Asymptotically Correct Approximation

The asymptotically correct approximation (ACA) was first introduced by Hobson [44] for the description of adsorption equilibrium on heterogeneous surfaces it has however become of wide use in the analysis of adsorption isotherms only after Cerofolini s investigation of the involved errors (which are of the same order as in the CA) and demonstration of its usefulness in determining the maximum adsorption energy [28]. The ACA can be extended to desorption kinetics by replacing the supposedly true desorption isotherm kinetics A (t,E) with their asymptotic limits. Since... 

Another LIA approach is the asymptotically correct approximation (ACA), as a first improvement over the CA (Cerofolini 1974), defined as... 

Cerofolini " has also examined the Hobson method (which he calls the asymptotically correct approximation , with Q = 0 and obtains a transform... 

The methods described in the two previous Sections, i.e. the condensation approximation and the asymptotically correct approximation, relate, through... 

While for patchwise heterogeneous surfaces lateral interactions can again be described simply by equation (4) with a kernel accounting for lateral interactions (for instance, Ross and Olivier considered in detail the case of the Hill-de Boer local isotherm), random heterogeneity requires a new analysis. Much work remains to be done in this field however, Rudzinski and his school have already extended both the condensation approximation and the asymptotically correct approximation in order to take random heterogeneity into account. The results obtained by the Polish scientists are encouraging as... 

A correct knowledge of the error structure is needed in order to have a correct summary of the statistical properties of the estimates. This is a difficult task. Measurement errors are usually independent, and often a known distribution, for example, Gaussian, is assumed. Many properties of least squares hold approximately for a wide class of distributions if weights are chosen optimally, that is, equal to the inverse of the variances of the measurement errors, or at least inversely proportional to them if variances are known up to a proportionality constant, that is, is equal or proportional to Zy, the N x N covariance matrix of the measurement error v. Under these circumstances, an asymptotically correct approximation of the covariance matrix of the estimation error 0 = 0 — 0 can be used to evaluate the precision of parameter estimates ... 

The various efforts to improve the effectiveness of the condensation-approximation method were made [5,6,9]. A more exact solution of the integral equation gives the asymptotically correct approximation method, developed by Hobson [118] for mobile adsorption and later refined by Cerofolini [66] for localized adsorption. In this treatment, the local isotherm is assumed to be a combination of a linear and a condensation isotherm. Hsu et al. [119] and Rudzinski et al. [109,110,120] adapted the Sommerfeld expansion method [121] to the solution of the integral equation in question. Although numerous modifications of the condensation-approximation method are known, aU improvements to this method make it more complicated and introduce additional numerical problems, but they do not change its... 

Another approximation which has had an important role in the development of the theory of adsorption on heterogeneous surfaces is the asymptotically correct approximation ... 

Hamel S, Casida ME, Salahub DR (2002b) Exchange-only optimized effective potential for molecules from resolution-of-the-identity techniques Comparison with the local density approximation, with and without asymptotic correction, J Chem Phys, 116 8276-8291... 

Further, there are asymptotically corrected XC kernels available, and other variants (for instance kernels based on current-density functionals, or for range-separated hybrid functionals) with varying degrees of improvements over adiabatic LDA, GGA, or commonly used hybrid DFT XC kernels [45]. The approximations in the XC response kernel, in the XC potential used to determine the unperturbed MOs, and the size of the one-particle basis set, are the main factors that determine the quality of the solutions obtained from (13), and thus the accuracy of the calculated molecular response properties. Beyond these factors, the quality of the... 

Earlier we also discussed the uncoupled HF approach to dispersion, where the sum over states is performed at the orbital level. Of course, this approach can also be applied with KS orbitals. However, HeBehnann and Jansen [193] found that the uncoupled sum-over-states approximation yields unacceptable errors. These are mainly due to neglect of the Coulomb and exchange-correlation kernels and are not substantially improved through an asymptotic correction of the exchange-correlation potential. 

The quality of the approximation is illustrated in Figure 9.6 for fhe case = 0.5. In the crossover region the interference oscillations are nicely reproduced, showing that their phases are correct. The next-to-leading order asymptotic correction makes a clear improvement in the description (upper curves). When = 0.1, there is almost no exponential region, while for large... 

In order to examine the mechanics of dislocation formation in a compliant film-substrate system, the energy of a dislocation at an arbitrary position in this structure in the absence of any other stress field is required. This energy is equivalent to the configurational force on the dislocation as a function of position through the thickness of the composite layer. In order to determine an approximate form for the critical thickness condition in Section 6.7.1, an ad hoc assumption on the variation of this force was made in (6.66). While the assumed variation of the force is asymptotically correct near either free surface and it has the obvious virtue of simplicity, the quality of the approximation is not evident. Thus, in this section, the variation of this force with position is examined in greater detail. 

Many methods have since been developed to asymptotically correct potentials to correct this unfortunate behavior. Any corrections to the ground-state potential are dissatisfying, however, as the resulting potential is not a functional derivative of an energy functional. Even mixing one approximation for fxcCr) and another for /xc has become popular in an attempt to rectify the problem. A more satisfying route to asymptotically correct... 

To demonstrate the power of QD analysis, we test two common approximations to the ground-state potential, both of which produce asymptotically correct potentials (exact exchange " (see the discussion on approximate functionals above and LB94 °). Exact exchange calculations are more CPU demanding than are traditional DFT calculations, but they are becoming pop-... 

Quantum Chemistry in Multiwavelet Bases Time-Dependent Density Functional Theory with Asymptotically Corrected Potentials in Local Density and Generalized Gradient Approximations. 

The solution given by Eq. (3.6.13) still requires numerical evaluation. An analytical solution has been obtained [34] that is exact for a infinite slab and is approximately correct for a particle of a general geometry. From this analytical solution, useful asymptotic and approximate solutions have been obtained for the maximum temperature rise. 

Figure 4.13 shows the comparison between the exact numerical solution and Eq. (4.3.73) for systems with = 1 and = 3, and with F = F = 3. It is seen that the approximate solution gives a satisfactory representation of the exact solution. The comparison has been made for the intermediate values of G, for which the difference is largest. For smaller and larger values of g, the agreement is better than shown in the figure. Furthermore, the approximate solution is exact slX X = I and is asymptotically correct as a -> 0 or a 00. 

It has been the author s contention that, at least for small molecules, the breakdown of the adiabatic approximation is rather minor for one-electron excitations compared to errors already present when common popular functionals are used in the adiabatic approximation. This contention has been largely confirmed by the general quality of excitation energies obtained by asymptotically-corrected functionals. Nevertheless it is known that, in con-... 

---