Kernel density function definition

The purpose of this work is to start from the basic equations of density functional theory to describe the changes in the energy associated with the transition from one ground-state to another, in terms of different sets of variables. In this process one will find the natural definitions of the hardness and softness kernels, the local hardness, the local softness, the global hardness and the global softness [23]. Then, we will proceed to establish their relation with ionization potentials and electron affinities, in order to confirm their behavior as a measure of chemical hardness or softness [14, 24]. Finally, this theoretical framework will be used to analyze the maximum hardness and the HSAB principles. 

To arrive at Eq. (180) we have used the definitions (145), (148), (171) and (175) of the density response functions. Furthermore, we have abbreviated the kernel of the (instantaneous) Coulomb interaction by w(x, x ) = 3(t — t )/ r — r. Finally, by inserting Eq. (180) into (168) one arrives at the time-dependent Kohn-Sham equations for the second-order density response ... 

Here sJc is the xc energy per particle of the homogeneous electron gas. By its very definition, the ALDA can be expected to be a good approximation only for nearly homogeneous densities, i.e., for functions n(r, t) that are slowly varying both spatially and temporally. It will turn out, however, that the ALDA gives rather accurate results even for rapidly varying densities (see Sects. 7 and 8). For the time-dependent xc kernel (153), Eq. (186) leads to... 

This quantity can be viewed as a generalization of Fukui s frontier MO concept and plays a key role in linking Frontier MO Theory and the HSAB principle. It can be interpreted either as the sensitivity of a system s chemical potential to an external perturbation at a particular point r, or as the change of the electron density p(r) at each point r when the total number of electrons is changed. The former definition has recently been implemented to evaluate this function,but the derivative of the density with respect to the number of electrons remains by far the most widely-used definition. The second order derivative of the energy with respect to the external potential is the linear response function xCi tO called the polarizability kernel... 

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