Relativistic quantum chemistry

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Relativistic quantum chemistry combines relativistic mechanics with quantum chemistry to explain elemental properties and structure, especially for the heavier elements of the periodic table. A prominent example of such an explanation is 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 and that do not consider relativity. Relativistic effects are important for the heavier elements with high atomic numbers. In the most common layout of the periodic table, these elements are shown in the lower area. Examples are the 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 relative to 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, in spite of 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 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 paper. Relativistic corrections were made to the Schrödinger equation to explain 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 is wrong for two reasons. First, electrons in s and p atomic orbitals travel at a significant fraction of the speed of light. Second, relativistic effects cause indirect consequences that are especially evident for d and f atomic orbitals.

Qualitative treatment

Relativistic γ as a function of velocity. For a small velocity, the ${\displaystyle E_{\rm {rel}}}$ (ordinate) is equal to ${\displaystyle E_{0}=mc^{2}}$ but as ${\displaystyle v_{\rm {e}}\to c}$ the ${\displaystyle E_{\rm {rel}}}$ goes to infinity.

 

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

${\displaystyle m_{\rm {rel}}={\frac {m_{\rm {e}}}{\sqrt {1-(v_{\rm {e}}/c)^{2}}}}}$

where ${\displaystyle \displaystyle m_{e},v_{e},c}$ 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 (${\displaystyle \displaystyle a_{0}}$) which is given by

${\displaystyle a_{0}={\frac {\hbar }{m_{\rm {e}}c\alpha }}}$

where ${\displaystyle \hbar }$ 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

${\displaystyle a_{\rm {rel}}={\frac {\hbar {\sqrt {1-(v_{\rm {e}}/c)^{2}}}}{m_{\rm {e}}c\alpha }}}$

Ratio of relativistic and nonrelativistic Bohr radii, as a function of electron velocity

It follows that

${\displaystyle {\frac {a_{\rm {rel}}}{a_{0}}}={\sqrt {1-(v_{\rm {e}}/c)^{2}}}}$

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. 

When the Bohr treatment is extended to hydrogenic atoms, the Bohr radius becomes

${\displaystyle r={\frac {n^{2}\hbar ^{2}4\pi \varepsilon _{0}}{m_{\rm {e}}Ze^{2}}}}$

where ${\displaystyle n}$ is the principal quantum number and Z is an integer for the atomic number. From quantum mechanics the angular momentum is given as ${\displaystyle mv_{\rm {e}}r=n\hbar }$. Substituting into the equation above and solving for ${\displaystyle v}$ gives

${\displaystyle r={\frac {mv_{e}rn\hbar 4\pi \varepsilon _{0}}{mZe^{2}}}}$

${\displaystyle 1={\frac {v_{\rm {e}}n\hbar 4\pi \varepsilon _{0}}{Ze^{2}}}}$

${\displaystyle v_{\rm {e}}={\frac {Ze^{2}}{n\hbar 4\pi \varepsilon _{0}}}}$

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

${\displaystyle v_{\rm {e}}={\frac {Z}{n}}}$

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

${\displaystyle {\frac {a_{\rm {rel}}}{a_{0}}}={\sqrt {1-\left({\frac {Z}{nc}}\right)^{2}}}}$

At this point one can see that for a low value of ${\displaystyle n}$ and a high value of ${\displaystyle Z}$ that ${\displaystyle {\frac {a_{\rm {rel}}}{a_{0}}}<1 annotation="">$. 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.

Periodic table deviations

The periodic table was constructed by scientists who noticed periodic trends in known elements of the time. Indeed, the patterns found in it is what gives the periodic table its power. Many of the chemical and physical differences between the 6th period (Cs–Rn) and the 5th period (Rb–Xe) arise from the larger relativistic effects for the former. These relativistic effects are particularly large for gold and its neighbors, platinum and mercury.

Mercury

Mercury (Hg) is a liquid down to −39°C. Bonding forces are weaker for Hg–Hg bonds than for its immediate neighbors such as cadmium (m.p. 321°C) and gold (m.p. 1064°C). The lanthanide contraction is a partial explanation; however, it does not entirely account for this anomaly. In the gas phase mercury is alone in metals in that it is quite typically found in a monomeric form as Hg(g). Hg₂²⁺(g) also forms and it is a stable species due to the relativistic shortening of the bond. 

Hg₂(g) does not form because the 6s² orbital is contracted by relativistic effects and may therefore only weakly contribute to any bonding; in fact Hg–Hg bonding must be mostly the result of van der Waals forces, which explains why the bonding for Hg–Hg is weak enough to allow for Hg to be a liquid at room temperature.

Au₂(g) and Hg(g) are analogous, at the least in having the same nature of difference, to H₂(g) and He(g). It is for the relativistic contraction of the 6s² orbital that gaseous mercury can be called a pseudo noble gas.

Color of gold and caesium

Spectral reflectance curves for aluminum (Al), silver (Ag), and gold (Au) metal mirrors

 

Alkali metal coloration: rubidium (silvery) versus caesium (golden)

The reflectivity of aluminum (Al), silver (Ag), and gold (Au) is shown in the graph above. The human eye sees electromagnetic radiation with a wavelength near 600 nm as yellow. Gold appears yellow because it absorbs blue light more than it absorbs other visible wavelengths of light; the reflected light reaching the eye is therefore lacking in blue compared to the incident light. Since yellow is complementary to blue, this makes a piece of gold under white light appear yellow to human eyes. 

The electronic transition from the 5d orbital to the 6s orbital is responsible for this absorption. An analogous transition occurs in silver, but the relativistic effects are smaller than in gold. While silver's 4d orbital experiences some relativistic expansion and the 5s orbital some contraction, the 4d-5s distance in silver is still much greater than the 5d-6s distance in gold. The relativistic effects increase the 5d orbital's distance from the atom's nucleus and decrease the 6s orbital's distance.

Caesium, the heaviest of the alkali metals which can be collected in quantities sufficient for viewing, has a golden hue, whereas the other alkali metals are silver-white. However, relativistic effects are not very significant at Z = 55 for caesium (not far from Z = 47 for silver). The golden colour of caesium comes from the decreasing frequency of light required to excite electrons of the alkali metals as the group is descended. For lithium through rubidium this frequency is in the ultraviolet, but for caesium it enters the blue–violet end of the spectrum; in other words, the plasmonic frequency of the alkali metals becomes lower from lithium to caesium. Thus caesium transmits and partially absorbs violet light preferentially while other colours (having lower frequency) are reflected; hence it appears yellowish.

Lead-acid battery

Without relativity, lead would be expected to behave much like tin, so tin-acid batteries should work just as well as the lead-acid batteries commonly used in cars. However, calculations show that about 10 V of the 12 V produced by a six-cell lead-acid battery arises purely from relativistic effects, explaining why tin-acid batteries do not work.

Inert pair effect

In Tl(I) (thallium), Pb(II) (lead), and Bi(III) (bismuth) complexes a 6s² electron pair exists. The inert pair effect is the tendency of this pair of electrons to resist oxidation due to a relativistic contraction of the 6s orbital.

Other effects

Additional phenomena commonly caused by relativistic effects are the following:

• Metallophilic interactions
• The stability of the gold anion, Au⁻, in compounds such as CsAu
• The crystal structure of lead, which is face-centered cubic instead of diamond-like
• The striking similarity between zirconium and hafnium
• The stability of the uranyl cation, as well as other high oxidation states in the early actinides (Pa-Am)
• The small atomic radii of francium and radium
• About 10% of the lanthanide contraction is attributed to the relativistic mass of high velocity electrons and the smaller Bohr radius that results
• In the case of gold, significantly more than 10% of its contraction is due to relativistically heavy electrons, and gold (element 79) is almost twice as dense as lead (element 82)