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I n the Classroom

Screening Percentages Based on Slater Effective Nuclear
Charge as a Versatile Tool for Teaching Periodic Trends
Kimberley A. Waldron,* Erin M. Fehringer, Amy E. Streeb, Jennifer E. Trosky, and Joshua J. Pearson
Department of Chemistry, Regis University, Denver, CO 80221; *

The solutions to the Schrödinger equation allow us to
understand thehydrogen atom in terms of its electron density at radial distances from the nucleus. Consideration of
the helium atom brings new complications. Each of helium’s
two electrons does not each “feel” two full units of nuclear
charge owing to the presence of the other electron. The interelectronic interaction between its two electrons must be estimated. One of the most common ways to approach this
problem isto assume that each electron in a given atom is
hydrogen-like and that it experiences interactions due to the
presence of other electrons. This can be accomplished using the
“variational method” in which an energy is arbitrarily set for
the system and that energy is minimized by the manipulation
of several parameters incorporated in the orbital wave functions.
J. C. Slater adopted this type ofapproach by developing
“Slater-type orbitals” (STOs), which approximate hydrogenlike orbitals. These (nodeless) orbitals are expressed by Slater
in terms of a wave function with polar coordinates (1, 2):
ψ (r,θ, φ) = Nnl r n 1e ξr Yml (θ, φ)


where Y are the spherical harmonics and ξ = Z */n *. Using
the variational method, the “effective” principal quantum
number n * and the “effective” atomicnumber Z * are varied
by Slater in a way that minimizes the energy of the system.
In order to obtain the best value of ξ , Slater developed a set of
rules that allows quantification of shielding by partitioning
the electrons into groups: electrons within the same group
or occupying “inner” groups all contribute to the shielding
(3). For example, rule four provides that an electron in a shelldirectly below the electron in question will “shield” that
electron by 85%. Summation of shielding contributions from
all electrons lying below the electron in question provides the
total shielding, S, which is then used to calculate the effective nuclear charge, Z *:
Z* = Z – S


Slater’s rules provide only approximate values for electron
energies in polyatomic atoms partly because they ignorethe
presence of nodes. Certainly, it is possible to obtain more
precise estimates using a more sophisticated computational































Figure 1. Slater Z* values for the (A) n = 2 elements and (B) group
1A elements.

technique such as the Hartree–Fock calculation (4 ), which
isalso a variational method. However, as an empirical method
based on ionization energies, Slater’s rules have proven useful
for predicting trends such as those for electronegativity and
ionic radius, and Slater’s optimized values of Z * and n *
should be useful for predicting ionization energies (in units
of electron volts) according to the following equation (5):
E/eV = (Z *)2(13.6 eV)/(n *)2

(3)Comparison with actual ionization energy data shows that
the correlation is a rough one. In addition, Slater’s Z * values
are a limited pedagogical tool for understanding shielding
trends throughout the periodic table, as we shall see.
Improving the Pedagogical Value of Slater’s Rules
At several points during an undergraduate curriculum
in chemistry, students are exposed to periodic trends inatomic
size and electronegativity. For introductory students this
information is presented in a format that depicts the periodic
table and indicates the “corners” of the table that represent
maxima and minima in the specific trend. Often, a simplified
“screening constant” is introduced. This is easily calculated by
assigning a value of 1.0 to each core electron and subtracting
this value from the...
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