mirror of
https://github.com/TREX-CoE/qmc-lttc.git
synced 2024-12-12 15:33:52 +01:00
Few more changes
This commit is contained in:
parent
029fd33f61
commit
490506964d
45
QMC.org
45
QMC.org
@ -33,7 +33,7 @@
|
||||
|
||||
* Introduction
|
||||
|
||||
This web site contains the QMC tutorial of the 2021 LTTC winter school
|
||||
This website contains the QMC tutorial of the 2021 LTTC winter school
|
||||
[[https://www.irsamc.ups-tlse.fr/lttc/Luchon][Tutorials in Theoretical Chemistry]].
|
||||
|
||||
We propose different exercises to understand quantum Monte Carlo (QMC)
|
||||
@ -45,7 +45,8 @@
|
||||
associated with a given wave function, and apply this approach to the
|
||||
hydrogen atom.
|
||||
Finally, we present the diffusion Monte Carlo (DMC) method which
|
||||
we use here to estimate the exact energy of the hydrogen atom and of the H_2 molecule.
|
||||
we use here to estimate the exact energy of the hydrogen atom and of the H_2 molecule,
|
||||
starting from an approximate wave function.
|
||||
|
||||
Code examples will be given in Python and Fortran. You can use
|
||||
whatever language you prefer to write the program.
|
||||
@ -61,41 +62,29 @@
|
||||
|
||||
* Numerical evaluation of the energy
|
||||
|
||||
In this section we consider the Hydrogen atom with the following
|
||||
In this section, we consider the hydrogen atom with the following
|
||||
wave function:
|
||||
|
||||
$$
|
||||
\Psi(\mathbf{r}) = \exp(-a |\mathbf{r}|)
|
||||
$$
|
||||
|
||||
We will first verify that, for a given value of $a$, $\Psi$ is an
|
||||
We will first verify that, for a particular value of $a$, $\Psi$ is an
|
||||
eigenfunction of the Hamiltonian
|
||||
|
||||
$$
|
||||
\hat{H} = \hat{T} + \hat{V} = - \frac{1}{2} \Delta - \frac{1}{|\mathbf{r}|}
|
||||
$$
|
||||
|
||||
To do that, we will check if the local energy, defined as
|
||||
To do that, we will compute the local energy, defined as
|
||||
|
||||
$$
|
||||
E_L(\mathbf{r}) = \frac{\hat{H} \Psi(\mathbf{r})}{\Psi(\mathbf{r})},
|
||||
$$
|
||||
|
||||
is constant.
|
||||
and check whether it is constant.
|
||||
|
||||
|
||||
The probabilistic /expected value/ of an arbitrary function $f(x)$
|
||||
with respect to a probability density function $p(x)$ is given by
|
||||
|
||||
$$ \langle f \rangle_p = \int_{-\infty}^\infty p(x)\, f(x)\,dx. $$
|
||||
|
||||
Recall that a probability density function $p(x)$ is non-negative
|
||||
and integrates to one:
|
||||
|
||||
$$ \int_{-\infty}^\infty p(x)\,dx = 1. $$
|
||||
|
||||
|
||||
The electronic energy of a system is the expectation value of the
|
||||
In general, the electronic energy of a system, $E$, can be rewritten as the expectation value of the
|
||||
local energy $E(\mathbf{r})$ with respect to the 3N-dimensional
|
||||
electron density given by the square of the wave function:
|
||||
|
||||
@ -104,8 +93,24 @@
|
||||
= \frac{\int \Psi(\mathbf{r})\, \hat{H} \Psi(\mathbf{r})\, d\mathbf{r}}{\int \left[\Psi(\mathbf{r}) \right]^2 d\mathbf{r}} \\
|
||||
& = & \frac{\int \left[\Psi(\mathbf{r})\right]^2\, \frac{\hat{H} \Psi(\mathbf{r})}{\Psi(\mathbf{r})}\,d\mathbf{r}}{\int \left[\Psi(\mathbf{r}) \right]^2 d\mathbf{r}}
|
||||
= \frac{\int \left[\Psi(\mathbf{r})\right]^2\, E_L(\mathbf{r})\,d\mathbf{r}}{\int \left[\Psi(\mathbf{r}) \right]^2 d\mathbf{r}}
|
||||
= \langle E_L \rangle_{\Psi^2}
|
||||
= \langle E_L \rangle_{\Psi^2}\,,
|
||||
\end{eqnarray*}
|
||||
where $\mathbf{r}$ is the vector of the 3N-dimensional electronic coordinates ($N=1$ for the hydrogen atom).
|
||||
|
||||
For a small number of dimensions, one can compute $E$ by evaluating the integrals on a grid. However,
|
||||
|
||||
The probabilistic /expected value/ of an arbitrary function $f(x)$
|
||||
with respect to a probability density function $p(x)$ is given by
|
||||
|
||||
$$ \langle f \rangle_p = \int_{-\infty}^\infty p(x)\, f(x)\,dx, $$
|
||||
|
||||
where probability density function $p(x)$ is non-negative
|
||||
and integrates to one:
|
||||
|
||||
$$ \int_{-\infty}^\infty p(x)\,dx = 1. $$
|
||||
|
||||
|
||||
|
||||
|
||||
** Local energy
|
||||
:PROPERTIES:
|
||||
|
Loading…
Reference in New Issue
Block a user