Hydrogen atom energies.

--This post is related to things we will cover in the near future, not the current HW..
To understand the origin and nature of chemical bonding, it is important to first understand that role of kinetic and potential energy in the formation of an atom.  The reason is: in a molecule or  covalently bonded crystal such as graphene, 2 or more atoms have decided it is preferable to be together than to be apart. Why is that? What is their motivation? The answer lies in energy. The energy is lower. Before we attempt to understand that, we need to understand the energy of one atom, for example, a hydrogen atom.


Here we will look at the origin of the size of a hydrogen atom and the role that potential and kinetic energy each play in establishing that size. Later, we will use similar considerations regarding K.E. and P.E. to understand the size of a molecule, the length of a bond in a crystal, and why molecules or crystals form at all.

   Consider a 3D wave-function of the form:  \( \psi_a = \frac{1}{\sqrt{\pi a^3}} e^{-r/a}\). This state is a function of r, not theta or phi, and you can show, via an integration in spherical coordinates, that the expectation value of the kinetic energy is:
K.E. = \(\hbar^2/2ma^2\) where  \(\hbar^2/2m = 0.076 eV \: nm^2\) and a is in nm.
Consider that \( \psi_a\) is a variational wave function and that "a" is the variational parameter.
Graph the K.E. as a function of a. What does that tell you?

For this same variational wave-function and a potential \(V(r) = \frac{-e^2}{4 \pi \epsilon_o r} \), the expectation value of the potential energy is:
P.E.= \( \frac{-e^2}{4 \pi \epsilon_o a} \),
where \( \frac{-e^2}{4 \pi \epsilon_o} = -1.44 eV\: nm \)
Graph the P.E. as a function of a. What does that tell you?

Combine the KE and the PE and find the optimal value of the total energy according to the variational principle.
a) What is the optimal value of the KE?
b) What is the optimal value of the PE?
c) What is the optimal value of the total E?
d) What is the optimal value of the a?
e) what do you learn from this. send me an email explaining and discussing what you have learned from this exercise.