Asymptotic behaviour

One of the most important deficiencies of the LDA exchange is that is does not have the correct asymptotic behaviour. Becke (1988a) recognized that it was... 

Becke proposed a widely used correction (B or B88) to tire LSDA exchange energy, which has the correct — asymptotic behaviour for the energy density (but not for the exchange potential). ... 

From the above discussion, it is apparent that the exponential asymptotic behaviour of KmU) characterizes the correlation between collisions rather than collision itself. Hence the quantity tm defined in Eq. (1.67) cannot be considered as a collision time. To determine the true duration of collision let us transform Eq. (1.63) to the integral-differential equation as was done in [51] ... 

Markovian perturbation theory as well as impact theory describe solely the exponential asymptotic behaviour of rotational relaxation. However, it makes no difference to this theory whether the interaction with a medium is a sequence of pair collisions or a weak collective perturbation. Being binary, the impact theory holds when collisions are well separated (tc < to) while the perturbation theory is broader. If it is valid, a new collision may start before the preceding one has been completed when To < Tc TJ = t0/(1 - y). 

Inequality (1.88) defines the domain where rotational relaxation is quasi-exponential either due to the impact nature of the perturbation or because of its weakness. Beyond the limits of this domain, relaxation is quasi-periodic, and t loses its meaning as the parameter for exponential asymptotic behaviour. The point is that, for k > 1/4, Eq. (1.78) and Eq. (1.80) reduce to the following ... 

Markovian theory of orientational relaxation implies that it is exponential from the very beginning but actually Eq. (2.26) holds for t zj only. If any non-Markovian equations, either (2.24) or (2.25), are used instead, then the exponential asymptotic behaviour is preceded by a short dynamic stage which accounts for the inertial effects (at t < zj) and collisions (at t < Tc). 

Fig. 2.4. The asymptotic behaviour of the IR spectrum beyond the edge of the absorption branch for CO2 dissolved in different gases (o) xenon (O) argon ( ) nitrogen ( ) neon (V) helium. The points are experimental data, the curves were calculated in [105] according to the quantum J-diffusion model and two vertical broken lines determine the region in which Eq. (2.58) is valid.

This is a rather general conclusion independent of the model of rotational relaxation. It is quite clear from Eq. (2.70) and Eq. (2.16) that the high-frequency asymptotic behaviour of both spectra is determined by the shape of g(co) ... 

In the present implementation, the unperturbed functions are not subject to any orthogonality constraint nor are required to diagonalize any model hamiltonian. This freedom yields a faster convergence of the variational expansion with the basis size and allows to obtain the phaseshift of the basis states without the analysis of their asymptotic behaviour. 

Figure 13. Detail of Figure 12, showing the asymptotic behaviour of the line A ni lf + jf l (dash-dotted line) with respect to the total cumulated uptake dashed line corresponds with the steady-state surface concentration A iii m. The evolution of the surface concentration F(t) = AficMfi o, t) is also shown... 

It was shown that the derived expression for k is equation 28. (Section III.D). If k-i = (/t2 +/t3 [ 151 ). at high [B] equation 28 may be transformed into equation 30, which is responsible for the plateau observed in some cases [e.g. the reactions of 2,4-dinitroanisole with cyclohexylamine in benzene (Figure 11) and in cyclohexane (not shown)]143,144 and it was also observed in the reactions with n-butylamine in benzene at 60 °C (the reactions at 80 °C show a slight curvature, tending to a farther asymptotic behaviour). In all the S Ar systems studied by other authors, in which fourth-order kinetics were found, the observation of a similar plateau in the plots of La/[B] vs [B] was not reported. 

In spite of the fact that we have introduced the factor of exp(—) in equation (47), our analytical expression for the dipole moment does not have a qualitatively correct asymptotic behaviour for the bond lengths r,—> 00. The function does not converge to the dipole moment of the NH2 fragment if we remove a hydrogen atom. However, neither does it diverge The calculated dipole moment values at large r, are around 2-3 D depending on which dissociation path we use. Obviously, the asymptotic behaviour of the dipole moment is of no importance for the simulations carried out in the present work we are only concerned with molecular states well below dissociation. 

An application of the variational principle to an unbounded from below Dirac-Coulomb eigenvalue problem, requires imposing upon the trial function certain conditions. Among these the most important are the symmetry properties, the asymptotic behaviour and the relations between the large and the small components of the wavefunction related to the so called kinetic balance [1,2,3]. In practical calculations an exact fulfilment of these conditions may be difficult or even impossible. Therefore a number of minimax principles [4-7] have been formulated in order to allow for some less restricted choice of the trial functions. There exist in the literature many either purely intuitive or derived from computational experience, rules which are commonly used as a guidance in generating basis sets for variational relativistic calculations. 

The Gaussian orbitals are very important in practical applications. In spite of their wrong asymptotic behaviour at both r —> 0 and r —> °o, nearly all molecular electronic structure calculation programs have been constructed using Gaussian sets of one-electron functions. In this example the Gaussian basis has been selected as... 

Results of similar accuracy as relativistic TFDW are found with a simple procedure based on near-nuclear correction which leave space for further improvements. For the reasons mentioned at the end of previous section the direct way to improve the present approach seems to be the refinement of the near nuclear corrections, a problem that we have just tackled with success in the non-relativistic framework [31,32]. The aim was to describe the near-nuclear region accurately by means of using the quantum mechanical exact asymptotic expression up to of the different ns eigenstates of Schodinger equation with a fit of the semiclassical potential at short distancies to the exact asymptotic behaviour (with four terms) of the potential near the nucleus. The result is that the density below Tq becomes very close to Hartree-Fock values and the improvement of the energy values is large (as an example, the energy of Cs+ is improved from the Ashby-Holzman result of-189.5 keV up to -205.6, very close to the HF value of -204.6 keV). This result makes us expect that a similar procedure in the relativistic framework may provide results comparable to Dirac-Fock ones. 

Although numerical values of and (p >, as well as p(0) can be easily obtained for near Hartree-Fock wavefunctions [16,17], some effort has been done in order to study the general features of these quantities as a function of the atomic number [18,19,20], The easiest way to estimate these comes from the TF model [21,22], and has the important property of providing the exact asymptotic behaviour at large Z,... 

We see how the difference among HF values and ours is going down when Z increases. This is because TF increases accuracy when Z grows. Therefore we have studied the asymptotic behaviour of p(0) with this method. It is well known that... 

In [18] Dimitrieva and Plindov found the asymptotic behaviour for the expectation values. Now we check this behaviour and the coefficients they give. For (r )... 

We can conclude that the present method of correcting TF calculations provides adequate estimations of expectation values for ground state atoms taking into account the simplicity of the model and it self-consistent nature, where no empirical parameters are used. It provides information about the asymptotic behaviour of quantities such as p(0) and (r 2) that cannot be evaluated with the standard semi classical approach and allow us to estimate momentum expectation values which are not directly related to the density in an exact way. 

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