Independent particle models

A more detailed model can be constructed for the nucleons in terms of a central potential that holds all the nucleons together plus a residual potential or residual interaction that lumps together all of the other nucleon-nucleon interactions. Other such important one-on-one interactions align the spins of unlike nucleons (p-n) and cause the pairing of like nucleons (p-p, n-n). The nucleons are then allowed to move independently in these potentials, that is, the Schrodinger equation is solved for the 

The central potential can be a simple harmonic oscillator potential/(r) kr2 or more complicated such as a Yukawa function f(r) (e a,/r) 1 or the Woods-Saxon function that has a flat bottom and goes smoothly to zero at the nuclear surface. The Woods-Saxon potential has the form 

A residual interaction that is also quite simple has been developed and applied with good results. Recall that the nucleon-nucleon force is attractive and very short ranged, so one might image that the nucleons must be in contact to interact. Thus, the simplest residual interaction is an attractive force that only acts when the nucleons touch or a 8 interaction (in the sense of a Kronecker 8 from quantum mechanics). This can be written as V(r, r%) = a where a is the strength of the interaction, and the 8 function only allows the force to be positive when the nucleons are at exactly the same point in space. In practice, the strength of the potential must be determined by comparison to experimental data. Notice, 

As we have seen, the nucleons reside in well-defined orbitals in the nucleus that can be understood in a relatively simple quantum mechanical model, the shell model. In this model, the properties of the nucleus are dominated by the wave functions of the one or two unpaired nucleons. Notice that the bulk of the nucleons, which may even number in the hundreds, only contribute to the overall central potential. These core nucleons cannot be ignored in reality and they give rise to large-scale, macroscopic behavior of the nucleus that is very different from the behavior of single particles. There are two important collective motions of the nucleus that we have already mentioned that we should address collective or overall rotation of deformed nuclei and vibrations of the nuclear shape about a spherical ground-state shape. 

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In Section II.C we gave a general discussion of the Coulomb correlation, and we will now define the correlation error in the independent-particle model in greater detail. It is convenient to study the first- and second-order density matrices and, according to the definitions (Eq. II.9) applied to the symmetryless case, we obtain... 

The general idea of using different orbitals for different spins" seems thus to render an important extension of the entire framework of the independent-particle model. There seem to be essential physical reasons for a comparatively large orbital splitting depending on correlation, since electrons with opposite spins try to avoid each other because of their mutual Coulomb repulsion, and, in systems with unbalanced spins, there may further exist an extra exchange polarization of the type emphasized by Slater. 

The system under study is assumed to consist of 2A, electrons, possibly in the presence of a nuclear framework. An orbital picture of the quantum behaviour of the system is then introduced on accepting the validity of an independent-particle model where each electron moves in the field of an effective potential v(r), which afterwards is left substantially unspecified. We emphasize, however, that the choice of v(F) is an essential step of any modeling. Besides semiempirical forms, effective potentials v[ (r)] functionally dependent on the electron numeral density n(r) are intuitively bound to play a prominent role in applications. 

The most widely used qualitative model for the explanation of the shapes of molecules is the Valence Shell Electron Pair Repulsion (VSEPR) model of Gillespie and Nyholm (25). The orbital correlation diagrams of Walsh (26) are also used for simple systems for which the qualitative form of the MOs may be deduced from symmetry considerations. Attempts have been made to prove that these two approaches are equivalent (27). But this is impossible since Walsh s Rules refer explicitly to (and only have meaning within) the MO model while the VSEPR method does not refer to (is not confined by) any explicitly-stated model of molecular electronic structure. Thus, any proof that the two approaches are equivalent can only prove, at best, that the two are equivalent at the MO level i.e. that Walsh s Rules are contained in the VSEPR model. Of course, the transformation to localised orbitals of an MO determinant provides a convenient picture of VSEPR rules but the VSEPR method itself depends not on the independent-particle model but on the possibility of separating the total electronic structure of a molecule into more or less autonomous electron pairs which interact as separate entities (28). The localised MO description is merely the simplest such separation the general case is our Eq. (6)... 

Fig. 2. Second-order contributions to intermolecular perturbation energies (schematic description by orbital excitations within the framework of the independent particle model AEpol and zI-Echt are represented by single excitations, AEms by correlated double excitations)...

The various methods used in quantum chemistry make it possible to compute equilibrium intermolecular distances, to describe intermolecular forces and chemical reactions too. The usual way to calculate these properties is based on the independent particle model this is the Hartree-Fock method. The expansion of one-electron wave-functions (molecular orbitals) in practice requires technical work on computers. It was believed for years and years that ab initio computations will become a routine task even for large molecules. In spite of the enormous increase and development in computer technique, however, this expectation has not been fulfilled. The treatment of large, extended molecular systems still needs special theoretical background. In other words, some approximations should be used in the methods which describe the properties of molecules of large size and/or interacting systems. The further approximations are to be chosen carefully this caution is especially important when going beyond the HF level. The inclusion of the electron correlation in the calculations in a convenient way is still one of the most significant tasks of quantum chemistry. 

Independent particle model and electron correlation expansion... 

In section 2, we provide a description of the methods employed in the present study the generation of Gaussian-type basis sets, the independent particle model and the treatment of electron correlation effects, and, the computational details. Results are presented and discussed in section 3. Section 4 contains our conclusions. 

J. L. Stuber and J. Paldus, Symmetry Breaking in the Independent Particle Model. In E. J. Brandas and E. S. Kryachko (Eds.) Fundamental World of Quantum Chemistry, A Tribute Volume to the Memory of Per-Olov Lowdin, Vol. 1. (Kluwer, Dordrecht, 2003), pp. 67-139. 

The major difficulty in wave function based calculations is that, starting from an independent-particle model, correlation between electrons of opposite spin must somehow be introduced into T. Inclusion of this type of electron correlation is essential if energies are to be computed with any degree of accuracy. How, through the use of multiconfigurational wave functions, correlation between electrons of opposite spin is incorporated into is the subject of Section 3.2.3. 

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. 

As was mentioned earlier, model function xpo corresponds to the zero-order approximation and the remaining part, Qxp, can be considered as a correction . If xpo is generated in an independent particle model (e.g., Hartree-Fock), then Qxp is often referred to as the correlation function [75, 76]. 

Applying Eq. (45) to the Hartree-Fock independent particle model we immediately... 

Since his appointment at the University of Waterloo, Paldus has fully devoted himself to theoretical and methodological aspects of atomic and molecular electronic structure, while keeping in close contact with actual applications of these methods in computational quantum chemistry. His contributions include the examination of stability conditions and symmetry breaking in the independent particle models,109 many-body perturbation theory and Green s function approaches to the many-electron correlation problem,110 the development of graphical methods for the time-independent many-fermion problem,111 and the development of various algebraic approaches and an exploration of convergence properties of perturbative methods. His most important... 

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