Page 285 - Essentials of physical chemistry

The Schrödinger Wave Equation 247

000 –645
+559 –559
–322 322
–559 +559 –322 322
000 645

000 –532
–461 461 –266 –266

–461 461
266 266
000 532
344
344 344
344 344

344
FIGURE 11.5 The 2Pz coefficients of benzene multiplied by 1000, obtained from a STO-3G SCF-MO
calculation using the program PCLOBE, see Chapter 17. Only the lowest five orbitals are shown. (From
Trindle, C. and Shillady, D., Electronic Structure Modeling: Connections between Theory and Software, CRC
Press, Boca Raton, FL, 2008. With permission.)

8
2
2
8p (9:10938215 10 31 kg)(2:22816 10 10 m) (2:99792458 10 m=s)
5(6:62606896 10 J s)
l p!p* ¼ 34
l p!p* ¼ 3:231217 10 7 m ffi 3231:217 10 10 m ¼ 3231 A ˚ :
Experimentally the leading edge of the ultraviolet spectrum of naphthalene occurs at about 3150 Å
but electronic spectra of large molecules are broadened by vibrational interactions as we will see in a
later chapter. Thus we will take the calculated p ! p* wavelength as qualitatively correct; quite
good considering the simplicity of the model. Overall as the aromatic molecules become larger, the
POR model is useful for qualitative reasoning. The POR model was first discussed by Platt [4] in
1949, expanded by Moffit [5] in 1954 and used for interpretation of magnetic circular dichroism
spectra by Michl [6] in 1978.

COMPARISON OF PIB AND POR APPLICATIONS
An instructive comparison of the PIB and POR models is given by the diagram in Figure 11.5. The main
pointforstudentsistoknowwhethertousethedegeneratelevelsinthePORmodelorthenon-degenerate
levels in the PIB model. The next important approximation is to estimate length or circumference using
an average of the –C¼C– bond length as 1.4 Å. From then on we have to realize these models mainly
apply to conjugated hydrocarbons. We should remember that the POR model provides understanding of
aromaticity. We have used these simple, but solvable, model systems to illustrate the postulates of
quantum mechanics and now we are ready to proceed to more realistic problems.

ADDITIONAL THEOREMS IN QUANTUM MECHANICS
We now have seen two problems with exact solutions in quantum mechanics. Actually there are
only a few such exact solutions remaining such as the harmonic oscillator, the rigid rotor, the
hydrogen atom, and the forced harmonic oscillator, and we need to know some general principles