Fermions electronic wave functions

As pointed out in the previous paragraph, the total wave function of a molecule consists of an electronic and a nuclear parts. The electrons have a different intrinsic nature from nuclei, and hence can be treated separately when one considers the issue of permutational symmetry. First, let us consider the case of electrons. These are fermions with spin and hence the subsystem of electrons obeys the Fermi-Dirac statistics the total electronic wave function... 

To understand how Kohn and Sham tackled this problem, we go back to the discussion of the Hartree-Fock scheme in Chapter 1. There, our wave function was a single Slater determinant SD constructed from N spin orbitals. While the Slater determinant enters the HF method as the approximation to the true N-electron wave function, we showed in Section 1.3 that 4>sd can also be looked upon as the exact wave function of a fictitious system of N non-interacting electrons (that is electrons which behave as uncharged fermions and therefore do not interact with each other via Coulomb repulsion), moving in the elfective potential VHF. For this type of wave function the kinetic energy can be exactly expressed as... 

An important difference between the BO and non-BO internal Hamiltonians is that the former describes only the motion of electrons in the stationary field of nuclei positioned in fixed points in space (represented by point charges) while the latter describes the coupled motion of both nuclei and electrons. In the conventional molecular BO calculations, one typically uses atom-centered basis functions (in most calculations one-electron atomic orbitals) to expand the electronic wave function. The fermionic nature of the electrons dictates that such a function has to be antisymmetric with respect to the permutation of the labels of the electrons. In some high-precision BO calculations the wave function is expanded in terms of basis functions that explicitly depend on the interelectronic distances (so-called explicitly correlated functions). Such... 

Answer. Orbitals are one-electron wave functions, ). The fact that electrons are fermions requires that each electron be described by a different orbital. The simplest form of a many-electron wave function, T(l, 2,..., Ne), is a simple product of orbitals (a Hartree product), 1(1) 2(2) 3(3) NfNe). However, the fact that electrons are fermions also imposes the requirement that the many-electron wave function be antisymmetric toward the exchange of any two electrons. All of the physical requirements, including the indistinguishability of electrons, are met by a determinantal wave function, that is, an antisymmetrized sum of Hartree products, ( 1,2,3,..., Ne) = 1(1) 2(2) 3(3) ( ). If (1,2,3,...,Ne) is taken as an approximation of (1,2,..., Ne), i.e., the Hartree-Fock approximation, and the orbitals varied so as to minimize the energy expectation value,... 

The approximation techniques described in the earlier sections apply to any (non-relativistic) quantum system and can be universally used. On the other hand, the specific methods necessary for modeling molecular PES that refer explicitly to electronic wave function (or other possible tools mentioned above adjusted to describe electronic structure) are united under the name of quantum chemistry (QC).15 Quantum chemistry is different from other branches of theoretical physics in that it deals with systems of intermediate numbers of fermions - electrons, which preclude on the one hand the use of the infinite number limit - the number of electrons in a system is a sensitive parameter. This brings one to the position where it is necessary to consider wave functions dependent on spatial r and spin s variables of all N electrons entering the system. In other words, the wave functions sought by either version of the variational method or meant in the frame of either perturbational technique - the eigenfunctions of the electronic Hamiltonian in eq. (1.27) are the functions D(xi,..., xN) where. r, stands for the pair of the spatial radius vector of i-th electron and its spin projection s to a fixed axis. These latter, along with the... 

The program has been tested on several QED and QCD motivated examples such as electron g — 2 in QED, gauge independence of the electron wave function renormalization in QED, a relation between the pole and the MS mass in QCD, exponentiation of the infra-red asymptotic of the heavy fermion propagator. Many details concerning these checks and other technical aspects of our calculations can be found in Ref. [11] which the interested reader should consult. 

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... 

Consider a model system of four electrons moving in an arbitrary electrostatic field generated by the nuclei in a molecule. For our purposes, it is not necessary to specify the number of these nuclei, their types, or positions only the general form of the electronic wavefunction is of interest. It is convenient to describe the motions of each electron separately by assigning them to one-electron functions, (l),(xi), where Xi is a vector of the coordinates (including spin) of electron 1. In addition, electrons are fermions, so the electronic wave-function must be antisymmetric with respect to interchange of the coordinates of any pair of electrons. A traditional and very useful starting point for such a four-electron wavefunction is the so-called Slater determinant... 

Answer. Orbitals are one-electron wave functions, (1). The fact that electrons are fermions requires that each electron be described by a different orbital. The simplest form of a many-electron wave function, T(l, 2,..., Ne), is a simple product of orbitals (a Hartree product), (1) 2(2) 3 (3) However, the fact that electrons... 

The next term, EX, is positive for all the molecular systems of interest for liquids. The name makes reference to the exchange of electrons between A and B. This contribution to AE is sometimes called repulsion (REP) to emphasize the main effect this contribution describes. It is a true quantum mechanical effect, related to the antisymmetry of the electronic wave function of the dimer, or, if one prefers, to the Pauli exclusion principle. Actually these are two ways of expressing the same concept. Particles with a half integer value of the spin, like electrons, are subjected to the Pauli exclusion principle, which states that two particles of this type cannot be described by the same set of values of the characterizing parameters. Such particles are subjected to a special quantum version of the statistics, the Fermi-Dirac statistics, and they are called fermions. Identical fermions have to be described with an antisymmetric wave function the opposite also holds identical particles described by an... 

Let us see the probability density that two identical fermions occupy the same position in space and, additionally, that they have the same spin coordinate xi,yi,zi,o-i) = (X2,) 2,Z2,o-2). We have /r(l, 1,3,4,...,Ai) = - /r(l, 1,3,4,. ..,N), hence i/r(l, 1,3,4,..., Af) = 0 and, of course, (1,1,3,4,..., N) = 0. Conclusion two electrons of the same spin coordinate (we will sometimes say "of the same spin ) avoid each other. This is called the exchange or Fermi hole around each electron. The reason for the hole is the antisymmetry of the electronic wave function, or in other words, the Pauli exclusion principle. ... 

The concept of orbitals, occupied by electron pairs, exists only in the mean held method. We will leave this idea in the future, and the Pauli exclusion principle should survive as a postulate of the antisymmetry of the electronic wave function (more generally speaking, of the wave function of fermions). 

Since electrons are fermions, the Pauli principle requires the electronic wave function to be antisymmetric. Therefore, all electronic wave function (except for one and two electron ground states) have at least one positive and one negative domain. This sign change due to the fermionic nature of the electrons makes the construction of accurate wave functions much more demanding than the construction of bosonic states where the ground states are positive and symmetric. 

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