Atomic Wavefunction Integration Question

[Kestrel]

Hello, I'm writing a program, and while I have a decent understanding of wavefunctions and atomic orbitals, this one appears to be a problem I can't beat.

I'm graphing various properties of atomic orbitals (wavefunction, wavefunction squared, radial probability distribution, and a cross section). I'm trying to find, roughly, the point at which the area under the curve reaches 90% (In otherwords, the integral of the function from 0 to x equals 90% the integral of the function from 0 to infinity). I could, I suppose, have the computer do a Riemann sum and calculate it that way, but I was wondering if there was a simpler way. The math is above me, forgive my ignorance, I'm a senior in high school with a semester of calculus under my belt.

Ideally, there would be a simple factor for each quantum number. For example, if you increase n by 1, the you would just multiply the 90% cutoff by 2, or some such number. Increase l by 1, and you would multiply the 90% cutoff point by 1.4. But I'm sure it's more complicated than that.

Any and all help would be greatly appreciated. If all else fails, I'll do a Riemann sum, but I just don't want to tax the computer any more if there's a simpler way.

Thanks in advance!
Scott

 

[gulsen]

If you have the wavefunction, you (or someone here) can calculate the integral analytically and you can simply search for that value. So, do you have the wavefunction (or at least Schrödinger equation? in it's differential form, for it might be solved)

Just for my curiosity, how are you going to make an inifinite Riemann sum?? Do you know where to cut the sum?

 

[denian]

I can understand your frustration with this problem. Atomic wavefunctions can be quite complex and difficult to calculate, especially when dealing with integrals. However, there are some methods that can make this process easier.

One approach is to use the Gaussian quadrature method, which is a numerical integration technique specifically designed for functions that are difficult to integrate using traditional methods. This method involves using a set of predetermined points and weights to approximate the integral, thus reducing the computational load on your computer.

Another approach is to use the Slater-type orbital (STO) basis set, which is a set of functions that can approximate the wavefunction of an atom. These basis functions are easier to integrate and can provide accurate results for your calculations.

In terms of your idea of using quantum numbers to simplify the calculation, it is not as straightforward as you described. While there are some relationships between quantum numbers and wavefunctions, it is not a simple multiplication or factor as you suggested. The best way to approach this problem is to use numerical methods or basis sets as mentioned above.

I hope this helps and good luck with your program! Keep exploring and learning about atomic wavefunctions, as they are fundamental to understanding the behavior of atoms and molecules.