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* Introduction * Introduction
This web site is the QMC tutorial of the LTTC winter school This web site contains the QMC tutorial of the 2021 LTTC winter school
[[https://www.irsamc.ups-tlse.fr/lttc/Luchon][Tutorials in Theoretical Chemistry]]. [[https://www.irsamc.ups-tlse.fr/lttc/Luchon][Tutorials in Theoretical Chemistry]].
We propose different exercises to understand quantum Monte Carlo (QMC) We propose different exercises to understand quantum Monte Carlo (QMC)
methods. In the first section, we propose to compute the energy of a methods. In the first section, we start with the computation of the energy of a
hydrogen atom using numerical integration. The goal of this section is hydrogen atom using numerical integration. The goal of this section is
to introduce the /local energy/. to familarize yourself with the concept of /local energy/.
Then we introduce the variational Monte Carlo (VMC) method which Then, we introduce the variational Monte Carlo (VMC) method which
computes a statistical estimate of the expectation value of the energy computes a statistical estimate of the expectation value of the energy
associated with a given wave function. associated with a given wave function, and apply this approach to the
Finally, we introduce the diffusion Monte Carlo (DMC) method which hydrogen atom.
gives the exact energy of the hydrogen atom and of the H_2 molecule. Finally, we present the diffusion Monte Carlo (DMC) method which
we use here to estimate the exact energy of the hydrogen atom and of the H_2 molecule.
Code examples will be given in Python and Fortran. You can use Code examples will be given in Python and Fortran. You can use
whatever language you prefer to write the program. whatever language you prefer to write the program.
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the wave functions considered here are real: for an $N$ electron the wave functions considered here are real: for an $N$ electron
system where the electrons move in the 3-dimensional space, system where the electrons move in the 3-dimensional space,
$\Psi : \mathbb{R}^{3N} \rightarrow \mathbb{R}$. In addition, $\Psi$ $\Psi : \mathbb{R}^{3N} \rightarrow \mathbb{R}$. In addition, $\Psi$
is defined everywhere, continuous and infinitely differentiable. is defined everywhere, continuous, and infinitely differentiable.
All the quantities are expressed in /atomic units/ (energies, All the quantities are expressed in /atomic units/ (energies,
coordinates, etc). coordinates, etc).