Wave functions exchange-correlation holes

Due to the antisymmetry requirement for the wave function (see Chapter 1), the holes have to satisfy some general (integral) conditions. The electrons with parallel spins have to avoid each other //j (ri,r2)dr2 = / xc (r, T2)dr2 = —1 (one electron disappears from the neighborhood of the other), while the electrons with opposite spins are not influenced by the Pauli exclusion principle /hxc (ri, r2)dr2 = /hxc (fi, r2)dr2 = 0. The exchange correlation hole is a sum of the exchange hole and the correlation hole hj" = -I- hg, ... 

Further, if the wave function depends also on the electron spins, spin variables over all electrons should also be integrated we will see this below, in the calculation of exchange hole. The expression in the curly brackets above is exactly the XC hole PxCM(r, r ) defined in Equation 7.17. A comparison with Equation 7.19a shows that adding the hole to the density is similar to subtracting the density of one electron p(r )/N from it. The hole thus represents a deficit of one electron from the density. This is easily verified by integrating p tM(V, r ) over the volume dr, which gives a value of — 1. However, the structure of the hole is not simple and this is because of the motion of different electrons correlated due to the Pauli exclusion principle and the Coulomb interaction between them. Finally we note that the product p(r)p cM(r, r ) is symmetric with respect to an exchange in the variables... 

As the Kohn-Sham wave function has a delocalized exchange hole, and therefore lacks the left-right correlation, is expected to display different bond midpoint features to In fact, Vs,km is ro for two-electron systems, so that Tc [p] and Vc reduce in this case to... 

Spin density is found in the molecular plane because of spin polarization, which is an effect arising from exchange correlation. The Fermi hole that surrounds the unpaired electron allows other electrons of the same spin to localize above and below the molecular plane slightly more than can electrons of opposite spin. Thus, if the unpaired electron is a, we would expect there to be a slight excess of density in the molecular plane as a result, the hyperfine splitting should be negative (see Section 9.1.3), and this is indeed the situation observed experimentally. An ROHF wave function, because it requires the spatial distribution of both spins in the doubly occupied orbitals to be identical, cannot represent this physically realistic situation. 

The concept of a spatial correlation is very rich in content, and its content is necessary and sufficient for energy assessment. It is useful to consider the spatial correlation in the approximation of the band model from all the wave functions contained in the Fermi sphere a determinant has to be formed as, also the determinants of weakly excited states a perturbation calculation has to be made with respect to 2e2/ x n — Xjn the reduction to the spatial correlation must be made the energy may be calculated. In fact the pertubation calculation has never been made for metals, while the spatial correlation was only calculated by Wigner and Seitz (1933) in the zero approximation. The result of their calculation is the well-known exchange hole, which may be represented in three-dimensional space if one electron is fixed in the coordinate origin. This may be done if the local space is homogenous i.e., has no... 

We note that this relation is not the same as Eq. (31). The electronic motion is thus correlated also for the HF wave function. A deeper analysis shows that for electrons with opposite spin will the second term (the exchange term) disappear when we integrate over the spin variable. This is not the case for electrons with the same spin (compare the exchange term in the HF energy expression). The electrons thus create a hole around themselves were other electrons with the same spin are forbidden. We call this Fermi correlation. Such a hole is not created for electrons with the opposite spin. There is a finite probability to find them in the same point in space but the magnitude of the exact wave function has a minimum for ri2 = 0 (ri2 being the distance between a pair of electrons). The derivative of the exact wave function is discontinuous at this point. We call this behavior a cusp. The wave function has the form... 

It is obvious that more sophisticated relativistic many-body methods should be used for correct treating the NEET effect. Really, the nuclear wave functions have the many-body character (usually, the nuclear matrix elements are parameterized according to the empirical data). The correct treating of the electron subsystem processes requires an account of the relativistic, exchange-correlation, and nuclear effects. Really, the nuclear excitation occurs by electron transition from the M shell to the K shell. So, there is the electron-hole interaction, and it is of a great importance a correct account for the many-body correlation effects, including the intershell correlations, the post-act interaction of removing electron and hole. 

In Equation 1.86, we found that if we try to put two electrons with the same spin at the same point, the wave function is equal to zero. It is quite easy to see in Equation 1.87 that if the two electrons approach each other, the determinant wave function tends to zero and is proportional to the distance between them, 6. (Set 1 = fitt and 2 = 2a = (5 + 5)a in Equation 1.87 and use the Taylor expansion to get Vji( 2) = Vji(h + 5). The result is a sum of two Slater determinants where one has two columns equal and the other one is proportional to 5.) This means that the density of electrons with the same spin, that is, the absolute square of the wave function, tends to zero as 5. If the position of an electron is assumed fixed, the probability density of electrons with the same spin tends to zero near to the fixed electron. The excluded probability density amounts to a full electron, as will be proven for a Slater determinant in Chapter 2. This hole is called the exchange hole. Electrons with the same spin are thus correlated in a Slater determinant. The correlation problem is the problem of accounting for a correlated motion between the electrons. 

---