Whilst the things we don’t know about quantum mechanics could fill a black hole, it’s still thought of as a glorious theory that swept in and revolutionised atomic theory. In a way it has, well, revolutionised one atom: hydrogen. Because that’s the only atom we know how to “solve”.

A quantum-mechanical solution to an atom is one that tells us where the electrons are. Electrons can’t have any energy level they want, they can only have certain values - values we call discrete and quantised energy levels; you may have heard them called orbitals or shells. We can find where the atoms are like a cop tracking some dodgy dealers. They move around a lot, but they have the same stopping points along their journey. If the cop knows these stopping points, she can lie in wait for the dealers to turn up.

Orbital theory and Newtonian physics together tell us that a charged particle constantly turning as it travels in a circle should radiate energy and slowly spiral towards the centre, but we know this doesn’t happen - otherwise every atom would become a miniature black hole. This is why the Bohr model of shells has been used since it was introduced in 1913. In this model, electrons are trapped in specific stationary energy states, like children playing musical chairs. Electrons are not allowed to leave their “chairs”, except to jump to new ones, absorbing or emitting the energy difference. This energy may be detected and measured as light. Practically, these measurements tell us where the energy states are, but we still can’t predict them theoretically.

The hydrogen atom is a two body system; there is a positively charged nucleus and a negatively charged electron, no more, no less. The Bohr model models the charge and gravity interactions between these two particles.

With a bit of maths, the model solutions are the quantum energy states.

At first glance, this seems enough - if we can solve it for hydrogen, the energy levels should be the same for all other kinds of atoms. But we’re pretty sure this isn’t true from experimental measurements and sensible chemical guesses.

Multi-electron systems may only be

Mathematicians look for theories as a way to explain the observations and predict further ones using our knowledge of a simpler system. Our current quantum mechanical model breaks when it gets above one electron.

However… the theory fits for other two body systems with exact solutions called Hydrogenic atoms. Hydrogenic atoms are positively charged elements that have had all but one electron removed, such as He

However, these systems are very complicated to make, and unstable. They don’t tell us much about real atoms, less about real compounds or chemistry. In fact, just looking at how complicated the hydrogen atom model is, and how simple hydrogen is compared with everything else, shows how much further science has left to go.

One model that might help "fix" the two particle limitation of the Bohr model is the Mean Field Theory model. This sums and averages all the effects of other particles, essentially modelling them as one giant average particle. This reduces the problem to a Bohr-like two body problem again, which is easy to solve. However, the Mean Field Theory is a laborious piece of maths and a mean feat to apply. Both theories have their pros and cons - the simplicity and speed of the Bohr model, but a limited range, may be expanded by the Mean Field Theory, but takes a long time, is hard work, and still relies on the Bohr or other atomic models at heart to give the final solution. Relativistic modelling of nuclear movement and corrections to ensure the electron or Mean Field Theory mega particle can't accidentally go inside the nucleus have also been developed to expand and optimise the Bohr model. Yet so far, no theory is precise, all-encompassing and straightforward...

The hydrogen atom consists of just one proton in the nucleus and one surrounding electron: a simple system to model. Image credit: TWDK / Rowena Fletcher-Wood |

#### What do we mean by “solve an atom”?

A quantum-mechanical solution to an atom is one that tells us where the electrons are. Electrons can’t have any energy level they want, they can only have certain values - values we call discrete and quantised energy levels; you may have heard them called orbitals or shells. We can find where the atoms are like a cop tracking some dodgy dealers. They move around a lot, but they have the same stopping points along their journey. If the cop knows these stopping points, she can lie in wait for the dealers to turn up.

Orbital theory and Newtonian physics together tell us that a charged particle constantly turning as it travels in a circle should radiate energy and slowly spiral towards the centre, but we know this doesn’t happen - otherwise every atom would become a miniature black hole. This is why the Bohr model of shells has been used since it was introduced in 1913. In this model, electrons are trapped in specific stationary energy states, like children playing musical chairs. Electrons are not allowed to leave their “chairs”, except to jump to new ones, absorbing or emitting the energy difference. This energy may be detected and measured as light. Practically, these measurements tell us where the energy states are, but we still can’t predict them theoretically.

There are 5 classically visible light lines in the hydrogen emission spectrum. Each line is formed by electrons moving from one energy level to another (specifically, moving down to level 2 from higher levels). These levels are often described as "electron shells" surrounding the nucleus, with energy states labelled outwards from the nucleus, starting at 1.Image credits: ©TWDK/R Fletcher-Wood/Jan Homann (CC BY-SA 3.0). Hydrogen emission spectrum by Jan Homann , via Wikimedia Commons. Shell model by R Fletcher-Wood / TWDK. |

#### Why hydrogen but nothing else?

The hydrogen atom is a two body system; there is a positively charged nucleus and a negatively charged electron, no more, no less. The Bohr model models the charge and gravity interactions between these two particles.

With a bit of maths, the model solutions are the quantum energy states.

#### Multi-electron systems or “everything else”

At first glance, this seems enough - if we can solve it for hydrogen, the energy levels should be the same for all other kinds of atoms. But we’re pretty sure this isn’t true from experimental measurements and sensible chemical guesses.

Multi-electron systems may only be

*approximated*using experimental evidence because electrons perturb other electrons (and nuclei perturb other nuclei). The distance of each electron from the nucleus affects the pulling power of the nucleus as a whole and so the spacing between the quantum energy states. Everything is interdependent. It’s complicated. It’s messy. And the electrons won’t hold still for a second, making it too unwieldy for a simple theoretical model.Mathematicians look for theories as a way to explain the observations and predict further ones using our knowledge of a simpler system. Our current quantum mechanical model breaks when it gets above one electron.

#### Hydrogenic Atoms

However… the theory fits for other two body systems with exact solutions called Hydrogenic atoms. Hydrogenic atoms are positively charged elements that have had all but one electron removed, such as He

^{+}or Li^{2+}. They have bigger nuclei than the hydrogen atom, and thus allow us to compare how the nuclear charge affects where the energy states are. We can add more protons and neutrons to the nucleus and still solve the equation, but not more free-moving electrons or any kind of bonding to other atoms.However, these systems are very complicated to make, and unstable. They don’t tell us much about real atoms, less about real compounds or chemistry. In fact, just looking at how complicated the hydrogen atom model is, and how simple hydrogen is compared with everything else, shows how much further science has left to go.

One model that might help "fix" the two particle limitation of the Bohr model is the Mean Field Theory model. This sums and averages all the effects of other particles, essentially modelling them as one giant average particle. This reduces the problem to a Bohr-like two body problem again, which is easy to solve. However, the Mean Field Theory is a laborious piece of maths and a mean feat to apply. Both theories have their pros and cons - the simplicity and speed of the Bohr model, but a limited range, may be expanded by the Mean Field Theory, but takes a long time, is hard work, and still relies on the Bohr or other atomic models at heart to give the final solution. Relativistic modelling of nuclear movement and corrections to ensure the electron or Mean Field Theory mega particle can't accidentally go inside the nucleus have also been developed to expand and optimise the Bohr model. Yet so far, no theory is precise, all-encompassing and straightforward...

*This article was written by TWDK's chemistry editor Rowena Fletcher-Wood, who has a PhD in environmental materials chemistry from the University of Birmingham. Rowena can be found on twitter as @RowenaFW*.
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