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Dear developers of ChemLLM:

Hi,there! There are several grammer errors about the below response, such as don' tkindly forget everything relatedwith and You probably won;t be able tell him. Maybe it is the format error from the convert from pdf to text file? I am just curious about this...

What is DFT(Density functional theory)? Please teach me through an example. I am a beginner in chemistry and physics.

I will try to explain it as simple (and hopefully correct)as possible:

The Schrödinger equation for quantum mechanics describes the behavior of electrons around nuclei, but solving this exact differential equations requires very powerful computers or even supercomputers with lots memory space(which are not available at all times). So we need some methods that can solve these types

of problems without using too much computational power/effort/time.So basically what density function theories do: they use approximations which allow us calculate electronic structure properties such like energy levels etc., by only knowing about electron densities instead directly calculating wave functions.(Source)

So let's say you have $n$ number(say 2-3 dozen numbers here just so i don't get lost while typing :) )electrons orbiting your nucleus.Now if someone asks how many neutron does my atom has? You probably won;t be able tell him right away because finding out their position would require more effort than needed(i.e measuring them one after another), therefore there must exist something called "probability distribution",a.k.a probability amplitude aka orbital cloud(a visual representation):

(source.)

Now since our goal was figuring-out probabilities rather then actual positions,and given enough time,money/powerful computer resources,you could find those values exactly via numerical integration techniques.But again,since doing calculations isn’t free(and sometimes impossible),

we want better ways. Luckily physicists came up pretty close solution - namely Born rule :

$$p(x)=\mathrm{\mid }\psi {\mathrm{\mid }}^{4}p(x)=\; |\backslash psi|^4$$ where p denotes probablity,$x-$space coordinate,

$\phi= \frac{1}{V}e^{-iEt/V}$is plane-wavefunction,i,e simplest type spherical harmonic expansion.

But now comes interesting part :since most atoms aren't isolated systems,

there exists interactions between different parts(electrostatic repulsion/electric field interaction).

This leads into Fermions interacting together forming molecules/blocks-of-electons-a-know-as-molecular-orbitals(MOs)(think shells+subshells ). Now when talking MOs,don' tkindly forget everything relatedwith shell filling rules(except maybe Hund;s Rule ),because usually speaking,this stuff doesn make any sense anymore.It'a lot easier thinking termslike bond order(bond strength)-stronger bonds mean higher occupation factor on corresponding molecular/orbaitl level;

respective energies also change accordingly