Dispersion second order

Second order dispersion reaction, using P5.08.06 or P5.08.10, or integrating the dispersion equation. 

The second-order dispersion energy is defined as the difference between the second-order polarization and induction energies, E = E j — E J. One can also use the following direct definition... 

Later Young186 proved that the multipole expansion of the second-order dispersion energy for the H2 molecule is divergent as well,... 

The induction-dispersion contribution, in turn, can be interpreted as the energy of the (second-order) dispersion interaction of the monomer X with the monomer Y deformed by the electrostatic field of the monomer Z (note that we have six such contributions). In particular, when X=A, Y=B, and Z=C the corresponding induction-dispersion contribution in terms of response functions is given by,... 

From the work of Casimir, Lifshitz, London and many others [229] we know that the perturbation expression for the dispersion interaction between separated systems can be related to the electric polarizabilities of the interacting species, and also to the correlation of fluctuating electric multipoles on the two systems. In the Present TDDFT context, a useful polarizability form for the second-order dispersion interaction was given by Zaremba and Kohn [231] who derived it directly from second-order perturbation theory ... 

Gutowski, M., Verbeek, J., van Lenthe, J. H., and Chalasinski, G., The impact of higher polarization functions on second-order dispersion energy. Partial wave expansion and damping phenomenon for He2, Chem. Phys. 111, 271-283 (1987). 

The simplest TVD schemes are constructed combining the first-order (and diffusive) upwind scheme and the second order dispersive central difference scheme. These TVD schemes are globally second order accurate, but reduce to first order accuracy at local extrema of the solution. 

Second-order dispersion in the pump beam broadens the pump pulse. As a result, an increase in the second-order dispersion parameter Dp causes an increase of visibility, but no change in the width of the dip. Second-order dispersion in the downconverted beams leads to a broadening of the dip, as well as asymmetry and oscillations at its borders. 

Second-order dispersion in an optical material (d and di are second-order dispersion parameters of the signal and idler beams in the material) through which downconverted photons propagate leads to asymmetry of the dip. The dip is particularly stretched to larger values of / (see Fig. 20) as a consequence of the deformation and lengthtening of the two-photon amplitude Au,i in a dispersive material. The higher the difference d — di of the dispersion parameters, the higher the asymmetry and the wider the dip moreover its minimum is shifted further to smaller values of / (see Fig. 20). 

Asymmetry of the dip caused by second-order dispersion in an optical material through which downconverted photons propagate can be suppressed... 

Additivity of the Second-order Dispersion Energy Non-additivity of the Third-order Dispersion Interaction... 

One further point to note is that a grating monochromator disperses light in a series of orders. Thus a monochromator setting at 500 mn will also pass light whose wavelength is 250 nm since the second order of dispersion of li t at X will occur at 2Xj. Placing an appropriate cut-off filter after a monochromator, i.e. a P3nrex filter in this example, can efficiently remove this second-order dispersion. 

The second-order dispersion interaction is exactly pairwise additive when it is calculated (as usual) in the multipole approximation, but not when penetration and exchange are included. The second-order penetration and exchange effects are of minor importance, however, and so they do not play a great role in the non-additivity. 

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