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

Question

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.
=------------------------------------------------------------------------------=
```

Answer 1

Let suppose that you want to do a geometry optimization. You need a criterion to said "the geometry is optimized". When the structure geometry is optimized, it will be in a local minimum of the potential energy surface. In such point the forces are zero over all the atoms. Due to precision and numerical errors, it will be very difficult to attain zero forces for all atoms. Instead, the user set up a threshold value $\delta$: if the forces are below such value, then the geometry optimization is said to converge. But depending on how low $\delta$ is, this process can take a lot of time (or even forever) so, you need to define a maximum value of steps (iterations) to be run.

In case of Self Consistent Field (SCF) methods, the electrons are "moved", and the system energy is calculated. In this case the criterion (normally) used is also defining a threshold value $\epsilon$ but instead for the energy, it is used for the difference between two consecutive energies values, i.e. $E_n-E_{n-1}$. If after "moving" all the electrons, $E_n-E_{n-1}<\epsilon$, it is said that the system converged. Here again, we need to define the maximum steps allowed.

In Molecular Mechanics (including Molecular Mechanics), only the first type of convergence is used. When working with ab initio, DFT or semiempirical methods, both type of convergence test are used.

At the end, your calculations stop because:

• attained the maximum number of steps: this is bad.
• attained the convergence criteria: this is good.

Note 1: the values of $\delta$ and $\epsilon$ strongly depends on what type of calculations you need to run and the type of your system. My recommendations is to look at the literature for inspiration.
Note 2: In general, you will need both convergence: forces and energies, but you can run a calculation for optimizing the geometry first, and then use the optimized geometry to run all the other calculations (as single point calculations).