Lesson 9 Gaussian and Slater Orbitals Case Study - joslynnlee/CHEM-454 GitHub Wiki
Background
We can first start with the Bohr model which looks at electrons as fixed orbits moving at a constant radius around the nucleus. But we know that electrons do not move in fixed orbits but rather random paths. We cannot know the individual location and movements of certain electrons but we can see where they spend a percentage of their time. An orbital is a probability of the presence of an electron at a given place and time. The use of atomic orbitals allows us to interpret molecular properties and charge distributions in terms of atomic properties and charges
Orbital representations can be classified as STO and GTO. Both of these representations are functions of the radius.
Slater type orbitals (STO)
In lecture we saw the contribution of alpha (α), its a typo that should have been zeta, ζ which is a constant related to the width of the orbital or we can also think of the effective charge of the nucleus for STOs.
At this point, the STO equation looks like a great solution to finding the basis set of a molecule. Although the STO equation is a wonderful approximation for the molecular orbitals, the problem lies in the calculations. You see, calculating the STO of a molecule requires enormous amounts of work. This uses way too much time, even for super computers. Instead, a scientist by the name of S. F. Boys developed a method of using a combination of Gaussian Type Orbitals in order to express the STO equation.
Gaussian type orbitals (GTO)
The GTOs is a function that is only the radial part of the orbit changes. Notice that the difference between the STO and GTO is in the "r." The GTO squares the "r" so that the product of the gaussian "primitives" (original gaussian equations) is another gaussian. By doing this, we have an equation we can work with and so the equation is much easier. However, the price we pay is loss of accuracy.
To compensate for this loss, we find that the more gaussian equations we combine, the more accurate our equation. In Basis Set Lab, you will be looking at a STO-3G basis set:
- The "STO" tells you that you are attempting to represent a STO equation for your molecule.
- The 3G tells you that in order to do this you are combining 3 gaussian primitives.
In this case, STO-NG, the number "N" before the "G" is the number of gaussian primitives that are used to simulate the STO equation. Remember, the bigger your N, the more accurate your results.
All basis set equations in the form STO-NG are considered to be "minimal" basis sets. The "extended" basis sets, then, are the ones that consider the higher orbitals of the molecule.
You will investigate how the Gaussian functions are computed, combined, and how the number of Gaussians affects the accuracy of the combined model.
Today's Practice
Let's go to the website: http://shodor.org/succeed-1.0/programs/compchem98/labs/sto/
For Section 1: For this, the Gaussian function is accurately describing the molecule. The defining characteristic of Gaussian type orbital is its radial part:
r is the length of the electron position vector r and α is a real parameter. The parameter α may be taken from tables of atomic orbital basis sets, which are often contained in quantum chemical computer programs, or can be downloaded from the web. Here for Hydrogen, the α = 0.4166
Interpreting the plot of radius verses Gaussian
The x-axis is the radius distance. r=0 is the nucleus. The Gaussian function becomes flat at the top and at larger values of r=3, the values fall off. Remember the y-axis is the function of the radius.
For Part 2
Instead of 'Gaussian Order Form' link go to: https://www.basissetexchange.org/
You will need to click on the hydrogen atom on the periodic table. Then you will select STO-3G from the list. Then in the Download Basis Set, you can select Gaussian or NWChem format. Then click on 'Get Basis Set'.
This is a basis set for hydrogen. The left list is alpha values, and the right list is coefficients (d) which will be used to sum up the three gaussians.
Check with Dr. Lee that you have to correct values.
For the 'g1', 'g2' and 'g3' use the following equation with the alpha from the Basis Set:
Once you have the 'g1', 'g2' and 'g3', these are the three gaussians. 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:
d1: coefficient for alpha 1 = gaussian 1
d2: coefficient for alpha 2 = gaussian 2
d3: coefficient for alpha 3 = gaussian 3
LCAO = d1g1 + d2g2 + d3g3
Questions: With 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.
For Part 3
You will compare the STO-3G approximation to the STO. The STO equation is:
The zeta, ζ , is the orbital coefficient and similar to alpha, α, in Gaussian curves. ζ for hydrogen is 1.24. Use this in the calculation for the radius range of 0 to 6.
Once you create a new graph with the LCAO and Slater, as well as the three gaussians. Think about how the gaussian and Slater approximations are related!
Instead of doing all the approximations, only generate the STO-6G approximation for hydrogen! Go back to the order form https://www.basissetexchange.org/ find the basis sets, both alpha and d values, for STO-6G.
Is there an increase in accuracy from STO-3G?
What to turn in?
Upload your excel sheet with the necessary plot for each part.
Write up a discussion addressing the questions in Part 2 and 3.