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Relativistic quantum chemistry

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Relativistic quantum chemistry invokes quantum chemical and relativistic mechanical arguments to explain elemental properties and structure, especially for heavy elements of the periodic table.

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] As per convention, "relativistic effects" are those discrepancies between values calculated by models considering and not considering relativity.[2] "Heavy elements" in this context refers, typically, to elements in the lower region of the Periodic Table where relativistic effects are important such as those elements found in the lanthanoid and actinoid series.[2]

Contents

[edit] History

Beginning in 1935 Bertha Swirles describes a relativistic treatment of a many-electron system,[3] in spite of 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."[4]

Unfortunately the theoretical chemists by and large agreed with Dirac's sentiment until the 1970s when relativistic effects began to become realized in heavy elements.[5] The Schrödinger equation had been developed without considering relativity in Schrödinger's famous 1926 paper.[6] Relativistic corrections were made to the Schrödinger equation (see Klein-Gordon equation ) in order 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 chemistry, most chemists were unfamiliar with relativistic quantum mechanics, and the focus at the time was on lighter elements typical for the organic chemistry focus of the time.[2]

Dirac's opinion on the role relativistic quantum mechanics would play for chemical systems is wrong for two reasons: the first being that s and p electrons 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 orbitals.[7]

Relativistic mass as a function of electron velocity. For a small velocity, the mrel (ordinate) is equal to m0 but as v_e\to c the mrel goes to infinity.

[edit] Qualitative Treatment

One of the most important and familiar result when considering relativity is the relativistic mass of the electron, it increases by

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

where \displaystyle m_e, v_e, c being the electron rest mass, velocity of the electron, and speed of light respectively. See the Figure to your right for an illustration of 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 a_0) which is given by

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

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

Bohr calculated that for a 1s electron of a hydrogen atom with an orbiting radius of 0.0529 nm that \alpha \approx 1/137. That is to say, namely, that the fine-structure constant shows the electron traveling nearly at 1/137 the speed of light.[8] One can extend this to a larger element by using the expression \alpha \approx Z/137. For gold with Z = 79 the 1s electron will be going (α = 0.58) 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%.

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

a_{rel}=\frac{\hbar \sqrt{1-(v_e/c)^2}}{m_e c \alpha}
Relativistic and unrelativistic Bohr fraction as a function of the electron velocity

It follows that

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

To your right this fraction of the relativistic and unrelativistic Bohr radius has been plotted as a function of the electron velocity. Notice how the relativistic model shows the radius decreasing for an ever-larger velocity.

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

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

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

v_e=\frac{Z}{n}

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

\frac{a_{rel}}{a_0}=\sqrt{1-(\frac{Z}{nc})^2}

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 be 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.[9]

[edit] Periodic 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 Row (Cs-Rn) and the 5th Row (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.

[edit] Mercury A Liquid

Mercury (Hg) is a liquid down to -39 Celsius (°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.[8] 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 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.[8]

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 pseudo noble gas.[8]

[edit] Color of Gold

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

The reflectivity of Au, Ag, Al is shown on the figure to your right. The human eye sees yellow as electromagnetic radiation with a wavelength near 600 nm. As is clear from the spectral reflectance curves for Au, the reason for seeing it yellow is that it is absorbing all the radiation not necessary for yellow and reflecting those wavelengths necessary to see yellow at an observer.

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, nonrelativistic gold would be white. The relativistic effects are raising the 5d orbital and lowering the 6s orbital.[10]

[edit] The inert pair effect

In Tl(I) (thallium), Pb(II) (lead), and Bi(III) (bismuth) complexes there is a 6s2 electron pair, the inert pair effect verbalizes the tendency for this pair of electrons to resist oxidation due to a relativistic contraction of the 6s orbital. [11]

[edit] Others

Some of the phenomena commonly attributed to relativistic effects are:

[edit] References

  1. ^ Kleppner, D. "A short history of atomic physics in the twentieth century", Reviews of Modern Physics, Vol. 71, No. 2 (1999).
  2. ^ a b c Kaldor, U. "Theoretical Chemistry and Physics of Heavy and Superheavy Elements", Kluser Academic Publishers 2003.
  3. ^ Swirles, B. "The Relativistic Self-Consistent Field", Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol. 152, No. 877 (Nov. 15, 1935), pp. 625-649.
  4. ^ Dirac, P.A.M. "Quantum Mechanics of Many-Electron Systems", Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol. 123, No. 792 (Apr. 6, 1929), pp. 714-733.
  5. ^ Pyykkö, P. "Relativistic Effects in Structural Chemistry", Chemical Reviews, Vol 88, Issue 3 (May 1, 2002), pp. 563-594.
  6. ^ Erwin Schrödinger, Annalen der Physik, (Leipzig) (1926), Main paper
  7. ^ Pyykkö, P. "Relativistic Effects in Structural Chemistry", Chemical Reviews, Vol 88, Issue 3 (May 1, 2002), pp. 563-594.
  8. ^ a b c d Norrby, Lars. "Why Is Mercury Liquid?", Journal of Chemical Education, Vol. 68, No. 2 (February, 1991).
  9. ^ Pitzer, K. "Relativistic Effects on Chemical Properties", Accounts of Chemical Research, Vol. 12, No. 8 (August, 1979), pp. 271-276.
  10. ^ Pyykko, P.; Desclaux, J. "Relativity and the periodic system of elements", Accounts of Chemical Research, Vol 12, No. 8 (May 2002), pp. 276-281.
  11. ^ Pyykkö, P. "Relativistic Effects in Structural Chemistry", Chemical Reviews, Vol 88, Issue 3 (May 1, 2002), pp. 563-594.

[edit] Further Reading

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