[Wien] Valence Broadening of Emission Spectrum

[Francisco Garcia]

Dear Gavin,

Thank you for your detailed and insightful response. The references you
provided, particularly J. Luitz's dissertation, were very helpful.

Thanks again!

FG

On Sat, Apr 19, 2025 at 4:16 PM Francisco Garcia <garcia.ff.000 at gmail.com>
wrote:

> Dear Prof. Blaha,
>
>
> I have two questions about the valence band emission spectra calculation
> in the subroutine valencebroadening.f: one question is about the usage of
> the parameter W and the other question is on how the Lorenztian convolution
> is done.
>
>
> (i) I thought W was a flag which determines which flavour of the
> broadening parameter gamma will be used (see the initial comments in the
> subroutine valencebroadening.f below). However, gamma appears to be a
> multiple of W in the emission calculation (please see below), which I find
> very confusing. Any reason(s) why?
>
>
>       subroutine
> ValenceBroadening(X,Y,yend,w,absorb,istep,wshift,E0,E1,E2,EF,delta,nimax)
> !     VALENCE BROADENING : the array y is broadened by convolution with a
> Lorentz-function.
> !     The result is in array yend.  Three different broadening schemes are
> available :
> !     - w=0 : the width of the Lorentz does not depend on energy
> !     - w=1 : the width of the Lorentz varies linearly with energy
> !     - w=2 : the width of the Lorentz varies quadratically with energy
> !     - w=3 : the width of the Lorentz is given by the scheme of Moreau et
> al.
>
> .
>
> .
>
> .
>
> !     EMISSION PART:
>                if(E0.NE.E2) then
>                   if (X(i1).gt.E0) then
>                      gamma=W*(1-((X(i1)-E0)/(EF-E0)))**2
>                   elseif (X(i1).gt.E1) then
>                      gamma=W
>                   else
>                      gamma=W+W*(1-((X(i1)-E2)/(E1-E2)))**2
>                   endif
>                else
>                   gamma=W*(1-((X(i1)-E0)/(EF-E0)))**2
>                endif
>             endif
>
>
>
> (ii) My second question is how the convolution of the Gaussian-broadened
> DOS with the Lorentzian was performed. In the subroutine
> valencebroadening.f, the Lorenztian convolution was computed as follows
> after setting gamma:
>
>             do i2=1,nimax
>
>                yend(i2)=yend(i2)+y(i1)/pi* &
>               (atan((X(i1)-X(i2)+delta)/gamma) &
>               -(atan((X(i1)-X(i2)-delta)/gamma)))
>
>             enddo
>
>
> It appears that an integral in the closed form was used to evaluate the
> convolution. I know that the integral of the Lorenztian can be obtained in
> a closed form: $$\int \frac{\gamma^2}{\pi(x^2+\gamma^2)} dx =
> \frac{\gamma}{\pi}} arctan(x / \gamma)$$. So that seems to be part of the
> explanation. But I am highly interested in how the above discretization was
> obtained from the convolution.
>
>
> Thank you Sir.
>
>
> FG
>
>
>
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