**B.G. Carlsson, J. Dobaczewski, J. Toivanen, P. Veselý**

We present solution of self-consistent equations for the NLO
nuclear energy density functional. We derive general expressions for
the mean fields expressed as differential operators depending on
densities and for the densities expressed in terms of derivatives of wave
functions. These expressions are then specified to the case of
spherical symmetry. We also present the computer program HOSPHE
(v1.00), which solves the self-consistent equations by using
the expansion of single-particle wave functions on the spherical
harmonic oscillator basis.

**PROGRAM SUMMARY**

*Manuscript Title:*
Solution of self-consistent equations for the NLO nuclear
energy density functional in spherical symmetry.
*Authors:*
B.G. Carlsson,
J. Dobaczewski,
J. Toivanen,
and
P. Veselý
*Program Title:* HOSPHE (v1.00)
*Journal Reference:*

*Catalogue identifier:*

*Licensing provisions:* none

*Programming language:* FORTRAN-90
*Operating system:* Linux

*RAM:* 50MB

*Number of processors used:* 1

*Keywords:*
Hartree-Fock, Skyrme interaction, nuclear energy density functional,
self-consistent mean-field

*PACS:* 07.05.Tp, 21.60.-n, 21.60.Jz

*Classification:* 17.22 Hartree-Fock Calculations

*External routines/libraries:* LAPACK, BLAS

*Nature of problem:*

The nuclear mean-field methods constitute principal tools of a
description of nuclear states in heavy nuclei. Within the Local Density Approximation
with gradient corrections up to NLO [1], the nuclear mean-field is
local and contains derivative operators up to sixth order. The
locality allows for an effective and fast solution of the
self-consistent equations.
*Solution method:*

The program uses the spherical harmonic oscillator basis to expand
single-particle wave functions of neutrons and protons for the
nuclear state being described by the NLO nuclear energy density
functional [1]. The expansion coefficients are determined by the
iterative diagonalization of the mean-field Hamiltonian, which
depends non-linearly on the local neutron and proton densities.
*Restrictions:*

Solutions are limited to spherical symmetry. The expansion
on the harmonic-oscillator basis does not allow for
a precise description of asymptotic properties of wave functions.
*Running time:*

50 sec. of CPU time for the ground-state of Pb described by
using the maximum harmonic-oscillator shell included in
the basis.
*References:*

- [1]
- B.G. Carlsson, J. Dobaczewski, and M. Kortelainen, Phys. Rev. C 78, 044326 (2008).

**LONG WRITE-UP**

- Introduction
- Overview of the method
- General forms of the NLO potentials, fields, and densities
- Building blocks
- Potentials
- Fields
- Fields for terms containing additional density dependence
- Rearrangement terms
- Densities

- The NLO potentials, fields, and densities in the spherical harmonic oscillator basis
- Spherical HO basis
- Densities in the spherical HO basis
- Potentials in the spherical HO basis
- Matrix elements of the Hamiltonian (60) in the spherical HO basis
- Matrix elements of the Hamiltonian (7) in the spherical HO basis
- The total potential energy in the spherical symmetry
- Direct Coulomb energy in the spherical symmetry
- Exchange Coulomb energy
- Numerical integration

- Overview of the software structure
- Description of the individual software components
- Differences between the notation in the code HOSPHE (v1.00) and in the text
- Description of input data
- Installation instructions
- Test run description
- Summary
- Acknowledgements
- Calculation of coefficients (72)
- Derivation of Eq. (98)
- Derivation of Eq. (113)
- Bibliography
- About this document ...