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Atomic orbitals - Visualisation using same-density surfaces

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Sommaire

Description

This article explains an orbital visualisation technique which consists of drawing the isovalue surface of the density function linked to the orbital's wavefunction.

S_{\alpha}=\left\lbrace \left( x,y,z\right)\in \R^{3} : \vert\psi_{n,l,m}\left( x,y,z\right) \vert^{2}=\alpha\right\rbrace

Programming

Algorithms

The algorithms used are described on the Marching Square and Marching Cube algorithms page.

Orbitals described by equations in complex numbers only depend on φ in relation to their phase factor, so their probability density isn't connected to φ. We shall therefore use the Marching squares algorithm to calculate the isodensity lines in the meridian planes and then around the z axis by rotating these lines. The colour used is chosen in relation to the phase.

Real orbitals are calculated by the Marching cube algorithm. Real orbital wave functions are in real numbers, so we shall colour the positive isosurfaces blue and the negative isosurfaces red.

These surfaces mark the boundary of a volume. Given that the wavefunction squared is a probability density function, the maximum possible volume is equal to 1. The volume defined by the isodensity surfaces therefore varies between 0 and 1. We would like to display 80% of the probability density for example, i.e. a volume equal to 0.8. To obtain this volume, we must adjust the isodensity value by choosing α such as \mbox{Volume}\left( S_{\alpha}\right) = 0,80.

Here are some examples of the values obtained for different probability density thresholds:

Visualisation orbitale seuil 0.JPG Visualisation orbitale seuil 10.JPG

Visualisation orbitale seuil 80.JPG

Visualisation orbitale seuil 90.JPG Visualisation orbitale seuil 100.JPG

Java applet

Examples

Choice of representatives

A few definitions

The conversion between cartesian variables and spherical variables gives us the following relationship between probability density functions:

\vert\psi_{n,\ell,m}\left( x,y,z\right) \vert^{2} = r^{2}.\sin\left(\theta\right).\vert\psi_{n,\ell,m}\left( r,\theta,\phi\right) \vert^{2}

Note the return of the Jacobian basis-change matrix (J = r^{2}.\sin\left(\theta\right)) which guarantees the conservation of elementary volumes: dx.dy.dz = J.dr.dθ.dφ


We shall now introduce Laguerre and Legendre polynomials: 

Legendre polynomials: L_{k}\left( x\right) = \dfrac{1}{k! 2^{k}}\dfrac{d^{k}}{{dx}^{k}}\left( \left( x^{2}-1\right) ^{k}\right)

and their derivatives L_{k}^{\left( \alpha\right) }\left( x\right) = \dfrac{d^{\alpha}}{{dx}^{\alpha}}L_{k}\left( x\right) 

Laguerre polynomials: P_{k}\left( x\right) =\dfrac{e^{x}}{k!}\dfrac{d^{k}}{{dx}^{k}}\left( e^{-x}x^{k}\right)

and their derivatives P_{k}^{\left( \alpha\right) }\left( x\right) = \dfrac{d^{\alpha}}{{dx}^{\alpha}}P_{k}\left( x\right)


We shall also define the following constant: 

Normalisation constant: C_{n,\ell,m} = \sqrt{\frac{2\ell+1}{4 \pi n^{4}} \dfrac{\left( n-\ell-1\right) !}{\left( n+\ell\right) !} \dfrac{\left( \ell-\vert m\vert\right) !}{\left( \ell+\vert m\vert\right) !} }

Representation in complex numbers 

The complex wave function: \psi_{n,\ell,m}\left( r,\theta,\phi\right)  = C_{n,\ell,m} . \left[ r^{\ell} . P_{n+\ell}^{\left( 2\ell+1\right) }\left( r\right) . e^{-\frac{r}{2}}\right]  . \left[ {\left( \sin \theta\right) }^{\vert m\vert} . L_{\ell}^{\left( \vert m\vert\right)} \left( \cos \theta\right)\right] . \left[ e^{i m \phi}\right]

Representation in real numbers 

The real wave function: \psi_{n,\ell,m}\left( r,\theta,\phi\right)  = C_{n,\ell,m} \cdot \left[ r^{\ell} \cdot P_{n+\ell}^{\left( 2\ell+1\right) }\left( r\right) \cdot e^{-\frac{r}{2}}\right]  \cdot \left[ {\left( \sin \theta\right) }^{\vert m\vert} \cdot L_{\ell}^{\left( \vert m\vert\right)} \left( \cos \theta\right)\right] \cdot\left[ {\left\{ \begin{array}{ll}
\sin\left( m \phi\right) & \mbox{si } m > 0 \\
\dfrac{1}{\sqrt{2}} & \mbox{si } m = 0 \\
\cos\left( m \phi\right) & \mbox{si } m < 0 
\end{array} \right. }\right]

The first few atomic orbitals

Complex orbitals

...

Real orbitals

SsAOReal.jpg

See also

Orbital Project

Atomic orbitals

Atomic orbitals - Visualisation using point clouds

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