Coulombic repulsion energy

In order to get an estimate of the order of magnitude of the correlation energy, Froman makes it plausible that the correlation energy should be roughly proportional to the total Coulomb repulsion energy of all the electrons within the system, and he suggests the formula... 

Technically, the time-independent Schrodinger equation (2) is solved for clamped nuclei. The Hamiltonian is broken into its electronic part, He, including the nuclear Coulomb repulsion energy, and the nuclear Hamiltonian HN. At this level, mass polarization effects are usually neglected. The wave function is therefore factorized as usual (r,X)= vP(r X)g(X). Formally, the electronic wave function d lnX) and total electronic energy, E(X), are obtained after solving the equation for each value of X ... 

At this point, we have reached the stage where we can describe the adatom-substrate system in terms of the ANG Hamiltonian (Muscat and Newns 1978, Grimley 1983). We consider the case of anionic chemisorption ( 1.2.2), where a j-spin electron in the substrate level e, below the Fermi level (FL) eF, hops over into the affinity level (A) of the adatom, whose j-spin electron resides in the lower ionization level (I), as in Fig. 4.1. Thus, the intra-atomic electron Coulomb repulsion energy on the adatom (a) is... 

As indicated in Equation 4.21, the interelectronic Coulomb repulsion energy functional J[p is written as the classical expression... 

When the effective on-site Coulomb repulsive energy (Geff) of the solid composed of Tt-radical molecules is smaller than the bandwidth (W), then the solid becomes a half-filled metal provided that the molecules stack uniformly without dimerization and can be described by a band picture. So far, no such radical molecules have been prepared. In order to decrease Ues and stabilize radical molecules chemically, a push-pull effect and an extension of the re-system have been implemented, though a large U ff and high reactivity (polymerization) are stiU crucial for the metallic transport. Table 2 summarizes selected organic conductors of neutral 7t-radical molecules. 

Now, in the TF-HK equation, all the potential terms can be set up by conventional plane-wave-basis teehniques with essentially linear scaling. However, for very large systems with more than 5000 nuclei, the computational cost associated with the nuelear-nuelear Coulomb repulsion energy becomes the major bottleneck." In this case, linear-scaling Ewald sirmmation techniques should be utilized. 

To put it crudely, this correlation ensures that electrons of the same spin cannot be in the same place at the same time. Therefore this type of correlation makes the Coulombic repulsion energy between electrons of the same spin smaller than that between electrons of opposite spin. This is the reason why Hand s rule states that, an electronic state in which two electrons occupy different orbitals with the same spin is lower in energy than an electronic state in which the electrons occupy the orbitals, but with opposite spins. 

Unfortunately, if a single configuration is used to approximate the many-electron wave function, electrons of opposite spin remain uncorrelated. The tacit assumption that electrons of opposite spin move independently of each other is, of course, physically incorrect, because, in order to minimize their mutual Coulombic repulsion energy, electrons of opposite spin do certainly tend to avoid each other. Therefore, a wave function, T, that consists of only one configuration will overestimate the Coulombic repulsion energy between electrons of opposite spin. 

However, there are two more types of one-electron operators in T. One of them, Jis called the Coulomb operator. The other, K,j, is called the exchange operator. Together, they replace the two-electron operators, e /ra, in FL, which give the Coulombic repulsion energy between each pair of electrons, k and 1. 

Jj — K,j operating on / gives the expression in Hartree-Fock (HF) theory for the effective Coulombic repulsion energy between an electron /,- and the pair of... 

The operator 2Jj computes the Coulombic repulsion energy between an electron in v[/, and the pair of electrons in /,-, assuming that the electrons in v[/, and in /,- move independently of each other. The operator ICj corrects the Coulombic repulsion energy, computed from 2Jj, for the fact that antisymmetrization of results in correlation between an electron in /,- and the electron of the same spin in v[/,. 

Functionals. The difference between the Fock operator, in wave function based calculations and the analogous operator in DFT calculations is that the Coulomb and exchange operators in T are replaced in DFT by a functional of the electron density. In principle, this functional should provide an exact formula for computing the Coulombic interactions between an electron in a KS orbital and all the other electrons in a molecule. To be exact, this functional must include corrections to the Coulombic repulsion energy, computed directly from the electron density, for exchange between electrons of the same spin and correlation between electrons of opposite spin. 

Metallic properties are also observed in columnar stacked complexes in which the intrachain separation is too long for conduction to be associated with the formation of a band from overlap of the metal dz2/pz orbitals. For these complexes the conduction process involves carriers present in the delocalized MOs extending over die whole complex. The conduction process in these compounds may be described by a hopping mechanism in which the on-site Coulomb repulsion energy has been reduced by the delocalization of the charge over the whole molecule. 

The third term, describing the interelectronic Coulomb repulsion energy, is given in density functional theory as ... 

A single value for (3 for all hydrocarbons is too much to hope from a simple theory. Experimental estimates of /i vary from 2.72 eV (to fit energies for the benzenoid hydrocarbons) to 3.48 eV (derived from the experimental absorption spectrum of naphthalene). The onsite Coulomb repulsion energy a is positive, but its value is not obtained directly from experiment, since it does not enter into the spacings of Hiickel eigenenergies. 

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