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\section*{Part a, Variational Monte Carlo studies of atoms}

The final aim of this project is to develop a diffusion Monte Carlo program which can be used to obtain ground state properties of atoms like He, Be, O, Ne, Si etc. If possible (time allowing) the hope is to be to be able to perform calculations
for important molecules 

The aim of the first part (part a) 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. The variational Monte Carlo part will include the basic ingredients for performing a diffusion Monte Carlo calculation. Although we confine ourselves to atoms and molecules,  
you should however make your code flexible enough to run for two-dimensional systems like electrons confined in so-called quantum dots or other fermionic systems in one, two and three dimensions.

We expect to finalize this part on March 22. Only a short report is required. We will start with the diffusion Monte Carlo part in the beginning of April. 
\section*{Exercise 1: Variational Monte Carlo calculations of the helium atom}

We will start with the simplest possible system beyond hydrogen, namely 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.

Your Monte Carlo moves are determined by
\begin{equation}
   {\bf R}' = {\bf R} +\delta \times r,
\end{equation}
where $r$ is a random number from the uniform distribution and $\delta$
a chosen step length.
In solving this exercise you need to devise an algorithm which finds
an optimal value of $\delta$ for the variational parameters $\alpha$ and $\beta$,
resulting in roughly $50\%$ accepted moves. 

Give a physical  interpretation of the best value of $\alpha$.
Make a plot of the variance as a function of the number of Monte Carlo
cycles.
\item[1b)]
Find closed form expressions for the local energy (see below) for the above 
trial wave function and explain shortly how this 
trial function satisfies 
the cusp condition when $r_1\rightarrow 0$ or
$r_2\rightarrow 0$ or  $r_{12}\rightarrow 0$. Show that
closed-form expression for the trial wave function is
\[ 
E_{L2} = E_{L1}+\frac{1}{2(1+\beta r_{12})^2}\left\{\frac{\alpha(r_1+r_2)}{r_{12}}(1-\frac{\mathbf{r}_1\mathbf{r}_2}{r_1r_2})-\frac{1}{2(1+\beta r_{12})^2}-\frac{2}{r_{12}}+\frac{2\beta}{1+\beta r_{12}}\right\},
\]
where
\[ 
E_{L1} = \left(\alpha-Z\right)\left(\frac{1}{r_1}+\frac{1}{r_2}\right)+\frac{1}{r_{12}}-\alpha^2.
\]

Compare the results of with and without the closed-form expressions (in terms of CPU time).
\item[1c)] 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[1d)]  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[1e)]  Repeat step 1c) 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 $r_i = \sqrt{r_{i_x}^2+r_{i_y}^2+r_{i_z}^2}$.


You can approximate the Slater determinant for the ground state of the Beryllium atom
by writing it out as
\begin{equation}
   \psi_{T}({\bf r_1},{\bf r_2}, {\bf r_3}, {\bf r_4}) \propto 
\left(\phi_{1s}({\bf r_1})\phi_{2s}({\bf r_2})-\phi_{1s}({\bf r_2})\phi_{2s}({\bf r_1})\right)
\left(\phi_{1s}({\bf r_3})\phi_{2s}({\bf r_4})-\phi_{1s}({\bf r_4})\phi_{2s}({\bf r_3})\right).
\end{equation}
Here you can see a simple code example which implements the above expression
\begin{lstlisting}
 for (i = 0; i < number_particles; i++) {
    argument[i] = 0.0;
    r_single_particle = 0;
    for (j = 0; j < dimension; j++) {
      r_single_particle  += r[i][j]*r[i][j];
    }
    argument[i] = sqrt(r_single_particle);
  }
// Slater determinant, no factors as they vanish in Metropolis ratio
wf  = (psi1s(argument[0])*psi2s(argument[1])
       -psi1s(argument[1])*psi2s(argument[0]))*
      (psi1s(argument[2])*psi2s(argument[3])
       -psi1s(argument[3])*psi2s(argument[2]));
\end{lstlisting}
For beryllium we can easily implement the explicit evaluation of the Slater determinant.  The above will serve as a useful check
for your function which computes the Slater determinat. 
The derivatives of the single-particle wave functions can be computed analytically and you should consider
using the closed form expression for the local energy (not mandatory, you can use numerical derivatives as well although a closed form expressions speeds up your code).

For the correlation part 
\[
\Psi_C=\prod_{i< j}g(r_{ij})= \exp{\left\{\sum_{i<j}\frac{ar_{ij}}{1+\beta r_{ij}}\right\}},
\]
we need to take into account whether electrons have equal or opposite spins since we have to obey the
electron-electron cusp condition as well.  For Beryllium, as an example,  you can fix electrons 1 and 2 to have spin up while
electrons 3 and 4 have spin down.
When the electrons have  equal spins 
\[
a= 1/4,
\]
while for opposite spins (as for the ground state of  helium)
\[
a= 1/2.
\] 

\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*{Part b, Variational Monte Carlo studies of molecules}

The 
H$_2$ molecule consists of two protons and two electrons 
with a ground state energy $E=-1.17460$ a.u. and equilibrium distance between the two hydrogen atoms
of $r_0=1.40$ Bohr radii.
We define our systems using the following variables.
Origo is chosen to be halfway between the two protons. The distance from 
proton 1 is defined as 
$-{\bf R}/2$ whereas proton 2 has a distance ${\bf R}/2$.
Calculations are performed for fixed distances ${\bf R}$ between the two protons.

Electron 1 has a distance $r_1$ from the chose origo, while  electron $2$
has a distance $r_2$. 
The kinetic energy operator becomes then
\be
   -\frac{\nabla_1^2}{2}-\frac{\nabla_2^2}{2}.
\ee
The distance between the two electrons is
$r_{12}=|{\bf r}_1-{\bf r}_2|$. 
The repulsion between the two electrons results in a potential energy term given by
\be
               +\frac{1}{r_{12}}.
\ee
In a similar way we obtain a repulsive contribution from the interaction between the two 
protons given by
\be
               +\frac{1}{|{\bf R}|},
\ee
where ${\bf R}$ is the distance between the two protons.
To obtain the final potential energy we need to include the attraction the electrons feel from the protons.
To model this, we need to define the distance between the electrons and the two protons.
If we model this along a 
chosen $z$-akse with electron 1 placed at a distance 
${\bf r}_1$ from a chose origo, one proton at $-{\bf R}/2$
and the other at  ${\bf R}/2$, 
the distance from proton 1 to electron 1 becomes
\be
{\bf r}_{1p1}={\bf r}_1+ {\bf R}/2,
\ee
and
\be
{\bf r}_{1p2}={\bf r}_1- {\bf R}/2,
\ee
from proton 2.
Similarly, for electron 2 we obtain
\be
{\bf r}_{2p1}={\bf r}_2+{\bf R}/2,
\ee
and
\be
{\bf r}_{2p2}={\bf r}_2-{\bf R}/2.
\ee
These four distances define the attractive contributions to the potential energy
\be
   -\frac{1}{r_{1p1}}-\frac{1}{r_{1p2}}-\frac{1}{r_{2p1}}-\frac{1}{r_{2p2}}.
\ee
We can then write the total Hamiltonian as 
\be
   \OP{H}=-\frac{\nabla_1^2}{2}-\frac{\nabla_2^2}{2}
   -\frac{1}{r_{1p1}}-\frac{1}{r_{1p2}}-\frac{1}{r_{2p1}}-\frac{1}{r_{2p2}}
               +\frac{1}{r_{12}}+\frac{1}{|{\bf R}|},
\ee
and if we choose ${\bf R}=0$ we obtain the helium atom.

In this part we will use a trial wave function of the form
\be
   \psi_{T}({\bf r_1},{\bf r_2}, {\bf R}) =
   \psi({\bf r}_1,{\bf R})\psi({\bf r}_2,{\bf R})
   \exp{\left(\frac{r_{12}}{2(1+\beta r_{12})}\right)},
\ee
with the following trial wave function 
\be
   \psi({\bf r}_1,{\bf R})=\left(\exp{(-\alpha r_{1p1})}
      +\exp{(-\alpha r_{1p2})}\right),
\ee
for electron 1 and
\be
   \psi({\bf r}_2,{\bf R})=\left(\exp{(-\alpha r_{2p1})}
      +\exp{(-\alpha r_{2p2})}\right).
\ee
The variational parameters are $\alpha$ and $\beta$.

One can show that in the limit where all distances approach zero that 
\be
    \alpha = 1+\exp{(-R/\alpha)},
    \label{eq:alpha}
\ee
resulting in $\beta$ as the only variational parameter.
\begin{enumerate}
\item Set up and algorithm and write a program which computes the 
expectation value of 
$\langle \OP{H} \rangle$
using the best variational Monte Carlo method 
from part a.
For each inter-proton distance  $R$ you must find the parameter 
$\beta$ which minimizes the energy. Plot the corresponding energy as function
of the distance $R$ between the protons.
\item Use thereafter the optimal parameter sets to compute the 
average distance
$\langle r_{12} \rangle$ between the electrons where the energy as function of
$R$ exhibits its minimum. Comment your results.
\item 
We modify now the approximation for the wave functions of electrons 1 and 2
by subtracting the two terms instead of adding up, viz
\be
   \psi({\bf r}_1,{\bf R})=\left(\exp{(-\alpha r_{1p1})}
      -\exp{(-\alpha r_{1p2})}\right),
\ee
for electron 1 
\be
   \psi({\bf r}_2,{\bf R})=\left(\exp{(-\alpha r_{2p1})}
      -\exp{(-\alpha r_{2p2})}\right),
\ee
for electron 2. Mathematically, this approach is equally viable as the previous one.
Repeat your calculations and see if you can obtain an energy minimum as 
function of  $R$. Comment your results.

\item Using your Slater determinants and Jastrow factors from part a, our final step consists in estimating the binding energy of the Be$_2$ molecule.
The last two references in the reference list may be useful.




\end{enumerate}

\section*{Brief summary on how ot write a report}

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++, Fortran2008 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*{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. 

\item Moskowitz and Kalos, Int.~Journal of Quantum Chemistry {\bf XX}, 1107 (1981).
Results for He and H$_2$.
\item Filippi, Singh and Umrigar, J.~Chemical Physics {\bf 105}, 123 (1996).   Useful results on
Be$_2$ to which you can benchmark your results against.


\end{enumerate}


\end{document}