Electron Density: Probability of Finding Electrons?

[ldv1452]

In an orbital with 2 electrons is electron density the probability of finding a particular electron in that region or the probability of finding either of the 2 electrons (the probability of finding electron 1 plus the probability of finding electron 2)?

 

[alxm]

You can define (and use) both kinds of density. Depends on the context. But usually when you talk about electronic density you're talking about the single-electron density, i.e. the probability of finding any electron at some position.

 

[ldv1452]

You can define (and use) both kinds of density. Depends on the context. But usually when you talk about electronic density you're talking about the single-electron density, i.e. the probability of finding any electron at some position.

So the probability of finding electron 1 is equal to the probability of finding electon 2 in a 2 electron orbital?

 

[sweet springs]

Hi. We cannot distinguish electrons in the same orbit (without introducing another parameter i.e. component of spin).
In another case of Be atom for an example, we neither cannot distinguish 1S electron and 2S electron in observation to find electron in a volume element.
Regards.

 

[alxm]

So the probability of finding electron 1 is equal to the probability of finding electon 2 in a 2 electron orbital?

Finding them "in an orbital" would the probability of the electron having a certain energy. Do you mean that, or the probability of finding the electron in a certain location? These are two different things.

As for the first alternative: Not necessarily; the two electrons have different spin (in fact, that's the only 'label' you can give them which will make any sense). An orbital isn't necessarily either occupied or not; you only have a probability of finding the electron in a given orbital. And the orbital populations (as it's called), aren't necessarily the same for the different spins; it depends on the system. If you have the same occupancies, then the electrons are indistinguishable. So naturally the probabilities will be the same.

However, when you're asking these questions, I suppose I should point out that this is where the limitations of the orbital picture come into play: Orbitals are single particle functions, meaning the probabilities are not correlated. If you look at the probability density u(x₁,x₂) that gives the probability that electron 1 is at x₁ while electron 2 is at x₂, when working with orbitals, then the two variables are independent of each other. The part of the probability function that depends on x₂ will not change when x₁ changes and vice-versa. In other words, the orbital picture assumes that electrons move independently of each other, that they're uncorrelated. (in statistics, two variables are uncorrelated if the probability of events A together with B is the product of P(A) and P(B))

In reality this is not true. The probability of where one electron is should naturally depend on where the other electron is. But this error is not so big that it limits the descriptive usefulness of the orbital picture, because the general shape and density is accurate to over 95%. To get better accuracy you have to view the electrons as being in multiple orbitals simultaneously. (or abandon the whole orbital approach) But if you just started to learn about orbitals now (as your questions would indicate), then you shouldn't worry about that just yet. That's more in-depth quantum chemistry. (In fact, the correlation problem is the central problem of QC).

 

[ldv1452]

Finding them "in an orbital" would the probability of the electron having a certain energy. Do you mean that, or the probability of finding the electron in a certain location? These are two different things.

As for the first alternative: Not necessarily; the two electrons have different spin (in fact, that's the only 'label' you can give them which will make any sense). An orbital isn't necessarily either occupied or not; you only have a probability of finding the electron in a given orbital. And the orbital populations (as it's called), aren't necessarily the same for the different spins; it depends on the system. If you have the same occupancies, then the electrons are indistinguishable. So naturally the probabilities will be the same.

However, when you're asking these questions, I suppose I should point out that this is where the limitations of the orbital picture come into play: Orbitals are single particle functions, meaning the probabilities are not correlated. If you look at the probability density u(x₁,x₂) that gives the probability that electron 1 is at x₁ while electron 2 is at x₂, when working with orbitals, then the two variables are independent of each other. The part of the probability function that depends on x₂ will not change when x₁ changes and vice-versa. In other words, the orbital picture assumes that electrons move independently of each other, that they're uncorrelated. (in statistics, two variables are uncorrelated if the probability of events A together with B is the product of P(A) and P(B))

In reality this is not true. The probability of where one electron is should naturally depend on where the other electron is. But this error is not so big that it limits the descriptive usefulness of the orbital picture, because the general shape and density is accurate to over 95%. To get better accuracy you have to view the electrons as being in multiple orbitals simultaneously. (or abandon the whole orbital approach) But if you just started to learn about orbitals now (as your questions would indicate), then you shouldn't worry about that just yet. That's more in-depth quantum chemistry. (In fact, the correlation problem is the central problem of QC).

Thank you. This is very interesting. And yes, I am new to learning all this so I'm sure it will take more background for me to fully understand these explanations, however, I am learning bit by bit with each question and explanation and it should all fit together better over time.