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@ -2110,10 +2110,11 @@ gfortran hydrogen.f90 qmc_stats.f90 vmc_metropolis.f90 -o vmc_metropolis
where $\psi(\mathbf{r},\tau) = \Psi(\mathbf{r},-i\,)$ where $\psi(\mathbf{r},\tau) = \Psi(\mathbf{r},-i\,)$
and and
\begin{eqnarray*} \begin{eqnarray*}
\psi(\mathbf{r},\tau) &=& \sum_k a_k \exp( -E_k\, \tau) \phi_k(\mathbf{r})\\ \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})\,. &=& \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 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 $E_0$, $k=0$ 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. 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
\[ \[
\frac{\partial \phi(\mathbf{r},t)}{\partial t} = D\, \Delta \phi(\mathbf{r},t). \frac{\partial \phi(\mathbf{r},t)}{\partial t} = D\, \Delta \phi(\mathbf{r},t).
@ -2167,6 +2168,11 @@ gfortran hydrogen.f90 qmc_stats.f90 vmc_metropolis.f90 -o vmc_metropolis
system by simulating the Schrödinger equation in imaginary time, by system by simulating the Schrödinger equation in imaginary time, by
the combination of a diffusion process and a branching process. the combination of a diffusion process and a branching process.
We note here that the ground-state wave function of a Fermionic system is
antisymmetric and changes sign.
I AM HERE
** Importance sampling ** Importance sampling
In a molecular system, the potential is far from being constant In a molecular system, the potential is far from being constant
@ -2189,18 +2195,18 @@ gfortran hydrogen.f90 qmc_stats.f90 vmc_metropolis.f90 -o vmc_metropolis
-\frac{\partial \Pi(\mathbf{r},\tau)}{\partial \tau} -\frac{\partial \Pi(\mathbf{r},\tau)}{\partial \tau}
= -\frac{1}{2} \Delta \Pi(\mathbf{r},\tau) + = -\frac{1}{2} \Delta \Pi(\mathbf{r},\tau) +
\nabla \left[ \Pi(\mathbf{r},\tau) \frac{\nabla \Psi_T(\mathbf{r})}{\Psi_T(\mathbf{r})} \nabla \left[ \Pi(\mathbf{r},\tau) \frac{\nabla \Psi_T(\mathbf{r})}{\Psi_T(\mathbf{r})}
\right] + E_L(\mathbf{r}) \Pi(\mathbf{r},\tau) \right] + (E_L(\mathbf{r})-E_T)\Pi(\mathbf{r},\tau)
\] \]
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). scheme presented in the previous section (VMC).
The new "potential" is the local energy, which has smaller fluctuations 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 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\, changing the number of particles according to $\exp\left[ -\delta t\,
\left(E_L(\mathbf{r}) - E_\text{ref}\right)\right]$ \left(E_L(\mathbf{r}) - E_T\right)\right]$
where $E_{\text{ref}}$ is a constant introduced so that the average where $E_T$ is the constant we had introduced above, which is adjusted to
of this term is close to one, keeping the number of particles rather the running average energy and is introduced to keep the number of particles
constant. reasonably constant.
This equation generates the /N/-electron density $\Pi$, which is the This equation generates the /N/-electron density $\Pi$, which is the
product of the ground state with the trial wave function. It product of the ground state with the trial wave function. It