Skip to main content
Chemistry LibreTexts

Homework 1 (Due 6/21/16 @ 10:00 a.m.)

  • Page ID
    52455
  • 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?