The Schrödinger equation is the equation that describes the wave function of a quantum particle, most often an electron. For a detailed explanation visit the Brilliant Cosmos page here. There are difficulties when it comes to solving this equation for more complicated scenarios than when the electron is just travelling through empty space. Added difficulty comes when attempting to convert the Schrödinger equation solutions out of their natural polar form and into a Cartesian one as to view the actual shape of the electron cloud in three dimensions. Although it may be necessary for students to learn about how to solve equations and draw graphs the truth is that in real life scientists just get computers to do all that for them.
There are many different programs that can solve differential equations like Schrödinger’s based on a set of functions that describe orbits called basis sets. The problem is that even the best computers begin to struggle when many particle systems are modelled and so small basis sets are what we are limited to. In order to sort this out computer scientists have designed an iterative method to solving the Schrödinger equation. This isn’t making the computers any more powerful or really any more efficient, instead it is just optimising these small basis sets so that they create the most accurate solution they can possibly produce. By performing a set of operations on the original functions they can be continually improved until the required accuracy has been reached. The paper has demonstrated applying this method towards a number of different scenarios each with solutions that relate to the Schrödinger equation or a similar differential and appears to be quite successful (although if the number of particles already maxes out the computer capabilities then this method finds convergence far from the true value). This research means that with given computer resources quantum systems can be modelled in the absolute optimum way. This hopefully means that quantum research using computer modelling will now be slightly more accessible as some models that were previously outside the capabilities of an institutions computers will now be solvable.