Concise Physical Chemistry

c17 JWBS043-Rogers September 13, 2010 11:28 Printer Name: Yet to Come

282 THE VARIATIONAL METHOD: ATOMS

17.10 SLATER-TYPE ORBITALS (STO)

In the absence of an exact solution of the Schrodinger equation for atoms beyond ̈

hydrogen in the periodic table, John Slater devised a set of empirical rules for writing

down approximate wave functions. Every entry in Table 17.1 is a function involving

the following: the radial distance in units of bohrs,r/a 0 ; a negative exponential

e−Zr/na^0 ,wherenis the principal quantum number, 1, 2, and 3 for H, He, and the first

two full rows of the periodic table; anda 0 is the Bohr radius.Zis the nuclear charge.

Slater wrote an approximate wave function involving only the radial part for the first

two full rows in the periodic table, ignoring the spherical harmonics. The function

takes the form

φ(r)=re

−−(naZ− 0 s)

An adjustable parametersis called thescreening constant, andZis also an adjustable

parameter called theeffective quantum number. For the first-, second-, and third-row

elements,Zsimply is 1, 2, and 3, though it becomes nonintegral later in the table.

What will concern us most is the screening constant, which can be arrived at by fitting

experimental data.

Slater’s rules for determining what are now calledSlater-type orbitals(STO) are

simple for the first three rows of the table. They become more complicated and

less reliable lower in the table. We have already seen that the screening constant

for the second electron in the helium atom is about 0.3. For 1selectrons in higher

atoms, Slater modified this slightly to 0.35. When a 1selectron screens apelectron,

screening is more effective than simple electron–electron screening in helium because

the probability density lobes of the 2porbitals lie outside of the 1sorbital. Slater chose

a screening constant of 0.85. Notice that the polynomial parts of the wave function

(Table 17.1) are gone. Only the “tail” of the wave function is represented because that

is the part involved in the first ionization potential and, for the most part, in chemical

bonding. For more detail on Slater’s rules, see Problem 15.79 in Levine (2000).

TABLE 17.1 Slater’s Rules

1. Array orbitals 1s, 2s, 2p, 3s, 3p, 3d,...


2. Consider only the orbital containing the electron in question and the one below it.


3. Electrons lower than (2) areinterior orbitals.


4. Orbitals higher than (2) areexterior orbitals.


5. Electrons designated s (1s, 2s, etc) have a screening constant of 0.30.


6. Electrons in the same orbital have a screening constant of 0.85.

Electrons are in the orbitalφ(r)=re

−(naZ−s)

(^0). For example, in bohrs and hartrees, the orbital
electron in helium hasφHe(r)=re
−(Zn−s)
=re
−(Zn−s)
=re
−(1(1)−.30)
= 1. 7 rdue to shielding.