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Author SHA1 Message Date
Anthony Scemama 52ed6eff8d VMC OK 2021-01-12 11:23:13 +01:00
Anthony Scemama 22f1eaf41b Fixed bugs 2021-01-12 10:55:00 +01:00
3 changed files with 146 additions and 74 deletions
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@ -743,11 +743,18 @@ print(f"E = {E} +/- {deltaE}")
: E = -0.4956255109300764 +/- 0.0007082875482711226
*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
subroutine uniform_montecarlo(a,nmax,energy)
implicit none
double precision, intent(in) :: a
integer , intent(in) :: nmax
integer*8 , intent(in) :: nmax
double precision, intent(out) :: energy
integer*8 :: istep
@ -772,7 +779,7 @@ end subroutine uniform_montecarlo
program qmc
implicit none
double precision, parameter :: a = 0.9
integer , parameter :: nmax = 100000
integer*8 , parameter :: nmax = 100000
integer , parameter :: nruns = 30
integer :: irun
@ -833,18 +840,18 @@ subroutine random_gauss(z,n)
! n is even
do i=1,n,2
z(i) = dsqrt(-2.d0*dlog(u(i)))
z(i+1) = z(i) + dsin( two_pi*u(i+1) )
z(i) = z(i) + dcos( 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) )
end do
else
! n is odd
do i=1,n-1,2
z(i) = dsqrt(-2.d0*dlog(u(i)))
z(i+1) = z(i) + dsin( two_pi*u(i+1) )
z(i) = z(i) + dcos( 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) )
end do
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 subroutine random_gauss
#+END_SRC
@ -907,7 +914,7 @@ print(f"E = {E} +/- {deltaE}")
#+END_SRC
#+RESULTS:
: E = -0.49507506093129827 +/- 0.00014164037765553668
: E = -0.49511014287471955 +/- 0.00012402022172236656
*Fortran*
@ -916,14 +923,14 @@ double precision function gaussian(r)
implicit none
double precision, intent(in) :: r(3)
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
subroutine gaussian_montecarlo(a,nmax,energy)
implicit none
double precision, intent(in) :: a
integer , intent(in) :: nmax
integer*8 , intent(in) :: nmax
double precision, intent(out) :: energy
integer*8 :: istep
@ -947,7 +954,7 @@ end subroutine gaussian_montecarlo
program qmc
implicit none
double precision, parameter :: a = 0.9
integer , parameter :: nmax = 100000
integer*8 , parameter :: nmax = 100000
integer , parameter :: nruns = 30
integer :: irun
@ -968,13 +975,9 @@ gfortran hydrogen.f90 qmc_stats.f90 qmc_gaussian.f90 -o qmc_gaussian
#+end_src
#+RESULTS:
: E = -0.49606057056767766 +/- 1.3918807547836872E-004
: E = -0.49517104619091717 +/- 1.0685523607878961E-004
** 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:
@ -991,6 +994,10 @@ gfortran hydrogen.f90 qmc_stats.f90 qmc_gaussian.f90 -o qmc_gaussian
*** 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
diffusion process characterized by a time-dependent density
@ -1107,20 +1114,22 @@ end subroutine drift
#+end_exercise
*Python*
#+BEGIN_SRC python :results output :tangle vmc.py
#+BEGIN_SRC python :results output
from hydrogen import *
from qmc_stats import *
def MonteCarlo(a,tau,nmax):
sq_tau = np.sqrt(tau)
# Initialization
E = 0.
N = 0.
sq_tau = np.sqrt(tau)
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):
d_old = drift(a,r_old)
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.
E += e_loc(a,r_new)
r_old = r_new
@ -1129,37 +1138,41 @@ def MonteCarlo(a,tau,nmax):
a = 0.9
nmax = 100000
tau = 0.001
tau = 0.2
X = [MonteCarlo(a,tau,nmax) for i in range(30)]
E, deltaE = ave_error(X)
print(f"E = {E} +/- {deltaE}")
#+END_SRC
#+RESULTS:
: E = -0.4112049153828464 +/- 0.00027934927432953063
: E = -0.4858534479298907 +/- 0.00010203236131158794
*Fortran*
#+BEGIN_SRC f90
subroutine variational_montecarlo(a,nmax,energy)
subroutine variational_montecarlo(a,tau,nmax,energy)
implicit none
double precision, intent(in) :: a
integer , intent(in) :: nmax
double precision, intent(in) :: a, tau
integer*8 , intent(in) :: nmax
double precision, intent(out) :: energy
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
double precision, external :: e_loc, psi, gaussian
sq_tau = dsqrt(tau)
! Initialization
energy = 0.d0
norm = 0.d0
call random_gauss(r_old,3)
do istep = 1,nmax
call random_gauss(r,3)
w = psi(a,r)
w = w*w / gaussian(r)
norm = norm + w
energy = energy + w * e_loc(a,r)
call drift(a,r_old,d_old)
call random_gauss(chi,3)
r_new(:) = r_old(:) + tau * d_old(:) + chi(:)*sq_tau
norm = norm + 1.d0
energy = energy + e_loc(a,r_new)
r_old(:) = r_new(:)
end do
energy = energy / norm
end subroutine variational_montecarlo
@ -1167,7 +1180,8 @@ end subroutine variational_montecarlo
program qmc
implicit none
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 :: irun
@ -1175,7 +1189,7 @@ program qmc
double precision :: ave, err
do irun=1,nruns
call gaussian_montecarlo(a,nmax,X(irun))
call variational_montecarlo(a,tau,nmax,X(irun))
enddo
call ave_error(X,nruns,ave,err)
print *, 'E = ', ave, '+/-', err
@ -1186,8 +1200,15 @@ end program qmc
gfortran hydrogen.f90 qmc_stats.f90 vmc.f90 -o vmc
./vmc
#+end_src
#+RESULTS:
: E = -0.48584030499187431 +/- 1.0411743995438257E-004
*** 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
probability density will suffer from a discretization error
@ -1227,27 +1248,36 @@ gfortran hydrogen.f90 qmc_stats.f90 vmc.f90 -o vmc
the move: set $\mathbf{r}_{n+1} = \mathbf{r}_{n}$, but *don't
remove the sample from the average!*
The /acceptance rate/ is the ratio of the number of accepted step
over the total number of steps. The time step should be adapted so
that the acceptance rate is around 0.5 for a good efficiency of
the simulation.
**** TODO Exercise
#+begin_exercise
Modify the previous program to introduce the accept/reject step.
You should recover the unbiased result.
Adjust the time-step so that the acceptance rate is 0.5.
#+end_exercise
*Python*
#+BEGIN_SRC python
#+BEGIN_SRC python :results output
from hydrogen import *
from qmc_stats import *
def MonteCarlo(a,tau,nmax):
E = 0.
N = 0.
accep_rate = 0.
sq_tau = np.sqrt(tau)
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):
eta = np.random.normal(loc=0., scale=1.0, size=(3))
r_new = r_old + tau * d_old + sq_tau * eta
chi = np.random.normal(loc=0., scale=1.0, size=(3))
r_new = r_old + tau * d_old + sq_tau * chi
d_new = drift(a,r_new)
d2_new = np.dot(d_new,d_new)
psi_new = psi(a,r_new)
@ -1257,76 +1287,118 @@ def MonteCarlo(a,tau,nmax):
q = psi_new / psi_old
q = np.exp(-argexpo) * q*q
if np.random.uniform() < q:
accep_rate += 1.
r_old = r_new
d_old = d_new
d2_old = d2_new
psi_old = psi_new
N += 1.
E += e_loc(a,r_old)
return E/N
N += 1.
E += e_loc(a,r_old)
return E/N, accep_rate/N
a = 0.9
nmax = 100000
tau = 0.1
tau = 1.0
X = [MonteCarlo(a,tau,nmax) for i in range(30)]
E, deltaE = ave_error(X)
print(f"E = {E} +/- {deltaE}")
E, deltaE = ave_error([x[0] for x in X])
A, deltaA = ave_error([x[1] for x in X])
print(f"E = {E} +/- {deltaE} {A} +/- {deltaA}")
#+END_SRC
#+RESULTS:
: E = -0.4951783346213532 +/- 0.00022067316984271938
: E = -0.49387078389332206 +/- 0.0033326460286729792 0.4983000000000001 +/- 0.006825097363627021
*Fortran*
#+BEGIN_SRC f90
subroutine variational_montecarlo(a,nmax,energy)
subroutine variational_montecarlo(a,tau,nmax,energy,accep_rate)
implicit none
double precision, intent(in) :: a
integer , intent(in) :: nmax
double precision, intent(out) :: energy
double precision, intent(in) :: a, tau
integer*8 , intent(in) :: nmax
double precision, intent(out) :: energy, accep_rate
integer*8 :: istep
double precision :: norm, sq_tau, chi(3), d2_old, prod, u
double precision :: psi_old, psi_new, d2_new, argexpo, q
double precision :: r_old(3), r_new(3)
double precision :: d_old(3), d_new(3)
double precision, external :: e_loc, psi
double precision :: norm, r(3), w
double precision, external :: e_loc, psi, gaussian
sq_tau = dsqrt(tau)
! Initialization
energy = 0.d0
norm = 0.d0
accep_rate = 0.d0
call random_gauss(r_old,3)
call drift(a,r_old,d_old)
d2_old = d_old(1)*d_old(1) + d_old(2)*d_old(2) + d_old(3)*d_old(3)
psi_old = psi(a,r_old)
do istep = 1,nmax
call random_gauss(r,3)
w = psi(a,r)
w = w*w / gaussian(r)
norm = norm + w
energy = energy + w * e_loc(a,r)
call random_gauss(chi,3)
r_new(:) = r_old(:) + tau * d_old(:) + chi(:)*sq_tau
call drift(a,r_new,d_new)
d2_new = d_new(1)*d_new(1) + d_new(2)*d_new(2) + d_new(3)*d_new(3)
psi_new = psi(a,r_new)
! Metropolis
prod = (d_new(1) + d_old(1))*(r_new(1) - r_old(1)) + &
(d_new(2) + d_old(2))*(r_new(2) - r_old(2)) + &
(d_new(3) + d_old(3))*(r_new(3) - r_old(3))
argexpo = 0.5d0 * (d2_new - d2_old)*tau + prod
q = psi_new / psi_old
q = dexp(-argexpo) * q*q
call random_number(u)
if (u<q) then
accep_rate = accep_rate + 1.d0
r_old(:) = r_new(:)
d_old(:) = d_new(:)
d2_old = d2_new
psi_old = psi_new
end if
norm = norm + 1.d0
energy = energy + e_loc(a,r_old)
end do
energy = energy / norm
accep_rate = accep_rate / norm
end subroutine variational_montecarlo
program qmc
implicit none
double precision, parameter :: a = 0.9
integer , parameter :: nmax = 100000
double precision, parameter :: tau = 1.0
integer*8 , parameter :: nmax = 100000
integer , parameter :: nruns = 30
integer :: irun
double precision :: X(nruns)
double precision :: X(nruns), accep(nruns)
double precision :: ave, err
do irun=1,nruns
call gaussian_montecarlo(a,nmax,X(irun))
call variational_montecarlo(a,tau,nmax,X(irun),accep(irun))
enddo
call ave_error(X,nruns,ave,err)
print *, 'E = ', ave, '+/-', err
call ave_error(accep,nruns,ave,err)
print *, 'A = ', ave, '+/-', err
end program qmc
#+END_SRC
#+begin_src sh :results output :exports both
gfortran hydrogen.f90 qmc_stats.f90 vmc.f90 -o vmc
./vmc
gfortran hydrogen.f90 qmc_stats.f90 vmc_metropolis.f90 -o vmc_metropolis
./vmc_metropolis
#+end_src
#+RESULTS:
: E = -0.49499990423528023 +/- 1.5958250761863871E-004
: A = 0.78861366666666655 +/- 3.5096729498002445E-004
* 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
$1s$ orbitals of the hydrogen atoms:

10
qmc_stats.f90
View File

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

4
qmc_uniform.f90
View File

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