Two-particle electrons

The electron- and spin-densities are the only building blocks of a much more powerful theory the theory of reduced density matrices. Such one-particle, two-particle,. .. electron- and spin-density matrices can be defined for any type of wavefunction, no matter whether it is of the HF type, another approximation, or even the exact wavefunction. A detailed description here would be inappropriate... 

Another result we can anticipate from electronic structure is the possible values for the total nuclear spin quantum number. For instance, the deuteron (a proton and a neutron) consists of two particles with an intrinsic spin of 1/2. Were these two particles electrons, we would know that the possible values for the total spin quantum number are 1 and 0. It turns out for the deuteron that of the two coupling possibilities, 1=1 and 1 = 0, the 1=1 spin coupling occurs for the ground state. This is a consequence of the interactions that dictate nuclear structure, which is outside the scope of this chapter. For chemical applications, the key information is simply that the deuteron is an / = 1 particle. 

Electrostatics is the study of interactions between charged objects. Electrostatics alone will not described molecular systems, but it is very important to the understanding of interactions of electrons, which is described by a wave function or electron density. The central pillar of electrostatics is Coulombs law, which is the mathematical description of how like charges repel and unlike charges attract. The Coulombs law equations for energy and the force of interaction between two particles with charges q and q2 at a distance rn are... 

Liquid Helium-4. Quantum mechanics defines two fundamentally different types of particles bosons, which have no unpaired quantum spins, and fermions, which do have unpaired spins. Bosons are governed by Bose-Einstein statistics which, at sufficiently low temperatures, allow the particles to coUect into a low energy quantum level, the so-called Bose-Einstein condensation. Fermions, which include electrons, protons, and neutrons, are governed by Fermi-DHac statistics which forbid any two particles to occupy exactly the same quantum state and thus forbid any analogue of Bose-Einstein condensation. Atoms may be thought of as assembHes of fermions only, but can behave as either fermions or bosons. If the total number of electrons, protons, and neutrons is odd, the atom is a fermion if it is even, the atom is a boson. 

Pauli proposed that two particles were emitted, and Fermi called the second one a neutrino, V. The complete process therefore is n — p -H e 9. Owing to the low probabiHty of its interacting with other particles, the neutrino was not observed until 1959. Before the j3 -decay takes place there are no free leptons, so the conservation of leptons requires that there be a net of 2ero leptons afterward. Therefore, the associated neutrino is designated an antineutrino, 9-, that is, the emitted electron (lepton) and antineutrino (antilepton) cancel and give a net of 2ero leptons. 

As the distance between the two particles varies, they are subject to these long-range r " attractive forces (which some authors refer to collectively as van der Waals forces). Upon very close approach they will experience a repulsive force due to electron-electron repulsion. This repulsive interaction is not theoretically well characterized, and it is usually approximated by an empirical reciprocal power of distance of separation. The net potential energy is then a balance of the attractive and repulsive components, often described by Eq. (8-16), the Lennard-Jones 6-12 potential. 

Corrections involving nuclei (with the nuclear spin I replacing the electron spin s) are analogous to the above one- and two-particle terms in eqs. (8.29-8.30), with the exception of those involving the nuclear mass, which disappears in the Bom-Oppenheimer approximation (which may be be considered as the Mnucieus oo limit). 

The notion of electrons in orbitals consists essentially of ascribing four distinct quantum numbers to each electron in a many-electron atom. It can be shown that this notion is strictly inconsistent with quantum mechanics (7). Definite quantum numbers for individual electrons do not have any meaning in the framework of quantum mechanics. The erroneous view stems from the original formulation of the Pauli principle in 1925, which stated that no two electrons could share the same four quantum numbers (8), This version of the principle was superseded by a new formulation that avoids any reference to individual quantum numbers for separate electrons. The new version due to the independent work of Heisenberg and Dirac in 1926 states that the wave function of a many-electron atom must be antisymmetrical with respect to the interchange of any two particles (9,10). 

Pauli s original version of the exclusion principle was found lacking precisely because it ascribes stationary states to individual electrons. According to the new quantum mechanics, only the atomic system as a whole possesses stationary states. The original version of the exclusion principle was replaced by the statement that the wavefunction for a system of fermions must be antisymmetrical with respect to the interchange of any two particles (Heisenberg [1925], Dirac [1928]). 

In 1926 Llewellyn Thomas proposed treating the electrons in an atom by analogy to a statistical gas of particles. Electron-shells are not envisaged in this model, which was independently rediscovered by Enrico Fermi two years later. For many years the Thomas-Fermi method was regarded as a mathematical curiosity without much hope of application since the results it yielded were inferior to those obtained by the method based on electron orbitals.17... 

The idea of constructing a good wave function of a many-particle system by means of an exact treatment of the two-particle correlation is also underlying the methods recently developed by Brueck-ner and his collaborators for studying nuclei and free-electron systems. The effective two-particle reaction operator and the self-consistency conditions introduced in this connection may be considered as generalizations of the Hartree-Fock scheme. 

If a charge exchange process, A + + B- A -f- B +, occurs when the distance between the two particles is large, we expect that no transfer of translational energy takes place in the reaction and that the same selection rules govern the ionization as in spectroscopic transitions. This means that if the molecule B is in a singlet state before the ionization, the ion B + will be formed in a doublet state after ionization of one electron without rearrangements of any other electrons, at least for small molecules. 

The interactions between electrons are inherently many-body forces. There are several methods in common use today which try to incorporate some, or all, of the many-body quantum mechanical effects. An important term is that of electronic exchange [57, 58]. Mathematically, when two particles in the many-body wavefunction are exchanged the wavefunction changes sign ... 

Positrons cannot be observed directly because, as Figure 22-6a illustrates, when a positron encounters an electron, the two particles annihilate each other, converting their entire mass into a pair of photons. The occurrence of positron emission can be inferred from the observation of such a pair of photons. Each photon produced in this process has a specific energy Epi ton = 9.87 X lO kJ/mol. Photons with such high energy are called y rays. 

We may express the single-particle wave function tpniqd fhe product of a spatial wave function 0n(r,) and a spin function % i). For a fermion with spin such as an electron, there are just two spin states, which we designate by a(i) for m = and f i) for Therefore, for two particles there are three... 

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