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VMC OK
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QMC.org
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QMC.org
@ -1248,27 +1248,36 @@ gfortran hydrogen.f90 qmc_stats.f90 vmc.f90 -o vmc
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the move: set $\mathbf{r}_{n+1} = \mathbf{r}_{n}$, but *don't
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remove the sample from the average!*
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The /acceptance rate/ is the ratio of the number of accepted step
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over the total number of steps. The time step should be adapted so
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that the acceptance rate is around 0.5 for a good efficiency of
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the simulation.
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**** TODO Exercise
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#+begin_exercise
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Modify the previous program to introduce the accept/reject step.
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You should recover the unbiased result.
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Adjust the time-step so that the acceptance rate is 0.5.
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#+end_exercise
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*Python*
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#+BEGIN_SRC python :results output
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from hydrogen import *
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from qmc_stats import *
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def MonteCarlo(a,tau,nmax):
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E = 0.
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N = 0.
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accep_rate = 0.
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sq_tau = np.sqrt(tau)
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r_old = np.random.normal(loc=0., scale=1.0, size=(3))
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d_old = drift(a,r_old)
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d2_old = np.dot(d_old,d_old)
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psi_old = psi(a,r_old)
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for istep in range(nmax):
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eta = np.random.normal(loc=0., scale=1.0, size=(3))
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r_new = r_old + tau * d_old + sq_tau * eta
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chi = np.random.normal(loc=0., scale=1.0, size=(3))
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r_new = r_old + tau * d_old + sq_tau * chi
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d_new = drift(a,r_new)
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d2_new = np.dot(d_new,d_new)
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psi_new = psi(a,r_new)
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@ -1278,75 +1287,111 @@ def MonteCarlo(a,tau,nmax):
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q = psi_new / psi_old
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q = np.exp(-argexpo) * q*q
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if np.random.uniform() < q:
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accep_rate += 1.
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r_old = r_new
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d_old = d_new
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d2_old = d2_new
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psi_old = psi_new
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N += 1.
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E += e_loc(a,r_old)
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return E/N
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N += 1.
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E += e_loc(a,r_old)
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return E/N, accep_rate/N
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a = 0.9
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nmax = 100000
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tau = 0.1
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tau = 1.0
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X = [MonteCarlo(a,tau,nmax) for i in range(30)]
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E, deltaE = ave_error(X)
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print(f"E = {E} +/- {deltaE}")
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E, deltaE = ave_error([x[0] for x in X])
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A, deltaA = ave_error([x[1] for x in X])
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print(f"E = {E} +/- {deltaE} {A} +/- {deltaA}")
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#+END_SRC
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#+RESULTS:
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: E = -0.4951783346213532 +/- 0.00022067316984271938
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: E = -0.49387078389332206 +/- 0.0033326460286729792 0.4983000000000001 +/- 0.006825097363627021
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*Fortran*
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#+BEGIN_SRC f90
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subroutine variational_montecarlo(a,nmax,energy)
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subroutine variational_montecarlo(a,tau,nmax,energy,accep_rate)
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implicit none
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double precision, intent(in) :: a
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integer , intent(in) :: nmax
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double precision, intent(out) :: energy
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double precision, intent(in) :: a, tau
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integer*8 , intent(in) :: nmax
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double precision, intent(out) :: energy, accep_rate
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integer*8 :: istep
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double precision :: norm, sq_tau, chi(3), d2_old, prod, u
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double precision :: psi_old, psi_new, d2_new, argexpo, q
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double precision :: r_old(3), r_new(3)
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double precision :: d_old(3), d_new(3)
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double precision, external :: e_loc, psi
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double precision :: norm, r(3), w
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double precision, external :: e_loc, psi, gaussian
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sq_tau = dsqrt(tau)
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! Initialization
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energy = 0.d0
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norm = 0.d0
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accep_rate = 0.d0
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call random_gauss(r_old,3)
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call drift(a,r_old,d_old)
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d2_old = d_old(1)*d_old(1) + d_old(2)*d_old(2) + d_old(3)*d_old(3)
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psi_old = psi(a,r_old)
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do istep = 1,nmax
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call random_gauss(r,3)
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w = psi(a,r)
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w = w*w / gaussian(r)
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norm = norm + w
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energy = energy + w * e_loc(a,r)
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call random_gauss(chi,3)
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r_new(:) = r_old(:) + tau * d_old(:) + chi(:)*sq_tau
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call drift(a,r_new,d_new)
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d2_new = d_new(1)*d_new(1) + d_new(2)*d_new(2) + d_new(3)*d_new(3)
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psi_new = psi(a,r_new)
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! Metropolis
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prod = (d_new(1) + d_old(1))*(r_new(1) - r_old(1)) + &
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(d_new(2) + d_old(2))*(r_new(2) - r_old(2)) + &
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(d_new(3) + d_old(3))*(r_new(3) - r_old(3))
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argexpo = 0.5d0 * (d2_new - d2_old)*tau + prod
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q = psi_new / psi_old
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q = dexp(-argexpo) * q*q
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call random_number(u)
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if (u<q) then
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accep_rate = accep_rate + 1.d0
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r_old(:) = r_new(:)
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d_old(:) = d_new(:)
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d2_old = d2_new
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psi_old = psi_new
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end if
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norm = norm + 1.d0
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energy = energy + e_loc(a,r_old)
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end do
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energy = energy / norm
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accep_rate = accep_rate / norm
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end subroutine variational_montecarlo
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program qmc
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implicit none
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double precision, parameter :: a = 0.9
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integer , parameter :: nmax = 100000
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double precision, parameter :: tau = 1.0
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integer*8 , parameter :: nmax = 100000
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integer , parameter :: nruns = 30
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integer :: irun
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double precision :: X(nruns)
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double precision :: X(nruns), accep(nruns)
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double precision :: ave, err
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do irun=1,nruns
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call gaussian_montecarlo(a,nmax,X(irun))
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call variational_montecarlo(a,tau,nmax,X(irun),accep(irun))
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enddo
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call ave_error(X,nruns,ave,err)
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print *, 'E = ', ave, '+/-', err
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call ave_error(accep,nruns,ave,err)
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print *, 'A = ', ave, '+/-', err
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end program qmc
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#+END_SRC
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#+begin_src sh :results output :exports both
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gfortran hydrogen.f90 qmc_stats.f90 vmc.f90 -o vmc
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./vmc
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gfortran hydrogen.f90 qmc_stats.f90 vmc_metropolis.f90 -o vmc_metropolis
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./vmc_metropolis
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#+end_src
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#+RESULTS:
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: E = -0.49499990423528023 +/- 1.5958250761863871E-004
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: A = 0.78861366666666655 +/- 3.5096729498002445E-004
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* TODO Diffusion Monte Carlo
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