In the previous exercise you learned how the value of the effective nuclear charge ( *Z _{eff}* ) affects the size of an orbital. You also had an opportunity to examine how

In 1930 J. C. Slater 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

*N _{0}* represents the number of

The effective nuclear charge is

For an electron in the fluorine 2s orbital, for example, *N _{0}* = 6 ( the electron of interest in the L shell is not counted),

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 *Z _{eff}* 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).

The Bohr model for the hydrogen atom predicts that the radius *r _{BM}* of an orbital with principal quantum number

The factor *a _{o}* 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:

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.

- Explore the usefulness of the Slater Rules for estimating
*Z*._{eff} - Test the usefulness of the Bohr model expression for orbital sizes using
*Z*in place of_{eff}*Z*.

Look at the various s orbitals for several multi-electron atoms and determine the best value of *Z _{eff}* for each orbital. The best value for

How well do the Slater rules work? Are the values of *Z _{eff}* obtained using the Slater rules similar to those you obtained?

How well does the Bohr model expression for orbital sizes work?

The radius listed below is the radius of the 90% isosurface. What is the best value for *k*? Is this value reasonable?

The value of *k* provides a sense of the size of the 90% isosurface compared with the radius provided by the Bohr model, which loosely describes the middle of the orbital.

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Effective Nuclear Charge

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