Slater's rules
In many electron systems the electron-electron interaction is complicated and there have been many attempts to introduce simplified methods for approximate calculations of different kinds. The most well known were introduced by Slater who derived rules for determination of screening constants s, which give an effective nuclear charge Ze that is experienced by each electron in the atom where
Ze = Z - s
The rules are the following:
|
Principle quantum number |
l quantum number |
Number of electrons in the shell |
s |
|
1 |
0 |
2 |
0,3 |
|
> 1 |
0,1 |
x,y,z |
0,35x + 0,85y + 1,00z |
|
3 |
2 |
x,y |
0,35xd + 1,00yd |
x = number of other electrons in the same shell as the screened electron
y = number of electrons in the shell with principle quantum number n - 1
z = number of electrons in shells with principle quantum number
n
- 2
xd = number of 3d-electrons
yd = number of electrons with principle quantum number
3
and l < 2
Example Calculate the binding energy for a 1s electron in helium.
We get Ze = Z - s = 2 - 0,3 = 1,7
The energy of the ion is Eion = - 2 a.u. = - 54,42 eV and the binding energy EB is obtained as the energy difference
The experimental value is 24,587 eV.
Exercise: Calculate the binding energy of the 3s electron in
the sodium atom and compare with the experimental value.
If we use the expression for the radius of the orbit obtained in the Bohr model with the effective nuclear charge along with the principle quantum number for the outermost electron we may estimate the atomic radius reff from the formula
Example Calculate the atomic radius for the neon atom.
We have s = 0,35x7 + 0,85x2 = 4,15 and thus Ze = 10 - 4,15 = 5,85. Hence we get reff = 4/5,85x0,529167 = 0,362 Å in reasonable agreement with the tabulated value, which is 0,354 Å.
Exercise:
Calculate the atomic radius of an oxygen atom.
See a picture representing the
Atomic radii
and study the dependence on the position in the periodic Table.
