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| '''Relativistic quantum chemistry''' invokes [[quantum chemistry|quantum chemical]] and [[relativistic mechanics|relativistic mechanical]] arguments to explain [[chemical element|elemental]] properties and structure, especially for the heavier elements of the [[periodic table]]. A prominent example of such an explanation would be the fact that the color of gold (in that it is not silvery like most other metals) is explained via such relativistic effects.
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| 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]].<ref name="Kleppner">{{cite journal|title=A short history of atomic physics in the twentieth century|doi=10.1103/RevModPhys.71.S78|url=http://www.cstam.org.cn/Upfiles/200732678933.pdf|year=1999|last1=Kleppner|first1=Daniel|journal=Reviews of Modern Physics|volume=71|issue=2|pages=S78|bibcode = 1999RvMPS..71...78K }}</ref> By convention, "relativistic effects" are those discrepancies between values calculated by models considering and not considering relativity.<ref>{{cite book|ref=harv|last1=Kaldor|first1=U.|last2=Wilson|first2=Stephen|title=Theoretical Chemistry and Physics of Heavy and Superheavy Elements|url=http://books.google.com/books?id=0xcAM5BzS-wC&pg=PA4|year=2003|publisher=Kluwer Academic Publishers|location=Dordrecht, Netherlands|isbn=1-4020-1371-X|page=4}}</ref> Relativistic effects are important for the heavier elements with high [[atomic number]]s. In the most common layout of the periodic table, these elements are shown in the lower area. Examples are the [[lanthanide]]s and [[actinide]]s.{{sfn|Kaldor|Wilson|2003|p=2}}
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| 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.{{Citation needed|date=July 2013}}
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| ==History==
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| Beginning in 1935 [[Bertha Swirles]] describes a relativistic treatment of a many-electron system,<ref name="Swirles">{{cite journal|doi=10.1098/rspa.1935.0211|title=The Relativistic Self-Consistent Field|year=1935|last1=Swirles|first1=B.|journal=Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences|volume=152|issue=877|pages=625 |bibcode = 1935RSPSA.152..625S }}</ref> in spite of [[Paul Dirac]]'s 1929 assertion that the only imperfections remaining in quantum mechanics
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| {{quote|"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."<ref name="Dirac">{{cite journal|doi=10.1098/rspa.1929.0094 |jstor=95222|url=http://gtwlx.jpkc.fudan.edu.cn/reference/ref4.pdf|format=free download pdf|title=Quantum Mechanics of Many-Electron Systems|year=1929|last1=Dirac|first1=P. A. M.|journal=Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences|volume=123|issue=792|pages=714|bibcode = 1929RSPSA.123..714D }}</ref>}}
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| Theoretical chemists by and large agreed with Dirac's sentiment until the 1970s, when relativistic effects began to become realized in heavy elements.<ref name="Pyykko">{{cite journal|doi=10.1021/cr00085a006|title=Relativistic effects in structural chemistry|year=1988|last1=Pyykko|first1=Pekka|journal=Chemical Reviews|volume=88|issue=3|pages=563}}</ref> The [[Schrödinger equation]] had been developed without considering relativity in Schrödinger's famous 1926 paper.<ref>[[Erwin Schrödinger]], ''Annalen der Physik, (Leipzig)'' (1926), [http://home.tiscali.nl/physis/HistoricPaper/Schroedinger/Schrodinger1926c.pdf Main paper]</ref> 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 line]]s 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.<ref>{{cite book|editor-last1=Kaldor|editor-first1=U.|editor-last2=Wilson|editor-first2=Stephen|title=Theoretical Chemistry and Physics of Heavy and Superheavy Elements|url=http://books.google.com/books?id=0xcAM5BzS-wC|year=2003|publisher=Kluwer Academic Publishers|location=Dordrecht, Netherlands|isbn=1-4020-1371-X}}</ref>{{page needed|date=July 2013}}
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| 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 orbital]]s.<ref name="Pyykko"/>
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| ==Qualitative treatment==
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| [[File:Lorentz factor.svg|thumb|right|Relativistic mass as a function of velocity. For a small velocity, the <math>m_{rel}</math> (ordinate) is equal to <math>m_0</math> but as <math>v_e\to c</math> the <math>m_{rel}</math> goes to infinity.]]
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| One of the most important and familiar results of relativity is that the [[Mass in special relativity|relativistic mass]] of the [[electron]] increases by
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| :<math>m_{rel}=\frac{m_{e}}{\sqrt{1-(v_e/c)^2}}</math>
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| where <math>\displaystyle m_e, v_e, c</math> 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.
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| This has an immediate implication on the [[Bohr radius]] (<math>\displaystyle a_0</math>) which is given by
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| :<math>a_0=\frac{\hbar}{m_e c \alpha}</math>
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| where <math>\hbar</math> is the [[reduced Planck's constant]] and α is the [[fine-structure constant]] (a relativistic correction for the [[Bohr model]]).
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| [[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#Physical_interpretations|fine-structure constant]] shows the electron traveling at nearly 1/137 the speed of light.<ref name="Norrby">{{cite journal|doi=10.1021/ed068p110|title=Why is mercury liquid? Or, why do relativistic effects not get into chemistry textbooks?|year=1991|last1=Norrby|first1=Lars J.|journal=Journal of Chemical Education|volume=68|issue=2|pages=110|bibcode = 1991JChEd..68..110N }}</ref> 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<sub>rel</sub> = 1.22m<sub>e</sub> and in turn putting this in for the Bohr radius above one finds that the radius shrinks by 22%.
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| If one substitutes in the relativistic mass into the equation for the Bohr radius it can be written
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| :<math>a_{rel}=\frac{\hbar \sqrt{1-(v_e/c)^2}}{m_e c \alpha}</math>
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| [[File:Bohrradiusfunctionofelectronvelocity.png|thumb|right| Ratio of relativistic and nonrelativistic Bohr radii, as a function of electron velocity]]
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| It follows that
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| :<math>\frac{a_{rel}}{a_0} =\sqrt{1-(v_e/c)^2} </math>
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| 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.
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| 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
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| :<math>L' = L \, \sqrt{1-v^2/c^2}</math>
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| so the radius of the 6s orbital shrinks to
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| :<math>a_{rel} = a_0 \, \sqrt{1-v^2/c^2}</math>
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| which is consistent with the result obtained by incorporating the increase of mass.
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| When the Bohr treatment is extended to [[hydrogen-like atom|hydrogenic-like atoms]] using the Quantum Rule, the Bohr radius becomes
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| :<math>r=\frac{n^2\hbar^2 4 \pi \epsilon_0}{m_eZe^2}</math>
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| where <math>n</math> is the [[principal quantum number]] and Z is an integer for the [[atomic number]]. From quantum mechanics the [[angular momentum]] is given as <math>mv_{e}r=n\hbar</math>. Substituting into the equation above and solving for <math>v</math> gives
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| :<math>r=\frac{mv_ern\hbar 4 \pi \epsilon_0}{mZe^2} </math>
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| :<math>1=\frac{v_en\hbar 4 \pi \epsilon_0}{Ze^2} </math>
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| :<math>v_e=\frac{Ze^2}{n\hbar 4 \pi \epsilon_0} </math>
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| From this point [[atomic units]] can be used to simplify the expression into
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| :<math>v_e=\frac{Z}{n}</math>
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| Substituting this into the expression for the Bohr ratio mentioned above gives
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| :<math>\frac{a_{rel}}{a_0}=\sqrt{1-\left(\frac{Z}{nc}\right)^2} </math>
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| At this point one can see that for a low value of <math>n</math> and a high value of <math>Z</math> that <math>\frac{a_{rel}}{a_0} < 1</math>. 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.<ref name="Pitzer">{{cite journal|doi=10.1021/ar50140a001|title=Relativistic effects on chemical properties|year=1979|last1=Pitzer|first1=Kenneth S.|journal=Accounts of Chemical Research|volume=12|issue=8|pages=271}}</ref>
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| ==Periodic table deviations==
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| 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 ([[Caesium|Cs]]-[[Radon|Rn]]) and the 5th period ([[Rubidium|Rb]]-[[Xenon|Xe]]) arise from the larger relativistic effects for the former. These relativistic effects are particularly large for [[gold]] and its neighbors, [[platinum]] and [[mercury (element)|mercury]].
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| ===Mercury===
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| [[Mercury (element)|Mercury]] (Hg) is a liquid down to −39 °[[Celsius|C]] (see [[melting point|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.<ref name="Norrby"/> In the gas phase mercury is alone in metals in that it is quite typically found in a monomeric form as Hg(g). Hg<sub>2</sub><sup>2+</sup>(g) also forms and it is a stable species due to the relativistic shortening of the bond.
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| Hg<sub>2</sub>(g) does not form because the 6s<sup>2</sup> 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.<ref name="Norrby"/>
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| Au<sub>2</sub>(g) and Hg(g) are analogous, at the least in having the same nature of difference, to H<sub>2</sub>(g) and He(g). It is for the relativistic contraction of the 6s<sup>2</sup> orbital that gaseous mercury can be called a pseudo noble gas.<ref name="Norrby"/>
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| ===Color of gold and caesium===
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| [[File:Image-Metal-reflectance.png|thumb|right|''Spectral reflectance curves'' for aluminum (Al), silver (Ag), and gold (Au) metal mirrors]]
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| 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 [[Absorption (electromagnetic radiation)|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.
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| 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.<ref name="PekkaPyykko">{{cite journal|doi=10.1021/ar50140a002|title=Relativity and the periodic system of elements|year=1979|last1=Pyykko|first1=Pekka|last2=Desclaux|first2=Jean Paul|journal=Accounts of Chemical Research|volume=12|issue=8|pages=276}}</ref>
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| 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 alkaline metals are silver-white, cesium metal has a distinctly golden hue.
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| ===Inert pair effect===
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| {{main|inert pair effect}}
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| In Tl(I) ([[thallium]]), Pb(II) ([[lead]]), and Bi(III) ([[bismuth]]) [[Complex (chemistry)|complexes]] there is a 6s<sup>2</sup> 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.<ref name="Pyykko"/>
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| ===Others===
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| Some of the phenomena commonly attributed to relativistic effects are:
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| *The stability of [[mercury(IV) fluoride]]
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| *[[Aurophilicity]]
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| *The stability of the gold anion, Au<sup>−</sup>, in compounds such as CsAu
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| *The crystal structure of [[lead]], which is [[face-centered cubic]] instead of diamond-like
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| *The striking similarity between [[zirconium]] and [[hafnium]]
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| *The stability of the [[uranyl cation]], as well as other high oxidation states in the early [[actinide]]s (Pa-Am)
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| *The small atomic radii of [[francium]] and [[radium]]
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| *About 10% of the [[lanthanide contraction]] is attributed to the relativistic mass of high velocity electrons and the smaller [[Bohr radius]] that results.
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| *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.
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| ==References==
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| {{reflist}}
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| ==Further reading==
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| *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}}
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| [[Category:Quantum chemistry]]
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| [[Category:Special relativity]]
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| {{Link GA|zh}}
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