Page 243 - Essentials of physical chemistry

Basic Spectroscopy 205

the interactions between the electrons change the weighting of the orbital functions to lower the
energy, then recalculate the electron interactions. This was done iteratively until convergence was
achieved. From thermodynamics, we know ‘‘energy goes down-hill’’ while ‘‘entropy tends to
increase.’’ For the SCF process, there is a powerful theorem shown in Chapter 16 called the
Variation Theorem that proves that lower energy approaches the true energy from above as long
as the numerical calculations are correct; even for a function which is a guess. All of this activity in
the 1930s was applied to one-center cases of atoms. Molecules larger than H 2 were beyond
treatment. In 1951, a breakthrough by C. C. J. Roothaan (Chapter 17) was made when a procedure
was derived as the linear combination of atomic orbitals to treat molecules. Even so, the applications
were limited to molecules of light elements, mostly organic compounds.
Rapid improvement in computers played a large role in the development of ‘‘quantum chemis-
try’’ in the 1960s but still there was a computational barrier against treating metal complexes
including all the electrons. However, the early computer programs for the atomic Hartree SCF
method were refined and very fast for one-center atomic problems. Thus, the emerging field of
quantum chemistry was ripe for an explosive development by K. H. Johnson [9] and his colleagues
in 1972. He used the one-center computer programs in what was called the multiple-scattering
X-alpha (MS-Xa) method with a ‘‘muffin-tin’’ treatment of molecules. Each atom was treated as a
spherical system set in an outer spherical field where electrons could roam and ‘‘exchange’’ with
each other as well as be scattered in the space between the atoms. Since electrons are indistinguish-
able, accurate calculations need a way to express the possibility of the interchange (exchange) of
electrons. A major innovation was to use an approximation invented=derived by J. C. Slater called
the ‘‘Xa’’ exchange formula, an empirical way to calculate the exchange energy that lowers the
calculated energy.

1

3 3
V Xa (~ r ) ffi 6a r(~ r ) ,
8p
where r( ~ r ) ¼ c * c(~ r ). With that approximation to allow indistinguishable electrons to exchange
positions, the rest of the calculation could use Coulomb’s law for electron–electron repulsion,
e q1 e q2 (Ze q )e q
. It was found necessary to represent
V ¼þ , and electron attraction to nuclei, V ¼
r 12 r 12
the kinetic energy of the electrons as proportional to the second derivative of the electron cloud
wave functions but this was all worked out from Schrödinger’s 1926 derivation. The Coulomb
interactions could be evaluated using numerical integration over a grid of points in a way similar to
the trapezoid rule for 1D integration and so we see Johnson’s scheme in Figure 9.11. The state-of-
the-art at that time was that the method was well established as long as the problem was for only one
atom at the center of the coordinate system. Johnson’s innovation was to merge these spherical
atoms into molecules. Although you will have to read later chapters to fully appreciate the Johnson
‘‘muffin-tin potential’’ and J. C. Slater’s contributions to quantum chemistry, we show you the
results here to offer a more modern treatment of electrons than the flat Bohr orbitals and to try to
explain the electronic spectrum of KMnO 4 .
In 1972, Johnson used Schrödinger’s orbital notation first derived in 1926 and that is probably
familiar to you from your freshman chemistry text. In Figure 9.12, we see the energy level scheme
for MnO 4 with the Mn valence shell orbital levels on the left and orbital levels for O on the right.

In the middle, we see the calculated levels for the complete MnO 4 tetrahedral complex (valence

shell). Here, we see the use of symmetry labels for a T d point group, the condensed spatial
description of a tetrahedral object. Point group theory is discussed in Chapter 18. This author was
at a conference where these results were presented (The Quantum Chemistry Symposium No. 6,
1972, held at Sanibel Island, Florida) and in spite of many experts in attendance, there was a hush
over the audience when Figures 9.13 and 9.14 were shown of the contours of electron orbital clouds
in the O–Mn–O bonds. At the time, this was truly amazing.