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


coordinates, etc).




** Energy and local energy




** Energy and local energy




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




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




* Numerical evaluation of the energy of the hydrogen atoms


* Numerical evaluation of the energy of the hydrogen atom




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


wave function:


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


$$




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






and check whether it is constant.


To do that, we will compute the local energy and check whether it is constant.




** Local energy


:PROPERTIES:


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:headerargs:f90: :tangle hydrogen.f90


:END:




You will now program all quantities needed to compute the local energy of the H atom for the given wave function.




Write all the functions of this section in a single file :


~hydrogen.py~ if you use Python, or ~hydrogen.f90~ is you use


Fortran.


@ 207,7 +206,7 @@ double precision function potential(r)


end function potential


#+END_SRC




*** Exercise 2


*** Exercise 2


#+begin_exercise


Write a function which computes the wave function at $\mathbf{r}$.


The function accepts a scalar =a= and a 3dimensional vector =r= as



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