Many-fermion system

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. 

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 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... 

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

REF80] G. Reffo, "Phenomenological Approach to Nuclear Level Densities," Theory and Applications of Moment Methods in Many-Fermion Systems (Plenum Press, New York, 1980). 

Moriond Meeting on Nucleosynthesis and its Implications to Nuclear and Particle Physics. Les Arcs, 1985 (Reidel) in press [WHI80] R.R.Whitehead, in Moment Method in Many Fermion System, eds. B.J. 

Nesbet, R.K. (1961). Approximate methods in the quantum theory of many-Fermion systems, Rev. Mod. Phys. 33, 28-36. 

Density Functional Theory of Many-Fermion Systems, Advances in Quantum... 

Now we turn to the topic of interest, that of the ground state of a many -Fermion system represented by a grand ensemble at chemical potential /i. The appropriate Rayleigh-Ritz principle for such a system is... 

Restricting the wave function by the form eq. (1.142) allows one to significantly reduce the calculation costs for all characteristics of a many-fermion system. Inserting eq. (1.142) into the energy expression (for the expectation value of the electronic Hamiltonian eq. (1.27)) and applying to it the variational principle with the additional condition of orthonormalization of the system of the occupied spin-orbitals 4>k (known in this context as molecular spin-orbitals) yields the system of integrodiffer-ential equations of the form (see e.g. [27]) ... 

In contrast to the regular review volumes with a rather eclectic content, the Editors tried instead in Volume 21 to focus interest on a single topic, in that case, on The Density-Functional Theory of Many-Fermion Systems. Since this experiment turned out very well, the Editors are now planning—in addition to the regular review volumes—a few more thematic volumes with contributions concentrating specifically on recent advances, and with specialists in the field as Guest Editors. 

J. Paldus, Diagrammatical Methods for Many-Fermion Systems. Lecture Notes, University of Nijmegen, 1981. 

S.B. Trickey spec, ed.. Density Fimctional Theory for Many-Fermion Systems, Adv. Quant. Chem. 21, Academic, San Diego, 1990. 

The field of atomic and molecular electronic structure was not the only one in which the correlation problem turned out to be of key importance. Indeed, this problem arises in any many-body quantum-mechanical system, particularly for many-fermion systems. For this very reason, much attention was devoted to it by solid-state and nuclear physicists alike. The important model systems that were intensely studied at that time were an electron gas in the former case and the model of infinite nuclear matter in the later case. 

For the sake of simplicity, let us consider the case where 4> = 1. (This would be directly applicable to the fermion sign problem [30], which arises in imaginary-time QMC sampling of many fermionic systems. Extension to general real-time QMC simulations where is a complex factor is straightforward.) In this case, = 1 always and the variance of the signal is controlled entirely by the size ( ). Under normal circumstances, ( ) can be very smalt, for the reasons that we have described earlier, and the simulation becomes unstable at long time. 

With the results obtained in this work, it becomes also possible to compare the nucleus with other many-fermion systems. This is schematically done in Fig. 7, where occupation numbers are shown for three representative systems. In the top part, occupation numbers for the Ne atom are shown. The first three sp levels. Is, 2s, and 2p, are basically full, whereas all the other orbitals have no occupation to speak of. This distribution reflects the relevance of the mean-field picture associated with a weak interaction between the particles which can be treated in HF approximation. This results in a jump in occupation of 1 at the Fermi energy. Occupation numbers for protons in Pb are shown in the middle part of Fig. 7. The occupation of the 3s 1/2 orbital (0.75) and the jump in occupation (0.65) correspond to experimental numbers [15,16], whereas the occupation of the deeply bound orbitals is inferred from the nuclear matter calculations discussed... 

For a quantum system with a given one particle density, FCr) is the only term which is sensitive to the nature (fermion or boson) of the particles. For a many-fermion system, T(r) czxi be formally expressed as the siun of two contributions, one of which accounting for the Pauli principle and the other not. However, another paatition scheme in which the total kinetic energy is written as the sum of the von Weizsacker term 7 pf (r)[26] and of a remaining non-von Weizsacker term T w r) term has been generally adopted[27, 28, 29, 30]. The von Weizsacker term ... 

This is a very useful expression for considering it associated with the mono-density operators when the many-fermionic systems are treated, although similar procedure applies for mixed (sample) states as well. There is immediate to see that for Af formally independent partitions the Hilbert space corresponding to the iV-mono-particle densities on pure states, we individually have, see Eqs. (4.176), (4.181), (4.182) and (4.184),... 

Maruhn J, Reinhard P, Suraud E (2010) Density functional theory. In Simple models of many-fermion systems. Springer, Berlin, pp 143-161... 

We have recently provided a strong evidence that the exact ground state of a many-fermion system, described by the Hamiltonian containing one- and two-body terms, may indeed be represented by the exponential cluster expansion employing a general two-body operator by connecting the problem with the Horn-Weinstein formula for the exact energy [152],... 

Wigner has put this statement in a more rigorous basis by saying that physically acceptable wavefunctions for microscopic systems must transform as irreducible representations (IR) of the symmetric (or permutation) group. As a consequence, the wavefunctions for quantum systems must be symmetric or antisymmetric under the permutation of any two identical particles of the system. For the case of many-fermion systems (particles with half-integer spin) the Pauh principle states that the wavefunctions must be antisymmetric. 

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