A fast and precise DFT wavelet code

Basis-set convergence

A fast and precise DFT wavelet code
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Contents

Running a wavelet computation on a CH4 molecule

The purpose of this lesson is to get familiar with the basic variables needed to run a wavelet computation in isolated boundary conditions. At the end of the lesson, one can run a wavelet run, check the amount of needed memory and understand the important part of the output.

Introduction: running the code

This lesson is based on this skeleton input.yaml file.

Beside this input file, BigDFT requires the atomic positions for the studied system and optionaly the pseudo-potential files. For the following tutorial, a methane molecule will be used. The position file is a simple XYZ file named posinp.xyz:

5  angstroemd0  # methane molecule
free
C        0           0           0
H       -0.63169789 -0.63169789 -0.63169789
H       +0.63169789 +0.63169789 -0.63169789
H       +0.63169789 -0.63169789 +0.63169789
H       -0.63169789 +0.63169789 +0.63169789

The pseudo-potential files are following the ABINIT structure and are of GTH or HGH types (see the pseudo-potential file] page on the ABINIT website for several LDA and GGA files and the page of M. Krack on the CP2K server for HGH pseudo for several functionals). The following files may be used for this tutorial: psppar.C and psppar.H.

Running BigDFT is done using the bigdft executable in a standard Unix way, the output being by default the standard output, it must be redirected to a file or to a pipe, like with the unix command tee:

user@garulfo:~/CH4/$ ls
bigdft    psppar.C     psppar.H     input.yaml    posinp.xyz
user@garulfo:~/CH4/$ ./bigdft | tee screenOutput
...

Warning, to run properly, the pseudo-potential files must be psppar.XX where XX is the symbol used in the position file. The other files can have user-defined names, as explained in "First runs with BigDFT" lesson.

If the code has been compiled with MPI capabilities (which is enabled by default), running BigDFT on several cores is as easy as run it as a serial job. There is no need to change anything in the input files. The following example shows how to run it on a Debian system with installed OpenMPI on a 4 core machine:

user@garulfo:~/CH4/$ ls
bigdft    psppar.C     psppar.H     input.yaml    posinp.xyz
user@garulfo:~/CH4/$ mpirun -np 4 ./bigdft | tee screenOutput
...

The wavelet basis set, a convergence study

The wavelet is a systematic basis set (as plane waves are), which means than one can increase arbitrarily the accuracy of the results by varying some parameters.

visualisation of the real space mesh

hgrid

The second and third lines of input.yaml are used to set up the basis set. In free boundary conditions, the basis set is characterised by a spatial expansion and a grid step, as shown in the side figure.

There is one float value on the second line describing the grid steps in the three space directions (i.e. x, y and z). These values are in bohr unit and typically range from 0.3 to 0.65. The harder the pseudo-potential, the lower value should be set up. These values are called hgrid.

crmult, frmult

The third line contains two float values that are two multiplying factors. They multiply quantities that are chemical species dependent. The first factor is the most important since it describes the spatial expansion of the basis set (in yellow on the figure beside). Indeed the basis set is defined as a set of real space points with non-zero values. These points are on a global regular mesh and located inside spheres centered on atoms. The first multiplying factor is called crmult for Coarse grid Radius MULTiplier. Increasing it means that further spatial expansion is possible for the wavefunctions. Typical values are 5 to 7.

Exercise: run BigDFT for the following values of hgrid and crmult and plot the total energy convergence versus hgrid. The final total energy (keyword EKS) can be retrieved at the end of the screen output, or using this command `grep FINAL screenOutput`, the value is in Hartree. A comprehensive explanation of the screen output will be given later in this tutorial.

Convergence rate
hgrid = 0.55bohr / crmult = 3.5
hgrid = 0.50bohr / crmult = 4.0
hgrid = 0.45bohr / crmult = 4.5
hgrid = 0.40bohr / crmult = 5.0
hgrid = 0.35bohr / crmult = 5.5
hgrid = 0.30bohr / crmult = 6.0

hgrid = 0.20bohr / crmult = 7.0

This precision plot shows the systematicity of the wavelet basis set: by improving the basis set, we improve the value of the total energy.

hgrid = 0.55bohr / crmult = 3.5  -->  -8.025214Ht
hgrid = 0.50bohr / crmult = 4.0  -->  -8.031315Ht
hgrid = 0.45bohr / crmult = 4.5  -->  -8.032501Ht
hgrid = 0.40bohr / crmult = 5.0  -->  -8.033107Ht
hgrid = 0.35bohr / crmult = 5.5  -->  -8.033239Ht
hgrid = 0.30bohr / crmult = 6.0  -->  -8.033300Ht

hgrid = 0.20bohr / crmult = 7.0  -->  -8.033319Ht

To go further, one can vary hgrid and crmult independently. This is shown in the previous figure with the grey line. The shape of the convergence curve shows that both these parameters should be modified simoultaneously in order to increase accuracy. Indeed, there are two kind of errors arising from the basis set. The first one is due to the fact the basis set can't account for quickly varying wavefunctions (value of hgrid should be decreased). The second error is the fact that the wavefunctions are constrained to stay inside the defined basis set (output values are zero). In the last case crmult should be raised.

Fine tuning of the basis set

The multi-scale property of the wavelets is used in BigDFT and a two level grid is used for the calculation. We've seen previously the coarse grid definition using the the multiplying factor crmult. The second multiplying value on this line of the input file is used for the fine grid and is called frmult. Like crmult, it defines a factor for the radii used to define the fine grid region where the number of degrees of freedom is indeed eight times the one of the coarse grid. It allows to define region near the atoms where the wavefunctions are allowed to vary more quickly. Typical values for this factor are 8 to 10. It's worth to note that even if the value of the multiplier is greater than crmult it defines a smaller region due to the fact that the units which are associated to these radii are significantly different.

The physical quantities used by crmult and frmult can be changed in the pseudo-potential by adding an additional line with two values in bohr. The two values that the code is using (either computed or read from the pseudo-potential files) are output in the following way in the screen output:

------------------------------------------------------------------ System Properties
Atom    N.Electr.  PSP Code  Radii: Coarse     Fine  CoarsePSP    Calculated   File
Si          4      10            1.80603  0.43563  0.93364         X
 H          1      10            1.46342  0.20000  0.00000         X

Analysing the output

The output of BigDFT is divided into four parts:

  • Input values are printed out, including a summary of the different input files (DFT calculation parameters, atom positions, pseudo-potential values...);
  • Input wavefunction creation, usually called "input guess";
  • The SCF (Self-Consistent Field) loop itself;
  • The post SCF calculations including the forces calculation and other possible treatment like a finite size effect estimation or a virtual states determination.

The system parameters output

All the read values from the different input files are printed out at the program startup. Some additional values are provided there also, like the memory consumption. Values are given for one process, which corresponds to one core in an MPI environment.

 #-------------------------------------------------------- Estimation of Memory Consumption
 Memory requirements for principal quantities (MiB.KiB):
   Subspace Matrix                     : 0.1 #    (Number of Orbitals: 4)
   Single orbital                      : 0.302 #  (Number of Components: 38568)
   All (distributed) orbitals          : 2.363 #  (Number of Orbitals per MPI task: 4)
   Wavefunction storage size           : 30.617 # (DIIS/SD workspaces included)
   Nonlocal Pseudopotential Arrays     : 0.59
   Full Uncompressed (ISF) grid        : 8.852
   Workspaces storage size             : 0.710
 Accumulated memory requirements during principal run stages (MiB.KiB):
   Kernel calculation                  : 102.802
   Density Construction                : 81.7
   Poisson Solver                      : 112.694
   Hamiltonian application             : 81.650
 Estimated Memory Peak (MB)            :  112
 Ion-Ion interaction energy            :  9.51544109841576E+00

The overall memory requirement needed for this calculation is thus: 112 MB

In this example, the memory requirement is given for one process run and the peak of memory will be in the initialisation during the Poisson solver kernel creation, while the SCF loop will reach 112MB during the Poisson solver calculation. For bigger systems, with more orbitals, the peak of memory is usually reached during the Hamiltonian application.

Exercise: run a small utility program provided with BigDFT called bigdft-tool to estimate the memory requirement of a run before submitting it to the queue system of a super-computer. It reads the same input file than the bigdft executable, and is thus convenient to validate inputs.

The executable take one mandatory argument that is the number of cores to run BigDFT on. Try several values from 1 to 6 and discuss the memory distribution.

user@garulfo:~/CH4/$ ls
bigdft-tool    psppar.C     psppar.H     input.yaml    posinp.xyz
user@garulfo:~/CH4/$ ./bigdft-tool -n 2
...

BigDFT distributes the orbitals over the available processes (the value W does not decrease anymore after 4 processes since there are only 4 bands in our example). This means that running a parallel job with more processors than orbitals will result in a bad speedup. The number of cores involved in the calculation might be however increased via OMP parallelisation, as it is indicated in "Scalability with MPI and OpenMP" lesson.

The input guess

The initial wavefunctions in BigDFT are calculated using the atomic orbitals for all the electrons of the s, p, d shells, obtained from the solution of the PSP self-consistent equation for the isolated atom.

#----------------------------------- Wavefunctions from PSP Atomic Orbitals Initialization
 Input Hamiltonian:
   Total No. of Atomic Input Orbitals  :  8
   Atomic Input Orbital Generation:
   -  {Atom Type: C, Electronic configuration: {s: [ 2.00], p: [ 2/3,  2/3,  2/3]}}
   -  {Atom Type: H, Electronic configuration: {s: [ 1.00]}}

The corresponding hamiltonian is then diagonalised and the n_band (norb in the code notations) lower eigenfunctions are used to start the SCF loop. BigDFT outputs the eigenvalues, in the following example, 8 electrons were used in the input guess and the resulting first fourth eigenfunctions will be used for a four band calculation.

   Input Guess Overlap Matrices: {Calculated:  Yes, Diagonalized:  Yes}
    #Eigenvalues and New Occupation Numbers
   Orbitals: [
 {e: -6.501330042644E-01, f:  2.0000},  # 00001
 {e: -3.636213579778E-01, f:  2.0000},  # 00002
 {e: -3.636197175998E-01, f:  2.0000},  # 00003
 {e: -3.636197175998E-01, f:  2.0000},  # 00004  <- Last InputGuess eval, H-L IG gap:  20.6959 eV
 {e:  3.947505813172E-01, f:  0.0000},  # 00005  <- First virtual eval
 {e:  3.947523909004E-01, f:  0.0000},  # 00006
 {e:  3.947523909004E-01, f:  0.0000},  # 00007
 {e:  5.960254618828E-01, f:  0.0000}] # 00008
   IG wavefunctions defined            :  Yes

The SCF loop

The SCF loop follows a direct minimisation scheme and is made of the following steps:

  • Calculate the charge density from the previous wavefunctions.
  • Apply the Poisson solver to obtain the Hartree potential from the charges and calculate the exchange-correlation energy and the energy of the XC potential thanks to the chosen functional.
  • Apply the resulting hamiltonian on the current wavefunctions.
  • Precondition the result and apply a steepest descent or a DIIS history method. This depends on idsx, not specified in the present input.yaml file. It is therefore set to the default value, which is 6 (see the bigdft output). To perform a SD minimisation one should add "idsx: 0" to the input.yaml file.
  • Orthogonalise the new wavefunctions.

Then, BigDFT outputs a summary of the parts of the energy:

Energies: {Ekin:  6.66763910179E+00, Epot: -1.04233602201E+01, Enl:  4.35158411655E-01, 
              EH:  1.51771192882E+01,  EXC: -3.08539890002E+00, EvXC: -4.03428148190E+00}, 

Finally the total energy and the square norm of the residue (gnrm) are printed out. The gnrm value is the stopping criterion. It can be chosen using gnrm_cv in the input.yaml file. The default value (1e-4) is used here and a good value can reach 1e-5.

iter:  6, EKS: -8.03335831460998051E+00, gnrm:  1.20E-03, D: -3.55E-05, 

Exercise: run `grep "EKS" screenOutput` and look at the convergence rate for our methane molecule.

The minimisation scheme coupled with DIIS (and thanks to the good preconditioner) is a very efficient way to obtain convergence for systems with a gap, even with a very small one. Usual run should reach the 1e-4 stop criterion within 15 to 25 iterations. Otherwise, there is an issue with the system, either there is no gap, or the input guess is too symmetric due to the LCAO diagonalization, specific spin polarization...

The post-SCF treatments

At the end of the SCF loop, a diagonalisation of the current hamiltonian is done to obtain Kohn-Sham eigenfunctions. The corresponding eigenvalues are also given.

The forces are then calculated.

Some other post-SCF may be done depending on the input.dft Media:

  • One can run an estimation of finite-size effects. This is explained in the manual (which is not yet completely updated to recent BigDFT versions).
  • One can run a Davidson treatment on the current hamiltonian to obtain the energies (and virtual wavefunctions) of the first unoccupied levels.

Exercise: Before going further, review the input.dft file to identify the meaning of the different lines as explained previously.

1st line, "0.450 0.450 0.450" hx, hy, hz are the grid spacing in the three directions.

2nd line, "5.0 9.0" crmult, frmult define the basis set real space expansion.

3rd line, "1" defines the exchange correlation functional, following the ABINIT numbering convention.

6th line, "1.e-04" is the stop criterion.

7th line, "50 10" the first value is the maximum number of SCF iteration and the second is the maximum number of restart after a fresh diagonalisation if convergence is not reached.

8th line, "6 6" the second value is the length of the DIIS history and should be put to 0 to use SD instead.

Exercise: run bigdft-tool when varying the DIIS history length and discuss the memory consumption.

Reducing the DIIS history is a good way to reduce the memory consumption when one cannot increase the number of processes. Of course this implies more iterations in SCF loops.


Adding a charge

BigDFT can treat charged system without the requirement to add a compensating background like in plane waves.

The additional charge to add to the system is set in the input.dft file at the fourth line. In the following example an electron has been added (-1):

-1 0.0 0.0 0.0 ncharge efield

Exercise: remove the last hydrogen atom in the previous methane example and modify input.dft to add an electron. Then run BigDFT for an electronic convergence.

One can notice that the total charge in the system is indeed -8 thanks to the additional charge. The convergence rate is still good for this CH3- radical since it is a closed shell system.

Running a geometry optimisation

In the previous charged example the geometry of the radical is kept the same than for the methane molecule, while it is likely to change. One can thus optimize the geometry with BigDFT.

To run geometry calculations (molecular dynamics, structure optimisations...) one should add another input file called input.geopt. The first line of this file contains the method to use. Here, we look for a local minimum so we can use the keyword LBFGS. The third line of this file contains the stopping criteria. There are two stopping criteria: the first being the number of loops (force evaluations) and the second is the maximum on forces. For isolated systems, the first criterion is well adapted while the second is good for periodic boundary conditions.

Exercise: take the CH3- radical posinp.xyz file, add the input.geopt and run a geometry optimisation.

The evolution of the forces during relaxation can be easily obtained running `grep FORCES screenOutput`. At each iteration, BigDFT outputs a file posoutXXX.xyz in the directory data with the geometry of the iteration XXX. You can visualize it using v_sim (select all files in the "Browser" tab).

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