Integrals polarization energy

The two-electron integrals in equations (16), (19) and (20) can be rewritten in terms of the densities p(AA r), p(SS r), and transition densitiesp(AA r) and p(SS r) that allow us to extract the physical meaning of these terms. The first one describes the electrostatic interaction between the unperturbed charge densities of A, p(AAlr) and the perturbed density of S, p(SS r), that is the contribution to the polarization energy of B in the field of A, and viceversa for A in the field of S. The second, (equation (20)), the dispersion term, contains contributions of the perturbed densities of the two systems, so that the numerators of equation (19) and equation (20) become ... 

The relaxation time for a given polarization mode (v) and wave vector (ic ) is calculated by i) integrating the energy autocorrelation function [71], or ii) fitting the energy correlation function with an exponential [72] or damped oscillatory function [73], as shown in Fig. 3. For each wave vector, the anharmonic frequency can also be extracted. 

Gavezzotti, A., Calculation of intermolecular interaction energies by direct numerical integration over electron densities. I. Electrostatic and polarization energies in molecular crystals,... 

In LSDA the spin polarization energy may also be expressed in terms of radial integrals (Gunnarsson 1976, 1977, Janak 1977, Brooks and Johansson 1983)... 

A. Gavezzotti,/. Phys. Chem. B, 106(16), 4145-4154 (2002). Calculation of Intermolecular Interaction Energies by Direct Numerical Integration over Electron Densities. I. Electrostatic and Polarization Energies in Molecular Crystals. 

Fig. 6. Free energies of hydration calculated, for a series of polar and non-polar solute molecules by extrapolating using (3) from a 1.6 ns trajectory of a softcore cavity in water plotted against values obtained using Thermodynamic Integration. The solid line indicates an ideal one-to-one correspondence. The broken line is a line of best fit through the calculated points.

Figure 2. The structural energy difference (a) and the magnetic moment (b) as a function of the occupation of the canonical d-band n corresponding to the Fe-Co alloy. The same lines as in Fig. 1 are used for the different structures. In (b) the concentration dependence of the Stoner exchange integral Id used for the spin-polarized canonical d-band model calculations is shown as a thin dashed line with the solid circles. The value of Id for pure Fe and Co, calculated from LSDA and scaled to canonical units, are also shown in (b) as solid squares.

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