Here's a video about resonance with multiple bonds (12min): Resonance, by Khan Academy, on YouTube

Here's another video about resonance with multiple bonds (6min): Resonance and Dot Structures, by readysetorgo, on YouTube

We previously discussed a different kind of resonance, which is the
tendency of some systems, such as the strings of instruments, to
vibrate easily at particular frequencies. Now we will talk about quantum mechanical resonance. First, it may be easier
to understand if you think about Fourier analysis. Perhaps you have
heard before that any function can be written as a sum of sin and cos
waves, like this:

f(x) = Σ_{n} a_{n}sin(nx) + Σ_{m}
b_{m}cos(mx)

When we try to find wavefunctions for real chemical systems, it is too
complicated to find an exact solution like the solutions for single
hydrogen atoms. Instead, we usually describe the real wavefunction ψ(x,y,z,t) using
series of functions (call them φ(x,y,z,t)) like a Fourier
transform.

ψ = Σ_{n} a_{n}φ_{n}(x,y,z,t)

Often the functions φ_{n} used
to build the real wavefunction ψ are
the hydrogen wavefunctions we talked about before, including s, p, d,
and f orbitals in each shell. There is a
principle that says that if you choose the coefficients
(a_{n}) so that the energy of the total wavefunction is
minimized, those are the best coefficients that get closest to the
real wavefunction ψ. In other words, real life finds the lowest
possible energy (highest stability), so the lowest energy we can find is the closest to
the real thing. If we imagine a system that might be described by
φ_{1} or φ_{2},

ψ = a φ_{1} + b φ_{2}

we can calculate the ratio a:b
that minimizes the energy of ψ. If a:b is very big,
φ_{1} is a good description of the system (at least
compared to φ_{2}). If a:b is very small,
φ_{2} is a good description. If a:b is close to 1, then
the real system is somewhere in between. The system is described as
resonating between the two states. This doesn't
mean that it alternates between them. It's like the difference between
blue and yellow stripes (alternating between colors), and green (resonance, a mix of
blue and yellow). The energy calculated for the combination will be
lower than either single energy, and this difference is called the
resonance energy.

A specific example might make this more clear. We can use
different Lewis structures to represent the trial wavefunctions
φ. For instance, imagine the formation of a bond between a H atom
and a H^{+} ion (H nucleus). We can consider 2 trial
wavefunctions corresponding to the following structures:

Structure 1: H_{A}^{+} •H_{B}

Structure 2:
H_{A}• H_{B}^{+}

If we calculate energy as a function of distance between the nuclei,
for either structure 1 or structure 2 we don't predict a bond to form
(instead, we expect the nucleus and atom to repel each other). If we
allow resonance between structure 1 and structure 2, then we find that
at a certain distance, 1.06 Å, the energy is a minimum. This
means a bond can form. The resonance between the 2 structures means
that the electron spends time near both nuclei. Since it has to move
back and forth (very quickly, but maybe in a random motion, we don't
know exactly), it must spend more of its time between
the nuclei. When it is between them, we can expect that both nuclei
are attracted to the electron, so it holds them together in a
bond.

In the hydrogen molecule, we can use a similar model. We use the 2 structures
H_{A}(1)H_{B}(2) +
H_{A}(2)H_{B}(1)
(where H_{A} and H_{B} are the 2 nuclei, and 1 and 2
are the electrons) to represent the normal covalent bond, H—H. We get closer to the experimental data when we
include the ionic structures
[H_{A}^{+}][:H_{B}^{–}] +
[:H_{A}^{–}][H_{B}^{+}]. At the
normal bond length, the Coulomb attraction between the ions makes
these structures stable enough to contribute about 2% to the full
description of the molecule.

Ionic resonance structures are much more important in cases where
the bond is between different elements. For instance, in HF, we expect
the structure [F^{–}][H^{+}] to be very
important, perhaps as important as H—F, because F pulls much
harder on electrons. We will keep discussing this in the next sections.

This work is licensed under a Creative Commons Attribution 4.0 International License. Last modified: Sun Apr 27 17:50:01 KST 2014