Coulomb-hole

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. 

It is clear that, for electrons with parallel spins, the auxiliary condition (Eq. II.2) gives rise to a correlation effect which very closely resembles the correlation effect coming from the Coulomb repulsion in the Hamiltonian for = 2 the Fermi hole replaces to a certain degree the Coulomb hole. This means that, if... 

The problem of finding the best approximation of this type and the best one-electron set y2t. . ., y>N is handled in the Hartree-Fock scheme. Of course, a total wave function of the same type as Eq. 11.38 can never be an exact solution to the Schrodinger equation, and the error depends on the fact that the two-electron operator (Eq. 11.39) cannot be exactly replaced by a sum of one-particle operators. Physically we have neglected the effect of the "Coulomb hole" around each electron, but the results in Section II.C(2) show that the main error is connected with the neglect of the Coulomb correlation between electrons with opposite spins. 

From Eq. 11.62 follows that the two-electron probability density does not show any "Coulomb hole" for pairs of either parallel or antiparallel spins, and this implies that the Hartree scheme is certainly affected by a large correlation error. 

First of all we note that the Fermi hole - which is due to the antisymmetry of the wave function - dominates by far the Coulomb hole. Second, another, very important property of the Fermi hole is that it, just like the total hole, integrates to -1... 

From equations (2-17) and (2-21) it is obvious that the Coulomb hole must be normalized to zero, i. e. the integral over all space contains no charge ... 

Exchange-Correlation Potential from the Fermi-Coulomb Hole.91... 

EXCHANGE-CORRELATION POTENTIAL FROM THE FERMI-COULOMB HOLE... 

If we wish to obtain the XC potential of Equation 7.15 as the work done in moving an electron in the field of its Fermi-Coulomb hole pxc(r, r ), we first calculate the field as... 

As noted above, the curl of the expression on the right-hand side of Equation 7.47 vanishes. However, it does not mean that the Coulombic and non-Coulombic components—the former is the electric field produced by the Fermi-Coulomb hole and the latter arises from the kinetic energy tensor—of this field also have vanishing curl. Thus the potential Wxc of Equation 7.38 may sometimes be path dependent [21]. 

To illustrate the convergence of the FCI principal expansion with respect to short-range electron correlation, we have in Fig. 1.1 plotted the ground-state He wavefunction with both electrons fixed at a distance of 0.5 ao from the nucleus, as a function of the angle 0i2 between the position vectors ri and r2 of the two electrons. The thick grey lines correspond to the exact nonrelativistic wavefunction, whereas the FCI wavefunctions are plotted using black lines. Clearly, the description of the Coulomb cusp and more generally the Coulomb hole is poor in the orbital approximation. In particular, no matter how many terms we include in the FCI wavefunction, we will not be able to describe the nondifferentiability of the wavefunction at the point of coalescence. 

Fermi hole + Coulomb hole = total hole... 

Fig. 8. Fermi hole, Coulomb hole and total hole in at various bond distances. In all plots the reference electron is placed at 0.3 bohr at the left of the right H atom...

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