Relativistic Quantum Chemistry - Qualitative Treatment

Qualitative Treatment

One of the most important and familiar results of relativity is that the relativistic mass of the electron increases by

where are the electron rest mass, velocity of the electron, and speed of light respectively. The figure at the right illustrates the relativistic effects on the mass of an electron as a function of its velocity.

This has an immediate implication on the Bohr radius which is given by

where is the reduced Planck's constant and α is the fine-structure constant (a relativistic correction for the Bohr model).

Arnold Sommerfeld calculated that, for a 1s electron of a hydrogen atom with an orbiting radius of 0.0529 nm, α ≈ 1/137. That is to say, the fine-structure constant shows the electron traveling at nearly 1/137 the speed of light. One can extend this to a larger element by using the expression v ≈ Zc/137 for a 1s electron where v is its radial velocity. For gold with (Z = 79) the 1s electron will be going (α = 0.58c) 58% of the speed of light. Plugging this in for v/c for the relativistic mass one finds that m_rel = 1.22m_e and in turn putting this in for the Bohr radius above one finds that the radius shrinks by 22%.

If one substitutes in the relativistic mass into the equation for the Bohr radius it can be written

It follows that

At right, the above ratio of the relativistic and nonrelativistic Bohr radii has been plotted as a function of the electron velocity. Notice how the relativistic model shows the radius decreasing with increasing velocity.

The same result is obtained when the relativistic effect of length contraction is applied to the radius of the 6s orbital. The length contraction is expressed as

so the radius of the 6s orbital shrinks to

which is consistent with the result obtained by incorporating the increase of mass.

When the Bohr treatment is extended to hydrogenic-like atoms using the Quantum Rule, the Bohr radius becomes

where is the principal quantum number and Z is an integer for the atomic number. From quantum mechanics the angular momentum is given as . Substituting into the equation above and solving for gives

From this point atomic units can be used to simplify the expression into

Substituting this into the expression for the Bohr ratio mentioned above gives

At this point one can see that for a low value of and a high value of that . This fits with intuition: electrons with lower principal quantum numbers will have a higher probability density of being nearer to the nucleus. A nucleus with a large charge will cause an electron to have a high velocity. A higher electron velocity means an increased electron relativistic mass, as a result the electrons will be near the nucleus more of the time and thereby contract the radius for small principal quantum numbers.