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Physical Review C

Dynamical basis generation method with an application to the four-nucleon problem

Guillermo H. Bozzolo, James P. Vary, Angel Ricardo Plastino

Published Jan 1, 1985• 4 citations
Publisher

Abstract

A variational method inspired by the Hartree-Fock approximation but not restricted to a single Slater determinant trial space is investigated. The physical motivation is that a method should attempt to find a subspace of collective states which are most strongly coupled to the ground state. This method attempts to do this by providing a systematic technique to generate basis states from the collective Hartree-Fock type of state. In the resulting basis space a residual diagonalization is easily performed. Results of a test with the four-nucleon problem with realistic effective nuclear Hamiltonians are shown.

DOI
10.1103/physrevc.31.207

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A variational method inspired by the Hartree-Fock approximation but not restricted to a single Slater determinant trial space is investigated. The physical motivation is that a method should attempt to find a subspace of collective states which are most strongly coupled to the ground state. This method attempts to do this by providing a systematic technique to generate basis states from the collective Hartree-Fock type of state. In the resulting basis space a residual diagonalization is easily performed. Results of a test with the four-nucleon problem with realistic effective nuclear Hamiltonians are shown.
The method of coherent correlated pairs is applied with a realistic effective Hamiltonian to the four-nucleon system. Three methods of solution are compared for ease of implementation and convergence rates. A Monte Carlo method yields the best upper bound within the coherent correlated pair space. The full variational treatment of the four-nucleon coherent correlated pair problem yields equations which have poor stability properties. An ansatz variational method proves to be rapidly convergent and yields results very close to those of the Monte Carlo method. The results for binding energy and the one body density distribution are compared with Hartree-Fock results using the same effective Hamiltonian in the same model space. As in previous soluble model applications the coherent correlated pair method provides a superior upper bound to the ground state energy.
A procedure is discussed that searches for the best description of the eigenstates of a Hamiltonian of a finite quantum many-body system in terms of a selected set of physically relevant configurations. The procedure resorts to iterative sequences of diagonalizations in spaces of very reduced size. Each diagonalization provides an energy-based importance measure that governs the selection of the configurations to be included in the states. The procedure is strictly variational and preserves the symmetries of the Hamiltonian throughout the iterative process. We report on some test applications to the Na8{}_{8}8​ metal cluster. A series of calculations is performed on a Hartree-Fock basis for a number of orbitals ranging from 5 to 20. In the case of 5 (10) orbitals, we compare the low-lying spectra generated with our procedure to complete (truncated) configuration interaction calculations. The procedure accurately reproduces the results in the complete space. In the most complex cases of 15 and 20 orbitals (the latter involving a many-particle space of 1011{}^{11}11 states), we construct the ground state and study its structure. The analysis highlights the efficiency of the method in generating the most appropriate model space for the description of individual eigenstates of the Hamiltonian.
We present a generalization of the Hartree-Fock (HF) method incorporating two-body correlations. The correlations are introduced by means of a unitary operator eiF{e}^{\mathrm{iF}}eiF, where FFF is a Hermitian one- and two-body operator to be determined by the variational principle. Assuming that the matrix elements of FFF are small so that powers of FFF higher than the second may be neglected, a set of linear equations determining FFF is obtained. The relation between our method and methods used to treat singular interactions is discussed. The method is applied to two recently suggested models and is compared with the HF method, second-order perturbation theory, and with the random-phase approximation. It is found that our method yields very good results within the range of validity of our approximations, which seems to call forth applications to more realistic problems. Possible applications to the study of collective phenomena in nuclei are indicated.
In this thesis a method for doing approximate calculations of the ground state of quantum mechanical many-body systems is developed and implemented. This method uses the Hartree-Fock method as a starting point and approximates the ground state as a linear combination of non-orthogonal Slater determinants. The Slater determinants are added consecutively by minimising their associated energy while keeping the previously found Slater determinants fixed. This method was tested on a quantum harmonic oscillator filled with interacting fermions and was found to give good approximations to the ground state of this system with only a few Slater determinants. Error estimation of the approximated ground state is done by calculating the energy variance. The energy variance was used in two different methods for improving upon the approximated energy, both of which was found to work well.
A variational principle is formulated to determine the single-particle states, their pairing, and the occupation number distribution for a trial state vector of the Bardeen, Cooper, Schrieffer type. The equations which are derived generalize those of the Hartree-Fock method obtained with a Slater determinant trial wave function. It is pointed out that in a suitable representation the vacuum state of a general quasi-particle transformation has such a trial form which exhibits directly the pairing of single-particle states. Another variational principle determines the excitation energies.Two coupling cases are distinguished: the commutative case in which the self-consistent densities and energies are related to quantities which all commute, and the more general noncommutative case. The latter is of importance in critical-field phenomena. The equations for the commutative case can be written in a matrix form which retains its validity in the more general noncommutative case. The simple matrix commutator equations appear as direct generalizations of the density matrix form of the Hartree-Fock equations.The equations for small oscillations have an equally simple form. Their connection with a diagonal representation of the quasi-particle energies is exhibited in a way which remains valid in the general coupling case. The "unphysical" solutions are excluded by the supplementary condition. The contact with the Green's function approach is established. The generalized matrix form of the Green's function equations shows especially clearly the symmetry properties of the method.
A theory of large amplitude collective motion of a many-particle system is presented, which is relevant, for example, to nuclear fission. The theory is a combination of techniques used in many areas of physics and mathematics. The starting point is the application of the time-dependent Schrödinger equation to generate invariant subspaces of the Hamiltonian in the Hartree–Fock approximation. This is a generalization of the group-theoretical device of generating orbits of a group in the construction of reduced representations. It is shown how solutions of the time-dependent Schrödinger equation can be expressed as instantaneous stationary states of a constrained static Hamiltonian. Thus contact is made with the traditional cranking models and constrained Hartree–Fock theories of large amplitude collective motion. The collective motion is quantized using the Hill–Wheeler–Griffin method of generator coordinates in a basis of generalized coherent states. One is thereby able to exploit much of the theory of harmonic oscillator coherent states, which have been so successfully used in the quantum theory of the laser. The resulting Schrödinger equation for the collective dynamics is expressed both in the Bargmann representation and in the more familiar Schrödinger representation. It is shown that solution of the Schrödinger equation in the small amplitude harmonic approximation reproduces the well-known RPA result. A pilot calculation for 28Si shows that application in large amplitude is also feasible.