## Effective Nuclear Charge

### Concepts

Slater Rules

In the previous exercise you learned how the value of the effective nuclear charge ( Zeff ) affects the size of an orbital. You also had an opportunity to examine how Zeff varies with the shell in an atom and how this effect is related to shielding or screening effects.

In 1930 J. C. Slate devised a simple set of guidelines for shielding or screening constants ( s ).

For a 1s electron, s = 0.3 .

For electrons in an s or p orbital with n > 1, the screening constant is given by

s = 1.00 N2 + 0.85 N1 + 0.35 N0

N0 represents the number of other electrons in the same shell, N1 represents the number of electrons in the next smaller shell ( n-1 ), and N2 is the number of electrons in other smaller shells ( n-2 and smaller ).

The effective nuclear charge is

Zeff = Z - s

For an electron in the fluorine 2s orbital, for example, N0 = 6 ( the electron of interest in the L shell is not counted), N1 = 2 (these are the 1s electrons), and N0 = 0 (there are no other inner shells). The Slater rules predict

s = 0.85 ( 2 ) + 0.35 ( 6 ) = 3.8

Zeff = 9 - 3.8 = 5.2

Is this value close to the value you obtained in the previous exercise?

The Slater rules indicate that electrons in the same shell as the electron of interest have a relatively small screening effect (they reduce the effective nuclear charge by 0.35 even though each electron possesses a full -1 charge). Electrons in the next inner shell (n-1) are very effective in shielding the nucleus, reducing Zeff by 0.85. Electrons that are even closer to the nucleus are completely effective in screening the nucleus (each electron shields a full -1 charge).

Orbital Size

The Bohr model for the hydrogen atom predicts that the radius rBM of an orbital with principal quantum number n is

rBM = ao n2 / Z

The factor ao is called the Bohr radius and has a value of 0.529 angstrom; it represents the radius of the inner-most orbital for the hydrogen atom

The display below presents the isosurface containing 90% of the electron density for an orbital. The radius for the Bohr model lies somewhere in the middle of the orbital, while the isosurface describes, in some sense, the outer limits of the orbital. Thus one might modify the Bohr formula to give the radius of an isosurface:

riso = k ao n2 / Zeff

The effective nuclear charge is used in place of the actual nuclear charge to account for the electron-electron repulsions.

The constant k is used to scale the Bohr radius from somewhere in the middle of the orbital to the limit described by the 90% isosurface. The exact value is not known from this simple reasoning, but a value of two or slightly larger would be plausible.

### Exercise

Objectives
• Explore the usefulness of the Slater Rules for estimating Zeff.
• Test the usefulness of the Bohr model expression for orbital sizes using Zeff in place of Z.

Look at the various s orbitals for several multi-electron atoms and determine the best value of Zeff for each orbital. The best value for Zeff is, in some sense, the value that produces the best size match.

Part 1

How well do the Slater rules work? Are the values of Zeff obtained using the Slater rules similar to those you obtained?

Part 2

How well does the Bohr model expression for orbital sizes work? What is the best value for k? Is this value reasonable?

Set Zeff

Li C F Na Si Cl
1s 1s 1s 1s 1s 1s
2s 2s 2s 2s 2s 2s
3s 3s 3s

Effective Nuclear Charge