Molecular orbitals don't get it

[shredder666]

Molecular orbitals... don't get it

Hi I'm new, to this forum and molecular science

Reading a bit about molecular orbitals, and I have to say the whole concept of anti bonding orbitals and bonding orbitals got me quite confused.

when atomic wave functions add you get bonding orbitals
when atomic wave functions subtract you get antibonding orbitals

well that's what my textbook said...

1.How can the same wave functions of let's say.. a diatomic molecular subtract and add at the same time?

2.How can/ does anti bonding orbitals (referred to as ABO from now on) and bonding orbitals (BO) exist at the same time?

3.What exactly is the ABO?

4.How can the ABO and the BO's exist at the same time without interacting with each other (like how atomic orbitals can hybridize) ?

Please try to answer any of these questions.

Thank you

 

[kanato]

Consider two hydrogen atoms in 1D, located at x = 0 and x = a. We will examine only simplified 1s orbitals:

$$\phi_1(x) = A e^{-|x|}$$
$$\phi_2(x) = A e^{-|x-a|}$$

We can choose A to be real and positive, so both wavefunctions are positive everywhere.
Now, how would electrons like to behave if there were two protons like this? The intuitive result is that the electron is going to want to be close to both protons, so it will wind up in something that looks like a linear combination of the atomic 1s orbitals. There are two such combinations:

$$\psi_b(x) = C (\phi_1(x) + \phi_2(x))$$
$$\psi_a(x) = C (\phi_1(x) - \phi_2(x)($$

You should get out a graphing calculator and plot both $$\psi_b$$ and $$\psi_a$$. Now $$\psi_b$$ is positive everywhere, and most of its weight is concentrated between the atoms. So $$\psi_b$$ is called bonding, because the electron acts like "glue" between the two atoms. $$\psi_a$$ goes to zero between the two atoms, and you actually see more weight "outside" the atoms. In state $$\psi_a$$ the electron is actually repelled from the central region, and you end up with two positively charged protons without much negative charge in between them, so they will be repelled from each other. This state is called antibonding because of this. So that is to answer question #3.

The rest of your questions are about what happens when you have multiple electrons. In a single atom problem, if I have multiple electrons, I put one electron in each state (two if you don't want to think about spin) starting from the lowest energy until I run out of electrons. In a molecular system, you do the same thing, but the lowest energy state in this example is the bonding $$\sigma$$ orbital formed from the two 1s states, and then the next lowest energy state is the antibonding $$\sigma^*$$ orbital formed from the two 1s states. In a hydrogen molecule, you will have two electrons in the bonding orbital, and if you go to $$H_2^-$$ you will have added an electron to the antibonding orbital. So that answers question #1 and #2.

To answer question #4, yes electrons in different states do interact with each other, but the details of the interactions are not all that important, and can usually be approximated by just having an effect on the energy levels of the various MO's.

 

[shredder666]

I don't have a graphing calculator ._.

Thank you for answering, is it possible for me to look at the general "shape" of the equation? I have no clue what you're talking about in the first 2 paragraphs lol. When I said I'm new, I'm REALLY new.

But nevertheless, I could make out a few things on my own. But I still have some confusions regarding this topic.

WHen you said "if there were two protons like this", I'm not sure you meant by "like this" does protons follow a similar wave function similar to an electron?

I'm more confused to why we're adding and subtracting, from what you said, anti-bonding orbitals and bonding orbitals DO exist at the same time, but then wouldn't that mean you're adding and subtracting the same two orbital equations at the same time? I do not get how that could work.

Oh yea and just for clarification, let's say we had an antibonding orbital that looked like an apple and an bonding orbital that looked like a pear, so the overall cloud would be an apple within a pear (or apple + pear)?

 

[PhaseShifter]

I'm more confused to why we're adding and subtracting, from what you said, anti-bonding orbitals and bonding orbitals DO exist at the same time, but then wouldn't that mean you're adding and subtracting the same two orbital equations at the same time? I do not get how that could work.

It's analogous to picking a coordinate system for a two-dimensional problem. The system can be described as either two molecular orbitals, or two atomic orbitals.

Any linear combination of the atomic orbitals should be cabable of being described as a linear combination of molecular orbitals and vice versa. So if you combine two atomic orbitals, you will get two molecular orbitals, if you combine three atomic orbitals you will get three molecular orbitals, etc.

Mathematically, you could combine them an infinite number of ways--but addition and subtraction are chosen (among other reasons) because they provide a wavefunction with maximum energy and a wavefunction of minimum energy due to the electrons being concentrated between the nuclei vs. out at the ends of the molecule.

 

[shredder666]

ok thanks

 

[kanato]

Well, if you don't have a graphing calculator, that's no excuse :) There's a lot of applets around on the web, like this one:
http://my.hrw.com/math06_07/nsmedia/tools/Graph_Calculator/graphCalc.html

When I said "if there were two protons like this," I was referring to the hydrogen nuclei in the arrangement I specified (one at x = 0 and one at x = a).

Also, to clarify, antibonding orbitals always have a nodal surface in between the atoms. So you can't have an antibonding orbital that looks like an apple.

 

[shredder666]

wait is the whole "subtracting and adding" thing similar to superposition?

also, the movement of electrons is always uncertain because it only has a boundary limit instead of a definite orbit, how does the variation of electron positions/movement affect the combined waves?

I have a hard time understanding what I just wrote so I'm going to say it this way.

the function of sin x has a neat and uniform motion, but I doubt electron orbital wave functions are like that (non-uniform), how does this non-uniformity affect the combined MOs?

 

[PhaseShifter]

wait is the whole "subtracting and adding" thing similar to superposition?

Yes.

also, the movement of electrons is always uncertain because it only has a boundary limit instead of a definite orbit, how does the variation of electron positions/movement affect the combined waves?

I have a hard time understanding what I just wrote so I'm going to say it this way.

the function of sin x has a neat and uniform motion, but I doubt electron orbital wave functions are like that (non-uniform), how does this non-uniformity affect the combined MOs?

Basically the wavefunctions not being simple sine waves just means the electrons don't have a well-defined linear momentum--but then again, that's you would expect. Electrons with a given amount of energy tend to go faster near the nuclei and slow down as they move away.

For a more detailed explanation you might want to check the quantum mechanics forum. I would be surprised if some lengthy explanation doesn't already exist there.