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Solutions
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QMC.org
@ -6,9 +6,10 @@
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#+LATEX_CLASS: report
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#+LATEX_HEADER_EXTRA: \usepackage{minted}
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#+HTML_HEAD: <link rel="stylesheet" title="Standard" href="worg.css" type="text/css" />
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#+OPTIONS: H:2 num:t toc:nil \n:nil @:t ::t |:t ^:t -:t f:t *:t <:t
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#+OPTIONS: H:4 num:t toc:t \n:nil @:t ::t |:t ^:t -:t f:t *:t <:t
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#+OPTIONS: TeX:t LaTeX:t skip:nil d:nil todo:t pri:nil tags:not-in-toc
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#+EXPORT_EXCLUDE_TAGS: solution
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# EXCLUDE_TAGS: Python solution
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# EXCLUDE_TAGS: Fortran solution
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#+BEGIN_SRC elisp :output none :exports none
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(setq org-latex-listings 'minted
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@ -54,7 +55,7 @@
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*Note*
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#+begin_important
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In Fortran, when you use a double precision constant, don't forget
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to put d0 as a suffix (for example 2.0d0), or it will be
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to put ~d0~ as a suffix (for example ~2.0d0~), or it will be
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interpreted as a single precision value
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#+end_important
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@ -68,32 +69,31 @@
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\Psi(\mathbf{r}) = \exp(-a |\mathbf{r}|)
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$$
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We will first verify that $\Psi$ is an eigenfunction of the Hamiltonian
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We will first verify that, for a given value of $a$, $\Psi$ is an
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eigenfunction of the Hamiltonian
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$$
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\hat{H} = \hat{T} + \hat{V} = - \frac{1}{2} \Delta - \frac{1}{|\mathbf{r}|}
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$$
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when $a=1$, by checking that $\hat{H}\Psi(\mathbf{r}) = E\Psi(\mathbf{r})$ for
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all $\mathbf{r}$. We will check that the local energy, defined as
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To do that, we will check if the local energy, defined as
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$$
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E_L(\mathbf{r}) = \frac{\hat{H} \Psi(\mathbf{r})}{\Psi(\mathbf{r})},
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$$
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is constant. We will also see that when $a \ne 1$ the local energy
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is not constant, so $\hat{H} \Psi \ne E \Psi$.
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is constant.
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The probabilistic /expected value/ of an arbitrary function $f(x)$
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with respect to a probability density function $p(x)$ is given by
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$$ \langle f \rangle_p = \int_{-\infty}^\infty p(x)\, f(x)\,dx $$.
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$$ \langle f \rangle_p = \int_{-\infty}^\infty p(x)\, f(x)\,dx. $$
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Recall that a probability density function $p(x)$ is non-negative
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and integrates to one:
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$$ \int_{-\infty}^\infty p(x)\,dx = 1 $$.
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$$ \int_{-\infty}^\infty p(x)\,dx = 1. $$
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The electronic energy of a system is the expectation value of the
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@ -114,8 +114,18 @@
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:header-args:f90: :tangle hydrogen.f90
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:END:
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Write all the functions of this section in a single file :
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~hydrogen.py~ if you use Python, or ~hydrogen.f90~ is you use
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Fortran.
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*** Exercise 1
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#+begin_exercise
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Find the theoretical value of $a$ for which $\Psi$ is an eigenfunction of $\hat{H}$.
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#+end_exercise
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*** Exercise 2
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#+begin_exercise
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Write a function which computes the potential at $\mathbf{r}$.
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The function accepts a 3-dimensional vector =r= as input arguments
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@ -127,7 +137,15 @@
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V(\mathbf{r}) = -\frac{1}{\sqrt{x^2 + y^2 + z^2}}
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$$
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*Python*
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**** Python
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#+BEGIN_SRC python :results none :tangle none
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import numpy as np
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def potential(r):
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# TODO
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#+END_SRC
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**** Python :solution:
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#+BEGIN_SRC python :results none
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import numpy as np
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@ -135,8 +153,16 @@ def potential(r):
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return -1. / np.sqrt(np.dot(r,r))
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#+END_SRC
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**** Fortran
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#+BEGIN_SRC f90 :tangle none
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double precision function potential(r)
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implicit none
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double precision, intent(in) :: r(3)
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! TODO
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end function potential
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#+END_SRC
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*Fortran*
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**** Fortran :solution:
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#+BEGIN_SRC f90
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double precision function potential(r)
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implicit none
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@ -145,7 +171,7 @@ double precision function potential(r)
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end function potential
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#+END_SRC
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*** Exercise 2
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*** Exercise 3
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#+begin_exercise
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Write a function which computes the wave function at $\mathbf{r}$.
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The function accepts a scalar =a= and a 3-dimensional vector =r= as
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@ -153,13 +179,28 @@ end function potential
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#+end_exercise
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*Python*
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**** Python
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#+BEGIN_SRC python :results none
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def psi(a, r):
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# TODO
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#+END_SRC
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**** Python :solution:
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#+BEGIN_SRC python :results none
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def psi(a, r):
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return np.exp(-a*np.sqrt(np.dot(r,r)))
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#+END_SRC
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*Fortran*
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**** Fortran
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#+BEGIN_SRC f90
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double precision function psi(a, r)
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implicit none
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double precision, intent(in) :: a, r(3)
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! TODO
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end function psi
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#+END_SRC
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**** Fortran :solution:
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#+BEGIN_SRC f90
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double precision function psi(a, r)
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implicit none
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@ -168,7 +209,7 @@ double precision function psi(a, r)
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end function psi
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#+END_SRC
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*** Exercise 3
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*** Exercise 4
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#+begin_exercise
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Write a function which computes the local kinetic energy at $\mathbf{r}$.
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The function accepts =a= and =r= as input arguments and returns the
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@ -205,13 +246,28 @@ end function psi
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-\frac{1}{2} \frac{\Delta \Psi}{\Psi} (\mathbf{r}) = -\frac{1}{2}\left(a^2 - \frac{2a}{\mathbf{|r|}} \right)
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$$
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*Python*
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**** Python
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#+BEGIN_SRC python :results none
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def kinetic(a,r):
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# TODO
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#+END_SRC
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**** Python :solution:
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#+BEGIN_SRC python :results none
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def kinetic(a,r):
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return -0.5 * (a**2 - (2.*a)/np.sqrt(np.dot(r,r)))
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#+END_SRC
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*Fortran*
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**** Fortran
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#+BEGIN_SRC f90
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double precision function kinetic(a,r)
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implicit none
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double precision, intent(in) :: a, r(3)
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! TODO
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end function kinetic
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#+END_SRC
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**** Fortran :solution:
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#+BEGIN_SRC f90
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double precision function kinetic(a,r)
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implicit none
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@ -221,11 +277,12 @@ double precision function kinetic(a,r)
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end function kinetic
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#+END_SRC
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*** Exercise 4
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*** Exercise 5
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#+begin_exercise
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Write a function which computes the local energy at $\mathbf{r}$.
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The function accepts =x,y,z= as input arguments and returns the
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local energy.
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Write a function which computes the local energy at $\mathbf{r}$,
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using the previously defined functions.
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The function accepts =a= and =r= as input arguments and returns the
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local kinetic energy.
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#+end_exercise
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$$
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@ -233,13 +290,28 @@ end function kinetic
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$$
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*Python*
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**** Python
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#+BEGIN_SRC python :results none
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def e_loc(a,r):
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#TODO
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#+END_SRC
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**** Python :solution:
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#+BEGIN_SRC python :results none
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def e_loc(a,r):
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return kinetic(a,r) + potential(r)
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#+END_SRC
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*Fortran*
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**** Fortran
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#+BEGIN_SRC f90
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double precision function e_loc(a,r)
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implicit none
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double precision, intent(in) :: a, r(3)
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! TODO
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end function e_loc
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#+END_SRC
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**** Fortran :solution:
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#+BEGIN_SRC f90
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double precision function e_loc(a,r)
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implicit none
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@ -254,47 +326,98 @@ end function e_loc
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:header-args:python: :tangle plot_hydrogen.py
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:header-args:f90: :tangle plot_hydrogen.f90
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:END:
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*** Exercise
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#+begin_exercise
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For multiple values of $a$ (0.1, 0.2, 0.5, 1., 1.5, 2.), plot the
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local energy along the $x$ axis.
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local energy along the $x$ axis. In Python, you can use matplotlib
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for example. In Fortran, it is convenient to write in a text file
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the values of $x$ and $E_L(\mathbf{r})$ for each point, and use
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Gnuplot to plot the files.
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#+end_exercise
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*Python*
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#+BEGIN_SRC python :results none
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**** Python
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#+BEGIN_SRC python :results none
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import numpy as np
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import matplotlib.pyplot as plt
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from hydrogen import e_loc
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x=np.linspace(-5,5)
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def make_array(a):
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y=np.array([ e_loc(a, np.array([t,0.,0.]) ) for t in x])
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return y
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plt.figure(figsize=(10,5))
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# TODO
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plt.tight_layout()
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plt.legend()
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plt.savefig("plot_py.png")
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#+end_src
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**** Python :solution:
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#+BEGIN_SRC python :results none
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import numpy as np
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import matplotlib.pyplot as plt
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from hydrogen import e_loc
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x=np.linspace(-5,5)
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plt.figure(figsize=(10,5))
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for a in [0.1, 0.2, 0.5, 1., 1.5, 2.]:
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y = make_array(a)
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y=np.array([ e_loc(a, np.array([t,0.,0.]) ) for t in x])
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plt.plot(x,y,label=f"a={a}")
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plt.tight_layout()
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plt.legend()
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plt.savefig("plot_py.png")
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#+end_src
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#+end_src
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#+RESULTS:
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#+RESULTS:
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[[./plot_py.png]]
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[[./plot_py.png]]
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**** Fortran
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#+begin_src f90
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program plot
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implicit none
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double precision, external :: e_loc
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*Fortran*
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#+begin_src f90
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double precision :: x(50), dx
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integer :: i, j
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dx = 10.d0/(size(x)-1)
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do i=1,size(x)
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x(i) = -5.d0 + (i-1)*dx
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end do
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! TODO
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end program plot
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#+end_src
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To compile and run:
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#+begin_src sh :exports both
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gfortran hydrogen.f90 plot_hydrogen.f90 -o plot_hydrogen
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./plot_hydrogen > data
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#+end_src
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To plot the data using gnuplot:
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#+begin_src gnuplot :file plot.png :exports both
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set grid
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set xrange [-5:5]
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set yrange [-2:1]
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plot './data' index 0 using 1:2 with lines title 'a=0.1', \
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'./data' index 1 using 1:2 with lines title 'a=0.2', \
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'./data' index 2 using 1:2 with lines title 'a=0.5', \
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'./data' index 3 using 1:2 with lines title 'a=1.0', \
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'./data' index 4 using 1:2 with lines title 'a=1.5', \
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'./data' index 5 using 1:2 with lines title 'a=2.0'
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#+end_src
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**** Fortran :solution:
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#+begin_src f90
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program plot
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implicit none
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double precision, external :: e_loc
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@ -323,20 +446,20 @@ program plot
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end do
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end program plot
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#+end_src
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#+end_src
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To compile and run:
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To compile and run:
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#+begin_src sh :exports both
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#+begin_src sh :exports both
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gfortran hydrogen.f90 plot_hydrogen.f90 -o plot_hydrogen
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./plot_hydrogen > data
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#+end_src
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#+end_src
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#+RESULTS:
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#+RESULTS:
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To plot the data using gnuplot:
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To plot the data using gnuplot:
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#+begin_src gnuplot :file plot.png :exports both
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#+begin_src gnuplot :file plot.png :exports both
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set grid
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set xrange [-5:5]
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set yrange [-2:1]
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@ -346,12 +469,12 @@ plot './data' index 0 using 1:2 with lines title 'a=0.1', \
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'./data' index 3 using 1:2 with lines title 'a=1.0', \
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'./data' index 4 using 1:2 with lines title 'a=1.5', \
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'./data' index 5 using 1:2 with lines title 'a=2.0'
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#+end_src
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#+end_src
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#+RESULTS:
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[[file:plot.png]]
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#+RESULTS:
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[[file:plot.png]]
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** Numerical estimation of the energy
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** TODO Numerical estimation of the energy
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:PROPERTIES:
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:header-args:python: :tangle energy_hydrogen.py
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:header-args:f90: :tangle energy_hydrogen.f90
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@ -476,7 +599,7 @@ gfortran hydrogen.f90 energy_hydrogen.f90 -o energy_hydrogen
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: a = 1.5000000000000000 E = -0.39242967082602065
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: a = 2.0000000000000000 E = -8.0869806678448772E-002
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** Variance of the local energy
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** TODO Variance of the local energy
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:PROPERTIES:
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:header-args:python: :tangle variance_hydrogen.py
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:header-args:f90: :tangle variance_hydrogen.f90
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@ -618,7 +741,7 @@ gfortran hydrogen.f90 variance_hydrogen.f90 -o variance_hydrogen
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: a = 2.0000000000000000 E = -8.0869806678448772E-002 s2 = 1.8068814270846534
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* Variational Monte Carlo
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* TODO Variational Monte Carlo
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Numerical integration with deterministic methods is very efficient
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in low dimensions. When the number of dimensions becomes large,
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@ -629,7 +752,7 @@ gfortran hydrogen.f90 variance_hydrogen.f90 -o variance_hydrogen
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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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** TODO 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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:header-args:f90: :tangle qmc_stats.f90
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@ -694,7 +817,7 @@ subroutine ave_error(x,n,ave,err)
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end subroutine ave_error
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#+END_SRC
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** Uniform sampling in the box
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** TODO Uniform sampling in the box
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:PROPERTIES:
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:header-args:python: :tangle qmc_uniform.py
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:header-args:f90: :tangle qmc_uniform.f90
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@ -816,7 +939,7 @@ gfortran hydrogen.f90 qmc_stats.f90 qmc_uniform.f90 -o qmc_uniform
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#+RESULTS:
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: E = -0.49588321986667677 +/- 7.1758863546737969E-004
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** Metropolis sampling with $\Psi^2$
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** TODO Metropolis sampling with $\Psi^2$
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:PROPERTIES:
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:header-args:python: :tangle qmc_metropolis.py
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:header-args:f90: :tangle qmc_metropolis.f90
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@ -1000,7 +1123,7 @@ gfortran hydrogen.f90 qmc_stats.f90 qmc_metropolis.f90 -o qmc_metropolis
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: E = -0.49478505004797046 +/- 2.0493795299184956E-004
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: A = 0.51737800000000000 +/- 4.1827406733181444E-004
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** Gaussian random number generator
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** TODO Gaussian random number generator
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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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@ -1045,7 +1168,7 @@ subroutine random_gauss(z,n)
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end subroutine random_gauss
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#+END_SRC
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** Generalized Metropolis algorithm
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** TODO Generalized 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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@ -1290,9 +1413,9 @@ gfortran hydrogen.f90 qmc_stats.f90 vmc_metropolis.f90 -o vmc_metropolis
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:header-args:f90: :tangle dmc.f90
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:END:
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** Hydrogen atom
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** TODO Hydrogen atom
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**** Exercise
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*** Exercise
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#+begin_exercise
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Modify the Metropolis VMC program to introduce the PDMC weight.
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@ -1439,7 +1562,7 @@ gfortran hydrogen.f90 qmc_stats.f90 vmc_metropolis.f90 -o vmc_metropolis
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: A = 0.78861366666666655 +/- 3.5096729498002445E-004
|
||||
|
||||
|
||||
** Dihydrogen
|
||||
** TODO Dihydrogen
|
||||
|
||||
We will now consider the H_2 molecule in a minimal basis composed of the
|
||||
$1s$ orbitals of the hydrogen atoms:
|
||||
|
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Reference in New Issue
Block a user