# GCM Solver (Ver. 3.0): A Mathematica Notebook for Diagonalization of the Geometric Collective Model (Bohr Hamiltonian) with Generalized Gneuss–Greiner Potential

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## Abstract

**:**

## 1. Introduction and Motivation

## 2. Physics Background

#### Matrix Elements

## 3. Program Explanation

`basismodules.nb`, namely the

`basis[Nph]`and

`subbasis[J,Nph]`. The first generates the full basis for a given number of phonons and the second extracts the subbasis corresponding to a given value of angular momentum. Notice that the output of these modules is not directly written into the code. A number of arrays and variables are used: for instance,

`dtot`gives the total dimension of the full basis, while

`Nst`and

`Nst2`, which are used throughout the program, give the subbasis’ dimension and the dimension upon trimming the last six phonon numbers.

`BasMat`, one type for each potential energy term and angular momentum. The matrices are always read for ${N}_{max}=40$. This is a practical solution, because matrices corresponding to bases of smaller dimensions, if needed, can be extracted by cutting the appropriate upper-left submatrix. The matrices are used to write the hamiltonian matrix, as a linear combination with certain real coefficients (although physically unjustified, nothing prevents us from using complex coefficients) that must be supplied by the user. The paper of Habs et al. [32] gives a list of fitted coefficientsfor a few medium mass nuclei. One can use some of these examples to run the code, keeping in mind that some of those coefficients have been fitted on old spectroscopic data and that modern experiments have discovered new states and changed some of the older assessments. Therefore, with those coefficients, one can, at most, hope to reproduce their results, while to get a better result one should select the states of interest, run again some multidimensional fitting procedure (least-squares, random steps, Monte Carlo inspired techniques, etc.) and diagonalize the matrix with new coefficients.

## 4. BasMat Library Description

`N40`folder contains subfolders, one for each set of values of the angular momentum, namely $L=0,2,3,4,5,6,7,8$ that allow going up to seniority $\ell =5$. For each pair of values $\{{N}_{max},L\}$, there is thus a subdirectory where a certain number of matrices are stored in Harwell–Boeing format (

`.rha`corresponding to real hermitian). This is a convenient and widely used format for storing sparse matrices. The library has been generated with another Mathematica notebook (not published) on a 24 processors Intel(R) Xeon(R) E5-2440 machine with 48 Gbyte RAM and Linux O.S. Although it is not easy to estimate the timing, due to multiple users and several processes running, it has taken about 2 h for the smallest matrix ($L=3$ with $dim=176$) and more than a week for one of the largest ($L=8$ with $dim=771$). Apparently, the time scales in a difficult-to-predict way that is not only controlled by the basis dimension, but also by the time it takes to calculate to a given precision integrals involving wavefunctions (polynomials) with growing value of L. The library matrices are read by the program depending on the chosen level of convergence. Higher maximum phonon numbers insure higher convergence, but at the price of higher computational costs. Therefore, it might be better to set smaller values of ${N}_{max}$ for trials and go for the higher values only at the moment of getting results. In those cases, although all matrix elements are read, only a submatrix corresponding to a smaller phonon number is used in computations. The preliminary readings and loading of databases is done in Part-1 of the code.

#### 4.1. Control of “Border Effects” Due to the Truncation of the Matrices

#### 4.2. Scaling Factor in the Radial Variable

`sfind[]`function that takes, as input, some of the coefficients.

## 5. Model Calculations and Examples

`coeff-habs.nb`and

`coeff-fortunato.nb`contain “best” sets of coefficients obtained from literature in the first case or from extensive trials with this program on state-of-the-art high-performance cpu’s in the second case. Of course, one can never be sure that the algorithm has hit the absolute minimum and these sets can be probably improved by running more and more random walk fitting procedures (for example, with a faster computer), changing the initial conditions, etc. Another source of discrepancies might be that some of the energies have been slightly re-determined since the times when the paper by Habs et al. was written.

`data-sets.nb`contains a few sets of experimental data that can be called in connection with the coefficients of the previous paragraph.

## 6. Playing Area and Fitting Subroutines

`data-sets.nb`file. If a particular dataset is wanted, but not contained in this file, it can be easily added by the user. The type and number of states to be compared can easily be changed by adapting the last line of the

`f[]`function. In the present version, the first ${0}^{+},{2}^{+},{4}^{+}$ and ${6}^{+}$ of the ground state band;, the ${2}^{+},{3}^{+}$ and ${5}^{+}$ of the gamma band; and the ${0}^{+}$ of the beta band are used. Notice that the 3 and 5 states are rather isolated (other occurrences of these values of J appear only at high energy), which is why they are used in place for example of the ${4}_{2}^{+}$ that is amidst or very close to other ${4}^{+}$ states and might be highly mixed with them. Often, one has to guess which excited ${0}^{+}$ state might be the bandhead of the beta band, and it might happen that is not always the lowest. Then, a random fitting procedure is given: starting from initial foul’s parameters, it performs a random walk in parameter space trying to find the minimum. Normall,y a few runs are enough to identify deep local minima, although with this technique one cannot be insured to have hit the lowest absolute minimum. The suggestion here is to run the routine from scratch a number of times (by experience, about 10 or 20 times) and try to seek the lowest minimum. When this is done, one can run a second minimization routine with a smaller variation of parameters to slowly minimize the sum of squared deviations. The final results are usually very good, although there might be cases where the whole procedure does not work. Finally, one can draw the potential energy surface as a contour plot or in 3D. We show in Figure 2 two examples, one for a spherical minimum and one with two almost coexisting minima in ${}^{180}$Pt (notice that the deformed minimum is lower). The program adjusts itself to plot things around the minimum, but of course the user can force any range upon the plotting routine.

`FindMinimum[]`command line.

## 7. Installation Tips

`GCMsolver_3.nb`,

`basismodules.nb`,

`data-sets.nb`,

`coeff-habs.nb`and

`coeff-fortunato.nb`(Supplementary Materials) in a directory and run the first under Mathematica. There might be an issue in setting directory and subdirectory names and paths, namely certain systems use the slashes and others backslashes. The user should adjust that according to the system in use.

## 8. Conclusions

## Supplementary Materials

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Output of the diagonalization routine for ${}^{128}$Xe. The first panel shows absolute energies in MeV, the second shows energies relative to the ground state and the third shows reduced energies, i.e., the energies of the second panel normalized with the energy of the first ${2}^{+}$ state.

**Figure 2.**Output of the contour plot drawing subroutine for ${}^{128}$ Xe from the

`habs[]`database and ${}^{180}$Pt from the

`fort[]`database. The plots are given as a function of $(\beta ,\gamma )$.

**Table 1.**Dimension of extended subbasis (second line) and trimmed subbasis (last line) for angular momentum used in the program (first line).

L | 0 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|

dim | 200 | 376 | 176 | 529 | 330 | 660 | 462 | 771 |

dim (trimmed) | 147 | 273 | 127 | 380 | 234 | 469 | 324 | 540 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Ferrari-Ruffino, F.; Fortunato, L.
GCM Solver (Ver. 3.0): A *Mathematica* Notebook for Diagonalization of the Geometric Collective Model (Bohr Hamiltonian) with Generalized Gneuss–Greiner Potential. *Computation* **2018**, *6*, 48.
https://doi.org/10.3390/computation6030048

**AMA Style**

Ferrari-Ruffino F, Fortunato L.
GCM Solver (Ver. 3.0): A *Mathematica* Notebook for Diagonalization of the Geometric Collective Model (Bohr Hamiltonian) with Generalized Gneuss–Greiner Potential. *Computation*. 2018; 6(3):48.
https://doi.org/10.3390/computation6030048

**Chicago/Turabian Style**

Ferrari-Ruffino, Fabrizio, and Lorenzo Fortunato.
2018. "GCM Solver (Ver. 3.0): A *Mathematica* Notebook for Diagonalization of the Geometric Collective Model (Bohr Hamiltonian) with Generalized Gneuss–Greiner Potential" *Computation* 6, no. 3: 48.
https://doi.org/10.3390/computation6030048