Fermions system

Since the total wave function must have the correct symmetry under the permutation of identical nuclei, we can determine the symmetiy of the rovi-bronic wave function from consideration of the corresponding symmetry of the nuclear spin function. We begin by looking at the case of a fermionic system for which the total wave function must be antisynmiebic under permutation of any two identical particles. If the nuclear spin function is symmetric then the rovibronic wave function must be antisymmetric conversely, if the nuclear spin function is antisymmebic, the rovibronic wave function must be symmetric under permutation of any two fermions. Similar considerations apply to bosonic systems The rovibronic wave function must be symmetric when the nuclear spin function is symmetric, and the rovibronic wave function must be antisymmetiic when the nuclear spin function is antisymmetric. This warrants... 

In this formulation, the electron density is expressed as a linear combination of basis functions similar in mathematical form to HF orbitals. A determinant is then formed from these functions, called Kohn-Sham orbitals. It is the electron density from this determinant of orbitals that is used to compute the energy. This procedure is necessary because Fermion systems can only have electron densities that arise from an antisymmetric wave function. There has been some debate over the interpretation of Kohn-Sham orbitals. It is certain that they are not mathematically equivalent to either HF orbitals or natural orbitals from correlated calculations. However, Kohn-Sham orbitals do describe the behavior of electrons in a molecule, just as the other orbitals mentioned do. DFT orbital eigenvalues do not match the energies obtained from photoelectron spectroscopy experiments as well as HF orbital energies do. The questions still being debated are how to assign similarities and how to physically interpret the differences. 

The most common description of relativistic quantum mechanics for Fermion systems, such as molecules, is the Dirac equation. The Dirac equation is a one-electron equation. In formulating this equation, the terms that arise are intrinsic electron spin, mass defect, spin couplings, and the Darwin term. The Darwin term can be viewed as the effect of an electron making a high-frequency oscillation around its mean position. 

The mixed and pure states of an A-particle fermion system can be described by positive and normalized operators,, which form a convex set... 

The behavior of a multi-particle system with a symmetric wave function differs markedly from the behavior of a system with an antisymmetric wave function. Particles with integral spin and therefore symmetric wave functions satisfy Bose-Einstein statistics and are called bosons, while particles with antisymmetric wave functions satisfy Fermi-Dirac statistics and are called fermions. Systems of " He atoms (helium-4) and of He atoms (helium-3) provide an excellent illustration. The " He atom is a boson with spin 0 because the spins of the two protons and the two neutrons in the nucleus and of the two electrons are paired. The He atom is a fermion with spin because the single neutron in the nucleus is unpaired. Because these two atoms obey different statistics, the thermodynamic and other macroscopic properties of liquid helium-4 and liquid helium-3 are dramatically different. 

Kim, S.P. and F.C. Khanna. TFD of Time Dependent Boson and Fermion Systems. quant-ph/0308053. 

One of the most amazing phenomena in quantum many-particle systems is the formation of quantum condensates. Of particular interest are strongly coupled fermion systems where bound states arise. In the low-density limit, where even-number fermionic bound states can be considered as bosons, Bose-Einstein condensation is expected to occur at low temperatures. The solution of Eq. (6) with = 2/j, gives the onset of pairing, the solution of Eq. (7) with EinP = 4/i the onset of quartetting in (symmetric) nuclear matter. At present, condensates are investigated in systems where the cross-over from Bardeen-Cooper-Schrieffer (BCS) pairing to Bose-Einstein condensation (BEC) can be observed, see [11,12], In these papers, a two-particle state is treated in an uncorrelated medium. Some attempts have been made to include the interaction between correlated states, see [7,13]. 

To construct the low energy effective theory of the fermionic system, we rewrite the fermion fields as... 

A systematic development of relativistic molecular Hamiltonians and various non-relativistic approximations are presented. Our starting point is the Dirac one-fermion Hamiltonian in the presence of an external electromagnetic field. The problems associated with generalizing Dirac s one-fermion theory smoothly to more than one fermion are discussed. The description of many-fermion systems within the framework of quantum electrodynamics (QED) will lead to Hamiltonians which do not suffer from the problems associated with the direct extension of Dirac s one-fermion theory to many-fermion system. An exhaustive discussion of the recent QED developments in the relevant area is not presented, except for cursory remarks for completeness. The non-relativistic form (NRF) of the many-electron relativistic Hamiltonian is developed as the working Hamiltonian. It is used to extract operators for the observables, which represent the response of a molecule to an external electromagnetic radiation field. In this study, our focus is mainly on the operators which eventually were used to calculate the nuclear magnetic resonance (NMR) chemical shifts and indirect nuclear spin-spin coupling constants. 

The original form of the DBC Hamiltonian was proposed for molecules. Here, we use it for general many-fermion systems. 

The first rigorous derivation of such a relativistic Hamiltonian for a two-fermion system that makes use of Feynman [13,14] formalism of QED was due to Bethe and Salpeter [30,31]. Recently, Broyles has extended it to many-eleetron atoms and molecules [32]. A detailed account of Broyle s derivation ean be found elsewhere [32,33] and will not be repeated here. Following Broyles, the stationary state many-fermion Hamiltonian based on QED ean be written as... 

We have focused on the prohlems associated with extending Dirac s one-fermion theory smoothly to many-fermion systems. A brief discussion of QED many-fermion Hamiltonians also was given. A comprehensive account of the problem of decoupling Dirac s four-component equation into two-component form and the serious drawbacks of the Pauli expansion were presented. The origins of the DSO and FC operators have been addressed. The working Hamiltonian which describe NMR spectra is derived. 

The Thomas-Fermi (TF) and related methods such as the Thomas-Fermi-Dirac (TFD) have played an important role in the study of complex fermionic systems due to their simplicity and statistical nature [1]. For atomic systems, they are able to provide some knowledge about general features such as the behaviour with the atomic number Z of different ground state properties [2,3]. 

Among the average properties which play a special role in the study of quantum fermionic systems are the radial expectation value (r ), the momentum expectation value (j9 ) and the atomic density at the nucleus p(0) = <5(r)>. These density-dependent quantities are defined by... 

The nonanalytic terms indicated in Eq. (5) are consequences of the singular logarithmic factors which contribute terms such as In(k ), In(k ) and so on to the long wavelength expansions of correlation functions describing interacting many-fermion systems in their ground states [7,8]. These terms are not... 

In 2001, Nakata and co-workers presented the results of realistic fermionic systems, like atoms and molecules, larger than previously reported for the variational calculation of the second-order reduced density matrix (2-RDM) [1]. 

Eq. (4)) [14—16], A key point here to understand the difference between these two formulations is that the dual SDP formulation (Eq. (4)) is not the dual of the primal SDP formulation (Eq. (1)). Both formulations produce two distinct pairs of primal and dual SDP problems, which mathematically describe the same fermionic system. Since their mathematical formulations differ, this implies differences in the computational effort to solve them. 

A g-density, pg, is non-N-representable if and only if there is no way to satisfy Eq. (2) without violating either the requirement that p, > 0 or the requirement that is antisymmetric. A non-A-representable g-density does not represent an A-fermion system. 

P5] S. B. Trickey, Ed., Density Functional Theory of Many-Fermion Systems, Adv. Quantum Chem. 21,1 05 (1990). 

In this paper a class of inequalities are derived for the elements of second-order reduced density matrices of fermion systems. 

We can word the results up to this point for an N-particle fermion system, using M-dimensional one-particle function basis, the elements of the second-order reduced density matrix in geminal basis are scalar products of ( ) piece of ( 2) dimensional vectors. 

Elements of second order reduced density matrix of a fermion system are written in geminal basis. Matrix elements are pointed out to be scalar product of special vectors. Based on elementary vector operations inequalities are formulated relating the density matrix elements. While the inequalities are based only on the features of scalar product, not the full information is exploited carried by the vectors D. Recently there are two object of research. The first is theoretical investigation of inequalities, reducibility of the large system of them. Further work may have the chance for reaching deeper insight of the so-called N-representability problem. The second object is a practical one examine the possibility of computational applications, associate conditions above with known methods and conditions for calculating density matrices. 

Figure 4.18 (a) Spectrum of the effective fermionic system that interacts with the bath modes k... 

Jayaprakash et al. (98, 99) presented an interacting TLK model, focusing on the effect of long- and short-range interactions between distinct facets and those between facets and curved surfaces. In this model the 2D statistics of steps is reduced to the 1D quantum fermion system, and the variation of the surface free energy (per projected area in the low-index plane) with angle 9 and temperature T is described by the equation... 

Orbital local-scaling transformation approach fermionic systems in the ground state 45... 

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