Fermion antisymmetry

One particular advantage of Slater determinants constructed from orthonormsd spin-orbitals is that matrix elements between determinants over operators such as H sure very simple. Only three distinct cases arise, as is well known and treated elsewhere. It is perhaps not surprising that the simplest matrix element formulas should be obtained from the treatment that exploits symmetry the least, as only the fermion antisymmetry has been accounted for in the determinants. As more symmetry is introduced, the formulas become more complicated. On the other hand, the symmetry reduces the dimension of the problem more and more, because selection rules eliminate more terms. We consider here the spin adaptation of Slater determinants. 

The three quark color states are restricted to the color singlet, 1=1, which together with the fermion antisymmetry principle leads to requirement that the flavor-ordinary spin space must be totally symmetric i. e., I I I I This, in turn, leads to the following relationships between the flavor space and the ordinary spin space ... 

The electronic ground state (ri,..., rjv) is a function of 3N variables which satisfies the fermion antisymmetry property ... 

Strangely enough, the universe appears to be eomprised of only two kinds of paitieles, bosons and fermions. Bosons are symmetrical under exehange, and fermions are antisymmetrieal under exehange. This bit of abstiaet physies relates to our quantum moleeular problems beeause eleetions are femiions. 

In short, the Slater determinantal moleculai orbital and only the Slater determinantal moleculai orbital satisfies the two great generalizations of quantum chemistry, uncertainty (indistinguishability) and fermion exchange antisymmetry. 

For an electronic wavefunction, antisymmetry is a physical requirement following from the fact that electrons are fermions. It is essentially a requirement that y agree with the results of experimental physics. More specifically, this requirement means that any valid wavefunction must satisfy the following condition ... 

Fermions are particles that have the properties of antisymmetry and a half-integral spin quantum number, among others. 

The orbital phase continuity conditions stem from the intrinsic property of electrons. Electrons are fermions, and are described by wavefnnctions antisymmetric (change plus and minus signs) with respect to an interchange of the coordinates of an pair of particles. The antisymmetry principle is a more fnndamental principle than Pauli s exclusion principle. Slater determinants are antisymmetric, which is why the overlap integral between t(a c) given above has a negative... 

However, billiard balls are a pretty bad model for electrons. First of all, as discussed above, electrons are fermions and therefore have an antisymmetric wave function. Second, they are charged particles and interact through the Coulomb repulsion they try to stay away from each other as much as possible. Both of these properties heavily influence the pair density and we will now enter an in-depth discussion of these effects. Let us begin with an exposition of the consequences of the antisymmetry of the wave function. This is most easily done if we introduce the concept of the reduced density matrix for two electrons, which we call y2. This is a simple generalization of p2(x1 x2) given above according to... 

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. 

Let us explore first the nature of the integrand E cl for the limiting cases. At X = 0 we are dealing with an interaction free system, and the only component which is not included in the classical term is due to the antisymmetry of the fermion wave function. Thus, E j° is composed of exchange only, there is no correlation whatsoever.19 Hence, the X = 0 limit of the integral in equation (6-25) simply corresponds to the exchange contribution of a Slater determinant, as for example, expressed through equation (5-18). Remember, that E ° can... 

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). 

Three distinct sets of linear mappings for the partial 3-positivity matrices in Eqs. (31)-(36) are important (i) the contraction mappings, which relate the lifted metric matrices to the 2-positive matrices in Eqs. (27)-(29) (ii) the linear interconversion mappings from rearranging creation and annihilation operators to interrelate the lifted metric matrices and (iii) antisymmetry (or symmetry) conditions, which enforce the permutation of the creation operators for fermions (or bosons). Note that the correct permutation of the annihilation operators is automatically enforced from the permutation of the creation operators in (iii) by the Hermiticity of the matrices. 

Electronic Wavefuntions Must be Constructed to Have Permutational Antisymmetry Because the N Electrons are Indistinguishable Fermions... 

BOSONS. Those elementary particles for which there is symmetry under intra-pair production. They obey Bose-Einstein statistics. Included are photons, pi mesons, and nuclei with an even number of particles. (Those particles for which there is antisymmetry fermions.) See Mesons Particles (Subatomic) and Photon and Photonics. 

PROBLEM 5.1.1. Given the requirement of overall antisymmetry, show that for a system of identical fermions each fermion must have its unique set of quantum number values, different from the values adopted by any other fermion. 

The total electronic wave function must be antisynunetric (change sign) with respect to the interchange of any two electron coordinates (since electrons are fermions, having a spin of 1/2). The Pauli principle which states that two electrons cannot have all quantum numbers equal, is a direct consequence of this antisymmetry... 

The commutation rules (3.132,3.136) are combined into a set of rules that constitute the algebra for ensuring antisymmetry in a many-fermion problem. 

The formation of molecules in a model with only Coulomb forces does of course rely strongly on the presence of both positive and negative particles, but the approximations made here have shifted all the non-trivial calculations to the electron treatment in (3.36). Because all electrons are fermion identical particles, quantum theory makes additional formal requirements of antisymmetry on their wavefunctions, which will be discussed below. 

SO that many types of solutions combining all the pair behaviours (symmetry or antisymmetry) are possible. However Pauh s exclusion principle states that the eigenstates which are relevant when dealing with electrons (i.e., fermions... 

The origin of the nonvauishing Joule-Thomson effect is the effective repulsive (Fermions) and attractive (Bosons) potential exerted on the gas molecules, which arises from the different ways in which quantum states can be occupied in sy.stems obeying Fermi-Dirac and Boso-Einstein statistics, respectively [17]. In other words, the effective fields are a consequence of whether Pauli s antisymmetry principle, which is relativistic in nature [207], is applicable. Thus, a weakly degenerate Fermi gas will always heat up ((5 < 0), whereas a weakly degenerate Bose gas will cool down (5 > 0) during a Joule-Thomson expansion. These conclusions remain valid even if the ideal quantum gas is treated relativistically, which is required to understand... 

The corresponding functions X , Xy then define what are known as the normal coordinates of vibration, and the Hamiltonian can be written in terms of these in precisely the form given by equation (A 1.1.691. with the caveat that each term refers not to the coordinates of a single particle, but rather to independent coordinates that involve the collective motion of many particles. An additional distinction is that treatment of the vibrational problem does not involve the complications of antisymmetry associated with identical fermions and the Pauli exclusion principle. Products of the normal coordinate functions nevertheless describe all vibrational states of the molecule (both ground and excited) in very much the same way that the product states of single-electron functions describe the electronic states, although it must be emphasized that one model is based on independent motion and the other on collective motion, which are qualitatively very different. Neither model faithfully represents reality, but each serves as an extremely useful conceptual model and a basis for more accurate calculations. 

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