mirror of https://gitlab.com/QEF/q-e.git
158 lines
4.6 KiB
Fortran
158 lines
4.6 KiB
Fortran
!
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! Copyright (C) 2009 Quantum ESPRESSO group
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! This file is distributed under the terms of the
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! GNU General Public License. See the file `License'
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! in the root directory of the present distribution,
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! or http://www.gnu.org/copyleft/gpl.txt .
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!
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subroutine radial_gradient(f,gf,r,mesh,iflag)
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!
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! This subroutine calculates the derivative with respect to r of a
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! radial function defined on the mesh r. If iflag=0 it uses all mesh
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! points. If iflag=1 it uses only a coarse grained mesh close to the
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! origin, to avoid large errors in the derivative when the function
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! is too smooth.
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!
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use kinds, only : DP
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implicit none
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integer, intent(in) :: mesh, iflag
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real(DP), intent(in) :: f(mesh), r(mesh)
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real(DP), intent(out) :: gf(mesh)
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integer :: i,j,k,imin,npoint
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real(DP) :: delta, b(5), faux(6), raux(6)
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!
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! This formula is used in the all-electron case.
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!
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if (iflag==0) then
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do i=2, mesh-1
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gf(i)=( (r(i+1)-r(i))**2*(f(i-1)-f(i)) &
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-(r(i-1)-r(i))**2*(f(i+1)-f(i)) ) &
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/((r(i+1)-r(i))*(r(i-1)-r(i))*(r(i+1)-r(i-1)))
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enddo
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gf(mesh)=0.0_dp
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!
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! The gradient in the first point is a linear interpolation of the
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! gradient at point 2 and 3.
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!
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gf(1) = gf(2) + (gf(3)-gf(2)) * (r(1)-r(2)) / (r(3)-r(2))
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return
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endif
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!
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! If the input function is slowly changing (as the pseudocharge),
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! the previous formula is affected by numerical errors close to the
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! origin where the r points are too close one to the other. Therefore
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! we calculate the gradient on a coarser mesh. This gradient is often
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! more accurate but still does not remove all instabilities observed
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! with the GGA.
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! At larger r the distances between points become larger than delta
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! and this formula coincides with the previous one.
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! (ADC 08/2007)
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!
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delta=0.00001_dp
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imin=1
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points: do i=2, mesh
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do j=i+1,mesh
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if (r(j)>r(i)+delta) then
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do k=i-1,1,-1
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if (r(k)<r(i)-delta) then
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gf(i)=( (r(j)-r(i))**2*(f(k)-f(i)) &
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-(r(k)-r(i))**2*(f(j)-f(i)) ) &
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/((r(j)-r(i))*(r(k)-r(i))*(r(j)-r(k)))
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cycle points
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endif
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enddo
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!
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! if the code arrives here there are not enough points on the left:
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! r(i)-delta is smaller than r(1).
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!
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imin=i
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cycle points
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endif
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enddo
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!
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! If the code arrives here there are not enough points on the right.
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! It should happen only at mesh.
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! NB: the f function is assumed to be vanishing for large r, so the gradient
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! in the last points is taken as zero.
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!
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gf(i)=0.0_DP
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enddo points
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!
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! In the first imin points the previous formula cannot be
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! used. We interpolate with a polynomial the points already found
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! and extrapolate in the points from 1 to imin.
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! Presently we fit 5 points with a 3rd degree polynomial.
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!
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npoint=5
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raux=0.0_DP
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faux=0.0_DP
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faux(1)=gf(imin+1)
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raux(1)=r(imin+1)
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j=imin+1
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points_fit: do k=2,npoint
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do i=j,mesh-1
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if (r(i)>r(imin+1)+(k-1)*delta) then
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faux(k)=gf(i)
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raux(k)=r(i)
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j=i+1
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cycle points_fit
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endif
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enddo
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enddo points_fit
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call fit_pol(raux,faux,npoint,3,b)
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do i=1,imin
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gf(i)=b(1)+r(i)*(b(2)+r(i)*(b(3)+r(i)*b(4)))
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enddo
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return
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end subroutine radial_gradient
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subroutine fit_pol(xdata,ydata,n,degree,b)
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!
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! This routine finds the coefficients of the least-square polynomial which
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! interpolates the n input data points.
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!
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use kinds, ONLY : DP
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implicit none
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integer, intent(in) :: n, degree
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real(DP), intent(in) :: xdata(n), ydata(n)
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real(DP), intent(out) :: b(degree+1)
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integer :: ipiv(degree+1), info, i, j, k
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real(DP) :: bmat(degree+1,degree+1), amat(degree+1,n)
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amat(1,:)=1.0_DP
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do i=2,degree+1
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do j=1,n
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amat(i,j)=amat(i-1,j)*xdata(j)
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enddo
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enddo
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do i=1,degree+1
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b(i)=0.0_DP
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do k=1,n
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b(i)=b(i)+ydata(k)*xdata(k)**(i-1)
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enddo
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enddo
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do i=1,degree+1
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do j=1,degree+1
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bmat(i,j)=0.0_DP
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do k=1,n
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bmat(i,j)=bmat(i,j)+amat(i,k)*amat(j,k)
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enddo
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enddo
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enddo
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!
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! This lapack routine solves the linear system that gives the
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! coefficients of the interpolating polynomial.
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!
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call DGESV(degree+1, 1, bmat, degree+1, ipiv, b, degree+1, info)
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if (info.ne.0) call errore('pol_fit','problems with the linear system', &
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abs(info))
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return
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end subroutine fit_pol
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