Coulomb interaction/integral computational quantum mechanics

Fig. 4.10. Electron momentum distributions for neon ( 75oi = 0.79 a.u. and /. 02 = 1.51 a.u.) subject to a linearly polarized monochromatic field with frequency ui = 0.057 a.u. and intensity I = 3.0 x 1014W/cm2, as functions of the electron momentum components parallel to the laser-field polarization. The left and the right panels correspond to the classical and to the quantum-mechanical model, respectively. The upper and lower panels have been computed for a contact and Coulomb-type interaction Vi2, respectively. In panels (a) and (d), and (h) and (e), the second electron is taken to be initially in a Is, and in a 2p state, respectively, whereas in panels (c) and (/) the spatial extension of the bound-state wave function has been neglected. The transverse momenta have been integrated over...

The last term can be easily calculated from the positions of the nuclei, the first term requires calculation of the one-electron integrals. Note that the resulting formula says that the forces acting on the nuclei follow from the classical Coulomb interaction involving the electronic density p, even if the electronic density has been (and must be) computed from quantum mechanics. 

The philosophical transition from the atomic prejudice to a view of intermolecular interaction in terms of diffuse electron density has its proper computational counterpart in full quantum mechanical calculations, which, however, cannot at present provide complete intermolecular energies because of limitations in the treatment of electron correlation, a major ingredient of the intermolecular interaction recipe. In a different perspective, the classical atom-atom force-field approach is widely applicable but entirely parametric and of scarce adherence to physical principles. The need is felt for an extension to represent in a more realistic manner the effects of diffuse electron clouds. This is done in the so-called semi-classical density sums (SCDS) or briefly. Pixel approach [9], which will now be described. The Pixel method is based on numerical integrations over molecular electron densities, and allows a separation of the total intermolecular cohesion energy into coulombic, polarization, dispersion, and repulsion contributions. 

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