A self-consistent procedure is presented for the determination of the properties of a many-fermion system, taking into account all two-particle correlations. We consider a system of $N$ fermions, interacting through two-body forces and write the wave function in the form, $\ensuremath{\Psi}={\ensuremath{\Psi}}_{0}+\ensuremath{\Sigma}{\mathrm{ij}}^{}{f}^{(2)}(\mathrm{ij})$, where ${\ensuremath{\Psi}}_{0}$ is a determinant of one-particle wave functions and ${f}^{(2)}(\mathrm{ij})$ is an antisymmetrized product of ($N\ensuremath{-}2$) one-particle functions and of one two-particle function. By introducing two-particle functions for each electron pair, all two-particle correlations are taken into account. It is shown that for the best one- and two-particle functions a system of coupled integrodifferential equations can be derived. These equations are derived by varying the expectation value of the Hamiltonian with respect to the one- and two-particle functions, taking into account the normalization and orthogonality as subsidiary conditions. After eliminating the Lagrangian multipliers, we have obtained the following result. We obtained $N$ one-particle equations for the $N$ one-particle wave functions and one-particle orbital parameters. These equations are characterized by a potential (and exchange operator) in which, besides the Hartree-Fock type potential terms, there are also the potentials arising from the two-particle functions, where the latter occur in diagonal, as well as in nondiagonal form. For the two-particle functions and the orbital parameters associated with them, we have obtained two-particle equations in which the equation for the function ${\ensuremath{\varphi}}_{\mathrm{ij}}$ contains all one-particle functions and all the other two-particle functions. It is shown that the system of coupled one- and two-particle equations can be solved with a self-consistent procedure. The method can be applied to systems with any number of particles.A self-consistent procedure is presented for the determination of the properties of a many-fermion system, taking into account all two-particle correlations. We consider a system of $N$ fermions, interacting through two-body forces and write the wave function in the form, $\ensuremath{\Psi}={\ensuremath{\Psi}}_{0}+\ensuremath{\Sigma}{\mathrm{ij}}^{}{f}^{(2)}(\mathrm{ij})$, where ${\ensuremath{\Psi}}_{0}$ is a determinant of one-particle wave functions and ${f}^{(2)}(\mathrm{ij})$ is an antisymmetrized product of ($N\ensuremath{-}2$) one-particle functions and of one two-particle function. By introducing two-particle functions for each electron pair, all two-particle correlations are taken into account. It is shown that for the best one- and two-particle functions a system of coupled integrodifferential equations can be derived. These equations are derived by varying the expectation value of the Hamiltonian with respect to the one- and two-particle functions, taking into account the normalization and orthogonality as subsidiary conditions. After eliminating the Lagrangian multipliers, we have obtained the following result. We obtained $N$ one-particle equations for the $N$ one-particle wave functions and one-particle orbital parameters. These equations are characterized by a potential (and exchange operator) in which, besides the Hartree-Fock type potential terms, there are also the potentials arising from the two-particle functions, where the latter occur in diagonal, as well as in nondiagonal form. For the two-particle functions and the orbital parameters associated with them, we have obtained two-particle equations in which the equation for the function ${\ensuremath{\varphi}}_{\mathrm{ij}}$ contains all one-particle functions and all the other two-particle functions. It is shown that the system of coupled one- and two-particle equations can be solved with a self-consistent procedure. The method can be applied to systems with any number of particles.

Physical ReviewOct 15, 196338 citations

Density functional theory of one-dimensional two-particle systems

H. L. Neal

We consider the density functional theory from the point of view of one-dimensional two-particle systems, where the particles are either spin-1/2 fermions or spin-0 bosons. The basic equations are derived and applied to the system of two coupled harmonic oscillators. An exact density functional solution is constructed from the exactly known single-particle density for this system. We demonstrate how this solution may be applied to other two-particle systems to calculate the exact single-particle density and energy.We consider the density functional theory from the point of view of one-dimensional two-particle systems, where the particles are either spin-1/2 fermions or spin-0 bosons. The basic equations are derived and applied to the system of two coupled harmonic oscillators. An exact density functional solution is constructed from the exactly known single-particle density for this system. We demonstrate how this solution may be applied to other two-particle systems to calculate the exact single-particle density and energy.

American Journal of PhysicsJun 1, 199818 citations

Many-Particle Systems

MASSIMO RUDAN

Springer eBooksSep 27, 20170 citations

Many-Particle Systems

Wolfgang Nolting

Springer eBooksJan 1, 20170 citations

Effect of the relativistic spin rotation on two-particle spin composition

Richard Lednicky, Vladimir Lvovich Lyuboshitz, Vladimir Lvovich Lyuboshitz

The effect of the relativistic spin rotation on two-particle spin states, conditioned by the setting of the spins of the particles in their rest frames and by the noncommutativity of the Lorentz transformations along noncollinear directions, is discussed. Particularly, the transition from the center-of-mass system (c.m.s.) of two spin-1∕2 particles to the laboratory is considered. When the vectors of the c.m.s. particle velocities are not collinear with the velocity vector of the c.m.s., the angles of the relativistic spin rotation for the two particles are different. As a result, the relative fractions of the singlet and triplet states in the relativistic system of two spin-1∕2 particles with a nonzero vector of relative momentum depend on the concrete frame in which the two-particle system is analyzed.The effect of the relativistic spin rotation on two-particle spin states, conditioned by the setting of the spins of the particles in their rest frames and by the noncommutativity of the Lorentz transformations along noncollinear directions, is discussed. Particularly, the transition from the center-of-mass system (c.m.s.) of two spin-1∕2 particles to the laboratory is considered. When the vectors of the c.m.s. particle velocities are not collinear with the velocity vector of the c.m.s., the angles of the relativistic spin rotation for the two particles are different. As a result, the relative fractions of the singlet and triplet states in the relativistic system of two spin-1∕2 particles with a nonzero vector of relative momentum depend on the concrete frame in which the two-particle system is analyzed.

arXiv:hep-ph/0302136Physical Review AMar 15, 20042 citations

A self-consistent procedure is presented for the determination of the properties of a many-fermion system, taking into account all two-particle correlations. We consider a system of

N

fermions, interacting through two-body forces and write the wave function in the form, $\ensuremath{\Psi}={\ensuremath{\Psi}}_{0}+\ensuremath{\Sigma}{\mathrm{ij}}^{}{f}^{(2)}(\mathrm{ij})$, where ${\ensuremath{\Psi}}_{0}$ is a determinant of one-particle wave functions and

{f}^{(2)}(\mathrm{ij})

is an antisymmetrized product of ($N\ensuremath{-}2$) one-particle functions and of one two-particle function. By introducing two-particle functions for each electron pair, all two-particle correlations are taken into account. It is shown that for the best one- and two-particle functions a system of coupled integrodifferential equations can be derived. These equations are derived by varying the expectation value of the Hamiltonian with respect to the one- and two-particle functions, taking into account the normalization and orthogonality as subsidiary conditions. After eliminating the Lagrangian multipliers, we have obtained the following result. We obtained

N

one-particle equations for the

N

one-particle wave functions and one-particle orbital parameters. These equations are characterized by a potential (and exchange operator) in which, besides the Hartree-Fock type potential terms, there are also the potentials arising from the two-particle functions, where the latter occur in diagonal, as well as in nondiagonal form. For the two-particle functions and the orbital parameters associated with them, we have obtained two-particle equations in which the equation for the function ${\ensuremath{\varphi}}_{\mathrm{ij}}$ contains all one-particle functions and all the other two-particle functions. It is shown that the system of coupled one- and two-particle equations can be solved with a self-consistent procedure. The method can be applied to systems with any number of particles.