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Alternative approach to building the mathematical model of the molecule:

We built a model of an atomic system just like if it was like our solar system. The only difference was that we apply the principles of quantum mechanics to atomic systems and the principles of classical mechanics to solar systems.

But there is another way of modelling a system.
If we observe a substance ( any substance in your environment ) it can be considered as a gas of electrons and nucleï. Just like we calculated a wave function for the electrons, we can calculate an electron density function for an inhomogenious electron gas corresponding to any substance.

Do not confused the gas mentioned here with the chemical notion of gasses, liquids or solids because in considering these concepts we look at the interactions between molecules or atoms, while now we focus on interactions of subatomic particles . This way a solid can be represented by a gas of elementary particles.

Here we are dealing with the subject of the quantum mechanical ensemble theory, where we elaborate on the grand canonical ensemble for molecular systems. This elaboration has resulted in the density functional theory for atoms and molecules.
This approach allows us to calculate the electron density and the energy of a system in its ground state at zero degrees Kelvin.
The approach is completely complementary the classical approach presented above.