Gaussian sampling
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QMC.org
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QMC.org
@ -550,7 +550,6 @@ Moreover, a Monte Carlo sampling will alow us to remove the bias due
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to the discretization of space, and compute a statistical confidence
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interval.
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** Computation of the statistical error
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:PROPERTIES:
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:header-args:python: :tangle qmc_stats.py
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@ -635,7 +634,6 @@ At every Monte Carlo step:
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Compute the energy of the wave function with $a=0.9$.
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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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@ -714,13 +712,57 @@ gfortran hydrogen.f90 qmc_stats.f90 qmc_uniform.f90 -o qmc_uniform
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: E = -0.49588321986667677 +/- 7.1758863546737969E-004
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** Gaussian sampling
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:PROPERTIES:
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:header-args:python: :tangle qmc_gaussian.py
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:header-args:f90: :tangle qmc_gaussian.f90
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:END:
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We will now improve the sampling and allow to sample in the whole
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3D space, correcting the bias related to the sampling in the box.
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Instead of drawing uniform random numbers, we will draw Gaussian
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random numbers centered on 0 and with a variance of 1. Now the
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equation for the energy is changed into
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random numbers centered on 0 and with a variance of 1.
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To obtain Gaussian-distributed random numbers, you can apply the
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[[https://en.wikipedia.org/wiki/Box%E2%80%93Muller_transform][Box Muller transform]] to uniform random numbers:
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\begin{eqnarray*}
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z_1 &=& \sqrt{-2 \ln u_1} \cos(2 \pi u_2) \\
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z_2 &=& \sqrt{-2 \ln u_1} \sin(2 \pi u_2)
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\end{eqnarray*}
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#+BEGIN_SRC f90 :tangle qmc_stats.f90
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subroutine random_gauss(z,n)
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implicit none
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integer, intent(in) :: n
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double precision, intent(out) :: z(n)
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double precision :: u(n+1)
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double precision, parameter :: two_pi = 2.d0*dacos(-1.d0)
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integer :: i
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call random_number(u)
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if (iand(n,1) == 0) then
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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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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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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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end if
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end subroutine random_gauss
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#+END_SRC
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Now the equation for the energy is changed into
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\[
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E = \frac{\int P(\mathbf{r}) \frac{\left[\Psi(\mathbf{r})\right]^2}{P(\mathbf{r})}\, \frac{\hat{H} \Psi(\mathbf{r})}{\Psi(\mathbf{r})}\,d\mathbf{r}}{\int P(\mathbf{r}) \frac{\left[\Psi(\mathbf{r}) \right]^2}{P(\mathbf{r})} d\mathbf{r}}
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@ -738,15 +780,14 @@ E \approx \frac{\sum_i w_i E_L(\mathbf{r}_i)}{\sum_i w_i}, \;\;
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w_i = \frac{\left[\Psi(\mathbf{r}_i)\right]^2}{P(\mathbf{r}_i)} \delta \mathbf{r}
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$$
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#+BEGIN_SRC 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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norm_gauss = 1./(2.*np.pi)**(1.5)
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def gaussian(r):
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return norm_gauss * np.exp(-np.dot(r,r)*0.5)
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#+END_SRC
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#+RESULTS:
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#+BEGIN_SRC python
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def MonteCarlo(a,nmax):
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E = 0.
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N = 0.
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@ -757,11 +798,7 @@ def MonteCarlo(a,nmax):
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N += w
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E += w * e_loc(a,r)
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return E/N
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#+END_SRC
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#+RESULTS:
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#+BEGIN_SRC python :results output
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a = 0.9
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nmax = 100000
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X = [MonteCarlo(a,nmax) for i in range(30)]
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@ -770,8 +807,67 @@ print(f"E = {E} +/- {deltaE}")
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#+END_SRC
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#+RESULTS:
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: E = -0.4952488228427792 +/- 0.00011913174676540714
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: E = -0.49507506093129827 +/- 0.00014164037765553668
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#+BEGIN_SRC f90
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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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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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double precision, intent(out) :: energy
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integer*8 :: istep
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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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energy = 0.d0
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norm = 0.d0
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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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end do
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energy = energy / norm
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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 , parameter :: nruns = 30
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integer :: irun
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double precision :: X(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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enddo
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call ave_error(X,nruns,ave,err)
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print *, 'E = ', 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 qmc_gaussian.f90 -o qmc_gaussian
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./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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** Sampling with $\Psi^2$
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We will now use the square of the wave function to make the sampling:
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@ -13,3 +13,31 @@ subroutine ave_error(x,n,ave,err)
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err = dsqrt(variance/dble(n))
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endif
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end subroutine ave_error
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subroutine random_gauss(z,n)
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implicit none
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integer, intent(in) :: n
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double precision, intent(out) :: z(n)
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double precision :: u(n+1)
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double precision, parameter :: two_pi = 2.d0*dacos(-1.d0)
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integer :: i
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call random_number(u)
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if (iand(n,1) == 0) then
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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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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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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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end if
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end subroutine random_gauss
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