Fully Anharmonic Vibrational Resonance Raman Spectrum of

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Fully Anharmonic Vibrational Resonance Raman Spectrum of Diatomic Systems Gustavo J. Costa, Antonio Carlos Borin, Rogério Custodio, and Luciano N. Vidal J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.7b01034 • Publication Date (Web): 09 Jan 2018 Downloaded from http://pubs.acs.org on January 12, 2018

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Journal of Chemical Theory and Computation

Fully Anharmonic Vibrational Resonance Raman Spectrum of Diatomic Systems Gustavo J. Costa,† Antonio C. Borin,‡ Rogério Custodio,¶ and Luciano N. Vidal∗,† †Departamento Acadêmico de Qu´ımica e Biologia, Universidade Tecnológica Federal do Paraná, Av. Dep. Heitor de Alencar Furtado, 5000, Curitiba/PR, 81280-340, Brazil. ‡Department of Fundamental Chemistry, Institute of Chemistry, University of São Paulo, NAP-Photo Tech the USP Consortium of Photochemical Technology, Av. Prof. Lineu Prestes, 748, São Paulo/SP, 05508-000, Brazil. ¶Instituto de Qu´ımica, Universidade Estadual de Campinas, R. Josué de Castro, 126, Campinas/SP, 13083-970, Brazil. E-mail: [email protected]

Abstract A study is presented on the resonance Raman (RR) spectrum based on fully anharmonic wavefunctions and energies obtained from ab initio multireference potential energy curves of diatomic systems. The vibrational problem is numerically solved using a variational stochastic method or the Cooley-Numerov method, as implemented in Le Roy’s LEVEL program. Anharmonic Franck-Condon and Herzberg-Teller integrals are numerically evaluated and the RR polarizability is calculated within the time-independent framework of the RR theory. At the harmonic level, the differential cross sections show faster convergence with respect to the number of intermediate vibrational states than what is obtained from anharmonic wavefunctions. Twice as many

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intermediate states are required to achieve the same convergence in the RR intensities as observed within the harmonic model. The anharmonic spectra evaluated for H2 , C2 and O2 molecules show that RR intensities are strongly affected by anharmonic effects. They differ from their harmonic counterparts not only in the position of the peaks, but also in the absolute and relative intensities.

1

Introduction

The resonance Raman (RR) effect is a form of inelastic light scattering observed when the system is illuminated by light of frequency near that of an electronic transition. 1,2 The intensity of RR scattered radiation depends on the square of (ro)vibronic polarizabilities which carry information from the excited electronic state in resonance with the incident radiation. 1,3 The RR polarizability is attainable from one of three general frameworks, namely: (i) The time-independent (TI) theory, also known as the sum over states approach, rooted on the Kramers-Heisenberg-Dirac dispersion formula 4,5 and the Herzberg-Teller vibronic theory; 1,3,6 (ii) the Transform Theory (TT), 7,8 based on the Optical Theorem, 9 relating the polarizability with the optical absorption through the Kramers-Kronig transform relations and (iii) the Time-dependent (TD) model by Heller et al., 10,11 in which the RR polarizability is written as a half-Fourier transform of the dipole moment cross-correlation function, that can be calculated by propagating a wavepacket of the initial state over the excited-state Potential Energy Surface (PES). 12,13 In the TT approach, the usually expensive task of computing lineshape functions is avoided by using Kramers-Kronig relations to obtain them from the absorption spectrum. 14–16 Similarly, the TD model may be used to overcome the computational bottleneck of evaluating a huge number of Franck-Condon (FC) and Herzberg-Teller (HT) integrals. 13 The need of information from two (or more) electronic states – in particular the equilibrium geometries of each state, their PES, transition moments and possibly their dependency on the nuclear coordinates – is commonplace in any vibronic spectroscopic calculation. In 2


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order to simplify the calculation of the RR spectrum, a number of approximations are commonly used to reduce the amount of data relative to the excited state, as well as to simplify the computation of FC and HT integrals. For instance, many RR theoretical studies neglect the HT vibronic coupling and adopt the independent mode, displaced harmonic oscillator (IMDHO) approximation, in which identical harmonic PES are assumed for both electronic states, differing only in their equilibrium minima. In addition, if the short-time dynamics (STD) of the wavepacket 17 is combined with IMDHO, the relative RR intensities will be obtained from a minimal amount of data, namely, the ground state harmonic frequencies and the excited-state energy gradient, evaluated at the ground equilibrium geometry. 18–21 However, the STD regime is well-suited only for pre-resonance regimes. 17 Another approach that also relies on STD is an extension of the far-from-resonance Placzek’s model 1 to include a finite lifetime into the electronic polarizability, thus preventing it from diverging at resonance frequencies. 21–23 On the other hand, RR spectroscopy has received several theoretical and computational improvements that are worth mentioning here. Within the TD model, recurrence relations that allow frequency changes between the ground and excited electronic states, the inhomogeneous broadening effects, 24 the inclusion of several excited states at the STD 25 and an intensity tracking algorithm for selection of high-intensity transitions 19 have been reported. A TD route to RR spectra with the inclusion of HT effects has been formulated based on vertical and adiabatic models of the excited state Hessian, 13 as well as an approach in which non-redundant delocalized internal coordinates are employed, enabling a more reliable description of flexible molecules. 26 Relevant contributions within the TI theory of RR were also given. Algorithms for the efficient selection of the vibrational states used in the calculation of FC and HT integrals, full account of differences between harmonic PES of the ground and excited states, HT effects, contribution from multiple excited states and solvation effects have also been reported. 27–30 A central aspect in vibronic calculations is the description adopted for the PES and

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most studies are grounded on the harmonic approximation which, in some cases, may be inadequate, such as in those systems where large amplitude motions are present (e.g. flexible molecules). Nevertheless, there has been an increasing number of studies that take the anharmonicity of the PES into account, in particular for UV-vis absorption, fluorescence and photoelectron spectroscopies. 31–40 Interesting approaches have been developed for the calculation of (an)harmonic FC integrals. In this regard, mention might be made of the procedure proposed by Luis et al., 41–43 in which those integrals are obtained by solving a set of homogeneous linear simultaneous equations whose coefficients are vibrational energy differences and matrix elements of the type hψue e |Vˆ g − Vˆ e |ψvee i. Here, ψue e , ψvee are vibrational wavefunctions of the excited electronic state and Vˆ g , Vˆ e are potential energy operators of the ground and excited states. These operators are written in the same set of vibrational coordinates by using the Duschinsky rotation 3 to relate normal modes of different electronic states. Anharmonic FC integrals are found when the second order perturbation theory is applied to that set of equations 41 or, instead, by using vibrational configuration interaction (VCI) wavefunctions, from vibrational self-consistent field (VSCF) reference, to describe ψue e and ψvee . 43 Another approach that allows for calculation of anharmonic vibronic spectra is based on the Raman wavefunction (RWF), 17 i.e. the half-Fourier transformed time-dependent wavepacket, obtained by solving the inhomogeneous Schrödinger equation (ISE), a linear differential equation that has been derived by Heller and co-workers. 17 In the time-independent route of Petrenko and Rauhut 44 for the RWF, the ISE is solved by an iterative method which avoids explicit calculation of FC integrals. That TI RWF formalism is illustrated by the simulation of the photoelectron spectra of ClO2 and other systems and the anharmonicity is included within the VSCF-VCI framework. As explained above, there are systems in which one (or some) degree of freedom is highly anharmonic, although the remaining ones are satisfactorily described by harmonic oscillators. A cost-effective way to simulate the vibronic spectra of such systems is achieved by using hybrid schemes, whereby one vibration is treated as being fully anharmonic and the

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others are modeled within the harmonic approximation. 45,46 For instance, the anharmonic photoelectron spectrum of systems as large as phenylanthracene could be simulated by means of a hybrid model 46 that minimizes the couplings between small amplitude motions and the large amplitude motion (LAM) using delocalized internal coordinates. The LAM is treated with a fully-numerical method based on the discrete variable representation theory. The anharmonic calculation of RR spectra has been seldom explored in the literature. Some of the few examples are the papers by Barone and co-workers, 13,28 based on the secondorder vibrational perturbation theory (VPT2), and by Ben-Nun and Mart´ınez, 47 who used the ab initio multiple spawning method and a correlation function approach to simulate the RR spectrum of ethylene. In the vibrational perturbation theory proposed by Barone, resonance terms of Fermi and Darling-Denninson types are treated using the so-called generalized VPT2 scheme. 48,49 The VPT2 energies, evaluated for both ground and excited electronic states, are then used to correct the Raman shifts whereas the corresponding RR intensities are partially corrected for anharmonicity by building RR polarizabilities using VPT2 energies and harmonic oscillator FC and HT integrals. The main advantage of the generalized VPT2 approach is that it can be combined with DFT and TD-DFT electronic structure methods to compute vibronic spectra of large systems. 50 While there are several studies of vibronic spectroscopies where the anharmonicity of the PES is taken into account, in the particular case of RR, that effect on the absolute and relative Raman intensities is not well known to date. Diatomic molecules represent a good starting point for anharmonic calculations as their electronic structure can be described with high accuracy at moderate computational cost. Therefore, we present in this work a TI route for the ab initio calculation of the vibrational RR spectrum based on multiconfiguration electronic structure methods, with the fully anharmonic vibrational structure being obtained numerically from a variational stochastic method 51 or from the iterative Cooley-Numerov 52 method. Results are presented for the H2 , C2 and O2 molecules, which can be considered as templates for assessing the following aspects of the RR spectrum: (i) Convergence of the

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differential cross sections with respect to the number of vibrational intermediate states, (ii) linear and quadratic HT contributions to RR intensities and (iii) dependency of the RR the intensities on the excitation energy – Raman excitation profiles. This paper is organized as follows: Firstly, the next section will detail the theoretical framework together with the computational aspects of the electronic structure, vibrational and vibronic calculations. After that, our numerical methodology and Fortran codes, developed for the vibrational and vibronic calculations, will be validated against analytical results at the harmonic level. Lastly, FC and HT spectra for the above-mentioned molecules will be presented and discussed, followed by the concluding remarks section.

2

Theory and computational aspects

The intensity of Raman-scattered radiation depends on the polarization states of incident and observed radiation, on the angle θ between the propagation direction of those beams, on the sample temperature and characteristics (e.g. its concentration) and varies as the fourth power on the wavenumber ν˜0 of the incident radiation. At room temperature most of the inelastically scattered radiation has its wavenumber ν˜R lower than ν˜0 (Raman Stokes scattering) as the ground vibrational state is usually more populated than upper vibrational states. Although different experimental setups may be used, Raman experiments are often performed with θ = 90◦ (right angle geometry) or θ = 180◦ (backscattering geometry), with the incident radiation being linearly polarized. The molecular contribution to the intensity of scattered radiation is commonly expressed through a differential scattering cross section dσ , dΩ

with Ω being the solid angle of the radiation collected for observation. If the sample is

constituted of freely rotating molecules and the incident radiation is linearly polarized and perpendicular to the scattering plane, the corresponding molecular differential cross section

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for the right angle scattering geometry will have the following general form: 1

dσ dΩ

= fi

π2 20

fi ν˜R4

1 45αf2 i + 7γf2i + 5δf2i 45

(1)

where 0 is the vacuum permittivity, fi is the fraction of molecules in thermal equilibrium at temperature T in the initial state |ii, αf i , γf i and δf i are rotational invariants for the polarizability αf i associated with the f ← i transition. The above differential cross section refers to scattered radiation recorded with no analyzer in the observed beam. Only Raman Stokes transitions will be considered in this work, that is ν˜R = ν˜0 − ν˜f i , with ν˜f i being the wavenumber associated with the f ← i transition. Using Eq.(1), the intensity If i of Raman scattered radiation, for a sample constituted of N molecules in thermal equilibrium at T , is given by the following product: If i = J N

dσ dΩ

(2) fi

with J being the irradiance of the incident radiation. 1 The squared isotropic, symmetric anisotropic and antisymmetric anisotropic polarizability invariants are evaluated as follows: 53 1 (ααα )S (αββ )S∗ 9 1 3(ααβ )S (ααβ )S∗ − (ααα )S (αββ )S∗ = 2 3 = (ααβ )A (ααβ )A∗ 2

αf2 i =

(3)

γf2i

(4)

δf2i

(5)

with ααβ being the αβ-Cartesian component of the αf i tensor. Einstein’s summation convention is adopted above, in which repeated Greek subscripts representing the Cartesian components are automatically summed over x, y, and z without an explicit summation sign. The symbol “∗” denotes the complex conjugate and S and A superscripts are used to refer to the symmetric and antisymmetric forms of αf i . These invariants are pure real numbers, 7



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since they are built by summing a polarizability component multiplied by its complex conjugate, while terms of the type (αaa )S (αbb )S∗ and (αbb )S (αaa )S∗ always appear in pairs like: (αaa )S (αbb )S∗ + (αbb )S (αaa )S∗ . When the incident radiation is in resonance with an excited electronic state (e.g. |er i), within the Born-Oppenheimer approximation, a good approximation for a pure vibronic polarizability is given by:

αab =

X hvfg |heg |ˆ µa |er i|vnr ihvnr |her |ˆ µb |eg i|vig i r vn

(6)

Er − Eg + Evn − Evi − E0 − iΓrvn

In the above formula, |eg i, |er i, Er and Eg are the ground and excited state electronic wavefunctions and respective energies, |vig i and |vfg i are the initial and final vibrational wavefunctions of the ground electronic state, |vnr i is a vibrational wavefunction of state |er i, Evn and Evi are vibrational energies and E0 is the energy of the incident photon. The pure vibronic half-width at half-maximum Γrvn of state |er i|vnr i was calculated from the corresponding radiative lifetime according to:

Γrvn

h = 4π

64π 4 (Er − Eg + Evn − Evi )3 3h4 c3

x,y,z X

g 2 hv |heg |ˆ µb |er i|vnr i i

(7)

b

At the FC level, the electric dipole electronic transition moment (e.g. heg |ˆ µa |er i) does not depend on nuclear coordinates, while at HT level it is expanded as a power function of the normal coordinate of the ground electronic state around the equilibrium geometry of that state. 3 For diatomic systems: heg |ˆ µa |er iQ = heg |ˆ µa |er i0 +

dheg |ˆ µa |er i dQ

1 Q+ 2 0

d2 heg |ˆ µa |er i dQ2

Q2 + · · ·

(8)

0

The FC approximation is obtained when the expansion (8) is truncated at the zeroth order term. The linear term of Eq.(8) is labelled in this work as HT1, the quadratic term as HT2 and high order HT corrections to heg |ˆ µa |er iQ are not considered here. Finally, if the 8


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differential scattering cross section is evaluated using αf i from Eq.(6), then the corresponding Raman shift is given by ν˜R = (Evf − Evi )/hc. Our Fortran code PLACZEK, 54,55 developed for infrared and Raman calculations, was extended to compute the differential cross section of Eq.(1) and quantities therein. The geometrical derivatives in Eq.(8) were evaluated by fitting the transition moment curve to a polynomial function whose coefficients where obtained from the least squares fitting technique. 56 The degree of the polynomial was allowed to vary from 1–5 and that with the minimum Mean Absolute Error was select to compute the first and second derivatives of the electronic transition moment. Additional care was taken regarding the sign of the transition moment components heg |ˆ µa |er i throughout the displaced geometries, as random changes can occur even if only small displacements from the reference geometry are made. The FC and HT integrals hvfg |Ql |vnr i and hvnr |Ql |vig i, with l = 0, 1, 2, were obtained from the numerical integration of arbitrary vibrational wavefunctions using the composite Simpson’s rule or from the cubic spline interpolation function. 57,58 For the sake of comparison, we have also implemented analytical formulae for harmonic oscillator wavefunctions. 59 Concerning the thermal population fi in Eq.(1), if the available set of vibrational energies is large enough to allow good convergence of the vibrational partition function qv (T ) (better than 0.1%), then fi is obtained from that qv (T ); otherwise, the harmonic oscillator qv (T ) is used. The vibrational wavefunctions and energies present in Eq.(6) were evaluated for Hamiltonians of the following type: 2 2 b = − ~ d + Vb H 2 dQ2

(9)

with Vb being an arbitrary potential energy operator. We have used two different approaches b (i) A Variational Stochastic Method (VSM), 51 which we have to find the eigenfunctions of H: implemented as a Fortran code with the Mersenne Twister pseudorandom number generator (MT19937) 60 or the LUXury RANdom pseudorandom number generator (RANLUX) 61

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replacing the built-in random number generator of the GNU’s Fortran compiler for better performance; and (ii) the Cooley-Numerov method implemented in the SCHRQ routine of Le Roy’s LEVEL program, 62 which is based on the Cooley-Cashion-Zare SCHR routine. The potential energy curves V (Q) and electronic transition moments of H2 , C2 and O2 were obtained from multiconfiguration ab initio wavefunctions. For H2 , Potential Energy Curves (PEC) 1 + for the X 1 Σ+ g and B Σu states were calculated at the CASSCF/d-aug-cc-pVQZ level, within

the D2h point group symmetry, with a full valence active space. The electric dipole transition moment was obtained from the single residues of the CASSCF linear response using 3 − the DALTON electronic structure code. 63 In the case of O2 , the X 3 Σ− g and B Σu states were

chosen based on the availability of experimental RR intensities. 64 Those states were studied at the MRCI/CASSCF(16,10)/aug-cc-pCV6Z level of theory, with the inclusion of relaxed Davidson correction, in such a way that core-valence correlation effects were taken into account. All calculations were done under D2h symmetry constraint, with the MOLPRO 65 software. Electronic transition moments were evaluated directly from the wavefunctions. For C2 , electronic structure data were taken from the literature. 66,67

3

Results and discussion

Before presenting the results of the RR spectra, we will first describe in more detail the procedure adopted to numerically evaluate the FC and HT integrals, in order to select the most efficient and accurate method and parameters for computing the RR cross sections.

3.1

Numerical methods

Our strategy to calculate FC and HT integrals for arbitrary vibrational wavefunctions was the adoption of numerical integration methods. For that purpose, we have selected the Simpson’s one-third rule and integration of the interpolating cubic spline function, wherein the corresponding Fortran routines were taken from ref. 57. When Simpson’s integration is used,

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some parameters related to convergence control must be specified, namely the relative and absolute integration errors ∆REL and ∆ABS . The accuracy of these integration methods was assessed by comparing numeric harmonic oscillator results against analytic data, evaluated using the Doktorov et al. formulae. 59 For instance, some FC and HT integrals for harmonic 3 − vibrational wavefunctions of the X 3 Σ− g and B Σu states of

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O2 are given in Table 1. The

corresponding vibrational functions associated with those integrals depend, in addition to the isotopic masses, on the equilibrium interatomic distance re and the harmonic wavenumber ωe , which were set equal to their MRCI(16,10)/aug-cc-pCV6Z values of 1.2062 ˚ A and −1 ˚ for the B 3 Σ− 1589.06 cm−1 for state X 3 Σ− u state. Upon g and of 1.5989 A and 723.05 cm

analysis of Table 1, it is possible to notice that both integration schemes are capable of delivering results that are in good agreement with the analytic data, even though those integrals with |vnr i = |0i were underestimated from Simpson’s integration when ∆REL = 10−03 and ∆ABS = 10−06 a.u. This result arises from the reduced number of points composing the integration grid (201 points). However, if that number is increased (e.g. to 801 points) good agreement with the analytic data is also achieved. For further calculations of FC and HT integrals, we have decided to obtain them from the cubic spline function due to its lower computation time and good agreement with the analytic data. As an additional accuracy check, one may compare harmonic oscillator differential cross sections calculated using numeric FC and HT integrals to those obtained from analytic integrals. Such cross sections for the fundamental transition of

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O2 are given in Table 2. They were evaluated using a set of

12 intermediate vibrational states and at three different vibronic levels: (i) Within the FC approximation, (ii) FC plus the HT1 correction (FCHT1 in Table 2) and (iii) FC plus HT1 and HT2 corrections (FCHT12 ). The results in Table 2 show good agreement between numeric and analytic data, with a convergence of two digits for the grid containing 201 points, and of four digits for the one containing 801 points.

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3.2

Electronic structure and vibrational calculations

The purpose of this section is to check the accuracy of the VSM and SCHRQ vibrational wavefunctions and energies against the experimental data, as well as to establish some necessary parameters to obtain converged VSM vibrational states. Before doing so, calculated electronic structure data for H2 and O2 will be presented and compared to available theoretical and experimental information in order to assess their accuracy. At the FC level, strong RR scattering is expected if the excitation energy falls within an absorption band of high intensity. Therefore, the B1 Σ+ u state of H2 was chosen due to the strongly allowed electronic transition between that and the ground state of the molecule. At the CASSCF(2,124)/daug-cc-pVQZ level, the corresponding transition moment is equal to 0.98332 a.u. and is in good agreement with the value of 0.98203 a.u. reported by Wolniewicz and Staszewska. 68 1 + The CASSCF(2,124)/d-aug-cc-pVQZ PEC of states X1 Σ+ g and B Σu (see Figure 1), in accor-

dance with the Wigner-Witner rule, 69 have different dissociation channels. This PEC yielded 81362.407 cm−1 for the energy of the H(2 S)←H(2 P0 ) atomic transition, while the experimental value was 82258.919 cm−1 . 70 As shown in Table 3, the CASSCF(2,124)/d-aug-cc-pVQZ spectroscopic constants agree well with the experimental data. The RR spectrum of O2 was studied within the spectral region known as the Schumann3 − Runge range, 64,71,72 from 205 nm to 175 nm, corresponding to the allowed X3 Σ− g ← B Σu

transition. For this spectral region, absolute experimental ro-vibronic RR differential cross 3 − sections are available. 64 The MRCI(16,10)/aug-cc-pCV6Z PEC of states X3 Σ− g and B Σu are

given in Figure 1. In general, the calculated spectroscopic constants for those states are in good agreement with the experimental values, as shown in Table 4. The only exception occurs in the dissociation energy of the B3 Σ− u state which was underestimated in our calculations. As illustrated in Figure 2, extrapolation of the ab initio PEC into the repulsion region was necessary in order to obtain the correct asymptotic behavior of the wavefunctions. They were extrapolated according to the V (Q) = A + B/Q function and constants A and B were found by requiring that V (Q) go through the first two data points. 73 After extrapolation, 12


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the correct asymptotic behavior was obtained (right panel of Figure 2). To assess the accuracy of the wavefunctions and their energies computed using our VSM code, those properties, evaluated for harmonic potentials, were compared to analytic data. Further, anharmonic energies were compared to experimental data, as illustrated in Figure 3 for H2 , where experimental values were obtained from the spectroscopic constants ωe , ωe xe and ωe ye . 74 Those VSM and SCHRQ energies are in good conformity with each other and lie close to the experimental values. The SCHRQ energies reported here were converged to 10−4 cm−1 . In VSM calculations, the number of points was typically fixed around 250 points, and about 106 steps were performed for each vibrational state to achieve energies converged to 1%. Fast optimization and better results were obtained from the MT19937 RNG and by guessing wavefunctions with random numbers. As a final check, the VSM and SCHRQ anharmonic energies for the ground electronic state of H2 were compared to those evaluated using Numerov’s method (Table 5) as implemented in the VIBROT program of the MOLCAS electronic structure software package. 75 Both VSM and SCHRQ energies agree well with Numerov data, although those from SCHRQ were closer to the Numerov energies and to the experimental data as well. Due to the large number of intermediate vibrational states required to achieve wellconverged RR cross sections and the slow convergence of VSM computations, fully anharmonic RR spectra were evaluated using only SCHRQ wavefunctions.

3.3

RR spectrum

Once the vibrational wavefunctions and their energies are known, together with electronic structure data (energies, transition moments and possibly their derivatives), the vibronic damping factor Γrvn can be calculated from Eq.(7) and the only additional information demanded to compute the RR spectrum are the sample’s temperature and excitation energy E0 . Unless otherwise specified, E0 was chosen to be equal to Er − Eg , i.e. the Te energy, thus resulting in the perfect cancelation of the (Er − Eg − E0 ) term in the denominator of Eq.(6). 13



The FC and HT integrals and vibrational energies, required to perform harmonic calculations, were obtained from the equilibrium interatomic distances and harmonic frequencies of the ground and excited electronic states PEC. Second derivatives of the electronic energy with respect to the interatomic distance were calculated from the cubic spline interpolation function. Within the TI approach of RR, αf i must be evaluated with a minimal number of intermediate vibrational states, Nvr , in order to achieve converged cross sections. The dependence of the fundamental and first two overtones differential cross sections of H2 on Nvr is illustrated in Table 6. For differential cross sections of harmonic wavefunctions, a convergence of two digits was attained with Nvr ≥ 15 at both FC and FCHT12 levels. In contrast, for the anharmonic overtone transitions, as many as 35 states were not enough to reach the same convergence as observed for harmonic wavefunctions. Such behavior is caused by non-zero FC and HT integrals of highly excited intermediate vibrational states. When those integrals are evaluated using harmonic wavefunctions, for large vnr , the corresponding wavefunction displays approximately regular oscillations extending up to regions of negative interatomic distances. Such oscillations create a cancelation effect on the resulting FC and HT integrals of highly excited intermediate states. Because anharmonic wavefunctions tend towards zero for small positive R, there are no regular oscillations of |vnr i overlapping with |vig i, and the same cancelation mechanism is not observed. A graphical explanation of the above arguments is given in Figure 4 for the h34r |0g i FC integral of H2 . The left panels display the harmonic wavefunctions |0g i and |34r i and their product h34r |0g i(R) summed over the interatomic distance R. As the integrand h34r |0g i(R) is approximately antisymmetric with respect to re of the ground electronic state, its summation involves positive and negative contributions of similar magnitude, thus resulting in a very low FC integral. The corresponding anharmonic |0g i, |34r i and h34r |0g i(R) are shown in the right panels of Figure 4. Here, the overlap between those anharmonic wavefunctions has a positive contribution to R < 1.4 bohr that does not cancel out after summing over R, and thus h34r |0g ianh = 0.025 whereas h34r |0g ihar = 0.000. An even slower convergence is observed in the anharmonic dif-

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ferential cross section of the 1 ← 0 transition of O2 . According to Figure 5, more than 80 intermediate vibrational states are required in order to have converged FC and FCHT12 cross sections, whereas for harmonic wavefunctions that number is lower than 30. This suggests how challenging it would be to calculate anharmonic RR spectra of polyatomic systems due to the substantial number of FC and HT integrals required to end up with converged cross sections. The importance of including the HT vibronic correction in order to properly describe the RR intensities of polyatomic systems, within the harmonic approximation, has already been demonstrated before. 29 For instance, the HT mechanism gives rise to bands that are not observed at the FC level. In diatomic molecules, molecular symmetry does not change during vibration; therefore, the HT1 and HT2 contributions can only change band intensities. In Table 7, the differential cross sections for some vf ← 0 transitions of H2 and O2 are given as a function of the vibronic model adopted (FC, FCHT1 or FCHT12 ). For H2 , stronger intensities are obtained at the FCHT1 and FCHT12 levels and, in addition, that enhancement is even more pronounced for anharmonic wavefunctions. For O2 , conversely, there is a decrease in the differential cross sections after the HT correction is introduced in both harmonic and anharmonic models. Interestingly, the relative RR intensities from harmonic wavefunctions are more sensitive to the adopted vibronic model than the anharmonic ones. For example, the harmonic cross section of the 2 ← 0 transition in O2 is twice as large as that of the 1 ← 0 transition at the FC level, and 4.1 times at the FCHT12 level, whilst for anharmonic wavefunctions, those ratios are of 4.9 and 4.7, respectively. The RR spectra of H2 obtained from harmonic and anharmonic wavefunctions at the FCHT12 level are compared in Figure 6. In spite of their simplicity, we see that the harmonic model overestimates Raman shifts and peak intensities of all transitions. Even more importantly, the relative intensities from those vibrational models are significantly different, e.g. the overtone 2 ← 0 is 42% stronger than the fundamental transition at the harmonic level and 5.2 times more intense at the anharmonic level. Nevertheless, the qualitative aspect

15



of the 3 ← 0 transition being the strongest, followed by 2 ← 0, 4 ← 0 and 1 ← 0 transitions, is recovered from the harmonic model. A stringent test to assess any spectroscopic model is to compare calculated absolute intensities to experimental data. The measured absorption spectrum of O2 shows that the 3 − 64 electronic transition B3 Σ− Bischel and Black reu ← X Σg is observed around 191 nm.

ported absolute Q-branch differential cross sections for the fundamental transition measured within the range 550–200 nm and fitted the resulting data to a function that matches Eq.(6), when the summation over vnr is limited to a single intermediate state. 76 Tomov et al. have used that function to extrapolate the differential cross section for λ0 = 193 nm, obtaining 2.01 × 10−31 m2 sr−1 for that property. 77 At the FCHT12 level, the harmonic value for the pure vibrational differential cross section (summed intensities of Q, O and S-branches) is 3.33 × 10−32 m2 sr−1 and the corresponding anharmonic cross section is 1.21 × 10−35 m2 sr−1 . Such low value for the anharmonic cross section is mainly caused by those FC and HT integrals involving the first |vn i states, which, in general, are one order of magnitude lower than the harmonic counterparts. Nevertheless, the anharmonic cross section is largely increased at some specific excitation wavelengths around 193 nm, as it is illustrated by the excitation profile of the 1 ← 0 transition given in Figure 7. According to Eq.(6), whenever the Er − Eg + Evn − Evi energy term approaches E0 , there will be resonance enhancement for the set of vf ← vi transitions. The poles of αf i for the vf ← 0 transitions are given in Table 8. For the excitation wavelength of 193.0 nm (51813 cm−1 ) the closest pole is that in which vn = 4, observed in 192.4 nm (51971 cm−1 ). On the other hand, the pole structure of both harmonic and anharmonic levels shows that the excitation wavelength λ0 =193.0 nm is in resonance with the vn = 2 pole. The excitation profile for the 1 ← 0 transition of O2 (Figure 7) shows that the resonance enhancement around each pole has a very narrow wavelength window. For λ0 =193.0 nm, both harmonic and anharmonic cross sections are not strictly in resonance with the intermediate state vn = 2. In systems displaying very sharp poles, like the RR scattering of the B3 Σ− u state of O2 , it may be necessary to consider

16


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the contribution of other excited states to αf i in order to properly describe their Raman spectra. Another important feature that should be attained in theoretical calculations of the RR spectra is the relative intensities. Zhang and Ziegler reported the RR of O2 obtained with an excitation wavelength of 190.69 nm, in which, according to Table 8, lies near the vn = 5 experimental pole. In order to improve the agreement between the theoretical poles and the experimental values, the harmonic and anharmonic spectra were calculated using the experimental Te energy given in Table 4. As shown in Figure 8, the relative intensities are poorly described at the harmonic level whereas at the anharmonic level the agreement with the experimental spectrum is significantly improved. Resonance enhancement in Raman scattering is strictly related to the small magnitude of the energy denominator in the polarizability expression:

∆E ≡ (Er − Eg + Evn − Evi − E0 )

(10)

That enhancement pattern of the Raman bands as a function of ∆E is well illustrated by the excitation profiles of vf ← 0 transitions, with vf = 1, 2, 3, in the anharmonic RR spectrum of C2 displayed in Figure 9. According to that figure, the differential cross sections can vary about ten orders of magnitude when E0 varies only 2000 cm−1 . Those differential cross sec1 tions were calculated for the X 1 Σ+ g and A Πu electronic states of dicarbon from the ab initio

multireference electronic structure data of references 66, 67. Therefrom Te = 8489.7 cm−1 and the absolute value of the transition moment is equal to 0.3482 a.u., at the ground state equilibrium geometry. The dependence of the electronic transition moment on interatomic distance is available from reference 66, and we have used those values to evaluate HT1 and HT2 corrections to the polarizability. The equilibrium interatomic distances and harmonic wavenumbers of those PEC are 1.2432 ˚ A, 1.3176 ˚ A, 1761.22 cm−1 and 1627.99 cm−1 , for the ground and excited states, respectively. Note that in Figure 9 all three excitation profiles have poles at the same E0 , which is because ∆E is common to the RR polarizabilities of

17



vf ← 0 transitions. The first pole arises when E0 matches the Te + E0r − E0g energy. Since Evn < Evi , with E0r = 729.02cm−1 and E0g = 919.55cm−1 , that pole occurs for ∆E lower than Te . The next two poles are observed when E0 = Te + E1r − E0g and E0 = Te + E2r − E0g . The harmonic excitation profiles are qualitatively similar to those of Figure 9, however their poles occur at different excitation wavenumbers. They are available as Supporting Information.

4

Conclusions

In this contribution, we have presented an ab initio multireference study of the resonance Raman (RR) spectrum of diatomic systems, wherein the anharmonicity of the potential energy curve is fully accounted for by numerically solving the vibrational Schrödinger equation using the Variational Stochastic Method (VSM) and Cooley-Numerov methods. Even though both methods are able to provide accurate solutions for the vibrational equation, due to the slow convergence observed for VSM when highly excited states are optimized, in the RR calculations we preferred to employ vibrational wavefunctions from the CooleyNumerov method. The RR polarizabilities were built using anharmonic Franck-Condon (FC) and Herzberg-Teller (HT) integrals that were numerically evaluated from the corresponding anharmonic wavefunctions. Due to a cancelation effect observed in harmonic FC and HT integrals of highly excited intermediate vibrational states, harmonic cross sections converge faster than anharmonic ones with respect to the number of intermediate vibrational states. Such cancelation mechanism does not occur in anharmonic FC and HT integrals. Therefore, the anharmonic calculation of RR spectra for polyatomic systems might require a much larger number of FC and HT integrals to achieve converged cross sections. The anharmonic spectra evaluated for H2 , C2 and O2 molecules show that RR intensities are strongly affected by anharmonic effects. They differ from their harmonic counterparts not only in the position of the peaks, but also in the absolute and relative intensities. In spite of the considerable

18


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computational task that would be needed to simulate the anharmonic RR spectra of polyatomic systems, our study indicates that the resulting spectra may be significantly improved regarding compliance with the experimental data.

Acknowledgement GJC thanks the Coordena¸cão de Aperfei¸coamento de Pessoal de N´ıvel Superior (Capes), for a masters scholarship. GJC and LNV thank Professors Rodolfo G. Begiato and Pedro A. M. Vazquez for their valuable support on numerical methods and computer programming. RC acknowledges financial support from Funda¸caõ de Amparo a` Pesquisa do Estado de São Paulo (FAPESP) - Center for Computational Engineering and Sciences, Grant 2013/08293-7, and Grant 2017/11485-6. ACB thanks the Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnológico (CNPq) for research fellowship, Grant 303352/2013-2, and the STI (Superintendência de Tecnologia da Informa¸caõ) of the University of São Paulo for services and computer time. The Computational Physics Lab facilities available at UTFPR-CT are also acknowledged.

Supporting Information Available The following is available as supporting information to this paper: • supporting-info.pdf: Radiative lifetimes and half-width at half-maximum, Γrvn , for H2 , C2 and O2 , evaluated for harmonic and anharmonic wavefunctions at the FCHT12 level. RR spectra and excitation profiles of C2 . • F77Code.txt: Source code of our Fortran routine developed to calculate the RR spectrum of diatomics from numeric wavefunctions. This code requires three input files, namely: (i) input.txt, (ii) X.txt and (iii) B.txt; The last two files may be named as you wish limited to the maximum size of 80 characters. 19



• input.txt: Input file of the F77Code for a sample calculation of the RR spectrum of the H2 molecule. • X.txt: LEVEL’s output file (fort.10) containing vibrational wavefunctions of the ground electronic state. • B.txt: LEVEL’s output file (fort.10) containing vibrational wavefunctions of the excited electronic state.

20


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Table 1: Harmonic oscillator FC and HT integrals (in a.u.) involving vibrational wavefunc3 − tions of the X 3 Σ− g and B Σu states of O2 . Integration by means of composite Simpson’s rule or from the cubic spline function was performed using 201 points. hvig |Ql |vnr i Analytic Simpsona Simpsonb h0|0i 0.000 0.000 0.000 h0|Q|0i 0.003 0.000 0.003 h0|Q2 |0i 0.096 0.000 0.095 h0|2i h0|Q|2i h0|Q2 |2i

0.002 0.047 1.342

0.002 0.047 1.339

h0|10i h0|Q|10i h0|Q2 |10i

0.254 2.476 37.979

0.254 2.477 37.958

0.002 0.047 1.339

Spline 0.000 0.003 0.095 0.002 0.047 1.339

0.254 0.254 2.477 2.477 37.958 37.956

Relative and absolute integration errors set equal to ∆REL = 10−03 and ∆ABS = 10−06 a.u., respectively. b Here, ∆REL = 10−06 and ∆ABS = 10−09 a.u. a

21



Table 2: Harmonic differential cross sections (in 10−31 m2 sr−1 ) for the 1 ← 0 transition of O2 at several vibronic levels. They were evaluated using 12 intermediate vibrational states, with T = 300 K and λ0 = 193 nm. Vib. level Analytic 201 pts 801 pts FC 3.0791 3.0817 3.0791 FCHT1 2.2694 2.2715 2.2694 FCHT12 2.2441 2.2463 2.2442

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Table 3: CASSCF(2,124)/d-aug-cc-pVQZ and experimental spectroscopic constants for H2 . Re /˚ A

Te /cm−1

ωe /cm−1

Be /cm−1

X1 Σ+ g Calc. Exp. 74

0.74200 0.74144

0 0

4399.990 4401.213

60.7622 60.8530

B1 Σ+ u Calc. Exp. 74

1.28426 1.29282

91638.8 91700.0

1367.410 1358.090

20.2830 20.0154

23



Table 4: MRCI(16,10)/aug-cc-pCV6Z and experimental spectroscopic constants for Re /˚ A

Te /cm−1

ωe /cm−1

De /eV

X Calc. Exp. 74,78

1.2062 1.2075

0 0

1589.06 1580.19

5.4359 5.2132

A3 ∆u Calc. Exp. 71,78

1.51 1.48

34759.7 34735.0

823.74 750.0

0.8790 0.9066

1.5989 1.6043

50833.19 49794.33

723.05 709.06

0.8383 1.0069

3

Σ− g

B3 Σ− u Calc. Exp. 71

24


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16

O2 .

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Table 5: CASSCF(2,124)/d-aug-cc-pVQZ vibrational energies (in cm−1 ) of the ground electronic state of H2 . The percent difference with respect to experimental values is given between parentheses. v 0 1 2 3 4 5 6 7 8 9

HO 2199.99( 1.4) 6599.98( 4.3) 10999.97( 7.4) 15399.97(10.6) 19799.95(14.1) 24199.94(17.8) 28599.93(21.8) 32999.92(26.0) 37399.92(30.6) 41799.90(35.4)

VSM 2020.31(-6.9) 6220.47(-1.7) 10214.66(-0.3) 13997.70( 0.6) 17562.14( 1.2) 20901.61( 1.8) 24012.43( 2.3) 26860.15( 2.6) 29454.07( 2.8) 31770.33( 2.9)

25

SCHRQ 2177.47( 0.3) 6335.40( 0.1) 10257.94( 0.1) 13949.94( 0.2) 17414.40( 0.4) 20652.24( 0.6) 23662.01( 0.8) 26439.51( 1.0) 28977.29( 1.2) 31263.80( 1.3)


Exp. 74 2170.27 6328.81 10244.68 13917.88 17348.40 20536.26 23481.44 26183.95 28643.78 30860.95


Table 6: Convergence of H2 differential cross sections (in 10−28 m2 sr−1 ) regarding the number of intermediate vibrational states Nvr . The excitation wavenumber is λ0 = 109.1 nm and T = 300 K. FC 1 ← 0a 2←0 3←0 Nvr Harm. Anharm. Harm. Anharm. Harm. Anharm. 5 2.84 0.29 3.72 0.98 3.95 1.30 10 3.58 0.16 4.00 1.28 3.97 1.61 15 3.58 0.21 4.02 1.41 3.96 1.48 20 3.58 0.25 4.02 1.39 3.96 1.42 25 3.58 0.26 4.02 1.36 3.96 1.42 30 3.58 0.27 4.02 1.34 3.96 1.43 35 3.58 0.27 4.02 1.34 3.96 1.44 FCHT12 5 5.19 0.83 7.83 2.92 8.60 4.19 10 5.94 0.55 8.40 3.60 8.73 4.99 15 5.94 0.63 8.43 3.83 8.73 4.71 20 5.94 0.69 8.43 3.80 8.73 4.62 25 5.94 0.71 8.43 3.76 8.73 4.62 30 5.94 0.72 8.43 3.74 8.73 4.63 35 5.94 0.72 8.43 3.73 8.73 4.63 a

Transition type: vfg ← vig

26


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Table 7: Differential cross sections (in m2 sr−1 ) of H2 and O2 as a function of the vibronic model adopted for the electronic transition moments. a

H2 FC FCHT1 FCHT12 O2 b FC FCHT1 FCHT12 a b

1←0 Harm. Anharm. 3.58×10−28 2.73×10−29 5.85×10−28 7.00×10−29 5.94×10−28 7.24×10−29 1←0 Harm. Anharm. Harm. Anharm. 5.34×10−32 1.56×10−35 1.49×10−32 7.82×10−36 1.47×10−32 7.43×10−36


Cross sections obtained with λ0 = 109.1 nm, T = 300 K and Nvr = 35. Here, λ0 = 196.7 nm, T = 300 K and Nvr = 120.

27




Table 8: Predicted poles (in nm) for the 1 ← 0 transition of O2 in the RR spectrum. vn 0 1 2 3 4 5 6

Harm. Anharm. Exp. 74 198.4 198.4 202.6 195.6 195.7 199.8 192.9 193.2 197.2 190.2 190.9 194.7 187.6 188.7 192.4 185.1 186.7 190.2 182.7 184.8 188.2

28


Page 28 of 48

Page 29 of 48

1

+

B Σu

-0.25

-0.50 2

2 0

Energy / Hartree

H( S)+H( P ) -0.75

-1.02

2

2

H( S)+H( S)

-1.08 -1.14

1

+

X Σg

-1.20 0.0

2.0

4.0

6.0 R / a.u.

8.0

10.0

-149.40 3 3

-149.60

3

Energy / Hartree

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


-

Σu

∆u Σg

-

-149.80

-150.00

3

1

3

3

O( P) + O( D) O( P) + O( P)

-150.20

-150.40 1.50

2.00

2.50

3.00 3.50 R / a.u.

4.00

4.50

5.00

Figure 1: Potential energy curves for H2 (top), CASSCF(2,124)/d-aug-cc-pVQZ level, and for O2 (bottom), MRCI(16,10)/aug-cc-pCV6Z level.

29


0.30

0.30

0.20

0.20

0.10

0.10 Ψv / a.u.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Ψv / a.u.


0.00

-0.10

0.00

-0.10

Ψ0 Ψ1

-0.20

0.00

Page 30 of 48

0.75

1.50

2.25 R / a.u.

3.00

3.75

-0.20

4.50

0.00

Ψ0 Ψ1 0.75

1.50

2.25 R / a.u.

3.00

3.75

Figure 2: VSM vibrational wavefunctions of the ground electronic state of H2 before (left) and after (right) extrapolation of the corresponding PEC in the repulsion region.

30


4.50

Page 31 of 48

4.0e+04

1

X Σg

1.4e+04

+

1

B Σu

+

1.2e+04 3.0e+04

1.0e+04 Ev /cm (Exp.)

2.0e+04

6.0e+03

4.0e+03

1.0e+04

0.0e+00 0.0e+00

8.0e+03

-1

-1

Ev /cm (Exp.)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Exact agreement VSM SCHRQ HO 5.0e+03

1.0e+04

1.5e+04 -1 Ev /cm (Calc.)

2.0e+04

2.5e+04

Exact agreement VSM SCHRQ HO

2.0e+03

3.0e+04

0.0e+00 0.0e+00

2.0e+03

4.0e+03

6.0e+03 -1 Ev /cm (Calc.)

8.0e+03

1.0e+04

Figure 3: Comparison of calculated vibrational energies for the electronic states X 1 Σ+ g and 1 + B Σu of H2 to experimental values. Harmonic energies are labeled as “HO”, whereas VSM and SCHRQ designate anharmonic energies.

31


1.2e+04


0.8 HO

SCHRG

HO

SCHRG

0.6

0.2

r

Σι ψ i . ψ i / a.u.

0.4

g

0.0 -0.2 -0.4 -0.6

g

0 r 34

0.2 ψ / a.u.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 32 of 48

0.1 0.0 -0.1 -0.2

0.4

0.8

1.2

1.6 R / a.u.

2.0

2.4

0.4

0.8

1.2

1.6 R / a.u.

2.0

2.4

Figure 4: Vibrational wavefunctions |0g i and |34r i of H2 (bottom panels) and their product summed along the interatomic distance (top panels). Labels “HO” and “SCHRQ” refer to harmonic and anharmonic wavefunctions, respectively.

32


Page 33 of 48

1.4e-35

2

(dσ/dΩ) / m sr

-1

Anharm.

1.0e-35 7.0e-36 3.5e-36

Harm.

3.6e-31

FC FCHT12

-1 2

(dσ/dΩ) / m sr

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


2.7e-31 1.8e-31 9.0e-32

0

10

20

30

40

50

60 Nvr

70

80

90

100

110

120

Figure 5: Convergence of the RR cross section for the 1 ← 0 transition of O2 with respect to the number of intermediate vibrational states.

33



FCHT12

Anharm.

8.0e-28

1

0

-1

6.0e-28

2

2

(dσ/dΩ) / m sr

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 34 of 48

0

3

0

4.0e-28

2.0e-28 4

0.0e+00

3000

6000

9000 12000 -1 Raman shift / cm

15000

0

18000

Figure 6: RR spectra of H2 computed at the FCHT12 level with λ0 = 109.1 nm, T = 300 K and Nvr = 35.

34


Page 35 of 48

193 nm

Anharm.

1e-32 1e-34

2

(dσ/dΩ) / m sr

-1

1e-33 1e-35 1e-36 1e-37 1e-38 1e-39 Harm.

-1

1e-29 1e-30

2

(dσ/dΩ) / m sr

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1e-31 1e-32 1e-33 1e-34 198

196

194

192 λ0 / nm

190

188

186

Figure 7: RR excitation profile for the 1 ← 0 transition of O2 obtained at the FCHT12 level with T = 300 K and Nvr = 120.

35



1

1 Exp.

0.6 0.4 0.2

0.6 0.4 0.2

0

0 Harm.

Anharm.

2

(dσ/dΩ) / m sr

-1

1.8e-33

-1

1.2e-31 2

Exp.

0.8 Relative Intensity

Relative Intensity

0.8

(dσ/dΩ) / m sr

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 36 of 48

9e-32 6e-32 3e-32 0

1.4e-33 9.2e-34 4.6e-34

1500

3000

4500


9000

10500

0

1500

3000

4500


9000

10500

Figure 8: RR spectra of O2 evaluated at the FCHT12 level with λ0 = 190.69 nm, T = 300 K and Nvr = 120.

36


Page 37 of 48

3

0

2

0

1e-30

2

(dσ/dΩ) / m sr

-1

1e-28 1e-32 1e-34 1e-36 1e-38

1e-30

2

(dσ/dΩ) / m sr

-1

1e-28 1e-32 1e-34 1e-36 1e-38

-1

1e-28 1

1e-30

0

2

(dσ/dΩ) / m sr

1e-32 1e-34 1e-36 1e-38 1e-40

8000

10000

4000

12000

14000 -1 E0/ cm

16000

18000

20000

Te r

v n= 0

3000

r

v n= 1 r

v n= 2

2000

-1

1000 ∆E / cm

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0 -1000 -2000 -3000 -4000 7500

8000

8500

9000

9500 10000 -1 E0 / cm

10500

11000

11500

12000

Figure 9: RR excitation profiles for the vfg ← 0g transitions of C2 (top) obtained at the FCHT12 level with T = 300 K and Nvr = 42. On the bottom, the energy denominator ∆E ≡ (Er − Eg + Evn − Evi − E0 ) is given as a function of the excitation energy E0 for three different intermediate states, with vig fixed as 0g .

37



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Fully Anharmonic Vibrational Resonance Raman Spectrum of

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