TREX/qmc-lttc
1
0
Fork 0
mirror of https://github.com/TREX-CoE/qmc-lttc.git synced 2024-12-21 11:53:58 +01:00
Code Issues Projects Releases Wiki Activity

Solutions

Browse Source
This commit is contained in:
Anthony Scemama 2021-02-04 17:01:38 +01:00
parent a258c1c564
commit efb48d936f
1 changed files with 7 additions and 109 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

116
QMC.org
View File

@ -892,7 +892,7 @@ gfortran hydrogen.f90 variance_hydrogen.f90 -o variance_hydrogen
./variance_hydrogen
#+end_src
**** Solution :solution:
**** Solution :solution2:
*Python*
#+BEGIN_SRC python :results none :exports both
#!/usr/bin/env python3
@ -1019,7 +1019,7 @@ gfortran hydrogen.f90 variance_hydrogen.f90 -o variance_hydrogen
: a = 1.0000000000000000 E = -0.50000000000000000 s2 = 0.0000000000000000
: a = 1.5000000000000000 E = -0.39242967082602065 s2 = 0.31449670909172917
: a = 2.0000000000000000 E = -8.0869806678448772E-002 s2 = 1.8068814270846534
* Variational Monte Carlo
Numerical integration with deterministic methods is very efficient
@ -1089,7 +1089,7 @@ subroutine ave_error(x,n,ave,err)
end subroutine ave_error
#+END_SRC
**** Solution :solution:
**** Solution :solution2:
*Python*
#+BEGIN_SRC python :results none :exports code
#!/usr/bin/env python3
@ -1265,7 +1265,7 @@ gfortran hydrogen.f90 qmc_stats.f90 qmc_uniform.f90 -o qmc_uniform
./qmc_uniform
#+end_src
**** Solution :solution:
**** Solution :solution2:
*Python*
#+BEGIN_SRC python :results output :exports both
#!/usr/bin/env python3
@ -1561,7 +1561,7 @@ gfortran hydrogen.f90 qmc_stats.f90 qmc_metropolis.f90 -o qmc_metropolis
./qmc_metropolis
#+end_src
**** Solution :solution:
**** Solution :solution2:
*Python*
#+BEGIN_SRC python :results output :exports both
#!/usr/bin/env python3
@ -1867,7 +1867,7 @@ subroutine drift(a,r,b)
end subroutine drift
#+END_SRC
**** Solution :solution:
**** Solution :solution2:
*Python*
#+BEGIN_SRC python :tangle hydrogen.py
def drift(a,r):
@ -1975,7 +1975,7 @@ gfortran hydrogen.f90 qmc_stats.f90 vmc_metropolis.f90 -o vmc_metropolis
./vmc_metropolis
#+end_src
**** Solution :solution:
**** Solution :solution2:
*Python*
#+BEGIN_SRC python :results output :exports both
#!/usr/bin/env python3
@ -2768,109 +2768,7 @@ gfortran hydrogen.f90 qmc_stats.f90 pdmc.f90 -o pdmc
And compute the ground state energy.
* Exam :noexport:
** Question 1
Consider the hydrogen atom. You are using Monte Carlo sampling to
compute the energy associated with a wave function $\Psi(\mathbf{r})$.
If you use a Gaussian with mean 0 and variance 1 (centered on the
nucleus) to generate the random samples, the correct weight
$w(\mathbf{r})$ involved in the expectation value of the energy
$\frac{1}{M}\sum_{i=1}^M E_L(\mathbf{r_i}) \times w(\mathbf{r_i})$ is:
A - $w(\mathbf{r})= \left|\Psi(\mathbf{r})\right|^2$
B - $w(\mathbf{r})= \left( 2 \pi \right)^{3/2} \exp \left( \frac{\mathbf{r}^2}{2} \right) \left|\Psi(\mathbf{r})\right|^2$
C - $w(\mathbf{r})= \frac{1}{\left( 2 \pi \right)^{3/2}} \exp \left( -\frac{\mathbf{r}^2}{2} \right) \left|\Psi(\mathbf{r})\right|^2$
D - $w(\mathbf{r})= \frac{1}{\left( 2 \pi \right)^{3/2}} \exp \left( -\frac{\mathbf{r}^2}{2} \right)$
** Question 2
In the exercises, you only considered "bosonic" wave functions.
Let's assume that you now deal with a system where the wave
function has nodes ($\Psi(\mathbf{r})=0$).
i ) Does the local energy diverge at the nodes?
ii) The drift $\nabla \Psi / \Psi$ diverge at the nodes. Does it
push the electrons towards the nodes or away from the nodes?
A - i) $E_L$ diverges, ii) $\nabla \Psi / \Psi$ pushes in the direction of the nodes
B - i) $E_L$ diverges, ii) $\nabla \Psi / \Psi$ pushes away from the nodes
C - i) $E_L$ is finite, ii) $\nabla \Psi / \Psi$ pushes in the direction of the nodes
D - i) $E_L$ is finite, ii) $\nabla \Psi / \Psi$ pushes away from the nodes
*Hint*: You can also think in one dimension if this helps you.
** Question 3
Consider the helium atom in its singlet ground state with the wave function
\[
\Psi(\mathbf{r}_1, \mathbf{r}_2) = \exp \left( - ( \mathbf{r}_1 +
\mathbf{r}_2 ) \right)
\].
When an electron approaches
i ) the nucleus or,
ii) the other electron,
the local energy diverges to:
A - i) $+\infty$ and ii) $-\infty$
B - i) $+\infty$ and ii) $+\infty$
C - i) $-\infty$ and ii) $-\infty$
D - i) $-\infty$ and ii) $+\infty$
*Hint 1* : Recall the expression of the Laplacian for the single
electron case (hydrogen).
*Hint 2* : Helium has $Z=2$.
** Question 4
Consider a 2-electron system.
We propose a move of the 2-electron configuration according to a uniform
distribution $[-\delta L/2, \delta L/2]$ in all directions.
What is the expression of the transition probability for
the i) forward and ii) reverse move?
A - i ) Forward : $T(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1}) = \frac{1}{(\delta L)^3}$
ii) Reverse : $T(\mathbf{r}_{n+1} \rightarrow \mathbf{r}_{n}) = -\frac{1}{(\delta L)^3}$
B - i ) Forward : $T(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1}) = \frac{1}{(\delta L)^6}$
ii) Reverse : $T(\mathbf{r}_{n+1} \rightarrow \mathbf{r}_{n}) = (\delta L)^6$
C - i ) Forward : $T(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1}) = \frac{1}{(\delta L)^6}$
ii) Reverse : $T(\mathbf{r}_{n+1} \rightarrow \mathbf{r}_{n}) = \frac{1}{(\delta L)^6}$
C - i ) Forward : $T(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1}) = -\frac{1}{(\delta L)^3}$
ii) Reverse : $T(\mathbf{r}_{n+1} \rightarrow \mathbf{r}_{n}) = \frac{1}{(\delta L)^3}$
** Question 5
If you run a single DMC calculation on the Li$^+$ ion in the singlet
ground state, which approximations impact the final energy:
- None
- The fixed-node approximation
- The time-step approximation
- The fixed-node approximation and the time-step approximation
* Schedule :noexport:
|------------------------------+---------|