2B_MatrixElements

This is a research project with the goal of implementing and evaluate different two-body matrix elements. The main base for the computation is the spherical harmonic oscillator and the matrix elements are coupled in J or JT schemes.

Table of contents

1. Installation
2. Theory
3. Some paragraph
   1. Sub paragraph
   2. Compute valence space using TBME_Runner
   3. Evaluating matrix elements individually
   4. Efficient computing of valence Spaces. TBME_SpeedRunner

Installation

Clone or download the repository. The package requirements are:

• Python 3, at least version 3.8 due the internal use of dictionaries and functions arguments (syntactic sugar).
• numpy, scipy & sympy
• pandas (for testing)
• matplotlib (for testing)

I.e, using pip for the libraries:

pip install pandas

To update Python into a newer version see i.e. How to Upgrade to Python 3.8 on Ubuntu

Theory

Many nuclear codes require require the implementation of a certain Hamiltonian two body interaction, for example, Interaction-Shell-model solves the Schr�dinger equation by direct diagonalization of these hamiltonians. Others, like Mean Field-HFB like codes implement internally (predefined) the two-body Hamiltonian to access larger valence spaces

In this suite, we develop the tools to implement them and program several potentials of common use. The main base of single-particle wave functions is the spherical harmonic oscillator (SHO) (but could be extended to another basis).

The suite have extensions to connect with taurus_vap code, also for the different types of Hamiltonians and for the center of mass correction of this program for no core calculations. See details of that code in:

Symmetry-projected variational calculations with the numerical suite TAURUS

Interactions are deal with the object oriented paradigm, in order to deal with their hierarchical complexity (coupling schemes, radial implementation, wave functions or specific constructions) and reuse certain processes. Details of these ideas and the parameters/definitions of currently implemented matrix elements are in docs/How_to_2B_MatrixElements.pdf.

Usage:

Compute a whole valence space using TBME_Runner, M.E. format � la Antoine

This program was originally designed to feed the interactions for a nuclear mean field code (taurus_vap), that solves the Hartree-Fock-Bogoliubov equations for an arbitrary two-body interaction. The requirements for the Hamiltonian files are based in another shell model code called Antoine, here we use its notation for the harmonic oscillator base.

The requirements for taurus_vap Hamiltonians are explained in docs/How_to_2B_MatrixElements. The class to do this computation is called TBME_Runner and it can only be set by the input file (see How to in the previous document).

from helpers import TBME_Runner

# Write an xml input: define HBarOmega, output filename, Forces ...

# Read and run the 
computation = TBME_Runner('my_input_file.xml')
computation.run()
 
# >> Computes and save the results (creates a /result folder) 

Another way to run it is by terminal, since TBME_Runner is set in main.py to run your input XML file when given as argument.

$ python3 main.py input.xml

Currently implemented two body interactions (All computed in JT scheme):

• Central interactions in the form of:

1. Gaussians (keyword potential='gaussian'): A * exp(-(r/a)^2)
2. r powers (potential='power'): A * (r/a)^N
3. 1/r form (potential='coulomb'): A * (a/r)
4. Exponential (potential='exponential'): A * exp(-(r/a))
5. Yukawa (potential='yukawa'): A * exp(-(r/a)) / (r/a)
6. Gaussian over power of r (potential='gaussian_power'): A * exp(-(r/a)^2) / (r/a)^N

where A, a and N are mandatory parameters to give when setting up. A=constant (in MeV), a=mu_length (in fm) and N=n_power.

• Coulomb interaction (J scheme)
• Brink-Boeker like interactions or gaussian potential series.
• Short range Spin-Orbit approximation potential
• Finite range Spin-Orbit and Finite range Tensor S12 interactions, dependent on arbitrary potentials and usual exchange potentials (Wigner, Barlett, Heisenberg, Majorana).
• Fixed Density dependent (Nuclear Fermi shell filling approximation), updated for a fixed core-nucleus with different oscillator b length and Z/N different from SHO arguments.
• Series of potentials, an extension of the Brink-Boeker sum of radial potential forms (any from the previous Central enumeration) to as many you want (I.e, to expand other potentials in terms of gaussians).
• Get them from a file, reuse previous computations to save time (also valid to multiply the results by a global factor).
• Kinetic two-body matrix element, necessary to evaluate the center of mass correction. Be careful with the internal setting of nucleon mass and HbarC constants in the code.
• Squared LS and ** L^2** finite range interactions, for realistic nucleon-nucleon potentials, also with Exchange operators.
• SDI / MSDI (Surface Delta Interaction / Modified SDI) interaction is Zero-range and isospin-dependant. The modified version append a repulsive constant for the diagonal matrix elements in order to correct the binding energy misbehaviour of the SDI (Brussaard_ & Glaudemans_ book (1977)).
• Multipole-Delta Multipole expansion of the dirac-delta interaction (explicit computation of the delta and sho-wf radial integral)
• Multipole-Moments in the form Q_l*Q_l (Q_l=C_l r^l Y_lm), i.e. QuadrupoleQuadrupole intractions.
• Density-Dependent Forces, set up for Gogny type parametrizations, in spherical symmetry and both for zero-range (also importable from rho/kappa wave functions data file) for D1S or the spherical-Fermi approximation for the finite-range for of the DG.
• Skyrme k-dependent components (IN PROGRESS)

---

Evaluating matrix elements individually

For educational, testing and other purposes, the elements are instanciable to be evaluated. The progress or steps can be printed or debbuged with suitable tools (See for example the XLog class for debugging the matrix element in tree format).

The interactions are classes, where the interaction parameters will be class attributes (statics). The matrix elements are therefore instances of an interaction class. The computation is done internally and then you get the value as an object @property.

from helpers.TBME_Runner import TBME_Runner
from helpers.WaveFunctions import QN_1body_jj, QN_2body_jj_JT_Coupling
from matrix_elements.CentralForces import CentralForce_JTScheme
from helpers.Enums import PotentialForms, SHO_Parameters, BrinkBoekerParameters

# 1. Define the parameters (Names standardized by enumerations) 
kwargs = {
    SHO_Parameters.A_Mass       : 4,
    SHO_Parameters.b_length     : 1.4989,
    SHO_Parameters.hbar_omega   : 18.4586,
    CentralMEParameters.potential : PotentialForms.Power,
    CentralMEParameters.mu_length : 1.0,
    CentralMEParameters.constant  : 1.0,
    CentralMEParameters.n_power   : 0,
}

# 2. Class setting (i.e, set parameters and turn on the XLog debugging)
CentralForce_JTScheme.setInteractionParameters(**kwargs)
CentralForce_JTScheme.turnDebugMode(True)

# 3. Instance the class for the matrix element <0s1/2 0s1/2|V|0s1/2 0s1/2 (J:3T:1)>
#    evaluation is done internally
me = CentralForce_JTScheme(
    	QN_2body_jj_JT_Coupling(QN_1body_jj(0,0,1), QN_1body_jj(0,0,1),1, 0),
    	QN_2body_jj_JT_Coupling(QN_1body_jj(0,0,1), QN_1body_jj(0,0,1),1, 0),
    )
    
# 4. get the m.e. value attribute and save the Log tree (Recoupling, Moshinsky series, etc) in a xml file.
print("me: ", me.value)
me.saveXLog('me_test')

Efficient computing of valence Spaces. TBME_SpeedRunner.

TBME_Runner was a first approach to evaluate the matrix elements, but evaluates the interactions one by one. That is: set an interaction, run the whole valence space for it and then repeat with the next. Nevertheless, most of the matrix elements, besides being in J or JT scheme, they internally are evaluated in the LS scheme. The new runner (TBME_SpeedRunne) invert the process: first it set each interaction (cannot be of the same type) and then evaluates the valence space, doing all the LS/LST evaluations in an intermediate step.

The conditions to run this class are:

• Can only evaluate one time each of matrix element. i.e If you want to evaluate two central gaussian potentials consider use the class Series of gaussians or evaluate them previously with TBME_Runner and import them with from a file.
• There is no limit in the files to import.
• Forces with no inner LS recoupling must be antisymmetrized and normalized. To be executed, implement a dummy method for L and S recoupling to skip them: _validKetTotalSpins _validKetTotalAngularMomentums:

def _validKetTotalSpins(self):
	return tuple()

• main method to run must be _run() and must perform only the normalization and the explicit antisymmetrization (if necessary, remeber that due 9j symmetry this is performed after the LS recoupling). SpeedRunner goes directly to evaluate method _LSCoupledME() defined for the interaction.

In most of the tests SpeedRunner saves nearly a 40% of the total computation time for more than 3 forces, not (1/N_forces)% since the only part of the process skipped is the LS recoupling. For most of the matrix elements, this part less expensive computationally than the inner LS matrix element operations.