Hardness kernel

The integral of the regional softness is a property built from the hardness kernel (more precisely its inverse) defined entirely in the region Equation 24.97 can be reformulated as... 

The linear response i plays a fundamental role. It can be evaluated using the Bethe-Salpter equation (Equation 24.82) where the screened response x is evaluated from Kohn-Sham equations (Equation 24.80). It is remarkable that any nonlinear response can be computed using the linear one and the hardness kernels [26,32]. For instance, y3(r, r1 r2, r3) (see diagram 52a in Ref. [26]) is... 

The hardness kernels in Equation 24.110 depend on the kinetic energy functional as well as on the electron-electron interactions. Thomas-Fermi models can be used to evaluate the kinetic part of these hardness kernels and can be combined with a band structure calculation of the linear response X -... 

Returning to the matter of the bounds of the softness spectrum [3], we have found it easier to address it through study of the hardness kernels [15], reciprocals to the softness kernels,... 

A similar functional chain-rule transformation gives the expression for the global hardness in terms of the hardness kernel ... 

The matrix of softness kernels is the inverse of the corresponding hardness kernel matrix ... 

The hardness matrix of equation (78a) can be expressed in terms of the subsystem hardness kernels using the following chain-rule transformation ... 

Hardness kernel is the inverse of softness kernel in the sense ... 

Local hardness cannot be obtained from hardness kernel by simple integration. But the relation exists as ... 

Local hardness integrates to give global hardness [76] in a way similar to that of hardness kernel ... 

M. Torrent-Sucarrat, M. Duran, M. Sola, Global Hardness Evaluation Using Simplified Models for the Hardness Kernel, J. Phys. Chem. A. 2002, 106,4632 638. 

The hardness hierarchy starts from the hardness kernel [9], ti (f, f ) defined as... 

The hardness kernel integrates to local hardness, ri(f), though not in the sense the softness kernel integrates to local softness [Eq. (20)]. In this case we have (but see below)... 

It is interesting to analyze the long range behavior of the generalized local hardness defined by Eq. (20). Because of the exponential decay of the electronic density, one can show that far away from the nuclei, the dominant term comes from the coulombic contribution to the hardness kernel [28], that is... 

The elements of the hardness matrix are thus identified as the hardness kernels. 

Nalcwajski recently proposed the Recursive Combination Rules [32] for the calculation of molecular hardness. Although derived along a different path, these are essentially equivalent with the EEM formalism. To demonstrate this, we proceed as follows. From the definition of the hardness kernel, we may write (Eqs. (7) and (18)) ... 

Numerical differences may originate from alternative approximations to the hardness kernel. The simple expression for the off-diagonal elements of the hardness matrix in EEM (Eq. (18)) stems from the fact that we use a spherical atom approximation and an atomic partitioning of the electron cloud. The two-center electron repulsion integral then reduces to the Coulombic form (4nfio)" e, ep/R p, which equals k/R,p in eV. 

A rehnement can be found in the Ohno formula [33] for calculating the hardness kernels ... 

An advantage of the Combination Rules is, that one is not restricted to only one definition for the calculation of the hardness kernels. Other, more refined approximations can be used (e.g. the Mataga-Nishimoto [34] formula or the formulas by Pariser and Ohno) which may enhance the accuracy. This is not the case in EEM, at least when we want to retain its internal consistency. For larger systems, EEM has the advantage that it enables the direct calculation of global and local properties while the Combination Rules have to be applied recursively. Also EEM is directly applicable to the solid state. 

A comparison between the three foregoing methods, together with the results of EEM calculations, is shown in Fig. 1, in which the global hardnesses for some homonuclear diatomic molecules are plotted against the experimentally observed values (1 — A) [36]. It must be stressed that in all cases the same parameterization has been used, namely the isolated-atom hardnesses r x [11]. The differences are only due to differences in the expressions. For the Combination Rules, the hardness kernels were obtained from Ohno s formula. 

The explicit forms for the hardness kernels can be obtained using known forms for the universal functional F [n]. Within the local density approximation... 

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