The MNDO method will be used because it is the oldest of the "NDO" methods. The CNDO/2 method is very similar, and the example we will look at will emphasize the similarity. The system to be examined is a regular hexagon of hydrogen atoms in which the H-H distance is 0.98316 Ångstrom. Of course, a regular hexagon of hydrogen atoms is not a stable system; the only reason we are using it here is to demonstrate the working of an SCF calculation. The optimized geometry was obtained by defining all bond lengths to be equal, constraining all bond angles to be 120 degrees and defining the system as being planar. We will need various reference data in order to follow the calculation. MOPAC contains a large data-set, BLOCK.F, of atomic and diatomic parameters for all the elements which have been parameterized. By reference to this source file we find that, for hydrogen:
G_{ss} = <φ_{s}φ_{s}|1/r|φ_{s}φ_{s}> |
= |
12.848 eV |
U_{ss} = <φ_{s}|H|φ_{s}> |
= |
-11.906276 eV |
ξ_{s} |
= |
1.331967 Bohr |
β_{s} |
= |
- 6.989064 eV |
E_{atom} |
= |
52.102 kcal/mol |
Atom | 1 | 2 | 3 | 4 | 5 | 6 |
1 | 0.0000 | |||||
2 | 0.9832 | 0.0000 | ||||
3 | 1.7029 | 0.9832 | 0.0000 | |||
4 | 1.9663 | 1.7029 | 0.9832 | 0.0000 | ||
5 | 1.7029 | 1.9663 | 1.7029 | 0.9832 | 0.0000 | |
6 | 0.9832 | 1.7029 | 1.9663 | 1.7029 | 0.9832 | 0.0000 |
The overlap integral of two Slater orbitals between two
hydrogen atoms is particularly simple:
At the optimum H-H distance of 0.9831571Å, this yields an overlap integral of 0.4643. The nearest-neighbor one-electron integral is thus
Atom | 1 | 2 | 3 | 4 | 5 | 6 |
1 | -51.7124 | |||||
2 | -3.2457 | -51.7124 | ||||
3 | -1.0970 | -3.2457 | -51.7124 | |||
4 | -0.6992 | -1.0970 | -3.2457 | -51.7124 | ||
5 | -1.0970 | -0.6992 | -1.0970 | -3.2457 | -51.7124 | |
6 | -3.2457 | -1.0970 | -0.6992 | -1.0970 | -3.2457 | -51.7124 |
On-diagonal one-electron integrals are more complicated than the off-diagonal
terms. The one-electron energy of an electron in an atomic orbital is the sum
of its kinetic energy and stabilization due to the positive nucleus of its own
atom, U_{ss} or U_{pp}, plus the stabilization due to all the other nuclei
in the system. Each electron on a hydrogen atom experiences a stabilization due
to the five other unipositive nuclei in the system. Within semiempirical theory
the electron-nuclear interaction is related to the electron-electron
interaction via
Atom | 1 | 2 | 3 | 4 | 5 | 6 |
1 | 12.8480 | |||||
2 | 9.6585 | 12.8480 | ||||
3 | 7.0635 | 9.6585 | 12.8480 | |||
4 | 6.3622 | 7.0732 | 9.6585 | 12.8480 | ||
5 | 7.0635 | 6.3622 | 7.0732 | 9.6585 | 12.8480 | |
6 | 9.6585 | 7.0635 | 6.3622 | 7.0732 | 9.6585 | 12.8480 |