Pauli antisymmetry principle

The Pauli exclusion principle requires that no two electrons can occupy the same spin-orbital is a consequence of the more general Pauli antisymmetry principle ... 

The Pauli antisymmetry principle is a requirement a many-electron wavefunction must obey. A many-electron wavefunction must be antisymmetric (i.e. changes sign) to the interchange of the spatial and spin coordinates of any pair of electrons i and/, that is ... 

The Pauli antisymmetry principle tells us that the wave function (including spin degrees of freedom), and thus the basis functions, for a system of identical particles must transform like the totally antisymmetric irreducible representation in the case of fermions, or spin (for odd k) particles, and like the totally symmetric irreducible representation in the case of bosons, or spin k particles (where k may take on only integer values). 

If x = rs denotes the space-spin variable, we recall from first principles (Magnasco, 2007,2009a) that, for a normalized N-electronwavefunction satisfying the Pauli antisymmetry principle, the one-electron density function is defined as ... 

The most uniformly successful family of methods begins with the simplest possible n-electron wavefunction satisfying the Pauli antisymmetry principle - a Slater determinant [2] of one-electron functions % r.to) called spinorbitals. Each spinorbital is a product of a molecular orbital xpt(r) and a spinfunction a(to) or P(co). The V /.(r) are found by the self-consistent-field (SCF) procedure introduced [3] into quantum chemistry by Hartree. The Hartree-Fock (HF) [4] and Kohn-Sham density functional (KS) [5,6] theories are both of this type, as are their many simplified variants [7-16],... 

When this is done these sums of signed products form a non-redundant set for the expansion of any function of the coordinates of n electrons which satisfies the Pauli (antisymmetry) principle. 

Thus two electrons with the same spin have zero probability of being found at the same point in three-dimensional space. (By the same spin, we mean the same value of m,.) Since i/f is a continuous function, Eq. (10.20) means that the probability of finding two electrons with the same spin close to each other in space is quite small. Thus the Pauli antisymmetry principle forces electrons of like spin to keep apart fi-om one another to describe this, one often speaks of Pauli repulsion between such electrons. This repulsion is not a real physical force, but a reflection of the fact that the electronic wave function must be antisymmetric with respect to exchange. 

The atomic Hamiltonian (t of (11.1) (which omits spin-orbit interaction) does not involve spin and therefore commutes with the total-spin operators 5 and S. The fact that commutes with ft is not enough to show that the atomic wave functions are eigenfunctions of 5 The Pauli antisymmetry principle requires that each tp must be an eigenfunction of the exchange operator with eigenvalue —1 (Section 10.3). Hence must also commute with if we are to have simultaneous eigenfunctions of H, S, and ic. Problem 11.16 shows that [. , 4fc] = 0, so the atomic wave functions are eigenfunctions of We have = S S + l)feV each atomic state can be characterized by a total-electronic-spin quantum number S. 

Note that, because of the algebraic properties of the second-quantization strings and the Pauli antisymmetry principle is satisfied for the product wave function FCI wave function and energy of the compound system can now be expanded as... 

Bohm and Schiitt presented quantum Monte Carlo calculations for the r-systems of these compounds. Electronic degrees of freedom are restricted by two quantum constrains. The first is the Pauli antisymmetry principle (PAP) which requires that many electron wave functions must change sign when the ordering of two electrons with the same spin is changed. The second is the Pauli exclusion principle (PEP) that prevents conformations with more than one electron of the same spin in the same atomic orbital. They carried out two sets of calculations. First, the Pauli exclusion principle was retained but the Pauli antisymmetry principle was not required, and in the second both principles were applied. 

In second quantization, the Pauli antisymmetry principle is incorporated through the algebraic properties of the creation and annihilation operators as discussed in Chapter 1. We note that, in density-functional theory (which bypasses the construction of the wave function and concentrates on the electron density), the fulfilment of the A -representability condition on the density represents a less trivial problem. A density is said to be N-representable if it can be derived from an antisymmetric wave function for N particles [1]. 

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