Page 36 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

THEORY OF ORBITAL OPTIMIZATION IN SCF AND MCSF CALCULATIONS 21

form of a power expansion

where the are numerical coefficients depending of l.

Substituting this form of into the eq.(5) gives a recursion relation which allows to
determine all the for any arbitrary choice of one of them. Choosing , one
gets

These expressions of the will allow us now to discuss the energy dependence of
and then to derive some consequences from this dependence.
1.2. THE VALLEY THEOREM
We first note that the choice made in deriving the eq.(7) simply consists in a
particular norm of (and thus of ). In fact the standard norm cannot
be used here since for most values of e the orbital is not square summable. The
choice is a convenient alternative for

Next we consider the value of . It implies the relation :

which is the well known ’Cusp’ theorem (see e.g. the ref.3).

An other aspect of the eq.(7) concerns the energy dependence of . In fact one
deduces from this equation that :

The meaning of the eq. (9) can be stated as : the energy dependence of vanishes
like near the origin (or even faster than since there is a partial cancellation
between the and terms). Therefore the energy dependence of vanishes like
or faster.
This statement will be referred to here as the ’Valley’ theorem. It constitutes the
formal basis of our description of the optimum orbitals in molecular systems.
In fact, the Valley theorem is a simple extension of the Cusp theorem. However, the
Cusp theorem provides only a local information (for r=0), while the Valley theorem