Homework 1

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Name: ______________________________

Section: _____________________________

Student ID#:__________________________

Q1

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

the helium atom
the \(H_2^+\) ion
the \(H_2\) molecule

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

8-2: Approximations of the Helium Atom
9.2: The \(H^+_2\) Prototypical Species

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?