mirror of https://gitlab.com/QEF/q-e.git
259 lines
6.3 KiB
Fortran
259 lines
6.3 KiB
Fortran
!
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! Copyright (C) 2004 PWSCF 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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!
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!---------------------------------------------------------------
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subroutine ascheq(nn,lam,e,mesh,dx,r,r2,sqr,vpot,ze2, &
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thresh,y,nstop)
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!---------------------------------------------------------------
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!
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! numerical integration of the radial schroedinger equation for
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! bound states in a local potential.
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! thresh determines the absolute accuracy for the eigenvalue
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!
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use kinds, only : DP
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implicit none
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integer :: mesh,lam, ierr
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integer:: nn,nstop,maxter,iter,l1,i,ik,ncross,n, &
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nstart,ns,n2,nst1,nst2,ndcr
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real(DP) :: ze2,ddx12,xl1,x4l6,x6l12,x8l20,eup,elw,b0e, &
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c1,c2,c3,c4,rr1,rr2,ymx,rap,rstart,di,expn, &
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fe,a0,a1,a2,sum0,f2,sum,sqlhf,f0,f1,dfe,de,eps,&
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yln,xp, sum1
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real(DP):: r(mesh),r2(mesh),sqr(mesh),vpot(mesh), y(mesh)
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real(DP),allocatable:: c(:), el(:), f(:)
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real(DP):: b(0:3),dx,e,thresh
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data maxter/50/
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!
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! set up constants and initialize
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!
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allocate(c(mesh),stat=ierr)
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allocate(f(mesh),stat=ierr)
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allocate(el(mesh),stat=ierr)
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iter=0
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ddx12=dx*dx/12.0_dp
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l1=lam+1
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sqlhf=(DBLE(lam)+0.5_dp)**2
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xl1=l1
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x4l6=4.0_dp*lam+6.0_dp
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x6l12=6.0_dp*lam+12.0_dp
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x8l20=8.0_dp*lam+20.0_dp
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ndcr=nn-lam-1
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!
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! set initial lower and upper bounds to the eigenvalue
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!
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eup=vpot(mesh)+sqlhf/r2(mesh)
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elw=eup
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do i=1,mesh
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elw=min(elw,vpot(i)+sqlhf/r2(i))
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enddo
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nstop=200
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if(eup.eq.elw) go to 900
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if(e.gt.eup) e=0.9_DP*eup+0.1_DP*elw
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if(e.lt.elw) e=0.9_DP*elw+0.1_DP*eup
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!
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! series developement of the potential near the origin
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!
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do i=1,4
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y(i)=vpot(i)-ze2/r(i)
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enddo
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call series(y,r,r2,b)
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!
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300 continue
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iter=iter+1
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nstop=300
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if(iter.gt.maxter) go to 900
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!
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! set up the f-function and determine the position of its last
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! change of sign
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! f < 0 (approximatively) means classically allowed region
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! f > 0 " " " forbidden "
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!
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f(1)=ddx12*(r2(1)*(vpot(1)-e)+sqlhf)
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do i=2,mesh
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f(i)=ddx12*(r2(i)*(vpot(i)-e)+sqlhf)
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if( f(i) .ne. sign(f(i),f(i-1)) ) ik=i
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enddo
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nstop=302
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if(ik.ge.mesh-2) go to 900
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do i=1,mesh
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f(i)=1.0_dp-f(i)
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enddo
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!
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y(:) = 0.0_dp
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!
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! determination of the wave-function in the first two points by
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! series developement
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!
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b0e=b(0)-e
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c1=0.5_dp*ze2/xl1
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c2=(c1*ze2+b0e)/x4l6
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c3=(c2*ze2+c1*b0e+b(1))/x6l12
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c4=(c3*ze2+c2*b0e+c1*b(1)+b(2))/x8l20
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rr1=(1.0_dp+r(1)*(c1+r(1)*(c2+r(1)*(c3+r(1)*c4))))*r(1)**l1
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rr2=(1.0_dp+r(2)*(c1+r(2)*(c2+r(2)*(c3+r(2)*c4))))*r(2)**l1
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y(1)=rr1/sqr(1)
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y(2)=rr2/sqr(2)
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!
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! start outward integration and count number of crossings
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!
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ncross=0
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ymx=0.0_dp
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do n=2,ik-1
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y(n+1)=((12.0_dp-10.0_dp*f(n))*y(n)-f(n-1)*y(n-1))/f(n+1)
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if ( y(n) .ne. sign(y(n),y(n+1)) ) ncross=ncross+1
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ymx=max(ymx,abs(y(n+1)))
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end do
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!
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! matching radius has been reached going out. if ncross is not
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! equal to ndcr, modify the trial eigenvalue.
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!
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if(ndcr < ncross) then
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!
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! too many crossings. e is an upper bound to the true eigen-
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! value. increase abs(e)
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!
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eup=e
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rap=(DBLE(ncross+l1)/DBLE(nn))**2
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e=(e-vpot(mesh))*rap+vpot(mesh)
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if(e.lt.elw) e=0.9_dp*elw+0.1_dp*eup
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go to 300
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else if (ndcr > ncross) then
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!
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! too few crossings. e is a lower bound to the true eigen-
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! value. decrease abs(e)
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!
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elw=e
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rap=(DBLE(ncross+l1)/DBLE(nn))**2
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e=(e-vpot(mesh))*rap+vpot(mesh)
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if(e.gt.eup) e=0.9_dp*eup+0.1_dp*elw
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go to 300
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end if
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!
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! prepare inward integration
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! charlotte froese can j phys 41,1895(1963)
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!
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! start at min( rmax, 10*rmatch )
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!
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nstart=mesh
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ns=10
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rstart=ns*r(ik)
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if(rstart.lt.r(mesh)) then
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do i=ik,mesh
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nstart=i
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if(r(i).ge.rstart) go to 403
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enddo
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403 nstart=nstart/2
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nstart=2*nstart+1
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end if
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!
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! set up a, l, and c vectors
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!
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n=ik+1
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el(n)=10.0_dp*f(n)-12.0_dp
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c(n)=-f(ik)*y(ik)
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n2=ik+2
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do n=n2,nstart
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di=10.0_dp*f(n)-12.0_dp
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el(n)=di-f(n)*f(n-1)/el(n-1)
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c(n)=-c(n-1)*f(n-1)/el(n-1)
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enddo
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!
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! start inward integration by the froese's tail procedure
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!
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expn=exp(-sqrt(12.0_dp*abs(1.0_dp-f(nstart-1))))
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y(nstart-1)=c(nstart-1)/(el(nstart-1)+f(nstart)*expn)
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y(nstart)=expn*y(nstart-1)
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do n=nstart-2,ik+1,-1
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y(n)=(c(n)-f(n+1)*y(n+1))/el(n)
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end do
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!
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! if necessary, improve the trial eigenvalue by the cooley's
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! procedure. jw cooley math of comp 15,363(1961)
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!
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fe=(12.0_dp-10.0_dp*f(ik))*y(ik)-f(ik-1)*y(ik-1)-f(ik+1)*y(ik+1)
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!
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! calculate the normalization
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!
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if(ymx.ge.1.0e10_dp) then
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do i=1,mesh
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y(i)=y(i)/ymx
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enddo
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end if
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a0=1.0_dp/DBLE(2*lam+3)
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a1=c1/DBLE(lam+2)
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a2=(c1*c1+c2+c2)/DBLE(2*lam+5)
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sum0=(a0+r(1)*(a1+r(1)*a2))*r(1)**(2*lam+3)
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nst2=nstart-2
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f2=r2(1 )*y(1 )*y(1 )
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sum=r(1)*f2/DBLE(2*l1+1)
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do n=1,nst2,2
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f0=f2
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f1=r2(n+1)*y(n+1)*y(n+1)
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f2=r2(n+2)*y(n+2)*y(n+2)
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sum=sum+f0+f2+4.0_DP*f1
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enddo
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sum=sum0+dx*sum/3.0_dp
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dfe=-y(ik)*f(ik)/dx/sum
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de=-fe*dfe
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eps=abs(de/e)
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if(abs(de).lt.thresh) go to 600
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if(eps.gt.0.25_dp) de=0.25_dp*de/eps
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if(de.gt.0.0_dp) elw=e
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if(de.lt.0.0_dp) eup=e
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e=e+de
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if(e.gt.eup) e=0.9_dp*eup+0.1_dp*elw
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if(e.lt.elw) e=0.9_dp*elw+0.1_dp*eup
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if(iter.lt.maxter) go to 300
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nstop=50
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600 continue
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!
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! normalize the eigenfunction and exit
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!
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do n=nstart,mesh-1
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y(n+1)=0.0_dp
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if(y(n).eq.0.0_dp) go to 601
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yln=log(abs(y(n)))
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xp=-sqrt(12.0_dp*abs(1.0_dp-f(n)))
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expn=yln+xp
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if(expn.lt.-80.0_dp) go to 601
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y(n+1)=sign(exp(expn),y(n))
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601 continue
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enddo
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sum1=0.0_dp
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do n=nstart,mesh-2,2
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f0=f2
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f1=r2(n+1)*y(n+1)*y(n+1)
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f2=r2(n+2)*y(n+2)*y(n+2)
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sum1=sum1+f0+f2+4.0_dp*f1
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enddo
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sum=sum+dx*sum1/3.0_dp
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sum=sqrt(sum)
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do n=1,mesh
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y(n)=sqr(n)*y(n)/sum
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enddo
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if(nstop.lt.100) go to 900
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nstop=0
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deallocate(el)
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deallocate(f )
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deallocate(c )
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return
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!
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! error exit
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!
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! 900 write(6,9000) nstop,nn,lam,elw,eup
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! 9000 format(5x,'error in ascheq: nstop =',i4,'. n l =',2i3,/ &
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! & 5x,'elw =',f15.10,' eup =',f15.10)
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900 continue
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deallocate(el)
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deallocate(f )
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deallocate(c )
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return
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end subroutine ascheq
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