Coulomb exchange

If we could evaluate ot, p, and S, which are called the coulomb, exchange, and overlap integrals respectively, we could compute E. 

Fh p) = Ec p) + Fxc p) + FM + FEAPhFT p) (3.15) (with subscripts C, XC, eN, Ext, and T denoting Coulomb, exchange-correlation, electron-nuclear attraction, external, and kinetic energies respectively). It is CTucial to remark that (3,15) is not the Kohn-Sham decomposition familiar in conventional presentations of DFT. There is no reference, model, nor auxiliary system involved in (3.15). From the construction presented above it is clear that in order to maintain consistency and to define functional derivatives properly all... 

Since the Coulomb, exchange, and correlation energies are all consequences of the interelectronic 1 /r12 operator in the Hamiltonian, one can define the exchange energy functional Ex [p] in the same manner as... 

Terms containing the W intermediates no longer contain a factor of The energy-independent, third-order term, Epp (oo), is a Coulomb-exchange matrix element determined by second-order corrections to the density matrix, where... 

Coulomb exchange effects are commonly introduced by means of the Dirac-Slater expression for the exchange energy of a electron gas ... 

Coulomb exchange is already taken into account in the construction of the zero-order effective Dirac equation, where the Coulomb source plays the role of the external potential. Hence, additional contributions of order Zaf could be connected only with the high-momentum Coulomb exchanges. Let us start by calculating the contribution of the skeleton Coulomb-Coulomb diagrams with on-shell external electron lines in Fig. 4.3, with the usual hope that the integrals would tell us themselves about any possible inadequacy of such an approximation. 

Direct calculation of the two Coulomb exchange photon contribution leads to the integral... 

The fluorescence and phosphorescence yields of various /8-diketones are found to be enhanced by coordination to Zn2+, Al3+ and Be2+ in an increasing order. This is ascribed to a ligand-ligand interaction in the excited and ground state, in the case of Zn, most probably arising through both Coulomb exchange and dipole-dipole interaction.748... 

The form of this equation makes explicit the fact that intermolecular forces do depend upon their vibrational states as well as on their electronic states. Due to the antisymmetrization of the global electronic wave function, Vaia2(R R12) contains Coulomb exchange terms and a direct term formed by the Coulomb multipole interactions and the infinite order perturbation electrostatic effects embodied in the reaction field potential [21, 22],... 

The first term, Vs, accounts for the Coulomb-exchange direct interaction between D and A (see Eq.10), and the second, Vexpucit, describes a solvent-mediated chromophore-chromophore contribution between the transition densities. In addition to this explicit medium effect (VexpuCit), we note that another implicit effect of the environment is included in the Vs term, due to changes on the transition densities upon solvation. It is useful to define a screening factor s, conceptually equivalent to the 1/n2 term in the Forster equation, so that V = sVs ... 

The Umn s of Eq. (3) contain two-particle interactions, including Coulomb, exchange, and dipole-dipole contributions, which are parameterized according to semi-empirical functional forms [61]. The parameters are adapted to PPV and are then transposed by scaling to other polymer species. 

This extension of the PCM is described in detail in [26], Here, it is sufficient to say that such an extension is an application of a nonequilibrium scheme within a QM perturbative linear response (PCM-LR) approach. The total electronic coupling, Kotai, is obtained as a sum of two terms, the direct (or Coulombic-exchange) coupling, implicitly modified by the medium (Vs), and the contribution involving the explicit solvent effect (T xpiicit) ... 

The configurations included can be therefore directly associated to specific energy contributions Coulomb-exchange, polarisation or induction (T0), extra polarisation (4 and ) and dispersion (4 6 ). 

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