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In my inorganic chemistry class, we were introduced to computational chemistry. We are told to use B3LYP theory in the program, but we never actually learned any of the computation yet. Could anyone explain simply what B3LYP is and how it fits into the DFT equation that looks like
$$ E_{\text{DFT}}[\rho] = T_{e}[\rho] + V_{ne}[\rho] + J[\rho] + E_{xc}[\rho] $$
'The level of theory' is a fancy word for scheme used to calculate energy of a molecule. There are quite many ways to do it, most very computationally expensive. However, quite recently it was proven that state energy depends on electron density distribution only, and details of correlation of electron movement are derivable from said distribution in all entirety. This gave rise to density functional methods, where instead of considering all details of electron movement, only electron density is considered. Theoretically this allows to lower computational requirements from fourth-seventh (depends on the method used) power of number of basis functions to only third power of said number. This is a big deal.
The problem is that exact and universal way to derive energy of a system of electrons from their distribution is unknown. Thus, a lot of various ways were tested and most successful ones made it into available software.
Most of such ways - functionals - split energy of the system into several parts. Some of them may be known exactly, such as energy of electron - nucleui interaction. Some, however, are not, such as energy of electron-electron interaction. Still, several border cases were considered in theoretical physics. Specifically, functional for electron gas is known and was used. This gave raise to so known Local Density Approximation. It performs tolerable in many cases.
An interesting option is to add so-known exact exchange into the mix. Essencially, exact exchange is an attempt to enforce Pauli principle by hand, i.e. that two electrons with same spin cannot occupy the same spot. The problem is, that part of it is already included in the base LDA, so that member is usually considered with reduced weight, say, 0.25.
Another possible way is to try to include members dependent on electron density gradient, to aknowledge that electron density in a molecule varies from point to point. This is known as GGA approach.
B3lyp is a functional, that includes exact exchange and GGA corrections in addition to LDA electron-electron and electron-nuclei energy. The weights of the parts were fit to reproduce geometry of a test suite of small molecules. As such use of b3lyp for calculations with heavier atoms is questionable.
Density functionals perform poorly when dispersion interactions play significant role, thought correction schemes for this are also known.
Further details are not important for you at the moment. However, consider grabbing some book on DFT and quantum chemistry in general. If you ever end in 'real' chemistry, this would come handy, as computational chemistry papers are present in big numbers and often offer significant insight.
