Page 50 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

THEORY OF ORBITAL OPTIMIZATION IN SCF AND MCSF CALCULATIONS 35

where is a spherical harmonic centered on the nucleus A and is a gaussian
centered on either of the two nuclei. A similar relation holds for with in the
place of , where M is the midpoint of the two nuclei.

We introduce now two unknown numerical constants and and we try to check
that the two conditions

directly by diagonalising the hamiltonian matrix.

To do that, we first guess starting values of and ; secondly we determine the
coefficients by minimising the quantity Q given by

where and are two set of points in the internal and external regions respectively;
thirdly we evaluate the energy of using the coefficients just determined. The
steps two and three are repeated with different values of and until the energy is
minimised.
It is seen that this process is essentially a least square fit of and by
and , subject to a minimum energy condition which allows to determine and
. Note that are related by the norm of so that there is in fact a single
parameter in this minimisation.

This calculation has been made here using the 4s basis set (which includes no po-
larisation p gaussian orbitals). The energy obtained in this way is very good : it
reproduces the energy obtained by diagonalisation (viz. -0.59088 H ; cf the Table 1)
with an error equal to 0.02 eV.
Concerning the expansion coefficients, the most significant comparison concerns the
values of the two orbitals : the one obtained by the fitting process just described and
the one obtained by diagonalising the matrix of the hamiltonian in the gaussian basis.
In fact we have found that the difference between these two orbitals never exceeds
3% in the internal region as well as in the external region.

We conclude that the description of the orbital by the proportionality between
and on the one hand and between and on the other hand is supported
by the present calculation and that it is indeed complete.
5. Conclusion

We have demonstrated formally that the optimum orbitals of any given molecular
system (canonical SCF orbitals or strongly occupied MCSCF orbitals that are closed
to the SCF ones) can be described very simply in the regions surrounding each nucleus