Relativistic quantum chemistry combinesrelativistic mechanics withquantum chemistry to calculateelemental properties and structure, especially for the heavier elements of theperiodic table. A prominent example is an explanation for the color ofgold: due to relativistic effects, it is not silvery like most other metals.[1]
The termrelativistic effects was developed in light of the history of quantum mechanics. Initially, quantum mechanics was developed without considering thetheory of relativity.[2] Relativistic effects are those discrepancies between values calculated by models that consider relativity and those that do not.[3] Relativistic effects are important for heavier elements with highatomic numbers, such aslanthanides andactinides.[4]
Relativistic effects in chemistry can be considered to beperturbations, or small corrections, to the non-relativistic theory of chemistry, which is developed from the solutions of theSchrödinger equation. These corrections affect the electrons differently depending on the electron speed compared with thespeed 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.[5]
Beginning in 1935,Bertha Swirles described a relativistic treatment of a many-electron system,[6] despitePaul 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 onlyCoulomb forces between the various electrons and atomic nuclei".[7]
Theoretical chemists by and large agreed with Dirac's sentiment until the 1970s, when relativistic effects were observed in heavy elements.[8] TheSchrödinger equation had been developed without considering relativity in Schrödinger's 1926 article.[9] Relativistic corrections were made to the Schrödinger equation (seeKlein–Gordon equation) to describe thefine structure of atomic spectra, but this development and others did not immediately trickle into the chemical community. Sinceatomic 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 theorganic chemistry focus of the time.[10]
Dirac's opinion on the role relativistic quantum mechanics would play for chemical systems has been largely dismissed for two main reasons. First, electrons ins andpatomic orbitals travel at a significant fraction of the speed of light. Second, relativistic effects give rise to indirect consequences that are especially evident ford andf atomic orbitals.[8]
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One of the most important and familiar results of relativity is that therelativistic mass of theelectron increases as
where are theelectron rest mass,velocity of the electron, andspeed of light respectively. The figure at the right illustrates this relativistic effect as a function of velocity.
This has an immediate implication on theBohr radius (), which is given by
where is thereduced Planck constant, and α is thefine-structure constant (a relativistic correction for theBohr model).
Bohr calculated that a1s orbital electron of a hydrogen atom orbiting at the Bohr radius of 0.0529 nm travels at nearly 1/137 the speed of light.[11] One can extend this to a larger element with anatomic numberZ by using the expression for a 1s electron, wherev is itsradial velocity, i.e., its instantaneous speed tangent to the radius of the atom. Forgold withZ = 79,v ≈ 0.58c, so the 1s electron will be moving at 58% of the speed of light. Substituting this in forv/c in the equation for the relativistic mass, one finds thatmrel = 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
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 decreases with increasing velocity.
When the Bohr treatment is extended tohydrogenic atoms, the Bohr radius becomeswhere is theprincipal quantum number, andZ is an integer for theatomic number. In theBohr model, theangular 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 a low value of and a high value of results in. 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, and as a result the electrons will be near the nucleus more of the time and thereby contract the radius for small principal quantum numbers.[12]
Mercury (Hg) is a liquid down to approximately −39 °C, itsmelting point. Bonding forces are weaker for Hg–Hg bonds than for their immediate neighbors such ascadmium (m.p. 321 °C) and gold (m.p. 1064 °C). Thelanthanide contraction only partially accounts for this anomaly.[11] Because the 6s2orbital is contracted by relativistic effects and may therefore only weakly contribute to any chemical bonding, Hg–Hg bonding must be mostly the result ofvan der Waals forces.[11][13][14]
Mercury gas is mostly monatomic, Hg(g). Hg2(g) rarely forms and has a low dissociation energy, as expected due to the lack of strong bonds.[15]
Au2(g) and Hg(g) are analogous with H2(g) and He(g) with regard to having the same nature of difference. The relativistic contraction of the 6s2 orbital leads to gaseous mercury sometimes being referred to as a pseudonoble gas.[11]
Thereflectivity ofaluminium (Al), silver (Ag), and gold (Au) is shown in the graph to the right. The human eye sees electromagnetic radiation with a wavelength near 600 nm as yellow. Goldabsorbs blue light more than it absorbs other visible wavelengths of light; the reflected light reaching the eye is therefore lacking in blue compared with the incident light. Since yellow iscomplementary 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 contraction, the 4d–5s distance in silver is 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. Due to the decreased 6s orbital distance, the electronic transition primarily absorbs in the violet/blue region of the visible spectrum, as opposed to the UV region.[16]
Caesium, the heaviest of thealkali metals that 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 atZ = 55 for caesium (not far fromZ = 47 for silver). The golden color 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 reaches the blue-violet end of the visible spectrum; in other words, theplasmonic frequency of the alkali metals becomes lower from lithium to caesium. Thus caesium transmits and partially absorbs violet light preferentially, while other colors (having lower frequency) are reflected; hence it appears yellowish.[17]
Without relativity,lead (Z = 82) would be expected to behave much liketin (Z = 50), so tin–acid batteries should work just as well as thelead–acid batteries commonly used in cars. However, calculations show that about 10 V of the 12 V produced by a 6-cell lead–acid battery arises purely from relativistic effects, explaining why tin–acid batteries do not work.[18]
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 resistoxidation due to a relativistic contraction of the 6s orbital.[8]
Additional phenomena commonly caused by relativistic effects are the following: