Particle, polarizable, energy

It is easy and instructive to verify that, under the conditions of validity of these equations, the interaction energy is not great compared with kT. Use the same parameters for particle polarizability as those used in estimating particle-particle interactions. 

When the Drude particles are treated adiabatically, a SCF method must be used to solve for the displacements of the Drude particle, d, similarly to the dipoles Jtj in the induced dipole model. The implementation of the SCF condition corresponding to the Born-Oppenheimer approximation is straightforward and the real forces acting on the nuclei must be determined after the Drude particles have attained the energy minimum for a particular nuclear configuration. In the case of N polarizable atoms with positions r, the relaxed Drude particle positions r + d5CF are found by solving... 

An important addition to the model was the inclusion of virtual particles representative of lone pairs on hydrogen bond acceptors [60], Their inclusion was motivated by the inability of the atom-based electrostatic model to treat interactions with water as a function of orientation. By distributing the atomic charges on to lone pairs it was possible to reproduce QM interaction energies as a function of orientation. The addition of lone pairs may be considered analogous to the use of atomic dipoles on such atoms. In the model, the polarizability is still maintained on the parent atom. In addition, anisotropic atomic polarizability, as described in Eq. (9-28), is included on hydrogen bond acceptors [65], Its inclusion allows for reproduction of QM polarization response as a function of orientation around S, O and N atoms and it facilitates reproduction of QM interaction energies with ions as a function of orientation. 

The main feature of the polarizability contribution to the energy shift is its logarithmic enhancement [26, 30]. The appearance of the large logarithm may easily be understood with the help of the skeleton integral. The heavy particle factor in the two-photon exchange diagrams is now described by the photon-nucleus inelastic forward Compton amplitude [31]... 

Another explanation must therefore be found. Now we know that besides forces of an electrical character there are others which act between atoms. Even the noble gases attract one another, although they are non-polar and have spherically symmetrical electronic structures. These so-called van der Waals forces cannot be explained on the basis of classical mechanics and London was the first to find an explanation of them with the help of wave mechanics. He reached the conclusion that two particles at a distance r have a potential energy which is inversely proportional to the sixth power of the distance, and directly proportional to the square of the polarizability, and to a quantity

excitation energies of the atom, so that... 

On pages 159 and 160 formulae have been derived for the energy of a dipole pi and a particle with polarizability a in the field of an ion a at a distance r. If the field of the ion is called F, these expressions become —piF and —- aF2. It was noted before that the factor [jl in the latter formula is due to the fact that the energy-required to polarize a particle by a field F is equal to half the energy of the particle in that field. 

The total energy of the polarizable particle in the field F is therefore —olF2 + JocF2 = — aF2 Energy of non-linear molecule AB2... 

Table 6.1 Contributions of the Keesom, Debye, and London potential energy to the total van der Waals interaction between similar molecules as calculated with Eqs. (6.6), (6.8), and (6.9) using Ctotal = Corient + Cind + Cdisp- They are given in units of 10-79 Jm6. For comparison, the van der Waals coefficient Cexp as derived from the van der Waals equation of state for a gas (P + a/V fj (Vm — b) = RT is tabulated. From the experimentally determined constants a and b the van der Waals coefficient can be calculated with Cexp = 9ab/ (47T21V ) [109] assuming that at very short range the molecules behave like hard core particles. Dipole moments /u, polarizabilities a, and the ionization energies ho of isolated molecules are also listed.

O Malley, T.F., Spruch, L. and Rosenberg, L. (1962). Low-energy scattering of a charged particle by a neutral polarizable system. Phys. Rev. 125 1300-1310. 

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