Relativistic quantum chemistry combines relativistic mechanics with quantum chemistry to calculate elemental properties and structure, especially for the heavier elements of the periodic table. A prominent example is an explanation for the color of gold: due to relativistic effects, it is not silvery like most other metals. The term relativistic effects was developed in light of the history of quantum mechanics. Initially, quantum mechanics was developed without considering the theory of relativity. Relativistic effects are those discrepancies between values calculated by models that consider relativity and those that do not. Relativistic effects are important for heavier elements with high atomic numbers, such as lanthanides and actinides. Relativistic effects in chemistry can be considered to be perturbations, or small corrections, to the non-relativistic theory of chemistry, which is developed from the solutions of the Schrödinger equation. These corrections affect the electrons differently depending on the electron speed compared with the speed of light. Relativistic effects are more prominent in heavy elements because only in these elements do electrons attain sufficient speeds for the elements to have properties that differ from what non-relativistic chemistry predicts.
History Beginning in 1935, Bertha Swirles described a relativistic treatment of a many-electron system, despite Paul Dirac's 1929 assertion that the only imperfections remaining in quantum mechanics "give rise to difficulties only when high-speed particles are involved and are therefore of no importance in the consideration of the atomic and molecular structure and ordinary chemical reactions in which it is, indeed, usually sufficiently accurate if one neglects relativity variation of mass and velocity and assumes only Coulomb forces between the various electrons and atomic nuclei". Theoretical chemists by and large agreed with Dirac's sentiment until the 1970s, when relativistic effects were observed in heavy elements. The Schrödinger equation had been developed without considering relativity in Schrödinger's 1926 article. Relativistic corrections were made to the Schrödinger equation (see Klein–Gordon equation) to describe the fine structure of atomic spectra, but this development and others did not immediately trickle into the chemical community. Since atomic spectral lines were largely in the realm of physics and not in that of chemistry, most chemists were unfamiliar with relativistic quantum mechanics, and their attention was on lighter elements typical for the organic chemistry focus of the time. Dirac's opinion on the role relativistic quantum mechanics would play for chemical systems has been largely dismissed for two main reasons. First, electrons in s and p atomic orbitals travel at a significant fraction of the speed of light. Second, relativistic effects give rise to indirect consequences that are especially evident for d and f atomic orbitals.
Qualitative treatment
One of the most important and familiar results of relativity is that the relativistic mass of the electron increases as
m rel = γ m e = m e 1 − ( v e / c ) 2 {\displaystyle m_{\text{rel}}=\gamma m_{e}={\frac {m_{e}}{\sqrt {1-(v_{e}/c)^{2}}}}}
where γ , m e , v e , c {\displaystyle \gamma ,m_{e},v_{e},c} are the Lorentz factor, electron rest mass, velocity of the electron, and speed of light respectively. The figure at the right illustrates this relativistic effect as a function of velocity. This has an immediate implication on the Bohr radius ( a 0 {\displaystyle a_{0}} ), which is given by
a 0 = ℏ m e c α , {\displaystyle a_{0}={\frac {\hbar }{m_{\text{e}}c\alpha }},}
where ℏ {\displaystyle \hbar } is the reduced Planck constant, and α is the fine-structure constant (a relativistic correction for the Bohr model). Bohr calculated that a 1s orbital electron of a hydrogen atom orbiting at the Bohr radius of 0.0529 nm travels at nearly 1/137 the speed of light. One can extend this to a larger element with an atomic number Z by using the expression v ≈ Z c 137 {\displaystyle v\approx {\frac {Zc}{137}}} for a 1s electron, where v is its radial velocity, i.e., its instantaneous speed tangent to the radius of the atom. For gold with Z = 79, v ≈ 0.58c, so the 1s electron will be moving at 58% of the speed of light. Substituting this in for v/c in the equation for the relativistic mass, one finds that mrel = 1.22me, and in turn putting this in for the Bohr radius above one finds that the radius shrinks by 22%. If one substitutes the "relativistic mass" into the equation for the Bohr radius it can be written
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