The meaning of the sign of the wavefunction in electronic structure

[jdedrick]

I’m hoping someone will help me fill in some holes in my understanding of the electronic wavefunction.

I understand the electronic wavefunction to be a *complex valued* function of the positions of all electrons in the system. But, most descriptions of atomic orbitals refer to the various “lobes” of high probability amplitude as being of positive or negative sign, and the nodes as being loci where the sign changes.

It is tempting to think that this is just shorthand for a description of some projection (e.g. the real part) if a complex phasor, and the nodes are those locations where the phase angle is 90 or 270 degrees. But this can’t be the case, since the probability density (which is the square of the phasor’s amplitude) also goes to zero at these locations. So the vector magnitude of the wavefunction must vary smoothly with position, and it must be this magnitude that is zero at the nodes. Is that correct?

So, my first question is, should the wavefunction generally, and orbitals specifically, be thought of as scalar fields which have positive and negative regions, or as complex numbers with smoothly varying phase vs. position. If they are complex numbers, what accounts for spatial nodes of the probability density – zero vector magnitude, or some particular phase?

From there I’d like to understand what happens to the sign/phase of the orbital when an atomic bond forms. If for example two singly-occupied 2p orbitals form a bond, does the initial sign/phase of the lobes of the orbitals that are closest to each other affect the interaction in any way? Is there meaning associated with whether a positive-valued lobe happens to abut a negative valued lobe of the other atom, vs. positive-to-positive? Is this in any way related to the bonding and antibonding orbitals that Molecular Orbital theorists talk about?

Thanks in advance for any insight you can impart.

 

[Redbelly98]

Welcome to PF

So, my first question is, should the wavefunction generally, and orbitals specifically, be thought of as scalar fields which have positive and negative regions, or as complex numbers with smoothly varying phase vs. position. If they are complex numbers, what accounts for spatial nodes of the probability density – zero vector magnitude, or some particular phase?

The wavefunction is a complex-valued function in space. The orbitals you see drawn in textbooks actually represent the probability density for the electrons. The probability density is equal to the square of the wavefunction's amplitude at any location.

The nodes occur where the wavefunction's amplitude (and therefore the probability) is zero.

From there I’d like to understand what happens to the sign/phase of the orbital when an atomic bond forms. If for example two singly-occupied 2p orbitals form a bond, does the initial sign/phase of the lobes of the orbitals that are closest to each other affect the interaction in any way? Is there meaning associated with whether a positive-valued lobe happens to abut a negative valued lobe of the other atom, vs. positive-to-positive? Is this in any way related to the bonding and antibonding orbitals that Molecular Orbital theorists talk about?

This question is outside my area of expertise, as I'm a physicist by training. Perhaps somebody else can answer this one.

 

[jdedrick]

RB98:

Thanks for the welcome, and for jumping in on this question!

The orbitals I'm referring to (do a Google search for "orbitron" for example) cannot be probability densities, since they have explicitly color-coded positive and negative regions.

Also, you use the word "amplitude" which seems to have a different meaning in quantum physics (where "probability amplitude" refers to a phasor) vs. general math, where amplitude is synonymous with magnitude. So I still don't know whether, at the locations where the probability is zero, is that because the "length" of the complex phasor went to zero, or because the phase angle went to 90 or 270 degrees?


Thanks, J

 

[kanato]

I understand the electronic wavefunction to be a *complex valued* function of the positions of all electrons in the system. But, most descriptions of atomic orbitals refer to the various “lobes” of high probability amplitude as being of positive or negative sign, and the nodes as being loci where the sign changes.

It is tempting to think that this is just shorthand for a description of some projection (e.g. the real part) if a complex phasor, and the nodes are those locations where the phase angle is 90 or 270 degrees. But this can’t be the case, since the probability density (which is the square of the phasor’s amplitude) also goes to zero at these locations. So the vector magnitude of the wavefunction must vary smoothly with position, and it must be this magnitude that is zero at the nodes. Is that correct?

Nodes specifically refer to where the wavefunction is zero, not just the real or imaginary part separately. The particular phase of the wavefunction at a particular point is irrelevant in this regard.

So, my first question is, should the wavefunction generally, and orbitals specifically, be thought of as scalar fields which have positive and negative regions, or as complex numbers with smoothly varying phase vs. position. If they are complex numbers, what accounts for spatial nodes of the probability density – zero vector magnitude, or some particular phase?

It's possible to chose atomic orbitals to be entirely real. For l > 0, m /= 0, this amounts to forming linear combinations of $$Y_{l,m} \pm Y_{l,-m}$$.

From there I’d like to understand what happens to the sign/phase of the orbital when an atomic bond forms. If for example two singly-occupied 2p orbitals form a bond, does the initial sign/phase of the lobes of the orbitals that are closest to each other affect the interaction in any way? Is there meaning associated with whether a positive-valued lobe happens to abut a negative valued lobe of the other atom, vs. positive-to-positive? Is this in any way related to the bonding and antibonding orbitals that Molecular Orbital theorists talk about?

Yes, atomic orbitals which combine in a bond to give a node between them are antibonding, and those which combine so that their densities add between the atoms are bonding. Antibonding orbitals are always higher in energy than bonding orbitals.

 

[genneth]

There is an assumption, usually implicit, that the many-body wavefunction can be factorised into single-electron wavefunctions (whilst respecting things like anti-symmetry). It is these single-electron wavefunctions which are usually depicted, and called orbitals. Now, by happy coincidence, these guys tend to be "real-ifiable", i.e. they can be chosen so that they are entirely real (the overall many-body wavefunction can just pick up any leftover phase). The reason is that these are bound orbits, so the average speed is zero. Recall that the momentum operator is -i*derivative, so you would only get a real velocity if there were variations in phase (and not just amplitude).

 

[jdedrick]

kanato & genneth:

Thanks to both of you - the picture is a lot clearer now. I'll try to repeat my understanding to see if I've absorbed it correctly: Since wavefunctions are linear things, valid ones can be superimposed to produce other equally valid wavefunctions/orbitals, and in the case of bound bound, single electron orbitals that can be done in such a way as to produce a result that is everywhere real.

As for bonding, my understanding from your responses and other reading I've done in the interim is that the signs of the constituent orbitals *are* relevant - bonding MO's result from constructive interference, and antibonding MO's from destructive interference. I still don't know whether it's proper to think of an atom "reorienting itself" in space in such a way as to expose the appropriately signed lobe, or whether some equivalent phase adjustment occurs. When an arbitrarily string of individual atoms binds into a polymer, is there some atomic-scale operation analogous to magnetic domain flipping that is required so that the polarity of the adjacent orbital lobes alternates, as would be required to form bonding MO's? Or am I just trying to wring more from a physically intuitive picture than is meaningful.

 

[kanato]

As for bonding, my understanding from your responses and other reading I've done in the interim is that the signs of the constituent orbitals *are* relevant - bonding MO's result from constructive interference, and antibonding MO's from destructive interference. I still don't know whether it's proper to think of an atom "reorienting itself" in space in such a way as to expose the appropriately signed lobe, or whether some equivalent phase adjustment occurs. When an arbitrarily string of individual atoms binds into a polymer, is there some atomic-scale operation analogous to magnetic domain flipping that is required so that the polarity of the adjacent orbital lobes alternates, as would be required to form bonding MO's? Or am I just trying to wring more from a physically intuitive picture than is meaningful.

Well, I think you are trying to wring more out of it than you should. You're talking about dynamics of the phase of a wavefunction during a chemical bonding process, which is a rather complicated subject (in part because the electrons on each atom become entangled with one another) and I don't think it's appropriate to think about it in terms of atomic orbitals dynamically reorienting themselves.

It's better to think about just the MO's as an alternate basis from the atomic orbitals, one which might arise from diagonalizing a certain molecular Hamiltonian rather than the hydrogen atom Hamiltonian.