mirror of
https://github.com/TREX-CoE/qmc-lttc.git
synced 2024-12-21 11:53:58 +01:00
Solutions
This commit is contained in:
parent
a258c1c564
commit
efb48d936f
114
QMC.org
114
QMC.org
@ -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
|
||||
@ -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
|
||||
@ -2768,108 +2768,6 @@ 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:
|
||||
|
||||
|
||||
Loading…
Reference in New Issue
Block a user