Correction schemes asymptotic corrections

In 1988, Becke proposed a gradient-corrected scheme for the exchange functional ( B88 ), ensuring that this should have the correct asymptotic limit (1/r) as oo [16] ... 

We will not carry this analysis further, since in writing the correction function on the form e(z,y) we have not explicitly included the dependence on derivatives of f. However, we note that this section provides reason for a classification of defect correction schemes by a new concept, called convergence order. A procedure which guarantees the asymptotic inequalities (within some disk)... 

As mentioned above, various correction schemes have been developed up to the present. However, there is room for further improvement in conventional correction schemes. Conventional hybrid functionals give poor excitation energies in TDDFT calculations as mentioned later. Asymptotic and SICs have little (or worse) effect on reproducibilities of molecular chemical properties. Recently, it has been proved that a long-range correction for exchange functionals obviously brings solutions to various... 

Table 20.7 summarizes mean absolute errors in calculated excitation energies of five typical molecules by TDDFT. The table also displays calculated results of asymptotically corrected AC [79] and LB [78] (AC-BOP and LBOP) and hybrid B3LYP [72] functionals, which are mentioned in the former section. The ab initio SAC-Cl [103] results are also shown to confirm the accuracies. The 6-311G- -- -(2d,2p) basis set was used in TDDFT calculations [104,105]. As the table indicates, the LC scheme clearly improves Rydberg excitation energies that are underestimated for pure BOP functional, at the same (or better) level as the AC scheme does. It should be noted that LC and AC schemes also provide improvements on valence excitation energies for all molecules. LC and AC results are comparable to SAC-CI results. The LB scheme clearly modifies... 

One formalism which has been extensively used with classical trajectory methods to study gas-phase reactions has been the London-Eyring-Polanyi-Sato (LEPS) method . This is a semiempirical technique for generating potential energy surfaces which incorporates two-body interactions into a valence bond scheme. The combination of interactions for diatomic molecules in this formalism results in a many-body potential which displays correct asymptotic behavior, and which contains barriers for reaction. For the case of a diatomic molecule reacting with a surface, the surface is treated as one body of a three-body reaction, and so the two-body terms are composed of two atom-surface interactions and a gas-phase atom-atom potential. The LEPS formalism then introduces adjustable potential energy barriers into molecule-surface reactions. 

In this nonvariational approach for the first term represents the potential of the exchange-correlation hole which has long range — 1/r asymptotics. We recognize the previously introduced splitup into the screening and screening response part of Eq. (69). As discussed in the section on the atomic shell structure the correct properties of the atomic sheU structure in v arise from a steplike behavior of the functional derivative of the pair-correlation function. However the WDA pair-correlation function does not exhibit this step structure in atoms and decays too smoothly [94]. A related deficiency is that the intershell contributions to E c are overestimated. Both deficiencies arise from the fact that it is very difficult to represent the atomic shell structure in terms of the smooth function p. Substantial improvement can be obtained however from a WDA scheme dependent on atomic shell densities [92,93]. In this way the overestimated intershell contributions are much reduced. Although this orbital-depen-... 

DPT schemes, which allow to calculate the electron affinities of atoms are based on the exact [59,60] and generalized (local) [61,62] exchange self-interaction-corrected (SIC) density functionals, treating the correlation separately in some approximation. Having better asymptotic behavior than GGA s, like in the improved SIC-LSD methods, one should obtain more... 

As it should be, at (3 = 0 this formula reduces to Eq. (4.174), which was obtained for a one-dimensional case. We remark, however, that in a tilted situation ((3 / 0) the coefficient D2 acquires a contribution independent on a that assumes the leading role. This effect is clearly due to admixing of transverse modes to the set of eigenfunctions of the system, and it is just it that causes so a significant discrepancy between the zero-derivative approximation and the correct asymptotic expansion for x(3 ) curves in Figure 4.12. Evaluation of the coefficient L>4 is done according to the same scheme and requires taking into account a number of the perturbation terms that makes it rather cumbersome. 

The main advantage of this approximation is that it is exact for two-electron systems (if the correct kf(r) = 0 is utilised in (4.15) before performing the functional differentiation (3.17) required for its application) and also correctly accounts for the self-interaction energies of individual closed shells if a shellpartitioning scheme is used [71]. Furthermore, the RWDA reproduces the asymptotic r proportionality of the exact x-only potential (although with the incorrect prefactor of 1/2 [103]). 

On the other hand, the highest occupied eigenvalues which reflect the asymptotic form of the density, and thus even in the x-only limit should be close to the ionisation potential, are too small by roughly 50% for the DFS and RLDA and by 30- % for the RWDA . Thus while the latter improves on the asymptotic form of in principle, the effect of this formal improvement on the physically relevant part of the asymptotic regime is rather limited. Note further, that for Yb all three schemes incorrectly predict the 4F7/2-orbital to be most weakly bound instead of the 6S1/2-orbital. The same deficiency has been observed for Cr and Cu in the nonrelativistic case [59]. These difficulties to reproduce the size and the ordering of the outermost eigenvalues are well known from the nonrelativistic case and are not related to the relativistic corrections in Ex [n]-... 

Numerical calculations using Kapuy s partitioning scheme have shown that for covalent systems the role of one-particle localization corrections in many-body perturbation theory is extremely important. For good quality results several orders of one-particle perturbations have to be taken into account, although the additional computational power requirement is much less in these cases than for the two-electron perturbative corrections. Another alternative for increasing the precision of the calculations is to estimate of the asymptotic behavior of the double power series expansion (24) from the first few terms by applying Canterbury approximants [31], which is a two-variable generalization of the well-known Pade approximation method. It has also been found [6, 7] that in more metallic-like systems the relative importance of the localization corrections decreases, at least in PPP approximation. 

Of course, we can never change a physical (or mathematical) problem by simply nondi-mensionalizing variables, no matter what the choice of scale factors. It is only when we attempt to simplify a problem by neglecting some terms compared with others on the basis of nondimensionalization that the correct choice of characteristic scales becomes essential. Fortunately, as we shall see, an incorrect choice of characteristic scales resulting in incorrect approximations of the equations or boundary conditions will always become apparent by the appearance of some inconsistency in the asymptotic-solution scheme. The main cost of incorrect scaling is therefore lost labor (depending on how far we must go to expose the inconsistency for a particular problem), rather than errors in the solution. 

The left-hand side is usually in a differential form. The right-hand side is times a function. The constant of integration in [Af, 4 - Nb ] cancels with 2H ° INca If this cancellation does not take place, the asymptotic scheme will not apply, since H, the correction, becomes comparable to. Table 7.2 provides some expressions for h ° and H... 

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