Theor Chem Acc (2007) 117:1029–1039DOI 10.1007/s00214-006-0221-2

REGULAR ARTICLE

A quantum mechanical polarizable continuum modelfor the calculation of resonance Raman spectra in condensed phase

Benedetta Mennucci · Chiara Cappelli ·Roberto Cammi · Jacopo Tomasi

Received: 14 July 2006 / Accepted: 13 October 2006 / Published online: 21 December 2006© Springer-Verlag 2006

Abstract In this paper we present two computationalstrategies to simulate resonance Raman spectra of sol-vated molecules within the framework of the polarizablecontinuum model (PCM). These two strategies refer totwo different theoretical approaches to the RR spectra,namely the transform theory and the short-time dynam-ics. The first is based on the explicit detemination ofthe mimimum geometries of ground and electronicallyexcited states, whereas the second only needs to knowthe Franck–Condon region of the excited state poten-tial energy surface. In both strategies we have appliedthe recent advances achieved in the QM description ofexcited state properties and geometries of solvated mol-ecules. In particular, linear response approaches suchas CIS and TDDFT, and their extensions to analyt-ical gradients, are here used to evaluate the quanti-ties required to simulate resonance Raman spectra. Themethods have been applied to the study of solvent effectson RRS of julolidine malononitrile (JM). The goodagreement found between the calculated and experi-mental RR spectra seems to confirm the reliability of thecomputational strategies based on the PCM description.

1 Introduction

Raman scattering is the result of the coupling betweenthe radiation and the components of the molecular

B. Mennucci (B) · C. Cappelli · J. TomasiDipartimento di Chimica, Università di Pisa,via Risorgimento 35, 56126 Pisa, Italye-mail: [email protected]

R. CammiDipartimento di Chimica, Università di Parma,Viale delle Scienze 17/A, 43100 Parma, Italy

polarizability which are modulated by molecularvibration. In Raman scattering measurements where thewavelength of the radiation is close to an electronic exci-tation of the molecule, the intensity of the signal can beenhanced by a factor of up to 104–106. This process isreferred to as resonance Raman scattering (RRS) [1].

One of the main interests in RRS is that this spec-troscopic technique yields information about electroni-cally excited state properties and structure. There are infact very few experimental measurements from whichone can obtain excited state data, by contrast therehave been important recent advances in QM theories todescribe excited state geometries and properties. It thusbecomes very useful to compare the results obtainedwith such new techniques with those extracted fromRRS.

Along this line, theoretical and computationalapproaches have to become more and more realisticso to include all the most important effects which candetermine the nature and the properties of the vari-ous electronically excited states. Among these effects, avery important role is obviously played by the solvent.It is in fact well known that polar solvents can largelyaffect the electronic nature of excited states, for exam-ple amplifying their charge-transfer character. Thesestrong electronic deformations in the solvated mole-cules are obviously accompanied by structural defor-mations which can lead to very different relaxed excitedstates with respect to the same systems in gas-phase.For all these reasons it is of large interest to developQM models which can determine properties and struc-tures of molecular excited states taking into accountthe possible effects of the solvent. Among the avail-able approaches, one of the most suited to be applied tothis kind of study is represented by continuum solvation

1030 Theor Chem Acc (2007) 117:1029–1039

models. These latter in fact can be (and they have been)extended to almost all the main QM description usedto describe excited states. In particular, the polarizablecontinuum model (PCM) developed in Pisa by Tomasiet al. [2–5] has shown to give reliable descriptions ofdifferent phenomena involving electronically excitedstates.

The extension of QM continuum models to excitedstates has been discussed in various papers (see, e.g.reviews such as [6,7] for an extended literature). Themain specificity is related to the fact that the electronicexcitation is a process involving not only the solutebut the entire solute–solvent system. As a consequence,the definition of the excited states of molecular sol-utes requires also the characterization of the solventdegrees of freedom. The difference of the characteris-tic time scale of the electronic degrees of freedom ofthe solute and the composite degrees of freedom ofthe solvent may lead to different excited state regimes,with two extreme situations, namely the “nonequilibri-um regime” in which the slow degrees of freedom of thesolvent are not equilibrated with the excited state elec-tronic redistribution upon vertical excitation processes,and the “equilibrium regime” in which the solvent isallowed to equilibrate i.e., reorganizes all its degrees offreedom including the slow ones. In polar solvent, thedifferent regimes may largely influence the properties ofthe solute excited states, and thus computational algo-rithm should allow for the use of both of them.

Recently, a further important specificity of the exten-sion of QM continuum models to describe excited stateshas been rigorously analyzed both from a formal anda numerical point of view [8,9]. In such an analysis,it has been shown that the application of these mod-els to either linear response (random phase approxi-mation, RPA, configuration interaction-singles CIS, andtime-dependent density functional theory,TDDFT) orstate specific approaches (complete active space self-consisting field, CASSCF, configuration interaction CI,etc.) may lead to differences due to an intrinsic nonlin-ear character of the solvent response operators used incontinuum models. The state specific (SS) approaches,which are based on the explicit calculation of the excitedstate wave function, properly take into account the var-iation of the solute–solvent interaction accompanyingthe change of the electronic density during an electronicexcitation, while the linear response methods introduceonly effects related to the corresponding transition den-sity. In order to reduce these intrinsic differences,recently we have presented a method in which a SScorrection is introduced in LR approaches [10]. Thismethod is based on the use of the relaxed density whichcan be obtained in LR approaches thanks to their exten-

sion to analytical gradients now available not only withinthe CIS version but also within TDDFT. In the presentpaper, these two techniques and their gradient exten-sions [11,12] will be applied to model RRS of solvatedsystems.

In particular, two different theoretical approacheswill be presented to describe RRS. The first is the trans-form theory (TT) approach by Peticolas and Rush [13]which allows the calculation of relative resonanceRaman intensities from Franck–Condon (FC) type scat-tering based on the optical absorption spectrum of a par-ticular compound and the differences in the equilibriumstructures between the ground and the excited state inresonance. The second, alternative, approach is based onthe formalism developed by Lee et al. [14,15] by recast-ing the original Kramers, Heisenberg, and Dirac (KHD)formalism [16,17], into a time-dependent formalism interms of wave-packet dynamics.

After a short introduction into the RRS theories inSect. 2 to clearly describe the specificities introduced inboth TT and STD by the presence of the solvent, resultsare presented for julolidine malononitrile in Sect. 3. Wewill focus on density-functional theory (DFT) meth-ods for the calculation of excited-state energies andstructures, in comparison to the Hartree–Fock (HF)/CISapproach. A discussion of the results and a conclusionare given in Sect. 4.

2 Theory for resonance Raman in solvated systems

In this section we first review the fundamental aspectsof TT and STD approaches and then we describe whichspecific aspects are involved when a PCM solvent isintroduced.

2.1 Transform theory

The transform theory of resonance Raman intensities isbased on the optical theorem, which connects the opticalabsorption with the imaginary part of the polarizabilitytensor components, and on the Kramers–Kronig rela-tions between the real and the imaginary part of thepolarizability tensor components.

Five are the “standard assumptions” made in the TTapproach [13]: (1) the Born–Oppenheimer approxima-tion is valid, (2) only one excited electronic state isimportant (from now on indicated with r), (3) ground-and excited-state potential energy surfaces are harmonic,(4) non-Condon effects are negligible, i.e., onlyFC-type scattering is important, (5) excited and ground-state normal coordinates differ only in their equilibrium

Theor Chem Acc (2007) 117:1029–1039 1031

positions, so that Duschinsky rotations are not impor-tant and the frequencies for ground and excited-statevibrations are the same (“independent mode, displacedharmonic oscillator model", IMDHO [18])

Within these approximations, the vibrational Raman-scattering cross section from an initial vibrational state ito a final state f of the electronic ground state is propor-tional to the square of the polarizability tensor whosecomponent ρσ is given by

αfiρσ (ω0) =

Me0r,ρMe

0r,σ

h̄

∑

k

⟨νf |νk〉 〈νi |νk〉ωki − ω0 − i�r

(1)

where the damping constant �r is the total homoge-neous linewidth (halfwidth at half-maximum) from bothpure dephasing and lifetime contributions, ω0 denotesthe laser excitation frequency and

ωki = ωr0+3N−6∑

j=1

vj�j

where �j is the harmonic vibrational frequency corre-sponding to normal mode j, h̄ωr0 is the energy differencebetween the v = 0 levels of the electronic ground andexcited state r.

In Eq. (1) the summation over intermediate states hasbeen restricted to vibrational levels of a single resonantelectronic state (the state r), and the Condon approxi-mation has been imposed such that the dipole momentmatrix elements have been separated into products of apurely electronic transition moment (Me

0r) and a vibra-tional overlap.

One important point in the transform-theoryapproach is that, for the “independent mode, displacedharmonic oscillator model,” the FC-type integrals areknown analytically in terms of the normal-mode dimen-sionless displacements �j of the excited state equilib-rium structure. As a result, the relative RRS intensitiescan be approximated as

i1←0j /i1←0

k � �2j /�

2k (2)

where we have neglected the frequency-dependent pre-factor which may be taken to be constant in a goodapproximation [19]. We note that these normal coordi-nate displacements correspond to dimensionless normalcoordinates qj, and thus

�j =3N−6∑

k

L−1jk �Rk (3)

where L−1jk are the elements of the L inverse matrix

determined from the solution of the ground statenormal-mode eigenvalue problem [20], and �Rk are the

changes in the internal coordinates upon excitation ofthe molecule into the relevant excited state.

The expression (2) states that relative RRS intensi-ties are completely determined once ground and excitedstate minimum geometries are known together with theground state vibrational normal modes.

2.2 TD theory: the short time dynamics approximation

An alternative, time-dependent formalism for the calcu-lation of resonance Raman intensities has been devel-oped by Lee et al. [14,15] by recasting the originalformalism developed by Kramers, Heisenberg, andDirac into a time-dependent formalism in terms of nucleiwave-packet dynamics. Although this time-dependentformulation is similar to the transform theory, it dif-fers in the interpretation of Raman scattering in termsof wave packets and the semiclassical propagation ofthem. This approach showed that many features of bothresonant and nonresonant Raman scattering could beunderstood in terms of short-time dynamics.

Within the time-dependent theory, the polarizabilitytensor becomes

αρσ =Me

0r,ρMe0r,σ

h̄

∞∫

0

dt 〈f |i(t)〉 exp[i (ωL+ωi− ωr0) t−g(t)

]

(4)

where ωi is the vibrational frequency (above the zeropoint) of the |i〉 vibrational level of the ground electronicstate, and

|i(t)〉 = exp (−iHvibt/h̄) |i〉 (5)

Hvib being the Hamiltonian for vibrational motion inexcited state. The quantity 〈f |i(t)〉 is a time-dependentvibrational overlap between the final vibrational stateand the initial state propagated for time t by the excitedstate vibrational Hamiltonian and g(t) is a lineshapefunction. It is interesting to note that in this time-domainformulation, contrary to the frequency dependentexpression, it is much easier to incorporate possiblesolvent effects in the dephasing. This can be obtainedfor example introducing a Brownian oscillator approachand thus at the end the lineshape function g(t) will incor-porate both lifetime decay and solvent-induced puredephasing [21].

The time dependence of |i(t)〉 represents the motionof the nuclei after the potential energy function is instan-taneously (on the time scale of vibrational motion)switched from that of the ground electronic state to thatof the excited state. The intensity of each Raman transi-tion depends on the overlap of |i(t)〉with a different finalstate. In general, those modes that undergo the largest

1032 Theor Chem Acc (2007) 117:1029–1039

excited-state geometry changes will exhibit the high-est intensities. In polyatomic molecules, each overlap〈f |i(t)〉 involves all vibrational coordinates.

Equation (4) implies that the time scale of the dynam-ics that contributes to the resonance Raman intensityshould extend for as long as the damping factor is non-negligible. In practice, though, in most fairly large mol-ecules the intensities are determined almost entirely bythe dynamics within less than a vibrational period (typi-cally only 10–20 fs or so). In this “short-time dynamics”(STD) limit, the relative RRS intensities can be approx-imated as

i1←0j /i1←0

k =(

∂Exel

∂qj

)2 / (∂Ex

el

∂qk

)2

(6)

where Exel is the electronic excitation energy, qj and qk

are the normal coordinates of modes j and k, respectivelyand the derivatives are computed at the ground-stateequilibrium position. This equation means that whenonly short-time dynamics is important the relative inten-sities are given by the gradient of the Franck–Condonvertical excited-state surface with respect to the normalcoordinates. In this case, no explicit knowledge of theexcited state minimum structure is required.

Such an expression can be related by the TT expres-sion (2) by defining the partial derivative of the excited-state electronic energy with respect to a ground-statenormal mode at the ground-state equilibrium positionin the IMDHO model [1],(

∂Exel

∂qj

)

gs= −�j�j (7)

TT and STD methods combined can be used to check theinternal consistency of the approximations made withrespect to the excited-state potential energy surface [22].

2.3 PCM for TT and STD descriptions

In the previous sections we have reviewed the theoret-ical expressions for the resonance Raman intensities inthe TT and STD approaches when a single excited elec-tronic state contributes to the optical response. Theseexpressions when applied to solvated systems shouldinclude, in principle, the changes in nuclear equilibriumgeometry (reorganization) along all solute vibrationalmodes plus a collective solvation coordinate, and alsoallow for the possibility of inhomogeneous broadeningof the electronic transition and broadening due to thefinite lifetime of the excited electronic state.

In the present study, however, the aspect of the broad-ening of the RR band will not be considered and thus theanalysis of the solvent effects will be applied to the final

approximate expressions of the TT and STD theories(Eqs. (2) and (6), respectively). In doing this, we cer-tainly cannot account for the presence of the solventon the bandshape but we include solvent effects on theposition and the shape of the ground and electronic statepotential energy surfaces. These effects are surely thedominant ones for a proper description of the RRS ofsolvated systems, especially when these latter are (likein the present paper) conjugated push–pull systems inwhich the fractional charge-transfer character, or degreeof bond-order alternation, is tunable by changing thepolarity of the solvent.

The main effects of the solvent on RR spectra (posi-tion of the peaks and their intensities) can thus beascribed to two different origins: one due to the sol-vent-induced changes in the geometry of both groundand excited states and the other due to the variationsinduced in the electronic distribution of both states.

In order to take into account these two effects, the QMdescription of the solute has to account for the solvent ateach step, namely the determination of the ground stateminimum geometry and of the corresponding vibra-tional normal modes and frequencies, and the determi-nation of the vertical excited state (in the STD scheme)or the relaxed excited state (in the TT scheme).

As far as concerns ground state geometry and vibra-tional modes and frequencies, PCM has been alreadysuccessfully applied to describe IR [23] and normalRaman [24] spectra of solvated systems. Here, we thusdo not repeat what already presented in those studiesbut we just recall that the interested reader can findall the details of the implementation of the PCM the-ory within those spectroscopies in the cited paper or inrecent reviews such as [6,25].

It is instead of interest to focus on the modelization ofthe PCM solvent effects on excited states. The applica-tion of PCM to TT and STD theories in fact involves thedetermination of portions of the PES for the solvatedexcited state.

As briefly discussed in Sect. 1, the new specific aspectintroduced in the modelization of excited state forma-tion and relaxation in solution is the dynamics of the sol-vent. In particular, in fast processes, such as electronicexcitations, electron transfers or ionizations, the time-scale of the change in the charge density of the solute isusually much smaller than the time-scale in which a polarsolvent fully relaxes to reach a new equilibrium state.During this relaxation, the solvent nuclear and molec-ular motions act as inertia on the solvation responseand a nonequilibrium regime is established. Due to themutual solute–solvent polarization, the new equilibriumis reached through changes of both solute and solvent,and an accurate description of the reorganization path

Theor Chem Acc (2007) 117:1029–1039 1033

should consider the evolution of their interaction and,possibly, the solute geometry relaxation.

If now apply this analysis to the TT and STD schemeswe obtain that, when relaxed excited states are consid-ered as in TT, a solvation equilibrated to the relaxedstate should be used while, when a FC portion of thePES has to be explored as in STD, nonequilibrium sol-vation is to be preferred. In this latter case, a parti-tion of the PCM effect in two separate components isdone, one related to the dynamic (or fast) electronicresponse of the solvent, and the other to the slower, orinertial, response connected to nuclear and/or molecularmotions inside the solvent [26–28]. According to whatwas said before, the dynamic component will dependon the instantaneous charge distribution of the soluteand on the optical dielectric constant. The inertial com-ponent, on the contrary, will still depend on the solutecharge distribution of the initial (ground) state.

The theory of the extension of PCM to excited statesand their derivative approaches has been already pre-sented both at CIS [11] and TDDFT level [12] and suc-cessfully applied to different organic molecules. Here,we only recall that for both methods, the energy func-tional to differentiate is the free energy G defined as

G = E− 12

⟨

∣∣∣VPCM∣∣∣

⟩(8)

where E is the eigenvalue of the effective Hamiltonian(H0 + VPCM) including the PCM operator VPCM. Thissolvent induced term represents the electrostatic inter-action between the solvent and the solute’s nuclei andelectrons. In the computational practice a boundary-element method BEM is applied by partitioning the cav-ity surface into discrete elements, called tesserae, and byrepresenting the solvent response with a collection ofapparent point charges, each one placed at the center ofa tessera. The detailed expression of the linear systemof equations defining such charges depends on the spe-cific version of the PCM method being used and it hasbeen previously published (see Ref. [6] for a completesurvey). Here we simply recall that such equations aredetermined by the form and shape of the cavity, by thedetails of the discretization of the surface and by thesolvent permittivity ε. By tuning the value of ε we candescribe the changes in the solvation of the excited statewhen passing from the Franck–Condon region of the sol-vent coordinate i.e., the nonequilibrium to a completelyrelaxed solvent. This is done by changing the value ofε used to compute the PCM charges from the opticalvalue ε∞ namely, the square of the refractive index tothe static bulk value ε0. Effects of these changes can besignificant for polar solvent for which ε∞ � ε0.

The QM treatment of the PCM operator VPCM isdelicate, as it depends on the solute total density andthus it induces a nonlinear character in the soluteSchrödinger equation. This nonlinearity for ground elec-tronic states is automatically solved by using standardself-consistent field iterative approaches developed forisolated systems. Passing to excited states, instead, twodifferent solvent-specific approaches have been devel-oped: (i) a general scheme in which the excited stateproblem described using state specific (SS) approaches(complete active space self-consisting field CASSCF ,CI, etc.) is iteratively solved in the presence of the corre-sponding solvent response and (ii) a simplified schemein which, following a linear response (LR) formalism,the solvent response is determined by transition densi-ties instead of state densities. Recently, we have shownthe intrinsic differences between the two schemes [8,9]and we have formulated a third hybrid scheme in whichthe computationally efficiency of LR approaches is com-bined to the more satisfactory description of the solventresponse of SS approaches [10]. This third scheme (alsoindicated as corrected LR, cLR) introduces a SS cor-rection in the original LR description using the PCMcharges produced by the relaxed density of each excitedstate which can be obtained in LR approaches thanksto their extension to analytical gradients. This correctedLR approach reduces the differences with respect toa real SS scheme and it allows for a more satisfactorydescription of the variation of the solute–solvent interac-tion accompanying the change of the electronic densityduring an electronic excitation.

In the following application to the simulation of RRspectra of a solvated push–pull molecule we will use bothan equilibrium PCM-TT and a nonequilibrium PCM-STD description. In the latter case, both the standardLR and the corrected LR method will be tested.

3 Resonance Raman of julolidine malononitrile

In this section we report the application of the meth-odologies for the study of RRS in solution reportedin the previous section to the evaluation of a relevantportion of the resonance Raman (RR) spectrum of julo-lidine malononitrile (JM) in solvents of different polar-ity. The experimental investigation of solvent effectson RR spectra of JM has been reported previously inRefs. [29,30].

JM belongs to the class of conjugated organic mol-ecules of interest for second-order nonlinear opticalapplications, due to the fact that it has reported to havelow-lying electronic transitions with a high degree ofintramolecular charge-transfer (CT) character and

1034 Theor Chem Acc (2007) 117:1029–1039

Fig. 1 Resonance Raman spectra of JM in different solvents aftersubtraction of fluorescence backgrounds. Asterisks label solventbands. Arrows mark a vibration whose intensity is particularlysolvent dependent. The spectra are taken from Ref. [30]

consequently is sensitive to changes in the local solventenvironment.

As it has been reported previously in the literature[29,30] experimental RR spectra of JM exhibit a markeddependence upon the solvation environment (see Fig. 1).

In the experimental RR spectrum, the most stronglysolvent-dependent vibration seems to be that at1612 cm−1. Such a vibration involves the stretching of thebonds defining the phenyl, which are aromatic in the neu-tral form but can be considered double bonds in thezwitterionic form. A strong resonance Raman intensityis experimentally evidenced for this band, suggestinglarge geometry change in this mode upon electronicexcitation.

In the following we shall test the predictive power ofPCM in reproducing the intensity pattern of the RRlines in the range 1550–1620 cm−1 by exploiting thetwo different methodologies described in the previoussection and assuming that CT lowest energy stronglyallowed transition is the only state determining the RRspectrum [31].

3.1 Computational details

The calculations of ground state energies, geometries,vibrational frequencies and normal modes were per-formed by exploiting Hartree–Fock (HF) and densityfunctional theory (DFT) with the B3LYP hybrid func-tional. The basis set chosen was 6-311G(d,p), e.g. thesame used in the previous study by Myers et al. [29,30].

For the calculation of excited states CIS and TDDFT(using the same basis set) were applied to the calcu-lation of excitation energies and geometries. Solventeffects were accounted for by exploiting the integralequation formalism (IEF) [4,5] version of the PCM asimplemented in a development version of the Gaussiancode [32].

Calculations of excitation energies were performedby using both the linear response (LR) and the “cor-rected LR” schemes (see Sect. 2.3). In all the calcula-tions in solution, a molecule-shaped cavity was used,made of interlocking spheres centered on heavy atomsand using for the radii of the spheres the default valuesimplemented in Gaussian.

3.2 Ground and excited state geometries

Before presenting results about the RRS intensities weanalyze ground and CT excited state geometries.

The variation in selected bond lengths (see Fig. 2 fortheir labeling) for JM ground state passing from cyclo-hexane to acetonitrile is reported in Fig. 3, where HFand DFT/B3LYP are compared.

Fig. 2 Scheme of JM structure with labelling of selected distances

-0.012

-0.008

-0.004

0.000

0.004

0.008

0.012

NC R1 R2 R3 R4 R5 R6 CN

B3lyp

HF

Fig. 3 Variation in selected bond lenghts (see Fig. 1) of JMground state passing from cyclohexane to acetonitrile. HF andDFT/B3LYP results are compared

Theor Chem Acc (2007) 117:1029–1039 1035

Fig. 4 Pictorialrepresentation of the neutraland the zwitterionicresonance forms for JM N

C

C

N

N

N+

C

C

N

N-

Table 1 B3lyp/6-311G(d,p) vibrational frequencies (cm−1)

Gas C6H12 CH2Cl2 CH3CN

M1 1589 1585 1581 1580M2 1604 1595 1586 1584M3 1651 1650 1648 1647

The solvent dependence of the ground statecoordinate changes reported in Fig. 3 are qualitativelyconsistent with a two-state model, pictorially repre-sented in Fig. 4.

More specifically, as solvent polarity increases, theground state displays a more zwitterion-like structure:with increasing solvent polarity and increasingzwitterionic character, we observe an alternate positiveand negative variations of the single and of the dou-ble bond lengths along the whole skeleton going fromthe amino to the cyano nitrogen. The expected solventdependence of the conjugated bond lengths is easilydeduced by inspection of the resonance forms in Fig. 4.This behavior is almost identical both at the HF andDFT level of description. Due to this equivalence, in thefollowing we shall limit the ground state description tothe DFT level.

In Table 1 we report the DFT/B3LYP vibrational fre-quencies of the modes corresponding to the region ofthe RR spectrum showing the largest solvent effects (seeFig. 1) and in Fig. 5 a pictorial view of the correspondingnormal modes.

The solvent effects on frequencies are not large. Thisis especially true for the mode labelled as “mode 3 (M3)”which corresponds to a “quinoidal” stretch of the julo-lidine ring (stretching of the phenyl bonds). Such aninsensitivity is confirmed by experimental RRS spec-tra. More pronounced solvent effects are found for theother two modes (M1 and M2) for which differences ofthe order of 10 and 20 cm−1, respectively, are calculatedpassing from gas phase to the most polar solvent. Thesedifferences well correlate with the differences found inthe bond lengths; for example, M2 mode for which thelargest solvent shift is obtained, can be characterized by

Fig. 5 Cartoon of normal modes corresponding to M1, M2 andM3 vibrations of JM based on B3LYP/6-311G(d,p) calculations

a stretching of the R4 bond for which in Fig. 3 we foundthe largest solvent sensitivity (together with NC).

Moving to the CT excited state geometry, the resultslargely depend on the QM level of calculations.

In Fig. 6 we report the ground-to-excited state varia-tions in the single and double bond of the JM skeleton asobtained at CIS or TDDFT level. Note that all the valuesreported in figure are obtained in both cases by consid-ering the DFT ground state; however, almost identicalresults are found by considering the HF ground state.At TDDFT level we could not locate the excited stateminimum for the isolated system.

The inspection of Fig. 6 clearly reveals a deficiency ofTDDFT in describing ground-to-excited state structuralchanges in JM. In fact, if we adopt the two-state pictureused to explain solvent effects on ground state geome-try, also here we should expect the typical alternationof positive and negative variations in single and dou-ble bonds which indicate an enhanced zwitterionic char-acter in the excited state. This alternation is correctlyreproduce by CIS but not by TDDFT. These findingsare not unexpected, as TDDFT, at least with the hybridB3LYP functional, has been reported to be unsuitablefor the description of charge-transfer conjugate systems

1036 Theor Chem Acc (2007) 117:1029–1039

Fig. 6 Variation in selectedbond lenghts (see Fig. 2) ofJM from ground to excitedstate in gas, cyclohexane andacetonitrile. Both CIS andTDDFT/B3LYP are reported

0.00

0.01

0.02

0.03

0.04

TDDFT

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

NC R1 R2 R3 R4 R5 R6 CN

GasC6H12

CH3CN

CIS

[33–35]. For this reason, in the following we will resortonly to CIS for the evaluation of RRS spectra.

3.3 Excited state displacements and RR intensities

In this section we report the results obtained for theadimensional ground-to-excited state displacements, �j,using the two different methodologies presented inSect. 2, i.e. TT which gives directly the displacementsby considering ground and excited state geometries andground state normal modes, and STD-IMDHO, whereinstead a �j value is indirectly obtained in terms of thegradient of the excitation energy in the FC region, i.e.no information on the excited state minimum geometryis required (see Eq. (7)).

The results obtained with the two theories arereported in Tables 2 and 3, respectively. In the case of TTthe displacements have been calculated by using Eq. (3)and taking care that both ground and excited states havethe same center of mass and their structures are orientedso to have coincidence in the principal axes of inertia. Inthe case of STD, for the evaluation of the excited stategradients we have tested both the LR and the correctedLR approximations (see Sect. 2.3 for more details).

The two alternative approaches, TT and STD (eitherin the LR or cLR version) give qualitatively similar

Table 2 Adimensional ground-to-excited state displacements, �j,obtained with the TT method at CIS level (using the DFT groundstate geometry) for the M1, M2 and M3 normal modes

Gas C6H12 CH2Cl2 CH3CN

M1 0.388 0.345 0.248 0.197M2 −0.523 −0.405 −0.201 −0.142M3 0.249 0.263 0.267 0.249

Table 3 Adimensional ground-to-excited state displacements, �jobtained with the STD method at CIS level (using the DFT groundstate geometry)

Gas C6H12 CH2Cl2 CH3CN

LR

M1 0.412 0.397 0.302 0.252M2 −0.538 −0.477 −0.243 −0.175M3 0.444 0.470 0.413 0.392

cLRM1 – 0.318 0.220 0.187M2 – −0.335 −0.139 −0.098M3 – 0.394 0.332 0.323

For solvated systems both the LR and the corrected LR approxi-mations are presented

results, although they resort to different approximationsand involve completely different assumptions on thesolvation regime (equilibrium vs. nonequilibrium, see

Theor Chem Acc (2007) 117:1029–1039 1037

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

M3M2M1

CH3CN

CH2Cl2

C6H12

Fig. 7 Variations in the dipole moment of the excited state (indebye) with respect to the three selected normal modes

Sect. 2.3). In both cases, a decrease of the displace-ment passing from the isolated to the solvated system isobserved for all the modes, being the largest changesfound for M2. Such a behavior is in line with whatalready pointed out for frequencies (see Table 1).

To support these structural data with related elec-tronic properties, in Fig. 7 we report the value of thevariation of the excited state dipole moment with respectto the selected normal coordinates (we recall here thatour approach relies on the assumption that frequencies,and normal coordinates, are the same for the groundand the excited state).

The observed trends in changing the solvent well cor-relate with what found for the adimensional ground-to-excited state displacements, �j: low variations (andlow solvent sensitivity) along the M3 mode and largervariations for the other two modes with M2 presentingthe largest solvent sensitivity.

If we now compare the results obtained for bothdisplacements and variations of dipole moments withexperimental RR spectra (see Fig. 1), it might be con-cluded that an opposite behavior is found for M3 forwhich experiments seems to show a large sensitivityto the solvent while very low sensitivity is found withPCM calculations. As a matter of fact, the disagreementbetween calculations and experiments is only apparentfor the reasons which follow.

In Figs. 8 and 9 simulated RRS spectra obtained bytransforming TT or STD(cLR) displacements into RRSintensities are shown. Such intensities were obtainedby using the approximated formula (2) and (6) and bysimulating the spectrum with lorentzian band shapes. Inaddition a scaling of the spectra was introduced, exactlyas done in the experiments [29]: this scaling is done so tohave approximately equal intensities of the strong line

1520 1560 1600 1640 1680 1720

Arb

itrar

y un

its

λ (nm)

CH3CN

CH2Cl

2

C6H

12

Gas

Fig. 8 TT resonance Raman intensities in gas and in varioussolvents. All spectra have been scaled to have approximately equalintensities in the band near 1580 cm−1

1520 1560 1600 1640 1680 1720

Arb

itary

uni

ts

λ (nm)

CH3CN

CH2Cl

2

C6H

12

Gas

Fig. 9 STD (cLR) resonance raman intensities in gas and in var-ious solvents. All spectra have been scaled to have approximatelyequal intensities in the band near 1580 cm−1

(corresponding to a combination of M1 and M2) in allthe solvents.

As it can be seen, the band corresponding to M3is now largely sensitive to the solvent and it correctlyincreases passing from gas to solution, and from apo-lar to polar solvent, exactly as found in the measuredRRS spectra. Both TT and STD approaches are ableto reproduce the experimental trend, although the rel-ative intensities of the two bands are not quantitativelyreproduced for the more polar solvents, where an over-estimation of the intensity of the M3 band is observed.From this analysis based on calculated RR spectra, itthus follows that the experimentally observed sensitiv-ity of the M3 band with the solvent is indeed a kind of an

1038 Theor Chem Acc (2007) 117:1029–1039

artifact induced by the other two adjacent bands whichboth significantly decrease passing from gas-phase (orapolar solvent) to polar solvents.

4 Conclusions

In this paper we have presented two computationalstrategies to simulate resonance Raman spectra of sol-vated molecules based on the transform theory andshort-time dynamics theory. In both strategies importantapproximations have been used, however the resultsobtained for the test case seem to confirm their reli-ability, at least to get the correct qualitative pictureof the solvent effects on the position and the inten-sity of the RR bands. These results have surely to beconfirmed on other systems before they can representa stated proof of the validity and the accuracy of thetheoretical methods and of the computational strate-gies. The main important goal of this study, however,was not the definition of a theory for resonance Ramanscattering in solution which covers all the aspects ofthe solvent effects, but instead an attempt to exploitthe recent advances achieved in the QM description ofexcited state properties and geometries of solvated mol-ecules. In particular, the recent implementation of ana-lytical gradients for time-dependent density functionaltheory excitation energies seems to provide a promisingroute to the geometry optimization of excited states forlarger molecules. Here, the application of the standardTDDFT/B3LYP has shown its limits but we are surethat the exploration of different functionals, especiallythe most recent ones accounting for long-range correc-tions [36–39] and thus most suited to properly describeCT excited states, would lead to better results.

In any case, both TT and STD theories may be nowtranslated into efficient computational approacheswhich can be easily applied also to larger molecularsystems and they can take into account solvent effects.In particular, QM continuum solvation methods suchas PCM seem to represent a promising way to cou-ple these theories with an accurate and efficient sol-vation model. Clearly, further extensions with respectto what presented here are not only possible but alsoalready initiated. The inclusion of solvent effects intothe shape of the RR band and not only on its positionand intensity, is one of these extensions. In this case,a real time-dependent picture of the solvent polariza-tion should be used; this type of approach has beenalready presented for TD stokes shifts and other relax-ation processes [10,40,41] and it should not be diffi-cult to reformulate it for the present problem. Anotherpossible research line is the extension of PCM to the

new method which has been developed by Schatz et al.[42] to describe the Raman scattering cross section. Thismethod is still based on a short-time approximation andit makes it possible to calculate both normal Raman scat-tering and resonance Raman scattering intensities fromthe geometrical derivatives of the frequency-dependentpolarizability (real or complex). As said above, PCMhas been already extended to treat NRS by includingsolvent effects in the real part of the frequency-depen-dent polarizability (and its derivatives) [24] and thus anextension to the complex part is surely feasible. Thisshould thus permit to treat RRS of solvated systems ina more general framework without requiring any of thestrong approximations used in both TT and STD, suchas assuming only one excited state to determine the RRspectrum.

Acknowledgment Financial support by the Italian MIUR(Ministero dell’Istruzione, Università e Ricerca), PRIN 2005, andby Gaussian Inc. is here acknowledged.

References

1. Myers AB (1996) Chem Rev 96:9112. Miertus S, Scrocco E, Tomasi J (1981) Chem Phys 55:1173. Cammi R, Tomasi J (1995) Comput Chem 16:14494. Cancès E, Mennucci B, Tomasi J (1997) J Chem Phys 107:30325. Mennucci B, Cancès E, Tomasi J (1997) J Phys Chem B

101:105066. Tomasi J, Mennucci B, Cammi R (2005) Chem Rev 105:29997. Mennucci B (2006) Theor Chem Acc 116:318. Cammi R, Corni S, Mennucci B, Tomasi J (2005) J Chem Phys

122:1045139. Corni S, Cammi R, Mennucci B, Tomasi J (2005) J Chem Phys

123:13451210. Caricato M, Mennucci B, Tomasi J, Ingrosso F, Cammi R,

Corni S, Scalmani G (2006) J Chem Phys 124:12452011. Cammi R, Mennucci B, Tomasi J. (2000) J Phys Chem A

104:563112. Scalmani G, Frisch M, Mennucci B, Tomasi J, Cammi R,

Barone V (2006) J Chem Phys 124:09410713. Peticolas WL, Rush T III (1995) J Comput Chem 16:126114. Lee S-Y, Heller EJ (1979) J Chem Phys 71:477715. Heller EJ, Sundberg RL, Tannor D (1982) J Phys Chem

86:182216. Kramers HA, Heisenberg W (1925) Z Phys 31:68117. Dirac PAM (1927) Proc R Soc Lond Ser A 114:71018. Blazej DC, Peticolas WL (1980) J Chem Phys 72:313419. Chan CK Page JBJ (1983) Chem Phys 79:523420. Wilson EB, Decius JC, Cross PC (1955) Molecular vibrations.

McGraw-Hill:New York21. Li B, Johnson AE, Mukamel S, Myers AB (1994) J Am Chem

Soc 116:1103922. Neugebauer J, Hess BA (2004) J Chem Phys, 120:1156423. Cammi R, Cappelli C, Corni S, Tomasi J (2000) J Phys Chem

A 104:987424. Cappelli C, Corni S, Tomasi J (2001) J Chem Phys 115:553125. Tomasi J, Cammi R, Mennucci B, Cappelli C, Corni S (2002)

Phys Chem Chem Phys 4:5697

Theor Chem Acc (2007) 117:1029–1039 1039

26. Aguilar MA, Olivares Del Valle FJ, Tomasi J (1993) J ChemPhys 98:7375

27. Cammi R, Tomasi J (1995) Int J Quant Chem Symp 29:46528. Mennucci B, Cammi R, Tomasi J (1998) J Chem Phys 109:279829. Moran AM, Egolf DS, Blanchard-Desce M, Myers Kelley A

(2002) J Chem Phys 116:254230. Myers Kelley A. (2005) Int J Quantum Chem 104:60231. Moran AM, Myers Kelley A, Tretiak S (2003) Chem Phys

Lett 367:29332. Frisch MJ et al (2004) GAUSSIAN, Development Version,

Revision D.02, Gaussian, Inc., Wallingford, CT33. Tozer DJ, Amos RD, Handy NC, Roos BO, Serrano-Andres L

(1999) Mol Phys 97:85934. Dreuw A, Weisman JL, Head-Gordon M (2003) J Chem Phys

119:2943

35. Bernasconi L, Sprik M, Hutter J (2003) J Chem Phys119:12417

36. Ikura H, Tsuneda T, Yanai T, Hirao K (2001) J Chem Phys115:3540

37. Yanai T, Tew DP, Handy NC (2004) Chem Phys Lett 393:5138. Tawada Y, Tsuneda T, Yanagisawa S, Yanai T, Hirao K (2004)

J Chem Phys 120:842539. Chiba M, Tsuneda T, Hirao K (2006) J Chem Phys 124:14410640. Ingrosso F, Mennucci B, Tomasi J (2003) J Mol Liq 108:2141. Caricato M, Ingrosso F, Mennucci B, Tomasi J (2005) J Chem

Phys 122:15450142. Jensen L, Zhao LL, Autschbach J, Schatz GC (2005) J Chem

Phys 123:174110