Perturbation theory frequency shifts

Many groups are now trying to fit frequency shift curves in order to understand the imaging mechanism, calculate the minimum tip-sample separation and obtain some chemical sensitivity (quantitative infonuation on the tip-sample interaction). The most conunon methods appear to be perturbation theory for considering the lever dynamics [103], and quantum mechanical simulations to characterize the tip-surface interactions [104]. Results indicate that the... 

It should be noted that there is a considerable difference between rotational structure narrowing caused by pressure and that caused by motional averaging of an adiabatically broadened spectrum [158, 159]. In the limiting case of fast motion, both of them are described by perturbation theory, thus, both widths in Eq. (3.16) and Eq (3.17) are expressed as a product of the frequency dispersion and the correlation time. However, the dispersion of the rotational structure (3.7) defined by intramolecular interaction is independent of the medium density, while the dispersion of the vibrational frequency shift (5 12) in (3.21) is linear in gas density. In principle, correlation times of the frequency modulation are also different. In the first case, it is the free rotation time te that is reduced as the medium density increases, and in the second case, it is the time of collision tc p/ v) that remains unchanged. As the density increases, the rotational contribution to the width decreases due to the reduction of t , while the vibrational contribution increases due to the dispersion growth. In nitrogen, they are of comparable magnitude after the initial (static) spectrum has become ten times narrower. At 77 K the rotational relaxation contribution is no less than 20% of the observed Q-branch width. If the rest of the contribution is entirely determined by... 

A transition linearly coupled to the phonon field gradient will experience, from the perturbation theory perspective, a frequency shift and a drag force owing to phonon emission/absorption. Here we resort to the simplest way to model these effects by assuming that our degree of freedom behaves like a localized boson with frequency (s>i. The corresponding Hamiltonian reads... 

To invoke the perturbation theory for a small anharmonic coupling coefficient, we use the Wick theorem for the coupling of the creation and annihilation operators of low-frequency modes in expression (A3.19). Retaining the terms of the orders y and y2, we are led to the following expressions for the shift AQ and the width 2T of the high-frequency vibration spectral line 184... 

The instantaneous OH frequency was calculated at each time step in an adiabatic approximation (fast quantal vibration in a slow classical bath ). We applied second-order perturbation theory, in which the instantaneous solvent-induced frequency shift from the gas-phase value is obtained from the solute-solvent forces and their derivatives acting on a rigid OH bond. This method is both numerically advantageous and allows exploration of sources of various solvent contributions to the frequency shift. 

In order to analyze these data, the frequency shift of geffectivecan be calculated by averaging over all orientations the anisotropic shift derived from a static spin Hamiltonian [67]. This treatment is based on the assumptions that molecular motion neither changes the spin precession rate nor perturbs the states and, thus, that the center of gravity of the spectrum is invariant even in presence of some motional averaging. For the allowed 11/2) <-> <-l/2 transition under perturbation theory, with expressions valid up to the third order, this shift is given by [47] ... 

A perturbation theory approach that simplifies the expression for second order shifts of resonance frequencies has been developed [59,60] for the analysis of the splitting pattern. This approach is valid only when the term K, defined... 

In strong magneticfields, the resonance frequencies are determined largely by the Zeeman interaction (A = Z in Tables 3.1.1 and 3.1.2). The other interactions can be treated as perturbations (cf. eqn (3.1.1)). The coupling to the rf field, the dipole-dipole interaction, the chemical shift, and the J coupling can be readily treated hy first-order perturbation theory. For the quadrupole interaction, this approximation holds true only for small quadrupole moments like those of Li and H. 

McRae87 88 has derived an expression for the solvent-induced frequency shift, from the second order perturbation theory, taking into account all the types of interactions suggested by Bayliss and McRae86. On the basis of a simple electrostatic model, the frequency shift, Av, is related to the refractive index and the static dielectric constant of the solvent by an equation consisting of four terms. The first term in the equation represents the contribution from dispersive interactions which give rise to a general red shift the second term represents the contribution from the solute dipole-induced solvent dipole interactions the third term accounts for the solute dipole-solvent dipole interactions and the fourth term represents the contribution from the... 

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