Coulomb correlation repulsion

It should be noted that several of the proposed functionals violate fundamental restrictions, such as predicting correlation energies for one-electron systems (for example P86 and PW91) or failing to have the exchange energy cancel the Coulomb self-repulsion (Section 3.3, eq. (3.32)). One of the more recent functionals which does not have these problems is due to Becke (B95), which has the form... 

Next, let us explore the consequences of the charge of the electrons on the pair density. Here it is the electrostatic repulsion, which manifests itself through the l/r12 term in the Hamiltonian, which prevents the electrons from coming too close to each other. This effect is of course independent of the spin. Usually it is this effect which is called simply electron correlation and in Section 1.4 we have made use of this convention. If we want to make the distinction from the Fermi correlation, the electrostatic effects are known under the label Coulomb correlation. 

Since two electrons of the same spin have a zero probability of occupying the same position in space simultaneously, and since t / is continuous, there is only a small probability of finding two electrons of the same spin close to each other in space, and an increasing probability of finding them an increasingly far apart. In other words the Pauli principle requires electrons with the same spin to keep apart. So the motions of two electrons of the same spin are not independent, but rather are correlated, a phenomenon known as Fermi correlation. Fermi correlation is not to be confused with the Coulombic correlation sometimes referred to without its qualifier simply as correlation . Coulombic correlation results from the Coulombic repulsion between any two electrons, regardless of spin, with the consequent loss of independence of their motion. The Fermi correlation is in most cases much more important than the Coulomb correlation in determining the electron density. 

The electron density distribution is determined by the electrostatic attraction between the nuclei and the electrons, the electrostatic repulsion between the electrons, the Fermi correlation between same spin electrons (due to the operation of the Pauli principle), and the Coulombic correlation (due to electrostatic repulsion). 

Even when a distortion cannot be directly detected, it can have deep consequences for the electronic properties of these materials. For example, there is a clear-cut competition between JT effect and the molecular Coulomb correlations. Hund s rules would tend to form a high-spin state that minimizes repulsion between electrons, while JT-split electronic levels imply a low-spin ground state. 

Non dynamical electron correlation is the part of the total correlation that is taken into account in a CASSCF calculation that correlates the valence electrons in valence orbitals. Physically, the non dynamical electron correlation is a Coulomb correlation that permits the electrons to avoid one another and reduce their mutual repulsion as much as possible with respect to a given zero order electronic structure defined by the Hartree-Fock wave function. In VB terms, the non dynamical correlation ensures a correct balance between the ionic and covalent components of the wave function for a given electronic system. The dynamical correlation is just what is still missing to get the exact nonrelativistic wave function. 

The early VB point of view was based solely on the purely covalent HL wave function. In this wave function the electrons are never allowed to approach each other and therefore their electron repulsion is minimized and their Coulomb correlation is at maximum. Thus, while the Hartree-Fock model has no electron correlation, giving equal weight to covalent and ionic structures, the early VB models overestimated the correlation. The true situation is about half-way in-between. In the same way as the Hartree-Fock wave function is improved by Cl, the purely covalent VB function can be improved by admixture of ionic structures as in eq 5, in which the coefficients X and p would be directly optimized in the VB framework. Both improved models thus lead to wave functions that are strictly equivalent and physically correct, even though their initial expressions appear entirely different. This... 

The pair-correlation density is a property that arises due to the Pauli and Coulomb correlations between electrons. Thus it can also be interpreted as the density p(r ) at r plus the reduction in this density at r due to the electron correlations. The reduction in density about an electron which occurs as a result of the Pauli exclusion principle and Coulomb repulsion is the quantum-mechanical Fermi-Coulomb hole charge distribution p (r, r ). Thus we may write the pair-correlation density as... 

A comparison of Equations (2.78) and (2.79) yields that in both approaches, MO (= Hartree-Fock) and VB, the Pauli exchange has been correctly included. The difference between the two is solely given by their amount of electronic correlation. In the MO approach, the electrons are completely uncorrelated (independent), and they may even go into the same atomic orbital, albeit with different spins, thus producing ionic states (H H+). The MO approach therefore does not take care of the energy penalty due to the Coulomb repulsion between the two electrons (see Section 2.9). Because the electronic Coulomb correlation has been completely ignored, the correlation energy may be defined as the difference between the correct energy and that of the Hartree-Fock solution, that is... 

In this analysis, we assume that an energy minimum due to coulomb correlation (van der Waals attraction) is absent and that state A is purely repulsive. 

This effect is known as exchange or Fermi correlation and is a direct consequence of the Pauli principle. The Fermi hole is in no way connected to the charge of electrons and applies equally to neutral fermions. This kind of correlation is included in the HF appro ich due to the antisymmetry of the Slater determinant [337]. The electrostatic repulsion of electrons (the l/ri2 term in the Hamiltonian) prevents the electrons from coming too close to each other and is known as Coulomb correlation. This effect is independent of the spin and is called simply electron correlation, and is completely neglected in the HF method. 

The determinantal form of the wavefunction discussed in Section 5.2.2 is used in the Hartree-Fock (HF) approach to solving the many-electron Schrodinger equation. The HF approach is considered to be an uncorrelated method, since it doesn t include Coulomb correlation, i.e., the correlation in electronic motion arising from the repulsive electrostatic electron-electron interactions. The correlation energy is therefore defined as the difference between the exact energy and the energy obtained by employing the Hartree-Fock approximation... 

Coulomb repulsion by occupying different r giom of space in the SCF picture Is 2s, both electrons reside in the same 2s region of space. In this particular example, the electrons undergo angular correlation to avoid one another. 

The first three terms in Eq. (10-26), the election kinetic energy, the nucleus-election Coulombic attraction, and the repulsion term between charge distributions at points Ti and V2, are classical terms. All of the quantum effects are included in the exchange-correlation potential... 

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