Chemical hardness matrix elements

The AIM electron-population displacements, d/V, are strongly coupled through the olf-diagonal hardness matrix elements //y y>,i Thus, a given displacement d/Vk strongly affects the chemical potentials of all AIM. This representation considers all AIM populational parameters as independent variables, which can be interpreted as projections of the populational vector (/V, d, + N2 2 + dm) onto the orthogonal system of populational axes associated with the constituent atoms, i.e., the AIM populational basis vectors ... 

Table X gives an idea of the strength of the various expansion methods, and it shows that, by using the principal term only, one can hardly expect to reach even the above-mentioned chemical margin, even if the wave function W gO(D) is actually very close in the helium case. This means that one has to rely on expansions in complete sets, and the construction of the modern electronic computers has fortunately greatly facilitated the numerical solution of secular equations of high order and the calculation of the matrix elements involved. For atoms, the development will probably go very fast, but, for small molecules one has first to program the conventional Hartree-Fock scheme in a fully self-consistent way for the computers, before the next step can be taken. For large molecules and crystals, the entire situation is much more complicated, and it will hence probably take a rather long time before one can hope to get a detailed understanding of the correlation phenomena in these systems.

For second energy derivatives that appear in the calculation of polarizabilities, chemical hardness, van der Waals coefficients, vibrational frequencies and other second order properties, the perturbed density matrix is required. McWeeny s self-consistent perturbation (SCP) theory (Diercksen and McWeeny 1966 Dodds et al. 1977 McWeeny 1962, 2001 McWeeny and Dier-cksen 1968 McWeeny et al. 1977) represents a direct approach for the calculation of this matrix. For the clarity of the presentation we assume perturbation-independent basis and auxiliary functions and restrict ourselves to closed-shell systems. Under these conditions the elements of the perturbed density matrix are given by the SCP formalism of McWeeny et al. (1977) ... 

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