Shielding and Effective Nuclear Charge

    An atom of ¹H consists of a single proton surrounded by an electron that resides in a spherical 1s orbital. Recall that orbitals represent probability distributions meaning that there is a high probability of finding this electron somewhere within this spherical region. This electron, being a negatively charged particle, is attracted to the positively charged proton.

    Now consider an atom of helium containing two protons (there are neutrons in the nucleus, too, but they are not pertinent to this topic) surrounded by two electrons both occupying the 1s orbital. In this case, and for all other many electron atoms, we need to consider not just the proton - electron attractions, but also the electron - electron repulsions. Because of this repulsion, each electron experiences a nuclear charge that is somewhat less than the actual nuclear charge. Essentially, one electron shields, or screens the other electron from the nucleus. The positive charge that an electron actually experiences is called the effective nuclear charge, Z_eff, and Z_eff is always somewhat less than the actual nuclear charge.

Z_eff = Z - S, where S is the shielding constant

The two figures represent different instantaneous positions for the two electrons in an atom of helium. The electrons both occupy a 1s orbital meaning that the time-averaged positions of the electrons can be represented by a sphere. At any moment, however, we could envision the two electrons as being on opposite sides of the nucleus in which case they poorly shield each other from the positive charge. At a different moment, one electron may be between the nucleus and the other electron, in which case the electron farther from the nucleus is rather effectively shielded from the positive charge.


    Considering helium, you might initially think that Z_eff would be one for each electron (i.e., each electron is attracted by two protons and shielded by one electron, 2 - 1 = 1), but Z_eff is actually 1.69. To determine Z_eff, we do not subtract the number of shielding electrons from the actual nuclear charge (that would always result in a value of Z_eff of one for any electron in any atom!), but rather we subtract the average amount of electron density that is between the electron we are concerned with and the nucleus. There are rules we can use to estimate the value of S and thus determine Z_eff, but we do not need to go into such details for this course.


Penetration

    An electron in an s orbital has a finite, albeit very small, probability of being located quite close to the nucleus. An electron in a p or d orbital on the other hand has a node (i.e., a region where there cannot be any electron density) at the nucleus. Comparing orbitals within the same shell, we say that the s orbital is more penetrating than the p or d orbitals, meaning that an electron in an s orbital has a greater chance of being located close to the nucleus than an electron in a p or d orbital. For this reason, electrons in an s orbital have a greater shielding power than electrons in a p or d orbital of that same shell. Also, because they are highly penetrating, electrons in s orbitals are less effectively shielded by electrons in other orbitals. For example, consider an atom of carbon whose electron configuration is 1s²2s²2p². The two electrons in the 1s orbital of C will do a better job of shielding the two electrons in the 2p orbitals than they will of shielding the two electrons in the 2s orbital. This means that for electrons in a particular shell, Z_eff will be greater for s electrons than for p electrons. Similarly, Z_eff is greater for p electrons than for d electrons. As a result, within a given shell of an atom, the s subshell is lower in energy than the p subshell which is in turn lower in energy than the d subshell.