Variational Monte Carlo

Computational physics simulation using the Variational Monte Carlo method to calculate the ground state energy for a system of $^4\text{He}$ at temperature $T = 0$ trapped in an external harmonic potential.

Theory

The hamiltonian used is

$$H = -\frac{\hbar^2}{2m} \sum_{i = 1}^N \nabla_i^2 + \frac{1}{2} m\omega^2 \sum_{i = 1}^N r_i^2 + \sum_{i < j} V(r_{ij})\ ,$$

where $V(r)$ is the classical Lennard-Jones potential:

$$V(r) = 4\varepsilon\left[\left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^{6}\right]\ .$$

The variational wave function used is

$$\Psi(r_1,\dots,r_N)=\exp{\left(-\frac{1}{2\alpha}\sum_{i = 1}^Nr_i^2-\frac{1}{2}\sum_{i < j}u_\beta(r_{ij})\right)}\ ,$$

with $u_\beta(r) = (\beta_1/r)^{\beta_2}$.

Compilation and execution

Compile in the repository directory using:

    make

An executable named run will be generated. Use

    ./run N n_steps alpha_start alpha_end alpha_step [alpha_saved]

to execute it, where N is the number of particles, n_steps is the number of Metropolis steps. alpha_saved is optional but it's needed to look at the observables values at a certain value of $\alpha$. For example, executing

    ./run 2 100000 25.0

will save the observable values of the simulation for $\alpha = 25\ \text{Å}^2$. Then, to plot the results one can use

    python3 plots/energy_plot.py N n_steps INT show

where INT and show are needed for plotting the interacting simulation results and to show the graph. If the simulation is run with multiple values of the parameter $\alpha$, one can look at the energy as a function of that, by running plots/variational_plot.py. At the moment, this can be achieved by modifying alpha_start and alpha_end in main.c.

Important alert!

At the moment, the simulation does not behave well, the energy in the MC simulation has huge peaks and the variational surface has got no minimum value. Please contact me if needed.