Electron correlation Fermi

Density functional theory-based methods ultimately derive from quantum mechanics research from the 1920 s, especially the Thomas-Fermi-Dirac model, and from Slater s fundamental work in quantum chemistry in the 1950 s. The DFT approach is based upon a strategy of modeling electron correlation via general functionals of the electron density. 

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 should be noted that the above conclusions have been reached on strictly electrostatic grounds a spin property has not been invoked for the two electrons. From the variation of i/i along the box it can be shown that the singlet state is of higher energy than the triplet because the two electrons are more crowded together for (S-state) than for (T-state). Thus there is less interelectronic repulsion m the T-state. The quantity 2J j. is a measure of the effect of electron correlation which reduces the repulsive force between the two electron (Fermi correlation energy). 

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. 

Sakurai, Y., Tanaka, Y., Bansil, A., Kaprzyk, S., Stewart, A.T. Nagashima, Y., Hyodo, T., Nanao, S., Kawata, H. and Shiotani, N. (1995) High-resolution Compton scattering study of Li asphericity of the Fermi surface and electronic correlation effects, Phys. Rev. Lett., 74, 2252-2255. 

Figure 9. The measured momentum density of an aluminium film. In the left panel we show the measured momentum density near the Fermi level (error bars), the result of the LMTO calculations (dashed line) and the result of these calculations in combination with Monte Carlo simulations taking into account the effects of multiple scattering (full line). In the central panel we show in a similar way the energy spectrum near zero momentum. In the right panel we again show the energy spectrum, but now the theory is that of an electron gas, taking approximately into account the effects of electron-electron correlation (dashed) and this electron gas theory plus Monte Carlo simulations (solid line).

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). 

Maxwell-Boltzmann particles are distinguishable, and a partition function, or distribution, of these particles can be derived from classical considerations. Real systems exist in which individual particles are indistinguishable. For example, individual electrons in a solid metal do not maintain positional proximity to specific atoms. These electrons obey Fermi-Dirac statistics (133). In contrast, the quantum effects observed for most normal gases can be correlated with Bose-Einstein statistics (117). The approach to statistical thermodynamics described thus far is referred to as wave mechanics. An equivalent quantum theory is referred to as matrix mechanics (134—136). 

The failure is not limited to metal-ammonia solutions nor to the linear Thomas-Fermi theory (19). The metals physicist has known for 30 years that the theory of electron interactions is unsatisfactory. E. Wigner showed in 1934 that a dilute electron gas (in the presence of a uniform positive charge density) would condense into an electron crystal wherein the electrons occupy the fixed positions of a lattice. Weaker correlations doubtless exist in the present case and have not been properly treated as yet. Studies on metal-ammonia solutions may help resolve this problem. But one or another form of this problem—the inadequate understanding of electron correlations—precludes any conclusive theoretical treatment of the conductivity in terms of, say, effective mass at present. The effective mass may be introduced to account for errors in the density of states—not in the electron correlations. 

A number of similarities [63] have been noted between properties of the organic superconductors such as (BEDT-TTF)2X, also known as (ET)2X, one of the ET salts, and those of the recently discovered cuprates, such as YBa2Cu307. Both have strong interactions in a plane with weak interactions out of the plane, giving a two-dimensional Fermi surface in both cases. The organics have portions of their Fermi surface that nest, and it appears now that this is also typical of the cuprates [64]. Both systems have a low density of carriers, with the result that screening is reduced, and therefore the electron-electron interactions are stronger than in an ordinary metal and electron-electron correlations are important in both cases. 

Another demonstration of the validity of these calculations is provided by BEDT-TTF-based salts. The calculated Fermi surface of these materials exhibit closed orbits characteristic of two-dimensional electronic interactions and this has been confirmed experimentally. For example, in the case of (BEDT-TTF)2I3, the calculated surface of these orbits (Fig. 21) [61] agrees well with the one measured by magnetic experiments [161]. However, the overall good agreement between calculation and experiment must not hide the fact that some qualitative discrepancies may arise in some cases. For example, (TMTTF)2X salts exhibit a resistivity minimum at a temperature at which no structural transition has yet been observed. The resistivity minimum is not explained by the one-electron band structure, and to account for this progressive electron localization, it is necessary to include in the calculations the effect of the electronic correlations [162]. Another difficulty has been met in the case of the semiconducting materials a -(BEDT-TTF)2X, for which the calculated band structure exhibits the characteristic features of a metal [93,97,100] and it is not yet understood... 

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