Understanding Atom Orbital Calculations: Wave Equations, Calculus, and Matrices

[Quantom]

I know they must use some sort of wave equation to calculate the exact orbitals of an atom, but I'm interested how they can be so absolutely sure that their calculated configurations are not wrong, what do they use calculus...matrices...etc?

 

[scorpion990]

They use the tools of... quantum mechanics.

Quantum mechanics states that all systems (an atom being a simple system with protons and neutrons surrounded by a collection of electrons) can be described by a mathematical function (the wavefunction). All wavefunctions must be solutions to a partial differential eigenfunction equation: the Schrödinger equation. Different "orbitals" are, for the most part, different solutions to this equation.

As for the math required to solve it... You'll need multivariable calculus to set up Schrödinger's equation in the coordinate system that it is usually solved in, and you'll need a knowledge of partial differential equations. Actually, most undergraduate quantum chemistry courses don't solve the hydrogen atom from start o finish completely =/ It's pretty involved...

As for your question: "I'm interested how they can be so absolutely sure that their calculated configurations are not wrong"
Well, we can't be absolutely sure of anything. Scientists in the 1900's thought that they knew absolutely everything that there is to know about nature, but the discovery of quantum mechanics proved them wrong. Keep in mind that quantum theory is just that... a theory.

However, current experiments, especially in spectroscopy, establish the possible reality of atomic orbitals beyond a reasonable doubt.

 

[Crosson]

Physicist always like to see chemist who are interested in learning some underlying physics on the side!

Here is a basic page with the equations that describe various hydrogen orbitals:

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

These solutions are derived from the laws of quantum mechanics. These might not be an exact description of nature, but they make the correct predictions for all the experiments that have been done so far.

 

[JPRitchie]

It should be noted that the s,p,d... etc. orbitals you refer to are solutions to the one-electron Schroedinger eqn. only. For atoms and molecules with more electrons, multi-electron systems, no exact closed solutions are known. Nonetheless, multi-electron systems can be approximated using linear combinations of hydrogenic orbitals (LCAO). Moreover, many properties of atomic and molecular systems can be explained, if not exactly predicted, using either AO's or LCAO's. Some of the strongest evidence of the fundamental role of atomic orbitals may be found in VSEPR theory. This qualitative theory predicts the structure of molecules of all kinds by considering repulsions between electron pairs occupying valence shell orbitals and hybrids. (This theory is normally described in any modern first year college chemisty text.) So, the existence of atomic orbitals in multi-electron systems has proven quite useful for a wide range of predictions, even if their existence hasn't been mathematically proved.

Regards,
Jim Ritchie

 

[colin9876]

why do the shells of atoms contain at max 2,8,8 electrons etc? seems a strange pattern
it would seem more logical to be something like 2,8,18 ... or something that increased as n squared or even n cubed??

 

[arunma]

Quantom: the solutions to the Schrödinger Equation for the Coulomb potential have a radial component and an angular component (the composite wavefunction is found by just multiplying both solutions). The angular solutions to the Schrödinger Equation are called spherical harmonics. The spherical harmonics are a family of functions defined by the quantum numbers l and m. You can look them up online, and if you have Mathematica, you can use it to plot them. Try plotting the absolute value squared for a couple of spherical harmonics and you'll see that the look exactly like the atomic orbitals.