Coulomb-hole contribution

The second term in eq. (4.9) is the Coulomb-hole contribution. Prom this equation, together, with the corresponding expression for I , we can write down the bonding matrix elements... 

Since the coiTelation between opposite spins has both intra- and inter-orbital contributions, it will be larger than the correlation between electrons having the same spin. The Pauli principle (or equivalently the antisymmetry of the wave function) has the consequence that there is no intraorbital conelation from electron pairs with the same spin. The opposite spin correlation is sometimes called the Coulomb correlation, while the same spin correlation is called the Fermi correlation, i.e. the Coulomb correlation is the largest contribution. Another way of looking at electron correlation is in terms of the electron density. In the immediate vicinity of an electron, here is a reduced probability of finding another electron. For electrons of opposite spin, this is often referred to as the Coulomb hole, the corresponding phenomenon for electrons of the same spin is the Fermi hole. 

With the KS theory Fermi and Coulomb holes defined by Eq. (27), the derivative vxc(r) can also be expressed in terms of its separate Pauli, Coulomb and correlation-kinetic contributions as... 

In Eq. (5), G is the one-particle Green s function, W is the screened Coulomb interaction and 5 = 0. The real part of the self-energy contains a screened exchange contribution, which requires an explicit calculation of the dielectric matrix of the system, and a Coulomb-hole term which takes into account the actual presence of the quasi-particle (excess electron or hole) in the system and its screening by the surrounding electrons. 

The Auger spectra of CF4 and SiF4 have been analyzed in terms of the concept of Coulomb localization. Contributions from localized two-hole states were used to explain the presence of six r.ither than three main peaks... 

Note that the self-interaction contribution to the Fermi-Coulomb hole charge is cancelled by the density, so that the pair-correlation density as defined by Eq. (17) is self-interaction free. 

The exchange-correlation hole represents the reduced probability of finding electron 2 at a position given that electron 1 is located at ti. The exchange part of /i c is called the Fermi hole, while the dynamical correlation gives rise to the Coulomb hole. Since exchange only occurs between electrons of the same spin, the total hole can also be written in terms of individual spin contributions. 

The XC-hole can be split into contributions from the exchange- or X-hole, which arises from the Fermion nature of an electron obeying the Pauli principle, and the correlation- or C-hole due to Coulomb repulsion within the pair of electrons. (The X-hole and C-hole are often referred to as Fermi hole and Coulomb hole, respectively). 

As the number of eigenstates available for coherent coupling increases, the dynamical behavior of the system becomes considerably more complex, and issues such as Coulomb interactions become more important. For example, over how many wells can the wave packet survive, if the holes remain locked in place If the holes become mobile, how will that affect the wave packet and, correspondingly, its controllability The contribution of excitons to the experimental signal must also be included [34], as well as the effects of the superposition of hole states created during the excitation process. These questions are currently under active investigation. 

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