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Fixed bugs
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QMC.org
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QMC.org
@ -743,11 +743,18 @@ print(f"E = {E} +/- {deltaE}")
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: E = -0.4956255109300764 +/- 0.0007082875482711226
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*Fortran*
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#+begin_note
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When running Monte Carlo calculations, the number of steps is
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usually very large. We expect =nmax= to be possibly larger than 2
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billion, so we use 8-byte integers (=integer*8=) to represent it, as
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well as the index of the current step.
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#+end_note
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#+BEGIN_SRC f90
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subroutine uniform_montecarlo(a,nmax,energy)
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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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integer*8 , intent(in) :: nmax
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double precision, intent(out) :: energy
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integer*8 :: istep
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@ -772,7 +779,7 @@ end subroutine uniform_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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integer*8 , parameter :: nmax = 100000
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integer , parameter :: nruns = 30
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integer :: irun
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@ -833,18 +840,18 @@ subroutine random_gauss(z,n)
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! n is even
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do i=1,n,2
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z(i) = dsqrt(-2.d0*dlog(u(i)))
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z(i+1) = z(i) + dsin( two_pi*u(i+1) )
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z(i) = z(i) + dcos( two_pi*u(i+1) )
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z(i+1) = z(i) * dsin( two_pi*u(i+1) )
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z(i) = z(i) * dcos( two_pi*u(i+1) )
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end do
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else
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! n is odd
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do i=1,n-1,2
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z(i) = dsqrt(-2.d0*dlog(u(i)))
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z(i+1) = z(i) + dsin( two_pi*u(i+1) )
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z(i) = z(i) + dcos( two_pi*u(i+1) )
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z(i+1) = z(i) * dsin( two_pi*u(i+1) )
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z(i) = z(i) * dcos( two_pi*u(i+1) )
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end do
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z(n) = dsqrt(-2.d0*dlog(u(n)))
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z(n) = z(n) + dcos( two_pi*u(n+1) )
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z(n) = z(n) * dcos( two_pi*u(n+1) )
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end if
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end subroutine random_gauss
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#+END_SRC
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@ -907,7 +914,7 @@ print(f"E = {E} +/- {deltaE}")
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#+END_SRC
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#+RESULTS:
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: E = -0.49507506093129827 +/- 0.00014164037765553668
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: E = -0.49511014287471955 +/- 0.00012402022172236656
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*Fortran*
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@ -916,14 +923,14 @@ double precision function gaussian(r)
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implicit none
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double precision, intent(in) :: r(3)
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double precision, parameter :: norm_gauss = 1.d0/(2.d0*dacos(-1.d0))**(1.5d0)
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gaussian = norm_gauss * dexp( -0.5d0 * dsqrt(r(1)*r(1) + r(2)*r(2) + r(3)*r(3) ))
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gaussian = norm_gauss * dexp( -0.5d0 * (r(1)*r(1) + r(2)*r(2) + r(3)*r(3) ))
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end function gaussian
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subroutine gaussian_montecarlo(a,nmax,energy)
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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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integer*8 , intent(in) :: nmax
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double precision, intent(out) :: energy
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integer*8 :: istep
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@ -947,7 +954,7 @@ end subroutine gaussian_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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integer*8 , parameter :: nmax = 100000
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integer , parameter :: nruns = 30
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integer :: irun
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@ -968,13 +975,9 @@ gfortran hydrogen.f90 qmc_stats.f90 qmc_gaussian.f90 -o qmc_gaussian
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#+end_src
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#+RESULTS:
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: E = -0.49606057056767766 +/- 1.3918807547836872E-004
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: E = -0.49517104619091717 +/- 1.0685523607878961E-004
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** Sampling with $\Psi^2$
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:PROPERTIES:
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:header-args:python: :tangle vmc.py
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:header-args:f90: :tangle vmc.f90
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:END:
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We will now use the square of the wave function to make the sampling:
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@ -991,6 +994,10 @@ gfortran hydrogen.f90 qmc_stats.f90 qmc_gaussian.f90 -o qmc_gaussian
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*** Importance sampling
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:PROPERTIES:
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:header-args:python: :tangle vmc.py
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:header-args:f90: :tangle vmc.f90
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:END:
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To generate the probability density $\Psi^2$, we consider a
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diffusion process characterized by a time-dependent density
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@ -1107,20 +1114,22 @@ end subroutine drift
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#+end_exercise
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*Python*
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#+BEGIN_SRC python :results output :tangle vmc.py
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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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sq_tau = np.sqrt(tau)
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# Initialization
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E = 0.
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N = 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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d_old = drift(a,r_old)
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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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r_new = r_old + tau * d_old + chi*sq_tau
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N += 1.
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E += e_loc(a,r_new)
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r_old = r_new
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@ -1129,37 +1138,41 @@ def MonteCarlo(a,tau,nmax):
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a = 0.9
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nmax = 100000
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tau = 0.001
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tau = 0.2
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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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#+END_SRC
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#+RESULTS:
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: E = -0.4112049153828464 +/- 0.00027934927432953063
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: E = -0.4858534479298907 +/- 0.00010203236131158794
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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)
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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(in) :: a, tau
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integer*8 , intent(in) :: nmax
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double precision, intent(out) :: energy
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integer*8 :: istep
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double precision :: norm, r_old(3), r_new(3), d_old(3), sq_tau, chi(3)
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double precision, external :: e_loc
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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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call random_gauss(r_old,3)
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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 drift(a,r_old,d_old)
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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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norm = norm + 1.d0
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energy = energy + e_loc(a,r_new)
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r_old(:) = r_new(:)
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end do
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energy = energy / norm
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end subroutine variational_montecarlo
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@ -1167,7 +1180,8 @@ 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 = 0.2
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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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@ -1175,7 +1189,7 @@ program qmc
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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))
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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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@ -1187,7 +1201,14 @@ gfortran hydrogen.f90 qmc_stats.f90 vmc.f90 -o vmc
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./vmc
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#+end_src
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#+RESULTS:
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: E = -0.48584030499187431 +/- 1.0411743995438257E-004
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*** Metropolis algorithm
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:PROPERTIES:
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:header-args:python: :tangle vmc_metropolis.py
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:header-args:f90: :tangle vmc_metropolis.f90
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:END:
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Discretizing the differential equation to generate the desired
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probability density will suffer from a discretization error
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@ -1236,7 +1257,7 @@ gfortran hydrogen.f90 qmc_stats.f90 vmc.f90 -o vmc
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#+end_exercise
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*Python*
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#+BEGIN_SRC python
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#+BEGIN_SRC python :results output
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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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@ -1325,8 +1346,14 @@ gfortran hydrogen.f90 qmc_stats.f90 vmc.f90 -o vmc
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./vmc
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#+end_src
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#+RESULTS:
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* TODO Diffusion Monte Carlo
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:PROPERTIES:
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:header-args:python: :tangle dmc.py
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:header-args:f90: :tangle dmc.f90
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:END:
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We will now consider the H_2 molecule in a minimal basis composed of the
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$1s$ orbitals of the hydrogen atoms:
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@ -27,17 +27,17 @@ subroutine random_gauss(z,n)
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! n is even
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do i=1,n,2
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z(i) = dsqrt(-2.d0*dlog(u(i)))
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z(i+1) = z(i) + dsin( two_pi*u(i+1) )
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z(i) = z(i) + dcos( two_pi*u(i+1) )
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z(i+1) = z(i) * dsin( two_pi*u(i+1) )
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z(i) = z(i) * dcos( two_pi*u(i+1) )
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end do
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else
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! n is odd
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do i=1,n-1,2
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z(i) = dsqrt(-2.d0*dlog(u(i)))
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z(i+1) = z(i) + dsin( two_pi*u(i+1) )
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z(i) = z(i) + dcos( two_pi*u(i+1) )
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z(i+1) = z(i) * dsin( two_pi*u(i+1) )
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z(i) = z(i) * dcos( two_pi*u(i+1) )
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end do
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z(n) = dsqrt(-2.d0*dlog(u(n)))
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z(n) = z(n) + dcos( two_pi*u(n+1) )
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z(n) = z(n) * dcos( two_pi*u(n+1) )
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end if
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end subroutine random_gauss
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@ -1,7 +1,7 @@
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subroutine uniform_montecarlo(a,nmax,energy)
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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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integer*8 , intent(in) :: nmax
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double precision, intent(out) :: energy
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integer*8 :: istep
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@ -26,7 +26,7 @@ end subroutine uniform_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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integer*8 , parameter :: nmax = 100000
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integer , parameter :: nruns = 30
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integer :: irun
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