Overview of Quantum Chemistry

Quantum Chemistry

Overview Materials

Overview

There are basically two different models of atomic structure, the Bohr model and the quantum model. In both models, we are often interested in evaluating the energy of a particular system.

Bohr Model:

h = Planck's constant
n = quantum number (orbit number)

Quantum Model:

H = E

where:

= psi, the wavefunction

E = the energy of the sytem

H = The "Hamiltonian", a nasty mathematical operator. The Hamiltonian equation is shown below.



The hamiltonian shown in the graphic above is the hamiltonian for the electrons. Notice that there are four terms (each term separated by a "+" or "-" sign):

1. a term to calculate the kinetic energy (k = 1/2 mv²) of the electrons
2. a term to describe the attraction of electrons to the nuclei. The little "r" term is the radius, or distance, of the electron from the nucleus. The large "R" term is the internuclear distance.
3. a term to describe the repulsive force of an electron to another electron
4. a term to describe the repulsive force of a nuclei with another nuclei.

Schroëdinger's equation is an eigenvalue equation. From it we obtain two interesting and important values: E, the energy of the system, and , (psi) the wavefunction. The wavefunction by itself has no significance. However, the square of the wavefunction can be used to find the probability of locating an electron in an atom. To find this probability for three-demensional space (x, y, and z coordinate system), you must use Schroëdinger's equation with a specified radius, the squared wavefunction value, and multiply the wavefunction by 4(pi)r² (the volume of a sphere).

As an example of what these wavefunctions look like, the graphic below shows the mathematical equations to calculate the wavefunctions for a one-electron system -- hydrogen -- at the level of the 1s, 2s, and 2p orbitals:



The important terms in these equations are the radius of the electron from the nucleus and the constant known as the Bohr radius, a value of 0.529 angstroms, or 0.528 Å. As an enrichment activity, you might consider modeling these equations using a simple spreadsheet or equation-solver such as MathCAD. The graphic below shows a section of my spreadsheet. I used a Bohr radius of 52.9 picometers, with the electron radius increasing by increments of 2 pm. If you graph the probabilities as a function of the radius, as shown below, you should see that the electron has the highest probability of being found at the "nodes" where Bohr predicted the orbitals should be:




Needless to say, calculating Schroëdinger's equation for systems with more than one electron is quite difficult. However, by using a number of approximations, coupled with the ever-increasing number-crunching of modern computers, it is now possible to "solve" Schroëdinger's equation for multiple-electron systems.