Traditional, non-relativistic quantum chemistry has been extremely successful in providing high quality, reliable predictions of the properties of ground state molecules containing first and second row elements. The transition metals pose a variety problems for quantum chemical investigators. While relativistic effects can be chemically significant for molecular species containing even third row transition metals, property predictions for even the ground state are often complicated by the presence of many close lying states. This makes it difficult to discuss relativistic and correlation effects separately for species containing these atoms. Atoms from groups IIIA-VIII-A, on the other hand, tend to have less complicated electronic structure, and, in general, their predicted electronic structure is less dependent upon the level of electron correlation employed. For these reasons, it is instructive to investigate the periodic trends of relativistic effects for atoms in this region of the period table.
One useful gauge of the magnitude of the well-established
relativistic s and p orbital
shrinkage and d orbital expansion
for the main group elements is given by the
difference in the expectation values of
R between the HF and DHF models for these elements.
These differences are illustrated in Figure 3.1
for the noble gas elements,
while Figures 3.2, 3.3 and 3.4
display these values for the main group elements of
the third, fourth, and fifth rows of the periodic table, respectively.
The reported values, 
, may be expressed as
Figure 3.1 clearly depicts the onset of chemically significant relativistic effects in the third row of the periodic table. The valence s and p orbital contraction for Kr should raise some concern for the ability of a non-relativistic Hamiltonian to properly describe the electronic structure for elements in or beyond the third row, while the extreme difference in the case of Rn gives strong evidence that completely non-relativistic techniques are not appropriate for systems containing elements in this region of the periodic table.
Figure 4: Difference between the
obtained with the non-relativistic
atomic orbitals and 
obtained with the
relativistic valence orbitals for the noble gas elements.
The horizontal trends of the third, fourth and fifth rows, illustrated in Figures 3.2, 3.3 and 3.4 support the ``gold maximum'' which is often observed for relativistic effects[37]. Those elements which are closer to the center of the periodic table clearly exhibit stronger s and p orbital contraction and d orbital expansion than do those which are closer to the noble elements.
Figure 5: Difference between the
obtained with the non-relativistic
atomic orbitals and 
obtained with the
relativistic valence orbitals for the third row group III-VII elements.
Figure 6: Difference between the
obtained with the non-relativistic
atomic orbitals and 
obtained with the
relativistic valence orbitals for the fourth row group III-VII elements.
Figure 7: Difference between the
obtained with the non-relativistic
atomic orbitals and 
obtained with the
relativistic valence orbitals for the fifth row group III-VII elements.
In these figures, it is interesting to note that the 
orbitals
display a greater orbital contraction than does the 
for some
elements. The overall contraction of the s shell is still greater than
that for the p shell, since the 
exhibit only a minute
contraction, and, in some cases, even a slight expansion.
Another test of the necessity of employing a relativistic description of the
electronic structure is the magnitude of the contribution of the small
component of the wavefunction. Many approximate, quasi-relativistic
methods employ only the large component of the wavefunction in the definition
of some contributions to the Hamiltonian, as is done in many implementations
of relativistic effective core potentials. Other methods, such as
Douglas-Kroll based methods, perform a decoupling of the large and small
component, and rely upon the decoupled large component to describe the
electronic wavefunction. The success of these methods is dependent upon
the assumption that the contributions due to the small component are much
smaller than those of the large component. For the valence atomic orbitals,
this assumption is typically a good one. This is illustrated by the
small component contribution to the valence s s-electron total probability
integral: 
. For the
heavier noble elements 
is given by:
0.000116 (Kr), 0.000117 (Xe), 0.000153 (Rn). Core
orbitals, on the other hand, tend to have much larger contributions from the
small component, as is illustrated by their significantly larger 
values: 0.0171 (Kr), 0.0397 (Xe), 0.109 (Rn).
Therefore, if accurate descriptions of the core orbitals are desired, fully
relativistic 4-component methods should be applied to systems containing
fourth and fifth row atoms.
Finally, spin-orbit effects in atomic systems can provide an especially
important gauge of the necessity for a relativistic treatment. The radial
expectation value differences reported in Tables I-IV clearly illustrate the
magnitude of the orbital splitting between the valence 
and
orbitals as well as the 
and 
orbitals.
Clearly this splitting will have significant implications on the chemical
behavior predicted for heavy elements by the DHF model as compared to the HF
model. The magnitude of the splitting follows the same general trends as
is observed for the the orbital contraction of
valence s and p orbitals. The effect is
highly pronounced for the fourth and fifth rows with the elements closest to
the transition metals exhibiting the strongest splitting. A similar trend
is observed for the DHF orbital energies as compared to the HF orbital
energies.