Does symmetry lower the energy of molecules?

Why does nature "prefer" symmetry? Why is symmetry energy-lowering? I keep hearing statements along the lines of this tossed around by chemists. Here are some examples:


<https://physics.stackexchange.com/questions/135671/nature-favours-symmetry>


<https://books.google.com/books?id=WkRy8b0F2HwC&pg=PA153&dq=nature+favors+symmetry&hl=en&sa=X&ei=V__fVPvpGsuxggSptIG4AQ&ved=0CDAQ6AEwAw>


[IBrF2 - van der Waals repulsions - and number of unique bond angles](https://chemistry.stackexchange.com/questions/14651/ibrf2-van-der-waals-repulsions-and-number-of-unique-bond-angles) (The more symmetric conformation of the molecule is probably preferred). 


So: 


1) Are any of the above statements true? If so, can you provide example(s)? 


2) I think the statement is false at least to some extent. Consider cyclobutadiene ... it's not a square; it's a *rectangle*. 




**Symmetry is preferred *if* it lowers energy, but it does not always.**


My high school chemistry teacher had a good rule of thumb. A chemical system will [almost] always tend towards the lowest energy configuration. (I insert "almost," because there are some cases where this isn't true - usually because of entropy.)


So yes, many things are symmetric and this usually lowers the energy.


You raise a good counter-example. Cyclobutadiene is not square. A square cyclobutadiene would be a di-radical, so it undergoes symmetry breaking (i.e., a Jahn-Teller distortion) and becomes a rectangle.


![enter image description here](https://i.stack.imgur.com/HDlHZ.jpg)


We see here that the rectangular distortion has lower total energy - one orbital goes down in energy and becomes doubly-occupied. The other goes up in energy and is now empty.


So what about benzene? Why is it symmetric, vs. a double-single-double.. bond alternation?


Well, in this case, the symmetric (aromatic) version is lower in energy.
(Figure, [Chemwiki](http://chemwiki.ucdavis.edu/Organic_Chemistry/Organic_Chemistry_With_a_Biological_Emphasis/Chapter_02%3A_Introduction_to_organic_structure_and_bonding_II/Section_2.1%3A_Molecular_orbital_theory%3A_conjugation_and_aromaticity))


![enter image description here](https://i.stack.imgur.com/nIV15.png)


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The [Jahn-Teller theorem](http://chemwiki.ucdavis.edu/Inorganic_Chemistry/Coordination_Chemistry/Coordination_Numbers/Jahn-Teller_Distortions) says almost opposite:



> 
> any non-linear molecular system in a degenerate electronic state will be unstable and will undergo distortion to form a system of lower symmetry and lower energy, thereby removing the degeneracy
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> 
> Why does nature "prefer" symmetry?
> 
> 
> Why is symmetry energy-lowering?
> 
> 
> Are any of the above statements true?
> 
> 
> 


Things are either "true" or "false". The condition is binary, there is no "somewhat true". In that sense, the above statements are false. There are many, many non-symmetric things in nature.


* The human heart is not symmetrically positioned within the human body.
* The narwhal's tusk is helically grooved and projects from the left side of the jaw.
* You are not symmetric with the person standing next to you.
* Like people, most snowflakes are different one from another (see [here](https://chemistry.stackexchange.com/questions/20338/why-is-it-that-every-snowflake-is-unique/20358#20358)).
* The [staggered](http://en.wikipedia.org/wiki/Alkane_stereochemistry) conformation ($\ce{D\_{3d}}$, lower symmetry) of ethane is lower in energy than the eclipsed conformation ($\ce{D\_{3h}}$, higher symmetry).
* [Cyclooctatetraene](http://en.wikipedia.org/wiki/Cyclooctatetraene) exists in a tub shaped conformation with 4-fold symmetry, rather than planar with 8-fold symmetry.
* [Cyclobutane](http://en.wikipedia.org/wiki/Cyclobutane) exists in a puckered $\ce{D\_{2d}}$ geometry rather than a planar geometry with 4-fold symmetry.
* [Cyclohexane](http://en.wikipedia.org/wiki/Cyclohexane) exists in a chair conformation with a 3-fold axis of symmetry rather than a planar configuration with 6-fold symmetry.


From these examples we see that higher symmetry is not always preferred. In those situations where a symmetric arrangement is preferred it is likely due to


* It requires less (DNA) code to create identical arms or arrangements on a living structure
* Electrostatic repulsions are often minimized by creating structures with higher symmetry (e.g. methane is tetrahedral rather than planar).


In those situations where the symmetric arrangement is not preferred evolutionary advantages or steric\electrostatic repulsions, hyperconjugation (ethane conformation), etc., outweigh other factors.


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