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me_numer.f90
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me_numer.f90
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module me_numer
!
! Numerical matrix elements based on the Numerov procedure
!
use accuracy
use lapack
use timer
implicit none
private
public ik, rk, out
public ME_numerov,simpsonintegral,simpsonintegral_rk,integral_rect_rk
!
!
real(rk) :: enermax= 70000.0_rk ! Upper limit for the energy search (1/cm)
!
real(rk), parameter :: enerdev= 0.1_rk ! In the i-th solution has been already defined at
! previouse steps, we solve the eigen-problem anyway
! but within +/- "enerdev" (cm-1) interval from
! the found solution
real(rk), parameter :: enerstep = 100.0_rk ! Energy interval needed for the step-wise search of the
! solution, when the iterative procedure does not help
!
real(rk), parameter :: enersmall= 0.000001_rk ! Small energy value to check whether the search interval not collapsed
!
real(rk), parameter :: thrsh_int = 10.0_rk**(-rk) ! Threshold in Numerov-integration
real(rk), parameter :: thrsh_dif = 0.1e-2_rk ! Threshold in numerical differentiation
real(rk), parameter :: thrsh_upper = 1.0e20_rk ! Upper limit for fncts in out-numerov integr.
!
integer(ik), parameter :: itermax= 500000 ! Maximal possible number of iterations (tries) for Numerov
integer(ik), parameter :: maxslots = 10000 ! Maximal possible number of slots for found energies
integer(ik), parameter :: epoints = 40 ! N of points used for extrapolation in lsq_fit
!
real(rk) :: rhostep ! mesh size
!
real(rk),parameter :: wave_init=1.0_rk ! initial arbitrary value at rho = rho_ref
!
!
integer(ik), parameter :: iterlimit=5000 ! Iteration limit in numerov integration
!
integer(ik) :: npoints ! number of grid points
real(rk) :: rho_b(2) ! rhomin..rhomax
integer(ik) :: iperiod ! the periodicity (can be negative for the reflecation)
!
integer(ik) :: verbose = 6 ! Verbosity level
!
logical :: periodic ! periodic boundary conditions: true or false
!
integer(ik) :: vmax,imin
!
integer(ik) :: io_slot ! unit numeber to store the numerov eigenvectors and their derivatives
!
integer(ik) :: imode ! the current mode
!
integer(ik) :: Nr = 4 ! 2*Nr+1 is the number of interpolation points
!
integer(ik) :: iparity !
!
character(len=cl),parameter :: boundary_condition = 'UNBOUND'
!
character(len=cl),parameter :: deriv_method = 'ML_diffs' ! We use this method to estimate d pvi_v / d rho
! where phi_v is a numerical eigenfunction from numerov
! deriv_method can be either 'd04aaf' of '5 points'
! '5 points'
real(rk) :: rho_switch = .0174532925199432957692369_rk ! the value of abcisse rho of the switch between regions (1 deg)
integer(ik) :: iswitch ! the grid point of switch
!
contains
!
! Matrix elements with Numerov-eigenfunctions
!
subroutine ME_numerov(vmax_,rho_b_,npoints_,numerpoints_,drho_,poten_,mu_rr_,icoord,iperiod_,&
enermax_,verbose_,energy,wavefunc)
!
integer(ik),intent(in) :: vmax_,npoints_,numerpoints_,iperiod_
real(rk),intent(out) :: energy(0:vmax_),wavefunc(0:npoints_,0:npoints_)
!
real(rk),intent(in) :: rho_b_(2)
real(rk),intent(in) :: poten_(0:npoints_),mu_rr_(0:npoints_),drho_(0:npoints_)
integer(ik),intent(in) :: icoord ! coordinate number for which the numerov is employed
real(rk),intent(in) :: enermax_ ! Upper limit for the energy search (1/cm)
integer(ik),intent(in) :: verbose_ ! Verbosity level
!
real(rk) :: rho
real(rk) :: h_t,sigma,sigma_t,rms,psipsi_t,characvalue,rhostep_,step_scale,fval,df_t
!
integer(ik) :: vl,vr,alloc,i,rec_len,i_,i1,i2
!
real(rk),allocatable :: phil(:),phir(:),dphil(:),dphir(:),phivphi(:),rho_kinet(:)
real(rk),allocatable :: f(:),poten(:),mu_rr(:),d2fdr2(:),dfdr(:),rho_(:)
character(len=cl) :: unitfname
real(rk),allocatable :: enerslot(:),enerslot_(:)
!
integer,parameter :: NEND = 500,mxfs=5,mxprm=6,mxfsp=16001,mxisp=16001
!
real(rk) :: BFCT,EFN,OVR,OVRCRT,RH,RMIN,VLIM
integer(ik) :: ifs,IWR,JP,OTMF,TMFTYP(mxfs)
real(rk) :: DER(0:mxprm-1),PSI(NEND),TMFPRM(0:mxprm-1,mxfs),VJ(mxfsp),z(mxisp)
!
if (verbose>=4) write (out,"(/'Numerov matrix elements calculations')")
!
! global variables
!
vmax = vmax_
npoints = numerpoints_
rho_b = rho_b_
iperiod = iperiod_
verbose = verbose_
imode=icoord
enermax = enermax_
!
periodic = .false.
if (iperiod>0) periodic = .true.
!
allocate(phil(0:npoints_),phir(0:npoints_),dphil(0:npoints_),dphir(0:npoints_), &
phivphi(0:npoints_),rho_kinet(0:npoints_),enerslot(0:maxslots), &
f(0:npoints),dfdr(0:npoints),d2fdr2(0:npoints),poten(0:npoints),mu_rr(0:npoints),stat=alloc)
if (alloc/=0) then
write (out,"('phi - out of memory')")
stop 'phi - out of memory'
endif
!
! numerov step size
rhostep = (rho_b(2)-rho_b(1))/real(npoints,kind=rk)
!
! real step size
rhostep_ = (rho_b(2)-rho_b(1))/real(npoints_,kind=rk)
!
! define the rho-type coordinate
!
rho_kinet(:) = drho_(:)
!
step_scale = rhostep/rhostep_
!
! interpolation
!
if (npoints==npoints_) then
!
poten = poten_
mu_rr = mu_rr_
!
else
!
allocate(rho_(0:npoints_),stat=alloc)
if (alloc/=0) stop 'rho_ - out of memory'
!
forall(i_ = 0:npoints_) rho_(i_) = rho_b(1)+real(i_,kind=rk)*rhostep_
!
do i = 0,npoints
!
rho = rho_b(1)+real(i,kind=rk)*rhostep
!
i_ = int( real(i,rk)*step_scale )
!
i1 = max(0,i_-Nr) ; i2 = min(npoints_,i_+Nr)
!
call polintrk(rho_(i1:i2),poten_(i1:i2),rho,fval,df_t)
poten(i) = fval
!
call polintrk(rho_(i1:i2),mu_rr_(i1:i2),rho,fval,df_t)
mu_rr(i) = fval
!
enddo
!
deallocate(rho_) !
endif
!
! Do some reporting
!
if (verbose>=3) then
write (out,"('vmax = ',i8)") vmax
write (out,"('icoord = ',i4)") icoord
write (out,"('rho_b (x) = ',2f12.4)") rho_b(1:2) !*180.0_rk/pi
write (out,"('rhostep (x) = ',2f12.4)") rhostep !*180.0_rk/pi
endif
!
call diff_2d_4points_rk(npoints,rho_b,mu_rr,periodic,0_ik,dfdr,d2fdr2)
!
f(:) = (d2fdr2(:)/mu_rr(:)-dfdr(:)**2/mu_rr(:)**2*0.5_rk)*0.5_rk
!
! Print out
!
if (verbose>=3) then
write(out,"('grid values (i,rho,rho_kinet,poten, mu_rr, f): ')")
do i_=0,npoints_,2
i = int( real(i_,rk)/step_scale )
rho = rho_b(1)+real(i_,kind=rk)*rhostep_
write(out,"(i8,2f14.6,3g14.6)") i_,rho,rho_kinet(i_),poten(i),mu_rr(i),f(i)
enddo
endif
!
inquire(iolength=rec_len) phil(:),dphil(:)
!
write(unitfname,"('Numerov basis set # ',i6)") icoord
call IOStart(trim(unitfname),io_slot)
!
open(unit=io_slot,status='scratch',access='direct',recl=rec_len)
!
! solve 1D Schroedinger equation with Numerov algorithm
!
iparity = 0
!
iswitch = max(1,int( rho_switch/rhostep_))
!
if (iperiod/=0) vmax = vmax/2
!
call numerov(npoints,npoints_,step_scale,poten,mu_rr,f,enerslot)
!
psi = 0
BFCT = 1.0
der = 0
OTMF = -1
!
TMFTYP = 0
!
EFN = 48000.0
BFCT = rhostep**2/mu_rr(1)
VJ(1:npoints+1) = poten(0:npoints)*BFCT
VLIM = poten(npoints)
!MU(ISOT(iset))*RH*RH/16.85762908d0
!
!call OVRLAP(BFCT,DER,EFN,OVR,OVRCRT,PSI,RH,RMIN,TMFPRM,VJ,&
! VLIM,z,ifs,IWR,JP,NEND,OTMF,TMFTYP)
!
if (iperiod/=0) then
allocate(enerslot_(0:maxslots),stat=alloc)
if (alloc/=0) then
write (out,"('phi - out of memory')")
stop 'phi - out of memory'
endif
!
iparity = 1
!
call numerov(npoints,npoints_,step_scale,poten,mu_rr,f,enerslot_)
!
do vl = vmax,0,-1
!
enerslot(2*vl ) = enerslot(vl)
if (2*vl+1<=vmax_) enerslot(2*vl+1) = enerslot_(vl)
!
enddo
!
vmax = vmax_
!
deallocate(enerslot_)
!
endif
!
! Matrix elements
!
sigma = 0.0_rk
rms = 0.0_rk
characvalue = maxval(enerslot(0:vmax))
energy(0:vmax) = enerslot(0:vmax)
if (trim(boundary_condition)/='UNBOUND') then
energy(0:vmax) = energy(0:vmax)-energy(0)
endif
!
do vl = 0,vmax
!
read (io_slot,rec=vl+1) (phil(i),i=0,npoints_),(dphil(i),i=0,npoints_)
!
wavefunc(:,vl) = phil(:)
!
do vr = vl,vmax
!
if (iperiod/=0.and.mod(abs(vl-vr),2)==1) cycle
!
if (vl==vr) then
phir = phil
dphir = dphil
else
read (io_slot,rec=vr+1) (phir(i),i=0,npoints_),(dphir(i),i=0,npoints_)
endif
!
! Here we prepare integrals of the potential
! <vl|poten|vr> and use to check the solution of the Schroedinger eq-n
! obtained above by the Numerov
!
phivphi(:) = phil(:)*poten_(:)*phir(:)
!
h_t = simpsonintegral_rk(npoints_,rho_b(2)-rho_b(1),phivphi)
!
! momenta-quadratic part
!
phivphi(:) =-dphil(:)*mu_rr_(:)*dphir(:)
!
psipsi_t = simpsonintegral_rk(npoints_,rho_b(2)-rho_b(1),phivphi)
!
! Add the diagonal kinetic part to the tested mat. elem-s
!
h_t = h_t - 0.5_rk*psipsi_t
!
!
phivphi(:) = phil(:)*phir(:)
!
psipsi_t = simpsonintegral_rk(npoints_,rho_b(2)-rho_b(1),phivphi)
!
! Count the error, as a maximal deviation sigma = | <i|H|j>-E delta_ij |
!
sigma_t = abs(h_t)
if (vl==vr) sigma_t = abs(h_t-enerslot(vl))
!
sigma = max(sigma,sigma_t)
rms = rms + sigma_t**2
!
! Now we test the h_t = <vl|h|vr> matrix elements and check if Numerov cracked
! the Schroedinger all right
if (vl/=vr.and.abs(h_t)>sqrt(small_)*abs(characvalue)*1e4.and.trim(boundary_condition)/='UNBOUND') then
write(out,"('ME_numerov: wrong Numerovs solution for <',i4,'|H|',i4,'> = ',f20.10)") vl,vr,h_t
stop 'ME_numerov: bad Numerov solution'
endif
!
if (vl==vr.and.abs(h_t-enerslot(vl))>sqrt(small_)*abs(characvalue)*1e4.and.trim(boundary_condition)/='UNBOUND') then
write(out,"('ME_numerov: wrong <',i4,'|H|',i4,'> (',f16.6,') =/= energy (',f16.6,')')") vl,vr,h_t,enerslot(vl)
stop 'ME_numerov: bad Numerov solution'
endif
!
! Reporting the quality of the matrix elemenst
!
if (verbose>=3.and.trim(boundary_condition)/='UNBOUND') then
if (vl/=vr) then
write(out,"('<',i4,'|H|',i4,'> = ',e16.2,'<-',8x,'0.0',5x,'; <',i4,'|',i4,'> = ',e16.2,'<-',8x,'0.0')") &
vl,vr,h_t,vl,vr,psipsi_t
else
write(out,"('<',i4,'|H|',i4,'> = ',f16.6,'<-',f16.6,'; <',i4,'|',i4,'> = ',f16.6)")&
vl,vr,h_t,enerslot(vl),vl,vr,psipsi_t
endif
endif
!
enddo
enddo
!
rms = sqrt(rms/real((vmax+1)*(vmax+2)/2,kind=rk))
!
if (verbose>=1) then
write(out,"('Maximal deviation sigma = | <i|H|j>-E delta_ij | is ',f18.8)") sigma
write(out,"('rms deviation is ',f18.8)") sqrt(rms)
endif
!
deallocate(phil,phir,dphil,dphir,phivphi,rho_kinet,enerslot,f,poten,mu_rr,d2fdr2,dfdr)
!
!
end subroutine ME_numerov
!
!********************************************************************!
! !
! numerov calculates bending functions and stores them !
! !
! !
!********************************************************************!
subroutine numerov(npoints,npoints_,step_scale,poten,mu_rr,f,enerslot)
!
integer(ik),intent(in) :: npoints,npoints_
real(rk),intent(in) :: step_scale
real(rk),intent(in) :: poten(0:npoints),mu_rr(0:npoints),f(0:npoints)
!
real(rk),intent(out) :: enerslot(0:maxslots)
integer(ik) :: v,i_
integer(ik) :: ierr,numnod,ntries,vrec,k0,ipoint,mtries
!
real(rk) :: eold,tsum
!
real(rk) :: pcout
!
real(rk),allocatable :: pot_eff(:),i0(:),phi_f(:),phi_t(:),psi_t(:),phi_f_(:),phi_d_(:)
!
real(rk) :: eguess,enerlow,enerupp,enerdelta,enershift,enermid
!
!real(rk) :: simpsonintegral
!
logical :: notfound
!
real(rk) :: potmin,oldphi,newphi,v_t(-2:2),amplit,amplit0,diff
!
real(rk) :: df_t1,df_t2,deltaE = 1.0_rk
!
integer(ik) :: alloc,i,iter,nm2,ic,imin_l,imin_r
integer(ik) :: icslots(0:maxslots),iright,ileft,ireflect
!
!
allocate(pot_eff(0:npoints),i0(0:npoints),phi_f(0:npoints),phi_t(0:npoints),psi_t(-30:npoints+30),phi_f_(0:npoints),&
phi_d_(0:npoints),stat=alloc)
if (alloc/=0) then
write (6,"('numerov: allocation is faild - out of memory')")
stop 'numerov: allocation is faild - out of memory'
endif
!
! fcoef = planck*avogno*1.0d+16/( 4.0d+00*pi*pi*vellgt )
!
! Check for the potential "1/0" probelm in mu_rr
!
do i=0,npoints
if ( abs(mu_rr(i))<small_*100.0_rk ) then
write(out,"('numerov: mu_rr is zero at i = ',i8)") i
stop 'numerov: mu_rr has a zero element'
endif
enddo
!
! before commencing the numerov-cooley numerical integration we
! store the following function in the f1 array:
!
! 0
! w = f (p ) + 2 i v (p )
! i 1 i 0 i
!
! in the renner-teller version of morbid, this function is
! extended with a term originating in the non-zero
! electronic angular momentum.
!
! -1 0 -1
! v is given in cm , and i is given in cm
! 0
!
! calculate effective potential w_i
!
pot_eff(:) = f(:) + 2.0_rk*( poten(:) )/mu_rr(:)
!
! Effective inertia moment
!
i0(:) = 2.0_rk/mu_rr(:)
!
! find minimum : position
!
potmin = safe_max
!
do i=0,npoints
!
if (pot_eff(i)<potmin) then
imin = i
potmin = pot_eff(i)
endif
!
enddo
! and value
!
potmin = pot_eff(imin)
!
if(trim(boundary_condition)=='UNBOUND') then
potmin = pot_eff(npoints)
imin = npoints
endif
!
if (imin<0.or.imin>npoints) then
write(out,"('numerov: pot_eff has no minimum',i8)")
stop 'numerov: pot_eff has no minimum'
endif
!
imin_l = minloc(poten(0:imin-1),dim=1)
imin_r = minloc(poten(imin+1:npoints),dim=1)+imin
!
ileft = npoints
iright = 0
!
! Print out
!
if (verbose>=5) then
write(out,"('grid values (poten, mu_rr, poten_eff): ')")
do i=0,npoints,2
write(out,"(i8,3f18.8)") i,poten(i),mu_rr(i),pot_eff(i)
enddo
write(out,"('potmin(eff) = ',f18.8,' at i = ',i8)") potmin,imin
endif
!
!
! Parameters for the Simson rule integration
!
! facodd=2.0_rk*rhostep/3.0_rk
! faceve=2.0_rk*facodd
!
! Here we start solving Schroedinger equation for the v-th eigenvalue
!
select case (trim(boundary_condition))
!
case default
!
! assign initial values
!
enerslot(0:maxslots)=poten(imin)-sqrt(safe_max)
icslots(0:maxslots) = npoints
!
enerlow=potmin/2.0_rk*mu_rr(imin)
enerupp=enermax
!
case ('UNBOUND')
!
! assign initial values
!
!enermax = maxval(poten(:))
!
enerlow=potmin/2.0_rk*mu_rr(imin)
!
deltaE = (enermax-enerlow)/(npoints+1)
!
do i=0,min(npoints,maxslots)
!
enerslot(i) = enerlow+real(i,rk)*deltaE
!
enddo
!
icslots(0:maxslots) = npoints/2
enerupp=enermax
!
!if (imin==npoints) imin = npoints/2
!
end select
!
do v=0,vmax+1
!
if (v/=0) enerlow = enerslot(v-1)
!
! Determine the highest undefined energy slot
i = v
do while (enerslot(i)<poten(imin).and.i<maxslots)
i = i + 1
enddo
!
if (enerslot(i)>=poten(imin)) then
enerupp=enerslot(i)
else
enerupp=enermax
endif
!
if (enerslot(v)>poten(imin)) then
!
enerlow=max(enerslot(v)-enerdev,poten(imin))
enerupp=min(enerslot(v)+enerdev,enermax)
ic = icslots(v)
!
endif
!
select case (trim(boundary_condition))
!
case ('UNBOUND')
!
ic = icslots(v)
enerupp=enerslot(v+1)
enerlow=max(enerslot(v)-deltaE,poten(imin))
enerlow = enerslot(v)
!
end select
!
enerdelta= enerstep
enershift = enerstep
ntries=0
mtries = 0
!
! start itererative procedure for searching the current solution
!
iter = 0
!
notfound = .true.
!
search_loop : do while(notfound.and.iter<itermax)
!
iter = iter + 1
!
enermid=(enerlow+enerupp)/2.0
!
!enerdelta = enerdelta*0.5_rk
!
eguess=enermid
!
eold=eguess
!
! Numerov procedure
!
ic = icslots(v)
!
if (ic/=npoints.and.enerslot(v)>=poten(imin).and.iter==1) eguess = enerslot(v)
!
if (verbose>=5) write(out,"('eguess = ',e14.7)") eguess
!
call numcoo ( v, pot_eff, i0, eguess, enerlow, ic, phi_f, pcout, ierr)
!
if (ierr>1) then
!
write(out,"('numerov: no solution found in numcoo, ierr = ',i8)") ierr
!
do i = 0,maxslots
if (enerslot(i)>poten(imin)) write (out,"(' v,ener = ',i8,f20.10)") i,enerslot(i)
enddo
!
stop 'numerov: no solution found in numcoo'
!
endif
!
!
if (ierr == 0) then
!
numnod=0
nm2=npoints-2
oldphi=phi_f(0)
newphi=phi_f(2)
!
do i=2,nm2
if ( oldphi*phi_f(i)<0.0 .or. &
( phi_f(i)==0.0_rk .and. newphi*oldphi<0.0_rk) ) numnod=numnod+1
newphi=phi_f(i+2)
oldphi=phi_f(i)
enddo
!
endif
!
if (ierr==0.and.iperiod<0) then
!
phi_t(:) = phi_f(:)/sqrt(mu_rr(:))
!
tsum = simpsonintegral_rk(npoints,rho_b(2)-rho_b(1),phi_t(:)**2)
!
phi_t(:)=phi_t(:)/sqrt(tsum)
!
ireflect = 1
if (iparity/=0) ireflect = -1
!
call diff_2d_4points_rk(Npoints,rho_b,phi_t,periodic,ireflect,psi_t(0:Npoints))
!
if ( (iparity==0.and.abs(psi_t(npoints ))>sqrt(thrsh_int)).or.&
(iparity==1.and.abs(phi_t(npoints ))>sqrt(thrsh_int))) then
!
if (verbose>=5) write(out,"(/'phi(N) and psi(N) = ',2g18.8,', energy = ',f12.4,', n = ',9i7)") &
phi_t(npoints),psi_t(npoints ),eguess,numnod
!
if ( numnod+1<maxslots.and.numnod>v) then
enerslot(numnod)=eguess
icslots(numnod) = ic
endif
!
numnod = maxslots+1
!
endif
!
endif
!
if (verbose>=5) write(out,"('v = ',i5,'; numnod = ',i8,', efound = ',f16.7,', ierr = ',i9)") v,numnod,eguess,ierr
!
select case (trim(boundary_condition))
!
case default
!
if ( ierr==0.and.numnod+1<maxslots.and.numnod>=v ) then
!
enerslot(numnod)=eguess
icslots(numnod) = ic
!
if (verbose>=6) then
!
write (out,"('v,numnod,ener = ',2i8,f20.10)") v,numnod,enerslot(numnod)
!
do i=0,npoints
!
write(out,"(i8,2f18.8)") i,phi_f(i),exp((rho_b(1)+rhostep*real(i,rk)))
!
enddo
!
endif
endif
!
if ( ierr == 0.and.v==numnod ) notfound = .false.
! if it cannot find a solution by Numerov procedure and ierr = 1, we can still try to
! devide the searching interval and repeate
! the Numerov procedure with a closer initial guess
! We do the same thing if the found solution is not what we need (/=v)
!
if (abs(enerupp-enerlow) < enersmall.and.(ierr == 1 .or. numnod /= v)) then
!
ntries=0
!
!enerdelta = enerdelta*0.5_rk
!
! Are we still at the begining?
!
if (v.eq.0) then
!
enerlow=poten(imin)
!
else
!
enerlow=enerslot(v-1)-0.001
!
endif
!
! Determine the highest undefined energy slot
i = v+1
do while (enerslot(i)<poten(imin).and.i<maxslots)
i = i + 1
enddo
!
if (enerslot(i)>=poten(imin)) then
enerupp=enerslot(i)
else
enerupp=enermax
endif
!
if (enermid>=enerupp) then
enerdelta = enerdelta*0.5_rk
enershift = 0.0_rk
endif
!
enerlow=enerlow+enershift
enershift=enershift+enerdelta
enerupp=enerlow
!
cycle search_loop
!
endif
!
if (ierr <=1 .and. numnod /= v) then
!
! We need to do something if the seacrh interval is collapsed
! this must have happend at ierr=1, and no solution found
!
if (mtries==0) then
!
mtries = 1
!
! Are we still at the begining?
!
if (v.eq.0) then
enerlow=poten(imin)
else
enerlow=enerslot(v-1)
endif
!
! Determine the highest undefined energy slot
i = v+1
do while (enerslot(i)<poten(imin).and.i<maxslots)
i = i + 1
enddo
!
if (enerslot(i)>=poten(imin)) then
enerupp=enerslot(i)
else
enerupp=enermax
endif
!
endif
!
if (numnod<v.or.ierr==1) then
enerlow=enermid
else
enerupp=enermid
endif
!
endif
!
! multiply the wavefunction with sqrt(irr) (eq. (6.4) of jensen)
!
phi_f(:) = phi_f(:)/sqrt(mu_rr(:))
!
phi_t(:) = phi_f(:)*phi_f(:)
!
! numerical intagration with simpson's rule #2
!
tsum = simpsonintegral_rk(npoints,rho_b(2)-rho_b(1),phi_t)
!
phi_f(:)=phi_f(:)/sqrt(tsum)
!
case ('UNBOUND')
!
if (verbose>=6) then
!
write (out,"('v,numnod,ener = ',2i8,f20.10)") v,numnod,enerslot(numnod)
!
do i=0,npoints
!
write(out,"(i8,2f18.8)") i,phi_f(i),exp((rho_b(1)+rhostep*real(i,rk)))
!
enddo
!
endif
!
if ( ierr == 0 ) notfound = .false.
!
! multiply the wavefunction with sqrt(irr) (eq. (6.4) of jensen)
!
phi_f(:) = phi_f(:)/sqrt(mu_rr(:))
!
oldphi = phi_f(0)
newphi = phi_f(1)
amplit = 0
amplit0 = 0
do i=2,npoints
oldphi=newphi
newphi=phi_f(i-1)
if ( oldphi<newphi.and.newphi<=phi_f(i) ) then
diff = abs(amplit-amplit0)
!
amplit0 = amplit
amplit = newphi
!
endif
enddo
!
phi_t(:) = phi_f(:)*phi_f(:)
!
! numerical intagration with simpson's rule #2
!
tsum = simpsonintegral_rk(npoints,rho_b(2)-rho_b(1),phi_t)
!
if (tsum>small_*100) then
phi_f(:)=phi_f(:)/sqrt(tsum)
endif
!
end select
!
enddo search_loop
!
if( notfound ) then
!
write(out,"('numerov: no solution found after ',i8,' iterations')") itermax
write (out,"(' v = ',i8,', eguess= ',e20.10)") v,eguess
!
do i = 0,maxslots
if (enerslot(i)>poten(imin)) write (out,"(' v,ener = ',i8,f20.10)") i,enerslot(i)
enddo
!
stop 'numerov: no solution found'
!
endif
!
if (ierr==0.and.periodic) then
if (verbose>=5) write(out,"(/'phi_t(0)-phi_t(npoints ) : ',3g18.8)") phi_t(0),phi_t(npoints ),phi_t(0)-phi_t(npoints )
if (verbose>=5) write(out,"(/'dphi_t(0)-dphi_t(npoints) : ',5g18.8)") phi_t(1),phi_t(npoints-1),df_t1,df_t2,df_t1-df_t2
endif
!
!if (iperiod<0) phi_f(:)=phi_f(:)/sqrt(2.0_rk)
!
!
if (verbose>=5) then
write (out,"('v,ener = ',i8,f20.10)") v,enerslot(numnod)
endif
!
if (verbose>=6) then
write(out,"(f18.8)") phi_f
endif
!
! direvatives of the wavefunctions method I
!
if(deriv_method == '5 points') then
!
do i=0,npoints ! --- np
!
do k0 = -2, 2
!
if (periodic) then
ipoint = min(max(i+k0,0),npoints)
v_t(k0) = phi_f(ipoint)
else
if (i+k0<0) then
v_t(k0) = phi_f(-(i+k0))
elseif(i+k0>npoints) then
v_t(k0) = phi_f(npoints)
else
v_t(k0) = phi_f(i+k0)
endif
endif
!
enddo
!
! 5-points expression
!
phi_t(i) = (-v_t( 2)/12.0_rk+2.0_rk/3.0_rk*v_t(1) &
+v_t(-2)/12.0_rk-2.0_rk/3.0_rk*v_t(-1) )/rhostep
!
!phi_t(i) = (v_t(1)-v_t(-1) )/(rhostep*2.0_rk)
!
enddo
!
! direvatives of the wavefunctions method II
!
elseif(deriv_method == 'ML_diffs') then
!
if (iperiod<0) then
!
ireflect = 1
if (iparity/=0) ireflect = -1
!
call diff_2d_4points_rk(Npoints,rho_b,phi_f,periodic,ireflect,phi_t)
!
else
!
ireflect = 0
if (iparity/=0) ireflect = -1
!
call diff_2d_4points_rk(Npoints,rho_b,phi_f,periodic,ireflect,phi_t)
!
endif
!
! direvatives of the wavefunctions method III
!
else
write(out,"('numerov: bad deriv_method ',a)") trim(deriv_method)
stop 'numerov: bad deriv_method'
endif
!
!
if (verbose>=5) then
!
do i=0,npoints
!
write(out,"(i8,2f18.8)") i,phi_f(i),phi_t(i)
!
enddo
!
endif
!
! Find the outmost points where the function is not zero
!
i = 0
!
do while (abs(phi_f(i))<100.0*sqrt(small_).and.i<npoints)
i = i + 1
enddo
!
ileft = min(ileft,i)
!
i = npoints
!
do while (abs(phi_f(i))<100.0*sqrt(small_).and.i>0)
i = i - 1
enddo
!
iright = max(iright,i)
!
! dump the eigenfunction
!
if (npoints_==npoints) then
phi_f_ = phi_f
phi_d_ = phi_t
else
do i_ = 0,npoints_
!
i = nint( real(i_,rk)/step_scale )
phi_f_(i_) = phi_f(i)
phi_d_(i_) = phi_t(i)
!
enddo
endif
!
vrec=v+1
!
if (iperiod/=0) then
!
vrec = 2*v+1
!
if (iparity/=0) vrec = 2*v+2
!
endif
!
write (io_slot,rec=vrec) (phi_f_(i_),i_=0,npoints_),(phi_d_(i_),i_=0,npoints_)
!
enddo
!
write(out,"(/' Outmost points of the nonvanishing wave-functions are: [',i6,'...',i6,'], i.e. [',f12.6,'...',f12.6,'] ')") &
ileft,iright,rho_b(1)+rhostep*real(ileft,rk),rho_b(1)+rhostep*real(iright,rk)
!
! report the found Numerov energies
!
write (out,"(/' Numerov-energies are:')")
!
enerlow = minval(enerslot(0:vmax),dim=1) !mask=(enerslot>=poten(imin)
!
do v=0,vmax
write (out,"(i8,f18.8)") v,enerslot(v)-enerlow
enddo
!
write (out,"('Zero-point-energy is :',f18.6)") enerlow
!
deallocate(pot_eff,i0,phi_f,phi_t,phi_f_,phi_d_)
!
end subroutine numerov
! subroutine solves eqn 6.5 , for the given value of kqua ,
! by the numerov - cooley technique.
!
! ( 0 )
! 2 ( 2 i )
! d ( 2 pp )
! ---phi (p) = ( f (p) + f (p) k + ----- ( v (p) - e ) ) phi (p)
! 2 b ( 1 2 _2 eff ) b
! dp ( h )
!
! for this the following defintions (equations 6.14 to 6.17) are
! needed :-
!
! ph = phi (p )
! i b i