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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_0-E_T)\, \tau)\sum_k a_k \exp( -(E_k-E_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 diffusion-branching 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
\[
\frac{\partial \phi(\mathbf{r},t)}{\partial t} = D\, \Delta \phi(\mathbf{r},t).
@ -2167,6 +2168,11 @@ gfortran hydrogen.f90 qmc_stats.f90 vmc_metropolis.f90 -o vmc_metropolis
system by simulating the Schrödinger equation in imaginary time, by
the combination of a diffusion process and a branching process.
We note here that the ground-state wave function of a Fermionic system is
antisymmetric and changes sign.
I AM HERE
** Importance sampling
In a molecular system, the potential is far from being constant
@ -2189,18 +2195,18 @@ gfortran hydrogen.f90 qmc_stats.f90 vmc_metropolis.f90 -o vmc_metropolis
-\frac{\partial \Pi(\mathbf{r},\tau)}{\partial \tau}
= -\frac{1}{2} \Delta \Pi(\mathbf{r},\tau) +
\nabla \left[ \Pi(\mathbf{r},\tau) \frac{\nabla \Psi_T(\mathbf{r})}{\Psi_T(\mathbf{r})}
\right] + E_L(\mathbf{r}) \Pi(\mathbf{r},\tau)
\right] + (E_L(\mathbf{r})-E_T)\Pi(\mathbf{r},\tau)
\]
The new "kinetic energy" can be simulated by the drifted diffusion
The new "kinetic energy" can be simulated by the drift-diffusion
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