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@ 2222,7 +2222,7 @@ gfortran hydrogen.f90 qmc_stats.f90 vmc_metropolis.f90 o vmc_metropolis


To this aim, we use the mixed estimator of the energy:




\begin{eqnarray*}


E(\tau) &=& \frac{\langle \psi(tau)  \hat{H}  \Psi_T \rangle}{\langle \psi(tau)  \Psi_T \rangle}\\


E(\tau) &=& \frac{\langle \psi(\tau)  \hat{H}  \Psi_T \rangle}{\langle \psi(\tau)  \Psi_T \rangle}\\


&=& \frac{\int \psi(\mathbf{r},\tau) \hat{H} \Psi_T(\mathbf{r}) d\mathbf{r}}


{\int \psi(\mathbf{r},\tau) \Psi_T(\mathbf{r}) d\mathbf{r}} \\


&=& \frac{\int \psi(\mathbf{r},\tau) \Psi_T(\mathbf{r}) E_L(\mathbf{r}) d\mathbf{r}}


@ 2332,19 +2332,19 @@ gfortran hydrogen.f90 qmc_stats.f90 vmc_metropolis.f90 o vmc_metropolis




 You estimate the energy as




\begin{eqnarray*}


E = \frac{\sum_{k=1}{N_{\rm MC}} E_L(\mathbf{r}_k) W(\mathbf{r}_k, k\delta t)}{\sum_{k=1}{N_{\rm MC}} W(\mathbf{r}_k, k\delta t)}


\end{eqnarray*}


\begin{eqnarray*}


E = \frac{\sum_{k=1}^{N_{\rm MC}} E_L(\mathbf{r}_k) W(\mathbf{r}_k, k\delta t)}{\sum_{k=1}^{N_{\rm MC}} W(\mathbf{r}_k, k\delta t)}


\end{eqnarray*}




 The result will be affected by a timestep error (the finite size of $\delta t$) and one


has in principle to extrapolate to the limit $\delta t \rightarrow 0$. This amounts to fitting


the energy computed for multiple values of $\delta t$.


has in principle to extrapolate to the limit $\delta t \rightarrow 0$. This amounts to fitting


the energy computed for multiple values of $\delta t$.




Here, you will be using a small enough timestep and you should not worry about the extrapolation.


Here, you will be using a small enough timestep and you should not worry about the extrapolation.


 The accept/reject step (steps 25 in the algorithm) is in principle not needed for the correctness of


the DMC algorithm. However, its use reduces significantly the timestep error.


the DMC algorithm. However, its use reduces significantly the timestep error.




PDMC algorithm is less stable than the branching algorithm: it


The PDMC algorithm is less stable than the branching algorithm: it


requires to have a value of $E_\text{ref}$ which is close to the


fixednode energy, and a good trial wave function. Its big


advantage is that it is very easy to program starting from a VMC



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