Nucleus independent particle

At this point one needs to remember that electrons are not completely independent particles, moving randomly in a positively charged background. They are a charged species and as such, they also interact with the positive ions (nucleus)... 

In order to improve the theoretical description of a many-body system one has to take into consideration the so-called correlation effects, i.e. to deal with the problem of accounting for the departures from the simple independent particle model, in which the electrons are assumed to move independently of each other in an average field due to the atomic nucleus and the other electrons. Making an additional assumption that this average potential is spherically symmetric we arrive at the central field concept (Hartree-Fock model), which forms the basis of the atomic shell structure and the chemical regularity of the elements. Of course, relativistic effects must also be accounted for as corrections, if they are small, or already at the very beginning starting with the relativistic Hamiltonian and relativistic wave functions. 

The variational method of quantum chemistry for the determination of the energy has a direct analog in QMC. This is a consequence of the capability of the MC method to perform integration, and should not be confused with MC integration of the integrals that arise in basis set expansion methods. An important branch of QMC is the development of compact and accurate wave functions characterized by explicit dependence on interparticle distances electron-electron and electron-nucleus that are typically written as a product of an independent particle function and a correlation function. Such wave functions lead one immediately to the VMC method for evaluation [3-5]. Wave functions constructed following VMC can also serve as importance functions for the more accurate DMC variant of QMC. 

Recent years have seen a growing interest in the simultaneous description of electronic and nuclear motion. The nonadiabatic coupling between the electronic and nuclear motion manifest itself in numerous and rather diverse phenomena. An independent particle model can be formulated in which the averaged interactions between the electrons, between the electrons and the nuclei and between the nuclei are described quantum mechanically. Multicomponent MBPT can then be used to formulate the corresponding correlation problem accounting for electron-electron interactions, electron-nucleus interactions and nucleus-nucleus interactions in either algebraic or diagrammatic terms. 

The unperturbed Hamiltonian operator is based on an independent particle model, that is, a model in which each particle, nucleus or electron, experiences an averaged interaction with the other particles in the system. The unperturbed Hamiltonian operator is a sum of a kinetic energy term and an effective potential energy term... 

In this chapter we review the recent history of and evidence for collective, moleculelike behavior of valence electrons in atoms and indicate some of the questions that will have to be explored in order to resolve the question of how well the electrons in atoms are described by independent-particle or collective models. We then turn the question around and ask whether atoms in a molecule could, under suitable circumstances, display independent-particle behavior, with their own one-particle angular momenta behaving like nearconstants of the motion. The larger question that emerges is then one of whether few-body systems—the valence electrons of an atom, the atoms that constitute a small polyatomic molecule, and perhaps others such as the nucleons in a nucleus, all of which have heretofore seemed nearly unrelated— share characteristics to the extent that we can devise a unifying picture of the dynamics of few-body systems that will expose their commonalities as well as their obvious differences. 

In the nonrelativistic case V v has to account for all interactions of the valence electron i with the nucleus and the (removed) core electron system, i.e., at the independent particle level for Coulomb and exchange interaction as well as... 

For many-electron atoms, the Schrodinger equation can be solved approximately, but very accurately nowadays, by transforming the difficult electron-electron repulsion terms into a spherically averaged form. Then, each electron can be considered to exhibit independent motion in the potential field due to the nucleus and the averaged repulsive interaction with the other electrons in the atom. This simplification leads to the Independent Particle Approximation preserving the orbital concept and first proposed by Hartree (1) for the simplest many-electron atom, helium, as the product... 

Here rjj is the distance between the electrons i and j. As an approximation to (2.17) we assume that every electron moves independently of the other electrons in an average field, generated by the nucleus and the other electrons (the independent particle model). The field is assumed to be central (dependent only on r). This is the central-field approximation. The assumption of a central field combined with the Pauli exclusion principle results in a shell structure for the electrons and successively heavier elements can be constructed using the building-up principle (the total energy is minimized). The atom can be characterized by its electron configuration, e.g. for the lowest state of sodium we have... 

The shell model describes the nucleus as a system of independent particles coupled by a residual interaction. This residual interaction is generally complicated, but in the case of particles with the same spin j it takes a particularly simple form. Figure 15 shows the schematic level scheme of a pair of 1 9/2 protons, compared with the experimentally observed level scheme of Po, which, in the shell model, is described as two 1 9/2 protons outside a closed jj, general... 

The first term accounts for the energy of the independent particles making up the nucleus, the second term describes the interaction between two pairs of particles, and the third term is required to keep the particle number in the nucleus correct. Thus the problem must be solved under the condition that the expectation value of the particle number coincides with the number of particles in the nucleus. 

Daudel s approach of event probability experienced in 2002 a revival in a study of Savin [61]. In this study, the radii r of spheres around the atomic nucleus such that the probability F (r) to find n electrons outside the sphere is maximal were analyzed. The highest outermost probability marked the radius of core-valence separation with electron populations in good agreement with the periodic table. A comparison to the event probabilities valid for independent particles was used to enhance the probability information. 

In the Breit Hamiltonian in (3.2) we have omitted all terms which depend on spin variables of the heavy particle. As a result the corrections to the energy levels in (3.4) do not depend on the relative orientation of the spins of the heavy and light particles (in other words they do not describe hyperfine splitting). Moreover, almost all contributions in (3.4) are independent not only of the mutual orientation of spins of the heavy and light particles but also of the magnitude of the spin of the heavy particle. The only exception is the small contribution proportional to the term Sio, called the Darwin-Foldy contribution. This term arises in the matrix element of the Breit Hamiltonian only for the spin one-half nucleus and should be omitted for spinless or spin one nuclei. This contribution combines naturally with the nuclear size correction, and we postpone its discussion to Subsect. 6.1.2 dealing with the nuclear size contribution. 

A considerable amount of evidence indicates that nuclear forces are charge-independent, i.e, the neutron-neutron, neutron-proton, and proton-proton forces are identical. The meson theory of nuclear forces, originated by Yukawa, postulates the atomic nucleus being held together by an exchange force in which particles, now called mesons, are exchanged between individual nucleons within the nucleus. 

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