What is the mathematical function for the atomic orbital?

[Coolamebe]

I keep seeing that "An atomic orbital is a mathematical function that describes the wave-like behaviour of either one electron or a pair of electrons in an atom" or something similar. However I can't find what the mathematical function is. Can someone tell me or link me to it please?

 

[chad kimsey]

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/Schrödinger.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 Schrödinger eq. starts there

 

[Coolamebe]

Thanks, I'm not too far in calculus, I've done some differential stuff (recently completed minima and maxima) and a little bit of integration.Thanks for all of the resources though!

 

[chad kimsey]

those links are simplified a decent amount, but without differential equations (solving them) it's even harder to understand. There are some really good sites that go into much more detail if youre interested look up "particle in a box" and jump around from some of the links that will most likely be there to explain other things. Heck, without practice or fooling with it on a regular basis, I don't understand all of it now. Lots of other pieces of info and findings lead up to this main equation and you have to understand or at least have seen it. But in the end, it gets too complicated to solve by hand quickly so using software designed to spit out energy, orbital densitiy probability, modeling dipole moments, etc. If you ever take physical chemistry its usually a senior class in chemical engineering bachelors degree. If you're really just curious look on amazon or somewhere for a physical chemistry book. It starts off with the ideal gas law, deriving non-ideal versions, then states of matter and phase diagrams, detailed explanation on MO theory and all kinds of things before you hit this subject. Its not too hard to follow reading for yourself if you know the mathematics. At first its just algebra and a bit of calc for minimizing etc. Later though, you must be able to create models, which creates differential equations to solve for that model.

 

[Khashishi]

While for a molecule, the orbitals are complicated, for the simpler problem which is the hydrogen atom, the orbitals aren't too difficult to write down. In this case, the orbitals are the product of a radial wavefunction and a spherical harmonic. If you search for those terms, you will find their forms.

 

[jtbell]

The Schrödinger equation is what you solve to get the actual functions for the orbitals. Here are the functions for some of the lower-energy orbitals for hydrogen:

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydwf.html#c3