Exchange Taylor expansion

Starting point is the Taylor expansion of the magnetic exchange parameters with respect to the components of the strain tensor e, leading to the so-called magnetoelastic Hamiltonian. 

Another common example arises in the expansion of the interaction energy of two atoms as a function of the inverse interatomic distance 1/Rab- This turns out to be a Taylor series which differ from the correct energy by a real contribution with zero Taylor expansion. The difference may be attributed to so-called exchange repulsion. In this case, it is nowadays known that die Taylor series has a zero convergence radius, so that the energy expression constitutes an example of an asymptotic series (to be defined in a moment) which is non-convergent for all Rab-... 

Recently there has been a great deal of interest in nonlinear phenomena, both from a fundamental point of view, and for the development of new nonlinear optical and optoelectronic devices. Even in the optical case, the nonlinearity is usually engendered by a solid or molecular medium whose properties are typically determined by nonlinear response of an interacting many-electron system. To be able to predict these response properties we need an efficient description of exchange and correlation phenomena in many-electron systems which are not necessarily near to equilibrium. The objective of this chapter is to develop the basic formalism of time-dependent nonlinear response within density functional theory, i.e., the calculation of the higher-order terms of the functional Taylor expansion Eq. (143). In the following this will be done explicitly for the second- and third-order terms... 

Density-gradient expansion (DGE). These are formal analogs of the three-dimensional Taylor expansion of the exchange-correlation energy in derivatives of the density ... 

Becke and Roussel [182] constructed a model exchange hole starting with the second-order Taylor expansion of the exact spherically averaged cr-spin hole [183]... 

In Equation 1.86, we found that if we try to put two electrons with the same spin at the same point, the wave function is equal to zero. It is quite easy to see in Equation 1.87 that if the two electrons approach each other, the determinant wave function tends to zero and is proportional to the distance between them, 6. (Set 1 = fitt and 2 = 2a = (5 + 5)a in Equation 1.87 and use the Taylor expansion to get Vji( 2) = Vji(h + 5). The result is a sum of two Slater determinants where one has two columns equal and the other one is proportional to 5.) This means that the density of electrons with the same spin, that is, the absolute square of the wave function, tends to zero as 5. If the position of an electron is assumed fixed, the probability density of electrons with the same spin tends to zero near to the fixed electron. The excluded probability density amounts to a full electron, as will be proven for a Slater determinant in Chapter 2. This hole is called the exchange hole. Electrons with the same spin are thus correlated in a Slater determinant. The correlation problem is the problem of accounting for a correlated motion between the electrons. 

The electromagnetic interaction between the sensitizer and activator is responsible for the energy transfer. Transfer via electric dipole-dipole interaction was first described by Forster ) and later Dexter ) expanded the treatment to include higher order electromagnetic and exchange interactions. The electrostatic interaction can be expressed as a multipole expansion using a Taylor s series about the sensitizer-activator separation Rja,... 

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