## Effective Nuclear Charge

### Concepts

By now you are well-acquainted with the properties of the orbitals for a hydrogen-like atom. A hydrogen-like atom is an atom that possesses only one electron. What are the properties of orbitals for multi-electron atoms?

The new feature that exists in multi-electron atoms is electron-electron repulsions. Unlike a hydrogen-like atom, for which analytical solutions of the Schrodinger equation are available, the Schrodinger equation cannot be solved analytically for multi-electron atoms owing to the electron-electron repulsion terms. With the aid of computers, however, numerical solutions have been obtained. Interestingly, it is possible to obtain a crude solution for a multi-electron atom by employing a relatively simple construct: employing the "effective" nuclear charge in place of the nuclear charge in the equations for a hydrogen-like atom.

The nuclear charge, Z, is the charge of the nucleus. The effective nuclear charge, Zeff, is the amount of positive charge on the nucleus perceived by an electron. Electrons intervening between the nucleus and an outer electron are said to shield or screen the outer electron from the nucleus so that the outer electron does not experience the full nuclear charge.

Consider the fluorine 2s orbital. By choosing an appropriate value for Zeff and employing this value in the equation for the hydrogen 2s orbital, one can obtain a crude approximation for the fluorine 2s orbital.

When Zeff = 1 (atomic number of hydrogen), the orbital is the original 2s orbital for the hydrogen atom.

When Zeff = 9 (atomic number of fluorine), the orbital is the 2s orbital expected for fluorine if there were no electron-electron repulsions.

The approximation for the fluorine 2s orbital using the hydrogen 2s equation is obtained with a value of Zeff somewhere between 1 and 9. A value of Zeff near 9 would indicate that there is very little shielding of a fluorine 2s electron by other electrons in the atom. A value of Zeff near 1 would indicate that there was almost complete shielding of a fluorine 2s electron by other electrons in the fluorine atom.

For a given electron, significant shielding is only provided by electrons in the same or smaller shells. To a first approximation, electrons in larger shells do not affect Zeff .

### Exercise

Objective
• Acquaint yourself with the concept of an effective nuclear charge and how this concept can be used to understand the properties of multi-electron atoms.

Employ the interactive display below to answer the following questions. The display compares s orbitals (1s or 2s) from the hydrogen and fluorine atoms. On the left is the hydrogen s orbital for a given shell, and the s orbital from the same shell for a fluorine atom is shown on the right. (The orbitals for the fluorine atom were obtain from numerical solutions of the Schrodinger equation.) The slider can be used to move the atoms together or apart; the VRML controls can be employed to zoom in or out.

When selecting a new pair of orbitals, please be patient. The VRML files are relatively large, and the download and processing of the files may take a several seconds (or longer).

In this exercise you will examine the effect of the effective nuclear charge on the size of an orbital. You may use the controls under the hydrogen orbital to control the value of Zeff used in displaying the hydrogen orbital.

Part 1

Examine the relative sizes of the hydrogen 2s and fluorine 2s orbitals when using Zeff = 1.

Are the two orbitals the same size? Is one orbital much larger than the other? Why is this behavior observed.

Part 2

Set the value of Zeff to 9 (the atomic number of fluorine). (You may need to use the slider or VRML zoom command to alter the display to obtain a good view of the orbitals.)

Are the two orbitals the same size? Is one orbital much larger than the other? Why is this behavior observed.

Part 3

Vary Zeff for the hydrogen 2s orbital until it is the same size as the fluorine 2s orbital.

When Zeff increases, what happens to the size of the hydrogen orbital? Why is this behavior observed?

What is the best estimate of the effective nuclear charge for an electron in the fluorine 2s orbital? Is this value plausible?

Part 4

Load the hydrogen 1s and fluorine 1s orbitals. Vary Zeff for the hydrogen 1s orbital until it is the same size as the fluorine 1s orbital.

What is the best estimate of the effective nuclear charge for an electron in the fluorine 1s orbital? Is this value plausible?

How does Zeff for a fluorine 2s electron compare with that for a fluorine 1s electron? Why are these two values different? Explain the relative values of Zeff in terms of the shielding provided by other electrons in the atom.

 Set Zeff: Hydrogen 1s / Fluorine 1s Hydrogen 2s / Fluorine 2s

Slater Rules and Orbital Size