This library implements accurate and efficient radial Schrödinger and Dirac finite element solvers. The formulation admits general potentials and meshes: uniform, exponential, or other. Additionally, a squared Hamiltonian approach has been used for the Dirac equation, which eliminates spurious states.

Detailed description of methods, convergence studies and implementation details may be found in the following article:

Čertík, Ondřej, et al. High-Order Finite Element Method for Atomic Structure Calculations. Computer Physics Communications, Volume 297, 2024, ISSN 0010-4655. https://www.sciencedirect.com/science/article/pii/S001046552300396X, http://arxiv.org/abs/2307.05856.

With the provided meshes, the solvers (both Schrödinger and Dirac) can converge to at least 1e-8 Ha accuracy (with double precision of approximately 16 significant digits) for all eigenvalues and total DFT energies for all atoms up to uranium (Z=92).

The converged nonrelativistic and relativistic results agree with dftatom to 1e-8 Ha accuracy, and with the NIST benchmarks to the stated accuracy of those benchmarks (2e-6 Ha in eigenvalues and 1e-6 Ha in total energies).

http://physics.nist.gov/PhysRefData/DFTdata/Tables/ptable.html

The accuracy is on par with dftatom and uses significantly less computationally expensive routines.

This program is packaged with fpm, the Fortran package manager.

# All of these can be passed the --profile=release flag
fpm build
fpm test
fpm run --profile=release conv -- 0 0 5

Where the parameters to conv are:

! <study_type> can be,
!       0: error as p is varied
!       1: error as rmax is varied
!       2: error as Ne is varied
!
! <equation> can be,
!       0: Schroedinger
!       1: Dirac
!
! For <study_type>
!       0, 1: 3rd parameter = Ne (Number of elements)
!       2   : 3rd parameter = p  (Polynomial order)

We can use an anaconda helper like micromamba (installation instructions are here.

We can now set up the tools needed. We support both fpm and meson as build systems.

# Global
micromamba install fpm meson  -c conda-forge
# Project Local
micromamba create -p ./tmp fpm meson -c conda-forge
micromamba activate ./tmp
# Optionally: blas lapack openmp gfortran
# Best obtained with a package manager
# Alternative
micromamba create -f environment.yml # creates fe
micromamba activate fe

fpm build
fpm run --profile=release conv -- 0 0 5

# release mode is the default
meson setup bbdir -Dwith_app=true
./bbdir/app/conv 0 0 5

Tests can be run by:

fpm test
# Or, changing to debug
meson setup bbdir -Dwith_tests=true --buildtype=debug
meson test -C bbdir
1/8 CoulombSchroed         OK              0.05s
2/8 DftSchroedFast         OK              0.06s
3/8 HarmonicSchroed        OK              0.11s
4/8 DftSchroed             OK              0.12s
5/8 HarmonicDirac          OK              0.47s
6/8 CoulombDirac           OK              0.49s
7/8 DftDiracFast           OK              1.95s
8/8 DftDirac               OK              6.45s

Ok:                 8   
Expected Fail:      0   
Fail:               0   
Unexpected Pass:    0   
Skipped:            0   
Timeout:            0   

Individual test binaries can also be executed, e.g.:

meson compile -C bbdir
./bbdir/testDftDirac

The API documentation is generated by doxygen with the doxyYoda theme.

To build a local copy and serve it consider:

bash scrpits/mkdoxydoc.sh

This program is MIT licensed, see the LICENSE file for details.