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@ -2758,350 +2758,7 @@ gfortran hydrogen.f90 qmc_stats.f90 pdmc.f90 -o pdmc
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: A = 0.98963533333333342 +/- 6.3052128284666221E-005
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** H_2
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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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$$
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\Psi(\mathbf{r}_1, \mathbf{r}_2) =
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\exp(-(\mathbf{r}_1 - \mathbf{R}_A)) +
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$$
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where $\mathbf{r}_1$ and $\mathbf{r}_2$ denote the electron
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coordinates and $\mathbf{R}_A$ and $\mathbf{R}_B$ the coordinates of
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the nuclei.
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* Old sections to be removed :noexport:
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:PROPERTIES:
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:header-args:python: :tangle none
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:header-args:f90: :tangle none
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:END:
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** Gaussian sampling :noexport:
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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.
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Now the sampling probability can be inserted into the equation of the energy:
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\[
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E = \frac{\int P(\mathbf{r})
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\frac{\left|\Psi(\mathbf{r})\right|^2}{P(\mathbf{r})}\,
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\frac{\hat{H} \Psi(\mathbf{r})}{\Psi(\mathbf{r})}\,d\mathbf{r}}{\int P(\mathbf{r}) \frac{\left|\Psi(\mathbf{r}) \right|^2}{P(\mathbf{r})} d\mathbf{r}}
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\]
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with the Gaussian probability
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\[
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P(\mathbf{r}) = \frac{1}{(2 \pi)^{3/2}}\exp\left( -\frac{\mathbf{r}^2}{2} \right).
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\]
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As the coordinates are drawn with probability $P(\mathbf{r})$, the
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average energy can be computed as
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$$
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E \approx \frac{\sum_i w_i E_L(\mathbf{r}_i)}{\sum_i w_i}, \;\;
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w_i = \frac{\left|\Psi(\mathbf{r}_i)\right|^2}{P(\mathbf{r}_i)} \delta \mathbf{r}
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$$
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*** Exercise
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#+begin_exercise
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Modify the program of the previous section to sample with
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Gaussian-distributed random numbers. Can you see an reduction in
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the statistical error?
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#+end_exercise
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**** Python
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#+BEGIN_SRC python :results output
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#!/usr/bin/env python3
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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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def MonteCarlo(a,nmax):
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E = 0.
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N = 0.
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for istep in range(nmax):
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r = np.random.normal(loc=0., scale=1.0, size=(3))
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w = psi(a,r)
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w = w*w / gaussian(r)
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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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a = 1.2
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nmax = 100000
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X = [MonteCarlo(a,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 = -0.49511014287471955 +/- 0.00012402022172236656
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**** Fortran
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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 * (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*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(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 = 1.2d0
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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 :: 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 = -0.49517104619091717 +/- 1.0685523607878961E-004
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** Improved sampling with $\Psi^2$ :noexport:
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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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$|\Psi(\mathbf{r},t)|^2$, which obeys the Fokker-Planck equation:
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\[
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\frac{\partial \Psi^2}{\partial t} = \sum_i D
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\frac{\partial}{\partial \mathbf{r}_i} \left(
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\frac{\partial}{\partial \mathbf{r}_i} - F_i(\mathbf{r}) \right) |\Psi(\mathbf{r},t)|^2.
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\]
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$D$ is the diffusion constant and $F_i$ is the i-th component of a
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drift velocity caused by an external potential. For a stationary
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density, \( \frac{\partial \Psi^2}{\partial t} = 0 \), so
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\begin{eqnarray*}
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0 & = & \sum_i D
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\frac{\partial}{\partial \mathbf{r}_i} \left(
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\frac{\partial}{\partial \mathbf{r}_i} - F_i(\mathbf{r}) \right) |\Psi(\mathbf{r})|^2 \\
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0 & = & \sum_i D
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\frac{\partial}{\partial \mathbf{r}_i} \left(
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\frac{\partial |\Psi(\mathbf{r})|^2}{\partial \mathbf{r}_i} -
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F_i(\mathbf{r})\,|\Psi(\mathbf{r})|^2 \right) \\
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0 & = &
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\frac{\partial^2 \Psi^2}{\partial \mathbf{r}_i^2} -
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\frac{\partial F_i }{\partial \mathbf{r}_i}|\Psi(\mathbf{r})|^2 -
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\frac{\partial \Psi^2}{\partial \mathbf{r}_i} F_i(\mathbf{r})
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\end{eqnarray*}
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we search for a drift function which satisfies
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\[
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\frac{\partial^2 \Psi^2}{\partial \mathbf{r}_i^2} = |\Psi(\mathbf{r})|^2 \frac{\partial F_i }{\partial \mathbf{r}_i} +
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\frac{\partial \Psi^2}{\partial \mathbf{r}_i} F_i(\mathbf{r})
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\]
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to obtain a second derivative on the left, we need the drift to be
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of the form
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\[
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F_i(\mathbf{r}) = g(\mathbf{r}) \frac{\partial \Psi^2}{\partial \mathbf{r}_i}
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\]
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\[
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\frac{\partial^2 \Psi^2}{\partial \mathbf{r}_i^2}
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= |\Psi(\mathbf{r})|^2 \frac{\partial g(\mathbf{r})}{\partial
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\mathbf{r}_i}\frac{\partial \Psi^2}{\partial
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\mathbf{r}_i} + |\Psi(\mathbf{r})|^2 g(\mathbf{r})
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\frac{\partial^2 \Psi^2}{\partial \mathbf{r}_i^2} + \frac{\partial
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\Psi^2}{\partial \mathbf{r}_i} g(\mathbf{r}) \frac{\partial
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\Psi^2}{\partial \mathbf{r}_i}
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\]
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$g = 1 / \Psi^2$ satisfies this equation, so
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\[
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F(\mathbf{r}) = \frac{\nabla |\Psi(\mathbf{r})|^2}{|\Psi(\mathbf{r})|^2} = 2 \frac{\nabla
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\Psi(\mathbf{r})}{\Psi(\mathbf{r})} = 2 \nabla \left( \log \Psi(\mathbf{r}) \right)
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\]
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In statistical mechanics, Fokker-Planck trajectories are generated
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by a Langevin equation:
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\[
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\frac{\partial \mathbf{r}(t)}{\partial t} = 2D \frac{\nabla
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\Psi(\mathbf{r}(t))}{\Psi} + \eta
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\]
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where $\eta$ is a normally-distributed fluctuating random force.
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Discretizing this differential equation gives the following drifted
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diffusion scheme:
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\[
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\mathbf{r}_{n+1} = \mathbf{r}_{n} + \delta t\, 2D \frac{\nabla
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\Psi(\mathbf{r})}{\Psi(\mathbf{r})} + \chi
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\]
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where $\chi$ is a Gaussian random variable with zero mean and
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variance $\delta t\,2D$.
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**** Exercise 2
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#+begin_exercise
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Sample $\Psi^2$ approximately using the drifted diffusion scheme,
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with a diffusion constant $D=1/2$. You can use a time step of
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0.001 a.u.
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#+end_exercise
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*Python*
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#+BEGIN_SRC python :results output
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#!/usr/bin/env python3
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from hydrogen import *
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from qmc_stats import *
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def MonteCarlo(a,dt,nmax):
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sq_dt = np.sqrt(dt)
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# Initialization
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E = 0.
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N = 0.
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r_old = np.random.normal(loc=0., scale=1.0, size=(3))
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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 + dt * d_old + chi*sq_dt
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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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return E/N
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a = 1.2
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nmax = 100000
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dt = 0.2
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X = [MonteCarlo(a,dt,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 = -0.4858534479298907 +/- 0.00010203236131158794
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*Fortran*
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#+BEGIN_SRC f90
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subroutine variational_montecarlo(a,dt,nmax,energy)
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implicit none
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double precision, intent(in) :: a, dt
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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_dt, chi(3)
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double precision, external :: e_loc
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sq_dt = dsqrt(dt)
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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 drift(a,r_old,d_old)
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call random_gauss(chi,3)
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r_new(:) = r_old(:) + dt * d_old(:) + chi(:)*sq_dt
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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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program qmc
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implicit none
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double precision, parameter :: a = 1.2d0
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double precision, parameter :: dt = 0.2d0
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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 :: ave, err
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do irun=1,nruns
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call variational_montecarlo(a,dt,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 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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* Project
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Change your PDMC code for one of the following:
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