Analytic parameterizations of the UEG local field factors

Maintainer: Aaron Kaplan

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This CC0-licensed repo contains Python3 libraries for computing the uniform electron gas (UEG) local field factors (LFFs) developed by Aaron Kaplan and Carl Kukkonen in:

A.D. Kaplan and C.A. Kukkonen, "QMC-consistent static spin and density local field factors for the uniform electron gas", arXiv:2303.08626 (2023). In review at Phys. Rev. B.

Please cite the previously listed paper when using this code or the results of our paper.

Basic use of the fitted LFFs

To use the fitted LFFs in Python code, type from AKCK_LFF.fitted_LFF import g_plus_new, g_minus_new The arguments of both g_plus_new and g_minus_new are, in order, the wavevector (units: inverse bohr), and the Wigner-Seitz (WS) radius / r_s (units: bohr).

Both g_plus_new and g_minus_new are wrappers around the parent structure simple_LFF. This function takes, in order, the wavevector (units: inverse bohr), WS radius (units: bohr), coefficients (dimensionless), and variant (string, either G₊ by setting var == "+"", or G_- by setting var == "-"). var controls which set of asymptotic coefficients are used.

To use the improved fit of the correlation spin stiffness in G_-, use the optional keyword argument acpars = 'AKCK'. By default, acpars = 'PW92', which uses the Perdew-Wang expression.

Detailed breakdown of the repo contents

These are listed in order of their appearance on the Github repo. Unless specified otherwise, the units used in the following are hartree atomic units: lengths are measured in bohr radii, and energies in hartree (1 hartree = 2 Ry).

Directories

• data_files

  • Files of the form CK_G*_rs_*.csv contain Quantum Monte Carlo (QMC) data on G₊ and G_- as a function of the wavevector (units: Fermi wavevector) from Ref. 1 (see References section below). Individual files are for a fixed value of the WS radius (units: bohr). When plus appears before _rs_ in the file name, data for G₊ is tabulated. Identically, when minus appears before _rs_ in the file name, data for G_- is tabulated. The value of the WS radius appears after _rs_ and before .csv.

  • Files of the form MCS_Gplus_rs_*.csv present QMC data for G₊ at a fixed value of the WS radius (same naming convention as before), but from Ref. 2.

• ec_data

  • Files of the form eps_c_*.csv tabulate computed correlation energies (units: hartree, from the adiabatic connection fluctuation dissipation theorem) as a function of the WS radius. Some keywords are:

    • COR: the wavevector-dependent exchange-correlation (xc) kernel of Corradini et al. [3]
    • NEW: the present wavevector-dependent density LFF, or xc kernel
    • RAD: the wavevector- and frequency-dependent Richardson-Ashcroft xc kernel [4]
    • RAS: the static limit of the Richardson-Ashcroft kernel [4], which is still wavevector-dependent
    • RPA: the random phase approximation, whereby the xc kernel is set to zero
    • rMCP07: the wavevector- and frequency-dependent revised MCP07 kernel of Kaplan et al. [5]
    • The suffix _GKQ indicates a result from globally-adaptive Gauss-Kronrod quadrature, whereas no suffix indicates a result using the RPA cutoffs

  • RPA_cutoffs.csv contain the numeric RPA cutoffs described in the work

  • eps_c_err.pdf is Fig. 2 of the main text

  • RPA_sanity.tex is Table S2

• figs contains the figures of the main text and supplemental material

• figs_from_fit contains plots similar to those of the manuscript, but generated immediately after the fitting procedure. These plots may use parameters in the LFFs that are not truncated at a fixed number of digits.

• fitted_LFF_pars contain the fitted parameters in the LFFs as both CSV- and LaTeX-formatted tables. Initial guesses for the parameters are contained in optpars_g*.csv.

• quad_grids is generated by gauss_quad.py, and contains grid points and weights for Gaussian quadrature

  • GKQ indicates Gauss-Kronrod quadrature
  • GLQ indicates Gauss-Legendre quadrature

• stiffness_refit contains the revised pars in the correlation spin stiffness (alphac_pars_rev.tex), and the susceptibility enhancement in tabular (chi_enhance.tex) and pictoral (suscep_enhance.pdf) form

Python files

• PW92.py contains both the Perdew-Wang parameterization of the UEG correlation energy [6] in ec_pw92, and the Pade approximant of the UEG on-top pair distribution function [7] in g0_unp_pw92_pade

• PZ81.py contains the Perdew-Zunger parameterization of the UEG correlation energy [8] in ec_pz81 and of the correlation spin-stiffness in alpha_c_pz81

• QMC_data.py contains QMC-computed UEG correlation energies from previous works. Citations are given throughout the file. This file is used in stiffness_refit.py.

• alda.py provides the adiabatic local density approximation to the xc kernel, which depends only on the WS radius. This file is modified from a previous repo maintained by the author, https://github.com/esoteric-ephemera/tc21/tree/master/dft Note that function alda thus requires a dictionary entry dv which takes the density n, Fermi wavevector kF, WS radius rs, and square-root of the WS radius rsh as keys. Possible optional keywords are x_only for exchange only (default: False), and parameterization of the kernel (default: PZ81, PW92 is another possible option).

• alpha_c_c1.py recomputes the value of the next-to-leading order term in the high-density expansion of the correlation spin-stiffness, as in Eq. (19). To call its functionality, use from AKCK_LFF.alpha_c_c1 import integrate_funs, and simply run integrate_funs().

• asymptotics.py provides the asymptotic expansion coefficients of G₊ in get_g_plus_pars, which takes only the WS radius as input, and of G_- in get_g_minus_pars, which takes the WS radius and relative spin polarization as input.

• corr.py computes the correlation energies of various density LFFs, as plotted in Fig. 2. To generate Fig. 2, run corr_plots(). corr_plots takes an optional keyword argument, gpl (list). To modify which kernels/LFFs are used, use the same keywords as in the ec_data/eps_c_*.csv files, described above.

• fit_LFF.py is the main fitting routine used here. Its usage syntax is from AKCK_LFF.fit_LFF import fitparser, fitparser(routine,manip,rs=None). Running this file requires secondary options

  • routine (string):
    • init: generates initial parameters for select values of the WS radius
    • manip: allows the user to manually manipulate the parameters in the LFF
    • main: main fitting routine which provides the parameters in the text
  • var (string) is the variant of the LFF, either + for G₊ or - for G_-.
  • optional keyword argument rs (float/int)
  • optional keyword argument acpars (string), defaults to 'PW92' to use the Perdew-Wang parameterization of the correlation spin stiffness. Use acpars = 'AKCK' to use the improved fit of the correlation spin stiffness.

• fit_RPA_cutoffs.py fits the numeric RPA cutoffs for the correlation energy, as described in the supplemental material. Run gen_RPA_cutoffs() (no arguments).

• fitted_LFF.py is described in the Basic use of the fitted LFFs section.

• g_corradini.py contains the density LFF G₊ of Corradini et al. [3] in g_corradini. This function requires the wavevector (unit: inverse bohr) and a density dictionary similar to alda as input.

• gauss_quad.py, modified from https://github.com/esoteric-ephemera/tc21/tree/master/, is an all-purpose numeric integrator using adaptive mesh refinement.

• mcp07_static.py contains the static limit of the modified CP07 kernel of Ruzsinszky et al. [9], in mcp07_static. This function takes the wavevector and density dictionary as inputs. An optional keyword, param (default PZ81) can use either the Perdew-Zunger [8] (PZ81) or Perdew-Wang [6] (PW92) parameterization of the UEG correlation energy as input.

• plot_LFFs.py generates plots of G₊ (gplus_plots) and G_- (gminus_plots) shown in the manuscript. Neither function takes arguments.

• rMCP07.py contains the wavevector- and frequency-dependent density LFF of Kaplan et al. [5] in g_rMCP07. Note that this function assumes that the frequency is purely imaginary. This function takes the wavevector, imaginary part of the frequency (entered as a real number), and density dictionary as input.

• ra_lff.py contains the various LFFs of Richardson and Ashcroft [4]. To use their density LFF G₊, call g_plus_ra; to use their spin LFF G_-, call g_minus_ra. Both take the wavevector (units: bohr) and imaginary part of the frequency (units: inverse hartree energy), and WS radius (units: bohr) as inputs. Both assume that the frequency is purely imaginary.

• stiffness_refit.py refits the correlation spin stiffness using the PW92 framework. To call its functionality, use fit_alpha_c_new() (takes no arguments)

• surf_plots.py produces the surface plots in the supplemental material. Call surf_plots() (takes no arguments).

References

1. C.A. Kukkonen and K. Chen, Phys. Rev. B 104, 195142 (2021). DOI: 10.1103/PhysRevB.104.195142

2. S. Moroni, D. M. Ceperley, and G. Senatore, Phys. Rev. Lett. 75, 689 (1995). DOI: 10.1103/PhysRevLett.75.689

3. M. Corradini, R. Del Sole, G. Onida, and M. Palummo, Phys. Rev. B 57, 14569 (1998). DOI: 10.1103/PhysRevB.57.14569

4. C. F. Richardson and N. W. Ashcroft, Phys. Rev. B 50, 8170 (1994). DOI: 10.1103/PhysRevB.50.8170

5. A. D. Kaplan, N. K. Nepal, A. Ruzsinszky, P. Ballone, and J. P. Perdew, Phys. Rev. B 105, 035123 (2022). DOI: 10.1103/PhysRevB.105.035123

6. J.P. Perdew and Y. Wang, Phys. Rev. B 45, 13244 (1992). DOI: 10.1103/PhysRevB.45.13244

7. J. P. Perdew and Y. Wang, Phys. Rev. B 46, 12947 (1992). DOI: 10.1103/PhysRevB.46.12947, and erratum Phys. Rev. B 56, 7018 (1997). DOI: 10.1103/PhysRevB.56.7018

8. J. P. Perdew and A. Zunger, Phys. Rev. B 23, 5048 (1981). DOI: 10.1103/PhysRevB.23.5048

9. A. Ruzsinszky, N. K. Nepal, J. M. Pitarke, and J. P. Perdew, Phys. Rev. B 101, 245135 (2020). DOI: 10.1103/PhysRevB.101.245135