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projects [2014/07/08 14:03] schunck created |
projects [2014/07/25 10:24] schunck [2) HF Problem in the Full Spherical Harmonic Oscillator Basis] |
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====== Computational Projects ====== | ====== Computational Projects ====== | ||
+ | As a starting point, we will ask every group of students to write a program (in the language of their choice) solving the Hartree-Fock equations in the spherical harmonic oscillator basis. To facilitate benchmarks, we chose to focus on systems of neutrons that are confined in a harmonic trap and are interacting via a schematic potential called the Minnesota potential. Spherical symmetry will be assumed throughout. Even this simplified system may require a significant effort, so we tried to make your life a little easier. | ||
+ | * We provide most of the necessary background about the harmonic oscillator in this {{HO_spherical.pdf|document}}; | ||
+ | * We give you, without demonstration, the matrix elements of the kinetic energy operator (see the same document); | ||
+ | * Upon request, we can make available to you various Fortran subroutines giving the nodes and weights of the Gauss-Laguerre quadrature. As an illustration, {{exp.cpp.gz|this program}} illustrates the various possible techniques to integrate numerically $\int_{0}^{\infty} xe^{-x}dx = 1$ | ||
+ | * We also provide [[https://wikihost.nscl.msu.edu/TalentDFT/doku.php?id=codes|DFT solvers]] that you can use as benchmarks. | ||
+ | And of course, we will be around to help you, so don't hesitate to ask if you have questions. | ||
+ | ==== 1) HF Problem in a Truncated Model Space ==== | ||
+ | |||
+ | In a first step, we will solve the HF equations for the system of N neutrons in a trap interacting with the Minnesota potential in a restricted basis consisting of only l=0 states. This simplifies tremendously the calculation of two-body matrix elements. We broke down the problem into several simple steps that are explained in this {{HF_truncated_v2.pdf|document}}. | ||
+ | |||
+ | ==== 2) HF Problem in the Full Spherical Harmonic Oscillator Basis ==== | ||
+ | |||
+ | Now that each group (hopefully) has a working HF code for the truncated S-wave model space model, we are ready to attack the general case. Since the generation of two-body matrix elements (TBMEs) is tedious for the general case, we have uploaded a F90 pack to do this for you, see {{:relcom2labsystem.tar.gz|here}}. We will give more explanations on how to use the code in the lectures. Finally, the following document outlines how the HF equations look in spherical symmetry for general model spaces, {{:hf_fullspherical.pdf|}}. We give below a description of what is contained in the files. | ||
+ | |||
+ | **Single-particle files** - The single-particle files are named **spM.dat** and **spJ.dat** for the M-scheme and coupled J scheme, respectively. The first line (starting with "Legend") gives the meaning of the numbers | ||
+ | * For the M-scheme: counter $n\ l\ 2j\ 2m_j\ 2t_z$ | ||
+ | * For the J-scheme: counter $n\ l\ 2j\ t_z\ 2n+l$ | ||
+ | |||
+ | **Interaction files** - The interaction files are named **VM-scheme.dat** and **VJ-scheme.dat**, respectively (for the M-scheme and the J-scheme). | ||
+ | * For the M-scheme: $a\ b\ c\ d\ v_{abcd}$ | ||
+ | * For the J-scheme: $t_z\ \pi\ 2J\ a\ b\ c\ d\ v_{abcd}$ | ||
+ | where $\pi$ is the parity. In each case, the a, b, c, d numbers refer to the counters listed in the single-particle files. | ||
+ | |||
+ | ==== 3) Possible Extensions for Week 3 and Later ==== | ||
+ | |||
+ | {{ :scheme.png?300 |}} | ||
+ | |||
+ | __References__ | ||
+ | * S. K. Bogner, R. J. Furnstahl, H. Hergert, M. Kortelainen, P. Maris, M. Stoitsov, and J. P. Vary, //Testing the density matrix expansion against ab initio calculations of trapped neutron drops//, [[https://journals.aps.org/prc/abstract/10.1103/PhysRevC.84.044306|Phys. Rev. C 84, 044306 (2011)]] |