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