# If chemical bonding is quantum mechanic, is saying they bend and vibrate like springs wrong?

I don't know much about quantum mechanics, but I do know that it's quite far removed from the easily observable and that anything "quantum" involves things that are discreet. So, how close is the analogy between molecular bending and vibration with that of springs? Is there maybe a better way to think, maybe a better analogy, to describe molecular bonds?

• The "nasty" fact is that there isn't any one model that creates an adequate analogy for chemical bonding. What model to pick depends on what phenomena you're modeling. So the spring model is useful in thinking about infrared spectra, but useless in pondering NMR spectra. Just adding "springs" to atoms though doesn't describe spring lengths or coil tension either. So "physical models" must be used with care.
– MaxW
Dec 8 '15 at 21:38
• I am reminded of this (unfortunately true) xkcd comic. Dec 8 '15 at 21:38
• @NicolauSakerNeto Those moments when you realise that a flaw of a visualisation was directly on the tip of your nose for years but you didn’t see it … Oh the pain ^^'
– Jan
Dec 8 '15 at 21:51

That's a good question, but remember, an analogy is just that: comparing dissimilar things on the basis of structure or function. Though there may not be a physical similarity, the analogy is effective in describing what happens, and can even be used to calculate outcomes, within limits.

In a similar way, the concept of a mass and a spring are used to describe an electrical R-L-C tuned circuit. Both have resonant frequencies and other properties that can be calculated from the physical properties of the components.

One of the difficulties in describing quantum phenomena is that until the physical properties are measured, they may be indeterminate. An electron is not in a specific place, but has a probability of being detected in a specific place. So though the analogy is not physically comparable, it helps understanding of the chemical bond, and can even produce useful results, such as IR spectra, and its use is valid.

This begs the question, though; perhaps there is another analogy that you might find more useful... Any suggestions from others are welcome!

I think there are two important points I would add to the good answer presented already. First is that this idea of modeling molecular vibrations as springs has been used to describe quite complex things in a fairly simple way. So, to some extent it is a good analogy. One example of this is an idea that Einstein actually came up with to find the heat capacity of solids. He treated every atom in a solid as a spring system which, as you point out, had to vibrate at discrete frequencies... and that's pretty much it. Using that idea alone he came up with an equation which predicts the heat capacity of solids quite accurately (gets worse at low temperatures). You can read about Einstein solids at that link if you want. It's kind of dense though...

One other thing is that molecular vibrations are clearly more complex than just a simple harmonic oscillator, but, as it happens, one can make corrections to the simple harmonic oscillator and end up with quite a good description of reality.

Specifically, the energy of a vibration is described by:$$E(v)=(v+\frac12)\omega-\omega_{xe}(v+\frac12)^2$$where $v$ is the vibrational energy level (things are discrete as you point out) $\omega$ is the vibrational frequency in wavenumbers, and $\omega_{xe}$ is the anharmonicity of the vibration. The interesting thing is this correction to the harmonic oscillator which the second term makes can be further expanded so that one has an infinite series which converges to the exact energy value of the vibration. So, in the sense that this can give the correct answer and is built on the idea of a simple harmonic oscillator, a spring is very much how atoms vibrate. Hope that is helpful.

That above equation coms from the Morse potential

Here is a picture comparing the two