mirror of
https://github.com/TREX-CoE/qmc-lttc.git
synced 2024-12-21 11:53:58 +01:00
OK up to VMC
This commit is contained in:
parent
a095fc0eec
commit
e8bf999e25
87
QMC.org
87
QMC.org
@ -49,7 +49,7 @@
|
||||
starting from an approximate wave function.
|
||||
|
||||
Code examples will be given in Python and Fortran. You can use
|
||||
whatever language you prefer to write the program.
|
||||
whatever language you prefer to write the programs.
|
||||
|
||||
We consider the stationary solution of the Schrödinger equation, so
|
||||
the wave functions considered here are real: for an $N$ electron
|
||||
@ -80,14 +80,17 @@
|
||||
= \frac{\int |\Psi(\mathbf{r})|^2\, E_L(\mathbf{r})\,d\mathbf{r}}{\int |\Psi(\mathbf{r}) |^2 d\mathbf{r}}
|
||||
\end{eqnarray*}
|
||||
|
||||
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 $E$.
|
||||
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 $E$.
|
||||
|
||||
To this aim, recall that the probabilistic /expected value/ of an arbitrary function $f(x)$
|
||||
with respect to a probability density function $P(x)$ is given by
|
||||
To this aim, recall that the probabilistic /expected value/ of an
|
||||
arbitrary function $f(x)$ with respect to a probability density
|
||||
function $P(x)$ is given by
|
||||
|
||||
$$ \langle f \rangle_p = \int_{-\infty}^\infty P(x)\, f(x)\,dx, $$
|
||||
$$ \langle f \rangle_P = \int_{-\infty}^\infty P(x)\, f(x)\,dx, $$
|
||||
|
||||
where a probability density function $p(x)$ is non-negative
|
||||
where a probability density function $P(x)$ is non-negative
|
||||
and integrates to one:
|
||||
|
||||
$$ \int_{-\infty}^\infty P(x)\,dx = 1. $$
|
||||
@ -95,13 +98,15 @@
|
||||
Similarly, we can view the the energy of a system, $E$, as the expected value of the local energy with respect to
|
||||
a probability density $P(\mathbf{r})$ defined in 3$N$ dimensions:
|
||||
|
||||
$$ E = \int E_L(\mathbf{r}) P(\mathbf{r})\,d\mathbf{r} \equiv \langle E_L \rangle_{\Psi^2}\,, $$
|
||||
$$ E = \int E_L(\mathbf{r}) P(\mathbf{r})\,d\mathbf{r} \equiv \langle E_L \rangle_{P}\,, $$
|
||||
|
||||
where the probability density is given by the square of the wave function:
|
||||
|
||||
$$ P(\mathbf{r}) = \frac{|\Psi(\mathbf{r})|^2}{\int |\Psi(\mathbf{r})|^2 d\mathbf{r}}\,. $$
|
||||
|
||||
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:
|
||||
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:
|
||||
|
||||
$$ E \approx \frac{1}{N_{\rm MC}} \sum_{i=1}^{N_{\rm MC}} E_L(\mathbf{r}_i) \,. $$
|
||||
|
||||
@ -148,7 +153,7 @@
|
||||
|
||||
#+begin_exercise
|
||||
Write a function which computes the potential at $\mathbf{r}$.
|
||||
The function accepts a 3-dimensional vector =r= as input arguments
|
||||
The function accepts a 3-dimensional vector =r= as input argument
|
||||
and returns the potential.
|
||||
#+end_exercise
|
||||
|
||||
@ -254,7 +259,7 @@ end function psi
|
||||
local kinetic energy.
|
||||
#+end_exercise
|
||||
|
||||
The local kinetic energy is defined as $$-\frac{1}{2}\frac{\Delta \Psi}{\Psi}.$$
|
||||
The local kinetic energy is defined as $$T_L(\mathbf{r}) = -\frac{1}{2}\frac{\Delta \Psi(\mathbf{r})}{\Psi(\mathbf{r})}.$$
|
||||
|
||||
We differentiate $\Psi$ with respect to $x$:
|
||||
|
||||
@ -281,7 +286,7 @@ end function psi
|
||||
|
||||
Therefore, the local kinetic energy is
|
||||
$$
|
||||
-\frac{1}{2} \frac{\Delta \Psi}{\Psi} (\mathbf{r}) = -\frac{1}{2}\left(a^2 - \frac{2a}{\mathbf{|r|}} \right)
|
||||
T_L (\mathbf{r}) = -\frac{1}{2}\left(a^2 - \frac{2a}{\mathbf{|r|}} \right)
|
||||
$$
|
||||
|
||||
*Python*
|
||||
@ -352,15 +357,28 @@ def e_loc(a,r):
|
||||
#+END_SRC
|
||||
|
||||
*Fortran*
|
||||
#+BEGIN_SRC f90 :tangle none
|
||||
|
||||
#+begin_note
|
||||
When you call a function in Fortran, you need to declare its
|
||||
return type.
|
||||
You might by accident choose a function name which is the
|
||||
same as an internal function of Fortran. So it is recommended to
|
||||
*always* use the keyword ~external~ to make sure the function you
|
||||
are calling is yours.
|
||||
#+end_note
|
||||
|
||||
#+BEGIN_SRC f90 :tangle none
|
||||
double precision function e_loc(a,r)
|
||||
implicit none
|
||||
double precision, intent(in) :: a, r(3)
|
||||
|
||||
double precision, external :: kinetic
|
||||
double precision, external :: potential
|
||||
|
||||
! TODO
|
||||
|
||||
end function e_loc
|
||||
#+END_SRC
|
||||
#+END_SRC
|
||||
|
||||
**** Solution :solution:
|
||||
*Python*
|
||||
@ -375,7 +393,8 @@ double precision function e_loc(a,r)
|
||||
implicit none
|
||||
double precision, intent(in) :: a, r(3)
|
||||
|
||||
double precision, external :: kinetic, potential
|
||||
double precision, external :: kinetic
|
||||
double precision, external :: potential
|
||||
|
||||
e_loc = kinetic(a,r) + potential(r)
|
||||
|
||||
@ -408,20 +427,38 @@ end function e_loc
|
||||
:header-args:f90: :tangle plot_hydrogen.f90
|
||||
:END:
|
||||
|
||||
#+begin_note
|
||||
The potential and the kinetic energy both diverge at $r=0$, so we
|
||||
choose a grid which does not contain the origin.
|
||||
#+end_note
|
||||
The program you will write in this section will be written in
|
||||
another file (~plot_hydrogen.py~ or ~plot_hydrogen.f90~ for
|
||||
example).
|
||||
It will use the functions previously defined.
|
||||
|
||||
In Python, you should put at the beginning of the file
|
||||
#+BEGIN_SRC python :results none :tangle none
|
||||
from hydrogen import e_loc
|
||||
#+END_SRC
|
||||
to be able to use the ~e_loc~ function of the ~hydrogen.py~ file.
|
||||
|
||||
In Fortran, you will need to compile all the source files together:
|
||||
#+begin_src sh :exports both
|
||||
gfortran hydrogen.f90 plot_hydrogen.f90 -o plot_hydrogen
|
||||
#+end_src
|
||||
|
||||
*** 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. 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.
|
||||
Gnuplot to plot the files. With Gnuplot, you will need 2 blank
|
||||
lines to separate the data corresponding to different values of $a$.
|
||||
#+end_exercise
|
||||
|
||||
#+begin_note
|
||||
The potential and the kinetic energy both diverge at $r=0$, so we
|
||||
choose a grid which does not contain the origin to avoid numerical issues.
|
||||
#+end_note
|
||||
|
||||
*Python*
|
||||
#+BEGIN_SRC python :results none :tangle none
|
||||
import numpy as np
|
||||
@ -3007,3 +3044,15 @@ gfortran hydrogen.f90 qmc_stats.f90 vmc.f90 -o vmc
|
||||
- [ ] Propose a project for the students to continue the
|
||||
programs. Idea: Modify the program to compute the exact energy of
|
||||
the H$_2$ molecule at $R$=1.4010 bohr. Answer: 0.17406 a.u.
|
||||
|
||||
* Schedule
|
||||
|
||||
|------------------------------+---------|
|
||||
| <2021-02-04 Thu 09:00-10:30> | Lecture |
|
||||
|------------------------------+---------|
|
||||
| <2021-02-04 Thu 10:45-11:10> | 2.1 |
|
||||
| <2021-02-04 Thu 11:10-11:30> | 2.2 |
|
||||
| <2021-02-04 Thu 11:30-12:15> | 2.3 |
|
||||
| <2021-02-04 Thu 12:15-12:30> | 2.4 |
|
||||
|------------------------------+---------|
|
||||
| | |
|
||||
|
||||
Loading…
Reference in New Issue
Block a user