MadSci Network: Chemistry

Re: Do the Slater rules for determining screening constants take into account

Date: Tue Oct 28 14:27:01 2003
Posted By: Todd Whitcombe, Associate Professor, Chemistry
Area of science: Chemistry
ID: 1065381516.Ch

What a good question! I will try to answer it, although a lot of what I 
am going to say could be found in text books (ie. Huheey, Inorganic 
Chemistry, 4 Ed.) and it may be easier to read it there or, hopefully, 
integrate my comments with the text to better explain it.

Do the Slater's Rules for screening constants take into account the 
stability of filled and half-filled orbitals?

Simple answer, no. Slater was working on development of his rules in the 
1920's and 30's, when there were no computers and quantum mechanics was 
just beginning. The ideas about orbitals and such were just being 
formulated. Slater's Rules were developed to explain the concept and, to 
a first approximation, allow for some rationalization of the observable 
trends in the periodic table. They work and they don't. They work in that 
they can help to explain such broadly observed phenomenon as the 
periodicity of ionization energy (which is, from Bohr, a function of 
nuclear charge squared and should increase exponentially) and ionic 
radii. They don't work in that Slater's Rules don't take into account 
some of the subtleties of the table, like half-filled and filled orbitals.

What other things did Clementi and Raimondi take into account when the 
calculated their values for effective nuclear charge?

Clementi and Raimondi did their work on effective nuclear charges in the 
early 1960s. By this time, there was a great deal of background work that 
had been done on orbitals and molecules. And the computer had been 
invented! This gave them the ability to incorporate self-consistent field 
(SCF) wave functions for the hydrogen to krypton atoms into their 
calculations. They didn't have to rely on Slater-type orbitals which, for 
simplicity of calculation, didn't contain nodes. They were thus able to 
go to a greater depth with a refined mathematical model, and this allowed 
them to clearly distinguish the s-orbitals from the p-orbitals in 
determining their set of rules. Specifically, they had a better model for 
dealing with electron penetration of the inner core.

How does the special stability of half-filled orbitals relate to the 
effective nuclear charge, and is it ever considered when calculating 

Again, the simple answer would be: it doesn't and no, it is not 
considered. But that doesn't help us much. As you have pointed out, the 
unexpected electronic configuration is a consequence of exchange energy 
and the calculation of the effective nuclear charge by SCF doesn't 
include that in considering the atom. The result is that Zeff is a linear 
progression across "Cr". But Clementi and Raimondi's model does take into 
account the filled orbitals. That is, their rules are able to distinguish 
the s-orbitals and p-orbitals, and appear to have steps that would 
accomodate the filling of all orbitals. In other words, they include some 
of the effects but not all.

One article I found on SCF points out that Clementi and Raimondi's 
results were still not great because it is difficult to "estimate 
repulsion from a wide-ranging electron by treating it as a fractional 
static electron at the nucleus, but it's better than nothing." They go on 
to point out that in more sophisticated SCF approaches, that each 
electron is treated as existing in a static electric field - due to the 
fixed nucleus and a static cloud of electron density given by the 
superposition of their assumed densities. This potential allows you to 
calculate a new set of densities, which are used as input to improve the 
orbitals, again and again until a self-consistency is reached. If the 
exchange integral is not included in the calculation, then all of the 
measurements will be missing this effect and it would appear that that is 
the case. I am not an expert on the latest measurements and calculations 
around effective nuclear charge, but I suspect that there has been little 
improvement on the SCF results and hence, the answer to the second part 
is "no".

I hope this helps! Good question and there is a fair amount of 
information out there. It is likely that your University has access to 
the original articles by Clementi and Raimondi (J. Chem. Phys. 1963, 38, 
2686; J. Chem. Phys. 1967, 47, 1300) and might have some of the 
more "recent" work by Froese-Fischer (Atomic Data 1972). And, of course, 
there is always the classic paper by Slater (Physical Rev. 1930). 

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