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@ -6,9 +6,10 @@
#+LATEX_CLASS: report
#+LATEX_HEADER_EXTRA: \usepackage{minted}
#+HTML_HEAD: <link rel="stylesheet" title="Standard" href="worg.css" type="text/css" />
#+OPTIONS: H:2 num:t toc:nil \n:nil @:t ::t |:t ^:t -:t f:t *:t <:t
#+OPTIONS: H:4 num:t toc:t \n:nil @:t ::t |:t ^:t -:t f:t *:t <:t
#+OPTIONS: TeX:t LaTeX:t skip:nil d:nil todo:t pri:nil tags:not-in-toc
#+EXPORT_EXCLUDE_TAGS: solution
# EXCLUDE_TAGS: Python solution
# EXCLUDE_TAGS: Fortran solution
#+BEGIN_SRC elisp :output none :exports none
(setq org-latex-listings 'minted
@ -54,7 +55,7 @@
*Note*
#+begin_important
In Fortran, when you use a double precision constant, don't forget
to put d0 as a suffix (for example 2.0d0), or it will be
to put ~d0~ as a suffix (for example ~2.0d0~), or it will be
interpreted as a single precision value
#+end_important
@ -68,32 +69,31 @@
\Psi(\mathbf{r}) = \exp(-a |\mathbf{r}|)
$$
We will first verify that $\Psi$ is an eigenfunction of the Hamiltonian
We will first verify that, for a given value of $a$, $\Psi$ is an
eigenfunction of the Hamiltonian
$$
\hat{H} = \hat{T} + \hat{V} = - \frac{1}{2} \Delta - \frac{1}{|\mathbf{r}|}
$$
when $a=1$, by checking that $\hat{H}\Psi(\mathbf{r}) = E\Psi(\mathbf{r})$ for
all $\mathbf{r}$. We will check that the local energy, defined as
To do that, we will check if the local energy, defined as
$$
E_L(\mathbf{r}) = \frac{\hat{H} \Psi(\mathbf{r})}{\Psi(\mathbf{r})},
$$
is constant. We will also see that when $a \ne 1$ the local energy
is not constant, so $\hat{H} \Psi \ne E \Psi$.
is constant.
The probabilistic /expected value/ of an arbitrary function $f(x)$
with respect to a probability density function $p(x)$ is given by
$$ \langle f \rangle_p = \int_{-\infty}^\infty p(x)\, f(x)\,dx $$.
$$ \langle f \rangle_p = \int_{-\infty}^\infty p(x)\, f(x)\,dx. $$
Recall that a probability density function $p(x)$ is non-negative
and integrates to one:
$$ \int_{-\infty}^\infty p(x)\,dx = 1 $$.
$$ \int_{-\infty}^\infty p(x)\,dx = 1. $$
The electronic energy of a system is the expectation value of the
@ -114,8 +114,18 @@
:header-args:f90: :tangle hydrogen.f90
:END:
Write all the functions of this section in a single file :
~hydrogen.py~ if you use Python, or ~hydrogen.f90~ is you use
Fortran.
*** Exercise 1
#+begin_exercise
Find the theoretical value of $a$ for which $\Psi$ is an eigenfunction of $\hat{H}$.
#+end_exercise
*** Exercise 2
#+begin_exercise
Write a function which computes the potential at $\mathbf{r}$.
The function accepts a 3-dimensional vector =r= as input arguments
@ -127,7 +137,15 @@
V(\mathbf{r}) = -\frac{1}{\sqrt{x^2 + y^2 + z^2}}
$$
*Python*
**** Python
#+BEGIN_SRC python :results none :tangle none
import numpy as np
def potential(r):
# TODO
#+END_SRC
**** Python :solution:
#+BEGIN_SRC python :results none
import numpy as np
@ -135,8 +153,16 @@ def potential(r):
return -1. / np.sqrt(np.dot(r,r))
#+END_SRC
**** Fortran
#+BEGIN_SRC f90 :tangle none
double precision function potential(r)
implicit none
double precision, intent(in) :: r(3)
! TODO
end function potential
#+END_SRC
*Fortran*
**** Fortran :solution:
#+BEGIN_SRC f90
double precision function potential(r)
implicit none
@ -145,7 +171,7 @@ double precision function potential(r)
end function potential
#+END_SRC
*** Exercise 2
*** Exercise 3
#+begin_exercise
Write a function which computes the wave function at $\mathbf{r}$.
The function accepts a scalar =a= and a 3-dimensional vector =r= as
@ -153,13 +179,28 @@ end function potential
#+end_exercise
*Python*
**** Python
#+BEGIN_SRC python :results none
def psi(a, r):
# TODO
#+END_SRC
**** Python :solution:
#+BEGIN_SRC python :results none
def psi(a, r):
return np.exp(-a*np.sqrt(np.dot(r,r)))
#+END_SRC
*Fortran*
**** Fortran
#+BEGIN_SRC f90
double precision function psi(a, r)
implicit none
double precision, intent(in) :: a, r(3)
! TODO
end function psi
#+END_SRC
**** Fortran :solution:
#+BEGIN_SRC f90
double precision function psi(a, r)
implicit none
@ -168,7 +209,7 @@ double precision function psi(a, r)
end function psi
#+END_SRC
*** Exercise 3
*** Exercise 4
#+begin_exercise
Write a function which computes the local kinetic energy at $\mathbf{r}$.
The function accepts =a= and =r= as input arguments and returns the
@ -205,13 +246,28 @@ end function psi
-\frac{1}{2} \frac{\Delta \Psi}{\Psi} (\mathbf{r}) = -\frac{1}{2}\left(a^2 - \frac{2a}{\mathbf{|r|}} \right)
$$
*Python*
**** Python
#+BEGIN_SRC python :results none
def kinetic(a,r):
# TODO
#+END_SRC
**** Python :solution:
#+BEGIN_SRC python :results none
def kinetic(a,r):
return -0.5 * (a**2 - (2.*a)/np.sqrt(np.dot(r,r)))
#+END_SRC
*Fortran*
**** Fortran
#+BEGIN_SRC f90
double precision function kinetic(a,r)
implicit none
double precision, intent(in) :: a, r(3)
! TODO
end function kinetic
#+END_SRC
**** Fortran :solution:
#+BEGIN_SRC f90
double precision function kinetic(a,r)
implicit none
@ -221,11 +277,12 @@ double precision function kinetic(a,r)
end function kinetic
#+END_SRC
*** Exercise 4
*** Exercise 5
#+begin_exercise
Write a function which computes the local energy at $\mathbf{r}$.
The function accepts =x,y,z= as input arguments and returns the
local energy.
Write a function which computes the local energy at $\mathbf{r}$,
using the previously defined functions.
The function accepts =a= and =r= as input arguments and returns the
local kinetic energy.
#+end_exercise
$$
@ -233,13 +290,28 @@ end function kinetic
$$
*Python*
**** Python
#+BEGIN_SRC python :results none
def e_loc(a,r):
#TODO
#+END_SRC
**** Python :solution:
#+BEGIN_SRC python :results none
def e_loc(a,r):
return kinetic(a,r) + potential(r)
#+END_SRC
*Fortran*
**** Fortran
#+BEGIN_SRC f90
double precision function e_loc(a,r)
implicit none
double precision, intent(in) :: a, r(3)
! TODO
end function e_loc
#+END_SRC
**** Fortran :solution:
#+BEGIN_SRC f90
double precision function e_loc(a,r)
implicit none
@ -255,46 +327,97 @@ end function e_loc
:header-args:f90: :tangle plot_hydrogen.f90
:END:
*** Exercise
#+begin_exercise
For multiple values of $a$ (0.1, 0.2, 0.5, 1., 1.5, 2.), plot the
local energy along the $x$ axis.
local energy along the $x$ axis. In Python, you can use matplotlib
for example. In Fortran, it is convenient to write in a text file
the values of $x$ and $E_L(\mathbf{r})$ for each point, and use
Gnuplot to plot the files.
#+end_exercise
*Python*
#+BEGIN_SRC python :results none
**** Python
#+BEGIN_SRC python :results none
import numpy as np
import matplotlib.pyplot as plt
from hydrogen import e_loc
x=np.linspace(-5,5)
def make_array(a):
y=np.array([ e_loc(a, np.array([t,0.,0.]) ) for t in x])
return y
plt.figure(figsize=(10,5))
# TODO
plt.tight_layout()
plt.legend()
plt.savefig("plot_py.png")
#+end_src
**** Python :solution:
#+BEGIN_SRC python :results none
import numpy as np
import matplotlib.pyplot as plt
from hydrogen import e_loc
x=np.linspace(-5,5)
plt.figure(figsize=(10,5))
for a in [0.1, 0.2, 0.5, 1., 1.5, 2.]:
y = make_array(a)
y=np.array([ e_loc(a, np.array([t,0.,0.]) ) for t in x])
plt.plot(x,y,label=f"a={a}")
plt.tight_layout()
plt.legend()
plt.savefig("plot_py.png")
#+end_src
#+end_src
#+RESULTS:
#+RESULTS:
[[./plot_py.png]]
[[./plot_py.png]]
**** Fortran
#+begin_src f90
program plot
implicit none
double precision, external :: e_loc
double precision :: x(50), dx
integer :: i, j
*Fortran*
#+begin_src f90
dx = 10.d0/(size(x)-1)
do i=1,size(x)
x(i) = -5.d0 + (i-1)*dx
end do
! TODO
end program plot
#+end_src
To compile and run:
#+begin_src sh :exports both
gfortran hydrogen.f90 plot_hydrogen.f90 -o plot_hydrogen
./plot_hydrogen > data
#+end_src
To plot the data using gnuplot:
#+begin_src gnuplot :file plot.png :exports both
set grid
set xrange [-5:5]
set yrange [-2:1]
plot './data' index 0 using 1:2 with lines title 'a=0.1', \
'./data' index 1 using 1:2 with lines title 'a=0.2', \
'./data' index 2 using 1:2 with lines title 'a=0.5', \
'./data' index 3 using 1:2 with lines title 'a=1.0', \
'./data' index 4 using 1:2 with lines title 'a=1.5', \
'./data' index 5 using 1:2 with lines title 'a=2.0'
#+end_src
**** Fortran :solution:
#+begin_src f90
program plot
implicit none
double precision, external :: e_loc
@ -323,20 +446,20 @@ program plot
end do
end program plot
#+end_src
#+end_src
To compile and run:
To compile and run:
#+begin_src sh :exports both
#+begin_src sh :exports both
gfortran hydrogen.f90 plot_hydrogen.f90 -o plot_hydrogen
./plot_hydrogen > data
#+end_src
#+end_src
#+RESULTS:
#+RESULTS:
To plot the data using gnuplot:
To plot the data using gnuplot:
#+begin_src gnuplot :file plot.png :exports both
#+begin_src gnuplot :file plot.png :exports both
set grid
set xrange [-5:5]
set yrange [-2:1]
@ -346,12 +469,12 @@ plot './data' index 0 using 1:2 with lines title 'a=0.1', \
'./data' index 3 using 1:2 with lines title 'a=1.0', \
'./data' index 4 using 1:2 with lines title 'a=1.5', \
'./data' index 5 using 1:2 with lines title 'a=2.0'
#+end_src
#+end_src
#+RESULTS:
[[file:plot.png]]
#+RESULTS:
[[file:plot.png]]
** Numerical estimation of the energy
** TODO Numerical estimation of the energy
:PROPERTIES:
:header-args:python: :tangle energy_hydrogen.py
:header-args:f90: :tangle energy_hydrogen.f90
@ -476,7 +599,7 @@ gfortran hydrogen.f90 energy_hydrogen.f90 -o energy_hydrogen
: a = 1.5000000000000000 E = -0.39242967082602065
: a = 2.0000000000000000 E = -8.0869806678448772E-002
** Variance of the local energy
** TODO Variance of the local energy
:PROPERTIES:
:header-args:python: :tangle variance_hydrogen.py
:header-args:f90: :tangle variance_hydrogen.f90
@ -618,7 +741,7 @@ gfortran hydrogen.f90 variance_hydrogen.f90 -o variance_hydrogen
: a = 2.0000000000000000 E = -8.0869806678448772E-002 s2 = 1.8068814270846534
* Variational Monte Carlo
* TODO Variational Monte Carlo
Numerical integration with deterministic methods is very efficient
in low dimensions. When the number of dimensions becomes large,
@ -629,7 +752,7 @@ gfortran hydrogen.f90 variance_hydrogen.f90 -o variance_hydrogen
to the discretization of space, and compute a statistical confidence
interval.
** Computation of the statistical error
** TODO Computation of the statistical error
:PROPERTIES:
:header-args:python: :tangle qmc_stats.py
:header-args:f90: :tangle qmc_stats.f90
@ -694,7 +817,7 @@ subroutine ave_error(x,n,ave,err)
end subroutine ave_error
#+END_SRC
** Uniform sampling in the box
** TODO Uniform sampling in the box
:PROPERTIES:
:header-args:python: :tangle qmc_uniform.py
:header-args:f90: :tangle qmc_uniform.f90
@ -816,7 +939,7 @@ gfortran hydrogen.f90 qmc_stats.f90 qmc_uniform.f90 -o qmc_uniform
#+RESULTS:
: E = -0.49588321986667677 +/- 7.1758863546737969E-004
** Metropolis sampling with $\Psi^2$
** TODO Metropolis sampling with $\Psi^2$
:PROPERTIES:
:header-args:python: :tangle qmc_metropolis.py
:header-args:f90: :tangle qmc_metropolis.f90
@ -1000,7 +1123,7 @@ gfortran hydrogen.f90 qmc_stats.f90 qmc_metropolis.f90 -o qmc_metropolis
: E = -0.49478505004797046 +/- 2.0493795299184956E-004
: A = 0.51737800000000000 +/- 4.1827406733181444E-004
** Gaussian random number generator
** TODO Gaussian random number generator
To obtain Gaussian-distributed random numbers, you can apply the
[[https://en.wikipedia.org/wiki/Box%E2%80%93Muller_transform][Box Muller transform]] to uniform random numbers:
@ -1045,7 +1168,7 @@ subroutine random_gauss(z,n)
end subroutine random_gauss
#+END_SRC
** Generalized Metropolis algorithm
** TODO Generalized Metropolis algorithm
:PROPERTIES:
:header-args:python: :tangle vmc_metropolis.py
:header-args:f90: :tangle vmc_metropolis.f90
@ -1290,9 +1413,9 @@ gfortran hydrogen.f90 qmc_stats.f90 vmc_metropolis.f90 -o vmc_metropolis
:header-args:f90: :tangle dmc.f90
:END:
** Hydrogen atom
** TODO Hydrogen atom
**** Exercise
*** Exercise
#+begin_exercise
Modify the Metropolis VMC program to introduce the PDMC weight.
@ -1439,7 +1562,7 @@ gfortran hydrogen.f90 qmc_stats.f90 vmc_metropolis.f90 -o vmc_metropolis
: 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: