Chemistry 3730 Fall 2000 Assignment 4

Due: Friday, Oct. 27, 10:00a.m.

1.

As discussed in class, we often use Gaussians to approximate hydrogenic basis functions in ab initio calculations. In the simplest case, we can use a single Gaussian. For the 1s orbital, we would have the unnormalized function

$$\varphi = e^{-\alpha r^2}.$$

Use the variational method to find an approximation to the ground-state energy of a hydrogenic atom with this Gaussian wavefunction. What is the percent error? [10 marks]

Hints and notes: This problem can be solved analytically. Maple will need to know that $\alpha$ is positive. Remember that, if you later want to give $\alpha$ a value, you will have to use assign(). You may find at some point that you want Maple to simplify an expression involving a square root. If so, Maple will also need to know that the quantities under the square root are positive. In Maple, e^u is exp(u) and $\pi$ is Pi.

The volume element in spherical polar coordinates is

$$dV = r^2\sin\theta\,dt\,d\theta\,d\phi.$$

The angles have the following ranges: $\theta\in[0,\pi]$, $\phi\in[0,2\pi)$. The Laplacian in spherical polar coordinates is

$$\nabla^2 = \displaystyle\frac{1}{r^2}\frac{\partial}{\partial... ...) + \frac{1}{r^2\sin^2\theta}\frac{\partial^2}{\partial\phi^2}$$

but if you think about it, you will realize that you don't need all these terms in this problem. The energy of a hydrogenic atom is given by

$$E_n = -\displaystyle\frac{Z^2e^4\mu}{32n^2\pi^2\epsilon_0^2\hbar^2}.$$

2.

(a)

Find the best basis set with which to compute the atomic orbitals of beryllium. The best basis set gives the lowest energy with the fewest basis functions. Your answer should include

i.

the basis set,

ii.

the calculated energy, and

iii.

the number of basis functions and Gaussians.

Differences in energy of less than 0.1kcal/mol (about 0.004eV) are not significant. [6 marks]

(b)

For your best basis set, does it help to add extra basis functions? If so, give an example. [2 marks]

3.

(a)

Calculate the ground-state energy of Fe³⁺. Give complete details of the calculation. Your grade on this question will be based on

i.

correctness,

ii.

finding a reasonably low energy,

iii.

the use of a sensible basis set, and

iv.

explanations of your computational procedures which are adequate to allow me to reproduce your results.

One bonus mark will be awarded for the lowest (reproducible) energy obtained. [10 marks]

Hints: You will need to do a UHF calculation. Make sure to use the correct spin multiplicity.

(b)

For your best basis set, what is the computed electronic configuration? Does this agree with your expectations for this ion? [3 marks]

Hint: You can get this information from the orbital window of HyperChem, but since you did a UHF calculation, you have to put the two sides of the orbital diagram together to get the configuration.