What are the different eigenstates of molecules that are most often used in chemistry?
What do you mean by different eigenstates of molecules? There is only one set of eigenstates of (the Hamiltonian of) a given molecule.
You mean the molecules only have energy eigenstates? What is an example of such eigenstates of molecules for example water?
Unfortunately, the question does not make sense.
I am guessing that OP do not understand the concept of eigenstates, eigenvalues, and what it means for a molecule to have them under the Hamiltonian.
"Energy eigenstates" does not make sense. Eigenvalues of a Hamiltonian of a given molecule are the energies. Each of these energies corresponds to each of the eigenstates of a Hamiltonian of a given molecule.
I understand them as pertain to only a single atom.. I don't know how to apply it to molecules. Can you give example of this "Each of these energies corresponds to each of the eigenstates of a Hamiltonian of a given molecule"... let's take the case of Sodium Chloride or salt.
I don't know if you are the one confused or I am. But "Energy Eigenstates" is simply the Eigenstates of the Hamiltonian.. see https://physics.stackexchange.com/questions/41070/what-is-an-energy-eigenstate-exactly
What I wanted to know are examples of energy eigenstates of the molecules. If I hit them with laser, what are the possible quantized energy eigenstates (or eigenstates of the Hamiltonian)?
Hold on, I'll write an answer after I have enough time, but to me "energy eigenstates" feels extremely awkward because I was taught this in Japanese. Direct translation barely make sense at all. But now with the link you've provided, it seems like the chances are, you do call it "energy eigenstates". Perhaps people call it "energy eigenstates" for convenience or it's a real term and has been given a definition, despite the possible confusion to the readers if they were taken literally.
Anyhow, if you are looking for energy of a molecules and the eigenstates of a molecule after absorption of a photon, then you'll have to think about excited states. That is when one (or more) of the electrons are excited into another orbital within a molecule. Calculation of excited states is not as easy as simply deriving energy of orbitals at ground state (which is also difficult already). I can only qualitatively explain if it was a simplified two level case. If you are talking about many-body system, then you'll need sophisticated calculation.
On a conceptual level it is in general the same thing as in the case of an isolated atom, just the calculations are quite ugly and don't yield nice results. So you will not find a closed-form expression for the energies of a molecule like there are for a hydrogen-like atom. Note, that we don't have closed-form solutions for any other atom - same reason, math gets ugly.
Sometimes, the Hamiltonian operator is also referred to as the energy operator. Thus its eigenvalues are called energy eigenstates in this sense.
Your question still seems vague to me, by examples do you mean you want a graphical example? If you do some computational chemistry calculation you will know that two eigenstates differing in only one level of energy can have a significantly different shape that giving only a particular example does not seem meaningful, at least to me. Anyway, it might be possible to find such graphical representation of a particular state of a particular molecule in scientific papers. Just need to warn you that due to the many-body nature of molecules you might not get a complete coordinate dependency of these states.
In molecules, often it's possible to separate the motion of nuclei and that of the electrons, it's called Born-Oppenheimer approximation. With this approximation in mind, it's possible to talk about the so-called electronic, vibrational, and rotational states. The energy levels between vibrational states in the same electronic level usually lie in infrared region and thus you need an infrared laser in order to probe these levels. Different electronic states are separated by photon energies in the visible to UV or even XUV range and therefore you need lasers with these frequencies for observing inter-electronic level transition. As for purely rotational states, if I remembered correctly you will need a laser in the microwave frequency region.
You probably meant eigenstates, not eigenvalues.
Yes that's a typo.
Separate names with a comma.