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@ -2222,7 +2222,7 @@ gfortran hydrogen.f90 qmc_stats.f90 vmc_metropolis.f90 -o vmc_metropolis
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To this aim, we use the mixed estimator of the energy:
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\begin{eqnarray*}
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E(\tau) &=& \frac{\langle \psi(tau) | \hat{H} | \Psi_T \rangle}{\langle \psi(tau) | \Psi_T \rangle}\\
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E(\tau) &=& \frac{\langle \psi(\tau) | \hat{H} | \Psi_T \rangle}{\langle \psi(\tau) | \Psi_T \rangle}\\
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&=& \frac{\int \psi(\mathbf{r},\tau) \hat{H} \Psi_T(\mathbf{r}) d\mathbf{r}}
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{\int \psi(\mathbf{r},\tau) \Psi_T(\mathbf{r}) d\mathbf{r}} \\
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&=& \frac{\int \psi(\mathbf{r},\tau) \Psi_T(\mathbf{r}) E_L(\mathbf{r}) d\mathbf{r}}
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@ -2332,19 +2332,19 @@ gfortran hydrogen.f90 qmc_stats.f90 vmc_metropolis.f90 -o vmc_metropolis
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- You estimate the energy as
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\begin{eqnarray*}
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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)}
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\end{eqnarray*}
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\begin{eqnarray*}
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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)}
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\end{eqnarray*}
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- The result will be affected by a time-step error (the finite size of $\delta t$) and one
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has in principle to extrapolate to the limit $\delta t \rightarrow 0$. This amounts to fitting
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the energy computed for multiple values of $\delta t$.
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has in principle to extrapolate to the limit $\delta t \rightarrow 0$. This amounts to fitting
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the energy computed for multiple values of $\delta t$.
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Here, you will be using a small enough time-step and you should not worry about the extrapolation.
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Here, you will be using a small enough time-step and you should not worry about the extrapolation.
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- The accept/reject step (steps 2-5 in the algorithm) is in principle not needed for the correctness of
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the DMC algorithm. However, its use reduces significantly the time-step error.
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the DMC algorithm. However, its use reduces significantly the time-step error.
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PDMC algorithm is less stable than the branching algorithm: it
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The PDMC algorithm is less stable than the branching algorithm: it
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requires to have a value of $E_\text{ref}$ which is close to the
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fixed-node energy, and a good trial wave function. Its big
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advantage is that it is very easy to program starting from a VMC
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