# Homework 1

Name: ______________________________

Section: _____________________________

Student ID#:__________________________

## Q1

Write out the complete time-independent Hamiltonians for (each term is explicitly given):

1. the helium atom
2. the $$H_2^+$$ ion
3. the $$H_2$$ molecule

You may want to refresh yourself on the basics of Chem 110A material:

## Q2

What is the Born-Oppenheimer approximation and what do we use it? When would it fail?

## Q3

Confirm that the two sp-orbitals are normalized and orthonormal.

## Q4

Show that the sp2 hybrid orbital

$| sp^2 \rangle = \dfrac{|s \rangle + \sqrt{2}|p \rangle }{\sqrt{3}}$

is normalized if the $$|s \rangle$$ and $$|p_z \rangle$$ atomic orbitals are also normalized.

## Q5

What is the average energy of a H atom $$|sp^3 \rangle$$ hybrid orbital is the energy of the $$|s \rangle$$ orbital is $$E_s$$ and the energy of the $$|p \rangle$$ orbital is $$E_p$$? Hint: this will require solving the variational energy:

$Energy = \dfrac{\langle sp^2 | \hat{H} | sp^2 \rangle }{ \langle sp^2 | sp^2 \rangle }$

## Q6

When one s and two p atomic orbitals are used to generate hybrid orbitals, how many hybrid orbitals will be generated?