Calculation of Absolute Resonance Raman Intensities - American

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Calculation of Absolute Resonance Raman Intensities: Vibronic Theory vs Short-Time Approximation† Krista A. Kane and Lasse Jensen* Department of Chemistry, PennsylVania State UniVersity, UniVersity Park, PennsylVania 16802 ReceiVed: June 30, 2009; ReVised Manuscript ReceiVed: August 10, 2009

We present the absolute resonance Raman scattering (RRS) intensities of uracil, rhodamine 6G (R6G), and iron(II) porphyrin with imidazole and CO ligands (FePImCO) calculated using density functional theory (DFT). The spectra are calculated using both the vibronic theory and the short-time approximation. We find that the absolute RRS intensities calculated using the short-time approximation are severely overestimated, as compared with results obtained using the vibronic theory. This issue is attributed to the sensitivity of the absolute RRS intensities to the adjustable damping factor within the short-time approximation. This is illustrated for uracil, for which the relative intensities were predicted accurately using the short-time approximation, but the absolute intensities were still overestimated. Although intensities comparable to that obtained with the vibronic theory could be obtained using the short-time approximation, it requires a large damping factor, roughly twice that estimated from the absorption spectrum, to be used in the simulations. Furthermore, we find that DFT underestimates the absolute RRS intensities for R6G as compared to experiments, which is most likely due to the neglect of solvent effects in the calculations. For R6G and FePImCO, vibronic effects are shown to enhance the low-frequency modes relatively more, improving the agreement with experiments. Introduction Resonance Raman scattering (RRS) is an important tool in analytical chemistry due to a higher sensitivity and selectivity than normal Raman scattering.1-11 This is the case for structural characterization of proteins due to a selective enhancement of the π f π* excitation in amides12,13 and for carbon nanotubes, in which the resonance-enhanced radial breathing mode depends on the diameter of the tube.14-16 The higher sensitivity arises when the wavelength of the radiation is close to an electronic excitation of the molecule, which can enhance the intensity of the signal by a factor of up to 104-106. At resonance, only vibrations that couple strongly to the excited state will be enhanced, leading to greater selectivity. In addition to this, RRS also enables information about the excited state to be determined. The theory of RRS is well-established and is generally based on either the vibronic theory of Albrecht and co-workers,1-3 or the time-dependent approach due to Heller.4-6,8 In the vibronic theory, the Born-Oppenheimer approximation is adapted to separate the vibronic states into products of electronic and vibrational states, followed by an expansion of the transition dipole moments into a Taylor series in the nuclear coordinates. This allows for the identification of different terms; that is, the A (Franck-Condon) and B (Herzberg-Teller) terms, contributing to the overall Raman intensities. A limitation of the vibronic model is that it contains sums over the intermediate vibrational states, which quickly become intractable for large molecules with many vibrational modes. For this reason, the “transform theory” method7,17-20 based on a Kramers-Kronig transform of the absorption spectrum to obtain the Raman profile is often adapted. In the approach of Heller, the sum overstates is avoided by using a time-dependent formalism in terms of wave packet dynamics. An important result is that when only short-time dynamics is important, the relative intensities are given by the †

Part of the “Barbara J. Garrison Festschrift”. * Corresponding author. E-mail: [email protected].

relative gradient of the Franck-Condon (vertical) excited-state surface with respect to the normal coordinates. Most theoretical studies of RRS rely on the independent mode displaced harmonic oscillator (IMDHO) model, which requires the calculation of the excited states displacements.21-24,24-26 This can now be done efficiently using electronic structure methods either from the gradient of the excited states or by finding the excited state minimum using geometry optimization. From the displacements, the RRS intensities can be calculated either directly using the excited state gradient method or from the vibronic theory or time-dependent formalism. Often, it is assumed that only one excited state is in resonance with the incident light; however, multiple excited states can be taken into account, as shown by Jarzecki using a weighted-gradient approximation.23 Recently, Jensen and Schatz presented a method based on geometrical derivatives of the real and imaginary frequency-dependent polarizabilities, which naturally includes multiple excited states.27 This method was used to study surface-enhanced Raman scattering (SERS).28-31 Often, the comparison between theory and experiments is performed using relative RRS intensities, which is sufficient to analyze and assign the RRS spectra. However, a more stringent test of the theoretical approach would be the direct calculation of absolute intensities. Furthermore, information about the absolute RRS cross section is crucial in determining accurate enhancement factors in SERS.32 This was illustrated by Jensen and Schatz33 in the calculation of the absolute RRS intensities of rhodamine 6G (R6G), a typical SERS chromophore. Gaff and Franzen also found good agreement between simulations and experimental RRS cross sections using Heller’s timedependent approach to calculate the absolute RRS intensities of aromatic amino acid models.25 In this work, we will present a comparison between the absolute RRS intensities calculated using the vibronic theory and the short-time approximation. We will present results for uracil, R6G, and an iron(II) porphyrin with imidazole and CO

10.1021/jp906152q  2010 American Chemical Society Published on Web 10/20/2009

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ligands (FePImCO). Uracil is selected, since it is expected that the short-time approximation is valid. Furthermore, for this molecule, we will compare results obtained using the excitedstate gradient, the polarizability gradient methods, and the vibronic model. For R6G, it is known that vibronic effects are important, and recent experimental absolute RRS intensities are available.24,34 FePImCO is a model of heme proteins, such as myoglobin or hemoglobin, with the CO ligand bound to the heme group, and for the porphyrin molecule, it is wellestablished that vibronic coupling effects are important.35 Comparison of the absolute RRS intensities obtained using each of these methods can provide insight into the accuracy of the short-time approximation.

vibrational level; |〈νg0j|νeuj〉|2 is the Franck-Condon factor; uj is the vibrational quantum number for normal mode, j; and Γ is the line width, equal to the half width at half-maximum of the absorption band. In contrast to the original transform theory, we will calculate Φ(ωL) from the sum-over-states approximation and not from a Kramers-Kronig transform of the experimental absorption spectra. This is also the approach taken by Guthmuller and Champagne.24,26 Within the IMDHO model, the Franck-Condon factors between the ground state, g, and the excited state, e, for the jth normal mode can be calculated analytically once the dimensionless displacement is known as38

|〈νg0j |νeuj〉| ) 2

Theory In a backscattering or forward-scattering geometry, the (normal and resonance) Raman intensites are given by differential cross sections (units of cm2/sr)36

I(180o) )

[ ]

Sj dσ (180o) ) Kj dΩ 45

(1)

where Sj is the scattering factor, which is a purely molecular property, in units of Å4/amu. The parameter Kj, which is independent of the experimental setup but depends on both the incident and scattered frequencies, is given by

Kj )

h 1 π2 υ˜ - υ˜ j)4 2 2( L 1 exp -hcυ ˜ j /kBT] [ ε0 8π cυ˜ j

(2)

where ε0 is the permittivity of vacuum, c is the speed of light in units of m/s, h is the Planck constant, kB is the Boltzmann constant, T is the temperature (taken to be 300 K), and υ˜ L and υ˜ j are the wavenumbers of the incident light and of the jth vibrational mode, respectively. In the following, we will compare three different ways of calculating the scattering factors: the vibronic theory, the excited state gradient method, and the polarizability gradient method. To calculate the RRS intensities using the vibronic theory, we will adopt the approach developed by Peticolas and Rush, assuming the IMDHO model.37 This approach was also recently adopted by Guthmuller and Champagne.24,26 The scattering factor for normal mode j is given by

Sj ) 12µ4

∆j2 |Φ(ωL) - Φ(ωL - ωj) | 2 2

(3)

where µ is the electronic transition dipole moment, ∆j is the dimensionless displacement, ωL is the frequency of the incident radiation, ωj is the frequency of the jth normal mode, and the function Φ(ωL) is given by: 3N-6

Φ(ωL) )

∑ u

∏ |〈νg0 |νeu 〉|2 j 3N-6

ω0 +

j

j

(4)

∑ ujωj - ωL - iΓ j

where ω0 is the frequency of the transition from the ground electronic state to the resonant electronic state for the zeroth

∆j2uj uj

2 uj !

e-∆j /2 2

(5)

In the IMDHO model, the dimensionless displacements are related to the excited state gradient by

( ) ∂E ∂qj

qj)0

) ωj(qj - ∆J)

|

qj)0

) -ωj∆j

(6)

where (∂E/∂qj)qj)0 is the partial derivative of the excited state electronic energy with respect to a ground state dimensionless normal mode at the ground state equilibrium position. All of the quantities invovled in eq 4 are determined from quantum chemical calculations, except for Γ, which is an adjustable parameter to account for homogeneous broadening. The multidimensional Franck-Condon factors needed to calculate Φ(ωL) are obtained using the two-dimensional array method of Ruhoff and Ratner.39 Finally, the absorption spectra can be calculated in a similar way:

A(ωL) ) ωLµ2Im Φ(ωL)

(7)

A simplification of the vibronic theory approach is the shorttime approximation or excited-state gradient method.4,10,22 Within the excited-state gradient method, the scattering factor for normal mode j is given by

( Γµ ) ω ∆ 4

Sj ) 12

2

2

j

j

(8)

where µ is the electronic transition dipole moment, Γ is an adjustable damping parameter, ωj is the frequency of normal mode j, and ∆j is the dimensionless displacement of normal mode j. This is a particularly simple model for calculating the RRS intensities, since it only requires the gradients of the excited states. The polarizability gradient method is also based on a short-time approximation but includes multiple excited states in the calculations.27 In this method, the scattering factor is given by

Sj ) 45Rj2 + 7β(R)j2

(9)

where the polarizability invariants are given by given by

1 j* Rj2 ) Re RjRRRββ 9

(

)

(10)

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(

j j* j* 3RRβ RRβ - RjRRRββ β(R)j ) Re 2 2

)

Kane and Jensen

(11)

These polarizability invariants are calculated by including a finite lifetime of the excited states within TDDFT.27 Computational Details All calculations in this study were done using the Amsterdam DensityFunctional(ADF)programsoftware.40,41 TheBecke-Perdew (BP86) XC-potential42,43 and a triple- ζ polarized Slater type (TZP) basis set have been used, except for the calculations for FePImCO, for which the TZ2P basis set has been used. The vibrational frequencies and normal modes were calculated within the harmonic approximation. For this reason, the BP86 functional has been chosen, since it usually gives harmonic frequencies close to experimental results without the use of scaling factors.44 The calculations of the excitation energies and oscillator strengths have employed the EXCITATION module of ADF.45-47 The dimensionless displacements needed for calculating the RRS intensities were obtained using the VIBRON module of ADF.40 The calculations of the RRS of uracil using the polarizability gradient method were performed following the method described in ref 48. In all of the calculations, the sum of the Franck-Condon factors was g0.96. Results and Discussion Several test systems are used to determine the accuracy and consistency of the vibronic model that we implemented. Initially, the normal Raman intensities of 2-bromo-2-methylpropane (2B2MP) are determined and compared to both experimental and theoretical results available in the literature. Subsequently, the RRS intensities of carbon disulfide (CS2) are calculated and compared with experimental results. The RRS intensities for the S2 state of uracil are determined using the short-time approximation and are compared with the intensities obtained using the polarizability gradient model.27 In principle, the intensities calculated using the short-time approximation and the polarizability gradient model should be comparable, since they are both computed using a single excited state. R6G and FePImCO are affected by vibronic coupling, and so these are ideal systems to investigate the accuracy of the intensities derived from the vibronic theory versus the short-time approximation. 2-Bromo-2-methylpropane. To establish the accuracy of our approach to determine absolute nonresonant Raman scattering intensities, we calculated the Raman intensities of 2B2MP in vacuum at 633 nm excitation. 2B2MP is a good benchmark system, since there is experimental data available in the gas phase with which to compare it.49 In addition, Le Ru et al.32 recently proposed this molecule as a good reference for determining absolute Raman intensities experimentally. The calculated differential cross sections for 2B2MP at 633 nm are compiled in Table 1 and compared with the experimental results. We see that our results are in excellent agreement with the experimental results, showing that DFT can predict accurate nonresonant Raman intensities. We are also in good agreement with the B3LYP/6-311++G(d,p) results of Le Ru et al.32 Carbon Disulfide. Myers et al.50 have reported absolute RRS total cross sections of CS2 measured in cyclohexane at 209 nm to be σ ) 77 × 10-23 cm2 for the fundamental mode at 653 cm-1. They reported a line width of Γ ) 0.025 eV in cyclohexane and a transition dipole moment of µ ) 3.08 au. We calculated the transition to be at 186 nm with a transition

TABLE 1: Comparison of DFT Results with Experimental Results49 for the Raman Active Modes of 2B2MP at 633 nma exptl49

DFT νb

pc

Sd

dσ/dΩe

dσ/dΩe

279 483 780

0.29 0.18 0.68

6.37 22.6 12.8

105 167 50.0

144 169 44.0

a All values correspond to the gas phase of 2B2MP. b Frequency in units of cm-1. c Depolarization ratio. d Scattering factor in units of Å4/amu. e Differential Raman cross section in units of 10-32 cm2/ sr.

dipole moment of µ ) 2.55 au, in reasonable agreement with the experimental result, especially considering that solvent effects are not included in the calculation. However, the calculated transition dipole moment is underestimated by around 20% as compared with the experimental value, which is significant, since the RRS intensities scale with µ4. Thus, to compare with the experimental results, we calculated the absolute RRS intensities of CS2 at 208 nm using the vibronic theory with Γ ) 0.0025 eV. To convert the differential cross section to total cross sections, we used σ ) (8π/3)((1 + 2F)/(1 + F)) (dσ/dΩ), where σ is the total cross section and F is the depolarization ratio.51 In converting from the differential cross section to the total cross section, the depolarization ratio was assumed to be 1/3, since only one excited state is considered.50 We obtained a value of σ ) 0.84 × 10-23 cm2 for the total cross section of CS2, which is about a factor of 9 lower than the experimental value. The underestimation of the transition dipole moment using TDDFT accounts for a factor of 2 of the difference, and local field corrections to the cross sections due to solvent effects account for about a factor of 3.32 Thus, corrected for these effects gives a σcorr ) 50 × 10-23 cm2, still about 30% lower than the experimental value. Therefore, it will be important to consider solvent effects to calculate accurate cross sections. Solvent effects on RRS spectra have recently been considered for the relative intensities,24,26,52 but so far, not for the absolute intensities. Uracil. The experimental absorption spectrum of uracil obtained in water is dominated by a transition at 259 nm.53 We calculate the transition to be at 262 nm in vacuum, which is in good agreement with the experiments. We calculated the RRS spectra of uracil at 262 nm using the three different methods, and the results are collected in Figure 1. The results from the two methods based on the short-time approximation (i.e., the polarizability gradient method and the excited state gradient method) are shown in Figure 1a and b, respectively. The damping parameter of Γ ) 0.1 eV used in the calculation was obtained by comparing the calculated absorption spectrum with the experimental spectrum. We see that the two spectra are in very good agreement with respect to both the relative intensities and the absolute values. The small differences in the vibrational frequencies is due to a slightly different numerical grid used in the two calculations. This illustrates that the two approaches are nearly identical when there is only one excited state in resonance with the incident light. In Figure 1c, we plot the RRS spectrum of uracil calculated using the vibronic theory and Γ ) 0.1 eV. We see that there is good agreement with the experimental spectrum (insert in Figure 1c).37 Uracil is not expected to be strongly influenced by vibronic coupling because the absorption spectrum is featureless, and we would expect the short-time approximation to be valid. Indeed, we find that the spectra calculated using the short-time approximation are in good agreement with the spectrum obtained

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Figure 1. Simulated RRS spectra of uracil using (a) the polarizability gradient method with Γ ) 0.1 eV, (b) the excited state gradient approach with Γ ) 0.1 eV, (c) the vibronic theory with Γ ) 0.1 eV, and (d) the excited state gradient model with Γ ) 0.5. RRS spectra have been broadened with a Lorentzian having a width of 20 cm-1. Insert in panel c is the experimental spectrum adapted from ref 37.

using the vibronic theory when considering the relative intensities. However, the vibronic theory predicts absolute intensities for uracil that are around a factor of 600 smaller. The reason for this is that the short-time approximation is very sensitive to the value of Γ used in the calculations, since the intensities scale as 1/Γ4. We find that for the RRS spectrum calculated using the excited state gradient method to be the same as that found from the vibronic theory, the damping factor must be 0.5 eV, as shown in Figure 1d. This value of Γ is much larger than expected from the absorption spectra. Therefore, the value of the damping factor must be selected very carefully for the shorttime approximation, even when the relative intensities are reasonable. Rhodamine 6G (R6G). Due to the importance of R6G as a single-molecule SERS probe, there have recently34 been several studies of its RRS spectrum both theoretically24,33 and experimentally. Jensen et al. used the polarizability gradient method to analyze the Raman scattering and RRS spectra of R6G and to estimate enhancement factors.33 Guthmuller and Champagne employed the vibronic theory to investigate the RRS spectra of R6G in vacuum and in ethanol for the S1 and S3 excited states.24 Recently, Shim et al. have experimentally obtained the RRS total cross sections of R6G using femtosecond stimulated Raman spectroscopy (FSRS).34 This represents the first measurement of absolute total RRS cross sections for R6G on resonance, since under traditional resonance Raman conditions, the spectrum is hidden due to the strong fluorescence. The calculated absorption spectrum of R6G in vacuum shows a strong band at 473 nm corresponding to the S0 f S1 transition.

The reported experimental absorption spectrum54 of R6G in ethanol has a wavelength of maximum absorption at 532 nm. In Figure 2a, we plot the simulated absorption spectra shifted to 532 nm to account for the solvent effects. Both the calculated and reported experimental absorption spectra have a weak vibronic shoulder, indicating the importance of vibronic coupling effects. We see that the excited state gradient method overestimates the absolute intensities by a factor of around 30. A much larger Γ ) 0.17 eV (see Figure 2d), is needed to make the intensities predicted by the short-time approximation comparable to that obtained from the vibronic theory. The vibronic effects cause the modes at 608 and 760 cm-1 to be enhanced relative to the strong peak at 1638 cm-1. This is in good agreement with the results of Guthmuller and Champagne24 and the experimental spectra34 (see insert in Figure 2c). The results of Guthmuller and Champagne24,55 show overall better agreement with the experimental spectrum, which is most likely due to the inclusion of solvent effects as well as using a hybrid functional. However, the results by Guthmuller and Champagne55 also show that the inclusion of vibronic coupling, solvent effects, and hybrid functionals are not sufficient to ensure a correct description of the intensities for the bands below 600 cm-1. This points to the importance of anharmonic corrections, which is currently unfeasible for a large molecule such as R6G. In Table 2, we compare the absolute RRS total cross sections calculated at 532 nm using the vibronic theory with those obtained experimentally for R6G in methanol at 532 nm by Shim et al.34 The simulated excitation energy has been shifted from 473 to 532 nm to account for solvent effects. Shim et al. utilized

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Figure 2. (a) Absorption spectrum of R6G calculated using the vibronic theory. Spectrum has been shifted from 473 nm to account for solvent effects. Simulated RRS spectrum of R6G at 532 nm calculated using the (b) excited state gradient method with Γ ) 0.07 eV, (c) vibronic theory with Γ ) 0.05 eV, and (d) excited state gradient method with Γ ) 0.17 eV. RRS spectra have been broadened with a Lorentzian having a width of 20 cm-1. Insert in panel c shows the experimental spectrum adapted from ref 34.

TABLE 2: Comparison of DFT Results (Gas Phase) at 532 nm Excitation with Reported Experimental Values34 for the RRS Cross Sections of R6G in Methanol at 532 nm Excitation (the Corresponding Transition)a exptl34

DFT νb

(dσ)/(dΩ)c

σd

νb

299 494 608 760 913 1112 1182 1343 1546 1638

4.82 1.55 5.80 1.94 1.20 1.97 2.52 4.74 3.91 13.2

0.0505 0.0162 0.0607 0.0203 0.0126 0.0206 0.0264 0.0468 0.0409 0.138

300 518 604 761 917 1116 1172 1356 1571 1647

σexp

d

0.6 ( 0.1 0.6 ( 0.1 4.1 ( 0.5 2.1 ( 0.3 0.5 ( 0.09 0.5 ( 0.08 1.4 ( 0.2 2.6 ( 0.3 1.6 ( 0.2 2.0 ( 0.3

a The simulated excitation energy has been shifted from 473 to 532 nm to account for solvent effects. b Frequency in units of cm. c RRS differential cross section in units of 10-26 cm2/sr. d RRS cross section in units of 10-23 cm2.

FSRS to obtain absolute total cross sections measured using methanol as an internal reference. To obtain the total cross sections, we converted the differential cross sections following the same approach as for CS2. From the table, we see that the calculated total cross sections are lower than the experimental values by a factor ranging from 10 to 70. This is consistent with what we found for CS2 and is likely a result of neglecting solvent effects in the calculations. However, shortcomings of the XC functional used in the calculations cannot be ruled out, since in particular, the relative intensities of the modes at 608 and 760 cm-1 are underestimated. Furthermore, since the RRS

Figure 3. Ball and stick model of FePImCO.

intensities scale as µ4, even slight underestimations of the transition dipole moments will have a significant impact on the calculated total cross sections. Experimentally, the oscillator strength for the S0 f S1 transition is found to be f ) 0.71.56 We calculate the oscillator strength to be f ) 0.60, which corresponds to an underestimation of the RRS intensities by a factor of ∼2. Iron(II) Porphyrin with Imidazole and CO Ligands (FePImCO). Here, we consider an iron(II) porphyrin with imidazoleandCOligands(FePImCO)asamodelofCO-myoglobin (see Figure 3). We use a structure with Cs symmetry in which the iron atom is in the plane of the porphyrin ring and the Fe-C-O bond angle is 180°, similar to what has been reported previously.57,58 The Fe-CO bond distance is 1.76 Å, the Fe-N (of the imidazole group) bond distance is 2.07 Å, and the Fe-N (of the porphyring ring) bond distance is 2.00 Å. The absorption spectrum of a metalloporphyrin is characterized by a weak Q and strong B (Soret) absorption band. These bands are typically explained in terms of Gouterman’s fourorbital model, where the Q and B bands are a result of transitions

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Figure 4. Simulated RRS spectrum of FePImCO for the B band at 415 nm calculated using (a) the vibronic theory with Γ ) 0.05 eV and (b) the excited state gradient method with Γ ) 0.1 eV. Simulated RRS spectrum of FePImCO for the Q band at 539 nm calculated using (a) the vibronic theory with Γ ) 0.05 eV and (b) the excited-state gradient method with Γ ) 0.1 eV. RRS spectra have been broadened with a Lorentzian having a width of 20 cm-1. Insert in panel a shows the experimental spectrum adapted from ref 61.

between the close-lying HOMO orbitals, a1u and a2u, and the doubly degenerate LUMO orbitals, eg.59,60 The interaction of the nearly degenerate a1u1eg1 and a2u1eg1 configurations leads to a high-lying state and also a low-lying state. The B band corresponds to the high-lying state, in which the transition dipoles of the two configurations add together. In contrast, the Q band corresponds to the low-lying state, in which the transition dipoles of the two configurations almost cancel. The B band typically occurs around 380-400 nm, and the Q band typically occurs between 500 and 600 nm.59,60 We find the strong B band to be at 415 nm and the Q band at 539 nm; both are in good agreement with the experiments. Figure 4a shows the RRS spectrum of FePImCO for the B band calculated using the vibronic theory approach with a damping factor of 0.05 eV at 415 nm excitation. The same spectrum calculated using the excited state gradient method with a damping factor of 0.05 eV at 415 nm excitation is shown in Figure 4b. For the Q band excitation, the RRS spectrum was calculated using the vibronic theory approach with a damping factor of 0.05 eV at 539 nm excitation, as shown in Figure 4c. The corresponding spectrum calculated using the excited state gradient approach is shown in Figure 4d. Again, we see that the excited state gradient method severely overestimates the absolute intensities. In the RRS spectra for the B band, the vibronic theory results in an enhancement of the relative intensity of the low-frequency peaks at 353, 527, and 722 cm-1. Experimentally, the B band RRS spectra for heme proteins show strong bands at 376, 674, and 1373 cm-1 due to the vibrations localized in the porphyrin ring.61 We find the strong modes to be at 353, 722, and 1327 cm-1, in good agreement with the experimental results (see insert in Figure 4a). The Fe-CO

stretch is calculated to be at 527 cm-1, in good agreement with the experimental results, where the mode is found to be at 507-512 cm-1.62 Similarly, the RRS spectrum for the Q band calculated using the vibronic theory results in an enhancement of the relative intensity of the low-frequency peaks compared to the high-frequency peaks. In fact, the peak at 353 cm-1 becomes even more intense than the band at 1327 cm-1, which was the most intense peak in the RRS spectrum calculated using the excited state gradient method. The RRS spectra of FePImCO demonstrate the importance of including vibronic coupling effects in the calculation of the RRS spectrum and also highlight the shortcomings of the short-time approximation for calculating absolute RRS intensities. Conclusions In this study, we have presented calculations of the absolute RRS intensities using DFT. In particular, we compared predictions obtained using the short-time approximation and the vibronic theory for uracil, R6G, and an iron(II) porphyrin with imidazole and CO ligands. We find that the absolute RRS intensities calculated using the short-time approximation are severely overestimated as compared to results obtained using the vibronic theory, the reason being that the intensities calculated using the short-time approximation are very sensitive to the damping factor, whereas the vibronic theory is less dependent on the damping factor. This was illustrated for uracil, in which the vibronic effects are expected to play only a minor role due to the broad, featureless absorption spectrum. However, even though the relative intensities were predicted accurately using the short-time approximation, the absolute intensities were

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