What determines the number of electrons in an atom

[thinktank1985]

Say for example we consider Helium.

It has 2 neutrons and 2 protons. Based on this, and using Schodingers equation we can calculate the energy states of the various orbitals of Helium (using self consistent field approach or by completely solving the Schodingers equation using finite difference). We also find the corresponding wave functions for these orbitals in this manner.

My quesion:

a)Since the hamiltonian is the summation of the potential and kinetic energies, the eigenvalues obtained needs a datum. So what is the datum of the energy values for the different orbitals that we obtained?

b)Can quantum mechanics proove why Helium in the normal state, has 2 electrons and not 7 or any other number?

 

[alxm]

. Based on this, and using Schodingers equation we can calculate the energy states of the various orbitals of Helium (using self consistent field approach or by completely solving the Schodingers equation using finite difference). We also find the corresponding wave functions for these orbitals in this manner.

Who's 'we'? Who solves the electronic S.E. with the finite difference method? I've never heard of anyone doing that.

Whether or not you have orbitals depends on the method you use, and there's only one real electronic Hamiltonian with one electronic wave function.

Since the hamiltonian is the summation of the potential and kinetic energies, the eigenvalues obtained needs a datum. So what is the datum of the energy values for the different orbitals that we obtained?

What do you mean, 'datum'? Are you asking what the energy values are measured against? You defined that when you defined your hamiltonian. As it were, you typically take stationary relative the coordinate system as zero kinetic energy, and infinite separation as zero Coulomb potential. Same as in classical mechanics.

Orbitals are not the same thing as energy eigenstates of the electronic Hamiltonian though, with the exception of one-electron atoms.

Can quantum mechanics proove why Helium in the normal state, has 2 electrons and not 7 or any other number?

A Helium nucleus can have zero or one or two or three or four, maybe even five electrons. It depends entirely on what the circumstances are. A third electron will prefer to be near a Helium atom than in a vacuum.

Anyway, you don't need quantum mechanics, you need Coulomb's law. Which says that it requires energy to separate two charges. So why would two otherwise-neutral atoms, in the absence of any dielectric stabilization, spontaneously separate into two charged species?

 

[thinktank1985]

Who's 'we'? Who solves the electronic S.E. with the finite difference method? I've never heard of anyone doing that.

http://en.wikipedia.org/wiki/Numerov's_method

What do you mean, 'datum'? Are you asking what the energy values are measured against? You defined that when you defined your hamiltonian. As it were, you typically take stationary relative the coordinate system as zero kinetic energy, and infinite separation as zero Coulomb potential. Same as in classical mechanics.

if I understand correctly you are saying that because the Hamiltonian is uses a certain datum(for the interaction potential), the energy eigenvalues also are determined according to the same datum.

Yes by datum, I was essentially asking what is the reference with respect to which the potential and kinetic energy (energy eigenvalue) is found.

 

[thinktank1985]

Anyway, you don't need quantum mechanics, you need Coulomb's law. Which says that it requires energy to separate two charges. So why would two otherwise-neutral atoms, in the absence of any dielectric stabilization, spontaneously separate into two charged species?

the energy eigenvalues of helium atom with 1 electron = -4/n^2*E0, where E0 = 13.6eV,

Now this doesn't include electron electron repulsion, so if n increases it just shows that the energy eigenvalue decreases in magnitude but is still <0, which means the system is stable.

If I consider electron electron interaction using SCF, and calculate the energy eigenvalue for 3 electrons will the energy eigenvalue >0. If so then it does show why helium has 2 electrons and not 3 or more in vacuum.

 

[Bob S]

A Helium nucleus can have zero or one or two or three or four, maybe even five electrons. It depends entirely on what the circumstances are.

Wow. Any references on this?

Bob S

 

[thinktank1985]

[/QUOTE]

Wow. Any references on this?

Bob S

I guess even though helium with 5 electrons may not be stable in vacuum or observable, fermi dirac statistics does show that theoretically that is possible

because of the following probability of occupying any eigenstate

f0(E)=1/(1+exp(E/kbT))

while the probability of existence of a helium atom with 3 or more electrons is very small, its not zero

p.s. I think the above equation applies to electonic band structure, consisting of a large number of atoms. I am not sure whether it would apply to single atoms. Is there any other variation of fermi function for probability of occupation of an orbital for a single atom in vacuum? But intuitively it seems something like function for single atoms should exist.

 

[Bob S]

...because of the following probability of occupying any eigenstate

f0(E)=1/(1+exp(E/kbT))

while the probability of existence of a helium atom with 3 or more electrons is very small, its not zero.

Does this eigenstate probability include the screening (shielding) by the two inner electrons in the 1s state?

Bob S

 

[cgk]

You can have any number of electrons on a helium nucleus as long as you can prevent them from flying away; by an external confining potential, for example.

Apart from that the course of action would be to just calculate the electronic system for one, two, three excess electrons, and to see whether or not it is table (with respect to the smaller one). Doing these kinds of calculations with first principles quantum chemistry is difficult, however, for various reasons, especially if the system is not stable.

There is also a nifty higher level theory for predicting the stability of dianions, which was recently proposed:

http://pubs.acs.org/doi/abs/10.1021/jp107177d

Maybe this explanation is more to your liking.