Frequency inner-shell vibration

This formula follows from the transition-state theory (TST) of unimole-cular reactions [42], Since it is commonly anticipated that vIS P vos, eqn. (22) predicts that the effective frequency is dominated by the inner-shell frequency even when the outer-shell barrier provides a substantial contribution to AG t. For metal-ligand, and other typical inner-shell vibrations, vis ss 1013s-1. Indeed, for the common circumstance where AG [> AG S, we expect that vn vis. This is intuitively reasonable since, on the basis of TST, we generally expect that the fastest motion along the reaction coordinate will control the frequency factor [28],... 

Good models for such studies are also metallocenes (M = Mn, Fe, Co) and Cr(CgHg)2 °, which were studied by Weaver and Gennett [148] in seven solvents. The authors compared the experimental data with two sets of calculated results. In the calculations of the first set of data, v was identified with the inner-shell vibration frequency V and it was assumed that the reaction is adiabatic (/c = l). In the second set the authors assumed that the frequency of surmounting the free energy barrier is controlled entirely by the dynamics of solvent reorganization. It was found that the second set of calculated data was much closer to the experimental results. 

The structural parameters and vibrational frequencies of three selected examples, namely, H2O, O2F2, and B2H6, are summarized in Tables 5.6.1 to 5.6.3, respectively. Experimental results are also included for easy comparison. In each table, the structural parameters are optimized at ten theoretical levels, ranging from the fairly routine HF/6-31G(d) to the relatively sophisticated QCISD(T)/6-31G(d). In passing, it is noted that, in the last six correlation methods employed, CISD(FC), CCSD(FC),..., QCISD(T)(FC), FC denotes the frozen core approximation. In this approximation, only the correlation energy associated with the valence electrons is calculated. In other words, excitations out of the inner shell (core) orbitals of the molecule are not considered. The basis of this approximation is that the most significant chemical changes occur in the valence orbitals and the core orbitals remain essentially intact. On... 

The nuclear frequency is related to the solvent and inner-shell reorganization energies as well as the corresponsing vibration frequencies. The electronic factor can be described on the basis of the Landau-Zener framework and is related to the electronic coupling matrix element... 

Finally, it is worth noting that the nature of the solvent-dependent ET dynamics is also predicted to be affected by the presence of inner-shell (reactant vibrational) contributions to the activation barrier [19]. As might be expected, the presence of higher-frequency vibrational contributions to the activation barrier can yield a marked attenuation in the degree to which overdamped solvent dynamics control the adiabatic barrier-crossing frequency [19]. The experimental exploration of such effects is limited in part by the paucity of redox couples suitable for solvent-dependent studies that exhibit known vibrational barriers, and complicated by the qualitatively similar behavior expected for nonadiabatic pathways. Nevertheless, there is some evidence that vibrational activation can indeed attenuate the role of solvent dynamics, although the theoretical predictions appear to overestimate the magnitude of this effect [10b,20]. 

The performance of DFT in the optimization of molecular geometry is summarized in Section 4. In Section 5, we present DFT results for the vibrational spectra of molecules. It is well known that HF harmonic vibrational frequencies are about 10% higher than the experimental harmonic frequencies. Vibrational frequencies computed at correlated levels of methods improve the agreement, but they are usually computationally too costly to be feasible for medium-sized molecules. Since DFT is computationally more economical than the HF method, the accuracy of DFT predicted vibrational frequencies is of great interest to us. Section 6 summarizes the DFT results for electron spectroscopy including core-electron binding energies (CEBEs) which are measured by the so-called electron spectroscopy for chemical analysis (ESCA), and inner-shell electron excitation spectra (ISEES). 

For adiabatic reaction pathways (i.e. Kel = 1) the nuclear frequency factor, vn (s 1), represents the rate at which reacting species in the vicinity of the transition state is transformed into products. This frequency will be influenced by a combination of the various motions associated with the passage of the system over the barrier, approximately weighted according to their relative contributions to the activation energy. These motions usually involve bond vibrations and solvent motion, associated with the characteristic inner- and outer-shell frequencies, vis and vos, respectively. A simple formula for vn which has been employed recently is [la, 7]... 

We now turn to the inner-sphere redox reactions in polar solvents in which the coupling of the electron with both the inner and outher solvation shells is to be taken into account. For this purpose a two-frequency oscillator model may the simplest to use, provided the frequency shift resulting from the change of the ion charges is neglected. The "adiabatic electronic surfaces of the solvent before and after the electron transfer are then represented by two similar elliptic paraboloids described by equations (199.11), where x and y denote the coordinates of the solvent vibrations in the outer and inner spheres, respectively. The corresponding vibration frequencies and... 

These vibrations are transmitted through the middle ear by three tiny bones known as the ossicles, being the hammer (malleus), anvil (incus) and stirrup (stapes). The hammer bone is fixed to the ear-drum and the stirrup to another membrane (the oval window) which separates the middle and inner parts of the ear. The section of the inner ear which receives sound waves is shaped like a snail s shell (the cochlea) and contains strands of tissue under varying tensions. These strands vibrate in response to sound waves of particular frequency which have entered the inner ear from the bones of the middle ear and produce nerve impulses in the auditory nerve which are then transmitted to the cortex of the brain. It is at this point that the signals are received as sound of a certain pitch, intensity and quality. 

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