For this project you can collaborate with fellow students and you can hand in a common report. This project (together with projects 3 and 4) counts 1/3 of the final mark.
The aim of this project is to use the Variational Monte Carlo (VMC) method to evaluate the ground state energy, the relative distance between two electrons and expectation values of the kinetic and potential energies of a quantum dots with \( N=2 \) electrons in three dimensions. It can be seen as a continuation of project 2, but instead of using eigenvalue solvers we will use a stochastic approach.
The systems we will focus on is thus a three-dimensional one, with two electrons confined to move in harmonic oscillator like traps. As in project 2, we have analytical results for the energy for specific frequencies, see Taut's article in the reference list below. Furthermore, for other frequencies, we have our results from project 2 which can be used to test our Monte Carlo results.
These systems are called quantum dots and constitute a lively research area in condensed matter physics and materials science, with applications spanning from the contruction of quantum circuits to applications to solar cells and nano-medicine. Although most studies are done for electrons in two dimensions, we will limit ourselves to the three-dimensional case since it allows us to compare our variational Monte Carlo (VMC) results with what we did in project 2.
Our aim here is to use the variational Monte Carlo method to study such systems, with an emphasis on understanding correlations due to the repulsive interaction between electrons. The advantage of the VMC approach is that we can carry out our calculations using cartesian coordinates.
The relevant background material can be found in chapter 14 of the lecture notes.
We consider a system of electrons confined in a pure three-dimensional isotropic harmonic oscillator potential, with an idealized total Hamiltonian given by $$ \begin{equation} \label{eq:finalH} \hat{H}=\sum_{i=1}^{N} \left( -\frac{1}{2} \nabla_i^2 + \frac{1}{2} \omega^2r_i^2 \right)+\sum_{i < j}\frac{1}{r_{ij}}, \end{equation} $$ where natural units (\( \hbar=c=e=m_e=1 \)) are used and all energies are in so-called atomic units a.u. It means that all distances \( r_i \) and \( r_{ij} \) are dimensionless.
We will study various trial wave functions for two electrons (\( N=2 \)) as functions of the oscillator frequency \( \omega \) using the above Hamiltonian. The Hamiltonian includes a standard harmonic oscillator part $$ \begin{equation*} \hat{H}_0=\sum_{i=1}^{N} \left( -\frac{1}{2} \nabla_i^2 + \frac{1}{2} \omega^2r_i^2 \right), \end{equation*} $$ and the repulsive interaction between two electrons given by $$ \begin{equation*} \hat{H}_1=\sum_{i < j}\frac{1}{r_{ij}}, \end{equation*} $$ with the distance between electrons given by \( r_{ij}=\sqrt{\mathbf{r}_1-\mathbf{r}_2} \). We define the modulus of the positions of the electrons (for a given electron \( i \)) as \( r_i = \sqrt{x_i^2+y_i^2+z_i^2} \).
For this project you can collaborate with fellow students and you can hand in a common report. This project (together with projects 3 and 4) counts 1/3 of the final mark.
The unperturbed wave function for the ground state of the two-electron system is given by $$ \begin{equation*} \Phi(\mathbf{r}_1,\mathbf{r}_2) = C\exp{\left(-\omega(r_1^2+r_2^2)/2\right)}, \end{equation*} $$ with \( C \) being a normalization constant and \( r_i = \sqrt{x_i^2+y_i^2+z_i^2} \). Note that the vector \( \mathbf{r}_i \) refers to the \( x \), \( y \) and \( z \) coordinates for a given particle. What is the total spin of this wave function? Find arguments for why the ground state should have this specific total spin.
Find closed form expressions for the local energy (see below) for the two trial wave functions presented here and explain shortly if these trial functions satisfy the so-called cusp condition when \( r_{12}\rightarrow 0 \). The first wave function $$ \Psi_{T1}(\mathbf{r}_1,\mathbf{r}_2) = C\exp{\left(-\alpha\omega(r_1^2+r_2^2)/2\right)}, $$ while the second trial function is $$ \Psi_{T2}(\mathbf{r}_1,\mathbf{r}_2) = C\exp{\left(-\alpha\omega(r_1^2+r_2^2)/2\right)} \exp{\left(\frac{r_{12}}{2(1+\beta r_{12})}\right)}, $$ where \( \alpha \) and \( \beta \) are variational parameters. Show that the first trial function gives an analytical expression for the local energy given by gives a closed-form expression $$ E_{L1} = \frac{1}{2}\omega^2\left( r_1^2+r_2^2\right)\left(1-\alpha^2\right) +3\alpha\omega. $$ Use this expression when developing your program. For the first trial function it will give you the exact analytical result when you exclude the Coulomb interaction. The exact result for the ground state is then \( 3\omega \). Adding the repulsive Coulomb interaction gives us then $$ E_{L1} = \frac{1}{2}\omega^2\left( r_1^2+r_2^2\right)\left(1-\alpha^2\right) +3\alpha\omega+\frac{1}{r_{12}}. $$ When you study the final energy for the first trial function, this is the result you should compare the second trial function with. The analytical expression for the second trial wave function (with \( E_{L1} \) now including the Coulomb repulsion) $$ E_{L2} = E_{L1}+\frac{1}{2(1+\beta r_{12})^2}\left\{\alpha\omega r_{12}-\frac{1}{2(1+\beta r_{12})^2}-\frac{2}{r_{12}}+\frac{2\beta}{1+\beta r_{12}}\right\} $$ The exact ground state energy for \( \omega =1 \) is \( 3.558 \) a.u. Check this result against your results from project 2 and remember to add the contribution from the center-of-mass energy, which is 1.5 a.u.
You will need to program both equations when computing the expectation value of the energy and the energy variance.
Plot the energy as a function of the variational parameter \( \alpha \) and discuss your results. Plot the variance as well and find the energy and variance minima as functions of \( \alpha \). Discuss your results.
Compute also the expectation value of the mean distance at the energy minimun \( r_{12}=\sqrt{\mathbf{r}_1-\mathbf{r}_2} \) between the two electrons for the optimal set of the variational parameters using \( \omega=0.01 \), \( \omega=0.5 \) and \( \omega=1.0 \). Comment your results. You will find it convenient to object orient your code.
Use thereafter the optimal value for \( \alpha \) as a starting point for computing the ground state energy of the two-electron quantum dot using the trial wave functions \( \psi_{T2}({\bf r_1},{\bf r_2}, {\bf r_{12}}) \). Use the analytical expression for the local energy. In this case you need to vary both \( \alpha \) and \( \beta \). The strategy here is to use \( \alpha \) from the previous exercise, [5c)] and then vary \( \beta \) in order to find the lowest energy as function of \( \beta \). Thereafter you change \( \alpha \) in order to see whether you find an even lower energy and so forth.
With the optimal parameters for the ground state wave function, compute the expectation values of the energy and the variance using \( \omega=0.01 \), \( \omega=0.5 \) and \( \omega=1.0 \). How important are the correlations introduced by the Jastrow factor? Comment your results. Compare the best VMC results wth those from project 2 for the same frequencies. Compute also the expectation value of the mean distance at the energy minimun \( r_{12}=\sqrt{\mathbf{r}_1-\mathbf{r}_2} \) between the two electrons for the optimal set of the variational parameters using \( \omega=0.01 \), \( \omega=0.5 \) and \( \omega=1.0 \). Comment your results and compare the results with and without the Jastrow factor.
The Hermite polynomials are the solutions of the following differential equation $$ \begin{equation} \frac{d^2H(x)}{dx^2}-2x\frac{dH(x)}{dx}+ (\lambda-1)H(x)=0. \label{eq:hermite} \end{equation} $$ The first few polynomials are $$ \begin{equation*} H_0(x)=1, \end{equation*} $$ $$ \begin{equation*} H_1(x)=2x, \end{equation*} $$ $$ \begin{equation*} H_2(x)=4x^2-2, \end{equation*} $$ $$ \begin{equation*} H_3(x)=8x^3-12x, \end{equation*} $$ and $$ \begin{equation*} H_4(x)=16x^4-48x^2+12. \end{equation*} $$ They fulfil the orthogonality relation $$ \begin{equation*} \int_{-\infty}^{\infty}e^{-x^2}H_n(x)^2dx=2^nn!\sqrt{\pi}, \end{equation*} $$ and the recursion relation $$ \begin{equation*} H_{n+1}(x)=2xH_{n}(x)-2nH_{n-1}(x). \end{equation*} $$
Here follows a brief recipe and recommendation on how to write a report for each project.
The preferred format for the report is a PDF file. You can also use DOC or postscript formats or as an ipython notebook file. As programming language we prefer that you choose between C/C++, Fortran2008 or Python. The following prescription should be followed when preparing the report: