Nuclear structure theory strives to build a comprehensive microscopic framework in which bulk nuclear properties, nuclear excitations, and nuclear reactions can all be described. Exotic radioactive nuclei are the critical new focus in this quest. The extreme isospin of these nuclei and their weak binding bring new phenomena that amplify important features of the nuclear many-body problem.
A proper theoretical description of such weakly bound systems requires a careful treatment of the asymptotic part of the nucleonic density. An appropriate framework for these calculations is Hartree-Fock-Bogoliubov (HFB) theory, solved in coordinate representation [1,2]. This method has been used extensively in the treatment of spherical nuclei [3] but is much more difficult to implement for systems with deformed equilibrium shapes. There have been three ways of implementing deformation effects into the coordinate-space HFB. The oldest method, the so-called two-basis method [4,5,6], is based on the diagonalization of the particle-particle part of the HFB Hamiltonian in the self-consistent basis, obtained by solving the HF problem with box boundary conditions. The disadvantage of this method is the appearance of a large number of positive-energy free-particle (box) states, which limits the number of discretized continuum states (the maximum single-particle energy taken in this method is usually less than 10MeV).
The second, very promising strategy, the so-called canonical-basis HFB method, utilizes the spatially localized eigenstates of the one-body density matrix without explicitly going to the quasiparticle representation [7,8,9]. Finally, an approach to axial coordinate-space HFB has recently been developed that uses a basis-spline method [10,11]. While precise, these two latter methods are not easy to implement and, because they are time-consuming, cannot be used in large-scale calculations in which a crucial factor is the ability to perform quick calculations for many nuclei.
In the absence of fast coordinate-space solutions to the deformed HFB equations, it is useful to consider instead the configuration-space approach, whereby the HFB solution is expanded in some single-particle basis. In this context, the basis of a harmonic oscillator (HO) turned out to be particularly useful. Over the years, many configuration-space HFB+HO codes have been developed, either employing Skyrme forces or the Gogny effective interaction [12,13,14,15,16], or using a relativistic Lagrangian [17] in the context of the relativistic Hartree-Bogoliubov theory. For nuclei at the drip lines, however, the HFB+HO expansion converges slowly as a function of the number of oscillator shells [3], producing wave functions that decay too rapidly at large distances.
A related alternative approach that has recently been proposed is to expand the quasiparticle HFB wave functions in a complete set of transformed harmonic oscillator (THO) basis states [18], obtained by applying a local-scaling coordinate transformation (LST) [19,20] to the standard HO basis. Applications of this HFB+THO methodology have been reported both in the non-relativistic [21] and relativistic domains [22]. In all of these calculations, specific global parameterizations were employed for the scalar LST function that defines the THO basis. There are several limitations in such an approach, however. For example, the minimization procedure that is needed in such an approach to optimally define the basis parameters is computationally very time-consuming, making it very difficult to apply the method systematically to nuclei across the periodic table.
Recently, a new prescription for choosing the THO basis has been proposed and employed in self-consistent large-scale calculations [23]. For a given nucleus, the new prescription requires as input the results from a relatively simple HFB+HO calculation, with no variational optimization. The resulting THO basis leads to HFB+THO results that almost exactly reproduce the coordinate-space HFB results for spherical nuclei [24]. Because the new prescription requires no variational optimization of the LST function, it can be applied in systematic studies of nuclear properties. In order to correct for the particle number nonconservation inherent to the HFB approach, the Lipkin-Nogami prescription for an approximate particle number projection, followed by an exact particle number projection after the variation has been implemented the code HFBTHO (v1.66p) [25,26].
The paper is organized as follows. Section 2 gives a brief summary of the HFB formalism. The implementation of the method to the case of the Skyrme energy density functional is discussed in Sec. 3, together with the overview of the THO method and the treatment of pairing. Section 4 describes the code HFBTHO (v1.66p). Finally, conclusions are given in Sec. 5.