Chemistry 3730 Fall 2000 Quiz 3

Name:

In questions requiring the use of Maple, outline your calculations in standard mathematical notation. In questions using HyperChem, give sufficient detail to allow me to reproduce your calculation. Note that information required for the solution of these problems is given on the last page of this paper.

1.

Suppose that a particle in one dimension experiences a potential

$$V(x) = V_\mathrm{max}(1-e^{-\alpha x^2})$$

where $V_\mathrm{max} = 10^{-20}\,\mathrm{J}$ and $\alpha = 10^{22}\,\mathrm{m^{-2}}$. The particle has a mass of $10^{-26}\,\mathrm{kg}$. Compute the variational energy using the trial wavefunction

$$\phi = e^{-\beta x^2}.$$

[10 marks]

Notes and hints: Maple will need to know that $\beta$ is a positive parameter. Don't forget that e^u is exp(u) in Maple. Once you have determined the proper value of $\beta$, you can use subs() to substitute it into your variational energy, or assign() to permanently assign the value. Be sure to state the units of the energy.

2.

There are two ways to calculate the ionization energy of an atom or molecule:

(a)

Calculate the ground state energies of the atom or molecule before and after removal of the electron. The ionization energy is the difference between these two energies.

(b)

According to Koopmans's theorem, the ionization energy is roughly equal to the negative of the energy of the orbital from which the electron is to be removed.

Using the 6-31G basis set, calculate the ionization energy of lithium using both methods. Compare your two answers to each other and to the experimental ionization energy (available through HyperChem's periodic table). [10 marks]

Note: In order for this to work properly, your two ab initio calculations should use identical computational methods.

Useful information

$h = 6.626\times 10^{-34}\,\mathrm{J/Hz}$

$E^\mathrm{var} = \displaystyle\frac{\langle\phi\vert\hat{H}\vert\phi\rangle}{\langle\phi\vert\phi\rangle}$

1eV = 23.06055kcal/mol