Inaccuarcies in the integration routines

[pmu2022]

Description of the Problem

If one performs LSMS with a converged potential and force some iteration by setting the RMS value to very low value, one should get stable energies. The Fermi, core and band energy will converge to an abs. error of about 1.0e-10. However, the Coulomb part of the total energy will in pathological cases only converge to about an abs. error of about 1e-5 per site. The current integration routines amplifiy the numerical error quite significantly. This starts to become a problem if one starts to compare deformed structures or structures with different volume.

By replacing the integration routines one can sigificantly improve the accuarcy.

Failure Logs

Comparison of the output of LSMS code for FCC Aluminium

LSMS_3: Program started
Using 4 MPI processes
Using 12 OpenMP threads
Reading input file 'i_lsms.lua'
Loaded input file!
System information:
===================
Number of atoms        :          4
Number of atomic types :          4
Performing Muffin-Tin (MT) calculation
communicated Parameters.
set global ids.
Band Energy = 4.306693 Ry           Fermi Energy = 0.501060 Ry
Total Energy = -1935.128843 Ry
RMS = 8.13266e-14
Band Energy = 4.306693 Ry           Fermi Energy = 0.501060 Ry
Total Energy = -1935.128858 Ry
RMS = 6.2226e-16
Band Energy = 4.306693 Ry           Fermi Energy = 0.501060 Ry
Total Energy = -1935.128849 Ry
RMS = 6.23632e-16
Band Energy = 4.306693 Ry           Fermi Energy = 0.501060 Ry
Total Energy = -1935.128869 Ry
RMS = 8.16995e-16
Band Energy = 4.306693 Ry           Fermi Energy = 0.501060 Ry
Total Energy = -1935.128863 Ry
RMS = 1.00189e-15
Band Energy = 4.306693 Ry           Fermi Energy = 0.501060 Ry
Total Energy = -1935.128883 Ry
RMS = 1.88232e-15
Band Energy = 4.306693 Ry           Fermi Energy = 0.501060 Ry
Total Energy = -1935.128834 Ry
RMS = 8.37203e-16
Band Energy = 4.306693 Ry           Fermi Energy = 0.501060 Ry
Total Energy = -1935.128854 Ry
RMS = 9.01121e-16
Band Energy = 4.306693 Ry           Fermi Energy = 0.501060 Ry
Total Energy = -1935.128838 Ry
RMS = 8.29867e-16
Band Energy = 4.306693 Ry           Fermi Energy = 0.501060 Ry
Total Energy = -1935.128848 Ry
RMS = 6.19569e-16
Writing new potentials.
Writing restart file.
Band Energy = 4.306692585315796 Ry
Fermi Energy = 0.501060346745968 Ry
Total Energy = -1935.128848338156331 Ry
timeScfLoop[rank==0] = 77.876099 sec
     number of iteration:10
timeScfLoop/iteration = 7.787610 sec
timeCalcChemPot[rank==0]/iteration = 0.242064 sec
timeCalcPotentialsAndMixing[rank==0]/iteration = 0.035397 sec
timeBuildLIZandCommList[rank==0]: 0.000728 sec
FOM Scale = 8077950976.000000
Energy Contour Points = 32
FOM = 1.03728e+09/sec
FOM * energyContourPoints = = 3.3193e+10/sec

LSMS_3: Program started
Using 4 MPI processes
Using 12 OpenMP threads
Reading input file 'i_lsms.lua'
Loaded input file!
System information:
===================
Number of atoms        :          4
Number of atomic types :          4
Performing Muffin-Tin (MT) calculation
communicated Parameters.
set global ids.
ezrho                       =     -1156.3309267973734222323400900 Ry
new ezrho                   =     -1156.3306251018768762151012197 Ry
ezrho                       =     -1156.3309387215751939947949722 Ry
new ezrho                   =     -1156.3306251018789225781802088 Ry
ezrho                       =     -1156.3309393748979800875531510 Ry
new ezrho                   =     -1156.3306251018786952045047656 Ry
ezrho                       =     -1156.3309296951908891060156748 Ry
new ezrho                   =     -1156.3306251018791499518556520 Ry
Band Energy = 4.306693 Ry           Fermi Energy = 0.501060 Ry
Total Energy = -1935.127611 Ry
RMS = 8.13266e-14
ezrho                       =     -1156.3309325557406737061683089 Ry
new ezrho                   =     -1156.3306251018796046992065385 Ry
ezrho                       =     -1156.3309379159788932156516239 Ry
new ezrho                   =     -1156.3306251018791499518556520 Ry
ezrho                       =     -1156.3309411621094113797880709 Ry
new ezrho                   =     -1156.3306251018775583361275494 Ry
ezrho                       =     -1156.3309370731285525835119188 Ry
new ezrho                   =     -1156.3306251018786952045047656 Ry
Band Energy = 4.306693 Ry           Fermi Energy = 0.501060 Ry
Total Energy = -1935.127611 Ry
RMS = 6.2226e-16
ezrho                       =     -1156.3309387215742845000931993 Ry
new ezrho                   =     -1156.3306251018768762151012197 Ry
ezrho                       =     -1156.3309379159782110946252942 Ry
new ezrho                   =     -1156.3306251018789225781802088 Ry
ezrho                       =     -1156.3309317613286566484021023 Ry
new ezrho                   =     -1156.3306251018800594465574250 Ry
ezrho                       =     -1156.3309317613257007906213403 Ry
new ezrho                   =     -1156.3306251018802868202328682 Ry
Band Energy = 4.306693 Ry           Fermi Energy = 0.501060 Ry
Total Energy = -1935.127611 Ry
RMS = 6.23632e-16
ezrho                       =     -1156.3309379159834406891604885 Ry
new ezrho                   =     -1156.3306251018791499518556520 Ry
ezrho                       =     -1156.3309417739890250231837854 Ry
new ezrho                   =     -1156.3306251018782404571538791 Ry
ezrho                       =     -1156.3309355713679451582720503 Ry
new ezrho                   =     -1156.3306251018802868202328682 Ry
ezrho                       =     -1156.3309444668936976086115465 Ry
new ezrho                   =     -1156.3306251018802868202328682 Ry
Band Energy = 4.306693 Ry           Fermi Energy = 0.501060 Ry
Total Energy = -1935.127611 Ry
RMS = 8.16995e-16
ezrho                       =     -1156.3309245151731374789960682 Ry
new ezrho                   =     -1156.3306251018809689412591979 Ry
ezrho                       =     -1156.3309426629396057251142338 Ry
new ezrho                   =     -1156.3306251018811963149346411 Ry
ezrho                       =     -1156.3309426629214158310787752 Ry
new ezrho                   =     -1156.3306251018814236886100844 Ry
ezrho                       =     -1156.3309444669059757870854810 Ry
new ezrho                   =     -1156.3306251018784678308293223 Ry
Band Energy = 4.306693 Ry           Fermi Energy = 0.501060 Ry
Total Energy = -1935.127611 Ry
RMS = 1.00189e-15
ezrho                       =     -1156.3309438475805563939502463 Ry
new ezrho                   =     -1156.3306251018796046992065385 Ry
ezrho                       =     -1156.3309428906359244138002396 Ry
new ezrho                   =     -1156.3306251018757393467240036 Ry
ezrho                       =     -1156.3309438475762362941168249 Ry
new ezrho                   =     -1156.3306251018796046992065385 Ry
ezrho                       =     -1156.3309438475803290202748030 Ry
new ezrho                   =     -1156.3306251018796046992065385 Ry
Band Energy = 4.306693 Ry           Fermi Energy = 0.501060 Ry
Total Energy = -1935.127611 Ry
RMS = 1.88232e-15
ezrho                       =     -1156.3309381599742664548102766 Ry
new ezrho                   =     -1156.3306251018780130834784359 Ry
ezrho                       =     -1156.3309302567847680620616302 Ry
new ezrho                   =     -1156.3306251018791499518556520 Ry
ezrho                       =     -1156.3309261012250317435245961 Ry
new ezrho                   =     -1156.3306251018791499518556520 Ry
ezrho                       =     -1156.3309302567809027095790952 Ry
new ezrho                   =     -1156.3306251018793773255310953 Ry
Band Energy = 4.306693 Ry           Fermi Energy = 0.501060 Ry
Total Energy = -1935.127611 Ry
RMS = 8.37203e-16
ezrho                       =     -1156.3309384179569860862102360 Ry
new ezrho                   =     -1156.3306251018780130834784359 Ry
ezrho                       =     -1156.3309302567681697837542742 Ry
new ezrho                   =     -1156.3306251018766488414257765 Ry
ezrho                       =     -1156.3309384179597145703155547 Ry
new ezrho                   =     -1156.3306251018777857098029926 Ry
ezrho                       =     -1156.3309384179651715385261923 Ry
new ezrho                   =     -1156.3306251018782404571538791 Ry
Band Energy = 4.306693 Ry           Fermi Energy = 0.501060 Ry
Total Energy = -1935.127611 Ry
RMS = 9.01121e-16
ezrho                       =     -1156.3309302567772647307720035 Ry
new ezrho                   =     -1156.3306251018789225781802088 Ry
ezrho                       =     -1156.3309302916939032002119347 Ry
new ezrho                   =     -1156.3306251018782404571538791 Ry
ezrho                       =     -1156.3309381599772223125910386 Ry
ezrho                       =     -1156.3309302567790837201755494 Ry
new ezrho                   =     -1156.3306251018786952045047656 Ry
new ezrho                   =     -1156.3306251018789225781802088 Ry
Band Energy = 4.306693 Ry           Fermi Energy = 0.501060 Ry
Total Energy = -1935.127611 Ry
RMS = 8.29867e-16
ezrho                       =     -1156.3309395155158654233673587 Ry
new ezrho                   =     -1156.3306251018786952045047656 Ry
ezrho                       =     -1156.3309302567895429092459381 Ry
new ezrho                   =     -1156.3306251018786952045047656 Ry
ezrho                       =     -1156.3309302567822669516317546 Ry
new ezrho                   =     -1156.3306251018768762151012197 Ry
ezrho                       =     -1156.3309395155133643129374832 Ry
new ezrho                   =     -1156.3306251018786952045047656 Ry
Band Energy = 4.306693 Ry           Fermi Energy = 0.501060 Ry
Total Energy = -1935.127611 Ry
RMS = 6.19569e-16
Writing new potentials.
Writing restart file.
Band Energy = 4.306692585315796 Ry
Fermi Energy = 0.501060346745968 Ry
Total Energy = -1935.127610643174421 Ry
timeScfLoop[rank==0] = 79.544595 sec
     number of iteration:10
timeScfLoop/iteration = 7.954459 sec
timeCalcChemPot[rank==0]/iteration = 0.158355 sec
timeCalcPotentialsAndMixing[rank==0]/iteration = 0.035670 sec
timeBuildLIZandCommList[rank==0]: 0.000665 sec
FOM Scale = 8077950976.000000
Energy Contour Points = 32
FOM = 1.01552e+09/sec
FOM * energyContourPoints = = 3.24968e+10/sec