One-body Potential Research Articles

Existence of a one-body barrier revealed in deep subbarrier fusion

Based on the adiabatic picture for heavy-ion reactions, in which the neck formation in the one-body system is taken into account, we propose a two-step model for fusion cross sections at deep sub-barrier energies. This model consists of the capture process in the two-body potential pocket, which is followed by the penetration of the adiabatic one-body potential to reach a compound state after the touching configuration. We describe the former process with the coupled-channels framework, while the latter with the Wentzel-Kramers-Brillouin (WKB) approximation by taking into account the coordinate dependent inertia mass. The effect of the one-body barrier is important at incident energies below the potential energy at the touching configuration. We show that this model well accounts for the steep fall-off phenomenon of fusion cross sections at deep sub-barrier energies for the $^{64}\mathrm{Ni}$+$^{64}\mathrm{Ni}$ and $^{58}\mathrm{Ni}$+$^{58}\mathrm{Ni}$ reactions.

Determinant quantum Monte Carlo study of the screening of the one-body potential near a metal-insulator transition

In this paper we present a determinant quantum monte carlo study of the two dimensional Hubbard model with random site disorder. We show that, as in the case of bond disorder, the system undergoes a transition from an Anderson insulating phase to a metallic phase as the onsite repulsion U is increased beyond a critical value U_c. However, there appears to be no sharp signal of this metal-insulator transition in the screened site energies. We observe that, while the system remains metallic for interaction values upto twice U_c, the conductivity is maximal in the metallic phase just beyond U_c, and decreases for larger correlation.

Correlated analytic ground-state densities for two-electron atomic ions from Be to Ne

Abstract In earlier work, we demonstrated that the three-parameter correlated variational ground-state wave function of Chandrasekhar for two-electron atomic ions led to a closed analytic ground-state electron density ρ ( r ) . Here, we determine the three parameters by imposing (i) Kato's cusp condition at the nucleus, (ii) the correct large-r behavior of ρ ( r ) involving the ionization potential, and (iii) the electrostatic potential created by ρ ( r ) at the nucleus, as given by the 1 / Z expansion of atomic theory. It is then found, from results obtained for Be through Ne, that the electron–electron coalescence condition is increasingly well accounted for as the atomic number Z increases. Finally, the one-body potential of density functional theory is extracted from ρ ( r ) .

Exchange force for two-level systems such as LiH and [formula omitted

Using the Dawson–March transformation, the Dirac density matrix γ( r, r′) can be written in terms of the density amplitude ρ ( r ) and a phase θ( r), where ρ( r) ≡ γ( r, r) is the ground-state electron density. Here, for systems such as LiH or H 3 - , it is shown how the force −∂ V( r)/∂ r corresponding to the one-body potential V( r) can be written, given the ground-state density ρ( r) from a high-level ab initio calculation or a diffraction experiment. The Hartree–Fock ground-state densities for LiH and H 3 - are utilized to calculate approximate phase angles θ( r). Finally, for H 3 - the force −∂ V( r)/∂ r corresponding to V( r) is calculated.

An ab initio study of the fcc and hcp structures of helium

The hexagonal close packed (hcp) and face centered cubic (fcc) structures of helium are studied by using a new ab initio computational model for large complexes comprising small subsystems. The new model is formulated within the framework of the energy incremental scheme. In the calculation of intra- and intersystem energies, model systems are introduced. To each subsystem associated is a set of partner subsystems defined by a vicinity criterion. In the independent calculations of intra- and intersystem energies, the calculations are performed on model subsystems defined by the subsystems considered and their partner subsystems. A small and a large basis set are associated with each subsystem. For partner subsystems in a model system, the small basis set is adopted. By introducing a particular decomposition scheme, the intermolecular potential is written as a sum of effective one-body potentials. The binding energy per atom in an infinite crystal of atoms is the negative value of this one-body potential. The one-body potentials for hcp and fcc structures are calculated for the following nearest neighbor distances (d0): 4.6, 5.1, 5.4, 5.435, 5.5, 5.61, and 6.1 a.u. The equilibrium distance is 5.44 a.u. for both structures. The equilibrium dimer distance is 5.61 a.u. For the larger distances, i.e., d0 > 5.4 a.u., the difference of the effective one-body potentials for the two structures is less than 0.2 microE(h). However, the hcp structure has the lowest effective one-body potential for all the distances considered. For the smallest distance the difference in the effective one-body potential is 3.9 microE(h). Hence, for solid helium, i.e., helium under high pressure, the hcp structure is the preferred one. The error in the calculated effective one-body potential for the distance d0 = 5.61 a.u. is of the order of 1 microE(h) (approximately 0.5%).

Exchange-correlation potential in terms of the idempotent Dirac density matrix of DFT

A formal proof has recently been proposed [I.A. Howard, N.H. March, Chem. Phys. Lett. 402 (2005) 1.] to show that the exchange-correlation potential V xc( r) of density functional theory (DFT) is, in principle, determined by the idempotent Dirac density matrix γ( r, r′) generated by the one-body potential V( r) = V ext( r) + V Hartree( r) + V xc( r). Here we turn this ‘existence’ theorem into a concrete result by expressing the Laplacian ∇ 2 V xc( r) solely in terms of γ( r, r′), which, although idempotent and therefore not many-electron in character, has the exact ground state density ρ( r) on its diagonal γ( r, r). As a specific illustration, we calculate ρ( r) by diffusion QMC and hence γ( r, r′) for the Be atom ground-state. V xc( r) is thereby obtained with satisfactory accuracy.

Dirac density matrix and the Legendre transform of the kinetic energy generated by one-dimensional model potentials

In density functional theory, the single-particle kinetic energy is still not known in an orbital-free form. The natural tool to calculate such kinetic energy for a given one-body potential V is the Dirac density matrix y. Here, by first taking solvable one-dimensional examples, and in particular the sech 2 (x) potential, forms for the Dirac density matrix are exhibited in terms of the potential.This in turn generates the Legendre transform of the single-particle kinetic energy. Finally, one-dimensional perturbation theory for the kinetic energy density, summed to all orders in V, is presented in the appendix.

Slater's Exchange Parameters α for Analytic and Variational Xα Calculations.

Recently, we formulated a fully analytical and variational implementation of a subset of density functional theory using Gaussian basis sets to express orbital and the one-body effective potential. The implementation, called the Slater-Roothaan (SR) method, is an extension of Slater's Xα method, which allows arbitrary scaling of the exchange potential around each type of atom in a heteroatomic system. The scaling parameter is Slater's exchange parameter, α, which can be determined for each type of atom by choosing various criteria depending on the nature of problem undertaken. Here, we determine these scaling parameters for atoms H through Cl by constraining some physical quantity obtained from the self-consistent solution of the SR method to be equal to its exact value. Thus, the sets of α values that reproduce the exact atomic energies have been determined for four different combinations of basis sets. A similar set of α values that is independent of a basis set is obtained from numerical calculations. These sets of α parameters are subsequently used in the SR method to compute atomization energies of the G2 set of molecules. The mean absolute error in atomization energies is about 17 kcal/mol and is smaller than that of the Hartree-Fock theory (78 kcal/mol) and the local density approximation (40 kcal/mol) but larger than that of a typical generalized gradient approximation (∼8 kcal/mol). A second set of α values is determined by matching the highest occupied eigenvalue of the SR method to the negative of the first ionization potential. Finally, the possibility of obtaining α values from the exact atomization energy of homonuclear diatomic molecules is explored. We find that the molecular α values show much larger deviation than what is observed for the atomic α values. The α values obtained for atoms in combination with an analytic SR method allow elemental properties to be extrapolated to heterogeneous molecules. In general, the sets of different α values might be useful for calculations of different properties using the analytic and variational SR method.

Effect of the one-body potential on interelectronic correlation in two-electron systems

The correlation energies of the helium isoelectronic sequence (IS) and of Hooke's IS are very similar and are both weakly increasing upon increasing the nuclear charge/force constant, respectively. However, their separation into radial and angular correlations shows interesting differences. First, for intermediate (and high) values of the force constant radial correlation in Hooke's IS is surprisingly low. Second, both systems exhibit a decrease in the relative contribution of radial versus angular correlation upon strengthening the one-body attractive potential; however, unlike the helium IS, in Hooke's IS the radial correlation energy increases in absolute value upon strengthening the attractive one-body potential. The contribution of radial correlation to the Coulomb hole is examined and the asymptotic behavior at both strong and weak attractive potentials is considered. Radial correlation in Hooke's IS is found to constitute about 9.3% of the total correlation energy when the spring constant approaches the limit k-->infinity, but 100% of the total correlation energy for k-->0. Our results highlight both the similarities and the differences between the helium and Hooke's ISs.

Predicting the single-proton and single-neutron potentials in asymmetric nuclear matter

We discuss the one-body potentials for protons and neutrons obtained from Dirac-Brueckner-Hartree-Fock calculations of neutron-rich matter, in particular their dependence upon the degree of proton/neutron asymmetry. The closely related symmetry potential is compared with empirical information from the isovector component of the nuclear optical potential.

A study of the adiabatic connection for two-electron systems

Some aspects of the adiabatic connection method are studied for two-particle spherically symmetric systems. Ground-state wave functions that are constrained by means of a set of moments to have the same density as a corresponding fully interacting system are obtained for noninteracting or partially interacting systems. Local one-body potentials that support these constrained wave functions are generated using a simple method. We examine an interacting two-particle system with a parameter-dependent one-body potential, which for a particular value of that parameter exhibits an intersection between the (3)S and the (3)P states, whereas the 2s and 2p eigenvalues of the corresponding Kohn-Sham potentials do not intersect along with the total energies. These results show that there do exist cases where occupying the orbitals from below in energy may not lead to the ground state, and that the inherent assumptions behind the adiabatic connection can sometimes be violated.

Weakly boundp3∕2neutrons and spin-response function in the many-body pair correlation of neutron drip line nuclei

Solving the simplified model of the Hartree-Fock Bogoliubov equation in coordinate space with the correct asymptotic boundary conditions, the spin-orbit splitting, occupation probabilities, effective pair-gap, mean square radius, and spin-response function are studied for weakly bound p neutrons. As the binding energy of p3∕2 neutrons becomes small or approaches zero, the spin-orbit splitting p3∕2−p1∕2 in the one-body potential drastically decreases and, at the same time, the effective pair-gap for the p neutrons becomes small, while the occupation probability of the p3∕2 level decreases only slightly. Consequently, in that limit low-lying broader spin response with almost constant amount of total strength appears with the peak moving toward very low excitation energies. (Less)

Regularity and chaos in interacting two-body systems

We study classical and quantum chaos for two interacting particles on the plane. This is the simplest nontrivial case which sheds light on chaos in interacting many-body systems. The system consists of a confining one-body potential, assumed to be a deformed harmonic oscillator, and a two-body interaction of Coulomb type. In general, the dynamics is mixed with regular and chaotic trajectories. The relative roles of the one-body field and the two-body interaction are investigated. Chaos sets in as the strength of the two-body interaction increases. However, the degree of chaoticity strongly depends on the shape of the one-body potential and, for some shapes of the harmonic oscillator, the dynamics remains regular for all values of the two-body interaction. Scaling properties are found for the classical as well as for the quantum mechanical problem.

Effective one-body potential of DFT plus correlated kinetic energy density for two-electron spherical model atoms

For three model problems concerning two-electron spin-compensated ground states with spherical density, the third-order linear homogeneous differential equation constructed for the determination of ρ ( r ) is used here in conjunction with the von Weizsäcker functional to characterize the one-body potential of density functional theory (DFT). Correlated von Weizsäcker-type terms are compared to the exact DFT functional.

Electronic Energy Spectrum of Two-Dimensional Solids and a Chain of C Atoms from a Quantum Network Model

The aim of this study is to explain how a quantum network can be used as simple model to calculate complex band structures. The paper contains an introduction, a mathematical exposure of the method, and applications to graphene, boron nitride, and polyacetylene chains. Using a quantum network is a simple, intuitive, and, yet, rather accurate way to obtain a band structure for a complex material. One focus here is to invoke physical and chemical intuition to construct the effective one-body potential along the wires of a quantum network.

Tomonaga-Luttinger model with an impurity for a weak two-body interaction

The Tomonaga-Luttinger model with impurity is studied by means of flow equations for Hamiltonians. The system is formulated within collective density fluctuations but no use of the bosonization formula is made. The truncation scheme includes operators consisting of up to four fermion operators and is valid for small electron-electron interactions. In this regime, the exact expression for the anomalous dimension is recovered. Furthermore, we verify the phase diagram of Kane and Fisher also for intermediate impurity strength. The approach can be extended to more general one-body potentials.

Propagator and Slater sum in one‐body potential theory

AbstractFor a one‐body potential V(r) generating eigenfunctions ψi(r) and corresponding eigenvalues ϵi, the Feynman propagator K(r, r′, t) is simply related to the canonical density matrix C(r, r}′, β) by β → it. The diagonal element S(r, r},β) of C is the so‐called Slater sum of statistical mechanics. Differential equations for the Slater sum are first briefly reviewed, a quite general equation being available for a one‐dimensional potential V(x). This equation can be solved for a sech2 potential, and some physical properties of interest such as the local density of states are derived by way of illustration. Then, the Coulomb potential –Ze2/r is next considered, and it is shown that what is essentially the inverse Laplace transform of S(r, β)/β can be calculated for an arbitrary number of closed shells. Blinder has earlier determined the Feynman propagator in terms of Whittaker functions and contact is here established with his work. The currently topical case of Fermion vapours which are harmonically confined is then treated, for both two and three dimensions. Finally, in an Appendix, a perturbation series for the Slater sum is briefly summarized, to all orders in the one‐body potential V(r). The corresponding kinetic energy is thereby accessible.

Momentum dependence of symmetry potential in asymmetric nuclear matter for transport model calculations

For transport model simulations of collisions between two nuclei which have $N/Z$ significantly different from unity one needs a one-body potential which is both isospin and momentum dependent. This work provides sets of such potentials.

Proposal for a nonlinear top-down toy model of the brain

Solutions to Newton's equations for particles in one-body potentials of form V 1(x i)=∑ p A (i) 2px i 2p , where p>0 and an integer, can be regarded as generators of infinite sequences of correlated frequencies {Ω m} . Simple nonlinear potentials can hence provide efficient ways of generating correlated information. Two-body potentials V 2( x i − x j ) can provide ways to communicate aspects of that information between the correlated frequency sequences. Temperature and noise can play a role in introducing a time scale across which the frequencies retain their identity. Introduction of explicit time dependence in the energy terms might be appropriate for constructing top down versions of toy models for the brain, something that is lacking at the present time. The richness of the nonlinear system along with the effects of heat baths, external noise and time dependence allows for the possibility of describing aging effects, processing of information “templates” in the brain and of the development of correlations between such “templates”. In short, nonlinearity, interactions, noise effects and introduction of time-dependent energies might allow for the construction of “top-down” models of the brain with the eventual goal of possibly unifying the neurological, molecular biological, biochemical and psychiatric approaches toward studying the brain.

Gamow shell model description of neutron-rich nuclei.

This work presents the first continuum shell-model study of weakly bound neutron-rich nuclei involving multiconfiguration mixing. For the single-particle basis, the complex-energy Berggren ensemble representing the bound single-particle states, narrow resonances, and the nonresonant continuum background is taken. Our shell-model Hamiltonian consists of a one-body finite potential and a zero-range residual two-body interaction. It is demonstrated that the residual interaction coupling to the particle continuum is important; in some cases, it can give rise to the binding of a nucleus.