Biomedical Engineering Reference
In-Depth Information
The self-consistent field functions for atoms with 2 to 36 electrons are computed with a
minimum basis set of Slater-type orbitals. The orbital exponents of the atomic orbitals are
optimized as to ensure the energy minimum. The analysis of the optimized orbital exponents
allows us to obtain simple and accurate rules for the 1s, 2s, 3s, 4s, 2p, 3p, 4p and 3d electronic
screening constants. These rules are compared with those proposed by Slater and reveal the
need for the screening due to the outside electrons. The analysis of the screening constants
(and orbital exponents) is extended to the excited states of the ground state configuration and
the positive ions.
What they did, starting fromSlater's ideas of 1s, 2s, 2p,... atomic orbitalswithmodified
orbital exponents (effective nuclear charges), was as follows for each atom in their set.
•
Decide on the electronic ground-state configuration and choose starting values of the
orbital exponents from Slater's rules.
•
Optimize each orbital exponent individually by the HF-LCAO procedure. At the end of
each optimization, the earlier optimized values will have changed and so...
•
Check for self-consistency amongst the orbital exponents and either exit or go back
one step.
There are better optimization procedures, as you will know from reading earlier chapters,
but that is how the early workers did things.
Some of the results of Clementi and Raimondi are given in Table 16.1. We call such basis
sets
single zeta
or
minimal
because they use exactly the same number of atomic orbitals as in
descriptive chemistry. For each atom there is just one 1s orbital, one 2s, three 2p and so on.
Table 16.1
Comparison of Slater's exponents with those of Clementi and Raimondi (CR)
Atom
CR 1s
exponent
Slater
value for 1s
CR 2s
exponent
CR 2p
exponent
Slater value
for 2s/2p
H
1
1
He
1.6875
1.70
Li
2.6906
2.70
0.6396
0.650
Be
3.6848
3.70
0.9560
0.975
B
4.6795
4.70
1.2881
1.2107
1.300
C
5.6727
5.70
1.6083
1.5679
1.625
N
6.6651
6.70
1.9237
1.9170
1.950
O
7.6579
7.70
2.2458
2.2266
2.275
F
8.6501
8.70
2.5638
2.5500
2.600
Ne
9.6241
9.70
2.8792
2.8792
2.925
16.8.2 Extension to Second-Row Atoms
Clementi (1964) extended this treatment to the second row in his 1964 paper, and he wrote
the following in his abstract.
The self-consistent field functions for the ground state of the first and second row atoms (from
He to Ar) are computed with a basis set in which two Slater-type orbitals (STO) are chosen
for each atomic orbital. The reported STOs have carefully optimized orbital exponents. The
total energy is not far from the accurate Hartree-Fock energy given by Clementi, Roothaan
and Yoshimine for the first row atoms and unpublished data for the second-row atoms. The
