CritPt / Challenge_41_main / In a real-space quantum Monte Carlo simulation of a crystal, only a finite…

# Problem setup:
In a real-space quantum Monte Carlo simulation of a crystal, only a finite number of N electrons can be simulated on a computer. This is typically done by tiling a few unit cells of the materials into a supercell with volume Ω. Using periodic boundary conditions, the simulation can access only a finite number of momenta, which are compatible with the supercell. As a result, the total energy calculated in the simulation will have a finite-size error, which is the difference between the result from an infinite system and that of a finite-size one. It is possible to correct this finite-size error if given access to the many-body wave function.
The density of the electron gas is typically parameterized using the Wigner-Seitz radius r_(s). When unpolarized, the Fermi wavevector k_(F)=√(2)/r_(s).

# Main problem:

After performing a diffusion Monte Carlo simulation of the unpolarized two-dimensional electron gas with N=122 electrons at density r_(s)=10 in a square box in the Fermi liquid phase using the Slater-Jastrow wave function, what should one add to the total energy per electron to remove finite-size effects? Report the answer in Hartree and keep two significant digits.

# Problem setup:
In a real-space quantum Monte Carlo simulation of a crystal, only a finite number of N electrons can be simulated on a computer. This is typically done by tiling a few unit cells of the materials into a supercell with volume $\Omega$. Using periodic boundary conditions, the simulation can access only a finite number of momenta, which are compatible with the supercell. As a result, the total energy calculated in the simulation will have a finite-size error, which is the difference between the result from an infinite system and that of a finite-size one. It is possible to correct this finite-size error if given access to the many-body wave function.
The density of the electron gas is typically parameterized using the Wigner-Seitz radius $r_s$. When unpolarized, the Fermi wavevector $k_F=\sqrt{2}/r_s$.

# Main problem:

After performing a diffusion Monte Carlo simulation of the unpolarized two-dimensional electron gas with $N=122$ electrons at density $r_s=10$ in a square box in the Fermi liquid phase using the Slater-Jastrow wave function, what should one add to the total energy per electron to remove finite-size effects? Report the answer in Hartree and keep two significant digits.