Adiabatic Time-dependent Hartree-Fock Theory Research Articles

TETRAHEDRAL SYMMETRY IN NUCLEI: NEW PREDICTIONS BASED ON THE COLLECTIVE MODEL

We apply the standard collective nuclear Hamiltonian derived in the literature with the help of the Adiabatic Time Dependent Hartree-Fock (ATDHF) theory. We use a schematic two-dimensional collective space spanned by the axial-quadrupole (α20) vs. tetrahedral (α32) deformations together with the potential energy surfaces calculated microscopically. We illustrate and discuss the collective-model solutions suggesting that the tetrahedral-symmetry nuclei may generate sizeable quadrupole transitions: this result is in contrast to the previous expectations and/or predictions.

Modeling the doubly excited state with time-dependent Hartree–Fock and density functional theories

Multielectron excited states have become a hot topic in many cutting-edge research fields, such as the photophysics of polyenes and in the possibility of multiexciton generation in quantum dots for the purpose of increasing solar cell efficiency. However, obtaining multielectron excited states has been a major obstacle as it is often done with multiconfigurational methods, which involve formidable computational cost for large systems. Although they are computationally much cheaper than multiconfigurational wave function based methods, linear response adiabatic time-dependent Hartree-Fock (TDHF) and density functional theory (TDDFT) are generally considered incapable of obtaining multielectron excited states. We have developed a real-time TDHF and adiabatic TDDFT approach that is beyond the perturbative regime. We show that TDHF/TDDFT is able to simultaneously excite two electrons from the ground state to the doubly excited state and that the real-time TDHF/TDDFT implicitly includes double excitation within a superposition state. We also present a multireference linear response theory to show that the real-time electron density response corresponds to a superposition of perturbative linear responses of the S(0) and S(2) states. As a result, the energy of the two-electron doubly excited state can be obtained with several different approaches. This is done within the adiabatic approximation of TDDFT, a realm in which the doubly excited state has been deemed missing. We report results on simple two-electron systems, including the energies and dipole moments for the two-electron excited states of H(2) and HeH(+). These results are compared to those obtained with the full configuration interaction method.

Adiabatic time-dependent Hartree-Fock theory with the Skyrme interaction.

A derivation for reducing the adiabatic time-dependent Hartree-Fock theory with the consistency condition to an efficacious form, in which the evolution of the single-particle states is given by a time-dependent Hartree-Fock-like equation while the consistency condition is given by a one-body equation, is discussed. The density-dependent Skyrme interaction is included in the derivation.

Technique for evaluating the optimal path in adiabatic time-dependent Hartree-Fock theory.

A method of achieving the consistent solution of the adiabatic time-dependent Hartree-Fock theory as the optimal path is discussed. The method is applied to evaluate the valley path in the soluble three-level model.

Canonicity condition in adiabatic time-dependent Hartree-Fock theory.

We show how the incorporation of the canonicity condition, which relates the generator of coordinate and momentum, gives the correct Lagrange multiplier needed to complete the proof that the solution of the adiabatic time-dependent Hartree-Fock theory with the consistency condition fulfills the constrained variation principle designating the valley and thus the canonicity condition is, in principle, a normalization requirement rather than a restriction on the adiabatic time-dependent Hartree-Fock path.

Comparison of quantised ATDHF and GCM theory with application to the 12C+20Ne system

A conceptional and numerical comparison of the one parameter generator coordinate method (GCM) and the quantised adiabatic time-dependent Hartree-Fock (ATDHF) theory is performed by applying both theories to the 12C+20Ne and 16O+16O system. Different parametrisations of the Skyrme interaction are used. The single-particle wavefunctions and the operators are represented on a three-dimensional grid in coordinate and momentum space. The collective path is evaluated in the gradient method, corresponding to GDM, and by solving the numerically more involved ATDHF equations.

Adiabatic time-dependent Hartree-Fock theory in the generalized valley approximation

A classical theory of large amplitude, adiabatic collective motion, developed and applied previously to test examples, is transcribed into the language of time-dependent Hartree-Fock theory in order to initiate the application of the new method to problems in nuclear physics. The formulation which emerges can be characterized as a generalized cranking theory, in the sense that the cranking operator cannot be chosen arbitrarily, as in the conventional formulation of this method, but is instead fully constrained by the formalism. A procedure for obtaining approximate solutions to the new equations is described and illustrated with several simplified many-body models. It is inferred that traditional cranking calculations can serve as starting points for more realistic models. This paper also includes additional discussion concerning the calculation of collective masses and the problem of local stability of collective motion, not covered in our previous work.

Uniqueness of the collective submanifold in the adiabatic theory of large amplitude collective motion

It is emphasized that the conditions for the existence of a collective submanifold which follow from adiabatic time-dependent Hartree-Fock theory are precisely the conditions for the existence of a manifold of solutions of Hamilton's equations confined to a surface of reduced dimensionality. A constructive procedure, valid in any number of dimensions and involving the concept of the multidimensional valley, is developed to determine whether a given system admits such a manifold. It is extended to include the idea of the approximate manifold, and an application to a generalized landscape model is described.

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 0 1 +, 0 4 +, 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.

Deformation effects in the12C-12C system

The quantized adiabatic time dependent Hartree-Fock (quantized ATDHF) theory is applied to the12C-12C-system. Especially the intrinsic oblate ground state deformation of this nucleus is considered. For four relevant mutual orientations of the two nuclei the optimal, i.e. maximally decoupled collective paths, associated to relative motion, are evaluated by solving a coupled set of nonlinear differential equations on three dimensional grids in coordinate and momentum space. In a second step, the constituents of the corresponding collective Hamiltonian are calculated. Besides the collective potentials this includes the evaluation of the appropriate mass parameters, of the quantum corrections with regard to rotation, translation and collective relative motion and of the centrifugal potentials, all of which are shown to be important. By means of generalized WKB methods, subbarrier fusion cross sections are evaluated for different assumptions on the reaction process and compared with experimental data. Finally, the collective Hamiltonian is calculated for a head-on-collision of two12C-chains.

Adiabatic time-dependent Hartree-Fock theory with the consistency condition

Adiabatic time-dependent Hartree-Fock (atdhf) theory, including the authors’ work in this field, has been summarised. In response to the criticism of Yamamuraet al the role of curvature in preventing the choice ofpure rpa mode as the solution near the static Hartree-Fock minimum has been discussed.

Quantized ATDHF calculations for the α + 16O 20Ne system

The quantized adiabatic time-dependent Hartree-Fock theory (quantized ATDHF.) is numerically applied to the α + 16O 20Ne system. Density-dependent interactions with finite-range direct Yukawa-tenns are used and the single-particle wave functions are represented on a three-dimensional grid in coordinate and momentum space. It is demonstrated that the evaluation of observables is independent on the actual choice of the collective coordinate. For the 20Ne ground state the importance of parity projection is shown. For the description of the dynamics of the α + 16O 20Ne system the collective potential, the mass parameter, rotational moments of inertia and quantum corrections (zero point energies) are evaluated microscopically. The collective Schrödinger equation is then solved for the ground-state rotational band and for the α- 16O elastic scattering. In both cases good agreement with experimental data is obtained.

Collective mass parameters and linear response techniques in three-dimensional grids

We discuss four prescriptions for evaluating a collective mass parameter suitable for translations, rotations and large amplitude collective motions. These are the adiabatic time dependent Hartree-Fock theory (ATDHF) and the generator coordinate method (GCM), both with and without curvature corrections. As practical example we consider the16O+16O collision using a recently developed density dependent interaction with direct Yukawa and Coulomb terms. We present a fast iteration scheme for solving the linear response equation in a three-dimensional coordinate or momentum space grid. As test cases we consider the rotational and translational inertia parameters for various distances between the heavy ions. We find that curvature corrections are negligible whereas the difference between ATDHF and GCM may amount to about 35% in case of rotation.

Dynamics of theO16+O16→S32fusion process

The collective mass and potential, the quantum corrections, and the inertia parameters of the /sup 16/O+ /sup 16/O..-->.. /sup 32/S system are evaluated by means of the quantized adiabatic time dependent Hartree-Fock theory in a three-dimensional coordinate and momentum lattice. The interaction used, consisting of a direct finite range Yukawa force and a density-dependent term, is fitted by static Hartree-Fock calculations (including center-of-mass corrections) to binding energies and elastic electron scattering form factors in the mass region of interest here. The result of the fit is a new interaction, still without exchange terms, which for light nuclei reproduces the binding energies, the diffraction radii, and the surface widths. The sub-barrier fusion cross section for /sup 16/O+ /sup 16/O..-->.. /sup 32/S calculated with this force in quantized adiabatic time dependent Hartree-Fock theory is in excellent agreement with experimental data in contrast to calculations using other interactions. For the evaluation of the fusion cross section above the barrier classical trajectory calculations are performed using a suitable phenomenological friction force. This yields agreement with experimental data up to energies of 40 MeV above the barrier. The quantum corrections and the dependence of the mass parameter on the fragment distance are shown to havemore » an important impact only below the barrier.« less

Three-dimensional nuclear dynamics in the quantized ATDHF approach

The quantized adiabatic time-dependent Hartree-Fock theory is numerically applied to the low energy large amplitude collective dynamics of heavy ion systems ranging from α + α to 16O + 16O. The problem is reduced to three successive steps. First, for the lowest mode the optimal, i.e., maximally decoupled, collective path {∥ φ q 〉} is evaluated by solving a coupled set of nonlinear differential equations for the single-particle wave functions ϕ q ( a) (r) of ∥ φ q 〉, depending on the collective coordinate q and three spatial coordinates. A density-dependent interaction with a direct finite range Yukawa-term is employed and three-dimensional coordinate- and momentum-grid techniques are used, including fast Fourier methods. In a second step the quantized collective Hamiltonian H c(q, d dq ) is extracted from {∥ φ q 〉} by means of generator coordinate techniques involving, besides q, a conjugate variable p. Starting from {∥φ〉} this procedure includes the numerical evaluation of the classical potential, V ( q), of the intertia parameter, M ( q), of the quantum corrections with regard to rotation, translation and collective q-motion, L ( q), and of the centrifugal potential. The third step consists of actually calculating the subbarrier fusion cross section by means of WKB methods applied to the collective Hamiltonian H c(q, d dq ) . The theoretical numbers are compared with results from Hartree-Fock calculations with quadrupole constraint, and with experimental data. The microscopic aspects of the dynamics, the relation to other theories, and the practical and conceptual problems arising from the quantized ATDHF theory are discussed in detail.

Quantized ATDHF calculations for subbarrier fusion of heavy ions

The quantized adiabatic time-dependent Hartree-Fock theory (quantized ATDHF) is applied to the subbarrier fusion process of 16 O + 16 O → 32 S . The corresponding optimal, i.e. maximally decoupled collective path is evaluated by means of three-dimensional coordinate and momentum space grid techniques using the Bonche-Koonin-Negele interaction completed by the Coulomb force. Collective mass parameters, potentials, quantum corrections and rotational moments of inertia are evaluated in dependence of the distance between the ions. The results are compared to experimental fusion data and to the outcome of Hartree-Fock with quadrupole constraint.

The mass parameter in the generator coordinate method

The collective kinetic energy for the fission mode is discussed in terms of the generator coordinate method. A comparison of the GCM mass formula with the cranking result and the result of the adiabatic time-dependent Hartree-Fock theory is established and the zero-point energies obtained in the GCM are examined in some detail. A numerical pilot study on the basis of a two-center shell model with pairing serves as an illustration of the general discussion.

Zero-Point Fluctuations of the Nuclear Shapes in the Adiabatic Time-Dependent Hartree-Fock Approach

Adiabatic time-dependent Hartree-Fock theory is used to describe the motion of the system around stable equilibrium. It is shown that after a proper quantization of the collective variables this method leads to matrix elements of operators between collective states which are identical to those obtained in the generator coordinate method. In particular both predict equal zero-point motion effects in the ground-state density and energy.

Adiabaticity in the Path-Integral Approach to nuclear large-amplitude vibrations

The concept of adiabaticity is implemented into the path integral approach (PIA) to largeamplitude collective motion: For bound vibrations this allows one to establish a clear connection between the path integral approach on one hand, and the quantized adiabatic time-dependent Hartree-Fock theory (quantized ATDHF) on the other hand. Assuming a collective path, {|φqpATDHF〉}, having the structure as suggested by ATDHF, i.e. |φqpATDHF〉 = exp(ipQ(q))|φqATDHF〉, with a 1p-1h operator Q(q), one can derive a set of equations for |φqATDHF〉 and Q(q) such that the multitude {φqpATDHF〉} is as close as possible to the multitude of periodic time-dependent Hartree-Fock solutions, φE(t)〉}, derived by PIA. These equations turn out to be virtually identical to the (time-independent) ATDHF equations. This feature allows one also to compare in detail the quantization prescriptions of PIA and of quantized ATDHF. If one ignores zero-point energy corrections and treats the collective quantized ATDHF hamiltonian, Hc(q, ∂/∂q), semiclassically by means of WKB methods, one obtains exactly the quantization rule of PIA in the pure classical limit. Furthermore the leading-order quantum corrections of PIA correspond approximately to a semiclassical treatment of Hc(q, ∂/∂q) including zero-point energy corrections.

Formally exact quantum variational principles for collective motion based on the invariance principle of the Schrödinger equation

Utilizing the concept of invariant collective subspace of a many-body system (or invariance principle of the time-dependent Schrödinger equation), we derive a number of formally exact variational principles to characterize the subspace. Previous studies based on time-dependent or adiabatic time-dependent Hartree-Fock theory are, in principle, contained as approximations.