diff --git a/Makefile b/Makefile
index cee7989df402a151448c00b0cf53645cf9803f5b..6f24ba0774e8babf1861a3aa666dd14daed2b2fa 100644
--- a/Makefile
+++ b/Makefile
@@ -8,9 +8,8 @@ FC = gfortran
# -Wsurprising -Waliasing -Wimplicit-interface -Wunused-parameter \
# -fwhole-file -fcheck=all -std=f2008 -pedantic -fbacktrace \
# -finit-real=nan -Wextra -Wconversion -pg
-
-FFLAGS = -march=native -mtune=native -ffast-math -O3 -funroll-loops
-
+#-O3 -funroll-loops
+FFLAGS = -march=native -mtune=native -O2
#FFLAGS = -O0
LIBS = -lblas -llapack
diff --git a/data.f90 b/data.f90
index 83b603e6ab25ad765dbc4a731d446d78dde23f1f..37358e00921d47fb8145b9e1a161b7367d92425b 100644
--- a/data.f90
+++ b/data.f90
@@ -20,6 +20,7 @@ module data
real(dp) :: DOutward,DPredOutward,DCorrOutward
real(dp), allocatable :: aOutward(:),aPredOutward(:),aCorrOutward(:)
real(dp) :: DInward,DPredInward,DCorrInward
+ real(dp) :: mix
real(dp), allocatable :: aInward(:),aPredInward(:),aCorrInward(:)
real(dp), allocatable :: orbs(:,:),orbsNorms(:)
integer, allocatable :: lambdas(:),ls(:),leigens(:),DeltaBeta(:)
diff --git a/documentation/documentation.aux b/documentation/documentation.aux
new file mode 100644
index 0000000000000000000000000000000000000000..a6ec9e8c06170a0c3e1e5ca46bb2e1afe23e3cdd
--- /dev/null
+++ b/documentation/documentation.aux
@@ -0,0 +1,3 @@
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diff --git a/documentation/documentation.log b/documentation/documentation.log
new file mode 100644
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+++ b/documentation/documentation.log
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+\usepackage[T1]{fontenc} % skanditavutus
+%\usepackage[ansinew]{inputenc} % skandien sy�� (windows)
+\usepackage[utf8]{inputenc} % 䤫k�� (linux)
+\usepackage[finnish]{babel} % suomenkielinen tavutus
+\usepackage[top=3.5cm, bottom=0.5cm, left=2.5cm, right=2.5cm]{geometry}
+%\usepackage{layout}
+\setlength{\voffset}{-0.75in}
+\setlength{\headsep}{5pt}
+\usepackage[]{graphicx}
+\usepackage{fancyhdr}
+\usepackage{amssymb,amsmath,amsthm}
+\usepackage{float}
+\usepackage{tikz}
+\usepackage{siunitx}
+\usepackage{physics}
+\pagestyle{fancy}
+\linespread{1.24} %riviv䬩 1.5
+\lhead{\vspace{-10cm}aRc}
+\rhead{\thepage}
+\renewcommand{\headrulewidth}{0.4pt}
+\renewcommand{\footrulewidth}{0.4pt}
+\newcommand{\R}{\mathbb{R}}
+\newcommand{\N}{\mathbb{N}}
+\newcommand{\Q}{\mathbb{Q}}
+\newcommand{\C}{\mathbb{C}}
+\begin{document}
+\begin{center}
+\huge{Documentation for the program}
+\end{center}
+
+This is a good place to start when reading the code of Levämäki's DFTATOM modification for Nagy's (2011) version of orbital-free
+density functional theory. The program can be seen in action in Levämäki 2015. You should have read that paper before reading this documentation
+or you will likely not understand anything. Also, to understand the technicalities of the integration methods, read Johnson's Atomic Structure Theory
+chapter 2. Here I will only describe the main points of the functioning of the program, so you don't need to deduce everything yourself.
+
+Basically, the program works by solving these equations:
+
+\begin{align}
+\begin{cases}
+P(\beta, r)&= \phi(\beta ,r),\ \ \ \ \ Q(\beta, r) = P'(\beta ,r)\\
+P'(\beta, r) &= Q(\beta,r)\\
+Q'(\beta ,r) &= 2[v_{ks}(r) + v_p(\beta ,r) - \mu ] P(\beta ,r) \\
+v'_p(\beta ,r) &= 4\dfrac{[\mu - v_p(\beta ,r)] P(\beta ,r) Q(\beta ,r) - \frac{\partial P}{\partial \beta}(\beta ,r)Q(\beta ,r) - P(\beta ,r) \frac{\partial Q}{\partial \beta} (\beta ,r)}{P^2(\beta ,r)}
+\end{cases}
+\end{align}
+
+using implicit Adams predictor-correctors and a two-point beta-grid (in the future, also a gamma grid).
+
+main Be.f90 is the main driver for the program. We need several different sorts of arrays for it to work. P and Q hold the solutions for the
+above differential equations (one for each point on the beta-grid). Pdot and Qdot hold the $\beta$-derivatives of the density. Pgdot and Qgdot hold the $\gamma$-derivatives of the
+density when they are implemented (that imposes additional terms on our equations shown soon). The $\beta$-grid variables NbgGrid determine at
+how many points are the $\beta$-derivatives evaluated --- more on that later. Here are the additional gamma-terms when they're
+working:
+
+\begin{align}
+\begin{cases}
+P(\beta, r)&= \phi(\beta ,r),\ \ \ \ \ Q(\beta, r) = P'(\beta ,r)\\
+P'(\beta, r) &= Q(\beta,r)\\
+Q'(\beta ,r) &= 2[v_{ks}(r) + v_p(\beta ,r) - \mu ] P(\beta ,r) \\
+v'_p(\beta ,r) &= 4\dfrac{[\mu - v_p(\beta ,r)] P(\beta ,r) Q(\beta ,r) - \frac{\partial P}{\partial \beta}(\beta ,r)Q(\beta ,r) - P(\beta ,r) \frac{\partial Q}{\partial \beta} (\beta ,r)}{P^2(\beta ,r)} - 2\bigg[ \dfrac{\frac{\partial P}{\partial \gamma} Q}{r^2 P^2} + \dfrac{\frac{\partial Q}{\partial \gamma}}{r^2 P} \bigg]\\
+&+ 2 \dfrac{\frac{\partial P}{\partial \gamma}}{r^3 P}
+\end{cases}
+\end{align}
+
+Many of the files are rather self-explanatory. hartreepot.f90 integrates the Poisson equation
+\begin{align}
+V_H '' + 2/r V_H' + 4\pi \rho = 0
+\end{align}
+to get the Hartree potential. The xc.f90 file simply returns exchange- and correlation energies. It is best not to touch either of these files,
+because these are well known problems and have been solved adequately in the code. energies.f90 likewise simply returns the energy values of
+the system after the density has been solved --- again, no need to touch it.
+
+The real magic happens in the OFDFT step in solverhovp.f90. This file contains the main logic. For this case of beryllium that is being investigated,
+it does the following things:
+
+\begin{itemize}
+\item Calculate 1s orbital energy by demanding that $v_P$ approaches 0 far from the nucleus
+\item Calculate $\mu$ with a shooting method, and this is the 2s orbital energy
+\end{itemize}
+1s orbital energy is calculated by repeatedly integrating outward from the nucleus towards the CTP-point, which is quite far from the nucleus.
+After every integration, it is checked whether $v_P$ has increased from the previous round. If it has, then we have found a minimum (it first
+decreases because of the initial guess, and after it hits a minimum, it starts increasing). This is near the correct value for 1s because we
+need to have the $v_P$ potential vanish at CTP. The integrations are done in AdamsOutwardCTP. Once we find this nearly correct value, it is refined
+with the bisection method. (All of this happens at the beginning of ofdft step, about until line 170).
+
+$\mu$ is solved with a shooting method. We first integrate inward from practical infinity (about 50 Bohr away), and stop at the CTP point. Then we
+integrate outward to the CTP point. We compare these solutions (done on lines 176-200 in solverhovp.f90) and force them to join at CTP. If it's
+not continuous, so on of the solutions has to be changed a lot to make them continuous at CTP, then we change $\mu$ and try again (lines 220-230).
+
+One important point about what happens inside the integration routines AdamsInward, Adamsoutward and AdamsOutwardCTP. The $\beta$-derivatives in our
+equations are determined in DiffBeta and stored in Pdot and Qdot, by solving the following system of equations:
+
+\begin{align}
+\begin{cases}
+\rho(\beta _1 ,r) &= \lambda _1 \exp[\beta _1 b_1]a_1(r) + \lambda _2 \exp[\beta _1 b_2] a_2(r)\\
+\rho (\beta _2 ,r) &=\lambda _1 \exp[\beta _2 b_1]a_1(r) + \lambda _2 \exp[\beta _2 b_2] a_2(r)
+\end{cases}
+\end{align}
+
+This is solved by calling a LAPACK routine. It is precisely this part of the code that is so slow to solve, because solving a system of equations
+scales as $\mathcal{O}(N^3)$. For example, if we had a Neon atom instead of Beryllium, we would have not only orbitals 1s and 2s, but also the
+2p orbital, so we would need 3 equations in (4). They would look like this:
+\begin{align}
+\rho (\beta _2 ,r) &=2 \exp[\beta _2 b_1]a_1(r) + 2 \exp[\beta _2 b_2] a_2(r) + 6(\exp [\beta _1 b_4] \exp[-2\gamma]a_4 +
+ \exp [\beta _1 b_5]\exp [-2\gamma]a_5
+\end{align}
+
+So then we must solve our equations on yet another grid! This is the part of the code that one must focus on to increase efficiency.
+
+The entire solverhovp.f90 is also wrapped in an outer do loop, which is a sort of sanity check. Because the solution is sensitive to the initial
+values, especially initial values of vP, we use the following procedure:
+
+\begin{enumerate}
+\item Do all the calculation steps described before
+\item Obtain the new density with the mix $\rho _{new} = \alpha \rho _{currentsolution} + (1-\alpha)\rho_{old}$
+\item Check this against known upper- and lower bounds of Weizsäcker energy (could also do this for vP energy) of the OLD density.
+\item If bounds are violated, we probably have a bad solution, so we need new starting values
+\end{enumerate}
+Why do we have a bad solution if the lower- and upper bounds are violated? The upper- and lower bounds are quite generous. Our density shouldn't
+change so rapidly as to violate those bounds. If it does, it is likely that the vP solution has somehow diverged and we need to try again!
+
+The bounds used are $\frac{1}{8} \int r^{-2} \rho_{new}\mathrm{dr}\cdot \dfrac{1}{1-\frac{ \int r^{-1}
+ \rho_{new}\mathrm{dr}^2}{ \int r^{-2} \rho_{new}\mathrm{dr}}}< T_{vw} <
+ \frac{1}{4}\int r^{-2}\rho _{new}\mathrm{dr} $
+\end{document}
\ No newline at end of file
diff --git a/main_Be b/main_Be
index 38b398d40df40f34cdbe65eeaa4be1dfec17a78f..61d14f158f95e37434373b6660f31d32853c510a 100755
Binary files a/main_Be and b/main_Be differ
diff --git a/main_Be.f90 b/main_Be.f90
index a5a26435ea1557aa567ddb375e753b32c50592d1..8741b8e7f79ffebc3199c818da8c988ceb6a8813 100644
--- a/main_Be.f90
+++ b/main_Be.f90
@@ -3,10 +3,10 @@ program main_Be
use constants, only: pi
use data
use init, only: initialize
- use derivatives, only: genDcoeffs,xDiff1,xDiff2,xDiff3
+ use derivatives, only: genDcoeffs,xDiff1,xDiff2,xDiff3, Weizsacker
use derivatives, only: rDiff1,rDiff2,rDiff3
use mesh, only: logmesh
- use trapezoidalrule, only: trapz7
+ use trapezoidalrule, only: trapz7, euler
use xc, only: LDAvX,LDAvC,LDAeX,LDAeC
use hartreepot, only: hartree
use solverhovp, only: ofdft_step
@@ -21,10 +21,12 @@ program main_Be
real(dp), allocatable :: RhoOld(:)
real(dp) :: muNew,L2Norm,e1s
real(dp), allocatable :: outArray(:,:)
+ real(dp), allocatable :: vwp(:), muar(:)
character(len=100) :: filename
real(dp) :: Etot,EXC,ECoul,ENuc,EKin
- real(dp) :: ConvTarget,mix
+ real(dp) :: ConvTarget
real(dp) :: BeforeLoop, AfterLoop
+ real(dp) :: xclb, vw, gkinetic, rhomu, paulik, vw_lb, vw_up, kin_up, vptest
! A program to solve the electronic structure of Be orbital-free
! with the accuracy of KS-DFT.
@@ -69,18 +71,18 @@ program main_Be
RhoOld = (8.0_dp*exp(-2.0_dp*R*Z)*R**2.0_dp*Z**3.0_dp+&
exp(-R*Z)*R**2.0_dp*Z**3.0_dp*(1.0_dp-(R*Z)/2.0_dp)**2.0_dp)
+ RhoNew(:,1) = RhoOld
vH=hartree(RhoOld)
vXC=LDAvX(RhoOld)+LDAvC(RhoOld)
vKS=vExt+vH+vXC
!
- ConvTarget=1.0e-8
+ ConvTarget=1.0e-5
! The main iteration loop can now begin
do iterMain=1,itersMainMax
print *,''
print *,'Iteration number ',iterMain
- print *,''
-
+
call cpu_time(BeforeLoop)
call ofdft_step(vKS,rhoNew,gNew,vPNew,muNew,e1s)
call cpu_time(AfterLoop)
@@ -95,8 +97,19 @@ program main_Be
! of the difference between new and old).
L2Norm=trapz7((RhoNew(:,1)-RhoOld)**2,1,Nmesh)
write(*,'(A18,E12.6)') 'Error in density: ',L2Norm
- print *, "STEP TIME", AfterLoop-BeforeLoop
+ write(*,'(A18,E12.6)') 'Step time: ',AfterLoop-BeforeLoop
RhoOld=RhoNew(:,1)
+ !xclb = 5.7425_dp * 2**(2.0_dp/3.0_dp)*trapz7(RhoNew(:,1)**(5.0_dp/3.0_dp),0, Nmesh)/(4.0_dp)
+ xclb = 2.871234 * trapz7(RhoNew(:,1) **(5/3),1,Nmesh)
+ paulik = trapz7(RhoNew(:,1)*vPNew(:,1),1,Nmesh)
+ gkinetic = trapz7(gNew(:,1),1,Nmesh)
+ vw = trapz7(RhoNew(:,1)*Weizsacker(RhoNew(:,1)),1,Nmesh)
+ vw_lb = 1.0_dp/8.0_dp * trapz7(RhoNew(:,1) * r**(-2), 1, Nmesh) *&
+ 1/(1-1.0_dp/Z*trapz7(RhoNew(:,1) * r**(-1),1,Nmesh)**2/trapz7(RhoNew(:,1) * r**(-2),1,Nmesh))
+ vw_up = 0.25_dp * trapz7(r**(-2) * RhoNew(:,1),1, Nmesh)
+ kin_up = (1 + (4*3.141592653_dp)**2 * (Z/2.0_dp)**(2.0_dp/3.0_dp)*9.0_dp/5**(2.0_dp/3.0_dp) )*vw
+ rhomu = trapz7(muNew*RhoNew(:,1),1,Nmesh)
+ vptest = xclb - vw -gkinetic+rhomu
! Check convergence
if (L2Norm<ConvTarget) then
write(*,'(A48)') '################################################'
@@ -106,10 +119,20 @@ program main_Be
call get_energies(Etot,EXC,ECoul,ENuc,EKin,RhoNew(:,1),gNew(:,1),vH,vExt,vPNew(:,1),muNew)
write(*,'(A29,F11.6)') 'Total energy =',Etot
write(*,'(A29,F11.6)') 'Kinetic energy =',EKin
+ write(*,'(A29,F11.6)') 'Kinetic energy upper bound =',kin_up
+ write(*,'(A29,F11.6)') 'Kinetic energy lower bound =',xclb
write(*,'(A29,F11.6)') 'Electron-electron energy =',ECoul
write(*,'(A29,F11.6)') 'Electron-nucleus energy =',ENuc
write(*,'(A29,F11.6)') 'Exchange-correlation energy =',EXC
write(*,'(A40)') '----------------------------------------'
+ write(*,'(A29,F11.6)') 'Von W kinetic energy =',vw
+ write(*,'(A29, F11.6)')'Von W lower bound =',vw_lb
+ write(*,'(A29, F11.6)')'Von W upper bound =',vw_up
+ write(*,'(A29, F11.6)')'VP lower bound =',vptest
+ write(*,'(A29,F11.6)') 'Pauli kinetic energy =',paulik
+ write(*,'(A29,F11.6)') 'g kinetic energy component =',gkinetic
+ write(*,'(A29,F11.6)') 'mu*rho kinetic energy =',rhomu
+ write(*,'(A40)') '----------------------------------------'
write(*,'(A3,F10.6)') '1s:',e1s
write(*,'(A3,F10.6)') '2s:',muNew
write(*,*) ''
@@ -120,13 +143,24 @@ program main_Be
write(*,*) ''
!write(*,'(A33,I3,A12)') 'Convergence has been achieved in ',iterMain,' iterations.'
!write(*,*) ''
+
call get_energies(Etot,EXC,ECoul,ENuc,EKin,RhoNew(:,1),gNew(:,1),vH,vExt,vPNew(:,1),muNew)
write(*,'(A29,F11.6)') 'Total energy =',Etot
write(*,'(A29,F11.6)') 'Kinetic energy =',EKin
+ write(*,'(A29,F11.6)') 'Kinetic energy upper bound =',kin_up
+ write(*,'(A29,F11.6)') 'Kinetic energy lower bound =',xclb
write(*,'(A29,F11.6)') 'Electron-electron energy =',ECoul
write(*,'(A29,F11.6)') 'Electron-nucleus energy =',ENuc
write(*,'(A29,F11.6)') 'Exchange-correlation energy =',EXC
write(*,'(A40)') '----------------------------------------'
+ write(*,'(A29,F11.6)') 'Von W kinetic energy =',vw
+ write(*,'(A29, F11.6)')'Von W lower bound =',vw_lb
+ write(*,'(A29, F11.6)')'Von W upper bound =',vw_up
+ write(*,'(A29, F11.6)')'VP lower bound =',vptest
+ write(*,'(A29,F11.6)') 'Pauli kinetic energy =',paulik
+ write(*,'(A29,F11.6)') 'g kinetic energy component =',gkinetic
+ write(*,'(A29,F11.6)') 'mu*rho kinetic energy =',rhomu
+ write(*,'(A40)') '----------------------------------------'
write(*,'(A3,F10.6)') '1s:',e1s
write(*,'(A3,F10.6)') '2s:',muNew
write(*,*) ''
@@ -136,10 +170,19 @@ program main_Be
! Save calculated quantities to disk.
write(filename,*) './results'
allocate(outArray(Nmesh,3))
+ allocate(vwp(Nmesh))
+ allocate(muar(Nmesh))
+ vwp = Weizsacker(RhoNew(:,1))
+ muar = muNew
outArray(:,1)=r
- outArray(:,2)=vPNew(:,1)
+ !outArray(:,2)= vwp(:)
+ outArray(:,3)=vPNew(:,1)
+ outArray(:,2) = Pdotsol(:,1)
+ print *, minval(vPNew(:,1)), minval(Pdotsol(:,1))
+ !outArray(:,3) = Qdotout(:,1)
+ !outArray(:,3) = Psol(:,1)
!outArray(:,3)=vPNew(:,2)
- outArray(:,3)=RhoNew(:,1)
+ !outArray(:,3)=RhoNew(:,1)
!outArray(:,3)=gNew(:,1)
!outArray(:,4)=vPNew(:,1)
!outArray(:,2)=orbs(1,:)
diff --git a/results b/results
index f66bf1e3bfe4432080e92799474c4b8406a037d5..4bc7a774e5a88476295008a2475eded4dc967399 100644
--- a/results
+++ b/results
@@ -1,2001 +1,2001 @@
- 9.9999999999999995E-008 0.0000000000000000 4.3805020717722154E-012
- 2.3667991994759664E-007 0.0000000000000000 2.4538527512234844E-011
- 3.7437613614517240E-007 0.0000000000000000 6.1396283993515657E-011
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diff --git a/solverhovp.f90 b/solverhovp.f90
index 1fe0be67907f0fe8280e13769abb1413f8dcf0b6..f700579ce12ea1139789df435854f1a5ec0959f5 100644
--- a/solverhovp.f90
+++ b/solverhovp.f90
@@ -11,6 +11,7 @@ contains
subroutine ofdft_step(vKS,rhoNew,gNew,vPNew,muNew,e1sOut)
use data
+ use derivatives, only: Weizsacker
real(dp), intent(in) :: vKS(:)
real(dp), intent(out) :: rhoNew(:,:),gNew(:,:),vPNew(:,:)
real(dp), intent(out) :: muNew,e1sOut
@@ -21,10 +22,30 @@ contains
real(dp) :: norm0,normfac0,Qdiff,Pint,mu,dmu,dmuTarget,e1sTarget
real(dp) ::gammaMin,gammaMax,gammaMid,gammaTarget,e1sShift
integer :: iterGamma,iterGammaMax
- real(dp) :: mumin,mumax,e1sMultiplier
+ real(dp) :: mumin,mumax,e1sMultiplier, gkinetic, rhomu, vw,xclb
+ real(dp) :: rhoOld(Nmesh), start = 10000.0_dp
integer, parameter :: prnt=0
!print *,'ofdft_step started'
+
+ ! Starting from TH potential
+ !e1s=-2.849007_dp
+ !mu=-0.216434_dp
+ !gamma=47.798473897919941_dp
+
+ ! Starting from H-like orbitals
+ !mu=-0.090_dp
+ !mu=-0.039249_dp
+ !e1s=-3.009855_dp
+ !e1s=-3.01_dp
+ !gamma=42.27944725494816_dp
+
+ !print *,'mu,dmu,gamma,ctp = ',mu,dmu,gamma,ctp
+
+
+ !stop
+ outer: do
+
Pin=0.0_dp
Pdotin=0.0_dp
Pout=0.0_dp
@@ -75,36 +96,20 @@ contains
e1sShift=1.0e-5_dp
e1sMultiplier=1.0_dp
- ! Starting from TH potential
- !e1s=-2.849007_dp
- !mu=-0.216434_dp
- !gamma=47.798473897919941_dp
-
- ! Starting from H-like orbitals
- !mu=-0.090_dp
- !mu=-0.039249_dp
- !e1s=-3.009855_dp
- !e1s=-3.01_dp
- !gamma=42.27944725494816_dp
-
- !print *,'mu,dmu,gamma,ctp = ',mu,dmu,gamma,ctp
-
- call AdamsOutwardCTP(Pout,Pdotout,Pgdotout,Qout,Qdotout,Qgdotout,orbs,vPout,mu,vKS,ctp,e1s,gamma)
- !call AdamsInward(Pin,Pdotin,Qin,Qdotin,vPin,mu,vKS,ctp)
+ call AdamsOutwardCTP(Pout,Pdotout,Pgdotout,Qout,Qdotout,Qgdotout,orbs,vPout,mu,vKS,ctp,e1s,gamma,start)
+ !call AdamsInward(Pin,Pdotin,Qin,Qdotin,vPin,mu,vKS,ctp)
vPmin=vPout(ctp,1)
if (prnt==1) then
print *,'e1s,vPmin = ',e1s,vPmin
end if
-
- !stop
-
+ !Outer loop to check for convergence issues
do itere=1,itereMax
e1s00=e1s0
vPmin00=vPmin0
e1s0=e1s
vPmin0=vPmin
e1s=e1s+step
- call AdamsOutwardCTP(Pout,Pdotout,Pgdotout,Qout,Qdotout,Qgdotout,orbs,vPout,mu,vKS,ctp,e1s,gamma)
+ call AdamsOutwardCTP(Pout,Pdotout,Pgdotout,Qout,Qdotout,Qgdotout,orbs,vPout,mu,vKS,ctp,e1s,gamma,start)
vPmin=vPout(ctp,1)
if (prnt==1) then
print *,'e1s,vPmin,ctp = ',e1s,vPmin,ctp
@@ -120,7 +125,7 @@ contains
end if
end do
e1smid=0.5_dp*(e1smin+e1smax)
- call AdamsOutwardCTP(Pout,Pdotout,Pgdotout,Qout,Qdotout,Qgdotout,orbs,vPout,mu,vKS,ctp,e1smid,gamma)
+ call AdamsOutwardCTP(Pout,Pdotout,Pgdotout,Qout,Qdotout,Qgdotout,orbs,vPout,mu,vKS,ctp,e1smid,gamma,start)
vPminMid=vPout(ctp,1)
do itere=1,itereMax
if ((e1sMax-e1sMin)<e1sTarget) then
@@ -131,7 +136,7 @@ contains
end if
if ((e1sMid-e1sMin)>=(e1sMax-e1sMid)) then
e1s=0.5_dp*(e1sMin+e1sMid)
- call AdamsOutwardCTP(Pout,Pdotout,Pgdotout,Qout,Qdotout,Qgdotout,orbs,vPout,mu,vKS,ctp,e1s,gamma)
+ call AdamsOutwardCTP(Pout,Pdotout,Pgdotout,Qout,Qdotout,Qgdotout,orbs,vPout,mu,vKS,ctp,e1s,gamma,start)
vPmin=vPout(ctp,1)
if (prnt==1) then
print *,'bisection: e1s,vPmin,ctp = ',e1s,vPmin,ctp
@@ -146,7 +151,7 @@ contains
end if
else
e1s=0.5_dp*(e1sMid+e1sMax)
- call AdamsOutwardCTP(Pout,Pdotout,Pgdotout,Qout,Qdotout,Qgdotout,orbs,vPout,mu,vKS,ctp,e1s,gamma)
+ call AdamsOutwardCTP(Pout,Pdotout,Pgdotout,Qout,Qdotout,Qgdotout,orbs,vPout,mu,vKS,ctp,e1s,gamma,start)
vPmin=vPout(ctp,1)
if (prnt==1) then
print *,'bisection: e1s,vPmin,ctp = ',e1s,vPmin,ctp
@@ -167,7 +172,7 @@ contains
!
do iterAdams=1,iterAdamsMax
!print *,'Adams iteration number ',iterAdams
- call AdamsOutward(Pout,Pdotout,Pgdotout,Qout,Qdotout,Qgdotout,orbs,vPout,mu,vKS,ctp,e1s,gamma)
+ call AdamsOutward(Pout,Pdotout,Pgdotout,Qout,Qdotout,Qgdotout,orbs,vPout,mu,vKS,ctp,e1s,gamma,start)
call AdamsInward(Pin,Pdotin,Qin,Qdotin,vPin,mu,vKS,ctp)
!print *,orbs(1,ctp-5:ctp+5)
!print *,''
@@ -232,80 +237,107 @@ contains
! Lastly we need to find the correct gamma
!
do iterGamma=1,iterGammaMax
- gammaMid=0.5_dp*(gammaMin+gammaMax)
- call AdamsOutward(Pout,Pdotout,Pgdotout,Qout,Qdotout,Qgdotout,orbs,vPout,mu,vKS,ctp,e1s,gammaMid)
- call AdamsInward(Pin,Pdotin,Qin,Qdotin,vPin,mu,vKS,ctp)
- ! Join the two solutions
- do j=1,NbgGrid
- fac(j)=Pout(ctp,j)/Pin(ctp,j)
- facdot(j)=Pdotout(ctp,j)/Pdotin(ctp,j)
- !print *,fac(j)
- end do
- Psol(:ctp,:)=Pout(:ctp,:)
- Pdotsol(:ctp,:)=Pdotout(:ctp,:)
- do j=1,NbgGrid
- Psol(ctp:,j)=Pin(ctp:,j)*fac(j)
- Pdotsol(ctp:,j)=Pdotin(ctp:,j)*facdot(j)
- end do
- vPsol(:ctp,:)=vPout(:ctp,:)
- vPsol(ctp:,:)=0.0
- ! Normalize
- norm0=trapz7(Psol(:,1)**2,1,Nmesh)
- normfac0=sqrt(real(Z,dp)/norm0)
- Psol=normfac0*Psol
- Pdotsol=normfac0*Pdotsol
- ! Construct orbitals
- orbs=Normfac0**2*orbs
- orbs(2,ctp:)=orbs(2,ctp)/Psol(ctp,1)**2*Psol(ctp:,1)**2
- do j=1,size(orbs,1)
- orbsNorms(j)=Trapz7(orbs(j,:),1,Nmesh)
- if (prnt==1) then
- print *,'ofdft_step: orbsNorms = ',orbsNorms(j)
- end if
- end do
- !print *,'gamma,orbsNorms = ',gammaMid,orbsNorms
- if (orbsNorms(1)<1.0_dp) then
- gammaMin=gammaMid
- else
- gammaMax=gammaMid
- end if
- if (abs(orbsNorms(1)-1.0_dp)<gammaTarget) then
- exit
- end if
- !if (orbsNorms(2)>1.0_dp) then
- ! gammaMin=gammaMid
- !else
- ! gammaMax=gammaMid
- !end if
- !if (abs(orbsNorms(2)-1.0_dp)<gammaTarget) then
- ! exit
- !end if
- end do
+ gammaMid=0.5_dp*(gammaMin+gammaMax)
+
+ call AdamsOutward(Pout,Pdotout,Pgdotout,Qout,Qdotout,Qgdotout,orbs,vPout,mu,vKS,ctp,e1s,gammaMid,start)
+ call AdamsInward(Pin,Pdotin,Qin,Qdotin,vPin,mu,vKS,ctp)
+ ! Join the two solutions
+ do j=1,NbgGrid
+ fac(j)=Pout(ctp,j)/Pin(ctp,j)
+ facdot(j)=Pdotout(ctp,j)/Pdotin(ctp,j)
+ !print *,fac(j)
+ end do
+ Psol(:ctp,:)=Pout(:ctp,:)
+ Pdotsol(:ctp,:)=Pdotout(:ctp,:)
+ do j=1,NbgGrid
+ Psol(ctp:,j)=Pin(ctp:,j)*fac(j)
+ Pdotsol(ctp:,j)=Pdotin(ctp:,j)*facdot(j)
+ end do
+ vPsol(:ctp,:)=vPout(:ctp,:)
+ vPsol(ctp:,:)=0.0
+ ! Normalize
+ norm0=trapz7(Psol(:,1)**2,1,Nmesh)
+ normfac0=sqrt(real(Z,dp)/norm0)
+ Psol=normfac0*Psol
+ Pdotsol=normfac0*Pdotsol
+ ! Construct orbitals
+ orbs=Normfac0**2*orbs
+ orbs(2,ctp:)=orbs(2,ctp)/Psol(ctp,1)**2*Psol(ctp:,1)**2
+ do j=1,size(orbs,1)
+ orbsNorms(j)=Trapz7(orbs(j,:),1,Nmesh)
+ if (prnt==1) then
+ print *,'ofdft_step: orbsNorms = ',orbsNorms(j)
+ end if
+ end do
+ !print *,'gamma,orbsNorms = ',gammaMid,orbsNorms
+ if (orbsNorms(1)<1.0_dp) then
+ gammaMin=gammaMid
+ else
+ gammaMax=gammaMid
+ end if
+ if (abs(orbsNorms(1)-1.0_dp)<gammaTarget) then
+ exit
+ end if
+ !if (orbsNorms(2)>1.0_dp) then
+ ! gammaMin=gammaMid
+ !else
+ ! gammaMax=gammaMid
+ !end if
+ !if (abs(orbsNorms(2)-1.0_dp)<gammaTarget) then
+ ! exit
+ !end if
+ end do
!
!print *,'ctp,r(ctp) = ',ctp,r(ctp)
!print * ,orbs(1,:)
!print *,'ofdft_step: e1s,mu = ',e1s,mu
- rhoNew=Psol**2
- gNew=2.0_dp*Psol*Pdotsol
- vPNew=vPsol
- muNew=mu
- e1sOut=e1s
+ rhoOld = rhoNew(:,1)
+ if(CheckLowerBounds(Psol(:,1)**2, rhoOld) .and.&
+ CheckUpperBounds(Psol(:,1)**2, rhoOld)) then
+ print *, "Bound checks succesful"
+ gNew=2.0_dp*Psol*Pdotsol
+ rhoNew=Psol**2
+ vPNew=vPsol
+ muNew=mu
+ e1sOut=e1s
+
+ exit outer
+ else if(CheckLowerBounds(Psol(:,1)**2, rhoOld) .eqv. .false.) then
+ print *, "Lower bound failed, recycling"
+ xclb = 2.871234 * trapz7(RhoOld(:) **(5/3),1,Nmesh)
+ vw = trapz7(RhoOld(:)*Weizsacker(RhoOld(:)),1,Nmesh)
+ gkinetic = trapz7(gNew(:,1),1,Nmesh)
+ rhomu = trapz7(mu*RhoOld(:),1,Nmesh)
+ start = xclb - vw -gkinetic+rhomu
+ !gNew=2.0_dp*Psol*Pdotsol
+ cycle outer
+ else if(CheckUpperBounds(Psol(:,1)**2, rhoOld) .eqv. .false.) then
+ print *, "Upper bound failed, recycling"
+ xclb = 2.871234 * trapz7(RhoOld(:) **(5/3),1,Nmesh)
+ vw = trapz7(RhoOld(:)*Weizsacker(RhoOld(:)),1,Nmesh)
+ gkinetic = trapz7(gNew(:,1),1,Nmesh)
+ rhomu = trapz7(muNew*RhoOld(:),1,Nmesh)
+ start = xclb - vw -gkinetic+rhomu
+ !gNew=2.0_dp*Psol*Pdotsol
+ cycle outer
+ end if
+ end do outer
end subroutine ofdft_step
- subroutine AdamsOutwardCTP(P,Pdot,Pgdot,Q,Qdot,Qgdot,orbs,vP,mu,vKS,ctp,e1s,gamma)
+ subroutine AdamsOutwardCTP(P,Pdot,Pgdot,Q,Qdot,Qgdot,orbs,vP,mu,vKS,ctp,e1s,gamma,start)
use data, only: Z,r,Dr,Nmesh,NbgGrid,AdamsOutwardOrder,Dbeta,Norbs,dx,LambdaOutward
use data, only: betaGrid,aOutward,aPredOutward,aCorrOutward,ctp0
real(dp), intent(out) :: P(:,:),Pdot(:,:),Q(:,:)
real(dp), intent(out) :: Qdot(:,:),vP(:,:),orbs(Norbs,Nmesh)
real(dp), intent(out) :: Pgdot(:, :), Qgdot(:,:)
- real(dp), intent(in) :: mu,vKS(:),e1s,gamma
+ real(dp), intent(in) :: mu,vKS(:),e1s,gamma,start
integer, intent(inout) :: ctp
real(dp) :: ccc,ddd,eCoeff,P00,Q00
real(dp), dimension(Nmesh,NbgGrid) :: CC,Pp,Qp
real(dp) :: DvP(NbgGrid),fvPOutward(AdamsOutwardOrder,NbgGrid)
integer :: i,j,PECEiterMax,sigma,endPoint,PECEiter,vPMaxIndex
real(dp) :: Ptmp,Qtmp,Delta,M11,M12,M21,M22
- !real(dp) :: test(size(vP(1,:),1))
+ real(dp) :: test(size(vP(1,:),1))
Pgdot = 0.0_dp
Qgdot = 0.0_dp
@@ -320,9 +352,9 @@ contains
eCoeff=mu-e1s
P00=ccc*r(1)*(1.0_dp-Z*r(1))
Q00=ddd*(ccc-2.0_dp*ccc*Z*r(1))
- !test=[0.0_dp,-1.0e-1_dp]
- !vP(1,:)=gamma*eCoeff/(gamma+exp(betaGrid(:)*eCoeff))
- vP(1,:)=0.0_dp
+ test=[0.0_dp,-1.0e-1_dp]
+ vP(1,:)=start!gamma*eCoeff/(gamma+exp(betaGrid(:)*eCoeff))
+ !vP(1,:)=0.0_dp
!vP(1,:)=gamma*eCoeff/(gamma+exp(betaGrid(:)*eCoeff))+test(:)
P(1,:)=sqrt(gamma*exp(betaGrid(:)*e1s)+exp(betaGrid(:)*mu))/sqrt(gamma+1.0_dp)*P00
Q(1,:)=sqrt(gamma*exp(betaGrid(:)*e1s)+exp(betaGrid(:)*mu))/sqrt(gamma+1.0_dp)*Q00
@@ -371,6 +403,8 @@ contains
end do
call DiffBeta(Pdot(i+1,:),Qdot(i+1,:),orbs(:,i+1),P(i+1,:),Q(i+1,:),e1s,mu)
call DiffGamma(Pdot(i,:),Pgdot(i,:),Qdot(i,:),Qgdot(i,:),orbs(:,i),P(i,:),Q(i,:),e1s,mu)
+
+
do PECEiter=1,PECEiterMax
do j=1,NbgGrid
fvPOutward(:,j)=Dr(i-AdamsOutwardOrder+2:i+1)*4*((mu-vP(i-AdamsOutwardOrder+2:i+1,j))*&
@@ -384,6 +418,7 @@ contains
(r(i-AdamsOutwardOrder+1:i)**2 * P(i-AdamsOutwardOrder+1:i,j)) +& !GAMMA 2 COMPLETE
Dr(i-AdamsOutwardOrder+1:i)*2*(Pgdot(i-AdamsOutwardOrder+1:i,j))/&
(r(i-AdamsOutwardOrder+1:i)**3 * P(i-AdamsOutwardOrder+1:i,j))
+
DvP(j)=MultistepOutward(fvPOutward(:,j),1,aCorrOutward)
vP(i+1,j)=vP(i,j)+DvP(j)
CC(i+1,j)=-2.0_dp*(mu-vKS(i+1)-vP(i+1,j))
@@ -426,13 +461,13 @@ contains
!end do
end subroutine AdamsOutwardCTP
- subroutine AdamsOutward(P,Pdot, Pgdot,Q,Qdot,Qgdot,orbs,vP,mu,vKS,ctp,e1s,gamma)
+ subroutine AdamsOutward(P,Pdot, Pgdot,Q,Qdot,Qgdot,orbs,vP,mu,vKS,ctp,e1s,gamma,start)
use data, only: Z,r,Dr,Nmesh,NbgGrid,AdamsOutwardOrder,Dbeta,Norbs,dx,LambdaOutward
use data, only: betaGrid,aOutward,aPredOutward,aCorrOutward
real(dp), intent(out) :: P(:,:),Pdot(:,:),Q(:,:)
real(dp), intent(out) :: Qdot(:,:),vP(:,:),orbs(Norbs,Nmesh)
real(dp), intent(out) :: Pgdot(:, :), Qgdot(:, :)
- real(dp), intent(in) :: mu,vKS(:),e1s,gamma
+ real(dp), intent(in) :: mu,vKS(:),e1s,gamma,start
integer, intent(in) :: ctp
real(dp) :: ccc,ddd,eCoeff,P00,Q00
real(dp), dimension(Nmesh,NbgGrid) :: CC,Pp,Qp
@@ -450,8 +485,8 @@ contains
Q00=ddd*(ccc-2.0_dp*ccc*Z*r(1))
test=[0.0_dp,-1.0e-1_dp]
!vP(1,:)=gamma*eCoeff/(gamma+exp(betaGrid(:)*eCoeff))
- vP(1,:)=0.0_dp
- !vP(1,:)=gamma*eCoeff/(gamma+exp(betaGrid(:)*eCoeff))+test(:)
+ !vP(1,:)=0.0_dp
+ vP(1,:)=start!gamma*eCoeff/(gamma+exp(betaGrid(:)*eCoeff))+test(:)
P(1,:)=sqrt(gamma*exp(betaGrid(:)*e1s)+exp(betaGrid(:)*mu))/sqrt(gamma+1.0_dp)*P00
Q(1,:)=sqrt(gamma*exp(betaGrid(:)*e1s)+exp(betaGrid(:)*mu))/sqrt(gamma+1.0_dp)*Q00
do j=1,NbgGrid
@@ -683,6 +718,44 @@ contains
end do
end function MultistepInwardImp
+ function CheckLowerBounds(Rho, RhoOld) result(bounds)
+ use data, only: r, Nmesh, mix
+ use derivatives, only: Weizsacker
+ real(dp), intent(in) :: rho(:), RhoOld(:)
+ real(dp) :: vw_true, vw_lower, RhoMix(Nmesh)
+ logical :: bounds
+
+ RhoMix(:)=(1.0_dp-mix)*RhoOld+mix*Rho(:)
+ vw_lower = 1.0_dp/8.0_dp * trapz7(RhoOld(:) * r**(-2), 1, Nmesh) *&
+ 1/(1-0.25_dp*trapz7(RhoOld(:) * r**(-1),1,Nmesh)**2&
+ /trapz7(RhoOld(:) * r**(-2),1,Nmesh))
+ !print *, vw_lower
+ vw_true = trapz7(RhoMix(:) * Weizsacker(RhoMix), 1, Nmesh)
+ !print *, vw_true
+ if (vw_lower < vw_true) then
+ bounds = .true.
+ else
+ bounds = .false.
+ end if
+ end function CheckLowerBounds
+
+ function CheckUpperBounds(Rho, RhoOld) result(bounds)
+ use data, only: r, Nmesh, mix
+ use derivatives, only: Weizsacker
+ real(dp), intent(in) :: rho(:), RhoOld(:)
+ real(dp) :: RhoMix(Nmesh)
+ real(dp) :: vw_true, vw_upper
+ logical :: bounds
+
+ RhoMix(:)=(1.0_dp-mix)*RhoOld+mix*Rho(:)
+ vw_true = trapz7(RhoMix(:) * Weizsacker(RhoMix), 1, Nmesh)
+ vw_upper = 0.25_dp * trapz7(r**(-2) * RhoOld(:),1, Nmesh)
+ if(vw_true < vw_upper) then
+ bounds = .true.
+ else
+ bounds = .false.
+ end if
+ end function CheckUpperBounds
subroutine DiffBeta(Pdot,Qdot,orbs,P,Q,e1s,mu)
use data, only: betaGrid,gammaGrid,NbgGrid,Norbs,leigens,lambdas
real(dp), intent(in) :: P(:),Q(:),e1s,mu
@@ -722,7 +795,7 @@ contains
Pdot(i)=0.5_dp*sum(energies(:)*A(i,:)*orbs(:))/P(i)
Qdot(i)=0.5_dp*(sum(energies(:)*A(i,:)*orbsPrime(:))-2.0_dp*Q(i)*Pdot(i))/P(i)
end do
-
+
!
!print *,Qdot
end subroutine DiffBeta
diff --git a/temp b/temp
new file mode 100644
index 0000000000000000000000000000000000000000..1bcb5eb2ba3c669a94d140f0f07b7c1a9f4062a8
--- /dev/null
+++ b/temp
@@ -0,0 +1,51 @@
+do iterGamma=1,iterGammaMax
+ gammaMid=0.5_dp*(gammaMin+gammaMax)
+
+ call AdamsOutward(Pout,Pdotout,Pgdotout,Qout,Qdotout,Qgdotout,orbs,vPout,mu,vKS,ctp,e1s,gammaMid)
+ call AdamsInward(Pin,Pdotin,Qin,Qdotin,vPin,mu,vKS,ctp)
+ ! Join the two solutions
+ do j=1,NbgGrid
+ fac(j)=Pout(ctp,j)/Pin(ctp,j)
+ facdot(j)=Pdotout(ctp,j)/Pdotin(ctp,j)
+ !print *,fac(j)
+ end do
+ Psol(:ctp,:)=Pout(:ctp,:)
+ Pdotsol(:ctp,:)=Pdotout(:ctp,:)
+ do j=1,NbgGrid
+ Psol(ctp:,j)=Pin(ctp:,j)*fac(j)
+ Pdotsol(ctp:,j)=Pdotin(ctp:,j)*facdot(j)
+ end do
+ vPsol(:ctp,:)=vPout(:ctp,:)
+ vPsol(ctp:,:)=0.0
+ ! Normalize
+ norm0=trapz7(Psol(:,1)**2,1,Nmesh)
+ normfac0=sqrt(real(Z,dp)/norm0)
+ Psol=normfac0*Psol
+ Pdotsol=normfac0*Pdotsol
+ ! Construct orbitals
+ orbs=Normfac0**2*orbs
+ orbs(2,ctp:)=orbs(2,ctp)/Psol(ctp,1)**2*Psol(ctp:,1)**2
+ do j=1,size(orbs,1)
+ orbsNorms(j)=Trapz7(orbs(j,:),1,Nmesh)
+ if (prnt==1) then
+ print *,'ofdft_step: orbsNorms = ',orbsNorms(j)
+ end if
+ end do
+ !print *,'gamma,orbsNorms = ',gammaMid,orbsNorms
+ if (orbsNorms(1)<1.0_dp) then
+ gammaMin=gammaMid
+ else
+ gammaMax=gammaMid
+ end if
+ if (abs(orbsNorms(1)-1.0_dp)<gammaTarget) then
+ exit
+ end if
+ !if (orbsNorms(2)>1.0_dp) then
+ ! gammaMin=gammaMid
+ !else
+ ! gammaMax=gammaMid
+ !end if
+ !if (abs(orbsNorms(2)-1.0_dp)<gammaTarget) then
+ ! exit
+ !end if
+ end do
\ No newline at end of file
diff --git a/trapezoidalrule.f90 b/trapezoidalrule.f90
index bcfec613ba6664ffb461cd4377cab9df1627bfcf..9f41db3d2a899ab2d10f9ad32a577285b83434bf 100644
--- a/trapezoidalrule.f90
+++ b/trapezoidalrule.f90
@@ -5,7 +5,7 @@ module trapezoidalrule
implicit none
private
- public trapz7
+ public trapz7, euler
contains
@@ -28,4 +28,15 @@ contains
s = s + sum(g(8:N-7))
end function trapz7
+ function euler(array, indMin, indMax) result(s)
+
+ real(dp), intent(in) :: array(:)
+ integer, intent(in) :: indMin, indMax
+ real(dp) :: s
+ real(dp) :: g(indMax-indMin+1)
+
+ g = array(indMin:indMax) * Dr(indMin:indMax)
+
+ s = sum(g)
+ end function euler
end module trapezoidalrule