rdiracint.f90

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! Copyright (C) 2002-2015 J. K. Dewhurst, S. Sharma and C. Ambrosch-Draxl.
! This file is distributed under the terms of the GNU Lesser General Public
! License. See the file COPYING for license details.
 
!BOP
! !ROUTINE: rdiracint
! !INTERFACE:
pure subroutine rdiracint(sol,kpa,e,nr,r,vr,nn,g0,g1,f0,f1)
! !INPUT/OUTPUT PARAMETERS:
!   sol  : speed of light in atomic units (in,real)
!   kpa  : quantum number kappa (in,integer)
!   e    : energy (in,real)
!   nr   : number of radial mesh points (in,integer)
!   r    : radial mesh (in,real(nr))
!   vr   : potential on radial mesh (in,real(nr))
!   nn   : number of nodes (out,integer)
!   g0   : major component multiplied by r (out,real(nr))
!   g1   : radial derivative of g0 (out,real(nr))
!   f0   : minor component multiplied by r (out,real(nr))
!   f1   : radial derivative of f0 (out,real(nr))
! !DESCRIPTION:
!   Integrates the radial Dirac equation from $r=0$ outwards. This involves
!   using the predictor-corrector method to solve the coupled first-order
!   equations (in atomic units)
!   \begin{align*}
!    \left(\frac{d}{dr}+\frac{\kappa}{r}\right)G_{\kappa}&=\frac{1}{c}
!    \{2E_0+E-V\}F_{\kappa}\\
!    \left(\frac{d}{dr}-\frac{\kappa}{r}\right)F_{\kappa}&=
!    -\frac{1}{c}\{E-V\}G_{\kappa},
!   \end{align*}
!   where $G_{\kappa}=rg_{\kappa}$ and $F_{\kappa}=rf_{\kappa}$ are the major
!   and minor components multiplied by $r$, respectively; $V$ is the external
!   potential; $E_0$ is the electron rest energy; $E$ is the eigen energy
!   (excluding $E_0$); and $\kappa=l$ for $j=l-\frac{1}{2}$ or $\kappa=-(l+1)$
!   for $j=l+\frac{1}{2}$.
!
! !REVISION HISTORY:
!   Created September 2002 (JKD)
!   Polynomial order fixed to 3, September 2013 (JKD)
!EOP
!BOC
implicit none
! arguments
real(8), intent(in) :: sol
integer, intent(in) :: kpa
real(8), intent(in) :: e
integer, intent(in) :: nr
real(8), intent(in) :: r(nr),vr(nr)
integer, intent(out) :: nn
real(8), intent(out) :: g0(nr),g1(nr),f0(nr),f1(nr)
! local variables
integer ir,ir0,i
real(8) ci,e0,t1,t2,t3,t4
! inverse speed of light
ci=1.d0/sol
! electron rest energy
e0=sol**2
t1=2.d0*e0+e
! determine the r -> 0 boundary values of F and G
t2=dble(kpa)/r(1)
t3=ci*(t1-vr(1))
t4=ci*(vr(1)-e)
f0(1)=1.d0
g0(1)=-t2/t4
g1(1)=t3-t2*g0(1)
! extrapolate to the first four points
g1(2:4)=g1(1)
f1(1:4)=0.d0
nn=0
do ir=2,nr
  t2=dble(kpa)/r(ir)
  t3=ci*(t1-vr(ir))
  t4=ci*(vr(ir)-e)
  ir0=max(ir-3,1)
! estimate the derivatives
  g1(ir)=poly3(r(ir0),g1(ir0),r(ir))
  f1(ir)=poly3(r(ir0),f1(ir0),r(ir))
  do i=1,8
! integrate to find wavefunction
    g0(ir)=poly4i(r(ir0),g1(ir0),r(ir))+g0(ir0)
    f0(ir)=poly4i(r(ir0),f1(ir0),r(ir))+f0(ir0)
! compute the derivatives
    g1(ir)=t3*f0(ir)-t2*g0(ir)
    f1(ir)=t4*g0(ir)+t2*f0(ir)
  end do
! check for overflow
  if (abs(g0(ir)) > 1.d100) then
! set the remaining points and return
    g0(ir+1:nr)=g0(ir); g1(ir+1:nr)=g1(ir)
    f0(ir+1:nr)=f0(ir); f1(ir+1:nr)=f1(ir)
    return
  end if
! check for node
  if (g0(ir-1)*g0(ir) < 0.d0) nn=nn+1
end do
return
 
contains
 
pure real(8) function poly3(xa,ya,x)
implicit none
! arguments
real(8), intent(in) :: xa(3),ya(3),x
! local variables
real(8) x0,x1,x2,y0,y1,y2
real(8) c1,c2,t0,t1,t2
! evaluate the polynomial coefficients
x0=xa(1)
x1=xa(2)-x0; x2=xa(3)-x0
y0=ya(1)
y1=ya(2)-y0; y2=ya(3)-y0
t0=1.d0/(x1*x2*(x2-x1))
t1=x1*y2; t2=x2*y1
c1=x2*t2-x1*t1
c2=t1-t2
t1=x-x0
! evaluate the polynomial
poly3=y0+t0*t1*(c1+c2*t1)
end function
 
pure real(8) function poly4i(xa,ya,x)
implicit none
! arguments
real(8), intent(in) :: xa(4),ya(4),x
! local variables
real(8) x0,x1,x2,x3,y0,y1,y2,y3
real(8) c1,c2,c3,t0,t1,t2,t3,t4,t5,t6
! evaluate the polynomial coefficients
x0=xa(1)
x1=xa(2)-x0; x2=xa(3)-x0; x3=xa(4)-x0
y0=ya(1)
y1=ya(2)-y0; y2=ya(3)-y0; y3=ya(4)-y0
t4=x1-x2; t5=x1-x3; t6=x2-x3
t1=x1*x2*y3; t2=x2*x3*y1; t3=x1*x3
t0=1.d0/(x2*t3*t4*t5*t6)
t3=t3*y2
c3=t1*t4+t2*t6-t3*t5
t4=x1**2; t5=x2**2; t6=x3**2
c2=t1*(t5-t4)+t2*(t6-t5)+t3*(t4-t6)
c1=t1*(x2*t4-x1*t5)+t2*(x3*t5-x2*t6)+t3*(x1*t6-x3*t4)
t1=x-x0
! integrate the polynomial
poly4i=t1*(y0+t0*t1*(0.5d0*c1+t1*(0.3333333333333333333d0*c2+0.25d0*c3*t1)))
end function
 
end subroutine
!EOC