Madelung term

Madelung term and, 1 162-171 permanent electrical multipole term Uq and, 1 176-177... 

The ionic theory thus has the advantage of isolating the quantum and other effects that are difficult to calculate into the potential C/iocai which is treated empirically (Section 3.1). The long-range effects which are important for crystal chemistry are then given by the Madelung term which can be readily calculated using classical electrostatic theory. 

The complementary potential Ad or generalized Madelung term is expanded in regular solid harmonics as... 

For most distortions, both overlap interactions and Madelung terms contribute to the elastic constants. However, for the distortion associated with C44 in the rocksalt structure, only the Madelung term enters. (There are no changes in nearest-neighbor distance to first order in the strain.) Thus the experimental values can be compared with the Madelung term, and by virtue of the Cauchy relation, c, 2 should take the same value. That contribution has been calculated by Kellerman (1940) and is Values of this expression arc given in Table... 

Um, see Madelung term Uu Uq, see Permanent electrical multipole term Uq... 

Madelung Term Lattice Energy Ionic Radius Lyotropic Number From Heat of formation ... 

It should be pointed out that the energy expressions in Eqs. (25) and (26) - especially the Madelung terms - are not the same, because the atomic spheres overlap in the ASA model whereas in the APW approach the muffin-tin spheres only touch. 

A fourth condition, to be discussed later, is a large interatomic Coulomb potential Madelung term) associated with the higher orbital. As kindly pointed out to me by Dr. Alan Williams, the first three conditions are also likely to occur only in polyatomic systems in atoms, if two eigenvalues are close from one another (condition i), the interelectronic repulsion terms tend to be ahke and therefore conditions (ii) and (iii) do not obtain. 

In other words, to the Hartree-Fock potential one must add a Madelung term. Because of this extra term, the configuration (... (4/) (2p) ) is more stable than (... (4/) (2p) ) provided the following condition obtains ... 

Equation [23d] has a nice interpretation. The energy of an electron in atomic orbital x is equal to the negative of its Mulliken electronegativity, (/ + A)/2, corrected for charge build-up on atom A, Q. Yaai to which it belongs, corrected in turn for the number of electrons in the x orbital itself (1 - P ), and then corrected by the Madelung terms (Coulomb attractions and repulsions) located throughout the molecule, Q/jY,uj-... 

In the atomic-sphere approximation the electrostatic interactions of (7.15), i.e. the Hartree term and (8.31,32), reduce to the interaction with the field -2Zt/r inside the sphere. In addition, the spheres interact via the Madelung term... 

The second step is to solve the Poisson equation (1.3) with the trial charge density which gives the electrostatic Hartree potential (7.27, 8.38). To this is added -2Z/r from the nuclei, the Madelung term (8.39) if more inequivalent atoms are included, and an exchange-correlation potential of some kind, e.g. that in [9.7], forming the effective one-electron trial potential (7.28, 8.39). 

We now calculate the ground-state energy of an electron gas interacting with the lattice charges through the modified Coulomb potential (131). The calculations follow the same steps as for the case of point charges. We shall therefore merely state the points at which differences must be introduced. This, of course, only happens in the contributions where the lattice plays some role, i.e., the Madelung term and the polarization term. 

In the same way as in Section IV-E, only the Madelung term and the polarization term are influenced by the replacement of the one-component lattice by a multicomponent one hence we need only study these contributions in the present section. ... 

Let US first calculate the Madelung term. Substituting (166) into the definition (34) of the lattice self-energy, we get ... 

If we neglect the Madelung term, the terms in crystals have... 

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