The applications are vast. First principles study of industrial processes, drug design and discovery, catalyst design, even simulation of biological systems can rely on quantum chemical calculations.
For the most part, the goal of a quantum chemical calculation is as follows: given a configuration of atoms (or more appropriately nuclei), output the multi-electron wave function and the electronic energy. It just so happens that electron wave functions follow the Schrodinger Equation, and the SE for a 100 or so electrons (like in a typical molecule) is impossible to solve exactly; the problem was of such great importance than the people who came up with the theory to allow for an estimated (but still useful) solution won the Nobel Prize for Chemistry.[2]
One of the most common tasks in quantum chemistry is called structure optimization. Suppose you create a water molecule in a molecular editor. You place two hydrogen atoms near an oxygen atom and connect them. Is this water? The formula is ($H_{2}O$), but real molecules do not exist in arbitrary shapes. The O–H bond length and the H–O–H bond angle have specific equilibrium values that arise from the underlying quantum mechanics of the electrons and nuclei.
For water, the O–H bond length is about 96 picometers, and the H–O–H bond angle is about 104.5°. In reality, the molecule vibrates around these values due to thermal motion and quantum zero-point energy, but the equilibrium geometry is close to these numbers.
If you want to simulate a reaction involving water, it is best to begin with this equilibrium configuration. You could manually set bond lengths and angles in your molecular editor. But imagine, a molecule with 50 or so atoms, or a biomolecule with a 1000 or so atoms, this would get ugly rather quickly!
[unfinished post]
[1] Machine Learning the Computational Cost of Quantum Chemistry, Heinen et al (2020)
[2] Walter Kohn and Arthur Sham
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