Monopole repulsion

Each two-electron integral is the sum of all the terms arising from the charge distribution representative of the first pair of atomic orbitals interacting with the charge distribution representative of the second pair of atomic orbitals. Thus in the simplest case, the (ssiss) interaction is represented by the repulsion of two monopoles, while a (pj pjjlp jjp jj), a much more complicated interaction,... 

Apart from these relativistic energy corrections, the treatment is non-relativistic regarding the theoretical formulation and one-electron wave function basis, d) The above treatment is also applied to the case of two core holes in the same main shell. The self-energies and hole-hole repulsion in Eq. (31) are then approximated by static monopole relaxation and screening, evaluated through the zlSCF method (cf. Sect. 3.6)... 

The last four terms depend only on the reference density, po, and represent the repulsive energy contribution, Flcp, discussed above. Thus, we just have to deal with the second-order terms. The second-order term in the charge density fluctuations dp(r), that is, the second term in Equation 5.51, is approximated by writing Ap as a superposition of atomic contributions, Ap0 = Apv. This approach decays quickly with the increasing distance from the corresponding center. To simplify the second term further, Elstner applied a monopole approximation ... 

Though this looks like an extensive quantity, its thermodynamic limit is infinite if Ai. = /dry rniy) is non-zero for both i = m and i = n. (The Coulomb repulsion of an infinite set of monopoles is infinite.) The problem can be removed by including the nuclear Coulomb attraction. Consider the charge distribution 5)n(r)... 

This transformation from the dipole to the monopole approximation in the calculations of the intermolecular forces has very significant consequences and represents an essential step in the improvement of the calculations. More recently further refinements have still been introduced in the calculations, prominent among which are the replacement of molecular polarizability by bond polarizabilities and the introduction of a supplementary short-range repulsion term generally in the form of a semiempirical function of the type proposed by Kitaygorodskii and used more extensively by Faidni and Simonetta. Details of these refinements should be looked for in the original papers. 

The nonbond interactions are summed over all partial atomic charges in pairs (separated by three or more bonds) taken as point monopoles in the Coulombic term. The dipole-induced dipole (London dispersion) interaction between polarizable atoms is contained in the last term, with the atomic overlap repulsion being included here (for algebraic convenience) as the 9th or 12th power. (CFF uses the 9th power, whereas the traditional Lennard-Jones choice is the 12th.)... 

Interactions become shorter-ranged and weaker as higher multipole moments become involved. When a monopole interacts with a monopole. Coulomb s law says u r) oc r But when a monopole interacts with a distant dipole, coulombic interactions lead to u r) oc r (see Equation (21.26)). Continuing up the multipole series, two permanent dipoles that are far apart interact as u(r) oc r Such interactions can be either attractive or repulsive, depending on the orientations of the dipoles. Table 24.2 gives typical energies of some covalent bonds, and Table 24.3 compares covalent to noncovalent bond strengths. 

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