more changes

QMC.org

@@ -60,7 +60,52 @@
   All the quantities are expressed in /atomic units/ (energies,
   coordinates, etc).
   
-* Numerical evaluation of the energy
+ ** Energy and local energy
+
+  For a given system with Hamiltonian $\hat{H}$ and wave function $\Psi$, we define the local energy as
+  
+  $$
+  E_L(\mathbf{r}) = \frac{\hat{H} \Psi(\mathbf{r})}{\Psi(\mathbf{r})},
+  $$
+
+  where $\mathbf{r}$ denotes the 3N-dimensional electronic coordinates.
+  
+  The electronic energy of a system, $E$, can be rewritten in terms of the 
+  local energy $E_L(\mathbf{r})$ as
+
+  \begin{eqnarray*}
+  E & = & \frac{\langle \Psi| \hat{H} | \Psi\rangle}{\langle \Psi |\Psi \rangle} 
+      =   \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}}  
+  \end{eqnarray*}
+   
+  For few dimensions, one can easily compute $E$ by evaluating the integrals on a grid but, for a high number of dimensions, one can resort to Monte Carlo techniques to compute $E$.
+  
+  To this aim, recall that 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 a probability density function $p(x)$ is non-negative
+  and integrates to one:
+
+  $$ \int_{-\infty}^\infty p(x)\,dx = 1. $$
+
+  Similarly, we can view the the energy of a system, $E$, as the expected value of the local energy with respect to 
+  a probability density $p(\mathbf{r}}$ defined in 3$N$ dimensions:
+  
+  $$ E =  \int E_L(\mathbf{r}) p(\mathbf{r})\,d\mathbf{r}} \equiv  \langle E_L \rangle_{\Psi^2}\,, $$
+  
+  where the probability density is given by the square of the wave function:
+  
+  $$ p(\mathbf{r}) = \frac{|Psi(\mathbf{r}|^2){\int \left |\Psi(\mathbf{r})|^2 d\mathbf{r}}\,. $$
+  
+  If we can sample configurations $\{\mathbf{r}\}$ distributed as $p$, we can estimate $E$ as the average of the local energy computed over these configurations:
+  
+  $$ E \approx \frac{1}{M} \sum_{i=1}^M E_L(\mathbf{r}_i} \,.
+  
+  * Numerical evaluation of the energy of the hydrogen atoms
 
   In this section, we consider the hydrogen atom with the following
   wave function:
@@ -78,40 +123,9 @@
 
   To do that, we will compute the local energy, defined as
 
-  $$
-  E_L(\mathbf{r}) = \frac{\hat{H} \Psi(\mathbf{r})}{\Psi(\mathbf{r})},
-  $$
-
+  
   and check whether it is constant.
 
-  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:
-
-  \begin{eqnarray*}
-  E & = & \frac{\langle \Psi| \hat{H} | \Psi\rangle}{\langle \Psi |\Psi \rangle} 
-      =   \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}\,,
-  \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:
    :header-args:python: :tangle hydrogen.py