Constrained Hartree-Fock Method Research Articles

Triaxial rigidity of Er166 and its Bohr-model realization

The triaxial nature of low-lying rotational bands of $^{166}\mathrm{Er}$ is presented from the viewpoint of the Bohr Hamiltonian and from that of many-fermion calculations by the Monte Carlo shell model and the constrained Hartree-Fock method with projections. A recently proposed novel picture of those bands suggests definite triaxial shapes of those bands, in contrast to the traditional view with the prolate ground-state band and the $\ensuremath{\gamma}$-vibrational excited band. Excitation level energies and $E2$ transitions can be described well by the Bohr Hamiltonian and by the many-fermion approaches, where rather rigid triaxiality plays vital roles, although certain fluctuations occur in shell-model wave functions. Based on the potential energy surfaces with the projections, we show how the triaxial rigidity appears and what the softness of the triaxiality implies. The excitation to the so-called double $\ensuremath{\gamma}$-phonon state is discussed briefly.

Self-consistent collective coordinate for reaction path and inertial mass

We propose a numerical method to determine the optimal collective reaction path for the nucleus-nucleus collision, based on the adiabatic self-consistent collective coordinate (ASCC) method. We use an iterative method combining the imaginary-time evolution and the finite amplitude method, for the solution of the ASCC coupled equations. It is applied to the simplest case, the $\alpha-\alpha$ scattering. We determine the collective path, the potential, and the inertial mass. The results are compared with other methods, such as the constrained Hartree-Fock method, the Inglis's cranking formula, and the adiabatic time-dependent Hartree-Fock (ATDHF) method.

Doubly-Magic 100Sn Nucleus with Delta Excitation under Compression

The energies and matter densities of finite nuclei under radial compression are investigated by using a constrained Hartree–Fock method with the Δ degree of freedom included. The results are presented for the doubly-magic nucleus 100 Sn in a effective baryon–baryon interaction. As the nucleus is compressed, the binding energy increases in a linear form approximately. The increase in binding energy is less in large model space systems. The gap between binding energy of only the nucleon and nucleon + Δ is reduced by increasing the model space. The number of Δ's increases as the model space decreases. It is found that as the nucleus is compressed to about 17 times the ordinary nuclear density, the Δ component is sharply increased to about 17.8% of all baryons in the system. This result is consistent with the values extracted from relativistic heavy-ion collisions. The number of Δ's increases as the radius decreases under compression. The single particle energy levels calculated and their behaviors under compression are examined too. A good agreement between results with effective Hamiltonian and the phenomenological shell model for the low lying single-particle spectra is obtained. The current results suggest that the radial density distribution is reduced at large model space for same radius. The peak of Δ radial density distribution is reduced as the model space increases for the same r rms . A considerable reduction in compressibility for the nucleus, and softening of the equation of state with the inclusion of the Δ's in the nuclear dynamics are suggested by the results.

Delta excitation calculation studies in the ground state of the compressed finite heavy doubly-magic nucleus ^{100}Sn

The energies and matter densities of finite nuclei under radial compression are investigated by using a constrained Hartree-Fock method with the \Delta degree of freedom included. The results are presented for the doubly-magic nucleus ^{100}Sn in an effective baryon-baryon interaction. It is found that as the nucleus is compressed to about three time of the ordinary nuclear density, the \Delta component is sharply increased to about 17% of all baryons in the system. This result is consistent with the values extracted from relativistic heavy-ion collisions. The single particle energy levels calculated and their behaviors under compression examined too. A good agreement between results with effective Hamiltonian and the phenomenological shell model for the low lying single-particle spectra obtained. A considerable reduction in compressibility for the nucleus, and softening of the equation of state with the inclusion of the \Delta 's in the nuclear dynamics are suggested by the results.

Wave functions derived from experiment. V. Investigation of electron densities, electrostatic potentials, and electron localization functions for noncentrosymmetric crystals.

The constrained Hartree-Fock method using experimental X-ray diffraction data is extended and applied to the case of noncentrosymmetric molecular crystals. A new way to estimate the errors in derived properties as a derivative with respect to added Gaussian noise is also described. Three molecular crystals are examined: ammonia [NH(3)], urea [CO(NH(2))(2)], and alloxan [(CO)(4)(NH)(2)]. The energetic and electrical properties of these molecules in the crystalline state are presented. In all cases, an enhancement of the dipole moment is observed upon application of the experimental constraint. It is found that the phases of the structure factors are robustly determined by the constrained Hartree-Fock model, even in the presence of simulated noise. Plots of the electron density, electrostatic potential, and the electron localization function for the molecules in the crystal are displayed. In general, relative to the Hartree-Fock model, there is a depletion of charge around hydrogen atoms and lone pair regions, and a build-up of charge within the molecular framework near nuclei, directed along the bonds. The electron localization function plots reveal an increase in the pair density between vicinal hydrogen atoms.

Shape coexistence in doubly-magic56Niby the Monte Carlo shell model

The structure of low-lying states of ${}^{56}\mathrm{Ni}$ is studied by the Monte Carlo shell model based on the quantum Monte Carlo diagonalization method. The coexistence of spherical yrast, prolate deformed, and other non-yrast states is described by the full $\mathrm{pf}$-shell calculations, by employing the FPD6 realistic residual interaction. To understand the properties of eigenstates thus obtained, we utilize a mean field analysis, such as a potential energy surface by constrained Hartree-Fock method.

Mean-field calculation of compressed finite nuclei withΔexcitations

The energies and matter densities of finite nuclei under radial compression are predicted by using a constrained Hartree-Fock method with the $\ensuremath{\Delta}$ degree of freedom included. The results are presented for ${}^{90}$Zr in a calculation within a model space of seven major oscillator shells. The main feature of this calculation is that the effective baryon-baryon interactions associated with the $\ensuremath{\Delta}$ are evaluated within a $G$-matrix approach based on a coupled-channel $\mathrm{NN}\ensuremath{\bigoplus}N\ensuremath{\Delta}\ensuremath{\bigoplus}\ensuremath{\pi}\mathrm{NN}$ model that can describe the $\mathrm{NN}$ data up to 1 GeV. It is found that as the nucleus is compressed to about 2--3 times of the ordinary nuclear density, the $\ensuremath{\Delta}$ component is sharply increased to about 10% of all baryons in the system. This result is consistent with the values extracted from relativistic heavy-ion collisions.

Analysis of Chemical Phenomena by Solving the Constrained Hartree–Fock Equation. III. Influence of Geometry Change on the Energy of π Electrons in Conjugated Hydrocarbons

Abstract The total energy of a conjugated unsaturated hydrocarbon can formally be partitioned into the energy of π electrons and the frame energy. This type of partitioning has been used to study the role of π electrons; to know the relationship between the π-electronic structure and its energy, the molecular geometry has been varied. This paper examines the influences of such geometry change on the energy of π electrons from a basic point of view. It was found that there are four types of geometrical influences which disguise the relationship between the π-electronic structure and its energy. Contrary to this, the constrained Hartree–Fock method was shown to disclose the relationship between π-electronic structure and its energy clearly. Application of the constrained Hartree–Fock method to D6h and D3h benzenes showed that in a D6h benzene π electrons favor the symmetrically uniform distribution independently of the bond length and the partitioning method, while in a D3h benzene the uniform distribution of π electrons is destroyed. Analysis of gradients of partitioned energies with respect to the bond order at a D3h-distorted benzene indicated that π electrons favor a more uniform delocalization, but this is opposed by one- and two-electron potentials.

Analysis of Chemical Phenomena by Solving the Constrained Hartree–Fock Equation. II. Relationship between π-Electronic Structure and Its Energy in Benzene

Abstract The constrained Hartree–Fock method provides quantum-mechanically acceptable wave functions of an arbitrary electronic structure without changing the molecular geometry. We have applied this method to the analysis of the behavior of π electrons in benzene finding that there are two types of resonance energy, simple resonance between double bonds and super-stabilization due to the topology of benzene, which were estimated as 54 and 168 kJ mol−1, respectively. According to our method, π electrons in benzene were found to have a propensity for uniform distribution. Namely, the D6h distribution of π electrons gives the minimum values concerning both the π-electronic and total energies, which are brought forth by lowering the kinetic energy of π electrons. The origin of delocalization of π electrons in benzene was attributed to relaxation of the kinetic-energy pressure of π electrons.

Configuration-constrained Hartree-Fock method —an illustrative example

The structure of solutions of the constrained Hartree-Fock method is studied within the time-dependent Hartree-Fock formalism. A new method for reaching various stationary Hartree-Fock points, not accessible by the usual adiabatic constrained Hartree-Fock method, is proposed.

Nonlinear resonance in the time-dependent Hartree-Fock manifold.

In order to try to open a new scope to explore the mutual dependence between the single-particle and collective modes of motion near to the level crossing region, a general method is developed to investigate the nonlinear resonant structure of the time-dependent Hartree-Fock (TDHF) manifold, without depending on the adiabatic assumption. By using the Lie canonical transformation method with the Deprit perturbation treatment, in this method, the maximal integrable-form representation of the TDHF manifold is introduced. This representation plays an essential role in exploring the nonlinear resonant structure of the TDHF manifold, which characterizes complex topology of the manifold. Aiming at relating the nonlinear resonance in the TDHF manifold with the dynamics between the single-particle and collective modes of motion near to the level crossing region, structure of the TDHF wave function in the nonlinear resonant region is investigated. It is clarified that an isolated nonlinear resonant region of the TDHF manifold is characterized by a local constant of motion (dynamical symmetry) and generates a new type of dynamical stable single-Slater-determinant states, which is topologically different from the TDHF states near the HF ground state, and cannot be reached by the conventional static Hartree-Fock method, constrained Hartree-Fock method, nor the adiabatic TDHF theories. One may expect that the appearance mechanism of the new dynamical stable single-Slater-determinant states gives us a new scope for understanding occurrence mechanism of a variety of collective sideband structure near to the level crossing region.

An examination of the effects of basis set and charge transfer in hydrogen-bonded dimers with a constrained Hartree-Fock method

Constrained Hartree–Fock methods in which orbitals are constructed from strictly local nonorthogonal subsets of the molecule's atomic orbital basis have been known for over a decade. These methods have been principally used to generate localized molecular orbitals and interpret intramolecular interactions. In this paper, constrained Hartree–Fock results from basis sets ranging from minimal to extensive are presented for hydrogen-bonded dimers in which individual molecular orbitals are constructed from atomic orbitals belonging to individual monomers. These calculations eliminate both basis set superposition errors (BSSE) as well as charge transfer between monomers. This allows one to examine the effects of basis sets on the electrostatic and polarization components to the energy unmasked from BSSE. The charge transfer components are also isolated by comparing results to unconstrained calculations near the Hartree–Fock limit, where BSSE is vanishingly small. Finally, at moderate intermolecular separations when charge transfer becomes negligible, the constrained results are compared to both counterpoise-corrected and unconstrained calculations.

SU(2) × SU(2) quasi-spin model and constrained Hartree-Fock method

In order to study the effectivity for the constrained Hartree-Fock (CHF) method, a quasi-spin model containing two kinds of interaction is proposed. The exact solution of the model Hamiltonian in SU(2) × SU(2) basis, the HF and some CHF solutions as well as the ground state phase transition are discussed.

Microscopic theory of heavy-ion interaction potentials

The authors develop a many-body theory for the real part of the heavy-ion interaction potential at energies near the Coulomb barrier. The theory is similar to the constrained Hartree-Fock method, but uses two-centre shell-model wavefunctions rather than self-consistent wavefunctions. This approximation allows them to calculate potential energy surfaces for very heavy systems. The interaction energy depends on two collective parameters: internuclear distance R and fragment deformation beta . As in most Hartree-Fock theories, they use a Skyrme force whose parameters are fitted to nuclei throughout the periodic table. The Coulomb energy is evaluated with high precision from the single-particle wavefunctions. They present potential energy surfaces for the reactions 40Ca+40Ca, 90Zr+90Zr and 238U+238U.

Core polarization effects in sd-shell nuclei and charge-symmetry breaking in the nuclear mean field.

The proton and neutron densities and nuclear potentials are calculated for the deformed sd-shell nuclei /sup 28/Si and /sup 32/S in the framework of the constrained Hartree-Fock method and compared with the results of the spherical nucleus /sup 40/Ca. Coulomb core polarization effects are responsible for differences in the proton and neutron Hartree-Fock potentials. These charge-asymmetry effects in the nuclear mean field must be taken into account when trying to decide whether the charge asymmetry observed in proton and neutron scattering is due to a genuine charge asymmetry in the nuclear force and, if possible, to put an upper limit on the magnitude of this effect. Our calculations indicate that there is no enhancement of the charge asymmetry due to nuclear deformation. The often proposed volume integral of the nuclear potential is found not to be an appropriate tool to investigate charge-asymmetry effects in nuclei due to a large cancellation in the integral.

Molecular-Dipole Deformation as a Collective Variable in Charge-Changing Ion-Atom Collisions

The molecular dipole moment along the internuclear line is treated as a collective coordinate to describe the sharing of electrons in slow atomic collisions. With use of the constrained Hartree-Fock method, potential energy surfaces for Li-F and ${(\mathrm{H}\mathrm{e}\ensuremath{-}\mathrm{H}\mathrm{e})}^{2+}$ are calculated as a function of the internuclear separation and the dipole deformation. This offers a conceptual framework to view a charge-changing process as the motion along a path in the energy surface. Consequences for the molecular orbitals are discussed.

Nuclear rotations studied by the time-dependent variational method

A general nuclear rotation including precession and wobbling motion is studied by the time-dependent variational method and a classical equation of motion is derived. The intrinsic wave function associated with the general rotational motion is constructed by making use of the constrained Hartree-Fock method and variables necessary in solving the equation are calculated. The method developed here is applied to a schematic extension of the Nilsson model.

On the basic homogeneity characteristic of atomic and molecular electronic energies

A homogeneity property of electronic energy with respect to atomic number, known to hold to a certain accuracy for neutral atoms, is extended and applied to neutral diatomic and polyatomic molecules. Two alternative hypotheses are examined. Homogeneity of the total energy, W, including repulsions between nuclei, leads to total Hartree–Fock energies at equilibrium proportional to the sum of orbital energies, and a linear first-order differential equation for the energy, for a diatomic molecule having the form R (dW/dR) +(3−k) W=Σεi, where k is the homogeneity parameter (about 7/3). A second, preferred hypothesis is homogeneity of the electronic energy E: ΣαZα(∂ E/∂ Zα)R=kE. It leads to a total energy equation in which appear both the sum of orbital energies and the repulsion between nuclei, and the differential equation R (dW/dR)+(3−k) W=Σεi+(2−k)(ZαZβ/R). The two hypotheses are tested on a number of atomic and molecular species, and their consequences are discussed. A constrained Hartree–Fock method in which the second hypothesis holds exactly is developed and shown to take the form of the extended Hückel method.

A microscopic description of nuclear molecules: Application to the 12C + 12C system

The interaction energy between two oblate 12C ions is calculated by means of the constrained Hartree-Fock method. The influence of the mutual orientation of the ions is investigated by considering two extreme configurations: an axial symmetric one where the two ions approach each with their symmetry axes aligned with the collision axis and a triaxial one where the axes of the fragments are perpendicular to the collision line. The corresponding potentials V 1 and V 2 display very distinct features. In particular the minima of the potentials occur for quite different interdistances. A method is devised for constructing from V 1 and V 2 the potentials and coupling factors between two ions rotating with definite angular momentum. Using these quantities in a coupled channel calculation, we explain the gross features of the elastic, single 2 + inelastic and double inelastic cross sections. The same calculation yields good agreement with the fusion data.

Choice of the constraining operator in the constrained hartree-fock method

The variation of the generator coordinate trial function with respect to the path of integration leads to a prescription for the choice of the constraining operator in constrained Hartree-Fock calculations. The resulting path follows a local valley in the deformation energy surface. The proposed scheme is applied to a three-level model with SU(3) symmetry.