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Anthony Scemama 2021-01-12 10:55:00 +01:00
parent b6d70add67
commit 22f1eaf41b
3 changed files with 75 additions and 48 deletions
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QMC.org
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@ -743,11 +743,18 @@ print(f"E = {E} +/- {deltaE}")
: E = -0.4956255109300764 +/- 0.0007082875482711226 : E = -0.4956255109300764 +/- 0.0007082875482711226
*Fortran* *Fortran*
#+begin_note
When running Monte Carlo calculations, the number of steps is
usually very large. We expect =nmax= to be possibly larger than 2
billion, so we use 8-byte integers (=integer*8=) to represent it, as
well as the index of the current step.
#+end_note
#+BEGIN_SRC f90 #+BEGIN_SRC f90
subroutine uniform_montecarlo(a,nmax,energy) subroutine uniform_montecarlo(a,nmax,energy)
implicit none implicit none
double precision, intent(in) :: a double precision, intent(in) :: a
integer , intent(in) :: nmax integer*8 , intent(in) :: nmax
double precision, intent(out) :: energy double precision, intent(out) :: energy
integer*8 :: istep integer*8 :: istep
@ -772,7 +779,7 @@ end subroutine uniform_montecarlo
program qmc program qmc
implicit none implicit none
double precision, parameter :: a = 0.9 double precision, parameter :: a = 0.9
integer , parameter :: nmax = 100000 integer*8 , parameter :: nmax = 100000
integer , parameter :: nruns = 30 integer , parameter :: nruns = 30
integer :: irun integer :: irun
@ -833,18 +840,18 @@ subroutine random_gauss(z,n)
! n is even ! n is even
do i=1,n,2 do i=1,n,2
z(i) = dsqrt(-2.d0*dlog(u(i))) z(i) = dsqrt(-2.d0*dlog(u(i)))
z(i+1) = z(i) + dsin( two_pi*u(i+1) ) z(i+1) = z(i) * dsin( two_pi*u(i+1) )
z(i) = z(i) + dcos( two_pi*u(i+1) ) z(i) = z(i) * dcos( two_pi*u(i+1) )
end do end do
else else
! n is odd ! n is odd
do i=1,n-1,2 do i=1,n-1,2
z(i) = dsqrt(-2.d0*dlog(u(i))) z(i) = dsqrt(-2.d0*dlog(u(i)))
z(i+1) = z(i) + dsin( two_pi*u(i+1) ) z(i+1) = z(i) * dsin( two_pi*u(i+1) )
z(i) = z(i) + dcos( two_pi*u(i+1) ) z(i) = z(i) * dcos( two_pi*u(i+1) )
end do end do
z(n) = dsqrt(-2.d0*dlog(u(n))) z(n) = dsqrt(-2.d0*dlog(u(n)))
z(n) = z(n) + dcos( two_pi*u(n+1) ) z(n) = z(n) * dcos( two_pi*u(n+1) )
end if end if
end subroutine random_gauss end subroutine random_gauss
#+END_SRC #+END_SRC
@ -907,7 +914,7 @@ print(f"E = {E} +/- {deltaE}")
#+END_SRC #+END_SRC
#+RESULTS: #+RESULTS:
: E = -0.49507506093129827 +/- 0.00014164037765553668 : E = -0.49511014287471955 +/- 0.00012402022172236656
*Fortran* *Fortran*
@ -916,14 +923,14 @@ double precision function gaussian(r)
implicit none implicit none
double precision, intent(in) :: r(3) double precision, intent(in) :: r(3)
double precision, parameter :: norm_gauss = 1.d0/(2.d0*dacos(-1.d0))**(1.5d0) double precision, parameter :: norm_gauss = 1.d0/(2.d0*dacos(-1.d0))**(1.5d0)
gaussian = norm_gauss * dexp( -0.5d0 * dsqrt(r(1)*r(1) + r(2)*r(2) + r(3)*r(3) )) gaussian = norm_gauss * dexp( -0.5d0 * (r(1)*r(1) + r(2)*r(2) + r(3)*r(3) ))
end function gaussian end function gaussian
subroutine gaussian_montecarlo(a,nmax,energy) subroutine gaussian_montecarlo(a,nmax,energy)
implicit none implicit none
double precision, intent(in) :: a double precision, intent(in) :: a
integer , intent(in) :: nmax integer*8 , intent(in) :: nmax
double precision, intent(out) :: energy double precision, intent(out) :: energy
integer*8 :: istep integer*8 :: istep
@ -947,7 +954,7 @@ end subroutine gaussian_montecarlo
program qmc program qmc
implicit none implicit none
double precision, parameter :: a = 0.9 double precision, parameter :: a = 0.9
integer , parameter :: nmax = 100000 integer*8 , parameter :: nmax = 100000
integer , parameter :: nruns = 30 integer , parameter :: nruns = 30
integer :: irun integer :: irun
@ -968,13 +975,9 @@ gfortran hydrogen.f90 qmc_stats.f90 qmc_gaussian.f90 -o qmc_gaussian
#+end_src #+end_src
#+RESULTS: #+RESULTS:
: E = -0.49606057056767766 +/- 1.3918807547836872E-004 : E = -0.49517104619091717 +/- 1.0685523607878961E-004
** Sampling with $\Psi^2$ ** Sampling with $\Psi^2$
:PROPERTIES:
:header-args:python: :tangle vmc.py
:header-args:f90: :tangle vmc.f90
:END:
We will now use the square of the wave function to make the sampling: We will now use the square of the wave function to make the sampling:
@ -991,6 +994,10 @@ gfortran hydrogen.f90 qmc_stats.f90 qmc_gaussian.f90 -o qmc_gaussian
*** Importance sampling *** Importance sampling
:PROPERTIES:
:header-args:python: :tangle vmc.py
:header-args:f90: :tangle vmc.f90
:END:
To generate the probability density $\Psi^2$, we consider a To generate the probability density $\Psi^2$, we consider a
diffusion process characterized by a time-dependent density diffusion process characterized by a time-dependent density
@ -1107,20 +1114,22 @@ end subroutine drift
#+end_exercise #+end_exercise
*Python* *Python*
#+BEGIN_SRC python :results output :tangle vmc.py #+BEGIN_SRC python :results output
from hydrogen import * from hydrogen import *
from qmc_stats import * from qmc_stats import *
def MonteCarlo(a,tau,nmax): def MonteCarlo(a,tau,nmax):
sq_tau = np.sqrt(tau)
# Initialization
E = 0. E = 0.
N = 0. N = 0.
sq_tau = np.sqrt(tau)
r_old = np.random.normal(loc=0., scale=1.0, size=(3)) r_old = np.random.normal(loc=0., scale=1.0, size=(3))
d_old = drift(a,r_old)
d2_old = np.dot(d_old,d_old)
psi_old = psi(a,r_old)
for istep in range(nmax): for istep in range(nmax):
d_old = drift(a,r_old)
chi = np.random.normal(loc=0., scale=1.0, size=(3)) chi = np.random.normal(loc=0., scale=1.0, size=(3))
r_new = r_old + tau * d_old + sq_tau * chi r_new = r_old + tau * d_old + chi*sq_tau
N += 1. N += 1.
E += e_loc(a,r_new) E += e_loc(a,r_new)
r_old = r_new r_old = r_new
@ -1129,37 +1138,41 @@ def MonteCarlo(a,tau,nmax):
a = 0.9 a = 0.9
nmax = 100000 nmax = 100000
tau = 0.001 tau = 0.2
X = [MonteCarlo(a,tau,nmax) for i in range(30)] X = [MonteCarlo(a,tau,nmax) for i in range(30)]
E, deltaE = ave_error(X) E, deltaE = ave_error(X)
print(f"E = {E} +/- {deltaE}") print(f"E = {E} +/- {deltaE}")
#+END_SRC #+END_SRC
#+RESULTS: #+RESULTS:
: E = -0.4112049153828464 +/- 0.00027934927432953063 : E = -0.4858534479298907 +/- 0.00010203236131158794
*Fortran* *Fortran*
#+BEGIN_SRC f90 #+BEGIN_SRC f90
subroutine variational_montecarlo(a,nmax,energy) subroutine variational_montecarlo(a,tau,nmax,energy)
implicit none implicit none
double precision, intent(in) :: a double precision, intent(in) :: a, tau
integer , intent(in) :: nmax integer*8 , intent(in) :: nmax
double precision, intent(out) :: energy double precision, intent(out) :: energy
integer*8 :: istep integer*8 :: istep
double precision :: norm, r_old(3), r_new(3), d_old(3), sq_tau, chi(3)
double precision, external :: e_loc
double precision :: norm, r(3), w sq_tau = dsqrt(tau)
double precision, external :: e_loc, psi, gaussian
! Initialization
energy = 0.d0 energy = 0.d0
norm = 0.d0 norm = 0.d0
call random_gauss(r_old,3)
do istep = 1,nmax do istep = 1,nmax
call random_gauss(r,3) call drift(a,r_old,d_old)
w = psi(a,r) call random_gauss(chi,3)
w = w*w / gaussian(r) r_new(:) = r_old(:) + tau * d_old(:) + chi(:)*sq_tau
norm = norm + w norm = norm + 1.d0
energy = energy + w * e_loc(a,r) energy = energy + e_loc(a,r_new)
r_old(:) = r_new(:)
end do end do
energy = energy / norm energy = energy / norm
end subroutine variational_montecarlo end subroutine variational_montecarlo
@ -1167,7 +1180,8 @@ end subroutine variational_montecarlo
program qmc program qmc
implicit none implicit none
double precision, parameter :: a = 0.9 double precision, parameter :: a = 0.9
integer , parameter :: nmax = 100000 double precision, parameter :: tau = 0.2
integer*8 , parameter :: nmax = 100000
integer , parameter :: nruns = 30 integer , parameter :: nruns = 30
integer :: irun integer :: irun
@ -1175,7 +1189,7 @@ program qmc
double precision :: ave, err double precision :: ave, err
do irun=1,nruns do irun=1,nruns
call gaussian_montecarlo(a,nmax,X(irun)) call variational_montecarlo(a,tau,nmax,X(irun))
enddo enddo
call ave_error(X,nruns,ave,err) call ave_error(X,nruns,ave,err)
print *, 'E = ', ave, '+/-', err print *, 'E = ', ave, '+/-', err
@ -1187,7 +1201,14 @@ gfortran hydrogen.f90 qmc_stats.f90 vmc.f90 -o vmc
./vmc ./vmc
#+end_src #+end_src
#+RESULTS:
: E = -0.48584030499187431 +/- 1.0411743995438257E-004
*** Metropolis algorithm *** Metropolis algorithm
:PROPERTIES:
:header-args:python: :tangle vmc_metropolis.py
:header-args:f90: :tangle vmc_metropolis.f90
:END:
Discretizing the differential equation to generate the desired Discretizing the differential equation to generate the desired
probability density will suffer from a discretization error probability density will suffer from a discretization error
@ -1236,7 +1257,7 @@ gfortran hydrogen.f90 qmc_stats.f90 vmc.f90 -o vmc
#+end_exercise #+end_exercise
*Python* *Python*
#+BEGIN_SRC python #+BEGIN_SRC python :results output
def MonteCarlo(a,tau,nmax): def MonteCarlo(a,tau,nmax):
E = 0. E = 0.
N = 0. N = 0.
@ -1325,8 +1346,14 @@ gfortran hydrogen.f90 qmc_stats.f90 vmc.f90 -o vmc
./vmc ./vmc
#+end_src #+end_src
#+RESULTS:
* TODO Diffusion Monte Carlo * TODO Diffusion Monte Carlo
:PROPERTIES:
:header-args:python: :tangle dmc.py
:header-args:f90: :tangle dmc.f90
:END:
We will now consider the H_2 molecule in a minimal basis composed of the We will now consider the H_2 molecule in a minimal basis composed of the
$1s$ orbitals of the hydrogen atoms: $1s$ orbitals of the hydrogen atoms:

10
qmc_stats.f90
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@ -27,17 +27,17 @@ subroutine random_gauss(z,n)
! n is even ! n is even
do i=1,n,2 do i=1,n,2
z(i) = dsqrt(-2.d0*dlog(u(i))) z(i) = dsqrt(-2.d0*dlog(u(i)))
z(i+1) = z(i) + dsin( two_pi*u(i+1) ) z(i+1) = z(i) * dsin( two_pi*u(i+1) )
z(i) = z(i) + dcos( two_pi*u(i+1) ) z(i) = z(i) * dcos( two_pi*u(i+1) )
end do end do
else else
! n is odd ! n is odd
do i=1,n-1,2 do i=1,n-1,2
z(i) = dsqrt(-2.d0*dlog(u(i))) z(i) = dsqrt(-2.d0*dlog(u(i)))
z(i+1) = z(i) + dsin( two_pi*u(i+1) ) z(i+1) = z(i) * dsin( two_pi*u(i+1) )
z(i) = z(i) + dcos( two_pi*u(i+1) ) z(i) = z(i) * dcos( two_pi*u(i+1) )
end do end do
z(n) = dsqrt(-2.d0*dlog(u(n))) z(n) = dsqrt(-2.d0*dlog(u(n)))
z(n) = z(n) + dcos( two_pi*u(n+1) ) z(n) = z(n) * dcos( two_pi*u(n+1) )
end if end if
end subroutine random_gauss end subroutine random_gauss

4
qmc_uniform.f90
View File

@ -1,7 +1,7 @@
subroutine uniform_montecarlo(a,nmax,energy) subroutine uniform_montecarlo(a,nmax,energy)
implicit none implicit none
double precision, intent(in) :: a double precision, intent(in) :: a
integer , intent(in) :: nmax integer*8 , intent(in) :: nmax
double precision, intent(out) :: energy double precision, intent(out) :: energy
integer*8 :: istep integer*8 :: istep
@ -26,7 +26,7 @@ end subroutine uniform_montecarlo
program qmc program qmc
implicit none implicit none
double precision, parameter :: a = 0.9 double precision, parameter :: a = 0.9
integer , parameter :: nmax = 100000 integer*8 , parameter :: nmax = 100000
integer , parameter :: nruns = 30 integer , parameter :: nruns = 30
integer :: irun integer :: irun