Parabolic Formula

RecknageP showed that a parabolic formula v=VQ(l+at+bfi)y is necessary to represent Regnaulfs results, whilst Bosscha used an exponential formula ... 

Recknagel2 showed that a parabolic formula v=v0(l+af+6 2), is necessary to represent Regnault s results, whilst Bosscha3 used an exponential formula v=v0eat. 

The trapezoidal rule, though more easily manipulated, is not quite so accurate as those rules based on the parabolic formula of Newton and Cotes. 

Often the parabolic formula is used to describe the penetration of carbonation ... 

If Eo-o is identified with the zero phonon transition from the experiment that includes zero point energy effects of the other (non active) modes the extra terms are included into E which also is determined from the spectra fitting the vertical transition energy to E - AE (vide infra). The shift of the excited state potential minimum with respect to that of the ground state, i.e. the equilibrium geometry distortion, is calculated from the parabolic formulas to be... 

If the ligand field parameter Dq of the relaxed (at equilibrium) excited state is available from the evaluation of excited state spectra a similar set of formulas can be derived. The ligand field parameters for the ground state and excited state equilibrium geometry Dq and Dq, respectively, (cf. Fig. 1) are related for the one-electron case d using the parabolic formula and Eq. (17) by... 

Parabolic Rule (Sintpson s Rule) This procedure consists of subdividing the intei val a [Pg.471]

Of special interest is the case of parabolic barrier (1.5) for which the cross-over between the classical and quantum regimes can be studied in detail. Note that the above derivation does not hold in this case because the integrand in (2.1) has no stationary points. Using the exact formula for the parabolic barrier transparency [Landau and Lifshitz 1981],... 

At high temperatures (/S -r 0) the centroid (3.53) collapses to a point so that the centroid partition function (3.52) becomes a classical one (3.49b), and the velocity (3.63) should approach the classical value Uci- In particular, it can be directly shown [Voth et al. 1989b] that the centroid approximation provides the correct Wigner formula (2.11) for a parabolic barrier at T > T, if one uses the classical velocity factor u i. A. direct calculation of Ax for a parabolic barrier at T > Tc gives... 

Although the correlation function formalism provides formally exact expressions for the rate constant, only the parabolic barrier has proven to be analytically tractable in this way. It is difficult to consistently follow up the relationship between the flux-flux correlation function expression and the semiclassical Im F formulae atoo. So far, the correlation function approach has mostly been used for fairly high temperatures in order to accurately study the quantum corrections to CLST, while the behavior of the functions Cf, Cf, and C, far below has not been studied. A number of papers have appeared (see, e.g., Tromp and Miller [1986], Makri [1991]) implementing the correlation function formalism for two-dimensional PES. 

The trend in the f-pressure is almost parabolic with band filling and this is typical for a transition metal (with d replaced by f). The physical basis was given by the Friedel who assumed that a rectangular density of states was being filled monotonically and thus was able to reproduce the parabolic trend in transition metal cohesive energies analytically. Pettifor has shown that the pressure formula can similarly be integrated analytically. 

A k-valent sphere, whose faces have gonality a orb, is called a ( a, b), k)-sphere (see Chapter 2). We call the parameters ( a, b), k) elliptic, parabolic, hyperbolic, according to the sign cfa(2, b, k). This sign has a consequence for the finiteness and growth of the number of graphs in the class of ( a, b], k)-spheres. Here, the link is provided by the Euler formula (1.1). 

Calculation of the pre-exponential factor in eqn. (7) is connected with the analysis of electron motion in parabolic coordinates. The first time such calculations were conducted was by Lanczos [12]. The formulae he obtained were cumbersome and we shall not give them here. The simple formula for the probability of ionization of a slightly excited atom is given in ref. 13 as... 

The formulas for the susceptibility of a harmonic oscillator, presented above, were first derived in Ref. 18 with neglect of correlation between the particles orientations and velocities. This derivation was based on an early version of the ACF method, in which the average perturbation theorem was not employed, so that the expression equivalent to Eq. (14c) was used. (The integrand of the latter involves the quantities perturbed by an a.c. field.) For a specific case of the parabolic potential, the above-mentioned theory is simple however, it becomes extremely cumbersome for more realistic forms of the potential well. 

A classical resonance-absorption theory [66, 67] was aimed to obtain the formulas applicable for calculation of the complex permittivity and absorption recorded in polar gases. In the latter theory a spurious similarity is used between, (i) an almost harmonic perturbed law of motion of a charge affected by a parabolic potential (ii) and the law of motion of a free rotor, this law being expressed in terms of the projection of a dipole moment onto the direction of an a.c. electric field. 

We have used the calculation scheme described in Section VII.B for the HC-CS model. Now we employ the composite HC-EB model. The only difference from the above-mentioned calculations concerns the formula for the spectral function Now we use Eq. (474) instead of Eq. (315). Setting vm to be close to Vr 200 cm-1, we estimate the steepness of the parabolic well as... 

The complicated function (41) may be used for calculation of the rate constant, if the tensors Q and Qf are known. So, the number of the reorganizing local vibrations should be small in order to calculate the change of the geometry and the force constants at electron transition. The potential energy of the vibration localized near acceptor may decline on the parabolic form, if the energy AE transmitted in the local vibration is sufficiently large. Then, the formulas (40) and (41) are useless, and it is necessary to simpliciter calculate the matrix elements (i/f(g ) iK<7f)) in the expression (9). 

---