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Relativistic quantum chemistry

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Title: Relativistic quantum chemistry  
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Relativistic quantum chemistry

Relativistic quantum chemistry invokes quantum chemical and relativistic mechanical arguments 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.[1] By convention, "relativistic effects" are those discrepancies between values calculated by models considering and not considering relativity.[2] 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.[3]

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 relativistic speeds.


  • History 1
  • Qualitative treatment 2
  • Periodic table deviations 3
    • Mercury 3.1
    • Color of gold and caesium 3.2
    • Inert pair effect 3.3
    • Others 3.4
  • References 4
  • Further reading 5


Beginning in 1935, Bertha Swirles described a relativistic treatment of a many-electron system,[4] 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."[5]

Theoretical chemists by and large agreed with Dirac's sentiment until the 1970s, when relativistic effects began to become realized in heavy elements.[6] The

  • P. A. Christiansen; W. C. Ermler; K. S. Pitzer. Relativistic Effects in Chemical Systems. Annual Review of Physical Chemistry 1985, 36, 407–432. doi:10.1146/annurev.pc.36.100185.002203

Further reading

  1. ^ Kleppner, Daniel (1999). "A short history of atomic physics in the twentieth century" (PDF). Reviews of Modern Physics 71 (2): S78.  
  2. ^ Kaldor, U.; Wilson, Stephen (2003). Theoretical Chemistry and Physics of Heavy and Superheavy Elements. Dordrecht, Netherlands: Kluwer Academic Publishers. p. 4.  
  3. ^ Kaldor & Wilson 2003, p. 2.
  4. ^ Swirles, B. (1935). "The Relativistic Self-Consistent Field". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 152 (877): 625.  
  5. ^ Dirac, P. A. M. (1929). "Quantum Mechanics of Many-Electron Systems" (free download pdf). Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 123 (792): 714.  
  6. ^ a b c Pyykko, Pekka (1988). "Relativistic effects in structural chemistry". Chemical Reviews 88 (3): 563.  
  7. ^ Erwin Schrödinger, Annalen der Physik, (Leipzig) (1926), Main paper
  8. ^ Kaldor, U.; Wilson, Stephen, eds. (2003). Theoretical Chemistry and Physics of Heavy and Superheavy Elements. Dordrecht, Netherlands: Kluwer Academic Publishers.  
  9. ^ a b c d Norrby, Lars J. (1991). "Why is mercury liquid? Or, why do relativistic effects not get into chemistry textbooks?". Journal of Chemical Education 68 (2): 110.  
  10. ^ Pitzer, Kenneth S. (1979). "Relativistic effects on chemical properties". Accounts of Chemical Research 12 (8): 271.  
  11. ^ Pyykko, Pekka; Desclaux, Jean Paul (1979). "Relativity and the periodic system of elements". Accounts of Chemical Research 12 (8): 276.  


  • The existence of mercury(IV) fluoride
  • Aurophilicity
  • 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, a lot more than 10% of its contraction is due to relativistically heavy electrons, and gold #79 is almost twice as dense as lead #82.

Some of the phenomena commonly attributed to relativistic effects are:


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

Inert pair effect

A similar effect occurs in caesium metal, the heaviest of the alkali metals which can be collected in quantities sufficient to allow viewing. Whereas the other alkali metals are silver-white, caesium metal has a distinctly golden hue.

The electronic transition responsible for this absorption is a transition from the 5d to the 6s level. An analogous transition occurs in Ag but the relativistic effects are lower in Ag so while the 4d experiences some expansion and the 5s some contraction, the 4d-5s distance in Ag is still much greater than the 5d-6s distance in Au because the relativistic effects in Ag are smaller than those in Au. Thus, non-relativistic gold would be white. The relativistic effects are raising the 5d orbital and lowering the 6s orbital.[11]

The reflectivity of Au, Ag, Al is shown on the figure to the right. The human eye sees electromagnetic radiation with a wavelength near 600 nm as yellow. As is clear from its reflectance spectrum, gold appears yellow because it absorbs blue light more than it absorbs other visible wavelengths of light; the reflected light (which is what we see) is therefore lacking in blue compared to the incident light. Since yellow is complementary to blue, this makes a piece of gold appear yellow (under white light) to human eyes.

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

Color of gold and caesium

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.[9]

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.[9]

Mercury (Hg) is a liquid down to −39 °C (see m.p.). 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.[9] 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.


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 (CsRn) and the 5th period (RbXe) arise from the larger relativistic effects for the former. These relativistic effects are particularly large for gold and its neighbors, platinum and mercury.

Periodic table deviations

At this point one can see that for a low value of n and a high value of Z that \frac{a_{rel}}{a_0} < 1. 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.[10]


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


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

v_e=\frac{Ze^2}{n\hbar 4 \pi \epsilon_0}
1=\frac{v_en\hbar 4 \pi \epsilon_0}{Ze^2}
r=\frac{mv_ern\hbar 4 \pi \epsilon_0}{mZe^2}

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

r=\frac{n^2\hbar^2 4 \pi \epsilon_0}{m_eZe^2}

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

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

a_{rel} = a_0 \, \sqrt{1-v^2/c^2}

so the radius of the 6s orbital shrinks to

L' = L \, \sqrt{1-v^2/c^2}

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

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.

\frac{a_{rel}}{a_0} =\sqrt{1-(v_e/c)^2}

It follows that

Ratio of relativistic and nonrelativistic Bohr radii, as a function of electron velocity
a_{rel}=\frac{\hbar \sqrt{1-(v_e/c)^2}}{m_e c \alpha}

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

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.[9] 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 mrel = 1.22me and in turn putting this in for the Bohr radius above one finds that the radius shrinks by 22%.

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

a_0=\frac{\hbar}{m_e c \alpha}

This has an immediate implication on the Bohr radius (\displaystyle a_0) which is given by

where \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.


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

Relativistic mass as a function of velocity. For a small velocity, the m_{rel} (ordinate) is equal to m_0 but as v_e\to c the m_{rel} goes to infinity.

Qualitative treatment

Dirac's opinion on the role relativistic quantum mechanics would play for chemical systems is wrong for two reasons: the first being that electrons in s and p atomic orbitals travel at a significant fraction of the speed of light and the second being that there are indirect consequences of relativistic effects which are especially evident for d and f atomic orbitals.[6]


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