By the time of Galileo, thinkers understood clearly that while the basic building blocks of matter were atoms (with unknown structure), the basic forms of matter were determined by the forms of molecules, arising from atoms fitting together in various possible structures. As we know today, when two atoms happen to approach closely, the individual electrons in the two atoms are affected and frequently adopt new arrangements in which, if the atoms come into contact (their probability distributions touch), the energy of the arrangement is lower than the energy of the two separate atoms, and a stable structure is formed. “Atomic bonds” are formed.   The binding energy is typically a few eV.

While any two atoms close together will at least weakly attract, the Pauli principle forbids the two  electron probability distributions from overlapping, because if they did, we would have two identical electrons in states with the same quantum numbers in the same region of space. If we make up a potential energy function which has a repulsive core that prevents this overlap, we have a form of potential energy which can be adapted for the interaction between two adjacent atoms in most molecules.  Of course the nuclei of the two atoms, being both positively charged, also repel.  So the strength of the repulsive core of the potential energy can be adjusted to take into account both effects.

The Coulomb force is an incredibly strong force, which we usually do not notice because ordinary matter is electrically neutral. However, on the atomic scale, any slight change in the arrangement of electrons in atoms, or change in the arrangement of atoms, can produce a substantial force that can create a highly stable bound system.

Molecules are bound quantum systems and so have a ground state and excited states. One simple model of excited states uses a quantum harmonic oscillator solution.

Note that for small oscillations the harmonic oscillato potential is a fair approximation to the realistic interaction potential between two atoms in a molecule. By expanding the molecular potential energy V(r) in a Taylor series about the minimum, we can get an estimate of the stiffness K for very small quantum oscillations about the equilibrium distance in terms of the second derivative of V(r) evaluated at the equilibrium point. Remember that the quantum harmonic oscillator has equally spaced levels, with spacing ℏω where ω is the square root of K divided by the reduced mass of the system. [We are working in the center of mass system of the molecule.]  By the way, this K has roughly the same value as that of a normal metal spring--- do you see why? It works out that typically while the electron excitations of the individual atoms are of course of the order of several eV, the level spacing of the vibrational states is smaller by a factor of 1/10 or more, because the reduced nuclear mass is much larger than the mass of an individual electron.

Want to understand Hooke's Law “from the bottom up?” Remember that F(r) = -∂V/∂r and you will immediately see that atoms resist being pulled apart by a force F = -Kr.  Much of our intuition about degrees of freedom of quantum systems comes from imagining a classical counterpart... such as two masses connected by a spring, suspended in empty space.  Such a classical system could vibrate, and could also rotate about its center of mass.  So it seemed reasonable that a diatomic molecule could do the quantum equivalent of the same, with energy levels due to the vibrational degrees of freedom, and energy levels due to the rotational degrees of freedom (although nothing in the quantum system is moving). The vibrational levels are closely spaced at around 0.1 eV so that transitions emit photons in the infrared region.  If we neglect vibration for the moment, the energy of a classical rotating rigid body is E = L²/(2ℑ), where ℑ is the rotational inertia of the system. The typical separation between rotational states is about 10⁻³ eV, so that the rotational states are basically built on top of individual vibrational states. In quantum physics, L² is an operator with eigenvalues ℏ²ℓ(ℓ + 1). Thus the eigen-energies are E_ℓ = ℏ²ℓ(ℓ + 1)/(2ℑ).

So built on the electronic excitations are closely spaced levels of vibration, and built upon each level of vibration are closely spaced levels of rotation. Thus molecules have an extremely complex band spectrum, where broad regions will absorb photons of a range of wavelengths. The more complex the molecule, the more complex and interwoven the many bands.

THE MAGIC WINDOW? It's hardly possible, based on what we can see around us astronomically, that life could exist on a planet without liquid water at the surface, and sunlight reaching the surface. So it's interesting that the water molecule has a window located at just the frequencies of the light from our sun. But this is not really surprising, visible light, with photon energies of a few eV, cannot be absorbed except in discrete transitions between electronic levels. The physics of molecular structure makes it impossible for either water, or most other common molecules in the universe, to absorb a range of frequencies at anywhere near the normal frequencies of starlight.

How about another aspect of molecular absorption of radiation, which has been well known and understood by physicists since 1896, which has been fairly consistently ignored, and which threatens to make large areas of the earth uninhabitable by humans and to largely destroy industrial civilization??? Physicists noticed around 135 years ago that the earth is much warmer than a simple calculation based on heat radiated by the earth to outer space versus solar heat absorbed by the earth would indicate. The culprit is a molecule, CO₂, which is a fairly minor constituent of the earth's atmosphere, but absorbs very strongly in the infrared. There are a number of molecules in the air that absorb in the infrared, such as H₂O, but they are regulated by natural processes. Since the early days of the Industrial Revolution brought on by steam power, burning of fossil fuels in the open air has exponentially increased the concentration of CO₂, and just as predicted and expected, the average temperature at the earth's surface has closely tracked the increased CO₂ levels observed. [The peak frequency of the radiation emitted by an object of surface temperature T depends  directly on T.  The surface of our sun is at 6000 K, while the earth's surface currently averages a temperature of about 288 K.]

At ultra-low temperatures, molecular bound states can form with binding energies so tiny that they would never be seen in nature on earth. By the way, how do you cool a system with a laser beam?