Pauli principle from antisymmetry

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. 

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. 

In Equation 6.16, K represent certain electronic configurations, and usually have the form of so-called Slater determinants that are in accord with the Pauli principle and the antisymmetry condition (the sign of K changes if the coordinates of two electrons are interchanged see Atkins [1983], Jensen [1998], Klessinger [1982], Levine [1991], Springborg [2000]). These //-electron Slater determinants are built from orbitals as one-electron functions t, the latter of which are multi-center molecular orbitals in the case of molecular orbital schemes (as opposed to, for example, valence-bond schemes where the one-electron functions are much more localized). 

Equation (1-180) can be obtained from a variational calculation of the energy of the dimer with a wave function that is product of determinantal wave functions of the monomers constructed from the optimal (selfconsistent) orbitals. Unfortunately, the optimization of the orbitals of one monomer in the electrostatic field of the other does not prevent the unphysical transfer of electrons from one system to the other, since the antisymmetry of the wave function of the dimer is not preserved, and consequently, the Pauli principle is not satisfied. This may lead to some unphysical results in the short range260. It should also be stressed that the interaction part of Eq. (1-180) cannot be obtained, as proposed in Ref. (261), by taking the expectation value of the interaction operator V with the product of determinantal wave functions of the monomers constructed from the optimal (selfconsistent) orbitals. This would result in an overcounting of the induction terms, already in the second order. 

In the development of the Slater method (Section 3.1) it was noted that the Pauli principle in the form (1.2.27) could always be satisfied by constructing the electronic wavefunction from determinants (i.e. antisymmetrized products) of spin-orbitals. In an earlier section, however, it was shown that for a two-electron system the antisymmetry principle could also be satisfied by writing the wavefunction as a product of individually symmetric or antisymmetric factors—one for spatial variables and the other for spin variables. Since, in the usual first approximation the Hamiltonian does not contain spin variables, it is natural to enquire whether a corresponding exact N-electron wavefunction might be written as a space-spin product in which the spatial factor is an exact eigenfunction of the spinless Hamiltonian (1.2.1). To investigate this possibility, we need a few basic ideas from group theory (Appendix 3). 

This theorem follows from the antisymmetry requirement (Eq. II.2) and is thus an expression for Pauli s exclusion principle. In the naive formulation of this principle, each spin orbital could be either empty or fully occupied by one electron which then would exclude any other electron from entering the same orbital. This simple model has been mathematically formulated in the Hartree-Fock scheme based on Eq. 11.38, where the form of the first-order density matrix p(x v xx) indicates that each one of the Hartree-Fock functions rplt y)2,. . ., pN is fully occupied by one electron. 

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... 

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 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 figure shows that, because two electrons with the same magnetic spin quantum number are forbidden from occupying the same position due to Pauli s principle, there is a repulsion between like-spin electrons known as the exchange repulsion, generating the so-called exchange (or Fermi) hole for the DC-DC (and also for the jS-jS) pair density this effect arises entirely from the antisymmetry of the wave function (see Section 2.11.3), and no such hole exists in the a-jS curve. ... 

From the form of Eq. (2.34), it is evident that the wave function vanishes if both electrons occupy the same spin orbital (i.e., if i = j). Thus the antisymmetry requirement immediately leads to the usual statement of the Pauli exclusion principle namely, that no more than one electron can occupy a spin orbital. 

Due to the antisymmetry requirement for the wave function (see Chapter 1), the holes have to satisfy some general (integral) conditions. The electrons with parallel spins have to avoid each other //j (ri,r2)dr2 = / xc (r, T2)dr2 = —1 (one electron disappears from the neighborhood of the other), while the electrons with opposite spins are not influenced by the Pauli exclusion principle /hxc (ri, r2)dr2 = /hxc (fi, r2)dr2 = 0. The exchange correlation hole is a sum of the exchange hole and the correlation hole hj" = -I- hg, ... 

So far, within the Born-Oppenheimer approximation, the theory is exact. However, we do not know the precise form of E [n] and V (r) = into which we have placed all the complicated many-body physics. As the name suggests, P [w] arises from a combination of two quantum mechanical effects electron exchange and correlation. Briefly, electron exchange arises because a many-body wave function must by antisymmetric under exchange of any two electrons since electrons are fermions. This antisymmetry of the wave function, which is simply a general expression of the Pauli exclusion principle, reduces the Coulomb energy of the electronic system by increasing the spatial separation between electrons of like... 

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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