Big Chemical Encyclopedia

Chemical substances, components, reactions, process design ...

Articles Figures Tables About

Fermi-correlation

The second bullet point describes a special type of correlation that prevents two electrons of like spin being found at the same point in space, and it applies whenever the particles are fermions. For that reason, it is described as Fermi correlation. [Pg.186]

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. [Pg.99]

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). [Pg.63]

This can only be true if p2 (xj, Xj) = 0. In other words, this result tells us that the probability of finding two electrons with the same spin at the same point in space is exactly zero. Hence, electrons of like spin do not move independently from each other. It is important to realize that this kind of correlation is in no way connected to the charge of the electrons but is a direct consequence of the Pauli principle. It applies equally well to neutral fermions and - also this is very important to keep in mind - does not hold if the two electrons have different spin. This effect is known as exchange or Fermi correlation. As we will show below, this kind of correlation is included in the Hartree-Fock approach due to the antisymmetry of a Slater determinant and therefore has nothing to do with the correlation energy E discussed in the previous chapter. [Pg.39]

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. [Pg.39]

It can easily be shown that the HF approximation discussed in Chapter 1 does include the Fermi-correlation, but completely neglects the Coulomb part. To demonstrate this, we analyze the Hartree-Fock pair density for a two-electron system with the two spatial orbitals ()> and < )2 and spin functions a and o2... [Pg.39]

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. [Pg.273]

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). [Pg.278]

The two-particle cumulant is a correlation increment. It describes Coulomb correlation, since the Fermi correlation is already contained in the description in terms of only. In terms of the cumulants, the energy expectation value can be written... [Pg.301]

The Laplacian of the electron density plays a dominant role throughout the theory.191 In addition, Bader has shown that the topology of the Laplacian recovers the Lewis model of the electron pair, a model that is not evident in the topology of the electron density itself. The Laplacian of the density thus provides a physical valence-shell electron pair repulsion (VSEPR) basis for the model of molecular geometry and for the prediction of the reaction sites and their relative alignment in acid-base reactions. This work is closely tied to earlier studies by Bader of the electron pair density, demonstrating that the spatial localization of electrons is a result of a corresponding localization of the Fermi correlation hole. [Pg.262]

The relationship between the Fermi correlation and the spatial localization of electrons is developed by defining, as McWeeny (1960) does, a correlation function/(rj, rj) in an expression which determines the extent to which the pair density deviates from a simple product of number densities,... [Pg.336]

In an electronic system it is essential to distinguish between the correlation of electrons with identical spin, Fermi correlation, from that of electrons with opposite spins. Coulomb correlation. The correlation term/(ri,r2) as defined in eqn (E7.10) will measure both types of correlation (McWeeny 1960). However, the limiting value of the correlation hole as expressed in eqn (E7.11) arises only from the correlation between electrons of the same spin, the Fermi hole. That part of/(r, r2) that refers to the correlation between electrons of opposite spin contributes zero when integrated over all space. If the co-... [Pg.336]

The effect of the Fermi correlation on the pair density as given in eqn ( 7.16) is so pronounced that even a single determinantal function yields a good description of the local properties of the Fermi hole. The addition of Coulomb correlation does not change its significant spatial features. One can view the net result of the Pauli principle as correcting for the self-pairing of... [Pg.337]

F(Sl, f2) is a measure of the total correlation which is contained within the region (f2). For a Hartree-Fock wave function, which includes only Fermi correlation, the magnitude of F(Sl, f2) is a measure of the total (integrated) Fermi hole of the N Si) particles that lies within the region 12. From eqns (E7.11) and (E7.19), the limiting value of the correlation is seen to be — N(S1). [Pg.338]

It is now easy to demonstrate that the fluctuation in the average electron population of a region 2 vanishes when the Fermi correlation for the N( i) electrons in 2 is totally contained in 2. Equation (E7.9) for A N, 2) can be reexpressed in terms of the average number of pairs in 2 as... [Pg.339]

The fluctuation attains its minimum value of zero under the same condition that D2( 2, 2) is reduced to a pure pair population, that is, when one particular event has a probability of unity. From eqn (E7.22) the limiting value of F( 2, 2) is equal to — JV( 2), a value which, as stated previously, implies that the total Fermi correlation hole for each electron in 2 is totally contained within the region 2. [Pg.339]

While it is possible to find regions of space bounded by surfaces such that the contained Fermi correlation is maximized to yield localized groupings of electrons, these regions correspond to atomic cores or localized atomic populations and not, in general, to localized pairs of bonded and non-bonded... [Pg.341]


See other pages where Fermi-correlation is mentioned: [Pg.20]    [Pg.43]    [Pg.190]    [Pg.280]    [Pg.297]    [Pg.151]    [Pg.186]    [Pg.190]    [Pg.3]    [Pg.26]    [Pg.174]    [Pg.71]    [Pg.50]    [Pg.57]    [Pg.220]    [Pg.250]    [Pg.251]    [Pg.252]    [Pg.252]    [Pg.299]    [Pg.336]    [Pg.343]   
See also in sourсe #XX -- [ Pg.186 ]

See also in sourсe #XX -- [ Pg.99 ]

See also in sourсe #XX -- [ Pg.588 ]

See also in sourсe #XX -- [ Pg.204 ]

See also in sourсe #XX -- [ Pg.186 ]

See also in sourсe #XX -- [ Pg.134 ]

See also in sourсe #XX -- [ Pg.82 ]

See also in sourсe #XX -- [ Pg.143 , Pg.495 ]

See also in sourсe #XX -- [ Pg.5 , Pg.124 ]




SEARCH



Electron correlation Fermi

Exchange-correlation potential Fermi hole

Fermi Sea Correlation Effects

Fermi operator Coulomb correlations

Fermi, generally correlation

Slater Determinants and Fermi Correlation

© 2024 chempedia.info