What does "The calculation has reached convergence" mean in Quantum ESPRESSO?

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I am a beginner in QE and I have a question. During I was reading about Output file, I saw a part where have it:

"The calculation goes on step by step until convergence is reached. Then we show all the energies of all bands (eigenvalues of the Kohn-Shan orbitals) for each k-point, the Fermi energy, the total energy, the contributions of all the terms to the total energy, and the calculation time used by each subroutine of the program."

Output file:

...
iteration #  9     ecut=   100.00 Ry     beta=0.70
Davidson diagonalization with overlap
ethr =  1.00E-13,  avg # of iterations =  1.0

negative rho (up, down):  1.151E-06 0.000E+00

total cpu time spent up to now is       28.4 secs

End of self-consistent calculation

k = 0.0000 0.0000 0.0000 (  3909 PWs)   bands (ev):

-17.8569  -5.9595  -1.3957  -1.3957   4.4242   9.2594   9.9561   9.9561

k = 0.0000 0.0550 0.0000 (  3909 PWs)   bands (ev):

-17.8034  -5.8955  -1.6155  -1.5115   4.4968   9.3266   9.9837  10.1886
...
-10.9411 -10.9411  -8.9101   1.6562   1.6562  12.4084  14.2298  14.2298

the Fermi energy is     1.6562 ev

!    total energy              =     -22.80493136 Ry
Harris-Foulkes estimate   =     -22.80493136 Ry
estimated scf accuracy    <          1.5E-15 Ry

The total energy is the sum of the following terms:

one-electron contribution =     -44.02527202 Ry
hartree contribution      =      24.36506286 Ry
xc contribution           =      -6.99363523 Ry
ewald contribution        =       3.85762303 Ry
Dispersion Correction     =      -0.00870488 Ry
smearing contrib. (-TS)   =      -0.00000512 Ry

convergence has been achieved in   9 iterations

Writing output data file grafeno.save

init_run     :      2.26s CPU      2.29s WALL (       1 calls)
electrons    :     20.10s CPU     20.24s WALL (       1 calls)

...

Parallel routines
fft_scatter  :      4.35s CPU      4.44s WALL (   23020 calls)

PWSCF        :    28.46s CPU        28.70s WALL

This run was terminated on:  11:51: 0   5Sep2017

=------------------------------------------------------------------------------=
JOB DONE.
=------------------------------------------------------------------------------=


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27 minutes ago, Scienc said:

I am a beginner in QE and I have a question. During I was reading about Output file, I saw a part where have it:

"The calculation goes on step by step until convergence is reached. Then we show all the energies of all bands (eigenvalues of the Kohn-Shan orbitals) for each k-point, the Fermi energy, the total energy, the contributions of all the terms to the total energy, and the calculation time used by each subroutine of the program."

Output file:

...
iteration #  9     ecut=   100.00 Ry     beta=0.70
Davidson diagonalization with overlap
ethr =  1.00E-13,  avg # of iterations =  1.0

negative rho (up, down):  1.151E-06 0.000E+00

total cpu time spent up to now is       28.4 secs

End of self-consistent calculation

k = 0.0000 0.0000 0.0000 (  3909 PWs)   bands (ev):

-17.8569  -5.9595  -1.3957  -1.3957   4.4242   9.2594   9.9561   9.9561

k = 0.0000 0.0550 0.0000 (  3909 PWs)   bands (ev):

-17.8034  -5.8955  -1.6155  -1.5115   4.4968   9.3266   9.9837  10.1886
...
-10.9411 -10.9411  -8.9101   1.6562   1.6562  12.4084  14.2298  14.2298

the Fermi energy is     1.6562 ev

!    total energy              =     -22.80493136 Ry
Harris-Foulkes estimate   =     -22.80493136 Ry
estimated scf accuracy    <          1.5E-15 Ry

The total energy is the sum of the following terms:

one-electron contribution =     -44.02527202 Ry
hartree contribution      =      24.36506286 Ry
xc contribution           =      -6.99363523 Ry
ewald contribution        =       3.85762303 Ry
Dispersion Correction     =      -0.00870488 Ry
smearing contrib. (-TS)   =      -0.00000512 Ry

convergence has been achieved in   9 iterations

Writing output data file grafeno.save

init_run     :      2.26s CPU      2.29s WALL (       1 calls)
electrons    :     20.10s CPU     20.24s WALL (       1 calls)

...

Parallel routines
fft_scatter  :      4.35s CPU      4.44s WALL (   23020 calls)

PWSCF        :    28.46s CPU        28.70s WALL

This run was terminated on:  11:51: 0   5Sep2017

=------------------------------------------------------------------------------=
JOB DONE.
=------------------------------------------------------------------------------=


I don't know the first thing about this program, but it sounds to me as if it may use some sort of method of iterative approximation that converges towards a final value. If that is so it may be programmed to stop iterating when the difference between successive iterations falls below a preset level, or something. Could that makes sense?

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1. Converged with respect to what?: wavefunction and charge density cutoffs.

The first thing we have to realize is that convergence is relative. If we study the converged total energy of a system with increasing wavefunction cutoff, it will continue to decrease as you increase the plane wave cutoff. Instead, we identify a set of quantities we wish to calculate - for instance dipoles, vibrational properties, populations, or relative energies, to name a few. We can then choose our cutoffs based on the point at which these properties are unchanging even if the total energy continues to decrease.

Edited by StringJunky
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1 hour ago, exchemist said:

I don't know the first thing about this program, but it sounds to me as if it may use some sort of method of iterative approximation that converges towards a final value. If that is so it may be programmed to stop iterating when the difference between successive iterations falls below a preset level, or something. Could that makes sense?

Thanks.

1 hour ago, StringJunky said:

Thank so much,

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