Course number NFPL551 (Charles University, Faculty of Mathematics and Physics).
Calculations were performed with the General Atomic and Molecular Electronic Structure System (GAMESS) that can be downloaded free of charge. Sample input files: unrestricted Hartree–Fock, restricted Hartree–Fock followed by CI.
To analyze the failure of the restricted Hartree–Fock approximation at large internuclear separations one can look at a simple two-orbital model, with orbital a localized around one nucleus and with orbital b localized around the other nucleus. The overlap of these orbitals can be neglected at large separations R, and hence they form an orthonormal basis of the model. In the formalism of second quantization the hamiltonian reads as
$$ \begin{multline} H= t\sum_{\sigma}\bigl(a^{\dagger}_{\sigma} b_{\sigma} + b^{\dagger}_{\sigma} a_{\sigma}\bigl) + U\bigl( a^{\dagger}_{\uparrow}a_{\uparrow} a^{\dagger}_{\downarrow}a_{\downarrow} + b^{\dagger}_{\uparrow}b_{\uparrow} b^{\dagger}_{\downarrow}b_{\downarrow} \bigr) \\ +V \sum_{\sigma\sigma'}a^{\dagger}_{\sigma} a_{\sigma} b^{\dagger}_{\sigma'} b_{\sigma'} -\frac{1}{2R}\sum_{\sigma}\bigl(a^{\dagger}_{\sigma} a_{\sigma} + b^{\dagger}_{\sigma} b_{\sigma}\bigl)\,, \end{multline} $$
where the first term describes the probability of an electron hopping from one nucleus to the other, the second term is the Coulomb repulsion the two electrons feel when they sit at the same nucleus, the third term is the Coulomb repulsion the electrons feel when they sit at different nuclei, and the last therm is the Coulomb attraction of the nuclei. The hopping probability t decreases to zero with increasing distance R, the factor V is approximately 1/R at large distances.
What have we learned?
For now I link just a Mathematica notebook (120kB) and its PDF version (270kB) with some calculations and graphs. The Mathematica file can be opened with a viewer that is available for free but it is an unbelievably huge piece of software for this little task.
To illustrate the variational Monte Carlo method, we calculate the ground-state energy of parahelium. A relatively simple trial wave function that has the correct electron-nucleus and electron-electron cusps read as
$$ \Psi_T \sim \biggl(1+\frac12 r_{12}\biggr) \Bigl[1+a\bigl(r_1^2+r_2^2\bigr)\Bigr] e^{-2(r_1+r_2)}\,. $$
The total energy can be also evaluated analytically with the aid of the Hylleraas method in this case; one gets E_{0}=−2.89536752 hartree for the optimal a=0.08786283656.
The efficiency of the variational Monte Carlo method depends on the choice of the proposal moves. It will be shown on parahelium ground state with the trial function shown above at the optimal a=0.08786283656.
The same setting as above but in combination with a more advanced technology that uses a stochastic implementation of the projection operator $\exp(-t H)$.
The data plotted in the figures above were obtained with a simple code (tar.gz, 100kB) that can be downloaded here, the package contains also its brief description (PDF, 84kB). It is meant for playing around and figuring out how the QMC algorithms work. For serious QMC simulations look at QMCPACK or QWalk which are grown-up and versatile pieces of software (and freely available too).