Exchange interaction radius

It was empirically found that, in a calculation of the electron structure of crystals with nonlocal exchange, the integration of the Hartree-Fock exchange potential over the direct lattice requires a so-called exchange-interaction radius to be introduced. The exchange-interaction radius cannot be arbitrarily large but must correspond to the used number of points k in the Brillouin zone. 

The theoretical model [130] was developed as an extension of the classical theory of dipolar broadening in dilute solid solutions in the absence of exchange interactions [16]. It was suggested in [130] how to determine the dipolar part in the line width by subtraction of the calculated input of Heisenberg exchange interaction of pairs of exchange-coupled ions. Equations were written for the three cubic lattices as a function of ion s concentration for various numbers of cationic sites included in a sphere of radius Rc with the assumption that clustering effects were absent. The results were compared well with experimental data on Cr3+ in MgO powders. 

The ionic radius concept is useful in deciding which ions are likely to be accommodated in a given lattice. It is usually safe to assume that ions of similar size and the same charge will replace one another without any change other than in the size of the unit cell of the parent compound. Limitations arise because there is always some exchange interaction between the electrons of neighbouring ions. 

For the number of shells in both structures, each lattice is related to the radius (R) of the nanoparticle [27-29]. Therefore, the value of R contains a number of shells and the size of a nanoparticle increases as the number of shells increases. The shells (R) and their numbers are only bounded to the nearest-neighbour pair exchange interactions (J) between spins. To provide the magnetization of the whole particle, each of the spin sites, which stand for the atomistic moments in the nanoparticle, are described by Ising spin variables that take on the values S1-= l, 0. For a core/surface (C/S) morphology, all spins in the nanoparticle are organized in three components that are core (C, filled circles), interface (or core-surface) (CS) and surface (S, empty circles) parts. The number of spins in these parts within the C/S-type nanoparticle are denoted byNc, Ncs and Ns, respectively. But, the total number of spins (N) in a C/S nanoparticle covers only C and S spin numbers, i.e. N =NC + Ns. On the other hand, the numbers of spin pairs for C, CS and S regions in 2D are defined by N [,=(N (.y(. /2)-Ncs,... 

The tris-bipyridine complexes on the other hand are encapsulated by the oxalate network. Thus in the co-doped systems a [Cr(bpy)3]3+ complex happening to sit in the first acceptor shell of a given donor is much closer to this donor than a [Cr(bpy)3]3+ complex sitting in the second shell, n-n overlap between ligand orbitals of the donor and an acceptor in the first shell ensure efficient energy transfer on the sub-microsecond timescale mediated by exchange interaction. Additionally, the relative orientation of donor and acceptor plays an important role for the n-n overlap. For acceptors further away, for which there is no exchange pathway, dipole-dipole interaction takes over. With a critical radius of the order of 11 A, this is much less efficient and the overall quantum efficiency is thus less than unity. 

Liu (1961) noted that the wave functions of the 4f electrons on different rare earth (R) atoms in the solid state do not usually overlap. This is because the radius of the 4f shell is almost 0.35 A and the wave functions are therefore zero on the Wigner-Seitz sphere. There can therefore be no direct exchange, and exchange interactions between different R-magnetic moments must be mediated by the conduction electrons. Liu points out that there are two possible interaction types. In the first type the 4f magnetic moment on the R-atom polarizes the sp conduction bands of the compound via a direct s-f exchange interaction given by... 

Ci44= -3.2, and Ci55= -3.2 [9]. Treatment of exchange interactions in Hartree-Fock LCAO calculations on the STO-2G level shows that for a-BN a cut-off radius of 10 a.u. is sufficient to reduce the error in total energy to less than 1 xIO"" a.u. per cell [10]. 

The sum (6.50) can be calculated for k kj, for example, by the Ewald method. However, for k = kj the series (6.50) appears to be divergent [95]. This divergence is the result of the general asymptotic properties of the approximate density matrix calculated by the summation over the special poits of BZ (see Sect. 4.3.3). The difficulties connected with the divergence of lattice sums in the exchange part have been resolved in CNDO calculations of solids by introduction of an interaction radius... 

The EHT method is noniterative so that the results of COM apphcation depend only on the overlap interaction radius. The more complicated situation takes place in iterative Mulliken-Riidenberg and self-consistent ZDO methods. In these methods for crystals, the atomic charges or the whole of the density matrix are calculated by summation over k points in the BZ and recalculated at each iteration step. The direct lattice summations have to be made in the surviving integrals calculation before the iteration procedure. However, when the nonlocal exchange is taken into account (as is done in the ZDO methods) the balance between direct lattice and BZ summations has to be ensured. This balance is automatically ensured in cychc-cluster calculations as was shown in Chap. 4. Therefore, in iterative MR and self-consistent ZDO methods the increase of the cyclic cluster ensures increasing accuracy in the direct lattice and BZ summation simultaneously. This advantage of COM is in many cases underestimated. 

Screened Coulomb hybrid functionals do not need to rely on the decay of the density matrix to allow calculations in extended systems [409]. The SR HF exchange interactions decay rapidly and without noticeable dependence on the bandgap of the system. The screening techniques do not rely on any truncation radius and provide much better control over the accuracy of a given calculation. In addition, the thresholds can be set very tightly, without resulting in extremely long alculations. 

In this formula, Z is not accessible to direct optical measurement and varies with distance R as exp(-R/L), where L is an effective average Bohr radius of donor and acceptor ions in excited and unexcited states. While the former two interactions are electrostatic in origin, the exchange interaction arises from the antisymmetry requirements of the electronic wavefunction for a system consisting of a donor and an acceptor. 

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