Coulomb exchange interaction

Asm is an antisymmetrizer operator between electrons from these two groups s and m which is usually expressed as a sum of the identity operator (1) and normalized permuting operator Pms Asm =l+pms. The total Hamiltonian is symmetric to any electron permutation. The interaction energy Vsm can be cast in terms of a direct Coulomb interaction and an exchange Coulomb interaction ... 

The LDA-I-U orbital-dependent potential (7.74) gives the energy separation between the upper valence and lower conduction bands equal to the Coulomb parameter U, thus reproducing qualitatively the correct physics for Mott-Hubbard insulators. To construct a calculation in the LDA-I-U scheme one needs to define an orbital basis set and to take into account properly the direct and exchange Coulomb interactions inside a partially filled d- f-) electron subsystem [439]. To realize the LDA-I-U method one needs the identification of regions in a space where the atomic characteristics of the electronic states have largely survived ( atomic spheres ). The most straightforward would be to use an atomic-orbital-type basis set such as LMTO [448]. 

The energy transfer rates between the ligands and the lanthanide ion were calculated for cryptate complexes. Their model includes the direct and exchange Coulomb interaction in the perturbation operator. Namely, the enei transfer rate is given as follows ... 

Flere we distinguish between nuclear coordinates R and electronic coordinates r is the single-particle kinetic energy operator, and Vp is the total pseudopotential operator for the interaction between the valence electrons and the combined nucleus + frozen core electrons. The electron-electron and micleus-micleus Coulomb interactions are easily recognized, and the remaining tenu electronic exchange and correlation... 

Coulomb integrals Jij describe the coulombic interaction of one charge density (( )i2 above) with another charge density (c )j2 above) exchange integrals Kij describe the interaction of an overlap charge density (i.e., a density of the form ( )i( )j) with itself ((l)i(l)j with ( )i( )j in the above). 

It is seen that J represents the Coulomb interaction of an electron in a Is orbital on nucleus A with nucleus B. K may be called a resonance or exchange integral, since both functions uu and Uub occur in it. 

One obvious drawback of the LDA-based band theory is that the self-interaction term in the Coulomb interaction is not completely canceled out by the approximate self-exchange term, particularly in the case of a tightly bound electron system. Next, the discrepancy is believed to be due to the DFT which is a ground-state theory, because we have to treat quasi-particle states in the calculation of CPs. To correct these drawbacks the so-called self-interaction correction (SIC) [6] and GW-approximation (GWA) [7] are introduced in the calculations of CPs and the full-potential linearized APW (FLAPW) method [8] is employed to find out the effects. No established formula is known to take into account the SIC. 

All of the examples of singlet energy transfer we have considered take place via the long-range resonance mechanism. When the oscillator strength of the acceptor is very small (for example, n-> n transitions) so that the Fdrster critical distance R0 approaches or is less than the collision diameter of the donor-acceptor pair, then all evidence indicates that the transfer takes place at a diffusion-controlled rate. Consequently, the transfer mechanism should involve exchange as well as Coulomb interaction. Good examples of this type of transfer have been provided by Dubois and co-workers.(47-49)... 

The first square bracket describes the final state of the system, where the acceptor is excited, and the donor is not. The second bracket represents the initial state with the donor excited and the acceptor not. However, notice that electron 1 starts on the donor, and after transfer ends up on the acceptor. For this reason this is called transfer by an exchange mechanism—the electrons of the D and A exchange. The interaction takes place by the usual Coulomb interaction, e1 j — F2, between a pair of electrons. 

That is, in the singlet-singlet transition both Coulomb interaction and exchange interaction are involved. However, when the distance between D and A is large, the exchange term can be ignored, and we can use the multipole expansion for e2/rij, that is,... 

Such an equation differs from Hartree s equation only by virtue of the extra exchange term, TBx. Whereas the electronic Coulomb interaction of the Hartree scheme is formulated as... 

Further, if the wave function depends also on the electron spins, spin variables over all electrons should also be integrated we will see this below, in the calculation of exchange hole. The expression in the curly brackets above is exactly the XC hole PxCM(r, r ) defined in Equation 7.17. A comparison with Equation 7.19a shows that adding the hole to the density is similar to subtracting the density of one electron p(r )/N from it. The hole thus represents a deficit of one electron from the density. This is easily verified by integrating p tM(V, r ) over the volume dr, which gives a value of — 1. However, the structure of the hole is not simple and this is because of the motion of different electrons correlated due to the Pauli exclusion principle and the Coulomb interaction between them. Finally we note that the product p(r)p cM(r, r ) is symmetric with respect to an exchange in the variables... 

The XC energy represents the correction to the Coulomb energy for the self-energy of an electron in a many-electron system. The latter is due to both the direct self-energy of the electron as well as the redistribution of electronic density around each electron because of the Pauli exclusion principle and the Coulomb interaction. As an example, we now discuss the case of Fermi hole and the exchange energy in Hartree-Fock (HF) theory [16]. For brevity, we restrict ourselves to closed-shell cases. 

In general, the (scarce) thermodynamic data for exchanges involving complexes leads us to conclude that the selectivity enhancement upon complexing is enthalpy driven and may be ascribed to enhanced charge dependent (primarily coulombic) interactions with the surface as compared with the aqueous ions. 

For allowed transitions on D and A the Coulombic interaction is predominant, even at short distances. For forbidden transitions on D and A (e.g. in the case of transfer between triplet states (3D + 3A —> 1D + 3A ), in which the transitions Ti —> S0 in D and So —> Ti in A are forbidden), the Coulombic interaction is negligible and the exchange mechanism is found, but is operative only at short distances (< 10 A) because it requires overlap of the molecular orbitals. In contrast, the Coulombic mechanism can still be effective at large distances (up to 80-100 A). 

Fig. 4.14. Schematic representation of the (A) Coulombic and (B) exchange mechanisms of excitation energy transfer. Cl Coulombic interaction EE electron exchange.

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