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All the quantities are expressed in /atomic units/ (energies,


coordinates, etc).




* Numerical evaluation of the energy


** Energy and local energy




For a given system with Hamiltonian $\hat{H}$ and wave function $\Psi$, we define the local energy as




$$


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


$$




where $\mathbf{r}$ denotes the 3Ndimensional electronic coordinates.




The electronic energy of a system, $E$, can be rewritten in terms of the


local energy $E_L(\mathbf{r})$ as




\begin{eqnarray*}


E & = & \frac{\langle \Psi \hat{H}  \Psi\rangle}{\langle \Psi \Psi \rangle}


= \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}}


\end{eqnarray*}




For few dimensions, one can easily compute $E$ by evaluating the integrals on a grid but, for a high number of dimensions, one can resort to Monte Carlo techniques to compute $E$.




To this aim, recall that 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 a probability density function $p(x)$ is nonnegative


and integrates to one:




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




Similarly, we can view the the energy of a system, $E$, as the expected value of the local energy with respect to


a probability density $p(\mathbf{r}}$ defined in 3$N$ dimensions:




$$ E = \int E_L(\mathbf{r}) p(\mathbf{r})\,d\mathbf{r}} \equiv \langle E_L \rangle_{\Psi^2}\,, $$




where the probability density is given by the square of the wave function:




$$ p(\mathbf{r}) = \frac{Psi(\mathbf{r}^2){\int \left \Psi(\mathbf{r})^2 d\mathbf{r}}\,. $$




If we can sample configurations $\{\mathbf{r}\}$ distributed as $p$, we can estimate $E$ as the average of the local energy computed over these configurations:




$$ E \approx \frac{1}{M} \sum_{i=1}^M E_L(\mathbf{r}_i} \,.




* Numerical evaluation of the energy of the hydrogen atoms




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


wave function:


@ 78,40 +123,9 @@




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




$$


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


$$






and check whether it is constant.




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:




\begin{eqnarray*}


E & = & \frac{\langle \Psi \hat{H}  \Psi\rangle}{\langle \Psi \Psi \rangle}


= \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}\,,


\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:


:headerargs:python: :tangle hydrogen.py



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