Relativistic Effects on Chemical Properties Donald R. McKelvey Indiana University of Pennsylvania, Indiana, PA 15705 Certain anomalous chemical properties have been noted for the heavy elements. Two classical examples are the unique properties of gold and mercury as compared to the other members of their families. A third example is the reluctance u i t h r 1)"-rtranitiw~~nct:~l, ~ i l ~ ~ lt hj ei t hi:l~c-~ jptli.ilh uuidntim > I J I ~'I'lie , . ~ r ~ l c r~r, ~x i~dla ~.t.utt i ~ ~ lor i the-t clt ments is two less thanthe group number. Thus, in Group IVA, the most stable oxidation state of lead is f 2 , while tin exhibits stable oxidation states of both +2 and +4. This phenomenon has been attributed to the so called "inert pair" effect. I shall discuss how these and other anomalous chemical properties may he explained on the basis of relativistic effects. In order to do this properly, I would like to trace the development of the relativistic wave equation, the Dirac equation. starting with the Bohr treatment of the hydrogen atom. For those readers who are unfamiliar with the Dirac equation, I would also like to summarize some of its major consequences. The Bohr Atom During the latter half of the 19th century it was difficult for scientists to understand why a heated gas of hydrogen atoms should emit electromagnetic radiation of only certain energies rather than a continuous spectrum of energies. There was no theory which was adequate in explaining the hydrogen spectrum until 1913 when Bohr formulated his theory. Bohr was able to derive an equation which correctly reproduced the spectral frequencies observed in the hydrogen spectrum. Bohr's treatment represented quite a triumph at the time even though it was unable to explain the spectra for elements with Z > 1and did not account for the so-called "fine structure" of the hydrogen spectrum. The Bohr-Sommerfeld Model In the Bohr-Sommerfeld model (1916) it was assumed that the electron traveled in an elliptical path with the nucleus located at one focus. This resulted in the introduction of a second quantum number since two polar coordinates, r a n d 0, were now needed to locate the position of the electron. Sommerfeld also took relativistic effects into consideration. An electron in a circular orbit has constant velocity, but an electron in an elliptical orbit has velocity that is different a t different positions, speeding up when it is near the nucleus and slowing down when it is far away. The relativistic mass of the electron changes with the changing velocity of the electron. By taking into account these relativistic effects, Sommerfeld was able to account for the fine structure in the hydrogen spectrum. However, there were still many aspects of atomic spectra which remained unexplained, most notably:
Schrodinger equation correctly predicted the hydrogen spectrum and the predictions could be extended to more complicated atomic systems. But there were still problems. The Schrodinger treatment by itself could not account for the fine structure in the hvdroeen atom. In fact. the Bohr-Som-
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abo& an axis through its center whiie at the same timecirculating about the nucleus. Thus, the electron would have an angular momentum due to its spinning in addition to the angular momentum which it possesses due to its circular motion ahbut the nucleus. Uhlenbeck and Goudsmit assigned a spin angular momentum of 'I& to the electron in order to explain the doublets observed in the suectra of the alkali metals. It was l'urll~cr:twllllit d l l l n t lht -pill :111:1lli,r ini.,tIIelitunl \c.rIc.r a.1cl)n,e t ~ u , ~ ~ i ~and i / chad d 1 % hc 3 l i-~ n %I t I ithcr \ r ~ t h t l r .xr~i~i,t an applied magnetic field. The value of the spin angular momentum was later revised to (&/2)h and it could have coma fourth quantum number, the spin quantum number (m, = was added to the three already ohtained from the Schriidinger equation. The Dirac Equation According to the postulates of quantum mechanics, the wave equation for ij/ is obtained from the classical expression for the total energy of the system by replacing the dynamical variables with operators and by the insertion of the wave function as the operand.
+
Dynarnicnl variable
Position:
112
Journal of Chemical Education
X,
y,z
Operator r,y,z
a
Total energy: E
ih-
at
According to Einstein's special theory of relativity, the energy of a part,iclr is given by
1) The relative intensities of the transitions: and 2) The spectra of elements with Z > 1. The Schrodinger Equation In 1926 the laws or postulates of quantum mechanics were developed by Erwin Schriidinger resulting in his famous equation. At about the same time, Werner Heisenherg, Max Born, and Pascual Jordan developed a treatment called matrix mechanics, which gave results which were later shown to be equivalent to those ohtained by Schriidinger. Solutions to the
.
Also, when the ~chrodingerequation was applied to the spectra of the alkali metals, it predicted singlets where doublets are actually observed. In 1925 George Uhlenbeck and Samuel A. Goudsmit showed that the Schrodinger treatment could be made to account correctlv for the fine structure of the hydrogen spectrum and
E"
cC2( p r Z + p Y+Pp z 2
+ mO2c2).
(1)
Taking E = i c (p,2 + p Y 2+ p Z 2t mo2c2)112
and applying the above postulate of quantum mechanics, one obtains:
Unfortunately, the group of mathematical symbols inside the radical has no meanine. Dirac solvedthe ~roblemhv assuminn the relationship
The probability density is given by
r
$11
+ moacZ=(a,p, + asp, + aipi + i3mo~)2.(3)
p , 2 + ~ y+ap i 2
The a's and P a r e terms which will be evaluated later. This then gives as the total energy,
Multiplying out the matrices gives
Dirac chose to use the negative root. He then applied the ahovementioned postulate of quantum mechanics to obtain
The Dirac velocity operators and aspect operator are usually given as follows:
This is the time dependent form of the Dirac equation. The a's are cclled the Dirac velocity operators and /3 is the Dirac aspect operator. The problem of evaluating these terms remains. In order to do this we return to eqn. (3) and multiply out the expression on the right side of the equation, obtaining
+ ~ Z C ( ~ P Z P ,+ U.Z~ZPZPI + arPplmo~ + a y Z ~ , 2+ %CZIP>P~+ ~ ~ ~ , 3 p , m ~ c + ~ , ~ Z P , P+Za,n,p,p, + a z Z ~ , Z+ a,O~,m,,c + Bazm,cpz + Oaymoe~,+ P a , r n , , ~+~p2mOZc2 ~ (6) + Uyazp,pz
Inspection of the right side of eqn. (6) reveals that if the eaualitv is to hold. then all of the cross uroduct terms. that is all the terms located off the diagonal, must add out. This may be accomnlished if each uair of terms svmmetricallv located about the diagonal adds u p to zero. For example, a,a,p,p, + a,a,p,p, = 0. Now p,p, = p,p, since the momentum operators are commuting operators. That is to say, it makes no difference whether $ is differentiated first with respect to x and then 2 or with respect to z first and then x . This requires that the a's and fl anticommute, that is s
Introducing the 4 X 4 matrices into the Dirac equation leads to four differential equations. Solutions to the equations give $1, $2, $3 and $a as functions of the polar coordinates, r, 4, and 0 and, unlike the Schrodinger equation, introduce all four quantum numbers. There is not, however, a one to one cor-
,
.
.
a,a, = -a,%, L # 1
uip = -0%
Furthermore, inspection of the diagonal reveals that in order for the equality to hold, the following must he true: cua2 = P2 = 1.
Dirac was ahle to show that these restrictions could not be satisfied if the a's and @ were scalar functions. He went on to show that they had to he at least 4 X 4 matrices and $ a 4component vector. It is usually written
later. The Dirac equation was solved for a hydrogen-like atom by C. G. Darwin in 1928, the same year that Dirac published his work. The equation is not solved here, and the reader is asked to refer to Bethe and Salpeter (1j for the solution to aparticle in a one-dimensional box. Consequences of the Dirac Treatment
The hydrogen spectrum When the Dirac theory is applied to the hydrogen atom, it correctly predicts the entire spectrum, including all fine structure. Well, almost. In 1947 Willis E. Lamb, Jr. and Robert C. Retherford were ahle to show that the 2s1/2 level was higher in energy than the 2p1/2 level by 1,060MHz. This amounts to about two parts per million when compared with the frequency of the Balmer-alpha line, which represents the energy difference hetween t h e n = 2 and n = 3 levels. The Dirar theory predicts that these states should have the same energy. This very small shift, called the Lamb shift, is explained in a theory called quantum electrodynamics. According to this theory there exist electromagnetic field fluctuations which are auantum mechanical in nature and the nucleus when they are close together. The shapes of the 2s1/2 and 2~112orbitals are such that the electron spends more
Table 1. Symbol
Name
The Dirac Quantum Numbers
Allowed Values
n
Principal
I , 2 . 3 . 4 . ..
!
Azimuthal
0.1.2 . . . ("-1)
i
Angular momentum
+I!*
rn
Magnetic
-j.-it1.j-1,)
%I
Property Principally Determined Size and energy of orbital. analogous to Schrodinger atom. Designated by the letters s. p. d. f etc. It no longer represents the orbital anguiar momentum and no longer determines the shape of the orbitals. Contributesto the orbital energy and shape. Uiually written as a subscript following the designation for .! Sometimes called the inner quantum number. The magnetic quantum number together with the angular momentum quantum number determines the orbital shapes.
Volume 60
Number 2
February 1983
113
time close to the nucleus when in the 2 s l orbital ~ thus resulting in its having a slightly higher energy. The Dirac treatment nealects these field fluctuations and assumes the electromagne&c fields to behave classically. Dirac quantum numbers: The Dirac quantum numbers are given in Table 1. The electron spin. An inspection of Table 1reveals that although " there are four Dirac auantum numbers. there is no spin quantum number. The concept of the electron spinning about an axis through its center is foreign to the Dirac treatment. All of the magnetic properties of the electron may he accounted for hv its circulational motion. Powell (3) explains "If a physical visualization of 'spin' be needed, it is perhaps better to think of it as the lower limit of circulational motion below which the electron will not drop. The electron appears to be a particle which doesn't like to move in straight lines-it insists on moving in helical spirals wherever it goes." The Pauli ~rincivle.The Pauli ~rincivleand the ensuingexclusion principle must be tacked onto the nonrelativistic version of auantum mechanics as additional postulates but are naturai consequences of a more gene& relativistic quantum mechanics (reference (21, page 308). Orbital shapes. The shapes of the orhitals are determined by j and m. Orbitals with the same values for j and m have the same shape. Thus, the sl/z andpyz orhitals hy necessity have m = fl/z and all have the same shape, namely they are all spherically symmetrical. The term "shape" is being used somewhat incorrectly here; "angular distribution" would be more accurate. Some organic chemists may he upset to learn that the three 2p orbitals are shaped like a sphere (2 p1/2). a douehnut 12 . '02i?. ", -.m = 31") and a ball that has been saueezed However, a set of muin around its equator (2 ~312,m = D orhitals havine van. tuallv-~ernendicular -, -. . " the shaoe of the 2 . m = 112 orbital can be constructed from a linear combination of the Diracp orbitals. All of the Dirac orhitals have electron densities which are symmetrical about the Z axis and all have a plane of symmetry perpendicular to the Z axis. None of the Dirac orbitals have nodal planes ( 3 , 4) and thus the question, "How does a p electron get from one lobe t,o the other if it cannot pass through the nodal plane?" can be answered by noting that according to the Dirac treatment there are no nodes. The reas
