- The parameterb(Eq.6.17) for a valencesAO.
- The parameterbfor a valencepAO.
- The parameterain the correction increment (f(RAB), (Eq.6.19) to the core–core
repulsion (Eq.6.18).
Some atoms have five parameters because for them MNDO takesbto be the
same forsandporbitals, and hydrogen has four parameters because MNDO does
not assign itporbitals.
We want the parameters that will give the best results, for a wide range of
molecules. What we mean by “results” depends on the molecular characteristics of
most interest to us. MNDO (and its siblings AM1 and PM3, below) was parameterized
[ 38 ] to reproduce heat of formation, geometry, dipole moment, and the first vertical
ionization energy (from Koopmans’ theorem; Section5.5.5). To parameterize MNDO
a training set of molecules (a “molecular basis set” is Dewar’s term – no connection
with a basis set of functions used to construct molecular orbitals) composed of small,
common molecules (e.g. methane, benzene, dinitrogen, water, methanol; 34 mole-
cules were used for the C, H, O, N set) was chosen and the six parameters above (Urr
etc.) were adjusted in an attempt to give the best values of the four molecular
characteristics (heat of formation, geometry, dipole moment, ionization energy).
Specifically, the objective was to minimizeY, the sum of the weighted squares of
the deviations from experiment of each of the four molecular characteristics:
Y¼
XN
i¼ 1Wi½YiðcalcÞ"YiðexpÞ^2 (6.20)Nis the number of molecules in the training set, andWiis a weighting factor
chosen to determine the relative importance of each characteristicYi. The actual
process of assigning values to the parameters is formally analogous to the problem of
geometry optimization (Section2.4). In geometry optimization we want the set of
atomic coordinates that correspond to a minimum (sometimes to a transition state)
on a potential energy hypersurface. In parameterizing a semiempirical method we
want the set of parameters that correspond to the minimum overall calculated
deviation of the chosen characteristics from their experimental values – the para-
meters that give the minimumY, above. Details of the parameterization process for
MNDO have been given by Dewar and coworkers [ 38 ] and by Stewart [ 41 ].
The results of MNDO calculations on 138 compounds limited to the elements C,
H, O, N were reported by Dewar and Thiel [ 38 ]. The absolute mean errors were: in
heat of formation, 26 kJ mol"^1 for all 138 compounds; in geometry, 0.014 A ̊ for
bond lengths for 228 bonds, 2for angles at C for acyclics (less for cyclic mole-
cules); in dipole moment, 0.30 D for 57 compounds; in ionization energy, 0.48 eV
for 51 compounds. To put the errors in perspective, typical values of these quan-
tities are, respectively, roughly"600 to 600 kJ mol"^1 , 1.0 to 1.5 A ̊, 0 to 3 D, and
10–15 eV. Although MNDO can reproduce these and other properties of a wide
variety of molecules [ 37 , 42 ], it is little-used nowadays, having been largely
superseded by AM1 and, perhaps to a somewhat lesser extent, PM3 (below).
406 6 Semiempirical Calculations