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Solutions
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QMC.org
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QMC.org
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@ -892,7 +892,7 @@ gfortran hydrogen.f90 variance_hydrogen.f90 -o variance_hydrogen
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./variance_hydrogen
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./variance_hydrogen
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#+end_src
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#+end_src
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**** Solution :solution:
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**** Solution :solution2:
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*Python*
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*Python*
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#+BEGIN_SRC python :results none :exports both
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#+BEGIN_SRC python :results none :exports both
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#!/usr/bin/env python3
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#!/usr/bin/env python3
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@ -1089,7 +1089,7 @@ subroutine ave_error(x,n,ave,err)
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end subroutine ave_error
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end subroutine ave_error
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#+END_SRC
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#+END_SRC
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**** Solution :solution:
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**** Solution :solution2:
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*Python*
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*Python*
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#+BEGIN_SRC python :results none :exports code
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#+BEGIN_SRC python :results none :exports code
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#!/usr/bin/env python3
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#!/usr/bin/env python3
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@ -1265,7 +1265,7 @@ gfortran hydrogen.f90 qmc_stats.f90 qmc_uniform.f90 -o qmc_uniform
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./qmc_uniform
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./qmc_uniform
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#+end_src
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#+end_src
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**** Solution :solution:
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**** Solution :solution2:
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*Python*
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*Python*
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#+BEGIN_SRC python :results output :exports both
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#+BEGIN_SRC python :results output :exports both
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#!/usr/bin/env python3
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#!/usr/bin/env python3
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@ -1561,7 +1561,7 @@ gfortran hydrogen.f90 qmc_stats.f90 qmc_metropolis.f90 -o qmc_metropolis
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./qmc_metropolis
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./qmc_metropolis
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#+end_src
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#+end_src
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**** Solution :solution:
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**** Solution :solution2:
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*Python*
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*Python*
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#+BEGIN_SRC python :results output :exports both
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#+BEGIN_SRC python :results output :exports both
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#!/usr/bin/env python3
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#!/usr/bin/env python3
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@ -1867,7 +1867,7 @@ subroutine drift(a,r,b)
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end subroutine drift
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end subroutine drift
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#+END_SRC
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#+END_SRC
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**** Solution :solution:
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**** Solution :solution2:
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*Python*
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*Python*
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#+BEGIN_SRC python :tangle hydrogen.py
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#+BEGIN_SRC python :tangle hydrogen.py
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def drift(a,r):
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def drift(a,r):
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@ -1975,7 +1975,7 @@ gfortran hydrogen.f90 qmc_stats.f90 vmc_metropolis.f90 -o vmc_metropolis
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./vmc_metropolis
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./vmc_metropolis
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#+end_src
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#+end_src
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**** Solution :solution:
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**** Solution :solution2:
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*Python*
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*Python*
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#+BEGIN_SRC python :results output :exports both
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#+BEGIN_SRC python :results output :exports both
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#!/usr/bin/env python3
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#!/usr/bin/env python3
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@ -2768,108 +2768,6 @@ gfortran hydrogen.f90 qmc_stats.f90 pdmc.f90 -o pdmc
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And compute the ground state energy.
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And compute the ground state energy.
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* Exam :noexport:
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** Question 1
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Consider the hydrogen atom. You are using Monte Carlo sampling to
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compute the energy associated with a wave function $\Psi(\mathbf{r})$.
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If you use a Gaussian with mean 0 and variance 1 (centered on the
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nucleus) to generate the random samples, the correct weight
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$w(\mathbf{r})$ involved in the expectation value of the energy
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$\frac{1}{M}\sum_{i=1}^M E_L(\mathbf{r_i}) \times w(\mathbf{r_i})$ is:
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A - $w(\mathbf{r})= \left|\Psi(\mathbf{r})\right|^2$
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B - $w(\mathbf{r})= \left( 2 \pi \right)^{3/2} \exp \left( \frac{\mathbf{r}^2}{2} \right) \left|\Psi(\mathbf{r})\right|^2$
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C - $w(\mathbf{r})= \frac{1}{\left( 2 \pi \right)^{3/2}} \exp \left( -\frac{\mathbf{r}^2}{2} \right) \left|\Psi(\mathbf{r})\right|^2$
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D - $w(\mathbf{r})= \frac{1}{\left( 2 \pi \right)^{3/2}} \exp \left( -\frac{\mathbf{r}^2}{2} \right)$
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** Question 2
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In the exercises, you only considered "bosonic" wave functions.
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Let's assume that you now deal with a system where the wave
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function has nodes ($\Psi(\mathbf{r})=0$).
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i ) Does the local energy diverge at the nodes?
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ii) The drift $\nabla \Psi / \Psi$ diverge at the nodes. Does it
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push the electrons towards the nodes or away from the nodes?
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A - i) $E_L$ diverges, ii) $\nabla \Psi / \Psi$ pushes in the direction of the nodes
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B - i) $E_L$ diverges, ii) $\nabla \Psi / \Psi$ pushes away from the nodes
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C - i) $E_L$ is finite, ii) $\nabla \Psi / \Psi$ pushes in the direction of the nodes
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D - i) $E_L$ is finite, ii) $\nabla \Psi / \Psi$ pushes away from the nodes
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*Hint*: You can also think in one dimension if this helps you.
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** Question 3
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Consider the helium atom in its singlet ground state with the wave function
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\[
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\Psi(\mathbf{r}_1, \mathbf{r}_2) = \exp \left( - ( \mathbf{r}_1 +
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\mathbf{r}_2 ) \right)
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\].
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When an electron approaches
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i ) the nucleus or,
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ii) the other electron,
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the local energy diverges to:
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A - i) $+\infty$ and ii) $-\infty$
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B - i) $+\infty$ and ii) $+\infty$
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C - i) $-\infty$ and ii) $-\infty$
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D - i) $-\infty$ and ii) $+\infty$
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*Hint 1* : Recall the expression of the Laplacian for the single
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electron case (hydrogen).
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*Hint 2* : Helium has $Z=2$.
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** Question 4
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Consider a 2-electron system.
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We propose a move of the 2-electron configuration according to a uniform
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distribution $[-\delta L/2, \delta L/2]$ in all directions.
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What is the expression of the transition probability for
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the i) forward and ii) reverse move?
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A - i ) Forward : $T(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1}) = \frac{1}{(\delta L)^3}$
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ii) Reverse : $T(\mathbf{r}_{n+1} \rightarrow \mathbf{r}_{n}) = -\frac{1}{(\delta L)^3}$
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B - i ) Forward : $T(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1}) = \frac{1}{(\delta L)^6}$
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ii) Reverse : $T(\mathbf{r}_{n+1} \rightarrow \mathbf{r}_{n}) = (\delta L)^6$
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C - i ) Forward : $T(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1}) = \frac{1}{(\delta L)^6}$
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ii) Reverse : $T(\mathbf{r}_{n+1} \rightarrow \mathbf{r}_{n}) = \frac{1}{(\delta L)^6}$
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C - i ) Forward : $T(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1}) = -\frac{1}{(\delta L)^3}$
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ii) Reverse : $T(\mathbf{r}_{n+1} \rightarrow \mathbf{r}_{n}) = \frac{1}{(\delta L)^3}$
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** Question 5
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If you run a single DMC calculation on the Li$^+$ ion in the singlet
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ground state, which approximations impact the final energy:
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- None
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- The fixed-node approximation
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- The time-step approximation
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- The fixed-node approximation and the time-step approximation
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* Schedule :noexport:
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* Schedule :noexport:
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