Solutions

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: