopy. Appl. Phys. Lett. 2012, 101, 073304. (12) Nagahara, T.; Fukushima, H.; Aida, M.; Ishibashi, T.-A. SelfAssembled Monolayers of Cyanoterphenyl Terminated Alkyl Disulfides Studied by Vibrationally and Electronically Doubly-Resonant Sum-Frequency Generation Spectroscopy. Chem. Phys. Lett. 2011, 506, 190−195. (13) Wu, D.; Deng, G.-H.; Guo, Y.; Wang, H.-f. Observation of the Interference between the Intramolecular IR−Visible and Visible−IR Processes in the Doubly Resonant Sum Frequency Generation Vibrational Spectroscopy of Rhodamine 6G Adsorbed at the Air/ Water Interface. J. Phys. Chem. A 2009, 113, 6058−6063. (14) Bozzini, B.; D’Urzo, L.; Mele, C.; Busson, B.; Humbert, C.; Tadjeddine, A. Doubly Resonant Sum Frequency Generation Spectroscopy of Adsorbates at an Electrochemical Interface. J. Phys. Chem. C 2008, 112, 11791−11795. (15) Huang, J. Y.; Shen, Y. R. Theory of Doubly Resonant Infrared− Visible Sum-Frequency and Difference-Frequency Generation from Adsorbed Molecules. Phys. Rev. A 1994, 49, 3973−3981. (16) Jensen, L.; Schatz, G. C. Resonance Raman Scattering of Rhodamine 6G as Calculated Using Time-Dependant Density Functional Theory. J. Phys. Chem. A 2006, 110, 5973−5977. (17) Shim, S.; Stuart, C. M.; Mathies, R. A. Resonance Raman CrossSections and Vibronic Analysis of Rhodamine 6G from Broadband Stimulated Raman Spectroscopy. ChemPhysChem 2008, 9, 697. (18) Guthmuller, J.; Champagne, B. Resonance Raman Spectra and Raman Excitation Profiles of Rhodamine 6G from Time-Dependent Density Functional Theory. ChemPhysChem 2008, 9, 1667−1669. (19) Milojevich, C. B.; Silverstein, D. W.; Jensen, L.; Camden, J. P. Probing Two-Photon Properties of Molecules: Large Non-Condon Effects Dominate the Resonance Hyper-Raman Scattering of Rhodamine 6G. J. Am. Chem. Soc. 2011, 133, 14590−14592. (20) Milojevich, C. B.; Silverstein, D. W.; Jensen, L.; Camden, J. P. Surface-Enhanced Hyper-Raman Scattering Elucidates the TwoPhoton Absorption Spectrum of Rhodamine 6G. J. Phys. Chem. C 2013, 117, 3046−3054. (21) Vallet, J. C.; Boeglin, A. J.; Lavoine, J. P.; Villaeys, A. A. Vibronic Mode Couplings in Adsorbed Molecules Analyzed by Doubly Resonant Sum-Frequency Generation. Phys. Rev. A 1996, 53, 4508− 4518. (22) Zalesny, R.; Bartkowiak, W.; Champagne, B. Ab Initio Calculations of Doubly Resonant Sum-Frequency Generation Second-Order Polarizabilities of LiH. Chem. Phys. Lett. 2003, 380, 549−555. (23) Zheng, R.-H.; Wei, W.-M.; Jing, Y.-Y.; Liu, H.; Shi, Q. Theoretical Study of Doubly Resonant Sum-Frequency Vibrational Spectroscopy for 1,1-Bi-2-naphthol Molecules on Water Surface. J. Phys. Chem. C 2013, 117, 11117−11123. (24) Heller, E. J. The Semiclassical Way to Molecular Spectroscopy. Acc. Chem. Res. 1981, 14, 368−375. (25) Heller, E. J.; Sundberg, R. L.; Tannor, D. Simple Aspects of Raman Scattering. J. Phys. Chem. 1982, 86, 1822−1833. (26) Lee, S.-Y.; Heller, E. J. Time-Dependent Theory of Raman Scattering. J. Chem. Phys. 1979, 71, 4777−4788. (27) Tannor, D. J.; Heller, E. J. Polyatomic Raman Scattering for General Harmonic Potentials. J. Chem. Phys. 1982, 77, 202−218. (28) Heller, E. J. Frozen Gaussians: A Very Simple Semiclassical Approximation. J. Chem. Phys. 1981, 75, 2923−2931.

ASSOCIATED CONTENT

S Supporting Information *

Detailed derivation of the time-dependent DR-SFG formalism, relevant normal mode images, and the DR-SFG excitation profile is presented. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the NSF Center for Chemical Innovation dedicated to Chemistry at the Space-Time Limit (CHE-082913). We acknowledge support received from Research Computing and Cyberinfrastructure, a unit of Information Technology Services at Penn State.

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