Gaussian and Slater Orbitals Case Study

Background

The traditional model of an atom involves visualizing the electrons as fixed in "orbits", moving at a constant radius around the nucleus. During the early part of this century, serious chemical and physical errors were found in that model, and a new idea took its place. Electrons do not move in fixed orbits, with a constant radius, the theory says, but in random paths, only maintaining some semblance of organization. This theory fits with other quantum theory by saying that we cannot predict the behavior of an individual electron precisely, but we can know general properties about it, and we can know certain things about the behavior of a group of electrons. The form that electrons take around the nucleus is a sort of "cloud". That is, we cannot know the individual locations and movements of certain electrons, but we know where they will spend a percentage of their time. This is the same as saying that if we could take a snapshot of the atom, the cloud is the region where the electrons would appear in almost every one of the snapshots. This cloud is called an "orbital".

The orbital can be represented mathematically by a "wavefunction". This is a mathematical abstraction which has no exact physical meaning. If the wavefunction is squared and multiplied by some factors, it represents actual probability of the presence of the electron at a given place and time. However, examining the graph of the wavefunction can help us understand how electron clouds are handled computationally.

The orbital representations that we will be working with are called Slater type orbitals (STO) and Gaussian type orbitals (GTO). These two are about equally hard to compute for simple atomic structures, but the STO's increase in mathematical complexity too much to be useful once several electons are introduced. GTO's are used as the next best approximation of the actual wavefunction. These approximations are used in ab initio methods to calculate mathematically the appearance of electron clouds and orbitals. However, once a molecule begins to be complex, (approximately more than 30 atoms), even the GTO approximation is too computationally expensive to be used.

Spreadsheets

To follow this study, you will need an Excel Spreadsheet. Please open one on your desktop now. If you would like a quick refresher on the use of spreadsheets, especially the commands we will be using, please look at the Basic Guide to Spreadsheets. If you are comfortable with these tools, please go ahead.

Study

Section 1

We will be modelling the simplest of atoms, hydrogen. As you may know, the most common form of hydrogen has only one proton as its nucleus and a single electron in the orbital. However, even this basic form requires complex calculations for a realistic model! To model the electron cloud(s), several numerical constants are needed. The set of constants that are used for a given atom, with a given level of complexity in the mathematical representation, is called the basis set. These are named by their level of complexity, for example, STO-G3 is a Slater type orbital using three gaussian curves in the approximation. Gaussian curves are similar in shape to the bell curve, or normal distribution, which you may have seen. A number of these curves are combined, or superimposed, to create another, more precise, approximation. You will investigate how the gaussian curves are computed, combined, and how the number of gaussians affects the accuracy of the combined model.
We will start by constructing the simple gaussian curve for hydrogen. The function for this is:

where alpha is 0.4166 and r is radius, or distance from the nucleus.

Start your spreadsheet by putting this constant in a cell somewhere around the upper left hand corner. Notice what cell it is in, because you will have to refer to it by name in your equations.
Use the cell next to it to put a label, such as "alpha".
Now we need to give the equations a list of radii to use in calculating the gaussian. Leave some cells at the top, and put a label in the left most column, "Radius".
We want to model the equations from -3 to 3 units away from the nucleus (nucleus=0, and units are in angstroms). So put a -3 in the cell below Radius. We want use steps of .01 units in the list of radii.
To avoid typing in all these numbers, (how many are there, anyway?) we're going to use a series command in Excel.
Highlight the -3 and enough cells below it to contain all the values we will have for radius.
Now go to Fill, but instead of using Down or Sideways, click on Series.
The series command will ask for a step size. Make that .01, and click OK.
Next to that list, we are going to generate values for a simple gaussian curve. So label it something clever like "gaussian" and put the formula in the first cell, making sure to refer to alpha and the radius by cell name in the appropriate places. Be careful of your parentheses; use lots of them, but make sure that they all match up properly!
You should get a value in the first cell which starts out ".00869" If you don't, go back and examine your formula more closely.
Fill the formula down the column.

Graphing

Now that we have this set of numbers, let's put it into a graph.

Highlight the lists "Radius" and "gaussian", including the titles if you want.
Now click on the button which has a bar graph and a wand in the toolbar.
This will ask you to click on the already highlighted area.
Follow the steps in this to create a line graph (second type of line graph), and use the labels appropriately.

Note the shape of this curve; this is a basic gaussian. What do you notice about the equation on either side of the nucleus? Are the two sides similar?

Part 2

Now we would like to use several gaussians to model the hydrogen orbital. To get all of the numerical constants required, we will have to use a basis set as a reference. These are stored in a large database, from which you can "order" a specific basis set. Look at the page, image and directions below to see how.
Go to the Gaussian Order Form. Another link to this site is on the online tools page. Your browser should look like this:

Fill in your order form to look like this one. Notice that we are ordering STO-3G, which means we will receive constants for 3 gaussians. The h stands for hydrogen. Don't put in an email!! When you're ready, click Submit. The result should be two columns of three numbers each. This is a basis set for hydrogen. The left list is alpha values, and the right list is coefficients which will be used to sum up the three gaussians.

Now that you have these constants, enter them (with corresponding labels) in the top of your spreadsheet. Next to the gaussian column, put three new columns, labelled something like "g1", etc. In the next cell down, enter the gaussian equation, using the values of alpha from the basis set and the radius as needed. Fill down the column, and repeat the procedure for the other two columns. Since we looked at the equations on both sides of the nucleus before, you should have observed that it is exactly the same on both sides. This is important! We are only interested in the positive side of the graph now, since we know that both sides will be exactly the same (and since negative and positive do not have any true physical meaning here!). So, change the first number under Radius to be 0. When you do this, the whole list should change to go from 0 to 6 instead of -3 to 3. Also, the values in your gaussian lists should change, since we are computing them for different radii now. Check your values against those of your classmates to make sure you are all getting correct answers.

No single one of these three gaussians is an approximation on its own; they must be combined. The way of combining them is called "Linear Combination of Atomic Orbitals" (LCAO). The other list of numbers in the basis set are coefficients for adding up the three gaussians. They are used in the following equation.

Create another column next to the three gaussians, labelled "LCAO". Put the equation, referring to the appropriate constants, in the first cell, and fill the column. Now put all three new gaussians and the LCAO on a graph. What do you notice about the shape of the three gaussians? How similar are they? How similar is each one to the simple gaussian we first created? How do they relate to the LCAO? Discuss these questions and compare results with you lab parter and classmates.

Part 3

Now we will compare this approximation to the STO. Below is the equation for this.

The "squiggle" is a Greek character standing for an orbital coefficient, similar to the constant for gaussian curves. For hydrogen, the "squiggle" is equal to 1.24. Put (and label!) this value on the top of your spreadsheet. Now create and fill another column, labelled Slater or STO, with this equation. Create a new graph with the LCAO and Slater, as well as the three gaussians, if you wish. Think about how the gaussian and Slater approximations are related!

To make this contrast clearer, we are going to examine several gaussian approximations and compare them to the STO. Set up your spreadsheet with the appropriate columns to do two-gaussian and six-gaussian approximations as well (ie, you will need enough columns for all of the new gaussians and columns for each sum). Now go back to the order form for basis sets and find the basis sets, both alpha and d values, for STO-2G and STO-6G. Put these values into your spreadsheet at the top, and set up each column with the neccessary equations. Now do both combinations, ie, the sums, of the 2G and 6G approximations. Then graph the 1 gaussian (the first one created), 2 G, 3 G (which was the second one we examined), 6G, and STO all on a single graph. Do not put all of the gaussians for each approximation on there, just the summation.

Remember that the STO is the most accurate approximation of the waveform. Discuss these questions and your results with your lab partner and classmates. How accurate is each of the gaussian approximations? Which approximation is the most accurate? Which one takes the most time to compute? How does accuracy relate to number of gaussians used? Which is more important, the time required or the accuracy? How should we decide which approximation to use for a specific project?

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