Atomic Orbitals

Definition of an atomic orbital

Atomic orbital (AO): a single-electron wavefunction (exact for H-like atoms)

\psi_{n\ell m_\ell}(r,\theta,\phi) = R_{n\ell}(r)\,Y_{\ell m_\ell}(\theta,\phi)

Probability distribution (full 3D):

|\psi|^2 dV = \big|R_{n\ell}(r)\big|^2 \cdot \big|Y_{\ell m_\ell}(\theta,\phi)\big|^2 \cdot r^2\sin\theta \, dr \, d\theta \, d\phi

(2)

Radial probability distribution:

P_r(r) = r^2\big|R_{n\ell}(r)\big|^2 \quad\text{with}\quad \int_{0}^{\infty} P_r(r) \, dr = 1

(3)

Angular probability density:

P_\Omega(\theta,\phi) = \big|Y_{\ell m_\ell}(\theta,\phi)\big|^2 \quad\text{with}\quad \int_{0}^{2\pi}\int_{0}^{\pi} P_\Omega \sin \theta \, d\theta \, d\phi = 1

(4)

Notes:

$s$ orbitals ( $\ell = 0$ ) are isotropic: $P_\Omega = \tfrac{1}{4\pi}$ .
$p, d, \ldots$ orbitals show angular nodes where $Y_{\ell m_\ell} = 0$ , setting the orbital shapes.

Radial profiles of atomic orbitals¶

When visualizing the radial probabilities, it is possible to either directly plot the square of the radial wavefunction ( $R_{nl}^2$ ) or the radial probability density ( $P_{nl}$ ):

{P_{nl}(r) = r^2 N_{nl}^2 R_{nl}^2(r)}

According to this expression, the most probable radius for an electron in the hydrogen-atom $1s$ orbital is $a_0$ (the Bohr radius). The figure above shows examples of $P_{nl}$ . Probability densities are useful, for example, in understanding charge distributions in atoms and molecules.

As the principal quantum number $n$ increases, the electron moves out to greater distances from the nucleus. The average distance for an electron in a given orbital (with quantum numbers $n$ and $l$ ) is given by (this is not the same as the most probable value):

{\langle r\rangle_{nl} = \int_0^\infty r \cdot P_{nl}(r) \, dr = \frac{n^2 a_0}{Z}\left\{ 1 + \frac{1}{2}\left[ 1 - \frac{l(l+1)}{n^2}\right]\right\}}

Note that the expectation value of $r$ and the most probable value of $r$ are not equal. The expectation value can be thought of like an average, and the most probable value like a maximum value.

For degenerate states with $l > 0$ , we have an additional degree of freedom in choosing how to represent the orbitals. In fact, any linear combination of a given set of $2l + 1$ orthogonal eigenfunctions corresponding to a degenerate set with orbital angular momentum $l$ is also a solution to the Schrödinger equation.

Two commonly used representations are the Cartesian form, which is real-valued (and, in the case of $l = 1$ , denoted $p_x$ , $p_y$ , and $p_z$ ), and the eigenfunctions of angular momentum ( $\hat{L}^2$ and $\hat{L}_z$ ), which are complex-valued and denoted $p_{-1}$ , $p_0$ , and $p_{+1}$ . Both polar and Cartesian forms represent the same physical electron density, just in different coordinate frames. The relation between the representations is:

{p_x = -\frac{1}{\sqrt{2}}\left( p_{+1} - p_{-1}\right) \propto \sin(\theta)\cos(\phi) \propto x}

{p_y = \frac{i}{\sqrt{2}}\left( p_{+1} + p_{-1}\right) \propto \sin(\theta)\sin(\phi) \propto y}

{p_z = p_0}

By combining $p_x$ , $p_y$ , and $p_z$ , the lobe of the orbital can be made to point in any direction. For $d$ -orbitals, we have five degenerate levels:

{d_{x^2 - y^2} = \frac{1}{\sqrt{2}}\left(d_{+2} + d_{-2}\right), \quad d_{xy} = -\frac{i}{\sqrt{2}}\left( d_{+2} - d_{-2}\right)}

{d_{xz} = -\frac{1}{\sqrt{2}}\left( d_{+1} - d_{-1}\right), \quad d_{yz} = \frac{i}{\sqrt{2}}\left( d_{+1} + d_{-1}\right)}

{d_{z^2} = d_0}