Angles between sp Hybrid Orbital
(This is not properly part of Chem 125, but if you're inquisitive, welcome.)

Suppose there are two hybrid orbitals on the same atom:

where s is an s orbital on a particular nucleus and pi and pj are two p orbitals on the same nucleus pointing in arbitrary directions.

Recall that the form of a 2p orbital is

where rho is the distance from electron to nucleus, and x is the coordinate in the direction of the p orbital. This can be any direction, not just the x, y, or z direction of some arbitrary cartesian coordinate system. The spwhatever hybrid orbital will point in the direction of this axis, the direction of its p orbital component.

Here's the constraint: the hybrids in use on any given center must be orthogonal to one another. Otherwise if both were occupied by two electrons, there would be too much electron density in one of their component orbitals (violating the Pauli Principle, which is an empirical statement of the way things always seem to be).

The hybrids are are said to be "orthogonal" when the integral of their product over all space is zero. Of course the integral of the product of sums is the sum of the integrals of products:


where θmn is the angle between the p orbitals

By substitution using the expressions for m and n from the top of this page we have the relationship between angle and hybridization

where θmn is angle between spm and spn hybrids

[While this relationship obviously holds true for the hybrid orbitals on a particular atom, the question of the angle between bonds is a little more subtle.  If Pauli requires the electrons of the bonds from the atom to be in orthogonal orbitals, we should consider not just the AOs on this individual atom, but also the AOs on the adjacent atoms that are involved in the two bonds.  These make a small contribution that is negligible for our purposes, so we can use the simple relationship above.]

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