Concise Physical Chemistry

c17 JWBS043-Rogers September 13, 2010 11:28 Printer Name: Yet to Come

PROBLEMS AND EXAMPLES 283

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FIGURE 17.2 Linear and angular momentum vectors.

17.11 SPIN–ORBIT COUPLING

An object moving in a straight line tends to conserve its linear momentumpby

continuing to move in a straight line. A spinning wheel tends to conserve its angular

momentumLby continuing to spin. Although electrons cannot be correctly described

by a deterministic circular path, they have anorbital angular momentumLand, as we

have seen, electrons have a property analogous to spin, so we anticipate (correctly) a

spin angular momentumS(Fig. 17.2). These momenta are vectors.

An atom has an angular momentum that is the sum of its electronic orbital and

spin angular momenta. Orbital and spin angular momentacoupleeither by vector

addition or vector subtraction according to whether they are in the same direction or

are opposed. This results in anomalous spectral splitting. The appearance of a pair

of sodium D lines where one line was expected is a result ofspin–orbit coupling.

The hydrogen spectrum also shows spectral splitting. For example, the 656.2 nm

(6562A) “line” of hydrogen is not really a line, but a ̊ doubletat 656.272 and

656.285 nm under high resolution.

In many-electron atoms, many vector combinations at different angles produce

complicated vector combinations ofLandSwhich result in complicated spectral

splittings. These patterns are not completely understood. Partial explanations valid

for lower atomic mass elements includeRussell–Saundersor LS coupling patterns

and spectral splittings.

PROBLEMS AND EXAMPLES

Example 17.1 A Mathcad© SCF Calculation

A calculation of the first three approximations to the SCF energyε(a,b)of the helium

atom is shown in File 17.3.

The first iteration of the SCF procedure in File 17.3 produces an approximation to

the first ionization potential of He−ε(a,b)=IP 1 (calc)=−(− 0 .812) hartrees—that

is, 10.2% too small. This is not very good, but it is a great improvement over the

>100% error we found when ther 12 term was ignored. Continuing the calculation

and substituting for the initial value ofb, we minimize to find a new value of

IP 1 (calc)= 0 .925, 2.4% in error, followed by IP 1 (calc)= 0 .889, 1.5% in error on