What is the currently accepted model of atomic structure? Is there a more recent model than the Bohr or Bohr-Sommerfeld that has been observed by physicists? - by Akshat Mahaja - Quora Question review

This document contains a review of the answer by Akshat Mahaja on the question in Quora: "What is the currently accepted model of atomic structure? Is there a more recent model than the Bohr or Bohr-Sommerfeld that has been observed by physicists?"
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Reflection

1. Answer Review

The structure of the atom is now a completely solved problem - so much so that the same model can be applied to describe the behaviour of molecules, rather than just single atoms, completely, as well as the behaviour of solid materials. The main refinement since Bohr-Sommerfeld is simply the application of proper quantum mechanics to the Bohr-Sommerfeld model. What follows below is a recap of the Bohr-Sommerfeld model, its weaknesses, and a description of the main improvements using full-fledged quantum mechanics.

1.1 Recap of the Bohr-Sommerfeld Atomic Model

You have an atom. The Bohr model says that: Sommerfeld improved this by:

1.2 Weaknesses

These models only worked for hydrogen and hydrogen-like (ionised atoms with the same net charge as the hydrogen nucleus) atoms. Even for these special cases, it couldn't explain some aspects of their behaviour. Hyperfine structure (or small observed energy shifts in the actual energy of an orbit), for instance, was impossible to explain with the Bohr-Sommerfeld model. Finally, the theory was very arbitrary. Why should the integral of the momentum over a coordinate be quantised? It fits the facts, but that's hardly a solid place to stand on when trying to answer deeper questions about the atom.

1.3 The Resolution

Enter quantum mechanics.

In quantum mechanics, you throw out the idea of orbits entirely. Electrons are no longer particles or waves that have a specified position or velocity, but are described by the square root of a probability distribution of position called wave functions (yes, I know how arbitrary that sounds, but trust me, there's a good reason for this). They are thus incapable of travelling around the atom in an actual path, since they don't have an actual position.

Instead, electrons can occupy states of energy, which completely determines the sort of probability distribution they have. These energies are completely determined by the total potential and kinetic energies of the system - mathematically, you can get it by solving the Schrödinger equation, which is a complicated differential equation. If you're interested in the math, you can read a simple example here.

To get the model of the atom, you:

The beauty of this is that these solutions to the Schrodinger equation tells you (mostly) everything you need to know about how the electrons in the atom behave.

Because electrons have no definite position, you might wonder what electrons might look like when they're in a definite energy state. Below you can see images of these states: the coloured regions are where the probability of finding the electron (for experts: the integral of the wavefunction modulus squared) is greater than 90%. In deference to classical mechanics and our old ways of thinking about electrons, we call these energy states orbitals.

The famous 1s, 2s, 3s, 3p ... orbitals of a hydrogen atom. In the pictures above, you'll notice a series of numbers labeling each orbital. These are called quantum numbers, and they define the exact nature of every energy state. Let's go through them one by one:

  1. The first, most important quantum number is n, the aptly-named principal quantum number. This quantum number determines the energy of the state - however, as you'll quickly see, there's a lot more to a state than just its energy.
  2. The second quantum number is L, the azimuthal quantum number. It can only take on values between 0 to n. In other words, if n = 2, the only acceptable values of L that correspond to a real state are L=0,1 . What is the significance of this number? Well, largely, it governs the orbital angular momentum of the electron and in turn determines the shape of the orbital (in other words, the regions you see in the images above).
  3. The third quantum number is m, the magnetic quantum number. It's very closely related to L : where L fixes the total angular momentum, \m fixes the angular momentum along the z -axis (or actually just any one particular axis). Obviously, m can take on only values between − L and L - what's not so obvious is that it can only take on integer values.
    • This number specifies which sub-orbital an electron occupies. You read that right - electrons can occupy orbitals within orbitals. In actual practice, of course, this just means the shape is altered further.
  4. The fourth and final quantum number is the spin quantum number, which is somewhat unique in that it does not affect the orbital at all.
    • To be even more precise, spin is not a number produced from orbital theory - it is something intrinsic to every particle. For electrons, it can only take on the values 1/2 or -1/2 ; for more exotic objects, such as photons or collections of particles, it can take on a much larger range of values.
    • Why does it matter? We'll see in a minute.

1.4 Resolutions of Old Paradoxes

All the old problems of the Bohr-Sommerfeld model vanished: This model works for all atoms, regardless of what they are. This is because the original Bohr model did not take into account electron-electron repulsion, while the modern model does (just add a repulsion potential to Schrodinger's equation!) It explained hyperfine structure very well. The secret lay in the discovery of quantum spin, which not even Schrodinger was aware of, and requires a full-blown treatment involving the much more fundamental Dirac equation. Quantum spin essentially means that electrons and nuclei have a magnetic component to them, the interaction between which leads to small energy differences between some electrons in an orbital and others. The resulting shift is very well explained theoretically. Finally, quantum mechanics is among the most logical and straightforward theories in physics, so much so that physicists call quantum mechanical calculations 'recipes'. It's very easy to know when you're on solid footwork in quantum mechanics. In many ways, it was QM's success with atomic theory that led finally to its overwhelming popularity. Prior to it, we were merely stumbling in the dark. Making It All Intuitive Schrodinger's equation and the resulting four quantum numbers are all very well, but we'd be in a rather sorry state if we had to go dig out the solutions to Schrodinger's equations for every question. Moreover, it turns out a lot of interesting questions in chemistry need more than just a description of atomic structure: they need a description of how electrons behave in this structure. To that end, therefore, modern chemistry admits three fundamental extensions: The Pauli principle: No two matter particles in the universe can have the exact same quantum numbers. In other words, you cannot have two electrons in the same orbital and suborbital with the same spin - they must have differing spins in order to live in the same orbital. Since there are only two possible values for the spin quantum number, at most two electrons can occupy a sub-orbital. Once that happens, the sub-orbital is considered full. The Aufbau principle: There is a distinct ordering in which electrons can fill up orbitals; or, an electron picks the lowest possible energy level available to it that is not already full. In other words, if you start off with an atom with no electrons and then start adding electrons one by one, the electrons aren't free to pick which orbital they want to live in - they have to follow certain rules. The diagram above lists out the order of each orbital in increasing energy. So let's use it: first we put in one electron. The lowest energy state available to it is the 1s orbital, so it goes there. We put in another electron, this time with a different spin from the first, and find that it too goes into 1s orbital with no fuss. Now we put in a third electron, but now we're in trouble: Pauli's principle prevents us from putting in more than two electrons into a state, so we have to put it in next highest state, which turns out to be 2s. On it goes. The ordering is (1s, 2s, 2p, 3s, 3p, ...) - just follow the first arrow until you hit the head, then jump to the tail of the next arrow. 3. Finally, there's Hund's rule of maximum multiplicity: if an electron can stay single in its suborbital, it will try its best to for as long as it can. In other words, you can only put one electron into each suborbital of an orbital at first - only when you run out of suborbitals can you start going back and pairing each electron up in each suborbital. The reason for this boils down to stability issues: when worked out this way, electrons are less likely to move to another orbital or sub-orbital. This is the method of maximum stability. And that's it. Congratulations! You now have a high schooler's understanding of the modern theory of atomic structure.

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Created: 1 May 2024

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