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OK up to VMC
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@ -49,7 +49,7 @@
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starting from an approximate wave function.
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Code examples will be given in Python and Fortran. You can use
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whatever language you prefer to write the program.
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whatever language you prefer to write the programs.
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We consider the stationary solution of the Schrödinger equation, so
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the wave functions considered here are real: for an $N$ electron
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@ -80,14 +80,17 @@
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= \frac{\int |\Psi(\mathbf{r})|^2\, E_L(\mathbf{r})\,d\mathbf{r}}{\int |\Psi(\mathbf{r}) |^2 d\mathbf{r}}
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\end{eqnarray*}
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For few dimensions, one can easily compute $E$ by evaluating the integrals on a grid but, for a high number of dimensions, one can resort to Monte Carlo techniques to compute $E$.
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For few dimensions, one can easily compute $E$ by evaluating the
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integrals on a grid but, for a high number of dimensions, one can
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resort to Monte Carlo techniques to compute $E$.
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To this aim, recall that 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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To this aim, recall that the probabilistic /expected value/ of an
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arbitrary function $f(x)$ with respect to a probability density
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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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where a probability density function $p(x)$ is non-negative
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where 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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@ -95,13 +98,15 @@
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Similarly, we can view the the energy of a system, $E$, as the expected value of the local energy with respect to
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a probability density $P(\mathbf{r})$ defined in 3$N$ dimensions:
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$$ E = \int E_L(\mathbf{r}) P(\mathbf{r})\,d\mathbf{r} \equiv \langle E_L \rangle_{\Psi^2}\,, $$
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$$ E = \int E_L(\mathbf{r}) P(\mathbf{r})\,d\mathbf{r} \equiv \langle E_L \rangle_{P}\,, $$
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where the probability density is given by the square of the wave function:
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$$ P(\mathbf{r}) = \frac{|\Psi(\mathbf{r})|^2}{\int |\Psi(\mathbf{r})|^2 d\mathbf{r}}\,. $$
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If we can sample $N_{\rm MC}$ configurations $\{\mathbf{r}\}$ distributed as $p$, we can estimate $E$ as the average of the local energy computed over these configurations:
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If we can sample $N_{\rm MC}$ configurations $\{\mathbf{r}\}$
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distributed as $P$, we can estimate $E$ as the average of the local
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energy computed over these configurations:
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$$ E \approx \frac{1}{N_{\rm MC}} \sum_{i=1}^{N_{\rm MC}} E_L(\mathbf{r}_i) \,. $$
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@ -148,7 +153,7 @@
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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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The function accepts a 3-dimensional vector =r= as input argument
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and returns the potential.
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#+end_exercise
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@ -254,7 +259,7 @@ end function psi
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local kinetic energy.
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#+end_exercise
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The local kinetic energy is defined as $$-\frac{1}{2}\frac{\Delta \Psi}{\Psi}.$$
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The local kinetic energy is defined as $$T_L(\mathbf{r}) = -\frac{1}{2}\frac{\Delta \Psi(\mathbf{r})}{\Psi(\mathbf{r})}.$$
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We differentiate $\Psi$ with respect to $x$:
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@ -281,7 +286,7 @@ end function psi
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Therefore, the local kinetic energy is
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$$
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-\frac{1}{2} \frac{\Delta \Psi}{\Psi} (\mathbf{r}) = -\frac{1}{2}\left(a^2 - \frac{2a}{\mathbf{|r|}} \right)
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T_L (\mathbf{r}) = -\frac{1}{2}\left(a^2 - \frac{2a}{\mathbf{|r|}} \right)
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$$
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*Python*
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@ -352,11 +357,24 @@ def e_loc(a,r):
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#+END_SRC
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*Fortran*
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#+begin_note
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When you call a function in Fortran, you need to declare its
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return type.
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You might by accident choose a function name which is the
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same as an internal function of Fortran. So it is recommended to
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*always* use the keyword ~external~ to make sure the function you
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are calling is yours.
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#+end_note
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#+BEGIN_SRC f90 :tangle none
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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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double precision, external :: kinetic
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double precision, external :: potential
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! TODO
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end function e_loc
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@ -375,7 +393,8 @@ 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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double precision, external :: kinetic, potential
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double precision, external :: kinetic
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double precision, external :: potential
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e_loc = kinetic(a,r) + potential(r)
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@ -408,20 +427,38 @@ end function e_loc
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:header-args:f90: :tangle plot_hydrogen.f90
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:END:
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#+begin_note
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The potential and the kinetic energy both diverge at $r=0$, so we
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choose a grid which does not contain the origin.
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#+end_note
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The program you will write in this section will be written in
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another file (~plot_hydrogen.py~ or ~plot_hydrogen.f90~ for
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example).
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It will use the functions previously defined.
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In Python, you should put at the beginning of the file
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#+BEGIN_SRC python :results none :tangle none
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from hydrogen import e_loc
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#+END_SRC
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to be able to use the ~e_loc~ function of the ~hydrogen.py~ file.
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In Fortran, you will need to compile all the source files together:
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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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#+end_src
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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. 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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Gnuplot to plot the files. With Gnuplot, you will need 2 blank
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lines to separate the data corresponding to different values of $a$.
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#+end_exercise
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#+begin_note
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The potential and the kinetic energy both diverge at $r=0$, so we
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choose a grid which does not contain the origin to avoid numerical issues.
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#+end_note
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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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@ -3007,3 +3044,15 @@ gfortran hydrogen.f90 qmc_stats.f90 vmc.f90 -o vmc
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- [ ] Propose a project for the students to continue the
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programs. Idea: Modify the program to compute the exact energy of
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the H$_2$ molecule at $R$=1.4010 bohr. Answer: 0.17406 a.u.
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* Schedule
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|------------------------------+---------|
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| <2021-02-04 Thu 09:00-10:30> | Lecture |
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|------------------------------+---------|
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| <2021-02-04 Thu 10:45-11:10> | 2.1 |
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| <2021-02-04 Thu 11:10-11:30> | 2.2 |
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| <2021-02-04 Thu 11:30-12:15> | 2.3 |
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| <2021-02-04 Thu 12:15-12:30> | 2.4 |
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|------------------------------+---------|
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| | |
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