A crude estimate of the importance of relativity in characterizing the
atomic orbital energies and spatial extent may be achieved by invoking the
Bohr model of the hydrogenic atom of charge Z.
In this classical picture, an electron in
an orbit designated by principal quantum number n would travel at a
radius 
and a velocity 
with a total energy 
given
(in a.u.) by
where m has been retained intentionally. Since the electron is participating in accelerated motion, the equations of special relativity cannot apply rigorously. However, it is possible to achieve an estimate of the changes which a relativistic Hamiltonian would introduce for this system by replacing m with the velocity-dependent mass
This relationship is illustrated in Figure 1.
Figure 1: Ratio of Relativistic Bohr Orbit to Non-Relativistic Bohr Orbit
The manifestation of this behavior is typically classified as a direct relativistic effect. Direct effects are most obvious for the s and p type orbitals since these orbitals tend to have a greater density near the nucleus, and hence experience a greater fraction of the full nuclear charge.
The orbital shrinking experienced by s and p orbitals will, in turn, have an affect on the higher angular momentum d and f orbitals. The greater density of the s and p orbitals in the nuclear vicinity effectively crowds out the d and f orbitals through orbital orthogonality and shielding effects. The resultant expansion of the d and f orbitals and the associated shift of the orbital energies is generally classified as an indirect relativistic effect. In atomic systems where this orbital shrinking and expansion is highly pronounced, non-relativistic methods will obviously be inadequate.
Another litmus test of the necessity of a relativistic treatment is the contribution of the small component to the total electron density predicted by the DHF method for a particular atomic orbital. The pattern of electron density and orthogonality relationships associated with a four-component wavefunction with appreciable large and small components are difficult to reproduce with a two-component wavefunction. The small component of the wavefunction tends to be largest very close to the nucleus and therefore plays a substantial role in the screening of nuclear charge for the higher energy orbitals. The contribution of the small component is usually greatest for orbitals with low principal quantum numbers. This is illustrated by Figure 2 which depicts the large and small components of the radial wavefunction for the s-like orbitals (j = 1/2, k = -1, l=0) of radon with principal quantum numbers n = 1,2,3.
Figure 2: Large and small component radial wavefunctions for
the n
orbitals of Rn, Xe and Kr.
Similarly, the small component to the 1s orbital of increasingly lighter elements tends to be smaller, and the profile of the small components tends to more closely match that of the large component. Therefore, the renormalized large component should give a closer estimate of the radial distribution of the electron density for the lighter elements. This may be observed the plots of the large and small components of the 1s-like DHF orbitals of Rn, Xe, and Kr depicted in Figure 3.
Figure 3: Large and small component radial wavefunctions for
the n
orbitals of Rn, Xe and Kr.
Spin-orbit (S-O) coupling is one of the most manifest of relativistic effects in chemistry. A simple picture of S-O coupling may be obtained by returning to the Bohr atom and employing the classical expression for the interaction of the electron's spin and the magnetic field which results from its orbital motion. An electron orbiting a nucleus may be said to experience the classical magnetic field
where E for a central potential 
is given by
The interaction energy, 
, of the intrinsic magnetic moment of the
electron with this magnetic field is given by
where the definition 
has been employed.
This classical interaction term turns out to be too large by a factor of 2.
The reason for this is directly related to the fact that our picture has
assumed that the central nucleus is stationary. This classical picture
provides a good model for the sorts of interactions which will give rise to
the atomic fine structure. The Dirac-Coulomb Hamiltonian accounts for this
sort of spin-orbit coupling in multi-electron atoms, but the interaction of
the spin of an electron with the orbital motion of another electron and the
interaction of one electron spin with that of another are not treated by the
DC Hamiltonian. In order to take these effects into account, the Breit
interaction term must be added. Fortunately, such interaction terms are
typically very small, and do not have a large effect on atomic electronic
structure, but they must be included for even qualitative predictions of the
atomic spectra of the heaviest atoms.