Introduction

Within W-SLDA Toolkit we provide a script for generating custom SLDA functional: extensions/sldae/sldae-dev-tool.c

Within the script user can provide its own formulas for:

double ground_state_energy (int dn, double _x, int id []);
double chemical_potential (int dn, double _x, int id []);
double inverse_effective_mass (int dn, double _x, int id []);
double pairing_gap (int dn, double _x, int id []);

In return, the script generates a source file readable by W-SLDA Engine. Once the functional is created, it becomes available for all code within the toolkit.

Example: SLDAE with the effective mass equal 1

The function for the computation of the effective mass has been modified to:

double
inverse_effective_mass (int dn, double _x, int id [])
{
    int ip0 [1] = {0}; double x0 = xm (dn, _x, ip0);
    // ...
    return x0;   
}

Next, the code is compiled and executed:

[gabrielw@dell sldae]$ gcc sldae-dev-tool.c -lm -lgsl -lgslcblas -o sldae-dev-tool       
[gabrielw@dell sldae]$ ./sldae-dev-tool 
 SLDAE DEV TOOLS
 [padé[4/4] interpolation of sldae parameters p = {beta, gamma, B, C}]
    p       a1 x + a2 x^2 + a3 x^3 + c x^4  
  ----- = ----------------------------------
  p_ufg   1 + b1 x + b2 x^2 + b3 x^3 + c x^4


 beta interpolation (beta_ufg = -0.481506)
         iter =  46      f(x) =  1.110e-16 
         a1 = +0.797400;
         a2 = +0.943961;
         a3 = +1.037043;
         b1 = +1.227062;
         b2 = +2.126210;
         b3 = +1.734795;
         c  = +0.523445;


 b_functional interpolation (b_functional_ufg = -0.481506)
         iter =  13      f(x) =  8.882e-16 
         a1 = +0.698112;
         a2 = +0.034678;
         a3 = +0.455138;
         b1 = +0.353905;
         b2 = +0.806444;
         b3 = +0.403679;
         c  = -0.031327;


 inverse_gamma interpolation (inverse_gamma_ufg = -0.094945)
         iter =  14      f(x) =  1.821e-14 
         a1 = +2.713374;
         a2 = +1.441196;
         a3 = +3.700251;
         b1 = +0.634303;
         b2 = +3.522427;
         b3 = +3.701347;
         c  = +1.877961;


 c_functional interpolation (c_functional_ufg = -10.532397)
         iter =  19      f(x) =  3.664e-15 
         a1 = +0.370145;
         a2 = +1.041299;
         a3 = +1.737347;
         b1 = +2.458364;
         b2 = +5.604947;
         b3 = +3.828124;
         c  = +1.085400;

Parameters written to: sldae_parameters.h
Creating file: sldae_parameters.txt

Once the functional is generated user should inspect content of file sldae_parameters.txt. It contains following columns:

# 1: lambda=a*k_F
# 2: ground_state_energy
# 3: inverse_effective_mass
# 4: chemical_potential
# 5: pairing_gap
# 6: beta_parameter
# 7: beta_parameter - PADE
# 8: b_functional
# 9: b_functional - PADE
#10: inverse_gamma_parameter
#11: inverse_gamma_parameter - PADE
#12: c_functional
#13: c_functional - PADE

In particular, it is important to confirm that PADE approximants correctly reproduce functions generated by analytical formulas. Note that W-SLDA Engine uses Pade approximants to deliver the highest performance.

[gabrielw@dell sldae]$ gnuplot 
gnuplot> plot "sldae_parameters.txt" u 1:6 w l, "" u 1:7 w l  
# and similar for other functions

Finally, file sldae_parameters.h should be moved to folder hpc-engine. From now new SLDAE functional will be used when FUNCTIONAL=SLDAE is selected in predefines.

Collection of own SLDAE-like functionals

The information about PADE approximants for functional coefficients loded to the engine through the file hpc-engine\sldae_functional.h, precisely:

// Select coefficients
#ifdef SLDAE_FORCE_A1
#include "sldae_parameters_A1.h"
#else
// default one
#include "sldae_parameters.h"
#endif 

User can modify this section to form that uploads a correct file with the desired set of coefficients.