Molecular vibrations independence

In the Bom-Oppenheimer picture the nuclei move on a potential energy surface (PES) which is a solution to the electronic Schrodinger equation. The PES is independent of the nuclear masses (i.e. it is the same for isotopic molecules), this is not the case when working in the adiabatic approximation since the diagonal correction (and mass polarization) depends on the nuclear masses. Solution of (3.16) for the nuclear wave function leads to energy levels for molecular vibrations (Section 13.1) and rotations, which in turn are the fundamentals for many forms of spectroscopy, such as IR, Raman, microwave etc. 

Although the idea of generating 2D correlation spectra was introduced several decades ago in the field of NMR [1008], extension to other areas of spectroscopy has been slow. This is essentially on account of the time-scale. Characteristic times associated with typical molecular vibrations probed by IR are of the order of picoseconds, which is many orders of magnitude shorter than the relaxation times in NMR. Consequently, the standard approach used successfully in 2D NMR, i.e. multiple-pulse excitations of a system, followed by detection and subsequent double Fourier transformation of a series of free-induction decay signals [1009], is not readily applicable to conventional IR experiments. A very different experimental approach is therefore required. The approach for generation of 2D IR spectra defined by two independent wavenumbers is based on the detection of various relaxation processes, which are much slower than vibrational relaxations but are closely associated with molecular-scale phenomena. These slower relaxation processes can be studied with a conventional... 

As the oscillators of the OPP model vibrate independently of each other, the frequencies are dispersionless, that is, independent of a wavevector q. For the internal modes of a molecular crystal, this tends to be a very good approximation. For the external modes, the dispersion can be pronounced, as shown in Figs. 2.1 and 2.2. In order to obtain the mean-square vibrational amplitudes for the latter, a summation over all phonon branches in the Brillouin zone must be performed. 

From a molecular viewpoint, we know that heat capacity is closely connected to internal modes of molecular vibration. According to the classical equipartition theorem (Sidebar 3.8), a nonlinear polyatomic molecule of Aat atoms has ftmodes = 3Aat — 6 independent internal modes of vibration, each of which would contribute equally to heat capacity... 

The complex quantity, y6br = e (y(3)r) + i Im (x r), represents the nuclear response of the molecules. The induced polarization is resonantly enhanced when the Raman shift wp — ws matches the frequency Qr of a Raman-active molecular vibration (Fig. 6.1A). Therefore, y(3)r provides the intrinsic vibrational contrast mechanism in CRS-based microscopies. The nonresonant term y6bnr represents the electronic response of both the one-photon and the two-photon electronic transitions [30]. Typically, near-infrared laser pulses are used to prevent the effect of two-photon electronic resonances. With input laser pulse frequencies away from electronic resonances, y(3)nr is independent of frequency and is a real quantity. It is important to realize that the nonresonant contribution to the total nonlinear polarization is simply a source for an unspecific background signal, which provides no chemical contrast in some of the CRS microscopies. While CARS detection can be significantly effected by the nonresonant contribution y6bnr [30], SRS detection is inherently insensitive to it [27, 29]. As will be discussed in detail in Sects. 6.3 and 6.4, this has major consequences for the image contrast mechanism of CARS and SRS microscopy, respectively. 

One of the features of transition state theory is that in principle it permits the calculation of absolute reaction rate constants and therefore the thermodynamic parameters of activation. There have been few successful applications of the theory to actual reactions, however, and agreement with experiment has not always been satisfactory. The source of difficulty is apparent when one realizes that there really is no way of observing any of the properties of the activated complex, for by definition its lifetime is of the order of a molecular vibration, or 10-14 sec. While estimates of the required properties can often be made with some confidence, there remains the uncertainty due to lack of independent information. 

The deformation of the nuclear skeleton of the molecules (atomic polarizability, aa, or mean molecular vibrational polarizability, av). This polarizability is independent of temperature. 

The coupling of the nuclear and electronic coordinates is probably not a major issue in the discussion of most outer-sphere electron-transfer reactions because (1) the nuclear motions, or molecular vibrations, at the donor and acceptor centers are independent of one another and (2) the D/A electronic coupling and the fraction of delocalized electron density are both small. One perspective on the classical, Marcus-Hush description of the electron-transfer reaction... 

NRVS data are commonly interpreted within a harmonic approximation, which describes molecular vibrations in terms of independent oscillations along a set... 

Later Bjerrum s theory was supported by the work of Kraus [138], who showed importance of the dielectric constant, and Atherton [139], who demonstrated the existence of ion pairs using electron spin resonance spectroscopy. The formation of ion pairs may be studied by various methods conductance studies, UV-visible spectrometry, IR spectrophotometry, partition, distribution, or solvent extraction. The lifetime of ion pairs was determined to be at least 10 sec, which is equivalent to about 10 molecular vibrations, demonstrating that ion pairs can be considered as independent species [140]. Today, the ion-pair formation as independent species is widely accepted. 

As early as 1939 Slater proposed an alternative approach that related the rate of a unimolecular reaction to the vibrations of the reacting molecule [55]. This theory was developed over the succeeding years, and is explained in Slater s famous 1959 book [56]. The theory is based on the familiar description of molecular vibrations in terms of normal modes [57]. If the vibrations of a molecule are assumed to be harmonic, they can be reduced to a set of independent harmonic... 

Unlike the classical spring model for molecular vibrations, there is not a continuum of energy levels. Instead, there are discrete energy levels described by quantum theory. The time-independent Schroedinger equation... 

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