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What is the difference between the carbon in the excited state, $\ce{[(1s)^2] (2s)^1 (2p)^3}$, and $\pu{sp^3}$ hybridization?

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    $\begingroup$ The difference is that excited state exists in nature, while sp3 hybridization exists in our heads. $\endgroup$ – Ivan Neretin Oct 4 '18 at 15:38
  • $\begingroup$ Those are two different (but related) concepts. This might get close as a homework question, please have a look here $\endgroup$ – Feodoran Oct 4 '18 at 16:36
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    $\begingroup$ This really shouldn't be closed as homework. This is one of the problems you'll have when hybridisation is taught as a law and not as a helpful tool. This is clearly a conceptual query, unfortunately based on a misconception. $\endgroup$ – Martin - マーチン Oct 4 '18 at 19:23
  • $\begingroup$ Whether hybridisation is real or not is not the point here. $\endgroup$ – Feodoran Oct 5 '18 at 6:26
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You are confusing different concepts here. The first one ($\ce{[(1s)^2] (2s)^1 (2p)^3}$) is an electron configuration. The other one ($\ce{sp^3}$) is an hybrid orbital.

An electron configuration describes the distribution of electrons in available orbitals (occupation). Hence it depends on a choice of orbitals, but that choice is not unique: One usually uses atomic orbitals for single atoms, and molecular orbitals for more than one atom. But one can also use localized orbitals or hybrid orbitals.

As a side note: An electron configuration is strictly speaking not an (excited) electronic state. You may say so in approximation, but there is more to it.

A hybrid orbital is constructed by a certain linear combination of atomic orbitals. It is a possible choice for the basis in an electron configuration.

As a second side note: Hybridization is a concept used in organic and inorganic chemistry to explain bonding situation. But it is just a model, a crutch, to help with interpretation and imagining things. In theoretical (quantum) chemistry it is rarely used, as there is not much value in using them. You get the very same results, if you just stick with the atomic orbitals (the one the hybrid orbitals are constructed from).

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