mirror of
https://github.com/TREX-CoE/qmc-lttc.git
synced 2024-12-21 11:53:58 +01:00
OK up to DMC
This commit is contained in:
parent
94971bc5ea
commit
dd40ff74d1
190
QMC.org
190
QMC.org
@ -1428,7 +1428,7 @@ gfortran hydrogen.f90 qmc_stats.f90 qmc_metropolis.f90 -o qmc_metropolis
|
||||
: E = -0.49515370205041676 +/- 1.7660819245720729E-004
|
||||
: A = 0.51713866666666664 +/- 3.7072551835783688E-004
|
||||
|
||||
** TODO Gaussian random number generator
|
||||
** Gaussian random number generator
|
||||
|
||||
To obtain Gaussian-distributed random numbers, you can apply the
|
||||
[[https://en.wikipedia.org/wiki/Box%E2%80%93Muller_transform][Box Muller transform]] to uniform random numbers:
|
||||
@ -1473,15 +1473,16 @@ subroutine random_gauss(z,n)
|
||||
end subroutine random_gauss
|
||||
#+END_SRC
|
||||
|
||||
** TODO Generalized Metropolis algorithm
|
||||
In Python, you can use the [[https://numpy.org/doc/stable/reference/random/generated/numpy.random.normal.html][~random.normal~]] function of Numpy.
|
||||
** Generalized Metropolis algorithm
|
||||
:PROPERTIES:
|
||||
:header-args:python: :tangle vmc_metropolis.py
|
||||
:header-args:f90: :tangle vmc_metropolis.f90
|
||||
:END:
|
||||
|
||||
One can use more efficient numerical schemes to move the electrons.
|
||||
But in that case, the Metropolis accepation step has to be adapted
|
||||
accordingly: the acceptance
|
||||
One can use more efficient numerical schemes to move the electrons,
|
||||
but the Metropolis accepation step has to be adapted accordingly:
|
||||
the acceptance
|
||||
probability $A$ is chosen so that it is consistent with the
|
||||
probability of leaving $\mathbf{r}_n$ and the probability of
|
||||
entering $\mathbf{r}_{n+1}$:
|
||||
@ -1499,19 +1500,19 @@ end subroutine random_gauss
|
||||
numbers. Hence, the transition probability was
|
||||
|
||||
\[
|
||||
T(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1}) & = &
|
||||
T(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1}) =
|
||||
\text{constant}
|
||||
\]
|
||||
|
||||
So the expression of $A$ was simplified to the ratios of the squared
|
||||
so the expression of $A$ was simplified to the ratios of the squared
|
||||
wave functions.
|
||||
|
||||
Now, if instead of drawing uniform random numbers
|
||||
choose to draw Gaussian random numbers with mean 0 and variance
|
||||
Now, if instead of drawing uniform random numbers we
|
||||
choose to draw Gaussian random numbers with zero mean and variance
|
||||
$\tau$, the transition probability becomes:
|
||||
|
||||
\[
|
||||
T(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1}) & = &
|
||||
T(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1}) =
|
||||
\frac{1}{(2\pi\,\tau)^{3/2}} \exp \left[ - \frac{\left(
|
||||
\mathbf{r}_{n+1} - \mathbf{r}_{n} \right)^2}{2\tau} \right]
|
||||
\]
|
||||
@ -1525,8 +1526,8 @@ end subroutine random_gauss
|
||||
To do this, we can add the drift vector
|
||||
|
||||
\[
|
||||
\frac{\nabla [ \Psi^2 ]}{\Psi^2} = 2 \frac{\nabla \Psi}{\Psi}
|
||||
\].
|
||||
\frac{\nabla [ \Psi^2 ]}{\Psi^2} = 2 \frac{\nabla \Psi}{\Psi}.
|
||||
\]
|
||||
|
||||
The numerical scheme becomes a drifted diffusion:
|
||||
|
||||
@ -1540,7 +1541,7 @@ end subroutine random_gauss
|
||||
The transition probability becomes:
|
||||
|
||||
\[
|
||||
T(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1}) & = &
|
||||
T(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1}) =
|
||||
\frac{1}{(2\pi\,\tau)^{3/2}} \exp \left[ - \frac{\left(
|
||||
\mathbf{r}_{n+1} - \mathbf{r}_{n} - \frac{\nabla
|
||||
\Psi(\mathbf{r}_n)}{\Psi(\mathbf{r}_n)} \right)^2}{2\,\tau} \right]
|
||||
@ -1553,14 +1554,30 @@ end subroutine random_gauss
|
||||
Write a function to compute the drift vector $\frac{\nabla \Psi(\mathbf{r})}{\Psi(\mathbf{r})}$.
|
||||
#+end_exercise
|
||||
|
||||
*Python*
|
||||
**** Python
|
||||
#+BEGIN_SRC python :tangle hydrogen.py :tangle none
|
||||
def drift(a,r):
|
||||
# TODO
|
||||
#+END_SRC
|
||||
|
||||
**** Python :solution:
|
||||
#+BEGIN_SRC python :tangle hydrogen.py
|
||||
def drift(a,r):
|
||||
ar_inv = -a/np.sqrt(np.dot(r,r))
|
||||
return r * ar_inv
|
||||
#+END_SRC
|
||||
|
||||
*Fortran*
|
||||
**** Fortran
|
||||
#+BEGIN_SRC f90 :tangle hydrogen.f90 :tangle none
|
||||
subroutine drift(a,r,b)
|
||||
implicit none
|
||||
double precision, intent(in) :: a, r(3)
|
||||
double precision, intent(out) :: b(3)
|
||||
! TODO
|
||||
end subroutine drift
|
||||
#+END_SRC
|
||||
|
||||
**** Fortran :solution:
|
||||
#+BEGIN_SRC f90 :tangle hydrogen.f90
|
||||
subroutine drift(a,r,b)
|
||||
implicit none
|
||||
@ -1579,14 +1596,38 @@ end subroutine drift
|
||||
(This is a necessary step for the next section).
|
||||
#+end_exercise
|
||||
|
||||
*Python*
|
||||
**** Python
|
||||
#+BEGIN_SRC python :results output :tangle none
|
||||
from hydrogen import *
|
||||
from qmc_stats import *
|
||||
|
||||
def MonteCarlo(a,nmax,tau):
|
||||
# TODO
|
||||
|
||||
# Run simulation
|
||||
a = 0.9
|
||||
nmax = 100000
|
||||
tau = 1.3
|
||||
X0 = [ MonteCarlo(a,nmax,tau) for i in range(30)]
|
||||
|
||||
# Energy
|
||||
X = [ x for (x, _) in X0 ]
|
||||
E, deltaE = ave_error(X)
|
||||
print(f"E = {E} +/- {deltaE}")
|
||||
|
||||
# Acceptance rate
|
||||
X = [ x for (_, x) in X0 ]
|
||||
A, deltaA = ave_error(X)
|
||||
print(f"A = {A} +/- {deltaA}")
|
||||
#+END_SRC
|
||||
|
||||
**** Python :solution:
|
||||
#+BEGIN_SRC python :results output
|
||||
from hydrogen import *
|
||||
from qmc_stats import *
|
||||
|
||||
def MonteCarlo(a,tau,nmax):
|
||||
E = 0.
|
||||
N = 0.
|
||||
def MonteCarlo(a,nmax,tau):
|
||||
energy = 0.
|
||||
accep_rate = 0.
|
||||
sq_tau = np.sqrt(tau)
|
||||
r_old = np.random.normal(loc=0., scale=1.0, size=(3))
|
||||
@ -1610,25 +1651,77 @@ def MonteCarlo(a,tau,nmax):
|
||||
d_old = d_new
|
||||
d2_old = d2_new
|
||||
psi_old = psi_new
|
||||
N += 1.
|
||||
E += e_loc(a,r_old)
|
||||
return E/N, accep_rate/N
|
||||
energy += e_loc(a,r_old)
|
||||
return energy/nmax, accep_rate/nmax
|
||||
|
||||
|
||||
# Run simulation
|
||||
a = 0.9
|
||||
nmax = 100000
|
||||
tau = 1.0
|
||||
X = [MonteCarlo(a,tau,nmax) for i in range(30)]
|
||||
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}\nA = {A} +/- {deltaA}")
|
||||
tau = 1.3
|
||||
X0 = [ MonteCarlo(a,nmax,tau) for i in range(30)]
|
||||
|
||||
# Energy
|
||||
X = [ x for (x, _) in X0 ]
|
||||
E, deltaE = ave_error(X)
|
||||
print(f"E = {E} +/- {deltaE}")
|
||||
|
||||
# Acceptance rate
|
||||
X = [ x for (_, x) in X0 ]
|
||||
A, deltaA = ave_error(X)
|
||||
print(f"A = {A} +/- {deltaA}")
|
||||
#+END_SRC
|
||||
|
||||
#+RESULTS:
|
||||
: E = -0.4949730317138491 +/- 0.00012478601801760644
|
||||
: A = 0.7887163333333334 +/- 0.00026834549840347617
|
||||
: E = -0.4951317910667116 +/- 0.00014045774335059988
|
||||
: A = 0.7200673333333333 +/- 0.00045942791345632793
|
||||
|
||||
*Fortran*
|
||||
**** Fortran
|
||||
#+BEGIN_SRC f90 :tangle none
|
||||
subroutine variational_montecarlo(a,tau,nmax,energy,accep_rate)
|
||||
implicit none
|
||||
double precision, intent(in) :: a, tau
|
||||
integer*8 , intent(in) :: nmax
|
||||
double precision, intent(out) :: energy, accep_rate
|
||||
|
||||
integer*8 :: istep
|
||||
double precision :: sq_tau, chi(3)
|
||||
double precision :: psi_old, psi_new
|
||||
double precision :: r_old(3), r_new(3)
|
||||
double precision :: d_old(3), d_new(3)
|
||||
double precision, external :: e_loc, psi
|
||||
|
||||
! TODO
|
||||
|
||||
end subroutine variational_montecarlo
|
||||
|
||||
program qmc
|
||||
implicit none
|
||||
double precision, parameter :: a = 0.9
|
||||
double precision, parameter :: tau = 1.0
|
||||
integer*8 , parameter :: nmax = 100000
|
||||
integer , parameter :: nruns = 30
|
||||
|
||||
integer :: irun
|
||||
double precision :: X(nruns), accep(nruns)
|
||||
double precision :: ave, err
|
||||
|
||||
do irun=1,nruns
|
||||
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_metropolis.f90 -o vmc_metropolis
|
||||
./vmc_metropolis
|
||||
#+end_src
|
||||
|
||||
**** Fortran :solution:
|
||||
#+BEGIN_SRC f90
|
||||
subroutine variational_montecarlo(a,tau,nmax,energy,accep_rate)
|
||||
implicit none
|
||||
@ -1637,7 +1730,7 @@ subroutine variational_montecarlo(a,tau,nmax,energy,accep_rate)
|
||||
double precision, intent(out) :: energy, accep_rate
|
||||
|
||||
integer*8 :: istep
|
||||
double precision :: norm, sq_tau, chi(3), d2_old, prod, u
|
||||
double precision :: 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)
|
||||
@ -1647,7 +1740,6 @@ subroutine variational_montecarlo(a,tau,nmax,energy,accep_rate)
|
||||
|
||||
! Initialization
|
||||
energy = 0.d0
|
||||
norm = 0.d0
|
||||
accep_rate = 0.d0
|
||||
call random_gauss(r_old,3)
|
||||
call drift(a,r_old,d_old)
|
||||
@ -1675,11 +1767,10 @@ subroutine variational_montecarlo(a,tau,nmax,energy,accep_rate)
|
||||
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
|
||||
energy = energy / dble(nmax)
|
||||
accep_rate = dble(accep_rate) / dble(nmax)
|
||||
end subroutine variational_montecarlo
|
||||
|
||||
program qmc
|
||||
@ -1709,8 +1800,8 @@ gfortran hydrogen.f90 qmc_stats.f90 vmc_metropolis.f90 -o vmc_metropolis
|
||||
#+end_src
|
||||
|
||||
#+RESULTS:
|
||||
: E = -0.49499990423528023 +/- 1.5958250761863871E-004
|
||||
: A = 0.78861366666666655 +/- 3.5096729498002445E-004
|
||||
: E = -0.49495906384751226 +/- 1.5257646086088266E-004
|
||||
: A = 0.78861366666666666 +/- 3.7855335138754813E-004
|
||||
|
||||
* TODO Diffusion Monte Carlo
|
||||
:PROPERTIES:
|
||||
@ -1733,10 +1824,10 @@ gfortran hydrogen.f90 qmc_stats.f90 vmc_metropolis.f90 -o vmc_metropolis
|
||||
from hydrogen import *
|
||||
from qmc_stats import *
|
||||
|
||||
def MonteCarlo(a,tau,nmax,Eref):
|
||||
E = 0.
|
||||
N = 0.
|
||||
accep_rate = 0.
|
||||
def MonteCarlo(a,nmax,tau,Eref):
|
||||
energy = 0.
|
||||
normalization = 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)
|
||||
@ -1747,8 +1838,8 @@ def MonteCarlo(a,tau,nmax,Eref):
|
||||
chi = np.random.normal(loc=0., scale=1.0, size=(3))
|
||||
el = e_loc(a,r_old)
|
||||
w *= np.exp(-tau*(el - Eref))
|
||||
N += w
|
||||
E += w * el
|
||||
normalization += w
|
||||
energy += w * el
|
||||
|
||||
r_new = r_old + tau * d_old + sq_tau * chi
|
||||
d_new = drift(a,r_new)
|
||||
@ -1761,18 +1852,19 @@ def MonteCarlo(a,tau,nmax,Eref):
|
||||
q = np.exp(-argexpo) * q*q
|
||||
# PDMC weight
|
||||
if np.random.uniform() < q:
|
||||
accep_rate += w
|
||||
accep_rate += 1
|
||||
r_old = r_new
|
||||
d_old = d_new
|
||||
d2_old = d2_new
|
||||
psi_old = psi_new
|
||||
return E/N, accep_rate/N
|
||||
return energy/normalization, accep_rate/nmax
|
||||
|
||||
|
||||
a = 0.9
|
||||
nmax = 10000
|
||||
tau = .1
|
||||
X = [MonteCarlo(a,tau,nmax,-0.5) for i in range(30)]
|
||||
E_ref = -0.5
|
||||
X = [MonteCarlo(a,nmax,tau,E_ref) for i in range(30)]
|
||||
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}\nA = {A} +/- {deltaA}")
|
||||
@ -1782,7 +1874,7 @@ print(f"E = {E} +/- {deltaE}\nA = {A} +/- {deltaA}")
|
||||
: E = -0.49654807434947584 +/- 0.0006868522447409156
|
||||
: A = 0.9876193891840709 +/- 0.00041857361650995804
|
||||
|
||||
*Fortran*
|
||||
**** Fortran
|
||||
#+BEGIN_SRC f90
|
||||
subroutine variational_montecarlo(a,tau,nmax,energy,accep_rate)
|
||||
implicit none
|
||||
@ -1924,7 +2016,7 @@ gfortran hydrogen.f90 qmc_stats.f90 vmc_metropolis.f90 -o vmc_metropolis
|
||||
the statistical error?
|
||||
#+end_exercise
|
||||
|
||||
*Python*
|
||||
**** Python
|
||||
#+BEGIN_SRC python :results output
|
||||
from hydrogen import *
|
||||
from qmc_stats import *
|
||||
@ -1954,7 +2046,7 @@ print(f"E = {E} +/- {deltaE}")
|
||||
#+RESULTS:
|
||||
: E = -0.49511014287471955 +/- 0.00012402022172236656
|
||||
|
||||
*Fortran*
|
||||
**** Fortran
|
||||
#+BEGIN_SRC f90
|
||||
double precision function gaussian(r)
|
||||
implicit none
|
||||
|
||||
@ -5,7 +5,7 @@ subroutine variational_montecarlo(a,tau,nmax,energy,accep_rate)
|
||||
double precision, intent(out) :: energy, accep_rate
|
||||
|
||||
integer*8 :: istep
|
||||
double precision :: norm, sq_tau, chi(3), d2_old, prod, u
|
||||
double precision :: 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)
|
||||
@ -15,7 +15,6 @@ subroutine variational_montecarlo(a,tau,nmax,energy,accep_rate)
|
||||
|
||||
! Initialization
|
||||
energy = 0.d0
|
||||
norm = 0.d0
|
||||
accep_rate = 0.d0
|
||||
call random_gauss(r_old,3)
|
||||
call drift(a,r_old,d_old)
|
||||
@ -43,11 +42,10 @@ subroutine variational_montecarlo(a,tau,nmax,energy,accep_rate)
|
||||
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
|
||||
energy = energy / dble(nmax)
|
||||
accep_rate = dble(accep_rate) / dble(nmax)
|
||||
end subroutine variational_montecarlo
|
||||
|
||||
program qmc
|
||||
|
||||
@ -1,9 +1,8 @@
|
||||
from hydrogen import *
|
||||
from qmc_stats import *
|
||||
|
||||
def MonteCarlo(a,tau,nmax):
|
||||
def MonteCarlo(a,nmax,tau):
|
||||
E = 0.
|
||||
N = 0.
|
||||
accep_rate = 0.
|
||||
sq_tau = np.sqrt(tau)
|
||||
r_old = np.random.normal(loc=0., scale=1.0, size=(3))
|
||||
@ -27,15 +26,22 @@ def MonteCarlo(a,tau,nmax):
|
||||
d_old = d_new
|
||||
d2_old = d2_new
|
||||
psi_old = psi_new
|
||||
N += 1.
|
||||
E += e_loc(a,r_old)
|
||||
return E/N, accep_rate/N
|
||||
return E/nmax, accep_rate/nmax
|
||||
|
||||
|
||||
# Run simulation
|
||||
a = 0.9
|
||||
nmax = 100000
|
||||
tau = 1.0
|
||||
X = [MonteCarlo(a,tau,nmax) for i in range(30)]
|
||||
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}\nA = {A} +/- {deltaA}")
|
||||
tau = 1.3
|
||||
X0 = [ MonteCarlo(a,nmax,tau) for i in range(30)]
|
||||
|
||||
# Energy
|
||||
X = [ x for (x, _) in X0 ]
|
||||
E, deltaE = ave_error(X)
|
||||
print(f"E = {E} +/- {deltaE}")
|
||||
|
||||
# Acceptance rate
|
||||
X = [ x for (_, x) in X0 ]
|
||||
A, deltaA = ave_error(X)
|
||||
print(f"A = {A} +/- {deltaA}")
|
||||
|
||||
Loading…
Reference in New Issue
Block a user