[d.lam.86]

Question 1:

Hi once again. I have been having great confusion on understanding how antibonding orbitals work but more so on how they exist. The textbook I have been reading doesn't fully explain how they exist. They only briefly define it as "molecular orbitals having zero values in the region between two nulclei and therefore concentrated in other regions". After having read that definition, I had come to the conclusion that antibonding orbitals are orbitals containing non-bonding valence electrons, but lo and behold, it isn't as I read further that they are also the existence of "non-bonding orbitals".

Could you please explain what use antibonding orbitals have in helping chemists understand more about orbital theory?

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Question 2:

Another strange thing about valence bond theory that I have noticed is that hybridized orbitals don't follow Hund's rule (placing electrons with same spin in one subshell before pairing them up afterwards) completely. For instance, in the ethylene molecule (C₂H₄), the hybridized C atom has singly-paired electrions in each sp² hybrid orbital and then one valence electron goes in a 2p orbital. Why does that occur? Why wouldn't it be possible for that single electron in the 2p orbital to pair up with one of the singly-occupied sp² hybrid orbitals as this feels more consistent with Hund's rule?

Thank you in advance!

[azmanam]

Let's start from the top.

In order to have molecules, two (or more) elements must bond.  the atomic orbitals from one element combine with the atomic orbitals from another element to form a (covalent, polar covalent, or ionic) bond.

Let's start simple.  Assume two singular hydrogen atoms.  Each hydrogen atom has one electron in its 1s orbital.  The 1s orbital is a sphere centered on the nucleus.  Now, what is an orbital?  The electron is zipping around the nucleus with some momentum, and at any given moment that electron is at some definite position (see the Heisenberg Uncertainty Principle for discussion of calculating momentum or position of an electron). 

Qualitatively, we say the electron orbits the nucleus; but what does that mean quantitatively?  There is a series of complicated mathematical equations (see wave functions and the Schrödinger equation) that allow us to determine mathematically where an electron is likely to be located.  Electrons can be located theoretically anywhere, but the solutions to those mathematical equations determine the probability of finding an electron in a given region of space.  For the 1s orbital of a hydrogen atom, the probability of finding an electron - as determined by those equations - is defined by a sphere centered on the nucleus (the 1s orbital).  We can say with a good deal of certainty that if one wanted to determine the position of an electron at any given time, it will be found in that region of space defined by the solutions to those equations - namely that sphere centered around the nucleus, the 1s orbital.

Things change, however, as we add more complexity to the system.  If two hydrogen atoms (complete with their one electron located in the probability density cloud (the sphere) defined by the solutions to those equations) approach each other and attempt to form a bond, the equations change, as do their solutions, and the region of space in which it is most probable to find an electron changes.

Couple of things to mention at this point.  One 1s orbital from one hydrogen atom and a second 1s orbital from another hydrogen atom approach and the electron from each atom will combine to form a sigma bond (single bond).  Another way of saying this is the electrons contained in two 1s atomic orbitals combine to form molecular orbitals in a sigma bond.  Molecular orbital theory dictates that the number of atomic orbitals in the beginning must equal the number of molecular orbitals at the end.  If two atomic orbitals were present at the beginning, two molecular orbitals must be present at the end.

So what does this physically mean?  We know that the two single electrons from each hydrogen atom will combine to form a sigma bond with two paired electrons.  Those two electrons come from each hydrogen atom's 1s orbital.  When those atoms combine, what we're really saying is those orbitals combine in what is called the linear combination of atomic orbitals (LCAO).  LCAO is how we manipulate the complex equations to allow us to find the solution for this new molecule.  LCAO involves both a) taking the equation for one of the atoms and adding it to the equation for the other atom and b) taking the equation for one of the atoms and subtracting it from the equation for the other atom.

The new solutions to these linearly combined equations defines the probability density clouds for the electron(s) in the new molecular orbitals.  The addition of the two equations can be thought of as analogous to constructive wave interference.  The constructive interference defines the region of space where it is most likely that the pair of electrons will be found.  For our hydrogen atom, the solution to the constructively combined equation will be a cylindrical region located between the two nuclei (the sigma bonding orbital).  This bonding molecular orbital is lower in energy that either of the two atomic orbitals.  It is in this region of space defined by the solution to the equation for the constructive interference of the 1s orbitals that the probability of finding the electrons is greatest.

The subtraction of the two equations can be thought of as analogous to destructive wave interference.  This destructive interference defines the region of space where it is less likely to find the electrons.  This results in sigma antibonding orbitals that look like two tops or two acorns on their sides with the points pointing towards the nuclei and the bulbs oriented away from the nuclei and each other.  Adding electrons to this region of space (the sigma* antibonding orbital) results in an overall destabilization of the molecule.  The presence of the antibonding orbitals is a necessary phenomenon as a result of molecular orbital theory, specifically the linear combination of atomic orbitals.

I hope that explained how antibonding orbitals come to exist.  To respond to your specific questions, the brief definition in your book is limited, at best.  It is true that antibonding orbitals have a node, or a region where the solution to the complex equations is zero.  When the solution to the probability equation is zero, there is zero probability of finding an electron in that region of space.  This node is the region of space in which the probability of finding an electron is zero.  It is located between the nuclei.  For simple diatomic molecules, and for valence orbitals of complex molecules, the antibonding orbitals typically do not contain electrons.  Nonbonding electrons are a different concept entirely.

For question 2, I see how you might suspect that to be the case.  But consider that it's not as if a singular carbon atom was just sitting around in sp2 hybridization and 4 hydrogen atoms and another singular carbon atom with sp2 hybridization happened to come along and form an ethylene molecule.  Singular carbon atoms exist with one 2s orbital and 3 2p orbitals and no hybrid orbitals.  Hybrid orbitals only arise as a product of bonding.  Thus, when the conditions are favorable for 2 carbon atoms to form a pi bond, one sp3 orbital on adjacent carbon atoms will rehybridize to a p orbital in an sp2 hybridized atom.  If one were to take away all the substitutents at that point, one would describe the 3 hybrid orbitals and 1 p orbital with single electrons as you describe.  But it doesn't start that way, we're just describing what the carbon atom looks like after it has hybridized.

Hope this helps some.

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Some references I used here:

http://www.ch.ic.ac.uk/vchemlib/course/mo_theory/main.html
http://www.science.uwaterloo.ca/~cchieh/cact/c120/mo.html
http://www.chemistryexplained.com/Ma-Na/Molecular-Orbital-Theory.html
http://en.wikipedia.org/wiki/Linear_combination_of_atomic_orbitals_molecular_orbital_method
http://user.mc.net/~buckeroo/LCAO.html
http://www.nyu.edu/classes/tuckerman/honors.chem/lectures/lecture_13/node4.html
http://id.mind.net/~zona/mstm/physics/waves/interference/constructiveInterference/InterferenceExplanation2.html
http://en.wikipedia.org/wiki/Interference#Constructive_and_destructive_interference
http://en.wikipedia.org/wiki/Antibonding
http://www.sparknotes.com/chemistry/bonding/molecularorbital/section1.html

[azmanam]

Here is actually a really fascinating pdf on atomic and molecular orbitals and explains graphically the addition and subtraction of the equations that I tried to explain in words.

http://www.upei.ca/~chem342/Resources/Reviews/Molecular_Orbital_Tutorial.pdf

[jebus197]

Thanks. I was searching the internet for a simple explanation of this topic and I came across this. I know it's an old topic, but the question I wanted to ask is why do the electrons in the diatomic hydrogen molecule 'decide' to occupy the lowest energy (1s orbital) state? I mean, is there some kind of preference, or mechanism that makes them do this? I'm OK with the idea that since there were two orbitals to start with, there should be two orbitals to finish with too, but why do the electrons 'choose' the lowest 1s orbital, over say another orbital. (Although in this case, clearly there can only be two.)

It's probably a dumb question, but is there some kind of rule that makes molecules want to do this?

[fledarmus]

That's sort of like asking why a bowling ball would "decide" to sit on the floor instead of floating six feet above the ground - the answer is that the energy is lower when it is sitting on the floor. It is possible to add energy to the system and get electrons to move up to higher energy states, but once the energy is removed, they will drop back down, emitting the energy they absorbed.

[jebus197]

Thanks, but could someone please explain in more detail? Is energy somehow lost when molecules are formed? If so what's the mechanism?

[Jeremy]

Bonding orbitals are lower in energy that atomic orbitals because the electrons are attracted to two nuclei.

[juanrga]

Thanks. I was searching the internet for a simple explanation of this topic and I came across this. I know it's an old topic, but the question I wanted to ask is why do the electrons in the diatomic hydrogen molecule 'decide' to occupy the lowest energy (1s orbital) state? I mean, is there some kind of preference, or mechanism that makes them do this? I'm OK with the idea that since there were two orbitals to start with, there should be two orbitals to finish with too, but why do the electrons 'choose' the lowest 1s orbital, over say another orbital. (Although in this case, clearly there can only be two.)

It's probably a dumb question, but is there some kind of rule that makes molecules want to do this?

This is a consequence of the second law of thermodynamics. For a system which can only interchange energy with the surrounds the law reads

dS >= dU/T

Now, consider a process that conserves entropy S,

0 >= dU/T

since T is positive, this implies that dU =< 0. That is, the system evolves towards states with a minimum of energy, because the state with higher energy is unstable.