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\section*{Introduction to numerical projects}

Here follows a brief recipe and recommendation on how to write a report for each
project.
\begin{itemize}
\item Give a short description of the nature of the problem and the eventual 
numerical methods you have used.
\item Describe the algorithm you have used and/or developed. Here you may find it convenient
to use pseudocoding. In many cases you can describe the algorithm
in the program itself.

\item Include the source code of your program. Comment your program properly.
\item If possible, try to find analytic solutions, or known limits
in order to test your program when developing the code.
\item Include your results either in figure form or in a table. Remember to
       label your results. All tables and figures should have relevant captions
       and labels on the axes.
\item Try to evaluate the reliabilty and numerical stability/precision
of your results. If possible, include a qualitative and/or quantitative
discussion of the numerical stability, eventual loss of precision etc. 

\item Try to give an interpretation of you results in your answers to 
the problems.
\item Critique: if possible include your comments and reflections about the 
exercise, whether you felt you learnt something, ideas for improvements and 
other thoughts you've made when solving the exercise.
We wish to keep this course at the interactive level and your comments can help
us improve it. We do appreciate your comments. 
\item Try to establish a practice where you log your work at the 
computerlab. You may find such a logbook very handy at later stages
in your work, especially when you don't properly remember 
what a previous test version 
of your program did. Here you could also record 
the time spent on solving the exercise, various algorithms you may have tested
or other topics which you feel worthy of mentioning.
\end{itemize}



\section*{Format for electronic delivery of report and programs}
%
The preferred format for the report is a PDF file. You can also
use DOC or postscript formats. 
As programming language we prefer that you choose between C++, Fortran95 or Python.
Finally, 
we recommend that you work together. Optimal working groups consist of 
2-3 students, but more people can collaborate. You can then hand in a common report. 



\section*{Project 1, Variational Monte Carlo studies of light atoms, deadline March 22}

The aim of this project is to use the Variational Monte
Carlo (VMC) method and evaluate 
the ground state energy of  the
helium, beryllium and neon atoms. You should however make your code flexible enough to run for two-dimensional systems like electrons confined in so-called quantum dots.


\section*{Exercise 1: Variational Monte Carlo calculations of the Helium atom}

We label $r_1$ the distance from electron 1 to the nucleus and similarly 
$r_2$ the distance between electron 2 and the nucleus.
The contribution to the potential energy from the interactions between the 
electrons and the nucleus is
\be
   -\frac{2}{r_1}-\frac{2}{r_2},
\ee 
and if we add the electron-electron repulsion with
$r_{12}=|{\bf r}_1-{\bf r}_2|$, the total potential energy 
$V(r_1, r_2)$ is
\be
 V(r_1, r_2)=-\frac{2}{r_1}-\frac{2}{r_2}+
               \frac{1}{r_{12}},
\ee
yielding the total Hamiltonian
\be
   \OP{H}=-\frac{\nabla_1^2}{2}-\frac{\nabla_2^2}{2}
          -\frac{2}{r_1}-\frac{2}{r_2}+
               \frac{1}{r_{12}},
\ee
and Schr\"odinger's equation reads
\be
   \OP{H}\psi=E\psi.
\ee
All equations are in so-called atomic units. The distances
$r_i$ and $r_{12}$ are dimensionless. To have energies in electronvolt
you need to multiply all results with 
$2\times E_0$,
where $E_0=13.6$ eV.
The experimental binding energy for helium in atomic units a.u. is $E_{\mathrm{He}}=-2.9037$ a.u..


\begin{enumerate}

\item[1a)] We want to perform  a Variational Monte Carlo calculation of the ground state of the helium atom.
In our first attempt we will use a brute force Metropolis sampling with a trial wave function which has the following form
\begin{equation}
   \psi_{T}({\bf r_1},{\bf r_2}, {\bf r_{12}}) = 
   \exp{\left(-\alpha(r_1+r_2)\right)}
   \exp{\left(\frac{r_{12}}{2(1+\beta r_{12})}\right)}, 
\label{eq:trial}
\end{equation}
with $\alpha$ and $\beta$ as variational parameters.

Your task is to perform a Variational Monte Carlo calculation
using the Metropolis algorithm to compute the integral
\begin{equation}
   \langle E \rangle =
   \frac{\int d{\bf r_1}d{\bf r_2}\psi^{\ast}_T({\bf r_1},{\bf r_2}, {\bf r_{12}})\OP{H}({\bf r_1},{\bf r_2}, {\bf r_{12}})\psi_T({\bf r_1},{\bf r_2}, {\bf r_{12}})}
        {\int d{\bf r_1}d{\bf r_2}\psi^{\ast}_T({\bf r_1},{\bf r_2}, {\bf r_{12}})\psi_T({\bf r_1},{\bf r_2}, {\bf r_{12}})}.
\end{equation}
You should parallelize your program.   Find the  energy minimum and compute also the mean distance
$r_{12}$ between the two electrons for the optimal set of the variational parameters.
A code for doing a VMC calculation for the helium atom can be 
found on the webpage of the course, see under programs.

You should also find a closed-form expression for the local energy. Compare the results of this calculation (in terms of CPU time) compared with a calculation which performs a brute numerical derivation.
\item[1b)] Introduce now importance sampling and study the dependence of the results as a function of the time step $\delta t$.  
Compare the results with those obtained under 1a) and comment eventual differences.
In performing the Monte Carlo analysis you should use blocking as a technique  to make the statistical analysis of the numerical data.
The code has to run in parallel. 
\item[1c)]  With the optimal parameters for the ground state wave function, compute the onebody density and
the charge density. Discuss your results and compare the results with those obtained with a pure hydrogenic wave functions. Run a Monte Carlo calculations without the Jastrow factor as well
and compute the same quantities. How important are the correlations induced by the Jastrow factor?

\item[1d)]  Repeat step 1b) by varying the energy using the 
conjugate gradient method to obtain the best possible set of parameters
$\alpha$ and $\beta$.

\end{enumerate}


\section*{Exercise 2: Variational Monte Carlo calculations of the Beryllium and  Neon atoms}
The previous exercise has prepared you for extending your calculational machinery  to other systems.
Here we will focus on the neon and beryllium atoms.
It is convenient to make modules or classes of trial wave functions, both many-body wave functions
and single-particle wave functions  and the quantum numbers  involved, such as spin, orbital momentum and principal
quantum numbers.

The new item you need to pay attention to is the calculation of the Slater Determinant. This is an additional complication
to your VMC calculations.
If we stick to hydrogen-like wave functions,
the trial wave function for Beryllium can be written as 
\begin{equation}
   \psi_{T}({\bf r_1},{\bf r_2}, {\bf r_3}, {\bf r_4}) = 
   Det\left(\phi_{1}({\bf r_1}),\phi_{2}({\bf r_2}),
   \phi_{3}({\bf r_3}),\phi_{4}({\bf r_4})\right)
   \prod_{i<j}^{4}\exp{\left(\frac{r_{ij}}{2(1+\beta r_{ij})}\right)}, 
\end{equation}
where $Det$ is a Slater determinant and the single-particle wave functions
are the hydrogen wave functions for the $1s$ and $2s$ orbitals. Their form
within the variational ansatz are given by
\begin{equation}
\phi_{1s}({\bf r_i}) = e^{-\alpha r_i},
\end{equation}
and 
\begin{equation}
\phi_{2s}({\bf r_i}) = \left(1-\alpha r_i/2\right)e^{-\alpha r_i/2}.
\end{equation}
For neon, the trial wave function can take the form
\begin{equation}
   \psi_{T}({\bf r_1},{\bf r_2}, \dots,{\bf r_{10}}) = 
   Det\left(\phi_{1}({\bf r_1}),\phi_{2}({\bf r_2}),
   \dots,\phi_{10}({\bf r_{10}})\right)
   \prod_{i<j}^{10}\exp{\left(\frac{r_{ij}}{2(1+\beta r_{ij})}\right)}, 
\end{equation}
In this case you need to include the $2p$ wave function as well.
It is given as
\begin{equation} 
\phi_{2p}({\bf r_i}) = \alpha {\bf r_i}e^{-\alpha r_i/2}.
\end{equation}
Observe that $r_i = \sqrt{r_{i_x}^2+r_{i_y}^2+r_{i_z}^2}$.


\begin{enumerate}
\item[(2a)]   Write a function which sets up the Slater determinant for beryllium and neon and can be generalized to
handle larger systems as well. 
Compute the ground state energies of neon and beryllium as you did for the helium atom
in 1d). 
The calculations should include  parallelization, blocking, importance sampling and energy minimization using the conjugate gradient approach.  
\item[2b)]  With the optimal parameters for the ground state wave function, compute again the onebody density and 
the charge density. Discuss your results and compare the results with those obtained with a pure hydrogenic wave functions. Run a Monte Carlo calculations without the Jastrow factor as well
and compute the same quantities. How important are the correlations induced by the Jastrow factor?


\end{enumerate}





















\section*{Literature}
\begin{enumerate}
\item B.~L.~Hammond, W.~A.~Lester and P.~J.~Reynolds, Monte Carlo methods
in Ab Inition Quantum Chemistry, World Scientific, Singapore, 1994, chapters
2-5 and appendix B.

\item B.H.~Bransden and C.J.~Joachain, Physics of Atoms and molecules,
Longman, 1986. Chapters 6, 7 and 9.
\item S.A.~Alexander and R.L.~Coldwell,
Int.~Journal of Quantum Chemistry, {\bf 63} (1997) 1001.  This article is available 
at the webpage of the course as the file jastrow.pdf under the project 1 link.
\item C.J.~Umrigar, K.G.~Wilson and J.W.~Wilkins, Phys.~Rev.~Lett.~{\bf 60}
(1988) 1719. 



\end{enumerate}


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