Isotropic Raman spectra

The quasi-classical theory of spectral shape is justified for sufficiently high pressures, when the rotational structure is not resolved. For isotropic Raman spectra the corresponding criterion is given by inequality (3.2). At lower pressures the well-resolved rotational components are related to the quantum number j of quantized angular momentum. At very low pressure each of the components may be considered separately and its broadening is qualitatively the same as of any other isolated line in molecular or atomic spectroscopy. 

The latter will be used below to describe broadening of resolved isotropic Raman spectra as well as their collapse and subsequent pressurenarrowing. 

The selection rules for isotropic Raman spectra ji = jf = j greatly simplify the formalism. The frequency matrix has only diagonal elements... 

Strekalov M. L., Burshtein A. I. Quantum theory of isotropic Raman spectra changes with gas density, Chem. Phys. 60, 133-48 (1981). 

Logan D. On the isotropic Raman spectra of isotopic binary mixtures. Mol Phys 1986 58 97-129. 

Fig. 20. Isotropic Raman spectra for the D-O stretching frequency, v, in 9.7 mol% HDO in H2O at various temperatures and densities [65].

Fig. 9. Simulated and observed isotropic Raman spectra of poly(ethylene oxide). Different conformational distributions for melt and aqueous solution are described in text.

A4. Isotropic Raman Spectra of pH2 and HD-Argon Mixtures in Liquid and Hypercritical Fluid. 

F Marsault-Herail, M Echargui. Isotopic dilution effects on the isotropic Raman spectra of methane in supercritical fluid. J Molecular Liquids 48 211, 1991. 

F Marsault-Herail, F Salmoun, J Dubessy, Y Garrabos. Isotropic Raman spectra of H2S in supercritical fluid. J Mol Liquids 62 251, 1994. 

The quantum theory of spectral collapse presented in Chapter 4 aims at even lower gas densities where the Stark or Zeeman multiplets of atomic spectra as well as the rotational structure of all the branches of absorption or Raman spectra are well resolved. The evolution of basic ideas of line broadening and interference (spectral exchange) is reviewed. Adiabatic and non-adiabatic spectral broadening are described in the frame of binary non-Markovian theory and compared with the impact approximation. The conditions for spectral collapse and subsequent narrowing of the spectra are analysed for the simplest examples, which model typical situations in atomic and molecular spectroscopy. Special attention is paid to collapse of the isotropic Raman spectrum. Quantum theory, based on first principles, attempts to predict the. /-dependence of the widths of the rotational component as well as the envelope of the unresolved and then collapsed spectrum (Fig. 0.4). 

In the conclusion of the present chapter we show how comparison of NMR and Raman scattering data allows one to test formulae (3.23) and (3.24) and extract information about the relative effectiveness of dephasing and rotational relaxation. In particular, spectral broadening in nitrogen caused by dephasing is so small that it may be ignored in a relatively rarefied gas when spectrum collapse proceeds. This is just what we are going to do in the next sections devoted to the impact theory of the isotropic Raman spectrum transformation. 

Fig. 3.6. Density transformation of nitrogen isotropic Raman spectrum normalized to a maximum [89] (gas density is given in amagat).

Fig. 3.10. Anomalous broadening of oxygen isotropic Raman spectrum in the vicinity of (but above) the critical point [137]. The notation is the same as in Fig. 3.5.

In quantum theory as in classical theory the isotropic Raman spectrum is expressed in terms of the average value of the polarizibility tensor a(0) = (1/3) Sp a randomly changing in time due to collisions ... 

Interestingly, this minor quantitative difference from 1 changes qualitatively the results obtained from (6.16). Everywhere within the limits established by (6.13), the Q-branch of the anisotropic spectrum broadens, in contrast to the narrowing of an isotropic Raman spectrum, described in preceding chapters. 

To obtain Raman spectra one needs the trajectories of the pq tensor elements of the chromophore s transition polarizability. Actually, for the isotropic Raman spectrum one needs only the average transition polarizability. This depends weakly on bath coordinates and this, together with the weak frequency dependence of the position matrix element, was included in our previous calculations [13, 98, 121]. For the VV and VH spectra, others have implemented... 

Rg. 3. Composition dependence of the isotropic Raman spectrum in the 0-H stretching region for aqueous sulfuric acid solutions. 

These collisions are instantaneous, (iii) The collision process is described by an S matrix. Its diagonal elements depend on rotational, but not vibrational, quantum numbers and its non-diagonal matrix elements vanish, (iv) Only a limited number of rotational sublevels are explicitely considered. In turn, the theory by Burshtein et al. conserves the assumptions (i,ii) but employs a different description of the collision process. The S matrix method is replaced by kinetic theoretical methods. The theory contains a collision strength parameter caracterizing the nature of the collision process. Starting from these premises these theories explain successfully the collapse of the Q-branch of the isotropic Raman spectrum through the motional narrowing (Fig.1). 

The last group of papers are relative to the computer-simulation study of the rotation-vibration correlations in pure liquids. The main authors are Levesque, Weis and Oxtoby (31,32) who examined liquid and HCl in much detail. A molecular dynamics simulation was carried out using 500 molecules for N2 and 256 for HCl periodic boundary conditions were imposed It was found that, in the isotropic Raman spectrum of liquid N2>intermolecular rotation-vibration interactions are mainly due to short range repulsion and dispersion forces the role of centrifugal forces seems secondary which is an unexpected result. The presence of important interference effects makes any partition illusory. Only intermolecular rotation-vibration correlation effects were examined in the case of infrared and anisotropic Raman spectra of liquid HCl. It results from this calculation that the simple product correlation function is indistinguishable from the total correlation function within the uncertainty of the simulation. This conclusion is similar to that reached by Tarjus and Bratos (12). It should not be forgotten, however, that the latter theory applies to diluted solutions whereas the function determined by Levesque, Weis and Oxtoby is an approximate correlation function of a pure liquid. 

Fig. 0.3. Raman spectrum of liquid oxygen [6]. The positions of the free rotator s. /-components are shown by vertical lines and the isotropic scattering contour is presented by the dashed line.

Chapter 3 is devoted to pressure transformation of the unresolved isotropic Raman scattering spectrum which consists of a single Q-branch much narrower than other branches (shaded in Fig. 0.2(a)). Therefore rotational collapse of the Q-branch is accomplished much earlier than that of the IR spectrum as a whole (e.g. in the gas phase). Attention is concentrated on the isotropic Q-branch of N2, which is significantly narrowed before the broadening produced by weak vibrational dephasing becomes dominant. It is remarkable that isotropic Q-branch collapse is indifferent to orientational relaxation. It is affected solely by rotational energy relaxation. This is an exceptional case of pure frequency modulation similar to the Dicke effect in atomic spectroscopy [13]. The only difference is that the frequency in the Q-branch is quadratic in J whereas in the Doppler contour it is linear in translational velocity v. Consequently the rotational frequency modulation is not Gaussian but is still Markovian and therefore subject to the impact theory. The Keilson-... 

Burshtein A. I., Storozhev A. V. The quantum theory of collapse of the isotropic Raman scattering spectrum, Chem. Phys. 135, 381-9 (1989). 

We first consider the intermolecular modes of liquid CS2. One of the details that two-dimensional Raman spectroscopy has the potential to reveal is the coupling between intermolecular motions on different time scales. We start with the one-dimensional Raman spectrum. The best linear spectra are based on time domain third-order Raman data, and these spectra demonstrate the existence of three dynamic time scales in the intermolecular response. In Fig. 3 we have modeled the one-dimensional time domain spectrum of CS2 for 3 cases (A) a single mode represented by the sum of three Brownian oscillators, (B) three Brownian oscillators, and (C) a distribution of 20 arbitrary Brownian oscillators. Case (A) represents the fully coupled, or isotropic case where the liquid is completely homogeneous on the time scales of the simulation. Case (B) deconvolutes the linear response into the three time scales that are directly evident in the measured response and is in the limit that the motions associated with each of the three timescales are uncoupled. Case (C) is an example where the liquid is represented by a large distribution of uncoupled motions. 

Figure 4.3-25 Raman spectrum of natural CO2 in the Fermi resonance region at a pressure of 40 kPa. Slitwidth 0.8 cm laser power 9 W at 514.5 nm, multiple pass cell. Upper spectrum experimental, middle spectrum calculated, lower spectrum isotropic contribution to calculated spectrum (Finsterholzl, 1982),...

A normal mode / is inactive or active in the Raman spectrum depending on whether (dal/dQi) and (dfijdQi) are zero. The symmetry of the normal mode may in most cases be used to determine which modes are active and which are isotropic (only (dal/dQi) different from zero). 

Pn [Eq- (1.125)] will have a value between 0 and y and Pp [Eq. (1.126)] will have a value between 0 and For the special case of an isotropic molecule a is zero so p and Pp are both zero for totally symmetric vibrations. If the vibrationally-distorted molecule is less symmetrical than the molecule in the equilibrium configuration, then al = 0 and is j and pp is in the Raman spectrum. Therefore, a measurement of the depolarization ratio provides a means of distinguishing totally symmetrical vibrations from the rest. See Fig. 1.35 for a polarized Raman spectrum of chloroform, and Fig. 1.36 for a polarized Raman spectrum of carbon tetrachloride, which is an isotropic molecule. 

Fig. 1.36. The polarized Raman spectrum of CCI4. The upper and lower curves correspond to 7 and /j., respectively. For the 459 cm" band the depolarization value is close to zero, expected for totally symmetrical vibrations in isotropic molecules. For the other bands at 218, 314, 762, 791 cm", the depolarization ratio is close to 0.75, expected for non-totally symmetric vibrations using polarized incident radiation from a laser source. Courtesy of J. R. Downey, Jr. and G. J. Janz, Rensselaer Polytechnic Institute, Troy, N.Y.

Spherical top molecules which are isotropic cannot have a pure rotational Raman spectrum. Asymmetric top molecules have no simple rotational energy equation. 

ROA and Raman intensity are proportional to the square of a tensor quantity, as expressed in Equation [1]. For Raman scattering only the square of the polarizability is needed, whereas ROA intensity arises from the product of the polarizability and an ROA tensor. The ROA tensor are approximately three orders of magnitude smaller than the polarizability, and hence an ROA spectrum is approximately three orders of magnitude smaller than its parent Raman spectrum. As noted above, the Greek subscripts of the tensor refer to the molecular axis system. However, for both Raman and ROA, linear combinations of products of tensors can be found that do not vary with the choice of the molecular coordinate frame. Such combinations are called invariants. All Raman intensities from samples of randomly oriented molecules can be expressed in terms of only three invariants, called the isotropic invariant, the symmetric... 

The contributions of the various geometries of three-dimensionally isotropic systems to the Raman spectrum are shown in Table 2.2. 

In the pioneering work the same information was extracted from the extremum position assuming it is independent of y [143]. This is actually the case when isotropic scattering is studied by the CARS spectroscopy method [134]. The characteristic feature of the method is that it measures o(ico) 2 not the real part of Ko(icu), as conventional Raman scattering does. This is insignificant for symmetric Lorentzian contours, but not for the asymmetric spectra observed in rarefied gas. These CARS spectra are different from Raman ones both in shape and width until the spectrum collapses and its asymmetry disappears. In particular, it turns out that... 

One possibility for this was demonstrated in Chapter 3. If impact theory is still valid in a moderately dense fluid where non-model stochastic perturbation theory has been already found applicable, then evidently the continuation of the theory to liquid densities is justified. This simplest opportunity of unified description of nitrogen isotropic Q-branch from rarefied gas to liquid is validated due to the small enough frequency scale of rotation-vibration interaction. The frequency scales corresponding to IR and anisotropic Raman spectra are much larger. So the common applicability region for perturbation and impact theories hardly exists. The analysis of numerous experimental data proves that in simple (non-associated) systems there are three different scenarios of linear rotator spectral transformation. The IR spectrum in rarefied gas is a P-R doublet with either resolved or unresolved rotational structure. In the process of condensation the following may happen. 

For oriented systems, the determination of molecular conformation is a complex problem because Raman spectra contain signals inherently due to both molecular conformation and orientation. To extract only the information relative to the conformation, one has to calculate a spectrum that is independent of orientation, in a similar way to the A0 structural absorbance of IR spectroscopy (Section 4). Frisk et al. [57] have shown that for a uniaxial sample aligned along the Z-axis, a spectrum independent of orientation (so-called isotropic spectrum), fso, can be calculated from the following linear combination of four polarized spectra [57]... 

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