What is a "hydrogen-like" or "hydrogenic" atom?

I'm studying some chemistry on my own in anticipation of the new school year and in my book, I came across the Rydberg equation for the first time. I worked through some examples and everything was fine until I came across this comment on the question, "[Calculate the wavelength of the radiation released when an electron moves from n=5 to n=2](https://chemistry.stackexchange.com/q/9355/402)":



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> For future reference: the Rydberg formula only works for hydrogen-like
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What is meant by "hydrogen-like"? I've heard that solving for multi-electron systems is (near) impossible, so I understand why hydrogen is used here, but I don't understand what "hydrogen-like" is.




A [hydrogen-like atom](https://en.wikipedia.org/wiki/Hydrogen-like_atom) (or ion) is simply any particle with a nucleus and one electron.




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That should be sufficient to answer the question at hand, but I thought I should say a bit more, as some of these answers are potentially confusing.


The *historical* reason why the Rydberg formula only works for hydrogen-like atoms is that it was originally formulated to explain the [spectral lines of hydrogen](https://en.wikipedia.org/wiki/Hydrogen_spectral_series). It was never intended to explain the spectra of multi-electron atoms.


The *physical* reason, though, is because the Rydberg formula uses energy levels that only depend on the principal quantum number $n$, which has to be a positive integer:


$$\bar{\nu} = Z^2\mathcal{R}\left(\frac{1}{n\_1^2} - \frac{1}{n\_2^2}\right) \qquad n\_1,n\_2 \in \mathbb{Z}^+$$


and nowadays we know that this is only true for hydrogen-like atoms;$^\*$ the energy levels of multi-electron atoms depend on both $n$ and $l$.$^\dagger$




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The $n$-dependency *was* later successfully rationalised by the Bohr model, but saying that "the Rydberg formula only works for hydrogen-like atoms because the Bohr model only works for them" is misleading and misses the point, as:


1. This implies that the Rydberg formula was derived from the Bohr model, which is not true; it was merely empirically determined, and the formula predated the Bohr model by 25 years.
2. The Bohr model simply does *not* work for hydrogen-like atoms. The fact that it reproduces the Rydberg formula should merely be considered a *serendipity*; Bohr arrived at the correct result by the wrong method.
3. It does not lend any real insight into the proper reason why the Rydberg formula does not apply to helium, etc. (which I briefly mentioned above).




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$^\*$ In fact, the energy levels of hydrogen are not only dependent on $n$ (due to various small effects such as – but not limited to – spin-orbit coupling, and hyperfine splitting). Wikipedia has a good overview of the topic [here](https://en.wikipedia.org/wiki/Fine_structure) and most QM textbooks have a chapter on the hydrogen atom, where they discuss these perturbations to the Hamiltonian and their effects on the energies. Not surprisingly, the inability to explain this was one of the failures of the Bohr model.


$^\dagger$ Of course, there is also a series of approximations here. The energy levels of multi-electron atoms are only approximately described by sums of orbital energies, so the transition energies are only approximately equal to a difference in energy between two orbitals.


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In the spirit of notable University Of Nottingham chemist [Sir Martyn Poliakoff](https://royalsociety.org/people/martyn-poliakoff-12107/)'s comment in the [Periodic videos](http://www.periodicvideos.com/index.htm) episode [Helium in Disguise](http://www.periodicvideos.com/videos/feature_hydrogen41.htm) (also [YouTube](https://youtu.be/7hdVjb2gRgQ)):



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> Again and again, for scientists, if they hear something surprising that makes them think in a different way, it’s really good.
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I'll add to the other excellent answers by listing some *less common hydrogen-like "atoms"*.


[The video](https://youtu.be/7hdVjb2gRgQ) is about **Muonic Heavy Hydrogen** or so-called ${}^{4.1}H$ is discussed at the end of this list. Yes, that's pronounced "hydrogen four-point-one." See the New Scientist article [Atomic disguise makes helium look like hydrogen](https://www.newscientist.com/article/dn20049-atomic-disguise-makes-helium-look-like-hydrogen/), and the primary article published in Science [Kinetic Isotope Effects for the Reactions of Muonic Helium and Muonium with H2](http://science.sciencemag.org/content/331/6016/448) Fleming, D. G. et al. Science 28 Jan 2011: Vol. 331, Issue 6016, pp. 448-450, DOI: 10.1126/science.1199421




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[**Positronium**](https://en.wikipedia.org/wiki/Positronium): (e+e−)



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> Positronium (Ps) is a system consisting of an [electron](https://en.wikipedia.org/wiki/Electron) and its [anti-particle](https://en.wikipedia.org/wiki/Antimatter), a [positron](https://en.wikipedia.org/wiki/Positron), bound together into an [exotic atom](https://en.wikipedia.org/wiki/Exotic_atom), specifically an [onium](https://en.wikipedia.org/wiki/Onium).
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It has Rydberg states with $E\_n \sim -(6.8eV)/n^2$ or half that of normal hydrogen because the reduced mass is half that for an electron bound to a much heavier item. The longer-lived triplet ${}^3S\_1$ state has a mean lifetime of about 142ns, and decays by electron-positron annihilation into three gamma-ray photons. It is sometimes studied by slowing and stopping positrons in powdered MgO where they capture an electron and tend to remain relatively unperturbed by the other atoms.




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[**Muonium**](https://en.wikipedia.org/wiki/Muonium): (μ+e−)



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> Muonium is an [exotic atom](https://en.wikipedia.org/wiki/Exotic_atom) made up of an [antimuon](https://en.wikipedia.org/wiki/Muon) and an [electron](https://en.wikipedia.org/wiki/Electron), which was discovered in 1960 **and is given the chemical symbol Mu**. During the muon's 2.2 µs lifetime, muonium can enter into compounds such as muonium chloride (MuCl) or sodium muonide (NaMu). Due to the mass difference between the antimuon and the electron, muonium (μ+e−) is more similar to [atomic hydrogen](https://en.wikipedia.org/wiki/Hydrogen_atom) (p+e−) than [positronium](https://en.wikipedia.org/wiki/Positronium) (e+e−). Its [Bohr radius](https://en.wikipedia.org/wiki/Bohr_radius) and ionization energy are within 0.5% of hydrogen, deuterium, and tritium, **and thus it can usefully be considered as an exotic light isotope of hydrogen**.
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[**True Muonium**](https://en.wikipedia.org/wiki/True_muonium): (μ+μ−)



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> True muonium or **muononium** is an exotic atom made up of an [antimuon](https://en.wikipedia.org/wiki/Muon) and a [muon](https://en.wikipedia.org/wiki/Muon). It is yet to be observed, but it may have been generated in the collision of electron and positron beams.
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[**Muonic hydrogen**](https://en.wikipedia.org/wiki/Exotic_atom#Muonic_atoms):



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> Negative muons can, however, form [muonic atoms](https://en.wikipedia.org/wiki/Exotic_atom#Muonic_atoms) (previously called mu-mesic atoms), by replacing an electron in ordinary atoms. Muonic hydrogen atoms are much smaller than typical hydrogen atoms because the much larger mass of the muon gives it a much more localized ground-state wavefunction than is observed for the electron.
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[**Muonic helium**](https://en.wikipedia.org/wiki/Muon#Muonic_atoms):



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> Muonic helium is created by substituting a muon for one of the electrons in helium-4. The muon orbits much closer to the nucleus, so muonic helium can therefore be regarded as an isotope of helium whose nucleus consists of two neutrons, two protons, and a muon, with a single electron outside. Colloquially, it could be called "helium 4.1", since the mass of the muon is slightly greater than 0.1 amu. **Chemically, muonic helium, possessing an unpaired valence electron, can bond with other atoms, and behaves more like a hydrogen atom than an inert helium atom.**
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> Muonic heavy hydrogen atoms with a negative muon may undergo nuclear fusion in the process of [muon-catalyzed fusion](https://en.wikipedia.org/wiki/Muon-catalyzed_fusion), after which the muon may leave the new atom to induce fusion in another hydrogen molecule. This process continues until the negative muon is trapped by a helium atom, and cannot leave until it decays.
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[![enter image description here](https://i.stack.imgur.com/AZGwu.jpg)](https://i.stack.imgur.com/AZGwu.jpg)


**above:** From [New Scientist](https://www.newscientist.com/article/dn20049-atomic-disguise-makes-helium-look-like-hydrogen/).


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Hydrogen-like atoms are atoms with a single electron "orbiting" a nucleus that has more than one nucleon. As @Xerxes pointed out in a comment, you can in principle have a nucleus made up of particles other than protons and neutrons (nucleons). [Positronium](https://en.wikipedia.org/wiki/Positronium) might be an extreme example of this.


Wikipedia actually has an entry about [Hydrogen-like atoms](https://en.wikipedia.org/wiki/Hydrogen-like_atom) which goes somewhat beyond what you asked.


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Hydrogen-like ions are ions that possess only one electron, just like a hydrogen atom.


The fact that these ions have only one ion in the outermost shell makes it simpler to analyze their radii and energies, as a simple electrostatic model can be used to describe them. Species having multiple electrons are difficult to study and are beyond the scope of the Bohr model. This is because the inter-electronic repulsions are hard to account for in electrodynamic interactions that make up the atom's bound system.


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Single electron systems are known as hydrogenic/hydrogen-like species like He+, Li2+, Be3+, etc.


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I think one key aspect that is overlooked in these replies is the Born-Oppenheimer approximation. The nucleus of any atom can be approximated as a point with very few relativistic corrections due to its mass.


An electron on the other hand is essentially massless (comparatively speaking. I know it is still a fermion and has mass). Once you put more than one electron in any orbital those relativistic corrections have consequences. So "hydrogen-like" means any atom that does not need Pauli or Dirac's insight to explain the deviations in the spectra i.e. the Bohr model.


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