3. Variational Monte Carlo Study of the He Atom

Perform a VMC calculation of the ground state energy of the He atom (two electrons around a nucleus of charge +2), using a wavefunction of the form

$$\Psi(\vec r_1,\vec r_2) = \phi(r_1) \phi(r_2) f(\vert\vec r_1 -\vec r_2\vert).$$ (8)

Here $\phi$ is thought of roughly as the 1s orbital, and this wavefunction represents one electron with spin up and one with spin down, with a Hydrogenic 1s orbital and a Jastrow-type (multiplicative) correlation function with the forms

$$\phi(r) = exp(-r/r_o);~~~f(r) = exp[ \frac{r}{a (1+b r)}],$$ (9)

where r_o, a, b are constants to be determined.

The nuclear cusp condition (i.e. cancelling of the divergences in the local energy $E(R) \equiv E(\vec r_1, \vec r_2)$), determines r_o, which in class was found to be r_o = a_B, the Bohr radius.

(i) The electron-electron cusp condition puts a constraint on a and b, and you should do the algebra necessary to determine this condition. This requires obtaining an analytic expresssion for the ``local kinetic energy.'' Provide this expression for the local kinetic energy, which will have to coded also, in the material that you hand in on Friday Oct 27. Make the choice that was discussed in class for the other condition that will determine the last constant

$$f(r=1 a_B) = 0.95 f(\infty).$$ (10)

(In a research setting, this last condition would be relaxed with the condition that the computed energy is minimum.)

(ii) Determine how you want to take your steps through coordinate space. The ``walk'' should be unbiased, and the acceptance ratio should be adjustable until it is not far from 50%. State your choice of trial steps clearly.

(iii) Perform the random walk for 1000 steps, save the local energy E(R) at each step and the cumulative average, and plot them vs. the step number. Also, determine your acceptance ratio. Adjust the step size, then rerun, until the acceptance ratio of moves is reasonable. Discuss what you have learned: does your code seem to be working, is the energy reasonable, can you estimate how many steps might be required to get a good value of the energy....?