Relativistic Effects in Chemistry
What is a relativistic effect? When scientists originally developed models to describe the behavior of atoms, the theory of relativity was not considered. This leads to slight discrepancies between values calculated with and without relativistic quantum mechanics, and these discrepancies have been termed relativistic effects. Relativistic effects are most prevalent in heavier metal atoms, especially in the late series transition metals, the lanthanides and the actinides.
This chart, obtained from a book by Kaldor and Wilson1, demonstrates the error between experimental values and the expected value based on both relativistic and nonrelativistic calculations. As you can see, the agreement between the two sets of calculations is fairly good for earlier metals like lithium and potassium. However, once you move down the periodic table, the predicted ionization potential is drastically different between the relativistic and nonrelativistic models. It is also evident that the model that takes relativistic effects into consideration is far closer to the experimental results than the model which neglects them. This demonstrates the importance of factoring in the relativistic effect when studying high atomic number elements.
What causes these differences? Ionization potential is not the only value that experiences relativistic discrepancies. Atomic mass among metals with a high nuclear charge is often higher than would be predicted without taking into consideration relativity. When there is a large nuclear charge, the velocity of electrons near the nucleus is increased.
$m_{rel} =\frac{m_e}{\sqrt{1-(\frac{\nu}{c})^2}}$
From the equation for relative mass of an electron, we know that an increase in velocity will lead to a larger mass, creating a heavier atom than originally predicted. This in turn leads to a relativistic contraction in the size of orbitals (since the Bohr radius and mass are inversely related). As the lower energy p and s orbitals get held more tightly to the nucleus, they become better shielding agents, decreasing the charge “felt” by the exterior d and f orbitals.
Figure 1. Depiction of shielding effect on valence electrons2
This shielding changes the effective nuclear charge of the exterior orbitals, which then impacts the energy levels depicted in the molecular orbital diagram of a particular metal.
How can we observe relativistic effects in practice? The shifts in Zeff for these molecular orbitals lead to some interesting properties and characteristics in certain transition metals. Gold is a good example of a late series metal that experiences changes due to the relativistic effect. The stabilization of s and p orbitals and destabilization of d orbitals leads to a smaller energy gap in the HOMO and LUMO of gold than would be otherwise calculated.
This energy gap between the 5d and 6s orbitals decreases to around 2.4eV, which lies in the visible light spectrum. Specifically, blue light is absorbed by gold which leads to the reflection of all other colors, resulting in the characteristic yellow-golden color that we see. Most other metals do not absorb visible light, which is why they appear silver or grey. Silver does not experience relativistic effects in the way that gold does, so despite being in the same group as gold, it has a wider energy gap and does not display golden coloration. Without the relativistic effect factored in, the calculated energy gap for gold would also absorb UV light, so the presence of color is visible confirmation that the relativistic effects hypothesized are in fact taking place. Relativistic effects also explain the anomalies surrounding the element mercury. Other metallic species (such as gold) have a diatomic gas phase, but mercury exists in the gas phase as a single molecule, much in the way that noble gases do. This is explained by the contraction of the 6s2 orbital of mercury due to relativistic effects, meaning that this orbital is unable to strongly participate in bonding. The lack of contribution by this orbital leads to much weaker bonding in mercury than you find in a nearly isoelectronic metal like Au, which does not have a filled 6s orbital and bonds more readily4. The idea that the orbital contraction is responsible for weaker bonding is further confirmed by the similar properties of Au2 and Hg22+ which have identical electronic configurations. The bond length of these compounds is shortened by 16% due to relativistic effects, which is not observed in the silver and cadmium analogues5. These deviations from expected behavior can be explained by relativistic effects. As a result, the weakened Hg bonding is responsible for both the monomeric gas phase as well as the liquid phase of mercury at room temperature, while none of the similar metallic species display these characteristics.
Figure 2. Energy-level diagram showing the relativistic effects of the energy of atomic orbitals in gold.3
Is the relativistic effect unique to transition metals? The discrepancies observed in the previous examples all involved the d-block of the periodic table. However, the relativistic effect can be observed in any element with a high atomic number. One example of how the relativistic effect impacts practical applications of chemistry is demonstrated by the lead-acid battery. Solid lead is used as an anode, lead dioxide is used as a cathode, and sulfuric acid serves as the electrolyte. This battery is widely used as a car battery and has origins over 150 years in the past. For a long time, it was not known why the lead acid battery was so successful while the same battery using tin rather than lead showed very little activity, since the elements of the same group tend to display highly similar characteristics. It was only after relativistic effects were discovered that this observation was fully explained. The ability of lead as an electrode material is overwhelmingly attributed to the increased 6s binding capability, which is directly due to the relativistic effects that are not present in the lighter metal, tin.
In summary, the relativistic effects are important to consider when you start to study heavier elements and a full chemical picture cannot be looked at without the inclusion of this phenomenon.
Questions:
1. How does the relativistic effect explain the yellow color of gold?
a. The energy level of the 5d orbital increases and the energy level of the 6s orbital decreases.
b. The energy level of the 5d orbital decreases and the energy level of the 6s orbital increases.
c. The energy level of the 5d orbital and the 6s orbital both increase.
d. The energy level of the 5d orbital and the 6s orbital both decrease.
Answer: a Because the 5d orbital is the HOMO and the 6s orbital is the LUMO, electronic transitions between the two will require less energy with these shifts and therefore will absorb light in the visible region.
2. Which of the following cannot be attributed to relativistic effects?
a. The existence of Hg as a liquid at room temperature.
b. Tin’s performance as an electrode in a battery.
c. A contraction in radius among heavier elements.
d. Shorter Au2(g) bond lengths than expected.
Answer: b Tin has a relatively low atomic number, so it does not experience relativistic effects as much as lead. Lead is only a useful electrode because of relativistic effects.
References:
1. Kaldor, U.; Wilson, Stephen (2003). “Theoretical Chemistry and Physics of Heavy and Superheavy Elements”. Dordrecht, Netherlands: Kluwer Academic Publishers. p. 4.
2. Something
3. Yam, V., Cheng,E.; “Highlights on the recent advances in gold chemistry—a photophysical perspective” 2008. Chem. Soc. Rev. 9.
4. Norby, L.; “Why is Mercury Liquid? Or, Why Do Relativistic Effects Not Get into Chemistry Textbooks?” 1991. J. Chem. Ed. 68(2) p.110.
5. Ziegler, T.; Snijders, J. G.; Baerends, E. J. J. Chem. Phys. 1981, 74, 1271
6. Ahuja, R., Blomqvist, A., Larsson, P., Pyykko, P., Zaleski-Ejgierd, P.; “Relativity and the Lead-Acid Battery” 2011. Phys. Rev. Lett. 106, 18301.