以一组统一的高斯函数为基函数的双原子分子的Hartree-Fock-Roothaan波函数(Ⅰ)——第一,二列元素的双原子氢化物AH,AH±和AH*

我们采用一组统一的基函数,从头计算第一、二列元素的双原子氢化物以及第一列元素的同核和异核双原子体系的波函数。本文是三篇一组文章的第一篇,得到了双原子氢化物的电子波函数以及轨道能量和总能量等物理量,原子核间距取实验值和(或)理论值。这些波函数是狭义Hartree-Fock方程的以Double Zeta收缩高斯型函数为基函数的展开式。这些态包括体系的基态AH、一些低激发态AH^*和正负离子态AH^±,A表示周期表中Li到F和Na到Cl的各种元素。计算限于闭壳层电子组态或只带一个没有填满的开壳层电子组态。作为例子,三种基态AH的电子波函数表报道于文中。 关键词：     

The p-40Ca and n-40Ca mean fields from the iterative moment approach

The real part V(r; E) of the p-⁴⁰Ca and n-⁴⁰Ca mean fields is extrapolated from positive towards negative energies by means of the iterative moment approach, which incorporates the dispersion relation between the real and imaginary parts of the mean field. The potential V(r; E) is the sum of a Hartree-Fock type component V^HF, (r; E) and a dispersive correction δV(r; E); the latter is due to the coupling of the nucleon to excitations of the ⁴⁰Ca core. The potentials V(r; E) and V_HF(r; E) are assumed to have Woods-Saxon shapes. The calculations are first carried out in the framework of the original version of the iterative moment approach, in which both the depth and the radius of the Hartree-Fock type contribution depend upon energy, while its diffuseness is constant and equal to that of V(r; E). The corresponding extrapolation towards negative energies is somewhat sensitive to the detailed parametrization of the energy dependence of the imaginary part of the mean field, which is the main input of the calculation. Moreover, the radius of the calculated Hartree-Fock type potential then increases with energy, in contrast to previous findings in ²⁰⁸Pb and ⁸⁹Y. A new version of the iterative moment approach is thus developed in which the radial shape of the Hartree-Fock type potential is independent of energy; the justification of this constraint is discussed. The diffuseness of the potential V(r; E) is assumed to be constant and equal to that of V_HF(r; E). The potential calculated from this new version is in good agreement with the real part of phenomenological optical-model potentials and also yields good agreement with the single-particle energies in the two valence shells. Two types of energy dependence are considered for the depth U_HF(E) of the Hartree-Fock type component, namely a linear and an exponential form. The linear approximation is more satisfactory for large negative energies (E < −30 MeV) while the exponential form is better for large positive energies (E > 50 MeV). This is explained by relating the energy dependence of U_HF(E) to the nonlocality of the microscopic Hartree-Fock type component. Near the Fermi energy the effective mass presents a pronounced peak at the potential surface. This is due to the coupling to surface excitations of the core and reflects the energy dependence of the potential radius. The absolute spectroscopic factors of low-lying single-particle excitations in ³⁹Ca, ⁴¹Ca, ³⁹K and ⁴¹Sc are found to be close to 0.8. The calculated p-⁴⁰Ca and n-⁴⁰Ca potentials are strikingly similar, although the two calculations have been performed entirely independently. The two potentials can be related to one another by introducing a Coulomb energy shift. Attention is drawn to the fact that the extrapolated energy dependence of the real part of the mean field at large positive energy sensitively depends upon the assumed behaviour of the imaginary part at large negative energy. Yet another version of the iterative moment approach is introduced, in which the radial shape of the HF-type component is independent of energy while both the radius and the diffuseness of the full potential V(r; E) depend upon E. This model indicates that the accuracy of the available empirical data is probably not sufficient to draw reliable conclusions on the energy dependence of the diffuseness of V(r; E).     

A direct determination of the density matrix in a nuclear system

We introduce an iterative steepest descent approach to determine directly the best one-body density matrix which minimizes the total energy of a nuclear system in the Hartree-Fock approximation, without solving the HF equations. An application is shown to the ²⁰Ne deformed nucleus with a Skyrme interaction. The possibility to introduce external shape constraints to describe general form of deformations is discussed.     

Calculation of the ground state energy of neutral atoms (Z ≤ 54) by the algebraic variant of the Hartree-Fock method

High-precision calculations of the ground-state energy of atoms He through Xe are performed in the algebraic approximation of the Hartree-Fock method. The orbital exponents of Slater-type basis functions are optimized using the second-order minimization methods, which allows the virial theorem to be fulfilled to within 10^?15–10^?17 for the first time. The energies of atoms calculated with rather limited basis sets are, in terms of accuracy, as good as the results obtained by using the numerical procedure for solving the Hartree-Fock equations.     

On classical hydrodynamics and collective motion as approximation to the nuclear many-body problem

Using time-dependent unitary transformations, one can cast a one-body equation of the time-dependent Hartree-Fock type into a form which is closely related to equations for a classical irrotational fluid. The hydrodynamic equation of state finds its counterpart in a stationary constrained field equation. The hydrodynamic equations in turn can be translated into a classical Hamiltonian formalism with an infinite number of generalised coordinates, which are given as all possible spatial moments of the density. The reduction to a few ones, the “natural collective coordinates” is possible by the choice of appropriate initial conditions. The lowest of the hydrodynamical frequencies can be calculated in closed form by harmonic approximations. For the quadrupole frequency a value ofω=31 A^?1/3 MeV/h is obtained. As expected, the value does not agree with the experiment, but rather is in between the characteristic frequency for theβ-vibration and the isoscalar giant quadrupole vibration.     

A new proposal for the discretization of the Griffin—Wheeler—Hartree—Fock equations

A polynomial expansion is proposed as a new way to discretize the Griffin-Wheeler-Hartree-Fock equations of the Generator Coordinate Hartree-Fock method. The implementation of the polynomial expansion in the Generator Coordinate Hartree-Fock method discretizes the Griffin-Wheeler-Hartree-Fock equations through a numerical mesh which is not equally spaced. This procedure makes the optimization of Gaussian exponents in the Generator Coordinate Hartree-Fock method more flexible and more efficient. The results obtained with the polynomial expansion for atomic Hartree-Fock energies show this technique is very powerful when employed in the design of compact and high accurate Gaussian basis sets used in ab initio non-relativistic (Hartree-Fock) and relativistic (Dirac-Fock) calculations.     

Die Hartree-Fock-Theorie mit Zwischenzuständen: Klärung ihrer Struktur bei Zeitumkehrinvarianz und Anwendung auf das Kleinsche Rotations-Vibrationsmodell

In an earlier paper we have extended the new Tamm-Dancoff method to the “New Tamm-Dancoff method with intermediate states”. This extension makes it possible to treat the effect of nearby levels in many body systems with Green's functions. In addition to well-known approximations, such as the Hartree-Fock theory and the Hartree-Bogoliubov theory, we obtain a series of new approximations. The “Hartree-Fock theory with intermediate states”, which is the subject of the present investigation, is one of these. By using time reversal invariance we have succeeded in clarifying its structure, and we give the solution procedure. The exchange terms in theN-particle intermediate states can be represented by an additional potentialY, which (as is the case for the generalized density matrixρ) has to be determined selfconsistently. In this way we have overcome the difficulties, that Kerman and Klein met in their “generalized Hartree-Fock approximation”, which has some close similarities with our Hartree-Fock theory with intermediate states. We demonstrate our method for the exactly soluble rotational-vibrational model of Klein et al. Hereby we show how to treat conservation laws and the degeneracy of levels. The Hartree-Fock equations with three intermediate states turn out to give analytical expressions for the energies and the matrix elements. These agree excellently with the exact values in the rotational part of the spectrum.     

Energy equation of state' of high-density plasmas

The equation of state for fully degenerate high-density plasmas is derived using a modified Thomas-Fermi model. Although the classical TF model is adequate to obtain the energy of an atom at very high densities it fails for low densities. A new version of this model for plasmas is presented which addresses this deficiency by including near-nucleus, exchange and correlation corrections. An analytic formula for the equation of stateE(n _i ) is obtained, valid for all densities (n _i <10²⁶ cm^–3). For low densities, Hartree-Fock results are reproduced with less than 1% error, and the classical result is recovered in the high-density limit.     

Iterative Solutions of the Schrödinger Equation

In the first example containing a long ranged potential, the long range part of the solution is obtained by an iterative Born-series type method. The convergence is illustrated for a case with the long range part of the potential given by C ₆/r ⁶. Accuracies of 1 : 10⁸ are achieved after 8 iterations. The second example iteratively calculates the solution of a non-linear Gross–Pitaevskii equation for condensed Bose atoms contained in a trap at low temperature.     

Accurate analytic presentation of solution for the spiked harmonic oscillator problem

High precision approximate analytic expressions of the ground state energies and wave functions for the spiked harmonic oscillator are found by first casting the correspondent Schrödinger equation into the nonlinear Riccati form and then solving that nonlinear equation analytically in the first iteration of the quasilinearization method (QLM). In the QLM the nonlinear differential equation is treated by approximating the nonlinear terms with a sequence of linear expressions. The QLM is iterative but not perturbative and gives stable solutions to nonlinear problems without depending on the existence of a smallness parameter. The choice of zero iteration is based on general features of exact solutions near the boundaries. Comparison of our approximate analytic expressions for binding energies and wave functions with the exact numerical solutions demonstrates their high accuracy in the wide range of parameters. The accuracy ranging between 10⁻³ and 10⁻⁷ for the energies and, correspondingly, 10⁻² and 10⁻⁷ for the wave functions in the regions, where they are not extremely small is reached. The derived formulas enable one to make accurate analytical estimates of how variation of different interactions parameters affects the correspondent physical systems.     

Two particle — two hole mixing in Hartree-Fock calculations

The constrained Hartree-Fock theory is used for studying the stability of the solutions against two particle — two hole excitations. Quadropole and hexadecapole deformation are imposed on the Hartree-Fock equations to get the true energy minimum and the equilibrium shape. Energy corrections are calculated using second-order perturbations and complete diagonalization. Applications are made to the ²⁰Ne and ²⁸Si nuclei.     

On the dynamics of statistical fluctuations in heavy ion collisions

We show how statistical fluctuations can be treated within the collective approach to heavy ion reactions. In the classical limit, the equation of motion for the distribution d in the collective variables Q_μ and their conjugate momenta P_μ turns out to be a Fokker-Planck equation. We briefly describe the connection of this equation to one of the Smoluchowski type for a distribution in Q_μ only, often used in heavy ion physics. For anharmonic motion our general Fokker-Planck equation is simplified to be linear in the deviations of the Q_μ mand P_μ from their mean values. The solution of this equation is discussed in terms of a simple Gaussian. The parameters of this Gaussian are determined completely by the first and second moments in Q_μ mand P_μ. The equations for the first moments are identical to the Newton equations including frictional forces. Those for the second moments are linear differential equations of first order and hence easily solvable. The whole derivation is completely analogous to that for the Newton equation reported recently. Here the starting point is the quantum mechanical von Neumann equation rather than the Heisenberg equations. As an intermediate result we obtain and discuss briefly a quantal equation for the reduced density operator d which includes frictional effects.     

Accurate analytic presentation of solution of the Schrödinger equation with arbitrary physical potential

High precision approximate analytic expressions of the ground state energies and wave functions for the arbitrary physical potentials are found by first casting the Schrödinger equation into the nonlinear Riccati form and then solving that nonlinear equation analytically in the first iteration of the quasilinearization method (QLM). In the QLM the nonlinear differential equation is treated by approximating the nonlinear terms by a sequence of linear expressions. The QLM is iterative but not perturbative and gives stable solutions to nonlinear problems without depending on the existence of a smallness parameter. The choice of zero iteration is based on general features of exact solutions near the boundaries. The approach is illustrated on the examples of the Yukawa, Woods-Saxon and funnel potentials. For the latter potential, solutions describing charmonium, bottonium and topponium are analyzed. Comparison of our approximate analytic expressions for binding energies and wave functions with the exact numerical solutions demonstrates their high accuracy in the wide range of physical parameters. The accuracy ranging between 10⁻⁴ and 10⁻⁸ for the energies and, correspondingly, 10⁻² and 10⁻⁴ for the wave functions is reached. The derived formulas enable one to make accurate analytical estimates of how variation of different interactions parameters affects correspondent physical systems.     

Semi-classical calculations of hot nuclei

We present semi-classical calculations of hot nuclei performed within the framework of the procedure recently introduced by Bonche, Levit and Vautherin in Hartree-Fock calculations, in order to take consistently into account the effects of continuum states. We use zero and second order Thomas-Fermi approximations for describing the kinetic and spin-orbit energies. After having introduced the ingredients of the formalism and given the main features of our resolution scheme of the Thomas-Fermi equations, we perform a detailed comparison of our calculations with Hartree-Fock results in order to test the accuracy of our model. We discuss the zero-temperature limit case where T² developments can be worked out. We show that, at low temperatures (T ≲ 2 MeV) the subtraction procedure is not indispensable, as expected from Hartree-Fock calculations. We also recall recent results obtained by using our formalism in estimating the temperature dependence of the level density parameter. As in Hartree-Fock calculations we find the existence of a limiting temperature T_lim beyond which the nucleus is unstable because of the Coulomb interaction. By comparing our theoretical values of T_lim with recent experimental data, we show that, in spite of the approximations in our model, this limiting temperature could be related to the actual disappearance of fusion-like processes in medium energy heavy-ion collisions.     

Iterative optimization based on an objective functional in frequency-space with application to jet-noise cancellation

In many technical applications, like supersonic jets, noise with a characteristic spectrum including certain dominant frequencies (e.g. jet-screech) is prevalent, and the elimination of sharp peaks in the acoustic spectrum is the aim of active or passive flow/noise control efforts. A mathematical framework for the optimization of control strategies is introduced that uses a cost objective in frequency-space coupled to constraints in form of partial differential equations in the time domain. An iterative optimization scheme based on direct and adjoint equations arises, which has been validated on two examples, the one-dimensional Burgers equation and the two-dimensional compressible Navier–Stokes equations. In both cases, the iterative scheme has proven effective and efficient in targeting and removing specified frequency bands in the acoustic spectrum. It is expected that this technique will find use in acoustic and other applications where the elimination or suppression of distinct frequency components is desirable.     

Few-electron semiconductor quantum dots with Gaussian confinement

We have performed Hartree-Fock calculations of the electronic structure of N ≤ 10 electrons in a quantum dot modeled with a confining Gaussian potential well. We discuss the conditions for the stability of N bound electrons in the system. We show that the most relevant parameter determining the number of bound electrons is V ₀ R ². Such a feature arises from widely valid scaling properties of the confining potential. Gaussian Quantum dots having N = 2, 5, and 8 electrons are particularly stable in agreement with the Hund rule. The shell structure becomes less and less noticeable as the well radius increases.      

Quantized ATDHF and angular momentum projection: Three-dimensional applications to heavy ion scattering

The quantized adiabatic time-dependent Hartree-Fock (qATDHF) theory is extended to the calculation of observables in nuclear phenomena using general many-body techniques including angular momentum projection. All calculations are performed using three-dimensional coordinate and momentum-grid techniques. The Bonche-Koonin-Negele interaction as well as several Skyrme-type forces has been used in the simple Hartree-Fock (HF) calculations of ¹²C and ²⁰Ne nuclei. Further, the maximally decoupled collective path is evaluated within qaTDHF, and the angular momentum projected kernels are inserted into the GCM formalism. As a test case, this formalism is applied to the phase shifts of elastic α-α scattering because they can be compared to experimental values. The same techniques are applied to the α-¹⁶O system and the results are compared with those obtained by solving the collective Schrödinger equation in a Gaussian Overlap Approximation, as in a previous publication. The GCM, extended to complex energies, is used to extract the widths of the resonance levels of the 0₁⁺, 0₄⁺, 0 bands in α-¹⁶O scattering. A model calculation, in which the structure of the nuclei is kept fixed to their HF ground state, is performed for the same α-¹⁶O system. We get quite different results with this “sudden” approximation than with the adiabatic calculation. The present calculations show that it is indeed possible to connect general and symmetry conserving many-body techniques to the qATDHF theory and to obtain in this way a purely microscopic framework which can be handled numerically, thus allowing evaluation of observables which are accessible to experiments.     

The gravitational field in the wave zone I. The radiating terms of the metric tensor

We consider an asymptotically flat space-time generated by a perfect fluid source of compact spatial support. Using the de Donder gauge conditions, the Einstein equations are reduced to a new form of Poisson-type equations. A formal iterative scheme is set up to solve these equations by expanding the components of the metric tensor in powers ofc ^–1. The coefficient of each power ofc ^–1 depends on the asymptotically retarded timeu andx, y, z and satisfies a Poisson-type equation. Assuming asymptotic flatness the solution is carried out in the first orders. The results are explicit expressions of the metric up to orderc ^–4 in terms of the source functions. These expressions hold over all space-time. A further expansion in powers ofr ^–1 gives the first terms of the metric that contribute to gravitational radiation.