Bonding and dissociation of the H_{2} 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
(configurationinteraction expansion as well as the diffusion Monte Carlo).
Large gaussian basis set is used to avoid any basisset bias.
Figure 2: Expectation value of S^{2} in the unrestricted Hartree–Fock
approximation. Note that ⟨S^{2}⟩ = 0 for a singlet state
and ⟨S^{2}⟩ = 2 for a triplet state.
Hartree–Fock approximation in a toy model of H_{2} 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 twoorbital 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 S^{2} in the unrestricted
Hartree–Fock approximation. For $ t > U'/2$
the UHF and RHF solutions coincide. The S^{2} operator
in the twoorbital model is
$$
S^2=S_+ S_=
\bigl(a^{\dagger}_{\uparrow} a_{\downarrow}
+b^{\dagger}_{\uparrow} b_{\downarrow}\bigr)
\bigl(a^{\dagger}_{\downarrow} a_{\uparrow}
+b^{\dagger}_{\downarrow} b_{\uparrow}\bigr)\,.
$$
What have we learned?

For large t (kinetic energy) the restricted Hartree–Fock
is a relatively good approximation.

When the onsite Coulomb repulsion is large compared to the kinetic energy,
the wave function in the form of a single determinant is not sufficient.
Restricted gives too large energy, unrestricted gives a reasonable energy but
it predicts magnetism where it should not be any: the UHF state is not an
eigenstate of S^{2} operator although
S^{2} commutes with the hamiltonian.
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
groundstate energy of parahelium. A relatively simple trial
wave function that has the correct electronnucleus and electronelectron
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.
Figure 5: Stochastic estimates of $\langle\Psi_TH\Psi_{T} \rangle$ 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 $\exp(t H)$.
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
grownup and versatile pieces of software (and freely available too).