Page 35 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

20 C. CHAVY ET AL.

1.1. HYDROGEN-LIKE ATOMS
In the case of hydrogen-like atoms the Schrödinger equation can be written as (in
atomic units) :

where T represents the kinetic energy operator, Z the nuclear charge, -Z/r the
Coulomb electron-nuclear attraction, e the energy and the orbital.

The solution of this equation can be factorised into the product of a radial part
and an angular part (spherical harmonic where the radial part depends of
the quantum number l but not of m (2).

Inserting this form of into the eq.(l) gives the equation to be satisfied by the
so called radial equation :

It can be demonstrated (2) that two linearly independent solutions of this equation
can be chosen in general (i.e. except for some values of e) in such a way that one
of them (the so called ’regular’ solution) is continuous at the origin and diverges at
infinity, and the other one (the so called ’irregular’ solution) diverges at the origin
and tends to zero at infinity.

Neither of these two solutions is square summable in general. However for some values
of e (the ’eigen values’) these two solutions coincide and can be accepted physically
for atoms since they both are continuous at the origin and they both tend to zero at
infinity.
It should be emphasized that we are not interested here specifically by these particular
values of e. On the contrary , what is useful here i.e. for the description of optimum
orbitals in molecules is to study the variation of the regular solution when e varies
continuously.

To solve that problem, we depart here from the development used for instance in (2)
and we write in the form :

(4)
Substituting this form of into the eq.(3) leads to :

(5)

But we are interested here only by the ’regular’ solution, and we can write in the