Its called the
Schrödinger Equation, and its derrivation is pretty complex. And even if you get that far, its not easy solving a non-linear second order differential equation. How far have you gone with mathematics; do you know calculus and diff. eq.? Here's a site that shows you the main equation modeling movement, position, and waveform of the electron. The equation is shown here, but this link has a tiny bit of info.
http://www.physlink.com/Education/AskExperts/ae329.cfm
http://www.thestargarden.co.uk/Schrodinger.html
It is used in pair with the
de Broglie wavelength and the
Planck relation, as well as shrodinger wave function. I posted a couple sites that show you the math and equations, but no real explanation as where they come from and all steps to show it works; but it gives you a general idea if you've got the knowledge for this kind of mathematics. The links i provided are referring to the "particle in a box" 1 dimensional model, like a vibrating string. Once solved, you can calculate position, orbital energy, as well as a couple others. However, this 1D model is far from the true nature of a 3D orbital; so you end up with a block set of second order diff. eq's where each electron has its own wave function, and the nasty part is each one tugs and applies electromagnetic forces on the other orbiting electrons. The simplest one to use these equations and graph the e- energy and get the waveforms is H2 because there are only 2 electrons. As you go up the periodic table, orbitals get more complex geometry, and more electron shells that repel each other. It would be impossible to model even a small molecule this way (stims from MO theory). Say acetic acid...2xC, 2xO, 4xH gives it (O=16) + (C=12) + (H=4) = 32 total electrons. So that means, a multi variable block of equations that look similar to the one above would have to be solved for 32 different wave equations! However, computer modeling can make millions of calculations a second, so there is good software to actually do and use this. HyperCHEM is excellent, it has all the above i mentioned plus much more for modeling, and the program gives very accurate conformations, energy values, you can show electron cloud density, all modeled in 3D. Modeling a reaction is even possible, but its tricky because you have to have every bank of rules that it calculates with set to the right kind of models, or it won't work. Look up particle in a box theory of electrons, the schrodinger eq. starts there
