Supplementary Information for the Class on Correlations in Many-Electron Systems

Bonding and dissociation of the H₂ molecule

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.

Figure 1: The restricted and unrestricted Hartree–Fock approximations are compared to the exact solution (configuration-interaction expansion as well as the diffusion Monte Carlo). Large gaussian basis set is used to avoid any basis-set bias.

Figure 2: Expectation value of S² in the unrestricted Hartree–Fock approximation. Note that ⟨S²⟩ = 0 for a singlet state and ⟨S²⟩ = 2 for a triplet state.

Hartree–Fock approximation in a toy model of H₂ molecule at large internuclear separations

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.

Figure 3: Exact ground state energy of the toy model compared to the restricted and unrestricted Hartree–Fock approximations. Note that the t axis is reversed for easier comparison with Figure 1. The wave function derived from UHF by projecting out the triplet component is a sum of two Slater determinants.

Figure 4: Expectation value of S² in the unrestricted Hartree–Fock approximation. For t > U'/2 the UHF and RHF solutions coincide. The S² operator in the two-orbital model is $$ S^2=S_+ S_-= \bigl(a^{\dagger}_{\uparrow} a_{\downarrow} +b^{\dagger}_{\uparrow} b_{\downarrow}\bigr) \bigl(a^{\dagger}_{\downarrow} a_{\uparrow} +b^{\dagger}_{\downarrow} b_{\uparrow}\bigr)\,. $$

Magnetic impurity in a metallic host

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.

Variational Monte Carlo (VMC)

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 E₀ = −2.89536752 hartree for the optimal a = 0.08786283656.

Figure 5: Stochastic estimates of $\langle\Psi_T|H|\Psi_{T} \rangle$ for several values of the variational parameter a. The minimum is found using a quadratic fit. The optimal a is not accurate due to large stochastic errors of individual energies. With longer simulation times one would find out that E(a) is more accurately described by a cubic function and the optimal a would come out accuratelly, eventually.

Efficiency of variational Monte Carlo simulations

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.

Figure 6: Independent Monte Carlo runs with identical number of steps but with varied size (and type) of the proposal moves. The error bars are ploted as a function of the ratio of the accepted moves to the total number of moves. Apparently, the simulation is inefficient when the ratio is small or when it is close to one. The location of the minimum depends on the type of the moves as well as on the particular system being simulated.

Diffusion Monte Carlo (DMC) method

The same setting as above but in combination with a more advanced technology that uses a stochastic implementation of the projection operator exp(−tH).

Figure 7: The simulation starts with the variational Monte Carlo (red) and at some point the DMC projection is switched on (blue). Plotted are samples of the local energy at individual steps (only each fifth step is explicitly shown). In a real application one would use the best trial function at hand to reach the highest efficiency, but here the VMC is intentionally detuned from optimum to accentuate the energy drop in DMC. Notice the substantially longer serial correlations in the DMC data compared to VMC.

Simple code to explore and play with QMC

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).