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* Introduction




This web site contains the QMC tutorial of the 2021 LTTC winter school


This website contains the QMC tutorial of the 2021 LTTC winter school


[[https://www.irsamc.upstlse.fr/lttc/Luchon][Tutorials in Theoretical Chemistry]].




We propose different exercises to understand quantum Monte Carlo (QMC)


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associated with a given wave function, and apply this approach to the


hydrogen atom.


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.


we use here to estimate the exact energy of the hydrogen atom and of the H_2 molecule,


starting from an approximate wave function.




Code examples will be given in Python and Fortran. You can use


whatever language you prefer to write the program.


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* Numerical evaluation of the energy




In this section we consider the Hydrogen atom with the following


In this section, we consider the hydrogen atom with the following


wave function:




$$


\Psi(\mathbf{r}) = \exp(a \mathbf{r})


$$




We will first verify that, for a given value of $a$, $\Psi$ is an


We will first verify that, for a particular value of $a$, $\Psi$ is an


eigenfunction of the Hamiltonian




$$


\hat{H} = \hat{T} + \hat{V} =  \frac{1}{2} \Delta  \frac{1}{\mathbf{r}}


$$




To do that, we will check if the local energy, defined as


To do that, we will compute the local energy, defined as




$$


E_L(\mathbf{r}) = \frac{\hat{H} \Psi(\mathbf{r})}{\Psi(\mathbf{r})},


$$




is constant.


and check whether it is constant.






The probabilistic /expected value/ of an arbitrary function $f(x)$


with respect to a probability density function $p(x)$ is given by




$$ \langle f \rangle_p = \int_{\infty}^\infty p(x)\, f(x)\,dx. $$




Recall that a probability density function $p(x)$ is nonnegative


and integrates to one:




$$ \int_{\infty}^\infty p(x)\,dx = 1. $$






The electronic energy of a system is the expectation value of the


In general, the electronic energy of a system, $E$, can be rewritten as the expectation value of the


local energy $E(\mathbf{r})$ with respect to the 3Ndimensional


electron density given by the square of the wave function:




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= \frac{\int \Psi(\mathbf{r})\, \hat{H} \Psi(\mathbf{r})\, d\mathbf{r}}{\int \left[\Psi(\mathbf{r}) \right]^2 d\mathbf{r}} \\


& = & \frac{\int \left[\Psi(\mathbf{r})\right]^2\, \frac{\hat{H} \Psi(\mathbf{r})}{\Psi(\mathbf{r})}\,d\mathbf{r}}{\int \left[\Psi(\mathbf{r}) \right]^2 d\mathbf{r}}


= \frac{\int \left[\Psi(\mathbf{r})\right]^2\, E_L(\mathbf{r})\,d\mathbf{r}}{\int \left[\Psi(\mathbf{r}) \right]^2 d\mathbf{r}}


= \langle E_L \rangle_{\Psi^2}


= \langle E_L \rangle_{\Psi^2}\,,


\end{eqnarray*}


where $\mathbf{r}$ is the vector of the 3Ndimensional electronic coordinates ($N=1$ for the hydrogen atom).




For a small number of dimensions, one can compute $E$ by evaluating the integrals on a grid. However,




The probabilistic /expected value/ of an arbitrary function $f(x)$


with respect to a probability density function $p(x)$ is given by




$$ \langle f \rangle_p = \int_{\infty}^\infty p(x)\, f(x)\,dx, $$




where probability density function $p(x)$ is nonnegative


and integrates to one:




$$ \int_{\infty}^\infty p(x)\,dx = 1. $$










** Local energy


:PROPERTIES:



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