Does there exist a mathematical model for determining the electron configuration of an atom? I mean the theory which would generalize the notion behind the informal elements of the Aufbau principle.
Yes, and this question filled a semester in my 4th year of Physics.
The Schrödinger's equation describes the problem exactly... with the caveat that it is impossible to solve exactly for more than one electron. A typical atomic physics course will then proceed to try to develop different methods and heuristics to solve this problem.
The basic idea of the shell model is as follows:
- Electrons are grouped at energy shells (the ones given by the solution of Schröodinger's equation)
- The complete shells behave similarly to an spherical envelope:
- The shells external to a given one do not affect it.
- They will partially screen the charge from the nucleus; i.e.: if you have $_3Li$, the third electron will only see charge 1, because the two electrons in the innermost shell will cover it.
- The electrons inside a shell behave more or less independently.
The last premise requires some explanation. If you solve the angular part of the equation for one electron, you get a family of possible solutions, the spherical harmonics. Each electron in a shell will follow one of them.
The number of spherical harmonics there can be is tightly linked to the rule you asked.
This model is not perfect, but it an be quite good. In order to improve it we can introduce corrections: each pair of electrons in the same shell will repel each other, so the atom will be a bit bigger, and the binding energy a bit smaller.
It must be noted at this point that the exact wavefunction is, in many cases, not computed. What we see in most experiments are the energy levels, and this is what the methods are focused on (and it also happens to be much much easier, lucky us).
Yes! There is such a model or equation governing the electron configuration of an atom!
It is Schrödinger's Wave Equation. You need to solve it for electrons surrounding an atom, and it is the reason why the Aufbau principle works. (If you want to do it yourself, I suggest starting with Hydrogen, as it is the easiest potential to work in. You'll need to know multivariable calculus, and a quantum mechanics book would be a good guide.) That equation even dictates which energy levels are filled when!
This is something that some physicists cover in an undergraduate quantum mechanics course. Otherwise, stick with Aufbau, as it's a "dumbing down" of the conclusions given by Schrödinger's Wave Equation.
(Due to this, you could even think of Chemistry as the study of things mostly subject to Quantum Mechanics, so your Chemistry class is a physics class! It's a silly way to think...)
I think if we understand the nature of electron and proton, we may be able find closest answer. we can use coulomb's equation for instance, F=Q^2*k/r^2 to see the behavior of electrons in an atom, and we can take care of repulsion forces, by introducing little bit of linear algebra. for instance how for does an electron needs to be from that and that electron to remain in appropriate motion. But then there is uncertainty principal, which says something like electrons don't have any specified position, if that is to be true, then our classical mechanics is somewhat not that much useful. the best model that i can think of is thinking of electrons as little point masses, and avoiding quantum mechanics, or rather using old newtonian physics.