# Electronic Bands of Liquid

DrDu suggested I posted this here.

Wiki says that "The electrons of a single, isolated atom occupy atomic orbitals. Each orbital forms at a discrete energy level. When multiple atoms join together to form into a molecule, their atomic orbitals combine to form molecular orbitals, each of which forms at a discrete energy level. As more atoms are brought together, the molecular orbitals extend larger and larger, and the energy levels of the molecule will become increasingly dense. Eventually, the collection of atoms form a giant molecule, or in other words, a solid. For this giant molecule, the energy levels are so close that they can be considered to form a continuum."

I'd like to understand the electronic band of liquid such as h20. What is the ranges of energy than an electron within the h20 molecule may have? Does it form any small bands or just discrete like separate atoms? I know thermal vibrations can break and reform hydrogen bonds very quickly. But they are uniform. Would the electrons wavefunction still somehow form energy bands? to what degree?

TeethWhitener
Gold Member
I love this question. A guy I work with and I occasionally have it out over a similar idea. Technically, "band theory" refers to crystals only. Electronic bands in solids are only well-defined when there is translational symmetry (look up Bloch's theorem for a rigorous explanation), and the bands are technically only continuous if the crystal is infinite. But the approximation is pretty good as long as the dimension of the crystal is much larger than the unit cell.

These technicalities make it a little difficult to talk precisely about what goes on in the electronic structure of a liquid or an amorphous solid, since there isn't really translational symmetry to speak of. But even though the theory might be significantly less precise, the idea is roughly the same. Each water molecule individually has discrete levels, but if you have two water molecules interacting, their energy levels will kind of look like a doubling of the levels. You can see this approximately by assuming that the total wavefunction of the dimer is a linear combination of the wavefunctions of the monomers (physically, this means that the interaction between monomers is weak). But this immediately gives us a splitting for each level because, for example, assuming $\psi_{A}$ and $\psi_{B}$ are the ground states of the two monomers, you get the symmetric and antisymmetric combinations: $$\psi^+_{AB} = \psi_A + \psi_B$$ and $$\psi^-_{AB} = \psi_A - \psi_B$$ as split "ground states" of the dimer. And this happens at each energy level for the monomer. Carrying this logic to a large number of monomers, you can (hopefully) see that the overall wavefunction will be split into zillions of little pieces, spread out around where the original monomer energy levels are. So even though it's not a "band structure," strictly speaking, for all intents and purposes it sure looks like one.

Of course, this is all within the approximation that the monomer wavefunctions form a decent basis set for the bulk liquid. This doesn't have to be true, especially in strongly correlated systems (where the interaction between the monomers is of comparable strength to the interactions within the monomers). But for most everyday substances, it's not a terrible starting point for thinking about their electronic structures.

I love this question. A guy I work with and I occasionally have it out over a similar idea. Technically, "band theory" refers to crystals only. Electronic bands in solids are only well-defined when there is translational symmetry (look up Bloch's theorem for a rigorous explanation), and the bands are technically only continuous if the crystal is infinite. But the approximation is pretty good as long as the dimension of the crystal is much larger than the unit cell.

These technicalities make it a little difficult to talk precisely about what goes on in the electronic structure of a liquid or an amorphous solid, since there isn't really translational symmetry to speak of. But even though the theory might be significantly less precise, the idea is roughly the same. Each water molecule individually has discrete levels, but if you have two water molecules interacting, their energy levels will kind of look like a doubling of the levels. You can see this approximately by assuming that the total wavefunction of the dimer is a linear combination of the wavefunctions of the monomers (physically, this means that the interaction between monomers is weak). But this immediately gives us a splitting for each level because, for example, assuming $\psi_{A}$ and $\psi_{B}$ are the ground states of the two monomers, you get the symmetric and antisymmetric combinations: $$\psi^+_{AB} = \psi_A + \psi_B$$ and $$\psi^-_{AB} = \psi_A - \psi_B$$ as split "ground states" of the dimer. And this happens at each energy level for the monomer. Carrying this logic to a large number of monomers, you can (hopefully) see that the overall wavefunction will be split into zillions of little pieces, spread out around where the original monomer energy levels are. So even though it's not a "band structure," strictly speaking, for all intents and purposes it sure looks like one.

Of course, this is all within the approximation that the monomer wavefunctions form a decent basis set for the bulk liquid. This doesn't have to be true, especially in strongly correlated systems (where the interaction between the monomers is of comparable strength to the interactions within the monomers). But for most everyday substances, it's not a terrible starting point for thinking about their electronic structures.

Could you please clarify this sentences: "You can see this approximately by assuming that the total wavefunction of the dimer is a linear combination of the wavefunctions of the monomers (physically, this means that the interaction between monomers is weak)."

If the interaction between monomers were strong.. why would it divert further from this approximation that the total wavefunction of the dimer is a linear combination of the wavefunctions of the monomers?

Also recall the hydrogen bonds form and reforms so many times in a second. So the pseudo energy bands can sort of like fluctuate due to the constant interchanges of the monomers?

TeethWhitener
Gold Member
Also recall the hydrogen bonds form and reforms so many times in a second. So the pseudo energy bands can sort of like fluctuate due to the constant interchanges of the monomers?
Right. The Born-Oppenheimer approximation works really well here, so the thermal rearrangements of the water molecules can, to a very good approximation, be separated from the motions of the electrons. The upshot is that the molecular motions smear out the electronic energy levels of the water. If we were examining bulk water spectroscopically, this effect would be visible as inhomogeneous (Gaussian) broadening, because the lifetime of an excited electronic state is much shorter than the timescale of the molecular motions. Thus the electrons in each water molecule would see a slightly different environment over the course of the excitation.
If the interaction between monomers were strong.. why would it divert further from this approximation that the total wavefunction of the dimer is a linear combination of the wavefunctions of the monomers?
I should clarify. In the limit of weak interactions between monomers, the ground state of the dimer can be closely approximated as the sum of the ground states of the two monomers. For strong interactions, the ground state of the dimer would include non-negligible contributions from higher energy levels of the monomers. In water, to a first approximation, we don't have to worry about that, because the interactions between water molecules are significantly weaker than the interactions within a water molecule.

Right. The Born-Oppenheimer approximation works really well here, so the thermal rearrangements of the water molecules can, to a very good approximation, be separated from the motions of the electrons. The upshot is that the molecular motions smear out the electronic energy levels of the water. If we were examining bulk water spectroscopically, this effect would be visible as inhomogeneous (Gaussian) broadening, because the lifetime of an excited electronic state is much shorter than the timescale of the molecular motions. Thus the electrons in each water molecule would see a slightly different environment over the course of the excitation.

I should clarify. In the limit of weak interactions between monomers, the ground state of the dimer can be closely approximated as the sum of the ground states of the two monomers. For strong interactions, the ground state of the dimer would include non-negligible contributions from higher energy levels of the monomers. In water, to a first approximation, we don't have to worry about that, because the interactions between water molecules are significantly weaker than the interactions within a water molecule.

Thanks for the information. Can the same description be valid in let's say plastic materials.. these synthetics are mostly polymers.. how large is the intermolecular distances between plastic polymers molecules and h2o molecules? which are more tightly packed? or are they the same degree? so plastics would still form electronic energy bands or are the electrons energy levels discrete due to perhaps their more larger intermolecular spacing than h2o?

TeethWhitener
Gold Member
Amorphous solids would work the same way. Particularly if the interactions between individual monomers are much weaker than interactions within them. However, there tend to be many examples of solids where intermolecular interactions do play an important role in electronic properties. Just off the top of my head, I imagine that any electroactive, photoactive, or conductive polymers would require a more careful treatement than the simple one I outlined above.

DrDu
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