[Pw_forum] Question for electron phonon coulping

[Stefano de Gironcoli]

Yanming ma wrote:

>> Do a calculation with a given set of k points and calculate el-ph as a
>> function of sigma
>
>
> In practise,
> I choose a 8x8x8 k mesh and a q point (say 0.5 0.5 0.0). I set nsig 
> =4, 8, 12, 16, 20, 24...respectively. I calculate the lambdas at this 
> q point with this k mesh and with different nsigs. Concerning the 
> lambda value, For example, for nsig=4 and 8, the final lambda values 
> are choosen as the final step gaussian broadening (step 4 for nsig=4, 
> step 8 for nsig=8).
> Then I can get a curve for nsig.vs.lambda at this q point.

You do not need to do a separate calculation for each value of nsig. 
with the original value of nsig=10 and the  step size of 0.01 (ryd) in a 
sigle run you get results for degauss1=0.01, 0.02, 0.03,,..., 0.10 (ryd).
The result of step 4 for nsig=4 is identical to step 4 for 
nsig=8,12,16,20,24 or any N>4 !
If you increase nsig to 24 (leaving the step size equal to 0.01) you can 
reach degauss1=0.24 if you like but, remember, one is interested in the 
limit of degauss1 going to zero.
NOTICE that I call degauss1 the value of the smearing used in the 
calculation of el-ph interaction which  will differ in general from 
degauss used in the scf/phonon calculation.

>
> Repeat the calculation with a larger k mesh, for example, 12x12x12.
>
> I compare the two curves and get a reliable nsig value, which the two 
> curves start to agree.
>
> if I repeat the calculation with more denser k mesh (say 16x16x16), I 
> will get a smaller reliable nsig by comparing the two more denser k mesh.
>
>> Trust only the curve in the region in which the calculation is converged
>> with respect to k-ponts summation.
>
> In practise, how can I decide the calculation is converged with 
> respect to k points summation?
> At a q point, with the same nsig, The calculated lambdas are same 
> within some errors for two K points mesh. Does this mean convergence? 

When for a given value of degauss1 (a given step in the nsig loop) the 
calculated lambda does not change anymore with increasing density of 
the  K-points mesh then you have reached convergence for that value of 
degauss1.

>
>> For sufficiently large set of k-points the SAFE region will approach
>> enough the limit of sigma going to zero
>> that you can stop the procedure.
>>
>
> I can not understand this paragraph. Sigma going to zero...?
>
SAFE meant converged with respect to k-point mesh.
The idea is that for sufficiently large set of K-points even the 
smallest values of degauss1are converged and one can use these values as 
approximations of the degauss1 going to zero limit.
How close one needs to go to the limit depends on the shape of the 
lambda vs degauss1curve.
Let's say 0.01-0.03 ryd or larger if the curve is sufficiently flat 
(horizontal).

As an example of this idea in a different context I include a figure 
where convergence with respect to k-points and degauss is computed for 
the total energy of lead.
In this figure one sees that for ngauss=1 the "converged" courve is 
sufficiently flat that you can usea value of degauss as large as 0.05 
ryd being within a franction of mryd from the exact limit (degauss=0 and 
infinite number of k-points). ngauss=0 converge to the same limit but 
with a stronger dependence on degauss.
This is the reason why in the calculation of lambda ngauss1 is always 
chosen equal to 1, although for el-ph interaction the dependence on 
degauss1 might be less nice.

So, do the calculation for several  set of k points and a given value of 
nsig (24 if you like). This will give you an entire curve 
lambda(degauss1) for each set of k-points.
Increse the k-point grid until the region of converged degauss1 extent 
sufficiently close to zero that you feel safe in extrapolating  that 
value to zero.
How much you need to go will depend on the shape of the function: 
whether it is very flat as the ngauss=1 case in the figure or with a 
large and changing slope as the negauss=0 case.

Once you are satisfied with the convergence of lambda at a given value 
of phonon wavevector q, then you still need to check convergence with 
respect to number of q-points (phonon wavevectors) you include in the 
calculation.
So you should compute el-ph interaction for a larger set of q-points and 
REPEAT the above procedure because the
convergence properties will in general depend on the q-points....

... and so on an so forth ...

Stefano de Gironcoli
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