Pauli, correlation principle

The fundamental laws which determine the behavior of an electronic system are the Schrodinger equation (Eq. II. 1) and the Pauli exclusion principle expressed in the form of the antisymmetry requirement (Eq. II.2). We note that even the latter auxiliary condition introduces a certain correlation between the movements of the electrons. 

Despite his numerous achievements, Mendeleyev is remembered mainly for the periodic table. Central to his concept was the conviction that the properties of the elements are a periodic function of their atomic masses. Today, chemists believe that the periodicity of the elements is more apparent when the elements are ordered by atomic number, not atomic mass. However, this change affected Mendeleyevs periodic table only slightly because atomic mass and atomic number are closely correlated. The periodic table does not produce a rigid rule like Paulis exclusion principle. The information one can extract from a periodic table is less precise. This is because its groupings contain elements with similar, but not identical, physical and chemical properties. 

Equation 4.49 defines the exchange or Fermi hole. It is as if an electron of a given spin digs a hole around itself in space in order to exclude another electron of the same spin from coming near it (Pauli exclusion principle). The integrated hole charge is unity, i.e., there is exactly one electron inside the hole. Likewise, the correlation energy functional can be defined as... 

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... 

Antisymmetrization results in T vanishing, not only if two electrons with the same spin occupy the same orbital, but also if electrons with the same spin have the same set of spatial coordinates. Thus, antisymmetrization of tp results not only in the Pauli exclusion principle but also in the correlation of electrons of the same spin. 

A consequence of the Pauli exclusion principle is that electrons with parallel spins tend to avoid one another it is said that about an electron there is a Fermi hole which other electrons tend to avoid. In consequence, four electrons with parallel spins occupying the four sp% orbitals of an atom tend to assume relative positions > corresponding to the corners of a tetrahedron about the nucleus.16 Hence the carbon atom in the state s22s2pz S may be described as tetrahedral. The effect of correlation is to increase the tetrahedral character by the assumption by the orbitals of some d, f, character, which concentrates them about the tetrahedral directions. 

Antisymmetrized function (10.8) has the property that if any two one-electron functions are identical, then xp is identically zero (satisfying the Pauli exclusion principle). Its second very important property if any two electrons lie at the same position, e.g., ri = r2 (and they also have parallel spins Si = S2), then P = 0. As the functions

spatial variables (r,9,with parallel spin are close together. Thus, unlike the single product function, the antisymmetrized sum of product functions (10.8) shows a certain degree of electron correlation. This correlation is incomplete - it arises by virtue of the Pauli exclusion principle rather than as a result of electrostatic repulsion, and there is no correlation at all between two electrons with antiparallel spins [16]. 

FJectron correlations are intimately associated with two assumptions (1) a fourth quantum number, the electron-spin quantum number s, and ( ) the Pauli exclusion principle. In order to account for spectral data, it is necessary to postulate that electrons spin about their own axis to create a magnetic moment (G2o). Whereas the magnetic moment associated with the angular momentum may have (2Z + 1) components mi in the direction of an external magnetic field H, the spin moment may have only two components corresponding to s = ms = 1/2. Classically the magnitude of the moment fia associated with an angular momentum p is... 

Vcore(v) is the potential associated with the interaction between valence and core electrons and VexchangeC-v) is the exchange potential between valence electrons. The exchange potential accounts for the repulsive interaction between electrons of like spin (Pauli exclusion principle) and electrons of either spin (correlation interaction). Both of these potentials are experienced in the bulk as well as at the surface. Vdipoie(v) is specific to the surface and arises from charge redistribution at the asymmetric surface. 

It is concluded that total nonadditive effects are dominated by SCF terms, which are directly attributed to electric polarization. However, since the polarization is constrained by the Pauli exclusion principle, classical models which neglect exchange phenomena may incur certain errors, especially in regions of strong overlap between electron clouds of the monomers. Correlation contributions to nonadditivity do appear small enough that they can be safely ignored, with efforts better concentrated on an accurate portrayal of the SCF phenomena. 

One of the main reasons for the good results obtained with the Hartree-Fock SCF method in electronic structure calculations for atoms and molecules is that the electrons keep away from each other due to the Pauli exclusion principle. This reduces the correlation between them, and provides a basis for the validity of the independent-particle model. The question arises as to the mechanisms that account for the validity of the SCF approximation in the vibrational case, which are obviously quite unrelated to the Pauli principle. 

On the other hand, around each electron from a point r, other electrons are excluded to delimit a hole at a point f with the definition of a pxc(r, r ). According to Pauli s principle, this hole has to occlude only one missing electron. From the correlation, it follows... 

We may now list some of the simple physical conditions that the exact exchange-correlation hole satisfies. A common decomposition of the hole is into its separate exchange and correlation contributions. The exchange (or Fermi) hole is the hole due to the Pauli exclusion principle, and obeys the exact conditions ... 

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