 # One-Electron Atoms¶

Just as the touchstone of chemistry is the periodic table, the touchstone of quantum chemistry is the atomic wavefunction. As we shall see, our intuition about many-electron atoms is built up from our knowledge of 1-electron atoms, mainly because many-electron atoms are mathematically intractable, while 1-electron atoms are not appreciably more complicated than an electron confined to a spherical ball. A detailed mathematical exposition on 1-electron atoms--which are often called hydrogenic atoms--is provided as a pdf. This is only a brief summary.

## Schrödinger Equation for One-Electron Atoms

Denoting the mass of the electron as $m_e$, the charge of the electron as $=e$, and the permittivity of free space as $\epsilon_0$, the Hamiltonian for the attraction of an electron to a atomic nucleus with atomic number $Z$ (and charge $+Ze$) at the origin, $(x,y,z) = (0,0,0)$, is: $$\hat{H}_{\text{1 el. atom}} = -\frac{\hbar^2}{2m_e} \nabla^2 - \frac{Z e^2}{4 \pi \epsilon_0 r}$$ where $r = \sqrt{x^2+y^2+z^2}$ is the distance of the electron from the nucleus. In atomic units, the Hamiltonian is: $$\hat{H}_{\text{1 el. atom}} = -\tfrac{1}{2} \nabla^2 - Zr^{-1}$$

Just as we did for the electron confined to a spherical ball, we rewrite the Schrödinger equation in spherical coordinates,

$$\left(-\frac{1}{2} \left( \frac{d^2}{dr^2} + \frac{2}{r} \frac{d}{dr}\right) + \frac{\hat{L}^2}{2r^2} - \frac{Z}{r} \right) \psi_{n,l,m_l}(r,\theta,\phi) = E_{n,l,m_l}\psi_{n,l,m_l}(r,\theta,\phi)$$

and use the fact the eigenvalues of the squared-magnitude of the angular momentum, $\hat{L}^2$ are the spherical harmonics $$\hat{L}^2 Y_l^{m} (\theta, \phi) = l(l+1)Y_l^{m} (\theta, \phi) \qquad l=0,1,2,\ldots m=0, \pm 1, \ldots, \pm l$$ and the technique of separation of variables to deduce that the wavefunctions of one-electron atoms have the form $$\psi_{n,l,m}(r,\theta,\phi) = R_{n,l}(r) Y_l^{m}(\theta,\phi)$$ where the radial wavefunction, $R_{n,l}(r)$ is obtained by solving the radial Schrödinger equation

$$\left(-\frac{1}{2} \left( \frac{d^2}{dr^2} + \frac{2}{r} \frac{d}{dr}\right) + \frac{l(l+1)}{2r^2} - \frac{Z}{r} \right) R_{n,l}(r) = E_{n,l}R_{n,l}(r)$$

## The Radial Equation for One-Electron Atoms¶

To solve the radial Schrödinger equation, we rewrite it as a homogeneous linear differential equation

$$\left( \frac{d^2}{dr^2} + \frac{2}{r} \frac{d}{dr} - \frac{l(l+1)}{r^2} + \frac{2Z}{r} + 2E_{n,l}\right) R_{n,l}(r) = 0$$

It is (quite a bit) more involved than the previous cases we have considered, but the same basic technique reveals that the eigenenergies are $$E_n = -\frac{Z^2}{2n^2}$$ and the radial wavefunctions are the product of an associated Laguerre polynomial and an exponential, $$R_{n,l}(r) \propto \left(\frac{2Zr}{n}\right)^l L_{n-1-l}^{2l+1}\left(\frac{2Zr}{n}\right) e^{-\frac{Zr}{n}}$$ with $$n=1,2,3,\ldots \\ l=0,1,\ldots,n-1 \\ m = 0,\pm 1, \pm2, \ldots, \pm l$$ ## Eigenenergies and Wavefunctions for One-Electron Atoms¶

The eigenenergies of the Hydrogenic wavefunctions do not depend on $m$ or $l$. So there are $(n+1)^2$ degenerate eigenfunctions, with energies $$E_n = -\frac{Z^2}{2n^2}$$ The energy eigenfunctions are: $$\psi_{nlm}(r,\theta,\phi) \propto \left(\frac{2Zr}{n}\right)^l L_{n-1-l}^{2l+1}\left(\frac{2Zr}{n}\right) e^{-\frac{Zr}{n}} Y_l^{m} (\theta, \phi)$$ These eigenfunctions are complex-valued, because the spherical harmonics are complex-valued. Like all other one-electron wavefunctions, these eigenfunctions are referred to as orbitals. For historical reasons, orbitals are labelled by their principle quantum number $n$ (which specifies their energy), their total angular momentum quantum number $l$, and the quantum number that specifies their angular momentum around the $z$ axis, $m$, $$\hat{L}_z Y_l^{m} (\theta, \phi) = \hbar m Y_l^{m} (\theta, \phi)$$ The $l$ quantum number is stored by a letter code that dates back to the pre-history of quantum mechanics, where certain spectral lines were labelled as sharp ($l=0$ indicated no spatial degeneracy that could be broken by an external field), principle ($l=1$ lines were still relatively sharp), diffuse ($l=2$ lines were quite diffuse due to the 5-fold degeneracy of d orbitals), and fundamental ($l=3$).

Note that the orbital images that appear above do not look that much like the usual orbital pictures, with the exception of the $m=0$ orbitals. This is because of the complex-valuedness. We often instead use the real spherical harmonics, which are defined simply as: \begin{align} S_l^{m>0}(\theta,\phi) &= \frac{1}{\sqrt{2}} \left(Y_l^{-m} (\theta, \phi) + (-1)^{m} Y_l^{m} (\theta, \phi) \right) \\ S_l^{m=0}(\theta,\phi) &= Y_l^{m=0} (\theta, \phi) \\ S_l^{m<0}(\theta,\phi) &= \frac{i}{\sqrt{2}} \left(Y_l^{-m} (\theta, \phi) - (-1)^{m} Y_l^{m} (\theta, \phi) \right) \end{align} The following animations shows one can take linear combinations of the (complex) spherical harmonics to form the $p_x$, $p_y$, etc. orbitals one generally uses in chemistry.  Using the orbitron, you can visualize the (real, Cartesian) spherical harmonics and the radial wavefunctions for hydrogenic orbitals.

## 🪞 Self-Reflection¶

• Using the conversion from atomic units to traditional chemical units of kJ/mol, what is the energy of the Hydrogen atom? How accurately, in atomic units, must one determine the energy of a one-electron atom in order to attain "chemical accuracy" of ~1 kJ/mol?
• Write a small Python script to evaluate the expectation value of the radius, $r$, for a one-electron atom.
• Test to confirm that the Heisenberg uncertainty principle for position and momentum holds for the ground state of a Hydrogenic atom.
• To what extent is the shape of the spherical harmonic intuitive, especially the doughnut shapes associated with an electron's angular momentum around the z axis.

## 🤔 Thought-Provoking Questions¶

• In one-electron atoms, the eigenenergies depend only on the principle quantum number, $n$, and not the angular momentum quantum number, $l$. Why are $s$ orbitals lower in energy than $p$ orbitals in real multielectron atoms, but not one-electron atoms? It turns out this is not an accidental degeneracy, but a hidden symmetry of the Hydrogen atom.
• Suppose electrons did not repel each other. Can you write the wavefunction for a many-electron atom in that case?
• Why do you think solving the Schrödinger equation for the one-electron molecule is more complicated than solving the Schrödinger equation for the one-electron atom?
• For what $Z$ is the energy of a one-electron atom comparable to the rest-mass energy of an electron, $mc^2$? For atomic numbers close to this value, relativistic effects become extremely important.

## 🔁 Recapitulation¶

• What are eigenfunctions for the $n=l+1$ state of a one-electron atom?
• What are the energy eigenfunctions and eigenvalues for a one-electron atom?
• How does the energy increase as the atomic number increases?

## 🔮 Next Up...¶

• Multielectron systems
• Approximate methods.

## 📚 References¶

My favorite sources for this material are:

There are also some excellent wikipedia articles: