Theory and method for calculating resonance Raman scattering from resonance polarizability derivatives

We present a method to calculate both normal Raman-scattering (NRS) and resonance Raman-scattering (RRS) spectra from the geometrical derivatives of the frequency-dependent polarizability. In the RRS case, the polarizability derivatives are calculated from resonance polarizabilities by including a finite lifetime of the electronic excited states using time-dependent density-functional theory. The method is a short-time approximation to the Kramers, Heisenberg, and Dirac formalism. It is similar to the simple excited-state gradient approximation method if only one electronic excited state is important, however, it is not restricted to only one electronic excited state. Since the method can be applied to both NRS and RRS, it can be used to obtain complete Raman excitation profiles. To test the method we present the results for the S2 state of uracil and the S4, S3, and S2 states of pyrene. As expected, the results are almost identical to the results obtained from the excited-state gradient approximation method. Comparing with the experimental results, we find in general quite good agreement which enables an assignment of the experimental bands to bands in the calculated spectrum. For uracil the inclusion of explicit waters in the calculations was found to be necessary to match the solution spectra. The calculated resonance enhancements are on the order of 10(4)-10(6), which is in agreement with experimental findings. For pyrene the method is also able to distinguish between the three different electronic states for which experimental data are available. The neglect of anharmonicity and solvent effects in the calculations leads to some discrepancy between theory and experiment.     

Ab initio calculation of resonance Raman cross sections based on excited state geometry optimization

A method for the calculation of resonance Raman cross sections is presented on the basis of calculation of structural differences between optimized ground and excited state geometries using density functional theory. A vibrational frequency calculation of the molecule is employed to obtain normal coordinate displacements for the modes of vibration. The excited state displacement relative to the ground state can be calculated in the normal coordinate basis by means of a linear transformation from a Cartesian basis to a normal coordinate one. The displacements in normal coordinates are then scaled by root-mean-square displacement of zero point motion to calculate dimensionless displacements for use in the two-time-correlator formalism for the calculation of resonance Raman spectra at an arbitrary temperature. The method is valid for Franck-Condon active modes within the harmonic approximation. The method was validated by calculation of resonance Raman cross sections and absorption spectra for chlorine dioxide, nitrate ion, trans-stilbene, 1,3,5-cycloheptatriene, and the aromatic amino acids. This method permits significant gains in the efficiency of calculating resonance Raman cross sections from first principles and, consequently, permits extension to large systems (>50 atoms).     

Analysis of time resolved femtosecond and femtosecond/picosecond coherent anti-Stokes Raman spectroscopy: application to toluene and Rhodamine 6G

The third-order polarization for coherent anti-Stokes Raman scattering (CARS) from a pure state is described by 48 terms in perturbation theory, but only 4 terms satisfy the rotating wave approximation. They are represented by Feynman dual time-line diagrams and four-wave mixing energy level diagrams. In time-resolved (tr) fs and fs/ps CARS from the ground vibrational state, one resonant diagram, which is the typical CARS term, with three field interactions-pump, Stokes, followed by probe-on the ket is dominant. Using the separable, displaced harmonic oscillators approximation, an analytic result is obtained for the four-time correlation function in the CARS third-order polarization. Dlott's phenomenological expression for off-resonance CARS from the ground vibrational state is derived using a three-state model. We calculated the tr fs and fs/ps CARS for toluene and Rhodamine 6G (R6G), initially in the ground vibrational state, to compare with experimental results. The observed vibrational features and major peaks for both tr fs and fs/ps CARS, from off-resonance (for toluene) to resonance (for R6G) pump wavelengths, can be well reproduced by the calculations. The connections between fs/ps CARS, fs stimulated Raman spectroscopy, and impulsive stimulated scattering for toluene and R6G are discussed.     

Resonance Raman scattering of rhodamine 6G as calculated using time-dependent density functional theory

In this work, we present the first calculation of the resonance Raman scattering (RRS) spectrum of rhodamine 6G (R6G) which is a prototype molecule in surface-enhanced Raman scattering (SERS). The calculation is done using a recently developed time-dependent density functional theory (TDDFT) method, which uses a short-time approximation to evaluate the Raman scattering cross section. The normal Raman spectrum calculated with this method is in good agreement with experimental results. The calculated RRS spectrum shows qualitative agreement with SERS results at a wavelength that corresponds to excitation of the S(1) state, but there are significant differences with the measured RRS spectrum at wavelengths that correspond to excitation of the vibronic sideband of S(1). Although the agreement with the experiments is not perfect, the results provide insight into the RRS spectrum of R6G at wavelengths close to the absorption maximum where experiments are hindered due to strong fluorescence. The calculated resonance enhancements are found to be on the order of 10(5). This indicates that a surface enhancement factor of about 10(10) would be required in SERS in order to achieve single-molecule detection of R6G.     

Quantum theory of (femtosecond) time-resolved stimulated Raman scattering

We present a complete perturbation theory of stimulated Raman scattering (SRS), which includes the new experimental technique of femtosecond stimulated Raman scattering (FSRS), where a picosecond Raman pump pulse and a femtosecond probe pulse simultaneously act on a stationary or nonstationary vibrational state. It is shown that eight terms in perturbation theory are required to account for SRS, with observation along the probe pulse direction, and they can be grouped into four nonlinear processes which are labeled as stimulated Raman scattering or inverse Raman scattering (IRS): SRS(I), SRS(II), IRS(I), and IRS(II). Previous FSRS theories have used only the SRS(I) process or only the "resonance Raman scattering" term in SRS(I). Each process can be represented by an overlap between a wave packet in the initial electronic state and a wave packet in the excited Raman electronic state. Calculations were performed with Gaussian Raman pump and probe pulses on displaced harmonic potentials to illustrate various features of FSRS, such as high time and frequency resolution; Raman gain for the Stokes line, Raman loss for the anti-Stokes line, and absence of the Rayleigh line in off-resonance FSRS from a stationary or decaying v=0 state; dispersive line shapes in resonance FSRS; and the possibility of observing vibrational wave packet motion with off-resonance FSRS.     

A Vibrational Raman Optical Activity Study of 1,1′‐Binaphthyl Derivatives

The Raman polarized and vibrational Raman optical activity (VROA) backward spectra are simulated for a series of 2,2′‐substituted 1,1′‐binaphthyl compounds presenting a variety of torsion angles between the two naphthalene rings. The substitution prevents free rotation along this torsion angle and the chirality of these compounds is thus called atropisomerism. However, the rotation is not completely frozen so that two different conformations, namely cisoid and transoid, are found and their Raman and VROA signatures are studied. As expected, the Raman spectra are not very sensitive whereas the VROA spectra present more complex patterns, which evolve as a function of the torsion angle between the two naphthalene groups. In particular, our analysis shows that some modes can be used as a probe for the determination of the torsion angle of these molecules in solution. The contributions of both invariants to the VROA backward intensity are also assessed.     

Theoretical determination of the vibrational Raman optical activity signatures of helical polypropylene chains.

Raman and vibrational Raman optical activity (VROA) spectra of helical conformers of polypropylene chains are simulated using ab initio methods to unravel the relationships between the vibrational signatures and the primary and secondary structures of the chains. For a polypropylene chain containing three units, conformational effects are shown to lead to more acute signatures for VROA than for Raman spectra. In addition to regular polypropylene chains, which can display right and left helicities with the same probability, chirality and therefore helicity are enforced by substituting one chain end with a phenyl group. The simulations predict that the threefold helical structures, which correspond to (TG)(N) conformations of the backbone, have a specific VROA backward signature in the form of an intense couplet around 1100 cm(-1). This couplet is associated with collective wagging and twisting motions, while most of its intensity comes from the anisotropic invariants combining normal coordinate derivatives of the electric dipole-electric dipole polarizability and of the electric dipole-magnetic dipole polarizability. A similar signature has already been found in model helical polyethylene chains, whereas it is very weak in forward VROA.     

Determination of absolute configurations from vibrational raman optical activity: (trans)-2,3-dimethylthiirane

Experimental and theoretical vibrational Raman optical activity (VROA) spectra of (2R,3R)-2,3-dimethylthiirane in the 200-1500 cm(-1) region are presented. The level of agreement obtained for the observed and predicted VROA signs suggests that the absolute configurations of chiral molecules can be determined confidently using VROA.     

UV near-resonance Raman spectroscopic study of 1,1'-bi-2-naphthol solutions

The normal and UV near-resonance Raman (UVRR) spectra of 1,1'-bi-2-naphthol (BN) in basic solution were measured and analyzed. Density functional theory (DFT) calculations were carried out to study the ground state geometry structure, vibrational frequencies nu, off-resonance Raman intensities I, and depolarization ratios rho of 1,1'-bi-2-naphtholate dianion (BN(2-)). On the basis of the calculated and experimental results of nu, I, and rho, the observed Raman bands were assigned in detail. The 1612 cm(-1) Raman band of BN in basic solution was found dramatically enhanced in the UV resonance Raman spectrum in comparison with the normal Raman spectrum. Analyzing the depolarization ratios of the 1366 and 1612 cm(-1) bands in the RR spectra manifests that both the symmetric and antisymmetric parts of transition polarizabilities contribute to the 1366 cm(-1) band, but that only the symmetric part contributes to the 1612 cm(-1) band.     

Inverse Raman bands in ultrafast Raman loss spectroscopy

Ultrafast Raman loss spectroscopy (URLS) is equivalent to anti-Stokes femtosecond stimulated Raman spectroscopy (FSRS), using a broadband probe pulse that extends to the blue of the narrow bandwidth Raman pump, and can be described as inverse Raman scattering (IRS). Using the Feynman dual time-line diagram, the third-order polarization for IRS with finite pulses can be written down in terms of a four-time correlation function. An analytic expression is obtained for the latter in the harmonic approximation which facilitates computation. We simulated the URLS of crystal violet (CV) for various resonance Raman pump excitation wavelengths using the IRS polarization expression with finite pulses. The calculated results agreed well with the experimental results of S. Umapathy et al., J. Chem. Phys. 133, 024505 (2010). In the limit of monochromatic Raman pump and probe pulses, we obtain the third-order susceptibility for multi-modes, and for a single mode we recover the well-known expression for the third-order susceptibility, χ(IRS) ((3)), for IRS. The latter is used to understand the mode dependent phase changes as a function of Raman pump excitation in the URLS of CV.     

Theoretical investigation of Raman optical activity signatures of Tröger's base

The Raman and VROA spectra of (S,S)-Tr?ger's base are simulated. We mainly discuss the peaks in the 1140-1400 cm(-1) wavenumber range where an intense VROA signature is found. In this range, nearly all of the Raman-active bands belong to the irreducible representation A (C(2) point group), whereas no such observation is made for the VROA spectrum. The vibrational normal modes associated with the peaks in this range mainly consist of wagging and twisting motions of the hydrogen atoms. From the atomic contribution patterns (ACPs) and the group coupling matrices (GCMs), one finds that the VROA backward-scattering intensities mainly arise from hydrogen and carbon atoms in the vicinity of the two chiral nitrogen atoms. The VROA signatures in the 1140-1400 cm(-1) range are therefore a fingerprint of the local chirality around the two chiral nitrogen centers.     

An analytical derivative procedure for the calculation of vibrational Raman optical activity spectra

We present an analytical time-dependent Hartree-Fock algorithm for the calculation of the derivatives of the electric dipole-magnetic dipole polarizability with respect to atomic Cartesian coordinates. Combined with analogous procedures to determine the derivatives of the electric dipole-electric dipole and electric dipole-electric quadrupole polarizabilities, it enables a fully analytical evaluation of the three frequency-dependent vibrational Raman optical activity (VROA) invariants within the harmonic approximation. The procedure employs traditional non-London atomic orbitals, and the gauge-origin dependence of the VROA intensities has, therefore, been assessed for the commonly used aug-cc-pVDZ and rDPS:3-21G basis sets.     

Vibrational Raman optical activity of pi-conjugated helical systems: hexahelicene and heterohelicenes

Helicenes constitute a special class of molecules combining helical conformation with pi-electron delocalization. These confer to helicenes specific chirooptical properties. In this article, we investigate the vibrational signatures thanks to the simulation of vibrational Raman optical activity (VROA) spectra. For that, four representative helicenes: hexahelicene, tetrathia-[7]-helicene, and its pyrrole and furan analogs have been simulated and interpreted using a recently implemented analytical scheme. Helicenes show intense VROA peaks attributed to their pi-conjugated structure and associated with collective vibrational modes. In hexahelicene, the dominant VROA features are due to vibrational modes involving motions of the carbon skeleton and H-wagging, but the intensity finds its source almost exclusively in the former. In the case of the three heterohelicenes, the previous statement is also verified, and on changing the heteroatoms, similar modes presenting comparable atomic contribution patterns have been highlighted, though the vibrational and electronic properties are modified. Some fingerprints could therefore be associated with the helicity of the system. In particular, in forward spectra, most of the VROA bands are positive for left-handed helicenes. Nevertheless, the spectral patterns are quite complex, and no easy rule-of-thumb could distinguish between the different heterohelicenes. Then, considering the fact that most of the contributions originate from the C atoms (group coupling matrices decomposition), it can be concluded that the major role of the heteroatom is restricted to modifying the geometry and the normal modes. At last, the small impact of the gauge-origin on the calculated spectra using a relatively modest basis set (rDPS:3-21G) is demonstrated here in the case of the tetrathia-[7]-helicene molecule presenting a C(2) symmetry. This further demonstrates the adequacy of this basis set for VROA calculations.     

Finite lifetime effects on the polarizability within time-dependent density-functional theory

We present an implementation for considering finite lifetime of the electronic excited states into linear-response theory within time-dependent density-functional theory. The lifetime of the excited states is introduced by a common phenomenological damping factor. The real and imaginary frequency-dependent polarizabilities can thus be calculated over a broad range of frequencies. This allows for the study of linear-response properties both in the resonance and nonresonance cases. The method is complementary to the standard approach of calculating the excitation energies from the poles of the polarizability. The real and imaginary polarizabilities can then be calculated in any specific energy range of interest, in contrast to the excitation energies which are usually solved only for the lowest electronic states. We have verified the method by investigating the photoabsorption properties of small alkali clusters. For these systems, we have calculated the real and imaginary polarizabilities in the energy range of 1-4 eV and compared these with excitation energy calculations. The results showed good agreement with both previous theoretical and experimental results.     

Ab initio calculation of structural,lattice dynamical,and thermal properties of cubic silicon carbide

We present first-principles calculations of the structural, lattice dynamical, and thermal properties as well as Raman results for cubic silicon carbide (3C SiC). The plane-wave pseudopotential approach to density functional theory (DFT ) in the local density approximation has been used to calculate the equilibrium properties of 3C SiC, i.e., the ground-state energy, the band structure, the valence electron density, the lattice constant, the bulk modulus, its pressure derivative, and the ionicity factor of the chemical bonds. The linear-response theory within DFT has been used to obtain the phonon frequencies, the eigenvectors, and the mean-square atomic displacements. Furthermore, we calculated the mode Grueisen parameters, the internal-strain parameter, the elastic constants, the Born effective charge, and the high-frequency dielectric constant. The specific heat at constant volume and at constant pressure, the thermal expansion coefficient, the temperature dependence of the lattice constant, and that of the isothermal and adiabatic bulk modulus have been derived within the quasi-harmonic approximation. Finally, the second-order Raman spectrum of 3C SiC has been calculated using phenomenological polarizability coefficients and ab initio frequencies and eigenvectors. © 1995 John Wiley & Sons, Inc.     

Using internal coordinates to describe photoinduced geometry changes in MLCT excited states

A resonance Raman intensity analysis of the metal-to-ligand charge-transfer (MLCT) transition for the rhenium compound Re(2-(2'-pyridyl)quinoxaline)(CO)(3)Cl (RePQX) is presented. Photoinduced geometry changes are calculated, and the results are presented using the vibrational normal modes and the redundant internal coordinates. A density functional theory calculation is used to determine the ground-state nonresonant Raman spectrum and a transformation matrix that transforms the redundant internal coordinates into the normal modes. The normal modes nu(37) (rhenium coordination sphere distortion) and nu(75) (ligand skeletal stretch) show the largest photoinduced geometry change (Delta = 1.0 and 0.7, respectively). A single carbonyl mode is enhanced in the resonance Raman spectra. Time-dependent density functional theory is used to calculate excited-state geometry changes, which are subsequently used to determine the signs of the photoinduced normal mode displacements. Transforming to internal coordinates reveals that all the CO bond lengths are displaced in the excited state. The Re-C and C-C ligand bond lengths are also displaced in the excited state. The results are discussed in terms of a simple one-electron picture for the electronic transition. Many bond angles and torsional coordinates are also displaced by the metal-to-ligand charge transfer, and most of these are associated with the rhenium coordination sphere. It is demonstrated that using internal coordinates presents a clear picture of the geometry changes associated with photoinduced electron transfer in metal polypyridyl systems.     

Basis set and density functional dependence of vibrational Raman optical activity calculations

We present an extensive investigation of the dependence of the scattering intensity difference of right and left circularly polarized light observed in vibrational Raman optical activity (VROA) on the choice of basis set and exchange-correlation functional. These dependencies are investigated for five molecules for which accurate experimental data are available: (S)-methyloxirane, (R)-epichlorhydrin, (S)-glycidol, (M)-spiro[2,2]pentane-1,4-diene, and (M)-sigma-[4]-helicene. Calculations are presented using the SVWN exchange-correlation functional (LDA), the BLYP exchange-correlation functional, and the B3LYP hybrid functional, using six different basis sets: the cc-pVDZ, cc-pVTZ, aug-cc-pVDZ, aug-cc-pVTZ, Sadlej's polarized basis set, and a minimal VROA basis set recently proposed by Zuber and Hug. It is demonstrated that results from pure gradient-corrected and hybrid functionals are comparable and that the aug-cc-pVDZ and aug-cc-pVTZ basis sets yield similar results. Furthermore, the combination of the small basis set by Zuber and Hug with an accurate force field represents the best compromise between computational accuracy and computational efficiency.     

FT-IR, FT-Raman, NMR and UV-vis spectra, vibrational assignments and DFT calculations of 4-butyl benzoic acid

The solid phase FTIR and FT-Raman spectra of 4-butyl benzoic acid (4-BBA) have been recorded in the regions 400-4000 and 50-4000cm(-1), respectively. The spectra were interpreted in terms of fundamentals modes, combination and overtone bands. The structure of the molecule was optimized and the structural characteristics were determined by density functional theory (DFT) using B3LYP method with 6-311++G(d,p) as basis set. The vibrational frequencies were calculated for monomer and dimer by DFT method and were compared with the experimental frequencies, which yield good agreement between observed and calculated frequencies. The infrared and Raman spectra were also predicted from the calculated intensities. (13)C and (1)H NMR spectra were recorded and (13)C and (1)H nuclear magnetic resonance chemical shifts of the molecule were calculated using the gauge independent atomic orbital (GIAO) method. UV-visible spectrum of the compound was recorded in the region 200-400nm and the electronic properties HOMO and LUMO energies were measured by time-dependent TD-DFT approach. The geometric parameters, energies, harmonic vibrational frequencies, IR intensities, Raman intensities, chemical shifts and absorption wavelengths were compared with the available experimental data of the molecule.     

Effective conjugation and Raman intensities in oligo(para-phenylene)s: a microscopic view from first-principles calculations

Electron-phonon coupling in oligo(para-phenylene)s is addressed in terms of the off-resonance Raman intensities of two distinct modes at 1220 and 1280 cm(-1). On the basis of Albrecht's theory, vibrational coupling and Raman intensities are calculated from first-principles quantum-chemical methods. A few-state model is used to evaluate the dependence of the mode intensities on oligomer length, planarity, and excitation wavelength. The link between electron delocalizationconjugation and Raman intensities is highlighted. Extending on prior studies, the present work focuses on providing an in-depth understanding of the origin of this correlation in addition to reproducing experimental findings. The model applied here allows us to interpret the results on a microscopic, quantum-mechanical basis and to relate the observed trends to the molecular orbital structure and nature of the excited states in this class of materials. We find quantitative agreement between the results of the calculations and those of measurements performed on oligo(para-phenylene)s of various chain lengths in the solid state and in solution.