Hyperconjugation in ethane conformers

I refer you to [this](http://www.princeton.edu/chemistry/macmillan/group-meetings/hyperconjugation.pdf) set of presentation slides. On slide 8, the author explains the preference for the staggered conformation of ethane by saying that there is a favourable hyperconjugative interaction between $\sigma^{}\_{\ce{C-H}}$ of the $\ce{C-H}$ bond in one methyl group and $\sigma^{\*}\_{\ce{C-H}}$ of another $\ce{C-H}$ bond in the other methyl group. 


However, I am puzzled by the diagram. Since both $\ce{C-H}$ bonds are similar in terms of bonding, their bonding and antibonding $\sigma$ molecular orbitals should have the same energies. Instead, the diagram shows that one of the $\sigma\_{\ce{C-H}}$ bonds is lower than the other in energy. How can this be right? I have shown a screenshot of the diagram below.


[![enter image description here](https://i.stack.imgur.com/J2cA8.png)](https://i.stack.imgur.com/J2cA8.png)


Also, how can a $\sigma\_{\ce{C-H}}$ donate electron density to a $\sigma^\*\_{\ce{C-H}}$ effectively? After all, the energy difference between them is rather significant since the orbital interaction between the atomic orbitals of $\ce{C}$ and $\ce{H}$ is large. 




I am not entirely sure what purpose the lower sigma bonding orbital in the MO-diagram serves, because an ethane molecular orbital would, as you have surmised correctly, have six degenerate $\sigma\_{\ce{C-H}}$ MOs and a $\sigma\_{\ce{C-C}}$ MO, lower in energy.


My best guess would be the position of the $\sigma\_{\ce{C-H}}$ on the right hand is quite arbitrary and the basic purpose of the diagram is to show the possible overlap between the $\sigma\_{\ce{C-H}}$ of one $\ce{C-H}$ bond and the $\sigma^{\*}\_{\ce{C-H}}$ of $\ce{C-H}$ bond anti to it - you don't really need to be worried about the $\sigma\_{\ce{C-H}}$ of *that anti bond* because it cannot overlap with the other $\sigma^{}\_{\ce{C-H}}$.


That brings us to the second part of your query. A $\sigma\_{\ce{C-H}}$ **can** donate electron density to an empty $\sigma^{\*}\_{\ce{C-H}}$ anti to it, because the latter is oriented correctly (anti) to accept the electron density from it. It's a question of orbital symmetry and not energy difference.


 5

Just to build on the very good answer from Sagnik, the purpose of the σC-H bond on the right side of the diagram is to show that the C-H bond is destabilized by electron density in its antibonding orbital. But the energy of the overall system is reduced.


The red orbital structure isn't in the *Angew.* paper either.


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