# Atomic Orbitals

**Prerequiresites:** Quantum Mechanics course

Electrons around a nucleus. Do they look like little well behaved planets orbiting a sun?

NOPE!

We get spread out blobs in special little patterns called orbitals. Here, we will look at their shapes and properties a bit. Today we will look at graphs in 1D and 2D, but the next post, Atomic Orbitals Pt. 2, uses a fancy, but slightly unstable plotting package, GLVisualize to generate some 3D plots.

The Hamiltonian for our problem is:

\begin{equation} {\cal H}\Psi(\mathbf{x}) =\left[ -\frac{\hbar}{2 m} \nabla^2 - \frac{Z e^2}{4 \pi \epsilon_0 r}\right]\Psi(\mathbf{x}) = E \Psi(\mathbf{x}) \end{equation} with \begin{equation} \nabla^2= \frac{1}{r^2}\frac{\partial}{\partial r} \left( r^2 \frac{\partial}{\partial r} \right)+ \frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta} \left( \sin \theta \frac{\partial}{\partial \theta} \right)+ \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2}{\partial \phi^2} \end{equation}

To solve this problem, we begin by guessing a solution with separated *radial* and *angular* variables,
\begin{equation}
\Psi(\mathbf{x}) = R(r) \Theta ( \theta,\phi)
\end{equation}

\begin{equation} \frac{E r^2 R(r)}{2r R^{\prime}(r) + r^2 R^{\prime \prime}(r)}= \frac{\left( \frac{1}{\sin \theta} \frac{\partial}{\partial \theta} \left( \sin \theta \frac{\partial \Theta(\theta,\phi)}{\partial \theta} \right)+ \frac{1}{\sin^2 \theta} \frac{\partial^2 \Theta(\theta,\phi)}{\partial \phi^2}\right) }{\Theta( \theta, \phi)} =C \end{equation}

Instead of going into the precise mechanisms of solving those two separate equations here, trust for now that they follow standard special functions, the associated *Legendre polynomial* and the generalized *Laguerre polynomial*. Try a standard quantum mechanics textbook for more information about this.

\begin{equation} Y^m_l(θ,ϕ) = (-1)^m e^{i m \phi} P^m_l (\cos(θ)) \end{equation} where $P^m_l (\cos (\theta))$ is the associated Legendre polynomial.

\begin{equation} R^{n,l} ( \rho ) = \rho ^l e^{- \rho /2} L^{2 l+1}_{n-l-1} ( \rho ) \end{equation} where $L^{2 l+1}_{n-l-1}(\rho)$ is the generalized Laguerre polynomial.

\begin{equation} \rho=\frac{2r}{n a_0} \end{equation}

\begin{equation} N=\sqrt{\left(\frac{2}{n}\right)^3 \frac{(n-l-1)}{2n(n+l)!}} \end{equation}

#### Cell to Evaluate

What’s below is a bunch of definitions that makes our calculations easier later on. Here I utilize the GNU scientific library, GSL imported above, to calculate the special functions.

### Programming Tip!

Even though it's not necessary, specifying the type of inputs to a function through `m::Int`

helps prevent improper inputs and allows the compiler to perform additional optimizations. Julia also implements *abstract types*, so we don't have to specify the exact type of Int. Real allows a numerical, non-complex type.

Type Greek characters in Jupyter notebooks via LaTeX syntax, e.g. \alpha+tab

The function `Orbital`

throws `DomainError()`

when `l`

or `m`

do not obey their bounds. Julia supports a wide variety of easy to use error messages.

#### Parameters

Grid parameters:
You might need to change `rmax`

to be able to view higher $n$ orbitals.

Remember that \begin{equation} 0<n \;\;\;\;\; \;\;\;\; 0 \leq l < n \;\;\;\;\; \;\;\;\; -l \leq m \leq l \;\;\;\;\; \;\;\;\; n,l,m \in {\cal Z} \end{equation}

Though I could create a wrapped up function with `Orbital(n,l,m)`

and evaluate that at each point, the below evaluation takes advantage of the separability of the solution with respect to spherical dimensions. The special functions, especially for higher modes, take time to calculate, and the fewer calls to GSL, the faster the code will run. Therefore, this implementation copies over radial and angular responses.

Don’t forget to check out Atomic Orbitals Pt. 2!