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@ 2110,10 +2110,11 @@ gfortran hydrogen.f90 qmc_stats.f90 vmc_metropolis.f90 o vmc_metropolis




where $\psi(\mathbf{r},\tau) = \Psi(\mathbf{r},i\,)$


and




\begin{eqnarray*}


\psi(\mathbf{r},\tau) &=& \sum_k a_k \exp( E_k\, \tau) \phi_k(\mathbf{r})\\


&=& \exp((E_0E_T)\, \tau)\sum_k a_k \exp( (E_kE_0)\, \tau) \phi_k(\mathbf{r})\,.


\begin{eqnarray*}


\end{eqnarray*}




For large positive values of $\tau$, $\psi$ is dominated by the


$k=0$ term, namely, the lowest eigenstate. If we adjust $E_T$ to the running estimate of $E_0$,


@ 2133,7 +2134,7 @@ gfortran hydrogen.f90 qmc_stats.f90 vmc_metropolis.f90 o vmc_metropolis


We can simulate this differential equation as a diffusionbranching process.






To this this, recall that the [[https://en.wikipedia.org/wiki/Diffusion_equation][diffusion equation]] of particles is given by


To see this, recall that the [[https://en.wikipedia.org/wiki/Diffusion_equation][diffusion equation]] of particles is given by




\[


\frac{\partial \phi(\mathbf{r},t)}{\partial t} = D\, \Delta \phi(\mathbf{r},t).


@ 2167,6 +2168,11 @@ gfortran hydrogen.f90 qmc_stats.f90 vmc_metropolis.f90 o vmc_metropolis


system by simulating the Schrödinger equation in imaginary time, by


the combination of a diffusion process and a branching process.




We note here that the groundstate wave function of a Fermionic system is


antisymmetric and changes sign.




I AM HERE




** Importance sampling




In a molecular system, the potential is far from being constant


@ 2189,18 +2195,18 @@ gfortran hydrogen.f90 qmc_stats.f90 vmc_metropolis.f90 o vmc_metropolis


\frac{\partial \Pi(\mathbf{r},\tau)}{\partial \tau}


= \frac{1}{2} \Delta \Pi(\mathbf{r},\tau) +


\nabla \left[ \Pi(\mathbf{r},\tau) \frac{\nabla \Psi_T(\mathbf{r})}{\Psi_T(\mathbf{r})}


\right] + E_L(\mathbf{r}) \Pi(\mathbf{r},\tau)


\right] + (E_L(\mathbf{r})E_T)\Pi(\mathbf{r},\tau)


\]




The new "kinetic energy" can be simulated by the drifted diffusion


The new "kinetic energy" can be simulated by the driftdiffusion


scheme presented in the previous section (VMC).


The new "potential" is the local energy, which has smaller fluctuations


when $\Psi_T$ gets closer to the exact wave function. It can be simulated by


changing the number of particles according to $\exp\left[ \delta t\,


\left(E_L(\mathbf{r})  E_\text{ref}\right)\right]$


where $E_{\text{ref}}$ is a constant introduced so that the average


of this term is close to one, keeping the number of particles rather


constant.


\left(E_L(\mathbf{r})  E_T\right)\right]$


where $E_T$ is the constant we had introduced above, which is adjusted to


the running average energy and is introduced to keep the number of particles


reasonably constant.




This equation generates the /N/electron density $\Pi$, which is the


product of the ground state with the trial wave function. It



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