Answer to Question #180668 in Physical Chemistry for Tom Wise

Question

How can you assume that S_ij = 0 for all i and j. For example S₁₂, how can the overlap integral of two adjacent P_z orbitals (one on the carbon and one on the adjacent oxygen) equal zero? Similarly for the other overlap integrals S₁₃=S₁₄=S₂₁=S₃₁=S₄₁ .

Answer 1

If the wave functions do not overlap, then the overlap integral is zero. The integral can also be zero if the wave functions have positive and negative aspects that cancel out. If the overlap integral is zero, then the wave functions are called orthogonal. As the distance of the functions get closer to R = 0, then the overlap integral gets closer to the maximum overlap integral of S = 1.

The wave functions of atomic orbitals decrease exponentially with distance. Orbital overlap is non-zero when two atoms are close together.

Overall, the full equation for the overlap integral, which is hard to calculate but is explained here, is

\[S(R)= \left \langle 1s_A | 1s_B \right \rangle = e^{-R/a_0} \left (1 + \dfrac{R}{a_0} + \dfrac{R^2}{3a_0^2} \right)\]